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# Path Following and Empirical Bayes Model Selection for Sparse Regression
Hua Zhou
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203
hua_zhou@ncsu.edu
Artin Armagan
SAS Institute, Inc
Cary, NC 27513
artin.armagan@sas.com
David B. Dunson
Department of Statistical Science
Duke University
Durham, NC 27708
dunson@stat.duke.edu
###### Abstract
In recent years, a rich variety of regularization procedures have been
proposed for high dimensional regression problems. However, tuning parameter
choice and computational efficiency in ultra-high dimensional problems remain
vexing issues. The routine use of $\ell_{1}$ regularization is largely
attributable to the computational efficiency of the LARS algorithm, but
similar efficiency for better behaved penalties has remained elusive. In this
article, we propose a highly efficient path following procedure for
combination of any convex loss function and a broad class of penalties. From a
Bayesian perspective, this algorithm rapidly yields maximum a posteriori
estimates at different hyper-parameter values. To bypass the inefficiency and
potential instability of cross validation, we propose an empirical Bayes
procedure for rapidly choosing the optimal model and corresponding hyper-
parameter value. This approach applies to any penalty that corresponds to a
proper prior distribution on the regression coefficients. While we mainly
focus on sparse estimation of generalized linear models, the method extends to
more general regularizations such as polynomial trend filtering after
reparameterization. The proposed algorithm scales efficiently to large $p$
and/or $n$. Solution paths of 10,000 dimensional examples are computed within
one minute on a laptop for various generalized linear models (GLM). Operating
characteristics are assessed through simulation studies and the methods are
applied to several real data sets.
Keywords: Generalized linear model (GLM); Lasso; LARS; Maximum a posteriori
estimation; Model selection; Non-convex penalty; Ordinary differential
equation (ODE); Regularization; Solution path.
## 1 Introduction
Sparse estimation via regularization has become a prominent research area over
the last decade finding interest across a broad variety of disciplines. Much
of the attention was brought by lasso regression (Tibshirani, 1996), which is
simply $\ell_{1}$ regularization, where $\ell_{\eta}$ is the $\eta$-norm of a
vector for $\eta>0$. It was not until the introduction of the LARS algorithm
(Efron et al., 2004) that lasso became so routinely used. This popularity is
attributable to the excellent computational performance of LARS, a variant of
which obtains the whole solution path of the lasso at the cost of an ordinary
least squares estimation. Unfortunately lasso has some serious disadvantages
in terms of estimation bias and model selection behavior. There has been a
rich literature analyzing lasso and its remedies (Fu, 1998; Knight and Fu,
2000; Fan and Li, 2001; Yuan and Lin, 2005; Zou and Hastie, 2005; Zhao and Yu,
2006; Zou, 2006; Meinshausen and Bühlmann, 2006; Zou and Li, 2008; Zhang and
Huang, 2008).
Motivated by disadvantages of $\ell_{1}$ regularization, some non-convex
penalties are proposed which, when designed properly, reduce bias in large
signals while shrinking noise-like signals to zero (Fan and Li, 2001; Candès
et al., 2008; Friedman, 2008; Armagan, 2009; Zhang, 2010; Armagan et al.,
2011). However, non-convex regularization involves difficult non-convex
optimization. As a convex loss function plus concave penalties is a difference
of two convex functions, an iterative algorithm for estimation at a fixed
regularization parameter can be constructed by the majorization-minimization
principle (Lange, 2010). At each iteration, the penalty is replaced by the
supporting hyperplane tangent at the current iterate. As the supporting
hyperplane majorizes the concave penalty function, minimizing the convex loss
plus the linear majorizing function (an $\ell_{1}$ regularization problem)
produces the next iterate, which is guaranteed to decrease the original
penalized objective function. Many existing algorithms for estimation with
concave penalties fall into this category (Fan and Li, 2001; Hunter and Li,
2005; Zou and Li, 2008; Candès et al., 2008; Armagan et al., 2011). Although
being numerically stable and easy to implement, their (often slow) convergence
to a local mode makes their performance quite sensitive to starting values in
settings where $p>>n$. Coordinate descent is another algorithm for
optimization in sparse regression at a fixed tuning parameter value and has
found success in ultra-high dimensional problems (Friedman et al., 2007; Wu
and Lange, 2008; Friedman et al., 2010; Mazumder et al., 2011). Nevertheless
the optimization has to be performed at a large number of grid points, making
the computation rather demanding compared to path following algorithms such as
LARS. For both algorithms the choice of grid points is tricky. When there are
too few, important events along the path are missed; when there are too many,
computation becomes expensive.
The choice of tuning parameter is another important challenge in
regularization problems. Cross-validation is widely used but incurs
considerable computational costs. An attractive alternative is to select the
tuning parameter according to a model selection criterion such as the Akaike
information criterion (AIC) (Akaike, 1974) or the Bayesian information
criterion (BIC) (Schwarz, 1978). These criteria choose the tuning parameter
minimizing the negative log-likelihood penalized by the model dimension. In
shrinkage estimation, however, the degrees of freedom is often unclear.
Intriguing work by Wang and Leng (2007) and Wang et al. (2007) extend BIC to
be used with shrinkage estimators. However, the meaning of BIC as an empirical
Bayes procedure is lost in such extensions. Zhang and Huang (2008) study the
properties of generalized information criterion (GIC) in a similar context.
In this paper, we address these two issues for the general regularization
problem
$\displaystyle\min_{\boldsymbol{\beta}}\,f(\boldsymbol{\beta})+\sum_{j\in{\cal
S}}P_{\eta}(|\beta_{j}|,\rho),$ (1)
where $f$ is a twice differentiable convex loss function, ${\cal
S}\subset\\{1,\ldots,p\\}$ indicates the subset of coefficients being
penalized (${\cal S}$ for shrink), and $P_{\eta}(t,\rho)$ is a scalar penalty
function. Here $\rho$ is the penalty tuning parameter and $\eta$ represents
possible parameter(s) for a penalty family. Allowing a general penalization
set ${\cal S}$ increases the applicability of the method, as we will see in
Section 4. Throughout this article we assume the following regularity
conditions on the penalty function $P_{\eta}(t,\rho)$: (i) symmetric about 0
in $t$, (ii) $P_{\eta}(0,\rho)>-\infty$, for all $\rho\geq 0$, (iii) monotone
increasing in $\rho\geq 0$ for any fixed $t$, (iv) non-decreasing in $t\geq 0$
for any fixed $\rho$, (v) first two derivatives with respect to $t$ exist and
are finite.
The generality of (1) is two-fold. First, $f$ can be any convex loss function.
For least squares problems,
$f(\boldsymbol{\beta})=\|\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\|_{2}^{2}/2$.
For generalized linear models (GLMs), $f$ is the negative log-likelihood
function. For Gaussian graphical models,
$f(\boldsymbol{\Omega})=-\log\det|\boldsymbol{\Omega}|+\text{tr}(\hat{\boldsymbol{\Sigma}}\boldsymbol{\Omega})$,
where $\hat{\boldsymbol{\Sigma}}$ is a sample covariance matrix and the
parameter $\boldsymbol{\Omega}$ is a precision matrix. Secondly, most commonly
used penalties satisfy the aforementioned assumptions. These include but are
not limited to
1. 1.
Power family (Frank and Friedman, 1993)
$\displaystyle P_{\eta}(|\beta|,\rho)$
$\displaystyle=\rho|\beta|^{\eta},\hskip 14.45377pt\eta\in(0,2].$
Varying $\eta$ from 2 bridges best subset regression to lasso ($l_{1}$)
(Tibshirani, 1996; Chen et al., 2001) to ridge ($l_{2}$) regression (Hoerl and
Kennard, 1970).
2. 2.
Elastic net (Zou and Hastie, 2005)
$\displaystyle P_{\eta}(|\beta|,\rho)$
$\displaystyle=\rho[(\eta-1)\beta^{2}/2+(2-\eta)|\beta|],\hskip
14.45377pt\eta\in[1,2].$
Varying $\eta$ from 1 to 2 bridges lasso regression to ridge regression.
3. 3.
Log penalty (Candès et al., 2008; Armagan et al., 2011)
$\displaystyle P_{\eta}(|\beta|,\rho)$
$\displaystyle=\rho\ln(\eta+|\beta|),\hskip 14.45377pt\eta>0.$
This penalty was called generalized elastic net in Friedman (2008) and log
penalty in Mazumder et al. (2011). Such a penalty is induced by a generalized
Pareto prior thresholded and folded at zero, with the oracle properties
studied by Armagan et al. (2011).
4. 4.
Continuous log penalty (Armagan et al., 2011)
$\displaystyle P(|\beta|,\rho)$ $\displaystyle=\rho\ln(\sqrt{\rho}+|\beta|).$
This version of log penalty was designed to guarantee continuity of the
solution path when the design matrix is scaled and orthogonal (Armagan et al.,
2011).
5. 5.
The SCAD penalty (Fan and Li, 2001) is defined via its partial derivative
$\displaystyle\frac{\partial}{\partial|\beta|}P_{\eta}(|\beta|,\rho)$
$\displaystyle=\rho\left\\{1_{\\{|\beta|\leq\rho\\}}+\frac{(\eta\rho-|\beta|)_{+}}{(\eta-1)\rho}1_{\\{|\beta|>\rho\\}}\right\\},\hskip
14.45377pt\eta>2.$
Integration shows SCAD as a natural quadratic spline with knots at $\rho$ and
$\eta\rho$
$\displaystyle P_{\eta}(|\beta|,\rho)$
$\displaystyle=\begin{cases}\rho|\beta|&|\beta|<\rho\\\
\rho^{2}+\frac{\eta\rho(|\beta|-\rho)}{\eta-1}-\frac{\beta^{2}-\rho^{2}}{2(\eta-1)}&|\beta|\in[\rho,\eta\rho]\\\
\rho^{2}(\eta+1)/2&|\beta|>\eta\rho\end{cases}.$ (2)
For small signals $|\beta|<\rho$, it acts as lasso; for larger signals
$|\beta|>\eta\rho$, the penalty flattens and leads to the unbiasedness of the
regularized estimate.
6. 6.
Similar to SCAD is the MC+ penalty (Zhang, 2010) defined by the partial
derivative
$\displaystyle\frac{\partial}{\partial|\beta|}P_{\eta}(|\beta|,\rho)$
$\displaystyle=\rho\left(1-\frac{|\beta|}{\rho\eta}\right)_{+}.$
Integration shows that the penalty function
$\displaystyle P_{\eta}(|\beta|,\rho)$
$\displaystyle=\left(\rho|\beta|-\frac{\beta^{2}}{2\eta}\right)1_{\\{|\beta|<\rho\eta\\}}+\frac{\rho^{2}\eta}{2}1_{\\{|\beta|\geq\rho\eta\\}},\hskip
14.45377pt\eta>0,$ (3)
is quadratic on $[0,\rho\eta]$ and flattens beyond $\rho\eta$. Varying $\eta$
from 0 to $\infty$ bridges hard thresholding ($\ell_{0}$ regression) to lasso
($\ell_{1}$) shrinkage.
The derivatives of penalty functions will be frequently used for the
development of the path algorithm and model selection procedure. They are
listed in Table 2 of Supplementary Materials for convenience.
Our contributions are summarized as follows:
1. 1.
We propose a general path seeking strategy for the sparse regression framework
(1). To the best of our knowledge, no previous work exists at this generality,
except the generalized path seeking (GPS) algorithm proposed in unpublished
work by Friedman (2008). Some problems with the GPS algorithm motivated us to
develop a more rigorous algorithm, which is fundamentally different from GPS.
Path algorithms for some specific combinations of loss and penalty functions
have been studied before. Homotopy (Osborne et al., 2000) and a variant of
LARS (Efron et al., 2004) compute the piecewise linear solution path of
$\ell_{1}$ penalized linear regression efficiently. Park and Hastie (2007)
proposed an approximate path algorithm for $\ell_{1}$ penalized GLMs. A
similar problem was considered by Wu (2011) who devises a LARS algorithm for
GLMs based on ordinary differential equations (ODEs). The ODE approach
naturally fits problems with piecewise smooth solution paths and is the
strategy we adopt in this paper. All of the aforementioned work deals with
$\ell_{1}$ regularization which leads to convex optimization problems. Moving
from convex to non-convex penalties improves the quality of the estimates but
imposes great difficulties in computation. The PLUS path algorithm of (Zhang,
2010) is able to track all local minima; however, it is specifically designed
for the least squares problem with an MC+ penalty and does not generalize to
(1).
2. 2.
We propose an empirical Bayes procedure for the selection of a good model and
the implied hyper/tuning parameters along the solution path. This method
applies to any likelihood model with a penalty that corresponds to a proper
shrinkage prior in the Bayesian setting. We illustrate the method with the
power family (bridge) and log penalties which are induced by the exponential
power and generalized double Pareto priors respectively. The regularization
procedure resulting from the corresponding penalties is utilized as a model-
search engine where each model and estimate along the path is appropriately
evaluated by a criterion emerging from the prior used. Yuan and Lin (2005)
took a somewhat similar approach in the limited setting of $\ell_{1}$
penalized linear regression.
3. 3.
The proposed path algorithm and empirical Bayes model selection procedure
extend to a large class of generalized regularization problems such as
polynomial trend filtering. Path algorithms for generalized $\ell_{1}$
regularization was recently studied by Tibshirani and Taylor (2011) and Zhou
and Lange (2011) for linear regression and by Zhou and Wu (2011) for general
convex loss functions. Using non-convex penalties in these general
regularization problems produces more parsimonious and less biased estimates.
Re-parameterization reformulates these problems as in (1) which is solved by
our efficient path algorithm.
4. 4.
A Matlab toolbox for sparse regression is released on the first author’s web
site. The code for all examples in this paper is available on the same web
site, observing the principle of reproducible research.
The remainder of the article is organized as follows. The path following
algorithm is derived in Section 2. The empirical Bayes criterion is developed
in Section 3 and illustrated for the power family and log penalties. Section 4
discusses extensions to generalized regularization problems. Various numerical
examples are presented in Section 5. Finally we conclude with a discussion and
future directions.
## 2 Path Following for Sparse Regressions
For a parameter vector $\boldsymbol{\beta}\in\mathbb{R}^{p}$, we use ${\cal
S}_{\cal Z}(\boldsymbol{\beta})=\\{j\in{\cal S}:\beta_{j}=0\\}$ to denote the
set of penalized parameters that are zero and correspondingly ${\cal
S}_{\bar{\cal Z}}(\boldsymbol{\beta})=\\{j\in{\cal S}:\beta_{j}\neq 0\\}$ is
the set of nonzero penalized parameters. ${\cal A}=\bar{\cal S}\cup{\cal
S}_{\bar{\cal Z}}$ indexes the current active predictors with unpenalized or
nonzero penalized coefficients. It is convenient to define the Hessian,
$\boldsymbol{H}_{\cal A}(\boldsymbol{\beta},\rho)\in\mathbb{R}^{|{\cal
A}|\times|{\cal A}|}$, of the penalized objective function (1) restricted to
the active predictors with entries
$\displaystyle H_{jk}$
$\displaystyle=\begin{cases}[d^{2}f(\boldsymbol{\beta})]_{jk}&j\in\bar{\cal
S}\\\
[d^{2}f(\boldsymbol{\beta})]_{jk}+\frac{\partial^{2}P(|\beta_{j}|,\rho)}{\partial|\beta_{j}|^{2}}1_{\\{j=k\\}}&j\in{\cal
S}_{\bar{\cal Z}}\end{cases}.$ (4)
Our path following algorithm revolves around the necessary condition for a
local minimum. We denote the penalized objective function (1) by
$h(\boldsymbol{\beta})$ throughout this article.
###### Lemma 2.1 (Necessary optimality condition).
If $\boldsymbol{\beta}$ is a local minimum of (1) at tuning parameter value
$\rho$, then $\boldsymbol{\beta}$ satisfies the stationarity condition
$\displaystyle\nabla_{j}f(\boldsymbol{\beta})+\frac{\partial
P(|\beta_{j}|,\rho)}{\partial|\beta_{j}|}\omega_{j}1_{\\{j\in{\cal
S}\\}}=0,\hskip 14.45377ptj=1,\ldots,p,$ (5)
where the coefficients $\omega_{j}$ satisfy
$\displaystyle\omega_{j}\in\begin{cases}\\{-1\\}&\beta_{j}<0\\\
[-1,1]&\beta_{j}=0\\\ \\{1\\}&\beta_{j}>0\end{cases}.$
Furthermore, $\boldsymbol{H}_{\cal A}(\boldsymbol{\beta},\rho)$ is positive
semidefinite.
###### Proof.
When the penalty function $P$ is convex, this is simply the first order
optimality condition for unconstrained convex minimization (Ruszczyński, 2006,
Theorem 3.5). When $P$ is non-convex, we consider the optimality condition
coordinate-wise. For $j\in\\{j:\beta_{j}\neq 0\\}$, this is trivial. When
$\beta_{j}=0$, $\beta_{j}$ being a local minimum implies that the two
directional derivatives are non-negative. Then
$\displaystyle d_{\boldsymbol{e}_{j}}h(\boldsymbol{\beta})$
$\displaystyle=\lim_{t\downarrow
0}\frac{h(\boldsymbol{\beta}+t\boldsymbol{e}_{j})-h(\boldsymbol{\beta})}{t}=\nabla_{j}f(\boldsymbol{\beta})+\frac{\partial
P(|\beta_{j}|,\rho)}{\partial|\beta_{j}|}\geq 0$ $\displaystyle
d_{-\boldsymbol{e}_{j}}h(\boldsymbol{\beta})$ $\displaystyle=\lim_{t\uparrow
0}\frac{h(\boldsymbol{\beta}+t\boldsymbol{e}_{j})-h(\boldsymbol{\beta})}{t}=-\nabla_{j}f(\boldsymbol{\beta})+\frac{\partial
P(|\beta_{j}|,\rho)}{\partial|\beta_{j}|}\geq 0,$
which is equivalent to (5) with $\omega_{j}\in[-1,1]$. Positive
semidefiniteness of $\boldsymbol{H}_{\cal A}$ follows from the second order
necessary optimality condition when restricted to coordinates in ${\cal A}$. ∎
We call any $\boldsymbol{\beta}$ satisfying (5) a _stationary point_ at
$\rho$. Our path algorithm tracks a stationary point along the path while
sliding $\rho$ from infinity towards zero. When the penalized objective
function $h$ is convex, e.g., $\eta\in[1,2]$ regime of the power family,
elastic net, or $d^{2}h$ is positive semidefinite, the stationarity condition
(5) is sufficient for a global minimum. When $h$ is not convex, the
minimization problem is both non-smooth and non-convex and there lacks an
easy-to-check sufficient condition for optimality. The most we can claim is
that the directional derivatives at any stationary point are non-negative.
###### Lemma 2.2.
Suppose $\boldsymbol{\beta}$ satisfies the stationarity condition (5). Then
all directional derivatives at $\boldsymbol{\beta}$ are non-negative, i.e.,
$\displaystyle d_{\boldsymbol{v}}h(\boldsymbol{\beta})=\lim_{t\downarrow
0}\frac{h(\boldsymbol{\beta}+t\boldsymbol{v})-h(\boldsymbol{\beta})}{t}\geq 0$
(6)
for any $\boldsymbol{v}\in\mathbb{R}^{p}$. Furthermore, if the penalized
objective function $h$ is convex, then $\boldsymbol{\beta}$ is a global
minimum.
###### Proof.
By definition of directional derivative and the stationarity condition (5),
$\displaystyle d_{\boldsymbol{v}}h(\boldsymbol{\beta})$ $\displaystyle=$
$\displaystyle d_{\boldsymbol{v}}f(\boldsymbol{\beta})+\sum_{j\in{\cal
S}:\beta_{j}\neq 0}v_{j}\left.\frac{\partial P_{\eta}(t,\rho)}{\partial
t}\right|_{t=|\beta_{j}|}\mathrm{sgn}(\beta_{j})+\sum_{j\in{\cal
S}:\beta_{j}=0}|v_{j}|\left.\frac{\partial P_{\eta}(t,\rho)}{\partial
t}\right|_{t=0}$ $\displaystyle=$
$\displaystyle\sum_{j}v_{j}\nabla_{j}f(\boldsymbol{\beta})+\sum_{j\in{\cal
S}:\beta_{j}\neq 0}v_{j}\left.\frac{\partial P_{\eta}(t,\rho)}{\partial
t}\right|_{t=|\beta_{j}|}\mathrm{sgn}(\beta_{j})+\sum_{j\in{\cal
S}:\beta_{j}=0}|v_{j}|\left.\frac{\partial P_{\eta}(t,\rho)}{\partial
t}\right|_{t=0}$ $\displaystyle=$ $\displaystyle\sum_{j\notin{\cal
A}}|v_{j}|\left(\mathrm{sgn}(v_{j})\cdot\nabla_{j}f(\boldsymbol{\beta})+\left.\frac{\partial
P_{\eta}(t,\rho)}{\partial t}\right|_{t=0}\right)$ $\displaystyle\geq$
$\displaystyle 0.$
Consider the scalar function $g(t)=h(\boldsymbol{\beta}+t\boldsymbol{v})$.
Convexity of $h$ implies that $g$ is convex too. Then the chord
$[g(t)-g(0)]/t=[h(\boldsymbol{\beta}+t\boldsymbol{v})-h(\boldsymbol{\beta})]/t$
is increasing for $t\geq 0$. Thus
$h(\boldsymbol{\beta}+\boldsymbol{v})-h(\boldsymbol{\beta})\geq
d_{\boldsymbol{v}}h(\boldsymbol{\beta})\geq 0$ for all $\boldsymbol{v}$,
verifying that $\boldsymbol{\beta}$ is a global minimum. ∎
We remark that, without convexity, non-negativeness of all directional
derivatives does not guarantee local minimality, as demonstrated in the
following example (Lange, 2004, Exercise 1.12). Consider the bivariate
function $f(x,y)=(y-x^{2})(y-2x^{2})$. Any directional derivative at origin
(0,0) is non-negative since $\lim_{t\to 0}[f(ht,kt)-f(0,0)]/t=0$ for any
$h,k\in\mathbb{R}$ and indeed, (0,0) is a local minimum along any line passing
through it. However (0,0) is not a local minimum for $f$ as it is easy to see
that $f(t,ct^{2})<0$ for any $1<c<2$, $t\neq 0$, and that $f(t,ct^{2})>0$ for
any $c<1$ or $c>2$, $t\neq 0$. Figure 1 demonstrates how we go down hill along
the parabola $y=1.4x^{2}$ as we move away from $(0,0)$. In this article, we
abuse terminology by the use of _solution path_ and in fact mean _path of
stationarity points_.
Figure 1: Contours of $f(x,y)=(y-x^{2})(y-2x^{2})$ and the parabola
$y=1.4x^{2}$ that passes through $(0,0)$, which is not a local minimum
although all directional derivatives at (0,0) are nonnegative.
### 2.1 Least squares with orthogonal design
Before deriving the path algorithm for the general sparse regression problem
(1), we first investigate a simple case: linear regression with orthogonal
design, i.e.,
$f(\boldsymbol{\beta})=\|\boldsymbol{y}-\boldsymbol{X}\boldsymbol{\beta}\|_{2}^{2}/2$
where
$\boldsymbol{x}_{j}^{t}\boldsymbol{x}_{k}=\|\boldsymbol{x}_{j}\|_{2}^{2}1_{\\{j=k\\}}$.
This serves two purposes. First it illustrates the difficulties
(discontinuities, local minima) of path seeking with non-convex penalties.
Secondly, the thresholding operator for orthogonal design is the building
block of the coordinate descent algorithm (Friedman et al., 2007; Wu and
Lange, 2008; Friedman et al., 2010; Mazumder et al., 2011) or iterative
thresholding, which we rely on to detect the discontinuities in path following
for the non-convex case.
For linear regression with orthogonal design, the penalized objective function
in (1) can be written in a component-wise fashion and the path solution is
$\displaystyle\hat{\beta}_{j}(\rho)=\text{argmin}_{\beta}\,\frac{a_{j}}{2}(\beta-
b_{j})^{2}+P_{\eta}(|\beta|,\rho)$ (7)
where
$a_{j}=\boldsymbol{x}_{j}^{t}\boldsymbol{x}_{j}=\|\boldsymbol{x}_{j}\|_{2}^{2}$
and
$b_{j}=\boldsymbol{x}_{j}^{t}\boldsymbol{y}/\boldsymbol{x}_{j}^{t}\boldsymbol{x}_{j}$.
The solution to (7) for some popular penalties is listed in Supplementary
Materials. Similar derivations can be found in (Mazumder et al., 2011).
Existing literature mostly assumes that the design matrix is standardized,
i.e., $a_{j}=1$. As we argue in forthcoming examples, in many applications it
is prudent not to do so. Figure 2 depicts the evolution of the penalized
objective function with varying $\rho$ and the solution path for
$a_{j}=b_{j}=1$ and the log penalty
$P_{\eta}(|\beta|,\rho)=\rho\ln(|\beta|+\eta)$ with $\eta=0.1$. At
$\rho=0.2191$, the path solution jumps from a local minimum 0 to the other
positive local minimum.
For the least squares problem with a non-orthogonal design, the coordinate
descent algorithm iteratively updates $\beta_{j}$ by (7). When updating
$\beta_{j}$ keeping other predictors fixed, the objective function takes the
same format with $a_{j}=\|\boldsymbol{x}_{j}\|_{2}^{2}$ and
$b_{j}=\boldsymbol{x}_{j}^{t}(\boldsymbol{y}-\boldsymbol{X}_{-j}\boldsymbol{\beta}_{-j})/\boldsymbol{x}_{j}^{t}\boldsymbol{x}_{j}$,
where $\boldsymbol{X}_{-j}$ and $\boldsymbol{\beta}_{-j}$ denote the design
matrix and regression coefficient vector without the $j$-th covariate. For a
general twice differentiable loss function $f$, we approximate the smooth part
$f$ by its Taylor expansion around current iterate $\boldsymbol{\beta}^{(t)}$
$\displaystyle f(\boldsymbol{\beta})\approx
f(\boldsymbol{\beta}^{(t)})+df(\boldsymbol{\beta}^{(t)})(\boldsymbol{\beta}-\boldsymbol{\beta}^{(t)})+\frac{1}{2}(\boldsymbol{\beta}-\boldsymbol{\beta}^{(t)})^{t}d^{2}f(\boldsymbol{\beta}^{(t)})(\boldsymbol{\beta}-\boldsymbol{\beta}^{(t)})$
and then apply thresholding formula (7) for $\beta_{j}$ sequentially to obtain
the next iterate $\boldsymbol{\beta}^{(t+1)}$.
$\begin{array}[]{cc}\includegraphics[width=198.7425pt]{orthogobjfun}&\includegraphics[width=198.7425pt]{orthogsolpath}\end{array}$
Figure 2: Log penalty with orthogonal design: $a=1$, $b=1$, $\eta=0.1$. Left:
Graphs of the penalized objective function
$a(\beta-b)^{2}/2+\rho\ln(\eta+|\beta|)$ at different $\rho$. Right: Solution
path. Here $ab\eta=0.1$ and $a(\eta+|b|)^{2}/4=0.3025$. Discontinuity occurs
somewhere between these two numbers.
### 2.2 Path following for the general case
The preceding discussion illustrates two difficulties with path seeking in
sparse regression using non-convex penalties. First the solution path may not
be continuous. Discontinuities occur when predictors enter the model at a non-
zero magnitude or vice versa. This is caused by jumping between local minima.
Second, in contrast to lasso, the solution path is no longer piecewise linear.
This prohibits making giant jumps along the path like LARS.
One strategy is to optimize (1) at a grid of penalty intensity $\rho$ using
coordinate descent. This has found great success with lasso and elastic net
regression (Friedman et al., 2007; Wu and Lange, 2008; Friedman et al., 2010).
The recent article (Mazumder et al., 2011) explores the coordinate descent
strategy with non-convex penalties. In principle it involves applying the
thresholding formula to individual regression coefficients until convergence.
However, determining the grid size for tuning parameter $\rho$ in advance
could be tricky. The larger the grid size the more likely we are to miss
important events along the path, while the smaller the grid size the higher
the computational costs.
In this section we devise a path seeking strategy that tracks the solution
path smoothly while allowing abrupt jumping (due to discontinuities) between
segments. The key observation is that each path segment is smooth and
satisfies a simple ordinary differential equation (ODE). Recall that the
active set ${\cal A}=\bar{\cal S}\cup{\cal S}_{\bar{\cal Z}}$ indexes all
unpenalized and nonzero penalized coefficients and $\boldsymbol{H}_{\cal A}$
is the Hessian of the penalized objective function restricted to parameters in
${\cal A}$.
###### Proposition 2.3.
The solution path $\boldsymbol{\beta}(\rho)$ is continuous and differentiable
at $\rho$ if $\boldsymbol{H}_{\cal A}(\boldsymbol{\beta},\rho)$ is positive
definite. Moreover the solution vector $\boldsymbol{\beta}(\rho)$ satisfies
$\displaystyle\frac{d\boldsymbol{\beta}_{\cal A}(\rho)}{d\rho}$
$\displaystyle=-\boldsymbol{H}_{\cal
A}^{-1}(\boldsymbol{\beta},\rho)\cdot\boldsymbol{u}_{\cal
A}(\boldsymbol{\beta},\rho),$ (8)
where the matrix $\boldsymbol{H}_{\cal A}$ is defined by (4) and the vector
$\boldsymbol{u}_{\cal A}(\boldsymbol{\beta})$ has entries
$\displaystyle
u_{j}(\boldsymbol{\beta},\rho)=\begin{cases}\frac{\partial^{2}P_{\eta}(|\beta_{j}|,\rho)}{\partial|\beta_{j}|\partial\rho}\text{sgn}(\beta_{j})&j\in{\cal
S}_{\bar{\cal Z}}\\\ 0&j\in\bar{\cal S}\end{cases}.$
###### Proof.
Write the stationarity condition (5) for active predictors as a vector
equation $k(\boldsymbol{\beta}_{\cal A},\rho)={\bf 0}$. To solve for
$\boldsymbol{\beta}_{\cal A}$ in terms of $\rho$, we apply the implicit
function theorem (Lange, 2004). This requires calculating the differential of
$k$ with respect to the dependent variables $\boldsymbol{\beta}_{\cal A}$ and
the independent variable $\rho$
$\displaystyle\partial_{\boldsymbol{\beta}_{\cal A}}k(\boldsymbol{\beta}_{\cal
A},\rho)$ $\displaystyle=\boldsymbol{H}_{\cal A}(\boldsymbol{\beta},\rho)$
$\displaystyle\partial_{\rho}k(\boldsymbol{\beta}_{\cal A},\rho)$
$\displaystyle=\boldsymbol{u}_{\cal A}(\boldsymbol{\beta}).$
Given the non-singularity of $\boldsymbol{H}_{\cal
A}(\boldsymbol{\beta},\rho)$, the implicit function theorem applies and shows
the continuity and differentiability of $\boldsymbol{\beta}_{\cal A}(\rho)$ at
$\rho$. Furthermore, it supplies the derivative (8). ∎
Proposition 2.8 suggests that solving the simple ODE segment by segment is a
promising path following strategy. However, the potential discontinuity along
the path caused by the non-convex penalty has to be taken care of. Note that
the stationarity condition (5) for inactive predictors implies
$\displaystyle\omega_{j}=-\frac{\nabla_{j}f(\boldsymbol{\beta})}{\frac{\partial}{\partial|\beta|}P(|\beta|,\rho)},\hskip
14.45377ptj\in{\cal S}_{\cal Z},$
and provides one indicator when the coefficient should escape to ${\cal
S}_{\bar{\cal Z}}$ during path following. However, due to the discontinuity, a
regression coefficient $\beta_{j}$, $j\in{\cal S}_{\cal Z}$, may escape with
$\omega_{j}$ in the interior of (-1,1). A more reliable implementation should
check whether an inactive regression coefficient $\beta_{j}$ becomes nonzero
using the thresholding formulae at each step of path following. Another
complication that discontinuity causes is that occasionally the active set
${\cal S}_{\cal Z}$ may change abruptly along the path especially when
predictors are highly correlated. Therefore whenever a discontinuity is
detected, it is advisable to use any nonsmooth optimizer, e.g., coordinate
descent, to figure out the set configuration and starting point for the next
segment. We pick up coordinate descent due to its simple implementation. Our
path following strategy is summarized in Algorithm 1.
Determine the first penalized predictor $j^{*}$ to enter model and the
corresponding $\rho_{\text{max}}$
Initialize ${\cal S}_{\cal Z}=\\{j^{*}\\}$, ${\cal S}_{\bar{\cal Z}}={\cal
S}\setminus\\{j^{*}\\}$, and
$\boldsymbol{\beta}(\rho_{\text{max}})=\text{argmin}_{\boldsymbol{\beta}_{\cal
S}={\bf 0}}f(\boldsymbol{\beta})$
repeat
Solve ODE $\frac{d\boldsymbol{\beta}_{\cal
A}(\rho)}{d\rho}=-\boldsymbol{H}_{\cal
A}(\boldsymbol{\beta},\rho)^{-1}\boldsymbol{u}_{\cal
A}(\boldsymbol{\beta},\rho)$ until (1) an active penalized predictor
$\beta_{j}$, $j\in{\cal S}_{\bar{\cal Z}}$, becomes 0, or (2) an inactive
penalized coefficient $w_{j}$, $j\in{\cal S}_{\cal Z}$, hits 1 or -1, or (3)
an inactive penalized predictor $\beta_{j}$, $j\in{\cal S}_{\cal Z}$, jumps
from 0 to a nonzero minimum, or (4) the matrix $\boldsymbol{H}_{\cal
A}(\boldsymbol{\beta},\rho)$ becomes singular.
if (1) or (2) then
Update sets $S_{\cal Z}$ and $S_{\bar{\cal Z}}$
else
Apply coordinate descent at current $\rho$ to determine $S_{\cal Z}$,
$S_{\bar{\cal Z}}$ and $\beta_{\cal A}$ for next segment
end if
until termination criterion is met
Algorithm 1 Path following for sparse regression.
Several remarks on Algorithm 1 are relevant here.
###### Remark 2.4 (Path following direction).
The ODE (8) is written in the usual sense and gives the derivative as $\rho$
increases. In sparse regression, we solve in the reverse direction and shall
take the opposite sign.
###### Remark 2.5 (Termination criterion).
Termination criterion for path following may depend on the specific likelihood
model. For linear regression, path seeking stops when the number of active
predictors exceeds the rank of design matrix $|{\cal
A}|>\mathrm{rank}(\boldsymbol{X})$. The situation is more subtle for logistic
or Poisson log-linear models due to separation. In binary logistic regression,
complete separation occurs when there exists a vector
$\boldsymbol{z}\in\mathbb{R}^{p}$ such that
$\boldsymbol{x}_{i}^{t}\boldsymbol{z}>0$ for all $y_{i}=1$ and
$\boldsymbol{x}_{i}^{t}\boldsymbol{z}<0$ for all $y_{i}=0$. When complete
separation happens, the log-likelihood is unbounded and the MLE occurs at
infinity along the direction $\boldsymbol{z}$. The log-likelihood surface
behaves linearly along this direction and dominates many non-convex penalties
such as power, log, MC+, and SCAD, which is almost flat at infinity. This
implies that the penalized estimate also occurs at infinity. Path seeking
should terminate whenever separation is detected, which may happen when
$|{\cal A}|$ is much smaller than the rank of the design matrix in large $p$
small $n$ problems. Separation occurs in the Poisson log-linear model too. Our
implementation also allows users to input the maximum number of selected
predictors until path seeking stops, which is convenient for exploratory
analysis of ultra-high dimensional data.
###### Remark 2.6 (Computational Complexity and Implementation).
Any ODE solver repeatedly evaluates the derivative (8). The path segment
stopping events (1)-(4) are checked during each derivative evaluation. Since
the Hessian restricted to the active predictors is always positive
semidefinite and the inactive penalized predictors are checked by
thresholding, the quality of solution along the path is as good as any fixed
tuning parameter optimizer such as coordinate descent (Mazumder et al., 2011).
Computational complexity of Algorithm 1 depends on the loss function, number
of smooth path segments, and the method for solving the ODE. Evaluating
derivative (8) takes $O(n|{\cal A}|^{2})$ flops for calculating the Hessian of
a GLM loss $\ell$ and takes $O(|{\cal A}|^{3})$ flops for solving the linear
system. Detecting jumps of inactive penalized predictor by thresholding takes
$O(|\tilde{\cal A}|)$ flops. The cost of $O(|{\cal A}|^{3})$ per gradient
evaluation is not as daunting as it appears. Suppose any fixed tuning
parameter optimizer is utilized for path following with a warm start. When at
a new $\rho$, assuming that the active set ${\cal A}$ is known, even the
fastest Newton’s method needs to solve the same linear system multiple times
until convergence. The efficiency of Algorithm 1 lies in the fact that no
iterations are needed at any $\rho$ and it adaptively chooses step sizes to
catch all events along the path. Algorithm 1 is extremely simple to implement
using software with a reliable ODE solver such as the ode45 function in Matlab
and the deSolve package in R (Soetaert et al., 2010). For instance, the rich
numerical resources of Matlab include differential equation solvers that alert
the user when certain events such as those stopping rules in Algorithm 1 are
fulfilled.
###### Remark 2.7 (Knots in SCAD and MC+).
Solving the ODE (8) requires the second order partial derivatives
$\frac{\partial^{2}}{\partial|\beta|^{2}}P(|\beta|,\rho)$ and
$\frac{\partial^{2}}{\partial|\beta|\partial\rho}P(|\beta|,\rho)$ of the
penalty functions, which are listed in Table 2. Due to their designs, these
partial derivatives are undetermined for SCAD and MC+ penalties at the knots:
$\\{\rho,\eta\rho\\}$ for SCAD and $\\{\eta\rho\\}$ for MC+. However only the
directional derivatives are needed, which are well-defined. Specifically we
use $\frac{\partial^{2}}{\partial|\beta|^{2}}P(|\beta|,\rho_{-})$ and
$\frac{\partial^{2}}{\partial|\beta|\partial\rho}P(|\beta|,\rho_{-})$. In
practice, the ODE solver rarely steps on these knots exactly due to numerical
precision.
Finally, switching the role of $\rho$ and $\eta$, the same argument leads to
an analogous result for path following in the penalty parameter $\eta$ with a
fixed regularization parameter $\rho$. In this article we focus on path
following in $\rho$ with fixed $\eta$ in the usual sense. Implications of the
next result will be investigated in future work.
###### Proposition 2.8 (Path following in $\eta$).
Suppose the partial derivative $\frac{\partial P_{\eta}(t,\rho)}{\partial
t\partial\eta}$ exists at all $t>0$ and $\rho$. For fixed $\rho$, the solution
path $\boldsymbol{\beta}(\eta)$ is continuous and differentiable at $\eta$ if
$\boldsymbol{H}_{\cal A}(\boldsymbol{\beta},\eta)$ is positive definite.
Moreover the solution vector $\boldsymbol{\beta}(\eta)$ satisfies
$\displaystyle\frac{d\boldsymbol{\beta}_{\cal A}(\eta)}{d\eta}$
$\displaystyle=$ $\displaystyle-\boldsymbol{H}_{\cal
A}^{-1}(\boldsymbol{\beta},\eta)\cdot\boldsymbol{u}_{\cal
A}(\boldsymbol{\beta},\eta),$
where the matrix $\boldsymbol{H}_{\cal A}$ is defined by (4) and the vector
$\boldsymbol{u}_{\cal A}(\boldsymbol{\beta})$ has entries
$\displaystyle
u_{j}(\boldsymbol{\beta},\eta)=\begin{cases}\frac{\partial^{2}P_{\eta}(|\beta_{j}|,\rho)}{\partial|\beta_{j}|\partial\eta}\text{sgn}(\beta_{j})&j\in{\cal
S}_{\bar{\cal Z}}\\\ 0&j\in\bar{\cal S}\end{cases}.$
## 3 Empirical Bayes Model Selection
In practice, the regularization parameter $\rho$ in sparse regression is tuned
according to certain criteria. Often we wish to avoid cross-validation and
rely on more efficient procedures. AIC, BIC and similar variants have
frequently been used. Recall that BIC arises from a Laplace approximation to
the log-marginal density of the observations under a Bayesian model. The
priors on the parameters are specifically chosen to be normal with mean set at
the maximum likelihood estimator and covariance that conveys the Fisher
information observed from one observation. This allows for a rather diffuse
prior relative to the likelihood. Hence the resulting maximum a posteriori
estimate is the maximum likelihood estimator. Often users plug in the
estimates from sparse regression into AIC or BIC to assess the quality of the
estimate/model. In this section we derive an appropriate empirical Bayes
criterion that corresponds to the exact prior under which we are operating.
All necessary components for calculating the empirical Bayes criterion fall
out nicely from the path following algorithm. A somewhat similar approach was
taken by Yuan and Lin (2005) to pick an appropriate tuning parameter for the
lasso penalized least squares noting that the underlying Bayesian model is
formed with a mixture prior – a spike at zero and a double exponential
distribution on $\beta_{j}\in\mathbb{R}$.
Conforming to previous notation, a model is represented by the active set
${\cal A}=\bar{\cal S}\cup{\cal S}_{\cal Z}$ which includes both un-penalized
and selected penalized regression coefficients. By Bayes formula, the
probability of a model ${\cal A}$ given data $\boldsymbol{y}$ is
$\displaystyle p({\cal A}|\boldsymbol{y})=\frac{p(\boldsymbol{y}|{\cal
A})p({\cal A})}{p(\boldsymbol{y})}.$
Assuming equal prior probability for all models, an appropriate Bayesian
criterion for model comparison is the marginal data likelihood
$p(\boldsymbol{y}|{\cal A})$ of model ${\cal A}$. If the penalty in the
penalized regression is induced by a proper prior $\pi(\beta)$ on the
regression coefficients, the marginal likelihood is calculated as
$\displaystyle p(\boldsymbol{y}|{\cal A})=\int\pi(\boldsymbol{\beta}_{\cal
A},\boldsymbol{y})\,d\boldsymbol{\beta}_{\cal
A}=\int\pi(\boldsymbol{y}|\boldsymbol{\beta}_{\cal A})\prod_{j\in{\cal
A}}\pi(\beta_{j})\,d\boldsymbol{\beta}_{\cal A}.$ (9)
In most cases the integral cannot be analytically calculated. Fortunately the
Laplace approximation is a viable choice in a similar manner to BIC, in which
$\pi(\beta_{j})$ is taken as the vaguely informative unit information prior.
We illustrate this general procedure with the log and power penalties. Note
that both the regularization parameter $\rho$ and penalty parameter $\eta$ are
treated as hyper-parameters in priors. Thus the procedure not only allows
comparison of models along the path with fixed $\eta$ but also models with
distinct $\eta$.
### 3.1 Log penalty
The log penalty arises from a generalized double Pareto prior (Armagan et al.,
2011) on regression coefficients
$\displaystyle\pi(\beta|\alpha,\eta)=\frac{\alpha\eta^{\alpha}}{2}(|\beta|+\eta)^{-(\alpha+1)},\hskip
14.45377pt\alpha,\eta>0.$
Writing $\rho=\alpha+1$ and placing a generalized double Pareto prior on
$\beta_{j}$ for active coefficients $j\in{\cal A}$ estimated at $(\rho,\eta)$,
the un-normalized posterior is given by
$\displaystyle\pi(\boldsymbol{\beta}_{\cal A},\boldsymbol{y}|\rho,\eta)$
$\displaystyle=$
$\displaystyle\left\\{\frac{(\rho-1)\eta^{\rho-1}}{2}\right\\}^{q}\exp\left\\{\ell(\boldsymbol{\beta}_{\cal
A})-\rho\sum_{j\in\mathcal{A}}\ln(\eta+|\beta_{j}|)\right\\},$
$\displaystyle=$
$\displaystyle\left\\{\frac{(\rho-1)\eta^{\rho-1}}{2}\right\\}^{q}\exp\left\\{-h(\boldsymbol{\beta}_{\cal
A})\right\\},$
where $q=|{\cal A}|$ and $h(\boldsymbol{\beta}_{\cal
A})=-\ell(\boldsymbol{\beta}_{\cal
A})+\rho\sum_{j\in\mathcal{A}}\ln(\eta+|\beta_{j}|)$. Then a Laplace
approximation to the integral (9) enables us to assess the relative quality of
an estimated model $\hat{\cal A}$ at particular hyper/tuning parameter values
$(\rho,\eta)$ by $\mathrm{EB}(\rho,\eta)=-\ln p(\boldsymbol{y}|\tilde{\cal
A})$. The following result displays the empirical Bayes criterion for the log
penalty and then specializes to the least squares case which involves unknown
variance. Note that, by the definition of double Pareto prior, $\rho>1$.
Otherwise the prior on $\beta_{j}$ is no longer proper.
###### Proposition 3.1.
For $\rho>1$, an empirical Bayes criterion for the log penalized regression is
$\displaystyle\mathrm{EB}_{\log}(\rho,\eta)\equiv-q\ln\left\\{\left(\frac{\pi}{2}\right)^{1/2}(\rho-1)\eta^{\rho-1}\right\\}+h(\tilde{\boldsymbol{\beta}})+\frac{1}{2}\log\det\boldsymbol{H}_{\cal
A}(\tilde{\boldsymbol{\beta}}),$
where $\tilde{\boldsymbol{\beta}}$ is the path solution at $(\rho,\eta)$,
${\cal A}={\cal A}(\tilde{\boldsymbol{\beta}})$ is the set of active
regression coefficients, $q=|{\cal A}|$, and $\boldsymbol{H}_{\cal A}$ is the
restricted Hessian defined by (4). Under a linear model, it becomes
$\displaystyle\mathrm{EB}_{\log}(\rho,\eta)\equiv-q\ln\left\\{\left(\frac{\pi\tilde{\sigma}^{2}}{2}\right)^{1/2}\left(\frac{\rho}{\tilde{\sigma}^{2}}-1\right)\eta^{\rho/\tilde{\sigma}^{2}-1}\right\\}+\frac{h(\tilde{\boldsymbol{\beta}})}{\tilde{\sigma}^{2}}+\frac{1}{2}\log\det\boldsymbol{H}_{\cal
A}(\tilde{\boldsymbol{\beta}}),$
where
$\tilde{\sigma}^{2}=\mathrm{argmin}_{\sigma^{2}}\left\\{-\frac{n-q}{2}\ln\sigma^{2}+\frac{q\rho}{\sigma^{2}}\ln\eta+q\ln\left(\frac{\rho}{\sigma^{2}}-1\right)-\frac{h(\tilde{\boldsymbol{\beta}})}{\sigma^{2}}\right\\}.$
###### Proof.
The Laplace approximation to the normalizing constant (9) is given by
$\ln p(\boldsymbol{y}|\tilde{\cal
A})\approx\ln\pi(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}},\boldsymbol{y}|\rho,\eta)+\frac{q}{2}\ln(2\pi)-\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}),$
where $\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}}=\mathrm{argmin}_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal
A}})=\mathrm{argmin}_{\boldsymbol{\beta}_{\tilde{\cal
A}}}-\ell(\boldsymbol{\beta}_{\tilde{\cal A}})+\rho\sum_{j\in\tilde{\cal
A}}\ln(\eta+|\beta_{j}|)$ and $[d^{2}h(\boldsymbol{\beta}_{\tilde{\cal
A}})]_{jk}=[-d^{2}\ell(\boldsymbol{\beta}_{\tilde{\cal
A}})]_{jk}+\rho(\eta+|\beta_{j}|)^{-2}1_{\\{j=k\\}}$ for $j,k\in\mathcal{A}$.
Then the empirical Bayes criterion is
$\displaystyle\mathrm{EB}(\rho,\eta)$ $\displaystyle=$ $\displaystyle-\ln
p(\boldsymbol{y}|\tilde{\cal A})$ $\displaystyle\approx$ $\displaystyle q\ln
2-q(\rho-1)\ln\eta-q\ln(\rho-1)+\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal
A}})-\frac{q}{2}\ln(2\pi)+\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}})$ $\displaystyle=$
$\displaystyle-q\ln\left\\{\left(\frac{\pi}{2}\right)^{1/2}(\rho-1)\eta^{\rho-1}\right\\}+\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})+\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}).$
Now consider the linear model with unknown variance $\sigma^{2}$,
$\displaystyle\pi(\boldsymbol{\beta}_{\cal
A},\boldsymbol{y}|\alpha,\eta,\sigma^{2})$ $\displaystyle=$
$\displaystyle(2\pi\sigma^{2})^{-n/2}\left(\frac{\alpha\eta^{\alpha}}{2}\right)^{q}$
$\displaystyle\times\exp\left\\{-\frac{\|\boldsymbol{y}-\boldsymbol{X}_{\cal
A}\boldsymbol{\beta}_{\cal
A}\|_{2}^{2}+2\sigma^{2}(\alpha+1)\sum_{j\in\mathcal{A}}\ln(\eta+|\beta_{j}|)}{2\sigma^{2}}\right\\}$
$\displaystyle=$
$\displaystyle(2\pi\sigma^{2})^{-n/2}\left\\{\frac{(\rho/\sigma^{2}-1)\eta^{\rho/\sigma^{2}-1}}{2}\right\\}^{q}\exp\\{-h(\boldsymbol{\beta}_{\cal
A})/\sigma^{2}\\},$
where $\rho=\sigma^{2}(\alpha+1)$ and $h(\boldsymbol{\beta}_{\cal
A})=\|\boldsymbol{y}-\boldsymbol{X}_{\cal A}\boldsymbol{\beta}_{\cal
A}\|_{2}^{2}/2+\rho\sum_{j\in\mathcal{A}}\ln(\eta+|\beta_{j}|)$. The Laplace
approximation to the normalizing constant is then given by
$\ln p(\boldsymbol{y}|\tilde{\cal
A},\sigma^{2})\approx\ln\pi(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}},\boldsymbol{y}|\tilde{\cal
A},\sigma^{2})+\frac{q}{2}\ln(2\pi)-\frac{1}{2}\log\det[\sigma^{-2}d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}})],$
which suggests the empirical Bayes criterion
$\displaystyle\mathrm{EB}(\eta,\rho|\sigma^{2})$ $\displaystyle\approx$
$\displaystyle\frac{n-q}{2}\ln(2\pi\sigma^{2})+q\ln
2-q\left(\frac{\rho}{\sigma^{2}}-1\right)\ln\eta-q\ln\left(\frac{\rho}{\sigma^{2}}-1\right)$
$\displaystyle+\frac{\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})}{\sigma^{2}}+\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}).$
Given $\tilde{\cal A}$, we can easily compute the value $\sigma^{2}$ that
minimizes the right-hand side
$\tilde{\sigma}^{2}=\mathrm{argmin}_{\sigma^{2}}\left\\{\frac{n-q}{2}\ln\sigma^{2}-\frac{q\rho}{\sigma^{2}}\ln\eta-q\ln\left(\frac{\rho}{\sigma^{2}}-1\right)+\frac{\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})}{\sigma^{2}}\right\\}.$
∎
### 3.2 Power Family
The power family penalty is induced by an exponential power prior on the
regression coefficients
$\displaystyle\pi(\beta|\rho,\eta)=\frac{\eta\rho^{1/\eta}}{2\Gamma(1/\eta)}e^{-\rho|\beta|^{\eta}},\hskip
14.45377pt\rho,\eta>0.$
The unnormalized posterior of the regression coefficients given a model $\cal
A$ can be written as
$\displaystyle\pi(\boldsymbol{\beta}_{\cal A},\boldsymbol{y}|\rho,\eta)$
$\displaystyle=$
$\displaystyle\left(\frac{\eta\rho^{1/\eta}}{2\Gamma(1/\eta)}\right)^{q}\exp\left\\{\ell(\boldsymbol{\beta}_{\cal
A})-\rho\sum_{j\in\mathcal{A}}|\beta_{j}|^{\eta}\right\\}$ (10)
$\displaystyle=$
$\displaystyle\left(\frac{\eta\rho^{1/\eta}}{2\Gamma(1/\eta)}\right)^{q}\exp\left\\{-h(\boldsymbol{\beta}_{\cal
A})\right\\}$
Again the Laplace approximation to the posterior $p(\boldsymbol{y}|\hat{\cal
A})$ yields the following empirical Bayes criterion for power family penalized
regression.
###### Proposition 3.2.
An empirical Bayes criterion for the power family penalized regression is
$\displaystyle\mathrm{EB}_{\mathrm{PF}}(\rho,\eta)\equiv-q\ln\frac{\sqrt{\pi}\eta\rho^{1/\eta}}{\sqrt{2}\Gamma(1/\eta)}+h(\tilde{\boldsymbol{\beta}})+\frac{1}{2}\log\det\boldsymbol{H}_{\cal
A}(\tilde{\boldsymbol{\beta}}).$
For linear regression, it becomes
$\displaystyle\mathrm{EB}_{\mathrm{PF}}(\rho,\eta)\equiv-q\ln\frac{\sqrt{\pi}\eta\rho^{1/\eta}}{\sqrt{2}\Gamma(1/\eta)}+\left(\frac{n-q}{2}-\frac{q}{\eta}\right)\left\\{1+\ln\frac{h(\tilde{\boldsymbol{\beta}})}{(n-q)/2+q/\eta}\right\\}+\frac{1}{2}\log\det\boldsymbol{H}_{\cal
A}(\tilde{\boldsymbol{\beta}}).$
###### Proof.
Given a certain model $\tilde{\cal A}$ observed at $(\rho,\eta)$, the Laplace
approximation to the normalizing constant is given by
$\ln p(\boldsymbol{y}|\tilde{\cal
A})\approx\ln\pi(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}},\boldsymbol{y}|\tilde{\cal A})+\frac{q}{2}\ln(2\pi)-\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}})$
where $\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}}=\mathrm{argmin}_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal
A}})=\mathrm{argmin}_{\boldsymbol{\beta}_{\tilde{\cal
A}}}-\ell(\boldsymbol{\beta}_{\tilde{\cal
A}})+\rho\sum_{j\in\mathcal{A}}|\beta_{j}|^{\eta}$ and
$[d^{2}h(\boldsymbol{\beta}_{\tilde{\cal
A}})]_{jk}=[-d^{2}\ell(\boldsymbol{\beta}_{\tilde{\cal
A}})]_{jk}+\rho\eta(\eta-1)|\beta_{j}|^{\eta-2}1_{\\{j=k\\}}$ for
$j,k\in\mathcal{A}$. Then
$\displaystyle\ln p(\boldsymbol{y}|\tilde{\cal A})$ $\displaystyle\approx$
$\displaystyle\frac{q}{2}\ln(2\pi)-q\ln
2+q\ln\eta+\frac{q}{\eta}\ln\rho-q\ln\Gamma(1/\eta)$
$\displaystyle-\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})-\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}),$
which yields
$\mbox{EB}(\rho,\eta)\equiv-q\ln\frac{\sqrt{\pi}\eta\rho^{1/\eta}}{\sqrt{2}\Gamma(1/\eta)}+\min_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})+\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}).$
Now consider the linear model case,
$\displaystyle\pi(\boldsymbol{\beta}_{\cal
A},\boldsymbol{y}|\rho,\eta,\sigma^{2})$ $\displaystyle=$
$\displaystyle(2\pi\sigma^{2})^{-n/2}\left(\frac{\eta\rho^{1/\eta}}{2\sigma^{2/\eta}\Gamma(1/\eta)}\right)^{q}\exp\left\\{-\frac{\|\boldsymbol{y}-\boldsymbol{X}_{\cal
A}\boldsymbol{\beta}_{\cal
A}\|_{2}^{2}/2+\rho\sum_{j\in\mathcal{A}}|\beta_{j}|^{\eta}}{\sigma^{2}}\right\\}$
$\displaystyle=$
$\displaystyle(2\pi\sigma^{2})^{-n/2}\left(\frac{\eta\rho^{1/\eta}}{2\sigma^{2/\eta}\Gamma(1/\eta)}\right)^{q}\exp\left\\{-h(\boldsymbol{\beta}_{\cal
A})/\sigma^{2}\right\\},$
where $h(\boldsymbol{\beta}_{\cal A})=\|\boldsymbol{y}-\boldsymbol{X}_{\cal
A}\boldsymbol{\beta}_{\cal
A}\|_{2}^{2}/2+\rho\sum_{j\in\mathcal{A}}|\beta_{j}|^{\eta}$. The Laplace
approximation to the normalizing constant at an estimated model $\tilde{A}$ is
then given by
$\ln p(\boldsymbol{y}|\tilde{\cal
A},\sigma^{2})\approx\ln\pi(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}},\boldsymbol{y}|\tilde{\cal
A},\sigma^{2})+\frac{q}{2}\ln(2\pi)-\frac{1}{2}\log\det[\sigma^{-2}d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}})],$
where $\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}}=\mathrm{argmin}_{\boldsymbol{\beta}_{\tilde{\cal
A}}}h(\boldsymbol{\beta}_{\tilde{\cal A}})$ and
$[d^{2}h(\boldsymbol{\beta}_{\tilde{\cal
A}})]_{jk}=\boldsymbol{x}^{\prime}_{j}\boldsymbol{x}_{k}+\rho\eta(\eta-1)|\tilde{\beta}_{j}|^{\eta-2}1_{\\{j=k\\}}$
for $j,k\in\tilde{\cal A}$. Then
$\displaystyle\ln p(\boldsymbol{y}|\tilde{\cal A},\sigma^{2})$
$\displaystyle\approx$ $\displaystyle
q\ln\frac{\sqrt{\pi}\eta\rho^{1/\eta}}{\sqrt{2}\Gamma(1/\eta)}-\left(\frac{n-q}{2}+\frac{q}{\eta}\right)\ln\sigma^{2}-\frac{h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}})}{\sigma^{2}}-\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}})-\frac{n}{2}\ln 2\pi.$
Plugging in the maximizing $\sigma^{2}$
$\tilde{\sigma}^{2}=\frac{h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}})}{(n-q)/2+q/\eta}$
and omitting the constant term $(n\ln 2\pi)/2$, we obtain
$\mbox{EB}(\rho,\eta)\equiv-q\ln\frac{\sqrt{\pi}\eta\rho^{1/\eta}}{\sqrt{2}\Gamma(1/\eta)}+\left(\frac{n-q}{2}+\frac{q}{\eta}\right)\left(1+\ln\frac{h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal
A}})}{(n-q)/2+q/\eta}\right)+\frac{1}{2}\log\det
d^{2}h(\tilde{\boldsymbol{\beta}}_{\tilde{\cal A}}).$
∎
## 4 Further Applications
The generality of (1) invites numerous applications beyond variable selection.
After reparameterization, many generalized regularization problems are subject
to the path following and empirical Bayes model selection procedure developed
in the previous two sections. In this section we briefly discuss some further
applications.
The recent articles (Tibshirani and Taylor, 2011; Zhou and Lange, 2011; Zhou
and Wu, 2011) consider the generalized $\ell_{1}$ regularization problem
$\displaystyle\min_{\boldsymbol{\beta}}f(\boldsymbol{\beta})+\rho\|\boldsymbol{V}\boldsymbol{\beta}\|_{1}+\rho\|\boldsymbol{W}\boldsymbol{\beta}\|_{+},$
where $\|\boldsymbol{a}\|_{+}=\sum_{i}\max\\{a_{i},0\\}$ is the sum of
positive parts of its components. The first regularization term enforces
equality constraints among coefficients at large $\rho$ while the second
enforces inequality constraints. Applications range from $\ell_{1}$ penalized
GLMs, shape restricted regressions, to nonparametric density estimation. For
more parsimonious and unbiased solutions, generalized sparse regularization
can be proposed
$\displaystyle\min_{\boldsymbol{\beta}}f(\boldsymbol{\beta})+\sum_{i=1}^{r}P(|\boldsymbol{v}_{i}^{t}\boldsymbol{\beta}|,\rho)+\sum_{j=1}^{s}P_{+}(\boldsymbol{w}_{j}^{t}\boldsymbol{\beta},\rho),$
(11)
where $P$ is a non-convex penalty function (power, double Pareto, SCAD, MC+,
etc.) and $P_{+}(t,\rho)=P(t,\rho)$ for $t\geq 0$ and $P(0,\rho)$ otherwise.
Devising an efficient path algorithm for (11) is hard in general. However,
when $\\{\boldsymbol{v}_{i}\\}$ and $\\{\boldsymbol{w}_{j}\\}$ are linearly
independent, it can be readily solved by our path algorithm via a simple
reparameterization. For ease of presentation, we only consider equality
regularization here. Let the matrix $\boldsymbol{V}\in\mathbb{R}^{r\times p}$
collect $\boldsymbol{v}_{i}$ in its rows. The assumption of full row rank of
$\boldsymbol{V}$ implies $r\leq p$. The trick is to reparameterize
$\boldsymbol{\beta}$ by
$\boldsymbol{\gamma}=\tilde{\boldsymbol{V}}\boldsymbol{\beta}$ where
$\tilde{\boldsymbol{V}}\in\mathbb{R}^{p\times p}$ is the matrix
$\boldsymbol{V}$ appended with extra rows such that $\tilde{\boldsymbol{V}}$
has full column rank. Then original coefficients $\boldsymbol{\beta}$ can be
recovered from the reparameterized ones $\boldsymbol{\gamma}$ via
$\boldsymbol{\beta}=(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{\gamma}$.
The reparameterized regularization problem is given by
$\displaystyle\min_{\boldsymbol{\gamma}}f[(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{\gamma}]+\sum_{j=1}^{r}P(|\gamma_{j}|,\rho)$
(12)
which is amenable to Algorithm 1. Note that $f$ remains convex and twice
differentiable under affine transformation of variables.
Regularization matrix $\boldsymbol{V}$ with full row rank appears in numerous
applications. For fused lasso, the regularization matrix
$\displaystyle\boldsymbol{V}_{1}=\left(\begin{array}[]{rrrrr}-1&1&\\\
&\ddots&\ddots\\\ &&-1&1\end{array}\right)$
has full row rank. In polynomial trend filtering (Kim et al., 2009; Tibshirani
and Taylor, 2011), order $d$ finite differences between successive regression
coefficients are penalized. Fused lasso corresponds to $d=1$ and the general
polynomial trend filtering invokes regularization matrix
$\boldsymbol{V}_{d}=\boldsymbol{V}_{d-1}\boldsymbol{V}_{1}$, which again has
full row rank. In Section 5.3, cubic trend filtering for logistic regression
is demonstrated on a financial data set. In all of these applications the
regularization matrix $\boldsymbol{V}$ is highly sparse and structured. The
back transformation
$(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{\gamma}$
in (12) is cheap to compute using a pre-computed sparse Cholesky factor of
$\boldsymbol{V}^{t}\boldsymbol{V}$. The design matrix in terms of variable
$\boldsymbol{\gamma}$ is
$\boldsymbol{X}(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}$. In
contrast to the regular variable selection problem, it shall not be assumed to
be centered and scaled.
## 5 Examples
Various numerical examples in this section illustrate the path seeking
algorithm and empirical Bayes model selection procedure developed in this
article. The first two classical data sets show the mechanics of the path
following for linear and logistic regressions and compare the model fit and
prediction performance under various penalties. The third example illustrates
the application of path algorithm and empirical Bayes procedure to cubic trend
filtering in logistic regression using a financial data set. The last
simulation example evaluates the computational efficiency of the path
algorithm in a large $p$ small $n$ setting. Run times are displayed whenever
possible to indicate the efficiency of our path following algorithm. The
algorithm is run on a laptop with Intel Core i7 M620 2.66GHz CPU and 8 GB RAM.
For reproducibility the code for all examples is available on the first
author’s web site.
### 5.1 Linear regression: Prostate cancer data
The first example concerns the classical prostate cancer data in (Hastie et
al., 2009). The response variable is logarithm of prostate specific antigen
(lpsa) and the seven predictors are the logarithm of cancer volume (lweight),
age, the logarithm of the amount of benign prostatic hyperplasia (lbph),
seminal vesicle invasion (svi), the logarithm of capsular penetration (lcp),
Gleason score (gleason), and percent of Gleason scores 4 or 5 (pgg45). The
data set contains 97 observations and is split into a training set of size 67
and a test set of 30 observations.
Figure 3 displays the solution paths of linear regression with nine
representative penalties on the training set. Discontinuities occur in the
paths from power family with $\eta=0.5$, continuous log penalty, and the log
penalty with $\eta=1$. In contrast, the lasso solution path, from either
enet(1) or power(1), is continuous and piecewise linear. Figure 3 also
illustrates the trade-off between continuity and unbiasedness. Using convex
penalties, such as enet $(\eta=1,1.5,2)$ and power $(\eta=1)$, guarantees the
continuity of solution path but causes bias in the estimates along the
solution path. For a non-convex penalty such as power $(\eta=0.5)$, estimates
are approximately unbiased once selected. However this can only be achieved by
allowing discontinuities along the path.
Figure 3: Solution paths for the prostate cancer data.
To compare the model fit along the paths, it is more informative to plot the
explained variation versus model dimension along the solution paths (Friedman,
2008). Upper panels of Figure 4 display such plots for the enet, power, and
log penalties at various penalty parameter values $\eta$. $y$-axis is the
proportion $R^{2}(\rho)/R^{2}(0)$, i.e., the $R^{2}$ from the path solutions
scaled by the maximum explained variation $R^{2}(0)$. Results for other
penalties (MC+, SCAD) are not shown for brevity. Non-convex penalties show
clear advantage in terms of higher explanatory power using fewer predictors.
The model fit of path solutions in the test set shows similar patterns to
those in Figure 2. To avoid repetition, they are not displayed here.
To evaluate the prediction performance, the prediction mean squared errors
(MSE) on the test set from the solution paths are shown in the lower panels of
Figure 4. Different classes of penalties all achieve the best prediction error
of 0.45 with 4-6 predictors. It is interesting to note the highly concave
penalties such as power ($\eta=0.2$) do not achieve the best prediction error
along the path. Lasso and moderately concave penalties are quite competitive
in achieving the best prediction error along the path. Convex penalties like
enet with $\eta>1$ tend to admit too many predictors without achieving the
best error rate.
$\begin{array}[]{cc}\includegraphics[width=180.67499pt]{prostate_R2_power}&\includegraphics[width=180.67499pt]{prostate_R2_log}\\\
\includegraphics[width=180.67499pt]{prostate_mse_power}&\includegraphics[width=180.67499pt]{prostate_mse_log}\end{array}$
Figure 4: Upper panels: $R^{2}$ vs model dimension from various penalties for
the prostate cancer data. Lower panels: Prediction mean square error (MSE) vs
model dimension from various penalties for the prostate cancer data.
### 5.2 Logistic regression: South Africa heart disease data
For demonstration of logistic regression, we again use the classical South
Africa heart disease data set in (Hastie et al., 2009). This data set has
$n=462$ observations measured on 7 predictors. The response variable is binary
(heart disease or not). We split the data set into a training set with 312
data points and a test set with 150 data points. Solution paths are obtained
for the training data set from various penalties and are displayed in Figure
5. Similar patterns are observed as those for the prostate cancer linear
regression example. The discontinuities for concave penalties such as power
($\eta=0.5$) and log penalty ($\eta=1$) lead to less biased estimates along
the paths. The plots of explained deviance versus model size for selected
penalties are given in the upper panels of Figure 6. Solutions from concave
penalties tend to explain more deviance with fewer predictors than lasso and
enet with $\eta>1$. Deviance plots for the test set show a similar pattern.
Prediction power of the path solutions is evaluated on the test data set and
the prediction MSEs are reported in the lower panels of Figure 6. The highly
concave penalties such as power $(\eta<1)$ and log penalty $(\eta=0.1)$ are
able to achieve the best prediction error rate 0.425 with 5 predictors. Convex
penalties and less concave ones perform worse in prediction power, even with
more than 5 predictors.
Figure 5: Solution paths for the South Africa heart disease data.
$\begin{array}[]{cc}\includegraphics[width=180.67499pt]{saheart_dev_power}&\includegraphics[width=180.67499pt]{saheart_dev_log}\\\
\includegraphics[width=180.67499pt]{saheart_mse_power}&\includegraphics[width=180.67499pt]{saheart_mse_log}\end{array}$
Figure 6: Upper panels: Negative deviance vs model dimension from various
penalties for the South Africa heart disease data. Lower panels: Prediction
mean square error (MSE) vs model dimension from various penalties for the
South Africa heart disease data.
### 5.3 Logistic regression with cubic trend filtering: M&A data
The third example illustrates generalized regularization with logistic
regression. We consider a merger and acquisition (M&A) data set studied in
recent articles (Zhou and Wu, 2011; Fan et al., 2011). This data set
constitutes $n=1,371$ US companies with a binary response variable indicating
whether the company becomes a leveraged buyout (LBO) target ($y_{i}=1$) or not
($y_{i}=0$). Seven covariates are recorded for each company. Table 1 lists the
7 predictors and their p-values in the classical linear logistic regression.
Predictors ‘long term investment’, ‘log market equity’, and ‘return S&P 500
index’ show no significance while the finance theory indicates otherwise.
Predictor | p-value
---|---
Cash Flow | 0.0019
Case | 0.0211
Long Term Investment | 0.5593
Market to Book Ratio | 0.0000
Log Market Equity | 0.5099
Tax | 0.0358
Return S&P 500 Index | 0.2514
Table 1: Predictors and their p-values from the linear logistic regression.
To explore the possibly nonlinear effects of these quantitative covariates, we
discretize each predictor into, say, 10 bins and fit a logistic regression.
The first bin of each predictor is used as the reference level and effect
coding is applied to each discretized covariate. The circles (o) in Figure 8
denote the estimated coefficients for each bin of each predictor and hint at
some interesting nonlinear effects. For instance, the chance of being an LBO
target seems to monotonically decease with market-to-book ratio and be
quadratic as a function of log market equity. Regularization can be utilized
to borrow strength between neighboring bins. The recent paper (Zhou and Wu,
2011) applies cubic trend filtering to the 7 covariates using $\ell_{1}$
regularization. Here we demonstrate a similar regularization using a non-
convex penalty. Specifically, we minimize a regularized negative logistic log-
likelihood of the form
$\displaystyle-l(\boldsymbol{\beta}_{1},\ldots,\boldsymbol{\beta}_{7})+\sum_{j=1}^{7}P_{\eta}(\boldsymbol{V}_{j}\boldsymbol{\beta}_{j},\rho),$
where $\boldsymbol{\beta}_{j}$ is the vector of regression coefficients for
the $j$-th discretized covariate. The matrices in the regularization terms are
specified as
$\displaystyle\boldsymbol{V}_{j}$
$\displaystyle=\left(\begin{array}[]{rrrrrrrr}-1&2&-1\\\ 1&-4&6&-4&1\\\
&1&-4&6&-4&1&\\\ &&&\ddots&\ddots&\ddots\\\ &&1&-4&6&-4&1&\\\
&&&&&-1&2&-1\end{array}\right),$
which penalizes the fourth order finite differences between the bin estimates.
Thus, as $\rho$ increases, the coefficient vectors for each covariate tend to
be piecewise cubic with two ends being linear, mimicking the natural cubic
spline. This is one example of polynomial trend filtering (Kim et al., 2009;
Tibshirani and Taylor, 2011) applied to logistic regression. Similar to semi-
parametric regressions, regularizations in polynomial trend filtering ‘let the
data speak for themselves’. In contrast, the bandwidth selection in semi-
parametric regression is replaced by parameter tuning in regularizations. The
number and locations of knots are automatically determined by the tuning
parameter which is chosen according to a model selection criterion. The left
panel of Figure 7 displayed the solution path delivered by the power penalty
with $\eta=0.5$. It bridges the unconstrained estimates (denoted by o) to the
constrained estimates (denoted by +). The right panel of Figure 7 shows the
empirical Bayes criterion along the path. The dotted line in Figure 8 is the
solution with smallest empirical Bayes criterion. It mostly matches the fully
regularized solution except a small ‘dip’ in the middle range of ‘log market
equity’. The classical linear logistic regression corresponds to the
restricted model where all bins for a covariate coincide. A formal analysis of
deviance indicates that the regularized model at $\rho=2.5629$ is significant
with respect to this null model with p-value 0.0023.
The quadratic or cubic like pattern in the effects of predictors ‘long term
investment’, ‘log market equity’, and ‘return S&P 500 index’ revealed by the
regularized estimates explain why they are missed by the classical linear
logistic regression. These patterns match some existing finance theories. For
instance, Log of market equity is a measure of company size. Smaller companies
are unpredictable in their profitability and extremely large companies are
unlikely to be an LBO target because LBOs are typically financed with a large
proportion of external debt. A company with a low cash flow is unlikely to be
an LBO target because low cash flow is hard to meet the heavy debt burden
associated with the LBO. On the other hand, a company carrying a high cash
flow is likely to possess a new technology. It is risky to acquire such firms
because it is hard to predict their profitability. The tax reason is obvious
from the regularized estimates. The more tax the company is paying, the more
tax benefits from an LBO.
$\begin{array}[]{cc}\includegraphics[width=216.81pt]{manda_solpath_power}&\includegraphics[width=216.81pt]{manda_ebcpath_power}\end{array}$
Figure 7: Regularized logistic regression on the M&A data. Left: The
trajectory of solution path from power penalty with $\eta=0.5$. Right:
Empirical Bayes criterion along the path. Vertical lines indicate the model
selected by the empirical Bayes criterion.
$\begin{array}[]{cc}\includegraphics[width=433.62pt]{manda_estimates}\end{array}$
Figure 8: Snapshots of the path solution to the regularized logistic
regression on the M&A data set. The best model according to empirical Bayes
criterion (dotted line) most matches the fully regularized solution (line with
crosses) except that it has a dip in the middle part of the ‘log market
equity’ variable.
### 5.4 GLM sparse regression: large $p$, small $n$
In all of the previous examples, the number of observations $n$ exceeds the
number of parameters $p$. Our final simulation example evaluates the
performance of the path following algorithm and empirical Bayes procedure in a
large $p$ small $n$ setup for various generalized linear models (GLM). In each
simulation replicate, $n=200$ independent responses $y_{i}$ are simulated from
a normal (with unit variance), Poisson, and binomial distribution with mean
$\mu_{i}$ respectively. The mean $\mu_{i}$ is determined by a $p=10,000$
dimensional covariate $\boldsymbol{x}_{i}$ through a link function
$g(\mu_{i})=\alpha+\boldsymbol{x}_{i}^{t}\boldsymbol{\beta}$ where $\alpha$ is
the intercept. Canonical links are used in the simulations. For the linear
model, $g(\mu)=\mu$. For the Poisson model, $g(\mu)=\ln\mu$. For the logistic
model, $g(\mu)=\log[\mu/(1-\mu)]$.
Numerous settings can be explored in this framework. For brevity and
reproducibility, we only display the results for a simple exemplary setup:
entries of covariate $\boldsymbol{x}_{i}$ are generated from iid standard
normal and the true regression coefficients are $\beta_{i}=3$ for
$i=1,\ldots,5$, $\beta_{i}=-3$ for $i=6,\ldots,10$, and $\alpha=\beta_{i}=0$
for $i=11,\ldots,10,000$. Results presented in Figure 9-12 are based on 100
simulation replicates. In each replicate, path following is carried out under
linear, Poisson, and logistic regression models coupled with power penalties
at $\eta=0.25,0.5,0.75,1$, representing a spectrum from (nearly) best subset
regression to lasso regression. Results for other penalties are not shown to
save space. Path following is terminated when at least 100 predictors are
selected or separation is detected in the Poisson or logistic models,
whichever occurs first. Results for the logistic sparse regression have to be
interpreted with caution due to frequent occurrence of complete separation
along solution paths. This is common in large $p$ small $n$ problems as the
chance of finding a linear combination of a few predictors that perfectly
predicts the $n=200$ binary responses is very high when there are $p=10,000$
candidate covariates. Therefore the results for logistic regression largely
reflect the quality of solutions when path following terminates at complete
separation.
Figure 9 displays the boxplots of run times at different combinations of the
GLM model and penalty value $\eta$. The majority of runs take less than one
minute across all models, except Poisson regression with $\eta=0.75$. The run
times in this setting display large variability with a median around 50
seconds. Logistic regression with non-convex penalties ($\eta=0.25,0.5,0,75$)
has shorter run times than lasso penalty ($\eta=1$) due to complete separation
at early stages of path following.
Figures 10 and 11 display the false positive rate (FPR) and false negative
rate (FNR) of the model selected by the empirical Bayes procedure at different
combinations of GLM model and penalty value $\eta$. FPR (type I error rate) is
defined as the proportion of false positives in the selected model among all
true negatives (9990 in this case). FNR (type II error rate) is defined as the
proportion of false negatives in the selected model among all true positives
(10 in this case). These two numbers give a rough measure of model selection
performance. For all three GLM models, power penalties with larger $\eta$
(close to convexity) tend to select more predictors, leading to significantly
higher FPR. On the other hand, the median FNR appears not significantly
improved in larger $\eta$ cases, although they admit more predictors. This
indicates the overall improved model selection performance of non-convex
penalties.
More interesting is the mean square error (MSE) of the parameter estimate
$\tilde{\boldsymbol{\beta}}$ at the model selected by the empirical Bayes
procedure. MSE is defined as
[$\sum_{j}(\tilde{\beta}_{j}-\beta)^{2}/p]^{1/2}$. Figure 12 shows that lasso
($\eta=1$) has risk properties comparable to the non-convex penalties,
although it is a poor model selector in terms of FPR and FNR.
We should keep in mind that these results are particular to the specific
simulation setting we presented here and may vary across numerous factors such
as pairwise correlations between the covariates, signal to noise ratio, sample
size $n$ and dimension $p$, penalty type, etc. We hope that the tools
developed in this article facilitate such comparative studies. The generality
of our method precludes extensive numerical comparison with current methods as
only a few software packages are available for the special cases of (1). In
supplementary materials, we compare the run times of our algorithm to that of
Friedman et al. (2010) for the special case of lasso penalized GLM.
$\begin{array}[]{c}\includegraphics[width=325.215pt]{n100-p10000-timing}\end{array}$
Figure 9: Run times of path following for GLM sparse regression with power
penalties from 100 replicates. Problem size is $n=200$ and $p=10,000$.
$\begin{array}[]{c}\includegraphics[width=289.07999pt]{n100-p10000-fpr}\end{array}$
Figure 10: False positive rate (FPR) of the GLM sparse model selected by the
empirical Bayes criterion.
$\begin{array}[]{c}\includegraphics[width=289.07999pt]{n100-p10000-fnr}\end{array}$
Figure 11: False negative rate (FNR) of the GLM sparse model selected by the
empirical Bayes criterion.
$\begin{array}[]{c}\includegraphics[width=289.07999pt]{n100-p10000-mse}\end{array}$
Figure 12: Mean square error (MSE) of the parameter estimate from the model
selected by the empirical Bayes criterion.
## 6 Discussion
In this article we propose a generic path following algorithm for any
combination of a convex loss function and a penalty function that satisfies
mild conditions. Although motivated by the unpublished work by Friedman
(2008), our algorithm turns out to be different from his general path seeking
(GPS) algorithm. Further research is needed on the connection between the two.
The ODE approach for path following tracks the solution smoothly and avoids
the need to choose a fixed step size as required by most currently available
regularization path algorithms.
Motivated by a shrinkage prior in the Bayesian setting, we derived an
empirical Bayes procedure that allows quick search for a model and the
corresponding tuning parameter along the solution paths from a large class of
penalty functions. All necessary quantities for the empirical Bayes procedure
naturally arise in the path algorithm.
Besides sparse regression, simple reparameterization extends the applicability
of the path algorithm to many more generalized regularization problems. The
cubic trend filtering example with the M&A data illustrates the point.
Our numerical examples illustrate the working mechanics of the path algorithm
and properties of different penalties. A more extensive comparative study of
the penalties in various situations is well deserved. The tools developed in
this article free statisticians from the often time consuming task of
developing optimization algorithms for specific loss and penalty combination.
Interested readers are welcome to use the SparseReg toolbox freely available
on the first author’s web site.
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## Supplementary Materials
### Derivatives of penalty functions
First two derivatives of commonly used penalty functions are listed in Table
2.
Penalty Function | $P_{\eta}(|\beta|,\rho)$ | $\frac{\partial P_{\eta}(|\beta|,\rho)}{\partial|\beta|}$
---|---|---
Power Family $\eta\in(0,2]$ | $\rho|\beta|^{\eta}$ | $\rho\eta|\beta|^{\eta-1}$
Elastic Net, $\eta\in[1,2]$ | $\rho[(\eta-1)\beta^{2}/2+(2-\eta)|\beta|]$ | $\rho[(\eta-1)|\beta|+(2-\eta)]$
Log, $\eta>0$ | $\rho\ln(\eta+|\beta|)$ | $\rho(\eta+|\beta|)^{-1}$
Continuous Log | $\rho\ln(\sqrt{\rho}+|\beta|)$ | $\rho(\sqrt{\rho}+|\beta|)^{-1}$
SCAD, $\eta>2$ | See (2) | $\rho\left\\{1_{\\{|\beta|\leq\rho\\}}+\frac{(\eta\rho-|\beta|)_{+}}{(\eta-1)\rho}1_{\\{|\beta|>\rho\\}}\right\\}$
MC+, $\eta\geq 0$ | See (3) | $\rho\left(1-\frac{|\beta|}{\rho\eta}\right)_{+}$
| $\frac{\partial^{2}P_{\eta}(|\beta|,\rho)}{\partial|\beta|^{2}}$ | $\frac{\partial^{2}P_{\eta}(|\beta|,\rho)}{\partial|\beta|\,\partial\rho}$
Power Family $\eta\in(0,2]$ | $\rho\eta(\eta-1)|\beta|^{\eta-2}$ | $\eta|\beta|^{\eta-1}$
Elastic Net, $\eta\in[1,2]$ | $\rho(\eta-1)$ | $(\eta-1)|\beta|+(2-\eta)$
Log, $\eta>0$ | $-\rho(\eta+|\beta|)^{-2}$ | $(\eta+|\beta|)^{-1}$
Continuous Log | $-\rho(\sqrt{\rho}+|\beta|)^{-2}$ | $(\sqrt{\rho}+\beta)^{-1}-\frac{1}{2}\sqrt{\rho}(\sqrt{\rho}+|\beta|)^{-2}$
SCAD, $\eta>2$ | $-(\eta-1)^{-1}1_{\\{|\beta|\in[\rho,\eta\rho)\\}}$ | $1_{\\{|\beta|<\rho\\}}+\eta(\eta-1)^{-1}1_{\\{|\beta|\in[\rho,\eta\rho)\\}}$
MC+, $\eta\geq 0$ | $-\eta^{-1}1_{\\{|\beta|<\rho\eta\\}}$ | $1_{\\{|\beta|<\rho\eta\\}}$
Table 2: Some commonly used penalty functions and their derivatives.
### Thresholding Formula for Least Squares with Orthogonal Design
We drop subscript $j$ henceforth to prevent clutter.
1. 1.
For the DP$(\eta)$ penalty, the objective function becomes
$\displaystyle\frac{a}{2}(\beta-b)^{2}+\rho\ln(\eta+|\beta|).$
Setting derivative to 0, we find that the optimal solution is given by
$\displaystyle\hat{\beta}(\rho)$
$\displaystyle=\begin{cases}\frac{(|b|-\eta)+[(|b|+\eta)^{2}-4\rho/a]^{1/2}}{2}\text{sgn}(b)&\rho\in[0,|ab\eta|]\\\
0\text{ or
}\frac{(|b|-\eta)+[(|b|+\eta)^{2}-4\rho/a]^{1/2}}{2}\text{sgn}(b)&\rho\in(|ab\eta|,a(\eta+|b|)^{2}/4)\\\
0&\rho\in[a(\eta+|b|)^{2}/4,\infty)\end{cases}.$
The ambiguous case reflects the difficulty with non-convex minimization and
has to be resolved by comparing the objective function values at the two
points. Moving from one local minimum at $0$ to the other non-zero one results
in a jump somewhere in the interval $[|ab\eta|,a(\eta+|b|)^{2}/4]$. Note that
at $\rho=|ab\eta|$, $\beta(\rho)=[(|b|-\eta)+||b|-\eta|]/2$, which is zero
when $\eta\geq|b|$. Therefore, the path is continuous whenever $\eta\geq|b|$.
2. 2.
For the continuous DP penalty, the objective function is
$\displaystyle\frac{a}{2}(\beta-b)^{2}+\rho\ln(\sqrt{\rho}+|\beta|)$
with the solution
$\displaystyle\hat{\beta}(\rho)$
$\displaystyle=\frac{(|b|-\sqrt{\rho})+[(|b|+\sqrt{\rho})^{2}-4\rho/a]^{1/2}}{2}\text{sgn}(b)$
for $\rho\leq a^{2}b^{2}$. Note at $\rho=a^{2}b^{2}$,
$\hat{\beta}(\rho)=[(b-ab)+|b-ab|]/2$, which is 0 when $a\geq 1$. Therefore,
when $a\geq 1$, the solution path is continuous
$\displaystyle\hat{\beta}(\rho)$
$\displaystyle=\begin{cases}\frac{(|b|-\sqrt{\rho})+[(|b|+\sqrt{\rho})^{2}-4\rho/a]^{1/2}}{2}\text{sgn}(b)&\rho\in[0,a^{2}b^{2}]\\\
0&\rho\in[a^{2}b^{2},\infty)\end{cases}.$
When $a<1$, the solution path is given by
$\displaystyle\hat{\beta}(\rho)$
$\displaystyle=\begin{cases}\frac{(|b|-\sqrt{\rho})+[(|b|+\sqrt{\rho})^{2}-4\rho/a]^{1/2}}{2}\text{sgn}(b)&\rho\in[0,\rho^{*}]\\\
0&\rho\in[\rho^{*},\infty)\end{cases}$
where $\rho^{*}>a^{2}b^{2}$ indicates the discontinuity point and shall be
determined numerically.
3. 3.
For power family, the objective function is
$\displaystyle\frac{a}{2}(\beta-b)^{2}+\rho|\beta|^{\eta}.$
For $\eta\in(0,1)$, the solution $\hat{\beta}(\rho)$ is the unique root of the
estimating equation
$\displaystyle a(\beta-b)+\rho\eta|\beta|^{\eta-1}\text{sgn}(\beta)=0$
or 0, whichever gives a smaller objective value. For $\eta=1$, it reduces to
the well-known soft thresholding operator for lasso
$\hat{\beta}(\rho)=\text{median}\\{\pm\rho/a+b,0\\}$. For the convex case
$\eta\in(1,2]$, the solution $\hat{\beta}(\rho)$ is always the (nonzero) root
of the estimating equation, i.e., no thresholding; only shrinkage occurs.
4. 4.
For elastic net, the objective function is
$\displaystyle\frac{a}{2}(\beta-b)^{2}+\rho[(\eta-1)\beta^{2}/2+(2-\eta)|\beta|]$
with a continuous path solution
$\displaystyle\hat{\beta}(\rho)=\text{median}\left\\{0,\frac{ab\pm\rho(2-\eta)}{a+\rho(\eta-1)}\right\\}.$
Again the lasso soft thresholding is recovered at $\eta=1$ and ridge shrinkage
is achieved at $\eta=2$.
5. 5.
For SCAD, the minimum of the penalized objective function over $[0,\rho]$ is
$\displaystyle\hat{\beta}_{1}(\rho)=\text{sgn}(b)\min\\{r,u_{1}\\}1_{r>0},$
where $r=(a|b|-\rho)/a$ and $u_{1}=\min\\{\rho,|b|\\}$, and the minimum over
$[\rho,\eta\rho]$ is
$\displaystyle\hat{\beta}_{2}(\rho)=\text{sgn}(b)\cdot\begin{cases}\rho
1_{\\{2r>\rho+u_{2}\\}}+u_{2}1_{\\{2r<\rho+u_{2}\\}}&a(\eta-1)<1\\\ \rho
1_{\\{\eta\rho\geq|b|\\}}+u_{2}1_{\\{\eta\rho<|b|\\}}&a(\eta-1)=1\\\ \rho
1_{\\{r\leq\rho\\}}+r1_{\\{r\in[\rho,u_{2}]\\}}+u_{2}1_{\\{r>u_{2}\\}}&a(\eta-1)>1\end{cases},$
where $u_{2}=\min\\{\eta\rho,|b|\\}$ and
$r=[ab(\eta-1)-\eta\rho]/[a(\eta-1)-1]$. When $|b|\leq\rho$, the solution is
$\hat{\beta}_{1}(\rho)$. When $|b|\in(\rho,\eta\rho]$, the solution is either
$\hat{\beta}_{1}(\rho)$ or $\hat{\beta}_{2}(\rho)$, whichever gives the
smaller penalized objective value. When $|b|>\eta\rho$, the solution is
$\hat{\beta}_{1}(\rho)$, $\hat{\beta}_{2}(\rho)$ or
$\hat{\beta}_{3}(\rho)=|b|$, whichever gives the smallest penalized objective
value.
6. 6.
For MC+, the path solution is either
$\displaystyle\hat{\beta}(\rho)=\text{sgn}(b)\cdot\begin{cases}b^{*}1_{\\{2r<b^{*}\\}}&a\eta<1\\\
b^{*}1_{\\{\rho<a|b|\\}}&a\eta=1\\\
\min\\{r,b^{*}\\}1_{\\{r>0\\}}&a\eta>1\end{cases},$
where $b^{*}=\min\\{\rho\eta,|b|\\}$ and $r=-(\rho-a|b|)/(a-\eta^{-1})$, or
$\hat{\beta}(\rho)=b$, whichever gives a smaller penalized objective value.
### Numerical Comparisons
Figure 13 displays the run times of lasso penalized GLM by the GLMNet package
in R (Friedman et al., 2010), which is the state-of-the-art method for
calculating the solution paths of GLM model with enet penalties. It applies
coordinate descent algorithm to a sequence of tuning parameters with warm
start. The simulation setup is same as in Section 5.4 and the top 100
predictors are requested from the path algorithm using its default setting. In
general GLMNet shows shorter run times than the ODE path algorithm (last
column of Figure 9). However a major difference is that GLMNet only computes
the solution at a finite number of tuning parameter values (100 by default),
while the ODE solver smoothly tracks the whole solution path.
$\begin{array}[]{c}\includegraphics[width=325.215pt]{glmnet_timing}\end{array}$
Figure 13: Run times of GLMNet for the lasso problems from 100 replicates.
Problem size is $n=200$ and $p=10,000$. Top 100 predictors are requested.
|
arxiv-papers
| 2012-01-17T15:18:25 |
2024-09-04T02:49:26.390708
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua Zhou and Artin Armagan and David B. Dunson",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1201.3528"
}
|
1201.3571
|
# A Generic Path Algorithm for Regularized Statistical Estimation
Hua Zhou
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203
E-mail: hua_zhou@ncsu.edu
Yichao Wu
Department of Statistics
North Carolina State University
Raleigh, NC 27695-8203
E-mail: wu@stat.ncsu.edu
###### Abstract
Regularization is widely used in statistics and machine learning to prevent
overfitting and gear solution towards prior information. In general, a
regularized estimation problem minimizes the sum of a loss function and a
penalty term. The penalty term is usually weighted by a tuning parameter and
encourages certain constraints on the parameters to be estimated. Particular
choices of constraints lead to the popular lasso, fused-lasso, and other
generalized $l_{1}$ penalized regression methods. Although there has been a
lot of research in this area, developing efficient optimization methods for
many nonseparable penalties remains a challenge. In this article we propose an
exact path solver based on ordinary differential equations (EPSODE) that works
for any convex loss function and can deal with generalized $l_{1}$ penalties
as well as more complicated regularization such as inequality constraints
encountered in shape-restricted regressions and nonparametric density
estimation. In the path following process, the solution path hits, exits, and
slides along the various constraints and vividly illustrates the tradeoffs
between goodness of fit and model parsimony. In practice, the EPSODE can be
coupled with AIC, BIC, $C_{p}$ or cross-validation to select an optimal tuning
parameter. Our applications to generalized $l_{1}$ regularized generalized
linear models, shape-restricted regressions, Gaussian graphical models, and
nonparametric density estimation showcase the potential of the EPSODE
algorithm.
Keywords: Gaussian graphical model, generalized linear model, lasso, log-
concave density estimation, ordinary differential equations, quasi-
likelihoods, regularization, shape restricted regression, solution path
## 1 Introduction
Regularization is a frequently used framework in statistics. Examples include
the lasso regression (Tibshirani, 1996; Chen et al., 2001) and the $l_{1}$
penalized generalized linear models (GLMs) among many others. For both the
lasso and the $l_{1}$ penalized GLMs, efficient solution path algorithms have
been proposed to ease the tuning of the regularization parameter (Osborne et
al., 2000; Efron et al., 2004; Park and Hastie, 2007). Yet extension to other
more general settings is nontrivial and has been an active research area.
In this article, we consider a general regularization framework
$\displaystyle\min_{\boldsymbol{\beta}\in\mathbb{R}^{p}}f(\boldsymbol{\beta})+\rho\|\boldsymbol{V}\boldsymbol{\beta}-\boldsymbol{d}\|_{1}+\rho\|\boldsymbol{W}\boldsymbol{\beta}-\boldsymbol{e}\|_{+},$
(1)
for which we propose an efficient exact path solver based on ordinary
differential equations (EPSODE). Here $f:\mathbb{R}^{p}\mapsto\mathbb{R}$ can
be any convex, smooth loss function of $\boldsymbol{\beta}\in\mathbb{R}^{p}$,
where $p>0$ is the dimensionality of the parameters. For any vector
$\boldsymbol{v}=(v_{i})$, $\|\boldsymbol{v}\|_{1}=\sum_{i}|v_{i}|$ denotes its
$l_{1}$ norm and $\|\boldsymbol{v}\|_{+}=\sum_{i}\max\\{v_{i},0\\}$ is the sum
of positive parts of its components. The EPSODE provides the exact solution
path to (1) as the tuning parameter $\rho$ varies.
### 1.1 Generality of (1)
The generality of (1) is two-fold. First $f$ can by any convex loss function.
For example, it can be the negative log-likelihood function of GLMs, negative
quasi-likelihood, the exponential loss function of the AdaBoost (Friedman et
al., 2000), or many other frequently used loss functions in statistics and
machine learning. Second we allow $\boldsymbol{V}$ and $\boldsymbol{W}$ to be
any regularization matrices of $p$ columns. This leads to broad applications.
In particular, the first regularization term
$\rho\|\boldsymbol{V}\boldsymbol{\beta}-\boldsymbol{d}\|_{1}$ encourages
equality constraints among parameters $\boldsymbol{\beta}$. When $\rho$ is
large enough, the minimizer $\boldsymbol{\beta}(\rho)$ of (1) satisfies
$\boldsymbol{V}\boldsymbol{\beta}(\rho)=\boldsymbol{d}$. For instance, when
$\boldsymbol{V}$ is the identity matrix and $\boldsymbol{d}={\bf 0}$, it
recovers the well-known lasso regression (Tibshirani, 1996; Chen et al.,
2001), which encourages sparsity of the estimates. When
$\displaystyle\boldsymbol{V}$
$\displaystyle=\left(\begin{array}[]{rrrrr}-1&1&\\\ &\ddots&\ddots\\\
&&-1&1\end{array}\right)$
and $\boldsymbol{d}={\bf 0}$, it corresponds to the fused-lasso penalty
(Tibshirani et al., 2005), which leads to smoothness among neighboring
regression coefficients. As we will show later, more complicated equality
constraints can be incorporated with properly designed $\boldsymbol{V}$ and
$\boldsymbol{d}$. On the other hand, the second regularization term
$\rho\|\boldsymbol{W}\boldsymbol{\beta}-\boldsymbol{e}\|_{+}$ enforces
regularization by inequality relations among regression coefficients. For
large enough $\rho$, the minimizer $\boldsymbol{\beta}(\rho)$ satisfies
$\boldsymbol{W}\boldsymbol{\beta}(\rho)\leq\boldsymbol{e}$. For instance,
setting $\boldsymbol{W}$ as the negative identity matrix and
$\boldsymbol{e}={\bf 0}$ encourages nonnegativity of the estimates, as
required in nonnegative least squares problems (Lawson and Hanson, 1987). In
the isotonic regression (Robertson et al., 1988; Silvapulle and Sen, 2005),
the estimates have to be nondecreasing. This can be achieved by the
regularization matrix
$\displaystyle\boldsymbol{W}$
$\displaystyle=\left(\begin{array}[]{rrrrr}1&-1&\\\ &\ddots&\ddots\\\
&&1&-1\end{array}\right)$
and $\boldsymbol{e}={\bf 0}$. More complicated constraints that occur in
shape-restricted regression and nonparametric regressions also can be
incorporated as we demonstrate in later examples.
In certain applications, both equality and inequality regularizations are
required. In that case, as shown in Section 2, at a large but finite $\rho$,
the minimizer $\boldsymbol{\beta}(\rho)$ coincides with the solution to the
following constrained optimization problem
$\displaystyle\min$ $\displaystyle f(\boldsymbol{\beta})$ s.t.
$\displaystyle\boldsymbol{V}\boldsymbol{\beta}=\boldsymbol{d}\mbox{ and
}\boldsymbol{W}\boldsymbol{\beta}\leq\boldsymbol{e}.$
Consequently EPSODE solves the linearly constrained estimation problem (1.1)
as a by-product. In this case, path following commences from the unconstrained
solution $\text{argmin}f(\boldsymbol{\beta})$ and ends at the constrained
solution to (1.1).
### 1.2 Previous Work
Several path algorithms have been devised for special cases of the general
regularization problem (1). For example, the homotopy method (Osborne et al.,
2000) and the least angle regression (LARS) procedure (Efron et al., 2004)
handle lasso penalized least squares problem. The solution path generated is
piecewise linear and illustrates the tradeoffs between goodness of fit and
sparsity. Rosset and Zhu (2007) give sufficient conditions for a solution path
to be piecewise linear and expand its applications to a wider range of loss
and penalty functions. Recently Tibshirani and Taylor (2011) devise a dual
path algorithm for generalized $l_{1}$ penalized least squares problems, which
is problem (1) with $f$ quadratic but without the second inequality
regularization term. Zhou and Lange (2011) consider (1) in full generality for
quadratic $f$. All these work concerns regularized linear regression for which
the solution path is piecewise linear. Several attempts have been made to path
following for regularized GLMs for which the solution path is no longer
piecewise linear. Park and Hastie (2007) propose a predictor-corrector
approach to approximate the lasso path for GLMs. Wu (2011) presents an
ordinary differential equation-based path algorithm which delivers the exact
path for lasso penalized GLMs. Friedman (2008) derives an approximate path
algorithm for any convex loss regularized by a separable, but not necessarily
convex penalty. Here a penalty function is called separable if its Hessian
matrix is diagonal. The separability restriction on the penalty term excludes
many important problems encountered in real applications.
Our proposed approach generalizes previous work in several aspects. First, it
works for any convex loss (or criterion) function. Second, it allows for any
type of regularization in terms of linear functions of parameters, equality or
inequality. Equality constrained regularizations include lasso, fused-lasso
and generalized $l_{1}$ penalty for example. Inequality constrained
regularizations are required in shape-restricted regression and nonparametric
log-concave density estimation. Last but not least, it is an exact path
algorithm.
### 1.3 A Motivating Example
For illustration, we consider a merger and acquisition (M&A) data set studied
in (Fan et al., 2011). This data set constitutes $n=1,371$ US companies with a
binary response variable indicating whether the company becomes a leveraged
buyout (LBO) target ($y_{i}=1$) or not ($y_{i}=0$). Seven covariates (1. cash
flow, 2. cash, 3. long term investment, 4. market to book ratio, 5. log market
equity, 6. tax, 7. return on S&P 500 index) are recorded for each company.
There have been intensive studies on the effects of these factors on the
probability of a company being a target for strategic mergers. Exploratory
analysis using linear logistic regression shows no significance in most
covariates.
To explore the possibly nonlinear effects of these quantitative covariates,
the varying-coefficient model (Hastie and Tibshirani, 1993) can be adopted
here. We discretize each predictor into, say, 10 bins and fit a logistic
regression. The first bin of each predictor is used as the reference level and
effect coding is applied to each discretized covariate. The circles (o) in
Figure 1 denote the estimated coefficients for each bin of each predictor and
hint at some interesting nonlinear effects. For instance, the chance of being
an LBO target seems to monotonically decease with market-to-book ratio and be
quadratic as a function of log market equity. Regularization can be utilized
to borrow strength between neighboring bins and gear solution towards clearer
patterns. To illustrate the flexibility of the regularization scheme (1), we
apply cubic trend filtering to 5 covariates (cash flow, cash, long term
investment, tax, return on S&P 500 index), impose the monotonicity (non-
increasing) constraint on the ‘market-to-book ratio’ covariate, and enforce
the concavity constraint on the ‘log market equity’ covariate. This can be
achieved by minimizing a regularized negative logistic log-likelihood of form
$\displaystyle-l(\boldsymbol{\beta}_{1},\ldots,\boldsymbol{\beta}_{7})+\rho\sum_{j\neq
4,5}\|\boldsymbol{V}_{j}\boldsymbol{\beta}_{j}\|_{1}+\rho\sum_{j=4,5}\|\boldsymbol{W}_{j}\boldsymbol{\beta}_{j}\|_{+},$
where $\boldsymbol{\beta}_{j}$ is the vector of regression coefficients for
the $j$-th discretized covariate. The matrices in the regularization terms are
specified as
$\displaystyle\boldsymbol{V}_{j}$
$\displaystyle=\left(\begin{array}[]{rrrrrrrr}-1&2&-1\\\ 1&-4&6&-4&1\\\
&1&-4&6&-4&1&\\\ &&&\ddots&\ddots&\ddots\\\ &&1&-4&6&-4&1&\\\
&&&&&-1&2&-1\end{array}\right)\text{ for }j=1,2,3,6,7,$
$\displaystyle\boldsymbol{W}_{4}$
$\displaystyle=\left(\begin{array}[]{rrrrrr}-1&1&\\\ &-1&1&\\\
&&\ddots&\ddots\\\ &&&-1&1&\\\ &&&&-1&1\end{array}\right),\mbox{ and }$
$\displaystyle\boldsymbol{W}_{5}$
$\displaystyle=\left(\begin{array}[]{rrrrrrr}1&-2&1\\\ &1&-2&1\\\
&&\ddots&\ddots&\ddots\\\ &&&1&-2&1&\\\ &&&&1&-2&1\end{array}\right).$
The equality constraint regularization matrix $\boldsymbol{V}_{j}$,
$j=1,2,3,6,7$, penalizes the fourth order finite differences between the bin
estimates. Thus, as $\rho$ increases, the coefficient vectors of covariates
1-3,6-7 tend to be piecewise cubic with two ends being linear, mimicking the
natural cubic spline. This is one example of the polynomial trend filtering
(Kim et al., 2009; Tibshirani and Taylor, 2011). Similar to semi-parametric
regressions, regularizations in polynomial trend filtering ‘let the data speak
for themselves’. In contrast, the bandwidth selection in semi-parametric
regressions is replaced by parameter tuning in regularizations. The number and
locations of knots are automatically determined by tuning parameter which is
chosen according to model selection criteria. In a similar fashion, the
coefficient vector gradually becomes monotone for covariate ‘market-to-book
ratio’ and concave for covariate ‘log market equity’. In addition, with $\rho$
large enough, we recover the corresponding constrained solution, which are
shown by the crosses (+) on solid lines in Figure 1. As noted above, our exact
path algorithm delivers the whole solution path bridging from the
unconstrained estimates (denoted by o) to the constrained estimates (denoted
by +). For example, the dotted lines in Figure 1 is a snapshot of the solution
at $\rho=0.6539$. Availability of the whole solution path renders model
selection along the path easy. For instance the regularization parameter
$\rho$ can be chosen by minimizing the cross-validation error or other model
selection criteria such as AIC, BIC, or $C_{p}$. Figure 2 displays the
solution path and the AIC and BIC along the path. It shows that both criteria
favor the fully regularized solution, namely the constrained estimates. The
whole solution path is obtained within seconds on a laptop using a Matlab
implementation of EPSODE.
The patterns revealed by the regularized estimates match some existing finance
theories. For instance, a company with low cash flow is unlikely to be an LBO
target because low cash flow is hard to meet the heavy debt burden associated
with the LBO. On the other hand, company carrying a high cash flow is likely
to possess a new technology. It is risky to acquire such firms because it is
hard to predict their profitability. The tax reason is obvious from the
regularized estimates. The more tax the company is paying, the more tax
benefits from an LBO. Log of market equity is a measure of company size.
Smaller companies are unpredictable in their profitability and extremely large
companies are unlikely to be an LBO target because LBOs are typically financed
with a large proportion of external debts. Interested readers are referred to
(Shivdasani and Wang, 2009) and references therein for related theories on
LBO.
This illustrative example demonstrates the flexibility of our novel path
algorithm. First, it can be applied to any convex loss function. In this
example, the loss function is the negative log-likelihood of a logistic model.
Second, it works for complicated regularizations like polynomial trend
filtering (equality constraints), monotonicity constraint, and concavity
constraint. More applications will be presented in Section 7 to illustrate the
potential of EPSODE.
$\begin{array}[]{cc}\includegraphics[width=433.62pt]{MandA_estimates}\end{array}$
Figure 1: Snapshots of the path solution to the regularized logistic
regression on the M&A data set.
$\begin{array}[]{cc}\includegraphics[width=180.67499pt]{MandA_solpath}&\includegraphics[width=180.67499pt]{MandA_AICBICpath}\end{array}$
Figure 2: Solution and AIC/BIC paths of the regularized logistic regression on
the M&A data set.
The rest of the paper is organized as follows. Section 2 reviews the exact
penalty method for optimization. Here the connections between constrained
optimization and regularization in statistics are made clear. Section 3
derives in detail the EPSODE algorithm for strictly convex loss function $f$.
Its implementation via the sweep operator and ordinary differential equations
are described in Section 4. An extension of EPSODE for $f$ convex but not
necessarily strictly convex is discussed in Section 5. Section 6 concerns
model selection along the path. Section 7 presents various applications of
EPSODE. Finally, Section 8 discusses the limitations of the path algorithm and
hints at future generalizations.
## 2 Exact Penalty Method for Convex Constrained Optimization
Consider the convex program
$\displaystyle\min$ $\displaystyle f(\boldsymbol{x})$ s.t. $\displaystyle
g_{i}(\boldsymbol{x})=0,1\leq i\leq r$ $\displaystyle
h_{j}(\boldsymbol{x})\leq 0,1\leq j\leq s,$
where the objective function $f$ is convex, equality constraint functions
$g_{i}$ are affine, and the inequality constraint functions $h_{j}$ are
convex. We further assume that $f$ and $h_{j}$ are smooth. Specifically we
require that $f$ and $h_{j}$ are continuously twice differentiable. To fix
notation, differential $df(\boldsymbol{x})$ is the row vector of partial
derivatives of $f$ at $\boldsymbol{x}$ and the gradient $\nabla
f(\boldsymbol{x})$ is the transpose of $df(\boldsymbol{x})$. The Hessian
matrix of $f(\cdot)$ is denoted by $d^{2}f(\boldsymbol{x})$.
Exact penalty method minimizes the function
$\displaystyle{\cal E}_{\rho}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
f(\boldsymbol{x})+\rho\sum_{i=1}^{r}|g_{i}(\boldsymbol{x})|+\rho\sum_{j=1}^{s}\max\\{0,h_{j}(\boldsymbol{x})\\}$
(4)
for $\rho\geq 0$. Classical results (Ruszczyński, 2006, Theorems 6.9 and 7.21)
state that for $\rho$ large enough, the solution to the optimization problem
(4) coincides with the solution to the original constrained convex program
(2). This justifies the exact penalty method as one way to solve constrained
optimization problems.
According to convex calculus (Ruszczyński, 2006, Theorem 3.5), the optimal
point $\boldsymbol{x}(\rho)$ of the function ${\cal E}_{\rho}(\boldsymbol{x})$
is characterized by the necessary and sufficient condition
$\displaystyle{\bf 0}$ $\displaystyle=$ $\displaystyle\nabla
f(\boldsymbol{x})+\rho\sum_{i=1}^{r}s_{i}\nabla
g_{i}(\boldsymbol{x})+\rho\sum_{j=1}^{s}t_{j}\nabla h_{j}(\boldsymbol{x})$ (5)
with coefficients satisfying
$\displaystyle s_{i}\in\begin{cases}\\{-1\\}&g_{i}(\boldsymbol{x})<0\\\
[-1,1]&g_{i}(\boldsymbol{x})=0\\\
\\{1\\}&g_{i}(\boldsymbol{x})>0\end{cases},\hskip 36.135pt\mbox{ and }\hskip
36.135ptt_{j}\in\begin{cases}\\{0\\}&h_{j}(\boldsymbol{x})<0\\\
[0,1]&h_{j}(\boldsymbol{x})=0\\\ \\{1\\}&h_{j}(\boldsymbol{x})>0\end{cases}.$
(6)
The sets defining possible values of $s_{i}$ and $t_{j}$ are the
subdifferentials of the functions $|x|$ and $x_{+}=\max\\{x,0\\}$. For path
following to make sense, we require uniqueness and continuity of the solution
$\boldsymbol{x}(\rho)$ to (4) as $\rho$ varies. The following lemma concerns
the continuity of the solution path and is the foundation of our path
algorithm.
###### Lemma 2.1.
1. 1.
(Uniqueness) If ${\cal E}_{\rho}$ is strictly convex, then its minimizer
$\boldsymbol{x}(\rho)$ is unique.
2. 2.
(Continuity) If ${\cal E}_{\rho}$ is strictly convex and coercive over an open
neighborhood of $\rho$, then the minimizer $\boldsymbol{x}(\rho)$ is
continuous at $\rho$.
3. 3.
(Continuity of $s_{i}$ and $t_{j}$) Furthermore, if the gradients $\\{\nabla
g_{i}(\boldsymbol{x}):g_{i}(\boldsymbol{x})=0\\}\cup\\{\nabla
h_{j}(\boldsymbol{x}):h_{j}(\boldsymbol{x})=0\\}$ of active constraints are
linearly independent at the solution $\boldsymbol{x}(\rho)$ over an open
neighborhood of $\rho$, then the coefficient paths $s_{i}(\rho)$ and
$t_{j}(\rho)$ are unique and continuous at $\rho$.
###### Proof.
The uniqueness of minimum under strict convexity is well-known (Ruszczyński,
2006). For continuity, suppose that the solution $\boldsymbol{x}(\rho)$ is not
continuous at $\rho$. Then there exists $\epsilon>0$ and a sequence
$\rho_{n}\to\rho$ such that
$\|\boldsymbol{x}(\rho_{n})-\boldsymbol{x}(\rho)\|\geq\epsilon$ for all $n$.
Since ${\cal E}_{\rho}$ is coercive, $\boldsymbol{x}(\rho_{n})$ is bounded and
there exists a subsequence of $\boldsymbol{x}(\rho_{n})$ that converges to
some point $\boldsymbol{y}$. Taking limits in the inequality ${\cal
E}_{\rho_{n}}[\boldsymbol{x}(\rho_{n})]\leq{\cal
E}_{\rho_{n}}(\boldsymbol{x})$ shows that ${\cal
E}_{\rho}(\boldsymbol{y})\leq{\cal E}_{\rho}(\boldsymbol{x})$ for all
$\boldsymbol{x}$, i.e., $\boldsymbol{y}=\boldsymbol{x}(\rho)$. This
contradicts with $\|\boldsymbol{y}-\boldsymbol{x}(\rho)\|\geq\epsilon$.
Therefore $\boldsymbol{x}(\rho)$ is continuous at $\rho$. For the continuity
of coefficients, under the linearly independence assumption, $s_{i}(\rho)$ and
$t_{j}(\rho)$ can be uniquely solved by the stationarity condition (5) given
solution vector $\boldsymbol{x}(\rho)$. Therefore continuity of $s_{i}(\rho)$
and $t_{j}(\rho)$ inherits from continuity of $\boldsymbol{x}(\rho)$. ∎
We remark that strict convexity only gives an easy-to-check sufficient
condition for uniqueness and continuity; it is not a necessary condition. A
convex but not strictly convex function can still have a unique minimum. The
absolute value function $|x|$ is such an example. When the loss function $f$
is strictly convex, then ${\cal E}_{\rho}$ is strictly convex for all
$\rho\geq 0$ and by Lemma 2.1 there exists a unique, continuous solution path
$\\{\boldsymbol{x}(\rho):\rho\geq 0\\}$. In Section 3 and 4, we derive the
path algorithm assuming that $f$ is strictly convex. When $f$ is convex but
not strictly convex, e.g., when $n<p$ in the least squares problems, the
solutions at smaller $\rho$ may not be unique. In that case, it is still
possible to obtain a solution path over the region of large $\rho$ where the
minimum of ${\cal E}_{\rho}$ is unique. In Section 5, we extend EPSODE to the
case $f$ is convex but may not be strictly convex. The third statement of
Lemma 2.1 implies that the active constraints ($g_{i}(\boldsymbol{x})=0$ or
$h_{j}(\boldsymbol{x})=0$) with interior coefficients must stay active until
the coefficients hit the end points of the permissible range, which in turn
implies that the solution path is piecewise smooth. This allows us to develop
a path following algorithm based on ODE.
## 3 The Path Following Algorithm
In this article, we specialize to the case where the constraint functions
$g_{i}$ and $h_{j}$ are affine, i.e., the gradient vectors $\nabla
g_{i}(\boldsymbol{x})$ and $\nabla h_{j}(\boldsymbol{x})$ are constant. This
leads to the regularized optimization problem formulated as (1) by defining
$g_{i}$ and $h_{j}$ as constraint residuals
$g_{i}(\boldsymbol{x})=\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}$ and
$h_{j}(\boldsymbol{x})=\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j}$. In
principle a similar path algorithm can be developed for the general convex
program where the inequality constraint functions $h_{j}$ are relaxed to be
convex. But that is beyond the scope of the current paper. In Sections 3 and
4, we assume that the loss function $f$ is strictly convex. This assumption is
relaxed in Section 5.
Our path following algorithm EPSODE works in a segment-by-segment fashion.
Along the path we keep track of the following index sets determined by signs
of constraint residuals
$\displaystyle{\cal N}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})=\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}<0\\},\hskip
36.135pt{\cal
N}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})=\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j}<0\\}$
$\displaystyle{\cal Z}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})=\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}=0\\},\hskip
36.135pt{\cal
Z}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})=\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j}=0\\}$
(7) $\displaystyle{\cal P}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})=\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}>0\\},\hskip
36.135pt{\cal
P}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})=\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j}>0\\}.$
Along each segment of the path, the set configuration is fixed. This is
implied by the continuity of both the solution and coefficient paths
established in Lemma 2.1. Throughout this article, we call the constraints in
${\cal Z}_{\text{E}}$ or ${\cal Z}_{\text{I}}$ active and others inactive.
Next we derive the ODE for the solution $\boldsymbol{x}(\rho)$ on a fixed
segment. Suppose we are in the interior of a segment. Let
$\boldsymbol{x}(\rho)$ be the solution of (4) indexed by the penalty parameter
$\rho$ and $\boldsymbol{x}(\rho+\Delta\rho)$ the solution when the penalty is
increased by an infinitesimal amount $\Delta\rho>0$. Then the difference
$\Delta\boldsymbol{x}(\rho)=\boldsymbol{x}(\rho+\Delta\rho)-\boldsymbol{x}(\rho)$
should minimize the increase in optimal objective value. That is, to the
second order, $\Delta\boldsymbol{x}$ is the solution to
$\displaystyle\min_{\Delta\boldsymbol{x}}$ $\displaystyle{\cal
E}_{\rho+\Delta\rho}(\boldsymbol{x}+\Delta\boldsymbol{x})-{\cal
E}_{\rho}(\boldsymbol{x})$ $\displaystyle\approx$ $\displaystyle
df(\boldsymbol{x})\cdot\Delta\boldsymbol{x}+\frac{1}{2}\Delta\boldsymbol{x}^{t}\cdot
d^{2}f(\boldsymbol{x})\cdot\Delta\boldsymbol{x}$
$\displaystyle+(\rho+\Delta\rho)\cdot\left[-\sum_{i\in{\cal
N}_{\text{E}}}\boldsymbol{v}_{i}+\sum_{i\in{\cal
P}_{\text{E}}}\boldsymbol{v}_{i}+\sum_{j\in{\cal
P}_{\text{I}}}\boldsymbol{w}_{j}\right]\cdot\Delta\boldsymbol{x}$
$\displaystyle+\Delta\rho\cdot\left[-\sum_{i\in{\cal
N}_{\text{E}}}g_{i}(\boldsymbol{x})+\sum_{i\in{\cal
P}_{\text{E}}}g_{i}(\boldsymbol{x})+\sum_{j\in{\cal
P}_{\text{I}}}h_{j}(\boldsymbol{x})\right]$ s.t.
$\displaystyle\boldsymbol{v}_{i}^{t}\cdot\Delta\boldsymbol{x}=0,i\in{\cal
Z}_{\text{E}},$
$\displaystyle\boldsymbol{w}_{j}^{t}\cdot\Delta\boldsymbol{x}=0,j\in{\cal
Z}_{\text{I}}.$
Note that the active constraints have to be kept active since the set
configuration is fixed along this segment by Lemma 2.1. This is why we have
these two sets of equality constraints. To ease notational burden, we define
$\displaystyle\boldsymbol{H}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
d^{2}f(\boldsymbol{x})$ (9) $\displaystyle\boldsymbol{u}_{\bar{\cal Z}}$
$\displaystyle=$ $\displaystyle-\sum_{i\in{\cal
N}_{\text{E}}}\boldsymbol{v}_{i}+\sum_{i\in{\cal
P}_{\text{E}}}\boldsymbol{v}_{i}+\sum_{j\in{\cal
P}_{\text{I}}}\boldsymbol{w}_{j}.$
This leads to the corresponding Lagrange multiplier problem
$\displaystyle\left(\begin{array}[]{cc}\boldsymbol{H}(\boldsymbol{x})&\boldsymbol{U}^{t}_{{\cal
Z}}\\\ \boldsymbol{U}_{{\cal Z}}&{\bf
0}\end{array}\right)\left(\begin{array}[]{c}\Delta\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\cal
Z}}\end{array}\right)=\left(\begin{array}[]{c}-\nabla
f(\boldsymbol{x})-(\rho+\Delta\rho)\boldsymbol{u}_{\bar{\cal Z}}\\\ {\bf
0}\end{array}\right),$
where the rows of the matrix $\boldsymbol{U}_{{\cal Z}}$ are the constant
differentials, $\boldsymbol{v}_{i}^{t}$, $i\in{\cal Z}_{\text{E}}$, and
$\boldsymbol{w}_{j}^{t}$, $j\in{\cal Z}_{\text{I}}(\boldsymbol{x})$, of the
active constraint functions. Denoting the inverse of matrix as
$\displaystyle\left(\begin{array}[]{cc}\boldsymbol{H}(\boldsymbol{x})&\boldsymbol{U}^{t}_{{\cal
Z}}\\\ \boldsymbol{U}_{{\cal Z}}&{\bf
0}\end{array}\right)^{-1}=\left(\begin{array}[]{cc}\boldsymbol{P}(\boldsymbol{x})&\boldsymbol{Q}(\boldsymbol{x})\\\
\boldsymbol{Q}^{t}(\boldsymbol{x})&\boldsymbol{R}(\boldsymbol{x})\end{array}\right),$
where
$\displaystyle\boldsymbol{P}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\boldsymbol{H}^{-1}(\boldsymbol{x})-\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\left[\boldsymbol{U}_{{\cal
Z}}\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\right]^{-1}\boldsymbol{U}_{{\cal
Z}}\boldsymbol{H}^{-1}(\boldsymbol{x})$
$\displaystyle\boldsymbol{Q}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\left[\boldsymbol{U}_{{\cal
Z}}\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\right]^{-1}$ (12) $\displaystyle\boldsymbol{R}(\boldsymbol{x})$
$\displaystyle=$ $\displaystyle-\left[\boldsymbol{U}_{{\cal
Z}}\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}(\boldsymbol{x})\right]^{-1},$
the solution of the difference vector $\Delta\boldsymbol{x}$ is
$\displaystyle\Delta\boldsymbol{x}$ $\displaystyle=$
$\displaystyle-\boldsymbol{P}(\boldsymbol{x})[\nabla
f(\boldsymbol{x})+(\rho+\Delta\rho)\boldsymbol{u}_{\bar{\cal Z}}]$
$\displaystyle=$ $\displaystyle-\boldsymbol{P}(\boldsymbol{x})[\nabla
f(\boldsymbol{x})+\rho\boldsymbol{u}_{\bar{\cal
Z}}(\boldsymbol{x})+\Delta\rho\cdot\boldsymbol{u}_{\bar{\cal Z}}]$
$\displaystyle=$
$\displaystyle-\boldsymbol{P}(\boldsymbol{x})[-\rho\boldsymbol{U}^{t}_{{\cal
Z}}\boldsymbol{r}_{{\cal Z}}+\Delta\rho\cdot\boldsymbol{u}_{\bar{\cal Z}}].$
Note $\boldsymbol{P}(\boldsymbol{x})\boldsymbol{U}^{t}_{{\cal Z}}={\bf 0}$.
Therefore
$\Delta\boldsymbol{x}=-\Delta\rho\cdot\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}$. This gives the direction for the infinitesimal update of solution vector
$\boldsymbol{x}(\rho)$. Taking limit in $\Delta\rho$ leads to the following
key result for developing the path algorithm.
###### Proposition 3.1.
Within interior of a path segment with set configuration (7), the solution
$\boldsymbol{x}(\rho)$ satisfies an ordinary differential equation (ODE)
$\displaystyle\frac{d\boldsymbol{x}(\rho)}{d\rho}=-\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}$ (13)
where the matrix $\boldsymbol{P}(\boldsymbol{x})$ and vector
$\boldsymbol{u}_{\bar{\cal Z}}$ are defined by (12) and (9).
Note that the right hand side of (13) is a constant vector in $\boldsymbol{x}$
when $f$ is quadratic and $g_{i}$ and $h_{j}$ are affine. Thus the
corresponding solution path is piecewise linear. This recovers the case
studied in (Zhou and Lange, 2011). The differential equation (13) holds on the
current segment until one of two types of events happens: an inactive
constraint becomes active or vice versa. The first type of event is easy to
detect – whenever a constraint function, $g_{i}(\boldsymbol{x})$, $i\in{\cal
N}_{\text{E}}\cup{\cal P}_{\text{E}}$, or $h_{j}(\boldsymbol{x})$, $j\in{\cal
N}_{\text{I}}\cup{\cal P}_{\text{I}}$, hits zero, we move that constraint to
the active set ${\cal Z}_{\text{E}}$ or ${\cal Z}_{\text{I}}$ and start
solving a new system of differential equations. To detect when the second type
of event happens, we need to keep track of the coefficients
$s_{i}(\boldsymbol{x})$ and $t_{j}(\boldsymbol{x})$ for active constraints.
Whenever the coefficient of an active constraint hits the boundary of its
permissible range in (6), the constraint has to be relaxed from being active
in next segment. It turns out the coefficients for active constraints admit a
simple representation in terms of current solution vector.
###### Proposition 3.2.
On a path segment with set configuration (7), the coefficients $s_{i}$ and
$t_{j}$ for active constraints are
$\displaystyle\boldsymbol{r}_{{\cal
Z}}(\rho)=\left(\begin{array}[]{c}\boldsymbol{s}_{{\cal
Z}_{\text{E}}}(\rho)\\\ \boldsymbol{t}_{{\cal
Z}_{\text{I}}}(\rho)\end{array}\right)=-\boldsymbol{Q}^{t}(\boldsymbol{x})\left[\frac{1}{\rho}\nabla
f(\boldsymbol{x})+\boldsymbol{u}_{\bar{\cal Z}}\right]$ (16)
where $\boldsymbol{x}=\boldsymbol{x}(\rho)$ is the solution at $\rho$ and the
matrix $\boldsymbol{Q}(\boldsymbol{x})$ is defined by (12).
###### Proof.
Stationarity condition (5) implies
$\displaystyle\boldsymbol{U}_{\cal Z}^{t}\boldsymbol{r}_{\cal
Z}=-\frac{1}{\rho}\nabla f(\boldsymbol{x})-\boldsymbol{U}_{\bar{\cal
Z}}^{t}\boldsymbol{r}_{\bar{\cal Z}}=-\frac{1}{\rho}\nabla
f(\boldsymbol{x})-\boldsymbol{u}_{\bar{\cal Z}}.$
Multiplying both sides by $\boldsymbol{Q}(\boldsymbol{x})$ gives (16). ∎
Given current solution vector $\boldsymbol{x}(\rho)$, the coefficients of the
active constraints are readily obtained from (16). Once a coefficient hits the
end points, we move that constraint from the active set to the inactive set
that matches the end point being hit. In next section, we detail the
implementation of the path algorithm.
## 4 Implementation: ODE and Sweeping Operator
Initialize $\rho=0$,
$\boldsymbol{\beta}(0)=\text{argmin}f(\boldsymbol{\beta})$ and its set
configuration (7).
repeat
Solve ODE (13) until an inactive constraint becomes active or the coefficient
(16) of an active constraint hits boundary.
Update the set configuration (7).
until ${\cal N}_{\text{E}}={\cal P}_{\text{E}}={\cal P}_{\text{I}}=\emptyset$
Algorithm 1 EPSODE: Solution path for regularization problem (1) with strictly
convex $f$.
Algorithm 1 summarizes EPSODE based on Propositions 3.1 and 3.2. It involves
solving ODEs segment by segment and is extremely simple to implement using
softwares with a reliable ODE solver such as the ode45 function in Matlab and
the deSolve package (Karline10RdeSolve) in R. There has been extensive
research in applied mathematics on numerical methods for solving ODEs, notably
the Runge-Kutta, Richardson extrapolation and predictor-corrector methods.
Some path following algorithms developed for specific statistical problems
(Park and Hastie, 2007; Friedman, 2008) turn out to be approximate methods for
solving the corresponding ODE. Wu (2011) first explicitly uses ODE to derive
an exact solution path for the lasso penalized GLM. The connection of path
following to ODE relieves statisticians from the burden of developing specific
path algorithms for a variety of regularization problems.
Any ODE solver repeatedly evaluates the derivative. Suppose the number of
parameters is $p$. Computation of the matrix-vector multiplications in (13)
and (16) has computation cost of order $O(p^{2})+O(p|{\cal Z}|)+O(|{\cal
Z}|^{3})$ if the inverse $H^{-1}$ of Hessian matrix of loss function $f$ is
readily available, where ${\cal Z}={\cal Z}_{\text{E}}\cup{\cal Z}_{\text{I}}$
and $|{\cal Z}|$ denotes its cardinality. Otherwise the computation cost is
$O(p^{3})+O(p|{\cal Z}|)+O(|{\cal Z}|^{3})$.
An alternative implementation avoids repeated matrix inversions by solving an
ODE for the matrices $\boldsymbol{P}$, $\boldsymbol{Q}$ and $\boldsymbol{R}$
themselves. The computations can be conveniently organized around the
classical sweep and inverse sweep operators of regression analysis (Dempster,
1969; Goodnight, 1979; Jennrich, 1977; Little and Rubin, 2002; Lange, 2010).
Suppose $\boldsymbol{A}$ is an $m\times m$ symmetric matrix. Sweeping on the
$k$th diagonal entry $a_{kk}\neq 0$ of $\boldsymbol{A}$ yields a new symmetric
matrix $\widehat{\boldsymbol{A}}$ with entries
$\displaystyle\hat{a}_{kk}$ $\displaystyle=$ $\displaystyle-\frac{1}{a_{kk}},$
$\displaystyle\hat{a}_{ik}$ $\displaystyle=$
$\displaystyle\frac{a_{ik}}{a_{kk}},\quad i\neq k$ $\displaystyle\hat{a}_{kj}$
$\displaystyle=$ $\displaystyle\frac{a_{kj}}{a_{kk}},\quad j\neq k$
$\displaystyle\hat{a}_{ij}$ $\displaystyle=$ $\displaystyle
a_{ij}-\frac{a_{ik}a_{kj}}{a_{kk}},\quad i,j\neq k.$
These arithmetic operations can be undone by inverse sweeping on the same
diagonal entry. Inverse sweeping on the $k$th diagonal entry sends the
symmetric matrix $\boldsymbol{A}$ into the symmetric matrix
$\check{\boldsymbol{A}}$ with entries
$\displaystyle\check{a}_{kk}$ $\displaystyle=$
$\displaystyle-\frac{1}{a_{kk}},$ $\displaystyle\check{a}_{ik}$
$\displaystyle=$ $\displaystyle-\frac{a_{ik}}{a_{kk}},\quad i\neq k$
$\displaystyle\check{a}_{kj}$ $\displaystyle=$
$\displaystyle-\frac{a_{kj}}{a_{kk}},\quad j\neq k$
$\displaystyle\check{a}_{ij}$ $\displaystyle=$ $\displaystyle
a_{ij}-\frac{a_{ik}a_{kj}}{a_{kk}},\quad i,j\neq k.$
Both sweeping and inverse sweeping preserve symmetry. Thus, all operations can
be carried out on either the lower or upper triangle of $\boldsymbol{A}$
alone, saving both computational time and storage. When several sweeps or
inverse sweeps are performed, their order is irrelevant.
At beginning ($\rho=0$) of the path following, we initialize a sweeping
tableau as
$\displaystyle\left(\begin{array}[]{c|c}\boldsymbol{H}^{-1}(\boldsymbol{x})&\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}^{t}\\\
\hline\cr*&\boldsymbol{U}\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}^{t}\end{array}\right),$
where the matrix $\boldsymbol{U}\in\mathbb{R}^{(r+s)\times p}$ holds all
constraint differentials $\boldsymbol{v}_{i}^{t}$ and $\boldsymbol{w}_{j}^{t}$
in rows. Further sweeping of diagonal entries corresponding to the active
constraints yields
$\displaystyle\left(\begin{array}[]{c|cc}\boldsymbol{P}(\boldsymbol{x})&\boldsymbol{Q}(\boldsymbol{x})&\boldsymbol{P}(\boldsymbol{x})\boldsymbol{U}_{\bar{\cal
Z}}^{t}\\\
\hline\cr*&\boldsymbol{R}(\boldsymbol{x})&\boldsymbol{Q}^{t}(\boldsymbol{x})\boldsymbol{U}_{\bar{\cal
Z}}^{t}\\\ &*&\boldsymbol{U}_{\bar{\cal
Z}}\boldsymbol{P}(\boldsymbol{x})\boldsymbol{U}_{\bar{\cal
Z}}^{t}\end{array}\right).$ (21)
Here we conveniently organized the columns of the swept active constraints
before those of un-swept ones. In practice the sweep tableau is not necessary
as in (21) and it is enough to keep an indicator vector recording which
columns are swept. The key elements for the path algorithm magically appear in
the sweep tableau (21)
$\displaystyle\frac{d\boldsymbol{x}(\rho)}{d\rho}$ $\displaystyle=$
$\displaystyle-\boldsymbol{P}(\boldsymbol{x})\boldsymbol{U}_{\bar{\cal
Z}}^{t}\boldsymbol{r}_{\bar{\cal Z}}$ $\displaystyle\boldsymbol{r}_{{\cal
Z}}(\rho)$ $\displaystyle=$
$\displaystyle-\boldsymbol{Q}^{t}(\boldsymbol{x})\boldsymbol{U}_{\bar{\cal
Z}}^{t}\boldsymbol{r}_{\bar{\cal
Z}}-\frac{1}{\rho}\boldsymbol{Q}^{t}(\boldsymbol{x})\nabla f(\boldsymbol{x}).$
Therefore path following procedure only involves solving ODE for the whole
sweep tableau (21) with sweeping or inverse sweeping at kinks between
successive segments. For this purpose we derive the ODE for the sweep tableau
(21). We adopt the convenient notations in (Magnus and Neudecker, 1999). For a
matrix function $F(\boldsymbol{X}):\mathbb{R}^{n\times
q}\to\mathbb{R}^{m\times p}$,
$\displaystyle
DF(\boldsymbol{X})=\frac{\partial\text{vec}F(\boldsymbol{X})}{\partial(\text{vec}\boldsymbol{X})^{t}}$
denotes the $mp\times nq$ Jacobian matrix. For example, Proposition 3.1 states
$D\boldsymbol{x}(\rho)=-\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}$.
###### Proposition 4.1 (ODE for Sweep Tableau).
On a segment of path with fixed set configuration, the matrices
$\boldsymbol{P}(\rho)$, $\boldsymbol{Q}(\rho)$ and $\boldsymbol{R}(\rho)$
satisfy the ordinary differential equations (ODE)
$\displaystyle D\boldsymbol{P}(\rho)$
$\displaystyle=[\boldsymbol{P}(\boldsymbol{x})\otimes\boldsymbol{P}(\boldsymbol{x})]\cdot[D\boldsymbol{H}(\boldsymbol{x})]\cdot\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}$ $\displaystyle D\boldsymbol{Q}(\rho)$
$\displaystyle=[\boldsymbol{Q}^{t}(\boldsymbol{x})\otimes\boldsymbol{P}(\boldsymbol{x})]\cdot[D\boldsymbol{H}(\boldsymbol{x})]\cdot\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}$ $\displaystyle D\boldsymbol{R}(\rho)$
$\displaystyle=[\boldsymbol{Q}^{t}(\boldsymbol{x})\otimes\boldsymbol{Q}^{t}(\boldsymbol{x})]\cdot[D\boldsymbol{H}(\boldsymbol{x})]\cdot\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}.$
###### Proof.
First consider
$\displaystyle\boldsymbol{R}(\boldsymbol{x})=-\left[\boldsymbol{U}_{{\cal
Z}}\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\right]^{-1}=-\boldsymbol{M}^{-1}(\boldsymbol{x}).$
By chain rule (Magnus and Neudecker, 1999, p91),
$\displaystyle D\boldsymbol{R}(\rho)$ $\displaystyle=$ $\displaystyle
D\boldsymbol{R}(\boldsymbol{M})\cdot D\boldsymbol{M}(\boldsymbol{H})\cdot
D\boldsymbol{H}(\boldsymbol{x})\cdot D\boldsymbol{x}(\rho)$ $\displaystyle=$
$\displaystyle[\boldsymbol{R}(\boldsymbol{x})\otimes\boldsymbol{R}(\boldsymbol{x})]\cdot
D\boldsymbol{M}(\boldsymbol{H})\cdot D\boldsymbol{H}(\boldsymbol{x})\cdot
D\boldsymbol{x}(\rho)$ $\displaystyle=$
$\displaystyle-[\boldsymbol{R}(\boldsymbol{x})\otimes\boldsymbol{R}(\boldsymbol{x})]\cdot\\{[\boldsymbol{U}_{\cal
Z}\boldsymbol{H}^{-1}(\boldsymbol{x})\otimes\boldsymbol{U}_{\cal
Z}\boldsymbol{H}^{-1}(\boldsymbol{x})]\cdot
D\boldsymbol{H}(\boldsymbol{x})\\}\cdot D\boldsymbol{x}(\rho)$
$\displaystyle=$
$\displaystyle[\boldsymbol{Q}^{t}(\boldsymbol{x})\otimes\boldsymbol{Q}^{t}(\boldsymbol{x})]\cdot[D\boldsymbol{H}(\boldsymbol{x})]\cdot\boldsymbol{P}(\boldsymbol{x})\boldsymbol{u}_{\bar{\cal
Z}}.$
Similar calculations yield formula for
$\boldsymbol{Q}(\boldsymbol{x})=-\boldsymbol{H}^{-1}(\boldsymbol{x})\boldsymbol{U}_{{\cal
Z}}^{t}\boldsymbol{R}(\boldsymbol{x})$ and
$\boldsymbol{P}(\boldsymbol{x})=\boldsymbol{H}^{-1}(\boldsymbol{x})-\boldsymbol{Q}(\boldsymbol{x})\boldsymbol{U}_{\cal
Z}\boldsymbol{H}^{-1}(\boldsymbol{x})$. ∎
Solving ODE for these matrices requires the $p^{2}$-by-$p$ Jacobian matrix of
the Hessian matrix $\boldsymbol{H}(\boldsymbol{x})=d^{2}f(\boldsymbol{x})$,
$\displaystyle
D\boldsymbol{H}(\boldsymbol{x})=\frac{\partial[\text{vec}\boldsymbol{H}(\boldsymbol{x})]}{\partial\text{vec}(\boldsymbol{x})^{t}}=\frac{\partial\text{vec}[df^{2}(\boldsymbol{x})]}{\partial\text{vec}(\boldsymbol{x})^{t}},$
which we provide for each example in Section 7 for convenience. When the
number of parameter $p$ is large, $D\boldsymbol{H}$ is a large matrix. However
there is no need to compute and store $D\boldsymbol{H}$ and we are only
required to compute the matrix vector multiplication
$D\boldsymbol{H}\cdot\boldsymbol{v}$ for any vector $\boldsymbol{v}$. In light
of the useful identity
$(\boldsymbol{B}^{t}\otimes\boldsymbol{A})\text{vec}(\boldsymbol{C})=\text{vec}(\boldsymbol{A}\boldsymbol{C}\boldsymbol{B})$,
evaluating the derivative for the whole tableau only involves multiplying
three matrices and incurs computational cost $O(p^{3})+O(p^{2}|{\cal
Z}|)+O(p|{\cal Z}|^{2})$.
Although we have presented the path algorithm as moving from $\rho=0$ to large
$\rho$, it can be applied in either direction. Lasso and fused-lasso usually
start from the constrained solution, while in presence of general equality
constraints, e.g., polynomial trend filtering, and/or inequality constraints,
the constrained solution is not readily available and the path algorithm must
be initiated at $\rho=0$.
## 5 Extension of EPSODE
So far we have assumed strictly convexity of the loss function $f$. This
unfortunately excludes many interesting applications, especially $p>n$ case of
the regression problems. In this section we briefly indicate an extension of
EPSODE to the case $f$ is convex but not necessarily strictly convex. In the
proof of Proposition 3.1, the infinitesimal change of solution
$\Delta\boldsymbol{x}$ is derived via minimizing the equality-constrained
quadratic program (3), the solution to which requires inverse of Hessian
$\boldsymbol{H}^{-1}$ and thus strict convexity of $f$. Alternatively we may
solve (3) via reparameterization. Let $\boldsymbol{U}_{\cal Z}$ hold the
active constraint vectors and $\boldsymbol{Y}\in\mathbb{R}^{p\times(p-|{\cal
Z}|)}$ be a null space matrix of $\boldsymbol{U}_{\cal Z}$, i.e., the columns
of $\boldsymbol{Y}$ are orthogonal to the rows of $\boldsymbol{U}_{\cal Z}$.
Then the infinitesimal change can be represented as
$\Delta\boldsymbol{x}=\boldsymbol{Y}\Delta\boldsymbol{y}$ for some vector
$\Delta\boldsymbol{y}\in\mathbb{R}^{p-|{\cal Z}|}$. Under this
reparameterization, the quadratic program (3) is equivalent to
$\displaystyle\min_{\Delta\boldsymbol{y}}\,\frac{1}{2}\Delta\boldsymbol{y}^{t}[\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}]\Delta\boldsymbol{y}+[df(\boldsymbol{x})+(\rho+\Delta\rho)\boldsymbol{u}_{\bar{\cal
Z}}^{t}]\boldsymbol{Y}\cdot\Delta\boldsymbol{y}$
with explicit solution
$\displaystyle\Delta\boldsymbol{y}=-[\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}[\nabla
f(\boldsymbol{x})+(\rho+\Delta\rho)\boldsymbol{u}_{\bar{\cal Z}}].$
Hence the infinitesimal change in $\boldsymbol{x}(\rho)$ is
$\displaystyle\Delta\boldsymbol{x}$
$\displaystyle=-\boldsymbol{Y}[\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}[\nabla
f(\boldsymbol{x})+(\rho+\Delta\rho)\boldsymbol{u}_{\bar{\cal Z}}]$
$\displaystyle=-\Delta\rho\cdot\boldsymbol{Y}[\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}\boldsymbol{u}_{\bar{\cal
Z}}].$
Again taking limit gives the following result in parallel to Proposition 3.1.
###### Proposition 5.1.
Within interior of a path segment with set configuration (7), the solution
$\boldsymbol{x}(\rho)$ satisfies an ordinary differential equation (ODE)
$\displaystyle\frac{d\boldsymbol{x}(\rho)}{d\rho}=-\boldsymbol{Y}[\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}\boldsymbol{u}_{\bar{\cal
Z}}$ (22)
where $\boldsymbol{Y}$ is a null space matrix of $\boldsymbol{U}_{\cal Z}$.
An advantage of (22) is that only non-singularity of the matrix
$\boldsymbol{Y}^{t}\boldsymbol{H}(\boldsymbol{x})\boldsymbol{Y}$ is required
which is much weaker than the non-singularity of $\boldsymbol{H}$. The
computational cost of calculating the derivative in (22) is $O((p-|{\cal
Z}|)^{3})+O(p(p-|{\cal Z}|))$, which is more efficient than (13) when
$p-|{\cal Z}|$ is small. However it requires the null space matrix
$\boldsymbol{Y}$, which is nonunique and may be expensive to compute.
Fortunately the null space matrix $\boldsymbol{Y}$ is constant over each path
segment and in practice can be calculated by QR decomposition of the active
constraint matrix $\boldsymbol{U}_{\cal Z}$. At each kink either one
constraint leaves ${\cal Z}$ or one enters ${\cal Z}$. Therefore
$\boldsymbol{Y}$ can be sequentially updated (Lawson and Hanson, 1987) and
need not to be calculated anew for each segment. Which version of (13) and
(22) to use depends on specific application. When the loss function $f$ is not
strictly convex, e.g., $p>n$ case in regression analysis, only (22) applies.
Interested readers are referred to (Nocedal and Wright, 2006) for a similar
dilemma in optimization methods.
## 6 Model Selection Along the Path
In applications such as penalized GLMs, the tuning parameter $\rho$ in the
regularization problem (1) is chosen by a model selection criterion such as
AIC, BIC, $C_{p}$, or cross-validation. The cross validation errors can be
readily computed using the solution path output by EPSODE. Yet the AIC, BIC,
and $C_{p}$ criteria require an estimate of the degrees of freedom of estimate
$\boldsymbol{\beta}(\rho)$. Specifically AIC and BIC are defined by
AIC
$\displaystyle=-\ell(\boldsymbol{\beta}(\rho))+\text{df}(\boldsymbol{\beta}(\rho))$
BIC $\displaystyle=-\ell(\boldsymbol{\beta}(\rho))+\frac{\log
n}{2}\text{df}(\boldsymbol{\beta}(\rho)),$
where $-\ell(\cdot)$ denotes the negative log-likelihood and
$\text{df}(\boldsymbol{\beta}_{\rho})$ is the degrees of freedom for estimate
$\boldsymbol{\beta}_{\rho}$. We propose to use
$\displaystyle\text{df}(\boldsymbol{\beta}(\rho))=p-|{\cal
Z}_{\text{E}}\cup{\cal Z}_{\text{I}}|$ (23)
as a measure of the degrees of freedom under GLMs. It is previously shown that
(23) is an unbiased estimate of the degrees of freedom for lasso penalized
least squares (Efron et al., 2004; Zou et al., 2007), generalized lasso
penalized least squares (Tibshirani and Taylor, 2011), and the least squares
version of the regularized problem (1) (Zhou and Lange, 2011). Using the same
degrees of freedom formula (23) for GLMs is justified by the local
approximation of GLM loglikelihood by weighted least squares. See (Park and
Hastie, 2007) for details.
## 7 Applications
In this section, we collect some representative regularized or constrained
estimation problems and demonstrate how they can be solved by path following.
For all applications, we list the first three derivatives of the loss function
$f$ in (1). In fact, the third derivative is only needed when implementing by
solving the ODE for the sweep tableau.
### 7.1 GLMs and Quasi-Likelihoods with Generalized $l_{1}$ Regularizations
The generalized linear model (GLM) deals with exponential families in which
the sufficient statistics is $Y$ and the conditional mean $\mu$ of $Y$
completely determines its distribution. Conditional on the covariate vector
$\boldsymbol{x}\in\mathbb{R}^{p}$, the response variable $y$ is modeled as
$\displaystyle
p(y|\boldsymbol{x};\boldsymbol{\beta},\sigma)\propto\exp\left\\{\frac{y\langle\boldsymbol{x},\boldsymbol{\beta}\rangle-\psi(\langle\boldsymbol{x},\boldsymbol{\beta}\rangle)}{c(\sigma)}\right\\},$
(24)
where the scalar $\sigma>0$ is a fixed and known scale parameter and the
vector $\boldsymbol{\beta}$ is the parameters to be estimated. The function
$\psi:\mathbb{R}\mapsto\mathbb{R}$ is the link function. When
$y\in\mathbb{R}$, $\psi(u)=u^{2}/2$ and $c(\sigma)=\sigma^{2}$, (24) is the
normal regression model. When $y\in\\{0,1\\}$, $\psi(u)=\ln(1+\exp(u))$ and
$c(\sigma)=1$, (24) is the logistic regression model. When $y\in\mathbb{N}$,
$\psi(u)=\exp(u)$, and $c(\sigma)=1$, (24) is the Poisson regression model.
The quasi-likelihoods generalize GLM without assuming a specific distribution
form of $Y$. Instead only a function relation between the conditional means
$\mu_{i}$ and variances $\sigma_{i}^{2}$, $\sigma_{i}^{2}=V(\mu_{i})$ for some
variance function $V$, is needed. Then the integral
$\displaystyle Q(\mu,y)=\int_{y}^{\mu}\frac{y-t}{\sigma^{2}V(t)}\,dt$
behaves like a log-likelihood function under mild conditions and is called the
quasi-likelihood. The quasi-likelihood includes GLMs as special cases with
appropriately chosen variance function $V(\cdot)$. Readers are referred to the
classical text (McCullagh and Nelder, 1983, Table 9.1) for the commonly used
quasi-likelihoods. By slightly abusing our notation, we assume a known link
function between the conditional mean $\mu_{i}$ and linear predictor
$\boldsymbol{x}_{i}^{T}\boldsymbol{\beta}$,
$\mu=\mu(\boldsymbol{x}_{i}^{T}\boldsymbol{\beta})$ and denote
$Q_{i}(\boldsymbol{\beta})=Q(\mu(\boldsymbol{x}_{i}^{T}\boldsymbol{\beta}),y_{i})$.
Then the quasi-likelihood with generalized $l_{1}$ regularization takes the
form
$\displaystyle-Q(\boldsymbol{\beta})+\rho\|\boldsymbol{V}\boldsymbol{\beta}-\boldsymbol{d}\|_{1}=-\sum_{i=1}^{n}Q_{i}(\boldsymbol{\beta})+\rho\|\boldsymbol{V}\boldsymbol{\beta}-\boldsymbol{d}\|_{1},$
(25)
which is a special case of the general form (1). Specific choices of the
regularization matrix $\boldsymbol{V}$ and constant vector $\boldsymbol{d}$
lead to lasso, fused-lasso, trend filtering, and many other applications.
For the path algorithm, we require the first two or three derivatives of the
complete quasi-likelihood. Denoting
$\boldsymbol{\eta}=\boldsymbol{X}\boldsymbol{\beta}$ with
$\boldsymbol{X}=(\boldsymbol{x}_{1}^{t},\boldsymbol{x}_{2}^{t},\cdots,\boldsymbol{x}_{n}^{t})^{t}$,
we have
$\displaystyle\nabla Q(\boldsymbol{\beta})$ $\displaystyle=$
$\displaystyle[D\boldsymbol{\mu}(\boldsymbol{\eta})]^{t}\boldsymbol{V}^{-1}(\boldsymbol{y}-\boldsymbol{\mu})/\sigma^{2}=\boldsymbol{X}^{t}[D\boldsymbol{\mu}(\boldsymbol{\eta})]\boldsymbol{V}^{-1}(\boldsymbol{y}-\boldsymbol{\mu})/\sigma^{2},$
$\displaystyle\boldsymbol{H}(\boldsymbol{\beta})=d^{2}Q(\boldsymbol{\beta})$
$\displaystyle=$
$\displaystyle[(\boldsymbol{y}-\boldsymbol{\mu})^{t}\boldsymbol{V}^{-1}\otimes\boldsymbol{X}^{t}]\cdot
D^{2}\boldsymbol{\mu}(\boldsymbol{\eta})\cdot\boldsymbol{X}/\sigma^{2},$ (26)
$\displaystyle D\boldsymbol{H}(\boldsymbol{\beta})=d^{3}Q(\boldsymbol{\beta})$
$\displaystyle=$
$\displaystyle[\boldsymbol{X}^{t}\otimes(\boldsymbol{y}-\boldsymbol{\mu})^{t}\boldsymbol{V}^{-1}\otimes\boldsymbol{X}^{t}]\cdot
D^{3}\boldsymbol{\mu}(\boldsymbol{\eta})\cdot\boldsymbol{X}/\sigma^{2},$
where $\boldsymbol{V}$ is a $n$-by-$n$ diagonal matrix with diagonal entries
$V(\mu(\boldsymbol{x}_{i}^{t}\boldsymbol{y}))$,
$D\boldsymbol{\mu}(\boldsymbol{\eta})$ is a $n$-by-$n$ diagonal matrix with
diagonal entries $\mu^{\prime}(\boldsymbol{x}_{i}^{t}\boldsymbol{\beta})$,
$D^{2}\boldsymbol{\mu}(\boldsymbol{\eta})$ is a $n^{2}$-by-$n$ matrix with
$(n(i-1)+i,i)$ entry equal to
$\mu^{\prime\prime}(\boldsymbol{x}_{i}^{t}\boldsymbol{\beta})$ for
$i=1,\ldots,n$ and 0 otherwise, and $D^{3}\boldsymbol{\mu}(\boldsymbol{\eta})$
is a $n^{3}$-by-$n$ matrix with $(n^{2}(i-1)+n(i-1)+i,i)$ entry equal to
$\mu^{\prime\prime\prime}(\boldsymbol{x}_{i}^{t}\boldsymbol{\beta})$ for
$i=1,\ldots,n$ and 0 otherwise. Note for GLM with canonical link, these
formulas simplify (Agresti, 2002, Section 4.6.4).
The most widely used $l_{1}$ regularization is the lasso penalty which imposes
sparsity on the regression coefficients. For numerical demonstration, we
revisit the M&A example introduced in Section 1 without discretizing each
predictor. We standardize each predictor first and consider the lasso
penalized linear logistic regression model. Figure 3 shows the lasso solution
path for each standardized predictor in the left panel and corresponding AIC
and BIC scores in the right panel. The order at which predictors enter the
model matches the more detailed patterns revealed by the varying coefficient
model in Figure 1. The almost monotone effects of the predictors ‘market-to-
book ratio’, ‘cash flow’, ’cash’, and ’tax’ can be captured by the usual
linear logistic regression and these covariates are picked up by lasso first.
The nonlinear effects shown in the other predictors are likely to be missed by
the linear logistic regression. For instance, the quadratic effects of ‘log
market equity’ shown in the regularized estimates in Figure 1 are missed by
both AIC and BIC criteria.
$\begin{array}[]{cc}\includegraphics[width=180.67499pt]{MandA_lassopath}&\includegraphics[width=180.67499pt]{MandA_lassoAICBIC}\end{array}$
Figure 3: M&A example revisited. Lasso solution path on the seven standardized
predictors.
All the generalized lasso problems studied in (Tibshirani and Taylor, 2011)
for Gaussian linear regression naturally generalize to GLMs or quasi-
likelihoods and are subject to the EPSODE path algorithm. This leads to
applications to lasso or fused-lasso penalized GLMs, outlier detections, trend
filtering, and image restoration for GLMs. For instance, cubic trend filtering
is performed on five predictors of the M&A example in Section 1. The graph-
guided penalized linear regression proposed in (Chen et al., 2010) can also be
generalized to GLMs or quasi-likelihoods. Suppose each node $i$ of a graph is
assigned a regression coefficient $\beta_{i}$. In graph penalized regression,
the objective function takes the form
$\displaystyle-\ell(\boldsymbol{\beta})+\lambda_{\text{G}}\sum_{i\sim
j}\left|\frac{\beta_{i}}{\sqrt{d_{i}}}-\text{sgn}(r_{ij})\frac{\beta_{j}}{\sqrt{d_{j}}}\right|+\lambda_{\text{L}}\sum_{j}|\beta_{j}|,$
(27)
where the set of neighboring pairs $i\sim j$ define the graph, $d_{i}$ is the
degree of node $i$, and $r_{ij}$ is the correlation coefficient between $i$
and $j$. This is simply a special case of (25) when the ratio
$\lambda_{\text{G}}/\lambda_{\text{L}}$ is fixed.
### 7.2 Shape-Restricted Regressions
Order-constrained regression has been an important modeling tool (Robertson et
al., 1988; Silvapulle and Sen, 2005). If $\boldsymbol{\beta}$ denotes the
parameter vector, monotone regression imposes isotone constraints
$\beta_{1}\leq\beta_{2}\leq\cdots\leq\beta_{p}$ or antitone constraints
$\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{p}$. In partially ordered
regression, subsets of the parameters are subject to isotone or antitone
constraints. In some other problems it is sensible to impose convex or concave
constraints. Note that if locations of regression parameters are at
irregularly spaced time points $t_{1}\leq t_{2}\leq\cdots\leq t_{p}$,
convexity translates into the constraints
$\displaystyle\frac{\beta_{i+2}-\beta_{i+1}}{t_{i+2}-t_{i+1}}\geq\frac{\beta_{i+1}-\beta_{i}}{t_{i+1}-t_{i}}$
for $1\leq i\leq p-2$. When the time intervals are uniform, the constraints
simplify to $\beta_{i+2}-\beta_{i+1}\geq\beta_{i+1}-\beta_{i}$,
$i=1,2,\cdots,p-1$. Concavity translates into the opposite set of
inequalities.
Most of previous work has focused on the linear regression problems because of
the computational and theoretical complexities in the generalized linear model
setting. The recent work (Rufibach, 2010) proposes an active set algorithm for
GLMs with order constraints. The EPSODE algorithm conveniently provides a
solution to the linearly constrained estimation problem (1.1). The relevant
derivatives of loss function are listed in (26). It is noteworthy that EPSODE
not only provides the constrained estimate but also the whole path bridging
the unconstrained estimate to the constrained solution. Availability of the
whole solution path renders model selection between the two extremes simple.
In the illustrative M&A example of Section 1, the bin predictors for the
‘market-to-book ratio’ are regularized by the antitone constraint and those
for the ‘log market equity’ covariate by the concavity constraint.
### 7.3 Gaussian Graphical Models
In recent years several authors (Friedman et al., 2008; Yuan, 2008) proposed
to estimate the sparse undirected graphical model by using lasso
regularizations to the log-likelihood function of the precision matrix, the
inverse of the variance-covariance matrix. Given an observed variance-
covariance matrix $\hat{\boldsymbol{\Sigma}}\in R^{p\times p}$, the negative
log-likelihood of the precision matrix
$\boldsymbol{\Omega}=\boldsymbol{\Sigma}^{-1}$ under normal assumption is
$\displaystyle
f(\boldsymbol{\Omega})=-\log\det\boldsymbol{\Omega}+\text{tr}(\hat{\boldsymbol{\Sigma}}\boldsymbol{\Omega})$
(28)
with the MLE solution $\hat{\boldsymbol{\Sigma}}^{-1}$ when
$\hat{\boldsymbol{\Sigma}}$ is non-degenerate. A zero in the precision matrix
implies conditional independence of the corresponding nodes. Graphical lasso
proposes to solve
$\displaystyle f(\boldsymbol{\Omega})+\rho\sum_{i<j}|\omega_{ij}|,$ (29)
where $\rho\geq 0$ is the tuning constant and $\omega_{ij}$ denotes the
$(i,j)$-element of $\boldsymbol{\Omega}$. It is well-known that the
determinant function is log-concave (Magnus and Neudecker, 1999). Therefore
the loss function $f$ (28) is convex and the EPSODE algorithm applies to (29).
Friedman et al. (2008) proposed an efficient coordinate descent procedure for
solving (29) at a fixed $\rho$. A recent attempt to approximate the whole
solution path is made by Yuan (2008). Again his path algorithm can be deemed
as a primitive predictor-corrector method for approximating the ODE solution.
With symmetry in mind, we parameterize $\boldsymbol{\Omega}$ in terms of its
lower triangular part by a $p(p+1)/2$ column vector $\boldsymbol{x}$ and let
$D\boldsymbol{\Omega}(\boldsymbol{x})=\frac{\partial\text{vec}\boldsymbol{\Omega}}{\partial(\text{vec}\boldsymbol{x})^{t}}$
be the corresponding $p^{2}$-by-$p(p+1)/2$ Jacobian matrix. Note
$D\boldsymbol{\Omega}(\boldsymbol{x})\cdot\boldsymbol{x}=\text{vec}\boldsymbol{\Omega}(\boldsymbol{x})$
and each row of $D\boldsymbol{\Omega}(\boldsymbol{x})$ has exactly one nonzero
entry which equals unity. We list here the first three derivatives of $f$. The
proof is straightforward using matrix calculus and omitted for brevity.
###### Lemma 7.1.
1. 1.
The derivatives for the Gaussian graphical model (28) with respect to
$\boldsymbol{\Omega}$ are
$\displaystyle Df(\boldsymbol{\Omega})$
$\displaystyle=df(\boldsymbol{\Omega})=[\text{vec}(-\boldsymbol{\Omega}^{-1}+\boldsymbol{\Sigma})]^{t}$
$\displaystyle D^{2}f(\boldsymbol{\Omega})$
$\displaystyle=d^{2}f(\boldsymbol{\Omega})=\boldsymbol{\Omega}^{-1}\otimes\boldsymbol{\Omega}^{-1}$
$\displaystyle D^{3}f(\boldsymbol{\Omega})$
$\displaystyle=-(\boldsymbol{I}_{n}\otimes\boldsymbol{K}_{nn}\otimes\boldsymbol{I}_{n})$
$\displaystyle\hskip
14.45377pt\cdot[\boldsymbol{\Omega}^{-1}\otimes\boldsymbol{\Omega}^{-1}\otimes\text{vec}(\boldsymbol{\Omega}^{-1})+\text{vec}(\boldsymbol{\Omega}^{-1})\otimes\boldsymbol{\Omega}^{-1}\otimes\boldsymbol{\Omega}^{-1}],$
where $\boldsymbol{K}_{nn}$ is the commutation matrix (Magnus and Neudecker,
1999).
2. 2.
The derivatives for the Gaussian graphical model (28) with respect to
$\boldsymbol{x}$ are
$\displaystyle Df(\boldsymbol{x})$ $\displaystyle=Df(\boldsymbol{\Omega})\cdot
D\boldsymbol{\Omega}(\boldsymbol{x})$
$\displaystyle\boldsymbol{H}(\boldsymbol{x})=D^{2}f(\boldsymbol{x})$
$\displaystyle=[D\boldsymbol{\Omega}(\boldsymbol{x})]^{t}\cdot
D^{2}f(\boldsymbol{\Omega})\cdot D\boldsymbol{\Omega}(\boldsymbol{x})$
$\displaystyle D\boldsymbol{H}(\boldsymbol{x})=D^{3}f(\boldsymbol{x})$
$\displaystyle=\\{[D\boldsymbol{\Omega}(\boldsymbol{x})]^{t}\otimes[D\boldsymbol{\Omega}(\boldsymbol{x})]^{t}\\}\cdot
D^{3}f(\boldsymbol{\Omega})\cdot D\boldsymbol{\Omega}(\boldsymbol{x}).$
When the covariance matrix $\hat{\boldsymbol{\Sigma}}$ is nonsingular, EPSODE
can be initiated either at $\rho=0$ or $\rho=\infty$. When
$\hat{\boldsymbol{\Sigma}}$ is singular, we start from $\rho=\infty$ and the
extended version of EPSODE (22) should be used. If starting at $\rho=0$, the
solution is initialized at $\hat{\boldsymbol{\Sigma}}^{-1}$; If starting at
$\rho=\infty$, the solution is initialized at
$\text{diag}(\hat{\sigma}_{ii}^{-1})$. Minimization of both the unpenalized
and penalized objective function has to be performed over the convex cone of
symmetric, positive semidefinite matrices, which is not explicitly
incorporated in our path following algorithm. The next result ensures the
positive definiteness of the path solution.
###### Lemma 7.2 (Positive definiteness along the path).
The path solution $\boldsymbol{\Omega}(\rho)$ minimizes (29) over the convex
cone of symmetric, positive semidefinite matrices.
###### Proof.
Both symmetry and the stationarity condition (5) are preserved by the path
following. Therefore the path solution $\boldsymbol{\Omega}(\rho)$ constitutes
a minimum of the penalized objective function (29). This implies that
$\text{det}(\boldsymbol{\Omega}(\rho))>0$ otherwise
$f(\boldsymbol{\Omega}(\rho))=\infty$, contradicting with the optimality of
$\boldsymbol{\Omega}(\rho)$. ∎
We illustrate the path algorithm by the classical example of 88 students’
scores on five math courses – mechanics, vector, algebra, analysis, and
statistics (Mardia et al., 1979, Table 1.2.1). Figure 4 displays the solution
path from EPSODE. The top three edges chosen by lasso are analysis-algebra,
statistics-algebra, and algebra-vector, matching the findings in (Yuan, 2008).
$\begin{array}[]{c}\includegraphics[width=252.94499pt]{MKBscore_lassopath}\end{array}$
Figure 4: Solution path of the 10 edges in lasso-regularized Gaussian
graphical model for the math score data. The top three edges chosen by lasso
are labeled.
### 7.4 Nonparametric Density Estimation
As part of the trend of statistics shifting from parametric models to semi- or
non-parametric models, nonparametric density estimation has attracted much
attention in recent years. The maximum likelihood estimation for nonparametric
density estimation often involves a nontrivial, high-dimensional constrained
optimization problem. In this section, we briefly demonstrate the
applicability of EPSODE to the maximum likelihood estimation of univariate
log-concave density. Extensions to multivariate log-concave density estimation
(Cule et al., 2009, 2010) and $k$-monotone density estimation (Balabdaoui and
Wellner, 2007) will be pursued elsewhere. Some algorithms have been
specifically crafted for log-concave density estimation, e.g., the iterative
convex minorant algorithm (ICMA) (Groeneboom and Wellner, 1992; Jongbloed,
1998) and more recently an active-set algorithm (Duembgen et al., 2007). It is
noteworthy that, besides providing an alternative solver for log-concave
density estimation, EPSODE offers the whole solution path between the
unconstrained and constrained solutions. For example, an “almost” log-concave
density estimate in the middle of the path can be chosen that minimizes cross-
validation or prediction error. This adds another dimension to the flexibility
of nonparametric modeling.
The family of log-concave densities is an attractive modeling tool. It
includes most of the commonly used parametric distributions as special cases.
Examples include normal, gamma with shape parameter $\geq 1$, and beta
densities with both parameters $\geq 1$. The survey paper (Walther, 2009)
gives a recent review. A probability density $g(\cdot)$ on $\mathbb{R}$ is
log-concave if its logarithm $\phi(x)=\ln g(x)$ is concave. Given iid
observations, from an unknown distribution of density $g(\cdot)$, with support
at points $x_{1}<\ldots<x_{n}$ with corresponding frequencies
$p_{1},\ldots,p_{n}$, it is well-known (Walther, 2002) that the nonparametric
MLE of $g$ exists, is unique and takes the form $\hat{g}=\exp(\hat{\phi})$
where $\hat{\phi}$ is continuous and piecewise linear on $[x_{1},x_{n}]$, with
the set of knots contained in $\\{x_{1},\ldots,x_{n}\\}$, and
$\hat{\phi}=-\infty$ outside the interval $[x_{1},x_{n}]$. This implies that
the MLE is obtained by minimizing the strictly convex function
$\displaystyle
f(\boldsymbol{\phi})=-\sum_{i=1}^{n}p_{i}\phi_{i}+\sum_{k=1}^{n-1}(x_{k+1}-x_{k})\int_{0}^{1}e^{(1-t)\phi_{k}+t\phi_{k+1}}\,dt.$
over
$\boldsymbol{\phi}=(\phi_{1},\phi_{2},\cdots,\phi_{n})^{t}\in\mathbb{R}^{n}$
subject to constraints
$\displaystyle\frac{\phi_{i+1}-\phi_{i}}{x_{i+1}-x_{i}}\leq\frac{\phi_{i}-\phi_{i-1}}{x_{i}-x_{i-1}},\hskip
7.22743pti=2,\ldots,n-1.$
The consistency of the MLE is proved by Pal et al. (2007) and the pointwise
asymptotic distribution of the MLE studied in (Balabdaoui et al., 2009).
Following Duembgen et al. (2007), we use notations
$\displaystyle\delta_{0}$
$\displaystyle=\delta_{n}=0,\,\delta_{i}=x_{i+1}-x_{i},\hskip
7.22743pti=1,\ldots,n-1$ $\displaystyle J(r,s)$
$\displaystyle=\int_{0}^{1}e^{(1-t)r+ts}\,dt=\begin{cases}\frac{e^{s}-e^{r}}{s-r}&r\neq
s\\\ e^{r}&r=s\end{cases}.$
Then the objective function becomes
$\displaystyle
f(\boldsymbol{\phi})=-\sum_{i=1}^{n}p_{i}\phi_{i}+\sum_{k=1}^{n-1}\delta_{k}J(\phi_{k},\phi_{k+1}).$
The path algorithm requires up to the third derivative of the objective
function $f$
$\displaystyle[\nabla f(\boldsymbol{\phi})]_{i}$
$\displaystyle=-p_{i}+\delta_{i-1}J_{01}(\phi_{i-1},\phi_{i})+\delta_{i}J_{10}(\phi_{i},\phi_{i+1})$
$\displaystyle[H(\boldsymbol{\phi})]_{ij}$
$\displaystyle=[d^{2}f(\boldsymbol{\phi})]_{ij}$
$\displaystyle=\begin{cases}\delta_{i-1}J_{11}(\phi_{i-1},\phi_{i})&j=i-1\\\
\delta_{i-1}J_{02}(\phi_{i-1},\phi_{i})+\delta_{i}J_{20}(\phi_{i},\phi_{i+1})&j=i\\\
\delta_{i}J_{11}(\phi_{i},\phi_{i+1})&j=i+1\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\frac{\partial[H(\boldsymbol{\phi})]_{i,i-1}}{\partial\phi_{k}}$
$\displaystyle=\begin{cases}\delta_{i-1}J_{21}(\phi_{i-1},\phi_{i})&k=i-1\\\
\delta_{i-1}J_{12}(\phi_{i-1},\phi_{i})&k=i\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\frac{\partial[H(\boldsymbol{\phi})]_{i,i}}{\partial\phi_{k}}$
$\displaystyle=\begin{cases}\delta_{i-1}J_{12}(\phi_{i-1},\phi_{i})&k=i-1\\\
\delta_{i-1}J_{03}(\phi_{i-1},\phi_{i})+\delta_{i}J_{30}(\phi_{i},\phi_{i+1})&k=i\\\
\delta_{i}J_{21}(\phi_{i},\phi_{i+1})&k=i+1\\\ 0&\text{otherwise}\end{cases}$
$\displaystyle\frac{\partial[H(\boldsymbol{\phi})]_{i,i+1}}{\partial\phi_{k}}$
$\displaystyle=\begin{cases}\delta_{i}J_{21}(\phi_{i},\phi_{i+1})&k=i\\\
\delta_{i}J_{12}(\phi_{i},\phi_{i+1})&k=i+1\\\ 0&\text{otherwise}\end{cases}.$
Interchanging the derivative and integral operators, justified by the
dominated convergence theorem, gives a useful representation for the partial
derivatives of $J$
$\displaystyle J_{ab}(r,s)$ $\displaystyle=\frac{\partial^{a+b}}{\partial
r^{a}\partial s^{b}}J(r,s)=\int_{0}^{1}(1-t)^{a}t^{b}e^{(1-t)r+ts}\,dt.$
We derive a recurrence relation for $J_{ab}(r,s)$ to facilitate its
computation.
###### Lemma 7.3.
$J_{ab}(r,s)$ satisfy following recurrence
1. 1.
For $r\neq s$,
$\displaystyle J_{00}(r,s)$ $\displaystyle=\frac{e^{s}-e^{r}}{s-r}$
$\displaystyle J_{10}(r,s)$
$\displaystyle=-\frac{e^{r}}{s-r}+\frac{e^{s}-e^{r}}{(s-r)^{2}}$
$\displaystyle J_{01}(r,s)$
$\displaystyle=\frac{e^{s}}{s-r}-\frac{e^{s}-e^{r}}{(s-r)^{2}}$ $\displaystyle
J_{11}(r,s)$
$\displaystyle=\frac{e^{s}+e^{r}}{(s-r)^{2}}-\frac{2(e^{s}-e^{r})}{(s-r)^{3}}$
$\displaystyle J_{ab}(r,s)$
$\displaystyle=\frac{a+b+s-r}{s-r}J_{a-1,b}(r,s)-\frac{a-1}{s-r}J_{a-2,b}(r,s)$
$\displaystyle J_{ab}(r,s)$
$\displaystyle=-\frac{a+b-s+r}{s-r}J_{a,b-1}(r,s)+\frac{b-1}{s-r}J_{a,b-2}(r,s).$
2. 2.
For $r=s$,
$\displaystyle
J_{ab}(r,s)=\frac{e^{r}a!b!}{(a+b+1)!}=\frac{a}{a+b+1}J_{a-1,b}=\frac{b}{a+b+1}J_{a,b-1}.$
###### Proof.
We recognize $J_{ab}(r,s)$ as the Kummer confluent hypergeometric function
multiplied by a constant
$\displaystyle J_{ab}(r,s)$
$\displaystyle=e^{r}B(a+1,b+1)\sum_{k=0}^{\infty}\frac{(b+1)_{(k)}}{(a+b+2)_{(k)}}\frac{(s-r)^{k}}{k!}$
$\displaystyle=e^{r}B(a+1,b+1)\,_{1}F_{1}(b+1,a+b+2\mid s-r)$
$\displaystyle=e^{s}B(a+1,b+1)\,_{1}F_{1}(a+1,a+b+2\mid r-s)$
$\displaystyle=\frac{e^{s}a!b!}{(a+b+1)!}\,_{1}F_{1}(a+1,a+b+2\mid r-s).$
Then the results follow from the well-known recurrence relation for Kummer
hypergeometric function
$\,{}_{1}F_{1}(x,y\mid z)=\frac{(1-y)(y+z-2)}{(x-y+1)z}\,_{1}F_{1}(x,y-1\mid
z)+\frac{(1-y)(2-y)}{(x-y+1)z}\,_{1}F_{1}(x,y-2\mid z).$
and symmetry $J_{ab}(r,s)=J_{ba}(s,r)$. ∎
To illustrate the path algorithm for this problem, we simulate $n=25$ points
from the extremal distribution Gumbel(0,1). Figure 5 displays the constrained
and unconstrained estimates of $\phi_{i}$ and the solution path bridging the
two.
$\begin{array}[]{cc}\includegraphics[width=180.67499pt]{gumbel_estimates}&\includegraphics[width=180.67499pt]{gumbel_solpath}\\\
\includegraphics[width=180.67499pt]{gumbel_cdf}\end{array}$ Figure 5: Log-
concave density estimation. $n=25$ points are generated from Gumbel(0,1)
distribution. Top left: Unconstrained and concavity-constrained estimates
$\phi$. Top right: Solution path. Bottom left: Empirical cdf and the cdf of
MLE density.
## 8 Conclusions
In this article we propose a generic path following algorithm EPSODE that
works for any regularization problems of form (1). The advantages are its
simplicity and generality. Path following only involves solving ODEs segment
by segment and is simple to implement using popular softwares such as R and
Matlab. Besides providing the whole regularization path, it also gives a
solver for linearly constrained optimization problems that frequently arise in
statistics. Our applications to shape-restricted regressions and nonparametric
density estimation are special cases in particular.
At least two extensions deserve further study. Current algorithm requires
sufficient smoothness (twice differentiable) in the loss function. This
precludes certain applications with non-smooth objective function, e.g., the
Huber loss in robust estimation and the loss function in quantile regression.
Generalization of our path algorithm to regularization of these loss functions
requires further research. Another restriction in our formulation is the
linearity in the regularization terms. In sparse regressions, several authors
have proposed nonlinear and non-convex penalties. The bridge regression (Frank
and Friedman, 1993; Fu, 1998) and SCAD penalties (Fan and Li, 2001) fall into
this category. As observed in (Friedman, 2008), when the penalty is not
convex, the solution path may not be continuous and poses difficulty in path
following, which strongly depends on the continuity and smoothness of the
solution path. Fortunately, in these problems, the discontinuities only occur
when new variables enter or leave the model. A promising strategy is to
initialize the starting point of next segment by solving an equality
constrained optimization problem. This again invites further investigation.
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|
arxiv-papers
| 2012-01-17T17:42:46 |
2024-09-04T02:49:26.404047
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua Zhou and Yichao Wu",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1201.3571"
}
|
1201.3593
|
# Path Following in the Exact Penalty Method
of Convex Programming
Hua Zhou
Department of Statistics
2311 Stinson Drive
North Carolina State University
Raleigh, NC 27695-8203
E-mail: hua_zhou@ncsu.edu
Kenneth Lange
Departments of Biomathematics,
Human Genetics, and Statistics
University of California
Los Angeles, CA 90095-1766
E-mail: klange@ucla.edu
###### Abstract
Classical penalty methods solve a sequence of unconstrained problems that put
greater and greater stress on meeting the constraints. In the limit as the
penalty constant tends to $\infty$, one recovers the constrained solution. In
the exact penalty method, squared penalties are replaced by absolute value
penalties, and the solution is recovered for a finite value of the penalty
constant. In practice, the kinks in the penalty and the unknown magnitude of
the penalty constant prevent wide application of the exact penalty method in
nonlinear programming. In this article, we examine a strategy of path
following consistent with the exact penalty method. Instead of performing
optimization at a single penalty constant, we trace the solution as a
continuous function of the penalty constant. Thus, path following starts at
the unconstrained solution and follows the solution path as the penalty
constant increases. In the process, the solution path hits, slides along, and
exits from the various constraints. For quadratic programming, the solution
path is piecewise linear and takes large jumps from constraint to constraint.
For a general convex program, the solution path is piecewise smooth, and path
following operates by numerically solving an ordinary differential equation
segment by segment. Our diverse applications to a) projection onto a convex
set, b) nonnegative least squares, c) quadratically constrained quadratic
programming, d) geometric programming, and e) semidefinite programming
illustrate the mechanics and potential of path following. The final detour to
image denoising demonstrates the relevance of path following to regularized
estimation in inverse problems. In regularized estimation, one follows the
solution path as the penalty constant decreases from a large value.
Keywords: constrained convex optimization, exact penalty, geometric
programming, ordinary differential equation, quadratically constrained
quadratic programming, regularization, semidefinite programming
11footnotetext: Research supported in part by USPHS grants GM53275 and MH59490
to KL, R01 HG006139 to KL and HZ, and NCSU FRPD grant to HZ.
## 1 Introduction
Penalties and barriers are both potent devices for solving constrained
optimization problems (Boyd and Vandenberghe, 2004; Forsgren et al., 2002;
Luenberger and Ye, 2008; Nocedal and Wright, 2006; Ruszczyński, 2006;
Zangwill, 1967). The general idea is to replace hard constraints by penalties
or barriers and then exploit the well-oiled machinery for solving
unconstrained problems. Penalty methods operate on the exterior of the
feasible region and barrier methods on the interior. The strength of a penalty
or barrier is determined by a tuning constant. In classical penalty methods, a
single global tuning constant is gradually sent to $\infty$; in barrier
methods, it is gradually sent to 0. Either strategy generates a sequence of
solutions that converges in practice to the solution of the original
constrained optimization problem.
Barrier methods are now generally conceded to offer a better approach to
solving convex programs than penalty methods. Application of log barriers and
carefully controlled versions of Newton’s method make it possible to follow
the central path reliably and quickly to the constrained minimum (Boyd and
Vandenberghe, 2004). Nonetheless, penalty methods should not be ruled out.
Augmented Lagrangian methods (Hestenes, 1975) and exact penalty methods
(Nocedal and Wright, 2006) are potentially competitive with interior point
methods for smooth convex programming problems. Both methods have the
advantage that the solution of the constrained problem kicks in for a finite
value of the penalty constant. This avoids problems of ill conditioning as the
penalty constant tends to $\infty$.
The disadvantage of exact penalties over traditional quadratic penalties is
lack of differentiability of the penalized objective function. In the current
paper, we argue that this impediment can be finessed by path following. Our
path following method starts at the unconstrained solution and follows the
solution path as the penalty constant increases. In the process, the solution
path hits, exits, and slides along the various constraint boundaries. The path
itself is piecewise smooth with kinks at the boundary hitting and escape
times. One advances along the path by numerically solving a differential
equation for the Lagrange multipliers of the penalized problem. In the special
case of quadratic programming with affine constraints, the solution path is
piecewise linear, and one can easily anticipate entire path segments (Zhou and
Lange, 2011b). This special case is intimately related to the linear
complementarity problem (Cottle et al., 1992) in optimization theory.
Homotopy (continuation) methods for the solution of nonlinear equations and
optimization problems have been pursued for many years and enjoyed a variety
of successes (Nocedal and Wright, 2006; Watson, 1986, 0001; Zangwill and
Garcia, 1981). To our knowledge, however, there has been no exploration of
path following as an implementation of the exact penalty method. Our modest
goal here is to assess the feasibility and versatility of exact path following
for constrained optimization. Comparing its performance to existing methods,
particularly the interior point method, is probably best left for later, more
practically oriented papers. In our experience, coding the algorithm is
straightforward in Matlab. The rich numerical resources of Matlab include
differential equation solvers that alert the user when certain events such as
constraint hitting and escape occur.
The rest of the paper is organized as follows. Section 2 briefly reviews the
exact penalty method for optimization and investigates sufficient conditions
for uniqueness and continuity of the solution path. Section 3 derives the path
following strategy for general convex programs, with particular attention to
the special cases of quadratic programming and convex optimization with affine
constraints. Section 4 presents various applications of the path algorithm.
Our most elaborate example demonstrates the relevance of path following to
regularized estimation. The particular problem treated, image denoising, is
typical of many inverse problems in applied mathematics and statistics Zhou
and Wu (2011). In such problems one follows the solution path as the penalty
constant decreases. Finally, Section 5 discusses the limitations of the path
algorithm and hints at future generalizations.
## 2 Exact Penalty Methods
In this paper we consider the convex programming problem of minimizing the
convex objective function $f(\boldsymbol{x})$ subject to $r$ affine equality
constraints $g_{i}(\boldsymbol{x})=0$ and $s$ convex inequality constraints
$h_{j}(\boldsymbol{x})\leq 0$. We will further assume that $f(\boldsymbol{x})$
and the $h_{j}(\boldsymbol{x})$ are twice differentiable. The differential
$df(\boldsymbol{x})$ is the row vector of partial derivatives of
$f(\boldsymbol{x})$; the gradient $\nabla f(\boldsymbol{x})$ is the transpose
of $df(\boldsymbol{x})$. The second differential $d^{2}f(\boldsymbol{x})$ is
the Hessian matrix of second partial derivatives of $f(\boldsymbol{x})$.
Similar conventions hold for the differentials of the constraint functions.
Exact penalty methods (Nocedal and Wright, 2006; Ruszczyński, 2006) minimize
the surrogate function
$\displaystyle{\mathcal{E}}_{\rho}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle
f(\boldsymbol{x})+\rho\sum_{i=1}^{r}|g_{i}(\boldsymbol{x})|+\rho\sum_{j=1}^{s}\max\\{0,h_{j}(\boldsymbol{x})\\}.$
(1)
This definition of ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is meaningful
regardless of whether the contributing functions are convex. If the program is
convex, then ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is itself convex. It is
interesting to compare ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ to the
Lagrangian function
$\displaystyle{\mathcal{L}}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
f(\boldsymbol{x})+\sum_{i=1}^{r}\lambda_{i}g_{i}(\boldsymbol{x})+\sum_{j=1}^{s}\mu_{j}h_{j}(\boldsymbol{x}),$
which captures the behavior of $f(\boldsymbol{x})$ near a constrained local
minimum $\boldsymbol{y}$. The Lagrangian satisfies the stationarity condition
$\nabla{\mathcal{L}}(\boldsymbol{y})={\bf 0}$; its inequality multipliers
$\mu_{j}$ are nonnegative and satisfy the complementary slackness conditions
$\mu_{j}h_{j}(\boldsymbol{y})=0$. In an exact penalty method one takes
$\displaystyle\rho$ $\displaystyle>$
$\displaystyle\max\\{|\lambda_{1}|,\ldots,|\lambda_{r}|,\mu_{1},\ldots,\mu_{s}\\}.$
(2)
This choice creates the favorable circumstances
$\displaystyle{\mathcal{L}}(\boldsymbol{x})$ $\displaystyle\leq$
$\displaystyle{\mathcal{E}}_{\rho}(\boldsymbol{x})\quad\mbox{for
all}\>\boldsymbol{x}$ $\displaystyle{\mathcal{L}}(\boldsymbol{z})$
$\displaystyle\leq$ $\displaystyle
f(\boldsymbol{z})\>=\>{\mathcal{E}}_{\rho}(\boldsymbol{z})\quad\mbox{for all
feasible}\>\boldsymbol{z}$ $\displaystyle{\mathcal{L}}(\boldsymbol{y})$
$\displaystyle=$ $\displaystyle
f(\boldsymbol{y})\>=\>{\mathcal{E}}_{\rho}(\boldsymbol{y})\quad\mbox{for}\;\boldsymbol{y}\;\mbox{optimal}$
with profound consequences. As the next proposition proves, minimizing
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is effective in minimizing
$f(\boldsymbol{x})$ subject to the constraints.
###### Proposition 2.1.
Suppose the objective function $f(\boldsymbol{x})$ and the constraint
functions are twice differentiable and satisfy the Lagrange multiplier rule at
the local minimum $\boldsymbol{y}$. If inequality (2) holds and
$\boldsymbol{v}^{*}d^{2}{\mathcal{L}}(\boldsymbol{y})\boldsymbol{v}>0$ for
every vector $\boldsymbol{v}\neq{\bf 0}$ satisfying
$dg_{i}(\boldsymbol{y})\boldsymbol{v}=0$ and
$dh_{j}(\boldsymbol{y})\boldsymbol{v}\leq 0$ for all active inequality
constraints, then $\boldsymbol{y}$ furnishes an unconstrained local minimum of
${\mathcal{E}}_{\rho}(\boldsymbol{x})$. For a convex program satisfying
Slater’s constraint qualification and inequality (2), $\boldsymbol{y}$ is a
minimum of ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ if and only if
$\boldsymbol{y}$ is a minimum of $f(\boldsymbol{x})$ subject to the
constraints. No differentiability assumptions are required for convex
programs.
###### Proof.
The conditions imposed on the quadratic form
$\boldsymbol{v}^{*}d^{2}{\mathcal{L}}(\boldsymbol{y})\boldsymbol{v}$ are well-
known sufficient conditions for a local minimum. Theorems 6.9 and 7.21 of the
reference Ruszczyński (2006) prove all of the foregoing assertions. ∎
As previously stressed, the exact penalty method turns a constrained
optimization problem into an unconstrained minimization problem. Furthermore,
in contrast to the quadratic penalty method (Nocedal and Wright, 2006, Section
17.1), the constrained solution in the exact method is achieved for a finite
value of $\rho$. Despite these advantages, minimizing the surrogate function
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is complicated. For one thing, it is no
longer globally differentiable. For another, one must minimize
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ along an increasing sequence $\rho_{n}$
because the Lagrange multipliers (2) are usually unknown in advance. These
hurdles have prevented wide application of exact penalty methods in convex
programming.
As a prelude to our derivation of the path following algorithm for convex
programs, we record several properties of
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ that mitigate the failure of
differentiability.
###### Proposition 2.2.
The surrogate function ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is increasing in
$\rho$. Furthermore, ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is strictly convex
for one $\rho>0$ if and only if it is strictly convex for all $\rho>0$.
Likewise, it is coercive for one $\rho>0$ if and only if is coercive for all
$\rho>0$. Finally, if $f(\boldsymbol{x})$ is strictly convex (or coercive),
then all ${\mathcal{E}}_{\rho}(\boldsymbol{x})$ are strictly convex (or
coercive).
###### Proof.
The first assertion is obvious. For the second assertion, consider more
generally a finite family $u_{1}(\boldsymbol{x}),\ldots,u_{q}(\boldsymbol{x})$
of convex functions, and suppose a linear combination
$\sum_{k=1}^{q}c_{k}u_{k}(\boldsymbol{x})$ with positive coefficients is
strictly convex. It suffices to prove that any other linear combination
$\sum_{k=1}^{q}b_{k}u_{k}(\boldsymbol{x})$ with positive coefficients is
strictly convex. For any two points $\boldsymbol{x}\neq\boldsymbol{y}$ and any
scalar $\alpha\in(0,1)$, we have
$\displaystyle u_{k}[\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}]\leq\alpha
u_{k}(\boldsymbol{x})+(1-\alpha)u_{k}(\boldsymbol{y}).$ (3)
Since $\sum_{k=1}^{q}c_{k}u_{k}(\boldsymbol{x})$ is strictly convex, strict
inequality must hold for at least one $k$. Hence, multiplying inequality (3)
by $b_{k}$ and adding gives
$\displaystyle\sum_{k=1}^{q}b_{k}u_{k}[\alpha\boldsymbol{x}+(1-\alpha)\boldsymbol{y}]$
$\displaystyle<$
$\displaystyle\alpha\sum_{k=1}^{q}b_{k}u_{k}(\boldsymbol{x})+(1-\alpha)\sum_{k=1}^{q}b_{k}u_{k}(\boldsymbol{y}).$
The third assertion follows from the fact that a convex function is coercive
if and only if its restriction to each half-line is coercive (Bertsekas, 2003,
Proposition 3.2.2). Given this result, suppose
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ is coercive, but
${\mathcal{E}}_{\rho^{\ast}}(\boldsymbol{x})$ is not coercive. Then there
exists a point $\boldsymbol{x}$, a direction $\boldsymbol{v}$, and a sequence
of scalars $t_{n}$ tending to $\infty$ such that
${\mathcal{E}}_{\rho^{\ast}}(\boldsymbol{x}+t_{n}\boldsymbol{v})$ is bounded
above. This requires the sequence $f(\boldsymbol{x}+t_{n}\boldsymbol{v})$ and
each of the sequences $|g_{i}(\boldsymbol{x}+t_{n}\boldsymbol{v})|$ and
$\max\\{0,h_{j}(\boldsymbol{x}+t_{n}\boldsymbol{v})$ to remain bounded above.
But in this circumstance the sequence
${\mathcal{E}}_{\rho}(\boldsymbol{x}+t_{n}\boldsymbol{v})$ also remains
bounded above. The final two assertions are obvious. ∎
## 3 The Path Following Algorithm
In this section, we take a different point of view. Instead of minimizing
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ for an increasing sequence $\rho_{n}$,
we study how the solution $\boldsymbol{x}(\rho)$ changes continuously with
$\rho$ and devise a path following strategy starting from $\rho=0$. For some
finite value of $\rho$, the path locks in on the solution of the original
convex program. In regularized statistical estimation and inverse problems,
the primary goal is to select relevant predictors rather than to find a
constrained solution. Thus, the entire solution path commands more interest
than any single point along it (Efron et al., 2004; Osborne et al., 2000;
Tibshirani and Taylor, 2011; Zhou and Lange, 2011b; Zhou and Wu, 2011).
Although our theory will focus on constrained estimation, readers should bear
in mind this second application area of path following.
The path algorithm relies critically on the first order optimality condition
that characterizes the optimum point of the convex function
${\mathcal{E}}_{\rho}(\boldsymbol{y})$.
###### Proposition 3.1.
For a convex program, a point $\boldsymbol{x}=\boldsymbol{x}(\rho)$ minimizes
the function ${\mathcal{E}}_{\rho}(\boldsymbol{y})$ if and only if
$\boldsymbol{x}$ satisfies the stationarity condition
$\displaystyle{\bf 0}$ $\displaystyle=$ $\displaystyle\nabla
f(\boldsymbol{x})+\rho\sum_{i=1}^{r}s_{i}\nabla
g_{i}(\boldsymbol{x})+\rho\sum_{j=1}^{s}t_{j}\nabla h_{j}(\boldsymbol{x})$ (4)
for coefficient sets $\\{s_{i}\\}_{i=1}^{r}$ and $\\{t_{j}\\}_{j=1}^{s}$.
These sets can be characterized as
$\displaystyle s_{i}\in\begin{cases}\\{-1\\}&g_{i}(\boldsymbol{x})<0\\\
[-1,1]&g_{i}(\boldsymbol{x})=0\\\
\\{1\\}&g_{i}(\boldsymbol{x})>0\end{cases}\hskip 14.45377pt\mbox{ and }\hskip
14.45377ptt_{j}\in\begin{cases}\\{0\\}&h_{j}(\boldsymbol{x})<0\\\
[0,1]&h_{j}(\boldsymbol{x})=0\\\ \\{1\\}&h_{j}(\boldsymbol{x})>0\end{cases}.$
(5)
At most one point achieves the minimum of
${\mathcal{E}}_{\rho}(\boldsymbol{y})$ for a given $\rho$ when
${\mathcal{E}}_{\rho}(\boldsymbol{y})$ is strictly convex.
###### Proof.
According to Fermat’s rule, $\boldsymbol{x}$ minimizes
${\mathcal{E}}_{\rho}(\boldsymbol{y})$ if and only if ${\bf 0}$ belongs to the
subdifferential $\partial{\mathcal{E}}_{\rho}(\boldsymbol{x})$ of
${\mathcal{E}}_{\rho}(\boldsymbol{y})$. To derive the subdifferential
displayed in equations (4) and (5), one applies the addition and chain rules
of the convex calculus. The sets defining the possible values of $s_{i}$ and
$t_{j}$ are the subdifferentials of the functions $|s|$ and
$t_{+}=\max\\{t,0\\}$, respectively. For more details see Theorem 3.5 and
ancillary material in the book (Ruszczyński, 2006). Finally, it is well known
that strict convexity guarantees a unique minimum. ∎
To speak coherently of solution paths, one must validate the existence,
uniqueness, and continuity of the solution $\boldsymbol{x}(\rho)$ to the
system of equations (1). Uniqueness follows from strict convexity as already
noted. Existence and continuity are more subtle.
###### Proposition 3.2.
If ${\mathcal{E}}_{\rho}(\boldsymbol{y})$ is strictly convex and coercive,
then the solution path $\boldsymbol{x}(\rho)$ of equation (1) exists and is
continuous in $\rho$. If the gradient vectors $\\{\nabla
g_{i}(\boldsymbol{x}):g_{i}(\boldsymbol{x})=0\\}\cup\\{\nabla
h_{j}(\boldsymbol{x}):h_{j}(\boldsymbol{x})=0\\}$ of the active constraints
are linearly independent at $\boldsymbol{x}(\rho)$ for $\rho>0$, then the
coefficients $s_{i}(\rho)$ and $t_{j}(\rho)$ are unique and continuous near
$\rho$ as well.
###### Proof.
In accord with Proposition 2.2, we assume that either $f(\boldsymbol{x})$ is
strictly convex and coercive or restrict our attention to the open interval
$(0,\infty)$. Consider a subinterval $[a,b]$ and fix a point $\boldsymbol{x}$
in the common domain of the functions ${\mathcal{E}}_{\rho}(\boldsymbol{y})$.
The coercivity of ${\mathcal{E}}_{a}(\boldsymbol{y})$ and the inequalities
$\displaystyle{\mathcal{E}}_{a}[\boldsymbol{x}(\rho)]$ $\displaystyle\leq$
$\displaystyle{\mathcal{E}}_{\rho}[\boldsymbol{x}(\rho)]\>\leq\>{\mathcal{E}}_{\rho}(\boldsymbol{x})\>\leq\>{\mathcal{E}}_{b}(\boldsymbol{x})$
demonstrate that the solution vector $\boldsymbol{x}(\rho)$ is bounded over
$[a,b]$. To prove continuity, suppose that it fails for a given
$\rho\in[a,b]$. Then there exists an $\epsilon>0$ and a sequence $\rho_{n}$
tending to $\rho$ such
$\|\boldsymbol{x}(\rho_{n})-\boldsymbol{x}(\rho)\|_{2}\geq\epsilon$ for all
$n$. Since $\boldsymbol{x}(\rho_{n})$ is bounded, we can pass to a subsequence
if necessary and assume that $\boldsymbol{x}(\rho_{n})$ converges to some
point $\boldsymbol{y}$. Taking limits in the inequality
${\mathcal{E}}_{\rho_{n}}[\boldsymbol{x}(\rho_{n})]\leq{\mathcal{E}}_{\rho_{n}}(\boldsymbol{x})$
demonstrates that
${\mathcal{E}}_{\rho}(\boldsymbol{y})\leq{\mathcal{E}}_{\rho}(\boldsymbol{x})$
for all $\boldsymbol{x}$. Because $\boldsymbol{x}(\rho)$ is unique, we reach
the contradictory conclusions
$\|\boldsymbol{y}-\boldsymbol{x}(\rho)\|_{2}\geq\epsilon$ and
$\boldsymbol{y}=\boldsymbol{x}(\rho)$.
Verification of the second claim is deferred to permit further discussion of
path following. The claim says that an active constraint
($g_{i}(\boldsymbol{x})=0$ or $h_{j}(\boldsymbol{x})=0$) remains active until
its coefficient hits an endpoint of its subdifferential. Because the solution
path is, in fact, piecewise smooth, one can follow the coefficient path by
numerically solving an ordinary differential equation (ODE). ∎
Our path following algorithm works segment-by-segment. Along the path we keep
track of the following index sets
$\displaystyle{\mathcal{N}}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})<0\\}\hskip
36.135pt{\mathcal{N}}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})<0\\}$
$\displaystyle{\mathcal{Z}}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})=0\\}\hskip
36.135pt{\mathcal{Z}}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})=0\\}$ (6)
$\displaystyle{\mathcal{P}}_{\text{E}}$
$\displaystyle=\\{i:g_{i}(\boldsymbol{x})>0\\}\hskip
36.135pt{\mathcal{P}}_{\text{I}}=\\{j:h_{j}(\boldsymbol{x})>0\\}$
determined by the signs of the constraint functions. For the sake of
simplicity, assume that at the beginning of the current segment $s_{i}$ does
not equal $-1$ or $1$ when $i\in{\mathcal{Z}}_{\text{E}}$ and $t_{j}$ does not
equal $0$ or $1$ when $j\in{\mathcal{Z}}_{\text{I}}$. In other words, the
coefficients of the active constraints occur on the interior of their
subdifferentials. Let us show in this circumstance that the solution path can
be extended in a smooth fashion. Our plan of attack is to reparameterize by
the Lagrange multipliers for the active constraints. Thus, set
$\lambda_{i}=\rho s_{i}$ for $i\in{\mathcal{Z}}_{\text{E}}$ and
$\omega_{j}=\rho t_{j}$ for $j\in{\mathcal{Z}}_{\text{I}}$. The multipliers
satisfy $-\rho<\lambda_{i}<\rho$ and $0<\omega_{j}<\rho$. The stationarity
condition now reads
$\displaystyle{\bf 0}$ $\displaystyle=$ $\displaystyle\nabla
f(\boldsymbol{x})-\rho\sum_{i\in{\mathcal{N}}_{\text{E}}}\nabla
g_{i}(\boldsymbol{x})+\rho\sum_{i\in{\mathcal{P}}_{\text{E}}}\nabla
g_{i}(\boldsymbol{x})+\rho\sum_{j\in{\mathcal{P}}_{\text{I}}}\nabla
h_{j}(\boldsymbol{x})$ $\displaystyle\hskip
36.135pt+\sum_{i\in{\mathcal{Z}}_{\text{E}}}\lambda_{i}\nabla
g_{i}(\boldsymbol{x})+\sum_{j\in{\mathcal{Z}}_{\text{I}}}\omega_{j}\nabla
h_{j}(\boldsymbol{x}).$
To this we concatenate the constraint equations $0=g_{i}(\boldsymbol{x})$ for
$i\in{\mathcal{Z}}_{\text{E}}$ and $0=h_{j}(\boldsymbol{x})$ for
$j\in{\mathcal{Z}}_{\text{I}}$.
For convenience now define
$\displaystyle\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})=\left[\\!\begin{array}[]{c}dg_{{\mathcal{Z}}_{\text{E}}}(\boldsymbol{x})\\\
dh_{{\mathcal{Z}}_{\text{I}}}(\boldsymbol{x})\end{array}\\!\right],\hskip
10.84006pt\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})=-\sum_{i\in{\mathcal{N}}_{\text{E}}}\nabla
g_{i}(\boldsymbol{x})+\sum_{i\in{\mathcal{P}}_{\text{E}}}\nabla
g_{i}(\boldsymbol{x})+\sum_{j\in{\mathcal{P}}_{\text{I}}}\nabla
h_{j}(\boldsymbol{x}).$
In this notation the stationarity equation can be recast as
$\displaystyle{\bf 0}$ $\displaystyle=$ $\displaystyle\nabla
f(\boldsymbol{x})+\rho\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})+\boldsymbol{U}_{{\mathcal{Z}}}^{t}(\boldsymbol{x})\left[\begin{matrix}\boldsymbol{\lambda}\\\
\boldsymbol{\omega}\end{matrix}\right].$
Under the assumption that the matrix
$\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})$ has full row rank, one can
solve for the Lagrange multipliers in the form
$\displaystyle\left[\begin{matrix}\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-[\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})\boldsymbol{U}_{{\mathcal{Z}}}^{t}(\boldsymbol{x})]^{-1}\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})\left[\nabla
f(\boldsymbol{x})+\rho\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})\right].$
(7)
Hence, the multipliers are unique. Continuity of the multipliers is a
consequence of the continuity of the solution vector $\boldsymbol{x}(\rho)$
and all functions in sight on the right-hand side of equation (7). This
observation completes the proof of Proposition 3.2.
Collectively the stationarity and active constraint equations can be written
as the vector equation ${\bf
0}=k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$. To solve
for $\boldsymbol{x}$, $\boldsymbol{\lambda}$ and $\boldsymbol{\omega}$ in
terms of $\rho$, we apply the implicit function theorem (Lange, 2004; Magnus
and Neudecker, 1999). This requires calculating the differential of
$k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$ with respect
to the underlying dependent variables $\boldsymbol{x}$,
$\boldsymbol{\lambda}$, and $\boldsymbol{\omega}$ and the independent variable
$\rho$. Because the equality constraints are affine, a brief calculation gives
$\displaystyle\partial_{\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega}}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$
$\displaystyle=$
$\displaystyle\left[\begin{matrix}d^{2}f(\boldsymbol{x})+\rho\sum_{j\in{\mathcal{P}}_{\text{I}}}d^{2}h_{j}(\boldsymbol{x})+\sum_{j\in{\mathcal{Z}}_{\text{I}}}\omega_{j}d^{2}h_{j}(\boldsymbol{x})&\boldsymbol{U}_{{\mathcal{Z}}}^{t}(\boldsymbol{x})\\\
\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})&{\bf 0}\end{matrix}\right]$
$\displaystyle\partial_{\rho}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$
$\displaystyle=$
$\displaystyle\left(\begin{matrix}\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})\\\
{\bf 0}\end{matrix}\right).$
The matrix
$\partial_{\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega}}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$
is nonsingular when its upper-left block is positive definite and its lower-
left block has full row rank (Lange, 2010, Proposition 11.3.2). Given that it
is nonsingular, the implicit function theorem applies, and we can in principle
solve for $\boldsymbol{x}$, $\boldsymbol{\lambda}$ and $\boldsymbol{\omega}$
in terms of $\rho$. More importantly, the implicit function theorem supplies
the derivative
$\displaystyle{d\over d\rho}\\!\left[\begin{matrix}\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-\partial_{\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega}}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)^{-1}\partial_{\rho}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho),$
(8)
which is the key to path following. We summarize our findings in the next
proposition.
###### Proposition 3.3.
Suppose the surrogate function ${\mathcal{E}}_{\rho}(\boldsymbol{y})$ is
strictly convex and coercive. If at the point $\boldsymbol{x}(\rho_{0})$ the
matrix
$\partial_{\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega}}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)$
is nonsingular and the coefficient of each active constraints occurs on the
interior of its subdifferential, then the solution path $\boldsymbol{x}(\rho)$
and Lagrange multipliers $\boldsymbol{\lambda}(\rho)$ and
$\boldsymbol{\omega}(\rho)$ satisfy the differential equation (8) in the
vicinity of $\boldsymbol{x}(\rho_{0})$.
In practice one traces the solution path along the current time segment until
either an inactive constraint becomes active or the coefficient of an active
constraint hits the boundary of its subdifferential. The earliest hitting time
or escape time over all constraints determines the duration of the current
segment. When the hitting time for an inactive constraint occurs first, we
move the constraint to the appropriate active set ${\mathcal{Z}}_{\text{E}}$
or ${\mathcal{Z}}_{\text{I}}$ and keep the other constraints in place.
Similarly, when the escape time for an active constraint occurs first, we move
the constraint to the appropriate inactive set and keep the other constraints
in place. In the second scenario, if $s_{i}$ hits the value $-1$, then we move
$i$ to ${\mathcal{N}}_{\text{E}}$; If $s_{i}$ hits the value $1$, then we move
$i$ to ${\mathcal{P}}_{\text{E}}$. Similar comments apply when a coefficient
$t_{j}$ hits 0 or 1. Once this move is executed, we commence path following
along the new segment. Path following continues until for sufficiently large
$\rho$, the sets ${\mathcal{N}}_{\text{E}}$, ${\mathcal{P}}_{\text{E}}$, and
${\mathcal{P}}_{\text{I}}$ are exhausted,
$\boldsymbol{u}_{\bar{\mathcal{Z}}}={\bf 0}$, and the solution vector
$\boldsymbol{x}(\rho)$ stabilizes. Our previous paper (Zhou and Lange, 2011b)
suggests remedies in the very rare situations where escape times coincide.
Path following simplifies considerably in two special cases. Consider convex
quadratic programming with objective function
$f(\boldsymbol{x})=\frac{1}{2}\boldsymbol{x}^{t}A\boldsymbol{x}+\boldsymbol{b}^{t}\boldsymbol{x}$
and equality constraints $\boldsymbol{V}\boldsymbol{x}=\boldsymbol{d}$ and
inequality constraints $\boldsymbol{W}\boldsymbol{x}\leq\boldsymbol{e}$, where
$\boldsymbol{A}$ is positive semi-definite. The exact penalized objective
function becomes
$\displaystyle{\mathcal{E}}_{\rho}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}A\boldsymbol{x}+\boldsymbol{b}^{t}\boldsymbol{x}+\rho\sum_{i=1}^{s}|\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}|+\rho\sum_{j=1}^{t}(\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j})_{+}.$
Since both the equality and inequality constraints are affine, their second
derivatives vanish. Both $\boldsymbol{U}_{\mathcal{Z}}$ and
$\boldsymbol{u}_{\bar{\mathcal{Z}}}$ are constant on the current path segment,
and the path $\boldsymbol{x}(\rho)$ satisfies
$\displaystyle{d\over d\rho}\\!\left[\begin{matrix}\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-\left(\begin{matrix}\boldsymbol{A}&\boldsymbol{U}_{\mathcal{Z}}^{t}\\\
\boldsymbol{U}_{\mathcal{Z}}&{\bf
0}\end{matrix}\right)^{-1}\left(\begin{matrix}\boldsymbol{u}_{\bar{\mathcal{Z}}}\\\
{\bf 0}\end{matrix}\right).$ (9)
This implies that the solution path $\boldsymbol{x}(\rho)$ is piecewise
linear. Our previous paper (Zhou and Lange, 2011b) is devoted entirely to this
special class of problems and highlights many statistical applications.
On the next rung on the ladder of generality are convex programs with affine
constraints. For the exact surrogate
$\displaystyle{\mathcal{E}}_{\rho}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle
f(\boldsymbol{x})+\rho\sum_{i=1}^{s}|\boldsymbol{v}_{i}^{t}\boldsymbol{x}-d_{i}|+\rho\sum_{j=1}^{t}(\boldsymbol{w}_{j}^{t}\boldsymbol{x}-e_{j})_{+},$
the matrix $\boldsymbol{U}_{\mathcal{Z}}$ and vector
$\boldsymbol{u}_{\bar{\mathcal{Z}}}$ are still constant along a path segment.
The relevant differential equation becomes
$\displaystyle{d\over d\rho}\\!\left[\begin{matrix}\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-\left(\begin{matrix}d^{2}f(\boldsymbol{x})&\boldsymbol{U}_{\mathcal{Z}}^{t}\\\
\boldsymbol{U}_{\mathcal{Z}}&{\bf
0}\end{matrix}\right)^{-1}\left(\begin{matrix}\boldsymbol{u}_{\bar{\mathcal{Z}}}\\\
{\bf 0}\end{matrix}\right).$ (10)
There are two approaches for computing the right-hand side of equation (10).
When $\boldsymbol{A}=d^{2}f(\boldsymbol{x})$ is positive definite and
$\boldsymbol{B}=\boldsymbol{U}_{\mathcal{Z}}$ has full row rank, the relevant
inverse amounts to
$\displaystyle\left(\begin{matrix}\boldsymbol{A}&\boldsymbol{B}^{t}\\\
\boldsymbol{B}&{\bf 0}\end{matrix}\right)^{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\boldsymbol{A}^{-1}-\boldsymbol{A}^{-1}\boldsymbol{B}^{t}[\boldsymbol{B}\boldsymbol{A}^{-1}\boldsymbol{B}^{t}]^{-1}\boldsymbol{B}\boldsymbol{A}^{-1}&\boldsymbol{A}^{-1}\boldsymbol{B}^{t}[\boldsymbol{B}\boldsymbol{A}^{-1}\boldsymbol{B}^{t}]^{-1}\\\
[\boldsymbol{B}\boldsymbol{A}^{-1}\boldsymbol{B}^{t}]^{-1}\boldsymbol{B}\boldsymbol{A}^{-1}&-[\boldsymbol{B}\boldsymbol{A}^{-1}\boldsymbol{B}^{t}]^{-1}\end{matrix}\right).$
The numerical cost of computing the inverse scales as
$O(n^{3})+O(|{\mathcal{Z}}|^{3})$. When $d^{2}f(\boldsymbol{x})$ is a
constant, the inverse is computed once. Sequentially updating it for different
active sets ${\mathcal{Z}}$ is then conveniently organized around the sweep
operator of computational statistics (Zhou and Lange, 2011b). For a general
convex function $f(\boldsymbol{x})$, every time $\boldsymbol{x}$ changes, the
inverse must be recomputed. This burden plus the cost of computing the entries
of $d^{2}f(\boldsymbol{x})$ slow the path algorithm for general convex
problems.
In many applications $f(\boldsymbol{x})$ is convex but not necessarily
strictly convex. One can circumvent problems in inverting
$d^{2}f(\boldsymbol{x})$ by reparameterizing (Nocedal and Wright, 2006). For
the sake of simplicity, suppose that all of the constraints are affine and
that $\boldsymbol{U}_{\mathcal{Z}}$ has full row rank. The set of points
$\boldsymbol{x}$ satisfying the active constraints can be written as
$\boldsymbol{x}=\boldsymbol{w}+\boldsymbol{Y}\boldsymbol{y}$, where
$\boldsymbol{w}$ is a particular solution, $\boldsymbol{y}$ is free to vary,
and the columns of $\boldsymbol{Y}\in\mathbb{R}^{n\times(n-|{\mathcal{Z}}|)}$
span the null space of $\boldsymbol{U}_{\mathcal{Z}}$ and hence are orthogonal
to the rows of $\boldsymbol{U}_{\mathcal{Z}}$. Under the null space
reparameterization,
$\frac{d\boldsymbol{x}}{d\rho}=\boldsymbol{Y}\frac{d\boldsymbol{y}}{d\rho}$.
Furthermore,
$\displaystyle\boldsymbol{Y}^{t}d^{2}f(\boldsymbol{x})\boldsymbol{Y}$
$\displaystyle=$ $\displaystyle
d_{\boldsymbol{y}}^{2}f(\boldsymbol{w}+\boldsymbol{Y}\boldsymbol{y})$
$\displaystyle\boldsymbol{Y}^{t}\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\nabla_{\boldsymbol{y}}\Big{[}-\rho\sum_{i\in{\mathcal{N}}_{\text{E}}}g_{i}(\boldsymbol{w}+\boldsymbol{Y}\boldsymbol{y})+\rho\sum_{i\in{\mathcal{P}}_{\text{E}}}g_{i}(\boldsymbol{w}+\boldsymbol{Y}\boldsymbol{y})+\rho\sum_{j\in{\mathcal{P}}_{\text{I}}}h_{j}(\boldsymbol{w}+\boldsymbol{Y}\boldsymbol{y})\Big{]}.$
It follows that equation (10) becomes
$\displaystyle\frac{d}{d\rho}\boldsymbol{y}$ $\displaystyle=$
$\displaystyle-[\boldsymbol{Y}^{t}d^{2}f(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}\boldsymbol{u}_{\bar{\mathcal{Z}}}$
$\displaystyle\frac{d}{d\rho}\boldsymbol{x}$ $\displaystyle=$
$\displaystyle-\boldsymbol{Y}[\boldsymbol{Y}^{t}d^{2}f(\boldsymbol{x})\boldsymbol{Y}]^{-1}\boldsymbol{Y}^{t}\boldsymbol{u}_{\bar{\mathcal{Z}}}.$
(11)
Differentiating equation (7) gives the multiplier derivatives
$\displaystyle\frac{d}{d\rho}\left[\begin{matrix}\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-(\boldsymbol{U}_{\mathcal{Z}}\boldsymbol{U}_{\mathcal{Z}}^{t})^{-1}\boldsymbol{U}_{\mathcal{Z}}\left(d^{2}f(\boldsymbol{x})\frac{d\boldsymbol{x}}{d\rho}+\boldsymbol{u}_{\bar{\mathcal{Z}}}\right).$
(12)
The obvious advantage of using equation (11) is that the matrix
$\boldsymbol{Y}^{t}d^{2}f(\boldsymbol{x})\boldsymbol{Y}$ can be nonsingular
when $d^{2}f(\boldsymbol{x})$ is singular. The computational cost of
evaluating the right-hand sides of equations (11) and (12) is
$O([n-|{\mathcal{Z}}|]^{3})+O(|{\mathcal{Z}}|^{3})$. When $n-|{\mathcal{Z}}|$
and $|{\mathcal{Z}}|$ are small compared to $n$, this is an improvement over
the cost $O(n^{3})+O(|{\mathcal{Z}}|^{3})$ of computing the right-hand side of
equation (8). Balanced against this gain is the requirement of finding a basis
of the null space of $\boldsymbol{U}_{\mathcal{Z}}$. Fortunately, the matrix
$\boldsymbol{Y}$ is constant over each path segment and in practice can be
computed by taking the QR decomposition of the active constraint matrix
$\boldsymbol{U}_{\mathcal{Z}}$. At each kink of the solution path, either one
constraint enters ${\mathcal{Z}}$ or one leaves. Therefore, $\boldsymbol{Y}$
can be sequentially computed by standard updating and downdating formulas
(Lawson and Hanson, 1987; Nocedal and Wright, 2006). Which ODE (8) or (11) is
preferable depends on the specific application. When the loss function
$f(\boldsymbol{x})$ is not strictly convex, for example when the number of
parameters exceeds the number of cases in regression, path following requires
the ODE (11). Interested readers are referred to the book (Nocedal and Wright,
2006) for a more extended discussion of range-space versus null-space
optimization methods.
For a general convex program, one can employ Euler’s update
$\displaystyle\left[\begin{matrix}\boldsymbol{x}(\rho+\Delta\rho)\\\
\boldsymbol{\lambda}(\rho+\Delta\rho)\\\
\boldsymbol{\omega}(\rho+\Delta\rho)\end{matrix}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{matrix}\boldsymbol{x}(\rho)\\\
\boldsymbol{\lambda}(\rho)\\\
\boldsymbol{\omega}(\rho)\end{matrix}\right]+\Delta\rho{d\over
d\rho}\left[\begin{matrix}\boldsymbol{x}(\rho)\\\
\boldsymbol{\lambda}(\rho)\\\ \boldsymbol{\omega}(\rho)\end{matrix}\right]$
to advance the solution of the ODE (8). Euler’s formula may be inaccurate for
$\Delta\rho$ large. One can correct it by fixing $\rho$ and performing one
step of Newton’s method to re-connect with the solution path. This amounts to
replacing the position-multiplier vector by
$\displaystyle\left[\begin{matrix}\boldsymbol{x}\\\ \boldsymbol{\lambda}\\\
\boldsymbol{\omega}\end{matrix}\right]-\partial_{\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega}}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho)^{-1}k(\boldsymbol{x},\boldsymbol{\lambda},\boldsymbol{\omega},\rho).$
In practice, it is certainly easier and probably safer to rely on ODE packages
such as the ODE45 function in Matlab to advance the solution of the ODE.
## 4 Examples of Path Following
Our examples are intended to illuminate the mechanics of path following and
showcase its versatility. As we emphasized in the introduction, we forgo
comparisons with other methods. Comparisons depend heavily on programming
details and problem choices, so a premature study might well be misleading.
###### Example 4.1.
Projection onto the Feasible Region
Finding a feasible point is the initial stage in many convex programs.
Dykstra’s algorithm (Dykstra, 1983; Deutsch, 2001) was designed precisely to
solve the problem of projecting an exterior point onto the intersection of a
finite number of closed convex sets. The projection problem also yields to our
generic path following algorithm. Consider the toy example of projecting a
point $\boldsymbol{b}\in\mathbb{R}^{2}$ onto the intersection of the closed
unit ball and the closed half space $x_{1}\geq 0$ (Lange, 2004). This is
equivalent to solving
minimize $\displaystyle\hskip
7.22743ptf(\boldsymbol{x})=\frac{1}{2}\|\boldsymbol{x}-\boldsymbol{b}\|^{2}$
subject to $\displaystyle\hskip
7.22743pth_{1}(\boldsymbol{x})=\frac{1}{2}\|\boldsymbol{x}\|^{2}-\frac{1}{2}\leq
0,\quad h_{2}(\boldsymbol{x})=-x_{1}\leq 0.$
The relevant gradients and second differentials are
$\displaystyle\nabla f(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\boldsymbol{x}-\boldsymbol{b},\hskip 14.45377pt\nabla
h_{1}(\boldsymbol{x})=\boldsymbol{x},\hskip 14.45377pt\nabla
h_{2}(\boldsymbol{x})=-\left(\,\begin{matrix}1\\\ 0\end{matrix}\,\right)$
$\displaystyle d^{2}f(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
d^{2}h_{1}(\boldsymbol{x})\;=\;\boldsymbol{I}_{2},\hskip
14.45377ptd^{2}h_{2}(\boldsymbol{x})={\bf 0}.$
Path following starts from the unconstrained solution
$\boldsymbol{x}(0)=\boldsymbol{b}$; the direction of movement is determined by
formula (8). For
$\boldsymbol{x}\in\\{\boldsymbol{x}:\|\boldsymbol{x}\|^{2}>1,x_{1}>0\\}$, the
path
$\displaystyle\frac{d}{d\rho}\boldsymbol{x}$ $\displaystyle=$
$\displaystyle-[(1+\rho)\boldsymbol{I}_{2}]^{-1}\boldsymbol{x}=-\frac{1}{1+\rho}\boldsymbol{x}$
heads toward the origin. For
$\boldsymbol{x}\in\\{\boldsymbol{x}:|x_{2}|>1,x_{1}=0\\}$, the path
$\displaystyle\frac{d}{d\rho}\left(\begin{array}[]{c}\boldsymbol{x}\\\
\omega_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle-\left(\begin{array}[]{ccc}1+\rho&0&-1\\\ 0&1+\rho&0\\\
-1&0&0\end{array}\right)^{-1}\left(\begin{array}[]{c}x_{1}\\\ x_{2}\\\
0\end{array}\right)=-\frac{1}{1+\rho}\left(\begin{array}[]{c}0\\\ x_{2}\\\
0\end{array}\right)$
also heads toward the origin. For
$\boldsymbol{x}\in\\{\boldsymbol{x}:\|\boldsymbol{x}\|^{2}>1,x_{1}<0\\}$, the
path
$\displaystyle\frac{d}{d\rho}\boldsymbol{x}$ $\displaystyle=$
$\displaystyle-[(1+\rho)\boldsymbol{I}_{2}]^{-1}\left(\begin{array}[]{c}x_{1}-1\\\
x_{2}\end{array}\right)=-\frac{1}{1+\rho}\left(\begin{array}[]{c}x_{1}-1\\\
x_{2}\end{array}\right).$
heads toward the point $(1,0)^{t}$. For
$\boldsymbol{x}\in\\{\boldsymbol{x}:\|\boldsymbol{x}\|^{2}=1,x_{1}<0\\}$, the
path
$\displaystyle\frac{d}{d\rho}\left(\begin{array}[]{c}\boldsymbol{x}\\\
\omega_{1}\end{array}\right)$ $\displaystyle=$
$\displaystyle-\left(\begin{array}[]{ccc}1+\omega_{1}&0&x_{1}\\\
0&1+\omega_{1}&x_{2}\\\
x_{1}&x_{2}&0\end{array}\right)^{-1}\left(\begin{array}[]{c}-1\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[]{c}-\frac{x_{2}^{2}}{1+\omega_{1}}\\\
\frac{x_{1}x_{2}}{1+\omega_{1}}\\\ -x_{1}\end{array}\right)$
is tangent to the circle. Finally, for
$\boldsymbol{x}\in\\{\boldsymbol{x}:\|\boldsymbol{x}\|^{2}<1,x_{1}<0\\}$, the
path
$\displaystyle\frac{d}{d\rho}\boldsymbol{x}$ $\displaystyle=$
$\displaystyle-\boldsymbol{I}_{2}^{-1}\left(\begin{array}[]{c}-1\\\
0\end{array}\right)=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$
heads toward the $x_{2}$-axis. The left panel of Figure 1 plots the vector
field $\frac{d}{d\rho}\boldsymbol{x}$ at the time $\rho=0$. The right panel
shows the solution path for projection from the points $(-2,0.5)^{t}$,
$(-2,1.5)^{t}$, $(-1,2)^{t}$, $(2,1.5)^{t}$, $(2,0)^{t}$, $(1,2)^{t}$, and
$(-0.5,-2)^{t}$ onto the feasible region. In projecting the point
$\boldsymbol{b}=(-1,2)^{t}$ onto $(0,1)^{t}$, the ODE45 solver of Matlab
evaluates derivatives at 19 different time points. Dykstra’s algorithm by
comparison takes about 30 iterations to converge (Lange, 2004).
$\begin{array}[]{cc}\includegraphics[width=162.6075pt]{halfdisc}&\includegraphics[width=162.6075pt]{halfdisc_traj}\end{array}$
Figure 1: Projection to the positive half disk. Left: Derivatives at $\rho=0$
for projection onto the half disc. Right: Projection trajectories from various
initial points.
###### Example 4.2.
Nonnegative Least Squares (NNLS) and Nonnegative Matrix Factorization (NNMF)
Non-negative matrix factorization (NNMF) is an alternative to principle
component analysis and is useful in modeling, compressing, and interpreting
nonnegative data such as observational counts and images. The articles (Berry
et al., 2007; Lee and Seung, 1999, 2001) discuss in detail estimation
algorithms and statistical applications of NNMF. The basic idea is to
approximate an $m\times n$ data matrix $\boldsymbol{X}=(x_{ij})$ with
nonnegative entries by a product $\boldsymbol{V}\boldsymbol{W}$ of two low
rank matrices $\boldsymbol{V}=(v_{ik})$ and $\boldsymbol{W}=(w_{kj})$ with
nonnegative entries. Here $\boldsymbol{V}$ and $\boldsymbol{W}$ are $m\times
r$ and $r\times n$ respectively, with $r\ll\min\\{m,n\\}$. One version of NNMF
minimizes the criterion
$\displaystyle f(\boldsymbol{V},\boldsymbol{W})$ $\displaystyle=$
$\displaystyle\|\boldsymbol{X}-\boldsymbol{V}\boldsymbol{W}\|_{\text{F}}^{2}=\sum_{i}\sum_{j}\Big{(}x_{ij}-\sum_{k}v_{ik}w_{kj}\Big{)}^{2},$
(17)
where $\|\cdot\|_{\text{F}}$ denotes the Frobenius norm. In a typical imaging
problem, $m$ (number of images) might range from $10^{3}$ to $10^{4}$, $n$
(number of pixels per image) might surpass $10^{4}$, and a rank $r=50$
approximation might adequately capture $\boldsymbol{X}$.
Minimization of the objective function (17) is nontrivial because it is not
jointly convex in $\boldsymbol{V}$ and $\boldsymbol{W}$. Multiple local minima
are possible. The well-known multiplicative algorithm (Lee and Seung, 1999,
2001) enjoys the descent property, but it is not guaranteed to converge to
even a local minimum (Berry et al., 2007). An alternative algorithm that
exhibits better convergence is alternating least squares (ALS). In updating
$\boldsymbol{W}$ with $\boldsymbol{V}$ fixed, ALS solves the $n$ separated
nonnegative least square (NLS) problems
$\displaystyle\min_{\boldsymbol{w}_{j}}\|\boldsymbol{x}_{j}-\boldsymbol{V}\boldsymbol{w}_{j}\|_{2}^{2}\quad\quad\text{
subject to }\boldsymbol{w}_{j}\geq 0,$ (18)
where $\boldsymbol{x}_{j}$ and $\boldsymbol{w}_{j}$ denote the $j$-th columns
of the corresponding matrices. Similarly, in updating $\boldsymbol{V}$ with
$\boldsymbol{W}$ fixed, ALS solves $m$ separated NNLS problems. The
unconstrained solution
$\boldsymbol{W}(0)=(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{X}$
of $\boldsymbol{W}$ for fixed $\boldsymbol{V}$ requires just one QR
decomposition of $\boldsymbol{V}$ or one Cholesky decomposition of
$\boldsymbol{V}^{t}\boldsymbol{V}$. The exact path algorithm for solving the
subproblem problem (18) commences with $\boldsymbol{W}(0)$. If
$\boldsymbol{W}(\rho)$ stabilizes with just a few zeros, then the path
algorithm ends quickly and is extremely efficient. For a NNLS problem, the
path is piecewise linear, and one can straightforwardly project the path to
the next hitting or escape time using the sweep operator (Zhou and Lange,
2011b). Figure 2 shows a typical piecewise linear path for a problem with
$r=50$ predictors. Each projection to the next event requires $2r^{2}$ flops.
The number of path segments (events) roughly scales as the number of negative
components in the unconstrained solution.
Figure 2: Piecewise linear paths of the regression coefficients for a NNLS
problem with 50 predictors.
###### Example 4.3.
Quadratically Constrained Quadratic Programming (QCQP)
Example 4.1 is a special case of quadratically constrained quadratic
programming (QCQP). In convex QCQP (Boyd and Vandenberghe, 2004, Section 4.4),
one minimizes a convex quadratic function over an intersection of ellipsoids
and affine subspaces. Mathematically, this amounts to the problem
$\displaystyle\text{minimize}\hskip 7.22743ptf(\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}\boldsymbol{P}_{0}\boldsymbol{x}+\boldsymbol{b}_{0}^{t}\boldsymbol{x}+c_{0}$
$\displaystyle\text{subject to}\hskip 7.22743ptg_{i}(\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\boldsymbol{a}_{i}^{t}\boldsymbol{x}-d_{i}=0,\hskip
7.22743pti=1,\ldots,r$ $\displaystyle h_{j}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{x}^{t}\boldsymbol{P}_{j}\boldsymbol{x}+\boldsymbol{b}_{j}^{t}\boldsymbol{x}+c_{j}\leq
0,\hskip 7.22743ptj=1,\ldots,s,$
where $\boldsymbol{P}_{0}$ is a positive definite matrix and the
$\boldsymbol{P}_{j}$ are positive semidefinite matrices. Our algorithm starts
with the unconstrained minimum
$\boldsymbol{x}(0)=-\boldsymbol{P}_{0}^{-1}\boldsymbol{b}_{0}$ and proceeds
along the path determined by the derivative
$\displaystyle\frac{d}{d\rho}\left[\begin{matrix}\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{\text{I}}}\end{matrix}\right]=-\left(\begin{matrix}\boldsymbol{P}_{0}+\rho\sum_{j\in{\mathcal{P}}_{\text{I}}}\boldsymbol{P}_{j}+\sum_{j\in{\mathcal{Z}}_{\text{I}}}\omega_{j}\boldsymbol{P}_{j}&\boldsymbol{U}_{{\mathcal{Z}}}^{t}(\boldsymbol{x})\\\
\boldsymbol{U}_{{\mathcal{Z}}}(\boldsymbol{x})&{\bf
0}\end{matrix}\right)^{-1}\left(\begin{matrix}\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})\\\
{\bf 0}\end{matrix}\right),$
where $\boldsymbol{U}_{\mathcal{Z}}(\boldsymbol{x})$ has rows
$\boldsymbol{a}_{i}^{t}$ for $i\in{\mathcal{Z}}_{\text{E}}$ and
$(\boldsymbol{P}_{j}\boldsymbol{x}+\boldsymbol{b}_{j})^{t}$ for
$j\in{\mathcal{Z}}_{\text{I}}$, and
$\displaystyle\boldsymbol{u}_{\bar{\mathcal{Z}}}(\boldsymbol{x})=-\sum_{i\in{\mathcal{N}}_{\text{E}}}\boldsymbol{a}_{i}+\sum_{i\in{\mathcal{P}}_{\text{E}}}\boldsymbol{a}_{i}+\sum_{i\in{\mathcal{P}}_{\text{I}}}(\boldsymbol{P}_{j}\boldsymbol{x}+\boldsymbol{b}_{j}).$
Affine inequality constraints can be accommodated by setting one or more of
the $\boldsymbol{P}_{j}$ equal to ${\bf 0}$.
As a numerical illustration, consider the bivariate problem
$\displaystyle\text{minimize}\hskip 7.22743ptf(\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\frac{1}{2}x_{1}^{2}+x_{2}^{2}-x_{1}x_{2}+\frac{1}{2}x_{1}-2x_{2}$
$\displaystyle\text{subject to}\hskip 7.22743pth_{1}(\boldsymbol{x})$
$\displaystyle=$
$\displaystyle\Big{(}x_{1}-\frac{1}{2}\Big{)}^{2}+x_{2}^{2}-1\;\leq\;0\;$ (19)
$\displaystyle h_{2}(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\Big{(}x_{1}+\frac{1}{2}\Big{)}^{2}+x_{2}^{2}-1\leq 0$
$\displaystyle h_{3}(\boldsymbol{x})$ $\displaystyle=$ $\displaystyle
x_{1}^{2}+\Big{(}x_{2}-\frac{1}{2}\Big{)}^{2}-1\leq 0.$
Here the feasible region is given by the intersection of three disks with
centers $(0.5,0)^{t}$, $(-0.5,0)^{t}$, and $(0,0.5)^{t}$, respectively, and a
common radius of 1. Figure 3 displays the solution trajectory. Starting from
the unconstrained minimum $\boldsymbol{x}(0)=(1,1.5)^{t}$, it hits, slides
along, and exits two circles before its journey ends at the constrained
minimum $(0.059,0.829)^{t}$. The ODE45 solver of Matlab evaluates derivatives
at 72 time points along the path.
###### Example 4.4.
Geometric Programming
Figure 3: Trajectory of the exact penalty path algorithm for a QCQP problem
(19). The solid lines are the contours of the objective function
$f(\boldsymbol{x})$. The dashed lines are the contours of the constraint
functions $h_{j}(\boldsymbol{x})$.
As a branch of convex optimization theory, geometric programming stands just
behind linear and quadratic programming in importance (Boyd et al., 2007;
Ecker, 1980; Peressini et al., 1988; Peterson, 1976). It has applications in
chemical equilibrium problems (Passy and Wilde, 1968), structural mechanics
(Ecker, 1980), digit circuit design (Boyd et al., 2005), maximum likelihood
estimation (Mazumdar and Jefferson, 1983), stochastic processes (Feigin and
Passy, 1981), and a host of other subjects (Boyd et al., 2007; Ecker, 1980).
Geometric programming deals with posynomials, which are functions of the form
$\displaystyle f(\boldsymbol{x})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}\prod_{i=1}^{n}x_{i}^{\alpha_{i}}\;\;=\;\;\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}\>=\>f(\boldsymbol{y}).$
(20)
In the left-hand definition of this equivalent pair of definitions, the index
set $S\subset\mathbb{R}^{n}$ is finite, and all coefficients
$c_{\boldsymbol{\alpha}}$ and all components $x_{1},\ldots,x_{n}$ of the
argument $\boldsymbol{x}$ of $f(\boldsymbol{x})$ are positive. The possibly
fractional powers $\alpha_{i}$ corresponding to a particular
$\boldsymbol{\alpha}$ may be positive, negative, or zero. For instance,
$x_{1}^{-1}+2x_{1}^{3}x_{2}^{-2}$ is a posynomial on $\mathbb{R}^{2}$. In
geometric programming, one minimizes a posynomial $f(\boldsymbol{x})$ subject
to posynomial inequality constraints of the form $h_{j}(\boldsymbol{x})\leq 1$
for $1\leq j\leq s$. In some versions of geometric programming, equality
constraints of monomial type are permitted (Boyd et al., 2007). The right-hand
definition in equation (20) invokes the exponential reparameterization
$x_{i}=e^{y_{i}}$. This simple transformation has the advantage of rendering a
geometric program convex. In fact, any posynomial $f(\boldsymbol{y})$ in the
exponential parameterization is log-convex and therefore convex. The concise
representations
$\displaystyle\nabla f(\boldsymbol{y})$ $\displaystyle=$
$\displaystyle\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}\boldsymbol{\alpha},\quad
d^{2}f(\boldsymbol{y})\;\;=\;\;\sum_{\boldsymbol{\alpha}\in
S}c_{\boldsymbol{\alpha}}e^{\boldsymbol{\alpha}^{t}\boldsymbol{y}}\boldsymbol{\alpha}\boldsymbol{\alpha}^{t}$
of the gradient and the second differential are helpful in both theory and
computation.
Without loss of generality, one can repose geometric programming as
$\displaystyle\text{minimize}\hskip 7.22743pt\ln f(\boldsymbol{y})$
$\displaystyle\text{subject to}\hskip 7.22743pt\ln g_{i}(\boldsymbol{y})$
$\displaystyle=$ $\displaystyle 0,\;\;1\leq i\leq r$ (21) $\displaystyle\ln
h_{j}(\boldsymbol{y})$ $\displaystyle\leq$ $\displaystyle 0,\;\;1\leq j\leq
s,$
where $f(\boldsymbol{y})$ and the $h_{j}(\boldsymbol{y})$ are posynomials and
the equality constraints $\ln g_{i}(\boldsymbol{y})$ are affine. In this
exponential parameterization setting, it is easy to state necessary and
sufficient conditions for strict convexity and coerciveness.
###### Proposition 4.5.
The objective function $f(\boldsymbol{y})$ in the geometric program (21) is
strictly convex if and only if the subspace spanned by the vectors
$\\{\boldsymbol{\alpha}\\}_{\boldsymbol{\alpha}\in S}$ is all of
$\mathbb{R}^{n}$; $f(\boldsymbol{y})$ is coercive if and only if the polar
cone $\\{\boldsymbol{z}:\boldsymbol{z}^{t}\boldsymbol{\alpha}\leq 0\;\mbox{for
all}\;\boldsymbol{\alpha}\in S\\}$ reduces to the origin ${\bf 0}$.
Equivalently, $f(\boldsymbol{y})$ is coercive if the origin ${\bf 0}$ belongs
to the interior of the convex hull of the set $S$.
###### Proof.
These claims are proved in detail in our paper (Zhou and Lange, 2011a). ∎
According to Propositions 2.1 and 3.2, the strict convexity and coerciveness
of $f(\boldsymbol{y})$ guarantee the uniqueness and continuity of the solution
path in $\boldsymbol{y}$. This in turn implies the uniqueness and continuity
of the solution path in the original parameter vector $\boldsymbol{x}$. The
path directions are related by the chain rule
$\displaystyle\frac{d}{d\rho}x_{i}(\rho)$ $\displaystyle=$
$\displaystyle\frac{dx_{i}}{dy_{i}}\frac{d}{d\rho}y_{i}(\rho)\;\;=\;\;x_{i}\frac{d}{d\rho}y_{i}(\rho).$
As a concrete example, consider the problem
minimize $\displaystyle\hskip
7.22743ptx_{1}^{-3}+3x_{1}^{-1}x_{2}^{-2}+x_{1}x_{2}$ subject to
$\displaystyle\hskip 7.22743pt\frac{1}{6}x_{1}^{1/2}+\frac{2}{3}x_{2}\leq
1,\;\;x_{1}>0,\;x_{2}>0.$
It is easy to check that the vectors $\\{(-3,0)^{t},(-1,-2)^{t},(1,1)^{t}\\}$
span $\mathbb{R}^{2}$ and generate a convex hull strictly containing the
origin ${\bf 0}$. Therefore, $f(\boldsymbol{y})$ is strictly convex and
coercive. It achieves its unconstrained minimum at the point
$\boldsymbol{x}(0)=(\sqrt[5]{6},\sqrt[5]{6})^{t}$, or equivalently
$\boldsymbol{y}(0)=(\ln 6/5,\ln 6/5)^{t}$. To solve the constrained
minimization problem, we follow the path dictated by the revised geometric
program (21). Figure 4 plots the trajectory from the unconstrained solution to
the constrained solution in the original $\boldsymbol{x}$ variables. The solid
lines in the figure represent the contours of the objective function
$f(\boldsymbol{x})$, and the dashed lines represent the contours of the
constraint function $h(\boldsymbol{x})$. The ODE45 solver of Matlab evaluates
derivatives at seven time points along the path.
Figure 4: Trajectory of the exact penalty path algorithm for the geometric
programming problem (4). The solid lines are the contours of the objective
function $f(\boldsymbol{x})$. The dashed lines are the contours of the
constraint function $h(\boldsymbol{x})$ at levels 1, 1.25, and 1.5.
###### Example 4.6.
Semidefinite Programming (SDP)
The linear semidefinite programming problem (Vandenberghe and Boyd, 1996)
consists in minimizing the trace function $\boldsymbol{X}\mapsto\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})$ over the cone of positive
semidefinite matrices $S_{+}^{n}$ subject to the linear constraints
$\mathop{\rm tr}\nolimits(\boldsymbol{A}_{i}\boldsymbol{X})=b_{i}$ for $1\leq
i\leq p$. Here $\boldsymbol{C}$ and the $\boldsymbol{A}_{i}$ are assumed
symmetric. According to Sylvester’s criterion, the constraint
$\boldsymbol{X}\in S_{+}^{n}$ involves a complicated system of inequalities
involving nonconvex functions. One way of cutting through this morass is to
focus on the minimum eigenvalue $\nu_{1}(\boldsymbol{X})$ of $\boldsymbol{X}$.
Because the function $-\nu_{1}(\boldsymbol{X})$ is convex, one can enforce
positive semidefiniteness by requiring $-\nu_{1}(\boldsymbol{X})\leq 0$. Thus,
the linear semidefinite programming problem is a convex program in the
standard functional form.
It simplifies matters enormously to assume that $\nu_{1}(\boldsymbol{X})$ has
multiplicity 1. Let $\boldsymbol{u}$ be the unique, up to sign, unit
eigenvector corresponding to $\nu_{1}(\boldsymbol{X})$. The matrix
$\boldsymbol{X}$ is parameterized by the entries of its lower triangle. With
these conventions, the following formulas
$\displaystyle-\frac{\partial}{\partial x_{ij}}\nu_{1}(\boldsymbol{X})$
$\displaystyle=$ $\displaystyle-\boldsymbol{u}^{t}\frac{\partial}{\partial
x_{ij}}\boldsymbol{X}\boldsymbol{u}$ (23)
$\displaystyle-\frac{\partial^{2}}{\partial x_{ij}\partial
x_{kl}}\nu_{1}(\boldsymbol{X})$ $\displaystyle=$
$\displaystyle-\boldsymbol{u}^{t}\frac{\partial}{\partial
x_{ij}}\boldsymbol{X}(\nu_{1}\boldsymbol{I}-\boldsymbol{X})^{-}\frac{\partial}{\partial
x_{kl}}\boldsymbol{X}\boldsymbol{u}$ (24)
$\displaystyle-\boldsymbol{u}^{t}\frac{\partial}{\partial
x_{kl}}\boldsymbol{X}(\nu_{1}\boldsymbol{I}-\boldsymbol{X})^{-}\frac{\partial}{\partial
x_{ij}}\boldsymbol{X}\boldsymbol{u}$ $\displaystyle=$
$\displaystyle-2\boldsymbol{u}^{t}\frac{\partial}{\partial
x_{ij}}\boldsymbol{X}(\nu_{1}\boldsymbol{I}-\boldsymbol{X})^{-}\frac{\partial}{\partial
x_{kl}}\boldsymbol{X}\boldsymbol{u}$
for the first and second partial derivatives of $-\nu_{1}(\boldsymbol{X})$ are
well known (Magnus and Neudecker, 1999). Here the matrix
$(\nu_{1}\boldsymbol{I}-\boldsymbol{X})^{-}$ is the Moore-Penrose inverse of
$\nu_{1}\boldsymbol{I}-\boldsymbol{X}$. The partial derivative of
$\boldsymbol{X}$ with respect to its lower triangular entry $x_{ij}$ equals
$\boldsymbol{E}_{ij}+1_{\\{i\neq j\\}}\boldsymbol{E}_{ji}$, where
$\boldsymbol{E}_{ij}$ is the matrix consisting of all 0’s excepts for a 1 in
position $(i,j)$. Note that
$\boldsymbol{u}^{t}\boldsymbol{E}_{ij}=u_{i}\boldsymbol{e}_{j}^{t}$ and
$\boldsymbol{E}_{kl}\boldsymbol{u}=u_{l}\boldsymbol{e}_{k}$ for the standard
unit vectors $\boldsymbol{e}_{j}$ and $\boldsymbol{e}_{k}$. The second partial
derivatives of $\boldsymbol{X}$ vanish. The Moore-Penrose inverse is most
easily expressed in terms of the spectral decomposition of $\boldsymbol{X}$.
If we denote the $i$th eigenvalue of $\boldsymbol{X}$ by $\nu_{i}$ and the
corresponding $i$th unit eigenvector by $\boldsymbol{u}_{i}$, then we have
$\displaystyle(\boldsymbol{X}-\nu_{1}\boldsymbol{I})^{-}$ $\displaystyle=$
$\displaystyle\sum_{i>1}\frac{1}{\nu_{i}-\nu_{1}}\boldsymbol{u}_{i}\boldsymbol{u}_{i}^{t}.$
Finally, the formulas
$\displaystyle\mathop{\rm
tr}\nolimits(\boldsymbol{A}_{i}\boldsymbol{X})-b_{i}$ $\displaystyle=$
$\displaystyle\sum_{k}(\boldsymbol{A}_{i})_{kk}x_{kk}+2\sum_{k}\sum_{l<k}(\boldsymbol{A}_{i})_{kl}x_{kl}-b_{i}$
$\displaystyle\frac{\partial}{\partial x_{kl}}[\mathop{\rm
tr}\nolimits(\boldsymbol{A}_{i}\boldsymbol{X})-b_{i}]$ $\displaystyle=$
$\displaystyle(\boldsymbol{A}_{i})_{kl}+1_{\\{k\neq
l\\}}(\boldsymbol{A}_{i})_{lk}$
express the linear constraints and their partial derivatives in terms of the
lower triangular entries of $\boldsymbol{X}$.
Initiating path following is problematic because $\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})$ has minimum $-\infty$. A good
strategy is to amend the surrogate function
${\mathcal{E}}_{\rho}(\boldsymbol{x})$ by adding the term
$\frac{\epsilon(\rho)}{2}\|\boldsymbol{X}\|_{\text{F}}^{2}$, where
$\epsilon(\rho)$ is a smooth positive function that decreases to 0. Taking
$\epsilon(\rho)=e^{-c\rho}$ for $c$ positive works well in practice. The new
surrogate function $\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})+\frac{\epsilon(\rho)}{2}\|\boldsymbol{X}\|_{\text{F}}^{2}$
is strictly convex and possesses a unique minimum for all $\rho\geq 0$. In
view of the identities
$\|\boldsymbol{X}\|_{\text{F}}^{2}=\sum_{i}\sum_{j}x_{ij}^{2}$ and
$\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})=\sum_{i}\sum_{j}c_{ij}x_{ij}$ for
$\boldsymbol{X}=(x_{ij})$ and $\boldsymbol{C}=(c_{ij})$, the initial condition
$\boldsymbol{X}(0)=-\epsilon(0)^{-1}\boldsymbol{C}$ is straightforward to
deduce.
Path following must be modified to accommodate the new surrogate function. In
the notation of (Magnus and Neudecker, 1999), let
$\boldsymbol{x}=\text{v}(\boldsymbol{X})$ be the $\frac{1}{2}n(n+1)$ vector
obtained from $\text{vec}(\boldsymbol{X})$ by eliminating all supradiagonal
entries, and let $\boldsymbol{D}$ be the $n^{2}\times\frac{1}{2}n(n+1)$
duplication matrix satisfying
$\text{vec}(\boldsymbol{X})=\boldsymbol{D}\boldsymbol{x}$. Applying the chain
rule to the obvious identities
$\|\boldsymbol{X}\|_{\text{F}}^{2}=\boldsymbol{x}\boldsymbol{D}^{t}\boldsymbol{D}\boldsymbol{x}$
and $\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})=\text{vec}(\boldsymbol{C})^{t}\boldsymbol{D}\boldsymbol{x}$,
one can extend the derivation of Proposition 3.3 and prove that
$\displaystyle{d\over d\rho}\\!\left[\begin{matrix}\boldsymbol{x}\\\
\boldsymbol{\lambda}_{{\mathcal{Z}}_{\text{E}}}\\\
\boldsymbol{\omega}_{{\mathcal{Z}}_{{\mathcal{I}}}}\end{matrix}\right]$
$\displaystyle=$
$\displaystyle-\left[\begin{matrix}\epsilon(\rho)\boldsymbol{D}^{t}\boldsymbol{D}-\boldsymbol{\omega}_{{\mathcal{Z}}_{{\mathcal{I}}}}d^{2}\nu_{1}(\boldsymbol{x})1_{\\{\nu_{1}(\boldsymbol{X})=0\\}}&\boldsymbol{U}_{{\mathcal{Z}}}^{t}\\\
\boldsymbol{U}_{{\mathcal{Z}}}&{\bf 0}\end{matrix}\right]^{-1}$
$\displaystyle\\!\times\\!\left(\begin{matrix}\frac{d\epsilon(\rho)}{d\rho}\boldsymbol{D}^{t}\boldsymbol{D}\boldsymbol{x}-\sum_{i\in{\mathcal{N}}_{\text{E}}}\boldsymbol{D}^{t}\text{vec}(\boldsymbol{A}_{i})+\sum_{i\in{\mathcal{P}}_{\text{E}}}\boldsymbol{D}^{t}\text{vec}(\boldsymbol{A}_{i})-\nabla\nu_{1}(\boldsymbol{x})1_{\\{\nu_{1}(\boldsymbol{X})<0\\}}\\\
{\bf 0}\end{matrix}\right).$
Path following proceeds until all constraints are satisfied and
$\epsilon(\rho)$ is negligible.
For didactic purposes, considering the problem of minimizing $\mathop{\rm
tr}\nolimits(\boldsymbol{C}\boldsymbol{X})$ subject to
$\displaystyle\mathop{\rm tr}\nolimits(\boldsymbol{A}_{1}\boldsymbol{X})$
$\displaystyle=$ $\displaystyle 1,\quad\mathop{\rm
tr}\nolimits(\boldsymbol{A}_{2}\boldsymbol{X})=2,\>\>\text{ and
}\>\>\>\boldsymbol{X}\in{\mathcal{S}}_{+}^{2},$
where
$\displaystyle\boldsymbol{C}=\left(\begin{matrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{matrix}\right),\quad\boldsymbol{A}_{1}=\left(\begin{matrix}1&0\\\
0&0\end{matrix}\right),\>\>\text{ and
}\>\>\>\boldsymbol{A}_{2}=\left(\begin{matrix}0&0\\\ 0&1\end{matrix}\right).$
Figure 5 displays the solution paths of the entries $x_{ij}$ of
$\boldsymbol{X}$ and the minimum eigenvalue $\nu_{1}$ . Here we use
$\epsilon(\rho)=e^{-\rho}$. The path starts with
$\boldsymbol{X}(0)=-\boldsymbol{C}$, hits, slides along, and exits various
constraints, and ends at the constrained solution
$\small\left(\begin{matrix}1&-\sqrt{2}\\\ -\sqrt{2}&2\end{matrix}\right)$.
Figure 5: Solution path of a semidefinite programming example.
###### Example 4.7.
Image Denoising
Image analysis is another fertile field for path following. Here we explore
how to restore or enhance images by removing noise. This example differs from
previous examples in that the fully constrained solution is trivial. The
solution path itself is the object of interest. Suppose that
${\boldsymbol{w}}=(w_{ij})\in\mathbb{R}^{m\times n}$ represents the recorded
gray levels across a 2D array of pixels from a noisy image with true gray
levels $\boldsymbol{u}=(u_{ij})$. The well-known denoising model of Rudin-
Osher-Fatemi (ROF) (Rudin et al., 1992) minimizes the total variation
regularized least squares criterion
$\displaystyle\frac{1}{2}\|{\boldsymbol{w}}-\boldsymbol{u}\|_{2}^{2}+\rho\text{TV}(\boldsymbol{u})$
(25) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{ij}(w_{ij}-u_{ij})^{2}+\rho\sum_{i,j}\sqrt{(u_{i+1,j}-u_{ij})^{2}+(u_{i,j+1}-u_{ij})^{2}}.$
The total variation penalty serves to smooth the reconstructed image and
preserve its edges. A similar effect can be achieved by replacing the
isotropic penalty $\text{TV}(\boldsymbol{u})$ by the anisotropic penalty
$\displaystyle\text{TV}_{1}(\boldsymbol{u})$ $\displaystyle=$
$\displaystyle\sum_{i,j}\Big{(}|u_{i+1,j}-u_{ij}|+|u_{i,j+1}-u_{ij}|\Big{)}.$
(26)
In this example we focus on path following for the anisotropic penalty and a
more general convex loss function $f(\boldsymbol{u})$. The objective function
is now
$\displaystyle f(\boldsymbol{u})+\rho\|\boldsymbol{D}\boldsymbol{u}\|_{1}.$
(27)
For instance, the amended loss function
$f(\boldsymbol{u})=\frac{1}{2}\|\boldsymbol{w}-\boldsymbol{K}\boldsymbol{u}\|_{2}^{2}$
with a Gaussian or motion blurring matrix $\boldsymbol{K}$ is appropriate in
many imaging problems. Poisson count data are relevant to image reconstruction
in X-ray and positron tomography (Lange, 2010) and to image denoising in
certain circumstances (Le et al., 2007). With Poisson noise, the least squares
criterion is replaced by a negative loglikelihood. The difference matrix
$\boldsymbol{D}$ captures the $\ell_{1}$ penalty (26). Note that the matrices
$\boldsymbol{w}$ and $\boldsymbol{u}$ are now viewed as vectors. For an
$m\times n$ 2D image, the difference matrix $\boldsymbol{D}$ has $2mn-m-n$
rows (penalties) and $mn$ columns (pixels). This matrix is very sparse, with
just $2(2mn-m-n)$ nonzero entries equal to $\pm 1$. When $m$ and $n$ are both
at least 2, $\boldsymbol{D}$ has more rows than columns and a reduced column
rank of $mn-1$.
For sufficiently large $\rho$, the minimum of the objective functions (25)
reduces to a constant vector (blank image) equal to the average value
$\bar{w}$ of the $w_{ij}$. The goal of image denoising is to find a $\rho$
such that the recovered image is judged satisfactory by visual inspection or
other more quantitative criteria. Notable computational advances in solving
this problem include Chambolle’s algorithm (Chambolle, 2004) and split Bregman
iteration (Goldstein and Osher, 2009). These methods minimize the objective
functions (25) and (27) for a fixed value of $\rho$. The web site of UCLA’s
Computational and Applied Math Group summarizes the most recent progress in
this area. In reality, outer iterations are almost always required to tune the
parameter $\rho$. Path following is an attractive option because it provides
the whole solution path at about the same computational cost as recovering the
solution for an individual $\rho$.
Although it is tempting to minimize the criterion (27) by path following, the
regularization matrix $\boldsymbol{D}$ has linearly dependent rows and
deficient rank. Because the assumptions of Proposition 3.2 are violated, the
multipliers $\boldsymbol{\lambda}_{\text{E}}$ of the active constraints in
equations (7) and (9) are not uniquely determined. One can intuitively
understand the difficulty by considering a square with four pixels. Whenever
any three constraints are active, the fourth is automatically active as well.
This constraint redundancy can be remedied by reparameterizing the model in
terms of neighboring pixel differences
$\boldsymbol{x}=\boldsymbol{D}\boldsymbol{u}$. Unfortunately, the rank
deficiency of $\boldsymbol{D}$ is also an issue. Adding the same constant to
all of the components of $\boldsymbol{u}$ yields exactly the same
$\boldsymbol{x}$. To circumvent this problem, we simply append a bottom row to
$\boldsymbol{D}$ with all entries 0 except for a 1 in the last position. If
$\boldsymbol{V}$ is the amended version of $\boldsymbol{D}$, then
$\boldsymbol{V}$ has full column rank, and the vector
$\boldsymbol{x}=\boldsymbol{V}\boldsymbol{u}$ uniquely determines the image.
Indeed, one can solve for $\boldsymbol{x}$ in the form
$\boldsymbol{u}=(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{x}$.
The bottom entry of $\boldsymbol{x}$ is obviously the gray level of the last
pixel of the image.
Despite the presence of the inverse of the huge $mn\times mn$ matrix
$\boldsymbol{V}^{t}\boldsymbol{V}\\!$, the transformation
$\boldsymbol{u}=(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}\boldsymbol{x}$
is not as daunting as it appears. First of all, multiplication by the sparse
matrix $\boldsymbol{V}^{t}$ is trivial. More importantly, the matrix
$\boldsymbol{V}^{t}\boldsymbol{V}$ is symmetric, banded, and extremely sparse.
To count its nonzero entries, note that except for diagonal entries, these
entries occur in the same positions as the nonzero entries of the adjacency
matrix of a corresponding graph with $2mn-m-n$ edges and $mn$ nodes. Because
an adjacency matrix has twice as many nonzero entries as edges, the matrix
$\boldsymbol{V}^{t}\boldsymbol{V}$ has at most $2(2mn-m-n)+mn=5mn-2m-2n$
nonzero entries. These occur within a band of width $\min\\{m,n\\}$ along the
main diagonal, depending on whether we stack columns or concatenate rows. The
most convenient way to solve equations of the kind
$\boldsymbol{V}^{t}\boldsymbol{V}\boldsymbol{a}=\boldsymbol{b}$ is to extract
the Cholesky decomposition $\boldsymbol{L}$ of
$\boldsymbol{V}^{t}\boldsymbol{V}$ and execute forward and backward
substitution. Although extraction of $\boldsymbol{L}$ is cheap for banded
matrices, it is even cheaper for banded matrices with just a handful of
nonzero entries per row. In our experience, the computational complexity of
extracting $\boldsymbol{L}$ scales linearly in the product $mn$. Since
$\boldsymbol{L}$ itself is sparse, forward and backward substitution are also
very cheap. For instance with a $256\times 256$ image, Matlab computes
$\boldsymbol{L}$ (a $65536\times 65536$ matrix) in 0.26 seconds on a laptop;
$\boldsymbol{L}$ contains just 1,971,395 nonzero entries. The sparsity of
$\boldsymbol{L}$ suggests that it be computed once and stored in compressed
format for all images of a given size. Many of its nonzero entries are close
to zero. Thus, a fairly light truncation of the non-diagonal entries of
$\boldsymbol{L}$ gives an even sparser matrix realizing nearly the same
transformation. Figure 6 displays the sparsity pattern of the matrix
$\boldsymbol{V}^{t}\boldsymbol{V}$ and its permuted Cholesky factor
$\boldsymbol{L}$ for $64\times 64$ images. Images of other sizes show similar
sparsity patterns.
$\begin{array}[]{cc}\includegraphics[width=162.6075pt]{adjmatrix}&\includegraphics[width=162.6075pt]{cholfactor}\end{array}$
Figure 6: Sparsity patterns of $\boldsymbol{V}^{t}\boldsymbol{V}$ and its
Cholesky decomposition $\boldsymbol{L}$ for 64-by-64 images.
The problem of minimizing the objective function
$\frac{1}{2}\|\boldsymbol{w}-\boldsymbol{K}\boldsymbol{u}\|_{2}^{2}+\rho\|\boldsymbol{D}\boldsymbol{u}\|_{1}$
in the transformed variable $\boldsymbol{x}$ turns out to coincide with lasso
penalized regression, for which an efficient path algorithm is known (Efron et
al., 2004; Osborne et al., 2000). Let us sketch how path following works in
the more general case. The objective function is
$f(\boldsymbol{B}\boldsymbol{x})+\rho\|\boldsymbol{x}_{-}\|_{1}$, where
$\boldsymbol{B}=(\boldsymbol{V}^{t}\boldsymbol{V})^{-1}\boldsymbol{V}^{t}$ and
$\boldsymbol{x}_{-}$ denotes the vector $\boldsymbol{x}$ with its last entry
deleted. The penalty contributions correspond to affine equality constraints
in constrained minimization. In path following, the penalty constant $\rho$
starts large and moves downward. The initial image is flat with gray level
determined by taking $\boldsymbol{x}_{-}={\bf 0}$ and adjusting the last entry
of $\boldsymbol{x}$ to minimize $f(\boldsymbol{B}\boldsymbol{x})$. Call this
point $\boldsymbol{x}_{\infty}$. The first escape time occurs at
$\rho_{\text{max}}=\max_{j}|(\boldsymbol{B}^{t}\nabla
f(\boldsymbol{B}\boldsymbol{x}_{\infty})_{j}|$. At this juncture path
following begins in earnest. Under the $\boldsymbol{x}$ parameterization, the
loss function has gradient $\boldsymbol{B}^{t}\nabla
f(\boldsymbol{B}\boldsymbol{x})$ and second differential
$\boldsymbol{B}^{t}d^{2}f(\boldsymbol{B}\boldsymbol{x})\boldsymbol{B}$.
Because $f(\boldsymbol{B}\boldsymbol{x})$ is not strictly convex, our previous
reparameterization from $\boldsymbol{x}$ to $\boldsymbol{y}$ variables is
needed. Based on equation (11), the path ODEs reduce to
$\displaystyle\frac{d}{d\rho}\boldsymbol{x}_{\bar{\mathcal{Z}}}$
$\displaystyle=$
$\displaystyle-(\boldsymbol{B}_{\bar{\mathcal{Z}}}^{t}d^{2}f(\boldsymbol{B}\boldsymbol{x})\boldsymbol{B}_{\bar{\mathcal{Z}}})^{-1}\text{sgn}(\boldsymbol{x}_{\bar{\mathcal{Z}}}),\hskip
14.45377pt\frac{d}{d\rho}\boldsymbol{x}_{\mathcal{Z}}={\bf 0},$
$\displaystyle\frac{d}{d\rho}\boldsymbol{\lambda}_{\mathcal{Z}}$
$\displaystyle=$
$\displaystyle-\boldsymbol{B}_{\mathcal{Z}}^{t}d^{2}f(\boldsymbol{B}\boldsymbol{x})\boldsymbol{B}_{\bar{\mathcal{Z}}}\frac{d}{d\rho}\boldsymbol{x}_{\bar{\mathcal{Z}}}.$
(28)
Observe that the updates of equation (11) drastically simplify because the
rows of the active constraint matrix $\boldsymbol{U}_{\mathcal{Z}}$ and the
columns of its null space matrix $\boldsymbol{Y}$ are populated by standard
Euclidean unit vectors. Furthermore, for the ROF model of image denoising,
$d^{2}f(\boldsymbol{B}\boldsymbol{x})$ is a diagonal matrix. Alternatively,
one can derive the ODE equations (28) from first principles by implicitly
differentiating the stationary conditions. Path following solves the coupled
ODEs (28) segment by segment.
For a quadratic loss function, the second differential is constant, and the
solution path is piecewise linear. Thus no ODE solving is involved. With a
blurring matrix $\boldsymbol{K}$, the second differential is
$\boldsymbol{B}^{t}d^{2}f(\boldsymbol{B}\boldsymbol{x})\boldsymbol{B}=\boldsymbol{B}^{t}\boldsymbol{K}^{t}\boldsymbol{K}\boldsymbol{B}$.
After each path extension, the path directions (28) yield the next event time
$\rho_{j}$ at which a nonzero component $x_{j}$ hits zero, or a multiplier
$\lambda_{j}$ of a zero component $x_{j}$ hits $\rho$ or $-\rho$. The path is
then extended to the closest of these event times. In deblurring or denoising,
the inverse of
$\boldsymbol{B}_{\bar{\mathcal{Z}}}^{t}\boldsymbol{K}^{t}\boldsymbol{K}\boldsymbol{B}_{\bar{\mathcal{Z}}}$
is best computed via a QR decomposition of
$\boldsymbol{B}_{\bar{\mathcal{Z}}}\boldsymbol{K}$. At each kink in the path,
$\boldsymbol{B}_{\bar{\mathcal{Z}}}\boldsymbol{K}$ changes by adding or
deleting a column of $\boldsymbol{B}\boldsymbol{K}$. As we mentioned earlier,
it is straightforward to update or downdate the QR decomposition (Lawson and
Hanson, 1987). In the original ROF model, traversing one time segment requires
about $O(p)$ operations for $p=mn$ total pixels. The whole process ends when
$T$ differences $x_{j}$ becomes nonzero. In practice, a large value of $T$
recovers too grainy an image, so $T$ is typically much smaller than $p$. The
total cost of computing the solution path is approximately $O(Tp)$, which is
comparable to the cost of start-of-art algorithms for minimization at a single
$\rho$.
Figure 7 illustrates denoising of a $112\times 91$ image of a lighthouse. The
corrupted image appears in the top-left corner of the figure. The $p=10,192$
pixels generate $20,182$ transformed variables. It takes our Matlab script
about one minute of desktop computing time to traverse $T=2,500$ segments
along the regularization path from $\rho=87.9881$ (blank image) to
$\rho=0.5206$ (a nearly optimal image). In the process, the lighthouse clearly
emerges from the fog of oversmoothing. Figure 7 displays selected snapshots
along the regularization path. We emphasize that path following based on
equation (27) reveals the entire path for the interval [0.5206,87.9881] of
$\rho$ values. In practice, one can accelerate path following by starting from
a $\rho$ nearer to the ultimate destination.
Figure 7: A noisy image and snapshots along the regularization path.
## 5 Discussion
Our path following algorithm for constrained convex optimization builds on but
differs from the tradition of path following in homotopy methods (Zangwill and
Garcia, 1981) and interior point programming Boyd and Vandenberghe (2004). The
paths encountered in the exact penalty method introduce the novelty of
piecewise differentiability, which can be effectively handled by tracking the
Langrange multipliers. Computational statisticians deserve credit for
exploring this difficult terrain (Efron et al., 2004; Osborne et al., 2000;
Zhou and Lange, 2011b; Zhou and Wu, 2011). To our knowledge we are the first
to make the connection to exact penalty methods.
Our algorithm enjoys the dual advantages of simplicity and generality. Given
the rich numerical resources of Matlab, it is straightforward to solve the
required ODEs segment by segment. Regardless of whether path following is
faster or slower than existing optimization methods, it supplies the whole
solution path. In regularized estimation, this level of detail offers
unprecedented insight into how penalties and predictors interact. Our example
on image denoising is a case in point.
In quadratic programming with affine equality and inequality constraints, the
solution path is piecewise linear (Zhou and Lange, 2011b). This permits path
following to take large steps. Furthermore, each step can be implemented very
efficiently by the sweep operator of computational statistics. Despite the
loss of these advantages in more complicated examples, the real culprit in
path-following deceleration in many applications is an excessive number of
constraints to be navigated. Our image denoising example suffers from this
defect. On the positive side of the ledger, in nonconvex problems path
following may well prove to be more reliable than competing methods in
separating global from local minima (Zhou and Lange, 2010).
Various extensions of path following are in order. First, the current
algorithm commences from the unconstrained solution. Our development relies on
the strict convexity and coerciveness of the objective function to ensure a
unique starting point. In principle, path initiation should work for any
problem with a unique unconstrained minimum. Similarly, path continuation
should be possible whenever the interior solution is well defined and
piecewise smooth. As the image denoising example suggests, reparametrization
can play an important role in correcting defects in strict convexity. Another
possibility is to amend the surrogate function
${\mathcal{E}}_{\rho}(\boldsymbol{x})$. In our semidefinite programming
example, we add the term $e^{-c\rho}\|\boldsymbol{X}\|_{\text{F}}^{2}$ to
enforce strict convexity and coerciveness. A similar tactic obviously works in
other examples.
A second generalization is to expand the list of penalty functions. For
instance, Euclidean penalties of the form
$\|\boldsymbol{M}\boldsymbol{x}+\boldsymbol{a}\|_{2}$ are useful in grouping
parameters in statistical problems. It should be straightforward to extend
path following to include such penalties. A third generalization is to remove
convexity restrictions altogether. As we have noted, the exact penalty method
applies equally to nonconvex programming. Path following in this setting is
nontrivial since the solution path is no longer necessarily continuous. This
poses a real challenge, and it is unclear to us whether one can construct a
theory as satisfying as that standing behind modern interior point methods. We
invite the optimization community to tackle this broader issue. In the
meantime, we are happy to share our Matlab code with interested researchers.
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|
arxiv-papers
| 2012-01-17T18:55:37 |
2024-09-04T02:49:26.416423
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hua Zhou and Kenneth Lange",
"submitter": "Hua Zhou",
"url": "https://arxiv.org/abs/1201.3593"
}
|
1201.3669
|
# A note on the values of the weighted $q$-Bernstein Polynomials and
modified$\ q$-Genocchi Numbers with weight $\alpha$ and $\beta$ via the
$p$-adic $q$-integral on $\mathbb{Z}_{p}$
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com and Mehmet
Acikgoz University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
(Date: January 8, 2012)
###### Abstract.
The rapid development of $q$-calculus has led to the discovery of new
generalizations of Bernstein polynomials and Genocchi polynomials involving
$q$-integers. The present paper deals with weighted $q$-Bernstein polynomials
and $q$-Genocchi numbers with weight $\alpha$ and $\beta$. We apply the method
of generating function and $p$-adic $q$-integral representation on
$\mathbb{Z}_{p}$, which are exploited to derive further classes of Bernstein
polynomials and $q$-Genocchi numbers and polynomials. To be more precise we
summarize our results as follows, we obtain some combinatorial relations
between $q$-Genocchi numbers and polynomials with weight $\alpha$ and $\beta$.
Furthermore we derive an integral representation of weighted $q$-Bernstein
polynomials of degree $n$ on $\mathbb{Z}_{p}$. Also we deduce a fermionic
$p$-adic $q$-integral representation of product weighted $q$-Bernstein
polynomials of different degrees $n_{1},n_{2},\cdots$ on $\mathbb{Z}_{p}$ and
show that it can be written with $q$-Genocchi numbers with weight $\alpha$ and
$\beta$ which yields a deeper insight into the effectiveness of this type of
generalizations. Our new generating function possess a number of interesting
properties which we state in this paper
###### Key words and phrases:
Genocchi numbers and polynomials, $q$-Genocchi numbers and polynomials, $q$
Genocchi numbers and polynomials with weight $\alpha$, Bernstein polynomials,
$q$-Bernstein polynomials, $q$-Bernstein polynomials with weight $\alpha$
###### 1991 Mathematics Subject Classification:
05A10, 11B65, 28B99, 11B68, 11B73.
## 1\. Introduction, Definitions and Notations
The $q$-calculus theory is a novel theory that is based on finite difference
re-scaling. First results in $q$-calculus belong to Euler, who discovered
Euler’s Identities for $q$-exponential functions and Gauss, who discovered
$q$-binomial formula. The systematic development of $q$-calculus begins from
F. H. Jackson who 1908 reintroduced the Euler Jackson $q$-difference operator
(Jackson, 1908). One of important branches of $q$-calculus is $q$-type special
orthogonal polynomials. Also $p$-adic numbers were invented by Kurt Hensel
around the end of the nineteenth century and these two branches of number
theory jointed with the link of $p$-adic $q$-integral and developed. In spite
of their being already one hundred years old, these special numbers and
polynomials, for instance $q$-Bernstein numbers and polynomials, $q$-Genocchi
numbers and polynomials and etc. are still today enveloped in an aura of
mystery within the scientific community. The $p$-adic integral was used in
mathematical physics, for instance, the functional equation of the $q$-zeta
function, $q$-stirling numbers and $q$-Mahler theory of integration with
respect to the ring $\mathbb{Z}_{p}$ together with Iwasawa’s $p$-adic $q$-$L$
functions. Professor T. Kim [29], also studied on $p$-adic interpolation
functions of special orthogonal polynomials. In during the last ten years, the
$q$-Bernstein polynomials and $q$-Genocchi polynomials have attracted a lot of
interest and have been studied from different angles along with some
generalizations and modifications by a number of researchers. By using the
$p$-adic invariant $q$-integral on $\mathbb{Z}_{p}$, Professor T. Kim in [26],
constructed $p$-adic Bernoulli numbers and polynomials with weight $\alpha$.
After Seo and first author in [9], extended Kim’s method for $q$-Genocchi
numbers and polynomials and also they defined $q$-Genocchi numbers and
polynomials with weight $\alpha$ and $\beta$. Our aim of this paper is to show
that a fermionic $p$-adic $q$-integral representation of product weighted
$q$-Bernstein polynomials of different degrees $n_{1},n_{2},\cdots$ on
$\mathbb{Z}_{p}$ can be written with $q$-Genocchi numbers with weight $\alpha$
and $\beta$.
Let $p$ be a fixed odd prime number. Throughout this paper we use the
following notations. By $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by $\left|p\right|_{p}=\frac{1}{p}$. In this paper we assume
$\left|q-1\right|_{p}<1$ as an indeterminate. In [23-25], let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For $f\in UD\left(\mathbb{Z}_{p}\right)$, the
fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is defined by T. Kim:
$\displaystyle I_{-q}\left(f\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q}}\sum_{\xi=0}^{p^{N}-1}q^{\xi}f\left(\xi\right)\left(-1\right)^{\xi}.\text{
}$
For $\alpha,k,n\in\mathbb{N}^{\ast}$ and $x\in\left[0,1\right]$, T. Kim et al.
defined weighted $q$-Bernstein polynomials as follows:
(1.2)
$B_{k,n}^{\left(\alpha\right)}\left(x,q\right)=\binom{n}{k}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}\text{,
\ (for detail, see [3, 27, 33, 34]). }$
In (1.2), we put $q\rightarrow 1$ and $\alpha=1$,
$\left[x\right]_{q^{\alpha}}^{k}\rightarrow x^{k},$
$\left[1-x\right]_{q^{-\alpha}}^{n-k}\rightarrow\left(1-x\right)^{n-k}$ and we
obtain the classical Bernstein polynomials (see[1], [2]),
where, $\left[x\right]_{q}$ is a $q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{, \ \ (see [2-28, 32-34]).}$
Note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$.
In previous paper [8], for $n\in\mathbb{N}^{\ast}$, modified $q$-Genocchi
numbers with weight $\alpha$ and $\beta$ are defined by Araci et al. as
follows:
$\displaystyle\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)}{n+1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{\beta}}}{\left[\alpha\right]_{q}^{n}\left(1-q\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}q^{\alpha\ell
x}\frac{1}{1+q^{\alpha\ell}}$ $\displaystyle=$
$\displaystyle\left[2\right]_{q^{\beta}}\sum_{m=0}^{\infty}\left(-1\right)^{m}\left[m+x\right]_{q^{\alpha}}^{n}.$
In the special case, $x=0$,
$g_{n,q}^{\left(\alpha,\beta\right)}\left(0\right)=g_{n,q}^{\left(\alpha,\beta\right)}$
are called the $q$-Genocchi numbers with weight $\alpha$ and $\beta$.
In [8], for $\alpha\in\mathbb{N}^{\ast}$ and $n\in\mathbb{N}$, $q$-Genocchi
numbers with weight $\alpha$ and $\beta$ are defined by Araci et al. as
follows:
(1.4) $g_{0,q}^{\left(\alpha,\beta\right)}=0,\text{ and
}g_{n,q}^{\left(\alpha,\beta\right)}\left(1\right)+g_{n,q}^{\left(\alpha,\beta\right)}=\left\\{\QATOP{\left[2\right]_{q^{\beta}},\text{
if }n=1,}{0,\text{ \ \ \ \ if }n>1.}\right.$
In this paper, we obtained some relations between the weighted $q$-Bernstein
polynomials and the modified $q$-Genocchi numbers with weight $\alpha$ and
$\beta$. From these relations, we derive some interesting identities on the
$q$-Genocchi numbers with weight $\alpha$ and $\beta$.
## 2\. On the $q$-Genocchi numbers and polynomials with weight $\alpha$ and
$\beta$
By the definition of $q$-Genocchi polynomials with weight $\alpha$ and
$\beta$, we easily get
$\displaystyle\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)}{n+1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left(\left[x\right]_{q^{\alpha}}+q^{\alpha
x}\left[\xi\right]_{q^{\alpha}}\right)^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}\left[x\right]_{q^{\alpha}}^{n-k}q^{\alpha
kx}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[\xi\right]_{q^{\alpha}}^{k}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}\left[x\right]_{q^{\alpha}}^{n-k}q^{\alpha
kx}\frac{g_{k+1,q}^{\left(\alpha,\beta\right)}}{k+1}.$
Therefore, we obtain the following Theorem:
###### Theorem 1.
For $n,\alpha,\beta\in\mathbb{N}^{\ast},$ we have
(2.1) $g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)=q^{-\alpha
x}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}g_{k,q}^{\left(\alpha,\beta\right)}\left[x\right]_{q^{\alpha}}^{n-k},$
Moreover,
(2.2) $g_{n,q}^{\left(\alpha,\beta\right)}\left(x\right)=q^{-\alpha
x}\left(q^{\alpha
x}g_{q}^{\left(\alpha,\beta\right)}+\left[x\right]_{q^{\alpha}}\right)^{n},$
by using the $umbral$(symbolic) convention
$\left(g_{q}^{\left(\alpha,\beta\right)}\right)^{n}=g_{n,q}^{\left(\alpha,\beta\right)}.$
By expression of (1), we get
$\displaystyle\frac{g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}\left(1-x\right)}{n+1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\beta\xi}\left[1-x+\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{-\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q^{-\beta}}}{\left(1-q^{-\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}q^{-\alpha\ell\left(1-x\right)}\frac{1}{1+q^{-\alpha\ell}}$
$\displaystyle=$ $\displaystyle\left(-1\right)^{n}q^{\alpha
n-\beta}\left(\frac{\left[2\right]_{q^{\beta}}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}q^{\alpha
lx}\frac{1}{1+q^{\alpha l}}\right)$ $\displaystyle=$
$\displaystyle\left(-1\right)^{n}q^{\alpha
n-\beta}\frac{g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)}{n+1}.$
Consequently, we obtain the following Theorem:
###### Theorem 2.
The following
(2.3)
$g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}\left(1-x\right)=\left(-1\right)^{n}q^{\alpha
n-\beta}g_{n+1,q}^{\left(\alpha,\beta\right)}\left(x\right)$
is true.
From expression of (2.2) and Theorem 1, we get the following Theorem:
###### Theorem 3.
The following identity holds
$g_{0,q}^{\left(\alpha,\beta\right)}=0,\text{ and
}q^{-\alpha}\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)^{n}+g_{n,q}^{\left(\alpha,\beta\right)}=\left\\{\QATOP{\left[2\right]_{q^{\beta}},\text{
if }n=1,}{0,\text{ \ \ \ \ \ if }n>1,}\right.$
with the usual convention about replacing
$\left(g_{q}^{\left(\alpha,\beta\right)}\right)^{n}$ by
$g_{n,q}^{\left(\alpha,\beta\right)}$.
For $n,\alpha\in\mathbb{N}$, by Theorem 3, we note that
$\displaystyle q^{2\alpha}g_{n,q}^{\left(\alpha,\beta\right)}\left(2\right)$
$\displaystyle=$
$\displaystyle\left(q^{\alpha}\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)+1\right)^{n}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}q^{k\alpha}\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)^{k}$
$\displaystyle=$
$\displaystyle\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)^{0}+nq^{\alpha}\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)^{1}$
$\displaystyle+\sum_{k=2}^{n}\binom{n}{k}q^{k\alpha}\left(q^{\alpha}g_{q}^{\left(\alpha,\beta\right)}+1\right)^{k}$
$\displaystyle=$ $\displaystyle
nq^{2\alpha}\left[2\right]_{q^{\beta}}-q^{\alpha}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
k}g_{k,q}^{\left(\alpha,\beta\right)}$ $\displaystyle=$ $\displaystyle
nq^{2\alpha}\left[2\right]_{q^{\beta}}+q^{\alpha}g_{n,q}^{\left(\alpha,\beta\right)},\text{
if }n>1.$
Consequently, we state the following Theorem:
###### Theorem 4.
For $n\in\mathbb{N}$, we have
$g_{n,q}^{\left(\alpha,\beta\right)}\left(2\right)={\
n\left[2\right]_{q^{\beta}}}+\dfrac{g_{n,q}^{\left(\alpha,\beta\right)}}{q^{\alpha}}.$
From expression of Theorem 2 and (2.3), we easily see that
$\displaystyle\left(n+1\right)q^{-\beta}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}q^{n\alpha-\beta}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[\xi-1\right]_{q^{\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}q^{n\alpha-\beta}g_{n+1,q}^{\left(\alpha,\beta\right)}\left(-1\right)=g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}\left(2\right).$
Thus, we obtain the following Theorem.
###### Theorem 5.
The following identity
$\left(n+1\right)q^{-\beta}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)=g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}\left(2\right)$
is true.
Let $n,\alpha\in\mathbb{N}$. By expression of Theorem 4 and Theorem 5, we get
$\displaystyle\left(n+1\right)q^{-\beta}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left({n+1}\right)q^{-\beta}{\left[2\right]_{q^{\beta}}}+q^{\alpha}g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}.$
For (2), we obtain corollary as follows:
###### Corollary 1.
For $n,\alpha\in\mathbb{N}^{\ast},$ we have
$\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{\beta}}\left(\xi\right)=\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n+1}.$
## 3\. Novel identities on the weighted $q$-Genocchi numbers
In this section, we develop modifed $q$-Genocchi numbers with weight $\alpha$
and $\beta$, namely, we derive interesting and worthwhile relations in
Analytic Number Theory.
For $x\in\mathbb{Z}_{p}$, the $p$-adic analogues of weighted $q$-Bernstein
polynomials are given by
(3.1)
$B_{k,n}^{\left(\alpha\right)}\left(x,q\right)=\binom{n}{k}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k},\text{
where }n,k,\alpha\in\mathbb{N}^{\ast}.$
By expression of (3.1), Kim et. al. get the symmetry of $q$-Bernstein
polynomials weight $\alpha$ as follows:
(3.2)
$B_{k,n}^{\left(\alpha\right)}\left(x,q\right)=B_{n-k,n}^{\left(\alpha\right)}\left(1-x,q^{-1}\right)\text{,
(for detail, see \cite[cite]{[\@@bibref{}{kim 19}{}{}]}).}$
Thus, from Corollary 1, (3.1) and (3.2), we see that
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}^{\left(\alpha\right)}\left(\xi,q\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}B_{n-k,n}^{\left(\alpha\right)}\left(1-\xi,q^{-1}\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n-l}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n-l+1}\right).$
For $n$, $k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with $n>k$, we
obtain
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}^{\left(\alpha\right)}\left(\xi,q\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n-l+1}\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n-l+1}\right),\text{
if }k>0.}\right.$
Let us take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ on the
weighted $q$-Bernstein polynomials of degree $n$ as follows:
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n}^{\left(\alpha\right)}\left(\xi,q\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[\xi\right]_{q^{\alpha}}^{k}\left[1-\xi\right]_{q^{-\alpha}}^{n-k}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}\frac{g_{l+k+1,q}^{\left(\alpha,\beta\right)}}{l+k+1}.$
Consequently, by expression of (3) and (3), we state the following Theorem:
###### Theorem 6.
The following identity holds
$\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}\frac{g_{l+k+1,q}^{\left(\alpha,\beta\right)}}{l+k+1}=\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n-l+1}\right),\text{
if }k>0.}\right.$
Let $n_{1},n_{2},k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with
$n_{1}+n_{2}>2k$. Then, we get
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}^{\left(\alpha\right)}\left(\xi,q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(\xi,q\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n_{1}+n_{2}-l}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}-l+1}\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}-l+1}\right),\text{
\ if }k\neq 0.}\right.$
Therefore, we obtain the following Theorem:
###### Theorem 7.
For $n_{1},n_{2},k\in\mathbb{N}^{\ast}$and $\alpha,\beta\in\mathbb{N}$ with
$n_{1}+n_{2}>2k,$ we have
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta\xi}B_{k,n_{1}}^{\left(\alpha\right)}\left(\xi,q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(\xi,q\right)d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}-l+1}\right),\text{
\ if }k\neq 0.}\right.$
By using the binomial theorem, we can derive the following equation.
$\displaystyle\int_{\mathbb{Z}_{p}}B_{k,n_{1}}^{\left(\alpha\right)}\left(\xi,q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(\xi,q\right)q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\left[\xi\right]_{q^{\alpha}}^{2k+l}q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\frac{g_{l+2k+1,q}^{\left(\alpha,\beta\right)}}{l+2k+1}.$
Thus, we can obtain the following Corollary:
###### Corollary 2.
For $n_{1},n_{2},k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with
$n_{1}+n_{2}>2k,$ we have
$\displaystyle\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\frac{g_{l+2k+1,q}^{\left(\alpha,\beta\right)}}{l+2k+1}$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}-l+1}\right),\text{
\ if }k\neq 0.}\right.$
For $\xi\in\mathbb{Z}_{p}$ and $s\in\mathbb{N}$ with $s\geq 2,$ let
$n_{1},n_{2},...,n_{s},k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with
$\sum_{l=1}^{s}n_{l}>sk$. Then we take the fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ for the weighted $q$-Bernstein polynomials of degree $n$ as
follows:
$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{\underbrace{B_{k,n_{1}}^{\left(\alpha\right)}\left(\xi,q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(\xi,q\right)...B_{k,n_{s}}^{\left(\alpha\right)}\left(\xi,q\right)}}q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\int_{\mathbb{Z}_{p}}\left[\xi\right]_{q^{\alpha}}^{sk}\left[1-\xi\right]_{q^{-\alpha}}^{n_{1}+n_{2}+...+n_{s}-sk}q^{-\beta\xi}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{l+sk}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n_{1}+n_{2}+...+n_{s}-sk}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}-l+1}\right),\text{
\ \ if }k\neq 0.}\right.$
So from above, we have the following Theorem:
###### Theorem 8.
For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with
$\sum_{l=1}^{s}n_{l}>sk$. Then we have
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\mathop{\displaystyle\prod}\limits_{i=1}^{s}B_{k,n_{i}}^{\left(\alpha\right)}\left(\xi\right)d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ if
}k=0,}{\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}-l+1}\right),\text{
\ \ if }k\neq 0.}\right.$
From the definition of weighted $q$-Bernstein polynomials and the binomial
theorem, we easily get
(3.6)
$\displaystyle\int_{\mathbb{Z}_{p}}\underset{s-times}{q^{-\beta\xi}\underbrace{B_{k,n_{1}}^{\left(\alpha\right)}\left(\xi,q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(\xi,q\right)...B_{k,n_{s}}^{\left(\alpha\right)}\left(\xi,q\right)}}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}q^{-\beta\xi}\left[\xi\right]_{q^{\alpha}}^{sk+l}d\mu_{-q^{\beta}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\frac{g_{l+sk+1,q}^{\left(\alpha,\beta\right)}}{l+sk+1}.$
Therefore, from (3.6) and Theorem 8, we get interesting Corollary as follows:
###### Corollary 3.
For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{N}$ with
$\sum_{l=1}^{s}n_{l}>sk.$ We have
$\displaystyle\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\frac{g_{l+sk+1,q}^{\left(\alpha,\beta\right)}}{l+sk+1}$
$\displaystyle=$
$\displaystyle\left\\{\QATOPD..{\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}+1},\text{
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if
}k=0,}{\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left(\left[2\right]_{q^{\beta}}+q^{\alpha-\beta}\frac{g_{n_{1}+n_{2}+...+n_{s}-l+1,q^{-1}}^{\left(\alpha,\beta\right)}}{n_{1}+n_{2}+...+n_{s}-l+1}\right),\text{
\ \ if }k\neq 0.}\right.$
## References
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* [34] Jung, N. S., Lee, H. Y., and Ryoo, C. S., Some Relations between Twisted ($h$,$q$)-Euler Numbers with Weight $\alpha$ and $q$-Bernstein Polynomials with Weight $\alpha$, Discrete Dynamics in Nature and Society, Volume 2011 (2011), Article ID 176296, 11 pages.
|
arxiv-papers
| 2012-01-17T23:37:02 |
2024-09-04T02:49:26.430279
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1201.3669"
}
|
1201.3872
|
High-precision Measurements of Ionospheric TEC Gradients with the Very Large Array VHF System
J. F. Helmboldt, 1 T. J. W. Lazio, 2 H. T. Intema, 3 & K. F. Dymond, 1
1US Naval Research Laboratory, Washington, DC, USA.
2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA.
3Jansky Fellow of the National Radio Astronomy Observatory, Charlottesville, VA, USA.
We have used a relatively long, contiguous VHF observation of a bright cosmic
radio source (Cygnus A) with the Very Large Array (VLA) to demonstrate the
capability of this instrument to study the ionosphere. This interferometer, and others like it, can observe ionospheric
total electron content (TEC) fluctuations on a much wider range of scales than is possible with many
other instruments.
We have shown that with a bright source, the VLA can measure differential TEC values between pairs of antennas ($\delta \mbox{TEC}$) with an precision of $3 \times 10^{-4}$ TECU. Here, we detail the data reduction and processing techniques used to achieve this level of precision. In addition, we demonstrate techniques for exploiting these high-precision $\delta \mbox{TEC}$ measurements to compute the TEC gradient observed by the array as well as small-scale fluctuations within the TEC gradient surface. A companion paper details specialized spectral analysis techniques used to characterize the properties of wave-like fluctuations within this data.
§ INTRODUCTION
The effects of the ionosphere have always been an obstacle for ground-based radio
frequency observations of astronomical sources. This is especially true for
interferometric observations used to make images of objects with relatively
small angular sizes. VHF and UHF interferometers such as the Very Large Array
(VLA) in New Mexico,
the Westerbork Synthesis Radio Telescope (WSRT) in the Netherlands, the Giant
Metrewave Radio Telescope (GMRT) in India, and the Australia Telescope Compact
Array (ATCA), among others, are all affected by the ionosphere in the same
way.The fundamental principles for the operation of an interferometer are well described in the literature <cit.>. Briefly, an interferometer measures the time-averaged correlation
of the complex electric fields measured at pairs of antennas pointed at
a particular object. These correlations, or “visibilities” provide a
measure of the spectrum of the sky brightness distribution at different
spatial frequencies. These frequencies are essentially the difference
between the position vectors of the two antennas normalized by the
wavelength of the observation in a coordinate system based on the position of the
object in the sky. These spatial frequencies are commonly referred to as
$u$, $v$, and $w$, and the coordinate system used is defined such that $u$ is
the spatial frequency in the east-west direction, $v$ corresponds to the
north-south direction, and $w$ gives the spatial frequency along the line of
sight to the object. Thus, as the object moves through the sky, the $u$,$v$,$w$
coordinates of a pair of antennas, or “baseline,” changes. The
visibility measured for a specific baseline at a particular time is given by
\begin{equation}
V_{\nu}(u,v,w) = \int \int I_{\nu}(l,m) e^{-2 \pi i (ul+vm+nw)} d \Omega
\end{equation}
where $\Omega$ denotes solid angle, $\nu$ the frequency of the observed signal,
and $I$ is the intensity on the sky at a
position given by the direction cosines $l$, $m$, and $n= \sqrt{1-l^2-m^2}$ which
are measured relative to the position of the observed object on the sky.Generally, interferometers are “fringe-stopped,” that is, the measured visibilities are
multiplied by a factor of $\mbox{exp}(-2 \pi i w)$ so that visibilities from a
source at the center of the field of view will have a phase of zero (i.e., the
source will not produce fringes). This is mainly done because with fringe-stopping and for small fields of view
(i.e., $n \simeq 1$), equation (1) becomes a simple two-dimensional Fourier transform such that
the observed visibilities, measured as a function of $u$ and $v$, may be converted to
maps of intensity on the sky using standard numerical methods.As signals from astronomical sources pass through the ionosphere, a phase term is added
given by
\begin{eqnarray}
\phi = \frac{e^2}{c m_e \nu} \int N_e(x) dx \\
\mbox{or, } \phi = 84.36 \left ( \frac{\nu}{\mbox{\scriptsize 100 MHz}} \right )^{-1} \left ( \frac{\mbox{\scriptsize TEC}}{\mbox{\scriptsize 1 TECU}} \right ) \mbox{ radians}
\end{eqnarray}
where $N_e$ is the electron density and $x$ denotes the path-length through the
ionosphere.Thus, the phase of the observed visibility for a baseline is altered by
the difference between the ionospheric phase terms observed by the two antennas, which
is proportional to the difference in the total electron content (TEC) observed along
the lines of sight from the two antennas to the observed object. To first order, if these
phase terms are not removed, then during the Fourier inversion process involved in making an image of the sky, the ionospheric phase terms have the effect of changing the apparent positions of objects
in the image plane. When higher order ionospheric effects begin to dominate,
objects begin to appear distorted in the image plane, and in extreme cases, almost disappear.Therefore, to make an image, one must remove these phase terms through a calibration process
which estimates the required phase corrections. Typically, after initial calibrations for
instrumental effects are performed, one usually uses some form of a procedure referred
to as “self-calibration” [Cornwell and Fomalont, 1999]. This involves dividing the visibilities by an
assumed sky model for the observed field of view which removes any contribution by the sky brightness distribution to the observed visibility phases. Following this, a linear fit is used to determine the complex gain for
each antenna. Since the interferometer only provides phase differences between antenna pairs,
the absolute phase of the complex gain for each antenna cannot be determined by this fitting
process. The general practice is to choose one reference antenna for the
interferometer and to set the phase for its complex gain to zero.Since for $N$ antennas, there are $N(N-1)/2$ baselines, this is generally
an over-determined problem and can be done over relatively short time periods, depending
on the brightness of the source. One may also use this calibration to make an image,
deconvolve the image to produce a better sky model, and then repeat the process until it
converges. This has been shown to be a rather robust procedure for determining the complex
antenna gains <cit.>.According to equations (2) and (3), the phase corrections obtained from the determined antenna
gains are essentially measurements of the difference between the TEC along the lines of sight of
a particular antenna and that of the reference antenna. These equations also demonstrate that
the effect of the ionosphere will be more substantial for VHF observations. Given the size
of available astronomical VHF interferometers (baselines ranging from $<1$ km to as large as
$\sim\! 30$ km), the robustness of the self-calibration procedure on relatively small time scales
(typically $\sim\! 1$ minute, but as small as a few seconds for extremely bright sources), and
the sensitivity of such interferometers to relatively small TEC variations (fluctuations in
differential TEC of as small as 0.001 TECU), these instruments are capable of studying TEC
fluctuations on substantially finer scales than many others available. Subsequently, previous
work has been done using astronomically motivated observations <cit.> as well as
observations geared toward studying the ionosphere <cit.> to
explore the phenomenology of the ionosphere on these fine scales.Much of this work has been performed with the VLA (latitude$= 34^{\circ} \: 04' \: 43.497''$ N and longitude$=107^{\circ} \: 37' \: 05.819''$ W). This is in part due to the fact that the
VLA is relatively well suited to the study of the ionosphere because its 27 antennas are distributed
in a “Y”-shape
which allows it to probe structures along three different directions. Arrays such as the WSRT and
ATCA are so-called “east-west” arrays because the antennas are aligned along a single east-west
axis and can therefore only observe ionospheric fluctuations along one dimension. The VLA is
also unique in that the antennas are moved from time to time among four different configurations,
referred to A, B, C, and D. The configurations go from larger and more spread out to smaller and more
compact. At its largest in the A configuration, each of the array's three arms is about 20 km long
with the shortest antenna separations being about 1.5 km. In the most compact configuration, D, the
arms are at most 0.6 km long and the shortest spacings are about 0.04 km. This allows the VLA the
ability to study the ionosphere over a wider range of physical scales than other similar
interferometers.Finally, the VLA had a somewhat unique VHF system in place that allowed
observations to be made simultaneously at 74 MHz and 327 MHz using a pair of dipole antennas
(one for each band) mounted near the prime focus of each antenna. Recently, the VLA electronics
and receivers were upgraded to establish the new Expanded VLA (EVLA) which does not include
the old VLA VHF system. However, a new and improved VHF system is now being developed and will
be available in the near future.Past work using the VLA and other similar instruments
has led to interesting results. These include discoveries such as the new class of magnetic-eastward-directed
(MED) waves, predominantly found at night, discovered by Jacobson and Erickson, 1992 as well
as larger statistical studies such as the measurements by Cohen and Röttgering, 2009 of the
dependence of differential ionospheric refraction on relatively large angular scales ($>10^{\circ}$) using data from a 74 MHz
all-sky survey which showed large dependences on time of day. However, there is much more to
be learned from these types of data, especially at smaller time and amplitude scales.
Therefore, we are embarking on a program utilizing
the VLA data archives (https://archive.nrao.edu) that seeks to push this type of
analysis to even finer scales. We will use previously unexplored data sets of the brightest
VHF objects. We will apply new techniques for calibration and the mitigation of radio frequency
interference (RFI) to other data sets to significantly improve their sensitivity to small amplitude
TEC fluctuations as well as fluctuations occurring on smaller time and spatial scales than could be
explored previously.Here, we describe the first step in this program, a thorough evaluation of
the ionospheric information contained within a single, relatively long VHF observation of one of
the brightest radio sources in the sky with the VLA. This data set provides the opportunity to
develop and establish techniques for processing, analyzing, and interpreting similar data in future
segments of our program. This paper focusses on the data selection and
calibration as well as the post-processing done on the phase information extracted from this
exemplar data set to measure TEC gradients with the array. In a companion paper [Helmboldt et al., 2011], we
detail new techniques for spectral analysis of these data.
§ DATA ACQUISITION AND PROCESSING
§.§ Phase Correction Determination
In the study presented in this paper, we have sought to explore the
kinds of ionospheric phenomena capable of being observed with the
at the lowest fluctuation levels and the smallest times scales possible.
To ensure that we were able to determine ionospheric phase
terms to the the level of accuracy needed at the smallest available time
intervals, including the resolution of any $2\pi$ ambiguities, we chose to
use a data set focused on Cygnus A (or “Cyg A”; also known as 3C405).
With a total intensity of more than 17000 Jy ($=1.7 \times 10^{-22}$
W m$^{-2}$ Hz$^{-1}$) at 74 MHz, Cyg A is one of the two brightest sources in the sky at frequencies below 100 MHz. Its brightness
distribution on the sky is also known relatively well, implying that
it can be used within the self-calibration procedure described briefly in 1 to solve for the phase corrections (both instrumental
and ionospheric) for each antenna over relatively short time intervals.The data set we have selected consists of simultaneous 74 MHz and 327 MHz
dual polarization observations of Cyg A with the VLA over a period
of more than 12 hours on 12-13 August, 2003 (VLA program number AK570). During this period, there was a moderate amount of geomagnetic activity ($K_p$ index $\approx \! 2$–4) and low solar activity ($F10.7\!=\!123$ SFU; 1 SFU$\!=\!10^{-22}$ W m$^2$ Hz$^{-1}$).
For this data, the VLA was in the A configuration with the VHF dipole system
available on all but three antennas (antenna numbers 11, 13, and 15; see 1 and Fig. <ref>). The exact
layout of the antennas is shown in Fig. <ref>. The data set contains a 35
minute scan (i.e., a contiguous block of observing time), and a longer, nearly
13 hour scan which, taken together, cover
a range of local time from roughly $17^{\mbox{\scriptsize h}}00^{\mbox{\scriptsize m}}$
on 12 August to $06^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ the next
morning. For the 74 MHz band, the total bandwidth was 1.5 MHz; it was
3 MHz for the 327 MHz band. For both bands, the time-averaged
visibilities were measured over intervals of 6.67 seconds simultaneously
in both RR and LL polarizations.The processing of VHF interferometric observations from the
VLA or other similar interferometers is, in the average case, a lengthy
and difficult process. One usually must deal with significant sources
of RFI, which typically appear much stronger
on shorter baselines. For the VLA, the field of view at 74 MHz is more than
ten degrees in diameter, and the ionospheric conditions may vary significantly
from one part of the field of view to another, which requires special
calibration techniques. One must also typically produce an image for the
entire field of view, regardless of which object(s) one is interested
in because the removal of sidelobes (i.e., secondary peaks of the impulse response in the image plane) produced by other objects is crucial
to minimizing noise in the final image. Fortunately, the unusually high
intensity of Cyg A makes it substantially brighter than any source of RFI
or any other object in the field of view. This makes the calibration
process much simpler and more straight forward.The calibration of the data was performed using standard tasks within the
Astronomical Image Processing System <cit.>.
The first step in the calibration was to determine the bandpass response
of each band (AIPS task ). This was done within one minute time
intervals by dividing the data by the visibilities measure within a few
central channels where the response typically peaks. The relative amplitude
and phase responses across each bandpass were then measured and interpolated
onto the full set of time steps. After trimming the first and last few channels
from each band where the response drops substantially, the data were corrected for
the bandpass responses.Following this, at 6.67 second intervals (i.e., the
shortest possible for these data), the data for each band and polarization
were used with model visibilities computed using images of Cyg A
presented by Lazio et al., 2006
(currently publicly available at http://lwa.nrl.navy.mil/tutorial/) to
compute the phase corrections for each antenna at each time step. This was
done with the AIPS task which does a series of consistency and
sanity checks as it determines the solutions needed to fit the measured
visibilities to the model and flags antennas and intervals that appear
spurious or of poor quality. We relaxed some of the criteria for these checks, namely
the minimum number of antennas (we used four) required and the minimum signal to noise
ratio (we used three) because of the relatively short time intervals used. As noted above,
an interferometer like the VLA can only measure relative phases, implying
that for this calibration to work, a reference antenna must be chosen whose
phase is arbitrary and is subsequently set to zero. Typically, the antenna
closest to the center of the VLA is not used because the effects of RFI tend
to be the worst for this antenna. However, as noted above, Cyg A is bright
enough that this is not a consideration for our data set, and we therefore
used this antenna as our reference (antenna 9; see Fig. <ref>).
§.§ Processing of Phase Corrections
Following the determination of phase corrections using standard AIPS
routines, several steps were taken to extract ionospheric information from
the phase data. This was done using ad hoc, python-based software.The phase corrections measured by the calibration process contain
contributions from several effects, the ionosphere being the largest,
especially at 74 MHz. In particular, the phase difference between two antennas within a single baseline is given by
\begin{equation}
\Delta \phi = \Delta \phi_{ion} + \Delta \phi_{inst} + \Delta \phi_{sour} + \Delta \phi_{amb}
\end{equation}
where $\Delta \phi_{ion}$ denotes the difference in the ionospheric phases along the lines-of-sight of the two antennas given by equation (3), $\Delta \phi_{instr}$ represents difference in the instrumental effects of the two antennas, $\Delta \phi_{sour}$ is the contribution to the phase difference from the structure of the observed source, and $\Delta \phi_{amb}$ is the contribution from $2\pi$ ambiguities. We can remove $\Delta \phi_{sour}$ by dividing the observed visibilities by a model of Cyg A and $\Delta \phi_{amb}$ by having short enough time sampling to “unwrap” the phases (see below).
The dual 74 and 327 MHz observing mode is very useful
for removing instrumental effects since the ionospheric phase simply scales
with wavelength [see equations (2) and (3)], whereas instrumental effects do not
necessarily. The instrumental components of the phase corrections include
errors in the delays added to individual antenna signals used to fringe-stop the visibilities (see 1) and offsets between the antenna pointing and the actual source position. These two effects and $2\pi$ ambiguities were dealt with in three separate steps.First, the AIPS task flags antennas that are spurious at each time step
so that the phase corrections are “missing” for some antennas at a relatively
small fraction ($\sim 2\mbox{\%}$) of time steps. Generally, a small amount of missing data is not problematic. However, to facilitate the use of fast Fourier transforms (FFTs) within our spectral analysis of the data presented in a companion paper which work best with evenly sampled data, these missing time steps were filled in. We have done this by linearly interpolating the real and imaginary parts of the complex
antenna gains, $g_{A}$, computed by for all antennas onto a common
time-step grid consisting of 7299 steps spaced by 6.67 seconds, and then
recomputing the phase corrections {$=\mbox{tan}^{-1}[\mbox{Im}(g_A)/\mbox{Re}
(g_A)]$}.Next, the time sampling of the data (6.67 seconds) was sufficiently short that
the phase, $\phi$, as a function of time for each antenna could be “unwrapped”
in the conventional way, i.e., by correcting phase jumps of more than $\pi$
radians by adding or subtracting $2 \pi$. There was one caveat to this process,
however. For each antenna, there were a few (ranging from zero to five) times
steps where, for one reason or another, the phase correction was either
spurious or represented a real and very short jump in the instrumental phase, appearing as sharp spikes in the unwrapped phases. Since there were
a total of 7299 time steps, a few short jumps in phase would not be an issue except for
their effect on the unwrapping process. Any of these spikes can cause an artificial
large phase jump if it is included in the unwrapping process, which we have
illustrated in the upper panel of Fig. <ref>.To combat this, we
wrote a simple algorithm that computes the difference between $\mbox{cos}(\phi)$ at
each time step and the value for the next time step, where $\phi$ is the wrapped
phase. Any time step where the absolute value of this difference was more than ten
times the standard deviation among all time steps for a give antenna was flagged and
not included in the unwrapping process. These empty time steps were then filled by
linearly interpolating the unwrapped phase data for the un-flagged time steps. It should be noted that these spikes only occupy 1–2 time steps (6.67–13.34 seconds) and that the surrounding data are otherwise well-behaved, making interpolation a reasonable and straightforward solution to eliminating instances of such spikes. An example of
how the data were flagged is illustrated in the middle panel of Fig. <ref>
while the resulting unwrapped phases are shown in the lower panel. From this result,
one can see that the spikes are not always completely removed from the data, but the
goal of eliminating their effect on the unwrapping process has been achieved.A number of instrumental effects can contribute to the phase corrections, including
errors in the fringe-stopping process (see 1) and offsets between the position of the source and that of the observed field center (i.e., pointing errors). These effects are generally stable
in time, changing insignificantly over periods of days <cit.>.
However, for
the VLA, the instrumental phase is known to occasionally have short jumps that one
must be wary of. We have found such a jump in our data occurring at a local time of
about $23^{\mbox{\scriptsize h}}06^{\mbox{\scriptsize m}}$ on 12 August. The jump is most obvious when one plots the difference
between the 74 MHz phase and the 327 MHz phase scaled by a factor of (327/74) so that
the ionospheric phases cancel out (note, this only gives us the scaled difference between the 327 and 74 MHz instrumental phases, not the instrumental phases themselves). We have plotted this difference as a function of
time for antenna 14 for both polarizations in Fig. <ref> to show the location
of what we will refer to as the “phase jump region.” While being fairly subtle in
the LL polarization, it is quite obvious in the RR polarization. The jump lasted
about 6 minutes and can be seen in the data for several antennas. To deal with this,
we have treated the phase jump region, as well as the time periods before and after
it, as separate scans, assuming that each scan has its
own instrumental phase. This basically resulted in us treating the data as if it
contained four scans instead of two.To remove instrumental effects, we have used a kind of continuum
subtraction process. Within this process, we have treated ionospheric fluctuations as features
superimposed on a smooth continuum consisting of the instrumental phases, which vary relatively slowly with time as well as any slowly varying component of the ionospheric phase. With the current data, we
unfortunately do not have the means to separate the slowly varying component of the ionospheric phase from the instrumental effects and can therefore only measure
fluctuations in TEC on relatively short ($<1$ hour; see below) time scales. In the future, with instruments with
larger bandwidths, it will be possible to use the wavelength dependence of the
phase corrections to separate these effects since the instrumental phases are $\propto \nu$ and the ionospheric phase is
$\propto \nu^{-1}$.We have chosen to perform our continuum subtraction process for each antenna, band,
and polarization
by smoothing the unwrapped phases with a one-hour-wide boxcar which appeared to
preserve any apparent fluctuations while giving a good representation of the continuum.
For the first scan and the phase jump region, we simply subtracted a single mean
value from all the phases since they are each shorter than an hour. We did the
same for the first and last hour of each of the remaining two scans so that the
same filter width would be used for all times steps.Following this, we found that the position offset component of the instrumental phases presented a problem for
this process near the edges of each scan. This is because, depending on the
antenna, these phases can vary significantly over one hour, especially at 327 MHz.
From the fringe-stopped (see 1) version of equation (1), one can see that a
position offset in the direction cosines $l$ and $m$ of $\Delta l$ and $\Delta m$
will produce an additional phase of $-2 \pi (u \Delta l + v \Delta m)$. Since
$u$ and $v$ are normalized by the observed wavelength, any such phase will be 4.4
times larger for the 327 MHz band. In addition, the offsets can be different for each band and this difference can vary with baseline. This is due to a number of factors including the fact that different model images were used for the bands which may not be exactly aligned and that Cyg A has a significant amount of resolved structure which larger baselines are more sensitive to, especially at 327 MHz where the angular resolution in the image plane is 4.4 times better than that at 74 MHz.We have demonstrated this in the upper panel of Fig. <ref> where we have plotted
$\phi_{74} - \phi_{327} (327/74)$ for antenna 3, LL polarization as a function of
time. With $\phi_{327}$ scaled by (327/74), the ionospheric phases are removed and
all that is left is the difference between the instrumental
phases for the two bands. The data follow a smooth curve which is inconsistent with
the known behavior of VLA instrumental phases, especially at 74 MHz. Furthermore, the curve that the
data follows is easily fit by a linear combination of the un-normalized versions
of the $u$ and $v$ coordinates (see the red curve in the upper panel of Fig.
<ref>). A single baseline (antenna 3 with
the reference antenna) will sweep out an ellipse in the $u$,$v$-plane because
of the rotation of the earth <cit.>. Therefore, this is exactly
what one would expect for a scenario where there is a single position offset
for each of the two bands during each scan.To show the effect of the time dependence of the position offset phase on
our continuum determination process, we have plotted the continuum-subtracted
version of $\phi_{74} - \phi_{327} (327/74)$ versus time in the middle panel of Fig. <ref>.
One can see from this plot that within the first scan and within the last hour of the
second and fourth scans where a single mean continuum value was subtracted from each,
the gradient of the position offset phases has introduced an artificial difference
which increases/decreases with time. Since the position offset phase is much larger
for the 327 MHz band, we have introduced the following additional step for the
continuum subtraction of the 327 MHz data. Within the first scan, the phase jump
region, and the first and last hours of the other two scans, we have fit a linear
combination of the un-normalized versions of $u$ and $v$ to $\phi_{74} - \phi_{327} (327/74)$
separately for each time range. Within each of these time periods, we used the
mean values for $\phi_{74}$, $\phi_{372}$, and the $u$,$v$ fit to construct a
time-variable continuum for the 327 MHz data. The benefits of this are illustrated
in the bottom panel of Fig. <ref> where we have plotted the continuum-subtracted
version of $\phi_{74} - \phi_{327} (327/74)$, this time, including the additional
computation for the time-variable 327 MHz continuum with the first/last hour of each
scan. One can see that the roughly linear features seen in the middle panel of
Fig. <ref> have been removed and that the remaining difference between the
74 MHz and 327 MHz continuum-subtracted phases is essentially noise.Following the application of the corrections detailed above, we used equation (3)
to convert the continuum subtracted phases for each antenna, band, and polarization
to values of differential TEC, or $\delta \mbox{TEC}$. Then, at each time step and
antenna, we computed the median $\delta \mbox{TEC}$ among the four values (i.e., two
bands and two polarizations) as well as the median absolute deviation (MAD) as an
estimate of the uncertainty in the median. To increase the reliability of the MAD
computations, we included with each time step the four nearest time steps (using
their own individual median values) for a total of 20 data points per time step.
In both computations (median and MAD), the median was
used to minimize the effects of any spurious data which remained.The resulting
$\delta \mbox{TEC}$ values are plotted as functions of time for each antenna in the
northern arm in Fig. <ref>, the southeastern arm in Fig. <ref>, and the southwestern arm
in Fig. <ref> along with the MAD values to illustrate the relatively high
precision to which $\delta \mbox{TEC}$ has been measured. The typical
$\delta \mbox{TEC}$ uncertainty, represented by the MAD computations, is about
$3 \times 10^{-4}$ TECU, demonstrating the remarkable ability of the VLA to detect extremely
small TEC fluctuations, even on time scales $<10$ seconds when an object as bright
as Cyg A is used.
§ MEASURING TEC GRADIENTS
§.§ General Approach
With the fully reduced $\delta \mbox{TEC}$ data, including robust estimates of
the uncertainties, we are in a position to explore a wide range of ionospheric phenomena.
First, we note that since the VLA measures differential TEC values between
antenna pairs, it is essentially only sensitive to changes in the TEC gradient.
Given the geometry of the array (see Fig. <ref>), we cannot numerically
compute the TEC gradient at each antenna location from our data, and measuring the full TEC gradient requires a somewhat ad hoc approach. Such measurements are crucial for any analysis of observed TEC fluctuations because without modification, the set of $\delta \mbox{TEC}$ time series can only be spectrally analyzed for specific assumed pattern models <cit.>.This is different from the normal mode of operation for radio interferometers in which standard techniques are used to invert and de-convolve sparsely sampled visibility data to make an image. As equation (1) demonstrates, the observed visibilities are functions of the differential antenna positions, $u$, $v$, and $w$. For small fields of view, the contribution of the $w$ term is negligible if fringe-stopping is applied (see 1). Thus, even for a Y-shaped array, reasonably good $u$,$v$-coverage can be obtained. This is improved further by the rotation of the earth which causes each baseline to sweep out an ellipse in the $u$,$v$-plane <cit.>. Using this fact to obtain better $u$,$v$ coverage is sometimes referred to as “earth rotation synthesis.”In contrast, the gradient of an arbitrary set of TEC fluctuations varies over the array as a function of the actual antenna positions projected onto the ionosphere. Improved spatial coverage can be obtained by exploiting the change in the apparent position of the observed source (i.e., rotation of the earth) and the movement of the fluctuations themselves, effectively converting temporal baselines into spatial ones. However, since the TEC fluctuations presumably have a distribution of speeds and directions, this is not as straightforward as in earth rotation synthesis. One must decompose the time series into temporal spectral modes and then analyze how the properties of each mode vary across the array to extract the size, speed, and direction of the dominant pattern(s) for that mode (such spectral techniques are detailed in a subsequent paper). Therefore, we must still contend with data that has been sampled in a Y-pattern which cannot be inverted in a straightforward manner. We have consequently developed two ad hoc techniques designed to provide measurements of the TEC gradient time series over the full array and along each of the VLA arms. Before implementing either technique, we first had to perform two
basic geometric corrections to the data so that the measured TEC gradients
would correspond to vertical TEC gradients as closely as possible. First, to ensure
that our characterization of the shape of the observed TEC surface is physically
meaningful, we needed to project the antenna pattern displayed in Fig. <ref>
onto the locations where the lines of sight of the antennas pass through the
ionosphere, or “pierce-points.” Second, we needed to compute the slant-to-vertical
TEC corrections for the line of sight to Cyg A as its apparent position on the sky
changed throughout the observation.
For a plane parallel approximation, both of these
corrections are relatively straightforward. However, since Cyg A was as low as
$12^{\circ}$ above the horizon during the observing run, a plane parallel
approximation was far from valid at all time steps. We have therefore computed
the two required geometric corrections using a spherical model detailed
in Appendix A.Within this model,
the ionosphere was approximated with a thin shell located at the height of the
maximum electron density, or “peak height” <cit.>. We
obtained estimates of the peak height as a function of time by using the International
Reference Ionosphere <cit.> software, inputting the date and time of our observations
and the latitude and longitude of the VLA. We then re-determined the peak heights using
the latitudes and longitudes of the pierce-points. We found that additional
iterations of this process only marginally changed the results and chose to use one
iteration only. The final peak heights used are plotted in the upper
panel of Fig. <ref> along with the projected separations from the array center for the
farthest antennas of each arm (antennas 1, 7, and 22; see Fig. <ref>) and the
corresponding slant-TEC corrections.
§.§ Polynomial Fits
After applying the geometric corrections to the antenna positions and
the $\delta \mbox{TEC}$ measurements, we sought to characterize the full two-dimensional TEC gradient observed by each antenna at each time step. Rather than assume a particular dominant structure (e.g., a plane wave), we simply assumed that since the array is smaller than many transient ionospheric phenomena, the observed TEC surface at any time step could be approximated with a low-order Taylor series. We examined many time steps and found
that a second order, two-dimensional Taylor series adequately approximated the amount of curvature in the TEC surface detected by
the VLA. This Taylor series
has the following form
\begin{equation}
\mbox{TEC} = p_{0}x + p_{1}y + p_{2}x^{2} + p_{3}y^{2} + p_{4}xy + p_{5}
\end{equation}
where $x$ and $y$ are the north-south and east-west antenna positions, respectively,
projected onto the surface of the ionosphere at the estimated peak height.
To maximize the amount of data used to constrain the parameters of each fit, we
used the difference between $\delta \mbox{TEC}$ for each of the 300 unique antenna
pairs at each time step. Thus, the form of equation (5) actually fit to the data
\begin{eqnarray}
\delta \mbox{TEC}_i - \delta \mbox{TEC}_j = p_0(x_i-x_j)+p_1(y_i-y_j) \nonumber \\
\end{eqnarray}
where the $i$ and $j$ subscripts denote the values for the $i^{\mbox{\scriptsize th}}$
and $j^{\mbox{\scriptsize th}}$ antennas, respectively. We also utilized some
standard sigma-clipping during the fitting process for each time step by computing
the rms of the fit residuals, rejecting all antenna pairs with absolute
residuals $>3 \, \mbox{rms}$, and repeating 50 times. As many as about 10 and
as few as zero were rejected for any given time step. We note that each time step
was fit independently to preserve the presence of any small-scale spatial/temporal
fluctuations.The fitted coefficients as a function of time are plotted in Fig. <ref>.
From these, one can see the same large amplitude and period
fluctuations at the beginning of the
observing run that are visible in the individual antenna data plotted in Fig.
<ref>–<ref>. Note that they are not quite as large here because of the
applied slant-TEC correction discussed above. Here, we can see that they are most
visible in the $p_0$ coefficient which is the
partial derivative of the TEC surface at the center of the array along the north-south
direction. With the
plots in Fig. <ref>, one can also see the same thing beginning to happen
near the end of the run toward dawn. This is qualitatively consistent with the
known behavior of medium-scale traveling ionospheric disturbances (MSTIDs) which
are prevalent near sunrise and sunset <cit.>. During the middle
of the night, the second-order terms become more significant.
§.§ Arm-based Approach
While the polynomial-based measurements provide useful information about the variation of the full two-dimensional TEC gradient, they neglect the ability of the VLA in its A configuration to detect
fluctuations on scales as small as a few kilometers. In principle, one could do this by simply increasing the order of the polynomials used. However, it is likely that the small-scale structures observed do not span the array. This implies that such fits would not yield accurate representations of the full TEC gradient at each antenna, especially those near the ends of the arms (see Fig. <ref>). Therefore, instead of using higher order polynomial fits, we have opted for an alternative approach to make full use of the data.This complementary method computes the projection of the TEC gradient (or, the spatial derivative of $\delta \mbox{TEC}$) along each VLA arm. The projected gradient was computed at each time step separately for the antennas of each arm using simple three-point Lagrangian interpolation. Given the typical $\delta \mbox{TEC}$ precision of $3 \times 10^{-4}$ TECU and the mean separation between antennas of 2.5 km, the precision of these projected TEC gradient measurements is typically about $2 \times 10^{-4}$ TECU km$^{-1}$. The time series of the projected gradient at each antenna is plotted in Fig. <ref>. We have also plotted in red the projected gradient computed using the polynomial coefficients plotted in Fig. <ref>. One can see that for the larger amplitude, longer period disturbances, the polynomial fits largely recover the structure observed using the data for individual antennas. However, during the middle of the night, there appears to be a significant amount of smaller-scale structure missed by the polynomial fits that can only be observed using the individual antenna gradients, especially for the shortest baselines near the center of the array.
§ DISCUSSION
Our exploration of a long, VHF observation of Cyg A with the VLA
has successfully demonstrated the power of this instrument to
characterize a variety of transient ionospheric phenomena. For this
observation, the typical $1 \sigma$ uncertainty in the $\delta \mbox{TEC}$
measurements was $3 \times 10^{-4}$ TECU, yielding more than an order of
magnitude better sensitivity to TEC fluctuations than can be
achieved with GPS-based relative TEC measurements
<cit.>.Large amplitude, long period waves reminiscent of MSTIDs are visible within the $\delta \mbox{TEC}$ data near dusk and dawn as well as other times intermittently throughout the night. The polynomial-based approach we have detailed in 3.2 appears to be able to recover the properties of the full two-dimensional TEC gradients associated with these relatively large disturbances as they passed over the array. This information can be used to estimate the size, speed and directions of such patterns down to scales of roughly half the size of the array ($\sim\! 20$ km). This is demonstrated in more detail in a subsequent paper describing the spectral analysis of these data.In addition, our approach of measuring the projected gradient at each antenna along each arm has shown that there are smaller-scale TEC fluctuations observed throughout the night, most prominently after midnight local time. Thus, the VLA can be used to simultaneously study fine-scale ionospheric dynamics. This may include a host of phenomena such as the small-scale distortions/structure within MSTID wavefronts, turbulent fluctuations from ion-neutral coupling within the lower ionosphere/thermosphere, and sporadic-E ($E_s$) layers. In the case of $E_s$, Coker et al., 2009 demonstrated with a combination of VLA data and optical observations that many of the small-scale fluctuations seen by the VLA during summer nighttime are likely associated with $E_s$ layers. Coker et al., 2009 showed that the TEC gradients caused by these layers are typically $\approx \! 0.001$ TECU km$^{-1}$ which is easily detectable using the arm-based gradient method. A specialized spectral analysis technique has also been developed for these measurements and will likewise be detailed in the companion manuscript to this paper.
§ GEOMETRIC CORRECTIONS
Two basic geometric corrections must be applied to the antenna positions
and $\delta \mbox{TEC}$ measurements so that they more accurately represent the
actual conditions within the ionosphere. Since we have used observations of
Cyg A that include times when it is relatively close to the horizon, we cannot
use a plane-parallel approximation. Instead, we have used a thin shell
approximation for the ionosphere where the shell is located at the height of
maximum electron density, $z_{\mbox{\scriptsize ion}}$, as computed by the IRI software for the dates and times
of the observations (see 3.1 and Fig. <ref>).
The full spherical corrections used are detailed below.First, the positions of the antennas on the ground must be converted to projected
positions within the ionosphere which, for a non-plane-parallel atmosphere,
change with the elevation of the observed source. For a spherical shell, we
may define a “pierce-point” for each antenna where its line of sight to the
source passes through the ionosphere. The positions of these pierce-points
relative to that for the center of the array can then be used as their projected
ionosphere positions. Fig. <ref> provides a schematic illustration
(not to scale) of how these positions are determined. We first define a set three
position vectors, $\mbox{\bf R}_{\mbox{\scriptsize \bf A}}$, $\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}$, and
$\mbox{\bf R}_{\mbox{\scriptsize \bf S}}$, which define the positions of the array center/antenna,
the ionosphere pierce-point, and the observed source, respectively, with the
center of the earth as the origin of the coordinate system.Next, we note that
the vast majority of astronomical sources, including Cyg A, are essentially
infinitely far away, which implies that the line of sight from the array center/antenna
location to the source is essentially parallel to that from the center of the
earth to the sources, or
$\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}-\mbox{\bf R}_{\mbox{\scriptsize \bf A}} \mbox{ }||\mbox{ } \mbox{\bf R}_{\mbox{\scriptsize \bf S}}$. If we define a “left-handed” coordinate system such that
$\mbox{\bf R}_{\mbox{\scriptsize \bf A}}$ points along the $z$-axis, then the source
position is given by
\begin{equation}
\frac{\mbox{\bf R}_{\mbox{\scriptsize \bf S}}}{|\mbox{\bf R}_{\mbox{\scriptsize \bf S}}|} = \mbox{cos}(h)\mbox{cos}(a) \hat{i} \nonumber + \mbox{cos}(h)\mbox{sin}(a) \hat{j} + \mbox{sin}(h) \hat{k}
\end{equation}
where $h$ is the angular elevation of the source, $a$ is the azimuthal angle measured
from north though east, and the $\hat{i}$ and $\hat{j}$ unit vectors point
toward the north and east, respectively, as viewed from the array/antenna.
Combining this with the assumption of parallel lines of sight to the source and
the fact that the length of $\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}$ is set to
$\mbox{R}_{\mbox{\scriptsize earth}} + z_{\mbox{\scriptsize ion}}$ yields the
following expression
\begin{equation}
\left ( \mbox{R}_{\mbox{\scriptsize earth}} + z_{\mbox{\scriptsize ion}} \right )^{2} =
\left [ r \; \mbox{cos}(h)\mbox{cos}(a) + x_{A} \right ]^{2} \nonumber
+ \left [ r \; \mbox{cos}(h)\mbox{sin}(a) + y_{A} \right ]^{2} \nonumber
+ \left [ r \; \mbox{sin}(h) + z_{A} + \mbox{R}_{\mbox \scriptsize earth} \right ]^{2}
\end{equation}
where $x_A$, $y_A$ and $z_A$ are the coordinates of the antenna relative to the
array center and $r=|\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}-\mbox{\bf R}_{\mbox{\scriptsize \bf A}}|$.
Since the antenna positions are known, $r$ is the only unknown variable. Equation
(A2) can then be rewritten as a quadratic equation and solved for $r$ keeping in
mind that $0 \leq r < \mbox{R}_{\mbox{\scriptsize earth}} + z_{\mbox{\scriptsize ion}}$
which allows one to compute the $x$, $y$, and $z$ coordinates of
$\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}$ for the array center and each antenna in
the current coordinate system. Following this, a coordinate rotation was performed
such that $\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}$ for the array center pointed
along the $z$-axis and the $x$ and $y$ axes pointed toward north and east, respectively,
as viewed from the location on the earth directly below the array center pierce-point. These rotated
coordinates were then taken to be the $x$ and $y$ antenna positions projected onto the
ionosphere thin shell for each time step. Fig. <ref> shows a graphical
representation of these computations for the array center (in black) and for an
exemplar antenna (in gray).The second correction deals with the fact that the path length through the ionosphere
is increased when the observed
source is closer to the horizon. For a thin spherical shell, it is increased by a
factor of $\mbox{sec}(\epsilon)$ where $\epsilon$ is the angle between the line of
sight from the VLA to the source and a line from the ionosphere pierce-point to the
location on the earth directly below it. In the schematic in Fig. <ref>, $\epsilon$ is the angle
between the position vectors $\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}$ and
$\mbox{\bf R}_{\mbox{\scriptsize \bf PP}}-\mbox{\bf R}_{\mbox{\scriptsize \bf A}}$.
Therefore, to compute the factor needed to correct our $\delta \mbox{TEC}$ measurements,
$\mbox{cos}(\epsilon)$, we simply computed the dot product between these two vectors
and divided by the product of their lengths,
$r(\mbox{R}_{\mbox{\scriptsize earth}} + z_{\mbox{\scriptsize ion}})$.Finally, while computing the above geometric corrections, we also computed estimates
of the apparent motion of Cyg A within the coordinate system of each time step. This
was done to estimate the degree of Doppler shifting of the temporal/spatial
frequencies of any detected wave phenomena. We did this
for each time step by recomputing the position of the array center pierce-point
for the two nearest time steps within the coordinate system of the current time step.
These positions were then used to numerically compute the time derivatives of the
$x$ and $y$ coordinates of the array center pierce-point to obtain the north-south
and east-west components of the sidereal velocity. These are plotted in Fig.
<ref> as functions of time along with a histogram for the azimuth angle
(measured north through east) of the sidereal velocity vector for the entire observing
run. One can see from this figure that the velocities were sometimes significant,
when the source was at lower elevations. In addition, while the motion is generally from
east to west, as one would naively assume, there is a significant spread in position
angle of more than $100^{\circ}$.
The authors would like to thank the referees for useful comments and suggestions. Basic research in astronomy at the Naval Research Laboratory is supported
by 6.1 base funding. The VLA was operated by the National Radio Astronomy Observatory which is a facility
of the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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A Method for Characterizing Transient Ionospheric Disturbances Using a Large Radiotelescope Array,
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Wavenumber-resolved Observations of Ionospheric Waves Using the Very Large Array Radiotelescope,
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Derivation of TEC and Estimation of Instrumental Biases from GEONET in Japan
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vol. 180, edited by G. B. Taylor, C. L. Carilli, and R. A. Perley, pp. 187–199,
ASP, San Francisco, Calif.
The layout of the VLA antennas during the observations of Cyg A. The reference
antenna (see 2) is highlighted in black.
Upper panel: the unwrapped phase, $\phi_{\mbox{\scriptsize U}}$ for antenna 4,
RR polarization, as a function of time (relative to midnight, 13 August)
when no attempt was made to “de-spike” the phase
data (see 2.2). Middle panel: the difference between
the cosine of the wrapped phase at a
time step i, $\mbox{cos}{\phi_{\mbox{\scriptsize i}}}$, and the next time step as a
function of time used to find spikes in the phase data. Flagged spikes are highlighted
with circles. Lower panel: the phase as a function of time, unwrapped with the flagged
spikes excluded.
As a function of time, the difference between the unwrapped 74 MHz phase,
$\phi_{74}$, and the 327 MHz phase, $\phi_{327}$, with the 327 MHz phase scaled such
that the ionospheric components of the two phases canceled out for the RR (upper)
and LL (lower) polarizations for antenna 14. The region where the instrumental
phases have changed relatively abruptly (i.e, the “phase jump” region first referred
to in 2.2) is shaded in gray.
Upper: As a function of time, the difference between the unwrapped 74 MHz phase,
$\phi_{74}$, and the 327 MHz phase, $\phi_{327}$, with the 327 MHz phase scaled such
that the ionospheric components of the two phases canceled out for antenna 3, LL
polarization. A fit to this data of a linear combination of the spatial frequencies
$u$ and $v$ [see equation (1)] which assumes
a single position offset (see 2.2) is also plotted in red. Middle: The same as
in the upper panel, but for the phases with the smoothed (with a one-hour wide box)
phases subtracted from each band. Lower: The same as the middle panel, but with a
fit of a linear combination of $u$ and $v$ used in the continuum subtraction of the
327 MHz phases at the edges of each scan (see the discussion in 2.2).
For each antenna in the northern arm (see Fig. <ref>) of the VLA, the
the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its
line of sight and that measured along the reference antenna's line of sight, $\delta
\mbox{TEC}$. The estimated uncertainty (see 2.2) is plotted in red in each panel.
For each antenna in the southeastern arm (see Fig. <ref>) of the VLA, the
the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its
line of sight and that measured along the reference antenna's line of sight, $\delta
\mbox{TEC}$. The estimated uncertainty (see 2.2) is plotted in red in each panel.
For each antenna in the southwestern arm (see Fig. <ref>) of the VLA, the
the difference between the TEC fluctuation (i.e., above or below the mean TEC) along its
line of sight and that measured along the reference antenna's line of sight, $\delta
\mbox{TEC}$. The estimated uncertainty (see 2.2) is plotted in red in each panel.
Upper: The height of the maximum electron density at the latitudes and
longitudes of the ionosphere pierce-points during the
observations as computed by the International Reference Ionosphere (IRI) software as a
function of local time. Middle: For each of the three furthest antennas in the array,
the distance of the ionosphere pierce-point of the antenna from that of the
array center assuming a thin-shell model at the heights plotted in the upper panel
as a function of time. Lower: The multiplicative factor used to correct the
observed slant-$\delta \mbox{TEC}$ values assuming a thin-shell model at the heights
plotted in the upper panel.
The fitted values of the coefficients from equation (6) as a
function of local time relative to midnight for 13 August, 2003.
The projection of the TEC gradient along each VLA arm (see Fig. <ref>) at each antenna as a function of local time (black curves). Plotted in red are the projected TEC gradients computed using the fitted polynomial coefficients plotted in Fig. <ref>. In each panel, the antenna number and the mean separation among the nearest antennas ($\Delta \mbox{r}$) are printed.
A schematic representation (not to scale) of the procedure detailed here for computing the
required geometric corrections. See the text of this appendix for a detailed
discussion of the different components of this schematic. Note, the image used
for Cyg A is a false-color image made with the VLA at 327 MHz <cit.>.
Upper: The estimated apparent velocity of Cyg A at the location within the
thin-shell ionosphere of the array center pierce-point in both the north-south (black)
and east-west (gray) directions as functions of time. Lower: A histogram for the
distribution of azimuth angles measured from north through east for the velocity
vectors plotted in the upper panel.
|
arxiv-papers
| 2012-01-18T18:27:43 |
2024-09-04T02:49:26.440373
|
{
"license": "Public Domain",
"authors": "J. F. Helmboldt, T. J. W. Lazio, H. T. Intema, and K. F. Dymond",
"submitter": "Joe Helmboldt",
"url": "https://arxiv.org/abs/1201.3872"
}
|
1201.3874
|
A New Technique for Spectral Analysis of Ionospheric TEC Fluctuations Observed with the Very Large Array VHF System: From QP Echoes to MSTIDs
J. F. Helmboldt, 1 T. J. W. Lazio, 2 H. T. Intema, 3 & K. F. Dymond, 1
1US Naval Research Laboratory, Washington, DC, USA.
2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA.
3Jansky Fellow of the National Radio Astronomy Observatory, Charlottesville, VA, USA.
We have used a relatively long, contiguous VHF observation of a bright cosmic
radio source (Cygnus A) with the Very Large Array (VLA) through the nighttime,
midlatitude ionosphere to demonstrate the phenomena observable with this instrument. In a companion paper, we showed that the VLA can detect fluctuations in total
electron content (TEC) with amplitudes of $\leq\!10^{-3}$ TECU and can measure TEC gradients with a precision of about $2 \times 10^{-4}$ TECU km$^{-1}$.
We detail two complementary techniques for producing spectral
analysis of these TEC gradient measurements. The first is able to track individual waves
with wavelengths of about half the size of the array ($\sim 20$ km) or more.
This technique was successful in detecting and characterizing many medium-scale
traveling ionospheric disturbances (MSTIDs) seen intermittently throughout the
night and has been partially validated using concurrent GPS measurements.
Smaller waves are also seen with this technique at nearly all times,
many of which move in similar directions as the detected MSTIDs. The second
technique allows for the detection and statistical description of the
properties of groups of waves moving in similar directions with wavelengths
as small as 5 km. Combining the results of both spectral techniques, we found
a class of intermediate and small scale waves which are likely the
quasi-periodic (QP) echoes that have been observed to occur within
sporadic-$E$ ($E_s$) layers. We find two distinct populations of these waves.
The members of one
population are coincident in time with MSTIDs and are consistent with
being generated within $E_s$ layers by the $E$–$F$ coupling instability.
The other population seems more influenced by the neutral wind, similar to the
predominant types of QP echoes found by the Sporadic-$E$ Experiments over Kyushu <cit.>. We have
also found that the spectra of background (i.e., isotropic) fluctuations
can be interpreted as the sum of two turbulent components with maximum
scales of about 300 km and 10 km.
§ INTRODUCTION
Ground-based remote sensing of the ionosphere is a rich field of study that has developed over several decades and now includes a host of different instruments including, but not limited to, GPS, CERTO [Bernhardt and Siefring, 2006], and other radio beacon receivers, ionosondes, and radar/HF arrays. A powerful yet relatively underused resource for remote sensing are radio-frequency arrays, particularly those that operate in the VHF regime. Designed as radio synthesis telescopes, they are chiefly used to observe cosmic sources. While such observations require detailed calibration schemes to remove the effects of the ionosphere, this calibration data is seldom used to actually study the ionosphere.These interferometers basically measure the time-averaged correlation of complex voltages from pairs of antennas which can be combined with Fourier methods to make relatively high angular resolution images of cosmic sources <cit.>. The correlated signals, or “visibilities” have an extra phase term added to them by the ionosphere proportional to the difference in the total electron content (TEC) between the two antennas' lines of sight. Because of the relatively large collecting area of the individual elements (usually dishes, dozens of meters across), and the brightness of many cosmic sources, these additional phase terms can typically be converted to differential TEC ($\delta \mbox{TEC}$) measurements with a precision of $\sim\!10^{-3}$ TECU or better. In addition, the range of scales that can be probed with such interferometers (dozens of meters to hundreds of kilometers) and the virtual ubiquity of target sources available on the sky make them valuable assets for the exploration of ionospheric dynamics from very fine to medium size scales.Consequently, radio arrays have been used, to a somewhat limited degree, for ionospheric studies. Because of the relative stability of its electronics, quiet radio frequency interference (RFI) environment, and available VHF system (bands at 74 and 330 MHz), the Very Large Array (VLA) has been used almost exclusively in these efforts. Seminal experiments were performed by Jacobson and Erickson, 1992, Jacobson and Erickson, 1992 using 330 MHz VLA observations of several sources to explore the environment of ionospheric waves above the VLA. Among other results, they discovered a new class of magnetic eastward directed waves that were later shown to actually be located within the plasmasphere [Hoogeveen and Jacobson, 1997]. Subsequent larger-scale investigations using data from the VLA Low-frequency Sky Survey <cit.>, a survey of the northern sky at 74 MHz, have shown that the median behavior of ionospheric fluctuations observed by the VLA over several years is essentially consistent with turbulence [Cohen and Röttgering, 2009]. Joint observations made with an all-sky optical camera and the VLA at 74 and 330 MHz by [Coker et al., 2009] demonstrated the VLA's ability to aid in the exploration of the interaction between gravity waves generated lower in the atmosphere and relatively small-scale phenomena such as sporadic-$E$. Recently, from a campaign using the VLA and the COSMIC satellite, Dymond et al., 2011 were able to detect a relatively rare instance of a southeastward-propagating traveling ionospheric disturbance (TID).In a companion paper [Helmboldt et al., 2011], we described in detail the methods for extracting ionspheric information from the calibration of VLA VHF system data. We demonstrated that when observing a bright cosmic source, the VLA VHF system was capable of achieving a typical $\delta \mbox{TEC}$ precision of $3 \times 10^{-4}$ TECU. Since arrays such as the VLA are essentially only sensitive to the TEC gradient, we also detailed techniques for measuring time series of the TEC gradient to a precision of about $2 \times 10^{-4}$ TECU km$^{-1}$. Here, we seek to develop this effort further by presenting new spectral analysis techniques for these TEC gradient measurements. We will demonstrate that these techniques are capable of detecting and characterizing several phenomena from medium ($\sim\!100$ km) to small ($\sim\!5$ km) scales while also providing a broader statistical description of the spectrum of TEC fluctuations observable with the VLA.
§ OBSERVATIONS AND TEC GRADIENT MEASUREMENTS
The data used in this analysis came from a roughly 12-hour VLA observation of one of the brightest cosmic radio sources, Cygnus A (or “Cyg A”; also known as 3C405) from a latitude and longitude of $34^{\circ} \: 04' \: 43.497''$ N and $107^{\circ} \: 37' \: 05.819''$ W (VLA program number AK570). The observations were conducted during the night of 12-13 August, 2003 simultaneously at 74 and 327 MHz, both operating in dual polarization. The VLA was in its “A” configuration spanning a circle with a diameter of about 40 km. The observations consisted of a 35-minute block of time, or “scan,” and a second 12-hour scan with a temporal sampling of 6.67 seconds. There was a moderate amount of geomagnetic activity ($K_p$ index $\approx \! 2$–4) and the amount of solar activity was fairly average ($F10.7\!=\!123$ SFU).Measurements of the TEC gradient toward Cyg A are described in detail in Helmboldt et al., 2011. In short, the ionosphere adds an extra phase term to the complex visibility measure for each pair of antennas which is proportional to the difference in the TEC along the two lines of sight, or $\delta \mbox{TEC}$. Thus, the interferometer is sensitive only to TEC gradients and not absolute TEC. However, the Y-shape of the array (northern, southeastern, and southwestern “arms”) makes measuring and performing Fourier inversions of the TEC gradient difficult. This is in contrast to the standardized procedure of inverting visibilities to produce images of cosmic sources. This is because the visibilities are functions of the differential antenna positions and the TEC gradient is a function of the actual antenna positions projected through the ionosphere. This is discussed in more detail by Helmboldt et al., 2011.Because of the consequences of the Y-shape of the VLA, Helmboldt et al., 2011 established two ad hoc approaches to measuring the TEC gradient. The first method estimates the full two-dimensional gradient at each antenna using a second-order, two-dimensional polynomial (Taylor series) on each time step. While insensitive to fine-scale structure significantly smaller than the array, Helmboldt et al., 2011 showed that this method is effective in recovering the gradient associated with relatively large, long-period disturbances visible at dusk, dawn, and intermittently throughout the night.The second method numerically computes the projection of the TEC gradient along each of the VLA arms at each antenna and time step. While little directional information can be obtained with these measurements, Helmboldt et al., 2011 showed that there are many significant fine-scale fluctuations, especially near midnight local time, that are detected with this method and missed/damped within the polynomial-based method. In the following section, we will detail the techniques we have developed to spectrally analyze the TEC gradient time series produced by these two methods.
§ SPECTRAL ANALYSIS
§.§ Method
To identify individual disturbances or sets of disturbances that passed over the array, we
have developed the following Fourier-based method which uses the polynomial-based TEC gradient measurements. This method is partially based
on that used by Jacobson and Erickson, 1992, but expands upon it to include some allowance
for wavefront distortions and multiple wave fronts observed simultaneously.The basic concept is to use both the apparent motion of Cyg A and the movements of the observed ionospheric disturbances to increase the spatial coverage of the VLA. In other words, we will essentially convert temporal baselines into spatial ones, sampling the observed phenomena with a relatively long, 40 km-wide “strip.” Since there are presumably several disturbances contributing to the TEC gradient an each time step with a range of speeds, the temporal sampling cannot be converted into the same set of spatial samples for all observed phenomena. Instead, we must separate the time series for each antenna into different Fourier modes and then analyze how the properties of each mode vary across the array to estimate the wavelength, speed, and direction of the contributing pattern(s).To achieve this, we first assumed that the TEC fluctuations could be
approximated as the sum of many individual oscillating modes, each having the
following form
\begin{equation}
\mbox{TEC}(t) = A(r_{\perp}) \mbox{exp} \left [ i \left ( - \vec{k} \cdot \vec{r} + 2 \pi \nu t \right ) \right ]
\end{equation}
where $\vec{k}$ and $\vec{r}$ are the wavenumber and position vectors, respectively,
$A(r_{\perp})$ is a complex amplitude which only varies perpendicular to $\vec{k}$
(i.e., wavefront distortions), and the temporal frequency, $\nu$, is assumed to be
$=v |\vec{k}|/(2 \pi)$ where $v$ is the wave speed.
For a single Fourier mode with temporal frequency, $\nu$, the Fourier transforms of the $x$ (N-S) and $y$ (E-W) partial
derivatives are given by
\begin{eqnarray}
F_{x}(\nu) \simeq \left [ i k_{\nu , x} + \frac{\partial}{\partial x} A_{\nu}(r_{\perp}) \right ] \mbox{e}^{- i \vec{k_{\nu}} \cdot \vec{r}} \\
F_{y}(\nu) \simeq \left [ i k_{\nu , y} + \frac{\partial}{\partial y} A_{\nu}(r_{\perp}) \right ] \mbox{e}^{- i \vec{k_{\nu}} \cdot \vec{r}}
\end{eqnarray}
From these equations, one can see that along a line parallel to $\vec{k}$, the $x$ and
$y$ dependences of the phases of $F_x$ and $F_y$ are identical since both
partial derivatives of $A_{\nu}(r_{\perp})$ are constant along this line. There
may be other lines in the $x,y$ plane where this is true. For instance, in
the absence of significant wavefront distortions, the $x$ and $y$ dependencies will
be the same everywhere. However, in the general case, the line parallel to
$\vec{k}$ is the only one where these dependencies are required to be equal according
to equations (2) and (3). We have exploited this fact in
our analysis to help determine the direction of $\vec{k}$ for different Fourier modes
and consequently, the wavenumbers and velocities.To illustrate this better, we have displayed an example of a wave, including wavefront distortions, in Fig. <ref>. If one imagines this wave drifting over the VLA at a constant speed, one can see that the time series measured at two separate points will have identical periods, but will be out of phase. For a constant speed, this phase difference will simply be proportional to the length of the separation between the points. However, if one examines the N-S and E-W partial derivatives of this wave along different position vectors, one can see that the wavefront distortions add additional variations in phase. We show this in the lower panels of Fig. <ref> where we have plotted the N-S and E-W partial derivatives, $F_x$ and $F_y$, along three different vectors shown on the image of the wave with color-coded arrows. In all cases, $F_x$ and $F_y$ are out of phase with one another. This phase difference is constant for the vector that is parallel to the wavenumber vector but varies with distance along the other two vectors. Therefore, when wavefront distortions are present, the time series phases will have the same spatial derivative for both $F_x$ and $F_y$ only when points along the direction of the wave are considered.While this derivation focuses on what one should expect for a single wave pattern, we must recognize that at a given temporal frequency, $\nu$, there will likely be contributions from several waves. This can result from several factors including the finite size of the temporal window used for the Fourier inversion to the existence of phenomena such as turbulent cascades which have distributions of spectral power that span a large range of frequencies. It this case, wave properties derived for a single mode using equations (2) and (3) will be weighted mean[Since the wave properties are derived from the phases of the combined Fourier transform of several phenomena, the derived wavenumbers and pattern speeds are not strictly weighted mean values but are proportional to the phases of the weighted mean Fourier transform of the phenomena.] values with more weight given to larger amplitude fluctuations. In addition, even for a single dominant wave, the actual wavenumber and speed for a particular value of $\nu$ may change over the span of time used in the Fourier inversion. Therefore, one should take the measured values for these properties to be averages over either the temporal window used for the Fourier transforms or the duration of the disturbance(s), whichever is shorter.Thus, we may identify instances where single dominant waves are present by deriving wave properties using equations (2) and (3) for several temporal frequencies at each time step. This can be done using Fourier transforms computed with a “sliding” window, one that is relatively wide but centered on each individual time step. The derived spectral power, direction, and speed can then be examined as functions of time and $\nu$. Any relatively large regions in the $t$,$\nu$ plane with relatively large power and uniform direction and speed are likely instances of dominant waves. The wave properties measured in all other locations in the $t$,$\nu$ plane are likely composite values from several waves and can be used to study wave properties on a statistical basis.
§.§ Derived Wave Properties
Using the polynomial-based TEC gradient fits, we calculated $F_x$ and $F_y$ from equations (2) and (3) as functions of time by computing the
discrete Fourier transform (DFT) of each polynomial coefficient within one-hour sliding
windows for frequencies up to 12 hr$^{-1}$ (or, periods $>5$ minutes). We chose
the one-hour
width because it was the same used to de-trend the $\delta \mbox{TEC}$ data
<cit.>. Because of the one-hour box width, we excluded the first 35 minute
scan from this analysis. We also only computed $F_x$ and $F_y$ for times ranging
from one half hour after the beginning of the second scan to one half hour before the
end of the observing run so that a full hour of data could be used for each time
step.The DFT of each polynomial coefficient was then used to compute $F_x$ and
$F_y$ at the locations of the antennas where the values of the polynomial fits
are most reliable. Upon inspection of this data, we found that the phases of
both $F_x$ and $F_y$ at a single value of $\nu$
were typically well approximated with planes across the array. Because of this,
at each time step and value of $\nu$, we fit planes to both these quantities.
We then checked the combined (in quadrature) rms
scatter about both of these fits against the rms difference between the phases
of $F_x$ and $F_y$ to see if the $x$ and $y$ dependencies of these
phases indeed differed across the array, indicating significant wavefront
distortions. In the vast majority ($>\!97$%) of cases, the rms difference between the
phases of the two partial derivative DFTs was larger than
the combined rms scatter about the two planar fits (a median of four times larger).
In these cases, we assumed that
the azimuth angle (measured clockwise from north) of the line where the two planes
intersected was the azimuth angle of $\vec{k}$ and that the derivative of the
DFT phase along this line gave $| \vec{k} |$. In the rare instance that there
was no evidence of wavefront distortions, we computed a mean $x$ and $y$ slope
for the DFT phases from the two planar fits and computed the direction and magnitude
of $\vec{k}$ from these slopes.The upper panel of Fig. <ref> displays the total power of the TEC
gradient in the direction of $\vec{k}$ as a function of local time and temporal
While seemingly random fluctuations are frequently apparent,
there are several instances where one can see significant
detections of waves above the background. Most of these waves are seen between
frequencies of 1 and 5 hr$^{-1}$, or periods of 12 to 60 minutes. This is roughly
consistent with what is typical for MSTIDs. To isolate these detections, we computed
a mask by measuring the median and median absolute deviation (MAD) within elliptical “annuli” around each
pixel that had dimensions of 1.5 hr in local time by 3 hr$^{-1}$ in $\nu$. Any pixel with a
value more than three times the MAD above the median for its annulus was considered
a detection above the background. We used this mask to display both the azimuth
angle for $\vec{k}$ and the wave speed for significant
detections in the middle and lower panels of Fig. <ref>, respectively.The plots in Fig. <ref> imply that the candidate MSTIDs, i.e. fluctuations
with periods $\sim 20$ minutes or greater, have estimated speeds between 100 and
200 m s$^{-1}$, consistent with typical MSTIDs detected at night during northern
hemisphere summer [Hernández-Pajares et al., 2006].
They also appear to move roughly westward/northwestward before midnight and toward the northeast after midnight. While Tsugawa et al., 2007 showed that summer nighttime MSTIDs in the northern hemisphere predominantly move toward the southwest, Hernández-Pajares et al., 2006 showed that near the west coast in California, they move almost due west and occasionally toward the northwest. Thus, the waves observed before midnight are somewhat typical MSTIDs. However, the northeastward directed waves seen after midnight are unusual and will be discussed in more detail below.
§.§ Mean Power Spectra
While the results displayed in Fig. <ref> are useful for identifying and examining instances of waves or groups of waves, the polynomial-based analysis can also be used to produce a more statistical description of the observed set of TEC fluctuations.
In Fig. <ref>, we have plotted the mean power within bins of
wavenumber, $k$, for one-hour blocks of time. We note that these wavenumbers
have not been corrected for any Doppler shifts caused by the motion of
the ionosphere pierce-points as the VLA tracked the apparent motion of Cyg A through the sky. However,
the results shown in Appendix A of Helmboldt et al., 2011 demonstrated that the equivalent velocities
at the estimated peak heights for the pierce-points are relatively small
($\sim 20 \mbox{ m s}^{-1}$), except near the beginning and end of the observing
run when Cyg A was rising and setting. At these times, the main features
detected in the data shown in Fig. <ref> generally move roughly
perpendicular to the pierce-point motions, implying that the Doppler shifts
were relatively small during these times as well.We have also plotted in Fig. <ref> what
we will refer to as “noise-equivalent” spectra. These spectra were
computed by performing the polynomial fits to each of the four
$\delta \mbox{TEC}$ measurements (two bands and two polarizations) and
computing the DFT of the difference between those fits and the fits to
the final $\delta \mbox{TEC}$. The wave analysis detailed above was then
applied to these residual DFTs to estimate the results one would achieve
if analyzing fluctuations that were simply noise. The noise-equivalent
spectra plotted in Fig. <ref> are average spectra computed
within each one-hour block and among the two bands and two polarizations.These spectra imply that this technique can indeed detect significant
power of structures as small as about half the size of the array (or,
$k \approx 0.3 \mbox{ km}^{-1}$). The MSTIDs visible in the upper panel
of Fig. <ref> are visible in most of the panels as significant
bumps above the background at wavenumbers ranging from 0.015–0.05 km$^{-1}$,
or wavelengths of about 130–500 km, which is again roughly consistent
with the known properties of MSTIDs. Few if any features are see at
higher wavenumbers where the spectrum ranges from a simple power-law,
reminiscent of turbulence, to relatively flat spectra, which may imply a
population of intermediate-scale waves.To explore these data further, we have computed mean power spectra
within bins of $\vec{k}$ azimuth angle. We have displayed these as
images in Fig. <ref> where we have divided the data
within the same wavenumber bin by the total power within that bin
over all azimuth angles
to enhance any detected features at higher wavenumbers.
These images reveal that there are often intermediate-scale waves
($k>0.1 \mbox{ km}^{-1}$, or wavelengths $<\!60$ km) detected moving in
some preferred direction.
Often, these waves appear to be moving in directions similar to
detected MSTIDs. This is especially apparent in the first three panels
where one can see that even when the MSTIDs have disappeared by a
local time of $-04^{\mbox{\scriptsize h}}$, the roughly northward moving
intermediate-scale waves persisted. The presence of strong MSTIDs do
not appear to be required for these intermediate-scale waves as they
seem to exist during all time periods. However, the MSTIDs broadly
appear to have the effect of “focussing” the intermediate-scale
waves into a particular direction.Those smaller-scale waves which are directed parallel to coincident MSTIDs may simply be small-scale structures within the MSTIDs themselves. Indeed, our spectral analysis method has indicated that significant wavefront distortions are common within the VLA data. However, the
smaller waves which do not travel in the same direction as MSTIDs may be located within the $E$ region, most likely within sporadic-$E$ ($E_s$) layers given the prevalence of this phenomenon during summer nighttime. In fact, Coker et al., 2009 showed that $E_s$ layers are likely the source of small-scale TEC fluctuations often observed with the VLA during summer nighttime. In addition, $\sim\!10$-km-sized wave-like fluctuations have frequently been observed within $E_s$ layers as so-called quasi-periodic (QP) echoes <cit.>.
If so, these waves may yield further insights into the coupling
between the midlatitude nighttime $E$ and $F$ layers
<cit.> as the MSTIDs are likely located within the
lower $F$ region <cit.>. This will be discussed further in 5.
§.§ Contemporaneous GPS and Ionosondes Data
§.§.§ GPS Data
To partially validate our technique for detecting and characterizing waves, we
have analyzed all available GPS data in the area for the time of our observations.
From 12–13 August, 2003, there were six dual frequency GPS receivers operating within
the state of New Mexico (station codes AZCN, NMSF, PIE1, SC01, TCUN, and WSMN)
with publicly available data in RINEX format. In contrast, there are currently more than 30 such stations in operation which bodes well for future studies conducted with the new VHF system being developed for the Expanded VLA (EVLA). Data for each of the six stations operating in 2003
were obtained and processed with standard GPS Toolkit [Tolman et al., 2004]
routines ( and )
to produce slant and vertical (for a height of 400 km) relative TEC values (i.e.,
no bias correction was done) for all available satellites.For each
station/satellite pair, the slant TEC data were de-trended according to
Hernández-Pajares et al., 2006 where they de-trended similar data by subtracting from each
time step the average of two values, each an interval $\tau$ before or after
the time step. They noted that for a wave with a period $T$, this technique
effectively multiplies the wave amplitude by a factor of
$2 \: \mbox{sin}^2(\pi \tau / T)$. They chose a value of $\tau = 300$s
based on physical arguments of the expected periods of MSTIDs. We have
instead chosen $\tau = 600$s so that the de-trending technique was optimized
for waves with periods similar to the MSTIDs detected within our
VLA data.After the GPS data were de-trended, they were corrected to
vertical TEC values using the corrections computed by within GPS Toolkit.
Within two-hour blocks, starting at a local time (relative to midnight, 13 August)
of $-06^{\mbox{\scriptsize h}}$,
we then identified station/satellite pairs which had relatively contiguous
data ($>\!90$% of the time interval covered) from one half hour before and one
half hour after the time block. We then computed the DFT of the de-trended
TEC data within a one-hour
sliding window to produce a spectrum at each time step within the two-hour
block.These spectra are displayed in Fig. <ref> within each two-hour block for the 12 station/satellite pairs with ionospheric “pierce-points” closest to that associated with Cyg A. Nearly all of these spectra show some level of wave activity with periods of about 12–60 minutes ($\nu\!=\!1$–5 hr$^{-1}$). Those station/satellite pairs with pierce-points closest to that of Cyg A in particular show very good agreement with the VLA-derived spectrum displayed in the upper panel of Fig. <ref>. This is especially true for the top three rows of Fig. <ref> for local times of $-06^{\mbox{\scriptsize h}}$ to $-04^{\mbox{\scriptsize h}}$ and $01^{\mbox{\scriptsize h}}$ to $03^{\mbox{\scriptsize h}}$.Unfortunately, the spacing among the GPS stations was too large to be
able to track individual waves. The closest separations among pierce-points
is $>\!100$ km when separations on the order of tens of kilometers are required
to overcome the effects of wavefront distortions <cit.>. This makes it difficult to use the GPS data to
directly validate the directional information we have determined for the
VLA-detected waves displayed in the middle panel of Fig. <ref>. However, the fact that the spectra shown in Fig. <ref> and <ref> observe many of the same waves is encouraging and at least partially validates our VLA-based approach.
§.§.§ Ionosondes Data
While there were no ionosondes operating in the immediate vicinity of the VLA, there were two relatively nearby at similar latitudes, one at Dyess Air Force Base (AFB) in west Texas ($32^{\circ} \: 30''$ N; $99^{\circ} \: 42'$ W), and one at Point Arguello in California ($35^{\circ} \: 36''$ N; $120^{\circ} \: 36'$ W). The contemporaneous data from these stations are useful for interpreting our VLA observations.For instance, northward or northeastward-directed MSTIDs, similar to those observed here at $01^{\mbox{\scriptsize h}}$–$03^{\mbox{\scriptsize h}}$ and $04^{\mbox{\scriptsize h}}$–$04^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$, have been observed over Japan with the SuperDARN Hokkaido HF radar and an airglow imager by Shiokawa et al., 2008. Like the northeastward-directed MSTIDs described here, these were detected at midlatitudes during nighttime. Shiokawa et al., 2008 noted that the roughly northward-directed MSTIDs were observed with the SuperDARN array most frequently in May and August. They also noted that the northeastward-directed waves were coincident in time with a drop in the height of the F-region as indicated with nearby ionosondes data.To demonstrate that this may also be the case for our VLA observations, we have plotted h$^\prime$F for both the Dyess AFB and Point Arguello ionosondes as functions of VLA local time in the upper panel of Fig. <ref>. The Point Arguello data show a substantial drop in h$^\prime$F at around $01^{\mbox{\scriptsize h}}$ at the start of the first observed instance of northeastward directed waves. The Dyess AFB data shows a decline in h$^\prime$F which begins around $02^{\mbox{\scriptsize h}}$ and reaches a minimum at about $04^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$, corresponding to the middle of the second period of northeastward MSTID activity. This indicates that it is plausible that we have observed a similar phenomenon with VLA over New Mexico as was found by Shiokawa et al., 2008 over Japan.The ionosondes data also serve to validate our claims that $E_s$ layers were present during our observations. In the lower panel of Fig. <ref>, we have plotted foEs for both ionosondes stations. Both stations observed E-region reflections with maximum frequencies between 2 and 6 MHz throughout the night, strongly indicating that $E_s$ layers were present. The most active time for the appearance of these layers seems to be between $-02^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ and $04^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ VLA local time, especially for the Point Arguello station. As evidenced by Fig. <ref> and <ref>, relatively small-scale fluctuations are visible within the VLA data throughout this time period. They are especially prominent from $-02^{\mbox{\scriptsize h}}$ to $01^{\mbox{\scriptsize h}}$ where their presence has made the mean spectra shown in Fig. <ref> virtually flat for wavenumbers $>\!0.13$ km$^{-1}$.
§ STATISTICAL WAVE PROPERTIES
§.§ Arm-based Spectra
While the Fourier analysis detailed in 3 has yielded interesting results,
it neglects the ability of the VLA in its A configuration to detect
fluctuations on scales as small as a few kilometers. To make full use
of the data, we have developed a complementary spectral analysis which
uses the projected TEC gradients measured along each arm which are sensitive to fluctuations with sizes as small as the shortest antenna spacings within the VLA.To analyze the projected gradients in a manner similar to the method used for the polynomial-based gradients, we first performed fast Fourier transforms (FFTs) of the times series for each antenna. This was done within a sliding window
of approximately one hour (512 time steps, or 56.9 minutes) centered on each time step. Then, for a single arm, time step, and temporal frequency,
we unwrapped the
phases of the FFTs along the arm and numerically computed the
derivatives of these phases with respect to distance along the arm
to obtain estimates of the (mean) projected wavenumber, $k_{proj}$, of each Fourier mode.
As noted in 3, our Fourier analysis of the polynomial fits revealed
that significant wavefront distortions were common. Because of this,
we expect that these $k_{proj}$ estimates are not reliable
for tracking individual disturbances along any of the arms. However, since the effects of
such distortions should average to zero over time, these estimates
are sufficient to perform a statistical analysis of the overall
population of waves seen by the VLA from a few to hundreds of
kilometers in wavelength.To begin such a statistical analysis, we binned the FFT and $k_{proj}$
data within one-minute intervals and computed the mean FFT power within
bins of $k_{proj}$ for each arm. The resulting spectra are displayed
in Fig. <ref> where positive projected wavenumbers correspond to directions outward from the center of the array, or “up” the arm and
negative projected wavenumbers correspond to inward directions, or “down” the arm.
Since these are projected wavenumbers, instances where there are
groups of waves moving in a particular direction will show up in
these plots as over-densities in the positive (negative) direction
for one arm, and in the negative (positive) direction for the other
two. In cases where a group of waves moves nearly perpendicular to
a particular arm, they will not show up at all in that arm's spectrum.
A good example of this can be seen between roughly $02^{\mbox{\scriptsize h}}$
and $03^{\mbox{\scriptsize h}}$ local time (highlighted with vertical white lines in the panels of Fig. <ref>). Here,
there is an obvious over-density on the positive side of the spectrum
for the southwestern arm and mild over-densities on the negative sides
of the spectra for the northern and southeastern arms. This implies
that these fluctuations are moving almost directly toward the southwest.
Part of this group of waves can also be seen the panel of the upper
right corner of Fig. <ref>, but in Fig. <ref>,
they can be seen
to extend to wavenumbers of at least 0.7 km$^{-1}$, or wavelengths
at least as small as 9 km.To establish the minimum scales on which this analysis can be
considered reliable, we have also computed noise-equivalent spectra
using the full two-band, two-polarization data as we did with the
polynomial-based Fourier analysis in 3. We have plotted in
Fig. <ref> mean spectra for each arm within one hour
blocks of time along with the corresponding noise-equivalent
spectra. In some instances, the spectra are always above the
noise-equivalent spectra. However, in general, the spectra reach
the noise level at projected wavelengths of about 5 km, or
$k_{proj}= \pm 1.26 \mbox{ km}^{-1}$ (see the vertical dotted lines in
Fig. <ref>). This is in keeping with the mean antenna
separation along the arms of about 2.5 km, implying an approximate
Nyquist sampling limit of 5 km. Because of this, we consider these
spectra most reliable for $|k_{proj}|\!<\!1.26 \mbox{ km}^{-1}$.We note that for most of these one-hour mean spectra, the relatively small-scale ($|k_{proj}|\!>0.1$) portions of the spectra that are above the noise level are roughly consistent with what is expected for $E_s$ layers. From the data presented by Coker et al., 2009, a typical time series of $E_s$ activity would yield values for the spectral power ranging from about $10^{-4}$ up to $10^{-3}$ mTECU$^2$ km$^{-2}$ hr$^2$. This is nearly exactly the range inhabited by the intermediate/small-scale regions of the one-hour mean spectra plotted in Fig. <ref>. In addition, the contemporaneous ionosondes data presented in 3.3.2 and Fig. <ref> show that $E_s$ layers were likely present during the VLA observations.For the remainder of the analysis, we found it useful to define
three size classes for the detected waves: (1) medium scale,
$|k_{proj}|<0.1 \mbox{ km}^{-1}$ which includes the full range of
MSTID sizes, (2) intermediate scale, $0.1<|k_{proj}|<0.3 \mbox{ km}^{-1}$,
which represents the remaining range of scales one can probe with
our previous polynomial-based approach, and (3) small scale,
$0.3<|k_{proj}|<1.26 \mbox{ km}^{-1}$ representing the scales that
cannot be probed with the polynomial fits.We have re-displayed the arm-based spectra in Fig. <ref>
with these three size classes marked. In this representation, we
have divided the spectra by an estimate of the spectra of background
fluctuations, i.e., those that do not move in a preferred direction.
We have estimated this background at each value of $|k_{proj}|$ and time step using a combination of the spectra from all three arms. In this process, only one of the two directions, up or down the arm, was used for each arm. In each case, the direction with the lowest power was used as it was more likely to be dominated by isotropic fluctuations. The median power among all three arms (within $\pm 1$ pixel in both time and $|k_{proj}|$) was then taken to be the value for the background. For displaying purposes only, we also applied a three-pixel
square median filter to the arm-based spectra and divided these by
the estimated background spectra and have shown them in Fig.
<ref>. In this representation, one can see that groups of
waves at virtually all detectable scales are seen moving in a variety
of preferred directions at different times.
§.§ Group Wave Properties
To exploit the arm-based spectra described above and displayed in Fig.
<ref>–<ref> to identify groups of waves moving in
a particular direction, we have developed the following method. To
begin, we computed a weighted mean value of $k_{proj}$ for each size
class, arm, and one-minute interval using the spectral power as the weight.
If there is a significant population of waves within any of the three
classes at a particular time step moving in a similar direction,
these weighted mean values will follow a cosine as a function of the
azimuth angle of the arms which will peak in the direction of the true
wavenumber vector, $\vec{k}$.To detect such groups of waves, we have
fit a simple cosine model to the weighted mean $k_{proj}$ values
at each time step, including the nearest two time steps so that the
fit would be better constrained and so some level of temporal
coherence would be imposed. We then compared the amplitude of the
fitted cosine with the rms scatter about the fit among the nine
values used (i.e., three arms and three time steps) to assess the
significance of the detection of a group of waves. We have plotted
the azimuth angles for $\vec{k}$ for both $3\sigma$ and $5\sigma$
detections for each of the three size classes in the panels of
Fig. <ref>. The $5\sigma$ detections for the medium
and intermediate scale waves have been re-plotted in the panels
of the smaller size class(es) for comparison. For reference,
we have also shaded in grey time ranges where MSTIDs where
detected using the polynomial-based technique as described in 3.In general, the $\vec{k}$ directions for the medium-scale class
agree with those seen for the detected MSTIDs displayed in Fig.
<ref>, which serves to partially validate our arm-based
method. Most of the time, the azimuth angles for the intermediate
and small classes roughly agree, indicating that they are likely
part of the same distribution of waves. Both often agree with the
azimuth angles determined for the medium class, but are at times
significantly different, most notably during two instances where
MSTIDs were detected.We have also determined approximate speeds for all of the detected
groups of waves which we have plotted in Fig. <ref>
with their values of
$|\vec{k}|$ taken from the above described cosine fits. The speeds
were computed using the mean temporal frequency within bins of
$k_{proj}$ that were within $\pm 0.1$ km$^{-1}$ for the small class
and $\pm 0.05$ km$^{-1}$ for the intermediate and medium classes
of the cosine fit-determined value of $|\vec{k}|$ from each arm. A spline fit
to the data as a function of arm azimuth angle was then used to
estimate the mean temporal frequency along the direction of $\vec{k}$
where waves, rather than background fluctuations, are more likely
to dominate. The speed was then
computed using this frequency and the value of $|\vec{k}|$ from the
cosine fit. The speeds agree in general with what can be seen in
Fig. <ref>, that the medium-scale class of waves are a
somewhat distinct population in terms of direction and speed whereas
the intermediate and small scale classes tend to have similar
directions and speeds.
§.§ Small-scale Waves
To further explore the nature of the small-scale waves in particular,
we have plotted the distribution of azimuth angles for the $5\sigma$
detections in Fig. <ref> for the small-scale class only.
From this it can be seen that the distribution has four distinct
peaks at angles of roughly $-155^{\circ}$, $-135^{\circ}$, $55^{\circ}$,
and $120^{\circ}$. For reference, we have also plotted the azimuth angles of the three VLA arms (both up and down the arms) as vertical dashed lines. From this, one can see that the group of waves with a peak azimuth angle of $55^{\circ}$ were moving almost directly up the southwestern arm. This indicates that this group of waves is likely small in extent as well as in wavelength. This is because for groups of waves substantially smaller than the array, the arm-based method we described above will be biassed toward waves traveling along one of the three arms because the number of antennas that “see” these waves will be maximized in this case.In contrast, groups of waves which span all or nearly all of the array will essentially be seen by all the antennas for any direction and will experience no such bias. This appears to be the case for the group of waves moving roughly southeastward and for the waves with a peak azimuth angle of $-155^{\circ}$. The peak of the distribution for the southeastward-directed waves is somewhat skewed toward the azimuth of the southeastern arm. However, the bulk of these waves have azimuth angles between that of the southeastern (up) and northern (down) arms, indicating that any influence of the arm-based bias on these waves is minimal. The group of waves moving toward an angle of $-135^{\circ}$ peak near but not at the azimuth angle of the southwestern arm, indicating that their measured distribution has been influenced by the arm-based bias, but to a lesser degree than those whose distribution peaks near $55^{\circ}$.Because of this bias, the paucity of waves directed either due west or east only indicates that there were few groups of small-wavelength waves moving in these directions which also spanned the array. However, the lack of small-scale waves seen directed northwest, due north, or due south (i.e., up the southeastern arm or up/down the northern arm) indicates a genuine absence of such phenomena and demonstrates that the population of small-scale waves was far from isotropic.In terms of the implied orientation of the wavefronts,
the peaks at $-135^{\circ}$ and $55^{\circ}$ are essentially the same
since they are roughly $180^{\circ}$ apart. They are both conspicuously
close to the required orientation for the $E_{s}$ layer instability
(or, “$E_s$LI”) described in detail by Cosgrove and Tsunoda, 2002. They
demonstrated that an $E_s$ layer is unstable against perturbations
with wavefronts aligned from northwest to southeast. In particular,
for no meridional wind component, the optimum orientation is 35$^{\circ}$
west of magnetic north. For the VLA, where the magnetic declination
is about 10$^{\circ}$, this corresponds to $\vec{k}$ azimuth angles
of either $-115^{\circ}$ or $65^{\circ}$. If there is a significant
meridional wind component, the optimum angle can be as much as
$15^{\circ}$ closer to due south, giving a range of optimum position
angles of $-130^{\circ}$ to $-115^{\circ}$ or $50^{\circ}$ to $65^{\circ}$.
This is roughly consistent with the locations of the two largest
peaks in Fig. <ref>. However, the exact peak azimuth angles of these two distributions are not precisely constrained given the bias toward the direction of the VLA arms described above. We can only say that they are generally moving either northeast or southwest.To illustrate the agreement with the predictions of the $E_s$LI model, we have
computed the azimuth angle (at the VLA) dependence of the $E_s$LI
assuming a magnetic dip angle of $45^{\circ}$ according to Cosgrove and Tsunoda, 2002,
including a range of peak azimuth angles of $15^{\circ}$. We have
plotted a scaled version of the growth rate in Fig. <ref>. From this
curve, one can see that the bulk of the two groups of waves with
azimuth angles near the optimum values for the $E_s$LI are contained
within the regions that are within $\sim 1/2$ of the maximum growth
rate.If we consider only those waves located in time ranges where MSTIDs
were detected, the agreement with the predictions of the $E_s$LI growth
rate are even better (see the red histograms in Fig. <ref>). In
fact, only a small fraction of such waves ($\sim\! 2$%) have azimuth
angles where the $E_s$LI growth rate is $\leq\! 0$. In contrast, for
all waves, about 16% lie outside this region. One can see from the red histograms in Fig. <ref> that the azimuth angle distribution for the southwest-directed waves coincident with MSTIDs is significantly skewed with a tail extending away from the azimuth angle of the southwestern VLA arm. This indicates that the arm-based bias discussed above has a larger effect on these waves than the other southwest-directed waves which are not seen with MSTIDs and have a more symmetric distribution. This further implies that these groups of waves and their northeast-directed counterparts are relatively small in extent as compared to the other detected groups of small-scale waves.We note that small-scale waves were detected during nearly all observed instances of MSTIDs. This indicates that the preference for small-scale waves aligned northwest to southeast during MSTID activity is not simply a product of the arm-based bias. In other words, there is no indication that there were periods of MSTID activity where small-scale waves were not detected because they were not directed along a VLA arm and their extent was too small to be detected with the arm-based method. This preference for northwest to southeast aligned, small-scale waves is again consistent
with the notion that these waves are associated with the $E_s$LI. This is because
according to Cosgrove and Tsunoda, 2004, the coupling between the Perkins instability
in the $F$ region and the
$E_s$LI is itself unstable.The growth rate for the coupled instability
is maximized for
instances where both the $F$ and $E_s$ layer perturbations are aligned
from northwest to southeast. From Fig. <ref>, we can see that
while the small-scale waves are so aligned, the MSTIDs are not necessarily.
However, we note that the range of allowed orientations for the Perkins
instability is much broader than that for the $E_s$LI as it depends
somewhat on the strength and orientation of the $F$ region neutral wind.We also note that models of this instability [Yokoyama et al., 2009, Yokoyama and Hysell, 2010] have
shown that random perturbations within an $E_s$ layer are enough to
form MSTIDs in the lower/bottom part of the $F$ region through the $E$-$F$ coupling
mechanism. The MSTIDs can then, through this same coupling, help
northwest to southeast aligned perturbations grow within the $E_s$
layer. Yokoyama et al., 2009 confirmed that this process works most
efficiently when conditions are right in the lower/bottom part of the $F$ region to
form northwest to southeast aligned MSTIDs. However, they also
showed that while weaker, MSTIDs of different orientations (in their
example, aligned north to south) could be formed through this
process. In this case, the $E_s$ layer waves that formed were
still aligned northwest to southeast due to the more rigid
directional constraints of the $E_s$LI. Therefore, the results
show here are quite consistent with the predictions of $E$–$F$
coupling in the nighttime, midlatitude ionosphere.The group of small-scale waves seen in Fig. <ref> to
have $\vec{k}$ pointed to the southeast cannot be explained
with the $E_s$LI. However, several authors have noted that QP echoes
are often found with no preferred direction, or with directions
inconsistent with the direction-dependent $E_s$LI
<cit.>. Yamamoto et al., 2005 found a tendency
for the fronts of QP echoes to be nearly perpendicular to
the direction of the neutral wind at the height of the $E_s$ layer(s). To
explore this possibility, we have plotted the neutral wind velocity azimuth angle profile
taken from publicly available GPI data-driven runs of the TIEGCM code
for the night of 13 August at $32.5^{\circ}$N and $110^{\circ}$W
at one hour intervals in Fig. <ref>.
In each panel, the $\vec{k}$ azimuth angles of each $5 \sigma$
detection of small-scale waves within the corresponding time bin
is plotted as a vertical dotted line.To form and maintain an $E_s$
layer, a zonal wind shear is required with the westward moving
wind at higher altitudes than the wind moving eastward. In the plots
in Fig. <ref>, this occurs at altitudes where the neutral
wind transitions from moving southeastward at lower altitudes
to moving southwestward at higher altitudes. This usually occurs
at heights between about 100 and 110 km which is typical for $E_s$
layers. The small-scale waves detected with $\vec{k}$ toward the southeast
are then consistent with having wavefronts nearly perpendicular to the
neutral wind at the $E_s$ layer. We also note that
the waves with $\vec{k}$ pointing to the southwest which are not coincident with MSTIDs may also be
consistent with this same scenario, depending on the altitude at which
the proposed
$E_s$ layer has formed.Yamamoto et al., 2005 concluded that these types
of QP echoes may be formed either by interactions with gravity
waves or via the Kelvin-Helmholtz instability. Since these waves were detected during the period of lowest MSTID activity (see Fig.
<ref> and <ref>), $E$–$F$ coupling likely has
little to do with the generation of these structures. Gravity waves or Kelvin-Helmholtz instabilities are much more
likely candidates for their generation mechanism.
§.§ Background Fluctuations
While our arm-based techniques have found evidence for groups of waves moving
in preferred directions on all size-scales, these results do not
exclude the existence of populations of structures moving
in a wide enough range of directions as to appear isotropic within
our analysis. To examine this possibility, we have looked to the background
spectra constructed using the arm spectra described in 4.1.
These spectra should be reasonably free of the presence of directed waves
and will provide insights into the quasi-isotropic set of background
fluctuations. Mean background spectra computed within one hour bins
are plotted in Fig. <ref>.While some of these spectra
may be reasonably approximated with single power-laws, many have the
appearance of “broken,” or two-component power-laws. Upon detailed
inspection, we found that both the inner and outer most regions of
the spectra were roughly $\propto k^{-5/3}$, but with different slopes
and offsets. This is intriguing since this is the dependence one
expects for turbulent fluctuations. In particular, for a thin
shell, Tatarski, 1961 predicted from Kolmogorov, 1941, Kolmogorov, 1941 that the spectrum of
refractive index fluctuations gives a spectrum of wavefront phase
variations $\propto k^{-11/3}$. Since we have examined the gradient
of TEC (or, wavefront phase) fluctuations, the fluctuations have
been effectively multiplied by a factor of $i |\vec{k}|$, and so the
power spectrum is altered by a factor of $k^2$, or $\propto k^{-5/3}$.
Thus, the general shapes of the background spectra suggest they may be
interpreted as the sums of two separate turbulent components.We have consequently fit a two-component turbulent model to each of the
background spectra plotted in Fig. <ref>. To allow
each spectrum to flatten above a maximum turbulent scale-length,
$\lambda_{max}$, we assumed the following form for each component
\begin{equation}
\Psi (k) = \left [ k^2 + \left ( \frac{2 \pi}{\lambda_{max}}\right )^2 \right ]^{- \frac{5}{6}}
\end{equation}
The resulting two-component fits are plotted with the spectra in Fig.
<ref>; the two components are plotted separately as well.
In general, one can see that each spectrum is well represented by
a component with a relatively large $\lambda_{max}$ (blue curves) and
one that flattens at much smaller scales (higher wavenumbers; green
curves). To illustrate this quantitatively, we have plotted in Fig.
<ref> the power at $\lambda_{max}$ (upper panel) and
$\lambda_{max}$ (lower panel) for each of the two components as functions
of local time. For both components, $\lambda_{max}$ varies somewhat with
time, but is typically $\sim 300$ km for the larger-scale turbulent
component and roughly 10 km for the smaller-scale component. While the
power at $\lambda_{max}$ seems relatively stable with time for the
larger component, the small-scale component shows a definite peak at
a local time of about $-00^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$.
This corresponds to the middle of the roughly three-hour lull in
MSTID activity that started around $-02^{\mbox{\scriptsize h}}$. Since we also detected small-scale waves consistent with QP-echoes generated by gravity waves during this time, it is possible that this increase in power is the result of the same gravity waves influencing turbulent processes in the lower ionosphere/thermosphere where the role of ion-neutral coupling is relatively strong.
§ DISCUSSION
Our exploration of a long, VHF observation of Cyg A with the VLA
has successfully demonstrated the power of this instrument to
characterize a variety of transient ionospheric phenomena. For this
observation, the typical $1 \sigma$ uncertainty in the $\delta \mbox{TEC}$
measurements was $3 \times 10^{-4}$ TECU, yielding more than an order of
magnitude better sensitivity to TEC fluctuations than can be
achieved with GPS-based relative TEC measurements
<cit.>. Through our new spectral-based analysis,
we have demonstrated the ability of the VLA to detect and characterize
individual instances of MSTIDs as well as smaller-scale structures
likely associated with $E_s$ layers. We note that since disturbances within the plasmasphere have been observed and in fact discovered with the VLA, it stands to reason that some of the phenomena we have detected may be located within the plasmasphere as well. However, the phenomena discovered by Jacobson and Erickson, 1992 have azimuth angles clustered around $102^\circ$ (i.e., near magnetic east for the VLA), and few if any of the wave-like structures we have detected meet this criterion. Therefore, the vast majority (if not all) of the phenomena we have observed were likely within the ionosphere.Among these phenomena are MSTIDs, small-scale wave-like phenomena consistent with $E_s$ layer disturbances, and turbulent fluctuations. Both the MSTIDs and the small-scale fluctuations were present intermittently throughout the night, and turbulent fluctuations were seen at all times, which is common within VHF observations with the VLA [Cohen and Röttgering, 2009]. The MSTIDs appear to change direction after local midnight from being directed generally westward to being directed toward the northeast (see below). There is also a noticeable change in the turbulent activity near midnight, namely the spectral power of these fluctuations on small ($\sim\!10$ km) scales peaks near midnight and gradually decreases toward dawn.
§.§ West/northwest-directed MSTIDs
Before midnight local time, we observed instances of westward/northwestward moving MSTIDs (see Fig. <ref>) with an additional group moving westward observed briefly near $02^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ (see Fig. <ref>). While atypical for what is observed for most of North America <cit.>, the directions of these waves are similar to what is typically observed near the west coast in California during summer nighttime <cit.>, especially those directed closer to due west.Small-scale waves were detected coincident with nearly all of these MSTIDs moving toward either the northeast or southwest. Given the difference in directions, it is unlikely that these are simply the signatures of wavefront distortions within the MSTIDs themselves. As discussed above, the amplitudes and orientations of these small-scale waves are consistent with them being generated within $E_s$ layers via the $E_s$LI. Coupling between the $E_s$LI and the Perkins instability in the F region <cit.> may be what has influenced the MSTIDs observed near dawn to move closer to northward as they were detected concurrently with northeastward-directed small-scale waves. In contrast, those waves seen near $02^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ were moving closer to due west and were accompanied by small-scale waves moving toward the southwest.
§.§ Northeastward-directed MSTIDs
Unusual, northeastward-directed MSTIDs were observed near $01^{\mbox{\scriptsize h}}$ to $02^{\mbox{\scriptsize h}}$ and $04^{\mbox{\scriptsize h}}$ to $04^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$. As discussed above, Shiokawa et al., 2008 observed similar phenomena over Japan at midlatitudes during the night when the height of the F region was seen to drop significantly. Similarly, we have found evidence within data from relatively nearby ionosondes that suggests that the F region may have experienced a similar drop in height during the instances of northeastward-directed MSTIDs observed with the VLA.Small-scale waves were also observed at the same times as these unusual MSTIDs. They were also mostly directed toward the northeast, suggesting that they may represent small-scale structures within the MSTIDs themselves. However, we note that (1) Fig. <ref> shows that they were moving significantly more slowly than the MSTIDs, (2) the distribution of azimuth angles seen in Fig. <ref> to peak near the azimuth angle of the VLA's southwestern arm indicates a bias that only affects groups of waves that span an area smaller than the array, and (3) toward the end of the night, the small-scale waves change direction toward the southwest while the MSTIDs remained northeastward directed. These factors seem to indicate that the small-scale waves are separate phenomena, similar to the other observed small-scale waves that are consistent with QP echoes generated within $E_s$ layers. It could be that coupling between these disturbances and the larger waves in the F region plays some role in the unusual direction of the observed MSTIDs. In any case, a more thorough, statistical examination of these phenomena seems warranted.
§.§ Southeast/southwest-directed QP Echoes
There are two distinct groups of small-scale waves which were detected with no coincident MSTIDs. They were predominantly directed toward the southeast (from $-02^{\mbox{\scriptsize h}}$ to $01^{\mbox{\scriptsize h}}$) with a shorter period of activity of waves moving toward the southwest (between $-03^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ and $-02^{\mbox{\scriptsize h}}45^{\mbox{\scriptsize m}}$). The southeastward-directed waves in particular seem to be somewhat different from the other observed small-scale waves. Given their orientations, they cannot have been generated via the $E_s$LI. They appear to be moving somewhat faster with
typical speeds of about 150 m s$^{-1}$ while most of the other small-scale waves have speeds between 30 and 100 m s$^{-1}$ (see Fig. <ref>). They also seem to be largely unaffected by the arm-based bias discussed in 4.3, indicating that as a group, the likely span an area larger than that of the array.Despite these differences, the southeastward-directed small-scale waves are generally consistent with QP echoes generated in $E_s$ layers, both in amplitude and wavelength. They are more specifically consistent with the QP echoes observed by the Sporadic-$E$ Experiments over Kyushu <cit.>, which were heavily influenced by the direction of the neutral wind at the height of the $E_s$ layer(s). The shift in direction at around $-02^{\mbox{\scriptsize h}}30^{\mbox{\scriptsize m}}$ from southwestward to southeastward may then be an indication of a change in $E_s$ layer height from above the wind sheer altitude to below it. This may imply the formation/dissipation of $E_s$ layers near this time.In addition, these waves occurred during the peak in small-scale turbulent activity as evidenced by the results shown in Fig. <ref>. This suggests that the same mechanism that generated these QP echoes also caused a significant turbulent disturbance in the lower ionosphere/thermosphere. Since the Rocky Mountains lie largely to the north and northwest of the VLA, this mechanism is likely gravity waves associated with wind flow over the Rocky Mountains.
The authors would like to thank the referees for useful comments and suggestions. Basic research in astronomy at the Naval Research Laboratory is supported
by 6.1 base funding. The VLA was operated by the National Radio Astronomy Observatory which is a facility
of the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
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To illustrate the technique used to extract wave properties described in 3, an example of a wave, including wavefront distortions, that might be observed by the VLA (shown as a red “Y”). The lower panels show the N-S (solid line) and E-W (dashed line) partial derivatives along each of the vectors plotted in the upper panel.
Upper: The power in the direction of the wavenumber vector, $\vec{k}$,
as a function of local time
(abscissa) and temporal frequency (ordinate) of Fourier modes within the $\delta \mbox{TEC}$
data. Middle: The azimuth angle of the wavenumber vector of each Fourier mode with power significantly larger than background levels (see 3.2). Lower: The velocity of each Fourier mode, using the same masking as in the middle panel. In all panels, the white contours are of spectral power.
Within one-hour bins, the mean power as a function of wavenumber, k, for the data
displayed in Fig. <ref>. The noise-equivalent spectra (see 3.3) are also plotted
as grey curves.
Within one-hour bins, the mean power within bins of wavenumber, k, and azimuth angle. The values within each bin have been normalized by the total power with all bins with the same wavenumber range to enhance the appearance of any detected features.
Fluctuation spectra as a function of local time and temporal frequency for contemporaneous GPS observations (see 3.4.1). Within each two-hour block, the spectra are sorted by the angular separation between their ionospheric pierce-points and that of Cyg A from top to bottom. In each panel, the station code and GPS satellite number are printed in the upper left and the pierce-point latitude and longitude relative to that of the Cyg A pierce point is printed in the upper right.
For two nearby ionosondes stations, Dyess Air Force Base ($32^{\circ} \: 30''$ N; $99^{\circ} \: 42'$ W) and Point Arguello ($35^{\circ} \: 36''$ N; $120^{\circ} \: 36'$ W), h$^\prime$F (upper) and foEs (lower) as functions of VLA local time.
For each of the VLA arms, the total power as a function of projected wavenumber,
$k_{proj}$, within one-minute bins. The region highlighted with vertical white lines in each panel is discussed in 4.1.
Within one-hour bins, the mean power within bins of projected wavenumber, $k_{proj}$,
for each of the VLA arms. The noise-equivalent spectrum (see 4.1) is also plotted in
each panel as a grey curve. The vertical dotted grey lines denote values of $k_{proj} = \pm 1.26 \mbox{ km}^{-1}$ which corresponds to the approximate Nyquist sampling limit for the average antenna separation of 2.5 km along the VLA arms (in A configuration).
The spectra from Fig. <ref> normalized by the estimated background spectrum and with the three wavenumber classes highlighted (see 4.1).
The wavenumber vector azimuth angle for $3\sigma$ and $5\sigma$ detections (see 4.2) for each of the three wavenumber classes. Time ranges where MSTIDs
were detected from the data displayed in the upper panel of Fig. <ref> are shaded
in grey.
The weighted mean wavenumber/wavelength (upper) and velocity (lower) for $5\sigma$ detections (see 4.2) for each of the three wavenumber classes.
The distribution of $\vec{k}$ azimuth angles for $5 \sigma$ detections of small-scale moving structures displayed in Fig. <ref> (shaded grey histograms). The same distribution for those detections that were coincident with MSTID detections (see Fig. <ref>) are displayed in red. The growth rate for the $E_{\mbox{\scriptsize s}}$ layer instability is plotted as a black curve (see right ordinate for scale), allowing for a range in meridional wind components <cit.>.
Within one-hour windows, the azimuth angle of the neutral wind velocity vector from TIEGCM as a function of altitude (points with cubic spline fits plotted as solid curves). In each panel, the vertical dotted lines represent the $\vec{k}$ azimuth angles for the $5 \sigma$ detections of small-scale moving structures. The light-grey shaded regions indicate the typical range in altitudes for $E_s$ layers, 100–110 km. Each dark-gray horizontal line indicates the altitude where the wind direction changes from eastward at lower altitudes to westward at higher altitudes, i.e., where the wind shear necessary for creating an $E_s$ layer exists.
Within one-hour bins, the mean estimated background fluctuation spectra (see 4.1 and 4.4). The curves represent two-component turbulence fits to the spectra (see 4.4; component 1 is the smaller-scale component; component 2 is the larger-scale component). Noise equivalent spectra are plotted as grey curves.
For the two-component turbulence fits displayed in Fig. <ref>, the power at the maximum wavelength (upper) and the maximum wavelength (lower) for each component as functions of local time [see 4.4 and equation (4)]. Here, turbulent component 1 (green curves in Fig. <ref>) refers to the smaller-scale turbulent component and turbulent component 2 (blue curves in Fig. <ref>) refers to the larger-scale component.
|
arxiv-papers
| 2012-01-18T18:38:22 |
2024-09-04T02:49:26.451055
|
{
"license": "Public Domain",
"authors": "J. F. Helmboldt, T. J. W. Lazio, H. T. Intema, and K. F. Dymond",
"submitter": "Joe Helmboldt",
"url": "https://arxiv.org/abs/1201.3874"
}
|
1201.3884
|
11institutetext: Research Institute of Health Science (IUNICS) and Department
of Mathematics and Computer Science, University of the Balearic Islands,
E-07122 Palma de Mallorca, Spain 11email: {arnau.mir,cesc.rossello}@uib.es
# Two results on expected values of imbalance indices of phylogenetic trees
Arnau Mir Francesc Rosselló
###### Abstract
We compute an explicit formula for the expected value of the Colless index of
a phylogenetic tree generated under the Yule model, and an explicit formula
for the expected value of the Sackin index of a phylogenetic tree generated
under the uniform model.
## 1 Introduction
A _phylogenetic tree_ is a representation of the shared evolutionary history
of a set of extant species. From the mathematical point a view, it is a leaf-
labeled rooted tree, with its leaves representing the extant species under
study, its internal nodes representing common ancestors of some of them, the
root representing the most recent common ancestor of all of them, and the arcs
representing direct descendance through mutations. In this paper we only
consider _binary_ phylogenetic trees, where every internal node has exactly
two children.
There are several stochastic models of the evolutionary processes that produce
phylogenetic trees. Two of the most popular are the Yule and the uniform
models. In the _Yule_ , or _Equal-Rate Markov_ , model [8, 23], starting with
a node, at every step a leaf is chosen randomly and uniformly, and it is
replaced by a _cherry_ (a phylogenetic tree consisting only of a root and two
leaves). Finally, the labels are assigned randomly and uniformly to the leaves
once the desired number of leaves is reached. Under this model of evolution,
different trees with the same number of leaves may have different
probabilities. More specifically, if $T$ is a binary phylogenetic tree with
$n$ leaves, and for every internal node $z$ we denote by $\kappa_{T}(z)$ the
number of its descendant leaves, then the probability of $T$ under the Yule
model is [4, 21]
$P_{Y}(T)=\frac{2^{n-1}}{n!}\prod_{v\
\mathrm{internal}}\frac{1}{\kappa_{T}(v)-1}.$
On the other hand, the main feature of the _uniform_ , or _Proportional to
Distinguishable Arrangements_ , model [17] is that all phylogenetic trees with
the same number of leaves have the same probability. From the point of view of
tree growth, this corresponds to a process where, starting with a node labeled
1, at the $k$-th step a new pendant arc, ending in the leaf labeled $k+1$, is
added either to a new root or to some edge (with all possible locations of
this new pendant arc equiprobable). Notice that this is not an explicit model
of evolution, only of tree growth.
The study of the probabilistic distributions of indices associated to
phylogenetic trees under different stochastic models of phylogenetic tree
growth has received a lot of interest in the last decades [12, 19]. The
ultimate goal of this line of research is to be able to take as null model
some stochastic model of phylogenetic tree growth and evaluate against it the
indices of a sample of phylogenetic trees reconstructed from data. Two of the
most popular indices used in this connection, measuring the degree of
symmetry, or _balance_ , of a tree, are Sackin’s [18] and Colless’ [6]
indices, which we define later in the main body of this paper, but there are
many other measures associated to phylogenetic trees that have been used in
this context, like for instance other imbalance indices [7, Chap. 33] or the
number of cherries of trees [20].
Several properties of the distributions of Sackin’s $S$ and Colless’ $C$
indices have been studied in the literature under different models [3, 9, 13,
14, 15, 16, 21]. In particular, their expected values have been studied under
the Yule and the uniform model. The results published so far on these expected
values have been the following. Let $S_{n}$ and $C_{n}$ be the random
variables defined by choosing a binary phylogenetic tree $T$ with $n$ leaves
and computing $S(T)$ or $C(T)$, respectively. Then:
* •
Under the Yule model,
* –
$E_{Y}(S_{n})=2n\sum\limits_{j=2}^{n}1/j$ [9].
* –
$E_{Y}(C_{n})=n\log(n)+(\gamma-1-\log(2))n+o(n)$ [3], where $\gamma$ is the
Euler constant.
* •
Under the uniform model,
* –
$E_{U}(S_{n})\sim\sqrt{\pi}n^{3/2}$ [22].
* –
$E_{U}(C_{n})\sim\sqrt{\pi}n^{3/2}$ [3].
And, for instance, these are the formulas used by the R package apTreeshape
[1] to compute the expected value of these indices for a given number of
leaves. Let us also mention that Rogers [15] found a recursive formula for the
moment-generating functions of $C$ and $S$, which allowed him to compute
$E_{Y}(C_{n})$ and $E_{U}(C_{n})$ for $n=1,\ldots,50$, but he did not obtain
any explicit formula for them.
In this paper we obtain explicit formulas for $E_{Y}(C_{n})$ and
$E_{U}(S_{n})$. Namely,
$E_{Y}(C_{n})=n\sum_{j=2}^{\lfloor n/2\rfloor}\frac{1}{j}+\delta_{odd}(n),$
where $\delta_{odd}(n)=1$ if $n$ is odd, and $\delta_{odd}(n)=0$ if $n$ is
even, and
$E_{U}(S_{n})=\frac{n}{2n-3}{}_{3}F_{2}\bigg{(}\begin{array}[]{l}2,\ 2,\
2-n\\\\[-2.15277pt] 1,\ 4-2n\end{array};2\bigg{)},$
where ${}_{3}F_{2}$ is a _hypergeometric function_ [2] that can be directly
computed with many software systems, like Mathematica or R. These formulas
thus contribute to our knowledge of the probability distributions of these
indices, and yield precise values which can be used is tests.
## 2 Preliminaries and notations
In this paper, by a _phylogenetic tree_ on a set $S$ of taxa we mean a binary
rooted tree with its leaves bijectively labeled in the set $S$. To simplify
the language, we shall always identify a leaf of a phylogenetic tree with its
label. We shall also use the term _phylogenetic tree with $n$ leaves_ to refer
to a phylogenetic tree on the set $\\{1,\ldots,n\\}$. We shall denote by
$L(T)$ the set of leaves of a phylogenetic tree $T$ and by $V_{int}(T)$ its
set of internal nodes.
Let $\mathcal{T}(S)$ be the set of isomorphism classes of phylogenetic trees
on a set $S$ of taxa, and set $\mathcal{T}_{n}=\mathcal{T}(\\{1,\ldots,n\\})$.
It is well known [7, Ch.3] that $|\mathcal{T}_{1}|=1$ and, for every
$n\geqslant 2$,
$|\mathcal{T}_{n}|=(2n-3)!!=(2n-3)(2n-5)\cdots 3\cdot 1.$
Whenever there exists a path from $u$ to $v$ in a phylogenetic tree $T$, we
shall say that $v$ is a _descendant_ of $u$. The _cluster_ of a node $v$ in
$T$ is the set $C_{T}(v)$ of its descendant leaves, an we shall denote by
$\kappa_{T}(v)$ the cardinal $|C_{T}(v)|$, that is, the number of descendant
leaves of $v$. The _depth_ $\delta_{T}(v)$ of a node $v$ in a phylogenetic
tree $T$ is the length (number of arcs) of the unique path from the root $r$
to $v$.
Given two phylogenetic trees $T_{1},T_{2}$ on disjoint sets of taxa
$S_{1},S_{2}$, respectively, we shall denote by $T_{1}\widehat{\ }\,T_{2}$ the
tree on $S_{1}\cup S_{2}$ obtained by connecting the roots of $T_{1}$ and
$T_{2}$ to a (new) common parent $r$. Every tree in $\mathcal{T}_{n}$ is
obtained as $T_{k}\widehat{\ }\,{}T_{n-k}$, for some subset
$S_{k}\subseteq\\{1,\ldots,n\\}$ with $k$ elements (with $1\leqslant
k\leqslant n-1$), some tree $T_{k}$ on $S_{k}$ and some tree $T_{n-k}$ on
$S_{k}^{c}=\\{1,\ldots,n\\}\setminus S_{k}$: actually, if we perform in this
order the choices necessary to produce a tree $T\in\mathcal{T}_{n}$ in this
way, we obtain every tree in $\mathcal{T}_{n}$ twice.
An _ordered $m$-forest_ on a set $S$ is an ordered sequence of $m$
phylogenetic trees $(T_{1},T_{2},\ldots,T_{m})$, each $T_{i}$ on a set $S_{i}$
of taxa, such that these sets $S_{i}$ are pairwise disjoint and their union is
$S$. Let $\mathcal{F}_{m,n}$ be the set of isomorphism classes of ordered
$m$-forests on $\\{1,\ldots,n\\}$. It is known (see, for instance, [11, Lem.
1]) that for every $n\geqslant m\geqslant 1$,
$|\mathcal{F}_{m,n}|=\frac{(2n-m-1)!m}{(n-m)!2^{n-m}}.$
## 3 Expected value of the Colless index under the Yule model
Let $T$ be a phylogenetic tree. For every $v\in V_{int}(T)$, the _balance
value_ of $v$ is $bal_{T}(v)=|\kappa_{T}(v_{1})-\kappa_{T}(v_{2})|$, where
$v_{1}$ and $v_{2}$ are its children. The _Colless index_ [6] of a
phylogenetic tree $T\in\mathcal{T}_{n}$ is
$C(T)=\sum_{v\in V_{int}(T)}bal_{T}(v).$
###### Lemma 1
If $T_{k}\in\mathcal{T}_{k}$ and $T_{n-k}\in\mathcal{T}_{n-k}$, then
1. (a)
$C(T_{k}\widehat{\ }\,{}T_{n-k})=C(T_{k})+C(T_{n-k})+|2n-k|$
2. (b)
$P_{Y}(T_{k}\widehat{\
}\,{}T_{n-k})=\dfrac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T_{n-k})$
where $P_{Y}$ denotes the probability of a phylogenetic tree under the Yule
model.
###### Proof
Assertion (a) is well known, and a direct consequence of the definition of
$C$. Assertion (b) is a direct consequence of the explicit probabilities of
$T_{k}$, $T_{n-k}$ and $T_{k}\widehat{\ }\,{}T_{n-k}$ under the Yule model,
and the fact that $V_{int}(T_{k}\widehat{\ }\,{}T_{n-k})=V_{int}(T_{k})\cup
V_{int}(T_{n-k})\cup\\{r\\}$, these unions being disjoint.
###### Lemma 2
Let $I:\bigcup_{n\in\mathbb{N}}\mathcal{T}_{n}\to\mathbb{R}$ be a mapping such
that, for every phylogenetic trees $T_{1},T_{2}$ on disjoint sets of taxa
$S_{1},S_{2}$, respectively,
$I(T_{1}\widehat{\ }\,T_{2})=I(T_{1})+I(T_{2})+f(|S_{1}|,|S_{2}|)$
for some mapping $f:\mathbb{N}\times\mathbb{N}\to\mathbb{R}$. For every
$n\geqslant 1$, let $I_{n}$ be the random variable that chooses a tree
$T_{n}\in\mathcal{T}_{n}$ and computes $I(T_{n})$, and let $E_{Y}(I_{n})$ be
its expected value under the Yule model. Then,
$E_{Y}(I_{n})=\frac{1}{n-1}\Big{(}2\sum_{k=1}^{n-1}E_{Y}(I_{k})+\sum_{k=1}^{n-1}f(k,n-k)\Big{)}$
###### Proof
We compute $E_{Y}(I_{n})$ using its very definition and Lemma 1.(b):
$\begin{array}[]{l}E_{Y}(I_{n})\displaystyle=\sum_{T_{n}\in\mathcal{T}_{n}}I(T_{n})\cdot
p_{Y}(T_{n})\\\
\quad\displaystyle=\sum_{k=1}^{n-1}\sum_{S_{k}\subset\\{1,\ldots,n\\}\atop|S_{k}|=k}\sum_{T_{k}\in\mathcal{T}(S_{k})}\sum_{T_{n-k}\in\mathcal{T}(S_{k})}I(T_{k}\widehat{\
}\,{}T_{n-k})\cdot p_{Y}(T_{k}\widehat{\ }\,{}T_{n-k})\\\
\quad\displaystyle=\frac{1}{2}\sum_{k=1}^{n-1}\binom{n}{k}\sum_{T_{k}\in\mathcal{T}_{k}}\sum_{T_{n-k}\in\mathcal{T}_{n-k}}(I(T_{k})+I(T_{n-k})\\\
\quad\displaystyle\qquad\qquad\qquad\qquad\qquad+f(k,n-k))\cdot\frac{2}{(n-1)\binom{n}{k}}P_{Y}(T_{k})P_{Y}(T_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\sum_{T_{k}}\sum_{T_{n-k}}(I(T_{k})+I(T_{n-k})+f(k,n-k))P_{Y}(T_{k})P_{Y}(T_{n-k})\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{(}\sum_{T_{k}}\sum_{T_{n-k}}I(T_{k})P_{Y}(T_{k})P_{Y}(T_{n-k})\\\
\quad\displaystyle\qquad\qquad\qquad\qquad+\sum_{T_{k}}\sum_{T_{n-k}}I(T_{n-k})P_{Y}(T_{k})P_{Y}(T_{n-k})\\\
\quad\displaystyle\qquad\qquad\qquad\qquad+\sum_{T_{k}}\sum_{T_{n-k}}f(k,n-k)P_{Y}(T_{k})P_{Y}(T_{n-k})\Big{)}\\\
\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}\Big{(}\sum_{T_{k}}I(T_{k})P_{Y}(T_{k})+\sum_{T_{n-k}}I(T_{n-k})P_{Y}(T_{n-k})+f(k,n-k)\Big{)}\\\
\end{array}$
$\begin{array}[]{l}\quad\displaystyle=\frac{1}{n-1}\sum_{k=1}^{n-1}(E_{Y}(I_{k})+E_{Y}(I_{n-k})+f(k,n-k))\\\
\quad\displaystyle=\frac{1}{n-1}\Big{(}2\sum_{k=1}^{n-1}E_{Y}(I_{k})+\sum_{k=1}^{n-1}f(k,n-k)\Big{)}\end{array}$
as we claimed.
Mappings $I$ satisfying the hypothesis in the previous lemma are a special
case of _binary recursive tree shape statistics_ in the sense of [10].
###### Theorem 3.1
Let $C_{n}$ be the random variable that chooses a tree $T\in\mathcal{T}_{n}$
and computes its Colless index $C(T_{n})$. Its expected value under the Yule
model is
$E_{Y}(C_{n})=n\sum_{j=2}^{\lfloor n/2\rfloor}\frac{1}{j}+\delta_{odd}(n),$
where $\delta_{odd}(n)=1$ if $n$ is odd, and $\delta_{odd}(n)=0$ if $n$ is
even.
###### Proof
To simplify the notations, we shall denote $E_{Y}(C_{n})$ simply by $E_{n}$.
By Lemmas 1.(a) and 2,
$E_{n}=\frac{1}{n-1}\Big{(}2\sum_{k=1}^{n-1}E_{k}+\sum_{k=1}^{n-1}|n-2k|\Big{)}.$
Now a simple computation shows that
$\sum_{k=1}^{n-1}|n-2k|=\left\\{\begin{array}[]{ll}\dfrac{n(n-2)}{2}&\mbox{ if
$n$ is even}\\\ \dfrac{(n-1)^{2}}{2}&\mbox{ if $n$ is odd}\end{array}\right.$
and therefore
$E_{n}=\frac{2}{n-1}\sum_{k=1}^{n-1}E_{k}+\left\\{\begin{array}[]{ll}\dfrac{n(n-2)}{2(n-1)}&\mbox{
if $n$ is even}\\\ \dfrac{n-1}{2}&\mbox{ if $n$ is odd}\end{array}\right.$
In order to obtain a recurrence of order one from this expression, we
distinguish the case when $n$ is even from the case when $n$ is odd.
* •
When $n$ is even
$E_{n}=\frac{2}{n-1}\sum_{k=1}^{n-1}E_{k}+\frac{n(n-2)}{2(n-1)},\quad
E_{n-1}=\frac{2}{n-2}\sum_{k=1}^{n-2}E_{k}+\dfrac{n-2}{2}$
and then
$\begin{array}[]{rl}E_{n}&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{k}+\frac{n(n-2)}{2(n-1)}\\\
&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{k}+\frac{n-2}{n-1}\cdot\dfrac{n-2}{2}+\frac{n-2}{n-1}\\\
&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{n-2}{n-1}E_{n-1}+\frac{n-2}{n-1}\\\
&\displaystyle=\frac{n}{n-1}E_{n-1}+\frac{n-2}{n-1}\end{array}$
* •
When $n$ is odd
$E_{n}=\frac{2}{n-1}\sum_{k=1}^{n-1}E_{k}+\frac{n-1}{2},\quad
E_{n-1}=\frac{2}{n-2}\sum_{k=1}^{n-2}E_{k}+\dfrac{(n-1)(n-3)}{2(n-2)}$
and then
$\begin{array}[]{rl}E_{n}&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{2}{n-1}\sum_{k=1}^{n-2}E_{k}+\frac{n-1}{2}\\\
&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{n-2}{n-1}\cdot\frac{2}{n-2}\sum_{k=1}^{n-2}E_{k}+\frac{n-2}{n-1}\cdot\dfrac{(n-1)(n-3)}{2(n-2)}+1\\\
&\displaystyle=\frac{2}{n-1}E_{n-1}+\frac{n-2}{n-1}E_{n-1}+1\\\
&\displaystyle=\frac{n}{n-1}E_{n-1}+1\end{array}$
So, in summary,
$E_{n}=\frac{n}{n-1}E_{n-1}+\left\\{\begin{array}[]{ll}\dfrac{n-2}{n-1}&\mbox{
if $n$ is even}\\\ 1&\mbox{ if $n$ is odd}\end{array}\right.$
In particular, if $n$ is even,
$\begin{array}[]{rl}E_{n}&\displaystyle=\frac{n}{n-1}E_{n-1}+\dfrac{n-2}{n-1}=\frac{n}{n-1}\Big{(}\frac{n-1}{n-2}E_{n-2}+1\Big{)}+\dfrac{n-2}{n-1}\\\
&\displaystyle=\frac{n}{n-2}E_{n-2}+2\end{array}$
Setting $x_{n}=E_{n}/n$, this equation becomes
$x_{n}=x_{n-2}+\frac{2}{n}$
whose solution (for even numbered terms) with $x_{2}=E_{2}/2=0$ is
$x_{n}=\sum_{i=2}^{n/2}\frac{1}{i}.$
Therefore, when $n$ is even,
$E_{n}=n\sum_{i=2}^{n/2}\frac{1}{i},$
and when $n$ is odd,
$E_{n}=\frac{n}{n-1}E_{n-1}+1=\frac{n}{n-1}\cdot(n-1)\sum_{i=2}^{(n-1)/2}\frac{1}{i}+1=1+n\sum_{i=2}^{\lfloor
n/2\rfloor}\frac{1}{i}$
as we claimed.
## 4 Expected value of the Sackin index under the uniform model
The _Sackin index_ [18] of a phylogenetic tree $T\in\mathcal{T}_{n}$ is
defined as the sum of the depths of its leaves:
$S(T)=\sum_{i=1}^{n}\delta_{T}(i).$
Alternatively,
$S(T)=\sum_{v\in V_{int}(T)}\kappa_{T}(v).$
Let $S_{n}$ be the random variable that chooses a tree $T\in\mathcal{T}_{n}$
and computes its Sackin index $S(T)$. Since, under the uniform model, all
trees in $\mathcal{T}_{n}$ have probability $1/((2n-3)!!)$, the expected value
of $S_{n}$ under the uniform model is
$\frac{\sum_{T\in\mathcal{T}_{n}}S(T)}{(2n-3)!!}.$
So, we need to compute the numerator in this fraction. Now, for every
$k=1,\ldots,n-1$, let
$c_{k,n}=|\\{T\in\mathcal{T}_{n}\mid\delta_{T}(1)=k\\}|.$
###### Lemma 3
For every $n\geqslant 3$,
$\displaystyle\sum_{T\in\mathcal{T}_{n}}S(T)=n\sum_{k=1}^{n-1}k\cdot c_{k,n}$
###### Proof
Notice that, for every $1\leqslant i\leqslant n$,
$|\\{T\in\mathcal{T}_{n}\mid\delta_{T}(i)=k\\}|=|\\{T\in\mathcal{T}_{n}\mid\delta_{T}(1)=k\\}|.$
Then
$\begin{array}[]{rl}\displaystyle\sum_{T\in\mathcal{T}_{n}}S(T)&\displaystyle=\sum_{T\in\mathcal{T}_{n}}\sum_{i=1}^{n}\delta_{T}(i)=\sum_{i=1}^{n}\sum_{T\in\mathcal{T}_{n}}\delta_{T}(i)\\\
&\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{n-1}k\cdot|\\{T\in\mathcal{T}_{n}\mid\delta_{T}(i)=k\\}|\\\
&\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{n-1}k\cdot|\\{T\in\mathcal{T}_{n}\mid\delta_{T}(1)=k\\}|=n\sum_{k=1}^{n-1}k\cdot
c_{k,n}\end{array}$
###### Lemma 4
For every $n\geqslant 2$ and $k=1,\ldots,n-1$,
$c_{k,n}=\frac{(2n-k-3)!k}{(n-k-1)!2^{n-k-1}}.$
###### Proof
To compute $c_{k,n}$ for $k\geqslant 1$, notice that every tree
$T\in\mathcal{T}_{n}$ such that $\delta(1)=k$ will have the form described in
Fig. 1. Therefore, it is determined by the ordered $k$-forest
$T_{1},T_{2},\ldots,T_{k}$ on $\\{2,\ldots,n\\}$, and thus
$c_{k,n}=|\mathcal{F}_{k,n-1}|=\frac{(2n-k-3)!k}{(n-k-1)!2^{n-k-1}}.$
$1$...$T_{k}$$T_{2}$$T_{1}$ Figure 1: The structure of a tree $T$ with
$\delta_{T}(1)=k$.
Now, recall that the (_generalized_) _hypergeometric function_ ${}_{p}F_{q}$
is defined [2] as
${}_{p}F_{q}\bigg{(}\begin{array}[]{rrr}a_{1},&\ldots,&a_{p}\\\\[-2.15277pt]
b_{1},&\ldots,&b_{q}\end{array};z\bigg{)}=\sum_{k\geqslant
0}\frac{(a_{1})_{k}\cdots(a_{p})_{k}}{(b_{1})_{k}\cdots(b_{q})_{k}}\cdot\frac{z^{k}}{k!},$
where $(a)_{k}:=a\cdot(a+1)\cdots(a+k-1)$. Many popular software systems, like
Mathematica or R, have implementations of these functions.
###### Theorem 4.1
The expected value of the random variable $S_{n}$ under the uniform model is
$E_{U}(S_{n})=\frac{n}{2n-3}{}_{3}F_{2}\bigg{(}\begin{array}[]{l}2,\ 2,\
2-n\\\\[-2.15277pt] 1,\ 4-2n\end{array};2\bigg{)}$
###### Proof
As we have already mentioned,
$E_{U}(S_{n})=\frac{\sum_{T\in\mathcal{T}_{n}}S(T)}{(2n-3)!!}=\frac{n}{(2n-3)!!}\sum_{k=1}^{n-1}k\cdot
c_{k,n}=\frac{n}{(2n-3)!!}\sum_{k=1}^{n-1}\frac{(2n-k-3)!k^{2}}{(n-k-1)!2^{n-k-1}}$
Now
$\begin{array}[]{rl}\displaystyle\frac{nk^{2}(2n-k-3)!}{(2n-3)!!(n-k-1)!2^{n-k-1}}&\displaystyle=\frac{nk^{2}(2n-k-3)!2^{n-2}(n-2)!}{(2n-3)!(n-k-1)!2^{n-k-1}}\\\\[8.61108pt]
&\displaystyle=\frac{nk^{2}(2n-k-3)!2^{n-2}(n-2)!k!}{(2n-3)!(n-k-1)!2^{n-k-1}k!}\\\\[8.61108pt]
&\displaystyle=\frac{nk^{2}2^{k-1}\binom{n-1}{k}}{(n-1)\binom{2n-3}{k}}\end{array}$
and thus
$\begin{array}[]{rl}E_{U}(S_{n})&=\displaystyle\frac{n}{n-1}\sum_{k=1}^{n-1}k^{2}2^{k-1}\cdot\frac{\binom{n-1}{k}}{\binom{2n-3}{k}}\\\
&=\displaystyle\frac{n}{n-1}\sum_{k=1}^{n-1}\frac{k^{2}2^{k-1}(n-1)(n-2)(n-3)\cdots(n-k)}{(2n-3)(2n-4)(2n-5)\cdots(2n-k-2)}\\\
&=\displaystyle\frac{n}{2n-3}\sum_{k=1}^{n-1}\frac{k^{2}2^{k-1}(n-2)(n-3)\cdots(n-k)}{(2n-4)(2n-5)\cdots(2n-k-2)}\\\
&=\displaystyle\frac{n}{2n-3}\sum_{k=1}^{n-1}\frac{k^{2}2^{k-1}(2-n)(2-n+1)\cdots(-n+k)}{(4-2n)(4-2n+1)\cdots(2-2n+k)}\\\
&=\displaystyle\frac{n}{2n-3}\sum_{k=0}^{n-2}\frac{(k+1)^{2}2^{k}(2-n)(2-n+1)\cdots(1-n+k)}{(4-2n)(4-2n+1)\cdots(3-2n+k)}\\\
&=\displaystyle\frac{n}{2n-3}\sum_{k\geqslant
0}\frac{((k+1)!)^{2}(2-n)(2-n+1)\cdots(1-n+k)\cdot
2^{k}}{(k!)^{2}(4-2n)(4-2n+1)\cdots(3-2n+k)}\\\
&=\displaystyle\frac{n}{2n-3}\sum_{k\geqslant
0}\frac{(2)_{k}(2)_{k}(2-n)_{k}}{(1)_{k}((4-2n)_{k}}\cdot\frac{2^{k}}{k!}=\dfrac{n}{2n-3}{}_{3}F_{2}\bigg{(}\begin{array}[]{l}2,\
2,\ 2-n\\\\[-2.15277pt] 1,\ 4-2n\end{array};2\bigg{)}\end{array}$
as we claimed.
## 5 Conclusion
In this paper we have obtained explicit formulas for the expected value of the
Sackin index under the uniform model and the Colless index under the Yule
model. These results add up to the already known expected value of the Sackin
index under the Yule model [9]. For any $n$, these expected values are easily
computed directly using for instance the software system R, and can be used
instead of their estimations in packages like apTreeshape [1] or SymmeTREE
[5].
It remains open the problem of finding an explicit formula for the expected
value of the Colless index under the uniform model.
Acknowledgements. The research reported in this paper has been partially
supported by the Spanish government and the UE FEDER program, through projects
MTM2009-07165 and TIN2008-04487-E/TIN. We thank G. Cardona and M. Llabrés for
several comments on this work.
## References
* [1] N. Bortolussi, E. Durand, M. Blum, O. François, apTreeshape: statistical analysis of phylogenetic tree shape. Bioinformatics 22 (2006), 363–364.
* [2] W.N. Bayley, Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics 32, Stechert-Hafner Service Inc. (1964).
* [3] M. G. B. Blum, O. François, S. Janson, The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance. Ann. Appl. Probab. 16 (2006), 2195–2214.
* [4] J. Brown, Probabilities of evolutionary trees. Syst. Biol. 43 (1994), 78–91.
* [5] K. M. Chan, B. R. Moore, SymmeTREE: Whole-tree analysis of differential diversification rates. Bioinformatics 21 (2005), 1709–1710.
* [6] D. H. Colless, Review of “Phylogenetics: the theory and practice of phylogenetic systematics”. Sys. Zool. 31 (1982), 100–104.
* [7] J. Felsenstein, Inferring Phylogenies. Sinauer Associates Inc., 2004.
* [8] E. Harding, The probabilities of rooted tree-shapes generated by random bifurcation. Adv. Appl. Prob. 3 (1971), 44–77.
* [9] M. Kirkpatrick, M. Slatkin, Searching for evolutionary patterns in the shape of a phylogenetic tree. Evolution 47 (1993), 1171–1181.
* [10] F. Matsen, Optimization Over a Class of Tree Shape Statistics. IEEE/ACM Trans. Comput. Biol. Bioinf. 4 (2007), 506–512.
* [11] A. Mir, F. Rosselló, The mean value of the squared path-difference distance for rooted phylogenetic trees. J. Math. Anal. Appl. 371 (2010), 168–176.
* [12] A. Mooers, S. B. Heard, Inferring evolutionary process from phylogenetic tree shape. Quart. Rev. Biol. 72 (1997) 31–54.
* [13] W. H. Mulder, Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees. J. Theor. Biol. 280 (2011), 139–145.
* [14] J. S. Rogers, Response of tree imbalance to number of terminal taxa. Sys. Biol. 42 (1993), 102–105.
* [15] J. S. Rogers, Central moments and probability distributions of Colless’s coefficient of tree imbalance. Evolution 48 (1994), 2026–2036.
* [16] J. S. Rogers, Central moments and probability distributions of three measures of phylogenetic tree imbalance, Sys. Biol. 45 (1996), 99–110.
* [17] D. E. Rosen, Vicariant Patterns and Historical Explanation in Biogeography. Syst. Biol. 27 (1978), 159–188.
* [18] M. J. Sackin, “Good” and “bad” phenograms. Sys. Zool. 21 (1972), 225–226.
* [19] K.T. Shao, R. Sokal, Tree balance. Sys. Zool. 39 (1990), 226–276.
* [20] M. Steel, A. McKenzie, Distributions of cherries for two models of trees. Math. Biosc. 164 (2000), 81–92.
* [21] M. Steel, A. McKenzie, Properties of phylogenetic trees generated by Yule-type speciation models. Math. Biosc. 170 (2001), 91–112.
* [22] L. Takács, A Bernoulli Excursion and Its Various Applications. Adv. Appl. Prob. 23 (1991), 557–585.
* [23] G. U. Yule, A mathematical theory of evolution based on the conclusions of Dr J. C. Willis. Phil. Trans. Royal Soc. (London) Series B 213 (1924), 21–87.
|
arxiv-papers
| 2012-01-18T19:16:27 |
2024-09-04T02:49:26.462409
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Arnau Mir, Francesc Rossello",
"submitter": "Francesc Rossell\\'o",
"url": "https://arxiv.org/abs/1201.3884"
}
|
1201.3967
|
# Advantages for controls imposed in a proper subset111This study is partially
supported by the National Natural Science Foundation of China under grants
10871154, 10831007, 10801041, 11161130003 and 11171264; and by the National
Basis Research Program of China (973 Program) under grant 2011CB808002.
Gengsheng Wang 222 School of Mathematics and Statistics, Wuhan University,
Wuhan, 430072, China. E-mail: wanggs62@yeah.net , Yashan Xu333 School of
Mathematical Sciences, Fudan University, Shanghai 200433, China. E-mail:
yashanxu@fudan.edu.cn
###### Abstract
In this paper, we study time optimal control problems for heat equations on
$\Omega\times\mathbb{R}^{+}$. Two properties under consideration are the
existence and the bang-bang properties of time optimal controls. It is proved
that those two properties hold when controls are imposed on some proper
subsets of $\Omega$; while they do not stand when controls are active on the
whole $\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with
Dirichlet boundary condition is revealed.
Keywords. time optimal control, heat equations, bang-bang property, property
of eigenfunctions of the Laplacian
2010 MSC. 34H15 49N20
## 1 Introduction.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$. Let $\omega$ be a non
empty and open subset of $\Omega$. Write $\chi_{\omega}$ for its
characteristic function. Consider the following controlled heat equation:
$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\partial_{t}y(x,t)-\triangle
y(x,t)=\chi_{\omega}(x)u(x,t)&\mbox{in}&\Omega\times\mathbb{R}^{+},\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr
y(x,t)=0&\mbox{on}&\partial\Omega\times\mathbb{R}^{+},\\\ \vskip 3.0pt plus
1.0pt minus 1.0pt\cr y(x,0)=y_{0}(x)&\mbox{in}&\Omega,\end{array}\right.$
(1.1)
where $u$ is a control function taken from a control constraint set and
$y_{0}$ is an initial state taken from $L^{2}(\Omega)$. The solution of (1.1)
corresponding to $u$ and $y_{0}$ will be treated as a function from
$\mathbb{R}^{+}$ to $L^{2}(\Omega)$ and denoted by $y(\cdot;u,y_{0})$.
The purpose of this study is to reveal the following fact: Some properties
hold for some time optimal control problems of (1.1) when $\omega$ is a proper
subset of $\Omega$, but do not stand when $\omega=\Omega$. Consequently, the
local control may be more effective than the global control for heat equations
in some cases.
We begin with introducing time optimal control problems. Let
$\bigr{\\{}\xi_{i}\bigl{\\}}^{\infty}_{i=1}$ be a complete set of
eigenfunctions for $-\Delta$ with Dirichlet boundary condition such that it
serves as a normalized orthonormal basis of $L^{2}(\Omega)$. Write
$\bigr{\\{}\lambda_{i}\bigl{\\}}^{\infty}_{i=1}$, with
$0<\lambda_{1}<\lambda_{2}\leq\cdots<+\infty$, for the corresponding set of
eigenvalues. Then, we take the following target set:
${\cal
S}_{m}=span\bigr{\\{}\,\xi_{m+1},\xi_{m+2},\cdots\bigl{\\}},\;\;\mbox{where}\;\;m\geq
2\;\;\mbox{is arbitrarily fixed}.$
Next, we define, for each natural number $k$ and each finite sequence of
positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, the following control
constraint set:
${\cal
U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}=\left\\{\sum\limits_{i=1}^{k}\alpha_{i}(\cdot)\xi_{i}\;\biggm{|}\;\mbox{each}\;\;\alpha_{i}(\cdot)\;\;\mbox{is
measurable
from}\;\;\mathbb{R}^{+}\;\;\mbox{to}\;\;\left[-\bar{a}_{i},\bar{a}_{i}\right]\right\\}.$
Consider the following time optimal control problem:
${({\cal P})}\hskip 28.45274pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;y(t;u,y_{0})\in{\cal S}_{m}\right\\},\;\;\mbox{where the infimum
is taken over all}\;\;u\in{\cal U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}.$
Two properties of Problem $(\mathcal{P})$ under consideration are as follows:
$(i)$ The existence of time optimal controls; $(ii)$ The bang-bang property:
any optimal control $u^{*}=\sum_{i=1}^{k}\alpha^{*}_{i}\xi_{i}$ satisfies that
for each $i$, $|\alpha^{*}_{i}(t)|=\bar{a}_{i}$ for almost every
$t\in(0,t^{*})$, where $t^{*}$ is the optimal time. In the case that $\Omega$,
$\omega$, $k$ and $y_{0}\notin\mathcal{S}$ are fixed, we say Problem
$(\mathcal{P})$ has optimal controls if for any finite sequence of positive
numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, it has optimal controls. When
$\omega=\Omega$, $y_{0}\notin\mathcal{S}_{m}$ and $k$ are given, Problem
$(\mathcal{P})$ has optimal controls if and only if there is a finite sequence
of positive numbers $\\{\bar{b}_{i}\\}_{i=1}^{k}$ such that the problem
$(\mathcal{P})$, with $\\{\bar{b}_{i}\\}_{i=1}^{k}$, has optimal controls (see
Remark 2.3).
The main results of this paper are broadly stated as follows: $(a)$ Suppose
that $\omega=\Omega$ and $y_{0}\notin\mathcal{S}_{m}$. Then, $k$ and $y_{0}$
are such that Problem $({\cal P})$ has no optimal control if and only if $k<m$
and $y_{0}$ satisfies
$\displaystyle\left(\mathop{\langle}y_{0},\xi_{k+1}\mathop{\rangle},\mathop{\langle}y_{0},\xi_{k+2}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}\neq
0;$ (1.2)
$(b)$ Suppose that $\omega=\Omega$ and $y_{0}\notin\mathcal{S}_{m}$. Assume
that either $k\geq m$ or $k<m$ and $y_{0}$ does not satisfy (1.2). Then, in
general, Problem $(\mathcal{P})$ does not hold the bang-bang property; $(c)$
Suppose that $\Omega$ and $\omega$ satisfy accordingly the following
conditions:
* •
(D1) The eigenvalues $\lambda_{1}\cdots\lambda_{m}$ are simple, i.e.,
$\lambda_{1}<\lambda_{2}<\cdots<\lambda_{m}$,
and
* •
(D2) $\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\neq
0\quad{\rm
for~{}all~{}}i\in\\{1,2,\cdots,m\\}\;\;\mbox{and}\;\;j\in\\{1,2,\cdots,k\\}.$
Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$ and each finite
sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, Problem
$(\mathcal{P})$ has optimal controls and holds the bang-bang property.
It is worth mentioning that for any fixed bounded domain $\Omega$, there are a
lot of open subsets $\omega$ in $\Omega$ such that
$\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\neq 0$ for
all $i,j=1,2,\cdots$ (see Theorem 4.2 for a new property of the eigenfunctions
$\\{\xi_{i}\\}_{i=1}^{\infty}$); while there are a lot of bounded domains
$\Omega$ such that the property $\bf(D1)$ holds (see Remark 4.1)).
## 2 Studies of Problem $(\mathcal{P})$ where $\Omega=\omega$
The following result is another version of Theorem 2.5 in [4]. It will be used
later.
###### Lemma 2.1.
Let $\hat{A}\in\mathbb{R}^{d\times d}$ and $\hat{B}\in\mathbb{R}^{d\times l}$,
where $d$ and $l$ are natural numbers. Suppose that
$rank\left(\hat{B},\hat{A}\hat{B},\hat{A}^{2}\hat{B},\cdots,\hat{A}^{d-1}\hat{B}\right)=d,$
(2.1)
and the spectrum of $\hat{A}$ belongs to the left half plane of $\mathbb{C}$.
Then, for each finite sequence of positive numbers $\\{b_{i}\\}_{i=1}^{l}$ and
each $w_{0}$ in $\mathbb{R}^{d}$, there are a $\hat{t}\geq 0$ and a control
$\hat{\beta}$ in the set:
$\bar{\cal
V}\triangleq\left\\{\beta=\left(\beta_{1},\cdots,\beta_{l}\right)^{T}\bigm{|}\;\mbox{each}\;\;\beta_{i}\;\mbox{is
measurable from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-b_{i},b_{i}]\right\\},$
(2.2)
such that the solution $\hat{w}(\cdot;\hat{\beta},w_{0})$ to the equation:
$\left\\{\begin{array}[]{ll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{\hat{w}}(t)=\hat{A}\hat{w}(t)+\hat{B}\hat{\beta}(t),&t\in\mathbb{R}^{+},\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hat{w}(0)=w_{0},\end{array}\right.$
(2.3)
reaches zero at $\hat{t}$.
###### Theorem 2.2.
Suppose $\omega=\Omega$ and $y_{0}\notin{\cal S}_{m}$. Then, $k$ and $y_{0}$
are such that Problem $({\cal P})$ has no optimal control if and only if $k<m$
and $y_{0}$ satisfies (1.2).
Proof. The proof will be organized in three steps as follows:
Step 1. Suppose that $k<m$ and $y_{0}$ satisfies (1.2). Then, for any finite
sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$, Problem $({\cal
P})$ has no optimal control.
Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Then each $u(\cdot)\in{\cal U}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$ can be expressed
as $u(t)=\sum\limits_{i=1}^{k}\alpha_{i}(t)\xi_{i}$. Write
$y(t;u,y_{0})=\sum\limits^{\infty}_{i=1}y_{i}(t)\xi_{i}$. Clearly, the
controlled equation (1.1) is equivalent to the following system:
$\displaystyle\dot{y}_{i}(t)+\lambda_{i}y_{i}(t)=\sum\limits_{j=1}^{k}\alpha_{j}(t)\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle},\;\;y_{i}(0)=\left<y_{0},\xi_{i}\right>,\qquad
i=1,2,\cdots.$
Write
$z(t)=\left(\begin{array}[]{c}y_{1}(t)\\\ y_{2}(t)\\\ \cdots\\\
y_{m}(t)\end{array}\right),\quad A=\left(\begin{array}[]{cccc}\lambda_{1}\\\
&\lambda_{2}\\\ &&\ddots\\\
&&&\lambda_{m}\end{array}\right),\quad\alpha(t)=\left(\begin{array}[]{c}\alpha_{1}(t)\\\
\alpha_{2}(t)\\\ \cdots\\\ \alpha_{k}(t)\end{array}\right),$
and
$B=\Bigr{(}\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}\Bigl{)}_{i,j}\in\mathbb{R}^{m\times
k}.$ (2.4)
Let
$U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}=[-\bar{a}_{1},\bar{a}_{1}]\times[-\bar{a}_{2},\bar{a}_{2}]\times\cdots\times[-\bar{a}_{k},\bar{a}_{k}].$
Consider the following time optimal control problem:
$(\widetilde{{\cal P}})\hskip 42.67912pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z(t;\alpha,z_{0})=0\right\\},\hskip 99.58464pt$
where the infimum is taken over all $\alpha$ from the control constraint set:
$\displaystyle{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}\triangleq\left\\{\alpha=(\alpha_{1},\cdots,\alpha_{k})^{T}\bigm{|}\;\mbox{each}\;\;\alpha_{i}\;\mbox{is
measurable
from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-\bar{a}_{i},\bar{a}_{i}]\right\\},$
and $z(\cdot;\alpha,z_{0})$ is the solution to the following equation:
$\left\\{\begin{array}[]{lll}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{z}(t)+Az(t)=B\alpha(t),\;\;t\in\mathbb{R}^{+},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr
z(0)=\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}.\end{array}\right.$
(2.5)
Clearly, Problems $(\mathcal{P})$ and $(\widetilde{{\cal P}})$ are equivalent,
i.e., $t^{*}$ and $u^{*}=\sum_{i=1}^{k}\alpha^{*}_{i}\xi_{i}$ are accordingly
the optimal time and an optimal control to Problem $(\mathcal{P})$ if and only
if $t^{*}$ and $(\alpha^{*}_{1},\cdots,\alpha^{*}_{k})^{T}$ are the optimal
time and an optimal control to Problem $(\widetilde{{\cal P}})$ respectively.
Since $\omega=\Omega$ and $k<m$, it follows from (2.4) that
$B=\left(\begin{array}[]{c}I_{k\times k}\\\ 0\end{array}\right)$ in this case.
Let $z_{1}(t)=(y_{1}(t),\cdots,y_{k}(t))^{T}$ and
$z_{2}(t)=(y_{k+1}(t),\cdots,y_{m}(t))^{T}$. Write
$A_{1}=\left(\begin{array}[]{cccc}\lambda_{1}\\\ &&\ddots\\\
&&&\lambda_{k}\end{array}\right)\;\;\mbox{and}\;\;\;\;A_{2}=\left(\begin{array}[]{cccc}\lambda_{k+1}\\\
&&\ddots\\\ &&&\lambda_{m}\end{array}\right).$
Then, Equation (2.5) can be written as
$\displaystyle\frac{d}{dt}\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)(t)+\left(\begin{array}[]{cc}A_{1}\\\
&A_{2}\end{array}\right)\left(\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right)(t)=\left(\begin{array}[]{c}I_{k\times k}\\\
0\end{array}\right)\alpha(t),$ (2.6)
together with the initial condition:
$(z_{1}(0),z_{2}(0))^{T}=\left(\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{k}\mathop{\rangle}\right)^{T},\left(\mathop{\langle}y_{0},\xi_{k+1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}\right)^{T}$
This, along with the condition (1.2), indicates that $z_{2}(t)\neq 0$, for
each $t>0$ and each control $\alpha$ in ${\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$. Consequently, Problem $(\mathcal{P})$ has
no time optimal control.
Step 2. Suppose that $k<m$ and $y_{0}$ does not satisfy (1.2). Then, Problem
$(\mathcal{P})$ has optimal controls.
Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Since $y_{0}$ does not satisfy (1.2), it holds that $z_{2}(0)=0$. Thus, it
follows from (2.6) that $z_{2}(t)=0$ for all $t\geq 0$. Hence, Problem
$(\widetilde{\cal P})$ shares the same optimal time and optimal controls with
the following time optimal control problem:
$(\widetilde{{\cal P}}_{1}):\hskip
14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z_{1}(t;\alpha)=0\right\\},$
where the infimum is taken over all $\alpha$ from ${\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{k}}$, and $z_{1}(\cdot;\alpha)$ is the solution
to the equation:
$\dot{z}_{1}(t)+A_{1}z_{1}(t)=I_{k\times
k}\alpha(t),\;\;t\in\mathbb{R}^{+},\;z_{1}(0)=\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{k}\mathop{\rangle}\right)^{T}.$
According to Lemma 2.1, Problem $(\widetilde{{\cal P}}_{1})$ has admissible
controls. Then, by the standard argument (see either Theorem 13 and the note
after it in Chapter III on Page 130 in [16] or Theorem 3.1 on Page 31 in [4]),
one can easily verify that Problem $(\widetilde{{\cal P}}_{1})$ has optimal
controls. Consequently, Problem $(\mathcal{P})$ has optimal controls.
Step 3. Suppose that $k\geq m$. Then, Problem $({\cal P})$ admits optimal
controls.
Let $\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers.
Since $B=\left(I_{m\times m},0_{m\times(k-m)}\right)$ in the case that $k\geq
m$, control variables $\alpha_{m+1}(\cdot),\cdots,\alpha_{k}(\cdot)$ play no
role in Equation (2.5) when $k>m$. Hence, in the case that $k\geq m$, the
effective controls in Problem $(\widetilde{P})$ have the form:
$\hat{\alpha}=(\alpha_{1}(\cdot),\cdots,\alpha_{m}(\cdot))^{T}$. Therefore,
Problem $(\widetilde{{\cal P}})$ shares the same optimal time and optimal
controls with the following time optimal control problem:
$(\widetilde{{\cal P}}_{2}):\hskip
14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z(t;\hat{\alpha})=0\right\\},$
where the infimum is taken over all
$\hat{\alpha}\triangleq(\alpha_{1},\cdots,\alpha_{m})^{T}$ from the control
constraint set:
$\displaystyle{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}\triangleq\left\\{\alpha=(\alpha_{1},\cdots,\alpha_{m})^{T}\bigm{|}\;\mbox{each}\;\;\alpha_{i}\;\mbox{is
measurable
from}\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-\bar{a}_{i},\bar{a}_{i}]\right\\},$
and $z(\cdot;\hat{\alpha})$ is the solution of the following equation:
$\left\\{\begin{array}[]{l}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\dot{z}(t)+Az(t)=I_{m\times m}\hat{\alpha}(t),\quad
t\in\mathbb{R}^{+},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
z(0)=z_{0}\triangleq\left(\mathop{\langle}y_{0},\xi_{1}\mathop{\rangle},\mathop{\langle}y_{0},\xi_{2}\mathop{\rangle},\cdots,\mathop{\langle}y_{0},\xi_{m}\mathop{\rangle}\right)^{T}.\end{array}\right.$
(2.7)
Then, by Lemma 2.1, using the same argument as that in Step 2, one can prove
that Problem $(\widetilde{{\cal P}})$ has optimal controls.
In summary, we complete the proof.
###### Remark 2.3.
From the proof of Theorem 2.2, one can easily verify the following: $(a)$
Suppose that $\omega=\Omega$. Let $y_{0}\notin\mathcal{S}_{m}$ and $k$ be
given. Then, Problem $(\mathcal{P})$ has optimal controls if and only if there
is a finite sequence of positive numbers $\\{\bar{b}_{i}\\}_{i=1}^{k}$ such
that the problem $(\mathcal{P})$, with $\\{\bar{b}_{i}\\}_{i=1}^{k}$, has
optimal controls. $(b)$ In the case that $\omega=\Omega$ and
$y_{0}\notin\mathcal{S}_{m}$, Problem $(\mathcal{P})$ has optimal controls,
provided either $k\geq m$ or $k<m$ and $y_{0}$ does not satisfy (1.2).
###### Theorem 2.4.
Let $\omega=\Omega$ and $y_{0}\notin{\cal S}_{m}$. Let
$\\{\bar{a}_{i}\\}_{i=1}^{k}$ be a finite sequence of positive numbers. For
each $i\in\\{1,\cdots,k\\}$, write
$T_{i}=\displaystyle\frac{1}{\lambda_{i}}\ln\left(1+\frac{\lambda_{i}}{\bar{a}_{i}}\left|\mathop{\langle}y_{0},\,\xi_{i}\mathop{\rangle}\right|\right).\qquad$
(2.8)
Then Problem $({\cal P})$, with $\\{\bar{a}_{i}\\}_{i=1}^{k}$, does not have
the bang-bang property, if either of the following conditions stands: $(i)$
$k\geq m$ and the numbers $T_{1},\cdots,T_{m}$ are not the same; $(ii)$ $k<m$,
$y_{0}$ does not satisfy (1.2) and the numbers $T_{1},\cdots,T_{k}$ are not
the same.
Proof. Simply write $(\mathcal{P})$ for the problem $({\cal P})$, with
$\\{\bar{a}_{i}\\}_{i=1}^{k}$. For each $i\in\\{1,\cdots k\\}$, define
$\displaystyle\widetilde{\alpha}_{i}(\cdot)=-\chi_{[0,T_{i}]}(\cdot)sgn\big{(}\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}\big{)}\bar{a}_{i}\triangleq\left\\{\begin{array}[]{cl}\vskip
3.0pt plus 1.0pt minus
1.0pt\cr\chi_{[0,T_{i}]}(\cdot)\bar{a}_{i},&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}<0,\\\
\vskip 3.0pt plus 1.0pt minus 1.0pt\cr
0,&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}=0,\\\ \vskip 3.0pt
plus 1.0pt minus
1.0pt\cr-\chi_{[0,T_{i}]}(\cdot)\bar{a}_{i},&\mbox{if}\;\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}>0.\end{array}\right.$
(2.12)
We first prove the following property $(\mathcal{H}_{1})$: When $k\geq m$,
$\widetilde{T}$ and $\widetilde{u}$ are the optimal time and an optimal
control to Problem $(\mathcal{P})$ respectively, where
$\widetilde{T}\triangleq\max\\{T_{1},T_{2},\cdots,T_{m}\\}\;\;\mbox{and}\;\;\widetilde{u}\triangleq\sum\limits_{i=1}^{m}\widetilde{\alpha}_{i}\xi_{i}.$
By the equivalence of Problems $(\mathcal{P})$ and $(\widetilde{{\cal
P}}_{2})$ (see Step 3 in the proof of Theorem 2.2), we need only to verify
that $\widetilde{T}$ and $\widetilde{\alpha}$ are the optimal time and an
optimal control to Problem $(\widetilde{{\cal P}}_{2})$ respectively, where
$\widetilde{\alpha}\triangleq(\widetilde{\alpha}_{1},\cdots,\widetilde{\alpha}_{m})^{T}$.
For this purpose, we observe from direct computation that for each
$i\in\\{1,\cdots,m\\}$, $T_{i}$ and $\widetilde{\alpha}_{i}(\cdot)$ are the
optimal time and the optimal control to the following time optimal control
problem:
$(P_{i}):\hskip 14.22636pt\displaystyle\inf\limits\left\\{t\geq
0\;\big{|}\;z_{i}(t;\alpha_{i})=0\right\\},$
where the infimum is taken over all $\alpha_{i}(\cdot)$ from the set of all
measurable functions from $R^{+}$ to $[-\bar{a}_{i},\bar{a}_{i}]$, and
$z_{i}(\cdot;\alpha_{i})$ solves the following equation:
$\dot{z}_{i}(t)+\lambda_{i}(t)z_{i}(t)=\alpha_{i}(t),\;\;z_{i}(0)=\mathop{\langle}y_{0},\xi_{i}\mathop{\rangle}.$
Clearly, $\widetilde{\alpha}\in{\cal V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}$ and
$(\left(z_{1}\left(\cdot;\widetilde{\alpha}_{1}\right),\cdots,z_{m}\left(\cdot;\widetilde{\alpha}_{m}\right)\right)^{T}$
is the solution $z\left(\cdot;\widetilde{\alpha}\right)$ to Equation (2.7)
with $\hat{\alpha}=\widetilde{\alpha}$. Since
$z_{i}\left(T_{i};\widetilde{\alpha}_{i}\right)=0$, it holds that
$\displaystyle
z_{i}\left(\widetilde{T};\widetilde{\alpha}_{i}\right)=0\;\;\mbox{ for
all}\;\;i\in\\{1,2,\cdots,m\\},\;\;\mbox{i.e.,}\;\;z\left(\widetilde{T};\widetilde{u}\right)=0.$
(2.13)
Hence, the optimal time to Problem $(\widetilde{{\cal P}}_{2})$ is not bigger
than $\widetilde{T}$. On the other hand, if
$\hat{\alpha}\triangleq(\hat{\alpha}_{1}\cdots,\hat{\alpha}_{m})^{T}\in{\cal
V}_{\\{\bar{a}_{i}\\}_{i=1}^{m}}$ and $\hat{T}>0$ are such that
$z\left(\hat{T};\hat{\alpha}\right)=0$, then it stands that
$z_{i}\left(\hat{T};\hat{\alpha}_{i}\right)=0\;\;\mbox{ for
all}\;\;i\in\\{1,\cdots,m\\}.$
By the optimality of $T_{i}$ to Problem $(P_{i})$, we see that $\hat{T}\geq
T_{i}$ for all $i\in\\{1,\cdots,m\\}$, from which, it follows that
$\hat{T}\geq\widetilde{T}$. Therefore, $\widetilde{T}$ is the optimal time to
Problem $(\widetilde{{\cal P}}_{2})$. Along with (2.13), this yields that
$\widetilde{\alpha}$ is an optimal control to this problem. Hence, the
property $(\mathcal{H}_{1})$ stands.
Since $y_{0}\notin{\cal S}_{m}$, it holds that $\widetilde{T}>0$. Because
$T_{1},\cdots,T_{m}$ are not the same, there is an
$i_{0}\in\\{1,2,\cdots,m\\}$ such that $T_{i_{0}}<\widetilde{T}$. Then, it
follows from (2.12) that $\widetilde{\alpha}_{i_{0}}(t)=0$ for all
$t\in(T_{i_{0}},\widetilde{T}]$. Thus, the optimal control $\widetilde{u}$
does not satisfy the bang-bang property.
Using the very similar argument to that in the proof of the property
$(\mathcal{H}_{1})$, one can easily show the following property
$(\mathcal{H}_{2})$: When $k<m$, $y_{0}$ does not satisfy (1.2), $\hat{T}$ and
$\hat{u}$ are the optimal time and an optimal control to Problem
$(\mathcal{P})$, where
$\hat{T}\triangleq\max\\{T_{1},T_{2},\cdots,T_{k}\\}\;\;\mbox{and}\;\;\hat{u}\triangleq\sum\limits_{i=1}^{k}\widetilde{\alpha}_{i}\xi_{i}.$
Then, by the property $(\mathcal{H}_{2})$, (2.12) and the assumptions that
$y_{0}\notin{\cal S}_{m}$ and the numbers $T_{1},\cdots,T_{k}$ are not the
same, one can easily show that the optimal control $\hat{u}$ does not satisfy
the bang-bang property. This completes the proof.
## 3 Studies of Problem $(\mathcal{P})$ where $\omega$ is a proper subset of
$\Omega$
###### Theorem 3.1.
Let $\Omega$ satisfy the condition $(D1)$. Suppose that $\omega$ holds the
condition $(D2)$. Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$
and each finite sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$,
Problem $(\mathcal{P})$ has optimal controls.
Proof. By the same way as that in Step 1 of the proof of Theorem 2.2, we
define the matrices $A$ and $B$, and the problem $(\widetilde{P})$. Write
$B_{ij}$ for the element in i-th row and j-th column of $B$, namely,
$B_{ij}=\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{j}\mathop{\rangle}$.
Let $B_{1}=(B_{11},\cdots,B_{m1})^{T}$. We first claim that
$rank(B_{1},AB_{1},A^{2}B_{1},\cdots,A^{m-1}B_{1})=m.$ (3.1)
In fact, since
$A^{j}B_{1}=\left(\begin{array}[]{ccc}\lambda_{1}\\\ &\ddots\\\
&&\lambda_{m}\end{array}\right)^{j}\left(\begin{array}[]{c}B_{11}\\\ \cdots\\\
B_{m1}\end{array}\right)=\left(\begin{array}[]{c}\lambda^{j}_{1}B_{11}\\\
\cdots\\\ \lambda_{m}^{j}B_{m1}\end{array}\right),$
it holds that
$\Bigr{|}(B_{1},AB_{1},\cdots,A^{m-1}B_{1})\Bigl{|}=\left|\begin{array}[]{cccc}B_{11}&\lambda_{1}B_{11}&\cdots&\lambda_{1}^{m-1}B_{11}\\\
B_{21}&\lambda_{2}B_{21}&\cdots&\lambda_{2}^{m-1}B_{21}\\\
\cdots&\cdots&\cdots&\cdots\\\
B_{m1}&\lambda_{m}B_{m1}&\cdots&\lambda_{m}^{m-1}B_{m1}\end{array}\right|,$
which is a determinant of Vandermonde’ type and equals to
$\prod_{i=1}^{m}B_{i1}\prod_{k>l}(\lambda_{k}-\lambda_{l})$. Because of
conditions $(D1)$ and $(D2)$, this determinant is not zero, which implies
(3.1).
Now, according to Lemma 2.1, Problem $(\widetilde{\mathcal{P}})$ has
admissible controls. Then, by the standard argument (see either Theorem 13 and
the note after it in Chapter III on Page 130 in [16] or Theorem 3.1 on Page 31
in [4]), one can easily show that Problem $(\widetilde{{\cal P}})$ admits
optimal controls. This, along with the equivalence of Problems $(\mathcal{P})$
and $(\widetilde{{\cal P}})$, completes the proof.
###### Remark 3.2.
From the proof of the above theorem,it follows that Theorem 3.1 still stands
when the condition $(D2)$ is replaced by the following condition:
* •
$(\widetilde{D}2)$
$\mathop{\langle}\chi_{\omega}\xi_{i}\,,~{}\xi_{1}\mathop{\rangle}\neq
0\quad{\rm for~{}all}\;\;i\in\\{1,2,\cdots,m\\}.$
Before studying the bang-bang property for Problem $(\mathcal{P})$ where
$\omega$ is a proper open subset of $\Omega$, we recall the general position
condition which plays an important role in the studies of the bang-bang
property for linear controlled ordinary differential equations. Let $\hat{A}$
and $\hat{B}$ be $m\times m$ and $m\times k$ matrices respectively. Let
$\hat{V}$ be a closed polyhedron in $\mathbb{R}^{k}$. We say that $\hat{V}$
satisfies the general position condition with respect to $(\hat{A},\hat{B})$,
if for each nonzero vector $v$, which is parallel to one of the edges of
$\hat{V}$, the vectors
$\hat{B}v,~{}\hat{A}\hat{B}v,~{}\cdots~{}\hat{A}^{m-1}\hat{B}v$
are linearly independent. Consider the following time optimal control problem:
$(\hat{P}):\hskip
56.9055pt\displaystyle\inf\limits\left\\{t:z(t;v,z_{0})=0\right\\},\hskip
99.58464pt$
where the infimum is taken over all measurable functions $v$ from
$\mathbb{R}^{+}$ to the polyhedron $\hat{V}$, and $z(\cdot;v,z_{0})$ is the
solution to the following equation:
$\dot{z}(t)+\hat{A}z(t)=\hat{B}v(t),\;\;t>0;\;\;z(0)=z_{0},$
with $z_{0}$ a non-zero vector in $\mathbb{R}^{m}$.
###### Lemma 3.3.
(see [16], [7]) Suppose that the closed polyhedron $\hat{V}$ satisfies the
general position condition with respect to $(\hat{A},\hat{B})$. Then any
optimal control $\bar{u}(t)$ to Problem $(\hat{P})$, if exists, takes values
on the vertices of $\hat{V}$ and has a finite number of switchings.
###### Theorem 3.4.
Let $\Omega$ satisfy the condition $(D1)$. Suppose that $\omega$ satisfies the
condition $(D2)$. Then, for each $k\geq 1$, each $y_{0}\notin\mathcal{S}_{m}$
and each finite sequence of positive numbers $\\{\bar{a}_{i}\\}_{i=1}^{k}$ ,
Problem $(\mathcal{P})$ holds the bang-bang property.
Proof. By the same way as that in Step 1 of the proof of Theorem 2.2, we
define the matrices $A$ and $B$, and the problem $(\widetilde{\mathcal{P}})$.
According to Lemma 3.3, Theorem 3.1 and the equivalence of Problems
$(\mathcal{P})$ and $(\widetilde{\mathcal{P}})$, it suffices to prove the
general condition of $U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}$ with respect to
$(A,B)$. Clearly, the later is equivalent to the statement that for each
$j\in\\{1,\cdots,k\\}$, the vectors $Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j}$ are
linearly independent, where $\\{e_{1},\cdots,e_{k}\\}$ is the standard basis
of $\mathbb{R}^{k}$.
Let $F_{j}=(Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j})$. It is clear that
$\begin{array}[]{rl}\vskip 3.0pt plus 1.0pt minus
1.0pt\cr|F_{j}|=&|(Be_{j},ABe_{j},\cdots,A^{m-1}Be_{j})|\\\ \vskip 6.0pt plus
2.0pt minus
2.0pt\cr=&\left|\begin{array}[]{cccc}B_{1j}&\lambda_{1}B_{1j}&\cdots&\lambda_{1}^{m-1}B_{1j}\\\
B_{2j}&\lambda_{2}B_{2j}&\cdots&\lambda_{2}^{m-1}B_{2j}\\\
\cdots&\cdots&\cdots&\cdots\\\
B_{mj}&\lambda_{m}B_{mj}&\cdots&\lambda_{m}^{m-1}B_{mj}\end{array}\right|\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr=&\prod_{i=1}^{m}B_{ij}\prod_{k>l}(\lambda_{k}-\lambda_{l}).\end{array}$
This, together with the conditions (D1) and (D2), yields that $|F_{j}|\neq 0$
for each $j\in\\{1,\cdots,k\\}$. Hence, $U_{\\{\bar{a}_{i}\\}^{k}_{i=1}}$
satisfies the general position condition with respect to $(A,B)$. This
completes the proof.
## 4 Further studies on the conditions $(D1)$ and $(D2)$
In this section, we first give a remark and a theorem, which reveal
accordingly some properties for eigenvalues and eigenfunctions of $-\Delta$
with Dirichlet boundary condition. From the remark, it follows that there are
a lot of $\Omega$ satisfying the property $\bf(D1)$. From the theorem, it
follows that for any bounded domain $\Omega$ in $\mathbb{R}^{n}$, there are a
lot of $\omega\subset\Omega$ where the property $\bf(D2)$ holds. We end this
section with another remark which provides an open problem.
###### Remark 4.1.
It is presented in [12] (see also [18], [14]) that there are a lot of $\Omega$
of class $C^{3}$ satisfies the condition $(D1)$ in the following sense: Let
$\Omega$ be a bounded open set of class $C^{3}$ in $\mathbb{R}^{n}$. For each
$\varepsilon\in(0,1)$, an $\varepsilon-$neighborhood of $\Omega$ is defined to
be the image $(I+\psi)(\Omega)$, where $I$ is the identity map over
$\mathbb{R}^{n}$ and $\psi\in C^{3}(\mathbb{R}^{n};\mathbb{R}^{n})$, with the
$C^{3}-$norm less than $\varepsilon$. For each bounded open set
$\widetilde{\Omega}$ of class $C^{3}$ in $\mathbb{R}^{n}$, Write
$\Delta_{\widetilde{\Omega}}$ for the self-adjoint operator in
$L^{2}(\widetilde{\Omega})$ generated by the Laplacian on $\widetilde{\Omega}$
with the homogeneous Dirichlet boundary condition. Then, for each
$\varepsilon\in(0,1)$, there is an $\varepsilon-$neighborhood of
$\Omega^{\varepsilon}$ such that $-\Delta_{\Omega^{\varepsilon}}$ has only
simple eigenvalues.
Before presenting the theorem, we introduce the following notations: for each
$x\in\mathbb{R}^{n}$ and each $\rho>0$, $B_{\rho}(x)$ stands for the open ball
in $\mathbb{R}^{n}$, centered at $x$ and of radius $\rho$;
$\overline{B_{\rho}(x)}$ denotes the closure of the ball $B_{\rho}(x)$; for
each $\rho>0$,
$\Omega^{\rho}\triangleq\Bigr{\\{}x\in\Omega\setminus\overline{\omega}\bigm{|}dist\left(\partial{B_{\rho}(x)},\partial\Omega\right)>0,~{}dist\left(\partial{B_{\rho}(x)},\partial\omega\right)>0\Bigl{\\}},$
where $dist\left(E_{1},E_{2}\right)\triangleq\inf\limits_{x_{1}\in
E_{1},x_{2}\in E_{2}}\|x_{1}-x_{2}\|_{\mathbb{R}^{n}}$ for any subsets $E_{1}$
and $E_{2}$ in $\mathbb{R}^{n}$.
###### Theorem 4.2.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ and $\omega$ be an open
subset of $\Omega$ such that
$\Omega\setminus\overline{\omega}\neq\varnothing$. Then, for any
$\varepsilon>0$, there exists an $\varepsilon_{0}\in(0,\varepsilon)$ such that
$\Omega^{\varepsilon_{0}}\neq\varnothing$ and for almost every
$\widetilde{x}\in\Omega^{\varepsilon_{0}}$,
$\displaystyle\displaystyle\Bigl{\langle}\chi_{\omega\cup
B_{\varepsilon_{0}}(\widetilde{x})}\xi_{i}\,,~{}\xi_{j}\Bigr{\rangle}\neq
0\quad{\rm for~{}all~{}}i,j\in\mathbb{N}.$ (4.1)
Proof. We recall that each eigenfunction $\xi_{i}$ belongs to
$C^{\infty}(\Omega)$ (see Page 335 in [5]). Let
$\displaystyle\varphi(x,\tau)=\xi_{i}(x)\displaystyle
e^{\sqrt{\lambda_{i}}\tau},\qquad(x,\tau)\in\Omega\times\mathbb{R}.$
It is obvious that
$\triangle_{x}\varphi(x,\tau)+\partial^{2}_{\tau}\varphi(x,\tau)=0,\qquad(x,\tau)\in\Omega\times\mathbb{R}.$
By the property of harmonic functions (see Page 6 in [8]), the function
$\varphi(\cdot,\cdot)$ is real analytic over $\Omega\times\mathbb{R}$. Thus,
each eigenfunction $\xi_{i}$ is real analytic over $\Omega$. Write
$D=\left\\{(x,\rho)\in\Omega\times\mathbb{R}^{+}\bigm{|}\overline{B_{\rho}(x)}\subset\Omega\right\\}.$
Then, for each pair $(i,j)$, we define a function $F_{i,j}(\cdot.\cdot)$ from
$D$ to $\mathbb{R}$ by setting:
$F_{i,j}(x,\rho)=\displaystyle\int_{B_{1}(0)}\xi_{i}(x+\rho\eta)\xi_{j}(x+\rho\eta)d\eta,\;\;(x,\rho)\in
D.$ (4.2)
Clearly, it is well defined. The rest of the proof will be carried out by the
following three steps:
Step 1. Suppose that $f$ is a real analytic function over $\Omega$. Define the
function $F:D\mapsto\mathbb{R}$ by
$F(x,\rho)=\displaystyle\int_{B_{1}(0)}f(x+\rho\eta)d\eta,\;\;(x,\rho)\in D.$
(4.3)
Then $F$ is real analytic over $D$.
We need only to explain that $F$ is real analytic in a small neighborhood of
$(x_{0},\rho_{0})$ for any point $(x_{0},\rho_{0})\in D$. First, there is a
neighborhood $U$ of $(x_{0},\rho_{0})$ in $\mathbb{R}^{n}\times\mathbb{R}^{+}$
such that $\overline{B_{\rho}(x)}\subset\Omega$ for any $(x,\rho)\in U$.
Hence, the function $f(x+\rho\eta)$ is real analytic in $(x,\rho,\eta)$ over
$U\times\\-{B_{1}(0)}$. Extend $f$ to a complex-valued function in
$(z,w,\eta)$ over a small neighborhood $U_{c}\times{B_{1}(0)}$ of
$U\times{B_{1}(0)}$ in ${\mathbb{C}}^{n}\times{\mathbb{C}}\times\\-{B_{1}(0)}$
by making use of the power series expansion. We then get $f_{c}(z+w\eta)$,
which is real analytic over $(z,w,\eta)\in U_{c}\times\\-{B_{1}(0)}$ and
holomorphic in $(z,w)\in U_{c}$ for each fixed $\eta\in\overline{{B}_{1}(0)}$.
Clearly, it holds that
$f_{c}(x+\rho\eta)=f(x+\rho\eta)\qquad{\rm for~{}all~{}}(x,\rho,\eta)\in
U\times B_{1}(0).$
Now we define a function $F_{c}:U_{c}\mapsto\mathbb{C}$ by setting:
$F_{c}(z,w)=\displaystyle\int_{B_{1}(0)}f(z+w\eta)d\eta,\;\;(z,w)\in U_{c},$
and define the operator $\bar{\partial}$ in the standard way:
$\displaystyle\bar{\partial}u(z,w)=\sum^{n}_{j=1}\frac{\partial
u}{\partial\bar{z}_{j}}d\bar{z}_{j}+\frac{\partial
u}{\partial\bar{w}}d\bar{w},$
where
$\displaystyle\frac{\partial}{\partial\bar{z}_{j}}=\frac{1}{2}\left(\frac{\partial}{\partial(Re(z_{j}))}+\sqrt{-1}\frac{\partial}{\partial(Im(z_{j}))}\right),\qquad\frac{\partial}{\partial\bar{w}}=\frac{1}{2}\left(\frac{\partial}{\partial(Re(w))}+\sqrt{-1}\frac{\partial}{\partial(Im(w))}\right)$
are the standard Cauchy-Riemann operators (see [9]). It follows from the
holomorphic property of $f_{c}$ in $(z,w)$ that
$\bar{\partial}F_{c}(z,w)=\displaystyle\int_{B_{1}(0)}\bar{\partial}f_{c}(z+w\eta)d\eta=\int_{B_{1}(0)}0~{}d\eta=0.$
Hence, $F_{c}$ is holomorphic in $U_{c}$. In particular, the function
$F_{c}(\cdot,\cdot)\Bigl{|}_{U_{c}\bigcap(\mathbb{R}^{n}\times\mathbb{R})}$ is
real analytic, i.e. $F(\cdot,\cdot)$ is analytic in $U$. Thus,
$F(\cdot,\cdot)$ is real analytic over $D$. Consequently, for each pair
$(i,j)$, the function $F_{i,j}(\cdot,\cdot)$ is real analytic over $D$.
Step 2. For each pair $(i,j)$, $F_{i,j}(\cdot,\cdot)$ is not identically a
constant over $D$.
By the unique continuation property of the eigenfunctions (see [10]), we see
that for each $i\in\\{1,2,\cdots\\}$,
$\xi_{i}(x)\neq 0\quad\mbox{for almost every}\quad x\in\Omega.$
Thus, it holds that for each pair $(i,j)$,
$(\xi_{i}\xi_{j})(x)\neq 0\quad\mbox{for almost every}\quad x\in\Omega.$
Since the function $(\xi_{i}\xi_{j})(\cdot)$ is continuous in $\Omega$ and
$\Omega\setminus\overline{\omega}\neq\varnothing$, there is an
$\hat{x}\in\Omega\setminus\overline{\omega}$ such that
$(\xi_{i}\xi_{j})(\hat{x})\neq 0$. Hence, when $\delta>0$ is small enough, the
function $\bigr{(}\xi_{i}\xi_{j}\bigl{)}(\cdot)$ is either positive or
negative over $B_{\delta}(\hat{x})$, and
$\overline{B_{\delta}(\hat{x})}\subset\Omega$. Now, it follows from the
definition of the function $F_{i,j}(\cdot,\cdot)$ that
$F_{i,j}(\hat{x},\delta_{1})\neq F_{i,j}(\hat{x},\delta_{2}),\;\;{\rm
when}~{}\delta_{1}\;\mbox{and}\;\delta_{2}\;\mbox{are different numbers
in}\;\;(0,\delta).$
Since $(\hat{x},\delta_{1})$ and $(\hat{x},\delta_{2})$ belong to $D$,
$F_{i,j}$ is not identically zero over $D$ for each pair $(i,j)$.
Step 3. To prove (4.1).
Since each $F_{i,j}$ is real analytic and is not identically a constant over
$D$, the set
$W_{i,j}\triangleq\left\\{(x,\rho)\in
D\bigm{|}F_{i,j}(x,\rho)+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}=0\right\\}$
is a real analytic subvariety with dimension at most $n$. Thus, the
$\mathbb{R}^{n+1}-$Lebesgue measure of the set
$W\triangleq\bigcup_{i,j}W_{i,j}$
is zero. Write $W^{\rho}=\left\\{x\in\Omega\bigm{|}(x,\rho)\in W\right\\}$.
Denote by $m(W^{\rho})$ the $\mathbb{R}^{n}-$Lebesgue measure of $W^{\rho}$.
According to Fubini’s Theorem,
$0=\displaystyle\int_{\Omega\times(0,\infty)}\chi_{W}(x,\rho)dxd\rho=\int^{\infty}_{0}\int_{\Omega}\chi_{W^{\rho}}(x)dxd\rho=\int^{\infty}_{0}m(W^{\rho})d\rho.$
Thus, there is a subset $E\subset(0,\infty)$ of zero measure such that
$m(W^{\rho})=0$ for each $\rho\in(0,\infty)\setminus E$. On the other hand,
since $\Omega\setminus\overline{\omega}\neq\varnothing$, there is
$\bar{\rho}>0$ such that $\Omega^{\rho}\neq\varnothing$ for all
$\rho\in(0,\bar{\rho}]$.
Now, for each $\varepsilon>0$, we arbitrarily take an $\varepsilon_{0}$ from
the set $(0,\min\\{\bar{\rho},\varepsilon\\})\setminus E$. Then, it stands
that $m\left(W^{\varepsilon_{0}}\right)=0$ and
$\Omega^{\varepsilon_{0}}\neq\varnothing$. Hence,
$\displaystyle m\left(\Omega^{\varepsilon_{0}}\setminus
W^{\varepsilon_{0}}\right)=m\left(\Omega^{\varepsilon_{0}}\right).$ (4.4)
Clearly, the statement that $x\in\Omega^{\varepsilon_{0}}\setminus
W^{\varepsilon_{0}}$ is equivalent to the statement that
$x\in\Omega^{\varepsilon_{0}}\;\;\mbox{and}\;\;F_{i,j}(x,\varepsilon_{0})+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}\neq
0\;\;\mbox{for all}\;\;i,j\in\mathbb{N}.$
This, together with (4.4), yields that
$\displaystyle
F_{i,j}(\widetilde{x},\varepsilon_{0})+\mathop{\langle}\chi_{\omega}\xi_{i},\xi_{j}\mathop{\rangle}\neq
0\;\;\mbox{for all}\;\;i,j\in\mathbb{N}\;\;\mbox{and for almost
every}\;\;\widetilde{x}\in\Omega^{\varepsilon_{0}}.$ (4.5)
Finally, by the definition of $\Omega^{\varepsilon_{0}}$, we see that
$B_{\varepsilon_{0}}(x)\subset\Omega\;\;\mbox{and}\;\;B_{\varepsilon_{0}}(x)\bigcap\omega=\varnothing\;\;\mbox{for
all}\;\;x\in\Omega^{\varepsilon_{0}}.$
Along with (4.5), these indicate (4.1). This completes the proof.
###### Remark 4.3.
Let $\\{a_{i}\\}_{i=1}^{\infty}\in
l^{2}_{+}\triangleq\left\\{\\{b_{i}\\}_{i=1}^{\infty}\in
l^{2}\;\big{|}\;b_{i}>0\;\;\mbox{for all}\;\;i\right\\}$. Consider the
following time optimal control problem $(P)$:
$\displaystyle\inf\limits\left\\{t:y(t;u,y_{0})=0\right\\}$, where the infimum
is taken over all $u$ from the set:
$U_{ad}=\left\\{u=\sum_{i=1}^{\infty}u_{i}(t)\xi_{i}\biggm{|}\;\mbox{each}\;\;u_{i}(\cdot)\;\;\mbox{is
measurable from}\;\;\mathbb{R}^{+}\;\;\mbox{to}\;\;[-a_{i},a_{i}]\right\\},$
and $y(\cdot;u,y_{0})$ is the solution to Equation (1.1). The set $U_{ad}$ is
called a control constraint set of the rectangular type. We say Problem $(P)$
has the bang-bang property if any optimal control
$u^{*}=\sum_{i=1}^{\infty}u^{*}_{i}(t)\xi_{i}$ satisfies that for each $i$,
$u^{*}_{i}(t)=a_{i}$ for a.e. $t\in(0,t^{*})$, where $t^{*}$ is the optimal
time.
It is not clear to us what conditions are needed to obtain the bang-bang
property for Problem $(P)$. With regard to this question, we would like to
mention the following: $(i)$ It is necessary to impose certain conditions on
$\\{a_{i}\\}_{i=1}^{\infty}\in l^{2}_{+}$ to ensure the existence of optimal
controls for Problem $(P)$ (see [11]); $(ii)$ When $U_{ad}$ is replaced by the
following control constraint sets of the ball type:
$\widetilde{U}_{ad}\triangleq\left\\{u(\cdot)\in
L^{\infty}(\mathbb{R}^{+};L^{2}(\Omega))\biggm{|}u(t)\in\widetilde{B}(0,r)\right\\},$
where $\widetilde{B}(0,r)$ is the ball in $L^{2}(\Omega)$, centered at the
origin and of radius $r>0$, the bang-bang property for the corresponding time
optimal control problem $(P)$ has been studied (see [6], [13], [19] and [15]).
## References
* [1]
* [2]
* [3]
* [4] L. C. Evans, Lecture notes: Version 0.2 for an undergraduate course ”An Introduction to Mathematical Optimal Control Theory”. Online:http://math.berkeley.edu/ evans/.
* [5] L. C. Evans, Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.
* [6] H. O. Fattorini, Time-optimal control of solutions of operational differenital equations. SIAM J. Control, Ser. A, 2(1964), 54-59.
* [7] H. O. Fattorini, Infinite-dimensional optimization and control theory. Encyclopedia of Mathematics and its Applications, 62., Cambridge University Press, Cambridge, 1999.
* [8] Qing Han; Fanghua Lin, Elliptic partial differential equations. Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997.
* [9] Steven G. Krantz, Function theory of several complex variables. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1982.
* [10] F. H. Lin, A uniqueness theorem for the parabolic equation, Comm. Pure Appl. Math., 43 (1990), 127-136.
* [11] Qi L$\ddot{u}$; Gengsheng Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim., 49(2011), 1124-1149.
* [12] A. M. Micheletti, Perturbazione dello spettro dell’operatore di Laplace, in relazione ad una variazione del campo. (Italian) Ann. Scuola Norm. Sup. Pisa, 26(1972), 151-169.
* [13] Victor J. Mizel; Thomas I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35(1997), 1204-1216.
* [14] Jaime H. Ortega; Enrique Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation. SIAM J. Control Optim., 39 (2000), 1585-1614 (electronic).
* [15] Kim Dang Phung; Gengsheng Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, To appear in J. Eur. Math. Soc.
* [16] L. S. Pontryagin; V. G. Boltyanskii; R. V. Gamkrelidze; E. F. Mishchenko, The mathematical theory of optimal processes., Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc: New York-London, 1962.
* [17] E. D. Sontag, Mathematical control theory: Deterministic finite-dimensional systems. Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998.
* [18] K. Uhlenbeck, Generic properties of eigenfunctions. Amer. J. Math., 98(1976), 1059-1078.
* [19] Gengsheng Wang, $L^{\infty}$-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim., 47(2008), 1701-1720.
|
arxiv-papers
| 2012-01-19T03:47:58 |
2024-09-04T02:49:26.470519
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Gengsheng Wang and Yashan Xu",
"submitter": "Yashan Xu",
"url": "https://arxiv.org/abs/1201.3967"
}
|
1201.4003
|
# PHYSICAL PARAMETERS OF THE RELATIVISTIC SHOCK WAVES IN GRBs: THE CASE OF 30
GRBs
SAŠA SIMIĆ Faculty of Science, Department of Physics, University of
Kragujevac
Radoja Domanovića 12, Kragujevac, Serbia, 34000
ssimic@kg.ac.rs LUKA Č. POPOVIĆ Astronomical Observatory, Volgina 7
Belgrade, Serbia, 11000
lpopovic@aob.bg.ac.rs
(18.01.2012 for publication in IJMPD)
###### Abstract
Using the modified internal shock wave model we fit the gamma ray burst (GRB)
light and spectral curves of 30 GRBs observed with BATSE. From the best
fitting we obtain basic parameters of the relativistic shells which are in
good agreement with predictions given earlier. We compare measured GRB
parameters with those obtained from the model and discuss connections between
them in the frame of the physical processes laying behind GRB events.
###### keywords:
Gamma rays: bursts; Gamma rays: theory; Shock waves
## 1 Introduction
The quest on resolving a Gamma Ray Burst (GRB) event consists of finding an
explanation for several parts e.g. spatial distribution of the event,
afterglow, spectral and light curve, collimation, etc. In the past decade the
GRB phenomenon has been thoroughly investigated both observationally see e.g.
[Bloom98]-[Ostlin08] and theoretically see e.g. [Meszaros97]-[Zitouni08].
However, a mystery of GRB phenomena lies in its heart, where the central
engine ejects material with relativistic energies and velocities. Due to the
high optical depth of the expanding material in the first phase of the GRB
event, structural observations of the central engine, that is located near the
core of the progenitor, are not possible. The only information that one can
obtain from the observations, in the first minute of a GRB event, is the
temporal variability of the $\gamma$-ray light curve. This usually shows
strong, but short fluctuations of the energy output with a typical time-scale
of the order of milliseconds to a couple of seconds [Norris96]. The numerical
simulations of [Kobayashi97] and [Nakar02] for the early phase of the
explosion, revealed that $\gamma$-ray light curve pulses replicate temporal
activity of the ’inner engine’. This can give information about the connection
between observations and physical processes occurring in the GRB core.
In order to make conclusion about the physical processes in the GRB core one
should assume the mechanism of GRB origin. The mechanism of pulse creation in
a GRB light curve is proposed to be connected with mutual interactions
(collisions) between faster and slower spreading shells see e.g. [Piran05], in
the so called internal shock model. Here we accepted slightly modified version
of the model with the additional assumption of the non-zero density
environment and also with different treatment of the slower material (shell)
accumulated at some distance from the GRB engine. In a previous paper
[Simic07], where we considered interaction of an incoming faster shell with
dense barrier, we demonstrated that the model is able to reproduce and fit
well the observed GRB light curves.
In this paper we apply the model to a sample of 30 GRB light curve pulses
observed with Burst and Transient Source Experiment (BATSE) in order to
discuss the physical parameters of relativistic shells.
The paper is organized as follow: in §2 we give a brief description of the
model and considered assumptions; in §3 we describe the observational sample
and fitting procedure; in §4 we discuss our results and finally in §5 we
outline our conclusions.
## 2 The model and assumptions
Here we describe the physical scenario and some approximations used in the
model of GRB light and spectral curves. First, we assume that the GRB engine
ejects an amount of relativistically expanding material, that spreads
isotropically from the center of explosion. The material is subsequently
ejected from time to time depending on the central engine activity. Here we
will not consider the nature of GRB progenitors, i.e. for the model it is not
relevant whether a GRB event is originated in the collapse of a massive star
[Woosley93] or in the process of merging of two compact objects – neutron
stars or black holes [Norris00], [Fryer99]. The most important for the model
is the assumption that the ejected material can mutually interact or interact
with the surrounding environment.The ejected material is probably irregular in
nature, with different initial parameters (mass $M_{\rm ej}$ and Lorentz
factor $\Gamma_{0}$).
During the expansion of the ejected material (closer to the GRB core), a slow
moving material is followed by a fast moving one, thus the faster moving
material will overtake the slower material and plunge into it. This
interaction produces the relativistic shells and shock waves that can
accelerate particles to very high speeds. The new formed relativistic shells
are probably with different velocities and could collide mutually producing
the observed GRB light curve pulses. The observed GRB light curves show vast
variety of pulses, ranging from very intense ones to those almost equal to
noise, and from symmetrical to highly asymmetrical ones. To model different
observational light curve pulse profiles, we consider here the modified
standard internal model, assuming that the accumulated matter is able to form
a slow moving barrier. This, allowed us to generate more diversity in light
curve pulse profiles.
In the shell interaction processes, the mass of a newly created shell is
approximately the sum of the two colliding shells and its Lorentz factor is
smaller than it was in the faster shell. It could be additionally reduced by
surrounding medium or further shell interactions, thus producing the
accumulation effect. Depending on the density and width of such created dense
barrier, further collision with it produces wider/thinner or higher/lower
pulses. Also, considering the energy density of expanding shells and barriers,
one can get more symmetrically shaped light curve pulses in case of the shells
with high energy and low energy barriers, and high asymmetrical pulse shapes
in the opposite case. This is the main mechanism that we use to explain high
temporal variability of the observed $\gamma$-ray light curves. From the
scenario described above, it follows that the selection of shell parameters
can have a random distribution in a given interval of values.
Similar as in [Kobayashi97], we consider that the ejected GRB material is
organized in an ultra-relativistic flow of well defined and collimated shells
with random initial energy. In contrary to [Piran05] and [Kobayashi97], where
null density hypothesis was used, we assume here that the surroundings regions
around the GRB central engine consist of at least small number of particles
with densities $n_{0}>1\ \rm cm^{-3}$. This allows us to analyze the density
of the moving shell and to model the density distribution and shape of the
barrier in a specific way (here we use Gaussian function, see Eq. 5 further in
the text). The assumption of $n_{0}\neq 0$ around the central engine seems to
be generally valid. If there are some kind of ejections from the central
engine, one can expect an amount of scattered material distributed to the
surrounding region, as e.g. in the collapsar model where a central star is
Wolf-Rayet type. Such star ejects a huge amount of material during its final
stage, therefore one can expect non zero density in its environment.
To describe the evolution of a relativistic shell, we adopt a phenomenological
model based on the [Huang98] that presents a system of the first order of
differential equations (where the distance $R$, Lorentz factor $\Gamma$ and
mass $m_{s}$ of the shell are included, see [Simic07]):
${\frac{{dR}}{{dt}}}=c\sqrt{\Gamma^{2}-1}{\left[{\Gamma+\sqrt{\Gamma^{2}-1}}\right]},$
(1)
${\frac{{d\Gamma}}{{dm_{s}}}}=-{\frac{{\Gamma^{2}-1}}{{M_{\rm
ej}+2(1-\xi)\Gamma m_{s}+\xi m_{s}}}},$ (2)
${\frac{{dm_{s}}}{{dt}}}=2\pi
nm_{p}(1-\cos\theta){\frac{{R^{2}}}{{\Gamma^{3}}}}\left({3\Gamma{\frac{{dR}}{{dt}}}-2R{\frac{{d\Gamma}}{{dt}}}}\right),$
(3)
where the parameter $\xi$ takes values from 0 in case of adiabatic expansion
to 1 in fully radiative case. $M_{\rm ej}$ and $\theta$ are the initial mass
and collimation angle of the shell, $n$ is the number density and $m_{p}$ is
the proton mass. Eqs. (1) - (3) are derived for an observer reference frame,
and they have to be solved simultaneously, together with the density equation.
The initial values of parameters and variables are highly dependent on
physical properties of the shocks.
The density of the barrier cerated from accumulating decelerated shells
(emitted by the central engine) could be described with (see [Blandford76]):
$\frac{n_{2}}{n_{1}}=\frac{\kappa_{2}\gamma_{2}+1}{\kappa_{2}-1}$ (4)
where $n_{2}$ and $n_{1}$ are number densities after and in front of the
shock, $\kappa_{2}$ is the ratio of the specific heat for the shocked fluid
and $\gamma_{2}$ is the Lorentz factor of the shocked fluid.
This equation gives a connection between density of perturbed and unperturbed
material. In the case of the ultra-relativistic expansion, the ratio of the
specific heats has a constant value of $\kappa=4/3$, then Eq. 4 can be reduced
to $\frac{n_{2}}{n_{1}}=4\gamma_{2}+3$. Also, in the relativistic regime, the
Lorentz factor of the shell $\Gamma$ is directly proportional to the Lorentz
factor of the shocked particles $\gamma_{2}$ [Blandford76].
In case of the collision of relativistic shells, a slower interacting shell
(or barrier in our model) presents a density perturbation in surrounding media
for the incoming faster shell. The barrier (or accumulated material) at a
distance $R_{c}$ from the central engine should have a distribution of the
density along the path of penetration of the faster shell. This can be
included in calculation, and here we assume a Gaussian density distribution
as:
$n=n_{0}\left({{\frac{{R_{0}}}{{R}}}}\right)^{s}(4\Gamma+3)\left({1+a\cdot\exp{\left[{-\left({{\frac{{R-R_{c}}}{{b}}}}\right)^{2}}\right]}}\right)$
(5)
where $a$ and $b$ describe the Gaussian intensity and width of the barrier and
$n_{0}$ is the density of the surrounding region. $R_{0}$ is the initial
position of the faster shell, $\Gamma$ is the Lorentz factor of the shell as
used in Eqs. (1-3). Note here that in Eq. 5 (as it was mentioned above)
$n_{0}>0$, and it is a crucial difference between our and the standard
internal shock (IS) model. In the standard IS model, the density of the
interacting shells is fixed by the mass loss rate from the central engine, the
Lorentz factor of the shells and the distance from the source where the
collisions take place. It is not directly related to the density of the
external medium. As a result, even in the absence of an external medium,
prompt emission will result from shocks taking place in the material ejected
from the source. Indeed there is some evidence of bursts occurring in very low
density environments which have a prompt emission but no detectable afterglow,
but as we mentioned above, one could expect that $n_{0}>0$ (especially close
to the accumulated material) in the central engine surroundings.
Generally, one can expect that the ejected material in form of the shell can
collide with the ISM which has a certain density distribution. In such a
highly relativistic physical system the relative motion of charged particles
of the ISM can generate an intense magnetic field in the reference frame of
the moving fluid. We calculate the magnetic field in a similar way as in
[Huang00], by assuming that the energy of the magnetic field is a certain
fraction, $\xi_{b}$, of the total energy of the relativistic shell. In the
comoving reference frame the magnetic field is calculated as:
$B^{{}^{\prime}}=\sqrt{8\pi\xi_{b}n_{0}\Gamma
m_{p}c^{2}(4\Gamma+3)\left({{\frac{{R_{0}}}{{R}}}}\right)^{s}\left({1+a\cdot
e^{-\left({{\frac{{R-R_{c}}}{{b}}}}\right)^{2}}}\right)},$ (6)
where the variables used in this equation are same as in the Eqs. 1-5.
The emission mechanism of shock waves is mainly based on the synchrotron
radiation, but for higher energy bands additional flux may be gained by the
Inverse Compton (IC) radiation [Piran05]. In the first approximation we
neglect the IC radiation.
We calculate the intensity of the radiation emitted by particles in the
relativistic shell using the formulae given by [Rybicki79]. Then the total
emitted flux can be calculated as e.g. in [Huang00a]. Note here that an
expanding shell contains relativistic electrons and baryons which contribute
to the synchrotron radiation. However, taking into account the difference in
velocities of these constituents, one can neglect the contribution of baryons
to the total emitted flux. Then in the comoving reference frame the total flux
is given as:
$P_{\nu}^{{}^{\prime}}=A\cdot\int_{\gamma_{emin}}^{\gamma_{emax}}\gamma_{e}^{-(p+1)}F(\nu^{{}^{\prime}}/\nu_{syn}^{{}^{\prime}})d\gamma_{e}$
(7)
where $A$, $\gamma_{emin}$ and $\gamma_{emax}$ are:
$A=\frac{\sqrt{3}e^{3}B^{{}^{\prime}}}{m_{e}c^{2}}\frac{m_{s}}{m_{p}};\gamma_{emin}=\xi_{e}\Gamma\frac{(p-2)}{(p-1)}\frac{mp}{me};\gamma_{emax}\rightarrow\infty;$
(8)
and
$F(x)=\int_{x}^{\infty}K_{5/3}(x)dx,$
where, $K_{5/3}$ is the Bessel function of the second order and
$\nu_{syn}^{{}^{\prime}}=3\gamma_{e}^{2}eB^{{}^{\prime}}/4\pi m_{e}c$ is the
critical frequency of the synchrotron radiation.
In Eq. 7 we neglect the effects of the surface curvature of an emitting shell,
since it has a small influence on the pulse shape.
In the case of ultra relativistic shells the cooling time is much shorter then
the dynamical (time of expansion) (see [Molinari07]). This is particulary
interesting in the gamma phase of explosion where shells are interacting with
each other.
With the model described above we are able to simulate collisions of
relativistic shells in the first phase of a GRB event, which produce the peaks
in the light curve [Simic07]. This model has been used to fit the light and
energy curves of the sample of 30 GRBs taken from the BATSE database.
## 3 Model vs. observations
In [Simic07] we demonstrated that the model is able to reproduce (simulate)
the observed light curves of GRBs. Moreover, the model can properly fit the
observed light curves. Here we selected 30 GRBs from the BATSE database and
fit them with the model in order to find the basic parameters of interacting
shells. In this section we describe the selected sample and the fitting
procedure.
### 3.1 The sample
From a large BATSE database (3rd channel, $E=100-300$ keV, for the light
curve) we select a sample of 30 GRBs using following criteria: (i) GRBs have
isolated light curves with the clear peak maximum. For proper application of
the model we avoid the complex pulse profiles. 111The process of pulse
creation is stochastic in nature and may result in a complex pulse shape, e.g.
it is often observed that two or more pulses are superposed; (ii) we avoid
small pulses because of their low temporal resolution; (iii) we include in the
sample as much as possible different GRBs (long and short lasting, with
different profiles, different intensity and different profile asymmetries).
In Table 3.1 we give a list and basic parameters of selected GRBs. The
parameters in the table are (from the first to the last column respectively):
the Full Width at Half Maximum of the intensity of light curve pulse (FWHM),
the time of peak intensity for the observed pulse ($t_{\rm peak}$), the total
duration of the pulse from the beginning to the end of its lower tail ($\Delta
t$), the maximal intensity ($J_{m}$) of the pulse measured in ${\rm
erg/cm^{2}sHz}$, and the asymmetry indicator ($w$) calculated as a ratio of
half-halfwidths before and after the maximum.
Parameters of the selected GRB light curves. From the first to the last column
the following parameters are given: Full width at half maximum (FWHM), time of
peak intensity $t_{\rm peak}$, duration of the pulse $\Delta t$, pulse
intensity ($J_{m}\rm[erg/cm^{2}sHz]$), asymmetry indicator $w_{s}$. GRB FWHM
[s] $\Delta t_{\rm peak}[s]$ $\Delta t[s]$ $J_{m}[\times 10^{-27}]$ $w_{s}$
GRB910629 0.55 0.4 1 13 0.6 GRB911104 0.85 0.5 1.3 13 0.5 GRB920715 0.5 0.45 1
3.2 0.4 GRB920720 0.75 0.6 1.4 40 0.7 GRB920808 0.5 0.25 0.7 11.3 0.4
GRB920811 0.2 0.25 0.3 3.5 0.7 GRB920830 2.75 1.9 6 7.8 0.6 GRB920912 0.4 0.58
1.2 5.4 0.7 GRB920924 0.4 0.4 0.8 4. 0.3 GRB921021 1.8 1 4 5.1 0.4 GRB921207
1.4 1.4 3 60. 0.6 GRB921208 1 0.85 2 2.25 0.8 GRB921222 1 0.85 2 2.25 0.8
GRB950129B 0.6 0.65 1.22 3.7 0.7 GRB950211B 1.5 0.95 3.5 6.5 0.4 GRB960111 1.2
0.5 2.6 9 0.5 GRB960207 0.33 0.25 0.6 9.2 0.8 GRB960229 0.55 0.35 1 5.4 0.8
GRB960311 0.4 0.4 0.7 6 0.6 GRB960409 2.4 2 5 10.5 0.7 GRB960418 0.9 0.8 1.5
4.7 0.4 GRB960524 0.55 0.4 1.2 6.2 0.8 GRB960528 2.2 0.55 4.2 4.2 0.6
GRB960530 5.5 3 11 5.7 0.6 GRB960613 1.6 1.8 4 6 0.3 GRB960617B 0.7 0.55 1.1
3.3 0.8 GRB970424 0.4 0.3 0.7 3.3 0.5 GRB991105 0.65 0.45 1.6 6 0.4 GRB991213
1.4 1 2.4 4 0.4 GRB000107 0.5 0.45 0.7 4.7 1.
Fig. 1 shows statistics of the measured parameters given in Table 3.1. As it
can be seen in Fig. 1 the values of observed parameters do not follow the
Gaussian distribution. Due to the relatively small number of GRBs in the
sample, here we are not able to discuss the power law indices of parameters
(for detailed studies of these GRB properties see [Norris96], [Hakkila08]).
Figure 1: Histograms of basic GRB characteristics for the sample of GRBs
presented in Table 3.1. On the Figures (a) to (d) we place distributions of
(a) the Full Width at Half Maximum of the intensity of light curve pulse
(FWHM), (b) the time of peak intensity for the observed pulse ($t_{\rm
peak}$), (c) the maximal intensity ($J_{m}$) of the pulse and (d) the
asymmetry indicator ($w$)
.
### 3.2 Fitting procedure
In order to follow the relativistic shell evolution and collision, we consider
the case of only one shell expanding from the GRB core. It propagates through
the surrounding media, which can contain a barrier with the mentioned Gaussian
profile. We fit with our model light and energy curves of GRBs given in Table
3.1. In order to find the best fitting we vary the parameters of the faster
shock and barrier. We consider the following parameters as free: the Lorentz
factor $\Gamma_{0}$, the total initial ejected mass of the shell $M_{ej}$, the
density of surrounding media $n_{0}$, the opening angle of the jet
$\theta_{m}$, the distance of collision $R_{c}$ and the parameters $a$ and $b$
which describe the shape (height and width) of the density barrier.
Additionally, we assumed that the barrier can move, thus we also put as a free
parameter the Lorentz factor of the barrier $\Gamma_{b}$. In Fig. 2 (left
panels) we show the best fit of three isolated pulses with different shapes:
GRB000508, GRB911104 and GRB911117. The light curves of these GRBs do not have
a standard form, i.e. the shape of pulses does not always follow the FRED
(Fast Rise Exponential Decay) behavior. As one can see from Fig. 2 the shapes
of the light and energy curves can be very well fitted with the model. In
right panels we show the best fit of the averaged spectral energy distribution
(ASED) taken from all four BATSE channels. Although the data for measured
counts in the energy channels are with large uncertainties, the fit of ASED
can be used for the confirmation of the validity of the GRB light curve fit,
since the same parameters are used to fit both curves.
The coefficients $\xi,\xi_{e}$ and $\xi_{b}$222$\xi$ describes fraction of
total shell energy that has been converted into radiation, $\xi_{e}$ fraction
of the total shell energy devoted to the electron plasma component and
$\xi_{b}$ fraction of total shell energy contained in the magnetic field.,
determine the radiation efficiency for expanding relativistic shell. The
synchrotron radiation is directly proportional to intensity of the magnetic
field, as well as the energy of the electron component of plasma.
Consequently, if one puts high values for those three components it will
increase the intensity of radiation and also the intensity of light curve
pulses. However, the conversion from the kinetic to radiative energy is the
process with relatively low efficiency (see e.g. [Eichler05]), therefore the
above coefficient must not exceed 10 to 20% [Piran05]. They are expected to be
smaller than 0.2. In order to reduce the number of free parameters in the
fitting procedure we fixed them to $\xi=0.1$, $\xi_{e}=0.2$, $\xi_{b}=0.2$
(see also [Zhang07]). Moreover, the different values of these parameters will
mainly affect the intensity of GRB light curve, but not the shape which is
mainly considered in the fitting procedure.
Also, the distribution of electrons in the shell follows a power law function
where the index of the electron distribution $p$ usually takes a value between
2 and 3 [Gallant99]. In the fitting we fix it to $p=2.5$ which well reproduce
the obtained values from the fits of the GRB light and energy curves. In the
fitting procedure we used $\chi^{2}$ minimization.
## 4 Results and discussion
Figure 2: The light (dashed line - left panels) curves and the averaged
spectral energy distribution in four BATSE channels I ch:(20-50)keV, II
ch:(50-100)keV, III ch:(100-300)keV, IV ch:$>$300keV (full circles - right
panels) for GRB911104, GRB911117 and GRB000508 (from top to bottom,
respectively) fitted with the model (solid lines - left panels and dashed
lines right panels).
The parameters of the shells and barriers for 30 GRBs obtained from the best
fittings are given in Table 4. The distribution of parameters are presented in
Fig. 3. As it can be seen from Table 4 (also see Fig. 3) the most of GRB light
curve pulses in our sample indicate an initial Lorentz factor around 95, with
the mean value of about 93. Cases with $\Gamma_{0}<90$ could be explained if
we consider the physical mechanism of the collisions. In the first place, the
initial value of the Lorentz factor directly determines the initial energy and
velocity of the incoming shell. So, for those shells (higher $\Gamma_{0}$)
produced light curve pulses will be more intense and short lasting. On the
contrary, smaller $\Gamma_{0}$ will produce low intensity and long lasting
pulses. This is in a good agreement with the observations of the pulse width -
luminosity correlation found in [Hakkila08].
The internal shell and barrier parameters obtained from the best fitting of
light and energy curves for the sample of 30 GRBs. In the last two rows we put
the mean and appropriate standard deviation of parameters. parameter
$\Gamma_{0}$ $M_{\rm ej}$ $\Gamma_{b}$ $n_{0}$ $\theta_{m}$ $R_{c}$ $\Delta R$
$n_{b}$ units - $\cdot 10^{-11}[M_{\rm sun}]$ - $[\rm cm^{-3}]$ $[\rm rad]$
$\cdot 10^{14}\rm[cm]$ $\cdot 10^{13}\rm[cm]$ $\cdot 10^{4}\rm[cm^{-3}]$
GRB910629 109 5.5 50 43 0.05 2.0 5.1 43. GRB911104 111 10. 43 63 0.06 2.5 5.1
56.7 GRB920715 103 1.3 51 33 0.06 2.2 5.8 33. GRB920720 90 25. 49 70 0.06 1.5
5.8 105. GRB920808 110 8. 50 33 0.07 1.9 7.8 8.3 GRB920811 111 1.1 65 65 0.06
1.2 2.2 58.5 GRB920830 71 13.5 45 37 0.1 3.5 25. 1.1 GRB920912 100 10. 74 70
0.05 2.0 3.5 4.2 GRB920924 125 2. 50 50 0.04 2.5 3.8 150. GRB921021 95 5. 65
20 0.08 3.5 21.7 0.6 GRB921207 73 53. 63 75 0.1 2.5 10. 150. GRB921208 71 3.8
49 70 0.07 1.9 7.3 4.9 GRB921222 87 2.5 70 20 0.06 2.1 7.8 6. GRB950129B 84 4.
70 20 0.05 1.8 7.5 100. GRB950211B 73 18. 56 90 0.05 1.7 8.3 4.5 GRB960111 95
10. 68 45 0.08 1.9 7.7 1.8 GRB960207 109 6.8 63 57 0.06 1.5 3.2 39.9 GRB960229
98 5.2 61 75 0.057 1.7 4.8 9.8 GRB960311 110 13. 77 19 0.04 1.9 5.2 9.5
GRB960409 75 18. 52 110 0.1 5.5 30. 5.5 GRB960418 75 3. 48 50 0.09 2.7 10.7
50. GRB960524 87 2.7 65 50 0.09 2.1 8.8 335. GRB960528 91 5.5 56 43 0.06 2.0
13.3 2.2 GRB960530 62 40. 46 31 0.055 3.8 30.1 0.6 GRB960613 87 13. 63 65 0.04
2.4 10.2 3.9 GRB960617B 103 3. 87 50 0.07 2.3 8.3 150. GRB970424 99 0.9 53 35
0.061 1.5 3. 105. GRB991105 99 6. 57 69 0.07 2.3 6.8 4.8 GRB991213 75 5. 55 10
0.05 2.1 9.3 9. GRB000107 115 4.3 90 10 0.05 1.5 3.3 10. Mean 93 10 60 50
0.064 2.26 9.4 48 Deviation 13.5 7.6 9.6 19.4 0.014 0.57 5.0 52
Figure 3: Histograms for obtained parameters from the best fit: $\Gamma_{0}$,
$M_{\rm ej}$, $\Gamma_{b}$, $n_{o}$, $\theta_{m}$, $R_{c}$ and $\Delta R$ (a-g
panels, respectively).
The total released isotropic energy of a GRB lies in the range from $8\times
10^{47}$ erg (GRB 980425 associated with supernova SN1998bw, see e.g. Ref
[Galama98], [Pian00]), to $2\times 10^{54}$ erg (GRB 990123 Ref.
[Andersen99]). Note here that using the conical jet model [Frail01] found that
the gamma-ray energies are clustered in a narrow interval around $5\times
10^{50}$ erg. The mass of the ejected shell in the most GRB light curve pulses
is in the interval of $\sim 10^{-11}-10^{-10}M_{\rm sun}$, with a mean value
of $10^{-10}M_{\rm sun}$. These could be used to calculate the initial energy
of a particular shell in the moment of the ejection, giving values in the
range from $10^{45}$ to $10^{47}$ergs. The sum of energies of all ejected
shells during the GRB event can indicate the total energy released by the
central engine in the particular GRB.
Additionally we assume that the barrier is moving, with the Lorentz factor
$\Gamma_{b}$. In the first approximation we take that its velocity is constant
until the collision. This parameter has multiple influence mainly on the
intensity and width of the created light curve pulse, as well as on the FRED
pulse shape. For the faster barriers interaction is long-lasting, but with a
low intensity. On the other hand, if the barrier is moving significantly
slower than the incoming shell, it will cause a more intense interaction
followed by the short-lasting and very strong pulses. The FRED shape is more
dominant in the former case.
The density of the ISM in the region around the central engine is assumed to
be homogeneous $(s=0)$, with a density approximately an order of magnitude
higher than at the distances where afterglow starts (see for example
[Kobayashi04]). We obtained values from several to few tens of particles per
cm-3 (see Table4) that is in the expected range $n_{0}\simeq(10^{0}-10^{3})$
cm-3. The distribution of $n_{0}$ (Fig. 3d) for our sample has a bell-like
shape with the maximal value around 60 cm-3. This values is more appropriate
to hypernova then to a merger scenario, because in the merger scenario one can
expect significantly smaller densities [Hakkila08].
Similar distribution is obtained for opening angle of relativistic shells
$\theta_{m}$, which is considered to be constant during the evolution. The
most probable values are around 0.05 radians ($\approx 3^{o}$) (see [Frail01]
and reference therein). Existence of this particular value is determined by
physical processes in the vicinity of the GRB central engine. Namely, if one
take a higher value of $\theta_{m}$, the resulting pulse is broader and a slow
decay feature is much more visible than in the case of smaller angles, where
the pulse is thinner and has a symmetrical shape. That is in a good agrement
with engaged physical processes during the shell expansion.
We suppose that the shell interaction occurs mainly close to the GRB engine,
(distance $R_{c}\sim 10^{14}\ \rm cm$), as it was proposed in the internal
shock scenario (see [Piran05]). This parameter has very small influence on the
shape of GRB pulses, but has an influence on the intensity of pulses. Pulses
produced in a collision closer to the GRB engine are more intense than ones
originating at larger distances.
The parameters of the Gaussian of the density distribution which describe the
structure of the barrier, the width at the half maximum $b$ and the density in
the central part $a$, both influence the shape of GRB light curve pulses. They
are translated into appropriate variables, number density $n_{b}$ and width
$\Delta R$ of the barrier, respectively, using the connections given by Eq. 9
and 10:
$n_{b}=n_{0}(1+a)$ (9)
$\Delta R=2b\sqrt{\ln\frac{2a}{a-1}}$ (10)
Note here that in the case of dense barrier one can expect $n_{b}/n_{0}>>1$
(see also Table 4), consequently $a>>1$ and Eq. 10 can be rewritten as $\Delta
R=2b\sqrt{\ln{2}}$. The influence of the barrier parameters on pulse profiles
is following: a barrier with narrow width and high number density will have a
strong interaction with fast shell, producing symmetrical and intense light
curve pulses. On the contrary, a barrier with larger width and lower number
density will cause small intensity pulses with high asymmetries.
In general, comparing the obtained values of parameters (Table 4) for
different GRBs, one can conclude that there is no significant difference
between them even when the shapes and durations of GRB pulses are different.
This suggests that the nature of GRBs is similar and that there should be no
big difference between the physical conditions of GRB progenitors (see
[Ghirlanda11]). On the other hand, the barrier density distribution can differ
from the Gaussian, that is assumed here, and it may reflect the values of
basic parameters. But in any case, one can expect that the density
distribution of the barrier has to be taken into account in the shock model.
### 4.1 Connection between the shell parameters
In order to find physical meaning of the obtained parameters, we explore
correlations between them. In Table 4 the parameters are divided in two
groups: one that describes a shell and other connected with the barrier. We
expect that parameters from those two groups are not in correlation, since
they are independent. The results are presented in Figs. 4 and 5, where we
separately denoted the long (t$>$2s) and short (t$<$2s) GRBs with open and
full circles, respectively. As one can see in Figs. 4 and 5 there are no
strong correlations between these quantities, but only in some cases, there is
a slight connections between different parameters, as e.g. $\theta_{m}$ vs.
$\Gamma_{0}$ (Fig.4a), $M_{\rm ej}$ vs. $\Gamma_{0}$ (Fig. 4b). In the case of
the barrier parameters a slight correlation can be found in $\Delta R$ vs.
$n_{b}$ (Fig. 5a). Additionally, we examine the correlation which may be
established between Lorentz factors of the incoming fast shock wave
$\Gamma_{0}$ and the moving barrier $\Gamma_{b}$, presented in Figure 5b.
Figure 4: Parameter dependance: $\theta_{m}$ vs. $\Gamma_{0}$ (a) and $M_{\rm
ej}$ vs. $\Gamma_{0}$ (b). The long (t$>2$s) GRBs are denoted with full circle
and short with open circle. Dashed line presents the border above which there
is no parameter values for this sample of GRBs.
Weak correlations or non-correlations can be noticed in each case particulary.
For example, with larger opening angle of the shock wave $\theta_{m}$, the
volume of the shell increases, that causes smaller initial Lorentz factors of
the shell and vice versa (see Fig. 4a). For the certain shell energy, the
increase of the space angle $\theta_{m}$ causes the increase of the ejected
mass $M_{\rm ej}$, and that causes the decrease of the initial $\Gamma_{0}$.
This is in a agrement with the trends presented in Figure 4b, where one can
see that for a higher value of $M_{\rm ej}$, the initial Lorentz factor tends
to be smaller. Of course, one can not expect to see obvious confirmation for
this conclusion with such small sample of examined GRBs, but rather signs of
trends for mentioned dependence.
Figure 5: Parameter dependance: $\Delta R$ vs. $n_{b}$ (a) and $\Gamma_{0}$
vs. $\Gamma_{b}$ (b), with $r$ designating the correlation coefficient. The
notation for short and long pulses is same as in Fig. 4.
In the case of the barrier parameters, there is an indication of the
connection between the particle number density $n_{b}$ and width of barrier
$\Delta R$ as it is shown in Fig. 5a, as $\Delta R\sim 1/n_{b}$. Physically
one can expect such situation, i.e. for a broader barrier, the density of
barrier tends to be smaller and vice versa.
In Fig. 5b the $\Gamma_{0}$ vs. $\Gamma_{b}$ is presented. There is no global
correlation between these Lorentz factors, but taking into account only short
pulses there is some indication that for a faster barrier the $\Gamma_{0}$ of
the shell is higher. The dashed and solid line in Fig. 5b present the linear
dependance of the given parameters, indicating that the incoming shell must
have higher $\Gamma_{0}$ than the slower moving barrier, providing just enough
necessary conditions for event of collision to happen.
### 4.2 Connection between shell parameters and observed pulse parameters
Additionally, we explore possible correlations between parameters obtained
from fitting the light curves and measured ones (given in §3.1). In Fig. 6 we
plot measured values against $\Gamma_{0}\cdot M_{\rm ej}$, which is
proportional to the energy of the incoming shell.
It is interesting that the pulse intensity for small energies
($\Gamma_{0}\cdot M_{\rm ej}<$0.2) shows nearly linear trend with the energy
(see Fig. 6c), but for $\Gamma_{0}\cdot M_{\rm ej}>0.2$ this trend is not
present.
Figure 6: Parameter dependance: FWHM, $\Delta t$, $J_{m}$ and $w$ vs.
$\Gamma_{0}\cdot M_{\rm ej}$ (scaled to the Max[$\Gamma_{0}\cdot M_{\rm
ej}$]), panels a - d, respectively. The notation for short and long pulses is
same as in Fig. 4 and 5.
In Figs 7a and b we present $\Delta t$ vs. $\Gamma_{0}$, $\Gamma_{b}$,
respectively and in Fig. 7c FWHM vs $\Delta R$. It is obvious that for a
faster barrier and shell the interaction will be shorter. Also, for a broader
barrier we obtain long lasting (wider) pulses as it is shown in Fig. 7c. There
is a correlation between FWHM and $\Delta R$, with correlation coefficients r
= 0.61 for short and r = 0.84 for long lasting GRBs. A linear relationship
between FWHM and $\Delta R$ is present as FWHM=a$\cdot\Delta R$ \+ b (where a
= 0.54 and b = 0.22 for short and a = 1.2 and b = 0.024 for long lasting GRBs,
see Fig. 7c).
Figure 7: Correlation of pulse and shell parameters. Case of $\Delta t$ vs.
$\Gamma_{0}$, $\Gamma_{b}$ (panels a, b), and FWHM vs. $\Delta R$, (panel c).
The notation for short and long pulses is same as in Fig. 4 and 5.
We found some connections between $n_{b}$ vs. $\Delta T$ and $n_{b}$ vs.
$J_{m}$, as presented in Fig. 8. This is expected, since for a barrier with
higher density there are more intense pulses (Fig. 8a). The smaller values of
density produce broader light curve pulses (Fig. 8b). It can be seen from Fig.
8a that short pulses have mainly smaller densities. This can be easily
understood analyzing the physical processes in the moment of collision. When
the particle number density is higher more particles are taking part in the
interaction, producing a higher gamma-ray flux. But if the $n_{b}$ decreases
the interaction is prolonged, caused by the stretching of the barrier
material, so long lasting pulses are produced.
Figure 8: Correlation of pulse and shell parameters. Case of $n_{b}$ on
$\Delta T$ (a) and $n_{b}$ on $J_{m}$ (b).
## 5 Conclusions
In this paper we extracted the basic parameters of internal shock waves during
the first phase of a GRB event by fitting 30 observed GRBs (from BATSE
database). To fit the observed GRB light curves we used the modified internal
shock model given in [Simic07], assuming the collision of a fast shell with a
slow moving (with respect to the velocity of the shell) barrier. We analyzed
the obtained parameters in order to find physical processes behind the GRB
origin, and we came to the following conclusions:
(i) Relativistic shell parameters obtained from the fitting of GRB light
curves are in a good agreement with expected ones and also with estimations
given earlier by other authors mentioned throughout the text.
(ii) The obtained values of internal shell physical parameters for GRBs with
different light curves are in the short interval, showing that the physical
processes behind the GRB creation are similar, i.e. there should be the
ejected mass that collides with surrounding regions - or accumulated slow
moving material.
Also, we analyzed possible connections between parameters obtained from the
best fitting of GRB light curves with measured ones. From this analysis we can
conclude:
(i) There is no correlation between parameters obtained from the best fitting
(Figs 4 and 5), only some indication that long GRBs have higher values of
Lorentz factor, and we found a slight trend between Lorentz factor of the
shell and moving barrier for short pulses.
(ii) There is a correlation between the intensity of pulses and the energy
density of the shell only for a low energy pulses ($\Gamma_{0}\cdot M_{\rm
ej}<0.2$, see Fig. 6c).
(iii) The FWHM of GRB light curve pulses is in the correlation with the width
of the barrier. Using this we give a relation between FWHM (that can be
measured from observed light curves) and $\Delta R$ that is a parameter of the
model (see Fig. 7b).
Finally, we can conclude that the modified internal shock model that assumes
also the barrier density distribution can well describe the first phase of the
GRB origin. Moreover, the obtained parameters for the internal shocks and
barriers well fit the physics of the GRBs.
## Acknowledgments
The work was supported by the Ministry of Education and Science of R. Serbia
through the project ”Astrophysical Spectroscopy of Extragalactic Objects”. We
would like to thank to the anonymous referee for very useful comments and
suggestions and D. Ilić for careful reading of this manuscript and comments.
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| 2012-01-19T10:07:59 |
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sasa Simi\\'c and Luka \\v{C}. Popovi\\'c",
"submitter": "Sasa Simic",
"url": "https://arxiv.org/abs/1201.4003"
}
|
1201.4004
|
# Cross-Over between universality classes in a magnetically disordered
metallic wire.
Guillaume Paulin Institut für Theoretische Physik, Universität zu K öln, Z
ülpicher Str. 77, 50937 K öln, Deutschland David Carpentier CNRS -
Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon,
46, Allée d’Italie, 69007 Lyon, France
###### Abstract
In this article we present numerical results of conduction in a disordered
quasi-1D wire in the possible presence of magnetic impurities. Our analysis
leads us to the study of universal properties in different conduction regimes
such as the localized and metallic ones. In particular, we analyse the cross-
over between universality classes occuring when the strength of magnetic
disorder is increased. For this purpose, we use a numerical Landauer approach,
and derive the scattering matrix of the wire from electron’s Green’s function
## 1 Introduction
Interplay between disorder and quantum interferences leads to one of the most
remarkable phenomenon in condensed matter : the Anderson localization of
waves. The possibility to probe directly the properties of this localization
with cold atoms[1, 2] have greatly renewed the interest on this fascinating
physics. In this paper, we focus on the particular situation where electrons
encounter two kinds of disorder: a usual scalar potential at the origin of
diffusion, and a magnetic potential, arising from a collection of frozen
random magnetic moments. This situation is naturally realized experimentally
in the study of transport properties of metallic spin glass wires [3, 4, 5,
6]. In these wires, the spins freeze at low temperatures when entering the
spin glass phase due to the frustrating magnetic couplings. In this glassy
phase, and neglecting any residual Kondo effect in this regime, the impurities
act effectively as a (weak) magnetic potential. We study numerically the
effect of both types of disorder on the statistical properties of the wire
conductance. In particular, we will focus on the experimentally relevant
crossover of (weak) localization properties of the wire as a function of the
magnetic disorder strength.
One dimensional disordered electronic systems are always localized. Following
the scaling theory [7] this implies that by increasing the length $L_{x}$ of
the wire for a fixed amplitude of disorder, its typical conductance ultimately
reaches vanishingly small values. The localization length $\xi$ separates
metallic regime for small length $L_{x}\ll\xi$ from the asymptotic insulating
regime. In the present paper, we focus on several universal properties of both
metallic and insulating regime of these wires in the simultaneous presence of
two kinds of disorder. The first type corresponds to scalar potentials induced
by the impurities, for which the system has time reversal symmetry (TRS) and
spin rotation degeneracy. In this class the Hamiltonian belongs to the so-
called Gaussian Orthogonal Ensemble (GOE) of the Random Matrix Theory
classification [8] (RMT), corresponding to the class AI in the modern
classification of Anderson universality classes (see e.g. [9]). If impurities
do have a spin, the TRS is broken as well as spin rotation invariance. The
Hamiltonian is then a unitary matrix, which corresponds in RMT to the Gaussian
Unitary Ensemble (GUE) with the breaking of Kramers degeneracy [10], and to
the Anderson class A [9]. However, for the experimentally relevant case of a
magnetic potential weaker than the scalar potential, the system is neither
described by the GUE class, nor by the GOE class, but extrapolates in between.
This intermediate regime, of particular relevance experimentally, is the main
object of study of the present paper. Moreover the present work paved the way
towards a numerical study of the correlation of conductances in the cross over
regime [11].
This paper is organized as follows : in section 2, the model and the numerical
method used will be described in details. In section 3 we identify the
localized and metallic regime of transport of the system. The localization
length will be determined by two different methods and the cross over between
both universality classes (GOE, GUE) will be highlighted. In section 4, the
insulating regime is studied with a particular focus on the statistical
distribution of the conductance, which allows us to highlight universal
behavior. In section 5, we turn to the study of the metallic regime, and
perform a careful analysis of each of the first three cumulants of the
statistical distribution of the conductance. We focus on the universal
properties of conductance fluctuations, and the non-analyticity of the
complete distribution is discussed. Finally section 6 is devoted to the
conclusion.
## 2 The model and the method
### 2.1 The model
In this paper, we study numerically the scaling of transport properties of
wires in the presence of magnetic and scalar disorders. We will focus on the
regime of phase coherent transport, reached experimentally at low temperature
(see in particular [6]). In this regime, the phase coherence length $L_{\phi}$
which phenomenologically accounts for inelastic scattering of electrons on
impurities [12] is larger than (or comparable with) the wire’s length $L_{x}$,
so that phase coherence for the propagating electrons is preserved in the
whole sample. Note that this phase coherence length $L_{\phi}$ includes in
particular a contribution from inelastic scattering on the non frozen magnetic
impurities through a Kondo dephasing, which is strongly reduced when entering
the magnetic glass phase [6].
We describe the behavior of electrons inside the disordered wire using a
tight-binding Anderson lattice model with two kinds of disorder potentials:
$\displaystyle\mathcal{H}=\sum_{<i,j>,s}t_{ij}c_{j,s}^{\dagger}c_{i,s}$
$\displaystyle+$
$\displaystyle\sum_{i,s}v_{i}c_{i,s}^{\dagger}c_{i,s}+J\sum_{i,s,s^{\prime}}{\vec{S}}_{i}.{\vec{\sigma}}_{s,s^{\prime}}c_{i,s}^{\dagger}c_{i,s^{\prime}}.$
(1)
$t_{ij}$ is the hopping term from site $i$ to $j$. In the following, $t_{ij}$
will take two different values: $t_{ij}=t_{//}$ in the longitudinal $x$
direction and $t_{ij}=t_{\perp}$ in the transverse $y$ direction. The scalar
disorder potential $V=\\{v_{i}\\}_{i}$ is diagonal in electron-spin space. We
choose the $v_{i}$ to be random scalars uniformly distributed in the interval
$[-W/2,W/2]$. In this work, we have chosen without loss of generality to fix
$t_{//}=1$ so that all energy scales are relative to the bandwidth $t_{//}=1$,
and the amplitude of disorder $W=0.6$. In eq. (1) $s,s^{\prime}$ label the
$SU(2)$ spin of electrons and the $\vec{S}_{i}$ account for spins of the
frozen magnetic impurities. A realistic choice for these frozen spins in a
spin glass phase consist in considering classical spins with random
orientations, thereby neglecting any small spatial correlation in a spin glass
configuration [13]. The coupling $J$ between the electron spins and the
magnetic impurities fixes the amplitude of the magnetic disorder. In this
work, we will monitor the behavior of the transport properties of the sample
as a function of this amplitude $J$, from $J=0$ to $J=0.4$. Indeed, variation
of the amplitude of magnetic disorder $J$ allows to extrapolate from GOE /
class AI for $J=0$ to GUE / class A for $J\neq 0$. We will also show in
section 5 that, as a bonus, the presence of the magnetic disorder allows also
an numerically easier settlement of the universal metallic regime of the wire.
For a given realization of both scalar $\\{v_{i}\\}_{i}$ and magnetic disorder
$\\{\vec{S}_{i}\\}_{i}$, the Landauer conductance of this model on a 2D square
lattice of size $L_{x}\times L_{y}$ is evaluated numerically using a recursive
Landaueur method described in details in the next paragraph.
### 2.2 The method
We have chosen to use a numerical method based on the lattice model (1) as
opposed to e.g. random matrix or Dorokhov-Mello-Pereyra-Kumar (DMPK) [14, 15]
method so as to provide a numerical study allowing for an easy comparison with
the experimental situation[6]. Moreover, this method allows for further
natural developments such as the study of the conductance change upon magnetic
impurities spin flipping, which would be difficult to reach by alternative
method. Starting from a lattice model such as (1), the natural method
providing the conductance of a finite size sample is based on the Landauer
formalism [16].
We consider a two-terminal setup, with electrodes connected to the wire at
$x=0$ and at $x=L_{x}$. These electrodes are described as semi-infinite
ribbons with the same transverse geometry as the sample, and described with
(1) but without randomness. Electrons are then confined in the transverse
$y$-direction via a potential that has the form:
$\displaystyle V(y)$ $\displaystyle=$ $\displaystyle 0\hskip
14.22636pt\mathrm{if}\hskip 14.22636pt0\leq y\leq L_{y}\ ;\ V(y)=\infty\hskip
14.22636pt\mathrm{otherwise}$ (2)
This confining potential in the $y$ direction leads to the appearance in the
electrodes of $N$ modes propagating in the $x$-direction. The complete wave
function of an electron in this tight binding lattice model reads then:
$\psi(x,y)=\phi_{n}(y)e^{\imath k_{x}x},$ (3)
where $k_{x}$ is the momentum of electrons in the longitudinal direction and
$\phi_{n}(y)=\sqrt{\frac{2}{N_{y}+1}}\sin\left(\frac{n\pi y}{N_{y}+1}\right).$
(4)
We used $N_{y}=L_{y}$ in units of lattice spacing. The group velocity of this
mode reads:
$v_{n}=2\frac{t_{//}}{\hbar}\sin\left(k_{x}\right).$ (5)
where $t_{//}$ is the longitudinal hopping amplitude. This velocity depends on
the momentum $k_{x}$ which is determined for a constant energy by the
dispersion relation
$E_{n}=\mu-2t_{\perp}\cos\left(\frac{n\pi}{N_{y}+1}\right)-2t_{//}\cos(k_{x}).$
(6)
At given energy $E-\mu$, we end up with the following relation for the
longitudinal part of the momentum of the electron:
$k_{x}=\arccos\left(\frac{\mu-E}{2t_{//}}-\frac{t_{\perp}}{t_{//}}\cos\left(\frac{n\pi}{N_{y}+1}\right)\right).$
(7)
To optimize the efficiency of the numerical study, we fix $t_{//}=2t_{\perp}$
and stay near the band center, avoiding in particular the presence of
fluctuating states studied in [17].
To compute the conductance of such a wire, the Landauer-Büttiker formalism of
coherent transport is used [18]. It allows to relate the dimensionless
conductance $g$ of a diffusive wire with the scattering matrix $T$:
$g=\sum_{\mathrm{modes}\,m,n}T_{mn},$ (8)
where $T_{mn}$ is the transmission coefficient between modes $m$ and $n$. The
dimensionless conductance $g$ is defined from the conductance $G$ of the
system as $g=G/(2e^{2}/h)$ when spin degeneracy is present ($J=0$), and
$g=G/(e^{2}/h)$ otherwise ($J\neq 0$).
Following Fisher and Lee[19] we relate the scattering matrix to the electronic
retarded Green’s functions of the system through:
$\displaystyle t_{mn}$ $\displaystyle=$
$\displaystyle\imath\hbar\sqrt{v_{n}v_{m}}\sum_{y,y^{\prime}=1}^{N_{y}}\phi_{n}(y)\mathcal{G}^{R}(y,x=0|y^{\prime},x=L_{x})\phi_{m}(y^{\prime}).$
(9)
with $T=Tr(t^{\dagger}t)$, $v_{n}$ and $\phi_{n}$ (resp. $v_{m}$ and
$\phi_{m}$) are the group velocity and the eigen wave function of propagating
mode $n$ (resp. $m$). Mode $n$ belongs to the left lead whereas mode $m$
belongs to the right one. In (9) $\mathcal{G}^{R}(y,x=0|y^{\prime},x=L_{x})$
represents the retarded Green’s function of an electron between the point
$(x=0,y)$ in the left contact between the system and the electrode and the
point $(x=L_{x},y^{\prime})$ in the right contact. Consequently, by
calculating the retarded Green’s functions of the system only between its both
sides, we are able to determine the dimensionless conductance of the wire. An
efficient method to calculate $\mathcal{G}^{R}(y,x=0|y^{\prime},x=L_{x})$,
which takes advantage of the quasi-one dimensional nature of the ribbon
consists in obtaining it recursively[20]. In figure 1 the principle of the
method is sketched: using a Dyson equation we deduce the boundary Green’s
function of a system of size $n+1$ from the corresponding Green’s function of
a subsystem of size $n$, and the exact Green’s function of the $n+1$ row. This
allows to perform matrix inversion only of the simple row system. At both
initial and final steps, we reconnect the system to the semi-infinite
electrodes (see fig. 2) described by a standard self energy:
$\mathcal{G}^{R}_{\mathrm{bound}}(y_{1},y_{2})=-\frac{1}{t_{//}}\sum_{n=1}^{N_{y}}\phi_{n}(y_{1})e^{\imath
k_{x}}\phi_{n}(y_{2}).$ (10)
The last step consists in combining the Green’s functions of the sample of
desired longitudinal size with the one of the right lead, which is also given
by equation (10).
Figure 1: Principle of recursive calculation of retarded Green’s functions of
the wire. Use of a Dyson: $G^{R}_{n+1}=G^{R}_{n}+G^{R}_{n}VG^{R}_{n+1}$. At
each step the longitudinal length $L_{x}$ is increased by one lattice spacing.
Figure 2: Boundary conditions: the wire is connected to two leads represented
by two semi infinite metallic wires.
The unitarity of the corresponding scattering matrix $\\{T_{mn}\\}_{m,n}$ is
used to monitor the accuracy of the numerical method. Such test yielded
typical relative error of order $10^{-4}$ for a system size $L_{x}=1600$ and
$L_{y}=40$. This method allows one to compute the conductance of a wire of
length $L_{x}$ and of width $L_{y}$ for any given configuration of scalar
disorder $V$ and for any configuration of frozen classical spins
$\\{\vec{S}_{i}\\}_{i}$.
In the next sections, we study the properties of this conductance for one
given configuration of magnetic disorder but for many different configurations
of scalar disorder. Universal properties are identified by varying the
transverse length $L_{y}$ from $10$ to $80$, with the aspect ratio
$L_{x}/L_{y}$ taken from $1$ to $6000$. Typical number of configurations of
scalar disorder $V$ we used were $N_{d}=5000$, with exceptions for the study
of the localization properties where for $L_{y}=10$ we sampled the conductance
distribution for $50000$ different configurations of disorder.
## 3 Localization length
### 3.1 Determination of $\xi$
We start our analysis by a determination of the parameters corresponding to
the metallic and localized regimes, through a careful determination of the
localization length of the system. While experimentally the only accessible
regime of a phase coherent wire is the metallic regime, numerically this
regime is difficult to reach and describe quantitatively, as opposed to the
localized regime. This is related to the extreme reduction in the number of
propagating modes in the numerical system which is associated with a
corresponding reduction of the localization length. Hence in order to clearly
identify the conditions to access universal properties of the transport in the
metallic regime, we start by a detailed determination of this localization
length in the experimentally relevant crossover situation. Afterwards we will
take the opportunity of the present study to describe other characteristics of
the localized regimes in the crossover situation, before turning to our main
interest : the universal metallic regime.
The localization length separates short wires of metallic behavior from a
insulating long wires. A first method to access to the localization length
$\xi$ from the conductance consists in considering the scaling behavior of the
typical conductance $g_{typ}$ defined as:
$g_{typ}=e^{\langle\log g\rangle},$ (11)
where $\langle\cdot\rangle$ represents the average over the different
configurations of scalar disorder $V$. This typical conductance decays
exponentially with the longitudinal length of the wire[21, 22] :
$g_{typ}\sim e^{-\frac{2L_{x}}{\xi}}$ (12)
in the regime of long wires $L_{x}\gg\xi$.
Figure 3: Evolution of $\langle\log g\rangle$ as a function of longitudinal
size for different transverse length ($L_{y}=10,20,30,40$) and $J=0$. The
linear part of the curve allows one to get the localization length $\xi(J=0)$
from the scaling form in the insulating regime $\langle\log
g\rangle=-\frac{2L_{x}}{\xi}$.
Figure 4: Evolution of $\langle\log g\rangle$ as a function of longitudinal
size for different transverse length ($L_{y}=10,20,30,40$) and $J=0.05$. The
linear part of the curve allows one to get the localization length
$\xi(J=0.05)$ from the scaling form in the insulating regime $\langle\log
g\rangle=-\frac{2L_{x}}{\xi}$.
Figure 5: Evolution of $\langle\log g\rangle$ as a function of longitudinal
size for different transverse length ($L_{y}=10,20,30,40$) and $J=0.2$. The
linear part of the curve allows one to get the localization length
$\xi(J=0.2)$ from the scaling form in the insulating regime $\langle\log
g\rangle=-\frac{2L_{x}}{\xi}$.
Figure 6: Evolution of $\langle\log g\rangle$ as a function of longitudinal
size for different transverse length ($L_{y}=10,20,30,40$) and $J=0.4$. The
linear part of the curve allows one to get the localization length
$\xi(J=0.4)$ from the scaling form in the insulating regime $\langle\log
g\rangle=-\frac{2L_{x}}{\xi}$.
Figures 3, 4, 5 and 6 show the behavior of $\langle\log g\rangle$ as a
function of longitudinal length $L_{x}$ for different widths $L_{y}$.
Different curves correspond to different values of magnetic disorder. The
linear fit of the large length part of the curves allow for a precise
determination of the corresponding localization length for each value of
$L_{y}$ and $J$.
A second method to determine this localization length consists in considering
the Lyapunov exponent $\gamma$ of the transfer matrix of the system, following
a standard random matrix theory approach[23, 21, 24]. This exponent can be
deduced from the conductance as
$\gamma(L_{x})=\frac{1}{2L_{x}}\log\left(1+\frac{1}{g(L_{x})}\right),$ (13)
and the localization length follows from its asymptotic behavior :
$\xi^{-1}=\lim_{L_{x}\to\infty}\gamma(L_{x}).$ (14)
On figure 7 and 8 we have plotted the Lyapunov exponent versus the inverse of
the longitudinal length for different values of magnetic disorder. Different
curves correspond to different widths of the wire.
Figure 7: Evolution of Lyapunov exponent with the inverse of longitudinal
length in semi-log plot. Circles correspond to $J=0$, squares to $J=0.05$,
diamonds to $J=0.2$ and triangles to $J=0.4$. The value of the transverse
length is $L_{y}=10$. The localization length can be extrapolated from the
value of $\gamma$ for $L_{x}\to\infty$.
Figure 8: Evolution of Lyapunov exponent with the inverse of longitudinal
length in semi-log plot. Circles correspond to $J=0$, squares to $J=0.05$,
diamonds to $J=0.2$ and triangles to $J=0.4$. The value of the transverse
length is $L_{y}=20$. The localization length can be extrapolated from the
value of $\gamma$ for $L_{x}\to\infty$.
With this method, a simple extrapolation of the curve is necessary to obtain
$\xi$, without any fit. the value of the inverse of the localization length
for an infinite wire. Nevertheless, we have found that this method shows less
accuracy than the preceding one: as shown on figures 7 or 8, the Lyapunov
exponent is still varying for the longest longitudinal length. We find that
both methods give fully compatible results while the Lyapunov exponent method
requires much larger system sizes than the typical conductance method for a
given required accuracy.
A first manifestation of the universality of the Anderson localization classes
appears through the dependance of $\xi$ on the transverse length $L_{y}$ (or
the number of propagating modes). It is expected to follow[21, 25]:
$\xi=(\beta L_{y}+2-\beta)l_{e},$ (15)
with $l_{e}$ the elastic mean free path and $\beta$ encodes the universal
class of the model : $\beta=1$ corresponds to the orthogonal universality
class GOE while $\beta=2$ for GUE. Note that this change in $\beta$ is
accompanied by an artificial doubling of the number of transverse modes
$N_{y}\equiv L_{y}\to 2N_{y}$ due to the breaking of Kramers degeneracy [21].
This effective factor 4 when breaking the spin rotation symmetry has been
discussed in ref. [26, 27] in details, when discussing the magnetic field
dependance of this localization length, in comparison with random matrix and
Non-linear sigma models. Comparison of numerical localization lengths for
different $J$ with (15) is shown in fig 9. Excellent agreement is found for
$J=0$ (GOE class, $\beta=1$) and with the GUE class for $J\geq 0.2$. For
intermediate values of $J\neq 0$ we observe a crossover between the two
extreme GOE and GUE laws, which cannot be described by eq. (15), and for which
no analytical work exists to our knowledge.
From these results, we also notice that the localization regime is reached for
much longer wires in the presence of magnetic impurities (GUE case) than
without (GOE): localization is hampered by the presence of these magnetic
impurities. As we will discuss below, this property helps in observing
numerically the universal weak localization regime and the associated
universal conductance fluctuations.
Figure 9: Evolution of localization length as a function of transverse
length. $l_{e}$ is the mean free path of the diffusive sample. Different
behavior of the localization length if $J=0$ or $J\neq 0$. Inset : Scaling of
the typical conductance $\langle\log g\rangle=-\frac{2L_{x}}{\xi}$.
### 3.2 The Insulating and Metallic regimes
The localization length discriminates between both insulating and metallic
regimes: the ribbon behaves indeed as a metal ($g\gg 1$) for lengths
$L_{x}\ll\xi$ and as an insulator ($g\ll 1$) if $L_{x}\gg\xi$. In both
asymptotic regimes the shape of the Probability Density Function (PDF) of the
conductance which is known: it is Log-normal for insulating wires[21] and
gaussian for metallic ones. By varying the longitudinal length we can study
the evolution of this PDF from a gaussian to a log-normal distribution, as
shown on figure 10. This plot is done for a given value of $L_{y}$ and $J$.
One can notice that in the metallic regime, the PDF is very well approximated
by a gaussian for relatively small wires : the gaussian regimes is easily
reached, whereas it takes length much larger than the localization length for
the distribution to become log-normal in the insulating regime. This point
will be discussed more precisely below on the cumulants of this PDF.
Figure 10: Evolution of the statistical distribution of the conductance for
different longitudinal sizes for $L_{y}=10$ and $J=0$. $N$ is the number of
disorder configurations used. Plain lines are gaussian (if $\langle
g\rangle>1$) or log-normal (if $\langle g\rangle<1$) interpolations of
numerical data.
The insulating regime is then characterized by $\langle g\rangle<1$ and the
metallic one by $\langle g\rangle>1$.
In order to characterize samples by the average $\langle g\rangle$, and in
particular plots higher cumulants as a function of $\langle g\rangle$, we now
turn to a short study of the behavior of this first cumulant as a function of
the system size. On figure 11 we have plotted $\langle g\rangle(L_{x})$ for
different values of magnetic disorder $J$ (hence different localization
lengths). These curves approximately collapse when plotted against the scaling
variable $L_{x}/\xi$, as show on fig. 12. we remind the reader that for $J=0$,
the average conductance is defined by $g=G/2G_{0}$, which explains why $J=0$
and $J=0.4$ curves coincide on fig. 11: according to figure 9, only for these
values of magnetic disorder universality classes are reached.
Figure 11: Evolution of average conductance versus longitudinal length for
different values of magnetic disorder for $L_{y}=10$ and $J=0,0.05,0.1,0.2$
and $0.4$. The line $\langle g\rangle=1$ is plotted as a frontier between
insulating and metallic regimes.
Figure 12: Evolution of average conductance versus the scaling variable
$L_{x}/\xi$, for $L_{y}=10$ and $J=0,0.05,0.1,0.2$ and $0.4$. All curves
nearly collapse in one single curve. The frontier between insulating and
metallic is drawn again.
This study of $\langle g\rangle(L_{x})$ allows to proceed in the study of
higher cumulants of the PDF of $g$ and test prediction of the single parameter
scaling of distributions[28].
## 4 Universal Insulating Regime
### 4.1 Probability Density Functions
In the insulating regime $L_{x}\geq\xi$, we expect a Log-normal conductance
statistical distribution, as seen previously. However in the weakly insulating
regime $\langle g\rangle\lesssim 1$ this log-normal asymptotic form is not
reached. Instead, as shown in fig. 10 we find a non-analytical behavior of
$P(g)$ in agreement with [29, 30, 31, 32].
Figure 13: Comparison of Probability density functions (PDF) of conductance
for $J=0$ (plain curves) and $J=0.2$ (dashed curves). Plots are performed for
different values of average conductance. (a): $\langle g\rangle(J=0)=0.84$ and
$\langle g\rangle(J=0.2)=0.79$. (b): $\langle g\rangle(J=0)=0.67$ and $\langle
g\rangle(J=0.2)=0.62$. (c): $\langle g\rangle(J=0)=0.45$ and $\langle
g\rangle(J=0.2)=0.42$. (d): $\langle g\rangle(J=0)=0.21$ and $\langle
g\rangle(J=0.2)=0.18$.
Figure 14: PDF of conductance for $\langle g\rangle<1(J=0.2)$ and $\langle
g\rangle>1(J=0)$ and Gaussian interpolations. $L_{y}=10$.
In order to study the dependance of this non-analyticity on the universality
class, we have plotted on figure 13 the distribution $P(g)$ for similar values
of $\langle g\rangle$ but different magnetic strengths $J=0$ (GOE) and $J=0.2$
(GUE). The shapes of these distributions are highly similar if $\langle
g\rangle\ll 1$, showing that distributions for $J=0$ and $J\neq 0$ reaches the
same Log-normal distribution at large system sizes, in agreement with the
super-universality scenario[33]. In the intermediate regime ($\langle
g\rangle\approx 1$), shapes are symmetry dependent. Moreover we find that the
non-analyticity appears for different values of conductance (close to $1$) and
the rate of the exponential decay [29] in the metallic regime seems to differ
from one class to the other (see for instance curves (a) or (b)). Finally, the
plot on figure 14 represents the distribution of conductance of mean value
just above and below the threshold $\langle g\rangle=1$. Plain lines represent
gaussian interpolations with a mean and a variance given by the first and the
second cumulant of each numerical conductance distribution. For $\langle
g\rangle>1$, the gaussian interpolation approximates very well the full
distribution while as soon as $\langle g\rangle<1$, the gaussian approximation
only applies in the tail $g\geq 1$ of the distribution of conductance[32]. The
shape of the distribution for $g<1$ (Figure 13) agrees qualitatively with the
numerical results of [34] (see in particular their fig. 4), Unfortunately, a
more accurate comparison proves to be difficult due to the lack of analytical
description of the distribution. To quantize further these results on the
whole distribution of $\log g$ we now turn to a quantitative study of the
second and third cumulant of this distribution.
### 4.2 Study of cumulants
This conductance distribution converges to the Log-normal only deeply in the
insulating regime, the convergence being very slow (much slower than in the
metallic regime). This qualitative result is confirmed by the study of
moments: in the insulating regime the second cumulant is expected to
follow[35, 25]:
$\langle\left(\log g-\langle\log g\rangle\right)^{2}\rangle=\langle(\log
g)^{2}\rangle_{c}=-2\langle\log g\rangle,$ (16)
Our numerical results are in agreement with this scaling with however very
slow convergence towards this law: corrections are measurable even for the
largest system size where the system is deeply in the localized state, as
shown on fig. 15. More precisely, we find that for the deep insulating regime
$\langle(\log g)^{2}\rangle_{c}=-1.88\langle\log g\rangle$ slope $-1.88$, with
a slight discrepancy with (16).
Figure 15: Plot of the variance of $\log g$ as a function of the mean for the
orthogonal ($J=0$) and unitary ($J=0.2$) case, for $L_{y}=10$ and $20$. The
slope of the linear fit is equal to $-1.88$. This plot shows also super-
universality as the behavior of the second cumulant does not depend neither on
geometry of the wire nor on the universality class.
This plot also shows that in the deep insulating regime the behavior of the
second cumulant as a function of the first one does not depend on the value of
magnetic disorder: both curves for $J=0$ and $J=0.2$ follow the same law. This
is in agreement with our previous result on statistical distributions: there
is a super-universal behavior in the deep insulating regime.
Finally we have studied the third cumulant of $\log g$ scaled in figure 16 as
a function of the first cumulant. The linear behavior for each value of
magnetic disorder in the deep insulating regime ($\langle\log g\rangle<-4$) is
in agreement with the single parameter scaling. We find that contrary to the
second cumulant the coefficient of proportionality between the skewness and
the average depends on the symmetry of disorder, which denotes a lack of
super-universality concerning this cumulant. For instance dots and diamonds
(which correspond to the case $J=0$) have the same behavior, as opposed to the
case $J=0.2$ (squares or triangles). A systematic study of this point with
even larger system sizes and other numerical methods would be of high interest
but is definitely beyond the scope of the present paper.
Figure 16: Plot of the skewness of $\log g$ as a function of the mean for the
orthogonal ($J=0$) and unitary ($J=0.2$) case, for $L_{y}=10$ and $20$.
The study of cumulants of the distribution of $\log g$ confirm the single
parameter scaling of the distribution, with a slight discrepancy concerning
the value of the coefficient of proportionality between second and first
cumulant. Moreover, super-universality has been highlighted concerning the
second cumulant but is lacking concerning the third one.
### 4.3 Comparison with exact results in the cross-over regime
For localization in wires connected to ideal contacts, exact formula for the
average conductance [36] and conductance fluctuations [37] have been derived
for the two universal orthogonal and unitary classes. These formula are of
particular interest in the intermediate regime between the metallic
($L_{x}\ll\xi$) and the deeply localized ($L_{x}\gg\xi$) regime. We have
numerically evaluated the formula (3.105) and (3.106) of ref. [37] and we
compare them with our numerical data in Fig. 17 and Fig. 18. We find an
excellent agreement between the $J=0$ data and the orthogonal exact formulae
on one hand, and the $J\neq 0$ data and the unitary formulae on the other
hand. This comparison naturally breaks down for small sizes where the quasi-1d
assumption for diffusion breaks down and a non universal regime takes place.
As shown on Fig. 18, the exact unitary behavior is recovered for $J=0.05$
beyond the cross-over length $L_{m}$ (see next section). Whether the scale
dependance of $\langle g\rangle$ and $\langle\delta g^{2}\rangle$ in the
cross-over regime ($J$ small such that $L_{m}\simeq\xi$) is amenable to an
exact formula along the lines of ref. [37] is an open question of high
interest.
Figure 17: Comparison of the numerical average conductance $\langle g\rangle$
as a function of the reduced length $L_{x}/\xi$ with the exact expressions of
ref. [37] for the orthogonal ($J=0$) and unitary ($J=0.2$) case, for various
transverse sizes $L_{y}$.
Figure 18: Comparison of the numerical average conductance fluctuations
$\langle\delta g^{2}\rangle$ as a function of the reduced length $L_{x}/\xi$
with the exact expressions of ref. [37] for the orthogonal ($J=0$) and unitary
($J=0.4$) case, for various transverse sizes $L_{y}$.
## 5 Universal Metallic Regime
We now focus on the universal metallic regime described by weak localization.
By definition weak localization corresponds to metallic diffusion, expected
for lengths of wire $l_{e}\ll L_{x}\ll\xi$. For this regime to be reached, we
thus need to increase $\xi$ through an increase of the number of transverse
modes $L_{y}$ with all other parameters fixed (see eq. (15)). Moreover, for a
fixed geometry, this regime will be easier to reach with magnetic impurities
than without. As we saw on fig. 13, the shape of the PDF of conductance is a
truncated gaussian in this regime. In the following we study its three first
cumulants quantitatively, starting with the variance.
### 5.1 Conductance Fluctuations and Universal Crossover
In the weak localization regime, the conductance fluctuations $\langle
g^{2}\rangle_{c}$ are expected to be independent on the size of the system,
and only depend on the universal localization class of the model. The figure
19 shows that for a suitable value of transverse length, the system reaches
the expect plateau in conductance fluctuations. The value of the plateau
identifies with the expected values ($1/15$ and $4/15$ for GUE and GOE
respectively) with a high accuracy. However, the presence of this plateau
depends strongly on the value of transverse length $L_{y}$ and on the magnetic
disorder. Figures 20 and 21 illustrates this point with further details: these
plots show the conductance fluctuations as a function of longitudinal size for
$J=0$ and $J=0.2$ and for two values of transverse length $L_{y}$. On the
first plot, for both values of magnetic disorder the universal plateau arises,
whereas it appears only for $J=0.2$ if $L_{y}=40$.
The evolution of this variance of the PDF of $g$ depends on the longitudinal
length through [12]:
$\displaystyle\langle\delta g^{2}\rangle=\langle g^{2}\rangle_{c}$
$\displaystyle=$
$\displaystyle\frac{1}{4}F\left(0\right)+\frac{3}{4}F\left(x\sqrt{\frac{4}{3}}\right)+\frac{1}{4}F\left(x\sqrt{2}\right)+\frac{1}{4}F\left(x\sqrt{\frac{2}{3}}\right),$
(17)
where $x=L_{x}/L_{m}$ and the scaling function $F(x)$ depends only on
dimension[12, 11]. The universality occurs in this equation in the two limit
$x=0$ corresponding to $J=0$ or $x\gg 1$ where $F(x)\to 0$. In both cases the
variance becomes geometry independent. Moreover this expression shows that the
whole crossover between the two classes is described by a universal crossover
function parametrized solely by the length $L_{m}$, called the magnetic
dephasing length scale [12]. In the figure 22, these conductance fluctuations
are plotted as a function of longitudinal length $L_{x}$ for different values
of $J$. A single parameter fit by (17) provides the determination of the
magnetic dephasing length $L_{m}$ as a function of the magnetic disorder $J$.
The Universal behavior is also highlighted for strong enough magnetic
disorder. The determination of this scattering length allows a precise study
of average conductance, and in particular the study of the classical part as
described in the next sub section below. Fig. 23 shows the scaling form of
these fluctuations (as a function of $L_{x}/L_{m}(J)$) in excellent agreement
with the theory (17). Moreover, for long wires (and large values of $J$)
conductance fluctuations are no longer $L_{x}$ dependent and equal to $1/15$,
in units of $G_{0}^{2}$. This is the so-called Universal Conductance
Fluctuations (UCF) regime which is precisely identified numerically in the
present work.
Figure 19: Second cumulant of $g$ as a function of first cumulant of $g$
showing the universal behavior in the metallic regime. Different curves
correspond to different transverse lengths ($L_{y}=10,40,80$) or magnetic
disorder ($J=0,0.2$).
Figure 20: Second cumulant of $g$ as a function of longitudinal length, for
$L_{y}=80$. Dots represent data for $J=0$ and squares for $J=0.2$. The value
of UCF is shown in each symmetry class (with or without magnetic disorder).
UCF regime is reached in both cases.
Figure 21: Second cumulant of $g$ as a function of longitudinal length, for
$L_{y}=40$. Dots represent data for $J=0$ and squares for $J=0.2$. The value
of UCF is shown in each symmetry class (with or without magnetic disorder. UCF
regime is reached in the case of magnetic impurities but not for scalar
impurities.
Figure 22: Variance of $g$ as a function of longitudinal size. UCF are shown.
Different curves correspond to different values of magnetic disorder $J$
($J=0.025\to 0.4$). Transverse length $L_{y}=40$. The only free parameter in
analytical fits is the magnetic length $L_{m}$.
Figure 23: Variance of $g$ as a function of $L_{x}/L_{m}$. Transverse length
$L_{y}=40$.
The plot of figure 19 confirms analytical results from [31] both qualitatively
in the shape of the curves and quantitatively in the values of fluctuations in
both universality classes. In our study, values of UCF are reached with a
maximal error of $1\%$ for GOE and $3\%$ for GUE with respect to the
analytical value of the UCF in the regime independent of $\langle g\rangle$
(i.e with much higher precision than e.g [38] and [30]).
Let us finally comment the work of Z. Qiao et al. [33] who have performed a
similar numerical Landuer study of 1D transport in various universality
classes, focusing mostly on the metallic regime. While both our study agree on
the UCF (although we have a higher accuracy for $\beta=1$), we did not find
evidence for a second universal conductance plateau. This result would
definitely deserves further study.
### 5.2 The average conductance
The main contribution to the average conductance is of classical origin.
However a weak localization correction must be added when the quantum behavior
of electrons is taken into account [12]. This quantum part manifests itself in
the magneto-conductance behavior of long wires (larger than the the phase
coherence length) where a weak magnetic field is sufficient to destroy this
quantum correction by dephasing the various diffusing path with respect to
each other[12]. We can write this average conductance as (without any magnetic
field):
$\langle g\rangle(J,L_{x},L_{m})=g_{class}(J,L_{x})+\delta
g_{WL}(L_{x},L_{m}),$ (18)
where $g_{class}$ is the classical part of the conductance and $\delta g_{WL}$
is the weak localization correction. For a quasi one dimensional system the
quantum correction reads the simple form[12]
$\delta
g_{WL}=\sum_{n=1}^{\infty}\left(\frac{-1/\pi^{2}}{n^{2}+2\left(\frac{L_{x}}{L_{m}}\right)^{2}}-\frac{3/\pi^{2}}{n^{2}+\frac{2}{3}\left(\frac{L_{x}}{L_{m}}\right)^{2}}\right).$
(19)
The knowledge of the magnetic length $L_{m}$ we gained in the previous study
of conductance fluctuations can now be used to completely characterize this
weak localization contribution to the conductance. By subtracting the
corresponding contribution to the average conductance, we obtain the classical
conductance, plotted in figure 24 as a function of longitudinal length.
Figure 24: Evolution of classical conductance $g_{class}=\langle
g\rangle-\delta g$ with the longitudinal length for $L_{y}=40$ and
$J=0.05,0.1,0.2,0.4$. Numerical data are well fitted using the conductivity as
the only free parameter. different data correspond to different values of
magnetic disorder. In inset is shown the set of numerical data for $J=0.2$.
The dotted line is the analytical fit without taking into account the contact
conductance. Plain line is the same analytical fit as in the main plot.
Taking into account the contact resistance[39] in the two terminal setup, the
expected expression for this classical conductivity reads
$g_{class}(J,L_{x})=\frac{1}{\frac{1}{L_{y}}+\frac{L_{x}}{\sigma_{0}(J)}}.$
(20)
Figure 24 shows this expression plotted for different values of magnetic
disorder. The corresponding fitting parameter $\sigma_{0}$ is plotted as a
function of $J$ in figure 25.
Figure 25: Evolution of conductivity $\sigma_{0}$ with magnetic disorder $J$
for $L_{y}=40$. Dots are numerical data. Plain line is the theory given by
Einstein relation and Matthiesen rule for the conductivity. Agreement is good
especially at low magnetic disorder.
To compare these results with theory, consider the Einstein relation which
links the (Einstein) conductivity to the diffusion constant:
$\sigma_{0}=se^{2}\rho_{0}(\varepsilon_{F})D,$ (21)
where $s$ is the spin degeneracy and $\rho_{0}(\varepsilon_{F})$ the
electronic density of states at the Fermi level. By definition, the diffusion
coefficient reads, for non magnetic impurities:
$D=v_{F}^{2}\tau_{e},$ (22)
with $v_{F}$ the Fermi velocity and $\tau_{e}$ the elastic scattering time. It
can be related to the scalar disorder by:
$\tau_{e}=\frac{1}{2\pi\rho_{0}n_{i}v_{0}^{2}}$ (23)
where $n_{i}$ is the impurity density and $v_{0}^{2}=W^{2}/12$. For more than
one diffusive process, it is compulsory to use the Matthiesen rule that
modifies scattering time $\tau_{e}$ in the following way:
$\frac{1}{\tau_{e}}\to\frac{1}{\tau_{e}}+\frac{1}{\tau_{m}},$ (24)
where $\tau_{m}=L_{m}^{2}/D$ and is related to the magnetic disorder:
$\tau_{m}=\frac{1}{2\pi\rho_{0}n_{i}J^{2}\langle S^{2}\rangle}.$ (25)
Using this allows one to get the $J$ dependance of the Einstein conductivity:
$\sigma_{0}(J)=\frac{\sigma_{0}(J=0)}{1+\frac{3}{W^{2}}J^{2}}.$ (26)
In figure 25 we compare this expression with numerical evaluation of the
conductivity. The good agreement between both curves provides an additional
check of the correct determination of the magnetic dephasing length $L_{m}$.
Including the magnetic disorder dependance of the diffusion coefficient
(through Matthiesen rule), we obtain a perturbative expression to second order
in $J$ for this magnetic dephasing length:
$L_{m}(J)=\sqrt{D(J)\tau_{m}(J)}\propto\frac{1}{J\sqrt{\frac{W^{2}}{12}+\frac{J^{2}}{4}}}.$
(27)
Figure 26: Evolution of magnetic dephasing length $L_{m}$ with magnetic
disorder $J$ for $L_{y}=40$. Dots are numerical data extracted from the study
of UCF and diamonds are numerical data extracted from the study of conductance
correlations for two different spin configurations. Plain and dotted lines are
analytical fits from perturbation theory at second order in $J$. Error bars
are smaller than dots and diamonds sizes.
On figure 26 we have plotted the numerical evaluation of the magnetic length
as a function of magnetic disorder and the corresponding fit with eq.(27). We
notice that it is also possible to obtain the magnetic length via the study of
correlations of conductance, i.e via the study of $\langle
g(V,\\{\vec{S}_{i}^{(1)}\\}_{i})g(V,\\{\vec{S}_{i}^{(2)}\\}_{i})\rangle_{c}$,
which goes beyond the scope of the present paper[11].
### 5.3 The third cumulant
Finally we consider the third cumulant of the distribution of conductance.
According to the analytical study of [31], this cumulant decays to zero in a
universal way as $\langle g\rangle$ increases. Here in figure 28 we find a
dependance of this decrease on the symmetry class: for GOE $\langle
g^{3}\rangle_{c}$ goes to zero in a monotonous way whereas it decreases,
changes its sign and then goes to zero in GUE case. For $\langle g\rangle>4$
numerical errors are dominant, then this part of the curve is irrelevant.
Moreover, for GUE this decrease seems to be universal whereas it depends on
the transverse length for GOE.
Figure 27: Plot of $\langle g^{3}\rangle_{c}$ as a function of $\langle
g\rangle$ in the metallic regime, averages are performed with various a number
of configurations $N_{des}$. Convergence curves are shown for $L_{y}=10$ and
$J=0$ or $J=0.2$.
On figure 27, is represented the convergence of the skewness when increasing
the number of configurations used to perform averages $N_{des}$ for both GOE
and GUE. Plots show a good enough convergence of averages to conclude that the
third cumulant of conductance is not zero for all values of $\langle
g\rangle$. Notice that the maximal number of averages is 50000.
Figure 28: Plot of $\langle g^{3}\rangle_{c}$ as a function of $\langle
g\rangle$ in the metallic regime for $L_{y}=10,80$ and $J=0,0.2$.
Moreover this fast vanishing of the third cumulant confirms the faster
convergence of the whole distribution towards the gaussian, compared to what
happens in the insulating regime. Based on our numerical results, we cannot
confirm nor refute the expected law $\langle g^{3}\rangle_{c}\propto 1/\langle
g\rangle^{n}$, with $n=2$ in GOE and $n=3$ in GUE [40, 41].
## 6 Conclusion
To conclude we have conducted extensive numerical studies of electronic
transport in the presence of random frozen magnetic moments. Comparing and
extending previous analytical and numerical studies, we have identified the
insulating and metallic regimes described by the universality classes GOE and
GUE. We have paid special attention to the dependance on this symmetry of
cumulants of the distribution of conductance in both metallic and insulating
universal regimes. In particular, we have identified with high accuracy the
domain of universal conductance fluctuations, and determined its extension in
the present model. We have also determined precisely the so-called magnetic
length $L_{m}$ which represents the elastic scattering length of the spin on
magnetic impurities. This length is of primary importance in experiments as it
controls the crossover between universality classes. This work paves the way
for further studies of transport in metals with frozen magnetic impurities as
we have clearly identified the range of the parameters to access the
experimentally relevant metallic diffusive regime. One possible extension
consists in considering evolution of the statistics of conductance as the
magnetic disorder is varied, e.g. by rotating or flipping the spins of
impurities. Comparing the conductance obtained in both spin configurations
mimics the experimental measurement of the conductance of a low temperature
canonical spin glass after two successive quenches [42, 11], without the
necessary restrictions of analytical approaches[11]. Experimentally, this
approach could give access to fundamental properties of a spin glass, that
have never been measured.
We thank X. Waintal for useful discussions. This work was supported by the ANR
grants QuSpins and Mesoglass. All numerical calculations were performed on the
computing facilities of the ENS-Lyon calculation center (PSMN).
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|
arxiv-papers
| 2012-01-19T10:22:15 |
2024-09-04T02:49:26.487842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guillaume Paulin and David Carpentier",
"submitter": "Guillaume Paulin",
"url": "https://arxiv.org/abs/1201.4004"
}
|
1201.4046
|
….
# X-rays as dominant excitation mechanism of [Fe ii] and $\rm H_{2}$ emission
lines in active galaxies
Oli L. Dors Jr.1, Rogemar A. Riffel2, Mónica V. Cardaci3,4,5, Guillermo F.
Hägele3,4,5, Ângela C. Krabbe1, Enrique Pérez-Montero6, Irapuan Rodrigues1
1 Universidade do Vale do Paraíba, Av. Shishima Hifumi, 2911, Cep 12244-000,
São José dos Campos, SP, Brazil
2 Universidade Federal de Santa Maria, Av. Roraima, 1000, Cep 97105-900, Santa
Maria, Brazil
3 Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET),
Argentina.
4Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de la La
Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina.
5 Departamento de Física Teórica, C-XI, Universidad Autónoma de Madrid, 28049
Madrid, Spain.
6 Instituto de Astrofísica de Andalucía (CSIC), PO Box 3004, 18080 Granada,
Spain E-mail:olidors@univap.br
###### Abstract
We investigate the excitation mechanisms of near-infrared [Fe ii] and $\rm
H_{2}$ emission lines observed in Active Galactic Nuclei (AGNs). We built a
photoionization model grid considering a two-component continuum, one accounts
for the Big Bump component peaking at $\rm 1Ryd$ and another represents the
X-ray source that dominates the continuum emission at high energies.
Photoionization models considering as ionizing source a spectral energy
distribution obtained from photometric data of the Sy 2 Mrk 1066 taken from
the literature were considered. Results of these models were compared with a
large sample of observational long-slit and Integral field Unit (IFU)
spectroscopy data of the nuclear region for a sample of active objects. We
found that the correlation between the observational [Fe
ii]$\lambda$1.2570$\,\mu$m/Pa$\beta$ vs.
H${}_{2}\lambda$2.1218$\,\mu$m/Br$\gamma$ is well reproduced by our models as
well as the relationships that involve the H2 emission line ratios observed in
the spectroscopic data. We conclude that the heating by X-rays produced by
active nuclei can be considered a common and very important mechanism of
excitation of [Fe ii] and $\rm H_{2}$.
###### keywords:
galaxies: Seyfert – galaxies: ISM – infrared: galaxies
††pagerange: X-rays as dominant excitation mechanism of [Fe ii] and $\rm
H_{2}$ emission lines in active galaxies–References
## 1 Introduction
The excitation of the Narrow Line Region of Seyfert (Sy) galaxies can reveal
how radiation and mass outflows from the nucleus interact with circumnuclear
gas. In particular, near-infrared (hereafter near-IR) observations are a
powerful tool to investigate this issue, because the obscuration –which can
affect the optical morphology of the emitting gas region– is less important at
these wavelengths (Mulchaey et al., 1996; Ferruit et al., 2000). Relevant
emission-lines in the near-IR include [Fe ii] $\lambda\,1.2570\,\mu$m and
$\lambda\,1.6440\,\mu$m, H i lines such as Pa$\beta$, and H2 at
$\lambda\,1.9576\,\mu$m, $\lambda\,2.1218\,\mu$m, and $\lambda\,2.3085\,\mu$m,
which can be used to map the gas kinematics and excitation (e.g. Riffel &
Storchi-Bergmann, 2011b; Riffel et al., 2010). Nevertheless, the dominant
excitation mechanisms of the [Fe ii] and H2 emission lines in the central
regions of active galaxies are still unclear and have been the subject of
several recent studies (e.g. Riffel et al., 2010, 2008, 2006; Storchi-Bergmann
et al., 2009; Hicks et al., 2009; Müller Sánchez et al., 2009; Ramos Almeida
et al., 2009; Rodríguez-Ardila et al., 2005; Rodríguez-Ardila et al., 2004;
Davies et al., 2007).
The H2 can be excited by two mechanisms: (i) fluorescent excitation through
absorption of soft-UV photons (912–1108 Å) in the Lyman and Werner bands,
existing both in star-forming regions and surrounding the Active Galactic
Nuclei (AGNs) (Black & van Dishoeck, 1987) and (ii) collisional excitation due
to the heating of the gas by shocks, the interaction of a radio jet with the
interstellar medium (Hollenbach & McKee, 1989), or by X-ray photons from the
central AGN (Maloney et al., 1996). Several studies based on intensity-line
ratios (Riffel et al., 2010; Storchi-Bergmann et al., 2009; Rodríguez-Ardila
et al., 2005; Rodríguez-Ardila et al., 2004) have shown that collisional
excitation processes dominate the H2 emission surrounding AGNs. However, which
is the dominant process is an open question. Veilleux et al. (1997), using J
and K-band spectra of a sample of 33 Sy 2 galaxies, found that shocks
associated with nuclear outflows are a likely source of both [Fe ii] and $\rm
H_{2}$ emission rather than circumnuclear starbursts, as suggested by Quillen
et al. (1999).
For the [Fe ii] emission, the [Fe ii] $\lambda\,1.2570\,\mu$m/Pa$\beta$ line
ratio is generally used to investigate the main mechanism of excitation. The
value of this line ratio is controlled by the quotient of the volumes of
partially and fully ionized gas regions, with [Fe ii] emission being excited
in the partially ionized gas (Mouri et al., 1990; Mouri et al., 1993;
Rodríguez-Ardila et al., 2005; Riffel et al., 2010, 2008, 2006; Storchi-
Bergmann et al., 2009). Such zones in AGNs are created by X-ray emission (e.g.
Simpson et al., 1996) and/or shock heating of the gas by mass outflows from
the nuclei which interact with the ambient clouds (e.g. Forbes & Ward, 1993).
This problem was addressed by Mouri et al. (2000), who compared the values of
the line ratios [Fe ii] $\lambda\,1.2570\,\mu$m/Pa$\beta$ and [O i]
$\lambda\,6300$ Å/H$\beta$ predicted by models, considering photoionization
and shock heating, with those observed in a sample of AGNs and Starburst
galaxies. These authors pointed out that in AGNs, X-ray heating is the most
important [Fe ii] excitation mechanism. However, Rodríguez-Ardila et al.
(2004), using near-IR spectroscopy of a sample of galaxies obtained with the
Infrared Telescope Facility, found that X-ray excitation is enough to explain
the H2 emission and part of the [Fe ii] emission observed in Sy 1 galaxies,
but fails to explain the emission of these elements in Sy 2. For these
objects, a combination of shocks and circumnuclear star-formation is required
to explain these emissions. Moreover, it is not clear whether the [Fe ii] and
H2 are excited by the same mechanism. Rodríguez-Ardila et al. (2004) found a
correlation between the [Fe ii] $\lambda\,1.2570\,\mu$m/Pa$\beta$ and the H2
$\lambda\,2.1218\,\mu$m/Br$\gamma$ ratios, indicating that both sets of lines
may be originated by a single dominant mechanism. However, high spatial
resolution spectroscopy data from Integral Field Unit (IFU) of active galaxies
indicate that the H2 and the [Fe ii] emitting gas have distinct flux
distributions and kinematics, with the former being considered a tracer of the
feeding of the AGN and the latter a tracer of its feedback (Riffel et al.,
2010, 2009, 2008; Storchi-Bergmann et al., 2009; Hicks et al., 2009; Müller
Sánchez et al., 2009). This result indicates that the lines of these elements
are formed in distinct regions. Although several works have investigated the
excitation origins of H2 and the [Fe ii], it is still unknown whether a common
mechanism can excite these elements. Fortunately, a large number of near-IR
data of AGNs are currently available in the literature, which enables an
extensive comparison with models yielding a more reliable conclusion about the
likely dominant excitation mechanism of these emission lines.
In this paper, we combined near-IR data of Sy galaxies obtained with IFU and
long-slit spectroscopy with photoionization models to investigate the origin
of the H2 and [Fe ii]. In Section 2, we describe the observational data used
in the analysis. The modelling procedures are presented in Sect. 3. In Sect.
4, the diagnostic diagrams used to compare the observational data with our
model predictions are described. Results and discussion are presented in
Sects. 5 and 6, respectively. A conclusion of the outcome is given in Sect. 7.
## 2 Observational data
We compiled from the literature observational data of the nuclear region of
active galaxies in the near-IR and optical spectral range obtained with long-
slit and IFU spectroscopy. The selection criterion was the presence of bright
infrared emission lines in their spectra. These data are described below.
### 2.1 Long-slit data
Near-IR emission line intensity ratios of 35 active galaxies were obtained
from Rodríguez-Ardila et al. (2004), Reunanen et al. (2002), Knop et al.
(2001), and Riffel et al. (2006). This sample comprises long-slit data of 13
Sy 1 and 21 Sy 2 galaxies, along with 1 Quasar. The intensities of the near-IR
[Fe ii] and H2 emission lines observed in these objects were compared with our
photoionization models. We also used the [O iii] $\lambda$ 5007 Å/H$\beta$ and
[O i] $\lambda$ 6300 Å/H$\alpha$ line intensity ratios of about 600 000
emission-line galaxies listed in the MPA/JHU Data catalogue of the Sloan
Digital Sky Survey DR7 release (available at http://www.mpa-
garching.mpg.de/SDSS/DR7/).
### 2.2 IFU data
For this study we selected two Sy 1 galaxies, Mrk 1157 and NGC 4151, and two
Sy 2 galaxies (ESO 428-G14 and Mrk 1066). All of them were previously observed
by our group using the IFU spectrographs of the Gemini telescopes. We selected
these objects because both J- and K-band spectroscopic data are available. The
observations of Mrk 1066, Mrk 1157, and NGC 4151 were performed using the
Near-IR Integral field Spectrograph (NIFS; McGregor et al., 2003) on Gemini
North, while ESO 428-G14 was observed with the Gemini Near Infra-Red
Spectrograph (GNIRS; Elias et al., 1998) on Gemini South.
## 3 Photoionization model
To analyse the [Fe ii] and $\rm H_{2}$ excitation mechanisms, we built a grid
of models using the photoionization code Cloudy/08 (Ferland et al., 1998), and
then we compared the line intensity ratios predicted by them with those
observed. The spectral energy distribution (SED) of the ionizing source used
as input for the Cloudy code was a two-component continuum ranging from $\sim
10^{15}$ Hz to $\sim 10^{21}$ Hz. The shape of this SED is similar to the one
observed in typical AGNs for that range. The first is the Big Bump component
peaking at $\rm 1\>Ryd$ with a high-energy and an infrared exponential cutoff
and the second one represents the X-ray source that dominates at high energies
and is characterized by a power law with an index $\alpha_{x}=-1$. Its
normalization was computed to produce the required value of the optical to
X-ray spectral index $\alpha_{ox}$. This index describes the continuum between
2 keV and 2500 Å (Zamorani et al., 1981). We assumed the default value of the
Cloudy code $\alpha_{ox}=-1.4$, because that is about the average of the
observed values, which are between -1.0 and -2.0, for the entire range of
observed luminosities of AGNs (Miller et al., 2011; Zamorani et al., 1981).
The cosmic ray emission was considered in the models as a second ionizing
source. Cosmic rays heat the ionized gas and produce secondary ionizations in
the neutral gas, which mostly increase the intensities of the $\rm H_{2}$
emission lines. We assumed a value of the $\rm H_{2}$ ionization rate of
$10^{-15}\>\rm s^{-1}$, which is about the same rate found by McCall et al.
(2003) for a galactic line of sight. It is worth noting that the value of the
cosmic ray rate must be estimated object by object. For example, Suchkov et
al. (1993) found for M 82 a cosmic ray rate several times larger than the one
in the Milk Way. Gamma ray observations of the starburst NGC 253 by Acero et
al. (2009) indicate a cosmic ray rate three orders of magnitude larger than
that for the Milky Way. Also, molecular data of star forming galaxies, such as
Arp 220, show evidence for extremely high cosmic ray rates yielded by the UV
emission from supernova remnants (Meijerink et al., 2011).
We computed a sequence of models assuming an electron density $N_{\rm
e}=10^{4}\rm\>cm^{-3}$, ionization parameter $U$ in the range $-4.0\leq\log
U\leq-1.0$ defined as $U=Q_{ion}/4\pi R^{2}_{\rm S}nc$, where $Q_{ion}$ is the
number of hydrogen ionizing photons emitted per second by the ionizing source,
$R_{\rm S}$ is the Strömgren radius (in cm), $n$ is the particle density (in
$\rm cm^{-3}$), and $c$ is the speed of light. The chosen range of these
values for $U$ is typical of narrow-line regions of Sy galaxies (e.g. Ferland
& Netzer 1983). The H2 emission lines are very dependent on the electron
density value assumed in the models. For example, when $N_{\rm e}$ varies from
$10^{3}$ to $10^{5}\>\rm cm^{-3}$, the logarithm of the H${}_{2}\,\lambda$
2.1218$\,\mu$m/Br$\gamma$ emission line intensity ratio span about 2.6 dex.
The value $N_{\rm e}=10^{4}\rm cm^{-3}$, assumed in our models, is a mean
value from those considered by Mouri et al. (2000). We considered in our
models three values of 12+log(O/H)= 8.38, 8.69, and 9.00, which correspond to
values of the metallicity 0.5, 1, and 2 times the solar value published by
Allende Prieto et al. (2001). The abundances of other metals in the nebula
were scaled linearly to the solar metal composition through the comparison of
the oxygen abundances, with the exception of the N and Fe abundances. The
nitrogen abundance was taken from the relation log(N/O)=log(0.034+120O/H) of
Vila- Costas & Edmunds (1993). The Fe/O abundance ratio has a large scatter
for a fixed O/H value (Izotov et al., 2006) and its value is uncertain because
the Fe and O abundances in grains are poorly known (Peimbert & Peimbert,
2010). Thus, we varied the Fe/O abundance ratio by about 0.7 dex on each
metallicity.
Table 1: Fe/O and O/H gas phase abundances assumed in the models. Metallicity (Z/Z⊙) | 12+log(O/H) | | log(Fe/O)
---|---|---|---
2 | 9.0 | | -1.47(a1) | -1.94(a2) | -2.24(a3)
1 | 8.69 | | -1.77(b1) | -2.24(b2) | -2.77(b3)
0.5 | 8.38 | | -2.15(c1) | -2.54(c2) | -2.76(c3)
The presence of internal dust was considered and the grain abundances (van
Hoof et al. 2001) were also linearly scaled with the oxygen abundance. To take
into account the depletion of refractory elements onto dust grains, the
abundances of Mg, Al, Ca, Ni, and Na were reduced by a factor of 10, and Si by
a factor of 2 relative to adopted abundances in each model in accordance with
Garnett et al. (1995). In Table 1, the O/H and Fe/O abundance values of the
gas phase assumed in the models are shown. The model of the $\rm H_{2}$
molecule described by Shaw et al. (2005) and the model of the $\rm Fe^{+}$ ion
described by Verner et al. (1999), which consider 371 energy levels, were
assumed in our computations. The outer radius of the modelled nebula is that
where the temperature falls below 1000 K.
## 4 Diagnostic Diagrams
We used four diagnostic diagrams containing predicted and observed emission
line ratios of the [Fe ii] , $\rm H_{2}$, [O iii], and [O i] which are
described below.
* •
[Fe ii] $\lambda$ 1.2570$\,\mu$m/Pa$\beta$ vs. H${}_{2}\,\lambda$
2.1218$\,\mu$m/Br$\gamma$ (Fig. 1) — Diagnostic diagram suggested by Larkin et
al. (1998) and Rodríguez-Ardila et al. (2004) to separate galaxies according
to their level of nuclear activity. Recently, Riffel et al. (2010) constructed
this diagram with spatially resolved IFU data of an AGN. Typical values for
the nucleus of Sy galaxies are $0.6\,\lesssim$ [Fe ii] $\lambda$
1.2570$\,\mu$m/Pa$\beta$ $\lesssim\,2.0$ and $0.6\,\lesssim$
H${}_{2}\,\lambda$ 2.1218$\,\mu$m/Br$\gamma$ $\lesssim\,2.0$ (Rodríguez-Ardila
et al., 2005). This [Fe ii]/Pa$\beta$ is very dependent on the Fe/O abundance
while the H2 emission lines are dependent on the ionization parameter.
* •
H${}_{2}\lambda$ 1.957$\,\mu$m/$\lambda$ 2.121$\,\mu$m and H${}_{2}\lambda$
2.033$\,\mu$m/$\lambda$ 2.223$\,\mu$m vs. H${}_{2}\lambda$
2.247$\,\mu$m/$\lambda$ 2.121$\,\mu$m (Fig. 2) — Mouri (1994) proposed these
diagrams to separate gas emission yielded by shocks from emission caused by
fluorescence. The drawback in using the H${}_{2}\lambda$
1.957$\,\mu$m/$\lambda$ 2.121$\,\mu$m ratio is that the H${}_{2}\lambda$
1.957$\,\mu$m may be affected by telluric bands of $\rm H_{2}O$ and $\rm
CO_{2}$, or blended with the [Si iv] $\lambda$ 1.963$\,\mu$m emission line
(Rodríguez-Ardila et al., 2005).
* •
[O iii] $\lambda\,5007$Å/H$\beta$ vs. [O i] $\lambda$ 6300Å/H$\alpha$ (Fig. 3)
— This diagram was suggested by Baldwin et al. (1981) to separate objects
according to their primary excitation mechanisms, i.e. (a) photoionization by
stars, (b) photoionization by a power law continuum source or (c) shock
heating. In particular, the [O i] $\lambda$ 6300 Å/H$\alpha$ line ratio is
greatly increased by the presence of shocking gas, even when it has low
velocities (e.g. Allen et al. 2008).
Figure 1: Diagnostic diagram showing the observational data taken from the
literature (see Sect. 2) and results from the grid of photoionization models
(see Sect.3). Solid lines connect curves of iso-$Z$, while dotted lines
connect curves of iso-$U$. The values of $\log U$ and $Z$ are indicated. The
three different lines for each $Z$ correspond to the different assumed values
of the Fe/O as indicated by the labels (see Table 1). Circles, squares, and
star represent Sy1, Sy2, and quasar data, respectively. The typical error bar
(not shown) of the emission line ratios is about 10 %. Figure 2: As in Fig. 1
for $\rm H_{2}$ emission lines. The arrow indicates the direction in which the
ionization parameter increases. Circles, squares, and star represent Sy1, Sy2,
and quasar data, respectively. The hatched area represents the region occupied
by shock model results from Hollenbach & McKee (1989).
## 5 Results
### 5.1 Integrated spectra
In Figs. 1 and 2 we show the first three diagnostic diagrams described above
containing the results of our grid of photoionization models and the data
sample. Sy 1, Sy 2 and quasar are represented by different symbols. For the
IFU data, the emission line ratios represented in these Figs. were estimated
by integrating the spaxels inside a central aperture of 0.5″$\times$0.5″ for
each galaxy, with exception of ESO428-G14 for which an aperture of
0.75″$\times$0.75″ was considered. These values are presented in Table 2.
In Fig. 1, we can see that almost all the observational ratios are within the
parameter space defined by our grid of photoionization models. A lower
metallicity than those assumed in our models is required to reproduce the data
of the galaxies out of the grid. The observed correlation between [Fe ii]
$\lambda$ 1.2570$\,\mu$m/Pa$\beta$ and H2 $\lambda$ 2.1218$\,\mu$m/Br$\gamma$
is explained by an increase in metallicity and ionization parameter.
Noteworthy that the parameter space defined by the models built using
$Z\,=\,0.5\,Z_{\odot}$ is almost completely contained in the one defined by
the models built using the solar metallicity.
Table 2: Integrated line ratio intensities of IFU data Object | [Fe ii] $\lambda$ 1.2570$\,\mu$m/Pa$\beta$ | H${}_{2}\,\lambda$ 2.1218$\,\mu$m/Br$\gamma$
---|---|---
ESO428-G14 | 0.75 | 1.10
Mrk 1066 | 0.52 | 0.96
Mrk1 157 | 0.73 | 2.24
NGC 4151 | 0.45 | 0.26
In the case of the diagnostic diagrams that only involve H2 line ratios (Fig.
2), the photoionization models are slightly dependent on the assumed
metallicities, covering almost the same parameter space, and strongly
dependent on variations in the ionization parameter. Taking into account the
observational error bars, our models are in good agreement with the observed
H${}_{2}\lambda$ 2.033$\,\mu$m/$\lambda$ 2.223$\,\mu$m and H${}_{2}\lambda$
2.247$\,\mu$m/$\lambda$ 2.121$\,\mu$m ratios (upper panel). On the other hand,
in the lower panel of Fig. 2 can be noticed larger dispersion of the
observational data which is not well reproduce by the models. This dispersion
could be the result of a contamination of the measurements of the
H${}_{2}\,\lambda$ 1.957 $\mu$m emission line intensities due to a blend with
the [Si iv] $\lambda$ 1.963$\,\mu$m line (as explained above). Therefore, the
predicted H${}_{2}\lambda$1.957$\,\mu$m intensities are somewhat lower than
the observed ones. In Fig. 2, we also show the area occupied by the
theoretical intensities of the line ratio H${}_{2}\lambda$
2.247$\,\mu$m/$\lambda$ 2.121$\,\mu$m from shock models performed by
Hollenbach & McKee (1989). These authors computed emission-line spectra of
steady interstellar shocks in molecular gas considering velocities from 30 to
150 km/s and particle densities of $10^{3}-10^{6}\>\rm cm^{-3}$. We can see
that most of the objects of our sample appear to have the X-rays as main
ionizing source while for the remaining ones a composite ionization by X-rays
and shock can be considered.
Fig. 3 shows the [O iii] $\lambda$ 5007 Å/H$\beta$ vs. [O i $\lambda$ 6300
Å/H$\alpha$ diagnostic diagram. In this Fig. we can see that the observational
data of AGNs are well describe by our models. If our models use the lower
values of the ionization parameter ($\log\,U\,<\,-3.5$; these models are not
shown), we can extend the parameter space to include the objects that have
values of the logarithm of the [O iii] $\lambda$ 5007 Å/H$\beta$ ratio lower
than zero. As in the case of the [Fe ii] $\lambda$ 1.2570$\,\mu$m/Pa$\beta$
and H2 $\lambda$ 2.1218$\,\mu$m/Br$\gamma$ diagnostic diagram (Fig. 1), the
parameter space of the models with solar metallicity almost contains that of
the $Z\,=\,0.5\,Z_{\odot}$ models.
Figure 3: [O iii $\lambda$ 5007 Å/H$\beta$ vs. [O i $\lambda$ 6300 Å/H$\alpha$
diagnostic diagram. The yellow line separates objects ionized by massive stars
from those containing active nucleus (Kewley et al., 2001). Blue, green and
red solid lines are as in Fig. 1. Points represent emission-line galaxies
listed in the MPA/JHU Data catalogue of the Sloan Digital Sky Survey DR7
release (see Sect. 2).
### 5.2 IFU data
We plot the [Fe ii] $\lambda$ 1.2570$\,\mu$m/Pa$\beta$ and H2 $\lambda$
2.1218$\,\mu$m/Br$\gamma$ diagnostic diagram for each spaxel of our four
objects with our model results (see upper panels of Figs. 4 and 5). In these
Figs., the spaxel data are separated by their ionization mechanism according
to the place in the diagnostic diagram, with different colours for each
mechanism. The different ionization mechanism zones are delimited in the Figs.
by dashed-lines, following the work of Rodríguez-Ardila et al. (2004). The
spaxels showing typical values of starbursts, Seyferts, and low-ionization
nuclear emission-line regions (LINERs) are represented by green open circles,
black filled circles, and red crosses, respectively. With the same colour
code, we show the spatial position of each spaxel in the IFU field of view
(see lower panels of Figs. 4 and 5). Our models completely represent the
region occupied by Seyfert and LINERs data.
Figure 4: Top panels: [Fe ii] $\lambda\,1.2570\,\mu$m/Pa$\beta$ vs. H2
$\lambda\,2.1212\,\mu$m/Br$\gamma$ line-ratio diagnostic diagram for Mrk 1157
(left) and ESO 428-G14 (right). The dashed lines delimit regions with ratios
typical of Starbursts (green open circles), Seyferts (black filled circles)
and LINERs (red crosses). Blue, green and red solid lines are as in Fig. 1.
Bottom panels: spatial position of each spaxel in the IFU field of view from
the diagnostic diagram.
Figure 5: As in Fig. 4 but for Mrk 1066 (left) and NGC 4151 (right).
## 6 Discussion
The excitation mechanism of the near-IR emission lines of the [Fe ii] and H2
in active galaxies have been the subject of several works. For example, Mouri
et al. (2000) compared results of models considering photoionization by X-rays
and shock heating with observational data of AGNs and starburst galaxies.
These authors built their models considering large ranges in shock velocities,
gas density, metallicity, and different ionizing continua. Mouri and
collaborators showed that the [Fe ii] emission is enhanced when a partially
ionized zone is produced by photoionization by X-rays (described by a power-
law) and shock heating. These two processes can be discriminated by the
electron temperature of the [Fe ii] region: 8000 K in heating by X-rays and
6000 K in shock heating. Comparing the electron temperature of the [Fe ii]
region estimated by Thompson (1995) for NGC 4151 ($8000\><\>T_{\rm
e}\><\>12000$ K) with their models, Mouri et al. (2000) showed that, at least
for this galaxy, it indicates that X-rays are the more important mechanism to
yield the [Fe ii] flux. These authors arrived to the same conclusion using the
[O i] $\lambda$ 6300 Å/H$\alpha$ vs. [Fe ii]$\lambda$1.2570$\,\mu$m]/Pa$\beta$
diagnostic diagram. A similar result was also obtained by Jackson & Beswick
(2007) by analysing J-band spectra of three Sy 2 galaxies.
For our models, we assumed an incident continuum whose shape is given by two
components, a Big Bump and an X-ray power law, varying the Fe/O abundance.
With these models that do not consider shock heating, we are able to explain
the observational data. Nevertheless, we do not exclude some contribution by
shock heating to the [Fe ii] emission. Comparing our models with the SDSS DR7
emission-line galaxies (Fig. 3), we are able to describe the [O i]/H$\alpha$
line ratio observed in AGNs. This diagram cannot be used to discriminate [Fe
ii] excitation mechanism, nevertheless, we must take into account that the [O
i]/H$\alpha$ line ratio is shock sensitive. Hence, although shock contribution
in the ionization of Fe cannot be ruled out, models considering a continuum
described by a Big Bump and an X-ray power law as the ionization source can
also reproduce the [Fe ii] emission lines as well as the behaviour of shock
sensitive emission lines such as [O i] $\lambda$ 6300 Å. Analysing IFU
observations of the Sy galaxy NGC 4151, Turner et al. (2002) found that the
[Fe ii] emission mainly arises in the visible narrow-line region in which the
dominant excitation mechanism is the photoionization by collimated X-ray
emission from the nucleus. Oliva et al. (2001) pointed out that in regions
where shocks are the dominant mechanism the iron-based grains are destroyed
but the phosphorus is not, yielding a larger [Fe ii] $\lambda\,1.2570$
$\mu$m/[P ii] $\lambda\,1.188$ $\mu$m line-ratio intensity than that observed
in the regions dominated by X-ray. In order to verify this, in Fig. 6 we show
a histogram containing this observed line intensity ratio for 17 Seyfert
galaxies, 5 Sy 1 and 12 Sy 2, taken from Jackson & Beswick (2007), Riffel et
al. (2006), and Oliva et al. (2001). It can be seen that the [Fe ii]/[P ii]
for Sy galaxies ranges from 1.5 to 6 (with a mean value of 2.7). The mean
value of this ratio is about 20 for SNRs, which indicate that the emitting gas
has recently passed through a fast shock (Oliva et al., 2001). Therefore,
these results confirm that shocks have little influence on the [Fe ii]
emission.
Figure 6: Histogram showing the [Fe ii]/[P ii] emission line intensity ratios
of a sample of objects collected from the literature.
Regarding the $\rm H_{2}$, this molecule can be excited via three distinct
mechanisms: (1) UV fluorescence, where photons with $\lambda\,>\,912$ Å are
absorbed by the $\rm H_{2}$ molecule and then re-emitted, resulting in the
population of various vibro-rotational levels, (2) shocks, where high-velocity
gas motions heat and accelerate this molecule; and (3) X-ray illumination,
where hard X-ray photons penetrate deep into molecular clouds, heating large
amounts of molecular gas resulting in the $\rm H_{2}$ emission (see Rodríguez-
Ardila et al., 2004, and references therein). Rodríguez-Ardila and
collaborators used the diagrams shown in Fig. 2 to compare observational data
of 22 objects with models considering a thermal emission, a non-thermal UV
excitation, a thermal UV excitation, and a mixture of thermal and low-density
fluorescence. These authors found that for 4 objects the excitation mechanism
is clearly thermal, while for the remaining objects a mixing with a non-
thermal process cannot be discarded, even though the results point out to a
dominant thermal mechanism.
To analyse the relative weight of the X-ray emission with respect to the other
model components (mainly with fluorescence and UV photons), not only for the
$\rm H_{2}$ emission but also for the [Fe ii], we made models fixing all
parameters with the exception of the $\alpha_{ox}$ value (see Fig. 7), which
is related to the X-ray power law normalization (see Section §3). We assumed
$Z$ = $Z$⊙ and log $U$ = $-$2.5 since the models built using the solar
metallicity and this value of the ionization parameter cover almost all the
parameter space occupied by the observational data (see Fig. 1). Taking into
account the $\alpha_{ox}$ definition (Tananbaum et al., 1979), which fixes the
Big Bump parameters, a decrement of the $\alpha_{ox}$ value implies that the
amount of the X-rays emitted by the source decreases. In Fig. 7 we can see
that our models with $\alpha_{ox}=-1.4$ reproduce well the observational AGN
data. Nevertheless, when we use lower values of this parameter, the ratios
predicted by the models go out of the region typically occupied by the AGNs
(Rodríguez-Ardila et al., 2004). Therefore, our models favour the scenario
suggested by Maloney et al. (1996), where the $\rm H_{2}$ molecule emission is
mainly governed by photons emitted at X-ray wavelengths from the central AGN.
This also can be inferred from the dependence of the $\rm H_{2}$ emission
lines on the ionization parameter $U$.
To verify if shock models can fit the observational data, we compared shock
model results by Hollenbach & McKee (1989) with our sample (see Fig. 2). Only
few observational points are located in the area occupied by these shock
models and, even in these cases, models considering X-rays also describe the
data.
On the other hand, varying in our models the $\rm H_{2}$ ionization rate by
cosmic rays by a factor of 200, we found that the H${}_{2}\,\lambda$
2.1218$\,\mu$m/Br$\gamma$ line ratio only increases by about 0.15 dex, which
shows that the additional ionization by cosmic rays has little influence on
the $\rm H_{2}$ emission lines.
Figure 7: Model results using solar metallicity, Ne = $10^{4}$ cm-3, log U =
-2.5 and varying only the $\alpha_{ox}$ parameter to see the influence of the
X-rays on the [Fe ii] $\lambda\,1.2570\,\mu$m/Pa$\beta$ and H2
$\lambda\,2.1218\,\mu$m/Br$\gamma$ emission line ratios. To delimit the region
occupied by the AGNs we follow Rodríguez-Ardila et al. (2004). Figure 8:
Spectral energy distribution at the Schwarzschild radio (10-5 pc) of the Sy 1
galaxy Mrk 1066 used as the photoionization source for some models of this
galaxy. We assumed a galaxy distance of 50 Mpc (Mould et al., 2000). The
photometric data were taken from Dressel & Condon (1978), Moshir et al.
(1990), de Vaucouleurs et al. (1991), Becker et al. (1991), Douglas et al.
(1996), Condon et al. (2002), Skrutskie et al. (2003), Braatz et al. (2004),
Guainazzi et al. (2005), Muñoz-Marín et al. (2007), and Cardamone et al.
(2007). Figure 9: Comparison between the grid model results shown in Fig. 1
(solid lines) and the models built considering the semi-empirical SED of Mrk
1066 shown in Fig. 8 (dashed lines).
A simple scenario where both [Fe ii] and $\rm H_{2}$ emissions are mainly due
to the X-ray continuum coming from the active nucleus has also been proposed
by other authors. For example, Blietz et al. (1994) and Knop et al. (2001)
showed that X-rays from the nucleus can heat the gas located in the narrow
line region driving the [Fe ii] and H 2 emission. Because 98 % of the iron is
tied up in dust grains, this process must free the iron through dust
destruction and yet not destroy the $\rm H_{2}$ molecules (Rodríguez-Ardila et
al., 2004). These authors computed the emergent [Fe ii] $\lambda$ 1.2570
$\,\mu$m and H${}_{2}\,\lambda$ 2.1218$\,\mu$m flux using the X-ray models by
Maloney et al. (1996) and compared their predictions with observational data
of seven objects. They found that X-ray heating can only explain a fraction of
the [Fe ii] and H2 emission, and they stated that the discrepancy found can be
alleviated if the emitting gas is located closer than the distance adopted in
their models. The X-ray data, provided by the XMM-Newton and Chandra space
telescopes, and their detailed analysis (see e.g. Piconcelli et al., 2005;
Longinotti et al., 2007; Bianchi et al., 2009; Krongold et al., 2009; Cardaci
et al., 2011; Corral et al., 2011, and references therein), provide
information about the continuum shape and the particular spectral features of
the AGNs in this wavelength range. For Mrk 1066, we compared the results
obtained using this simple scenario that only involves a continuum modelled by
a Big Bump and an X-ray power law with those obtained using its intrinsic SED.
We built the observational SED taking the photometric data from the NASA/IPAC
Extragalactic Database (NED), following Cardaci et al. (2009). To enhance the
number of points of the SED as needed by Cloudy, we performed a linear
interpolation among the semi-empirical points (see Fig. 8). We built a new
grid of photoionization models under the same assumptions of abundances,
ionization parameters and density, but only for one value of the Fe/O ratio
for each metallicity. In Fig. 9 the predictions of our models using the SED of
Mrk 1066 and the model results presented in Fig. 1 assuming the same Fe/O
abundance as the Mrk 1066 models are shown. The model results derived using
the two different ionizing sources are mostly in agreement.
The semi-empirical SED of Mrk 1066 includes not only the range covered by the
Cloudy model but also the radio and IR wavelengths. Hence, the agreement
between solid and dashed lines in Fig. 9 only indicates that the assumed
multicomponent model is a good representation of the AGN continuum when
studying the [Fe ii] and H2 emission.
Recent resolved integral field spectroscopy of the central region of active
galaxies shows that the ionized (in particular the [Fe ii] emitting gas) and
the molecular (traced by the H2 emission) gas have distinct flux distributions
and kinematics. The molecular component is more restricted to the plane of the
galaxies and the ionized one extends to high latitudes above it, which is in
most cases co-spatially with the radio jet (e.g., Riffel et al., 2006, 2008,
2009, 2010; Riffel & Storchi-Bergmann, 2011a, b; Storchi-Bergmann et al.,
2009; Storchi-Bergmann et al., 2010). Usually the [Fe ii] has an enhancement
in flux and velocity dispersion in regions surrounding the radio structure,
suggesting that the radio jet plays an important role in the [Fe ii] emission.
Our models are able to reproduce the [Fe ii] emission of active galaxies
without considering shock excitation by the radio jet. Thus, the enhancement
in the [Fe ii] flux in the vicinity of radio structures can be interpreted as
being due to an enhancement in the gas density, caused by the interaction of
the radio jet with the emitting gas, and mainly excited by X-rays from the
central engine.
The main exciting mechanism of infrared emission lines of ESO 428-G14, Mrk
1157, Mrk 1066 and NGC 4151 was discussed by Riffel et al. (2006, 2008, 2010)
and Storchi-Bergmann et al. (2009), respectively. However, these authors did
not reach conclusive results. For example, Riffel et al. (2006) suggested that
the [Fe ii] excitation in ESO 428-G14 is mainly due to shocks. Nevertheless,
the detailed analysis performed in the present work confronting our models
with the IFU data shows that X-rays are a more reliable dominant excitation
mechanism even in the case of ESO 428-G14.
## 7 Conclusions
In this work we show that a photoionization model grid built by adopting a
continuum source characterized by two components, one accounting for the Big
Bump component peaking at $\rm 1\>Ryd$ and the other describing the X-rays
emission, is able to reproduce the [Fe ii] and H2 infrared emission lines of a
sample of AGNs. Testing the influence of the X-rays on the intensity of these
emission lines, we found that a decrement in the X-ray content of the
continuum source translates into a weakening of these lines, and the models
are no longer compatible with the observations. This implies that the heating
by the X-ray emission from the active nuclei can be considered as the most
important mechanism of excitation for the IR emission lines of these elements.
## Acknowledgments
We are grateful to the referee, Neal Jackson, for a thorough reading of the
manuscript and for suggestions that greatly improved its clarity.
Based on observations obtained at the Gemini Observatory, which is operated by
the Association of Universities for Research in Astronomy, Inc., under a
cooperative agreement with the NSF on behalf of the Gemini partnership: the
National Science Foundation (United States), the Science and Technology
Facilities Council (United Kingdom), the National Research Council (Canada),
CONICYT (Chile), the Australian Research Council (Australia), Ministério da
Ciência e Tecnologia (Brazil) and south-eastCYT (Argentina). This research has
made use of the NASA/IPAC Extragalactic Database (NED) which is operated by
the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration. OLD and ACK
are grateful to the FAPESP for support under grant 2009/14787-7 and
2010/01490-3. MC and GH are grateful to the Spanish Ministerio de Ciencia e
Innovación for support under grant AYA2010-21887-C04-03, and the Comunidad de
Madrid under grant S2009/ESP-1496 (ASTROMADRID). EPM is grateful to the
Spanish Ministerio de Ciencia e Innovación for support under grant
AYA2010-21887-C04-02, and the Junta de Andalucía under grant TIC114.
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|
arxiv-papers
| 2012-01-19T12:52:53 |
2024-09-04T02:49:26.498186
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Oli L. Dors Jr, Rogemar A. Riffel, Monica V. Cardaci, Guillermo F.\n Hagele, Angela C. Krabbe, Enrique Perez-Montero, Irapuan Rodrigues",
"submitter": "Oli Luiz Dors Junior",
"url": "https://arxiv.org/abs/1201.4046"
}
|
1201.4068
|
11institutetext: CERN, Geneva, Switzerland
# RF engineering basic concepts: the Smith chart
F. Caspers
###### Abstract
The Smith chart is a very valuable and important tool that facilitates
interpretation of S-parameter measurements. This paper will give a brief
overview on why and more importantly on how to use the chart. Its definition
as well as an introduction on how to navigate inside the chart are
illustrated. Useful examples show the broad possibilities for use of the chart
in a variety of applications.
## 0.1 Motivation
With the equipment at hand today, it has become rather easy to measure the
reflection factor $\Gamma$ even for complicated networks. In the “good old
days” though, this was done measuring the electric field strength111The
electrical field strength was used since it can be measured considerably more
easily than the magnetic field strength. at a coaxial measurement line with a
slit at different positions in the axial direction (Fig. 1).
DUTfromgeneratormovable electric field
probe[algebraic=true]07.2sin(2*x)$U_{\text{min}}$$U_{\text{max}}$ Figure 1:
Schematic view of a measurement set–up used to determine the reflection
coefficient as well as the voltage standing wave ratio of a device under test
(DUT) [1]
A small electric field probe, protruding into the field region of the coaxial
line near the outer conductor, was moved along the line. Its signal was picked
up and displayed on a microvoltmeter after rectification via a microwave
diode. While moving the probe, field maxima and minima as well as their
position and spacing could be found. From this the reflection factor $\Gamma$
and the Voltage Standing Wave Ratio (VSWR or SWR) could be determined using
the following definitions:
* •
$\Gamma$ is defined as the ratio of the electrical field strength $E$ of the
reflected wave over the forward travelling wave:
$\Gamma=\frac{E\text{ofreflectedwave}}{E\text{offorwardtravelingwave}}\hskip
5.69046pt.$ (1)
* •
The VSWR is defined as the ratio of maximum to minimum measured voltage:
$\text{VSWR}=\frac{U_{\text}{max}}{U_{\text{min}}}=\frac{1+|\Gamma|}{1-|\Gamma|}\hskip
5.69046pt.$ (2)
Although today these measurements are far easier to conduct, the definitions
of the aforementioned quantities are still valid. Also their importance has
not diminished in the field of microwave engineering and so the reflection
coefficient as well as the VSWR are still a vital part of the everyday life of
a microwave engineer be it for simulations or measurements.
A special diagram is widely used to visualize and to facilitate the
determination of these quantities. Since it was invented in 1939 by the
engineer Phillip Smith, it is simply known as the Smith chart [2].
## 0.2 Definition of the Smith chart
The Smith chart provides a graphical representation of $\Gamma$ that permits
the determination of quantities such as the VSWR or the terminating impedance
of a device under test (DUT). It uses a bilinear Moebius transformation,
projecting the complex impedance plane onto the complex $\Gamma$ plane:
$\Gamma=\frac{Z-Z_{\text{0}}}{Z+Z_{\text{0}}}\hskip
14.22636pt\text{with}\hskip 14.22636ptZ=R+\text{j}\,X\hskip 5.69046pt.$ (3)
As can be seen in Fig. 2 the half-plane with positive real part of impedance
$Z$ is mapped onto the interior of the unit circle of the $\Gamma$ plane. For
a detailed calculation see Appendix .6.
[linecolor=green,linewidth=0.03](15,9)(12,9)(15,6)[linecolor=green,linewidth=0.03](15,7.5)(13.8,8.4)(15,6)[linecolor=green,linewidth=0.03](15,12)(10.2,8.4)(15,6)[linecolor=darkyellow,linewidth=0.03](15,3)(15,6)(12,3)[linecolor=darkyellow,linewidth=0.03](15,4.5)(15,6)(13.8,3.6)[linecolor=darkyellow,linewidth=0.03](15,0)(15,6)(10.2,3.6)Im
($\Gamma$)Re ($\Gamma$)$X$= Im ($Z$)$R$= Re ($Z$) Figure 2: Illustration of
the Moebius transform from the complex impedance plane to the $\Gamma$ plane
commonly known as Smith chart
### 0.2.1 Properties of the transformation
In general, this transformation has two main properties:
* •
generalized circles are transformed into generalized circles (note that a
straight line is nothing else than a circle with infinite radius and is
therefore mapped as a circle in the Smith chart);
* •
angles are preserved locally.
Figure 3 illustrates how certain basic shapes transform from the impedance to
the $\Gamma$ plane.
[linecolor=green,linewidth=0.03](15,9)(12,9)(15,6)[linecolor=green,linewidth=0.03](15,7.5)(13.8,8.4)(15,6)[linecolor=green,linewidth=0.03](15,12)(10.2,8.4)(15,6)[linecolor=darkyellow,linewidth=0.03](15,3)(15,6)(12,3)[linecolor=darkyellow,linewidth=0.03](15,0)(15,6)(10.2,3.6)[linecolor=darkyellow,linewidth=0.03](15,4.5)(15,6)(13.8,3.6)[linecolor=darkyellow,linewidth=0.03](15,0)(15,6)(10.2,3.6)Im
($\Gamma$)Re ($\Gamma$)[linestyle=dashed,arrows=->](13.5,7.5)(12,9)(15,6)$X$=
Im ($Z$)$R$= Re ($Z$) Figure 3: Illustration of the transformation of basic
shapes from the $Z$ to the $\Gamma$ plane
### 0.2.2 Normalization
The Smith chart is usually normalized to a terminating impedance
$Z_{\text{0}}$ (= real):
$z=\frac{Z}{Z_{\text{0}}}\hskip 5.69046pt.$ (4)
This leads to a simplification of the transform:
$\Gamma=\frac{z-1}{z+1}\hskip 14.22636pt\Leftrightarrow\hskip
14.22636ptz=\frac{1+\Gamma}{1-\Gamma}\hskip 5.69046pt.$ (5)
Although $Z$ = 50 $\Omega$ is the most common reference impedance
(characteristic impedance of coaxial cables) and many applications use this
normalization, any other real and positive value is possible. Therefore it is
crucial to check the normalization before using any chart.
Commonly used charts that map the impedance plane onto the $\Gamma$ plane
always look confusing at first, as many circles are depicted (Fig. 4). Keep in
mind that all of them can be calculated as shown in Appendix .6 and that this
representation is the same as shown in all previous figures — it just contains
more circles.
Figure 4: Example of a commonly used Smith chart
### 0.2.3 Admittance plane
The Moebius transform that generates the Smith chart provides also a mapping
of the complex admittance plane ($Y=\frac{1}{Z}$ or normalized
$y=\frac{1}{z}$) into the same chart:
$\Gamma=-\frac{y-\text{1}}{y+\text{1}}=-\frac{Y-Y_{\text{0}}}{Y+Y_{\text{0}}}=-\frac{1/Z-1/Z_{\text{0}}}{1/Z+1/Z_{\text{0}}}=\frac{Z-Z_{\text{0}}}{Z+Z_{\text{0}}}=\frac{z-\text{1}}{z+\text{1}}\hskip
5.69046pt.$ (6)
Using this transformation, the result is the same chart, but mirrored at the
centre of the Smith chart (Fig. 5).
(12,0)(12,12)[linecolor=green,linewidth=0.03](15,9)(12,9)(15,6)[linecolor=green,linewidth=0.03](15,7.5)(13.8,8.4)(15,6)[linecolor=green,linewidth=0.03](15,12)(10.2,8.4)(15,6)[linecolor=darkyellow,linewidth=0.03](15,3)(15,6)(12,3)[linecolor=darkyellow,linewidth=0.03](15,4.5)(15,6)(13.8,3.6)[linecolor=darkyellow,linewidth=0.03](15,0)(15,6)(10.2,3.6)Im
($\Gamma$)Re ($\Gamma$)$B$= Im ($Y$)$G$= Re
($Y$)$\Gamma=-\frac{Y-Y_{\text{0}}}{Y+Y_{\text{0}}}\text{with}Y=G+\text{j}\,B$
Figure 5: Mapping of the admittance plane into the $\Gamma$ plane
Often both mappings, the admittance and the impedance plane, are combined into
one chart, which looks even more confusing (see last page). For reasons of
simplicity all illustrations in this paper will use only the mapping from the
impedance to the $\Gamma$ plane.
## 0.3 Navigation in the Smith chart
The representation of circuit elements in the Smith chart is discussed in this
section starting with the important points inside the chart. Then several
examples of circuit elements will be given and their representation in the
chart will be illustrated.
### 0.3.1 Important points
There are three important points in the chart:
1. 1.
Open circuit with $\Gamma=1,z\rightarrow\infty$
2. 2.
Short circuit with $\Gamma=-1,z=0$
3. 3.
Matched load with $\Gamma=0,z=1$
They all are located on the real axis at the beginning, the end, and the
centre of the circle (Fig. 6).
[linecolor=green,linewidth=0.03](15,9)(12,9)(15,6)[linecolor=green,linewidth=0.03](15,7.5)(13.8,8.4)(15,6)[linecolor=green,linewidth=0.03](15,12)(10.2,8.4)(15,6)[linecolor=darkyellow,linewidth=0.03](15,3)(15,6)(12,3)[linecolor=darkyellow,linewidth=0.03](15,4.5)(15,6)(13.8,3.6)[linecolor=darkyellow,linewidth=0.03](15,0)(15,6)(10.2,3.6)Im
($\Gamma$)Re ($\Gamma$)matched loadshort circuitopen circuit Figure 6:
Important points in the Smith chart
The upper half of the chart is inductive, since it corresponds to the positive
imaginary part of the impedance. The lower half is capacitive as it is
corresponding to the negative imaginary part of the impedance.
Concentric circles around the diagram centre represent constant reflection
factors (Fig. 7). Their radius is directly proportional to the magnitude of
$\Gamma$, therefore a radius of 0.5 corresponds to reflection of 3 dB (half of
the signal is reflected) whereas the outermost circle (radius = 1) represents
full reflection.
$\left|\Gamma\right|$= 1$\left|\Gamma\right|$= 0.75$\left|\Gamma\right|$=
0.5$\left|\Gamma\right|$= 0.25$\left|\Gamma\right|$= 0 Figure 7: Illustration
of circles representing a constant reflection factor
Therefore matching problems are easily visualized in the Smith chart since a
mismatch will lead to a reflection coefficient larger than 0 (Eq. (7)).
$\text{Powerintotheload=forwardpower-
reflectedpower:}P=\frac{1}{2}\left(\left|a\right|^{2}-\left|b\right|^{2}\right)=\frac{\left|a\right|^{2}}{2}\left(1-\left|\Gamma\right|^{2}\right)\hskip
5.69046pt.$ (7)
In Eq. (7) the European notation222The commonly used notation in the US is
power = $\left|a\right|^{2}$. These conventions have no impact on S parameters
but they are relevant for absolute power calculation. Since this is rarely
used in the Smith chart, the definition used is not critical for this paper.
is used, where power = $\frac{\left|a\right|^{2}}{2}$. Furthermore
$(1-\left|\Gamma\right|^{2})$ corresponds to the mismatch loss.
Although only the mapping of the impedance plane to the $\Gamma$ plane is
used, one can easily use it to determine the admittance since
$\Gamma(\frac{1}{z})=\frac{\frac{1}{z}-1}{\frac{1}{z}+1}=\frac{1-z}{1+z}=\left(\frac{z-1}{z+1}\right)\text{or}\Gamma(\frac{1}{z})=-\Gamma(z)\hskip
5.69046pt.$ (8)
In the chart this can be visualized by rotating the vector of a certain
impedance by 180∘ (Fig. 8).
Impedance $z$Reflection $\Gamma$Admittance $y=\frac{1}{z}$Reflection -$\Gamma$
Figure 8: Conversion of an impedance to the corresponding emittance in the
Smith chart
### 0.3.2 Adding impedances in series and parallel (shunt)
A lumped element with variable impedance connected in series is an example of
a simple circuit. The corresponding signature of such a circuit for a variable
inductance and a variable capacitor is a circle. Depending on the type of
impedance, this circle is passed through clockwise (inductance) or
anticlockwise (Fig. 9).
[linewidth=0.2,linecolor=darkred,arrows=<-](5.0625,8.29)(5,10)(4,9.6)[linewidth=0.2,linecolor=darkgreen,arrows=->](5.0625,8.29)(4,9.6)(3.45,8.8)[coilwidth=0.5cm,coilheight=0.7](2.4,4)(6,4)Series
$L$$Z$Series $C$$Z$ Figure 9: Traces of circuits with variable impedances
connected in series
If a lumped element is added in parallel, the situation is the same as for an
element connected in series mirrored by 180∘ (Fig. 10). This corresponds to
taking the same points in the admittance mapping.
(4.2,0)(4.2,12)[linewidth=0.2,linecolor=darkred,arrows=<-](5.0625,8.29)(5,10)(4,9.6)(11.8,0)(11.8,12)
[linewidth=0.2,linecolor=darkgreen,arrows=->](5.0625,8.29)(4,9.6)(3.45,8.8)[coilwidth=0.5cm,coilheight=0.7](4.2,0.6)(4.2,4)Shunt
$L$$Z$Shunt $C$$Z$ Figure 10: Traces of circuits with variable impedances
connected in parallel
Summarizing both cases, one ends up with a simple rule for navigation in the
Smith chart:
For elements connected in series use the circles in the impedance plane. Go
clockwise for an added inductance and anticlockwise for an added capacitor.
For elements in parallel use the circles in the admittance plane. Go clockwise
for an added capacitor and anticlockwise for an added inductance.
This rule can be illustrated as shown in Fig. 11
[linewidth=0.2,linecolor=darkred,arrows=<-](5.0625,8.29)(5,10)(4,9.6)[linewidth=0.2,linecolor=darkred,arrows=->](5.0625,8.29)(4,9.6)(3.45,8.8)(4,9.6)-90[linewidth=0.2,linecolor=darkgreen,arrows=<-](5.0625,8.29)(5,10)(4,9.6)[linewidth=0.2,linecolor=darkgreen,arrows=->](5.0625,8.29)(4,9.6)(3.45,8.8)Series
$L$Series $C$Shunt $C$Shunt $L$ Figure 11: Illustration of navigation in the
Smith chart when adding lumped elements
### 0.3.3 Impedance transformation by transmission line
The S matrix of an ideal, lossless transmission line of length $l$ is given by
$S=\left[\begin{array}[]{cc}0&\text{e}^{-\text{j}\beta l}\\\
\text{e}^{-\text{j}\beta l}&0\\\ \end{array}\right]$ (9)
where $\beta=\frac{2\pi}{\lambda}$ is the propagation coefficient with the
wavelength $\lambda$ ($\lambda=\lambda_{\text{0}}$ for
$\epsilon_{\text{r}}=1$).
When adding a piece of coaxial line, we turn clockwise on the corresponding
circle leading to a transformation of the reflection factor
$\Gamma_{\text{load}}$ (without line) to the new reflection factor
$\Gamma_{\text{in}}=\Gamma_{\text{load}}\text{e}^{-\text{j}2\beta l}$.
Graphically speaking, this means that the vector corresponding to
$\Gamma_{\text{in}}$ is rotated clockwise by an angle of 2$\beta l$ (Fig. 12).
[linewidth=0.05,arrows=<-](3.732,3.762)(5,3.6)(3.5,5)$\Gamma_{\text{load}}$$2\beta
l$$\Gamma_{\text{in}}$ Figure 12: Illustration of adding a transmission line
of length $l$ to an impedance
The peculiarity of a transmission line is that it behaves either as an
inductance, a capacitor, or a resistor depending on its length. The impedance
of such a line (if lossless!) is given by
$Z_{\text{in}}=\text{j}Z_{\text{0}}\tan(\beta l)\hskip 5.69046pt.$ (10)
The function in Eq. (10) has a pole at a transmission line length of
$\lambda/4$ (Fig. 13).
[algebraic=true,fillstyle=solid]03.8tan(0.35*x)+4[algebraic=true,fillstyle=solid]5.28.975979tan(0.35*x)+4Im
($Z$)Re ($Z$)inductivecapacitive$\frac{\lambda}{4}$$\frac{\lambda}{2}$ Figure
13: Impedance of a transmission line as a function of its length $l$
Therefore adding a transmission line with this length results in a change of
$\Gamma$ by a factor $-$1:
$\Gamma_{\text{in}}=\Gamma_{\text{load}}\text{e}^{-\text{j}2\beta
l}=\Gamma_{\text{load}}\text{e}^{-\text{j}2(\frac{2\pi}{\lambda})l}\stackrel{{\scriptstyle
l=\frac{\lambda}{4}}}{{=}}\Gamma_{\text{load}}\text{e}^{-\text{j}\pi}=-\Gamma_{\text{load}}\hskip
5.69046pt.$ (11)
Again this is equivalent to changing the original impedance $z$ to its
admittance $1/z$ or the clockwise movement of the impedance vector by 180∘.
Especially when starting with a short circuit (at $-$1 in the Smith chart),
adding a transmission line of length $\lambda/4$ transforms it into an open
circuit (at $+$1 in the Smith chart).
A line that is shorter than $\lambda/4$ behaves as an inductance, while a line
that is longer acts as a capacitor. Since these properties of transmission
lines are used very often, the Smith chart usually has a ruler around its
border, where one can read $l/\lambda$ — it is the parametrization of the
outermost circle.
### 0.3.4 Examples of different 2-ports
In general, the reflection coefficient when looking through a 2-port
$\Gamma_{\text{in}}$ is given via the S-matrix of the 2-port and the
reflection coefficient of the load $\Gamma_{\text{load}}$:
$\Gamma_{\text{in}}=\text{S}_{11}+\frac{\text{S}_{12}\text{S}_{21}\Gamma_{\text{load}}}{1-\text{S}_{22}\Gamma_{\text{load}}}\hskip
5.69046pt.$ (12)
In general, the outer circle of the Smith chart as well as its real axis are
mapped to other circles and lines.
In the following three examples different 2-ports are given along with their
S-matrix, and their representation in the Smith chart is discussed. For
illustration, a simplified Smith chart consisting of the outermost circle and
the real axis only is used for reasons of simplicity.
#### Transmission line $\lambda/16$
The S-matrix of a $\lambda/16$ transmission line is
$\text{S}=\left[\begin{array}[]{cc}0&\text{e}^{-\text{j}\frac{\pi}{8}}\\\
\text{e}^{-\text{j}\frac{\pi}{8}}&0\\\ \end{array}\right]$ (13)
with the resulting reflection coefficient
$\Gamma_{\text{in}}=\Gamma_{\text{load}}\text{e}^{-\text{j}\frac{\pi}{4}}\hskip
5.69046pt.$ (14)
This corresponds to a rotation of the real axis of the Smith chart by an angle
of 45∘ (Fig. 14) and hence a change of the reference plane of the chart (Fig.
14). Consider, for example, a transmission line terminated by a short and
hence $\Gamma_{\text{load}}=-1$. The resulting reflection coefficient is then
equal to $\Gamma_{\text{in}}=\text{e}^{-\text{j}\frac{\pi}{4}}$.
$z=0$$z=1$$z=\infty$[c]increasing terminating[c](0,-0.5)resistor Figure 14:
Rotation of the reference plane of the Smith chart when adding a transmission
line
#### Attenuator 3 dB
The S-matrix of an attenuator is given by
$\text{S}=\left[\begin{array}[]{cc}0&\frac{\sqrt{2}}{2}\\\
\frac{\sqrt{2}}{2}&0\\\ \end{array}\right]\hskip 5.69046pt.$ (15)
The resulting reflection coefficient is
$\Gamma_{\text{in}}=\frac{\Gamma_{\text{load}}}{2}\hskip 5.69046pt.$ (16)
In the Smith chart, the connection of such an attenuator causes the outermost
circle to shrink to a radius of 0.5333An attenuation of 3 dB corresponds to a
reduction by a factor 2 in power. (Fig. 15).
$z=0$$z=1$$z=\infty$ Figure 15: Illustration of the appearance of an
attenuator in the Smith chart
#### Variable load resistor
Adding a variable load resistor (0 $<z<\infty$) is the simplest case that can
be depicted in the Smith chart. It means moving through the chart along its
real axis (Fig. 16).
$z=0$$z=1$$z=\infty$ Figure 16: A variable load resistor in the simplified
Smith chart. Since the impedance has a real part only, the signal remains on
the real axis of the $\Gamma$ plane
## 0.4 Advantages of the Smith chart — a summary
* •
The diagram offers a compact and handy representation of all passive
impedances444Passive impedances are impedances with positive real part. from 0
to $\infty$. Impedances with negative real part such as reflection amplifier
or any other active device would show up outside the Smith chart.
* •
Impedance mismatch is easily spotted in the chart.
* •
Since the mapping converts impedances or admittances ($y=\frac{1}{z}$) into
reflection factors and vice versa, it is particularly interesting for studies
in the radio frequency and microwave domain. For reasons of convenience,
electrical quantities are usually expressed in terms of direct or forward
waves and reflected or backwards waves in these frequency ranges instead of
voltages and currents used at lower frequencies.
* •
The transition between impedance and admittance in the chart is particularly
easy: $\Gamma(\text{y=}\frac{1}{z})=-\Gamma(z)$.
* •
Furthermore the reference plane in the Smith chart can be moved very easily by
adding a transmission line of proper length (Section 0.3.4).
* •
Many Smith charts have rulers below the complex $\Gamma$ plane from which a
variety of quantities such as the return loss can be determined. For a more
detailed discussion see Appendix .7.
## 0.5 Examples for applications of the Smith chart
In this section two practical examples of common problems are given, where the
use of the Smith chart greatly facilitates their solution.
### 0.5.1 A step in characteristic impedance
Consider a junction between two infinitely short cables, one with a
characteristic impedance of 50 $\Omega$ and the other with 75 $\Omega$ (Fig.
17).
Junction between a50 $\Omega$ and a 75 $\Omega$ cable(infinitely short
cables)[c](0,0.1)$a_{1}$[c](0,-0.35)$b_{1}$[c](0,0.1)$a_{2}$[c](0,-0.35)$b_{2}$
Figure 17: Illustration of the junction between a coaxial cable with 50
$\Omega$ characteristic impedance and another with 75 $\Omega$ characteristic
impedance respectively. Infinitely short cables are assumed – only the
junction is considered
The waves running into each port are denoted with $a_{i}$ ($i=1,2$) whereas
the waves coming out of every point are denoted with $b_{i}$. The reflection
coefficient for port 1 is then calculated as
$\Gamma_{1}=\frac{Z-Z_{1}}{Z+Z_{1}}=\frac{75-50}{75+50}=0.2\hskip 5.69046pt.$
(17)
Thus the voltage of the reflected wave at port 1 is 20% of the incident wave
($a_{2}=a_{1}$ $\cdot$ $0.2$) and the reflected power at port 1 is 4%555Power
is proportional to $\Gamma^{2}$ and thus 0.22 = 0.04.. From conservation of
energy, the transmitted power has to be 96% and it follows that $b_{2}^{2}$ =
0.96.
A peculiarity here is that the transmitted energy is higher than the energy of
the incident wave, since $E_{\text{incident}}$ = 1, $E_{\text{reflected}}$ =
0.2 and therefore $E_{\text{transmitted}}$ = $E_{\text{incident}}$ \+
$E_{\text{reflected}}$ = 1.2. The transmission coefficient $t$ is thus $t$ = 1
+ $\Gamma$. Also note that this structure is not symmetric (S${}_{11}\neq$
S22), but only reciprocal (S${}_{21}=$ S12).
The visualization of this structure in the Smith chart is easy, since all
impedances are real and thus all vectors are located on the real axis (Fig.
18).
$V_{1}=a+b=1.2$$b=+0.2$$I_{1}Z=a-b$$-b$Incident wave $a=1$ Figure 18:
Visualization of the two-port formed by the two cables of different
characteristic impedance
As stated before, the reflection coefficient is defined with respect to
voltages. For currents its sign inverts and thus a positive reflection
coefficient in terms of voltage definition becomes negative when defined with
respect to current.
For a more general case, e.g., $Z_{1}$ = 50 $\Omega$ and $Z_{2}$ = 50 +
$\text{j}$80 $\Omega$, the vectors in the chart are depicted in Fig. 19.
$a=1$$b$$V_{1}=a+b$$-b$$I_{1}Z=a-b$[algebraic=true]00.30.1*sin(20.944*x)[c](0,0.2)$I_{1}$[c](0,0.2)$a$[c](0,-0.6)$b$$V_{1}$$Z_{\text{G}}=50\Omega$$z=1$$Z=50+$j$80\Omega$(load
impedance)$z=1+$j$1.6$ Figure 19: Visualization of the two-port depicted on
the left in the Smith chart
### 0.5.2 Determination of the $Q$ factors of a cavity
One of the most common cases where the Smith chart is used is the
determination of the quality factor of a cavity. This section is dedicated to
the illustration of this task.
A cavity can be described by a parallel $RLC$ circuit (Fig. 20)
[algebraic=true]00.30.1*sin(20.944*x)[coilwidth=0.3,coilheight=0.5,coilarm=0.01](1,1.6)(1,3.6)$Z_{\text{G}}$[coilwidth=0.3,coilheight=0.5,coilarm=0.01](1.6,1.6)(1.6,3.6)$R$[coilwidth=0.3,coilheight=0.5,coilarm=0.01](5.3,1.8)(5.3,3.4)$L$$C$$V_{\text{beam}}$$V_{0}$$Z_{\text{input}}$$Z_{\text{shunt}}$
Figure 20: The equivalent circuit that can be used to describe a cavity. The
transformer is hidden in the coupling of the cavity ($Z\approx 1$ M$\Omega$,
seen by the beam) to the generator (usually $Z=50\,\Omega$)
where the resonance condition is given when:
$\omega L=\frac{1}{\omega C}\hskip 5.69046pt.$ (18)
This leads to the resonance frequency of
$\omega_{\text{res}}=\frac{1}{\sqrt{LC}}\text{\hskip 28.45274ptor\hskip
28.45274pt}f_{\text{res}}=\frac{1}{2\pi}\frac{1}{\sqrt{LC}}\hskip 5.69046pt.$
(19)
The Impedance $Z$ of such an equivalent circuit is given by
$Z(\omega)=\frac{1}{\frac{1}{R}+\text{j}\omega C+\frac{1}{\text{j}\omega
L}}\hskip 5.69046pt.$ (20)
The 3 dB bandwidth $\Delta f$ refers to the points where Re($Z$) = Im($Z$)
which corresponds to two vectors with an argument of 45∘ (Fig. 21) and an
impedance of $|Z_{(-3\text{dB})}|=0.707R=R/\sqrt{2}$.
Re ($Z$)Im
($Z$)$45^{\circ}$$f=f^{\,-}_{(-3\text{dB})}$$f=f_{(\text{res})}$$f=f^{\,+}_{(-3\text{dB})}$$f=0$$f\rightarrow\infty$
Figure 21: Schematic drawing of the 3 dB bandwidth in the impedance plane
In general, the quality factor $Q$ of a resonant circuit is defined as the
ratio of the stored energy $W$ over the energy dissipated in one cycle $P$:
$Q=\frac{\omega W}{P}\hskip 5.69046pt.$ (21)
The $Q$ factor for a resonance can be calculated via the 3 dB bandwidth and
the resonance frequency:
$Q=\frac{f_{\text{res}}}{\Delta f}\hskip 5.69046pt.$ (22)
For a cavity, three different quality factors are defined:
* •
$Q_{0}$ (unloaded $Q$): $Q$ factor of the unperturbed system, i. e., the stand
alone cavity;
* •
$Q_{\text{L}}$ (loaded $Q$): $Q$ factor of the cavity when connected to
generator and measurement circuits;
* •
$Q_{\text{ext}}$ (external $Q$): $Q$ factor that describes the degeneration of
$Q_{0}$ due to the generator and diagnostic impedances.
All these $Q$ factors are connected via a simple relation:
$\frac{1}{Q_{\text{L}}}=\frac{1}{Q_{0}}+\frac{1}{Q_{\text{ext}}}\hskip
5.69046pt.$ (23)
The coupling coefficient $\beta$ is then defined as
$\beta=\frac{Q_{0}}{Q_{\text{ext}}}\hskip 5.69046pt.$ (24)
This coupling coefficient is not to be confused with the propagation
coefficient of transmission lines which is also denoted as $\beta$.
In the Smith chart, a resonant circuit shows up as a circle (Fig. 22, circle
shown in the detuned short position). The larger the circle, the stronger the
coupling. Three types of coupling are defined depending on the range of $beta$
(= size of the circle, assuming the circle is in the detuned short position):
[linewidth=0.045](3.9,0.9)(6.9,3.9)(0.9,3.9)[linewidth=0.045](3.9,6.9)(0.9,3.9)(6.9,3.9)Locus
of Im ($Z$) = Re ($Z$)f0f5f6f4f3f1f2 Figure 22: Illustration of how to
determine the different $Q$ factors of a cavity in the Smith chart
* •
Undercritical coupling ($0<\beta<1$): The radius of resonance circle is
smaller than 0.25. Hence the centre of the chart lies outside the circle.
* •
Critical coupling ($\beta=1$): The radius of the resonance circle is exactly
0.25. Hence the circle touches the centre of the chart.
* •
Overcritical coupling ($1<\beta<\infty$): The radius of the resonance circle
is larger than 0.25. Hence the centre of the chart lies inside the circle.
In practice, the circle may be rotated around the origin due to the
transmission lines between the resonant circuit and the measurement device.
From the different marked frequency points in Fig. 22 the 3 dB bandwidth and
thus the quality factors $Q_{0}$, $Q_{\text{L}}$ and $Q_{\text{ext}}$ can be
determined as follows:
* •
The unloaded $Q$ can be determined from f5 and f6. The condition to find these
points is Re($Z$) = Im($Z$) with the resonance circle in the detuned short
position.
* •
The loaded $Q$ can be determined from f1 and f2. The condition to find these
points is $\left|\text{Im}(\text{S}_{11})\right|\rightarrow$ max.
* •
The external $Q$ can be calculated from f3 and f4. The condition to determine
these points is $Z$ = $\pm\text{j}$.
To determine the points f1 to f6 with a network analyzer, the following steps
are applicable:
* •
f1 and f2: Set the marker format to Re(S11) + j Im(S11) and determine the two
points, where Im(S11) = max.
* •
f3 and f4: Set the marker format to $Z$ and find the two points where
$Z=\pm$j.
* •
f5 and f6: Set the marker format to $Z$ and locate the two points where
Re($Z$) = Im($Z$).
## .6 Transformation of lines with constant real or imaginary part from the
impedance plane to the $\Gamma$ plane
This section is dedicated to a detailed calculation of the transformation of
coordinate lines form the impedance to the $\Gamma$ plane. The interested
reader is referred to Ref. [3] for a more detailed study.
Consider a coordinate system in the complex impedance plane. The real part $R$
of each impedance is assigned to the horizontal axis and the imaginary part
$X$ of each impedance to the vertical axis (Fig. 23).
[showorigin=false,tickstyle=bottom,labelsep=5pt]->(5.5,5.5)Re($z$)Im($z$)$z=3.5+$j$3$
Figure 23: The complex impedance plane
For reasons of simplicity, all impedances used in this calculation are
normalized to an impedance $Z_{\text{0}}$. This leads to the simplified
transformation between impedance and $\Gamma$ plane:
$\Gamma=\frac{z-1}{z+1}\hskip 5.69046pt.$ (25)
$\Gamma$ is a complex number itself: $\Gamma=a+$j$c$. Using this identity and
substituting $z=R+$ j$X$ in equation (25) one obtains
$\Gamma=\frac{z-1}{z+1}=\frac{R+\text{j}X-1}{R+\text{j}X+1}=a+\text{j}c\hskip
5.69046pt.$ (26)
From this the real and the imaginary part of $\Gamma$ can be calculated in
terms of $a$, $c$, $R$ and $X$:
$\displaystyle\text{Re:}a(R+1)-cX$ $\displaystyle=$ $\displaystyle R-1;$ (27)
$\displaystyle\text{Im:}c(R+1)+aX$ $\displaystyle=$ $\displaystyle X.$ (28)
### .6.1 Lines with constant real part
To consider lines with constant real part, one can extract an expression for
$X$ from Eq. (A.4):
$X=c\frac{1+R}{1-a}$ (29)
and substitute this into Eq. (A.3):
$a^{2}+c^{2}-2a\frac{R}{1+R}+\frac{R-1}{R+1}=0\hskip 5.69046pt.$ (30)
Completing the square, one obtains the equation of a circle:
$\left(a-\frac{R}{1+R}\right)^{2}+c^{2}=\frac{1}{(1+R)^{2}}\hskip 5.69046pt.$
(31)
From this equation the following properties can be deduced:
* •
The centre of each circle lies on the real $a$ axis.
* •
Since $\frac{R}{1+R}\geq 0$, the centre of each circle lies on the positive
real $a$ axis.
* •
The radius $\rho$ of each circle follows the equation
$\rho=\frac{1}{(1+R)^{2}}\leq 1$.
* •
The maximal radius is 1 for $R$ = 0.
#### Examples
Here the circles for different $R$ values are calculated and depicted
graphically to illustrate the transformation from the $z$ to the $\Gamma$
plane.
1. 1.
$R$ = 0: This leads to the centre coordinates (ca/cc) =
$\left(\frac{0}{1+0}/0\right)=(0/0)$, $\rho=\frac{1}{1+0}=1$
2. 2.
$R$ = 0.5: (ca/cc) = $\left(\frac{0.5}{1+0.5}/0\right)=(\frac{1}{3}/0)$,
$\rho=\frac{1}{1+0.5}=\frac{2}{3}$
3. 3.
$R$ = 1: (ca/cc) = $\left(\frac{1}{1+1}/0\right)=(\frac{1}{2}/0)$,
$\rho=\frac{1}{1+1}=\frac{1}{2}$
4. 4.
$R$ = 2: (ca/cc) = $\left(\frac{2}{1+2}/0\right)=(\frac{2}{3}/0)$,
$\rho=\frac{1}{1+2}=\frac{1}{3}$
5. 5.
$R$ = $\infty$: (ca/cc) = $\left(\frac{\infty}{1+\infty}/0\right)=(1/0)$,
$\rho=\frac{1}{1+\infty}=0$
This leads to the circles depicted in Fig. 24.
$R=0$$R=\frac{1}{2}$$R=1$$R=2$$R=\infty$Im ($\Gamma$)Re
($\Gamma$)[showorigin=false,tickstyle=bottom,labelsep=5pt]<->(0,0)(-1.3,-1.3)(1.3,1.3)$R=0$$R=\frac{1}{2}$$R=1$$R=2$[showorigin=false,tickstyle=bottom,labelsep=5pt]<->(0,0)(-0.6,-2.4)(2.4,2.4)Im
($z$)Re ($z$) Figure 24: Lines of constant real part transformed into the
$\Gamma$ plane
### .6.2 Lines with constant imaginary part
To calculate the circles in the Smith chart that correspond to the lines of
constant imaginary part in the impedance plane, the formulas (A.3) and (A.4)
are used again. Only this time an expression for $R$ and $R$ \+ 1 is
calculated from Eq. (A.3)
$R=\frac{a+1-cX}{1-a}\hskip 14.22636pt\text{and}\hskip
14.22636pt1+R=\frac{2-cX}{1-a}$ (32)
and substituted into Eq. (A.4):
$a^{2}-2a+1+c^{2}-2\frac{c}{X}=0\hskip 5.69046pt.$ (33)
Completing the square again leads to the equation of a circle:
$(a-1)^{2}+\left(c-\frac{1}{X}\right)^{2}=\frac{1}{X^{2}}\hskip 5.69046pt.$
(34)
Examining this equation, the following properties can be deduced:
* •
The centre of each circle lies on an axis parallel to the imaginary axis at a
distance of 1.
* •
The first coordinate of each circle centre is 1.
* •
The second coordinate of each circle centre is $\frac{1}{X}$. It can be
smaller or bigger than 0 depending on the value of $X$.
* •
No circle intersects the real a axis.
* •
The radius $\rho$ of each circle is $\rho=\frac{1}{\left|X\right|}$.
* •
All circles contain the point (1/0).
#### Examples
In the following, examples for different $X$ values are calculated and
depicted graphically to illustrate the transformation of the lines with
constant imaginary part in the impedance plane to the corresponding circles in
the $\Gamma$ plane.
1. 1.
$X$ = -2: (ca/cc) = $\left(1/\frac{1}{-2}\right)=(1/-0.5)$,
$\rho=\frac{1}{\left|-2\right|}=0.5$
2. 2.
$X$ = -1: (ca/cc) = $\left(1/\frac{1}{-1}\right)=(1/-1)$,
$\rho=\frac{1}{\left|-1\right|}=1$
3. 3.
$X$ = -0.5: (ca/cc) = $\left(1/\frac{1}{-0.5}\right)=(1/-2)$,
$\rho=\frac{1}{\left|-2\right|}=2$
4. 4.
$X$ = 0: (ca/cc) = $\left(1/\frac{1}{0}\right)=(1/\infty)$,
$\rho=\frac{1}{\left|0\right|}=\infty$ = real $a$ axis
5. 5.
$X$ = 0.5: (ca/cc) = $\left(1/\frac{1}{0.5}\right)=(1/2)$,
$\rho=\frac{1}{\left|-2\right|}=2$
6. 6.
$X$ = 1: (ca/cc) = $\left(1/\frac{1}{1}\right)=(1/1)$,
$\rho=\frac{1}{\left|1\right|}=1$
7. 7.
$X$ = 2: (ca/cc) = $\left(1/\frac{1}{2}\right)=(1/0.5)$,
$\rho=\frac{1}{\left|2\right|}=0.5$
8. 8.
$X$ = $\infty$: (ca/cc) = $\left(1/\frac{1}{\infty}\right)=(1/0)$,
$\rho=\frac{1}{\left|\infty\right|}=0$
A graphical representation of the circles corresponding to these values is
given in Fig. 25.
[showorigin=false,tickstyle=bottom,labelsep=5pt]<->(0,0)(-1.2,-1.2)(1.2,1.2)[linecolor=cyan](15,9)(12,9)(15,6)[linecolor=red](15,7.5)(13.8,8.4)(15,6)[linecolor=green](15,12)(10.2,8.4)(15,6)[linecolor=blue](15,3)(15,6)(12,3)[linecolor=darkred](15,4.5)(15,6)(13.8,3.6)[linecolor=darkgreen](15,0)(15,6)(10.2,3.6)Im
($\Gamma$)Re
($\Gamma$)$X=2$$X=1$$X=\frac{1}{2}$$X=-\frac{1}{2}$$X=-1$$X=-2$$X=\infty$$X=0$Im
($z$)R = Re ($z$)$X=2$$X=1$$X=\frac{1}{2}$$X=0$$X=-\frac{1}{2}$$X=-1$$X=-2$
Figure 25: Lines of constant imaginary part transformed into the $\Gamma$
plane
## .7 Rulers around the Smith chart
Some Smith charts provide rulers at the bottom to determine other quantities
besides the reflection coefficient such as the return loss, the attenuation,
the reflection loss etc. A short instruction on how to use these rulers as
well as a specific example for such a set of rulers is given here.
### .7.1 How to use the rulers
First, one has to take the modulus (= distance between the centre of the Smith
chart and the point in the chart referring to the impedance in question) of
the reflection coefficient of an impedance either with a conventional ruler
or, better, using a compass. Then refer to the coordinate denoted as CENTRE
and go to the left or for the other part of the rulers to the right (except
for the lowest line which is marked ORIGIN at the left which is the reference
point of this ruler). The value in question can then be read from the
corresponding scale.
### .7.2 Example of a set of rulers
A commonly used set of rulers that can be found below the Smith chart is
depicted in Fig. 26.
Figure 26: Example for a set of rulers that can be found underneath the Smith
chart
For further discussion, this ruler is split along the line marked centre, to a
left (Fig. 27) and a right part (Fig. 28) since they will be discussed
separately for reasons of simplicity.
Figure 27: Left part of the rulers depicted in Fig. 26 Figure 28: Right part
of the rulers depicted in Fig. 26
The upper part of the first ruler in Fig. 27 is marked SWR which refers to the
Voltage Standing Wave Ratio. The range of values is between one and infinity.
One is for the matched case (centre of the Smith chart), infinity is for total
reflection (boundary of the SC). The upper part is in linear scale, the lower
part of this ruler is in dB, noted as dBS (dB referred to Standing Wave
Ratio). Example: SWR = 10 corresponds to 20 dBS, SWR = 100 corresponds to 40
dBS (voltage ratios, not power ratios).
The second ruler upper part, marked as RTN.LOSS = return loss in dB. This
indicates the amount of reflected wave expressed in dB. Thus, in the centre of
the Smith chart nothing is reflected and the return loss is infinite. At the
boundary we have full reflection, thus a return loss of 0 dB. The lower part
of the scale denoted as RFL.COEFF. P = reflection coefficient in terms of
POWER (proportional $|\Gamma|^{2}$). There is no reflected power for the
matched case (centre of the Smith chart), and a (normalized) reflected power =
1 at the boundary.
The third ruler is marked as RFL.COEFF,E or I. With this, the modulus (=
absolute value) of the reflection coefficient can be determined in linear
scale. Note that since we have the modulus we can refer it both to voltage or
current as we have omitted the sign, we just use the modulus. Obviously in the
centre the reflection coefficient is zero, while at the boundary it is one.
The fourth ruler is the voltage transmission coefficient. Note that the
modulus of the voltage (and current) transmission coefficient has a range from
zero, i.e., short circuit, to +2 (open = 1+$|\Gamma|$ with $|\Gamma|$=1). This
ruler is only valid for $Z_{\text{load}}$ = real, i.e., the case of a step in
characteristic impedance of the coaxial line.
The upper part of the first ruler in Fig. 28, denoted as ATTEN. in dB assumes
that an attenuator (that may be a lossy line) is measured which itself is
terminated by an open or short circuit (full reflection). Thus the wave
travels twice through the attenuator (forward and backward). The value of this
attenuator can be between zero and some very high number corresponding to the
matched case. The lower scale of this ruler displays the same situation just
in terms of VSWR. Example: a 10 dB attenuator attenuates the reflected wave by
20 dB going forth and back and we get a reflection coefficient of $\Gamma$ =
0.1 (= 10% in voltage).
The upper part of the second ruler, denoted as RFL.LOSS in dB refers to the
reflection loss. This is the loss in the transmitted wave, not to be confused
with the return loss referring to the reflected wave. It displays the relation
$P_{\text{t}}=1-|\Gamma|^{2}$ in dB. Example: If $|\Gamma|=1/\sqrt{2}=$ 0.707,
the transmitted power is 50% and thus the loss is 50% = 3 dB.
The third ruler (right), marked as TRANSM.COEFF.P refers to the transmitted
power as a function of mismatch and displays essentially the relation
$P_{\text{t}}=1-|\Gamma|^{2}$. Thus in the centre of the Smith chart there is
a full match and all the power is transmitted. At the boundary there is total
reflection and for a $\Gamma$ value of 0.5, for example, 75% of the incident
power is transmitted.
## References
* [1] H. Meinke, F.–W. Gundlach, Taschenbuch der Hochfrequenztechnik, Springer Verlag, Berlin, 1992
* [2] P. Smith, Electronic Applications of the Smith Chart, Noble Publishing Corporation, 2000
* [3] M. Paul, Kreisdiagramme in der Hochfrequenztechnik, R. Oldenburg Verlag, Muenchen, 1969
* [4] O. Zinke, H. Brunswig, Lehrbuch der Hochfrequenztechnik, Springer Verlag, Berlin – Heidelberg, 1973
|
arxiv-papers
| 2012-01-19T14:31:35 |
2024-09-04T02:49:26.508579
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Caspers (CERN)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1201.4068"
}
|
1201.4215
|
2011 Vol. No. XX, 000–000
11institutetext: College of Science, China University of Petroleum, Qingdao
266555, China; hdchen@upc.edu.cn
22institutetext: Key Laboratory of Solar Activity, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, China;
33institutetext: Harvard-Smithsonian Center for Astrophysics, MA 02138, USA;
Received [year] [month] [day]; accepted [year] [month] [day]
# Kinematics of an untwisting solar jet in polar coronal hole observed by
SDO/AIA
H. Chen 1122 J. Zhang 22 S. Ma 1133
###### Abstract
Using the multi-wavelength data from the Atmospheric Imaging Assembly (AIA) on
board the Solar Dynamics Observatory (SDO) spacecraft, we study a jet occurred
in coronal hole near the northern pole of the Sun. The jet presented distinct
helical upward motion during ejection. By tracking six identified moving
features (MFs) in the jet, we found that the plasma moved at an approximately
constant speed along the jet’s axis, meanwhile, they made a circular motion in
the plane transverse to the axis. Inferred from linear and trigonometric
fittings to the axial and transverse heights of the six tracks, the mean
values of axial velocities, transverse velocities, angular speeds, rotation
periods, and rotation radiuses of the jet are 114 km s-1, 136 km s-1, 0.81
s-1, 452 s, and 9.8 $\times$ 103 km respectively. As the MFs rose, the jet
width at the corresponding height increased. For the first time, we derived
the height variation of the longitudinal magnetic field strength in the jet
from the assumption of magnetic flux conservation. Our results indicate that,
at the heights of 1 $\times$ 104 $\sim$ 7 $\times$ 104 km from jet base, the
flux density in the jet decreased from about 15 to 3 G as a function of
B=0.5(R/R${}_{\sun}$-1)-0.84 (G). A comparison was made with the other results
in previous studies.
###### keywords:
sun: activity — sun: chromosphere — sun: magnetic fields — sun: flare — sun:
rotation
## 1 Introduction
Solar jets are small-scale plasma ejections along straight or slightly curved
coronal fields (e.g., Shibata et al. 1994; Li et al. 1996; Chae et al. 1999).
They can be observed as emission in Ultraviolet (UV; e.g., Schmieder et al.
1988; Chen et al. 2008), Extreme-ultraviolet (EUV; e.g., Schmahl 1981;
Alexander & Fletcher 1999; Kamio et al. 2007; Kim et al. 2007; Chifor et al.
2008a, 2008b; Kamio et al. 2009; Yang et al. 2011a; Tian et al. 2011), soft
X-ray (SXR; e.g., Shibata et al. 1992; Zhang et al. 2000; Cirtain et al. 2007;
Moore et al. 2011) and white light (WL; e.g., Wang et al. 1998a; Liu et al.
2005b). The detailed statistical properties of X-ray jets was studied by
Shimojo et al. (1996) and more recently by Savcheva et al. (2007). In
morphology, surges are very similar to jets, but they appear as absorption
features when observed on solar disk. Sometimes, surges are observed to be
associated with filament formation (e.g., Liu et al. 2005a), filament eruption
(e.g., Chen et al. 2009a; Guo et al. 2010; Moore et al. 2010; Hong et al.
2011) and even coronal mass ejections (CMEs, e.g., Liu et al. 2005b; Jiang et
al. 2008). Generally speaking, surges and jets are the different appearances
in different wavelengths of the same phenomenon. In the following context, we
use the term “jets” refers to both surges and jets.
Helical or twisted structures in jets have been reported by many authors
(e.g., Dizer 1968; Shibata et al. 1992; Canfield et al. 1996; Wilhelm et al.
2002; Jibben & Canfield 2004; Jiang et al. 2007; Liu et al. 2009; Shen et al.
2011; Liu et al. 2011; Curdt & Tian 2011). By using line of sight velocity
field (e.g., Xu et al. 1984; Gu et al. 1996; Jibben & Canfield 2004) and
stereoscopic (e.g., Patsourakos et al. 2008; Nisticò et al. 2009)
observations, some researchers confirmed that the rotation motions in some
jets are real. Xu et al. (1984) proposed a double-pole diffusion model to
explain the rotating motion of a surge. However, in consideration of the close
relationship between jets and photospheric magnetic flux activities, such as
magnetic flux emergence, convergence, and cancelation etc (e.g., Roy 1973;
Wang & Shi 1993; Shimojo et al. 1998; Zhang et al. 2000; Liu & Kurokawa 2004;
Chen et al. 2008), more authors incline to think that the spinning of jets
results from the relaxation of magnetic twist, which occurs when twisted
photospheric magnetic loop reconnects with ambient open fields (e.g., Shibata
& Uchida 1986; Shibata et al. 1994; Canfield et al. 1996; Patsourakos et al.
2008; Nisticò et al. 2009; Kamio et al. 2010; He et al. 2010). Recently,
three-dimensional simulations by Pariat et al. (2009) show that high-level
magnetic stress due to twisting motion can lead to an explosive release of
energy via reconnection, which will produce massive, high-speed jets driven by
nonlinear Alfvén wave. And if the stress is constantly applied at the
photospheric boundary, this mechanism would generate recurrent untwisting
quasi-homologous jets (e.g., Pariat et al. 2010; Asai et al. 2001; Chen et al.
2008; Yang et al. 2011b). More recently, the simulations by Díaz et al. (2011)
indicate that the speed of the flow along the field lines of twisted magnetic
flux tubes may be super-Alfvénic and the twisted tube is subject to the kink
instability, which could explain the behaviour of super-Alfvénic jets and the
disruption of some observed jets.
As mentioned above, so far, the main observational methods to investigate the
spining of jets are focused on the analysis of line of sight velocity field
and stereoscopic observations, or taking advantage of the technique of time-
distance analysis (e.g., Liu et al. 2009). Lower temporal and spatial
resolutions of these observations or the limitation of the technique used in
these studies can not make the exact kinematics of jet clear. For example, the
tracks or stripes in the time-distance slit images can not stand for the real
motion of jet plasma along the slit direction due to the perpendicular
velocity. The Atmospheric Imaging Assembly (AIA; Lemen et al. 2011) on the
Solar Dynamics Observatory (SDO; Schwer et al. 2002) images the solar
atmosphere in 10 wavelengths with 12 s high temporal resolution. The
instrument observes solar plasma from photosphere to low corona with a full-
disk field of view and the pixel size is about 0.6. High-resolution AIA
intensity images can reveal the fine structures in jets, which provides us an
opportunity to track the motions of some moving features (MFs) in jets. Using
this new method, we study the detailed kinematics of one AIA 304 Å jet, which
has been investigated by Shen et al. (2011) mainly using the technique of
time-distance slit images. One aim of this paper is to compare the results
from the two different methods.
In addition, the measurement of coronal magnetic field strength is a long-
standing unresolved problem in solar physics (e.g., West et al. 2011). Due to
thermal broadening and polarization effect, usual methods for measuring
coronal flux density, such as Zeeman splitting of spectral lines and Hanle
effect, become complicated. Focusing on the stronger active region fields, Lin
et al. (2004) measured the magnetic flux density 100$\arcsec$ ($\sim$7
$\times$ 104 km ) above an active region to be 4 G, which is smaller than the
results (10$\sim$33 G) presented by an earlier work of Lin et al. (2000). Some
indirect methods, such as photospheric extrapolation techniques (e.g., Wang &
Sheeley 1992; Liu & Lin 2008), radio techniques (e.g., Ramesh et al. 2010),
and coronal seismology (e.g., Uchida 1970; Roberts et al. 1984; Chen et al.
2011; West et al. 2011; Gopalswamy & Yashiro 2011), are applied to estimate
the coronal magnetic field. In this study, in combination with the
observations of the studied jet, we try to provide a new technique to estimate
the magnetic field of the higher part of the jet. This would give valuable
insight into the coronal magnetic field structure.
In the next section, we briefly describe the observations and data used in our
study. This is followed by a detailed study of the kinematics of the jet and
an estimation of the longitudinal magnetic flux density in the jet. Finally,
we give the summaries and discussions.
Figure 1: Panel a: AIA 1600 Å image displaying the brightening patches (BPs)
at the base of jet. Panels b-j: negative AIA 304 Å images showing the detailed
evolution of the jet. The images have been clockwisely rotated by 18$\degr$
from the northern pole of the Sun, which is the same for all AIA images in
Fig. 2 and 3. The field of view (FOV) of 304 Å images is 132$\times$ 216\. The
dashed box in panel b indicates the FOV of panel a, which is 48$\times$ 79.
## 2 Data and Observations
On 2010 August 21, a jet occurred at the northeast limb (E0N81) of the Sun
(Shen et al. 2011). The observation from Extreme UltraViolet Imager (EUVI;
Wuelser et al. 2004) of Sun Earth Connection Coronal and Heliospheric
Investigation (SECCHI; Howard et al. 2008) on board the spacecraft B of Solar
Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008) indicates that
the jet is rooted in coronal hole (e.g., Zhang et al. 2007) near the northern
pole of the Sun. The detailed evolution of the jet was observed by the AIA on
SDO, which provides multiple simultaneous high-resolution full-disk images up
to 0.5R${}_{\sun}$ above the solar limb with 1.2 arcsec spatial resolution and
12 s cadence. All the ten bandpasses have been employed in the observations of
this jet activity. In this paper, we mainly used the channels centered at 304
Å, 1600 Å, 171 Å, 193 Å, and 211 Å (Level 1.5 images) with the temperature
responses peak at 0.05 MK, 0.1 MK, 0.6 MK, 1.5 MK (also 20 MK), and 2.0 MK,
respectively (Lemen et al. 2011). We did not perform any de-rotation since the
rotation effect will not influence our results significantly.
## 3 Results
### 3.1 General Evolution of the Jet
Figure 1 shows the morphology and general evolution of the jet at AIA 304Å
(reversed color table). Since the projected direction of the jet’s axis is
about 18anti-clockwise from the northern pole of the Sun, all the AIA images
in this paper have been rotated the same angle clockwisely for better showing.
According to the AIA observations, the jet took place at about 06:07 UT, when
a brightening patch BP1 (see panel a of Fig. 1) firstly began to appear at one
(eastern) side of the root and gradually evolved into a apparent inverted “Y”
structure in 171 Å images. Since then, dense plasma began to flow out from BP1
and expanded westwardly. From AIA 1600 Å images, we can see that another
brightening patch BP2 (in panel a of Fig. 1) appeared at the opposite
(western) side of the base region at about 06:18 UT and peaked at 06:23 UT. In
combination with 304 Å observations, it seems that the main mass of jet was
ejected from above BP2 rather than BP1 after BP2 appeared. We consider that
this location change of jet footpoint has a close association with the
magnetic reconnection occurred at the jet base.
Figure 2: Panels a and c are AIA 304 Å and 171 Å intensity images,
respectively. They have a same FOV of 150$\times$ 216\. The white narrow boxes
indicate the positions of the slits S1-S3, which have a FOV of 74$\times$ 4\.
Panel b: slit images from AIA 304 Å channel along S1-S3, respectively. Panel
d: slit images along S2 from AIA 171 Å, 193 Å, and 211 Å channels,
respectively. The two white dashed lines indicate the time 06:34:08 and
06:32:24 UT when the AIA 304 Å and 171 Å intensity images (in panels a and c,
respectively) were recorded. The arrows in panels b and d point to some
stripes, which indicate the transverse motions of the plasma across the jet.
As the plasma was ejected outwards, the jet also spun clockwise as viewed from
its footpoints. Because of the movements along both axial and transverse
(rotation) direction, the jet appeared as upward helical structures. Some fine
twisted threads with a mean width of a few arcsecs can be identified clearly,
which are indicated by the white arrows in panels e and f of Fig. 1. According
to the charity definition of jet in Jibben and Canfield (2004), the jet we
studied here is a right-hand jet. As time went on, these threads gradually
unwound, and one big bifurcation (indicated by the arrow in panel g of Fig. 1)
appeared at about 06:35 UT from the bottom and spread upward along the body of
the jet. At about 06:45 UT, after reaching a maximum height of 17.9 $\times$
104 km, the material began to fall back along almost the axial direction
without any transverse motion.
Interestingly, we note that another jet (indicated by the arrows in panels i
and j of Fig. 1) occurred at about 06:42 UT, when the first has not completely
disappeared yet. Its feet are very close to the feet of the first jet, which
means they likely originated from the same source region. However, the
ejection directions of the two jets are not very consistent with each other.
Similar phenomenon has also been reported by Chen et al. (2008). In recent
simulation of Pariat et al. (2010), their results show that the drifting
directions are different for recurrent jets, even if the underlying magnetic
system and the driving motion remain constant.
Figure 3: Panel a: one AIA 304 Å image overlaid with the tracks of MF1-MF6
(plus, asterisk, triangle, diamond, square, and $\times$, respectively). The
time in the parentheses are the start and end tracking time of the
corresponding MFs. The FOV of panel a is 168$\times$ 144\. The yellow dashed
box indicates the FOV of the slit image in panel b, which is about 14$\times$
52\. Panel b: slit images showing MF2. The blue asterisks indicate the
positions of MF2 at different time.
### 3.2 Helical Upward Motion
A remarkable character of this jet is its distinct transverse rotating motion.
In Figure 2, we show this transverse motion in detail. Two AIA intensity
images in 304 Å and 171 Å wavelengths are given in panels a and c of Fig. 2,
respectively. The three white narrow boxes (74$\times$ 4) in the 304 Å images
mark three slits S1-S3 from top to bottom, which are perpendicular to the jet
axis. The heights of S1-S3 from the jet base are about 2.36 $\times$ 104, 4.43
$\times$ 104, and 6.50 $\times$ 104 km, respectively. In panel b of Fig. 2, we
display the time-distance diagrams at 304 Å along slits S1, S2 and S3 from top
to bottom, respectively. As shown in these time-distance diagrams, it can be
seen that there are many stripe structures, which indicate the transverse
motion of the plasma in the jet. In total, fifteen stripes can be clearly
identified in panel b, among which several typical ones are indicated by the
white arrows.
We performed linear fittings to all the fifteen time-distance tracks, and
found that the transverse velocities of these features range from 70 to 200 km
s-1 with a mean value of 134 km s-1. Using the same method, a more detailed
investigation on the transverse motion of this jet has been done by Shen et
al. (2011). According to their results, the total average transverse velocity
of the jet is 123 km s-1. In this paper, using the observations from other
bandpasses, we further extended this study. In panel d of Fig. 2, the time-
distance diagrams from 171 Å, 193 Å, and 211 Å intensity images are plotted
from top to bottom, respectively. Because the transverse rotation is not very
clear in 171 Å, 193 Å, and 211 Å observations along S1 and S3, the time-
distance diagrams at slit S2 are shown alone. Similarly, from the slit images
in panel d, the transverse helical motion features can be identified clearly,
some of which are marked by the black arrows. Furthermore, in morphology,
these transverse motion features observed in the three EUV wavelengths are
very similar. According to the linear fitting results, the mean transverse
velocity of these features is around 140 km s-1, which is similar to the
result from 304 Å slit images.
Figure 4: Time variations of the relative axial (circle) and transverse (plus)
heights of MF1-MF6. The red and blue solid lines are the results of linear and
trigonometric fittings to the axial and transverse heights, respectively.
Here, va, vt, T, $\omega$, and A represent the axial velocity, transverse
velocity, rotation period, angular speed and rotation radius of the MFs,
respectively. Figure 5: Height variation of axial magnetic field strength in
the jet. The $\times$ (red), square (blue), and triangle (purple) symbols
correspond to the data derived from the tracks of MF6, MF5, and MF3,
respectively. The dashed and dash-dotted lines are the results from Verth et
al. (2011) and Dulk & McLean (1978), respectively. The solid line is the fit
to our observational data. The dotted line indicates the position of solar
surface.
To reveal the kinematics of the jet more clearly, we tracked six moving
features (MFs) that can be clearly identified in the jet. Considering the
freezing-in effect of plasma-magnetic field coupling, we assume that each MF
moved along the same magnetic field line during ejection. In panel a of Figure
3, one AIA 304 Å image at 06:38:56 UT is overlaid with the tracks of MF1-MF6
(marked with plus, asterisk, triangle, diamond, square, and $\times$ symbols,
respectively). The start and ending tracking time of each MF are shown in the
corresponding parentheses. It can be seen from Fig.3 that most of MFs’ tracks
appear like helical lines, which indicates the MFs made helical upward motions
in the jet.As an example, we show the evolution of MF2 in panel b of Fig.3 The
blue asterisks in panel b indicate the positions of MF2 at different time.
In Figure 4, we plot the time profiles of axial (circle) and transverse (plus)
heights of the six MFs. Note that the time and heights in each panel are the
values relative to the initial time and heights of respective MF. According to
the different distributions of the axial and transverse heights, we can see
that all the MFs seemed to move at a approximately constant speed along the
jet’s axis and make a circular motion across the jet at the meantime. We
performed linear (red solid line) and trigonometric (blue solid line) fittings
to the axial and transverse heights of each MF, respectively. It can be found
that the observational data are fitted very well, which provide further
evidence of the helical motion of the jet. The axial velocity (va), transverse
velocity (vt), angular speed ($\omega$), rotation period (T), and rotation
radius (A) of each MF are derived from the linear and trigonometric fitting
results and shown in the corresponding panel of Fig. 4. The mean values of va,
vt, $\omega$, T and A are 114 km s-1, 136 km s-1, 0.81s-1 (or 14.1 $\times$
10-3 rad s-1), 452 s, and 9.8 $\times$ 103 km , respectively. In comparison
with the results from Shen et al. (2011), we found most of the results are
similar except the mean axial velocities, which are 114 and 171 km s-1 in our
and their studies, respectively. This difference maybe results from the
limitation of our MFs sample number.
In additon, from the fitting results in our study, there seems to be no
obvious correlation between va and vt. For example, the transverse velocities
of MF1, MF2 and MF3 are about 150 km s-1; however, the axial velocity (153 km
s-1) of MF1 is much bigger than those (93 and 82 km s-1) of MF2 and MF3. Of
course, for more accurate statistical relationship between the axial and
transverse velocity of jet plasma, further study based on more samples is
needed. Using these results, we roughly estimate the twist spreading into the
outer corona during ejection, which may be restored in the photospheric flux
rope before. The AIA 304 Å movie shows that the total spinning period of the
jet is approximately from 06:16 UT to 06:42 UT ($\sim$26 minutes). Assuming
that the jet made an uniform circular motion, the total twist restored can be
yielded by dividing the total spinning time by the mean rotation period (452
s), which is about 3.6 turns. In contrast to the result presented by Shen et
al. (2011), ours is bigger..
### 3.3 Axial Magnetic Field Strength in the Jet
As mentioned at the beginning, in this study, we try to provide a new method
to estimate the longitudinal magnetic field in the jet. Our basic idea is that
assuming the jet plasma flows in the same flux tube during ejection, the
magnetic flux across the transverse section of the jet would keep constant,
i.e.
$B_{o}S_{o}=BS$ (1)
So, if we can determine the photospheric magnetic field strength (Bo),
transverse area (So) of the flux tube at the jet base and the transverse area
(S) of the flux tube at a certain height, then the axial magnetic field
strength (B) at the corresponding height can be derived from equation (1).
We describe the determinations of Bo, So, and S as below. First, we
approximately think the flux tube, i.e. the channel along which the jet
material flows, as a axisymmetric cylinder with a increasing radius (r). So,
the transverse area So and S can be represented by $\pi$$r_{o}^{2}$ and
$\pi$$r^{2}$, respectively. Due to dispersion, it is difficult to measure the
jet’s radius (ro) at the base directly. Thus, ro is estimated by the size of
the brightening patch BP2 appeared at the jet base in AIA 1600 Å image, which
implies ro is about 2.6 $\times$ 103 km. By tracking the axial heights of the
MFs at different time, we measured the width of jet at the corresponding
heights, which is twice the size of r. As for Bo, since the jet occurred at
solar rim and there is no available photospheric magnetic field data, here, we
take the mean photospheric flux density value ($\sim$500 G) from Chen et al.
(2008) as Bo. Considering the similar spatial and temporal scales of jets in
this study and Chen et al. (2008), the used value (500 G) of Bo should be
reasonable.
Using the corresponding data from the tracks of MFs, the axial magnetic field
strength (B) along the jet were derived from equation (1) and its variation
with height (from the base of jet) is shown in Figure 5. Note that we only
used the data from three tracks of MF3 (purple triangle), MF5 (blue square),
and MF6 (red $\times$). This is because not only the axial heights of the
three MFs but also their evolution time have better succession than the
other’s (see Fig. 3). The errors of B mainly result from the uncertainty in
measurements of ro and r and increase as the heights decline. From Fig. 5, it
can be seen that B decreases with increasing height. Especially, it falls
quickly at the lower height. According to our results, B decreases half (from
about 15$\pm$4 to 7$\pm$2 G) from the height of 1.1 $\times$ 104 km to 2.8
$\times$ 104 km, with a mean drop rate of 4 G per 104 km. As the height keeps
increasing, B gradually declines to about 3$\pm$1 G at the height of 7
$\times$ 104 km. From the heights of 2.8 $\times$ 104 to 7 $\times$ 104 km,
the mean decline rate of B is about 1 G per 104 km, which is only one fourth
of that below 2.8 $\times$ 104 km.
Taking advantage of the same relationship of the magnetic flux density with
the width of flux tube, i.e. B $\sim$ 1/r2, Verth et al. (2011) studied the
magnetic field strength along a solar spicule. Their results (dashed line) are
shown in Fig. 5. Obviously, the flux density derived in Verth et al. (2011)
drops more quickly at the typical heights (from photosphere to 7 $\times$ 103
km) of a spicule. As a comparison, we also show the results (dash-dotted line
in Fig. 5) from the empirical active region magnetic field model (Dulk &
McLean 1978), which is given by B=0.5(R/R${}_{\sun}$-1)-1.5 G. On the whole,
the field strength values from the model are about six times of our results.
In consideration of the different magnetic field structures between above
active region and in the coronal hole and the certain errors of measurements,
we think our results are reasonable. By revising slightly to the empirical
formula presented by Dulk & McLean (1978), we found a new formula
$B=0.5(R/R_{\sun}-1)^{-0.84}~{}~{}G$ (2)
fits our observations well, wherein, R is the distance from the solar center.
In Fig. 5, the fitting results are indicated by the black solid line across
the colored symbols.
## 4 SUMMARY
In this paper, we present a detailed study of a jet which showed a distinct
transverse rotating motion during its ejection. The observational results
appear to be consistent with an untwisting model of magnetic reconnection
(e.g., Shibata & Uchida 1986; Pariat et al. 2009; 2010). By tracking six
identified features moving helically in the jet, we found that the jet plasma
moved at an approximately constant velocity along the axial direction and made
a circular motion in the plane perpendicular to the jet axis. we derived the
axial velocity, transverse velocity, angular speed, rotation period and
rotation radius for each MF. Their mean values are 114 km s-1, 136 km s-1,
0.81s-1, 452 s, and 9.8 $\times$ 103 km , respectively. By comparison with the
other study using different method (Shen et al. 2011), we found most of the
results are similar. For more accurate kinematics of jet plasma, a more
extensive statistical investigation work is expected in the future.
On assumption of the magnetic flux conservation in the same flux tube, we made
an estimation of the field strength of the jet occurred in the polar coronal
hole. Our results show that the longitudinal flux density of the jet at the
heights of 1 $\times$ 104 $\sim$ 7 $\times$ 104 km from solar surface,
decreased from about 15 to 3 G. Comparing with the result from Dulk & McLean
(1978), a new formula of B=0.5(R/R${}_{\sun}$-1)-0.84 (G) fits our estimated
data well. It should be noted that the Bo used in our study is just an
estimated value, which maybe leads to a major error of the absolute value of
B. However, it would not significantly affect the height variation of B. On
the other hand, since almost all of the direct (Lin et al. 2000; Lin et al.
2004) or indirect (Cho et al. 2007; Ramesh et al. 2010; West et al. 2011)
measurements of coronal fields strength in previous studies are mainly focused
on the stronger fields above or at least emanate from major active regions,
our results could offer helpful information about the magnetic field
structures above mini active region newly-emerged in coronal hole.
The formation of the moving features (MFs) in the jet is also an interesting
question. Similar features in jets can be seen in some other observations
(e.g., Jiang et al. 2007; Liu et al. 2009; Liu et al. 2011). Although we know
that both local density enhancement and temperature enhancement might be
responsible for the existence of the MFs. However, why and how the local
density or temperature enhancement takes place are still obscure at present
time. We suggest that three possible mechanisms might contribute to the
formation of MFs. First, the formation of the MFs might be associated with the
successive occurrence of magnetic reconnection at the jet base. Second, the
intrinsic (sausage or kink) instability in the mass flow as described in Chen
et al. (2009b) and Díaz et al.(2011) might be another possible candidate. In
morphology, the MFs are similar with the plasma blobs observed in coronal
streamers (e.g., Sheeley et al. 1997; Wang et al. 1998b; Wang et al. 2000;
Song et al. 2009). The simulations by Chen et al.(2009b) reveal that the
sausage-kink instability of coronal streamers may lead to the formation of the
plasma blobs. At this point, we think that the production mechanism of MFs in
jets may be similar to that of the plasma blobs. In addition, the simulations
of Díaz et al.(2011) support that the kink instability in the mass flow can
result in the disruption observed in solar jets. At last, the intrinsic
unevenness of plasma density in the photospheric twisted flux tube might also
be a possible formation mechanism of the MFs in solar jets.
###### Acknowledgements.
The authors sincerely thank the anonymous referee for very helpful and
constructive comments that improved this paper. We are grateful to all the
members of the Solar Magnetism and Activity group of National Astronomical
Observatory of CAS for invaluable help. We acknowledge the AIA team for the
easy access to calibrated data. The AIA data are courtesy of SDO (NASA) and
the AIA consortium. This work was supported by the National Natural Science
Foundation of China (11103090, G11025315, 40890161, 10921303, 40825014, and
40890162), the CAS project KJCX2-YW-T04, the National Basic Research Program
of China under grant G2011CB811403, and Shandong Province Natural Science
Foundation (ZR2011AQ009).
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|
arxiv-papers
| 2012-01-20T06:33:51 |
2024-09-04T02:49:26.519347
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Huadong Chen, Jun Zhang and Suli Ma",
"submitter": "Huadong Chen",
"url": "https://arxiv.org/abs/1201.4215"
}
|
1201.4258
|
# Resonant tunneling diode based on graphene/h-BN heterostructure
V. Hung Nguyen, F. Mazzamuto, A. Bournel, and P. Dollfus V. Hung Nguyen is
with the Institut d’Electronique Fondamentale, CNRS, Universit$\acute{e}$
Paris-Sud, UMR8622, 91405 Orsay, France and also with the Center for
Computational Physics, Institute of Physics, VAST, P.O. Box 429 Bo Ho, Hanoi
10000, Vietnam (e-mail: viet-hung.nguyen@u-psud.fr).F. Mazzamuto, A. Bournel
and P. Dollfus are with the Institut d’Electronique Fondamentale, CNRS,
Universit$\acute{e}$ Paris-Sud, UMR8622, 91405 Orsay, France.
###### Abstract
In this letter, we propose the resonant tunneling diode (RTD) based on a
double-barrier graphene/boron nitride (BN) heterostructure as device suitable
to take advantage of the elaboration of atomic sheets containing different
domains of BN and C phases within a hexagonal lattice. The device operation
and performance are investigated by means of a self-consistent model within
the non-equilibrium Green’s function formalism on a tight-binding Hamiltonian.
This RTD exhibits a negative differential conductance effect which involves
the resonant tunneling through both the electron and hole bound states of the
graphene quantum well. It is shown that the peak-to-valley ratio can reach the
value of 4 at room temperature for gapless graphene and the value of 13 for a
bandgap of 50 meV.
###### Index Terms:
Graphene, Boron nitride substrate, Resonant tunneling diode.
## I Introduction
The benefits of the extraordinary intrinsic transport properties of graphene
[1] are usually hindered by the defects of the supporting insulating
substrate. Indeed, graphene on various substrates such as SiO2 [2], SiC [3] or
other high-$\kappa$ insulators [4] shows a reduced carrier mobility, which is
usually attributed to surface roughness, charge surface states or surface
optical phonons. Recently, it has been shown that graphene reported on
hexagonal boron nitride (h-BN) has higher mobility than on any other substrate
[5, 6, 7]. A mobility of 275 000 cm2/Vs at low temperature and 125 000 cm2/Vs
at room temperature, i.e. as high as for suspended graphene, has even been
reported [8]. It is due to the fact that the surface of h-BN is flat, with a
low density of charged impurities, does not have dangling bonds and is
relatively inert [5]. Hence, with the same atomic structure as graphene and a
1.8$\%$ higher lattice constant [9] h-BN is becoming a very promising
candidate as high bandgap insulating material to make possible reaching the
ballistic transport regime in graphene devices even at room temperature and
exploiting the peculiarities of graphene properties inherent in the massless
and chiral character of charge carriers. Additionally, it has been shown that
it is possible to obtain atomic sheets containing different domains of h-BN
and C phases with various compositions [10], which offers the perspective of
designing new heterostructures and devices [11].
In the field of quantum devices, graphene resonant tunnelling diodes (RTDs)
have been proposed and investigated previously [12, 13]. However, while the
operation of graphene nanoribbon (GNR) RTDs is strongly dependent on the
device shape [12], the 2D graphene structure studied in [13] assumes that a
bandgap opens in graphene epitaxially grown on SiC, which is a debated
question, in addition to the substrate-induced mobility reduction. In this
work, we thus propose a new graphene RTD based on graphene/h-BN
heterostructures, where the two barriers are assumed to be formed by large
bandgap h-BN zones, as schematized in Fig. 1. Though it may be useful and
possible in case of Bernal stacking on BN substrate [14], the bandgap in
graphene is not necessary for this device to operate correctly.
To investigate the electrical characteristics of this RTD, we have developed a
self-consistent simulation code based on the Green’s function formalism [15]
to solve the tight-binding Hamiltonian suggested in [16].
Figure 1: Simulated device wherein the graphene sheets are deposited on a h-BN
substrate and two barriers are thus formed by the BN layers between the
graphene zones.
## II Physical model and results
In the structure schematized in Fig. 1, the BN barriers are inserted in a
monolayer graphene sheet reported on a h-BN substrate of thickness 10 nm and
dielectric constant $\epsilon_{r}$ = 3.5 [5]. We assume the transverse width
along OY direction to be much larger than the typical device length along the
transport direction, i.e. a few tens of nanometers, so that the extension
along the transverse direction can be considered through Bloch periodic
boundary conditions [17]. The length of the heavily doped regions at both
device ends and the BN barrier thickness are assumed to be 17 nm and 1.3 nm,
respectively. Our approach is based on the self-consistent solution of the 2D
Poisson and Schrödinger equations. The nearest-neighbor tight-binding
Hamiltonian with parameters determined in [16] is solved using the Green’s
function method [15] in the ballistic approximation. In this simulation, the
null Neumann conditions are applied at the 2D-domain boundaries (dashed line
in Fig. 1). In the case of Bernal stacking, graphene reported on a h-BN
substrate was shown to exhibit a finite bandgap, but this bandgap vanishes
when graphene is misaligned with respect to h-BN lattice [14]. Therefore,
unless otherwise stated, the zero bandgap of graphene was assumed in our
study. All simulations were performed at room temperature.
Figure 2: (a) Self consistent data of potential energy and (b) corresponding
electron density at different applied biases. The doped concentration in two
device ends $N_{D}=5\times 10^{16}m^{-2}$. The vertical-dashed lines indicate
the h-BN regions. Figure 3: (a) Diagram showing the LDOS and the transmission
coefficient as a function of energy at $V_{bias}=0.4$ V and (b) I-V
characteristics of the simulated device (for both cases of zero and finite
bandgaps). The figures in (a) are plotted for a transversal momentum
$k_{y}\neq K_{y}$ (momentum at the Dirac point). The corresponding
electron/hole potential profiles are superimposed on the LDOS (solid lines).
The doped concentration $N_{D}=5\times 10^{16}m^{-2}$.
In Fig. 2, the self-consistent solutions of potential and electron density are
plotted for different biases applied to the device with well width $D$ = 4.3
nm and doping concentration in access regions $N_{D}=5\times 10^{16}$ m-2. The
low doping in the central zone generates a potential barrier which controls
the threshold voltage of the device. Differently from the conventional RTDs
[18], Fig. 2(b) shows that the local charge density is not symmetrical even at
zero bias. This feature can be understood as a consequence of the device
asymmetry between the left graphene section coupled to nitrogen atoms and the
right section bounded to boron atoms of the h-BN lattice. Besides, it is shown
that the electron density accumulated in the quantum well increases when
increasing the bias voltage from 0 V to 0.4 V and decreases when further
raising the bias from 0.4 V to 0.6 V. This feature is nothing but an effect of
resonant tunneling [18] as described below.
We now go to explore the electrical behavior of the device. In Fig. 3, we
display the I-V characteristics together with a diagram showing the LDOS and
the corresponding transmission coefficient at $V_{bias}$ = 0.4 V. Indeed, the
diagram in Fig. 3(a) shows that the large bandgap ($\approx 3.9$ eV [16]) in
the h-BN barriers generates the quantization of carrier states in the graphene
quantum well, which gives rise to resonant tunneling effects. As a
consequence, the I-V curves can exhibit a strong negative different
conductance (NDC) in both cases of gapless and gapped graphene. In principle,
the valley current in semiconductor RTDs occurs at a finite bias when there is
no available state in the emitter region for tunneling via the confined states
in the quantum well. This feature is essentially due to the finite bandgap of
the emitter. The results obtained here are thus explained as follows. Though
the energy bandgap of graphene may be actually zero for the transverse
momentum mode corresponding to the Dirac (or K) point (i.e., $k_{y}\equiv
K_{y}=2\pi/3\sqrt{3}a_{c}$), a finite energy gap
$\hat{E}_{g}(k_{y})=2t_{CC}\left|1-2cos\left(a_{c}k_{y}\sqrt{3}/2\right)\right|$
still appears for the other modes (see in ref. [19]), where $t_{CC}=2.5$ eV
[16] and $a_{c}$ is the carbon-carbon distance. For instance,
$\hat{E}_{g}(k_{y})\approx 0.11$ eV for $k_{y}=K_{y}+0.008\pi/a_{c}\sqrt{3}$
as seen in the diagram of Fig. 3(a). Therefore, since the total current
results from the contribution of many transverse momentum modes, the valley
current occurs even in gapless graphene devices as seen in Fig. 3(b) when the
tunneling corresponding to finite values of $\left|k_{y}-K_{y}\right|$ is
suppressed at high bias. Moreover, because of the zero (or small) bandgap of
graphene, it also shows that the current valley in this device is much
narrower than that in conventional RTDs [18]. Besides, when introducing a
finite bandgap [12, 14] in graphene, both the valley and peak currents are
reduced while the peak-to-valley current ratio (PVR) increases. The latter
reaches 4 and 13 for zero bandgap and $E_{g}$ = 50 meV, respectively.
Figure 4: I-V characteristics (a) for different doped concentrations in two
contacts and (b) for different well widths $D$. $D=4.3$ nm in (a) and
$N_{D}=5\times 10^{16}$ m-2 in (b).
Especially, from Fig. 3, we find an interesting feature: differently from
other RTDs [12, 18] where only the electron states in the quantum well (QW)
are resonant, in this device, both electron and hole states can contribute to
the current, as a consequence of the chirality of carriers in graphene. This
point has been also discussed in [20] and was shown to reduce the NDC effect
in single potential barrier graphene structures. Practically, as we can
understand from Fig. 3(a), the confined hole state in the QW makes the
tunneling from hole states in the emitter to electron states in the collector
possible. This band-to-band (BTB) process contributes to the rapid increase of
current beyond $V_{bias}$= 0.25 V, in addition to the normal resonant
tunneling of electrons via the electron state in the QW. The vanishing of the
latter process is responsible for the valley regime in the I-V
characteristics. When further increasing $V_{bias}$, the BTB tunneling through
the first electron level in the QW occurs, which leads to the rapid re-
increase of current and a narrow current valley seen in Fig. 3(b).
To evaluate the role of device parameters on the RTD operation, we display in
Figs. 4(a) and (b) the I-V characteristics obtained for different doping
concentrations $N_{D}$ in the access regions and for different well widths
$D$, respectively. In fact, when increasing $N_{D}$, the energy spacing
between the Fermi level (zero-energy point) and the flat potential energy in
the device ends at zero bias increases. This results in shifting the valley
region to high bias. Besides, the enhancement of BTB resonant tunneling at
high bias makes both the peak and valley currents higher when increasing
$N_{D}$. It finally reduces the PVR which reaches $\sim$ 3.9, 3.8, 2.2 and 1.8
for $N_{D}=2.5\times 10^{16}$, $5.0\times 10^{16}$, $7.5\times 10^{16}$, and
$10^{17}$ $m^{-2}$, respectively. Fig. 4(b) shows that when increasing the
well width $D$, the current valley moves to the low bias while the NDC effect
generally weakens. This feature can be straightforwardly understood as the
influence of $D$ on the position and number of energy levels in the QW. In the
case of large $D$ (e.g., 6.4 nm in Fig. 4(b)), the multi-level contribution
gives rise to a quite complex I-V curve with low PVR.
Finally, though not shown here, the overall current appears to be strongly
reduced when increasing the BN barrier thickness, as a consequence of the
large bandgap of h-BN. It therefore suggests that to observe well the effects
discussed above, a barrier thickness smaller than 2 nm is mandatory.
## III Conclusion
A simulation study of graphene RTDs based on graphene/h-BN double-barrier
structures has been performed by means of the self-consistent solution of the
2D Poisson and Schr dinger equations within the tight-binding. It was shown
that the resonant tunneling and the resulting NDC behavior may involve both
the electron and hole bound states of the graphene quantum well. The
sensitivity of the electrical characteristics to the device parameters as the
doping density of access zones, the QW thickness and the barrier thickness has
been analyzed. Though the chiral band-to-band tunneling tends to reduce the
width of the valley region in the I-V characteristics and to induce a rapid
re-increase of current, the PVR of NDC effect can reach the value of 4 at room
temperature for gapless graphene and the value of 13 for a small bandgap of 50
meV which may result from Bernal stacking of graphene on h-BN. This work
suggests that the engineering of graphene/h-BN structures is opening a new
route for high-performance graphene devices.
## Acknowledgment
This work was partially supported by the French ANR through the projects
NANOSIM-GRAPHENE (ANR-09-NANO-016) and MIGRAQUEL (ANR-10-BLAN-0304).
## References
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|
arxiv-papers
| 2012-01-20T10:44:18 |
2024-09-04T02:49:26.528137
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nguyen Viet Hung, Fulvio Mazzamuto, Arnaud Bournel, Philippe Dollfus",
"submitter": "Viet Hung Nguyen",
"url": "https://arxiv.org/abs/1201.4258"
}
|
1201.4402
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-229 LHCb-PAPER-2011-040
First observation of the decays $\kern
3.73305pt\overline{\kern-3.73305ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$
The LHCb Collaboration 111Authors are listed on the following pages.
First observations of the Cabibbo suppressed decays $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$ are
reported using 35 $\rm pb^{-1}$ of data collected with the LHCb detector.
Their branching fractions are measured with respect to the corresponding
Cabibbo favored decays, from which we obtain ${\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-})/{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-})=(5.9\pm 1.1\pm 0.5)\times 10^{-2}$ and
${\cal{B}}(B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-})/{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-})=(9.4\pm 1.3\pm 0.9)\times 10^{-2}$, where the
uncertainties are statistical and systematic, respectively. The
$B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$ decay is particularly interesting,
as it can be used in a similar way to $B^{-}\rightarrow D^{0}K^{-}$ to measure
the CKM phase $\gamma$.
To be submitted to Physical Review Letters.
The LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, A.
Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R. Currie46, C.
D’Ambrosio37, P. David8, P.N.Y. David23, I. De Bonis4, S. De Capua21,k, M. De
Cian39, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, L. Del Buono8, C. Deplano15, D.
Derkach14,37, O. Deschamps5, F. Dettori24, J. Dickens43, H. Dijkstra37, P.
Diniz Batista1, F. Domingo Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil
Suárez36, D. Dossett44, A. Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, S. Easo45, U. Egede49, V. Egorychev30, S. Eidelman33, D. van
Eijk23, F. Eisele11, S. Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch.
Elsasser39, D. Elsby55, D. Esperante Pereira36, L. Estève43, A.
Falabella16,14,e, E. Fanchini20,j, C. Färber11, G. Fardell46, C. Farinelli23,
S. Farry12, V. Fave38, V. Fernandez Albor36, M. Ferro-Luzzi37, S. Filippov32,
C. Fitzpatrick46, M. Fontana10, F. Fontanelli19,i, R. Forty37, M. Frank37, C.
Frei37, M. Frosini17,f,37, S. Furcas20, A. Gallas Torreira36, D. Galli14,c, M.
Gandelman2, P. Gandini51, Y. Gao3, J-C. Garnier37, J. Garofoli52, J. Garra
Tico43, L. Garrido35, D. Gascon35, C. Gaspar37, N. Gauvin38, M. Gersabeck37,
T. Gershon44,37, Ph. Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D.
Golubkov30, A. Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa
Gándara35, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening51, S. Gregson43, B. Gui52, E. Gushchin32,
Yu. Guz34, T. Gys37, G. Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S.
Hansmann-Menzemer11, R. Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44,
T. Hartmann56, J. He7, V. Heijne23, K. Hennessy48, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, E. Hicks48, K. Holubyev11, P. Hopchev4, W.
Hulsbergen23, P. Hunt51, T. Huse48, R.S. Huston12, D. Hutchcroft48, D.
Hynds47, V. Iakovenko41, P. Ilten12, J. Imong42, R. Jacobsson37, A. Jaeger11,
M. Jahjah Hussein5, E. Jans23, F. Jansen23, P. Jaton38, B. Jean-Marie7, F.
Jing3, M. John51, D. Johnson51, C.R. Jones43, B. Jost37, M. Kaballo9, S.
Kandybei40, M. Karacson37, T.M. Karbach9, J. Keaveney12, I.R. Kenyon55, U.
Kerzel37, T. Ketel24, A. Keune38, B. Khanji6, Y.M. Kim46, M. Knecht38, R.
Koopman24, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K. Kreplin11, M.
Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K. Kruzelecki37, M.
Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37,
G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E. Lanciotti37, G.
Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55, R. Le Gac6, J.
van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J. Lefrançois7, O.
Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles48, R. Lindner37,
C. Linn11, B. Liu3, G. Liu37, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar35,
N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F. Machefert7,
I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S. Malde51,
R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U. Marconi14,
R. Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51, A. Martín
Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G. McGregor50, R.
McNulty12, M. Meissner11, M. Merk23, J. Merkel9, R. Messi21,k, S.
Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23, F. Muheim46, K.
Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35, J. Mylroie-
Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M. Nedos9, M.
Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, N.
Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
Orlandea28, J.M. Otalora Goicochea2, P. Owen49, K. Pal52, J. Palacios39, A.
Palano13,b, M. Palutan18, J. Panman37, A. Papanestis45, M. Pappagallo47, C.
Parkes50,37, C.J. Parkinson49, G. Passaleva17, G.D. Patel48, M. Patel49, S.K.
Paterson49, G.N. Patrick45, C. Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos
Alvarez36, A. Pellegrino23, G. Penso22,l, M. Pepe Altarelli37, S.
Perazzini14,c, D.L. Perego20,j, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, A. Petrella16,37, A.
Petrolini19,i, A. Phan52, E. Picatoste Olloqui35, B. Pie Valls35, B.
Pietrzyk4, T. Pilař44, D. Pinci22, R. Plackett47, S. Playfer46, M. Plo
Casasus36, G. Polok25, A. Poluektov44,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell51, J. Prisciandaro38, V. Pugatch41, A.
Puig Navarro35, W. Qian52, J.H. Rademacker42, B. Rakotomiaramanana38, M.S.
Rangel2, I. Raniuk40, G. Raven24, S. Redford51, M.M. Reid44, A.C. dos Reis1,
S. Ricciardi45, K. Rinnert48, D.A. Roa Romero5, P. Robbe7, E. Rodrigues47,50,
F. Rodrigues2, P. Rodriguez Perez36, G.J. Rogers43, S. Roiser37, V.
Romanovsky34, M. Rosello35,n, J. Rouvinet38, T. Ruf37, H. Ruiz35, G.
Sabatino21,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail47, B. Saitta15,d,
C. Salzmann39, M. Sannino19,i, R. Santacesaria22, C. Santamarina Rios36, R.
Santinelli37, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C. Satriano22,m,
A. Satta21, M. Savrie16,e, D. Savrina30, P. Schaack49, M. Schiller24, S.
Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba18,l, M.
Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N. Serra39, J. Serrano6,
P. Seyfert11, M. Shapkin34, I. Shapoval40,37, P. Shatalov30, Y. Shcheglov29,
T. Shears48, L. Shekhtman33, O. Shevchenko40, V. Shevchenko30, A. Shires49, R.
Silva Coutinho44, T. Skwarnicki52, A.C. Smith37, N.A. Smith48, E. Smith51,45,
K. Sobczak5, F.J.P. Soler47, A. Solomin42, F. Soomro18, B. Souza De Paula2, B.
Spaan9, A. Sparkes46, P. Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39,
S. Stoica28, S. Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39,
V.K. Subbiah37, S. Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26,
S. T’Jampens4, E. Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E.
Tournefier4,49, M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda
Garcia37, A. Ukleja27, P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R.
Vazquez Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M.
Veltri17,g, B. Viaud7, I. Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A.
Vollhardt39, D. Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S.
Wandernoth11, J. Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D.
Websdale49, M. Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P.
Williams44,45, M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W.
Witzeling37, S.A. Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z.
Yang3, R. Young46, O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52,
W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
56Physikalisches Institut, Universität Rostock, Rostock, Germany, associated
to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
The standard model (SM) of particle physics provides a good description of
nature up to the TeV scale, yet many issues remain unresolved [1], including,
but not limited to, the hierarchy problem, the preponderance of matter over
antimatter in the Universe, and the need to explain dark matter. One of the
main objectives of the LHC is to search for new physics beyond the Standard
Model (SM) either through direct detection or through interference effects in
$b$\- and $c$-hadron decays. In the SM, the Cabibbo-Kobayashi-Maskawa (CKM)
matrix [2, *Kobayashi:1973fv] governs the strengths of weak charged-current
interactions and their corresponding phases. Precise measurements on the CKM
matrix parameters may reveal deviations from the consistency that is expected
in the SM, making study of these decays a unique laboratory in which to search
for physics beyond the standard model.
The most poorly constrained of the CKM parameters is the weak phase
$\gamma\equiv{\rm arg}\left(-{V_{\rm ub}^{*}V_{\rm ud}\over V_{\rm
cb}^{*}V_{\rm cd}}\right)$. Its direct measurement reaches a precision of
$10^{\circ}-12^{\circ}$ [4, 5]. Two promising methods of measuring this phase
are through the time-independent and time-dependent analyses of
$B^{-}\rightarrow D^{0}K^{-}$ [6, *Dunietz:1992ti, *Atwood:1994zm,
*Atwood:1996ci, 10, *Gronau:1991dp, 12] and $B^{0}_{s}\rightarrow
D_{s}^{\mp}K^{\pm}$ [13, 14], respectively. Both approaches can be extended to
higher multiplicity modes, such as $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, $B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-}$ [15] and $B^{0}_{s}\rightarrow
D_{s}^{\mp}K^{\pm}\pi^{+}\pi^{-}$, which could provide a comparable level of
sensitivity. The last two decays have not previously been observed.
In this Letter, we report first observations of the Cabibbo-suppressed (CS)
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$
decays, where $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$ and $D^{0}\rightarrow
K^{-}\pi^{+}$, where charge conjugation is implied throughout this Letter.
These signal decays are normalized with respect to the topologically similar
Cabibbo-favored (CF) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$
decays, respectively. For brevity, we use the notation $X_{d}$ to refer to the
recoiling $\pi^{-}\pi^{+}\pi^{-}$ system in the CF decays and $X_{s}$ for the
$K^{-}\pi^{+}\pi^{-}$ system in the CS decays.
The analysis presented here is based on 35 $\rm pb^{-1}$ of data collected
with the LHCb detector in 2010\. For these measurements, the most important
parts of LHCb are the vertex detector (VELO), the charged particle tracking
system, the ring imaging Cherenkov detectors (RICH) and the trigger. The VELO
is instrumental in separating particles coming from heavy quark decays and
those emerging directly from $pp$ interactions, by providing an impact
parameter (IP) resolution of about $16\,\upmu\rm m$ \+ 30$\,\upmu\rm
m$/$p_{\rm T}$ (transverse momentum, $p_{\rm T}$ in GeV/$c$). The tracking
system measures charged particles’ momenta with a resolution of
$\sigma_{p}/p\sim 0.4\%(0.6\%)$ at 5 (100) GeV/$c$. The RICH detectors are
important to identify kaons and suppress the large backgrounds from pions
misidentified as kaons. Events are selected by a two-level trigger system. The
first level is hardware-based, and requires either a large transverse energy
deposition in the calorimeter system, or a high $p_{\rm T}$ muon or pair of
muons detected in the muon system. The second level, the high-level trigger,
uses simplified versions of the offline software to reconstruct decays of
$b$\- and $c$-hadrons both inclusively and exclusively. Candidates passing the
trigger selections are saved and used for offline analysis. A more detailed
description of the LHCb detector can be found elsewhere [16]. In this analysis
the signal and normalization modes are topologically identical, allowing loose
trigger requirements to be made with small associated uncertainty. In
particular, we exploit the fact that $b$-hadrons are produced in pairs in $pp$
collisions, and include events that were triggered by the decay products of
either the signal $b$-hadron or the other $b$-hadron in the event. This
requirement increases the efficiency of our trigger selection by about 80%
compared to the trigger selections requiring the signal $b$-hadron to be
responsible for triggering the event, as was done in Ref. [17].
The selection criteria used to reconstruct the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$
final states are described in Ref. [17]. The Cabibbo suppression results in
about a factor of 20 lower rate. To improve the signal-to-background ratio in
the CS decay modes, additional selection requirements are imposed, and they
are applied to both the signal and normalization modes. The $B$ meson
candidate is required to have $p_{\rm T}>4$ GeV/$c$, ${\rm IP}<60~{}\,\upmu\rm
m$ with respect to its associated primary vertex (PV), where the associated PV
is the one having the smallest impact parameter $\chi^{2}$ with respect to the
track. We also require the flight distance $\chi^{2}>144$, where the
$\chi^{2}$ is with respect to the zero flight distance hypothesis, and the
vertex $\chi^{2}/{\rm ndf}<5$, where ndf represents the number of degrees of
freedom in the fit. The last requirement is also applied to the vertices
associated with $X_{d}$ and $X_{s}$. Three additional criteria are applied
only to the CS modes. First, to remove the peaking backgrounds from
$B\rightarrow DD_{s}^{-},~{}D_{s}^{-}\rightarrow K^{-}\pi^{+}\pi^{-}$, we veto
events where the invariant mass, $M(X_{s})$, is within 20 MeV/$c^{2}$ of the
$D_{s}$ mass. Information from the RICH is critical to reduce background from
the CF decay modes. This suppression is accomplished by requiring the kaon in
$X_{s}$ to have $p<100$ GeV/$c$ (above which there is minimal $K$/$\pi$
separation from the RICH), and the difference in log-likelihoods between the
kaon and pion hypotheses to satisfy $\Delta{\rm ln}\mathcal{L}(K-\pi)>8$. The
latter requirement is determined by optimizing $N_{S}/\sqrt{N_{S}+N_{B}}$,
where we assume 100 signal events ($\sim 1/20$ of the CF decay yields) prior
to any particle identification (PID) selection requirement, and the
combinatorial background yield, $N_{B}$, is taken from the high $B$-mass
sideband (5350-5580 MeV/$c^{2}$). We also make a loose PID requirement of
$\Delta{\rm ln}\mathcal{L}(K-\pi)<10$ on the pions in $X_{s}$ and $X_{d}$.
Selection and trigger efficiencies are determined from simulation. Events are
produced using pythia [18] and long-lived particles are decayed using evtgen
[19]. The detector response is simulated with geant4 [20]. The
$DK^{-}\pi^{+}\pi^{-}$ final states are assumed to include 50%
$DK_{1}(1270)^{-}$ and 20% $DK_{1}(1400)^{-}$, with smaller contributions from
$DK_{2}(1430)^{-}$, $DK^{*}(1680)^{-}$, $D\bar{K}^{*}(892)^{0}\pi^{-}$ and
$D_{1}(2420)K^{-}$. The resonances included in the simulation of the $X_{d}$
system are described in Ref. [17]. The relative efficiencies, including
selection and trigger, but not PID selection, are determined to be
$\epsilon_{\kern 1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}}/\epsilon_{\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}}=1.05\pm 0.04$ and $\epsilon_{B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-}}/\epsilon_{B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}}=0.942\pm 0.036$, where the uncertainties are
statistical only. The efficiencies have a small dependence on the contributing
resonances and their daughters’ masses, and we therefore do not necessarily
expect the ratios to be equal to unity. Moreover, the additional selections on
the CS modes contribute to small differences between the signal and
normalization modes’ efficiencies.
The PID efficiencies are determined in bins of track momentum and
pseudorapidity ($\eta$) using the $D^{0}$ daughters from
$D^{*\pm}\rightarrow\pi_{s}^{\pm}D^{0}$, $D^{0}\rightarrow K^{-}\pi^{+}$
calibration data, where the particles are identified without RICH information
using the charge of the soft pion, $\pi_{s}$. The kinematics of the kaon in
the $X_{s}$ system are taken from simulation after all offline and trigger
selections. Applying the PID efficiencies to the simulated decays, we
determine the efficiencies for the kaon to pass the $\Delta{\rm
ln}\mathcal{L}(K-\pi)>8$ requirement to be $(75.9\pm 1.5)\%$ for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $(79.2\pm 1.5)\%$ for $B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-}$.
Invariant mass distributions for the normalization and signal modes are shown
in Fig. 1. Signal yields are determined through unbinned maximum likelihood
fits to the sum of signal and several background components. The signal
distributions are parametrized as the sum of two Gaussian functions with
common means, and shape parameters, $\sigma_{\rm core}$ and $f_{\rm core}$
that represent the width and area fraction of the narrower (core) Gaussian
portion, and $r_{w}\equiv\sigma_{\rm wide}/\sigma_{\rm core}$, which is the
ratio of the wider to narrower Gaussian width.
Figure 1: Invariant mass distributions for (a) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$, (b) $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$, (c) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and (d) $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$
candidates from 35 $\rm pb^{-1}$ of data for all selected candidates. Fits as
described in the text are overlaid.
The CF modes are first fit with $f_{\rm core}$ and $r_{w}$ constrained to the
values from simulation within their uncertainties, while $\sigma_{\rm core}$
is left as a free parameter since simulation underestimates the mass
resolution by $\sim$10%. For the CF decay mode fits, the background shapes are
the same as those described in Ref. [17]. The resulting signal shape
parameters from the CF decay fits are then fixed in subsequent fits to the CS
decay modes. For $\sigma_{\rm core}$, the values from the CF decay fits are
scaled by width correction factors ($\sim$0.95) obtained from MC simulations.
For the CS decays, invariant mass shapes of specific peaking backgrounds from
other $b$-hadron decays are determined from MC simulation. The largest of
these backgrounds comes from $D^{(*)}\pi^{-}\pi^{+}\pi^{-}$ decays, where one
of the $\pi^{-}$ passes the $\Delta{\rm ln}\mathcal{L}(K-\pi)>8$ requirement
and is misidentified as a $K^{-}$. To determine the fraction of events in
which this occurs, we use measured PID fake rates ($\pi$ faking $K$) obtained
from $D^{*\pm}$ calibration data (binned in $(p,\eta$)), and apply them to
each $\pi^{-}$ in $D\pi^{-}\pi^{+}\pi^{-}$ simulated events. A decay is
considered a fake if either pion has $p<100$ GeV/$c$, and a randomly generated
number in the interval from [0, 1] is less than that pion’s determined fake
rate. The pion’s mass is then replaced by the kaon’s mass, and the invariant
mass of the $b$-hadron is recomputed. The resulting spectrum is then fitted
using a Crystal Ball [21] lineshape and its parameters are fixed in fits to
data. Using this method, we find the same cross-feed rate of $(4.4\pm 0.7)\%$
for both $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$
into $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$,
respectively, where the uncertainty includes both statistical and systematic
sources. A similar procedure is used to obtain the
$D^{*}\pi^{-}\pi^{+}\pi^{-}$ background yields and shapes. The background
yields are obtained by multiplying the observed CF signal yields in data by
the cross-feed rates and the fraction of background in the region of the mass
fit (5040$-$5580 MeV/$c^{2}$).
We also account for backgrounds from the decays $B\rightarrow
DD_{s}^{-},~{}D_{s}^{-}\rightarrow K^{-}K^{+}\pi^{-}$, where the $K^{+}$ is
misidentified as a $\pi^{+}$. The yields of these decays are lower, but are
offset by a larger fake rate since the PID requirement on the particles
assumed to be pions is significantly looser ($\Delta{\rm
ln}\mathcal{L}(K-\pi)<10$). Using the same technique as described above, the
fake rate is found to be $(24\pm 2)\%$. The fake yield from this source is
then computed from the product of the measured yield of $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}D_{s}^{-}$ in data
($161\pm 14({\rm stat})$) and the above fake rate. The $B^{-}\rightarrow
D^{0}D_{s}^{-}$ yield was not directly measured, but was determined from known
branching fractions [22] and efficiencies from simulation. Additional
uncertainty due to these extrapolations is included in the estimated
$B^{-}\rightarrow D^{0}D_{s}^{-}$ background yield.
The last sources of background, which do not contribute to the signal regions,
are from $D^{*}K^{-}\pi^{+}\pi^{-}$, where the soft pion or photon from the
$D^{*}$ is lost. The shapes of these low mass backgrounds are taken from the
fitted $D^{*}\pi^{-}\pi^{+}\pi^{-}$ shapes in the $D\pi^{-}\pi^{+}\pi^{-}$
mass fits, and the yield ratios
$N(D^{*}K^{-}\pi^{+}\pi^{-})/N(DK^{-}\pi^{+}\pi^{-})$, are constrained to be
equal to the ratios obtained from CF mode fits with a 25% uncertainty.
The combinatorial background is assumed to have an exponential shape. A
summary of the signal shape parameters and the specific $b$-hadron backgrounds
used in the CS signal mode fits is given in Table 1.
The fitted yields are $2126\pm 69$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $1630\pm 57$ and $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ events. For the CS modes, we find $90\pm 16$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $130\pm 17$ $B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-}$ signal decays. The CS decay signals have
significances of 7.2 and 9.0, respectively, calculated as $\sqrt{-2{\rm
ln}({\mathcal{L}_{0}}/{\mathcal{L}_{\rm max}}})$, where ${\mathcal{L}_{\rm
max}}$ and ${\mathcal{L}_{0}}$ are the fit likelihoods with the signal yields
left free and fixed to zero, respectively. In evaluating these significances,
we remove the constraint on
$N(D^{*}K^{-}\pi^{+}\pi^{-})/N(DK^{-}\pi^{+}\pi^{-})$, which would otherwise
bias the $D^{*}K^{-}\pi^{+}\pi^{-}$ yield toward zero and inflate
${\mathcal{L}_{0}}$. Varying the signal or background shapes or normalizations
within their uncertainties has only a minor impact on the significances. We
therefore observe for the first time the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$
decay modes.
Table 1: Summary of parameters used in the CS mass fits. Values without uncertainties are fixed in the CS mode fits, and values with uncertainties are included with a Gaussian constraint with central values and widths as indicated. Parameter | $~{}D^{+}K^{-}\pi^{+}\pi^{-}~{}$ | $~{}D^{0}K^{-}\pi^{+}\pi^{-}~{}$
---|---|---
Mean mass (MeV/$c^{2}$) | $5276.3$ | $5276.5$
$\sigma_{\rm core}$ (MeV/$c^{2}$) | $15.7$ | $17.5$
$f_{\rm core}$ | 0.88 | 0.93
$\sigma_{\rm wide}/\sigma_{\rm core}$ | 3.32 | 2.82
$N(D\pi\pi\pi)$ | $63\pm 10$ | $48\pm 8$
$N(D^{*}\pi\pi\pi)$ | $47\pm 9~{}$ | $107\pm 18$
$N(DD_{s})$ | $23\pm 3~{}$ | $38\pm 8$
$N(D^{*}K\pi\pi)/N(DK\pi\pi)$ | $0.62\pm 0.16$ | $~{}1.86\pm 0.46$
The ratios of branching fractions are given by
${{\cal{B}}(H_{b}\rightarrow
H_{c}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(H_{b}\rightarrow
H_{c}\pi^{-}\pi^{+}\pi^{-})}={Y^{\rm CS}\over Y^{\rm CF}}\times\epsilon_{\rm
tot}^{\rm rel},$
where $Y^{\rm CF}$ ($Y^{\rm CS}$) are the fitted yields in the CF (CS) decay
modes, and $\epsilon_{\rm tot}^{\rm rel}$ are the products of the relative
selection and PID efficiencies discussed previously. The results for the
branching fractions are
$\displaystyle{{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(5.9\pm 1.1\pm 0.5)\times
10^{-2},$ $\displaystyle{{\cal{B}}(B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-})\over{\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-})}$ $\displaystyle=(9.4\pm 1.3\pm 0.9)\times
10^{-2},$
where the first uncertainties are statistical and the second are from the
systematic sources discussed below.
Most systematic uncertainties cancel in the measured ratios of branching
fractions; only those that do not are discussed below. One source of
uncertainty comes from modeling of the $K^{-}\pi^{+}\pi^{-}$ final state. In
Ref. [17], we compared the $p$ and $p_{\rm T}$ spectra of $\pi^{\pm}$ from
$X_{d}$, and they agreed well with simulation. We have an insufficiently large
data sample to make such a comparison in the CS signal decay modes. The
departure from unity of the efficiency ratios obtained from simulation are due
to differences in the $p_{\rm T}$ spectra between the $X_{d}$ daughters in CF
decays and the $X_{s}$ daughters in the CS decays. These differences depend on
the contributing resonances and the daughters’ masses. We take the full
difference of the relative efficiencies from unity (4.6% for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and 6.1% for $B^{-}$) as a
systematic uncertainty.
The kaon PID efficiency includes uncertainties from the limited size of the
data set used for the efficiency determination, the limited number of events
in the MC sample over which we average, and possible systematic effects
described below. The statistical precision is taken as the RMS width of the
kaon PID efficiency distribution obtained from pseudo-experiments, where in
each one, the kaon PID efficiencies in each ($p$, $\eta$) bin are fluctuated
about their nominal values within their uncertainties. This contributes 1.5%
to the overall kaon PID efficiency uncertainty. We also consider the
systematic error in using the $D^{*}$ data sample to determine the PID
efficiency. The procedure is tested by comparing the kaon PID efficiency using
a MC-derived efficiency matrix with the efficiency obtained by directly
requiring $\Delta{\rm ln}\mathcal{L}(K-\pi)>8$ on the kaon from $X_{s}$ in the
signal MC. The relative difference is found to be $(3.6\pm 1.9)\%$. We take
the full difference of 3.6% as a potential systematic error. The total kaon
PID uncertainty is 3.9%.
The fit model uncertainty includes 3% systematic uncertainty in the yields
from the normalization modes [17]. The uncertainties in the CS signal fits are
obtained by varying each of the signal shape parameters within the uncertainty
obtained from the CF mode data fits. The signal shape parameter uncertainties
are 2.7% for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and 2.5% for
$B^{-}$. For the specific $b$-hadron background shapes, we obtain the
uncertainty by refitting the data 100 times, where each fit is performed with
all background shapes fluctuated within their covariances and subsequently
fixed in the fit to data (1%). The uncertainties in the yields from the
assumed exponential shape for the combinatorial background are estimated by
taking the difference in yields between the nominal fit and one with a linear
shape for the combinatorial background (2%). In total, the relative yields are
uncertain by 4.5% for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and
4.4% for $B^{-}$.
The limited number of MC events for determining the relative efficiencies
contributes 4.1% and 3.8% to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and $B^{-}$ branching fraction
ratio uncertainties, respectively. Other sources of uncertainty are
negligible. In total, the uncertainties on the ratio of branching fractions
are 8.6% for $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ and 9.3% for
$B^{-}$.
We have also looked at the substructures that contribute to the CS final
states. Figure 2 shows the observed distributions of (a) $K^{-}\pi^{+}\pi^{-}$
invariant mass, (b) $M(D^{0}\pi^{+}\pi^{-})-M(D^{0})$ invariant mass
difference, (c) $K^{-}\pi^{+}$ invariant mass, and (d) $\pi^{+}\pi^{-}$
invariant mass for $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$. We show events
in the $B$ mass signal region, defined to have an invariant mass from
5226$-$5326 MeV/$c^{2}$, and events from the high-mass sideband (5350$-$5550
MeV/$c^{2}$), scaled by the ratio of expected background yields in the signal
region relative to the sideband region. An excess of events is observed
predominantly in the low $K^{-}\pi^{+}\pi^{-}$ mass region near 1300$-$1400
MeV/$c^{2}$, and the number of signal events decreases with increasing mass.
In Fig. 2(b) there appears to be an excess of $\sim$10 events in the region
around 550$-$600 MeV/$c^{2}$, which suggests contributions from
$D_{1}(2420)^{0}$ or $D_{2}^{*}(2460)^{0}$ meson decays. These decays can also
be used for measuring the weak phase $\gamma$ [23]. This yield, relative to
the total, is similar to what was observed in $B^{-}\rightarrow
D^{0}\pi^{-}\pi^{+}\pi^{-}$ decays [17]. Figures 2(c) and (d) show significant
enhancements at the $\bar{K}^{*0}$ and $\rho^{0}$ masses, consistent with
decays of excited strange states, such as the $K_{1}(1270)^{-}$,
$K_{1}(1400)^{-}$ and $K^{*}(1410)^{-}$. Similar distributions are observed
for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$, except that no excess of events is observed near
550$-$600 MeV/$c^{2}$ in the $M(D^{0}\pi^{+}\pi^{-})-M(D^{0})$ invariant mass
difference.
Figure 2: Invariant masses within the $B^{-}\rightarrow
D^{0}K^{-}\pi^{+}\pi^{-}$ system. Shown are (a) $M(K^{-}\pi^{+}\pi^{-})$ , (b)
$M(D\pi^{+}\pi^{-})-M(D)$, (c) $M$($K^{-}\pi^{+})$, and (d)
$M(\pi^{+}\pi^{-})$ . The points with error bars correspond to the signal
region, and the hatched histograms represent the scaled sideband region.
In summary, we report first observations of the Cabibbo-suppressed decay modes
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$ and
measurements of their branching fractions relative to $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow
D^{+}\pi^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}\pi^{-}\pi^{+}\pi^{-}$.
The $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$ decay is particularly
interesting because, with more data, it can be used to measure the weak phase
$\gamma$, using similar techniques as for $B^{-}\rightarrow D^{0}K^{-}$ and
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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|
arxiv-papers
| 2012-01-20T21:37:29 |
2024-09-04T02:49:26.542330
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R.B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.J. Back, D.S.\n Bailey, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J. Blouw,\n S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi,\n A. Borgia, T.J.V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier,\n C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F. Constantin,\n A. Contu, A. Cook, M. Coombes, G. Corti, G.A. Cowan, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, S. De Capua, M. De Cian, F.\n De Lorenzi, J.M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, L. Est\\`eve, A. Falabella, E.\n Fanchini, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, M. Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J.\n Garofoli, J. Garra Tico, L. Garrido, D. Gascon, C. Gaspar, N. Gauvin, M.\n Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, G. Haefeli, C.\n Haen, S.C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew, J.\n Harrison, P.F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J.A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R.S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C.R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M.\n Karbach, J. Keaveney, I.R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y.M. Kim, M. Knecht, R. Koopman, P. Koppenburg, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J.H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde,\n R.M.D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A. Papanestis,\n M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel,\n S.K. Paterson, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrella, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S.\n Playfer, M. Plo Casasus, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig\n Navarro, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S. Rangel, I.\n Raniuk, G. Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, K.\n Rinnert, D.A. Roa Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez\n Perez, G.J. Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T.\n Ruf, H. Ruiz, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, C. Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R.\n Santinelli, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M.\n Savrie, D. Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, A.C. Smith, N.A.\n Smith, E. Smith, K. Sobczak, F.J.P. Soler, A. Solomin, F. Soomro, B. Souza De\n Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp,\n S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah,\n S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Topp-Joergensen, N. Torr, E. Tournefier, M.T. Tran, A.\n Tsaregorodtsev, N. Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer,\n V. Vagnoni, G. Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi,\n J.J. Velthuis, M. Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona, J.\n Visniakov, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams,\n F.F. Wilson, J. Wishahi, M. Witek, W. Witzeling, S.A. Wotton, K. Wyllie, Y.\n Xie, F. Xing, Z. Xing, Z. Yang, R. Young, O. Yushchenko, M. Zavertyaev, F.\n Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A.\n Zvyagin",
"submitter": "Steven R. Blusk",
"url": "https://arxiv.org/abs/1201.4402"
}
|
1201.4406
|
# Fundamental solution of the Laplacian in the hyperboloid model of hyperbolic
geometry
H S Cohl1,2 and E G Kalnins3 1Information Technology Laboratory, National
Institute of Standards and Technology, Gaithersburg, MD, USA 2Department of
Mathematics, University of Auckland, Auckland, New Zealand 3Department of
Mathematics, University of Waikato, Hamilton, New Zealand hcohl@nist.gov
###### Abstract
Due to the isotropy of $d$-dimensional hyperbolic space, one expects there to
exist a spherically symmetric fundamental solution for its corresponding
Laplace-Beltrami operator. The $R$-radius hyperboloid model of hyperbolic
geometry ${\mathbf{H}}_{R}^{d}$ with $R>0$, represents a Riemannian manifold
with negative-constant sectional curvature. We obtain a spherically symmetric
fundamental solution of Laplace’s equation on this manifold in terms of its
geodesic radius. We give several matching expressions for this fundamental
solution including a definite integral over reciprocal powers of the
hyperbolic sine, finite summation expression over hyperbolic functions, Gauss
hypergeometric functions, and in terms of the associated Legendre function of
the second kind with order and degree given by $d/2-1$ with real argument
greater than unity. We also demonstrate uniqueness for a fundamental solution
of Laplace’s equation on this manifold in terms of a vanishing decay at
infinity.
###### pacs:
02.30.Em, 02.30.Gp, 02.30.Jr, 02.40.Ky
###### ams:
31C12, 32Q45, 33C05, 35A08, 35J05
## 1 Introduction
We compute closed-form expressions of a spherically symmetric Green’s function
(fundamental solution) of the Laplacian (Laplace-Beltrami operator) on a
Riemannian manifold of constant negative sectional curvature, namely the
hyperboloid model of hyperbolic geometry. Useful background material relevant
for this paper can be found in Vilenkin (1968) [25], Thurston (1997) [23], Lee
(1997) [18] and Pogosyan & Winternitz (2002) [22].
This paper is organized as follows. In section 2 we describe the hyperboloid
model of hyperbolic geometry and its corresponding metric, global geodesic
distance function, Laplacian and geodesic polar coordinate systems which
parametrize points in this model. In section 3 for the hyperboloid model of
hyperbolic geometry, we show how to compute radial harmonics in a geodesic
polar coordinate system and derive several alternative expressions for a
radial fundamental solution of the Laplacian on the $d$-dimensional $R$-radius
hyperboloid with $R>0$. In section 4 we prove that our derived fundamental
solution is unique in terms of a vanishing decay at infinity.
Throughout this paper we rely on the following definitions. For
$a_{1},a_{2},\ldots\in{\mathbf{C}}$, if $i,j\in{\mathbf{Z}}$ and $j<i$ then
$\sum_{n=i}^{j}a_{n}=0$ and $\prod_{n=i}^{j}a_{n}=1$. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and the set
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$ The set ${\mathbf{R}}$ represents
the real numbers.
## 2 The hyperboloid model of hyperbolic geometry
Hyperbolic space in $d$-dimensions is a fundamental example of a space
exhibiting hyperbolic geometry. It was developed independently by Lobachevsky
and Bolyai around 1830 (see Trudeau (1987) [24]). It is a geometry analogous
to Euclidean geometry, but such that Euclid’s parallel postulate is no longer
assumed to hold.
There are several models of $d$-dimensional hyperbolic geometry including the
Klein (see Figure 1), Poincaré (see Figure 2), hyperboloid, upper-half space
and hemisphere models (see Thurston (1997) [23]).
Figure 1: This figure is a graphical depiction of stereographic projection
from the hyperboloid model to the Klein model of hyperbolic space. Figure 2:
This figure is a graphical depiction of stereographic projection from the
hyperboloid model to the Poincaré model of hyperbolic space.
The hyperboloid model for $d$-dimensional hyperbolic space is closely related
to the Klein and Poincaré models: each can be obtained projectively from the
others. The upper-half space and hemisphere models can be obtained from one
another by inversions with the Poincaré model (see section 2.2 in Thurston
(1997) [23]). The model we will be focusing on in this paper is the
hyperboloid model.
The hyperboloid model, also known as the Minkowski or Lorentz models, are
models of $d$-dimensional hyperbolic geometry in which points are represented
by the upper sheet (submanifold) $S^{+}$ of a two-sheeted hyperboloid embedded
in the Minkowski space ${\mathbf{R}}^{d,1}$. Minkowski space is a
$(d+1)$-dimensional pseudo-Riemannian manifold which is a real finite-
dimensional vector space, with coordinates given by ${\bf
x}=(x_{0},x_{1},\ldots,x_{d})$. It is equipped with a nondegenerate, symmetric
bilinear form, the Minkowski bilinear form
$[{\bf x},{\mathbf{y}}]=x_{0}y_{0}-x_{1}y_{1}-\ldots-x_{d}y_{d}.$
The above bilinear form is symmetric, but not positive-definite, so it is not
an inner product. It is defined analogously with the Euclidean inner product
for ${\mathbf{R}}^{d+1}$
$({\bf x},{\mathbf{y}})=x_{0}y_{0}+x_{1}y_{1}+\ldots+x_{d}y_{d}.$
The variety $[{\bf x},{\bf x}]=x_{0}^{2}-x_{1}^{2}-\ldots-x_{d}^{2}=R^{2}$,
for ${\bf x}\in{\mathbf{R}}^{d,1}$, using the language of Beltrami (1869) [3]
(see also p. 504 in Vilenkin (1968) [25]), defines a pseudo-sphere of radius
$R$. Points on the pseudo-sphere with zero radius coincide with a cone. Points
on the pseudo-sphere with radius greater than zero lie within this cone, and
points on the pseudo-sphere with purely imaginary radius lie outside the cone.
For $R\in(0,\infty)$, we refer to the variety $[{\bf x},{\bf x}]=R^{2}$ as the
$R$-radius hyperboloid ${\mathbf{H}}_{R}^{d}$. This variety is a maximally
symmetric, simply connected, $d$-dimensional Riemannian manifold with
negative-constant sectional curvature (given by $-1/R^{2}$, see for instance
p. 148 in Lee (1997) [18]), whereas Euclidean space ${\mathbf{R}}^{d}$
equipped with the Pythagorean norm, is a space with zero sectional curvature.
For a fixed $R\in(0,\infty),$ the $R$-radius hypersphere
${\mathbf{S}}_{R}^{d}$, is an example of a space (submanifold) with positive
constant sectional curvature (given by $1/R^{2}$). We denote unit radius
hyperboloid by ${\mathbf{H}}^{d}:={\mathbf{H}}_{1}^{d}$ and the unit radius
hypersphere by ${\mathbf{S}}^{d}:={\mathbf{S}}_{1}^{d}$.
In our discussion of a fundamental solution for the Laplacian in the
hyperboloid model of hyperbolic geometry, we focus on the positive radius
pseudo-sphere which can be parametrized through subgroup-type coordinates,
i.e. those which correspond to a maximal subgroup chain $O(d,1)\supset\ldots$
(see for instance Pogosyan & Winternitz (2002) [22]). There exist separable
coordinate systems which parametrize points on the positive radius pseudo-
sphere (i.e. such as those which are analogous to parabolic coordinates, etc.)
which can not be constructed using maximal subgroup chains (we will no longer
discuss these).
Geodesic polar coordinates are coordinates which correspond to the maximal
subgroup chain given by $O(d,1)\supset O(d)\supset\ldots$. What we will refer
to as standard geodesic polar coordinates correspond to the subgroup chain
given by $O(d,1)\supset O(d)\supset O(d-1)\supset\cdots\supset O(2).$ Standard
geodesic polar coordinates (see Olevskiĭ (1950) [20]; Grosche, Pogosyan &
Sissakian (1997) [13]), similar to standard hyperspherical coordinates in
Euclidean space, can be given by
$\left.\begin{array}[]{rcl}x_{0}&=&R\cosh r\\\\[0.56917pt] x_{1}&=&R\sinh
r\cos\theta_{1}\\\\[2.84544pt] x_{2}&=&R\sinh
r\sin\theta_{1}\cos\theta_{2}\\\\[2.84544pt] &\vdots&\\\\[0.56917pt]
x_{d-2}&=&R\sinh r\sin\theta_{1}\cdots\cos\theta_{d-2}\\\\[2.84544pt]
x_{d-1}&=&R\sinh r\sin\theta_{1}\cdots\sin\theta_{d-2}\cos\phi\\\\[2.84544pt]
x_{d}&=&R\sinh r\sin\theta_{1}\cdots\sin\theta_{d-2}\sin\phi,\\\\[2.84544pt]
\end{array}\quad\right\\}$ (1)
where $r\in[0,\infty)$, $\phi\in[0,2\pi)$, and $\theta_{i}\in[0,\pi]$ for
$i\in\\{1,\ldots,d-2\\}$.
The isometry group of the space ${\mathbf{H}}_{R}^{d}$ is the pseudo-
orthogonal group $SO(d,1),$ the Lorentz group in $(d+1)$-dimensions.
Hyperbolic space ${\mathbf{H}}_{R}^{d}$, can be identified with the quotient
space $SO(d,1)/SO(d)$. The isometry group acts transitively on
${\mathbf{H}}_{R}^{d}$. That is, any point on the hyperboloid can be carried,
with the help of a Euclidean rotation of $SO(d-1)$, to the point
$(\cosh\alpha,\sinh\alpha,0,\ldots,0),$ and a hyperbolic rotation
$\left.\begin{array}[]{rcl}x_{0}^{\prime}&=&-x_{1}\sinh\alpha+x_{0}\cosh\alpha\\\\[2.84544pt]
x_{1}^{\prime}&=&-x_{1}\cosh\alpha-x_{0}\sinh\alpha\\\\[2.84544pt]
\end{array}\quad\right\\}$
maps that point to the origin $(1,0,\ldots,0)$ of the space. In order to study
a fundamental solution of Laplace’s equation on the hyperboloid, we need to
describe how one computes distances in this space.
One may naturally compare distances on the positive radius pseudo-sphere
through analogy with the $R$-radius hypersphere. Distances on the hypersphere
are simply given by arc lengths, angles between two arbitrary vectors, from
the origin, in the ambient Euclidean space. We consider the $d$-dimensional
hypersphere embedded in ${\mathbf{R}}^{d+1}$. Points on the hypersphere can be
parametrized using hyperspherical coordinate systems. Any parametrization of
the hypersphere ${\mathbf{S}}_{R}^{d}$, must have $({\bf x},{\bf
x})=x_{0}^{2}+\ldots+x_{d}^{2}=R^{2}$, with $R>0$. The distance between two
points on the hypersphere ${\bf x},{{\bf x}^{\prime}}\in{\mathbf{S}}_{R}^{d}$
is given by
$d({\bf x},{{\bf x}^{\prime}})=R\gamma=R\cos^{-1}\left(\frac{({\bf x},{{\bf
x}^{\prime}})}{({\bf x},{\bf x})({{\bf x}^{\prime}},{{\bf
x}^{\prime}})}\right)=R\cos^{-1}\left(\frac{1}{R^{2}}({\bf x},{{\bf
x}^{\prime}})\right).$ (2)
This is evident from the fact that the geodesics on ${\mathbf{S}}_{R}^{d}$ are
great circles (i.e. intersections of ${\mathbf{S}}_{R}^{d}$ with planes
through the origin) with constant speed parametrizations (see p. 82 in Lee
(1997) [18]).
Accordingly, we now look at the geodesic distance function on the
$d$-dimensional positive radius pseudo-sphere ${\mathbf{H}}_{R}^{d}.$
Distances between two points on the positive radius pseudo-sphere are given by
the hyperangle between two arbitrary vectors, from the origin, in the ambient
Minkowski space. Any parametrization of the hyperboloid
${\mathbf{H}}_{R}^{d}$, must have $[{\bf x},{\bf x}]=R^{2}$. The geodesic
distance between two points ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ is given by
$d({\bf x},{{\bf x}^{\prime}})=R\cosh^{-1}\left(\frac{[{\bf x},{{\bf
x}^{\prime}}]}{[{\bf x},{\bf x}][{{\bf x}^{\prime}},{{\bf
x}^{\prime}}]}\right)=R\cosh^{-1}\left(\frac{1}{R^{2}}[{\bf x},{{\bf
x}^{\prime}}]\right),$ (3)
where the inverse hyperbolic cosine with argument $x\in(1,\infty)$ is given by
(see (4.37.19) in Olver et al. (2010) [21])
$\cosh^{-1}x=\log\left(x+\sqrt{x^{2}-1}\right).$
Geodesics on ${\mathbf{H}}_{R}^{d}$ are great hyperbolas (i.e. intersections
of ${\mathbf{H}}_{R}^{d}$ with planes through the origin) with constant speed
parametrizations (see p. 84 in Lee (1997) [18]). We also define a global
function $\rho:{\mathbf{H}}^{d}\times{\mathbf{H}}^{d}\to[0,\infty)$ which
represents the projection of global geodesic distance function (3) on
${\mathbf{H}}_{R}^{d}$ onto the corresponding unit radius hyperboloid
${\mathbf{H}}^{d}$, namely
$\rho(\widehat{\bf x},{\widehat{\bf x}^{\prime}}):=d({\bf x},{{\bf
x}^{\prime}})/R,$ (4)
where $\widehat{\bf x}={\bf x}/R$ and ${\widehat{\bf x}^{\prime}}={{\bf
x}^{\prime}}/R$.
### 2.1 The Laplacian on the hyperboloid model
Parametrizations of a submanifold embedded in either a Euclidean or Minkowski
space is given in terms of coordinate systems whose coordinates are
curvilinear. These are coordinates based on some transformation that converts
the standard Cartesian coordinates in the ambient space to a coordinate system
with the same number of coordinates as the dimension of the submanifold in
which the coordinate lines are curved.
On a $d$-dimensional Riemannian manifold $M$ (a manifold together with a
Riemannian metric $g$), the Laplace-Beltrami operator (Laplacian)
$\Delta:C^{p}(M)\to C^{p-2}(M),$ $p\geq 2$, in curvilinear coordinates
${\mathbf{\xi}}=(\xi^{1},\ldots,\xi^{d})$ is given by
$\Delta=\sum_{i,j=1}^{d}\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial\xi^{i}}\left(\sqrt{|g|}g^{ij}\frac{\partial}{\partial\xi^{j}}\right),$
(5)
where $|g|=|\det(g_{ij})|,$ the infinitesimal distance is given by
$ds^{2}=\sum_{i,j=1}^{d}g_{ij}d\xi^{i}d\xi^{j},\ $ (6)
and
$\sum_{i=1}^{d}g_{ki}g^{ij}=\delta_{k}^{j},$
where $\delta_{i}^{j}\in\\{0,1\\}$ is the Kronecker delta defined for all
$i,j\in{\mathbf{Z}}$ such that
$\delta_{i}^{j}:=\left\\{\begin{array}[]{ll}\displaystyle 1&\qquad\mathrm{if}\
i=j,\\\\[2.84544pt] \displaystyle 0&\qquad\mathrm{if}\ i\neq
j.\end{array}\right.$
For a Riemannian submanifold, the relation between the metric tensor in the
ambient space and $g_{ij}$ of (5) and (6) is
$g_{ij}({\mathbf{\xi}})=\sum_{k,l=0}^{d}G_{kl}\frac{\partial
x^{k}}{\partial\xi^{i}}\frac{\partial x^{l}}{\partial\xi^{j}}.$
On ${\mathbf{H}}_{R}^{d}$ the ambient space is Minkowski, and therefore
$G_{ij}=\mathrm{diag}(1,-1,\ldots,-1)$.
The set of all geodesic polar coordinate systems on the hyperboloid correspond
to the many ways one can put coordinates on a hyperbolic hypersphere, i.e.,
the Riemannian submanifold $U\subset{\mathbf{H}}_{R}^{d}$ defined for a fixed
${{\bf x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ such that $d({\bf x},{{\bf
x}^{\prime}})=b=const,$ where $b\in(0,\infty)$. These are coordinate systems
which correspond to subgroup chains starting with $O(d,1)\supset
O(d)\supset\cdots$, with standard geodesic polar coordinates given by (1)
being only one of them. (For a thorough description of these see section X.5
in Vilenkin (1968) [25].) They all share the property that they are described
by $(d+1)$-variables: $r\in[0,\infty)$ plus $d$-angles each being given by the
values $[0,2\pi)$, $[0,\pi]$, $[-\pi/2,\pi/2]$ or $[0,\pi/2]$ (see Izmest’ev
et al. (1999, 2001) [16, 17]).
In any of the geodesic polar coordinate systems, the global geodesic distance
between any two points on the hyperboloid is given by (cf. (3))
$d({\bf x},{{\bf x}^{\prime}})=R\cosh^{-1}\bigl{(}\cosh r\cosh
r^{\prime}-\sinh r\sinh r^{\prime}\cos\gamma\bigr{)},$ (7)
where $\gamma$ is the unique separation angle given in each hyperspherical
coordinate system. For instance, the separation angle in standard geodesic
polar coordinates (1) is given by the formula
$\displaystyle\cos\,\gamma=\cos(\phi-\phi^{\prime})\prod_{i=1}^{d-2}\sin\theta_{i}{\sin\theta_{i}}^{\prime}+\sum_{i=1}^{d-2}\cos\theta_{i}{\cos\theta_{i}}^{\prime}\prod_{j=1}^{i-1}\sin\theta_{j}{\sin\theta_{j}}^{\prime}.$
(8)
Corresponding separation angle formulae for any geodesic polar coordinate
system can be computed using (2), (3), and the associated formulae for the
appropriate inner-products. Note that by making use of the isometry group
$SO(d,1)$ to map ${{\bf x}^{\prime}}$ to the origin, then $\rho=Rr$ for
${\mathbf{H}}_{R}^{d}$ and in particular $\rho=r$ for ${\mathbf{H}}^{d}$.
Hence, for the unit radius hyperboloid, there is no distinction between the
global geodesic distance and the $r$-parameter in a geodesic polar coordinate
system. For the $R$-radius hyperboloid, the only distinction between the
global geodesic distance and the $r$-parameter is the multiplicative constant
$R$.
The infinitesimal distance in a geodesic polar coordinate system on this
submanifold is given by
$ds^{2}=R^{2}(dr^{2}+\sinh^{2}r\ d\gamma^{2}),$ (9)
where an appropriate expression for $\gamma$ in a curvilinear coordinate
system is given. If one combines (1), (5), (8) and (9), then in a particular
geodesic polar coordinate system, Laplace’s equation on ${\mathbf{H}}_{R}^{d}$
is given by
$\Delta f=\frac{1}{R^{2}}\left[\frac{\partial^{2}f}{\partial r^{2}}+(d-1)\coth
r\frac{\partial f}{\partial
r}+\frac{1}{\sinh^{2}r}\Delta_{{\mathbf{S}}^{d-1}}f\right]=0,$ (10)
where $\Delta_{{\mathbf{S}}^{d-1}}$ is the corresponding Laplace-Beltrami
operator on the unit radius hypersphere ${\mathbf{S}}^{d-1}$.
## 3 A Green’s function in the hyperboloid model
### 3.1 Harmonics in geodesic polar coordinates
Geodesic polar coordinate systems partition the $R$-radius hyperboloid
${\mathbf{H}}_{R}^{d}$ into a family of $(d-1)$-dimensional hyperbolic
hyperspheres, each with a radius $r\in(0,\infty),$ on which all possible
hyperspherical coordinate systems for ${\mathbf{S}}^{d-1}$ may be used (see
for instance Vilenkin (1968) [25]). One then must also consider the limiting
case for $r=0$ to fill out all of ${\mathbf{H}}_{R}^{d}$. In geodesic polar
coordinates one can compute the normalized hyperspherical harmonics in this
space by solving the Laplace equation using separation of variables which
results in a general procedure which is given explicitly in Izmest’ev et al.
(1999, 2001) [16, 17]. These angular harmonics are given as general
expressions involving trigonometric functions, Gegenbauer polynomials and
Jacobi polynomials.
The harmonics in geodesic polar coordinate systems are given in terms of a
radial solution multiplied by the angular harmonics. The angular harmonics are
eigenfunctions of the Laplace-Beltrami operator on ${\mathbf{S}}^{d-1}$ with
unit radius which satisfy the following eigenvalue problem
$\Delta_{{\mathbf{S}}^{d-1}}Y_{l}^{K}(\widehat{\bf
x})=-l(l+d-2)Y_{l}^{K}(\widehat{\bf x}),$
where $\widehat{\bf x}\in{\mathbf{S}}^{d-1}$, $Y_{l}^{K}(\widehat{\bf x})$ are
normalized hyperspherical harmonics, $l\in{\mathbf{N}}_{0}$ is the angular
momentum quantum number, and $K$ stands for the set of $(d-2)$-quantum numbers
identifying degenerate harmonics for each $l$. The degeneracy
$(2l+d-2)\frac{(d-3+l)!}{l!(d-2)!}$
(see (9.2.11) in Vilenkin (1968) [25]), tells you how many linearly
independent solutions exist for a particular $l$ value and dimension $d$. The
hyperspherical harmonics are normalized such that
$\int_{{\mathbf{S}}^{d-1}}Y_{l}^{K}(\widehat{\bf
x})\overline{Y_{l^{\prime}}^{K^{\prime}}(\widehat{\bf
x})}d\omega=\delta_{l}^{l^{\prime}}\delta_{K}^{K^{\prime}},$
where $d\omega$ is the Riemannian (volume) measure (see for instance section
3.4 in Grigor’yan (2009)[12]) on ${\mathbf{S}}^{d-1}$ which is invariant under
the isometry group $SO(d)$ (cf. (11)), and for $x+iy=z\in{\mathbf{C}}$,
$\overline{z}=x-iy$, represents complex conjugation. The generalized Kronecker
delta $\delta_{K}^{K^{\prime}}$ (cf. (2.1)) is defined such that it equals 1
if all of the $(d-2)$-quantum numbers identifying degenerate harmonics for
each $l$ coincide, and equals zero otherwise.
Since the angular solutions (hyperspherical harmonics) are well-known (see
Chapter IX in Vilenkin (1968) [25]; Chapter 11 in Erdélyi et al. (1981) [7]),
we will now focus on the radial solutions on ${\mathbf{H}}_{R}^{d}$ in
geodesic polar coordinates, which satisfy the following ordinary differential
equation (cf. (10)) for all $R\in(0,\infty),$ namely
$\frac{d^{2}u}{dr^{2}}+(d-1)\coth
r\frac{du}{dr}-\frac{l(l+d-2)}{\sinh^{2}r}u=0.$
Four solutions to this ordinary differential equation
$u_{1\pm}^{d,l},u_{2\pm}^{d,l}:(1,\infty)\to{\mathbf{C}}$ are given by
${\displaystyle u_{1\pm}^{d,l}(\cosh
r)=\frac{1}{\sinh^{d/2-1}r}P_{d/2-1}^{\pm(d/2-1+l)}(\cosh r)},$
and
${\displaystyle u_{2\pm}^{d,l}(\cosh
r)=\frac{1}{\sinh^{d/2-1}r}Q_{d/2-1}^{\pm(d/2-1+l)}(\cosh r)},$
where $P_{\nu}^{\mu},Q_{\nu}^{\mu}:(1,\infty)\to{\mathbf{C}}$ are associated
Legendre functions of the first and second kind respectively (see for instance
Chapter 14 in Olver et al. (2010) [21]).
Due to the fact that the space ${\mathbf{H}}_{R}^{d}$ is homogeneous with
respect to its isometry group, the pseudo-orthogonal group $SO(d,1)$, and
therefore an isotropic manifold, we expect that there exist a fundamental
solution of Laplace’s equation on this space with spherically symmetric
dependence. We specifically expect these solutions to be given in terms of
associated Legendre functions of the second kind with argument given by $\cosh
r$. This associated Legendre function naturally fits our requirements because
it is singular at )$r=0$ and vanishes at infinity, whereas the associated
Legendre functions of the first kind, with the same argument, are regular at
$r=0$ and singular at infinity.
### 3.2 Fundamental solution of the Laplacian
In computing a fundamental solution of the Laplacian on
${\mathbf{H}}_{R}^{d}$, we know that
$-\Delta{\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})=\delta_{g}({\bf
x},{{\bf x}^{\prime}}),$
where $g$ is the Riemannian metric on ${\mathbf{H}}_{R}^{d}$ and
$\delta_{g}({\bf x},{{\bf x}^{\prime}})$ is the Dirac delta function on the
manifold ${\mathbf{H}}_{R}^{d}$. The Dirac delta function is defined for an
open set $U\subset{\mathbf{H}}_{R}^{d}$ with ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ such that
$\int_{U}\delta_{g}({\bf x},{{\bf
x}^{\prime}})d\mathrm{vol}_{g}=\left\\{\begin{array}[]{ll}\displaystyle
1&\qquad\mathrm{if}\ {{\bf x}^{\prime}}\in U,\\\\[2.84544pt] \displaystyle
0&\qquad\mathrm{if}\ {{\bf x}^{\prime}}\notin U,\end{array}\right.$
where $d\mathrm{vol}_{g}$ is the Riemannian (volume) measure, invariant under
the isometry group $SO(d,1)$ of the Riemannian manifold
${\mathbf{H}}_{R}^{d}$, given (in standard geodesic polar coordinates) by
$d\mathrm{vol}_{g}=R^{d}\sinh^{d-1}r\,dr\,d\omega:=R^{d}\sinh^{d-1}r\,dr\,\sin^{d-2}\theta_{d-1}\cdots\sin\theta_{2}d\theta_{1}\cdots
d\theta_{d-1}.$ (11)
Notice that as $r\to 0^{+}$ that $d\mathrm{vol}_{g}$ goes to the Euclidean
measure, invariant under the Euclidean motion group $E(d)$, in spherical
coordinates. Therefore in spherical coordinates, we have the following
$\delta_{g}({\bf x},{{\bf
x}^{\prime}})=\frac{\delta(r-r^{\prime})}{R^{d}\sinh^{d-1}r^{\prime}}\frac{\delta(\theta_{1}-\theta_{1}^{\prime})\cdots\delta(\theta_{d-1}-\theta_{d-1}^{\prime})}{\sin\theta_{2}^{\prime}\cdots\sin^{d-2}\theta_{d-1}^{\prime}}.$
In general since we can add any harmonic function to a fundamental solution of
the Laplacian and still have a fundamental solution, we will use this freedom
to make our fundamental solution as simple as possible. It is reasonable to
expect that there exists a particular spherically symmetric fundamental
solution ${\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})$ on the
hyperboloid with pure radial $\rho(\widehat{\bf x},{\widehat{\bf
x}^{\prime}}):=d({\bf x},{{\bf x}^{\prime}})/R$ (cf. (4)) and constant angular
dependence (invariant under rotations centered about the origin), due to the
influence of the point-like nature of the Dirac delta function. For a
spherically symmetric solution to the Laplace equation, the corresponding
$\Delta_{{\mathbf{S}}^{d-1}}$ term vanishes since only the $l=0$ term
survives. In other words, we expect there to exist a fundamental solution of
Laplace’s equation such that ${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=f(\rho)$.
We have proven that on the $R$-radius hyperboloid ${\mathbf{H}}_{R}^{d}$, a
Green’s function for the Laplace operator (fundamental solution for the
Laplacian) can be given as follows.
###### Theorem 3.1.
Let $d\in\\{2,3,\ldots\\}.$ Define
${\mathcal{I}}_{d}:(0,\infty)\to{\mathbf{R}}$ as
${\mathcal{I}}_{d}(\rho):=\int_{\rho}^{\infty}\frac{dx}{\sinh^{d-1}x},$
${\bf x},{{\bf x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$, and
${\mathcal{H}}_{R}^{d}:({\mathbf{H}}_{R}^{d}\times{\mathbf{H}}_{R}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{H}}_{R}^{d}\\}\to{\mathbf{R}}$ defined such
that
${\mathcal{H}}_{R}^{d}({\bf x},{\bf
x}^{\prime}):={\displaystyle\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}R^{d-2}}{\mathcal{I}}_{d}(\rho)},$
where $\rho:=\cosh^{-1}\left([\widehat{\bf x},{\widehat{\bf
x}^{\prime}}]\right)$ is the geodesic distance between $\widehat{\bf x}$ and
${\widehat{\bf x}^{\prime}}$ on the pseudo-sphere of unit radius
${\mathbf{H}}^{d}$, with $\widehat{\bf x}={\bf x}/R,$ ${\widehat{\bf
x}^{\prime}}={{\bf x}^{\prime}}/R$, then ${\mathcal{H}}_{R}^{d}$ is a
fundamental solution for $-\Delta$ where $\Delta$ is the Laplace-Beltrami
operator on ${\mathbf{H}}_{R}^{d}$. Moreover,
$\displaystyle{\mathcal{I}}_{d}(\rho)=\left\\{\begin{array}[]{ll}\displaystyle(-1)^{d/2-1}\frac{(d-3)!!}{(d-2)!!}\Biggl{[}\log\coth\frac{\rho}{2}+\cosh\rho\sum_{k=1}^{d/2-1}\frac{(2k-2)!!(-1)^{k}}{(2k-1)!!\sinh^{2k}\rho}\Biggr{]}&\mathrm{if}\
d\ \mathrm{even},\\\\[17.07182pt]
\left\\{\begin{array}[]{l}\displaystyle(-1)^{(d-1)/2}\Biggl{[}\frac{(d-3)!!}{(d-2)!!}\\\\[11.38092pt]
\displaystyle\hskip
34.14322pt+\left(\frac{d-3}{2}\right)!\sum_{k=1}^{(d-1)/2}\frac{(-1)^{k}\coth^{2k-1}\rho}{(2k-1)(k-1)!((d-2k-1)/2)!}\Biggr{]},\\\\[12.80365pt]
\mathrm{or}\\\\[0.0pt]
\displaystyle(-1)^{(d-1)/2}\frac{(d-3)!!}{(d-2)!!}\left[1+\cosh\rho\sum_{k=1}^{(d-1)/2}\frac{(2k-3)!!(-1)^{k}}{(2k-2)!!\sinh^{2k-1}\rho}\right],\end{array}\right\\}&\mathrm{if}\
d\ \mathrm{odd}.\end{array}\right.$
$\displaystyle=\frac{1}{(d-1)\cosh^{d-1}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right),$
$\displaystyle=\frac{1}{(d-1)\cosh\rho\,\sinh^{d-2}\rho}\,{}_{2}F_{1}\left(\frac{1}{2},1;\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right),$
$\displaystyle=\frac{e^{-i\pi(d/2-1)}}{2^{d/2-1}\Gamma\left(d/2\right)\sinh^{d/2-1}\rho}\,Q_{d/2-1}^{d/2-1}(\cosh\rho),$
where $!!$ is the double factorial, ${}_{2}F_{1}$ is the Gauss hypergeometric
function, and $Q_{\nu}^{\mu}$ is the associated Legendre function of the
second kind.
In the rest of this section, we develop the material in order to prove this
theorem.
Due to the fact that the space ${\mathbf{H}}_{R}^{d}$ is homogeneous with
respect to its isometry group $SO(d,1)$, and therefore an isotropic manifold,
without loss of generality, we are free to map the point ${{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ to the origin. In this case the global
distance function $\rho:{\mathbf{H}}^{d}\times{\mathbf{H}}^{d}\to[0,\infty)$
coincides with the radial parameter in geodesic polar coordinates, and we may
interchange $r$ with $\rho$ accordingly (cf. (7) with $r^{\prime}=0$) in our
representation of a fundamental solution of Laplace’s equation on this
manifold. Since a spherically symmetric choice for a fundamental solution of
Laplace’s equation is harmonic everywhere except at the origin, we may first
set $g=f^{\prime}$ in (10) and solve the first-order equation
$g^{\prime}+(d-1)\coth\rho\ g=0,$
which is integrable and clearly has the general solution
$g(\rho)=\frac{df}{d\rho}=c_{0}\sinh^{1-d}\rho,$ (14)
where $c_{0}\in{\mathbf{R}}$ is a constant which depends on $d$. Now we
integrate (14) to obtain a fundamental solution for the Laplacian in
${\mathbf{H}}_{R}^{d}$
${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=c_{0}{\mathcal{I}}_{d}(\rho)+c_{1},$ (15)
where
${\mathcal{I}}_{d}(\rho):=\int_{\rho}^{\infty}\frac{dx}{\sinh^{d-1}x},$ (16)
and $c_{0},c_{1}\in{\mathbf{R}}$ are constants which depend on $d$. This
definite integral result is mentioned in section II.5 of Helgason (1984) [14]
and as well in Losev (1986) [19] . Notice that we can add any harmonic
function to (15) and still have a fundamental solution of the Laplacian since
a fundamental solution of the Laplacian must satisfy
$\int_{{\mathbf{H}}_{R}^{d}}(-\Delta\varphi)({{\bf
x}^{\prime}}){\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})\
d\mathrm{vol}_{g}^{\prime}=\varphi({\bf x}),$
for all $\varphi\in{\mathcal{D}}({\mathbf{H}}_{R}^{d}),$ where ${\mathcal{D}}$
is the space of test functions, and $d\mathrm{vol}_{g}^{\prime}$ is the
Riemannian (volume) measure on ${\mathbf{H}}_{R}^{d}$, in the primed
coordinates. In particular, we notice that from our definition of
${\mathcal{I}}_{d}$ (16) that
$\lim_{\rho\rightarrow\infty}{\mathcal{I}}_{d}(\rho)=0.$
Therefore it is convenient to set $c_{1}=0$ leaving us with
${\mathcal{H}}_{R}^{d}({\bf x},{{\bf
x}^{\prime}})=c_{0}{\mathcal{I}}_{d}(\rho).$ (17)
In Euclidean space ${\mathbf{R}}^{d}$, a Green’s function for Laplace’s
equation (fundamental solution for the Laplacian) is well-known and is given
in the following theorem (see Folland (1976) [8]; p. 94, Gilbarg & Trudinger
(1983) [9]; p. 17, Bers et al. (1964) [4], p. 211).
###### Theorem 3.2.
Let $d\in{\mathbf{N}}$. Define
${\mathcal{G}}^{d}({\bf x},{\bf
x}^{\prime})=\left\\{\begin{array}[]{ll}\displaystyle\frac{\Gamma(d/2)}{2\pi^{d/2}(d-2)}\|{\bf
x}-{\bf x}^{\prime}\|^{2-d}&\qquad\mathrm{if}\ d=1\mathrm{\ or\ }d\geq
3,\\\\[10.0pt] \displaystyle\frac{1}{2\pi}\log\|{\bf x}-{\bf
x}^{\prime}\|^{-1}&\qquad\mathrm{if}\ d=2,\end{array}\right.$
then ${\mathcal{G}}_{R}^{d}$ is a fundamental solution for $-\Delta$ in
Euclidean space ${\mathbf{R}}^{d}$, where $\Delta$ is the Laplace operator in
${\mathbf{R}}^{d}$.
Note most authors only present the above theorem for the case $d\geq 2$ but it
is easily-verified to also be valid for the case $d=1$ as well.
The hyperboloid ${\mathbf{H}}_{R}^{d}$, being a manifold, must behave locally
like Euclidean space ${\mathbf{R}}^{d}$. Therefore for small $\rho$ we have
$e^{\rho}\simeq 1+\rho$ and $e^{-\rho}\simeq 1-\rho$ and in that limiting
regime
${\mathcal{I}}_{d}(\rho)\approx\int_{\rho}^{1}\frac{dx}{x^{d-1}}\simeq\left\\{\begin{array}[]{ll}-\log\rho&\qquad\mathrm{if}\
d=2,\\\\[5.69046pt] {\displaystyle\frac{1}{\rho^{d-2}}}&\qquad\mathrm{if}\
d\geq 3,\end{array}\right.$
which has exactly the same singularity as a Euclidean fundamental solution for
Laplace’s equation. Therefore the proportionality constant $c_{0}$ is obtained
by matching locally to a Euclidean fundamental solution of Laplace’s equation
${\mathcal{H}}_{R}^{d}=c_{0}{\mathcal{I}}_{d}\simeq{\mathcal{G}}^{d},$
near the singularity located at ${\bf x}={{\bf x}^{\prime}}$.
We have shown how to compute a fundamental solution of the Laplace-Beltrami
operator on the hyperboloid in terms of an improper integral (16). We would
now like to express this integral in terms of well-known special functions. A
fundamental solution ${\mathcal{I}}_{d}$ can be computed using elementary
methods through its definition (16). In $d=2$ we have
${\mathcal{I}}_{2}(\rho)=\int_{\rho}^{\infty}\frac{dx}{\sinh
x}=\frac{1}{2}\log\frac{\cosh\rho+1}{\cosh\rho-1}=\log\coth\frac{\rho}{2},$
and in $d=3$ we have
${\mathcal{I}}_{3}(\rho)=\int_{\rho}^{\infty}\frac{dx}{\sinh^{2}x}=\frac{e^{-\rho}}{\sinh\rho}=\coth\rho-1.$
This exactly matches up to that given by (3.27) in Hostler (1955) [15]. In
$d\in\\{4,5,6,7\\}$ we have
$\displaystyle{\mathcal{I}}_{4}(\rho)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\log\coth\frac{\rho}{2}+\frac{\cosh\rho}{2\sinh^{2}\rho},$
$\displaystyle{\mathcal{I}}_{5}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{3}(\coth^{3}\rho-1)-(\coth\rho-1),$
$\displaystyle{\mathcal{I}}_{6}(\rho)$ $\displaystyle=$
$\displaystyle\frac{3}{8}\log\coth\frac{\rho}{2}+\frac{\cosh\rho}{4\sinh^{4}\rho}-\frac{3\cosh\rho}{8\sinh^{2}\rho},\quad\mathrm{and}$
$\displaystyle{\mathcal{I}}_{7}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{5}(\coth^{5}\rho-1)-\frac{2}{3}(\coth^{3}\rho-1)+\coth\rho-1.$
Now we prove several equivalent finite summation expressions for
${\mathcal{I}}_{d}(\rho)$. We wish to compute the antiderivative
$\mathfrak{I}_{m}:(0,\infty)\to{\mathbf{R}}$, which is defined as
$\mathfrak{I}_{m}(x):=\int\frac{dx}{\sinh^{m}x},$
where $m\in{\mathbf{N}}$. This antiderivative satisfies the following
recurrence relation
$\mathfrak{I}_{m}(x)=-\frac{\cosh
x}{(m-1)\sinh^{m-1}x}-\frac{(m-2)}{(m-1)}\mathfrak{I}_{m-2}(x),$ (18)
which follows from the identity
$\frac{1}{\sinh^{m}x}=\frac{\cosh x}{\sinh^{m}x}\cosh
x-\frac{1}{\sinh^{m-2}x},$
and integration by parts. The antiderivative $\mathfrak{I}_{m}(x)$ naturally
breaks into two separate classes, namely
$\displaystyle\int\frac{dx}{\sinh^{2n+1}x}$ $\displaystyle=$
$\displaystyle(-1)^{n+1}\frac{(2n-1)!!}{(2n)!!}$ (19) $\displaystyle\hskip
28.45274pt\times\left[\log\coth\frac{x}{2}+\cosh
x\sum_{k=1}^{n}\frac{(2k-2)!!(-1)^{k}}{(2k-1)!!\sinh^{2k}x}\right]+C,$
and
$\int\frac{dx}{\sinh^{2n}x}=\left\\{\begin{array}[]{l}\displaystyle(-1)^{n+1}\frac{(2n-2)!!}{(2n-1)!!}\cosh
x\sum_{k=1}^{n}\frac{(2k-3)!!(-1)^{k}}{(2k-2)!!\sinh^{2k-1}x}+C,\qquad\mbox{or}\\\\[15.6491pt]
\displaystyle(-1)^{n+1}(n-1)!\sum_{k=1}^{n}\frac{(-1)^{k}\coth^{2k-1}x}{(2k-1)(k-1)!(n-k)!}+C,\end{array}\right.$
(20)
where $C$ is a constant. The double factorial
$(\cdot)!!:\\{-1,0,1,\ldots\\}\to{\mathbf{N}}$ is defined by
$n!!:=\left\\{\begin{array}[]{ll}\displaystyle n\cdot(n-2)\cdots
2&\quad\mathrm{if}\ n\ \mathrm{even}\geq 2,\\\\[2.84544pt] \displaystyle
n\cdot(n-2)\cdots 1&\quad\mathrm{if}\ n\ \mathrm{odd}\geq 1,\\\\[2.84544pt]
\displaystyle 1&\quad\mathrm{if}\ n\in\\{-1,0\\}.\end{array}\right.$
Note that $(2n)!!=2^{n}n!$ for $n\in{\mathbf{N}}_{0}$. The finite summation
formulae for $\mathfrak{I}_{m}(x)$ all follow trivially by induction using
(18) and the binomial expansion (cf. (1.2.2) in Olver et al. (2010) [21])
$(1-\coth^{2}x)^{n}=n!\sum_{k=0}^{n}\frac{(-1)^{k}\coth^{2k}x}{k!(n-k)!}.$
The formulae (19) and (20) are essentially equivalent to (2.416.2–3) in
Gradshteyn & Ryzhik (2007), except (2.416.3) is not defined for the integrand
$1/\sinh x$. By applying the limits of integration from the definition of
${\mathcal{I}}_{d}(\rho)$ in (16) to (19) and (20) we obtain the following
finite summation expressions for ${\mathcal{I}}_{d}(\rho)$
$\displaystyle{\mathcal{I}}_{d}(\rho)=\left\\{\begin{array}[]{ll}\displaystyle(-1)^{d/2-1}\frac{(d-3)!!}{(d-2)!!}\Biggl{[}\log\coth\frac{\rho}{2}+\cosh\rho\sum_{k=1}^{d/2-1}\frac{(2k-2)!!(-1)^{k}}{(2k-1)!!\sinh^{2k}\rho}\Biggr{]}&\mathrm{if}\
d\ \mathrm{even},\\\\[17.07182pt]
\left\\{\begin{array}[]{l}\displaystyle(-1)^{(d-1)/2}\Biggl{[}\frac{(d-3)!!}{(d-2)!!}\\\\[11.38092pt]
\displaystyle\hskip
34.14322pt+\left(\frac{d-3}{2}\right)!\sum_{k=1}^{(d-1)/2}\frac{(-1)^{k}\coth^{2k-1}\rho}{(2k-1)(k-1)!((d-2k-1)/2)!}\Biggr{]},\\\\[12.80365pt]
\mathrm{or}\\\\[0.0pt]
\displaystyle(-1)^{(d-1)/2}\frac{(d-3)!!}{(d-2)!!}\left[1+\cosh\rho\sum_{k=1}^{(d-1)/2}\frac{(2k-3)!!(-1)^{k}}{(2k-2)!!\sinh^{2k-1}\rho}\right],\end{array}\right\\}&\mathrm{if}\
d\ \mathrm{odd}.\end{array}\right.$ (27)
Moreover, the antiderivative (indefinite integral) can be given in terms of
the Gauss hypergeometric function
$\int\frac{d\rho}{\sinh^{d-1}\rho}=\frac{-1}{(d-1)\cosh^{d-1}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right)+C,$
(29)
where $C\in{\mathbf{R}}$. The Gauss hypergeometric function
${}_{2}F_{1}:{\mathbf{C}}^{2}\times({\mathbf{C}}\setminus-{\mathbf{N}}_{0})\times\\{z\in{\mathbf{C}}:|z|<1\\}\to{\mathbf{C}}$
can be defined in terms of the infinite series
${}_{2}F_{1}(a,b;c;z):=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}n!}z^{n}$
(see (15.2.1) in Olver et al. (2010) [21]), and elsewhere in $z$ by analytic
continuation. (see (2.1.5) in Andrews, Askey & Roy 1999), The Pochhammer
symbol (rising factorial) $(\cdot)_{l}:{\mathbf{C}}\to{\mathbf{C}}$ is defined
by
$(z)_{n}:=\prod_{i=1}^{n}(z+i-1),$
where $l\in{\mathbf{N}}_{0}$. Note that
$(z)_{l}=\frac{\Gamma(z+{l})}{\Gamma(z)},$
for all $z\in{\mathbf{C}}\setminus-{\mathbf{N}}_{0}$. The gamma function
$\Gamma:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$ (see Chapter 5
in Olver et al. (2010) [21]), which is ubiquitous in special function theory,
is an important combinatoric function which generalizes the factorial function
over the natural numbers. It is naturally defined over the right-half complex
plane through Euler’s integral (see (5.2.1) in Olver et al. (2010) [21])
$\Gamma(z):=\int_{0}^{\infty}t^{z-1}e^{-t}dt,$
$\mbox{Re}\ \\!z>0$. Some properties of the gamma function, which we will find
useful are included below. An important formula which the gamma function
satisfies is the duplication formula (i.e., (5.5.5) in Olver et al. (2010)
[21])
$\Gamma(2z)=\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma\left(z+\frac{1}{2}\right),$
(30)
provided $2z\not\in-{\mathbf{N}}_{0}$,
The antiderivative (29) is verified as follows. By using
$\frac{d}{dz}{}_{2}F_{1}(a,b;c;z)=\frac{ab}{c}{}_{2}F_{1}(a+1,b+1;c+1;z)$
(see (15.5.1) in Olver et al. (2010) [21]), and the chain rule, we can show
that
$\displaystyle\frac{d}{d\rho}\frac{-1}{(d-1)\cosh^{d-1}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right)$
$\displaystyle=$
$\displaystyle\frac{\sinh\rho}{\cosh^{d}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right)$
$\displaystyle+\frac{d\sinh\rho}{(d+1)\cosh^{d+2}\rho}\,{}_{2}F_{1}\left(\frac{d+1}{2},\frac{d+2}{2};\frac{d+3}{2};\frac{1}{\cosh^{2}\rho}\right).$
The second hypergeometric function can be simplified using Gauss’ relations
for contiguous hypergeometric functions, namely
$z\,{}_{2}F_{1}(a+1,b+1;c+1;z)=\frac{c}{a-b}\bigl{[}{}_{2}F_{1}(a,b+1;c;z)-{}_{2}F_{1}(a+1,b;c;z)\bigr{]}$
(see p. 58 in Erdélyi et al. (1981) [6]), and
${}_{2}F_{1}(a,b+1;c;z)=\frac{b-a}{b}{}_{2}F_{1}(a,b;c;z)+\frac{a}{b}\,{}_{2}F_{1}(a+1,b;c;z)$
(see (15.5.12) in Olver et al. (2010) [21]). By doing this, the term with the
hypergeometric function cancels leaving only a term which is proportional to a
binomial through
${}_{2}F_{1}(a,b;b;z)=(1-z)^{-a}$
(see (15.4.6) in Olver et al. (2010) [21]), which reduces to
$1/\sinh^{d-1}\rho$. By applying the limits of integration from the definition
of ${\mathcal{I}}_{d}(\rho)$ in (16) to (29) we obtain the following Gauss
hypergeometric representation
${\mathcal{I}}_{d}(\rho)=\frac{1}{(d-1)\cosh^{d-1}\rho}\,{}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right).$
(31)
Using (31), we can write another expression for ${\mathcal{I}}_{d}(\rho)$.
Applying Eulers’s transformation
${}_{2}F_{1}(a,b;c;z)=(1-z)^{c-a-b}{}_{2}F_{1}\left(c-a,c-b;c;z\right)$
(see (2.2.7) in Andrews, Askey & Roy (1999) [2]), to (31) produces
${\mathcal{I}}_{d}(\rho)=\frac{1}{(d-1)\cosh\rho\,\sinh^{d-2}\rho}\,{}_{2}F_{1}\left(\frac{1}{2},1;\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right).$
Our derivation for a fundamental solution of Laplace’s equation on the
$R$-radius hyperboloid ${\mathbf{H}}_{R}^{d}$ in terms of the associated
Legendre function of the second kind is as follows. By starting with (31) and
the definition of the associated Legendre function of the second kind
$Q_{\nu}^{\mu}:(1,\infty)\to{\mathbf{C}},$ namely
$Q_{\nu}^{\mu}(z):=\frac{\sqrt{\pi}e^{i\pi\mu}\Gamma(\nu+\mu+1)(z^{2}-1)^{\mu/2}}{2^{\nu+1}\Gamma(\nu+\frac{3}{2})z^{\nu+\mu+1}}\,{}_{2}F_{1}\left(\frac{\nu+\mu+2}{2},\frac{\nu+\mu+1}{2};\nu+\frac{3}{2};\frac{1}{z^{2}}\right),$
for $|z|>1$ and $\nu+\mu+1\notin-{\mathbf{N}}_{0}$ (see (8.1.3) in Abramowitz
& Stegun (1972) [1]), we derive
${}_{2}F_{1}\left(\frac{d-1}{2},\frac{d}{2};\frac{d+1}{2};\frac{1}{\cosh^{2}\rho}\right)=\frac{2^{d/2}\Gamma\left(\frac{d+1}{2}\right)\cosh^{d-1}\rho}{\sqrt{\pi}e^{i\pi(d/2-1)}(d-2)!\sinh^{d/2-1}\rho}Q_{d/2-1}^{d/2-1}(\cosh\rho).$
(32)
We have therefore verified that the harmonics computed in section 3.1, namely
$u_{2+}^{d,0},$ give an alternate form of a fundamental solution for the
Laplacian on the hyperboloid. Using the duplication formula for gamma
functions (30), (31), and (32), we derive
${\mathcal{I}}_{d}(\rho)=\frac{e^{-i\pi(d/2-1)}}{2^{d/2-1}\Gamma\left(d/2\right)\sinh^{d/2-1}\rho}Q_{d/2-1}^{d/2-1}(\cosh\rho).$
Notice that our chosen fundamental solutions of the Laplacian on the
hyperboloid have the property that they tend towards zero at infinity (even
for the $d=2$ case, unlike Euclidean fundamental solutions of the Laplacian).
Therefore these Green’s functions are positive (see Grigor’yan (1983) [10];
Grigor’yan (1985) [11]) and hence ${\mathbf{H}}_{R}^{d}$ is not parabolic.
Note that as a result of our proof, we see that the relevant associated
Legendre functions of the second kind for $d\in\\{2,3,4,5,6,7\\}$ are (cf.
(LABEL:sumgradryzhikIn))
$\displaystyle Q_{0}(\cosh\rho)=\log\coth\frac{\rho}{2},$
$\displaystyle\frac{1}{\sinh^{1/2}\rho}Q_{1/2}^{1/2}(\cosh\rho)=i\sqrt{\frac{\pi}{2}}(\coth\rho-1),$
$\displaystyle\frac{1}{\sinh\rho}Q_{1}^{1}(\cosh\rho)=\log\coth\frac{\rho}{2}-\frac{\cosh\rho}{\sinh^{2}\rho},$
$\displaystyle\frac{1}{\sinh^{3/2}\rho}Q_{3/2}^{3/2}(\cosh\rho)=3i\sqrt{\frac{\pi}{2}}\left(-\frac{1}{3}\coth^{3}\rho+\coth\rho-\frac{2}{3}\right),$
$\displaystyle\frac{1}{\sinh^{2}\rho}Q_{2}^{2}(\cosh\rho)=3\,\log\coth\frac{\rho}{2}-2\frac{\cosh\rho}{\sinh^{4}\rho}-3\frac{\cosh\rho}{\sinh^{2}\rho},\quad\
\mathrm{and}$
$\displaystyle\frac{1}{\sinh^{5/2}\rho}Q_{5/2}^{5/2}(\cosh\rho)=15i\sqrt{\frac{\pi}{2}}\left(\frac{1}{15}\coth^{5}\rho-\frac{2}{3}\coth^{3}\rho+\coth\rho-\frac{8}{15}\right).$
The constant $c_{0}$ in a fundamental solution for the Laplace operator on the
hyperboloid (17) is computed by locally matching up the singularity to a
fundamental solution for the Laplace operator in Euclidean space, Theorem 3.2.
The coefficient $c_{0}$ depends on $d$. It is determined as follows. For
$d\geq 3$ we take the asymptotic expansion for $c_{0}{\mathcal{I}}_{d}(\rho)$
as $\rho$ approaches zero and match this to a fundamental solution of
Laplace’s equation for Euclidean space given in Theorem 3.2. This yields
$\displaystyle c_{0}=\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}}.$ (33)
For $d=2$ we take the asymptotic expansion for
$c_{0}{\mathcal{I}}_{2}(\rho)=c_{0}\log\coth\frac{\rho}{2}\simeq
c_{0}\log\|{\bf x}-{{\bf x}^{\prime}}\|^{-1}$
as $\rho$ approaches zero, and match this to
$\displaystyle{\mathcal{G}}^{2}({\bf x},{{\bf
x}^{\prime}})=(2\pi)^{-1}\log\|{\bf x}-{{\bf x}^{\prime}}\|^{-1},$ therefore
$\displaystyle c_{0}=(2\pi)^{-1}$. This exactly matches (33) for $d=2$. The
derivation that ${\mathcal{I}}_{d}(\rho)$ is an fundamental solution of the
Laplace operator on the hyperboloid ${\mathbf{H}}^{d}_{R}$ and the functions
for ${\mathcal{I}}_{d}(\rho)$ are computed above.
The sectional curvature of a pseudo-sphere of radius $R$ is $-1/R^{2}$. Hence
using results in Losev (1986) [19], all equivalent expressions in Theorem 3.1
can be used for a fundamental solution of the Laplace-Beltrami operator on the
$R$-radius hyperboloid ${\mathbf{H}}_{R}^{d}$ (cf. section 2), namely (where
$R$ is now a free parameter)
${\mathcal{H}}_{R}^{d}({\bf x},{\bf
x}^{\prime}):={\displaystyle\frac{\Gamma\left(d/2\right)}{2\pi^{d/2}R^{d-2}}{\mathcal{I}}_{d}\left(\rho\right)}.$
The proof of Theorem 3.1 is complete.
Furthermore, due to a theorem proved in [19], all equivalent expressions for
${\mathcal{I}}_{d}(\rho)$ in Theorem 3.1 represent upper bounds for a
fundamental solution of the Laplace-Beltrami operator on non-compact
Riemannian manifolds with negative sectional curvature not exceeding
$-1/R^{2}$ with $R>0$.
We would also like to mention that a similar computation for a fundamental
solution of Laplace’s equation on the positive-constant sectional curvature
compact manifold, the $R$-radius hypersphere, has recently been computed in
Cohl (2011) [5].
## 4 Uniqueness of fundamental solution in terms of decay at infinity
It is clear that in general a fundamental solution of Laplace’s equation in
the hyperboloid model of hyperbolic geometry ${\mathcal{H}}_{R}^{d}$ is not
unique since one can add any harmonic function
$h:{\mathbf{H}}_{R}^{d}\to{\mathbf{R}}$ to ${\mathcal{H}}_{R}^{d}$ and still
obtain a solution to
$-\Delta{\mathcal{H}}_{R}^{d}({\bf x},{{\bf x}^{\prime}})=\delta_{g}({\bf
x},{{\bf x}^{\prime}}),$
since $h$ is in the kernel of $-\Delta$.
###### Proposition 4.1.
There exists precisely one $C^{\infty}$-function
$H:({\mathbf{H}}_{R}^{d}\times{\mathbf{H}}_{R}^{d})\setminus\\{({\bf x},{\bf
x}):{\bf x}\in{\mathbf{H}}_{R}^{d}\\}\to{\mathbf{R}}$ such that for all ${{\bf
x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$ the function $H_{{\bf
x}^{\prime}}:{\mathbf{H}}_{R}^{d}\setminus\\{{{\bf
x}^{\prime}}\\}\to{\mathbf{R}}$ defined by $H_{{\bf x}^{\prime}}({\bf
x}):=H({\bf x},{{\bf x}^{\prime}})$ is a distribution on
${\mathbf{H}}_{R}^{d}$ with
$-\Delta H_{{\bf x}^{\prime}}=\delta_{g}(\cdot,{{\bf x}^{\prime}})$
and
$\lim_{d({\bf x},{{\bf x}^{\prime}})\to\infty}H_{{\bf x}^{\prime}}({\bf
x})=0,$ (34)
where $d({\bf x},{{\bf x}^{\prime}})$ is the geodesic distance between two
points ${\bf x},{{\bf x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$.
Proof. Existence: clear. Uniqueness. Suppose $H$ and $\tilde{H}$ are two such
functions. Let ${{\bf x}^{\prime}}\in{\mathbf{H}}_{R}^{d}$. Define the
$C^{\infty}$-function $h:{\mathbf{H}}_{R}^{d}\setminus\\{{{\bf
x}^{\prime}}\\}\to{\mathbf{R}}$ by $h=H_{{\bf x}^{\prime}}-\tilde{H}_{{\bf
x}^{\prime}}.$ Then $h$ is a distribution on ${\mathbf{H}}_{R}^{d}$ with
$-\Delta h=0$. Since ${\mathbf{H}}_{R}^{d}$ is locally Euclidean one has by
local elliptic regularity that $h$ can be extended to a $C^{\infty}$-function
$\hat{h}:{\mathbf{H}}_{R}^{d}\to{\mathbf{R}}$. It follows from (34) for $H$
and $\tilde{H}$ that
$\lim_{d({\bf x},{{\bf x}^{\prime}})\to\infty}\hat{h}({\bf x})=0.$ (35)
The strong elliptic maximum/minimum principle on a Riemannian manifold for a
bounded domain $\Omega$ states that if $u$ is harmonic, then the
supremum/infimum of $u$ in $\Omega$ coincides with the supremum/infimum of $u$
on the boundary $\partial\Omega$. By using a compact exhaustion sequence
$\Omega_{k}$ in a non-compact connected Riemannian manifold and passing to a
subsequence ${\bf x}_{k}\in\partial\Omega_{k}$ such that ${\bf
x}_{k}\to\infty$, the strong elliptic maximum/minimum principle can be
extended to non-compact connected Riemannian manifolds with boundary
conditions at infinity (see for instance section 8.3.2 in Grigor’yan (2009)
[12]). Taking $\Omega_{k}\subset{\mathbf{H}}_{R}^{d}$, the strong elliptic
maximum/minimum principle for non-compact connected Riemannian manifolds
implies using (35) that $\hat{h}=0$. Therefore $h=0$ and $H({\bf x},{{\bf
x}^{\prime}})=\tilde{H}({\bf x},{{\bf x}^{\prime}})$ for all ${\bf
x}\in{\mathbf{H}}_{R}^{d}\setminus\\{{{\bf x}^{\prime}}\\}$.
By Proposition 4.1, for $d\geq 2$, the function ${\mathcal{H}}_{R}^{d}$ is the
unique normalized fundamental solution of Laplace’s equation which satisfies
the vanishing decay (34).
## Acknowledgements
Much thanks to Simon Marshall, A. Rod Gover, Tom ter Elst, Shaun Cooper,
George Pogosyan, Willard Miller, Jr., and Alexander Grigor’yan for valuable
discussions. I would like to express my gratitude to Carlos Criado Cambón in
the Facultad de Ciencias at Universidad de Málaga for his assistance in
describing the global geodesic distance function in the hyperboloid model. We
would also like to acknowledge two anonymous referees whose comments helped
improve this paper. I acknowledge funding for time to write this paper from
the Dean of the Faculty of Science at the University of Auckland in the form
of a three month stipend to enhance University of Auckland 2012 PBRF
Performance. Part of this work was conducted while H. S. Cohl was a National
Research Council Research Postdoctoral Associate in the Information Technology
Laboratory at the National Institute of Standards and Technology,
Gaithersburg, Maryland, U.S.A.
## References
## References
* [1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C., 1972.
* [2] G. E. Andrews, R. Askey, and R. Roy. Special functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999.
* [3] E. Beltrami. Essai d’interprétation de la géométrie noneuclidéenne. Trad. par J. Hoüel. Annales Scientifiques de l’École Normale Supérieure, 6:251–288, 1869.
* [4] L. Bers, F. John, and M. Schechter. Partial differential equations. Interscience Publishers, New York, N.Y., 1964.
* [5] H. S. Cohl. Fundamental solution of laplace’s equation in hyperspherical geometry. Symmetry, Integrability and Geometry: Methods and Applications, 7(108), 2011.
* [6] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vol. I. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
* [7] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher transcendental functions. Vol. II. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981.
* [8] G. B. Folland. Introduction to partial differential equations. Number 17 in Mathematical Notes. Princeton University Press, Princeton, 1976.
* [9] D. Gilbarg and N. S. Trudinger. Elliptic partial differential equations of second order. Number 224 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin etc., second edition, 1983.
* [10] A. A. Grigor’yan. Existence of the Green function on a manifold. Russian Mathematical Surveys, 38(1(229)):161–162, 1983.
* [11] A. A. Grigor’yan. The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds. Matematicheskie Zametki, 128(170)(3):354–363, 446, 1985.
* [12] A. A. Grigor’yan. Heat kernel and analysis on manifolds, volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, 2009.
* [13] C. Grosche, G. S. Pogosyan, and A. N. Sissakian. Path-integral approach for superintegrable potentials on the three-dimensional hyperboloid. Physics of Particles and Nuclei, 28(5):486–519, 1997.
* [14] S. Helgason. Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, volume 113 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1984.
* [15] L. Hostler. Vector spherical harmonics of the unit hyperboloid in Minkowski space. Journal of Mathematical Physics, 18(12):2296–2307, 1977.
* [16] A. A. Izmest’ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz. Contractions of Lie algebras and separation of variables. The $n$-dimensional sphere. Journal of Mathematical Physics, 40(3):1549–1573, 1999.
* [17] A. A. Izmest’ev, G. S. Pogosyan, A. N. Sissakian, and P. Winternitz. Contractions of Lie algebras and the separation of variables: interbase expansions. Journal of Physics A: Mathematical and General, 34(3):521–554, 2001\.
* [18] J. M. Lee. Riemannian manifolds, volume 176 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
* [19] A. G. Losev. Harmonic functions on manifolds of negative curvature. Matematicheskie Zametki, 40(6):738–742, 829, 1986.
* [20] M. N. Olevskiĭ. Triorthogonal systems in spaces of constant curvature in which the equation $\Delta_{2}u+\lambda u=0$ allows a complete separation of variables. Matematicheskiĭ Sbornik, 27(69):379–426, 1950. (in Russian).
* [21] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. Cambridge University Press, Cambridge, 2010.
* [22] G. S. Pogosyan and P. Winternitz. Separation of variables and subgroup bases on $n$-dimensional hyperboloids. Journal of Mathematical Physics, 43(6):3387–3410, 2002.
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|
arxiv-papers
| 2012-01-20T22:08:09 |
2024-09-04T02:49:26.551504
|
{
"license": "Public Domain",
"authors": "Howard S. Cohl and Ernie G. Kalnins",
"submitter": "Howard Cohl",
"url": "https://arxiv.org/abs/1201.4406"
}
|
1201.4415
|
# Spectroscopy and Thermometry of Drumhead Modes in a Mesoscopic Trapped-Ion
Crystal using Entanglement
Brian C. Sawyer brian.sawyer@boulder.nist.gov Joseph W. Britton Time and
Frequency Division, National Institute of Standards and Technology, Boulder,
CO 80305 Adam C. Keith Department of Physics, North Carolina State
University, Raleigh, NC 27695 C.-C. Joseph Wang James K. Freericks
Department of Physics, Georgetown University, Washington, DC 20057 Hermann
Uys Council for Scientific and Industrial Research, Pretoria, South Africa
Michael J. Biercuk Centre for Engineering Quantum Systems, School of Physics,
The University of Sydney, NSW Australia John J. Bollinger Time and Frequency
Division, National Institute of Standards and Technology, Boulder, CO 80305
###### Abstract
We demonstrate spectroscopy and thermometry of individual motional modes in a
mesoscopic 2D ion array using entanglement-induced decoherence as a method of
transduction. Our system is a $\sim$400 $\mu$m-diameter planar crystal of
several hundred 9Be+ ions exhibiting complex drumhead modes in the confining
potential of a Penning trap. Exploiting precise control over the 9Be+ valence
electron spins, we apply a homogeneous spin-dependent optical dipole force to
excite arbitrary transverse modes with an effective wavelength approaching the
interparticle spacing ($\sim$20 $\mu$m). Center-of-mass displacements below 1
nm are detected via entanglement of spin and motional degrees of freedom.
###### pacs:
52.27.Jt, 52.27.Aj, 03.65.Ud, 03.67.Bg
Studies of quantum physics at the interface of microscopic and mesoscopic
regimes have recently focused on the observation of quantum coherent phenomena
in optomechanical systems O’Connell et al. (2010); Kippenberg and Vahala
(2008); Teufel et al. (2009). The realization of quantum coherence in
mechanical oscillations involving many particles behaving approximately as a
continuum provides exciting insights into the quantum-classical transition.
Previous work has shown that crystals of cold, trapped ions behave as atomic-
scale nanomechanical oscillators Biercuk et al. (2010); Jost et al. (2009);
Brown et al. (2011), with the benefits of in-situ tunable motional modes and
exploitable single-particle quantum degrees of freedom (e.g. valence electron
spin). Our system of hundreds of crystallized ions in a Penning trap provides
a bottom-up approach to studying mesoscopic quantum coherence. In this
context, the relevant particle numbers are sufficiently small to permit
excellent quantum control without sacrificing continuum mechanical features.
Beyond these capabilites, trapped ions have long provided a laboratory
platform for studying diverse physical phenomena including: strongly-coupled
one-component plasmas (OCPs) Ichimaru (1982); Dubin and O’Neil (1999); quantum
computation Hanneke et al. (2009); Monz et al. (2009) and simulation
Friedenauer et al. (2008); Kim et al. (2010); Islam et al. (2011); Britton et
al. (2012); Lanyon et al. (2011); dynamical decoupling Biercuk et al. (2009a);
and atomic clocks and precision measurement Rosenband et al. (2008).
Figure 1: (color online) (a) Calculated structure of selected transverse
eigenmodes ($\vec{b}_{m}$) for a 2D crystal of 331 9Be+ ions. Mode
frequencies, $\omega_{m}$, decrease as effective wavelength gets shorter. The
arbitrary color scale indicates relative ion displacement amplitude. One
example of an ion spin state with similar symmetry is given below each of the
four highest-frequency eigenmodes. The symbol $\times$($\bullet$) denotes
spin-projection into (out of) the plane. Interaction between these spin and
mode configurations mediated by the spin-dependent optical dipole force (ODF)
leads to excitation of the corresponding eigenmode. (b) Illustration of a
single plane of 9Be+ within the Penning trap. Two 313-nm beams intersect at
the ion cloud to form a traveling wave of beat frequency $\mu_{R}$ and
effective wavevector $\overrightarrow{\Delta k}$ along the direction of the
trap magnetic field. The electric field intensity is uniform in the plane, but
the spin-dependent induced AC Stark shift permits excitation of transverse
modes of arbitrary wavelength.
In this Letter, we present an experimental and theoretical study of motional
drumhead modes in a 2D crystal of 9Be+ ions confined within a Penning trap. We
excite _inhomogeneous_ modes of arbitrary wavelength (see Fig. 1(a)) through
application of a _homogeneous_ , spin-state-dependent optical dipole force
(ODF) to a large-scale spin superposition. Distinct drumhead modes are
entangled with the 9Be+ valence electron spins by tuning a beat frequency
($\mu_{R}$) between two ODF lasers near a mode resonance. This spin-motion
entanglement is detected as a $\mu_{R}$-dependent decoherence of ion spins
whose magnitude conveys the specific mode temperature.
Previous global mode studies on 2D planar ion arrays were restricted to modes
with wavelengths on the order of the cloud size Heinzen et al. (1991);
Bollinger et al. (1993); Weimer et al. (1994); Tinkle et al. (1994); Dantan et
al. (2010). By contrast, the short-wavelength modes studied here are of
particular interest due to their increased sensitivity to strong-correlation
corrections Kriesel et al. (2002); Castro et al. (2010) compared to those with
long wavelength, which are well-described by fluid theory. Thermometry of
large Coulomb crystals has thus far been limited to determination of global
temperature through Doppler profile measurements Jensen et al. (2004), which
give a minimum sensitivity of $\sim$0.5 mK in 9Be+. Our temperature
measurement is mode-specific and may be employed below the Doppler cooling
limit, providing an alternative to Raman sideband thermometry Monroe et al.
(1995).
The Penning trap used for this work is detailed in a previous publication
Biercuk et al. (2009b). Application of static voltages to a stack of
cylindrical electrodes provides harmonic confinement along $\hat{z}$ (the trap
symmetry axis) with a 9Be+ center-of-mass (COM) oscillation frequency of
$\omega_{1}/2\pi=795$ kHz that is independent of the number of trapped ions.
The trap resides within the room-temperature bore of a superconducting magnet,
and radial confinement is achieved via the Lorentz force generated by rotation
of the ion cloud through the static, homogeneous magnetic ($B$) field of
$\sim$4.46 T oriented along $\hat{z}$. Application of a time-dependent
quadrupole ‘rotating wall’ potential permits phase-stable control of the
rotation frequency ($\omega_{r}$), and thus the confining radial force of the
trap Hasegawa et al. (2005); Huang et al. (1998). In the limit of a weak
rotating wall potential, the harmonic trap potential in a frame rotating at
$\omega_{r}$ is Dubin and O’Neil (1999)
$\displaystyle q\Phi_{\text{trap}}(r,z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}M\omega_{1}^{2}\left(z^{2}+\beta r^{2}\right),$ (1)
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{\omega_{r}(\Omega_{c}-\omega_{r})}{\omega_{1}^{2}}-\frac{1}{2}$
(2)
where $M$ ($q$) is the mass (charge) of a single 9Be+, $\Omega_{c}=2\pi\times
7.6$ MHz is the cyclotron frequency, and $z$ ($r$) is axial (radial) distance
from the trap center. We set the rotation frequency, $\omega_{r}$, such that
the radial confinement is weak relative to transverse confinement ($\beta\ll
1$), resulting in a single ion plane.
Figure 2: (color online) (a) Pulse sequence used for excitation and detection
of transverse motional modes. Global spin rotations are performed with
microwaves at $\sim$124 GHz, while the state-dependent optical dipole force is
applied in each arm of the spin echo for a duration $\tau$. We implement
$\pi$-pulse times ($t_{\pi}$) as short as 65 $\mu$s. (b) Measured (points with
statistical error bars) and fit (solid blue line) probability of detecting
$|\\!\\!\uparrow\rangle$ ($P_{\uparrow}$) at the end of the spin echo
sequence. Frequency-dependent decoherence is due to entanglement of spins with
the axial COM mode ($\omega_{1}/2\pi=795$ kHz) as a function of ODF detuning
$\delta_{1}\equiv\mu_{R}-\omega_{1}$ in a cloud of $190\pm 8$ ions. Each point
is an average of 90 experimental runs. The fit provides a mode temperature of
$2.3\pm 0.5$ mK, whose error includes a 5% uncertainty in ODF beam angle,
$\theta_{\text{R}}$. For comparison, the lower (upper) dashed line is
calculated assuming 0.4 mK (4.0 mK). (c) Illustrated phase-space trajectories
of state $|\\!\\!\uparrow\rangle_{N}$ at different detunings, $\delta_{1}$, in
a frame rotating at $\omega_{1}$. Axis labels represent COM momentum
($p_{z}\propto\text{Im}[\alpha_{j1}]$) and position
($z\propto\text{Re}[\alpha_{j1}]$).
The $m_{J}=\pm 1/2$ projections of the Be+ ${}^{2}S_{1/2}$ ground state are
split by $\sim$124 GHz and serve as $|\\!\\!\uparrow\rangle$ and
$|\\!\\!\downarrow\rangle$ ‘qubit’ states, respectively. Global spin rotations
are performed by injecting 124-GHz radiation through a waveguide attached to
the side of the trap. The 9Be+ ions are Doppler laser cooled with laser beams
directed both parallel and perpendicular to $\hat{z}$. Both beams are tuned to
the ${}^{2}S_{1/2}(m_{J}=+1/2)$–${}^{2}P_{3/2}(m_{J}=+3/2)$ transition at
$\sim$313 nm to cool ion motion below 1 mK. This same transition is used for
ion detection and projective spin-state measurement. Discrimination of
$|\\!\\!\uparrow\rangle$ (bright) from $|\\!\\!\downarrow\rangle$ (dark) is
performed with a fidelity $>99$% Biercuk et al. (2009b).
The axial and radial confining potentials are tuned to yield a planar ion
configuration. Due to mutual Coulomb repulsion and the low ion temperature,
the ions’ minimum-energy configuration is a 2D crystal with triangular order
Mitchell et al. (1998). Ion spacing is $\sim$20 $\mu$m, and individual ions
can be resolved using stroboscopic imaging at $\omega_{r}$. The planar array
of $N$ ions exhibits $3N$ motional modes, $N$ of which are drumhead
oscillations transverse to the crystal plane (see Fig. 1(a)). As with 1D ion
strings, the frequencies of these transverse modes decrease with decreasing
effective wavelength due to screening of confining electric fields by nearby
ions. The transverse eigenvectors ($\vec{b}_{m}$, $m\in[1,N]$) and
corresponding eigenfrequencies ($\omega_{m}$) are obtained by first
numerically calculating the zero-temperature 2D ion configuration in the
presence of the Penning trap potentials. Applying a Taylor expansion of the
combined trap and Coulomb potential about each ion equilibrium position, we
diagonalize the $N\times N$ stiffness matrix whose eigenvalues and unit
eigenvectors are $\omega_{m}$ and $\vec{b}_{m}$, respectively Zhu et al.
(2006); Kim et al. (2009). The relative displacement amplitude of an ion $j$
is given by the $j$th element of $\vec{b}_{m}$, denoted as $b_{jm}$, where
$\sum_{m}\left|b_{jm}\right|^{2}=\sum_{j}\left|b_{jm}\right|^{2}=1$.
To excite transverse modes in our 2D Coulomb crystal, we employ a spin-
dependent ODF generated by interfering two off-resonant laser beams at the ion
cloud position. This is depicted schematically in Fig. 1(b). The two ODF beams
are produced from a single beam using a 50/50 beamsplitter and subsequently
pass through separate acousto-optic modulators that allow fast ($\sim$1
$\mu$s) switching and impart a relative detuning $\mu_{R}$. The beams
intersect at an angle of $\theta_{\text{R}}=4.8^{\circ}\pm 0.2^{\circ}$ at the
ion cloud position, and their relative alignment is adjusted to orient the
effective wavevector ($\overrightarrow{\Delta k}$) of the resulting standing
($\mu_{R}=0$) or traveling ($\mu_{R}\neq 0$) wave to within $\sim
0.05^{\circ}$ of $\hat{z}$. The common wavelength (313.133 nm) and unique
linear polarizations of the beams are chosen such that the AC Stark shift from
the interfering beams on state $|\\!\\!\uparrow\rangle$ is equal in magnitude
and opposite in sign to that on $|\\!\\!\downarrow\rangle$ EPA . The result of
the interference between these two beams is a spin-dependent force on each
ion, $j$ ($F_{\uparrow,j}=-F_{\downarrow,j}\equiv F_{j}$). The Hamiltonian for
this interaction is
$\hat{H}_{\text{ODF}}=-\sum_{j=1}^{N}F_{j}\hat{z}_{j}(t)\cos{(\mu_{R}t)}\hat{\sigma}^{z}_{j}$,
where $\hat{z}_{j}(t)$ is the time-dependent position operator and
$\hat{\sigma}^{z}_{j}$ is the $z$-component Pauli operator for ion $j$ Britton
et al. (2012). The elliptical beam waists (100 $\mu\text{m}\times 1000$
$\mu$m, with the major axis oriented parallel to the ion plane EPA ) are
sufficiently large to generate an approximately uniform ODF with variation
below 10% across the $\sim$400 $\mu$m-diameter planar ion crystal. Typical
ODFs for this work are $F_{j}\sim 10^{-23}$ N along $\hat{z}$.
Figure 2(a) illustrates the experimental control sequence for microwaves
(black line) and ODF lasers (shaded regions) used to coherently excite
transverse modes of motion. Ions are first prepared in the ‘bright’ state
$|\\!\\!\uparrow\rangle_{N}\equiv\prod_{j=1}^{N}|\\!\\!\uparrow_{j}\rangle$
via optical pumping Biercuk et al. (2009b). The sequence of microwave pulses
in Fig. 2(a) comprises a spin echo (SE) Hahn (1950) that, in the absence of
the ODF beams, rotates the ions to the ‘dark’ state
$|\\!\\!\downarrow\rangle_{N}$ with $>$99% fidelity. The SE cancels low-
frequency precession about $\hat{z}$ due to ODF laser intensity and magnetic
field fluctuations as well as microwave phase noise Uys et al. (2009); Biercuk
et al. (2009a). The spin-dependent ODF is applied in each arm of the SE for a
duration $\tau$.
The initial microwave pulse rotates each spin by $\pi/2$ to produce the state
$\prod_{j=1}^{N}\frac{1}{\sqrt{2}}\left(|\\!\\!\uparrow_{j}\rangle-|\\!\\!\downarrow_{j}\rangle\right)$,
which is a superposition of all possible ($2^{N}$) spin permutations.
Importantly, it is the creation of this state that permits subsequent
excitation of arbitrary transverse modes with our homogeneous, spin-dependent
ODF. By tuning $\mu_{R}$ near a mode of frequency $\omega_{m}$, the spin-
dependent ODF excites those components of the spin superposition with
approximately the same symmetry as the eigenvector $\vec{b}_{m}$. A subset of
these eigenvectors and associated spin states are illustrated in Fig. 1(a).
Depending on experimental parameters, the spin states may be entangled with
different motional states at the end of the control sequence of Fig. 2(a).
Upon measurement of the spin state (performing a trace over the motion),
entanglement is manifested as spin decoherence that varies with $\mu_{R}$. We
observe this as a decrease in the length of the spins’ Bloch vector and a
concomitant increase in the probability ($P_{\uparrow}$) of measuring state
$|\\!\\!\uparrow\rangle$ averaged over all ions.
Figure 2(b) gives experimental and theoretical results for a sweep of
$\mu_{R}$ near the COM frequency, $\omega_{1}$, with $\tau=500$ $\mu$s and
$\delta_{1}=(\mu_{R}-\omega_{1})$. On resonance ($\delta_{1}=0$), the pulse
sequence leads to excitation (de-excitation) of the COM mode in the first
(second) arm. When the product $|\delta_{1}\tau/2\pi|=l$ is a non-zero
integer, each spin state traverses $l$ full loops in phase space over $\tau$
(see Fig. 2(c)). At intermediate detunings, the spin and motion remain
entangled at the end of the pulse sequence, producing the lineshape of Fig.
2(b). These motional excitations are described by the spin-dependent
displacement operator
$\hat{U}(\tau)=\prod_{j,m}\exp{\left[(\alpha_{jm}\hat{a}^{{\dagger}}_{m}-\alpha^{\ast}_{jm}\hat{a}_{m})\hat{\sigma}^{z}_{j}\right]}$
Kim et al. (2009); Monroe et al. (1996); EPA , where $\alpha_{jm}(\tau)$ is
the coherently-driven complex displacement amplitude for ion $j$ of mode $m$,
and $\hat{a}^{{\dagger}}_{m}(\hat{a}_{m})$ is the creation (annihilation)
operator for mode $m$. Accounting for both arms of the pulse sequence, we
obtain EPA
$\alpha_{jm}=\frac{F_{j}b_{jm}}{\hslash(\mu_{R}^{2}-\omega_{m}^{2})}\sqrt{\frac{\hslash}{2M\omega_{m}}}\left[\omega_{m}(1-\cos{\phi})+i\mu_{R}\sin{\phi}-e^{i\omega_{m}\tau}\left\\{\omega_{m}\left[\cos{(\mu_{R}\tau)}-\cos{(\mu_{R}\tau+\phi)}\right]-i\mu_{R}\left[\sin{(\mu_{R}\tau)-\sin{(\mu_{R}\tau+\phi)}}\right]\right\\}\right],$
(3)
where $\hslash$ is Planck’s constant, $F_{j}$ is the ODF magnitude on ion $j$,
and $\phi=(\tau+t_{\pi})(\mu_{R}-\omega_{m})$ accounts for phase evolution of
the ODF drive relative to that of the mode.
Figure 3: (color online) (a) Measured (lower) and calculated (offset)
probabilities for measuring $|\\!\\!\uparrow\rangle$ after the spin echo
sequence as a function of ODF beat frequency for a sweep of $\mu_{R}$ over the
first five transverse modes with $250\pm 15$ ions. The modes at $\omega_{2}$
and $\omega_{3}$ are split due to distortion of the ion cloud boundary by the
rotating wall potential. Panels (b) and (c) give results of wider sweeps with
$\omega_{r}/2\pi=$ 43.2 kHz and 44.7 kHz, respectively, in a crystal of
$345\pm 25$ ions. Frequency-dependent deviation from P${}_{\uparrow}\sim 0.1$
is due to spin-motional entanglement, while the background is due to
spontaneous emission from the ODF beams. The histogram (red bars) shown below
each experimental curve depicts the density of calculated eigenmodes at the
given $\omega_{r}$. Histogram bins are 10 kHz wide and plotted with an
arbitrary vertical scale. As described in Fig. 1(a), the highest-frequency
feature is that of the COM mode and the $\sim$50 lowest-frequency eigenmodes
include nearest-neighbor ions oscillating out of phase. Features at
$\omega_{r}$ and precise harmonics thereof (shaded in light green) are due to
spin-motion entanglement with in-plane degrees of freedom excited by the small
($\sim 10^{-3}F_{j}$) component of ODF perpendicular to $\hat{z}$.
Although the coherently driven, spin-dependent displacements ($\alpha_{jm}$)
are independent of the initial motional state (assuming Lamb-Dicke confinement
Britton et al. (2012)), the spin-motion entanglement signal in Fig. 2(b)
sensitively depends on this initial state. This can be qualitatively
understood in terms of the spatial structure of a harmonic oscillator Fock
state, $|n_{m}\rangle$, of mode $m$. A state $|n_{m}\rangle$ exhibits $n$
wavefunction nodes and therefore, as $n$ increases, a fixed spin-dependent
displacement results in less wavefunction overlap between different spin
components due to the increasing spatial frequency of $|n_{m}\rangle$
wavefunctions. This leads to larger decoherence and greater displacement
sensitivity as the average mode occupation, $\bar{n}_{m}$, is increased for a
given mode. We fit the experimental measurements in Fig. 2(b) using theory
that attributes a thermal state of motion to each mode $m$ characterized by
mode occupation $\bar{n}_{m}\sim k_{B}T_{m}(\hslash\omega_{m})^{-1}$ and
temperature $T_{m}$. Neglecting spin-spin correlation contributions, we find
the probability $P^{(j)}_{\uparrow}$ of detecting ion $j$ in state
$|\\!\\!\uparrow\rangle$ at the end of the pulse sequence to be EPA
$P_{\uparrow}^{(j)}=\\!\\!\frac{1}{2}\left[1-e^{-2\Gamma\tau}\exp{\left(-2\sum_{m}|\alpha_{jm}|^{2}(2\bar{n}_{m}+1)\right)}\right].$
(4)
Here $\Gamma$ accounts for decoherence due to spontaneous emission induced by
the ODF lasers over the duration $2\tau$, and is responsible for the
background level of $P_{\uparrow}\sim 0.1$ observed in all experimental data
presented here Uys et al. (2010). The total detection probability
$P_{\uparrow}$ is obtained by averaging all $P_{\uparrow}^{(j)}$.
For interaction with the COM mode ($b_{j1}=\frac{1}{\sqrt{N}},\forall
j\in[1,N]$), $\alpha_{j1}$ is obtained from Eq. (3) through measurement of the
ODF laser intensities Britton et al. (2012) and trapped-ion number, while
$\Gamma$ is determined from decoherence observed with $\mu_{R}$ detuned far
from any modes. As such, the only parameter of Eq. (4) not measured directly
is $\bar{n}_{1}$, which is varied to fit experimental data as in Fig. 2(b),
where we obtain $\bar{n}_{1}=60\pm 13$ ($T_{1}=2.3\pm 0.5$ mK).
We note that a detectable phase-space displacement is obtained with a very
small amplitude of $|\alpha_{jm}|$. For example, in Fig. 2(b), the 20%
decrease in the Bloch vector at $\delta_{1}\tau/2\pi\backsimeq\pm 1.4$
corresponds to a spin-state-dependent excitation of the COM mode with a mean
excursion of $\sim$0.6 nm in each arm of the pulse sequence. This shift is
less than 0.2% of the wavefunction spread of a single ion in the planar array.
Our sensitivity to displacements improves with increasing mode temperature
provided that the ODF is adjusted to avoid full decoherence
($P_{\uparrow}=0.5$) at the detuning of interest.
Figure 3(a) shows the result of a sweep of $\mu_{R}$ over five transverse
modes and corresponding theory. The theoretical spectrum (offset for clarity)
is generated assuming $T_{1}=10$ mK and $T_{m>1}=0.4$ mK, with $T_{1}$
obtained from a fit. The large COM temperature of Fig. 3(a) is produced by
quickly switching off the $\hat{z}$-oriented Doppler cooling beam on a time
scale of $\sim$$2\pi\omega_{1}^{-1}$. In this case, sudden loss of radiation
pressure from the cooling light induces a COM oscillation amplitude of
$\sim$50 nm that we detect as an elevated $\bar{n}_{1}$. A more adiabatic
reduction of the cooling beam intensity yields $\bar{n}_{1}\sim 26$
($T_{1}\sim 1$ mK). For modes other than the COM, we must additionally
calculate the $b_{jm}$ values for the trap potentials and ion number in a
given experiment. For these modes, we find temperatures consistent with the
Doppler cooling limit of 0.43 mK.
To measure the full spectrum of transverse modes, we repeat the sequence of
Fig. 2(a) for $30\text{ kHz}\leq\mu_{R}/2\pi\leq 800\text{ kHz}$ with $\tau=1$
ms. With the exception of the COM mode, the frequencies of the remaining $N-1$
modes depend sensitively on our choice of crystal rotation frequency,
$\omega_{r}$ Weimer et al. (1994). Figures 3(b)-(c) show the result of these
experimental runs for $\omega_{r}/2\pi=43.2$ kHz and 44.7 kHz, respectively.
For this ion number of $345\pm 25$, the single-plane configuration is stable
over the range $42.2~{}\text{kHz}\lesssim\omega_{r}/2\pi\lesssim
45.2~{}\text{kHz}$. Histograms of calculated mode density versus
$\mu_{R}/2\pi$ are plotted below each experimental curve with an arbitrary
vertical scale and bin width of 10 kHz. The distribution of eigenfrequencies
narrows as $\omega_{r}$ is decreased; weaker radial confinement (see Eq. (2))
leads to lower ion densities and reduced screening of trap potentials, thereby
moving the frequency of the shortest-wavelength mode toward that of the COM.
This behavior is clearly visible in Figs. 3(b)-(c). Additionally, we find
quantitative agreement between the measured spectrum and that generated from
numerical calculation of the transverse eigenmodes under the given
experimental conditions, documenting coupling to both short- and long-
wavelength modes. The sharp features of Figs. 3(b)-(c) shaded in light green
reflect excitation of in-plane resonances at harmonics of $\omega_{r}$ due to
a very small component of the ODF ($\sim$$10^{-3}F_{j}$) along the ion plane.
These spectral features may be reduced through more careful alignment of
$\overrightarrow{\Delta k}$ to $\hat{z}$, but their strong response suggests
an elevated motional temperature perpendicular to $\hat{z}$.
In summary, we have used entanglement of spin and motional degrees of freedom
to map the full transverse mode spectrum of a mesoscopic 2D ion array. This
technique provides a tool for sensitively and accurately measuring the
temperature and displacement amplitude of individual drumhead modes,
facilitating identification of mode-specific heating mechanisms and the
resulting non-equilibrium energy distributions. Coherent, spin-dependent
excitation of transverse modes is the basis for engineering quantum spin-spin
interactions with trapped ions Friedenauer et al. (2008); Kim et al. (2010);
Zhu et al. (2006); Porras and Cirac (2004, 2006); Islam et al. (2011); Britton
et al. (2012); Kim et al. (2009), making mode characterization a critical
element of such experiments. Future work will include investigation of low-
frequency in-plane modes at frequencies smaller than $\omega_{r}$. A predicted
subset of these modes includes in-plane shearing motion whose restoring force
is due exclusively to strong correlations.
This work was supported by the DARPA-OLE program and NIST. B. C. Sawyer is
supported by a NRC fellowship funded by NIST. M. J. Biercuk and J. J.
Bollinger acknowledge partial support from the ARC Centre of Excellence for
Engineered Quantum Systems, CE110001013. A. C. Keith was supported by the NSF
under grant number DMR-1004268. J. K. Freericks was supported by the McDevitt
endowment bequest at Georgetown University. We thank D.H.E. Dubin, D. Porras,
K.-K. Ni, D. Slichter, and S. Manmana for comments on the manuscript. This
manuscript is a contribution of NIST and not subject to U.S. copyright.
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Supplementary Material:
Spectroscopy and Thermometry of Drumhead Modes in a Mesoscopic Trapped-Ion
Crystal using Entanglement
Figure 4: Sketch of ODF laser beam setup. a) The ODF laser beams lie in the
$y$-$z$ plane at angles $\pm\theta_{R}/2$ with respect to the $y$-axis. b)
View looking in the $-\hat{y}$ direction. The beams are linearly polarized but
with different polarization angles relative to vertical polarization.
## Optical Dipole Force Details
Figure 4 shows a simple sketch of the optical dipole force (ODF) laser beam
set-up. As discussed below, the frequency as well as the beam polarizations
were chosen to null the AC Stark shift from an individual beam and to produce
a state-dependent force which is equal in magnitude but opposite in sign for
the $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ qubit
states ($F_{\uparrow}=-F_{\downarrow}$ ). The off-resonant laser beam
frequency was detuned from the cycling transition
$\left(\left|\uparrow\right\rangle\rightarrow\left|P_{3/2},m_{J}=3/2\right\rangle\right)$
by $\Delta_{R}\simeq-63.8$ GHz. This gives detunings of $15.6$ GHz and 26.1
GHz respectively from the
$\left|\uparrow\right\rangle\rightarrow\left|P_{3/2},m_{J}=1/2\right\rangle$
and
$\left|\downarrow\right\rangle\rightarrow\left|P_{3/2},m_{J}=-1/2\right\rangle$
transitions. Laser beam waists were $w_{z}\simeq 100\>\mu$m in the vertical
(z-direction) and $w_{x}\simeq 1$ mm in the horizontal direction. Here we
define the waist as the distance from the center of the beam over which the
electric field intensity decreases by $1/e^{2}$ (i.e. $I(z)\sim
e^{-(z/w_{z})^{2}}\>$). With the small $2.4^{o}$ incident angle each beam
makes with respect to the plane of the crystal, this provided greater than 90%
uniform electric field intensity across ion crystal arrays with $N<250$.
We used linearly polarized laser beams. Let
$\begin{array}[]{ccc}\vec{E}_{U}\left(\vec{r},t\right)&=&\hat{\epsilon}_{U}E_{U}\cos\left(\vec{k}_{U}\cdot\vec{r}-\omega_{U}t\right)\\\
\vec{E}_{L}\left(\vec{r},t\right)&=&\hat{\epsilon}_{L}E_{L}\cos\left(\vec{k}_{L}\cdot\vec{r}-\omega_{L}t\right)\end{array}$
denote the electric fields of the upper and lower ODF beams. If $\phi_{p}$ is
the angle of the laser beam electric-field polarization with respect to
vertical polarization $\left(\hat{\epsilon}\cdot\hat{x}=0\right)$, then the AC
Stark shift of the qubit states when illuminated by a single beam can be
written
$\begin{array}[]{c}\Delta_{\uparrow,\,acss}=A_{\uparrow}\cos^{2}\left(\phi_{p}\right)+B_{\uparrow}\sin^{2}\left(\phi_{p}\right)\\\
\Delta_{\downarrow,\,acss}=A_{\downarrow}\cos^{2}\left(\phi_{p}\right)+B_{\downarrow}\sin^{2}\left(\phi_{p}\right)\end{array}$
where $A_{\uparrow}$($A_{\downarrow}$) is the Stark shift of the
$\left|\uparrow\right\rangle$($\left|\downarrow\right\rangle$) state for a
$\pi$-polarized beam ($\hat{\epsilon}$ parallel to the $\hat{z}$-axis) and
$B_{\uparrow}$($B_{\downarrow}$) is the Stark shift of the
$\left|\uparrow\right\rangle$($\left|\downarrow\right\rangle$) state for a
$\sigma$-polarized beam ($\hat{\epsilon}$ perpendicular to the
$\hat{z}$-axis). (Here we neglect the small $\sigma$ polarization
($\propto\sin(2.4^{o})$) that exists when $\phi_{p}=0$.) The Stark shift of
the qubit transition is
$\Delta_{acss}=\left(A_{\uparrow}-A_{\downarrow}\right)\cos^{2}\left(\phi_{p}\right)+\left(B_{\uparrow}-B_{\downarrow}\right)\sin^{2}\left(\phi_{p}\right)\>.$
(5)
If $A_{\uparrow}-A_{\downarrow}$ and $B_{\uparrow}-B_{\downarrow}$ have
opposite signs, there is an angle which makes $\Delta_{acss}=0$. For a laser
detuning of $\Delta_{R}=-63.8$ GHz, $\Delta_{acss}=0$ at $\phi_{p}\simeq\pm
65{}^{o}$.
With $\Delta_{acss}=0$ for each ODF laser beam, we exploit the freedom to
choose their polarization in order to obtain a state-dependent force.
Specifically we choose $\vec{E}_{U}$ to have a polarization given by
$\phi_{p,u}=65{}^{o}$ and $\vec{E}_{L}$ to have a polarization given by
$\phi_{p,l}=-65{}^{o}$. In this case the interference term in the expression
for the electric field intensity $\left(\vec{E}_{U}+\vec{E}_{L}\right)^{2}$
produces a polarization gradient which results in spatially dependent AC Stark
shifts
$\begin{array}[]{c}\left(A_{\uparrow}\cos^{2}\left(\phi_{p}\right)-B_{\uparrow}\sin^{2}\left(\phi_{p}\right)\right)2\sin\left(\delta
k\cdot z+\mu_{R}t\right)\\\
\left(A_{\downarrow}\cos^{2}\left(\phi_{p}\right)-B_{\downarrow}\sin^{2}\left(\phi_{p}\right)\right)2\sin\left(\delta
k\cdot z+\mu_{R}t\right)\end{array}$
for the qubit levels. Here $\delta
k\equiv\left|\vec{k}_{U}-\vec{k}_{L}\right|=2k\sin\left(\frac{\theta_{R}}{2}\right)$
is the wave vector difference between the two ODF laser beams,
$\mu_{R}=\omega_{U}-\omega_{L}$ is the ODF beat note, and
$\phi_{p}=\left|\phi_{p,u}\right|=\left|\phi_{p,l}\right|$. The spatially
dependent AC Stark shift produces a state-dependent force
$F_{\uparrow,\downarrow}(z,t)=F_{\uparrow,\downarrow}\cos\left(\delta k\cdot
z-\mu_{R}t\right)$ where
$\begin{array}[]{c}F_{\uparrow}=2\,\delta
k\left(A_{\uparrow}\cos^{2}\left(\phi_{p}\right)-B_{\uparrow}\sin^{2}\left(\phi_{p}\right)\right)\\\
F_{\downarrow}=2\,\delta
k\left(A_{\downarrow}\cos^{2}\left(\phi_{p}\right)-B_{\downarrow}\sin^{2}\left(\phi_{p}\right)\right)\end{array}\>.$
In general $F_{\uparrow}\neq-F_{\downarrow}.$ We operate at $\Delta_{R}=-63.8$
GHz where for $\Delta_{acss}=0$ we also obtain
$F_{\uparrow}=-F_{\downarrow}\equiv F$.
For a given $\phi_{p,u}\,$, $\phi_{p,l}\,$, and $\Delta_{R}$ we use straight
forward atomic physics along with well known values for the energy levels and
matrix elements of 9Be+ to calculate $F$ as a function of the electric field
intensity
$I_{R}=\frac{c\epsilon_{o}}{2}\left|E_{L}\right|^{2}=\frac{c\epsilon_{o}}{2}\left|E_{U}\right|^{2}$
at the center of the laser beams. For $\theta_{R}=4.8^{o}$ and $I_{R}=1\>$
W/cm2 , $F=1.5\times 10^{-23}$ N.
## Wave Front Alignment
Figure 5: Sketch of the 1D optical lattice wave fronts (red lines) generated
by the ODF laser beams. These wave fronts need to be aligned with with the ion
planar array (represented by the blue dots). Here
$\lambda_{R}=2\pi/|\overrightarrow{\Delta k}|\approx 3.7\,\mu\mbox{m}$ and
$\theta_{err}$ denotes the angle of misalignment. $d_{0}\sim 400\>\mu$m is the
typical array diameter for $N\sim 200$ ions. With the wave front alignment
technique discussed in the text we obtain $\theta_{\mbox{err}}<0.05^{\circ}$.
Figure 6: Top-view image of the spatially inhomogeneous fluorescence from a
single ion plane produced by the AC Stark from a static ($\mu_{R}=0)$ optical
dipole force lattice with misaligned wave fronts. Dark bands are regions of
high standing wave electric field intensity (parallel to the dashed yellow
line). The bright horizontal feature bisecting the center of the image is
fluorescence from the weak Doppler laser cooling beam directed perpendicular
to the magnetic field. The image was obtained by subtracting a background
image with the ODF beams off.
The ODF laser beams produce a 1D optical lattice characterized by the
effective wave vector $\delta\vec{k}$ and beat note $\mu_{R}$. In the previous
section we assumed that $\delta\vec{k}\parallel\hat{z}$, or equivalently that
the wave fronts of the lattice were aligned perpendicular to the
$\hat{z}$-axis (magnetic field axis). If the wave fronts are not normal to the
$\hat{z}$-axis as sketched in Fig. 5, then the time dependence of the optical
dipole force seen by an ion in the rotating frame depends on the $(x,y)$
position of the ion. This complicates the interaction generated by the optical
dipole force and is avoided by careful alignment.
We used top-view images (images of the ion resonance fluorescence scattered
along the magnetic field) from a single plane to measure a misalignment of the
ODF wave fronts. For this measurement we set $\mu_{R}=0$ (stationary 1D
lattice) and detune the frequency of the ODF laser beams approximately 0.5 GHz
below the
$\left|\uparrow\right\rangle\rightarrow\left|{}^{2}P_{3/2}\>m_{J}=+3/2\right\rangle$
Doppler cooling transition. This small detuning generates sufficiently large
AC Stark shifts on the cooling transition to measurably change the ion scatter
rate from the Doppler cooling laser. With the Doppler cooling laser on and the
ODF beams turned off we observe a spatially uniform, time-averaged image of a
rotating planar crystal. With the ODF beams on, ions located in regions of
high electric field intensity at the anti-nodes of the optical lattice are
Stark shifted out of resonance with the Doppler cooling laser. This is what
produced the dark bands in the top-view image shown in Fig. 6. From images
like this we determine how to move the ODF beams to align the wave fronts
normal to $\hat{z}$. Improved alignment is indicated by a longer wavelength
fringe pattern. With this technique we have aligned the ODF wave fronts with
the planar array to better than $\theta_{err}\lesssim 0.05^{o}$.
Images like that shown in Fig. 6 were typically obtained with 1 s integration.
This means the imprint of the 1D lattice on the planar arrays was stable
during the integration time and indicates a phase stability of our 1D lattice
of better than 1s. We note that direct fluorescence imaging of the 1D lattice,
for example by tuning the ODF laser resonant with the Doppler cooling
transition, is not viable. Even at low powers, resonantly scattered photons
across the large horizontal waist of the ODF beams apply a large torque,
causing the rotation frequency and radius of the array to rapidly change,
typically driving the ions into very large radial orbits.
## Spin-Motion Entanglement Produced by the Spin-Dependent Optical Dipole
Force
With the wave vector $\delta\vec{k}$ of the 1D optical lattice aligned
parallel to $\hat{z}$, the optical dipole force generated by the lattice is
independent of the ion position and can be written
$F_{\uparrow}(t)=-F_{\downarrow}(t)\equiv F\cos\left(\mu_{R}t\right)$ (6)
where $\mu_{R}$ is the frequency difference between the ODF laser beams. More
generally we allow for the possibility that the ODF laser intensity could be
diferent for each ion, resulting in a different spin-dependent force $F_{j}$
for each ion $j$,
$F_{j\uparrow}(t)=-F_{j\downarrow}(t)\equiv F_{j}\cos\left(\mu_{R}t\right)\>.$
(7)
In the experimental set-up, the variation in $F_{j}$ is less than 20%. The ODF
interaction with the ion spins can be written as
$H_{ODF}=-\sum_{j=1}^{N}F_{j}\cos\left(\mu_{R}t\right)\hat{z}_{j}\hat{\sigma}_{j}^{z}\>.$
(8)
Here $\hat{z}_{j}$ is the axial position operator for the $j^{th}$ ion, which
can be written in terms of the axial normal modes
$\left(\vec{b}_{m},\omega_{m}\right)$ of the planar array,
$\hat{z}_{j}=\sum_{m=1}^{N}b_{jm}\sqrt{\frac{\hslash}{2M\omega_{m}}}\left(\hat{a}_{m}e^{-i\omega_{m}t}+\hat{a}_{m}^{\dagger}e^{i\omega_{m}t}\right)\>.$
(9)
The eigenvectors are normalized so that
$\sum_{m}\left|b_{jm}\right|^{2}=\sum_{j}\left|b_{jm}\right|^{2}=1$. Both the
eigenvectors $\vec{b}_{m}$ and eigenfrequencies $\omega_{m}$ are calculated by
solving for the ion equilibrium positions and diagonalizing the stiffness
matrix obtained by Taylor expansion of the potential about the ion equilibrium
positions Britton et al. (2011).
The Hamiltonian $H_{ODF}$ of Eq. (8) is time dependent. The evolution operator
for $H_{ODF}$ is obtained from a second order expansion of the Magnus formula
Zhu and Wang (2003); Kim et al. (2009)
$\hat{U}_{ODF}\left(t\right)=\exp\left[\frac{-i}{\hslash}\int_{0}^{t}H_{ODF}(t^{\prime})dt^{\prime}-\frac{1}{2\hslash^{2}}\int_{0}^{t}dt_{2}\int_{0}^{t_{2}}\left[H_{ODF}\left(t_{2}\right),H_{ODF}\left(t_{1}\right)\right]dt_{1}\right]\>.$
(10)
Higher order terms do not contribute as the commutator
$\left[H_{ODF}(t_{2}),H_{ODF}(t_{1})\right]$ commutes with
$H_{ODF}(t^{\prime})$. Following the discussion of Ref. Kim et al. (2009),
$U_{ODF}(t)$ can be written
$\begin{array}[]{ccc}\hat{U}_{ODF}(t)&=&\exp\left[\sum_{j}\left(\sum_{m}\left(\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m}\right)\hat{\sigma}_{j}^{z}\right)+i\sum_{j,k}J_{j,k}(t)\hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}\right]\\\
&=&\exp\left[\sum_{j}\left(\sum_{m}\left(\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m}\right)\hat{\sigma}_{j}^{z}\right)\right]\cdot\exp\left[i\sum_{j,k}J_{j,k}(t)\hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}\right]\\\
&\equiv&\hat{U}_{SM}(t)\cdot\hat{U}_{SS}(t)\end{array}\,.$ (11)
The first term $U_{SM}(t)$ describes spin-dependent displacements
$\alpha_{jm}(t)$ of the normal modes $m$ where, for the $\cos(\mu_{R}t)$ time
dependence of the interaction in Eq. (8),
$\alpha_{jm}(t)=\frac{F_{j}b_{jm}z_{0m}}{\hslash\left(\mu_{R}^{2}-\omega_{m}^{2}\right)}\left[\omega_{m}-e^{i\omega_{m}t}\left(\omega_{m}\cos(\mu_{R}t)-i\mu_{R}\sin(\mu_{R}t)\right)\right]\>.$
(12)
Here $z_{0m}=\sqrt{\hslash/(2M\omega_{m})}$ . The second term
$\hat{U}_{SS}(t)$ describes an effective spin-spin interaction where the
pairwise coupling $J_{j,k}(t)$ is given by
$J_{j,k}(t)=\frac{F_{j}F_{k}}{2\hslash^{2}}\sum_{m}\frac{b_{jm}b_{km}z_{0m}^{2}}{\mu_{R}^{2}-\omega_{m}^{2}}\left\\{\frac{\omega_{m}\sin(\mu_{R}-\omega_{m})t}{\mu_{R}-\omega_{m}}+\frac{\omega_{m}\sin(\mu_{R}+\omega_{m})t}{\mu_{R}+\omega_{m}}-\frac{\omega_{m}\sin(2\mu_{R}t)}{2\mu_{R}}-\omega_{m}t\right\\}\,.$
(13)
For now we assume $\hat{U}_{SS}(t)$ can be neglected. We will discuss the
validity of this assumption at the end of this section.
The interaction
$\hat{U}_{SM}(t)=\exp\left[\sum_{j}\left(\sum_{m}\left(\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m}\right)\hat{\sigma}_{j}^{z}\right)\right]$
generates spin-motion entanglement that is the subject of this study. The
commutator
$\begin{array}[]{ccc}\left[\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m},\alpha_{km}(t)\hat{a}_{m}^{\dagger}-\alpha_{km}^{\ast}(t)\hat{a}_{m}\right]&=&\alpha_{jm}(t)\alpha_{km}^{\ast}(t)-\alpha_{jm}^{\ast}(t)\alpha_{km}(t)\\\
&=&0\end{array}$
because $\alpha_{jm}(t)\alpha_{km}^{\ast}(t)$ is real. Therefore we can write
$\hat{U}_{SM}(t)$ as a product of individual spin displacements
$\hat{U}_{SM}(t)=\prod_{j,m}\exp\left(\left(\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m}\right)\hat{\sigma}_{j}^{z}\right)\>,$
(14)
which is Eq. (4) of the Letter. By neglecting the spin-spin entanglement
($\hat{U}_{SS}(t)$) we can independently calculate the evolution of each spin
$j$.
Figure 7: Pulse sequences described in this supplemental material. a) Ramsey
pulse sequence consisting of two $\pi/2$ rotations with an intermediate arm of
duration $\tau$ during which the ODF is applied. b) The spin echo sequence
repeated from Fig. 2(a) of the Letter which consists of two arms of duration
$\tau$.
We now calculate the spin motion entanglement generated by $\hat{U}_{SM}(t)$
during the free precession period of a Ramsey sequence shown in Fig. 7(a). The
calculation for the spin-echo sequence of Fig. 7(b) used in the experiments is
identical except for a more complicated expression for the
$\alpha_{jm}(t)^{\prime}$s (see next section). Each spin $j$ is prepared in
state $\left|\uparrow\right\rangle$ at the start of the sequence. If an ODF is
not applied during the free precession period, the spin is rotated to the dark
$\left|\downarrow\right\rangle$ state by the final $\pi/2$ pulse of the
sequence. With the application of a spin-dependent ODF, in general the spin is
entangled with the motion at the end of the Ramsey sequence. We detect this
spin-motion entanglement by measuring the probability of finding spin $j$ in
the $\left|\uparrow\right\rangle$ state. Let
$\hat{U}_{SM}^{(j)}(t)=\exp\left(\sum_{m}\left(\alpha_{jm}(t)\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}(t)\hat{a}_{m}\right)\hat{\sigma}_{j}^{z}\right)$
denote the evolution of spin $j$ by the spin-dependent ODF. By re-writing
$\hat{U}_{SM}^{(j)}(t)=\cosh\left(\sum_{m}\left(\alpha_{jm}\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}\hat{a}_{m}\right)\right)+\sinh\left(\sum_{m}\left(\alpha_{jm}\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}\hat{a}_{m}\right)\right)\hat{\sigma}_{j}^{z}$
we calculate
$P_{\uparrow,SM}^{(j)}=\left\langle\left(\sinh\left(\sum_{m}\left(\alpha_{jm}\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}\hat{a}_{m}\right)\right)\right)^{\dagger}\sinh\left(\sum_{m}\left(\alpha_{jm}\hat{a}_{m}^{\dagger}-\alpha_{jm}^{\ast}\hat{a}_{m}\right)\right)\right\rangle_{th}$
(15)
where $P_{\uparrow,SM}^{(j)}$ denotes the probability of measuring the
$\left|\uparrow\right\rangle$ state for spin $j$ produced by the
$\hat{U}_{SM}(t)$ interaction, and $\left\langle\;\right\rangle_{th}$ denotes
an expectation value averaged over a thermal (Maxwell-Boltzmann) distribution
of modes. We evaluate Eq. (15) by writing the $\sinh$ functions in exponential
form. It is then necessary to evaluate expressions of the form $\left\langle
e^{\hat{A}}e^{\hat{B}}\right\rangle_{th}$ where $\hat{A}$ and $\hat{B}$ are
operators which are linear in the raising and lowering operators
$\hat{a}_{m}^{\dagger}$ and $\hat{a}_{m}$. In this case we can make use of the
result Ashcroft and Mermin (1976)
$\left\langle
e^{\hat{A}}e^{\hat{B}}\right\rangle_{th}=e^{(1/2)\left\langle\hat{A}^{2}+2\hat{A}\hat{B}+\hat{B}^{2}\right\rangle_{th}}$
to obtain
$P_{\uparrow,SM}^{(j)}=\frac{1}{2}\left[1-\exp\left(-2\sum_{m}\left|\alpha_{jm}(t)\right|^{2}\left(2\bar{n}_{m}+1\right)\right)\right]\>.$
(16)
Here $\bar{n}_{m}\simeq k_{B}T_{m}/(\hslash\omega_{m})$ is the mean occupation
number of a Maxwell-Boltzmann distribution characterized by temperature
$T_{m}$. We measure the probability of detecting $\left|\uparrow\right\rangle$
averaged over all the ions $\left(\sum_{j}P_{\uparrow}^{(j)}\right)/N$.
The simple result of Eq. (16) was obtained under the assumption that we could
neglect $\hat{U}_{SS}(t)$ in Eq. (11). In general $\hat{U}_{SS}(t)$ will
contribute to the measured $P_{\uparrow}^{(j)}$. This can be straight
forwardly estimated when $\mu_{R}$ is tuned close to the COM mode
$\omega_{1}$. In this case the resulting pair-wise interaction coefficients
are identical for all ion pairs $J_{j,k}(t)\simeq J(t)$ with
$J(t)=\frac{F^{2}}{2\hslash^{2}}\cdot\frac{z_{01}^{2}}{N\left(\mu_{R}^{2}-\omega_{1}^{2}\right)}\left\\{\frac{\omega_{1}\sin(\mu_{R}-\omega_{1})t}{\mu_{R}-\omega_{1}}+\frac{\omega_{1}\sin(\mu_{R}+\omega_{1})t}{\mu_{R}+\omega_{1}}-\frac{\omega_{1}sin(2\mu_{R}t)}{2\mu_{R}}-\omega_{1}t\right\\}\>.$
For small detunings $\left|\mu_{R}-\omega_{1}\right|\ll\omega_{1}$, $J(t)$ is
approximately bounded by $\left|J(t)\right|\lesssim J\cdot t$ where
$J=\frac{F^{2}}{2\hslash^{2}}\cdot\frac{z_{01}^{2}}{N\left(\mu_{R}^{2}-\omega_{1}^{2}\right)}\omega_{1}.$
The fully connected, uniform Ising interaction
$\exp\left[iJ\left(\sum_{j,k}\hat{\sigma}_{j}^{z}\hat{\sigma}_{k}^{z}\right)t\right]$
obtained by coupling through the COM mode is identical to the single-axis
twisting interaction analyzed by Kitagawa and Ueda Kitagawa and Ueda (1993).
We use the expressions given in Ref. Kitagawa and Ueda (1993) to calculate
$P_{\uparrow,SS}^{(j)}$, the probability of measuring spin $j$ in the
$\left|\uparrow\right\rangle$ state at the end of the Ramsey sequence due to
the $\hat{U}_{SS}(t)$ interaction,
$P_{\uparrow,SS}^{(j)}\simeq\frac{1}{2}\left[N8(Jt)^{2}\right]\>.$ (17)
This expression is valid for short times $t$ where $P_{\uparrow,SS}^{(j)}$ is
small.
We obtain strong spin-motion entanglement for small detunings
$\left|\mu_{R}-\omega_{1}\right|\ll\omega_{1}$. The magnitude of the
coherently driven amplitude $\alpha_{j,m=1}(t)$ in the expression for
$\hat{U}_{SM}(t)$ (Eq. (14)) and $P_{\uparrow,SM}^{(j)}$ (Eq. (16)) is
maximized for a detuning $\left|\mu-\omega_{1}\right|\simeq\pi/t$ where
$\left|\alpha_{j,1}\right|_{max}=\left|\alpha_{j,1}\left(t\simeq\frac{\pi}{\left|\mu_{R}-\omega_{1}\right|}\right)\right|\simeq\frac{Fz_{01}}{\hslash\sqrt{N}\left|\mu_{R}^{2}-\omega_{1}^{2}\right|}2\omega_{1}\>.$
The above expression neglects terms of order
$\left(\mu_{R}-\omega_{M}\right)/\omega_{M}$. Inserting
$\left|\alpha_{j,1}\right|_{max}$ into Eq. (16) and assuming the exponent is
small gives
$P_{\uparrow,SM}^{(j)}\simeq\frac{1}{2}\left[2\left|\alpha_{j,1}\right|_{max}^{2}\left(2\bar{n}_{1}+1)\right)\right]\>.$
(18)
We compare $P_{\uparrow,SS}^{(j)}$ (Eq. (17)) with $P_{\uparrow,SM}^{(j)}$
(Eq. (18)),
$\frac{P_{\uparrow,SS}^{(j)}}{P_{\uparrow,SM}^{(j)}}\simeq\frac{N\cdot
8\left(Jt\right)^{2}}{2\left|\alpha_{j,1}\right|_{max}^{2}(2\bar{n}_{1}+1)}\simeq\frac{F^{2}}{4\hslash^{2}}\cdot\frac{z_{01}^{2}}{2\bar{n}_{1}+1}t^{2}\>.$
(19)
For the work reported here $F\sim 10^{-23}$ N,
$z_{01}=\sqrt{\hslash/\left(2M\omega_{1}\right)}\sim 30$ nm, and
$\bar{n}_{1}\sim 10$ (Doppler cooling limit). For a typical interaction time
$t\lesssim 10^{-3}$ s we calculate
$P_{\uparrow,SS}^{(j)}/P_{\uparrow,SM}^{(j)}\lesssim 0.1$. Therefore for small
detunings satisfying
$\left|\mu_{R}-\omega_{1}\right|\lesssim(2\pi)/t\ll\omega_{1}$ we expect the
spin-motion entanglement signature generated by $\hat{U}_{SM}(t)$ to dominate
contributions due to $\hat{U}_{SS}(t)$. We note that the spin-motion
entanglement signature $\left(P_{\uparrow,SM}^{(j)}\right)$ decreases with
temperature. For ground state cooling it may not be possible to neglect
$\hat{U}_{SS}(t)$.
We do not estimate $P_{\uparrow,SS}^{(j)}$ for $\mu_{R}$ tuned close to modes
$\omega_{M}$ other than the COM mode $\omega_{1}$. Therefore we do not know if
it is a good approximation to neglect $\hat{U}_{SS}(t)$ when resonantly
coupling to non-COM modes. However, we experimentally observe that neglecting
$\hat{U}_{SS}(t)$ gives a good description of our experimental measurements
for $\mu_{R}$ tuned close to the tilt ($\omega_{2}$ and $\omega_{3}$) and the
next lower frequency modes ($\omega_{4}$ and $\omega_{5}$).
## Spin Echo Sequence with Decoherence
To calculate $\alpha_{jm}(t)$ for the full spin echo sequence used in the
experiment (see Fig. 7(b)), we must account for the accumulated phase
difference between the ODF drive and oscillating ion cloud over the first arm
and intermediate microwave $\pi$-pulse of combined duration $(\tau+t_{\pi})$.
This requires derivation of $\alpha_{jm}(t)$ for an ODF interaction with an
arbitrary phase offset, $\phi$, given by the more general
$H_{ODF}(\phi)=-\sum_{j=1}^{N}F_{j}\cos\left(\mu_{R}t+\phi\right)\hat{z}_{j}\hat{\sigma}_{j}^{z},$
(20)
where $\phi=(\tau+t_{\pi})(\mu_{R}-\omega_{m})=(\tau+t_{\pi})\delta_{m}$.
Following the previous derivation of $\alpha_{jm}(t)$ for $\phi=0$ (Eq. (12)),
we obtain
$\alpha_{jm}(t,\phi)=\frac{F_{j}b_{jm}z_{0m}}{\hslash\left(\mu_{R}^{2}-\omega_{m}^{2}\right)}\left[\omega_{m}\cos(\phi)-i\mu_{R}\sin(\phi)-e^{i\omega_{m}t}\left\\{\omega_{m}\cos(\mu_{R}t+\phi)-i\mu_{R}\sin(\mu_{R}t+\phi)\right\\}\right]\>.$
(21)
We will now define a new $\alpha^{\text{SE}}_{jm}$ that may be substituted for
$\alpha_{jm}$ in Eq. (14) to calculate $P_{\uparrow}$ for the full spin echo
sequence exhibiting arm durations of $\tau$:
$\alpha^{\text{SE}}_{jm}=\alpha_{jm}(\tau,\phi=0)-\alpha_{jm}(\tau,\phi),$
(22)
where the above expression is given explicitly in Eq. (6) of the Letter.
To justify implementation of Eq. (22), it is useful to calculate
$P_{\uparrow}$ for a single spin undergoing both the Ramsey and spin echo
sequences. To simplify notation, we define the displacement operator
$\hat{D}(\alpha_{jm})=\exp(\alpha_{jm}\hat{a}^{{\dagger}}_{m}-\alpha^{\ast}_{jm}\hat{a}_{m})$
which is applied separately to
$|\\!\\!\uparrow_{j}\rangle\otimes|\psi_{m}\rangle$ and
$|\\!\\!\downarrow_{j}\rangle\otimes|\psi_{m}\rangle$, where
$|\psi_{m}\rangle$ is an arbitrary motional state of mode $m$. Assuming the
state is initialized to $|\\!\\!\uparrow_{j}\rangle\otimes|\psi_{m}\rangle$,
we calculate the result of the Ramsey sequence,
$P_{\uparrow}^{(j)\text{Ramsey}}$, to be
$P_{\uparrow}^{(j)\text{Ramsey}}=\frac{1}{4}\langle\psi_{m}|\left\\{\hat{D}^{{\dagger}}\left(\alpha_{jm}(\tau,\phi)\right)-\hat{D}^{{\dagger}}\left(-\alpha_{jm}(\tau,\phi)\right)\right\\}\left\\{h.c.\right\\}|\psi_{m}\rangle,$
(23)
where $\left\\{h.c.\right\\}$ denotes the Hermitian conjugate of the first
bracketed expression. Here the arbitrary phase $\phi$ has no physical
significance since its value is common to all displacements, and we have once
again made the assumption that $F_{j\uparrow}=-F_{j\downarrow}$. However, the
spin echo result given by $P^{(j)\text{SE}}_{\uparrow}$ is
$\displaystyle P^{(j)\text{SE}}_{\uparrow}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\langle\psi_{m}|\left\\{\hat{D}^{{\dagger}}\left(-\alpha_{jm}(\tau,\phi)\right)\hat{D}^{{\dagger}}\left(\alpha_{jm}(\tau,0)\right)-\hat{D}^{{\dagger}}\left(\alpha_{jm}(\tau,\phi)\right)\hat{D}^{{\dagger}}\left(-\alpha_{jm}(\tau,0)\right)\right\\}\left\\{h.c.\right\\}|\psi_{m}\rangle$
(24) $\displaystyle=$
$\displaystyle\frac{1}{4}\langle\psi_{m}|\left\\{\hat{D}^{{\dagger}}\left(\alpha^{\text{SE}}_{jm}\right)-\hat{D}^{{\dagger}}\left(-\alpha^{\text{SE}}_{jm}\right)\right\\}\left\\{h.c.\right\\}|\psi_{m}\rangle.$
(25)
We obtain Eq. (26) from Eq. (25) using the multiplicative properties of
$\hat{D}$ and neglecting overall phase factors that leave
$P^{(j)\text{SE}}_{\uparrow}$ unchanged. Note that Eq. (26) is identical to
Eq. (24) after an appropriate redefinition of $\alpha_{jm}$.
Finally, the derivation of Eq. (16) neglected the effects of spontaneous
emission from the ODF laser beams. Decoherence of the Bloch vector due to
spontaneous emission from off-resonant light is well studied in our system Uys
et al. (2010). The qubit levels are closed under spontaneous light scattering;
that is, spontaneous light scattering does not optically pump an ion to a
different ground state level outside of the two qubit levels. In the presence
of off-resonant laser light, the decrease in the Bloch vector due to
spontaneous scattering during the arms of a spin-echo sequence is
$P_{\uparrow,spon}^{(j)}=\frac{1}{2}\left[1-\exp\left(-\Gamma\cdot
2\tau\right)\right]\>.$
Here $\Gamma\equiv\left(\Gamma_{Ram}+\Gamma_{el}\right)/2$ has contributions
from both Raman scattering and elastic Rayleigh scattering that can be
calculated from the laser beam parameters. With the spin echo sequence, we
account for spontaneous emission by modifying Eq. (16) as follows
$P_{\uparrow,SM}^{(j)\text{SE}}=\frac{1}{2}\left[1-e^{-\Gamma
2\tau}\exp\left(-2\sum_{m}\left|\alpha_{jm}^{\text{SE}}\right|^{2}\left(2\bar{n}_{m}+1\right)\right)\right],$
where $\tau$ is the length of time of a single arm of the spin-echo sequence.
## References
* Britton et al. (2011) J. W. Britton et al., Nature 484, 489 (2012).
* Zhu and Wang (2003) S.-L. Zhu and Z. D. Wang, Phys. Rev. Lett. 91, 187902 (2003).
* Kim et al. (2009) K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 103, 120502 (2009).
* Ashcroft and Mermin (1976) N. W. Ashcroft and N. D. Mermin, _Solid State Physics_ (Saunders College, Philadelphia, 1976).
* Kitagawa and Ueda (1993) M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
* Uys et al. (2010) H. Uys, M. J. Biercuk, A. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, and J. J. Bollinger, Phys. Rev. Lett. 105, 200401 (2010).
|
arxiv-papers
| 2012-01-21T00:33:01 |
2024-09-04T02:49:26.560180
|
{
"license": "Public Domain",
"authors": "Brian C. Sawyer, Joseph W. Britton, Adam C. Keith, C.-C. Joseph Wang,\n James K. Freericks, Hermann Uys, Michael J. Biercuk, and John J. Bollinger",
"submitter": "Brian Sawyer",
"url": "https://arxiv.org/abs/1201.4415"
}
|
1201.4431
|
# Minimum Variability Time Scales of Long and Short GRBs
G. A. MacLachlan1, A. Shenoy1, E. Sonbas2,3, K. S. Dhuga1, B. E. Cobb1, T. N.
Ukwatta1,3,4, D. C. Morris1,3,5, A. Eskandarian1, L. C. Maximon1, and W. C.
Parke1
1Department of Physics, The George Washington University, Washington, D.C.
20052, USA.
2University of Adiyaman, Department of Physics, 02040, Adiyaman, Turkey.
3NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA.
4Department of Physics and Astronomy, Michigan State University, East Lansing,
MI 48824, USA.
5Department of Physics, University of Virgin Islands, Virgin Islands, USA.
E-mail: maclach@gwu.edu (GAM)
###### Abstract
We have investigated the time variations in the light curves from a sample of
long and short Fermi/GBM Gamma ray bursts (GRBs) using an impartial wavelet
analysis. The results indicate that in the source frame, that the variability
time scales for long bursts differ from that for short bursts, that
variabilities on the order of a few milliseconds are not uncommon, and that an
intriguing relationship exists between the minimum variability time and the
burst duration.
###### keywords:
Gamma-ray bursts
## 1 Introduction
The prompt emission from Gamma-ray Bursts (GRBs) shows complex time profiles
that have eluded a generally accepted explanation. Fenimore & Ramirez-Ruiz
(2000) reported a correlation between variability of GRBs and the peak
isotropic luminosity. The existence of the variability-luminosity correlation
suggests that the prompt emission light curves have embedded temporal
information related to the microphysics of GRBs. Several models have been
proposed to explain the observed temporal variability of GRB light curves.
Leading models such as the internal shock model (Kobayashi, Piran, and Sari,
1997) and the photospheric model (Ryde, 2004) link the rapid variability
directly to the activity of the central engine. Others invoke relativistic
outflow mechanisms to suggest that local turbulence amplified through Lorentz
boosting leads to causally disconnected regions which in turn act as
independent centers for the observed prompt emission. Within more recent
models, both Morsony et al. (2010) and Zhang & Yan (2011) argue that the
temporal variability may show two different scales depending on the physical
mechanisms generating the prompt emission.
In order to further our understanding of the prompt emission phase of GRBs and
to explicitly test some of the key ingredients in the various models, it is
clearly important to extract the variability for both short and long gamma-ray
bursts in a robust and unbiased manner. It is also clear that the chosen
methodology should not only be mathematically rigorous but also be
sufficiently flexible to apply to transient phenomena with multiple time
scales and a wide dynamic range. A wide dynamic range is naturally provided by
the bimodal separation of GRB duration occurring at $T_{90}=2$ sec as observed
by Kouveliotou et al. (1993) to distinguish between long and short duration
GRBs.
In this paper, we extract variability time scales for GRBs using a method
based on wavelets. The technique for such a temporal analysis is universal,
and has the advantage over Fourier analysis that transients and frequency
correlations can be more easily picked out in the data. Results presented
herein were compared with Bhat et al. (2012) who gave pulse parameters for
approximately 400 pulses obtained from 34 GRBs. It was shown by MacLachlan et
al. (2012) that the minimum variablility time scale tracks the rise times of
pulses very well for over three orders of magnitude in time scale. The
relation of minimum variability time scales to pulse parameters has been
extended to four orders of magnitude by Sonbas et al. (2012) who applied the
present technique to analyze X-ray flares.
The time scales being investigated here have power densities very near to that
of the noise in the data which makes extracting these time scales nontrivial.
A somewhat older but still interesting discussion of extracting signal in a
noisy environment can be found in Scargle (1982) and a more recent discussion
found in Kostelich & Schreiber (1993). The technique we offer is not
necessarily new but is different from previous published wavelet analyses
(Fritz & Bruch, 1998; Walker et al., 2000; Tamburini et al., 2009; Anzolin et
al., 2010)111Note that Fritz & Bruch (1998) analyzed optical data and
Tamburini et al. (2009); Anzolin et al. (2010) X-ray data from Cataclysmic
Variables. in that we apply a number of modifications suggested by various
authors (Addison, 2002; Coifman, 1995; Percival, 2000; Strang and Nguyen,
1997) as explained further in Sec 4.
The layout of the paper is as follows: the source of the data is described in
section 2; the main aspects of the wavelet methodology are outlined in section
3; in section 4 we provide the details of the data analysis; in section 5 we
present and discuss our main findings; finally, in section 6, we summarize our
conclusions.
## 2 Data
The Gamma-Ray Burst Monitor (GBM) on board Fermi observes GRBs in the energy
range 8 keV to 40 MeV. The GBM is composed of 12 thallium-activated sodium
iodide (NaI) scintillation detectors (12.7 cm in diameter by 1.27 cm thick)
that are sensitive to energies in the range of 8 keV to 1 MeV, and two bismuth
germanate (BGO) scintillation detectors (12.7 cm diameter by 12.7 cm thick)
with energy coverage between 200 keV and 40 MeV. The GBM detectors are
arranged in such a way that they provide a significant view of the sky (Meegan
et al., 2009).
In this work, we have extracted light curves for the GBM NaI detectors over
the entire energy range (8 keV - 1 MeV, also including the overflow beyond 1
MeV). Typically, the brightest three NaI detectors were chosen for the
extraction. Lightcurves for both long and short GRBs were extracted at a time
binning of 200 microseconds. The long GRBs were extracted over a duration
starting from 20 seconds before the trigger and up to about 50 seconds after
the $T_{90}$ for the burst without any background subtraction. For short GRBs,
durations were chosen to be 20 seconds before the trigger and 10 seconds after
the $T_{90}$. The $T_{90}$ durations were obtained from the Fermi GBM-Burst
Catalog (Paciesas et al., 2012).
Figure 1: GBM GRB080925775. Preburst portion of the light curve, used for
background removal, is shown in gray. The burst portion, from which a time
scale is extracted, is shown in black.
## 3 Methodology
We report on our model-independent statistical investigation of the
variability of Fermi/GBM long and short GRBs. We extract this information by
using a fast wavelet transformation to encode GRB light curves into a wavelet
representation and then compute a statistical measure of the variance of
wavelet coefficients over multiple time-scales.
### 3.1 Minimum Variability Time Scales
It is often the case when multiple processes are present that one process will
dominate the others at certain time scales but those same processes may
exchange dominance at other time scales. A wavelet technique is useful in
these situations because the variances of wavelet coefficients are sensitive
to whichever processes dominate the light curve at a given time scale.
Moreover, the technique can be used to classify those dominant processes as
well as provide a means to determine the characteristic time scale,
$\tau_{\beta}$, for which the processes exchange dominance. Determination of
$\tau_{\beta}$ helps in the development of theoretical models and the
understanding of observational data. Indeed, if there is a transition from a
time-scaling region to that of white noise then there is a smallest
variability time for the physical processes present.
### 3.2 Wavelet Transforms
Wavelet transformations have been shown to be a natural tool for multi-
resolution analysis of non-stationary time-series (Flandrin, 1989, 1992;
Mallat, 1989). Wavelet analysis is similar to Fourier analysis in many
respects but differs in that wavelet basis functions are well-localized,
_i.e._ have compact support, while Fourier basis functions are global. Compact
support means that outside some finite range the amplitude of wavelet basis
functions goes to zero or is otherwise negligibly small (Percival, 2000). In
principle, a wavelet expansion forms a faithful representation of the original
data, in that the basis set is orthonormal and complete.
#### 3.2.1 Discrete Dyadic Wavelet Transforms
Given the discrete nature of the data, we employ a discrete wavelet analysis.
The rescaled-translated nature of the wavelet basis functions make the wavelet
transform well-localized in both frequency and time, which results in an
insensitivity to background photon counts expressed by polynomial fits. The
level of insensitivity, formally known as the vanishing moment condition, can
be adjusted by the choice of wavelet basis function. By construction, the
discrete wavelet transform is a multi-resolution operation (Mallat, 1989).
Such wavelets, denoted $\psi_{j,k}(t)$, form a dyadic basis set, i.e. wavelets
in the set have variable widths and variable central time positions.
The wavelet analysis employed in this study, as with the fast Fourier
transform, begins with a light curve with $N$ elements,
$X_{i}=\\{X_{0}\mathellipsis X_{N-1}\\},$ (1)
where $N$ is an integer power of two. The light curve is convolved with a
scaling function, $\phi_{j,k}(t_{i})$, and wavelet function,
$\psi_{j,k}(t_{i})$ which are rescaled and translated versions of the original
scaling and wavelet functions $\phi(t_{i})=\phi_{0,0}$, and
$\psi(t_{i})=\psi_{0,0}$. Translation is indexed by $k$ and rescaling is
indexed by $j$. The rescaling and translation relation is given by
$\psi_{j,k}(t)=2^{-j/2}\psi(2^{-j}t-k).$ (2)
The precise forms of the scaling and wavelet functions are not unique. The
choices are made according to the features one wishes to exploit (Percival,
2000; Addison, 2002). The scaling function acts as a smoothing filter for the
input time-series and the wavelet function probes the time-series for detail
information at some time scale, $\Delta t$, which is twice that of the finest
binning of the data, $T_{\rm bin}$. In the analysis, the time scale is doubled
$\Delta t\rightarrow 2\Delta t$
and the transform is repeated until
$\Delta t=NT_{\rm bin}.$
In this analysis we choose the Haar (Addison, 2002) scaling/wavelet basis
because it has the smallest possible support, has one vanishing moment, and is
equivalent to the Allan variance (Howe & Percival, 1995), allowing for a
straightforward interpretation.
#### 3.2.2 The Haar Wavelet Basis
Convolving the light curve, $X$, with the scaling functions yields
approximation coefficients,
$a_{j,k}=\langle\phi_{j,k},X\rangle.$ (3)
Interrogating $X$ with the wavelet basis functions yields scale and position
dependent detail coefficients,
$d_{j,k}=\langle\psi_{j,k},X\rangle,$ (4)
It is interesting to point out that for the trivial $N=2$ case the Haar
wavelet transform and the Fourier transform are identical.
### 3.3 Logscale Diagrams and Scaling
Logscale diagrams are useful for identifying scaling and noise regions.
Construction of a logscale diagram for each GRB proceeds from the variance of
detail coefficients (Flandrin, 1992),
$\beta_{j}=\frac{1}{n_{j}}\sum_{k=0}^{n_{j}-1}|d_{j,k}|^{2},$ (5)
where the $n_{j}$ are the number of detail coefficients at a particular scale,
$j$. A plot of $\log_{2}$ variances versus scale, $j$, takes the general form
$\log_{2}\ \beta_{j}=\alpha j+{\rm constant},$ (6)
and is known as a logscale diagram. A linear regression is made of each
logscale diagram and the slope parameter, $\alpha$, (depicting a measure of
scaling) is estimated. White-noise processes appear in logscale diagrams as
flat regions while non-stationary processes appear as sloped regions with the
following condition on the slope parameter, $\alpha>1$ (Abry et al., 2003;
Percival, 2000; Flandrin, 1989).
## 4 Data Analysis
### 4.1 Background Subtraction
We now present a method for removing photometric background due to noise not
intrinsic to the GRB so that physical variability arising from the GRB remains
for further analysis. Background subtraction for a statistical analysis of
variability via wavelet transforms should proceed in the space variances as
opposed to a traditional flat or linear subtraction of counts. This owes to
the fact that Haar detail coefficients are insensitive to polynomial trends in
the signal up to first order. Subtraction of a flat or linear background from
a light curve is an operation under which the wavelet transform is invariant
(as are Fourier transforms) apart from the mean signal coefficient.
The GRB light curves show power at various variablity time scales. Most often,
there is a region of the logscale diagram (log-power verses log-varibility
time) with a single slope, indicating scaling in the power over those
variability-times, and a flat region at the shortest variability times,
indicating the presence of white-noise. Some of this white-noise may be
intrinsic to the GRB. Some may be attributed to instrumental noise and to
background emissions from sources not including the GRB in question. We
therefore express the variability of the burst, $\beta_{j}^{\rm burst}$, at
time scales $j$ as comprising of individual variances: a scaling component,
$\beta^{\rm scaling}$; an intrinsic noise component, $\beta^{\rm noise}$; and
a background component, $\beta^{\rm background}$. The variability of the burst
can then be described as a linear combination of the component variances so
long as the components have vanishing covariances. In this event we write,
$\beta_{j}^{\rm burst}=\beta_{j}^{\rm scaling}+\beta_{j}^{\rm
noise}+\beta_{j}^{\rm background}.$ (7)
The minimum variability time scale, $\tau_{\beta}$, is identified from a
logscale diagram by the octave, $j_{\rm{intersection}}$, of the intersection
of the flat intrinsic noise domain, $\beta_{j}^{\rm noise}$, with the sloped
scaling domain, $\beta_{j}^{\rm scaling}$,
$\tau_{\beta}\equiv T_{\rm bin}\times 2^{j_{\rm intersection}}.$ (8)
In practice, the octave at which the intersection occurs is determined by
equating the polynomial fits to the flat intrinsic noise domain and the sloped
scaling domain and solving for $j_{\rm intersection}$. The uncertainty in
$\tau_{\beta}$ is determined by propagating the uncertainty in the parameters
from the fits to the $\beta_{j}$ which in turn follow from a bootstrap
procedure described in Sec. 4.1.2 and Sec. 4.1.3. It is at this time scale,
$\tau_{\beta}$, that a structured physical process appears to give way to one
that is stochastic and unstructured. Clearly one seeks to remove
$\beta_{j}^{\rm background}$ from Eq. 7 to arrive at the cleanest possible
signal,
$\beta_{j}^{\rm burst}\rightarrow\beta_{j}^{\rm clean}\equiv\beta_{j}^{\rm
burst}-\beta_{j}^{\rm background}=\beta_{j}^{\rm scaling}+\beta_{j}^{\rm
noise}.$ (9)
In order to estimate the variance of the background during the burst, we will
assume that the variance obtained from a preburst portion of the light curve
can serve as an acceptable surrogate for the background variance. That is,
$\beta_{j}^{\rm preburst}\equiv\beta_{j}^{\rm background},$ (10)
and then the background is removed from the signal according to the relation,
$\log_{2}(\beta_{j}^{\rm clean})=\log_{2}(\beta_{j}^{\rm burst}-\beta_{j}^{\rm
preburst}).$ (11)
A simple algebraic manipulation of Eq. 11 gives a form,
$\log_{2}(\beta_{j}^{\rm clean})=\log_{2}(\beta_{j}^{\rm
burst})+\log_{2}\left(1-\frac{\beta_{j}^{\rm preburst}}{\beta_{j}^{\rm
burst}}\right).$ (12)
For long GRBs, the preburst is defined relative to a 0 s trigger time as T-20
s to T-5 s and for short GRBs the preburst is defined to be from T-15 s to T-1
s. Here T is the trigger time of the burst.
#### 4.1.1 Statistical Uncertainties
We have considered the statistical uncertainties in the light curve by a
typical bootstrap approach in which the square root of the number of counts
per bin is used to generate an additive poisson noise. A new poisson noise is
considered for each iteration through the bootstrap process. More significant
contributions to the uncertaintaies are discussed in Sec. 4.1.2 and Sec.
4.1.3.
#### 4.1.2 Circular Permutation
Spurious artifacts due to incidental symmetries resulting from accidental
misalignment (Percival, 2000; Coifman, 1995) of light curves with wavelet
basis functions are minimized by circularly shifting the light curve against
the basis functions. Circular shifting is a form of translation invariant de-
noising (Coifman, 1995). It is possible a shift will introduce additional
artifacts by moving a different symmetry into a susceptible location. Thus,
our approach is to circulate the signal through all possible values, or at
least a representative sampling, and then take an average over the cases which
do not show spurious correlations.
#### 4.1.3 Reverse-Tail Concatenation
Both discrete Fourier and discrete wavelet transformations imply an overall
periodicity equal to the full time-range of the input data. This can be
interpreted to mean that for a series of $N$ elements,
$\\{X_{0},X_{1}\mathellipsis X_{N-1}\\}$ then $X_{0}$ is made a surrogate for
$X_{N}$ and $X_{1}$ is made a surrogate for $X_{N+1}$, and so forth. This
assumption may lead to trouble if $X_{0}$ is much different from $X_{N-1}$. In
this case, artificially large variances may be computed. Reverse-tail
concatenation minimizes this problem by making a copy of the series which is
then reversed and concatenated onto the end of the original series resulting
in a new series with a length twice that of the original. Instead of matching
boundary conditions like,
$X_{0},X_{1},\ldots,X_{N-1},X_{0},$ (13)
we match boundaries as,
$X_{0},X_{1},\ldots X_{N-1},X_{N-1},\ldots,X_{1},X_{0}.$ (14)
Note that the series length has thus artificially been increased to $2N$ by
reversing and doubling of the original series. Consequently, the wavelet
variances at the largest scale in a logscale diagram reflect this redundancy.
This is the reason the wavelet variances at the largest scale are excluded
from least-squares fits of the scaling region.
Another difficulty in wavelet expansions is that the initialization procedure
of the multi-resolution algorithm may pollute the detail coefficients at the
finest scale (see Strang and Nguyen, 1997; Abry et al., 2003). For this reason
we follow the advice of Abry et al. (2003) and discard the detail coefficients
at the finest scale.
### 4.2 Simulation
Figure 2: In the left hand panel are simulated light curves and noise
processes: an _ideal_ fBm process (green) and a _background_ Poisson noise
(black). The sum of the fBm and Poisson processes is shown in red and is
labeled _observed_. The _observed_ light curve (red) is the sum of the fBm and
Poisson noise. The right hand panel shows the results of the background
subtraction procedure. Red and black points show the logscale diagrams
corresponding to the _observed_ light curve and _background_ , respectively.
The green data shows the logscale diagram for the _ideal_ light curve and the
blue data are the logscale with background removed. The agreement between
green and blue data demonstrates the merit of the background removal
procedure. Figure 3: Results of signal to noise sensitivity test. We
generated 1000 simulated light curves with expected $j_{\rm{intersection}}$
equal to 6 and 7 for various brightness. We show in gray the region along the
$\xi$-axis where we find the GRBs analyzed in this paper, $0.3<\xi<0.74$. We
note that in the region covered by the GRBs presented herein, brightness does
not affect $j_{\rm{intersection}}$ greatly.
The efficacy of this background subtraction method and the sensitivity to
signal to noise was tested using simulated data in the form of fractional
Brownian motion (fBm) time series that were first discussed by Mandelbrot
(1968). One advantage of using fBms for simulation of time series data is that
short duration, statistically significant fluctuations which trigger the
identification of a minimum variability time scale arise naturally as part of
the random process which produces them. Another is that fBms have a scaling
parameter, $\alpha$, which is easily varied.
The outline of the simulation procedure begins with using the numerical
computing environment MATLAB to produce 1000 realizations of fBms with scaling
parameter $\alpha$ randomly chosen from the range $1.0\leq\alpha\leq 2.0$ by
using a uniform random number generator. The fBms were then acted upon by a
Poisson operator which transformed each time series into a Poisson-distributed
series but left other properties of the fBm intact, e.g., $\alpha$.
The fBms were then combined with a Poisson noise with variance, $\lambda_{B}$.
These Poisson noises were regarded as intrinsic to the GRB. Another set of
Poisson noises with variances, $\lambda_{I}$, were generated and these noise
signals were interpreted as external _background_ meant to be removed by the
subtraction procedure.
The idealized light curves were then combined with external backgrounds
resulting in pseudo-_observed_ light curves shown in red on the left panel of
Fig. 2. The pseudo-observed light curves and the external background noises
were transformed into wavelet coefficients and wavelet variances were computed
according to Eq. 5. The variances of the pseudo-observed light curve (labeled
_actual_) and the background are plotted in the right panel of Fig. 2 in red
and black, respectively. The background was subtracted from the pseudo-
observed light curve as detailed in Eq. 12 and the resulting corrected
variances are plotted in blue in the right panel of Fig. 2. The corrected
variances are to be compared to the variances of the ideal light curve which
are plotted in green.
##### Signal to Noise Sensitivity
We define a brightness parameter, $\xi$, such that
$\xi\equiv 2^{-\delta/2},$ (15)
where
$\delta\equiv\log_{2}\lambda_{B}-\log_{2}\lambda_{I}.$ (16)
Ideally the octave, $j_{\rm{intersection}}$, which is related to the minimum
variability time scale, $\tau_{\beta}$, as defined in Eq. 8 is completely
determined by $\xi$, $\lambda_{I}$, $\alpha$, and the standard deviation of
the increments of the fBm, $\sigma$. We fixed the values of $\lambda$,
$\sigma$, and varied $\alpha$ and $\xi$ so that the expected values of
$j_{\rm{intersection}}$ are $\langle j_{\rm{intersection}}\rangle=\\{6,7\\}.$
The time series thus produced were then analyzed as described in Sec. 4.1. The
results of the simulation are given in Fig. 3. The horizontal red and blue
lines show the expected $j_{\rm{intersection}}$ for octaves 6 and 7,
respectively, and are given as a guide. Brightness, $\xi$, increases to the
right. We show in gray the region along the $\xi$-axis where we find the GRBs
analyzed in this paper based on the background noise level and the estimated
noise level instrinsic to the GRB. We have noted by our own experience that
for $\xi<0.1$ the technique discussed in this paper performs poorly. However,
in the range of signal to noise sampled by the GRBs used here, $0.3<\xi<0.74$,
the background subtraction technique does not suffer from a large systematic
response to variations in brightness, as can be seen in Fig. 3.
##### Flux Sensitivity
We also investigated the reliability of the analysis as a function of flux by
removing randomly selected counts from the original simulated signal component
while leaving the background noise level undisturbed.The analysis is then
repeated for the newly _dimmed_ simulated light curves and comparision is made
to the original un-dimmed version. This brightness comparison is similar to
the one described in Norris et al. (1995) but with a different normalization.
In Norris et al. (1995) light curves were normalized by peak intensities. In
this study the simulated light curves were normalized by signal power at the
time scales specified by the dyadic partitioning of the wavelet transform.
Dimming of the simulated light curves was done by removing 0-10% and
reanalyzing then repeating by removing 10-20% and so forth up to 70-80%. We
also considered the effect of larger variations in count removal, i.e.,
removing 0-25% 25-50% and 50-75%. We find that a decrease in flux has
essentially the same effect on $\tau_{\beta}$ as increasing the noise level.
However, the largest effect was by the wider bite of counts. For example, one
can expect more accurate results from this analysis by removing 30-40% of
counts on a bin to bin basis than by removing 25-50%. We conclude that flux
related effects are more serious when it varies widely throughout the duration
of the light curve. In all cases we have studied removing 80% or more of the
counts in the signal was a reliable way to make the method fail. Fortunately
when the method fails in this way it does so in an obvious way, i.e., the
white noise signal power coefficients in the logscale diagram become highly
irregular. As discussed in Sec. 4.3, we test for this effect in the GRB data
by performing a chi-squared test on the white noise signal power coefficients
and rejecting any GRB that fails.
In summary, 1000 simulated light curves were generated and background noise
was added. The light curves with background noise were then denoised using the
same algorithm applied to actual GRB data in which preburst data were used as
a surrogate for background. The simulated background subtracted variances were
then compared to the variances of the ideal light curves, i.e., light curves
without external background noise. Signal to noise effects on the reliability
of the method were also considered and found either to be small compared to
our quoted errors or large enough that $\tau_{\beta}$ could not be determined.
In the case of the latter the GRB was removed from the analysis. The results
indicate that the background subtraction method is robust and gives confidence
that external background noise can be subtracted from the GRB light curves
with the assumption that preburst data can serve as a surrogate for background
noise.
Figure 4: A histogram of minimum variability time scales, in the observer
frame, for long and short GRBs. It is clear that the distribution of long GRBs
is displaced from the distribution of short GRBs.
### 4.3 Selection Criteria
We analyzed 122 GRBs (61 long and 61 short) listed in the Fermi GBM-Burst
Catalog (Paciesas et al., 2012) for the first two years of the GBM mission. As
discussed in Sec. 4.2, the signal-to-background ratio is a factor to be
considered in recovering the intrinsic light curve (see Eq. 12). We required
the following condition on the ratio of variances,
$\frac{\beta_{j}^{\rm preburst}}{\beta_{j}^{\rm burst}}<0.75,$ (17)
for one or more octaves, $j$. In addition, we also required that the first
order polynomial fits to the noise region and to the scaling region each had a
$\chi^{2}$/d.f. that was less than 2. This reduced the sample to 14 short GRBs
(Tab. 1) and 46 long GRBs (Tab. 2) for a total of 60 and it is these GRBs
which are used to create Figs. 4, 5, and 8. For boosting into the source frame
(Figs. 6 and 7) a known $z$ is obviously required and this cut further reduced
the data set to 2 short GRBs and 16 long GRBs for a total of 18 GRBs
considered in the source frame (see Tab. 3).
Figure 5: Minimum variability time scale versus $T_{90}$ in the Observer
frame. Figure 6: Minimum variability time scale versus $T_{90}$ in Source
frame. The correction for time dilation shortens $T_{90}$ and decreases the
minimum variability time scale of each burst. Figure 7: Minimum variability
time scale versus $T_{90}$ with symbol size determined by luminosity (larger
symbols for higher luminosity). No obvious relation between minimum
variability time scale and luminosity is apparent. See Fig. 6 for error bars.
Figure 8: The ratio of duration-to-minimum variability time scale
($T_{90}/\tau_{\beta}$) versus $T_{90}$.
## 5 RESULTS and DISCUSSION
For a large sample of short and long GBM bursts, we have used a technique
based on wavelets to determine the minimum time scale ($\tau_{\beta}$) at
which scaling processes dominate over random noise processes. The
$\tau_{\beta}$ is the intersection of the scaling region (red-noise) of the
spectrum in the logscale diagram with that of the ‘flat’ portion representing
the (white-noise) random noise component. This transition time scale is the
shortest resolvable variability time for physical processes intrinsic to the
GRB. Histograms of the extracted $\tau_{\beta}$ values for long and short GRBs
are shown in Fig. 4. We make two observations regarding these histograms: (1)
There is a clear temporal offset in the extracted mean $\tau_{\beta}$ values
for long and short GRBs. We believe this is the first clear demonstration of
this temporal difference. Walker et al. (2000), who studied the temporal
variability of long and short bursts using the BATSE data set did not report a
systematic difference between the two types of bursts. (2) The two histograms
are quite broad and very similar in dispersion. While the difference in the
mean $\tau_{\beta}$ is understandable (a point we discuss further elsewhere)
the similarity of the dispersion is somewhat surprising since the progenitors
and the environment for the two types of bursts are presumably very different.
The comparison is qualitative at best however because the $\tau_{\beta}$ scale
has not been corrected for redshift ($z$), an effect that impacts the long
bursts more than the short bursts. In passing we note that the dispersion of
the $\tau_{\beta}$ histogram (for long bursts) is in agreement with the
results of Ukwatta et al. (2011) who performed a power density spectral
analysis of a large sample of Swift long GRBs. In that work the authors
extracted threshold frequencies and related them to a variability scale.
In Fig. 5 we show a log-log plot of $\tau_{\beta}$ versus $T_{90}$ (the
duration of the bursts); long GRBs are indicated by circles, the short ones by
squares and both time scales are with respect to the observer frame. As in the
histograms above, the fact that short GRBs, in general, tend to have smaller
$\tau_{\beta}$ values compared to long GRBs, is evident in this figure. Also
shown in the figure (as a dash line) is the trajectory of $\tau_{\beta}$ equal
to $T_{90}$. As we expect, no long GRBs exhibit a $\tau_{\beta}$ longer than
$T_{90}$ although interestingly a few short GRBs of extremely short duration
appear to be approaching the limit of equality. In addition to establishing a
characteristic time scale for short and long bursts, this figure also hints at
a positive correlation between this time and the duration of bursts. We note
that the $\tau_{\beta}$ scale spans approximately two decades for both sets of
GRBs and that the two groups are fairly well clustered in the
$\tau_{\beta}$-$T_{90}$ plane. A closer examination of the two groups,
however, indicates that a correlation between $\tau_{\beta}$ and $T_{90}$, if
present, is marginal at best. This is certainly true for the short-GRB group,
especially given the large uncertainties in the $T_{90}$s for these bursts.
The situation for the long-burst group on the other hand is not immediately
clear. In order to explore this further we cast the $\tau_{\beta}$ and the
$T_{90}$ time scales into the source frame by applying the appropriate $(1+z)$
factor to the GRBs for which the z is known. Unfortunately the z is not
available for the majority of the short GRBs but we note that the correction
is the same for both axes and is, to first order, small for the short GRBs
since the mean z for this group is $<0.8$. The corrected results for long-GRBs
are shown as a log-log plot in Fig. 6. We see from this figure (and Fig. 5)
the appearance of a very intriguing feature: A plateau region in which the
$\tau_{\beta}$ is essentially independent of $T_{90}$ and a scaling region in
which the $\tau_{\beta}$ appears to increase with $T_{90}$, with the
transition occurring around $T_{90}$ on the order of a few seconds.
If one assumes a positive correlation between luminosity and variability as
suggested by a number of authors, then one might expect smaller $\tau_{\beta}$
values for higher luminosity bursts compared to those of lower luminosity. To
investigate this, the data (in Fig. 6) are re-plotted in Fig. 7 in which the
size of each datum symbol has been modulated by the gamma-ray luminosity of
the burst, i.e., a large symbol implies a high luminosity and a small symbol a
low luminosity. We see from Fig. 7 that no obvious connection between
$\tau_{\beta}$ and luminosity is evident.
Under the assumption that the $\tau_{\beta}$ is a measure proportional to the
smallest causally-connected structure associated with a GRB light curve, it is
then possible to interpret the scaling trend in terms of the internal shock
model in which the basic units of emission are assumed to be pulses that are
produced via the collision of relativistic shells emitted by the central
engine. Indeed, we note that Quilligan et al. (2002) in their study of the
brightest BATSE bursts with $T_{90}$ $>2$ sec explicitly identified and fitted
distinct pulses and demonstrated a strong positive correlation between the
number of pulses and the duration of the burst. More recent studies (Bhat &
Guiriec, 2011; Bhat et al., 2012; Hakkila & Cumbee, 2008; Hakkila & Preece,
2011) provide further evidence for the pulse paradigm view of the prompt
emission in GRBs. In our work we have not relied on identifying distinct
pulses but instead have used the multi-resolution capacity of the wavelet
technique to resolve the smallest temporal scale present in the prompt
emission. If the smallest temporal scale is made from pulse emissions from the
smallest structures, then we can get a measure of the number of pulses in a
given burst through the ratio $T_{90}$/$\tau_{\beta}$ . In the simple model in
which a pulse is produced every time two shells collide then the ratio,
$T_{90}$/$\tau_{\beta},$ should show a correlation with the duration of the
burst. A plot of this ratio versus $T_{90}$ is shown for a sample of short and
long bursts in Fig. 8. The correlation is apparent.
It is now widely accepted that the progenitors for the two classes of GRBs are
quite distinct i.e., the merger of compact objects in the case of short GRBs
and the collapse of rapidly rotating massive stars in the case of long GRBs.
Formation of an accretion disk in the two cases is posed in a number of models
but important factors such as the size of the disk, the mass of the disk, the
strength of the magnetic field, in addition to the magnitude of the accretion
rate during the prompt phase, remain largely uncertain. With contributions
from intrinsic variability of the central engine or nearby shock-wave
interactions within a jet, we should not be surprised to observe a systematic
difference in the extracted variability time scales for long and short bursts,
since the progenitors have different spatial scales. Knowing the variability
timescales, we can estimate the size of an assumed emission region. From Fig.
5, we note that the smallest temporal-variability scale for the short bursts
is approximately 3 ms and that for the long bursts is approximately 30 ms:
These times translate to emission scales of approximately $10^{8}$ and
$10^{9}$ cm respectively. Our variability times and size scales are generally
consistent with the findings of Walker et al. (2000) although these authors
also reported observing time scales as small as few microseconds. We find no
evidence for variability times as low as a few microseconds.
Morsony et al. (2010) modeled the behavior of a jet propagating through the
progenitor and the surrounding circumstellar material and showed that the
resulting light curves exhibited both short-term and long–term variability.
They attribute the long-term variability, at the scale of few seconds, to the
interaction of the jet with the progenitor. The short-term scale, at the level
of milliseconds, they attribute to the variation in the activity of the
central engine itself. Alternatively, Zhang & Yan (2011) consider a model in
which the prompt emission is the result of a magnetically powered outflow
which is self-interacting and triggers rapid turbulent reconnections to power
the observed GRBs. This model also predicts two variability components but
interestingly and in contrast to the findings of Morsony et al. (2010) , it is
the slow component that is associated with the activity of the central engine,
and the fast component is linked to relativistic magnetic turbulence. While we
are not in a position to distinguish between these two models it is intriguing
nonetheless to note (see Fig. 5) that indeed there do appear to be two
distinct time domains for the $\tau_{\beta}$: a plateau region dominated
primarily by short bursts although it includes some long bursts too, and a
scaling region (i.e., a hint of a correlation between $\tau_{\beta}$ and
$T_{90}$) that is comprised solely of long bursts. In addition, we observe
that the time scale in the plateau region is the order of milliseconds whereas
that for the scaling region is approaching seconds.
There is considerable dispersion in the extracted $\tau_{\beta}$. The
variation is evident for both short and long-duration GRBs. The main cause of
this dispersion is not fully understood but one factor that may play a
significant role is angular momentum. As Lindner et al. (2010) note, the basic
features of the prompt emission can be understood in terms of accretion that
results via a simple ballistic infall of material from a rapidly rotating
progenitor. Material with low angular momentum will radially accrete across
the event horizon whereas the material with sufficient angular momentum will
tend to circularize outside the innermost stable circular orbit and form an
accretion disk. Simulations that go beyond the simple radial infall model
(Lindner et al., 2010, 2011) suggest that the formation of the disk leads to
an accretion shock that traverses outwards through the infalling material. If
one assumes that the initiation of such an accretion shock and the subsequent
emission of the prompt gamma-rays are associated with a particular time scale,
the variability of this scale then (as given by the dispersion in
$\tau_{\beta}$ for example) may reflect the different dynamics (initial
angular momentum and the mass of the black hole) of each GRB in our sample. In
the case of long GRBs, the mass of the central black hole can vary by an order
of magnitude thus potentially explaining a large part of the dispersion seen
in the $\tau_{\beta}$. However a similar dispersion for short bursts is
difficult to reconcile using the same arguments since the mass range for the
central black hole in standard merger models (at least for NS-NS mergers) is
expected to be significantly smaller.
## 6 CONCLUSIONS
We have studied the temporal properties of a sample of prompt-emission light
curves for short and long-duration GRBs detected by the Fermi/GBM mission. By
using a technique based on wavelets we have extracted the variability
timescales for these bursts. Our main results are summarized as follows:
a) Both short and long-duration bursts indicate a temporal variability at the
level of a few milliseconds. Variability of this order appears to be a common
feature of GRBs. This finding is consistent with the work of Walker et al.
(2000). However, unlike these authors we do not find evidence of variability
at a time scale of few microseconds.
b) In general the short-duration bursts have a variability time scale that is
significantly shorter than long-duration bursts. In addition, the
$\tau_{\beta}$ values seem not to depend in any obvious way on the luminosity
of the bursts. The dispersion over different GRBs in the extracted time scale
for short-duration bursts is an order of magnitude within the smallest
variability time, that time being approximately 3 milliseconds. The dispersion
for the long-duration bursts is somewhat larger. The origin of the dispersion
in either case is not known, although we should expect that the size of the
initial angular momentum and the mass of the system play significant roles. We
note in passing that the 3 millisecond time scale may not be a physical lower
limit and may be a result of signal to noise and the set of GRBs used in this
analysis. We remind the reader that our light curve resolution was 200 $\mu$s
and if a strong enough signal within a range of time scales between 0.5 - 3
milliseconds were present we would expect our technique to be sensitive to it.
c) The ratio of $T_{90}/\tau_{\beta}$ appears to be positively correlated with
the minimum variability time scale. This suggests further support for the
pulse paradigm view of the prompt emission as being the result of shell
collisions. In this respect, the minimum variability time scale is likely
related to key pulse parameters such as risetimes and/or widths.
d) For short-duration bursts, the variability parameter $\tau_{\beta}$ shows
negligible dependence on the duration of the bursts (characterized by
$T_{90}$). In contrast, the long-duration bursts indicate evidence for two
variability time scales: a plateau region (at the shortest time scale) which
is essentially independent of burst duration and a scaling region (at the
higher time scale) that shows a positive correlation with burst duration. The
transition between the two regions occurs around $T_{90}$ on the order of a
few seconds in the source frame.
## ACKNOWLEDGEMENTS
The NASA grant NNX11AE36G provided partial support for this work and is
gratefully acknowledged. The authors, in particular GAM and KSD, acknowledge
very useful discussions with Jon Hakkila and Narayan Bhat early in the
manuscript development. GAM and KSD also acknowledge helpful correspondences
with Jeffery Scargle.
Table 1: Short GRBs (Observer Frame). GRB | $T_{90}$[sec] | $\delta T_{90}$[sec] | $\tau_{\beta}$ [sec] | $\delta\tau^{-}_{\beta}$ [sec] | $\delta\tau^{+}_{\beta}$ [sec]
---|---|---|---|---|---
080723913 | 0.192 | 0.345 | 0.0307 | 0.0192 | 0.0510
081012045 | 1.216 | 1.748 | 0.0052 | 0.0024 | 0.0044
081102365 | 1.728 | 0.231 | 0.0258 | 0.0100 | 0.0165
081105614 | 1.280 | 1.368 | 0.0306 | 0.0147 | 0.0282
081107321 | 1.664 | 0.234 | 0.0504 | 0.0129 | 0.0173
081216531 | 0.768 | 0.429 | 0.0138 | 0.0037 | 0.0050
090108020 | 0.704 | 0.143 | 0.0241 | 0.0064 | 0.0088
090206620 | 0.320 | 0.143 | 0.0143 | 0.0063 | 0.0112
090227772 | 1.280 | 1.026 | 0.0053 | 0.0009 | 0.0011
090228204 | 0.448 | 0.143 | 0.0028 | 0.0005 | 0.0005
090308734 | 1.664 | 0.286 | 0.0120 | 0.0040 | 0.0059
090429753 | 0.640 | 0.466 | 0.0285 | 0.0115 | 0.0193
090510016 | 0.960 | 0.138 | 0.0049 | 0.0009 | 0.0011
100117879 | 0.256 | 0.834 | 0.0331 | 0.0122 | 0.0192
Table 2: Long GRBs (Observer Frame). GRB | $T_{90}$[sec] | $\delta T_{90}$[sec] | $\tau_{\beta}$ [sec] | $\delta\tau^{-}_{\beta}$ [sec] | $\delta\tau^{+}_{\beta}$ [sec]
---|---|---|---|---|---
080723557 | 58.369 | 1.985 | 0.0440 | 0.0113 | 0.0151
080723985 | 42.817 | 0.659 | 0.1894 | 0.0557 | 0.0789
080724401 | 379.397 | 2.202 | 0.0741 | 0.0208 | 0.0290
080804972 | 24.704 | 1.460 | 0.4306 | 0.1336 | 0.1937
080806896 | 75.777 | 4.185 | 0.4189 | 0.1471 | 0.2268
080807993 | 19.072 | 0.181 | 0.0232 | 0.0096 | 0.0164
080810549 | 107.457 | 15.413 | 0.1353 | 0.0648 | 0.1243
080816503 | 64.769 | 1.810 | 0.1067 | 0.0428 | 0.0715
080817161 | 60.289 | 0.466 | 0.1919 | 0.0402 | 0.0509
080825593 | 20.992 | 0.231 | 0.0775 | 0.0138 | 0.0168
080906212 | 2.875 | 0.767 | 0.1011 | 0.0182 | 0.0222
080916009 | 62.977 | 0.810 | 0.2266 | 0.0630 | 0.0872
080925775 | 31.744 | 3.167 | 0.1748 | 0.0425 | 0.0562
081009140 | 41.345 | 0.264 | 0.1095 | 0.0170 | 0.0201
081101532 | 8.256 | 0.889 | 0.0948 | 0.0302 | 0.0444
081125496 | 9.280 | 0.607 | 0.2182 | 0.0504 | 0.0656
081129161 | 62.657 | 7.318 | 0.0912 | 0.0292 | 0.0429
081215784 | 5.568 | 0.143 | 0.0319 | 0.0043 | 0.0050
081221681 | 29.697 | 0.410 | 0.2701 | 0.0641 | 0.0841
081222204 | 18.880 | 2.318 | 0.1956 | 0.0533 | 0.0732
081224887 | 16.448 | 1.159 | 0.2055 | 0.0356 | 0.0431
090102122 | 26.624 | 0.810 | 0.0347 | 0.0111 | 0.0164
090131090 | 35.073 | 1.056 | 0.0733 | 0.0169 | 0.0220
090202347 | 12.608 | 0.345 | 0.1444 | 0.0575 | 0.0954
090323002 | 135.170 | 1.448 | 0.1598 | 0.0436 | 0.0599
090328401 | 61.697 | 1.810 | 0.0682 | 0.0139 | 0.0175
090411991 | 14.336 | 1.086 | 0.0673 | 0.0391 | 0.0935
090424592 | 14.144 | 0.264 | 0.0249 | 0.0031 | 0.0036
090425377 | 75.393 | 2.450 | 0.1346 | 0.0369 | 0.0508
090516137 | 118.018 | 4.028 | 0.4938 | 0.2063 | 0.3544
090516353 | 123.074 | 2.896 | 0.7992 | 0.5686 | 1.9711
090528516 | 79.041 | 1.088 | 0.1314 | 0.0320 | 0.0423
090618353 | 112.386 | 1.086 | 0.2631 | 0.0536 | 0.0673
090620400 | 13.568 | 0.724 | 0.1667 | 0.0422 | 0.0564
090626189 | 48.897 | 2.828 | 0.0498 | 0.0078 | 0.0093
090718762 | 23.744 | 0.802 | 0.1621 | 0.0482 | 0.0686
090809978 | 11.008 | 0.320 | 0.2436 | 0.0515 | 0.0652
090810659 | 123.458 | 1.747 | 0.7319 | 0.3027 | 0.5161
090829672 | 67.585 | 2.896 | 0.0678 | 0.0141 | 0.0177
090831317 | 39.424 | 0.572 | 0.0266 | 0.0103 | 0.0169
090902462 | 19.328 | 0.286 | 0.0223 | 0.0026 | 0.0029
090926181 | 13.760 | 0.286 | 0.0435 | 0.0061 | 0.0070
091003191 | 20.224 | 0.362 | 0.0300 | 0.0051 | 0.0062
091127976 | 8.701 | 0.571 | 0.0395 | 0.0059 | 0.0069
091208410 | 12.480 | 5.018 | 0.0621 | 0.0180 | 0.0254
100414097 | 26.497 | 2.073 | 0.0418 | 0.0074 | 0.0090
Table 3: Long and Short GRBs ($T_{90}$ and $\tau_{\beta}$ in Observer Frame). Luminosities are taken from references given in footnotes. GRB | $z$ | $T_{90}$[sec] | $\delta T_{90}$[sec] | $\tau_{\beta}$ [sec] | $\delta\tau^{-}_{\beta}$ [sec] | $\delta\tau^{+}_{\beta}$ [sec] | $L_{iso}$ [ergs/s] | $\delta L_{iso}^{-}$ [ergs/s] | $\log$ $\delta L_{iso}^{+}$ [ergs/s]
---|---|---|---|---|---|---|---|---|---
080804972 | 2.204 | 24.704 | 1.460 | 0.4306 | 0.1336 | 0.1937 | 111Nava et al. 2011 (A&A 530, A21 (2011))3.58$\cdot 10^{52}$ | 5.82$\cdot 10^{51}$ | 7.85$\cdot 10^{51}$
080810549 | 3.350 | 107.457 | 15.413 | 0.1353 | 0.0648 | 0.1243 | 222GCN #81009.59$\cdot 10^{52}$ | 1.28$\cdot 10^{52}$ | 1.28$\cdot 10^{52}$
080916009 | 4.350 | 62.977 | 0.810 | 0.2266 | 0.0630 | 0.0872 | 333Ghirlanda et al. 2011 (arXiv:1107.4096)1.04$\cdot 10^{54}$ | 8.79$\cdot 10^{52}$ | 8.79$\cdot 10^{52}$
081222204 | 2.770 | 18.880 | 2.318 | 0.1956 | 0.0533 | 0.0732 | 111Nava et al. 2011 (A&A 530, A21 (2011))1.26$\cdot 10^{53}$ | 7$\cdot 10^{51}$ | 6$\cdot 10^{51}$
090102122 | 1.547 | 26.624 | 0.810 | 0.0347 | 0.0111 | 0.0164 | 333Ghirlanda et al. 2011 (arXiv:1107.4096)8.71$\cdot 10^{52}$ | 5.6$\cdot 10^{51}$ | 5.6$\cdot 10^{51}$
090323002 | 3.570 | 135.170 | 1.448 | 0.1598 | 0.0436 | 0.0599 | 111Nava et al. 2011 (A&A 530, A21 (2011))6.87$\cdot 10^{53}$ | 6.55$\cdot 10^{53}$ | 4.45$\cdot 10^{52}$
090328401 | 0.736 | 61.697 | 1.810 | 0.0682 | 0.0139 | 0.0175 | 444GCN #90571.79$\cdot 10^{52}$ | 1.42$\cdot 10^{51}$ | 1.11$\cdot 10^{51}$
090424592 | 0.544 | 14.144 | 0.264 | 0.0249 | 0.0031 | 0.0036 | 555Ukwatta, T. N., et al. 2010, _ApJ._ , 711, 10731.62$\cdot 10^{52}$ | 4$\cdot 10^{50}$ | 5$\cdot 10^{50}$
090510016 | 0.903 | 0.960 | 0.138 | 0.0049 | 0.0009 | 0.0011 | 333Ghirlanda et al. 2011 (arXiv:1107.4096)1.78$\cdot 10^{53}$ | 1.2$\cdot 10^{51}$ | 1.2$\cdot 10^{51}$
090516353 | 4.100 | 123.074 | 2.896 | 0.7992 | 0.5686 | 1.9711 | 666GCN #94158.17$\cdot 10^{52}$ | 2.85$\cdot 10^{52}$ | 6.1$\cdot 10^{51}$
090618353 | 0.540 | 112.386 | 1.086 | 0.2631 | 0.0536 | 0.0673 | 555Ukwatta, T. N., et al. 2010, _ApJ._ , 711, 10738.47$\cdot 10^{51}$ | 1.17$\cdot 10^{51}$ | 3.4$\cdot 10^{50}$
090902462 | 1.822 | 19.328 | 0.286 | 0.0223 | 0.0026 | 0.0029 | 333Ghirlanda et al. 2011 (arXiv:1107.4096)5.89$\cdot 10^{53}$ | 9.71$\cdot 10^{51}$ | 9.71$\cdot 10^{51}$
090926181 | 2.106 | 13.760 | 0.286 | 0.0435 | 0.0061 | 0.0070 | 333Ghirlanda et al. 2011 (arXiv:1107.4096)7.40$\cdot 10^{53}$ | 1.45$\cdot 10^{52}$ | 1.45$\cdot 10^{52}$
091003191 | 0.897 | 20.224 | 0.362 | 0.0300 | 0.0051 | 0.0062 | 111Nava et al. 2011 (A&A 530, A21 (2011))4.53$\cdot 10^{52}$ | 3.71$\cdot 10^{51}$ | 6.55$\cdot 10^{51}$
091127976 | 0.490 | 8.701 | 0.571 | 0.0395 | 0.0059 | 0.0069 | 777GCN #102043.70$\cdot 10^{51}$ | 1.38$\cdot 10^{50}$ | 1.06$\cdot 10^{50}$
091208410 | 1.063 | 12.480 | 5.018 | 0.0621 | 0.0180 | 0.0254 | 111Nava et al. 2011 (A&A 530, A21 (2011))1.45$\cdot 10^{52}$ | 1.48$\cdot 10^{51}$ | 3.45$\cdot 10^{51}$
100117879 | 0.920 | 0.256 | 0.834 | 0.0331 | 0.0122 | 0.0192 | 111Nava et al. 2011 (A&A 530, A21 (2011))2.63$\cdot 10^{52}$ | 5.01$\cdot 10^{51}$ | 1.08$\cdot 10^{52}$
100414097 | 1.368 | 26.497 | 2.073 | 0.0418 | 0.0074 | 0.0090 | 888GCN #105951.00$\cdot 10^{53}$ | 1.58$\cdot 10^{52}$ | 7.6$\cdot 10^{51}$
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|
arxiv-papers
| 2012-01-21T04:01:32 |
2024-09-04T02:49:26.569769
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. A. MacLachlan, A. Shenoy, E. Sonbas, K. S. Dhuga, B. Cobb, T. N.\n Ukwatta, D. C. Morris, A. Eskandarian, L. C. Maximon, and W. C. Parke",
"submitter": "Glen MacLachlan",
"url": "https://arxiv.org/abs/1201.4431"
}
|
1201.4475
|
# Biholomorphic Convex Mappings of Order $\alpha$ on the Unit Ball in Hilbert
Spaces
Ming-Sheng Liu
School of Mathematical Sciences, South China Normal University,
Guangzhou, 510631 Guangdong, P.R.China
Email:liumsh@scnu.edu.cn
Yu-Can Zhu
Department of Mathematics, Fuzhou University, Fuzhou,350002 Fujian, P.R.China
E-mail:zhuyucan@fzu.edu.cn Corresponding author.
Abstract. In this paper, we first introduce the concept of biholomorphic
convex mapping of order $\alpha$ on the unit ball $B$ in a complex Hilbert
space $X$. Next we provide some sufficient conditions that a locally
biholomorphic mapping $f$ is a biholomorphic convex mapping of order $\alpha$
and give an Alexander’s theorem between the subclass of convex mappings and
the subclass of starlike mappings on $B$ in Hilbert space. We also obtain the
order of starlikeness of biholomorphic convex mappings of order $\alpha$ on
$B$ in Hilbert spaces. Finally, we construct some concrete examples of
biholomorphic convex mappings of order $\alpha$ on $B$ in Hilbert spaces by
means of a linear operator.
Keywords. Biholomorphic convex mapping; Biholomorphic starlike mapping;
Locally biholomorphic mapping; biholomorphic convex mapping of order $\alpha$.
2000 MR Subject Classification 32H02, 30C45
## 1 Introduction
The holomorphic functions of one complex variable which map the unit disk onto
starlike or convex domains have been extensively studied. These functions are
easily characterized by simple analytic or geometric conditions. In moving to
higher dimensions, several difficulties arise. In the case of one complex
variable, the following well known theorems had been established(cf. [3]).
Theorem A Suppose that $\alpha\in[0,1)$ and
$f(z)=z+\sum\limits_{n=2}^{\infty}a_{n}z^{n}$ is a holomorphic function on the
unit disk $U=\\{z\in\mathbb{C}:|z|<1\\}$ in the complex plane $\mathbb{C}$.
(1) If $\sum\limits_{n=2}^{\infty}n^{2}|a_{n}|\leq 1$, then $f$ is a convex
function in the unit disk $U$.
(2) If $\sum\limits_{n=2}^{\infty}n(n-\alpha)|a_{n}|\leq 1-\alpha$, then $f$
is a convex function of order $\alpha$ in $U$.
Theorem B Suppose that $\alpha\in[0,1)$ and
$f(z)=z+\sum\limits_{n=2}^{\infty}a_{n}z^{n}$ is a holomorphic function on the
unit disk $U$ in the complex plane $\mathbb{C}$.
(1) If $\sum\limits_{n=2}^{\infty}n|a_{n}|\leq 1$, then $f$ is a starlike
function in the unit disk $U$.
(2) If $\sum\limits_{n=2}^{\infty}(n-\alpha)|a_{n}|\leq 1-\alpha$, then $f$ is
a starlike function of order $\alpha$ in $U$.
Roper and Suffridge established the n-dimensional version of Theorem A(1), and
we[9] established the n-dimensional version of Theorem B(1)(2) as follows.
Theorem C(Roper and Suffridge[12]) Let
$f(z)=z+\sum\limits_{k=2}^{\infty}A_{k}(z^{k})$ be a holomorphic mapping on
the unit ball $B_{2}^{n}$. If $\sum\limits_{k=2}^{\infty}k^{2}\|A_{k}\|\leq
1$, then $f(z)$ is a convex mapping on $B_{2}^{n}$.
Theorem D(Liu and Zhu[9]) Suppose that $\alpha\in[0,1)$. Let
$f(z)=z+\sum\limits_{k=2}^{\infty}A_{k}(z^{k})$ be a holomorphic mapping on
the unit ball $B$ in Hilbert space. If
$\sum\limits_{k=2}^{\infty}(k-\alpha)\|A_{k}\|\leq A(\alpha)$, where
$A(\alpha)$ is defined by
$A(\alpha)=\left\\{\begin{array}[]{lll}\frac{(2-\alpha)\sqrt{1-2\alpha}}{\sqrt{5-2\alpha}},&0\leq\alpha\leq\frac{1}{4},\\\
\frac{(2-\alpha)(1-\alpha)}{2+\alpha},&\frac{1}{4}<\alpha\leq\frac{2}{5},\\\
\alpha,&\frac{2}{5}<\alpha<\frac{1}{2},\\\
1-\alpha,&\frac{1}{2}\leq\alpha<1.\end{array}\right.$
Then $f(z)$ is a starlike mapping of order $\alpha$ on $B$ in Hilbert space.
A problem is naturally proposed: can we establish the n-dimensional version
for Theorem A(2)?
## 2 Preliminaries
In order to state and prove our main results, we recall some definitions and
notations. Suppose that $X$ is a complex Hilbert space with inner product
$\langle\cdot,\cdot\rangle$ and norm
$\|\cdot\|=\sqrt{\langle\cdot,\cdot\rangle}$, and $G$ is a domain in $X$. A
mapping $f:G\to X$ is said to be holomorphic on $G$, if for any $z\in G$,
there exists a linear operator $Df(z):X\to X$ such that
$\lim_{h\to 0}\frac{\|f(z+h)-f(z)-Df(z)h\|}{\|h\|}=0.$
The linear operator $Df(z)$ is called the Fr$\acute{e}$chet derivative of $f$
at $z\in G$.
If $f$ is holomorphic on $G$, then for every $k=1,2,\cdots$, and every
$z_{0}\in G$, there is a bounded symmetric $k-$ linear operator
$D^{k}f(z_{0}):X\times X\times\cdots\times X\to X$ such that
$f(z)=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}D^{k}f(z_{0})((z-z_{0})^{k})$
for all $z$ in some neighborhood of $z_{0}$, where
$D^{0}f(z_{0})((z-z_{0})^{0})=f(z_{0})$ and
$D^{k}f(z_{0})((z-z_{0})^{k})=D^{k}f(z_{0})(z-z_{0},z-z_{0},\cdots,z-z_{0})$
for $k\geq 1$.
A mapping $f:G\to X$ is said to be biholomorphic on $G$ if $f$ is holomorphic
on $G$, $f(G)$ is a domain, and the inverse $f^{-1}$ exists and is holomorphic
on $f(G)$. A mapping $f:G\to X$ is said to be locally biholomorphic on $G$, if
for any $z\in G$, there exists a neighborhood $U$ of $z$ such that $f|_{U}$ is
biholomorphic on $U$. Then $f$ is locally biholomorphic on $G$ if and only if
its Fréchet derivative $Df(z)$ has a bounded inverse at each $z\in G$.
The unit ball in $X$ is $B=\\{z\in X:\|z\|<1\\}$. Let $N(B)$ denote the class
of all local biholomorphic mappings $f:B\rightarrow X$ such that
$f(0)=0,Df(0)=I$, where $I$ is the identity operator in $X$. A biholomorphic
mapping $f:B\to X$ is called a biholomorphic starlike mapping if $tf(B)\subset
f(B)$ for $0\leq t\leq 1$ with $f(0)=0$. Let $S^{\ast}(B)$ be the subclass of
$N(B)$ consisting of starlike mappings on $B$. Then $f\in S^{\ast}(B)$ if and
only if $f$ is locally biholomorphic such that
$\mbox{Re}\langle Df(z)^{-1}f(z),z\rangle>0$
for all $z\in B\backslash\\{0\\}$(cf.[1, 3, 6]).
A mapping $f\in N(B)$ is called starlike of order $\alpha\in(0,1)$ on $B$ if
$\bigg{|}\langle
Df(z)^{-1}f(z),z\rangle-\frac{1}{2\alpha}\|z\|^{2}\bigg{|}<\frac{1}{2\alpha}\|z\|^{2}\quad\mbox{for
all }\quad z\in B\backslash\\{0\\},$
Let $S^{\ast}(B,\alpha)$ denote the class of starlike mappings of order
$\alpha$ on $B$ for $0<\alpha<1$ and let $S^{\ast}(B,0)\equiv S^{\ast}(B)$. It
is obvious that $S^{\ast}(B,\alpha)\subset S^{\ast}(B)$ for $0\leq\alpha<1$.
A biholomorphic mapping $f:B\to X$ is said biholomorphic convex mapping if
$(1-t)f(z_{1})+tf(z_{2})\in f(B)$
for all $z_{1},z_{2}\in B$ and $0\leq t\leq 1$. The class of all biholomorphic
convex mappings on $B$ with $f(0)=0,Df(0)=I$ is denoted by $K(B)$.
We[16] obtained a necessary and sufficient condition that a locally
biholomorphic mapping was a biholomorphic convex mapping on B in the Hilbert
space $X$ as follows.
Theorem A(Zhu and Liu[16]). Let $f:B\to X$ be a locally biholomorphic mapping.
Then $f$ is a biholomorphic convex mapping on $B$ if and only if
$\displaystyle\|x\|^{2}-\mbox{Re}\Big{\langle}Df(z)^{-1}D^{2}f(z)(x,x),z\Big{\rangle}>0$
(2.1)
for $z\in B\setminus\\{0\\}$ and $x\in X\setminus\\{0\\}$ with
$\mbox{Re}\langle x,z\rangle=0$.
Remark 1. Theorem A had improved a result of Hamada and Kohr[4]. Setting
$X=C^{n}$ in Theorem A, we also obtain Theorem 2 in [2].
Corollary 1. Let $0\leq\alpha<1$ and $f:B\to X$ be a locally biholomorphic
mapping. If $f$ satisfies the following inequality
$\displaystyle\|x\|^{2}-\mbox{Re}\Big{\langle}Df(z)^{-1}D^{2}f(z)(x,x),z\Big{\rangle}>\alpha\cdot\|x\|^{2}$
(2.2)
for $z\in B\setminus\\{0\\}$ and $x\in X\setminus\\{0\\}$ with
$\mbox{Re}\langle x,z\rangle=0$. Then $f$ is a biholomorphic convex mapping on
$B$.
We call such mapping $f$, which satisfies the hypothesis of Corollary 1, a
biholomorphic convex mapping of order $\alpha$ on $B$. We let $K(B,\alpha)$
denote the subclass of all biholomorphic convex mappings of order $\alpha$ on
$B$ with $f(0)=0,Df(0)=I$.
In this paper, we provide some sufficient conditions for biholomorphic convex
mapping of order $\alpha$ and an Alexander’s theorem between the subclass of
convex mappings and the subclass of starlike mappings on $B$ in Hilbert space.
We also obtain the order of starlikeness of biholomorphic convex mappings of
order $\alpha$ on $B$ in Hilbert spaces. Finally, we introduce a linear
operator in purpose to construct some concrete examples of biholomorphic
convex mappings on $B$ in Hilbert spaces. From these, we give some examples of
biholomorphic convex mappings on $B$ in Hilbert spaces.
## 3 Main Results
We first establish some sufficient conditions for biholomorphic convex mapping
of order $\alpha$ on $B$.
Theorem 1. Suppose that $0\leq\alpha<1$ and $f:B\to X$ is a locally
biholomorphic mapping. If $f$ satisfies
$\displaystyle\Big{\|}Df(z)^{-1}D^{2}f(z)(x,x)\Big{\|}\leq 1-\alpha$ (3.1)
for $z\in B\setminus\\{0\\}$ and $x\in X$ with $\|x\|=1$ and $\mbox{Re}\langle
x,z\rangle=0$, then $f$ is a biholomorphic convex mapping of order $\alpha$ on
$B$.
Proof. Since $f:B\to X$ is a locally biholomorphic mapping, for any $z\in
B\setminus\\{0\\}$ and $x\in X\setminus\\{0\\}$ with $\|x\|=1$ and
$\mbox{Re}\langle x,z\rangle=0$, we have
$\begin{array}[]{lll}\|x\|^{2}-\mbox{Re}\Big{\langle}Df(z)^{-1}D^{2}f(z)(x,x),z\Big{\rangle}&\geq&\|x\|^{2}-|\langle
Df(z)^{-1}D^{2}f(z)(x,x),z\rangle|\\\
&\geq&\|x\|^{2}-\|Df(z)^{-1}D^{2}f(z)(x,x)\|\|z\|\\\
&>&\|x\|^{2}-\|Df(z)^{-1}D^{2}f(z)(x,x)\|\\\
&=&\|x\|^{2}-\|Df(z)^{-1}D^{2}f(z)(\frac{x}{\|x\|},\frac{x}{\|x\|})\|\|x\|^{2}.\end{array}$
Notice that $\|\frac{x}{\|x\|}\|=1$, we conclude from (3.1) that
$\begin{array}[]{lll}\|x\|^{2}-\mbox{Re}\Big{\langle}Df(z)^{-1}D^{2}f(z)(x,x),z\Big{\rangle}&>&\|x\|^{2}-\|Df(z)^{-1}D^{2}f(z)(\frac{x}{\|x\|},\frac{x}{\|x\|})\|\|x\|^{2}\\\
&\geq&\alpha\cdot\|x\|^{2}.\end{array}$
Hence by Corollary 1, we obtain that $f\in K(B,\alpha)$, and the proof is
complete.
Corollary 2. Suppose that $0\leq\alpha<1$ and $f:B\to X$ is a locally
biholomorphic mapping with $\|Df(z)-I\|\leq c<1$ for each $z\in B$, where $I$
is the identity operator in $X$. If $f$ satisfies
$\|D^{2}f(z)(x,x)\|\leq(1-c)(1-\alpha)$
for all $x\in X$ with $\|x\|=1$ and $z\in B\setminus\\{0\\}$ such that
$\mbox{Re}\langle x,z\rangle=0$, then $f$ is a biholomorphic convex mapping of
order $\alpha$ on $B$.
Proof. Since $\|Df(z)-I\|\leq c<1$ for any $z\in B$, we obtain that
$Df(z)=I-(I-Df(z))$ is an invertible linear operator(see [13], P192), and
$\|Df(z)^{-1}\|\leq\frac{1}{1-\|I-Df(z)\|}\leq\frac{1}{1-c}$
for all $z\in B$.
Thus for any $x\in X$ with $\|x\|=1$ and $z\in B\setminus\\{0\\}$ such that
$\mbox{Re}\langle x,z\rangle=0$, by the hypothesis of Corollary 2, we have
$\displaystyle\|Df(z)^{-1}D^{2}f(z)(x,x)\|$ $\displaystyle\leq$
$\displaystyle\|Df(z)^{-1}\|\|D^{2}f(z)(x,x)\|$ $\displaystyle\leq$
$\displaystyle\frac{1}{1-c}\cdot(1-c)(1-\alpha)=1-\alpha.$
Hence by Theorem 1, we obtain that $f$ is a biholomorphic convex mapping of
order $\alpha$ on $B$, and the proof is complete.
Remark 2. Setting $\alpha=0$ in Theorem 1, we get Corollary 1 in [16]; Setting
$\alpha=0$ in Corollary 2, we get Corollary 2 in [16].
Theorem 2. Let $0\leq\alpha<1$ and
$f(z)=z+\sum\limits_{k=2}^{+\infty}A_{k}(z^{k}):B\to X$ be a holomorphic
mapping. If $f$ satisfies
$\displaystyle\sum\limits_{k=2}^{+\infty}k(k-\alpha)\|A_{k}\|\leq 1-\alpha$,
then $f\in K(B,\alpha)$.
Proof. By direct calculating the Fr$\acute{e}$chet derivatives of $f(z)$, we
obtain
$\displaystyle Df(z)=I+\sum\limits_{k=2}^{+\infty}kA_{k}(z^{k-1},\cdot),$
$\displaystyle
D^{2}f(z)(x,x)=\sum\limits_{k=2}^{+\infty}k(k-1)A_{k}(z^{k-2},x^{2})$
and
$\|Df(z)-I\|\leq\sum\limits_{k=2}^{+\infty}k\|A_{k}\|\leq\frac{1}{2-\alpha}\sum\limits_{k=2}^{+\infty}k(k-\alpha)\|A_{k}\|\leq\frac{1-\alpha}{2-\alpha}<1$
for $z\in B,\,x\in X$. Hence we obtain that $Df(z)=I-(I-Df(z))$ is an
invertible linear operator(see [13],P192), and
$\displaystyle\|D^{2}f(z)(x,x)\|$ $\displaystyle\leq$
$\displaystyle\sum\limits_{k=2}^{+\infty}(k^{2}-k)\|A_{k}\|\|z\|^{k-2}\|\|x\|^{2}$
$\displaystyle\leq$ $\displaystyle
1-\alpha+\alpha\sum\limits_{k=2}^{+\infty}k\|A_{k}\|-\sum\limits_{k=2}^{+\infty}k\|A_{k}\|$
$\displaystyle=$
$\displaystyle(1-\sum\limits_{k=2}^{+\infty}k\|A_{k}\|)(1-\alpha)$
for $z\in B\setminus\\{0\\}$ and $\|x\|=1$ with $\mbox{Re}\langle
x,z\rangle=0$. By Corollary 2 for $c=\sum\limits_{k=2}^{+\infty}k\|A_{k}\|$,
we obtain that $f\in K(B,\alpha)$, and the proof is complete.
Remark 3. Setting $X=\mathbb{C}^{n},\alpha=0$ in Theorem 2, we may obtain
Theorem 2.1 in [12]. Our proof is more simple than theirs. Setting
$X=\mathbb{C}$ in Theorem 2, we get Theorem A(2).
Example 1. Let $0\leq\alpha<1$ and $A$ be a symmetric bilinear operator from
$X\times X$ to $X$ with $\|A\|\leq\frac{1-\alpha}{4-2\alpha}$. If we let
$f(z)=z+A(z,z)$, then $f\in K(B,\alpha)$.
Proof. Some straightforward computations yield the relations
$Df(z)=I+2A(z,\cdot),\quad D^{2}f(z)(x,y)=2A(x,y)$
for $z\in B,x,y\in X$. It implies
$Df(0)=I,\quad D^{2}f(0)(\cdot,\cdot)=2A(\cdot,\cdot)\quad\mbox{and}\quad
D^{k}f(0)=0$
for $k=3,4,\cdots$. Hence we obtain
$\sum\limits_{k=2}^{+\infty}\frac{k(k-\alpha)\|D^{k}f(0)\|}{k!}=(2-\alpha)\|D^{2}f(0)\|=2(2-\alpha)\|A\|\leq
1-\alpha.$
By Theorem 2, we conclude that $f\in K(B,\alpha)$, and the proof is complete.
Example 2. Let $0\leq\alpha<1$, $0<|a|\leq 1/2$ and $u\in X$ with $\|u\|=1$.
Then
$f(z)=z+a\langle z,u\rangle^{2}u\in
K(B,\alpha)\Longleftrightarrow|a|\leq\frac{1-\alpha}{4-2\alpha}.$
Proof. Let $c=1-\frac{2|a|}{1-\alpha}$. If
$0<|a|\leq\frac{1-\alpha}{4-2\alpha}$, then we have
$\frac{1-\alpha}{2-\alpha}\leq c<1$ and $2|a|=(1-c)(1-\alpha)\leq c$. Short
computations yield the relations
$\displaystyle Df(z)=I+2a\langle z,u\rangle\langle\cdot,u\rangle u,\quad
D^{2}f(z)(x,x)=2a\langle x,u\rangle^{2}u.$ (3.2)
It implies
$\|Df(z)-I\|\leq 2|a|\|z\|<2|a|\leq c,\quad\|D^{2}f(z)(x,x)\|\leq
2|a|=(1-c)(1-\alpha)$
for all $x\in X$ with $\|x\|=1$ and $z\in B$ such that $\mbox{Re}\langle
x,z\rangle=0$.
By Corollary 2, we obtain that $f\in K(B,\alpha)$.
Conversely, we shall prove that $0<|a|\leq\frac{1-\alpha}{4-2\alpha}$ when
$f\in K(B,\alpha)$.
If not, then $|a|>\frac{1-\alpha}{4-2\alpha}$. Let $\theta=\arg
a,\,x=ie^{-i\theta}u$ and $z_{0}=-re^{-i\theta}u$ for
$\frac{1-\alpha}{(4-2\alpha)|a|}<r<1$, where $u\in X$ with $\|u\|=1$. Then
$\|x\|=1,z_{0}\in B\setminus\\{0\\}$ and $\mbox{Re}\langle
x,z_{0}\rangle=\mbox{Re}\\{-ir\\}=0$.
Some straightforward computations from (3.2) yield the relations
$\displaystyle Df(z_{0})^{-1}=I+\frac{2|a|r}{1-2|a|r}\langle\cdot,u\rangle
u,\quad D^{2}f(z_{0})(x,x)=-2|a|e^{-i\theta}u$ $\displaystyle
Df(z_{0})^{-1}D^{2}f(z_{0})(x,x)=-\frac{2|a|e^{-i\theta}}{1-2|a|r}u.$
Hence we obtain
$\|x\|^{2}-\mbox{Re}\Big{\langle}Df(z_{0})^{-1}D^{2}f(z_{0})(x,x),z_{0}\Big{\rangle}=\frac{1-4|a|r}{1-2|a|r}<\alpha.$
This contradicts (2.2), and the proof is complete.
Next, we provide an Alexander’s theorem between the subclass of convex
mappings and the subclass of starlike mappings on $B$ in Hilbert space. In the
case of one complex variable, Alexander’s theorem told us that $f(z)$ is a
convex function on the unit disc $U$ if and only if $zf^{\prime}(z)$ is a
starlike function on the unit disc $U$. This theorem is no longer true in
several complex variables(see [3]). However, we have the following Alexander’s
theorem.
Theorem 3(Alexander’s Theorem). Suppose that $0\leq\alpha<1$ and $A(\alpha)$
is defined by Theorem D. Let
$SK(B,\alpha)=\\{f(z):f(z)=z+\sum\limits_{k=2}^{\infty}A_{k}(z^{k})\in
H(B)\mbox{ such that }\sum\limits_{k=2}^{\infty}k(k-\alpha)\parallel
A_{k}\parallel\leq A(\alpha)\\},$
and
$SS^{*}(B,\alpha)=\\{f(z):f(z)=z+\sum\limits_{k=2}^{\infty}A_{k}(z^{k})\in
H(B)\mbox{ such that }\sum\limits_{k=2}^{\infty}(k-\alpha)\parallel
A_{k}\parallel\leq A(\alpha)\\}.$
Then $SK(B,\alpha)\subset K(B)$, $SS^{*}(B,\alpha)\subset S^{*}(B)$, and
$f(z)\in SK(B,\alpha)$ if and only if $Df(z)(z)\in SS^{*}(B,\alpha)$.
Notice that $A(\alpha)\leq 1-\alpha$ for $0\leq\alpha<1$, by applying Theorem
D and Theorem 2, we can prove this theorem easily.
Now we establish a result on the order of starlikeness of function class
$K(B,\alpha)$.
Theorem 4. Suppose that $0\leq\alpha<1$. Then $K(B,\alpha)\subset
S^{*}(B,\beta)$, where
$\beta=\beta(\alpha)=\frac{2\alpha-1+\sqrt{(2\alpha-1)^{2}+8}}{4}.$
In order to prove the above theorem, we need the following lemmas.
Lemma 1.([10]) Let $g(z)=a+a_{1}z+\cdots$ is analytic in $U$ and
$g(z)\not\equiv a$. If there exists $z_{0}\in U\backslash\\{0\\}$ such that
$|g(z_{0})|=\max\limits_{|z|\leq|z_{0}|}|g(z)|$, then there exists real number
$t\geq\frac{|g(z_{0})|-|a|}{|g(z_{0})|+|a|}$ such that
$z_{0}g^{{}^{\prime}}(z_{0})=t\ g(z_{0})$.
Lemma 2. Let $f\in N(B),\,0<\rho<1$. If there exists $z_{0}\in
B\backslash\\{0\\}$ such that
${\rm Re}\ \frac{\|z\|^{2}}{\langle
Df(z_{0})^{-1}f(z_{0}),z_{0}\rangle}=\rho,$
and ${\rm Re}\ \frac{\|z\|^{2}}{\langle Df(z)^{-1}f(z),z\rangle}\geq\rho$ for
all $\|z\|<\|z_{0}\|$. Then there exist real numbers $\theta,\ t,\,m$ such
that:
(i) $\langle h(z_{0}),z_{0}\rangle=\frac{\|z_{0}\|^{2}}{2\rho}\
(1+e^{i\theta})$, where $h(z)=Df(z)^{-1}f(z)$ ;
(ii) $\langle Dh(z_{0})(z_{0}),z_{0}\rangle=\frac{\|z_{0}\|^{2}}{2\rho}\
[1+(1+t)e^{i\theta}]$, where $t\geq\frac{1-|2\rho-1|}{1+|2\rho-1|}$ ;
(iii) $e^{i\theta}\overline{Dh(z_{0})}(z_{0})+e^{-i\theta}h(z_{0})=mz_{0}$,
where $m=\frac{2\cos\theta+2+t}{2\rho}$.
Proof. Since $h:B\to X$ is a holomorphic mapping and $h(0)=0,Dh(0)=I$, and
$\bigg{|}\langle
h(z_{0}),z_{0}\rangle-\frac{\|z_{0}\|^{2}}{2\rho}\bigg{|}=\frac{\|z_{0}\|^{2}}{2\rho},$
(3.3) $\bigg{|}\langle
h(z),z\rangle-\frac{\|z\|^{2}}{2\rho}\bigg{|}\leq\frac{\|z\|^{2}}{2\rho},\quad\|z\|<\|z_{0}\|.$
(3.4)
Let
$\psi(\xi)=\frac{2\rho}{\|\xi z_{0}\|^{2}}\langle h(\xi z_{0}),\xi
z_{0}\rangle-1=\frac{2\rho}{\|z_{0}\|^{2}}\langle\frac{h(\xi
z_{0})}{\xi},z_{0}\rangle-1,$
then $\psi(\xi)$ is analytic in $\overline{U}$ and $|\psi(\xi)|\leq
1=|\psi(1)|$ for $\xi\in\overline{U}$, $\psi(0)=2\rho-1$. By Lemma 1, there is
a real number $t\geq\frac{1-|2\rho-1|}{1+|2\rho-1|}$ such that
$\psi^{{}^{\prime}}(1)=t\psi(1)$.
Let $\psi(1)=e^{i\theta}$ for some real number $\theta$, then we obtain (i)
holds, and
$\psi^{{}^{\prime}}(1)=\frac{2\rho}{\|z_{0}\|^{2}}\langle
Dh(z_{0})(z_{0})-h(z_{0}),z_{0}\rangle=te^{i\theta},$
which implies (ii) holds.
From (3.3) and (3.4), we obtain that
${\rm Re}[e^{-i\theta}\langle
h(z),z\rangle]\leq\frac{\|z_{0}\|^{2}}{2\rho}(1+\cos\theta)={\rm
Re}[e^{-i\theta}\langle h(z_{0}),z_{0}\rangle],\quad\|z\|<\|z_{0}\|.$ (3.5)
Let $r=\|z_{0}\|,B(r)=\\{z\in X:\|z\|<r\\}$, then the tangent hyperplane of
$\partial B(r)$ at $z_{0}$ is
$T_{z_{0}}(\partial B(r))=\\{b\in X:\langle b,z_{0}\rangle=0\\}.$
For any tangent vector $a\in T_{z_{0}}(\partial B(r))$ with $\|a\|=1$, set
$\gamma(t)=\sqrt{1-t^{2}}\ z_{0}+t\ ra$, then $\|\gamma(t)\|=r$ for
$t\in(-1,1)$ and $\gamma(0)=z_{0},\gamma^{\prime}(0)=ra$. Let
$\varphi(t)={\rm Re}[e^{-i\theta}\langle h(\gamma(t)),\gamma(t)\rangle].$
From (3.5), we obtain $\varphi(t)\leq\varphi(0)$ for $t\in(-1,1)$, so that
$\varphi(0)=\max\limits_{t\in(-1,1)}\varphi(t)$. Hence
$\displaystyle 0=\varphi^{\prime}(0)$ $\displaystyle=$ $\displaystyle{\rm
Re}[e^{-i\theta}\langle Dh(z_{0})ra,z_{0}\rangle+e^{-i\theta}\langle
h(z_{0}),ra\rangle]=r{\rm Re}\langle v,a\rangle,$
where $v=e^{i\theta}\overline{Dh(z_{0})}(z_{0})+e^{-i\theta}h(z_{0})$. This
implies that $v$ is a normal vector of $\partial B(r)$ at $z_{0}$, thus there
exists a real number $m$ such that $v=mz_{0}$. Since
$\displaystyle m\|z_{0}\|^{2}$ $\displaystyle=$ $\displaystyle{\rm Re}\langle
z_{0},mz_{0}\rangle={\rm Re}\langle z_{0},v\rangle$ $\displaystyle=$
$\displaystyle{\rm Re}\ \langle
z_{0},e^{i\theta}\overline{Dh(z_{0})}(z_{0})\rangle+{\rm Re}\ \langle
z_{0},e^{-i\theta}h(z_{0})\rangle$ $\displaystyle=$ $\displaystyle{\rm Re}\
[e^{-i\theta}\langle Dh(z_{0})(z_{0}),z_{0}\rangle]+{\rm Re}\
\langle[e^{-i\theta}\langle h(z_{0}),z_{0}\rangle]$ $\displaystyle=$
$\displaystyle\frac{\|z_{0}\|^{2}}{2\rho}\
(\cos\theta+1+t)+\frac{\|z_{0}\|^{2}}{2\rho}\ (\cos\theta+1),$
we obtain that $m=\frac{2\cos\theta+2+t}{2\rho}$, and this completes the proof
of Lemma 2.
Proof of Theorem 4. Let $h(z)=[Df(z)]^{-1}f(z)$ with $f(z)\in K(B,\alpha)$,
and $g(z)=\frac{\|z\|^{2}}{\langle h(z),z\rangle}$, then
$g(0)=1>\beta=\beta(\alpha)$.
Suppose $f\not\in S^{\ast}(B,\beta)$, then by the continuity of $g(z)$, there
exists $z_{0}\in B\setminus\\{0\\}$ such that ${\rm Re}\ g(z_{0})=\beta$ and
${\rm Re}\ g(z)\geq\beta$ for all $\|z\|<\|z_{0}\|$. Thus it follows from
Lemma 2 that there exist real numbers $\theta$, $t\geq\frac{1-\beta}{\beta}$
and $m=\frac{2\cos\theta+2+t}{2\beta}$ such that (i)-(iii) of Lemma 2 hold.
Let $b=e^{-i\theta}h(z_{0})-\frac{z_{0}}{2\beta}(1+\cos\theta)$, then it
follows from Lemma 2(i) that
${\rm Re}\ \langle b,z_{0}\rangle={\rm Re}\ [e^{-i\theta}\langle
h(z_{0}),z_{0}\rangle]-\frac{\|z_{0}\|^{2}}{2\beta}(1+\cos\theta)=0,$
so that
${\rm Re}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(b,b),z_{0}\rangle<(1-\alpha)\
\|b\|^{2}.$ (3.6)
Let $b_{1}=iz_{0}$, then ${\rm Re}\ \langle b_{1},z_{0}\rangle={\rm
Re}[i\|z_{0}\|^{2}]=0$, so we conclude from the fact $f(z)\in K(B,\alpha)$
that
${\rm
Re}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(iz_{0},iz_{0}),z_{0}\rangle<(1-\alpha)\
\|z_{0}\|^{2}.$ (3.7)
On the other hand, by Lemma 2, we have
$\|b\|^{2}=\|h(z_{0})\|^{2}-\frac{\|z_{0}\|^{2}}{4\beta^{2}}(1+\cos\theta)^{2},$
(3.8)
and
$\|h(z_{0})\|\geq\frac{\|z_{0}\|}{2\beta}|1+e^{i\theta}|=\frac{\|z_{0}\|}{2\beta}\sqrt{2+2\cos\theta}$,
and
$\displaystyle m\ e^{-i\theta}\langle h(z_{0}),z_{0}\rangle$ $\displaystyle=$
$\displaystyle\langle e^{-i\theta}h(z_{0}),mz_{0}\rangle$ $\displaystyle=$
$\displaystyle\langle
e^{-i\theta}h(z_{0}),e^{i\theta}\overline{Dh(z_{0})}(z_{0})\rangle+\langle
e^{-i\theta}h(z_{0}),e^{-i\theta}h(z_{0})\rangle$ $\displaystyle=$
$\displaystyle e^{-i2\theta}\ \langle
Dh(z_{0})h(z_{0}),z_{0}\rangle+\|h(z_{0})\|^{2},$
so that
$\displaystyle{\rm Re}\ [e^{-i2\theta}\ \langle
Dh(z_{0})h(z_{0}),z_{0}\rangle]$ $\displaystyle=$ $\displaystyle m{\rm Re}\
[e^{-i\theta}\langle h(z_{0}),z_{0}\rangle]-\|h(z_{0})\|^{2}$ $\displaystyle=$
$\displaystyle\frac{m\|z_{0}\|^{2}}{2\beta}(1+\cos\theta)-\|h(z_{0})\|^{2}.$
By computing the Frechet derivatives for both sides of equation
$Df(z)h(z)=f(z)$, we obtain
$[Df(z)]^{-1}D^{2}f(z)(h(z),h(z))+Dh(z)h(z)=h(z),$
and
$[Df(z)]^{-1}D^{2}f(z)(h(z),z)+Dh(z)(z)=z,$
thus we can obtain from the above equalities that
$\displaystyle{\rm
Re}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(b,b),z_{0}\rangle={\rm
Re}[e^{-i2\theta}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(h(z_{0}),h(z_{0})),z_{0}\rangle$
$\displaystyle\hskip 85.35826pt-\frac{1+\cos\theta}{\beta}{\rm
Re}[e^{-i\theta}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(h(z_{0}),z_{0}),z_{0}\rangle]$
$\displaystyle\hskip 85.35826pt+\frac{(1+\cos\theta)^{2}}{4\beta^{2}}{\rm
Re}[e^{-i\theta}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(z_{0},z_{0}),z_{0}\rangle]$
$\displaystyle\hskip 85.35826pt=-{\rm Re}[e^{-i2\theta}\ \langle
Dh(z_{0})h(z_{0}),z_{0}\rangle]+{\rm Re}[e^{-i2\theta}\ \langle
h(z_{0}),z_{0}\rangle]$ $\displaystyle\hskip
85.35826pt+\frac{1+\cos\theta}{\beta}\\{{\rm Re}[e^{-i\theta}\langle
Dh(z_{0})(z_{0}),z_{0}\rangle]-{\rm Re}[e^{-i\theta}\langle
z_{0},z_{0}\rangle]\\}$ $\displaystyle\hskip
85.35826pt-\frac{(1+\cos\theta)^{2}}{4\beta^{2}}{\rm
Re}[e^{-i\theta}\langle[Df(z_{0})]^{-1}D^{2}f(z_{0})(iz_{0},iz_{0}),z_{0}\rangle]$
$\displaystyle\hskip
85.35826pt\geq-\frac{m\|z_{0}\|^{2}}{2\beta}(1+\cos\theta)+\|h(z_{0})\|^{2}-\frac{\|z_{0}\|^{2}}{2\beta}(1+\cos\theta)$
$\displaystyle\hskip
85.35826pt+\frac{\|z_{0}\|^{2}}{2\beta^{2}}(1+\cos\theta)(\cos\theta+1+t)+\frac{(1+\cos\theta)^{2}}{4\beta^{2}}(\alpha-1)\|z_{0}\|^{2}$
$\displaystyle\hskip 85.35826pt\geq(1-\alpha)\
\|b\|^{2}+\alpha\|h(z_{0})\|^{2}-\alpha\frac{\|z_{0}\|^{2}}{4\beta^{2}}(2+2\cos\theta)$
$\displaystyle\hskip
85.35826pt+\frac{1+\cos\theta}{4\beta^{2}}\|z_{0}\|^{2}(2\alpha-2\beta+\frac{1-\beta}{\beta})$
$\displaystyle\hskip 85.35826pt\geq(1-\alpha)\ \|b\|^{2},$
which contradicts (3.6). Hence $f\in S^{\ast}(B,\beta)$, and the proof is
complete.
Remark 4. Setting $X=\mathbb{C}^{n}$ in Theorem 4, we obtain the related
result in [7, 8, 15]
By applying the growth theorem[5] of starlike mappings of order $\rho$ and
Theorem 4, we have the following corollary.
Corollary 3. Let
$0\leq\alpha<1,\beta=\beta(\alpha)=\frac{2\alpha-1+\sqrt{(2\alpha-1)^{2}+8}}{4}$.
If $f(z)\in K(B,\alpha)$, then for $z\in B$, we have
$\frac{\|z\|}{(1+\|z\|)^{2(1-\beta)}}\leq\|f(z)\|\leq\frac{\|z\|}{(1-\|z\|)^{2(1-\beta)}},$
and $f(B)\supset\frac{1}{2^{2(1-\beta)}}B$.
Remark 5. Setting $\alpha=0$ in Corollary 3, we obtain the growth theorem of
convex mappings[1, 3].
Finally, we introduce a linear operator[16] in purpose to construct some
concrete examples of biholomorphic convex mappings of order $\alpha$ on $B$ in
a Hilbert space $X$.
Let
$H(U)=\\{f:U\rightarrow\mathbb{C}\mbox{ are analytic in }U\mbox{ with
}f(0)=0,f^{\prime}(0)=1\\},$
then
$f\in K(\alpha)\Longleftrightarrow f\in H(U)\mbox{ and
}\mbox{Re}\bigg{\\{}1+\frac{zf^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{\\}}>\alpha\quad\mbox{for
all}\quad z\in U.$
Let
$SK(B,\alpha)=\\{f\in N(B):\|Df(z)^{-1}D^{2}f(z)(\cdot,\cdot)\|\leq
1-\alpha\mbox{ for all }z\in B\\}.$
From Theorem 1, we have $SK(B,\alpha)\subset K(B,\alpha),SK(U,\alpha)\subset
K(\alpha)$ and
$SK(U,\alpha)=\bigg{\\{}f\in
H(U):\bigg{|}\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}\bigg{|}\leq
1-\alpha\mbox{ for all }z\in U\bigg{\\}}.$
Let $m$ be a positive integer and $\dim X\geq m\geq 2$. Then there exist
$u_{1},u_{2},\cdots,u_{m}\in X$ with $\|u_{j}\|=1(j=1,2,\cdots,m)$ such that
$\langle u_{j},u_{k}\rangle=0(j\neq k)$. For $g_{1},g_{2},\cdots,g_{m}\in
H(U)$, we define the operator $\Phi$ as
$\displaystyle\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})(z)=z-\sum\limits_{j=1}^{m}\langle
z,u_{j}\rangle u_{j}+\sum\limits_{j=1}^{m}g_{j}(\langle z,u_{j}\rangle)u_{j}$
(3.9)
for $z\in B$.
Theorem 5. Suppose that
$0\leq\alpha<1,\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})$ is
defined by (3.9), where $g_{1},g_{2},\cdots,g_{m}\in H(U)$ are locally
univalent functions on $\Delta$.
(1) If $\Phi_{u_{1},\cdots,u_{m}}(g_{1},\cdots,g_{m})\in K(B,\alpha)$, then
$g_{1},g_{2},\cdots,g_{m}\in K(\alpha)$.
(2) If
$h(\xi)=\left\\{\begin{array}[]{lll}\frac{1-(1-\xi)^{2\alpha-1}}{2\alpha-1},&\alpha\neq\frac{1}{2}\\\
-\ln(1-\xi),&\alpha=\frac{1}{2}\end{array}\right.$, then $h\in K(\alpha)$, but
$\Phi_{u_{1},\cdots,u_{m}}(h,\cdots,h)\notin K(B,\alpha)$.
(3) $\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})\in
SK(B,\alpha)$ if and only if $g_{1},g_{2},\cdots,g_{m}\in SK(U,\alpha)$.
Proof. Let
$f(z)=\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})(z)$, where
$g_{1},g_{2},\cdots,g_{m}\in H(U)$ are locally univalent functions on $U$. By
some straightforward computations, we obtain
$\displaystyle Df(z)=I-\sum\limits_{j=1}^{m}\langle\cdot,u_{j}\rangle
u_{j}+\sum\limits_{j=1}^{m}g^{\prime}_{j}(\langle
z,u_{j}\rangle)\langle\cdot,u_{j}\rangle u_{j},$ $\displaystyle
Df(z)^{-1}=I-\sum\limits_{j=1}^{m}\bigg{(}1-\frac{1}{g^{\prime}_{j}(\langle
z,u_{j}\rangle)}\bigg{)}\langle\cdot,u_{j}\rangle u_{j},$ $\displaystyle
D^{2}f(z)(x,x)=\sum\limits_{j=1}^{m}g^{\prime\prime}_{j}(\langle
z,u_{j}\rangle)[\langle x,u_{j}\rangle]^{2}u_{j}$
for $z\in B$ and $x\in X$. Hence we have
$\displaystyle
Df(z)^{-1}D^{2}f(z)(x,x)=\sum\limits_{j=1}^{m}\frac{g^{\prime\prime}_{j}(\langle
z,u_{j}\rangle)}{g^{\prime}_{j}(\langle z,u_{j}\rangle)}[\langle
x,u_{j}\rangle]^{2}u_{j}.$ (3.10)
(1) Assume that
$f=\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})\in K(B)$, for
every $\xi\in U\setminus\\{0\\}$ and $k$ fixed, we let $z=\xi u_{k}$ and
$x=i\xi u_{k}$, then $\mbox{Re}\langle x,z\rangle=\mbox{Re}\\{i|\xi|^{2}\\}=0$
and $z\in B\setminus\\{0\\}$. Note that $\langle u_{j},u_{k}\rangle=0(j\neq
k)$, from (3.10), we obtain
$\displaystyle\|x\|^{2}-\mbox{Re}\langle Df(z)^{-1}D^{2}f(z)(x,x),z\rangle$
$\displaystyle=$ $\displaystyle|\xi|^{2}+|\xi|^{2}\mbox{Re}\bigg{\\{}\frac{\xi
g^{\prime\prime}_{k}(\xi)}{g^{\prime}_{k}(\xi)}\bigg{\\}}=|\xi|^{2}\mbox{Re}\bigg{\\{}1+\frac{\xi
g^{\prime\prime}_{k}(\xi)}{g^{\prime}_{k}(\xi)}\bigg{\\}}>\alpha\|x\|^{2}=\alpha|\xi|^{2}$
for $\xi\in U\setminus\\{0\\}$. This implies $g_{k}\in K(\alpha)$ for
$k=1,2,\cdots,m$.
(2) A simple computation yields
$\mbox{Re}\bigg{\\{}1+\frac{\xi
h^{\prime\prime}(\xi)}{h^{\prime}(\xi)}\bigg{\\}}=\alpha+(1-\alpha)\mbox{Re}\bigg{\\{}\frac{1+\xi}{1-\xi}\bigg{\\}}>\alpha$
for all $\xi\in U$. It follows $h\in K(\alpha)$.
Let
$\displaystyle\sqrt{\frac{1-\alpha}{2}}<a<1,\displaystyle\frac{\sqrt{1-\alpha}}{\sqrt{2}a}<r<1$,
$z=ru_{1}+a\sqrt{1-r^{2}}u_{2}\quad\mbox{and}\quad
x=a\sqrt{1-r^{2}}u_{1}-ru_{2}.$
Then we have $\|x\|^{2}=a^{2}(1-r^{2})+r^{2}>0,\mbox{Re}\langle x,z\rangle=0$
and
$0<\|z\|^{2}=r^{2}+a^{2}(1-r^{2})<1.$
Notice that $\langle x,u_{j}\rangle=0(j\geq 3)$, from (3.10) , we obtain
$\displaystyle\|x\|^{2}$ $\displaystyle-$ $\displaystyle\mbox{Re}\langle
DF(z)^{-1}D^{2}F(z)(x,x),z\rangle$ $\displaystyle=$
$\displaystyle\|x\|^{2}-\mbox{Re}\bigg{\\{}\frac{2}{1-\langle
z,u_{1}\rangle}[\langle x,u_{1}\rangle]^{2}\langle
u_{1},z\rangle+\frac{2}{1-\langle z,u_{2}\rangle}[\langle
x,u_{2}\rangle]^{2}\langle u_{2},z\rangle\bigg{\\}}$ $\displaystyle=$
$\displaystyle
a^{2}(1-r^{2})+r^{2}-\bigg{\\{}\frac{2r}{1-r}a^{2}(1-r^{2})+\frac{2a\sqrt{1-r^{2}}}{1-a\sqrt{1-r^{2}}}r^{2}\bigg{\\}}$
$\displaystyle<$ $\displaystyle 1-2r^{2}a^{2}<\alpha,$
where $F=\Phi_{u_{1},u_{2},\cdots,m}(h,h,\cdots,h)$. By Corollary 1, we have
$F\not\in K(B,\alpha)$.
(3) Assume that $g_{1},g_{2},\cdots,g_{m}\in
SK(U,\alpha),f=\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})$,
from (3.10), we obtain
$\displaystyle\Big{\|}Df(z)^{-1}D^{2}f(z)(x,x)\Big{\|}$ $\displaystyle=$
$\displaystyle\bigg{\|}\sum\limits_{j=1}^{m}\frac{g^{\prime\prime}_{j}(\langle
z,u_{j}\rangle)}{g^{\prime}_{j}(\langle z,u_{j}\rangle)}[\langle
x,u_{j}\rangle]^{2}u_{j}\bigg{\|}$ (3.11) $\displaystyle\leq$
$\displaystyle\sum\limits_{j=1}^{m}\bigg{|}\frac{g^{\prime\prime}_{j}(\langle
z,u_{j}\rangle)}{g^{\prime}_{j}(\langle z,u_{j}\rangle)}\bigg{|}|\langle
x,u_{j}\rangle|^{2}$ $\displaystyle\leq$
$\displaystyle(1-\alpha)\sum\limits_{j=1}^{m}|\langle x,u_{j}\rangle|^{2}$
for $z\in B$ and $x\in X$.
Fix $x\in X$, let $x_{0}=\sum\limits_{j=1}^{m}\langle x,u_{j}\rangle u_{j}$, a
simple computation yields
$\langle x-x_{0},u_{j}\rangle=\langle
x,u_{j}\rangle-\sum\limits_{k=1}^{m}\langle x,u_{k}\rangle\langle
u_{k},u_{j}\rangle=\langle x,u_{j}\rangle-\langle x,u_{j}\rangle=0,$
for $j=1,2,\cdots,m$. This leads to $\langle x-x_{0},x_{0}\rangle=0$. Hence we
conclude that
$\displaystyle\|x\|^{2}$ $\displaystyle=$
$\displaystyle\|(x-x_{0})+x_{0}\|^{2}=\|x-x_{0}\|^{2}+\|x_{0}\|^{2}$ (3.12)
$\displaystyle=$ $\displaystyle\|x-x_{0}\|^{2}+\sum\limits_{j=1}^{m}|\langle
x,u_{j}\rangle|^{2}$ $\displaystyle\geq$
$\displaystyle\sum\limits_{j=1}^{m}|\langle x,u_{j}\rangle|^{2}.$
From (3.11) and (3.12), we obtain
$\Big{\|}Df(z)^{-1}D^{2}f(z)(x,x)\Big{\|}\leq(1-\alpha)\sum\limits_{j=1}^{m}|\langle
x,u_{j}\rangle|^{2}\leq(1-\alpha)\|x\|^{2}\leq 1-\alpha$
for all $z\in B$ and $x\in X$ with $\|x\|=1$. Since $X$ is a Hilbert space, by
the result in [13](see P.342), we have
$\displaystyle\|Df(z)^{-1}D^{2}f(z)(\cdot,\cdot)\|$ $\displaystyle=$
$\displaystyle\sup_{\|x\|=1,\ \|y\|=1}\|Df(z)^{-1}D^{2}f(z)(x,y)\|$
$\displaystyle=$ $\displaystyle\sup_{\|x\|=1}\|Df(z)^{-1}D^{2}f(z)(x,x)\|\leq
1-\alpha.$
It follows that $f\in SK(B,\alpha)$.
Conversely, suppose that
$f=\Phi_{u_{1},u_{2},\cdots,u_{m}}(g_{1},g_{2},\cdots,g_{m})\in SK(B,\alpha)$.
For every $\xi\in U$ and $k$ fixed ($1\leq k\leq m$), we let $z=\xi u_{k}$ and
$x=u_{k}$, then we have $z\in B,\langle z,u_{k}\rangle=\xi$ and $\|x\|=1$.
Note that $\langle u_{j},u_{k}\rangle=0(j\neq k)$ and $\|u_{k}\|=1$, from
(3.10), we obtain
$\bigg{|}\frac{g^{\prime\prime}_{k}(\xi)}{g^{\prime}_{k}(\xi)}\bigg{|}=\|Df(z)^{-1}D^{2}f(z)(x,x)\|\leq\|Df(z)^{-1}D^{2}f(z)(\cdot,\cdot)\|\|x\|^{2}\leq
1-\alpha$
for $\xi\in U$. That is, $g_{k}\in SK(U,\alpha)$ for $k=1,2,\cdots,m$, and the
proof is complete.
Remark 6. Let $X=\mathbb{C}^{n},\alpha=0$. If we choose
$u_{1}=(1,0,\cdots,0),u_{2}=(0,1,\cdots,0),\cdots$,
$u_{n}=(0,0,\cdots,1)\in\mathbb{C}^{n}$, then we have
$z=\sum\limits_{j=1}^{n}\langle z,u_{j}\rangle u_{j}$ for
$z\in\mathbb{C}^{n}$. From Theorem 5, we obtain a result, which is Theorem 3
and Theorem 4 in [14] for case $p=2$. Part (2) was obtained by Roper and
Suffridge [11], [12] using a different method.
Example 3. Let $0\leq\alpha<1,\dim X\geq m\geq 2$ and
$\lambda_{j}\in\mathbb{C}$ with $\lambda_{j}\neq 0(j=1,2,\cdots,m)$, then
$f(z)=z-\sum\limits_{j=1}^{m}\langle z,u_{j}\rangle
u_{j}+\sum\limits_{j=1}^{m}\frac{e^{\lambda_{j}\langle
z,u_{j}\rangle}-1}{\lambda_{j}}u_{j}\in
K(B,\alpha)\Longleftrightarrow|\lambda_{j}|\leq 1-\alpha(j=1,2,\cdots,m),$
where $u_{j}\in X$ with $\|u_{j}\|=1$ such that $\langle
u_{j},u_{k}\rangle=0(j,k=1,2,\cdots,m,j\neq k)$.
Proof. If $|\lambda_{j}|\leq 1(j=1,2,\cdots,m)$, setting
$g_{j}(\xi)=\displaystyle\frac{e^{\lambda_{j}\xi}-1}{\lambda_{j}}$, then we
have that $g_{j}$ is analytic on $U$ with $g_{j}(0)=0,g^{\prime}_{j}(0)=1$
such that
$\displaystyle\frac{g^{\prime\prime}_{j}(\xi)}{g^{\prime}_{j}(\xi)}=\lambda_{j}$
for $\xi\in U(j=1,2,\cdots,m)$. Hence $g_{j}\in SK(U)$. From Theorem 5, we
obtain $f\in SK(B,\alpha)$.
Conversely, we shall prove that $|\lambda_{j}|\leq 1-\alpha$ for all
$j=1,2,\cdots,m$ when $f$ is a biholomorphic convex mapping of order $\alpha$
on $B$.
If not, then there exists $k$ such that $|\lambda_{k}|>1-\alpha$. Let
$\displaystyle\frac{1-\alpha}{|\lambda_{k}|}<r<1,\theta=\arg\lambda_{k},z_{0}=-re^{-i\theta}u_{k}$
and $x=ie^{-i\theta}u_{k}$, then $\|x\|=1,\mbox{Re}\langle
x,z_{0}\rangle=\mbox{Re}\\{-ir\\}=0$. Using the fact that $\langle
u_{j},u_{k}\rangle=0(j\neq k)$, from (3.10), we obtain
$\displaystyle Df(z_{0})^{-1}D^{2}f(z_{0})(x,x)$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{m}\frac{g^{\prime\prime}_{j}(\langle
z_{0},u_{j}\rangle)}{g^{\prime}_{j}(\langle z_{0},u_{j}\rangle)}[\langle
x,u_{j}\rangle]^{2}u_{j}$ $\displaystyle=$ $\displaystyle\lambda_{k}[\langle
x,u_{k}\rangle]^{2}u_{k}=-|\lambda_{k}|e^{-i\theta}u_{k}.$
Hence we have
$\|x\|^{2}-\mbox{Re}\langle
Df(z_{0})^{-1}D^{2}f(z_{0})(x,x),z_{0}\rangle=1-r|\lambda_{k}|<\alpha,$
which contradicts (2.2). This completes the proof.
## References
* [1] S. Gong, Convex and Starlike Mappings in Several Complex Variables, Scince Press/Kluwer Academic Publishers, 1998.
* [2] S. Gong, S. Wang, Q. Yu, Biholomorphic convex mappings of ball in $C^{n}$, Pacific J. Math. 161(1993), 287-306.
* [3] I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimensions, Dekker, New York, 2003.
* [4] H. Hamada, G. Kohr, $\Phi$-like and convex mappings in infinite dimensional spaces, Rev. Roumaine Math. Pures Appl. 47(2002), 315-328.
* [5] Hidetaka Hamada, Gabriela Kohr and Piotr Liczberski, Starlike mappings of order $\alpha$ on the unit ball in complex Banch spaces, Glasnik Matematicki, 36(1)(2001), 39-48.
* [6] G. Kohr, Certain partial differential inequalities and applications for holomorphic mappings defined on the unit ball of $\mathbb{C}^{n}$, Ann. Univ. Mariae Curie-Sklodowska, Sect.A, 50(1996),87-94.
* [7] G. Kohr, On some alpha convex mappings on the unit ball of $\mathbb{C}^{n}$, Demonstratio Math., 31(1)(1998), 209-222.
* [8] Hao Liu and Ke-Ping Lu, Two subclasses of starlike mappings of several complex variables(in Chinese), Chin. Ann. of Math., 21A(5)(2000), 533-546.
* [9] M. S. Liu and Y. C. Zhu, The Radius of Convexity and the Sufficient Condition for Starlike Mappings, accept by Bulletin of the Malaysian Mathematical Sciences Society (2010.4.5)
* [10] S.S. Miller and P.T. Mocanu, Differential subordinations:Theorey and Applications, Pure and Applied Mathematics No.225, New York:Marcel Dekker Inc., 2000.
* [11] K. Roper, T. Suffridge, Convex mappings on the unit ball of $C^{n}$, J. d’Analyse Math., 65(1995), 333-347.
* [12] K. Roper, T. J. Suffridge, Convexity properties of holomorphic mappings in $C^{n}$, Trans. Amer. Math. Soc. 351(1999), 1803-1833.
* [13] A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, Wiley, New York, 1980.
* [14] Y. C. Zhu, Biholomorphic convex mappings on $B_{p}^{n}$(in Chinese), Chin. Ann. of Math. 24A(3)(2003), 269-278.
* [15] Yu-Can Zhu, $\alpha$-convex mappings on $B_{p}$(in Chinese), Acta Math. Scientia, 25A(1)(2005), 84-92.
* [16] Y. C. Zhu and M. S. Liu, Criteria for biholomorphic convex mappings on the unit ball in Hilbert spaces, J. Math. Anal. Appl. 322(2006), 495-511.
|
arxiv-papers
| 2012-01-21T14:22:32 |
2024-09-04T02:49:26.579295
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ming-Sheng Liu and Yu-Can Zhu",
"submitter": "Ming-Sheng Liu",
"url": "https://arxiv.org/abs/1201.4475"
}
|
1201.4563
|
# Spin Hall effect in a Kagome lattice driven by Rashba spin-orbit interaction
Moumita Dey Theoretical Condensed Matter Physics Division, Saha Institute of
Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti santanu@post.tau.ac.il School of Chemistry, Tel Aviv
University, Ramat-Aviv, Tel Aviv-69978, Israel S. N. Karmakar Theoretical
Condensed Matter Physics Division, Saha Institute of Nuclear Physics,
Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
###### Abstract
Using four-terminal Landauer-Büttiker formalism and Green’s function
technique, in this present paper, we calculate numerically spin Hall
conductance (SHC) and longitudinal conductance of a finite size Kagome lattice
with Rashba spin-orbit (SO) interaction both in presence and absence of
external magnetic flux in clean limit. In the absence of magnetic flux, we
observe that depending on the Fermi surface topology of the system SHC changes
its sign at certain values of Fermi energy. Unlike the infinite system (where
SHC is a universal constant $\pm\frac{e}{8\pi}$), here SHC depends on the
external parameters like SO coupling strength, Fermi energy, etc. We show that
in the presence of any arbitrary magnetic flux, periodicity of the system is
lost and the features of SHC tends to get reduced because of elastic
scattering. But again at some typical values of flux ($\phi=\frac{1}{2}$,
$\frac{1}{4}$, $\frac{3}{4}\ldots$, etc.) the system retains its periodicity
depending on its size and the features of spin Hall effect (SHE) reappears.
Our predicted results may be useful in providing a deeper insight into the
experimental realization of SHE in such geometries.
###### pacs:
73.23.-b, 72.25.Dc, 71.70.Ej
## I Introduction
Rapid progress in spin based information processing and storage devicing
technologies has been metamorphosed into an emerging field called
‘spintronics’ wolf , revolutionizing nanotechnology with the plethora of
concepts which are particularly aimed at the exquisite control and
manipulation of spin degree of freedom in semiconductor structures i.e., using
the spin as a career of classical or quantum information. Despite being a few
decades old topic semiconductor spintronics has reignited interest to
investigate the role of SO interaction in generating pure spin current which
is the central theme of this newborn branch of condensed matter physics.
Originating from the relativistic correction to the Schrödinger equation, SO
interaction provides an all electrical way to generate and manipulate spin
current in a far precise way rather than the usual magnetic field based spin
control. Longitudinal flow of unpolarized charge current through a sample with
SO coupling can induce non-equilibrium spin accumulation at the lateral edges
of the sample in transverse direction, and therefore, a pure spin current is
established if connected through ideal leads in transverse direction. This is
the basic phenomenon of Spin Hall Effect. The main source of SO coupling in
mesoscopic systems comes from either magnetic impurities (extrinsic type) or
from structural or bulk inversion asymmetry in the confining potential of the
system (intrinsic type) yielding Rashba or Dresselhaus type SO interaction
rashba ; dressel ; winkler .
Few years back some theoretical proposals were made on the existence of
intrinsic SHE in hole doped mura1 ; mura2 or electron doped sinova semi-
conducting systems where SO interaction strength is strong enough to split the
Bloch energy bands for up and down spin electrons. In this case a pure spin
current is predicted to flow along $Y$ direction, in response to the
longitudinal electric field along $X$ direction through the infinite
homogeneous system, essentially capturing the essence of a semi-classical
effect. Sinova et al. predicted an universal value of Spin Hall conductivity
(equal to $\pm\frac{e}{8\pi}$) for two-dimensional electron gas (2DEG) in the
clean limit being independent of the SO coupling
Figure 1: (Color online). Four-probe set-up for measuring spin Hall effect. An
unpolarized charge current is allowed to pass through the longitudinal leads,
$1$ and $4$, and because of Rashba SO interaction pure spin current (no charge
current is associated with it, as they are voltage probes) flows through the
transverse leads, $2$ and $3$.
strength and electron density. The intrinsic SHE is much different from the
extrinsic effect proposed by Hirsch hirsch , where up and down spin electrons
get deflected in opposite directions due to spin dependent scattering off
impurities. However, the magnitude of spin Hall current in the intrinsic case
is expected to be several orders of magnitude larger than the extrinsic one.
These theoretical anticipations have also been verified experimentally. SHE in
$2$-dimensional ($2$D) hole wunder or electron sih gases have attracted a
tremendous attention in research community. Furthermore, in some recent
experiments wunder ; kato the existence of spin accumulation on the lateral
edge of a two-terminal $2$D hole gas or a $3$D n-type semiconducting system
has also been detected.
Later on various theoretical investigations have been done to illustrate SHE
in mesoscopic 2DEG and ring geometries focusing on different aspects of this
intriguing phenomena. In 2004 Sinova et al. studied SHE by measuring dc
voltage drop in response to a longitudinal dc current sinova1 . In 2005, Sheng
et al. further investigated SHE in a finite size square lattice sheng to
explore the non-quantized nature of spin Hall conductances which also depends
on various physical parameters such as SO coupling strength, electronic Fermi
energy, and disorder strength, etc. In the same year, Nikolic and co-workers
observed niko1 quasi-periodic oscillations in spin Hall conductances due to
spin sensitive quantum interference effect in a finite width mesoscopic ring.
In another work, again they studied SHE in 2DEG in detail niko2 both for
ordered and disordered cases. After that, in 2007, Nicolik et al. made a
comparative study between extrinsic and intrinsic SHEs in disordered
mesoscopic multi-terminal systems, again considering a square lattice topology
niko3 .
Till date a wealth of literature has been formed studying SHE. But the focus
of most of those works were on p- or n-doped semiconductors. Even the
numerical calculations for finite size systems were based on either square
lattice li ; moca1 or ring moca2 geometries. Later, discovery of a new class
of spin Hall insulators mura3 by Murakami et al. establishes the fact that
apart from the SO interaction strength the lattice structure itself plays a
very significant role in determining spin Hall conductances through its band
structure. In 2009, Liu et al. observed more exotic SHE in Kagome kagome and
Honeycomb honey lattices because of their fascinating topology. But their
analysis were based on Kubo formalism sinova , which requires an infinite
homogeneous system in clean limit. Hence a deeper insight into the
experimental detection of such effect in this kind of geometry demands a
quantitative prediction of spin Hall conductances in finite size systems. This
is the main motivation behind our work.
This kind of geometry is interesting not only from the theoretical point of
view, but it has also profound experimental significance expt1 ; expt2 ; expt3
. This type of lattice can be easily fabricated by modern patterning technique
pattern , or observed in reconstructed semiconductor surfaces recons . The
presence of a transverse magnetic field has a deeper impact on electronic
properties in a kagome lattice which has also been reflected in our SHE study.
In 1994, Nori et al. have studied quantum interference effect due to
electronic motion on kagome lattice in presence of a perpendicular magnetic
field nori1 . Later, in 2002, they have investigated both analytically and
numerically the mean field superconducting-normal phase boundaries in several
$2$D networks considering a transverse magnetic field. They have determined
the transition temperature as a function of magnetic flux $\phi$ passing
through the smallest triangular plaquettes using the lattice path integral
technique nori6 .
In the present work we explore different aspects of SHE in a Kagome lattice
geometry with Rashba type SO interaction in clean limit using Landauer-
Büttiker formalism. To the best of our knowledge no such finite size spin Hall
conductance calculation has been done for a kagome lattice geometry so far.
The paper is organized as follows. After presenting a brief introduction and
motivation in Section I, in Section II, we describe the model and theoretical
formulation to obtain the SHC and longitudinal conductance. The numerical
results are illustrated in Section III. Finally, in Section IV, we summarize
our results.
## II Theoretical formulation
### II.1 Model and Hamiltonian
We start describing our model which corroborates with the experimental set-up
for observing the SHE, where a four-probe mesoscopic bridge is used for
detection of pure spin current. Four ideal leads are attached to the central
region (see Fig. 1) which is a finite size kagome lattice with Rashba type SO
interaction. An unpolarized charge current is allowed to pass through the
longitudinal leads (lead-$1$ and lead-$4$) inducing spin Hall current in the
transverse direction (lead-$2$ and lead-$3$).
A discrete lattice model is used to describe the finite size kagome lattice
and also the side attached leads within the framework of tight-binding
approximation assuming only nearest-neighbor coupling. The Hamiltonian
representing the entire system can be written as a sum of three terms,
$H=H_{kag}+H_{leads}+H_{tun}.$ (1)
The first term represents the Hamiltonian for the finite size kagome lattice
and it reads as,
$H_{kag}=\sum_{i}\mbox{\boldmath$c$}_{i}^{{\dagger}}\mbox{\boldmath$\epsilon$}\mbox{\boldmath$c$}_{i}+\sum_{\langle
ij\rangle}\mbox{\boldmath$c$}_{i}^{{\dagger}}\mbox{\boldmath$\tilde{t}$}_{ij}\mbox{\boldmath$c$}_{j}+\sum_{\langle
ij\rangle}i\,t_{so}\mbox{\boldmath$c$}_{i}^{{\dagger}}[\vec{\sigma}\times\hat{d}_{ij}]_{z}\mbox{\boldmath$c$}_{j}$
(2)
where,
$\mbox{\boldmath$c$}^{\dagger}_{i}=\left(\begin{array}[]{cc}c_{i,\uparrow}^{\dagger}&c_{i,\downarrow}^{\dagger}\end{array}\right);$
$\mbox{\boldmath$c$}_{i}=\left(\begin{array}[]{c}c_{i,\uparrow}\\\
c_{i,\downarrow}\end{array}\right);$
$\mbox{\boldmath$\epsilon$}=\left(\begin{array}[]{cc}\epsilon&0\\\
0&\epsilon\end{array}\right)$ and
$\mbox{\boldmath$\tilde{t}$}_{ij}=\tilde{t}_{ij}\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right).$
Here, $\epsilon$ is the on-site potential energy and for a perfectly ordered
system it is set equal to $0$ for all the atomic sites.
$c_{i,\sigma}^{{\dagger}}$ and $c_{i,\sigma}$ correspond to the creation and
annihilation operators, respectively, of an electron with spin $\sigma$ at the
$i$-th site of the conductor. $\tilde{t}_{ij}$ represents the isotropic
hopping strength between nearest-neighbor sites in presence of magnetic field.
The effect of the magnetic field $\vec{B}$ ($=\vec{\nabla}\times\vec{A}$) is
incorporated in the hopping term $\tilde{t}_{ij}$ through the Peierl’s phase
factor and it can be written as,
$\tilde{t}_{ij}=te^{-i\frac{2\pi}{\phi_{0}}\int\limits^{\vec{r}_{j}}_{\vec{r}_{i}}\vec{A}.\vec{dl}}$
where $t$ is the hopping strength in absence of magnetic field.
$\phi_{0}(=hc/e)$ is the elementary flux quantum. The specific choice of the
vector potential $\vec{A}$ in this case and the exact calculation of the
Peierl’s phase factor are discussed in the subsequent sub-section.
Figure 2: (Color online). Schematic view of a Kagome lattice in presence of a
perpendicular magnetic field where the structural unit (dashed region) and the
co-ordinate axes are shown. The length and width of the lattice strip are
determined by two parameters, viz, $N_{x}$ and $N_{y}$. Here, $N_{x}$
represents the number of structural units whereas $N_{y}$ is expressed as
$(n_{max}+1)/2$, where $n_{max}$ represents the total number of horizontal
lines parallel to $X$ axis. For this geometry $n_{max}=3$.
The third term corresponds to the Rashba type SO coupling in the system
through spin dependent nearest-neighbor hopping term which introduces spin
flipping in the system. The quantity $t_{so}$ estimates the strength of the
Rashba SO interaction originated due to structural inversion asymmetry of the
confining potential and different band offsets at the heterostructure quantum
well interface. In this term, $\vec{\sigma}$ is the spin angular momentum of
the electron and $\hat{d}_{ij}$ is the unit vector along the direction from
$i$-th site to $j$-th site.
The four metallic leads attached to the conductor are considered to be semi-
infinite and ideal, i.e., without any disorder and SO interaction. The leads
are described by a similar non-interacting single particle Hamiltonian as
written below.
$H_{leads}=\sum_{\alpha=1,2,3,4}H_{\alpha}$ (3)
where,
$H_{\alpha}=\sum\limits_{n}\epsilon_{l}c_{n}^{{\dagger}}c_{n}+\sum_{\langle
mn\rangle}t_{l}c^{{\dagger}}_{m}c_{n}.$ (4)
Similarly, the conductor-to-lead coupling is described by the following
Hamiltonian.
$H_{tun}=\sum_{\alpha=1,2,3,4}H_{tun,\alpha}$ (5)
Here,
$H_{tun,\alpha}=t_{c}[c^{{\dagger}}_{i}c_{m}+c^{{\dagger}}_{m}c_{i}]$ (6)
In the above expression, $\epsilon_{l}$ and $t_{l}$ stand for the site energy
and nearest-neighbor hopping between the sites of the leads. The coupling
between the leads and the conductor is defined by the hopping integral
$t_{c}$. In Eq. 6, $i$ and $m$ belong to the boundary sites of the kagome
ribbon and leads, respectively. The summation over $\alpha$ is due to
incorporation of the four side-attached leads.
### II.2 Calculation of the Peierl’s phase factor
Now we proceed to evaluate the Peierl’s phase factor in the term
$\tilde{t}_{ij}$.
We choose the vector potential $\vec{A}$ in the form,
$\vec{A}=-By\>\hat{x}+\frac{By}{\sqrt{3}}\>\hat{y}=(-1,\frac{1}{\sqrt{3}},0)\,By.$
(7)
This specific choice is followed from a literature schreiber , and the purpose
of doing that is solely the simplification of the factor
$\int\vec{A}.\vec{dl}$ along a particular direction ($\zeta$ axis in this
case).
With this particular choice of $\vec{A}$ we determine $\tilde{t}_{ij}$ for
three different types of hopping paths in the kagome lattice as follows.
$\bullet\>\underline{\textbf{Case\;1:}}$ Our choice of gauge ensures that the
component of $\vec{A}$ along $\zeta$ (see Fig. 2) axis is zero. Therefore,
$\tilde{t}_{ij}=t$, for an electron moving along $\zeta$ axis ($+$ve, $-$ve or
its parallel direction). $\bullet\>\underline{\textbf{Case\;2:}}$ If we
consider the motion along $X$ axis, in general for the $n$-th line (Fig. 2) we
can write the hopping integral as,
$\displaystyle\tilde{t}_{ij}$ $\displaystyle=$ $\displaystyle
t\,e^{\frac{i8\pi
n\phi}{\phi_{0}}}\>\mbox{\scriptsize{(hopping\;along\;+ve\;X\;axis)}}$ (8)
$\displaystyle=$ $\displaystyle t\,e^{\frac{-i8\pi
n\phi}{\phi_{0}}}\>\mbox{\scriptsize{(hopping\;along\;-ve\;X\;axis)}}$
where, $\phi$ is the flux through a smallest triangle of the lattice, and
$n=0,1,2,3,\ldots,(2N_{y}-1)$. $\bullet\>\underline{\textbf{Case\;3:}}$
Finally, we consider the hopping along $k$-th site to $i$-th site and all its
parallel directions (see Fig. 3).
It can be shown by straightforward algebra that for an upward pointing
triangle ($\bigtriangleup ijk$) the modified hopping strengths are given by,
$\displaystyle\tilde{t}_{k\rightarrow
i}=t\,e^{-\frac{i8\pi\phi}{\phi_{0}}(n+\frac{1}{4})}\hskip
5.69046pt\mbox{and}\hskip 5.69046pt\tilde{t}_{i\rightarrow
k}=t\,e^{\frac{i8\pi\phi}{\phi_{0}}(n+\frac{1}{4})}$ (9)
The value of $n$ belongs to the base line of the triangle.
Similarly, for a downward pointing triangle
Figure 3: (Color online). Upward and downward pointing triangles labeled with
proper site indices.
($\bigtriangleup ik^{\prime}j^{\prime}$) the modified hopping integrals can be
written as,
$\displaystyle\tilde{t}_{i\rightarrow
k^{\prime}}=te^{-\frac{i8\pi\phi}{\phi_{0}}(n+\frac{3}{4})}\hskip
4.26773pt\mbox{and}\hskip 4.26773pt\tilde{t}_{k^{\prime}\rightarrow i}$
$\displaystyle=$ $\displaystyle
te^{\frac{i8\pi\phi}{\phi_{0}}(n^{\prime}-\frac{1}{4})}$ (10)
Here, $n^{\prime}=n+1$, as for a downward pointing triangle the sites
$j^{\prime}$ and $k^{\prime}$ does not belong to the same value of $n$.
Following the above prescription we incorporate the effect of magnetic field
quite easily in our lattice geometry. But there are other ways through which
we can introduce the effect of magnetic field in lattice models. For example,
Nori et al. have used a different approach of lattice path integral technique
nori1 ; nori6 ; nori2 ; nori3 ; nori4 ; nori5 to describe the effect of
magnetic field in different lattice structures. Particularly, this technique
allows us to evaluate the physical quantities in terms of explicit functions
for a continuously tunable flux.
### II.3 Expressions of Longitudinal ($G_{L}$) and spin Hall conductances
($G_{SH}$)
According to spin Hall phenomenology, in our model pure spin current is
predicted to flow through the transverse leads, due to the flow of charge
current through the longitudinal leads. Hence, the linear response
longitudinal (4-probe) and Spin Hall conductances are defined as
$G_{L}=\frac{I^{q}_{4}}{V_{1}-V_{4}}$ (11)
and
$G_{sH}=\frac{\hbar}{2e}\frac{I^{s}_{2}}{V_{1}-V_{4}}$ (12)
where, $I_{4}^{q}$ and $I_{2}^{s}$ are the charge current and spin current
flowing through the lead-4 and lead-2, respectively. $V_{m}$ $(m=1,2,3$ and
$4)$ is the potential at the $m$-th lead.
Now, following Landauer-Büttiker multi-probe formalism the charge and spin
currents flowing through the lead $m$ with potential $V_{m}$, can be written
in terms of spin resolved transmission probabilities as pareek ,
$I_{m}^{q}=\frac{e^{2}}{h}\sum_{n,\sigma^{\prime},\sigma}(T_{nm}^{\sigma\sigma^{\prime}}V_{m}-T_{mn}^{\sigma\sigma^{\prime}}V_{n})$
(13)
$I_{m}^{s}=\frac{e^{2}}{h}\sum_{n,\sigma^{\prime}}\left[(T^{\sigma^{\prime}\sigma}_{nm}-T^{\sigma^{\prime}-\sigma}_{nm})V_{m}+(T^{-\sigma\sigma^{\prime}}_{mn}-T^{\sigma\sigma^{\prime}}_{mn})V_{n}\right]$
(14)
Considering linear transport regime, at absolute zero temperature the linear
conductance $(G_{pq})$ is obtained using Landauer conductance formula land ,
$G_{pq}^{\sigma\sigma^{\prime}}=\frac{e^{2}}{h}T_{pq}^{\sigma\sigma^{\prime}}(E_{F}).$
(15)
Using Landauer conductance formula spin current through $m$-th lead can be re-
written in terms of spin resolved conductances as,
$\displaystyle I^{s}_{m}$ $\displaystyle=$
$\displaystyle\sum_{n}[(G_{nm}^{\uparrow\uparrow}+G_{nm}^{\downarrow\uparrow}-G_{nm}^{\uparrow\downarrow}-G_{nm}^{\downarrow\downarrow})V_{m}$
(16)
$\displaystyle+(G_{mn}^{\downarrow\uparrow}+G_{mn}^{\downarrow\downarrow}-G_{mn}^{\uparrow\uparrow}-G_{mn}^{\uparrow\downarrow})V_{n}]$
Equation 16 can be simplified in terms of two quantities defined as follows.
$\displaystyle G^{in}_{mn}$ $\displaystyle=$ $\displaystyle
G_{mn}^{\uparrow\uparrow}+G_{mn}^{\uparrow\downarrow}-G_{mn}^{\downarrow\uparrow}-G_{mn}^{\downarrow\downarrow}$
$\displaystyle G^{out}_{mn}$ $\displaystyle=$ $\displaystyle
G_{mn}^{\uparrow\uparrow}+G_{mn}^{\downarrow\uparrow}-G_{mn}^{\uparrow\downarrow}-G_{mn}^{\downarrow\downarrow}$
(17)
Physically the term $\sum_{n}G_{nm}^{out}V_{m}$ is the total spin current
flowing from the $m$-th lead with voltage $V_{m}$ to all other $n$ leads,
while the term $\sum_{n}G_{mn}^{in}V_{n}$ defines the total spin current
flowing into the $m$-th lead from the all other $n$ leads having potential
$V_{n}$.
Therefore, the spin current through lead $m$ becomes,
$I_{m}^{s}=\sum_{n}\left[G_{nm}^{out}V_{m}-G_{mn}^{in}V_{n}\right]$ (18)
Hence, following spin Hall phenomenology, in our set-up since the transverse
leads are voltage probes, the net charge currents through lead-$2$ and
lead-$3$ are zero i.e., $I_{2}^{q}=I_{3}^{q}=0$. On the other hand, as the
currents in the various leads depend only on voltage differences among them,
we can set one of the voltages to zero without any loss of generality. Here,
we set $V_{4}=0$. So, from Eq. 18 we have,
$I^{s}_{2}=\left(G_{12}^{out}+G_{32}^{out}+G_{42}^{out}\right)V_{2}-G_{23}^{in}V_{1}-G_{21}^{in}V_{1}$
(19)
Hence, the expression of spin Hall conductance becomes
$G_{sH}=\frac{\hbar}{2e}\left[(G_{12}^{out}+G_{32}^{out}+G_{42}^{out})\frac{V_{2}}{V_{1}}-G_{23}^{in}\frac{V_{3}}{V_{1}}-G_{21}^{in}\right]$
(20)
This is the most general expression of SHC, but it can further be simplified
if we assume that the leads are connected to a geometrically symmetric ordered
bridge, so, $\frac{V_{3}}{V_{1}}=\frac{V_{2}}{V_{1}}=0.5$. Now, in absence of
external magnetic flux only the Rashba SO interaction does not break the time
reversal symmetry (TRS). In general, for a time reversal invariant system
$G_{pq}^{\sigma\sigma^{\prime}}=G_{qp}^{-\sigma^{\prime}-\sigma}$, which is
equivalent to write $G_{pq}^{in}=-G_{qp}^{out}$. Therefore, obeying TRS the
expression for the spin Hall conductance can be written in terms of spin
resolved transmission probabilities in a compact form as,
$G_{sH}=\frac{e}{8\pi}\left[T_{42}^{out}+2\;T_{32}^{out}+3\;T_{12}^{out}\right]$
(21)
One important point to be noted here is that unlike the charge current spin
current is a vector quantity, which immediately gives rise to the three
different components of spin Hall conductances ($G_{sH}^{x}$, $G_{sH}^{y}$ and
$G_{sH}^{z}$) and they are defined as follows.
$\displaystyle G_{sH}^{x}=I_{2}^{x}/(V_{1}-V_{4})$ $\displaystyle
G_{sH}^{y}=I_{2}^{y}/(V_{1}-V_{4})$ $\displaystyle
G_{sH}^{z}=I_{2}^{z}/(V_{1}-V_{4})$ (22)
Using Eq. 21 all the three different components of SHC can be evaluated, only
the choice of basis while constructing the matrices is important. For example,
if we are working in $\sigma_{z}$ diagonal representation (i.e., the $Z$ axis
chosen to be the spin quantization axis), then Eq. 21 gives the $z$-component
of SHC. Similarly, other components can also be evaluated by a simple unitary
transformation to the basis set. In the present work we are working only with
the $z$-component of SHC.
In a similar way from Eqs. 11 and 13 the expression of longitudinal
conductance can be written as
$G_{L}=\frac{e^{2}}{h}\left[T_{41}+0.5\;T_{42}+0.5\;T_{43}\right].$ (23)
### II.4 Evaluation of the Transmission Probability by Green’s function
technique
To obtain the transmission probability of an electron through such a four-
probe mesoscopic bridge system, we use Green’s function formalism. Within the
regime of coherent transport and in the absence of Coulomb interaction this
technique is well applied.
The single particle Green’s function operator representing the entire system
for an electron with energy $E$ is defined as,
$G=\left(E-H+i\eta\right)^{-1}$ (24)
where, $\eta\rightarrow 0^{+}$.
Following the matrix forms of $H$ and $G$ the problem of finding $G$ in the
full Hilbert space of $H$ can be mapped exactly to a Green’s function
$G_{kag}^{eff}$ corresponding to an effective Hamiltonian in the reduced
Hilbert space of the conductor (i.e., the kagome lattice itself) and we have,
$\mbox{\boldmath${\mathcal{G}}$=$G_{kag}^{eff}$}=\left(\mbox{\boldmath$E-H_{kag}-\sum\limits_{\alpha,\sigma}\Sigma_{\alpha}^{\sigma}$}\right)^{-1}$
(25)
where,
$\mbox{\boldmath$\Sigma_{\alpha}^{\sigma}$}=\mbox{\boldmath$H_{tun,\alpha}^{{\dagger}}G_{\alpha}H_{tun,\alpha}$}$
(26)
These $\Sigma_{\alpha}$ ($\alpha=1,2,3$ and $4$) are the contact self-energies
introduced to incorporate the effect of coupling of the conductor to the
attached ideal leads. It is evident from Eq. 26 that the form of the self-
energies are independent of the conductor itself through which spin
transmission is studied.
Following Lee and Fisher’s expression for the probability of an electron to
transmit from lead q with spin $\sigma^{\prime}$ to lead p with spin $\sigma$
can be written as lee ,
$T_{pq}^{\sigma\sigma^{\prime}}=\mbox{Tr}\mbox{\boldmath[$\Gamma_{p}^{\sigma}\mathcal{G}^{r}\Gamma_{q}^{\sigma^{\prime}}\mathcal{G}^{a}$]}.$
(27)
$\Gamma_{k}^{\sigma}$’s are the coupling matrices representing the coupling
between the kagome lattice and the leads, and they are mathematically defined
by the relation,
$\mbox{\boldmath$\Gamma_{k}^{\sigma}$}=i\left[\mbox{\boldmath$\Sigma_{k}^{\sigma}-\Sigma_{k}^{\sigma{\dagger}}$}\right]$
(28)
Here, $\Sigma_{k}^{\sigma}$ and $\Sigma_{k}^{\sigma{\dagger}}$ are the
retarded and advanced self-energies associated with the $k$-th lead,
respectively.
It is shown in literature by Datta et al. datta1 ; datta2 that the self-
energy can be expressed as,
$\mbox{\boldmath${\Sigma^{\sigma}_{k}}$}=\mbox{\boldmath$\Lambda_{k}^{\sigma}$}-i\mbox{\boldmath$\Delta_{k}^{\sigma}$}.$
(29)
The real part of self-energy describes the shift of the energy levels and the
imaginary part corresponds to the broadening of the levels. The finite
imaginary part appears due to incorporation of the semi-infinite leads having
continuous energy spectrum. Therefore, the coupling matrices can easily be
obtained from the self-energy expression and is expressed in the form,
$\mbox{\boldmath$\Gamma_{k}^{\sigma}$}=-2\,{\mbox{Im}}(\mbox{\boldmath$\Sigma_{k}^{\sigma}$}).$
(30)
### II.5 Evaluation of the Self-Energy
Finally, it remains the evaluation of the self-energies for the finite-width,
multi-channel, square lattice leads. Now for the semi-infinite longitudinal
leads (leads $1$ and $4$) as the translational invariance is preserved in $X$
direction only, the wave function amplitude at any arbitrary site $m$ of the
leads can be written as, $\phi_{m}\propto e^{ik_{x}m_{x}a}\sin(k_{y}m_{y}a)$,
with energy
$E=2t_{L}[\cos(k_{x}a)+\cos(k_{y}a)]$ (31)
In Eq. 31, $k_{x}$ is continuous, while $k_{y}$ has discrete values given by,
$k_{y}(i)=\frac{i\pi}{(M+1)a}$ (32)
Here, $i=1,2,3\ldots M$. $M$ is the total number of transverse channels in the
leads, and in our case $M=2N_{y}$.
The self-energy matrices are constructed in the reduced Hilbert space of the
conductor itself. These matrices have non-zero elements only for the sites on
the edge layer of the sample coupled to the leads and it is given by,
$\Sigma^{r}_{1(4)}(m,n)=\frac{2}{M+1}\sum_{k_{y}}\sin(k_{y}m_{y}a)\Sigma^{r}(k_{y})\sin(k_{y}n_{y}a)$
(33)
$\Sigma^{r}(k_{y})$ is the self-energy of each transverse channel and for a
specific value of $k_{y}$ it becomes,
$\Sigma^{r}(k_{y})=\frac{t_{c}^{2}}{2t_{L}^{2}}\left[E-\epsilon(k_{y})-i\sqrt{4t_{L}^{2}-(E-\epsilon(k_{y}))^{2}}\right]$
(34)
with $\epsilon(k_{y})=2t_{L}\cos(k_{y}a)$, when the energy lies within the
band, i.e., $|E-\epsilon(k_{y})|<2t_{L}$; and
$\Sigma^{r}(k_{y})=\frac{t_{c}^{2}}{2t_{L}^{2}}\left[E-\epsilon(k_{y})\mp\sqrt{(E-\epsilon(k_{y}))^{2}-4t_{L}^{2}}\right]$
(35)
when the energy lies outside the band. Here the $-ve$ sign appears for
$E>\epsilon(k_{y})+2|t_{L}|$ and $+ve$ sign comes for
$E<\epsilon(k_{y})-2|t_{L}|$.
The self-energy matrices for the other two leads (leads $2$ and $3$) are also
constructed in a similar way. The only difference in this case is that the
translational invariance is preserved along $Y$ direction, and accordingly,
the finite values of $k_{x}$ are chosen.
## III Results and discussion
We start analyzing the numerical results by referring to the values of
different parameters used for our calculation. Throughout the presentation we
set $\epsilon=\epsilon_{l}=0$, and fix all the hopping integrals ($t$, $t_{l}$
and $t_{c}$) at the value $1$. The energy scale is measured in unit of $t$ and
choose the unit where $c=h=e=1$. The Rashba coupling strength $t_{so}$ is also
scaled in unit of $t$, and it is usually chosen as $t_{so}\lesssim t$. The
magnetic flux $\phi$ is measured in unit of elementary flux quantum $\phi_{0}$
(=$hc/e$).
### III.1 Variation of spin Hall conductances with Fermi energy
In Fig. 4 we plot the $z$-component of spin Hall conductance $(G_{sH}^{z})$ as
a function of Fermi energy $(E_{F})$ for two different values of Rashba
coupling strengths and different system sizes. This figure demonstrates the
fact that unlike the bulk or infinite system, in this case the spin Hall
conductance does not have a universal value ($\pm\frac{e}{8\pi}$ as predicted
in the case of an infinite 2DEG), rather it depends explicitly on the system
parameters like Fermi energy ($E_{F}$), strength of Rashba SO interaction
($t_{so}$), system size, etc. The non-zero spin Hall conductance observed in
Fig. 4 is a consequence of the fact that in presence of the Rashba SO
interaction in the conductor, the up and down spin electrons flow in opposite
transverse directions even in the absence of external magnetic flux, leading
to a pure spin current, in response to the flow of charge current along the
longitudinal direction.
It has already been shown in literature that geometries like kagome lattice,
graphene flakes, etc., exhibit some unique features of the spin Hall
conductance due to their fascinating structures.
Figure 4: (Color online). Variation of $G_{sH}^{z}$ with Fermi energy in the
absence of external magnetic flux for two different sets of parameter values
(shown inside the figure).
Following Kubo formalism, Liu et al. have analyzed the variation of conserved
spin Hall conductance with respect to the Fermi energy for an infinite kagome
lattice kagome . Considering a two-band approximation and treating the Rashba
SO coupling as perturbation, they have analyzed the presence of various spin
Hall plateaus (at $\pm\frac{e}{8\pi}$ and $\pm\frac{e}{4\pi}$) with Berry
phase interpretation. It has been shown that when the Fermi energy lies within
the range $-2<E_{F}<0$, the contribution to spin Hall conductivity due to the
conventional part ($\sigma_{xy}^{s0}$) is $-1$ (in units of $\frac{e}{8\pi}$).
In this case, by expanding the Hamiltonian around the $\Gamma$ point upto
first order in the Rashba coefficient $t_{so}$ they have established the
similarity of the Rashba Hamiltonian with that of a semiconductor 2DEG, and by
straightforward analytical calculation spin Hall conductivity is obtained as
$-\frac{e}{8\pi}$. But with the increase in electron filling, within the range
$0<E_{F}<1$, expansion of the Hamiltonian around the K point (‘K-valley’
Hamiltonian) exhibits a Dirac type spectrum with linear dependence of energy
on momentum, and accordingly, the value of conventional spin Hall conductivity
gets the value $2$. Furthermore the anti-symmetry at the band center is
completely due to the particle-hole symmetry of the employed tight-binding
Hamiltonian. The variation in sign and magnitude of spin Hall conductivity in
the range ($-2<E_{F}<1$) is associated with the change in Fermi surface
topology surrounding the high symmetry $\Gamma$ and K Brillouin Zone (BZ)
points. The analysis is completely suitable for an infinite system. Here, in
our work we try to investigate numerically whether the essence of these
predicted features in a realistic, finite size, small scale, mesoscopic system
are still present or not.
In this figure (Fig. 4) we observe that for a particular system size with
finite Rashba coupling strength, the spin Hall conductance
Figure 5: (Color online). Variation of four-probe longitudinal conductance as
a function of Fermi energy in the absence of external magnetic flux.
is nearly an antisymmetric function with respect to the band center i.e.,
$G^{z}_{sH}=0$ at $E_{F}=1$, reflecting the particle-hole symmetry of the
tight-binding Hamiltonian. It is a consequence of the fact that the spin
current is defined as the difference between spin resolved charge currents
($I_{\uparrow}$ and $I_{\downarrow}$), and the spin current carried by
negatively charged electrons at $E_{F}<1$ can be interpreted as the
propagation of positively charged holes with opposite spin in opposite
direction. Apart from that, sign reversals take place also at $E_{F}=0$ and at
$E_{F}=2$ as already predicted for the infinite system. The oscillatory
behavior in SHC for a smaller system size is entirely the finite size effect.
The feature of sign reversals in SHC gets prominent with the increase of
system size. Hence, our numerical results for finite sized systems provide all
the essential features of SHC pattern observed in an infinite kagome lattice.
### III.2 Variation of longitudinal conductance with Fermi energy
In Fig. 5 we explore the variation of four-probe longitudinal conductance in
presence of finite Rashba interaction strength ($t_{so}\neq 0$) as a function
of Fermi energy. It exhibits quite a similar behavior to our previous
investigation kagomeold of two-terminal longitudinal conductance. The
addition of the two other semi-infinite transverse finite width leads allows
some extra phase-breaking paths and thereby lifting the transmission zeros and
hence broadening the conductance peaks. The conductance spectrum reveals
itself the energy eigenstates of the finite size system. The presence of
Rashba coupling does not affect the energy spectrum in a significant way apart
from shifting of the energy eigenvalues a little.
### III.3 Spin Hall conductance as a function of Rashba coupling strength
The dependence of spin Hall conductance of a finite size kagome lattice on the
Rashba coupling strength at the typical energy $E_{F}=-0.8$ is illustrated in
Fig. 6. Unlike the results predicted by using linear
Figure 6: (Color online). Z-component of spin Hall conductance as a function
of Rashba coupling strength in the absence of magnetic flux for a typical
system size and at a particular Fermi energy.
response theory, in this case $G_{sH}^{z}\rightarrow 0$ as $t_{so}\rightarrow
0$, and $G_{sH}^{z}$ shows non-negligible, non-zero values within the range
$0.1<t_{so}<5$. The strength of Rashba SO interaction can be tuned externally
by using a gate voltage. Although the physically accessible range of $t_{so}$
(in unit of $t$) is $0.001$ to $1$, but here we plot the spectrum for a wider
range, a part of which may be beyond the experimental reach till date. For a
sufficiently large SO coupling, $G_{sH}^{z}$ again drops to $0$, because large
SO coupling in the conductor forms a large potential barrier for the incident
electrons, yielding a very small probability to transmit through the
conductor.
### III.4 Effect of Magnetic field
#### III.4.1 Energy-flux characteristics
Figure 7 depicts the energy-flux characteristics of a finite size ($N_{x}=6$
and $N_{y}=2$) kagome lattice both in the presence and absence of Rashba type
SO coupling. The energy eigenvalues are obtained by diagonalizing the
Hamiltonian and the energy-flux spectrum is often called the Hofstader
spectrum. Form Fig. 7(a) it is clearly visible that a highly degenerate level
exists at $E=-2t$ when $\phi$ is set equal to $0$, and the application of a
very small non-zero flux starts to break the degeneracy. The presence of even
a very small magnetic flux affects the phase of the electronic wave function
and thus tends to destroy the quantum interference and eventually breaks the
flat band kimura . Again at the half flux-quantum ($\phi=\phi_{0}/2$), the
flat band re-appears because of the interference effect but at the energy
$E=2t$. The energy levels are periodic
Figure 7: (Color online). Energy-flux characteristics of a finite size kagome
lattice in (a) absence and (b) presence of Rashba type SO coupling. The
position of the highly degenerate energy level at $E=-2$ is slightly shifted
due to Rashba interaction but the degeneracy is not broken.
in $\phi$, showing $\phi_{0}$ (which is set to $1$ in our calculation) flux-
quantum periodicity and the energy spectrum is mirror symmetric about
$\phi=0.5$. The presence of a small non-zero SO interaction does not affect
the degeneracy much which is clearly noticed from Fig. 7(b).
#### III.4.2 Effect of magnetic flux on spin Hall conductance
In Fig. 8 we plot the nature of SHC as a function of Fermi energy for
different values of magnetic flux $\phi$, where (a), (b) and
Figure 8: (Color online). Variation of $G_{sH}^{z}$ as a function of Fermi
energy for three different values of flux, where (a) $\phi=0$ (b) $\phi=0.32$
and (c) $\phi=0.5$.
(c) correspond to $\phi=0$, $0.32$ and $0.5$, respectively. From the spectra
it is observed that by applying an arbitrary magnetic flux ($\phi=0.32$) the
features of SHE tend to get diminished, but interestingly at
$\phi=\phi_{0}/2$, all the above mentioned features re-appear being just the
inverse of the case for $\phi=0$, in accordance with the change of band
structure with the application of magnetic flux. These features can be
illustrated as follows.
If we write the Schrödinger equations for three different sites in a unit cell
(Fig. 9) of an infinite kagome lattice schreiber , we get,
$\displaystyle E\psi(MA,NA,1)$ $\displaystyle=$
$\displaystyle\psi(MA,NA,2)+\psi(MA,(N+1)A,2)+$ $\displaystyle
e^{\frac{i8\pi\phi}{\phi_{0}}(N+\frac{1}{4})}\psi(MA,NA,3)+$ $\displaystyle
e^{\frac{-i8\pi\phi}{\phi_{0}}(N+\frac{3}{4})}\psi((M-1)A,(N+1)A,3)$
$\displaystyle E\psi(MA,NA,2)$ $\displaystyle=$ $\displaystyle\psi(MA,NA,1)+$
$\displaystyle\psi((M-1)A,(N-1)A,1)+$ $\displaystyle e^{\frac{-i8\pi
N\phi}{\phi_{0}}}\psi((M-1)A,NA,3)+$ $\displaystyle e^{\frac{i8\pi
N\phi}{\phi_{0}}}\psi(MA,NA,2)$ $\displaystyle E\psi(MA,NA,3)$
$\displaystyle=$ $\displaystyle e^{\frac{-i8\pi
N\phi}{\phi_{0}}}\psi(MA,NA,2)+$ (36) $\displaystyle e^{\frac{i8\pi
N\phi}{\phi_{0}}}\psi((M+1)A,NA,2)+$ $\displaystyle
e^{\frac{-i8\pi\phi}{\phi_{0}}(N+\frac{1}{4})}\psi(MA,NA,1)+$ $\displaystyle
e^{\frac{-i8\pi\phi}{\phi_{0}}(N-\frac{1}{4})}\psi(MA,(N-1)A,1)$
where, $\psi(MA,NA,j)$ denotes the wave amplitude at a particular site
($MA,NA,j$) of the unit cell. Here, $M$ indicates the $M$-th triangle along
the $X$ axis, $N$ indicates $N$-th triangle along the $\zeta$ axis and $j=1,2$
and $3$ represents the vertex in the triangle.
Here, for any arbitrary flux, translational invariance is lost along all
directions and the hopping integrals become different. As a result the sharp
features like the sign reversal about the symmetry points of the Fermi surface
get reduced because of the elastic scattering, which is
Figure 9: (Color online). Unit cell configuration of an infinite kagome
lattice.
visible in Fig. 8(b). Very interestingly, again at $\phi=\phi_{0}/2$,it can be
shown from the above equations that the translational invariance is again
restored along both the $X$ and $\zeta$ directions which can be deducted from
the above equations, and therefore, scattering effect gets completely
suppressed and SHC re-appears but with inverse manner due to the change in
band structure.
It can also be observed that for $\phi=\frac{n}{8m}$ ($n$ and $m$ being
integers), translational invariance is retained (increasing the unit cell
dimension) and hence system size has an important role as we are dealing with
a finite size kagome lattice. In this case, some anisotropy is introduced in
the system through the hopping terms. Therefore, the SHC pattern changes
accordingly due to the changes in the Fermi surface topology. In our case
($N_{x}=N_{y}=8$) we observe that for $\phi=\frac{1}{4}$ and $\frac{3}{4}$ a
regular pattern in SHC is obtained along with the sign reversal at the
Figure 10: (Color online). Variation of $G_{sH}^{z}$ as a function of Fermi
energy for (a) $\phi=\frac{1}{4}$ (b) $\phi=\frac{3}{4}$.
band-center (see Fig. 10). The application of an external magnetic flux breaks
the time reversal symmetry (TRS), but still SHE is still observed though
reduced due to the elastic scattering.
The reason is that in the presence of an external magnetic field along the $Z$
direction, an ordinary Hall voltage appears along the transverse edges of the
sample and since the transverse leads are voltage probes no charge current
will flow through them. Therefore the tendency of accumulation of charges of
both spins somehow tries to reduce the features of SHE, which is entirely an
effect due to the spin accumulation along the transverse edges and the flow of
spin current in that direction, if leads are connected.
Before we end this section, we would like to point out that in a kagome
lattice geometry Nori et al. nori6 have described several physical phenomena
in the presence of a transverse magnetic field. They have analyzed the
formation of cusps in the variation of superconducting transition temperature
as a function of magnetic field in this geometry at
$\phi=\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{3}{4},\frac{7}{8}$, etc.,
considering lower order approximation in the lattice path integral method
which directly reflects the lattice topology on the electronic properties.
Though our present analysis is quite different from their analysis, but here
we also establish that the SHE exactly reappears at these typical values of
magnetic flux $\phi$ which reflects the nature of the lattice model.
## IV Closing remarks
In conclusion, in the present paper we have studied different aspects of
mesoscopic spin Hall effect induced by Rashba type SO interaction in a kagome
lattice geometry attached to four finite width probes both in the presence and
absence of external magnetic flux. We have evaluated spin Hall conductance
(SHC) and longitudinal conductance for a finite size system in the clean limit
using four-terminal Landauer-Büttiker formalism and Green’s function
technique. In the absence of magnetic flux, we have observed that due to the
change in Fermi surface topology SHC changes its sign at certain values of
Fermi energy, along with the band center. Unlike the infinite system (where
SHC is a universal constant $\pm\frac{e}{8\pi}$), here SHC depends on the
external parameters like SO coupling strength, Fermi energy, etc. We have
shown that in presence of arbitrary magnetic flux, periodicity of the system
is lost and the features of SHC get suppressed because of the weak elastic
scattering, but not lost completely. On the other hand, at some typical values
of flux ($\phi=\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}\ldots$, etc.) the
system retains its periodicity again depending on the system size and the
features of spin Hall effect (SHE) re-appears.
It is also important to note that in our theoretical model we have included
the effect of magnetic field through the phase factor in the hopping term and
ignored the Zeeman term in the Hamiltonian since in this geometry, the Zeeman
coupling strength ($\sim 0.03$ meV) is much smaller than the hopping integral
($\sim 1.5$ meV) which has been discussed in the earlier work kimura .
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|
arxiv-papers
| 2012-01-22T14:49:03 |
2024-09-04T02:49:26.590641
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Moumita Dey, Santanu K. Maiti and S. N. Karmakar",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1201.4563"
}
|
1201.4633
|
# Interplay of magnetic field and geometry in magneto-transport of mesoscopic
loops with Rashba and Dresselhaus spin-orbit interactions
Shreekantha Sil Department of Physics, Visva-Bharati, Santiniketan, West
Bengal-731 235, India Santanu K. Maiti santanu@post.tau.ac.il School of
Chemistry, Tel Aviv University, Ramat-Aviv, Tel Aviv-69978, Israel Arunava
Chakrabarti Department of Physics, University of Kalyani, Kalyani, West
Bengal-741 235, India
###### Abstract
Electronic transport in closed loop structures is addressed within a tight-
binding formalism and in the presence of both the Rashba and Dresselhaus spin-
orbit interactions. It has been shown that any one of the spin-orbit (SO)
fields can be estimated precisely if the other one is known, by observing
either the transmission resonance or anti-resonance of unpolarized electrons.
The result is obtained through an exact analytic calculation for a simple
square loop, and through a numerically exact formulation for a circular ring.
The sensitivity of the transport properties on the geometry of the
interferometer is discussed in details.
###### pacs:
73.23.Ad, 71.70.Ej
## I Introduction
Spintornics is a recent field of utmost research interest that include
magnetic memory circuits, quantum computers zutic ; datta ; ando ; aharony1
magnetic nano-structures and quasi one-dimensional semiconductor rings which
have been acknowledged as ideal candidates for investigating the effects of
quantum coherence in low-dimensions, and have been examined as the prospective
quantum devices moldo ; engels ; vlaminck ; ando1 . A central mechanism
governing the physics in the meso- and nano-length scales are the Rashba and
Dresselhaus spin-orbit interactions which result from a structural inversion
asymmetry rashba , and bulk inversion asymmetry dresselhaus ; meier
respectively. The effects are pronounced in quantum rings formed at the
interface of two narrow gap semiconductors, as already discussed in the
literature koga ; premper .
Needless to say, an accurate estimation of the spin-orbit interaction (SOI)
strengths is crucial in the field of spintronics. The Rashba spin-orbit
interaction (RSOI) can be controlled by a gate voltage placed in the vicinity
of the sample premper ; cmhu ; grundler and hence, can be ‘measured’.
Comparatively speaking, reports on the techniques of measurement of the
Dresselhaus spin-orbit interaction (DSOI) are relatively few premper ; cmhu ;
grundler . Very recently we have put forward an idea of estimating the DSOI
strength provided the RSOI is known by measuring a minimum in the Drude weight
santanu . A minimum in the Drude weight appears only when the strengths of the
RSOI and DSOI are exactly identical.
From a closer look at the Rashba and Dresselhaus SO interactions it turns out
that both the interactions are equivalent to the $SU(2)$ gauge field which
introduces a phase in the wave function. In this communication we describe the
role of this phase and considering its effect on quantum interference we
develop a simple idea about how one of the two SOI’s can be estimated while
the other is known. This quantum interference effect in presence of SO
interactions has not been described in our previous work santanu and the
present analysis may provide a basic understanding of designing switching
devices for spintronic applications in near future. A simple version of a
quantum ring, in the form of a loop with a rhombic geometry is considered for
an analytical attack on the problem, while numerically exact results are
provided for circular rings with and without disorder and with a magnetic flux
threading these polygonal structures. We adopt a tight-binding formalism in
contrast to a recently proposed scheme where a continuous version of the model
is presented ramaglia to consider the combined effect of the RSOI and the
DSOI. When strengths of
Figure 1: (Color online). Schematic view of a mesoscopic square loop subjected
to RSOI and DSOI and connected to the leads (source and drain) at its two
extremities.
the RSOI and DSOI are equal, the end to end transmission across the rhombic
loop is shown to be equal to unity and vanishes when the loop is threaded by a
magnetic flux $\phi$ equal to the half flux-quantum ($\phi_{0}/2$). Thus, one
can estimate the DSOI by observing the peak when $\phi=0$ or dip when
$\phi=\phi_{0}/2$ in the transmission (conductance) spectrum when the RSOI is
known, and vice versa.
The idea is extended to the case of rings with circular geometry, where we
have evaluated the transmission coefficient numerically. The essential
differences in the cases of a rhombic loop and a circular ring are discussed
to highlight the sensitivity of the results on the loop geometry.
In what follows we describe the procedure and the results. In section II, we
present the model and the method. In section III we discuss the sensitivity of
the results on the geometry of the closed loop structures, and in section IV
we draw our conclusions.
## II The model and the method
The Hamiltonian: Let us consider the rhombic loop depicted in Fig. 1 which is
threaded by a magnetic flux $\Phi$. Each side of the loop is of length $L$,
and the loop contains $N$ number of equispaced atomic sites with ‘lattice
constant’ $a$. Within a tight-binding framework the Hamiltonian for this
network in the presence of the RSOI and DSOI reads,
$\mbox{\boldmath$H$}=\mbox{\boldmath$H_{0}$}-i\alpha\mbox{\boldmath$H_{R}$}+i\beta\mbox{\boldmath$H_{D}$}$
(1)
where,
$\mbox{\boldmath$H_{0}$}=-\sum_{i}\left(\mbox{\boldmath$c_{i}^{\dagger}t$}\mbox{\boldmath$c_{i+1}$}e^{i\phi}+\mbox{\boldmath$c_{i+1}^{\dagger}t$}\mbox{\boldmath$c_{i}$}e^{-i\phi}\right)$
(2)
The Rashba and Dresselhaus spin-orbit parts of the Hamiltonian, viz, $H_{R}$
and $H_{D}$, are given by,
$H_{R}$ $\displaystyle=$
$\displaystyle\sum_{i}\left(\sin\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\mbox{\boldmath$c_{i+1}$}\right.-\left.\cos\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\mbox{\boldmath$c_{i+1}$}\right)e^{i\phi}+h.c.$
$H_{D}$ $\displaystyle=$
$\displaystyle\sum_{i}\left(-\cos\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\mbox{\boldmath$c_{i+1}$}\right.+\left.\sin\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\mbox{\boldmath$c_{i+1}$}\right)e^{i\phi}+h.c.$
(3)
where, $i$ refers to the atomic sites in the arms of the loop.
$\phi=2\pi\Phi/N\phi_{0}$, and $\phi_{0}=hc/e$, the fundamental flux-quantum.
The other operators in Eq. 3 are as follows.
$c_{i}$=$\left(\begin{array}[]{c}c_{i,\uparrow}\\\
c_{i,\downarrow}\end{array}\right)\hskip 5.69046pt\mbox{and}\hskip 5.69046pt$
$t$=$t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$.
Here the site energy of an electron at any $i$-th site is assumed to be zero
throughout the geometry. $t$ is the nearest-neighbor hopping integral.
$\alpha$ and $\beta$ are the isotropic nearest-neighbor transfer integrals
which measure the strengths of Rashba and Dresselhaus SOI, respectively.
$\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the Pauli spin matrices.
$c_{i,\sigma}^{\dagger}$ ($c_{i,\sigma}$) is the creation (annihilation)
operator of an electron at the site $i$ with spin $\sigma$
($\uparrow,\downarrow$).
Eigenvalues and eigenfunctions of the Hamiltonian: These are obtained by
adopting a $\bf{k}$-space description of the Hamiltonian given in Eq. 3, viz,
${\bf H}=\sum_{k}{\bf c_{k}^{\dagger}H_{k}c_{k}}$. Using discrete Fourier
transform $c_{k}=\frac{1}{\sqrt{N}}\sum_{n}c_{n}\exp(-i{\bf k}.n{\bf a})$, the
Hamiltonian matrix reads,
$\mbox{\boldmath$H_{k}$}=\left(\begin{array}[]{cc}\epsilon_{k}&\gamma_{k}+i\delta_{k}\\\
\gamma_{k}-i\delta_{k}&\epsilon_{k}\end{array}\right)\\\ $ (4)
where,
$\displaystyle\epsilon_{k}$ $\displaystyle=$ $\displaystyle-2t\cos(ka+\phi)$
$\displaystyle\gamma_{k}$ $\displaystyle=$ $\displaystyle
2\left(\alpha\sin\theta+\beta\cos\theta\right)\sin(ka+\phi)$
$\displaystyle\delta_{k}$ $\displaystyle=$ $\displaystyle
2\left(\alpha\cos\theta+\beta\sin\theta\right)\sin(ka+\phi)$
While writing the above expressions, we have set the lattice constant $a=1$.
The energy eigenvalues are obtained from Eq. 4, and are given by,
$E_{k_{\pm}}=\epsilon_{k}\pm\sqrt{\gamma_{k}^{2}+\delta_{k}^{2}}$.
Let us denote the left vertex of the loop in Fig. 1 as the ‘origin’ $(0,0)$. A
simple but lengthy algebra now allows one to write the wave function for an
energy $E$ at a distance $L$ along any arm of the rhombic loop as,
$|\Phi_{E}(L,a)\rangle=\mathcal{R}_{E}(L,\alpha,\beta,\theta)|\Phi_{E}(0)\rangle$
(5)
where, the elements of the matrix $\mathcal{R}_{E}(L,\alpha,\beta,\theta)$
are,
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{ik_{+}La}+e^{ik_{-}La}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{i(k_{+}La+\nu_{k+})}-e^{i(k_{-}La+\nu_{k-})}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{i(k_{+}La-\nu_{k+})}-e^{i(k_{-}La-\nu_{k-})}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{ik_{+}La}+e^{ik_{-}La}\right)$ (6)
Here, $\nu_{k\pm}=\tan^{-1}(\delta_{k\pm}/\gamma_{k\pm})$. $k_{\pm}$ are the
wave vectors corresponding to the energy values
$\epsilon_{k}\pm\sqrt{\gamma_{k}^{2}+\delta_{k}^{2}}$, as already mentioned.
Transmission of unpolarized electrons: Let us now assume that the electrons
enter the loop at the point $A$ through the source, and are drained out at
$B$. In addition, without any loss of generality, and to get a relatively
simple set of equations, we take our loop to be a square one with
$\theta=\pi/4$. For an electron traveling in the loop in the clockwise sense
with a specified energy $E$, the wave vector $k_{\pm}$ are the solutions of
the equation $E=E_{k_{\pm}}$, and can be explicitly written as,
$\displaystyle k_{\pm}a$ $\displaystyle=$
$\displaystyle\cos^{-1}\xi_{\pm}(E)-\phi$ (7)
where,
$\displaystyle\xi_{\pm}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\left[-\frac{E}{2t\sqrt{1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\right.$
(8) $\displaystyle\pm$
$\displaystyle\left.\frac{\alpha+\beta}{t}\sqrt{1-\frac{E^{2}}{4t^{2}(1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\right]$
In a similar manner, we need to work out the wave vectors $k^{\prime}_{\pm}$
for the electrons with the same energy $E$, and traveling in the counter-
clockwise sense. The result is,
$\displaystyle k^{\prime}_{\pm}a$ $\displaystyle=$
$\displaystyle\cos^{-1}\xi^{\prime}_{\pm}(E)-\phi$ (9)
where,
$\displaystyle\xi^{\prime}_{\pm}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\left[-\frac{E}{2t\sqrt{1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\right.$
(10) $\displaystyle\pm$
$\displaystyle\left.\frac{|\alpha-\beta|}{t}\sqrt{1-\frac{E^{2}}{4t^{2}(1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\right]$
The probability that the electrons travel in the clockwise or the counter-
clockwise sense are assumed to be equal. The transmission amplitude is given
by the matrix,
$\tau=\left(\begin{array}[]{cc}\tau_{\uparrow\uparrow}&\tau_{\uparrow\downarrow}\\\
\tau_{\downarrow\uparrow}&\tau_{\downarrow\downarrow}\end{array}\right)$ (11)
which, for $\theta=\pi/4$, simplifies to,
$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)\right.$
(12)
$\displaystyle+\left.\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)\right]$
$\mathcal{R}_{E}$ matrices can be cast in to the forms,
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)$ $\displaystyle=$
$\displaystyle\frac{e^{i\phi
L}}{2}\left(\mathcal{A}_{1}I+\frac{\mathcal{B}_{1}\sigma_{x}}{\sqrt{2}}-\frac{\mathcal{B}_{1}\sigma_{y}}{\sqrt{2}}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)$ $\displaystyle=$
$\displaystyle\frac{e^{i\phi
L}}{2}\left(\mathcal{A}_{2}I+\frac{\mathcal{B}_{2}\sigma_{x}}{\sqrt{2}}+\frac{\mathcal{B}_{2}\sigma_{y}}{\sqrt{2}}\right)$
with,
$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\exp(ik_{+}La)+\exp(ik_{-}La)$ $\displaystyle\mathcal{A}_{2}$
$\displaystyle=$
$\displaystyle\exp(ik_{+}^{\prime}La)+\exp(ik_{-}^{\prime}La)$
$\displaystyle\mathcal{B}_{1}$ $\displaystyle=$
$\displaystyle\exp(ik_{+}La)-\exp(ik_{-}La)$ $\displaystyle\mathcal{B}_{2}$
$\displaystyle=$
$\displaystyle\exp(ik_{+}^{\prime}La)-\exp(ik_{-}^{\prime}La)$ (14)
The transmission amplitude given by Eq. 12 can now be explicitly written as,
$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[(2\mathcal{A}_{1}\mathcal{A}_{2}+\sqrt{2}(\mathcal{A}_{2}\mathcal{B}_{1}+\mathcal{A}_{1}\mathcal{B}_{2})\sigma_{x}\right.$
(15) $\displaystyle+$
$\displaystyle\left.\sqrt{2}(\mathcal{A}_{1}\mathcal{B}_{2}-\mathcal{A}_{2}\mathcal{B}_{1})\sigma_{y})\right.\cos
2\phi L$ $\displaystyle+$
$\displaystyle\frac{i}{4}(\sigma_{x}\sigma_{y}-\sigma_{y}\sigma_{x})\mathcal{B}_{1}\mathcal{B}_{2}\sin
2\phi L$
The coefficient of transmission for an incoming up-spin electron is
$T_{\uparrow}=|\tau_{\uparrow\uparrow}+\tau_{\downarrow\uparrow}|^{2}$, and
the transmission coefficient for an incoming down-spin electron is
$T_{\downarrow}=|\tau_{\downarrow\downarrow}+\tau_{\downarrow\uparrow}|^{2}$,
so that the final transmission coefficient for spin unpolarized electrons
turns out to be,
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}(T_{\uparrow}+T_{\downarrow})$ (16) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{|\mathcal{A}_{1}\mathcal{A}_{2}|^{2}}{8}+\frac{|\mathcal{A}_{2}\mathcal{B}_{1}+\mathcal{A}_{1}\mathcal{B}_{2}|^{2}}{16}+\frac{|\mathcal{A}_{1}\mathcal{B}_{2}-\mathcal{A}_{2}\mathcal{B}_{1}|^{2}}{16}\right]$
$\displaystyle=$
$\displaystyle\cos^{2}\left(\frac{(k^{\prime}_{+}-k^{\prime}_{-})}{2}La\right)+\sin^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right)$
$\displaystyle\times\cos^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right),\hskip
14.22636pt\mbox{for}\hskip 4.26773pt\phi=0$ $\displaystyle=$ $\displaystyle
4\sin^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right)\sin^{2}\left(\frac{(k^{\prime}_{+}-k^{\prime}_{-})}{2}La\right),$
$\displaystyle\mbox{for}\hskip 4.26773pt\phi=\pi/4L$
When $\alpha=\beta$, from Eq. 10 we observe that
$\xi^{\prime}_{+}=\xi^{\prime}_{-}$ or $k^{\prime}_{+}=k^{\prime}_{-}$ and it
gives $\mathcal{B}_{2}=0$. Therefore, for $\phi=0$, the transmission
coefficient $T=1$, and, $T=0$ for $\phi=\pi/4L$. Thus we get perfect
transmission for $\phi=0$ while a vanishing of transmission coefficient for
$\phi=\pi/4L$.
## III Effect of loop geometry
Electronic transport turns out to be sensitive to the geometry of the
mesoscopic loop. To this end, we have numerically calculated the
Figure 2: (Color online). Schematic view of a mesoscopic ring subjected to
RSOI and DSOI and threaded by a magnetic flux $\Phi$. The ring is
symmetrically connected to the leads (source and drain).
two-terminal spin transport in a ring geometry threaded by a magnetic flux
$\Phi$ (Fig. 2). The role of $\Phi$ will be discussed later. Here, for the
time being, we ignore the flux.
Figure 3: (Color online). Two-terminal transmission coefficient of a $40$-site
ordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). $\Phi$ is set at $0$.
Figure 4: (Color online). Two-terminal transmission coefficient of a $40$-site
disordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). The results have been averaged over $60$ disorder configurations.
$\Phi$ is fixed at $0$.
For a ring like structure, the azimuthal angle keeps on changing as one
traverses the perimeter of the ring. This generates an effective site
dependent hopping integral in the Hamiltonian santanu . As a result,
scattering takes place as the electron travels across the sites on the ring.
The scattering becomes a maximum when $\alpha=\beta$ santanu , and naturally,
the two-terminal transport is expected to show up a minimum at $\alpha=\beta$.
We use exact numerical methods. In Fig. 3 we show the variation of the two-
terminal transmission
Figure 5: (Color online). Two-terminal transmission coefficient of a $40$-site
ordered ring for different values of RSOI ($\alpha$) and the DSOI ($\beta$).
$\Phi$ is set equal to $0.5$.
Figure 6: (Color online). Two-terminal transmission coefficient of a $40$-site
disordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). The results have been averaged over $60$ disorder configurations.
$\Phi$ is fixed at $0.5$.
coefficient for an ordered ($W=0$, $W$ measures the strength of disorder) ring
of $40$ sites. Similar observations are presented in Fig. 4 for a $40$-site
ring with random diagonal disorder. The results in the latter case have been
averaged over $60$ disorder configurations. In both the figures the
transmission minimum as $\alpha$ equals $\beta$ are obvious. It is interesting
to note that the random disorder does not destroy the minima, which speaks for
the robustness of the results from the standpoint of experiments.
Before we end this section, it should be appreciated that, in an experiment
the transmission minimum is not easy to locate, and hence an inaccuracy in the
value of the SOI might be introduced. This difficulty may be circumvented if
the transmission minimum becomes precisely equal to zero. This is easily
achieved if the ring is threaded by a magnetic flux which is set equal to half
the flux-quantum $\phi_{0}=hc/e$. In Figs. 5 and 6 the results are presented
for a $40$-site ordered ring and a randomly disordered ring of the same size.
The flux threading the rings is set at $\Phi=\phi_{0}/2$. The two-terminal
transmission coefficient exhibits clear zeros in both the cases as soon as the
DSOI equals the strength of the RSOI. Once again, the transmission zero in
this special situation is independent of the disorder configuration.
## IV Conclusion
In conclusion, we have presented exact analytical results to show that the
spin-orbit interactions present in a mesoscopic sample can be measured by
observing two-terminal transmission resonance in a simple square network. The
transmission coefficient is shown to be exactly equal to unity when the Rashba
and the Dresselhaus interactions become equal in strength. Thus, knowing the
Rashba interaction, for example, a determination of the Dresselhaus term is
possible. For a multi-site ring, the transmission maximum observed for the
single square loop gets converted in to transmission minimum. This happens
again when the Rashba and the Dresselhaus interactions are equal. With a
magnetic flux equal to half the flux-quantum, the transmission minimum becomes
an exact transmission zero, and hence facilitates a possible experimental
measurement.
## References
* (1) I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
* (2) S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).
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* (4) A. Aharony, O. E.-Wohlman, Y. Tokura, and S. Katsumoto, Phys. Rev. B 78, 125328 (2008).
* (5) V. Moldoveanu and B. Tanatar, Phys. Rev. B 81, 035326 (2010).
* (6) G. Engels, J. Lange, T. Schapers, and H. Luth, Phys. Rev. B 55, R1958 (1997).
* (7) V. Vlaminck and M. Bailleul, Science 322, 410 (2008).
* (8) K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008).
* (9) E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).
* (10) G. Dresselhaus, Phys. Rev. 100, 580 (1955).
* (11) L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Schön, and K. Ensslin, Nature Physics 3, 650 (2007).
* (12) T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 89, 046801 (2002).
* (13) J. Premper, M. Trautmann, J. Henk, and P. Bruno, Phys. Rev. B 76, 073310 (2007).
* (14) C.-M. Hu, J. Nitta, T. Akazaki, H. Takayanagi, J. Osaka, P. Pfeffer, and W. Zawadaki, Phys. Rev. B 60, 7736 (1999).
* (15) D. Grundler, Phys. Rev. Lett. 84, 6074 (2000).
* (16) S. K. Maiti, M. Dey, S. Sil, A. Chakrabarti, and S. N. Karmakar, Europhys. Lett. 95, 57008 (2011).
* (17) V. M. Ramaglia, V. Cataudella, G. De Fillipis, and C. A. Perroni, Phys. Rev. B 73, 155328 (2006).
|
arxiv-papers
| 2012-01-23T06:54:27 |
2024-09-04T02:49:26.599746
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shreekantha Sil, Santanu K. Maiti and Arunava Chakrabarti",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1201.4633"
}
|
1201.4648
|
¡html¿¡head¿ ¡meta http-equiv=”content-type” content=”text/html;
charset=ISO-8859-1”¿
¡title¿CERN-2011-007¡/title¿
¡/head¿
¡body¿
¡h1¿¡a href=”http://cas.web.cern.ch/cas/Denmark-2010/Aarhus-advert.html”¿CAS -
CERN Accelerator School: RF for Accelerators¡/a¿¡/h1¿
¡h2¿Ebeltoft, Denmark, 8 - 17 Jun 2010¡/h2¿
¡h2¿Proceedings - CERN Yellow Report
¡a href=”https://cdsweb.cern.ch/record/1231364”¿CERN-2011-007¡/a¿¡/h2¿
¡h3¿editor: R. Bailey¡/h3¿
These proceedings present the lectures given at the twenty-fourth specialized
course organized by the CERN Accelerator School (CAS). The course was held in
Ebeltoft, Denmark, from 8-17 June, 2010 in collaboration with Aarhus
University, with the topic ’RF for Accelerators’ While this topic has been
covered by CAS previously, early in the 1990s and again in 2000, it was
recognized that recent advances in the field warranted an updated course.
Following introductory courses covering the background physics, the course
attempted to cover all aspects of RF for accelerators; from RF power
generation and transport, through cavity and coupler design, electronics and
low level control, to beam diagnostics and RF gymnastics. The lectures were
supplemented with several sessions of exercises, which were completed by
discussion sessions on the solutions.
¡h2¿Lectures¡/h2¿
¡p¿ LIST:arXiv:1201.2345¡br¿
LIST:arXiv:1111.4354¡br¿
LIST:arXiv:1201.2346¡br¿
LIST:arXiv:1201.4068¡br¿
LIST:arXiv:1112.3177¡br¿
LIST:arXiv:1112.3201¡br¿
LIST:arXiv:1112.3203¡br¿
LIST:arXiv:1112.3209¡br¿
LIST:arXiv:1112.3221¡br¿
LIST:arXiv:1112.3226¡br¿
LIST:arXiv:1201.3202¡br¿
LIST:arXiv:1111.4897¡br¿
LIST:arXiv:1201.1154¡br¿
LIST:arXiv:1201.2593¡br¿
LIST:arXiv:1201.2597¡br¿
LIST:arXiv:1201.2598¡br¿
LIST:arXiv:1201.2600¡br¿
LIST:arXiv:1201.3247¡br¿
LIST:arXiv:1112.3232¡br¿
LIST:arXiv:1112.2176¡br¿ ¡/p¿ ¡/body¿¡/html¿
|
arxiv-papers
| 2012-01-23T08:36:38 |
2024-09-04T02:49:26.605325
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Bailey (ed. CERN)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1201.4648"
}
|
1201.4671
|
11institutetext: Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands
Utrecht University, P.O.Box 80000, 3508 TA Utrecht, The Netherlands
# Azimuthal correlations in Pb–Pb and pp collisions measured with the ALICE
detector
You Zhou, (for the ALICE Collaboration) 1122 yzhou@nikhef.nl; you.zhou@cern.ch
1122
###### Abstract
We present results from the measurements of azimuthal correlations of charged
particles in $\sqrt{s_{{}_{NN}}}$ = 2.76 TeV Pb–Pb collisions and
$\sqrt{s_{{}_{NN}}}$ = 7 TeV pp collisions. In addition, the comparison of the
experimental measurements in pp collisions with those from Pythia and Phojet
simulations are presented.
heavy–ion collisions anisotropic flow azimuthal correlations
###### pacs:
25.75.Gz, 25.75.Ld, 05.70.Fh
††articletype: Editorial
## I Introduction
The study of azimuthal correlations is one of the most important tools to
probe the properties of the medium generated in heavy–ion collisions.
Experimentally, these azimuthal correlations are not determined solely by
anisotropic flow JYO-PRD but also have other contributions, usually refered
to as non–flow which are not correlated to the participant plane ART-arXiv .
Anisotropic flow, especially the second order harmonic $v_{2}$ (elliptic
flow), has been systematically studied from SPS to LHC energies SPS-v2 ;
RHIC-v2 ; LHC-v2 . Recently it has been argued that fluctuations in the
initial matter distribution give rise to odd harmonics like $v_{3}$
(triangular flow) BA-PRC . In this contribution, we report the anisotropic
flow for charged particles measured in $\sqrt{s_{{}_{NN}}}=$ 2.76 TeV Pb–Pb
collisions. We also discuss azimuthal correlation measurements in pp
collisions compared to simulations from Pythia and Phojet.
## II Anisotropic flow in Pb–Pb collisions
Figure 1: $v_{2}$, $v_{3}$ and $v_{4}$ $p_{t}$-integrated flow as a function
of centrality. Full and open blue squares show the $v_{3}\\{2\\}$ and
$v_{3}\\{4\\}$, respectively. The full circle and full diamond are symbols for
$v_{3/\Psi_{\mathrm{RP}}}$ and $v^{2}_{3/\Psi_{2}}$. In addition, the
hydrodynamic calculations BHA-PRC for $v_{3}$ and AMPT simulations JX-PRC
for $v_{2}$, $v_{3}$ and $v_{4}$ are shown by dash lines and full gray
markers. ALICE data points taken from ALICE-V3 .
Figure 2: $v_{2}$, $v_{3}$, $v_{4}$, $v_{5}$ as a function of transverse
momentum and for three event centralities. The full and open symbols are for
$|\Delta\eta|>$0.2 and $|\Delta\eta|>$1.0, respectively. (a) 30-40$\%$
centrality percentile compared to hydrodynamic model calculations BS-arXiv ,
(b) 0-5$\%$ centrality percentile, (c) 0-2$\%$ centrality percentile. Figures
taken from ALICE-V3 .
In this contribution, we report on the study the azimuthal correlations via 2–
and 4–particle cumulants AB-QC . In Fig. 1 we observe that the $v_{3}$
measurements from the 2– and 4–particle cumulants differ from zero; the
$v_{3}\\{4\\}$ is a factor of 2 smaller than $v_{3}\\{2\\}$ which can be
understood if $v_{3}$ originates predominantly from event–by–event
fluctuations of the initial spatial geometry RSB-PRC . At the same time, we
investigate the correlation between $\Psi_{3}$ and the reaction plane
$\Psi_{\mathrm{RP}}$ as well as the correlations between $\Psi_{3}$ and
$\Psi_{2}$, evaluated by
$v_{3/\Psi_{\mathrm{RP}}}=\langle\cos(3\phi-3\Psi_{\mathrm{RP}})\rangle$ and
$v_{3/\Psi_{2}}^{2}=\langle\cos(3\phi_{1}+3\phi_{2}-2\phi_{3}-2\phi_{4}-2\phi_{5})\rangle/v_{2}^{3}$,
respectively. We observe that $v_{3/\Psi_{\mathrm{RP}}}$ and
$v_{3/\Psi_{2}}^{2}$ are consistent with zero within uncertainties. Based on
these results, we conclude that $v_{3}$ develops as a correlation of all
particles with respect to the third order participant plane $\Psi_{3}$, while
there is no (or very weak) correlation between $\Psi_{\mathrm{RP}}$ (or
$\Psi_{2}$) and $\Psi_{3}$. The centrality dependence of $v_{3}$ is compared
to hydrodynamic calculations. The data are described well by calculations
based on Glauber initial conditions and $\eta/s=0.08$, while underestimated by
the MC–KLN initial conditions and $\eta/s=0.16$ BHA-PRC . The comparison
suggests that $\eta/s$ of the produced matter is small. Finally, the data are
described well by the AMPT model calculations, with only a slight
overestimation of $v_{2}\\{2\\}$ in the most central collisions JX-PRC .
To further constrain the properties of the system, we compare the
$p_{t}$–differential flow of $v_{2}$ and $v_{3}$ to hydrodynamic calculations
in Fig. 2(a). We find that the hydrodynamic calculations with Glauber initial
conditions can describe the elliptic and triangular differential flow
measurements, although not for higher $p_{t}$. However, the $v_{2}(p_{t})$
measurements seem to suggest $\eta/s$=0 while for $v_{3}(p_{t})$ the
hydrodynamic calculations with $\eta/s$=0.08 provide a better description.
Currently there is no hydrodynamic calculation which simultaneously describes
the $p_{t}$–differential $v_{2}$ and $v_{3}$ measurements at LHC energies with
the same value for $\eta/s$. In central collisions 0-5$\%$ we observe that the
higher harmonics $v_{3}$ and $v_{4}$ exceed $v_{2}$ and become the dominant
harmonics at intermediate $p_{t}$. This occurs already at lower $p_{t}$ for
more central collisions 0-2$\%$. In AMPT simulations, it is observed that the
initial geometrical fluctuations leads to anisotropic collective expansions
even at an impact parameter of b=0 GL-PRL .
Figure 3: Cumulants for charged particles in 7 TeV pp collisions. (a)
2-particle cumulant ; (b) 4-particle cumulant. The shadow areas represent the
results for Pythia (purple) and Phojet (pink).
## III Anisotropic flow or non–flow in pp collisions?
At LHC energies relatively high multiplicity events are observed in pp
collisions LHC-pp . Some theoretical work predict elliptic flow magnitudes up
to 0.2 in pp collisions at LHC energies hydro-pp . It is interesting to
investigate whether collective effects appear in such events and if we can
test those predictions. The 2– and 4–particle cumulant when dominated by
anisotropic flow, correspond to: QC$\\{2\\}=v^{2}$, QC$\\{4\\}=-v^{4}$.
Therefore if the measured azimuthal correlations are dominated by anisotropic
flow, they should show the typical flow signature (+,–) which has been
observed in Pb–Pb collisions AB-QM . Figure 3 presents the 2– and 4–particle
cumulant as a function of the measured uncorrected multiplicity, defined as
the number of charged particle tracks which pass our track selection. We
observe that the measured QC$\\{4\\}$ is positive in the currently measured
multiplicity range, which suggests that its dominant contribution is not
coming from anisotropic flow. Also we find that both QC$\\{2\\}$ and
QC$\\{4\\}$ decrease with increasing multiplicity, which is a typical
behaviour for non–flow. In addition, we notice that both Pythia and Phojet can
qualitatively describe the trend and sign of the QC$\\{2\\}$ and QC$\\{4\\}$.
However, both of them do overestimate the strength of the azimuthal
correlation measurements.
## IV Conclusion
The azimuthal correlations of charged particles measured in
$\sqrt{s_{{}_{NN}}}$ = 2.76 TeV Pb–Pb collisions are presented. Our results
constrain the corresponding models. The analyses with 2– and 4–particle
cumulant in $\sqrt{s_{{}_{NN}}}$ = 7 TeV pp collisions show that such
azimuthal correlations are not dominated by anisotropic flow in the
multiplicity range presented.
## References
* (1) J.Y. Ollitrault, Phys. Rev. D 46 229 (1992)
* (2) S.A. Voloshin, A.M. Poskanzer and R. Snellings, in Landolt-Boernstein, Relativistic Heavy Ion Physics, Vol. 1/23, p 5-54 (Springer-Verlag, 2010)
* (3) C. Alt et al. (NA49 Collaboration), Phys. Rev. C 68, 034903 (2003)
* (4) K.H. Ackermann et al. (STAR Collaboration), Phys. Rev. Lett. 86, 402 (2001)
* (5) K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 105, 252302 (2010)
* (6) B. Alver and G. Roland, Phys. Rev. C 81, 054905 (2010)
* (7) A. Bilandzic, R. Snellings and S. Voloshin, Phys. Rev. C 83, 044913 (2011)
* (8) K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 107, 032301 (2011)
* (9) B. Schenke, S. Jeon, and C. Gale, Phys. Lett. B 59, 702 (2011)
* (10) R.S. Bhalerao, M. Luzum and J.Y. Ollitrault, Phys. Rev. C 84, 034910 (2011)
* (11) B. Alver et al., Phys. Rev. C 82, 034901 (2010)
* (12) J. Xu and C.M. Ko, Phys. Rev. C 84, 044907 (2011)
* (13) G.L. Ma and X.N. Wang, Phys. Rev. Lett. 86 3496 (2001)
* (14) K. Aamodt et al. (ALICE Collaboration), Eur. Phys. J. C 68, 89 (2010); Eur. Phys. J. C 68, 345 (2010)
* (15) J. Casalderrey-Solana and U. A. Wiedemann, Phys. Rev. Lett. 104, 102301 (2010); E. Avsar et. al., Phys. Lett. B 702, 394 (2011)
* (16) A. Bilandžić (for ALICE Collaboration), J. Phys. G 38, 124052 (2011)
|
arxiv-papers
| 2012-01-23T10:41:07 |
2024-09-04T02:49:26.609320
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "You Zhou (for the ALICE Collaboration)",
"submitter": "You Zhou",
"url": "https://arxiv.org/abs/1201.4671"
}
|
1201.4721
|
-titleHadron Collider Physics symposium (HCP 2011)
11institutetext: Royal Holloway, University of London.
# A Search for Heavy Resonances in the Dilepton Channel
Daniel Hayden On behalf of the ATLAS Collaboration daniel.hayden@cern.ch
###### Abstract
There are many extensions to the Standard Model of particle physics which
predict the addition of a U(1) symmetry, and/or extra spatial dimensions,
which give rise to new high mass resonances such as the Z′ and Randall-Sundrum
graviton. The LHC provides a unique opportunity to explore the TeV scale where
these phenomena may become apparent, and can be searched for using the
precision tracking and high energy resolution calorimetry of the ATLAS
detector. This poster presents the search for high mass resonances in the
dilepton channel, and was conducted with an integrated luminosity of 1.08/1.21
fb-1 in the dielectron/dimuon channel respectively, at a centre of mass energy
$\sqrt{s}$ = 7 TeV.
## 1 Introduction
There are many possible extensions to the Standard Model (SM) predicted at the
TeV energy scale which may be visible at the LHC. Many of these extensions
predict extra U(1) symmetry with an associated spin-1 particle Theory:ZP1
Theory:ZP2 . In its simplest form this U(1) symmetry can be arbitrarily added
to the existing SM gauge group, resulting in SU(3) $\times$ SU(2) $\times$
U(1) from the SM, and additionally U(1)′ for the Sequential Standard Model
Z${}^{\prime}_{SSM}$. More rigorously motivated models proceed via the
decomposition of Grand Unified Theories such as E6 $\rightarrow$
SO(10)$\times$U(1)ψ $\rightarrow$ SU(5) $\times$ U(1)χ $\times$ U(1)ψ leading
to Z′($\theta$) = Z${}^{\prime}_{\chi}$cos$\theta$ \+
Z${}^{\prime}_{\psi}$sin$\theta$, where the mixing angle $\theta$ determines
the coupling to fermions and results in various possible models with specific
Z′ states. Other extensions of the SM seek to answer questions such as the
hierarchy problem where the relative weakness of gravity compared to the other
forces of nature can be explained with the use of warped extra dimensions in
theories such as the Randall-Sundrum model Theory:G . A feature of this theory
would be a massive spin-2 particle called the graviton (G∗) which should be
observable at the LHC and have a mass/width that depends on the curvature of
the warped dimension, k, and the reduced Planck scale, $\overline{M}_{Pl}$,
leading to another parameter of interest, the coupling k/$\overline{M}_{Pl}$.
Both of the new particles mentioned would appear as resonances in the dilepton
invariant mass spectrum measured by the ATLAS detector ATLAS , and these
results comprise a search using the detector in this endeavor.
## 2 Dilepton Resonance Search
The search for dilepton resonances was conducted in both the electron and muon
channels separately, which were then combined to give the final result. To
identify candidate events from data, each analysis selected high energy
electron/muon pairs. The main background to a Z′/G∗ search in these channels
is from Drell-Yan, with smaller contributions from $t\bar{t}$, W+jets,
diboson, and QCD events 111QCD events here are defined as semi-leptonic decays
of b and c quarks in the dimuon sample, or at least one electron coming from
photon conversions, semi-leptonic heavy quark decays or a hadronic fake, in
the dielectron sample.. These SM background contributions were estimated using
Monte Carlo (MC) simulation, except for QCD which was estimated from data
using a reverse identification selection sample for electrons, and a non-
isolated sample for muons.
For both the dielectron and dimuon channel analyses, a data quality
requirement is made to ensure parts of the ATLAS detector important for
e/$\gamma$ or $\mu$ analysis respectively are working optimally. The events
are also required to have at least one primary vertex with greater than two
tracks, and pass a single electron trigger with a transverse energy (ET)
greater than 20 GeV or equivalently for the dimuon analysis, a muon trigger
with transverse momentum (pT) greater than 22 GeV.
For an event to be accepted by the analysis in the dielectron channel, an
event must contain at least two electron candidates with ET $>$ 25 GeV and
$|$$\eta$$|$ $<$ 2.47, also excluding the region between the barrel and endcap
calorimeters 1.37 $\leq$ $|$$\eta$$|$ $\leq$ 1.52. The electron candidates
that pass these criteria must have been reconstructed from electromagnetic
cells clusters with an associated charged particle track from the inner
detector. Shower shape variables and hadronic calorimeter leakage, along with
information from the inner detector is then used to strengthen the
identification of the electron candidates. A hit in the first layer of the
pixel detector is required to suppress background from photon conversions.
From the remaining electron candidates the highest ET pair is selected and the
higher ET electron required to pass an isolation threshold of less than 7 GeV
in a cone of 0.2 around the cluster ($\Delta$R =
$\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$) to reduce the QCD background.
Finally, the invariant mass of the selected pair must be greater than 70 GeV
to be accepted by the analysis, and no opposite charge requirement is made to
minimise the impact of possible charge mis-identification.
In the dimuon channel, two oppositely charged muons are required. Each muon
must have pT $>$ 25 GeV, and pass quality criteria from the inner detector as
well as having at least three hits in each of the inner, middle, and outer
layers of the muon spectrometer to improve momentum resolution. Muons are
discarded if they have hits in both the barrel and endcap regions because of
residual misalignment. To suppress the cosmic ray background the $z$ position
of the primary vertex is required to be less than 200 mm, and muon tracks must
have a transverse impact parameter $|$$d_{0}$$|$ $<$ 0.2 mm, also being within
1 mm of the primary vertex along the beam-line. To reduce the QCD background
in the muon channel, each muon is required to be isolated such that
$\sum{p_{T}}$($\Delta$R $<$ 0.3)/pT $<$ 0.05. The two highest $p_{T}$ muons
passing this selection form a pair and are required to have an invariant mass
greater than 70 GeV to be accepted by the dimuon analysis.
The dilepton analysis was performed with an integrated luminosity of 1.08 fb-1
in the electron channel, and 1.21 fb-1 in the muon channel. The results of
this analysis can be found in Exotics:EPS , and the main kinematic plots of
interest, namely the invariant mass spectrum for both the electron and muon
channel, are presented in Figure 1.
Figure 1: Invariant mass spectrum for the electron (top) and muon (bottom)
channel dilepton resonance search. Various possible Z${}^{\prime}_{SSM}$
signals are overlayed to show how an expected signal would manifest itself.
## 3 Statistical Analysis
Any excess in the observed data over the SM prediction can be quantified using
a Log Likelihood Ratio (LLR) test:
$LLR=-2ln\frac{{\cal L}(data|N_{sig}+N_{bkg})}{{\cal L}(data|N_{bkg})}$ (1)
In this dataset the greatest excesses give $p$-values of 54% and 24% for the
dielectron and dimuon channels respectively. Therefore as no significant
excess is observed, limits are set on the cross section times branching ratio
($\sigma$B) for the Z${}^{\prime}_{SSM}$ and G∗ decaying to leptons, at 95%
confidence level using the Bayesian Analysis Toolkit (BAT) BAT . BAT
constructs a binned likelihood, combining the electron and muon channel
searches and accounting for observed ($n$) and expected ($\mu$) events with
associated nuisance parameters ($\theta$) on a bin by bin basis:
${\cal L}(data|\sigma
B,\theta_{i})=\prod_{l=1}^{N_{channel}}\prod_{k=1}^{N_{bin}}\frac{\mu_{lk}^{n_{lk}}e^{-\mu_{lk}}}{n_{lk!}}\prod_{i=1}^{N_{sys}}G(\theta_{i},0,1)$
(2)
Employing Bayesian statistics (assuming a flat positive prior so that
$\pi$($\sigma$B) = 1) and treating the nuisance parameters as Gaussian priors,
Markov Chain Monte Carlo is used to reduce the likelihood (${\cal
L}^{\prime}$) and obtain the marginalised posterior probability, which is then
solved for ($\sigma$B)95:
$0.95=\frac{\int_{0}^{{\sigma B}_{95}}{\cal L}^{\prime}(\sigma B)\pi(\sigma
B)d(\sigma B)}{\int_{0}^{\infty}{\cal L}^{\prime}(\sigma B)\pi(\sigma
B)d(\sigma B)}$ (3)
The resulting limits on the Z′/G∗ $\sigma$B are converted into mass exclusion
limits using the theoretical dependence of $\sigma$B as a function of
resonance mass. The $\sigma$B limits are presented in Figure 2. Table 1
summarises the excluded mass values for the models considered. The results
presented here represent a large step forward in the search for heavy dilepton
resonances, exceeding previous experiments’ mass exclusion limits for Z′/G∗
resonances in the dilepton channel. With a total integrated luminosity of
$\sim$5 fb-1 recorded by the ATLAS detector in 2011, this search will soon be
updated probing even further into the TeV scale regime in search of new
physics beyond the current SM.
Figure 2: 95% confidence level $\sigma$B limits for various Z′ models (top),
and RS graviton k/Mpl couplings (bottom).
| E6 Z′ Models
---|---
Model | Z${}^{\prime}_{\psi}$ | Z${}^{\prime}_{N}$ | Z${}^{\prime}_{\eta}$ | Z${}^{\prime}_{I}$
Mass limit [TeV] | 1.49 | 1.52 | 1.54 | 1.56
| E6 Z′ Models |
---|---|---
Model | Z${}^{\prime}_{S}$ | Z${}^{\prime}_{\chi}$ | Z${}^{\prime}_{SSM}$
Mass limit [TeV] | 1.60 | 1.64 | 1.83
G∗ Coupling k/MPl | 0.01 | 0.03 | 0.05 | 0.10
---|---|---|---|---
Mass limit [TeV] | 0.71 | 1.03 | 1.33 | 1.63
Table 1: 95% confidence level lower mass exclusion limits for various Z′
models and RS graviton k/Mpl couplings, decaying to two leptons (dielectron or
dimuon).
## References
* (1) D. London and J. L. Rosner, Extra Gauge Bosons in E(6), Phys. Rev D34 (1986) 1530.
* (2) P. Langacker, The Physics of Heavy Z′ Gauge Bosons, Rev. Mod. Phys 81 (2009) 1199.
* (3) L. Randall and R. Sundrum, A large mass hierarchy from a small extra dimension, Phys. Rev. Lett 83 (1999) 3370.
* (4) ATLAS Collaboration, The ATLAS Experiment at the CERN Large Hadron Collider, JINST 3 S08003 (2008) .
* (5) The ATLAS Collaboration, Search for high-mass dilepton resonances in pp collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, Phys. Rev. Lett 107, 272002 (2011) .
* (6) A Caldwell, D. Kollar, K. Kröninger, BAT - The Bayesian Analysis Toolkit, Computer Physics Communication 180 (2009) 2197.
|
arxiv-papers
| 2012-01-23T14:11:24 |
2024-09-04T02:49:26.614439
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel Hayden (for the ATLAS Collaboration)",
"submitter": "Daniel Hayden",
"url": "https://arxiv.org/abs/1201.4721"
}
|
1201.4748
|
-titleHadron Collider Physics Symposium 2011 11institutetext: LPSC-Grenoble, CNRS/IN2P3, UJF, INPG
# Search for extra dimensions in the diphoton final state with ATLAS
Quentin Buat On behalf of the ATLAS Collaboration quentin.buat@cern.ch
###### Abstract
The large difference between the Planck scale and the electroweak scale, known
as the hierarchy problem, has been addressed in some models through the
existence of extra spatial dimensions. A search for evidence of extra spatial
dimensions has been performed, through an analysis of the diphoton final state
in data recorded in 2011 with the ATLAS detector at the CERN Large Hadron
Collider. The analysis uses a dataset of 2.12 $fb^{-1}$ of proton-proton
collisions at $\sqrt{s}=7$ TeV. The diphoton invariant mass spectrum is
observed to be in good agreement with the expected Standard Model (SM)
background. We set 95$\%$ CL lower limits on the scale related to virtual
graviton exchange process in the context of the Arkani-Hamed, Dimopoulos,
Dvali model (ADD) and on the lightest Kaluza Klein excitation mass in the
context of the Randall-Sundrum model (RS).
## 1 Introduction
Recently, there has been great interest in models which address the hierarchy
problem through the existence of extra spatial dimensions. In this analysis we
search for evidence of extra spatial dimensions in the context of the ADD
ADDPaper and RS RSPaper models. This analysis briefly summarized here, is
described in more detail in GravPaper .
In the ADD context, _n_ flat extra spatial dimensions are postulated with a
compactification radius R. The gravity is the only field that propagates in
those extra dimensions and acquires Kaluza Klein (KK) modes. In the ADD
context, resolving the hierarchy problem implies small values of 1/R. This
leads to an almost continuous KK spectrum of the graviton mass.
Experimentally, an ADD signal will contribute through virtual graviton
exchange to the diphoton invariant mass spectrum. This process can be
parametrized as a function of the number of extra dimensions _n_ and an
ultraviolet cutoff ($M_{s}$). Several formalisms exist in the literature to
define $M_{s}$, referred to here as GRW GRW , HLZ HLZ and Hewett Hewett .
The RS model postulates a 5-dimensional space-time bounded by two (3+1) branes
with the SM particles localized on one of the branes. The fifth dimension has
a ”warped” geometry which allows to naturally generate TeV scales from the
Planck scale. The fundamental Planck scale ($M_{Pl}$) on one brane is related
to the apparent scale ($\Lambda_{\pi}$) on the other brane by the relation
$\Lambda_{\pi}=\overline{M}_{Pl}\exp{(-k\pi r_{c})}$ where $k$ is the
curvature scale of the extra dimension, $r_{c}$ the compactification radius
and $\overline{M}_{Pl}=\frac{M_{Pl}}{\sqrt{8\pi}}$. The observed hierarchy of
scales can be naturally reproduced by this model with $kr_{c}\approx 11$. In
the minimal RS model, gravitons are the only particles that propagate in the
bulk. Consequently a series of massive KK excitations is predicted with a mass
splitting of the order of 1 TeV. Finally the RS model can be expressed in
terms of the coupling $k/\overline{M}_{Pl}$ and the mass of the lightest KK
excitation ($m_{G}$). Experimentally the presence of a RS signal can be probed
as a narrow resonance in the diphoton invariant mass spectrum.
## 2 Photon reconstruction and identification
A complete description of the ATLAS detector can be found in DetectorPaper .
In this analysis we look for final states with two photons. Therefore we rely
on the inner tracker and the calorimetric system of the detector. The tracking
system of ATLAS is composed of layers of silicon-based and straw-tube
detectors surrounded by a 2 T magnetic field for momentum measurements. It
allows to measure the tracks of the 30% of photons converting in $e^{+}e^{-}$
pairs before reaching the calorimeter. The photon energy deposit is measured
by the Liquid Argon (LAr) electromagnetic calorimeter, covering the region
$0<|\eta|<1.37$ and $1.52<|\eta|<2.37$. It is segmented in three longitudinal
layers. Most of the energy of the electromagnetic (EM) shower is recovered by
the second layer which has a granularity of
$\Delta\eta\times\Delta\phi=0.025\times 0.025$. The first layer with a thinner
segmentation is designed to reject jets dominated by neutral hadrons such as a
$\pi^{0}$ or a $\eta$. Indeed, such a hadron may decay into two photons which
can be identified by resolving the two maxima in the first layer of the LAr
calorimeter. In the central region ($|\eta|<1.81$), a presampler is used to
evaluate the energy lost in the upstream material. The hadronic calorimeter,
composed of scintillating tiles-iron sampling for the central part
($|\eta|<1.7$) and of liquid-argon-copper/tungsten sampling for $|\eta|>1.7$
is used to measure the leakage of energy of the photon candidates. A complete
description of the reconstruction and identification of photons with ATLAS can
be found in ATLASphotons .
## 3 Trigger and Data selection
The events are recorded using a trigger requiring at least two photon
candidates with transverse energy $E_{T}^{\gamma}>20$ GeV and satisfying
identification requirements based on the leakage of energy into the hadronic
calorimeter and the transverse width of the EM shower. This trigger is fully
efficient for high mass diphoton events passing the offline selection
requirements. Events are required to have at least one primary collision
vertex, with at least three reconstructed tracks. Selected events have at
least two photon candidates, each with $E_{T}^{\gamma}>25$ GeV and lying
outside the transition region between the barrel and endcap calorimeters. In
addition, the two photons are required to satisfy standard quality criteria
and to lie outside detector regions where their energy is not measured in an
optimal way. The two highest $E_{T}^{\gamma}$ photon candidates have to
satisfy a set of identification requirements on the hadronic leakage and on
the lateral width of the EM shower. The requirements on the EM shower use the
thin granularity of the first sampling to achieve a high purity of the
selected photon sample. The isolation transverse energy $E_{T}^{iso}$ for each
photon is calculated ATLASphotons by summing over the cells within a cone of
radius $\Delta R=\sqrt{(\eta-\eta^{\gamma})^{2}+(\phi-\phi^{\gamma})^{2}}<0.4$
around the direction of the photon. Then the energy deposit of the photon
itself is subtracted as well as the soft jet activity of the underlying event
ambientenergy . In addition, $E_{T}^{iso}$ is corrected for the leakage of the
photon energy into the isolation ring. To further reduce the jet background,
an isolation cut of $E_{T}^{iso}<5$ GeV is applied on the two leading photons.
For all events the two photons with the highest $E_{T}^{\gamma}$ values are
considered and the diphoton invariant mass has to exceed 140 GeV. A total of
6846 events are selected.
## 4 Monte Carlo Simulation Studies
Monte Carlo (MC) simulations were performed to study the detector response to
various scenarios of ADD and RS models as well as to perform studies of the SM
background.
The SM diphoton background was simulated with PYTHIA 6.424 pythia and
MRST2007LOMOD MRST2007lomod parton distribution functions (PDFs). The PYTHIA
events were reweighted as a function of the diphoton invariant mass to the
differential cross section predicted by the NLO calculation of DIPHOX 1.3.2
diphox . The reweighting factor decreased smoothly from $\approx 1.6$ for
$m_{\gamma\gamma}=140$ GeV to 1 for large masses.
SHERPA 1.2.3 sherpa with CTEQ6L cteq66 PDFs was used to simulate the various
ADD scenarios. The ADD MC samples were used to determine the signal acceptance
($A$) and selection efficiency ($\epsilon$). The acceptance varied for the
various ADD implementations and fell from typical values of $\approx 20$% for
$M_{S}=1.5$ TeV down to $\approx 15$% for $M_{S}=3$ TeV. The selection
efficiency, for events within the detector acceptance, was found to be
$\approx 70$%.
RS model MC signal samples were produced with PYTHIA 6.424 for a range of
$m_{G}$ and $k/\overline{M}_{Pl}$ values, using the MRST2007LOMOD PDFs. The
products of $A\times\epsilon$ for the RS signal models were in the range
$\approx(53-60)$%, slowly rising with increasing graviton mass. The
reconstructed shape of the graviton resonance was modeled by a Breit-Wigner
(BW) lineshape convoluted with a double-sided Crystal Ball function to
describe the detector response. The natural width of the BW was fixed
according to the expected theoretical value, which varies as the square of
$k/\overline{M}_{Pl}$. As an example, the value of the width for
$k/\overline{M}_{Pl}=0.1$ increases from $\approx 8$ GeV for $m_{G}=800$ GeV
to $\approx 30$ GeV for $m_{G}=2200$ GeV. Fitting this mass dependence
provides a parametrization used to describe signals with any values of $m_{G}$
and $k/\overline{M}_{Pl}$.
## 5 Background evalution
The SM $\gamma\gamma$ production is the largest background of this analysis.
The invariant mass spectrum shape was determined by PYTHIA reweighted by
DIPHOX NLO cross-section predictions.
The second significant background arises from a different physics object
(electron or jet) misidentified as a photon. This background, called reducible
background, is dominated by the production of $\gamma$ \+ jet or dijet events.
Studies have shown that the Drell Yan contribution can be neglected for events
with $m_{\gamma\gamma}>140$ GeV. The shape of the reducible background is
determined by a data-driven technique. By reverting the identification
criteria on the leading and/or subleading photon, we get three different
control samples enriched in $\gamma$ \+ jet, jet + $\gamma$ and dijet events.
The invariant mass spectra of the three samples are consistent and they are
merged to determine the reducible background shape. This shape is extrapolated
to high diphoton masses where the data control sample are statistically
limited.
The fraction of each background has been determined by a two dimensional
template fit method using the photon isolation variable, described in more
detail in SMdiphoton . Finally the background expectation is normalized to the
data in the invariant mass range [140, 400] GeV where the presence of any
possible ADD and RS signal has been excluded by previous searches.
## 6 Systematic uncertainties
The systematic uncertainty on the background estimation arises from three
different sources. The irreducible background uncertainty is obtained by
varying the scales of the models and the PDFs in DIPHOX while the reducible
background uncertainty is obtained by fitting the three control samples
described in section 5 with the functionnal form. The uncertainty on the
background estimation varies from $\approx$2% for the low mass region to
$\approx$20% at a mass of $\approx$2 TeV. For the signal, the PDFs uncertainty
is of 10-15% for ADD models and 5-10 % for RS models and the uncertainty on
the signal yields has been evaluated to be of 6.7 % for both ADD and RS
models.
## 7 Results and Interpretation
Figure 1 shows the observed invariant mass spectrum of diphoton events, with
statistical significance of the bin-by-bin difference between data and the
expected background at the bottom. This difference is measured in standard
deviations based on Poisson statistics. The small variations show the good
agreement between data and the estimated SM background. This is confirmed by
an analysis using the BUMPHUNTER bumphunter tool, which yields a probability
of 0.28, given the background only hypothesis, to observe discrepancies as
large as the one observed in the bin by bin comparison. In the absence of any
significant deviation we set 95% CL on the signal cross sections, using a
bayesian approach bayes with a flat prior. In the context of ADD models the
signal search region was chosen as $m_{\gamma\gamma}>1.1$ TeV by optimizing
the expected limit. The observed (expected) limit on the cross section due to
new physics is 2.49 (1.94) fb. This result can be translated into 95% CL lower
limits on $M_{s}$ for different numbers of extra dimensions and formalisms.
Table 1 summarizes the observed limits value. In the context of the RS models,
the observed invariant mass spectrum was compared to templates of the expected
background and various signal parametrizations. The limit is set as 95% CL on
the product of the cross-section $\times$ Branching ratio ($\sigma B$) as a
function of $m_{G}$. Then the cross-section limit is converted into a mass
limit using the theoretical dependence. Table 2 shows the limit in the
diphoton final state for various values of $k/\overline{M}_{Pl}$. By combining
with previously published ATLAS results zprime from the dilepton final state,
we obtain a 95% CL limit on $\sigma B$ as a function of $m_{G}$ shown by the
top plot of figure 2. The result is also interpreted in the plane of
$k/\overline{M}_{Pl}$ versus $m_{G}$ (bottom plot of figure 2).
Figure 1: Observed invariant mass spectrum, superimposed with the predicted SM background and examples of signals for ADD and RS models. The bin-by-bin significance of the difference between data and background is shown in the lower panel. Table 1: 95% CL limits on $M_{S}$ (in TeV) for various implementations of the ADD model, using both LO (k-factor = 1) and NLO (k-factor = 1.70) theory cross section calculations. k-factor | GRW | Hewett | HLZ
---|---|---|---
Value | | Pos | Neg | $n=3$ | $n=4$ | $n=5$ | $n=6$ | $n=7$
1 | 2.73 | 2.44 | 2.16 | 3.25 | 2.73 | 2.47 | 2.30 | 2.17
1.70 | 2.97 | 2.66 | 2.27 | 3.53 | 2.97 | 2.69 | 2.50 | 2.36
Table 2: 95% CL lower limits on $m_{G}$, for various values of $k/\overline{M}_{Pl}$. The $G\rightarrow\gamma\gamma$ channel alone and the combination with the dilepton results of Ref. zprime are shown, using both LO (k-factor = 1) and NLO (k-factor = 1.75) theory cross section calculations. k-Factor Value | Channel(s) Used | 95% CL Limit [TeV]
---|---|---
$k/\overline{M}_{Pl}$ Value
0.01 | 0.03 | 0.05 | 0.1
1 | $G\to\gamma\gamma$ | 0.74 | 1.26 | 1.41 | 1.79
$G\to\gamma\gamma$/$ee$/$\mu\mu$ | 0.76 | 1.32 | 1.47 | 1.90
1.75 | $G\to\gamma\gamma$ | 0.79 | 1.30 | 1.45 | 1.85
$G\to\gamma\gamma$/$ee$/$\mu\mu$ | 0.80 | 1.37 | 1.55 | 1.95
Figure 2: (Top) Expected and observed 95% CL limits from the combination of
$G\rightarrow\gamma\gamma/ee/\mu\mu$ channels on $\sigma B$ as a function of
the graviton mass. The theory bands are drawn assuming a k-factor of 1.75.
(Bottom) The RS results interpreted in the plane of $k/\overline{M}_{Pl}$
versus graviton mass, and including recent results from other experiments
TevatronRS ; CMSggnew . The region above the curve is excluded at 95% CL.
## References
* (1) N. Arkani-Hamed, S. Dimopoulos, and G.R. Dvali, Phys. Lett. B429, 263 (1998).
* (2) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999).
* (3) ATLAS Collaboration, arXiv:1112.2194 (2011); submitted to Phys. Lett. B.
* (4) G. Giudice, et al., Nucl. Phys. B544, 3 (1999).
* (5) T. Han, et al., Phys. Rev. D59, 105006 (1999).
* (6) J. Hewett, Phys. Rev. Lett. 82, 4765 (1999).
* (7) ATLAS Collaboration, JINST 3, S08003 (2008).
* (8) ATLAS Collaboration, Phys. Rev. D83, 052005 (2011).
* (9) M. Cacciari, et al., JHEP 04, 065 (2010).
* (10) T. Sjöstrand et al., Comput. Phys. Commun. 135, 238 (2001).
* (11) A. Sherstnev and R. S. Thorne, Eur. Phys. J. C55, 553 (2008).
* (12) T. Binoth, et al., Eur. Phys. J. C16, 311 (2000).
* (13) T. Gleisberg et al., JHEP 02, 007 (2009).
* (14) P. M. Nadolsky, et al., Phys. Rev. D78, 013004 (2008).
* (15) ATLAS Collaboration, Phys. Rev. D85, 012003 (2012).
* (16) G. Choudalakis, arXiv:1101.0390 (2011).
* (17) A. Caldwell,et al., Comput. Phys. Commun. 180, 2197 (2009).
* (18) ATLAS Collaboration, Phys. Rev. Lett. 107, 272002 (2011)
* (19) D0 Collaboration, Phys. Rev. Lett. 104, 241802 (2010); CDF Collaboration, Phys. Rev. Lett. 107, 051801 (2011).
* (20) CMS Collaboration, arXiv:1112.0688 (2011) (submitted to Phys. Rev. Lett.).
|
arxiv-papers
| 2012-01-23T15:58:06 |
2024-09-04T02:49:26.619952
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quentin Buat",
"submitter": "Quentin Buat",
"url": "https://arxiv.org/abs/1201.4748"
}
|
1201.5000
|
-titleHadron Collider Physics Symposium 2011 11institutetext: Laboratoire de l’Accélérateur Linéaire, Orsay, France
# Photon polarisation in $b\rightarrow s\gamma$ using $B\rightarrow$ K∗e+e- at
LHCb
Michelle Nicol (on behalf of the LHCb collaboration) 11 nicol@lal.in2p3.fr
###### Abstract
The $b\rightarrow s\gamma$ transition proceeds through flavour changing
neutral currents, and thus is particularly sensitive to the effects of new
physics. An overview of the method to measure the photon polarisation at the
LHCb experiment via an angular analysis of $B\rightarrow K^{*}e^{+}e^{-}$ at
low $q^{2}$ is presented. The status of the $B\rightarrow K^{*}\mu^{+}\mu^{-}$
analysis with 309 pb-1 of $pp$ collisions at $\sqrt{s}$=7 TeV at LHCb is also
given.
## 1 Introduction
Although the branching ratio of $b\rightarrow s\gamma$ has been measured to be
consistent with Standard Model (SM) predictions, new physics could still be
present and detectable through the analysis of details of the decay process.
In particular, the photon from the b is predominantly left handed in the SM,
whereas additional right handed currents can arise in certain new physics
models, such as the Left-Right symmetric models, or in some supersymmetric
models nonSM . Access to the polarisation information is available via an
angular analysis of $B\rightarrow K^{*}e^{+}e^{-}$.
Hadronic form factors render theoretical prediction over the whole $q^{2}$
(the dilepton invariant mass squared) range difficult. However, it has been
shown that these uncertainties are controllable at low $q^{2}$, where the
photon term dominates, and certain asymmetries providing information on the
photon polarisation can be formed. RefJ
## 2 $B\rightarrow K^{*}\mu^{+}\mu^{-}$ status at LHCb
With 309 pb-1 of $pp$ collisions at $\sqrt{s}$=7 TeV, collected in three
months during the first half of 2011, the forward backward asymmetry of the
dilepton system, $A_{FB}$ has been measured muon using $B\rightarrow
K^{*}\mu^{+}\mu^{-}$ events, (as is shown in Fig. 1), along with $F_{L}$, the
K∗ longitudinal polarisation (Fig. 1); an input required for the photon
polarisation measurement. These observables have been measured as being in
good agreement with SM predictions, SM , implying a SM like Wilson Coefficient
$C_{7}$, but still allowing for the existence of $C_{7}^{{}^{\prime}}$ (right
handed currents). As stressed above, the measurement is most sensitive at low
$q^{2}$. It would therefore be preferable to perform the analysis using
electrons. However, experimentally it is more challenging to observe electrons
than muons, primarily due to the fact that muons provide a very clean
signature to trigger on. With 309 pb-1 of LHCb data, $B\rightarrow
K^{*}\mu^{+}\mu^{-}$ in the $q^{2}$ range 0-2 GeV has been observed, as is
shown, along with other $q^{2}$ ranges, in Fig. 2. With the rest of the 2011
data, one can expect to see a $B\rightarrow K^{*}e^{+}e^{-}$ signal.
Figure 1: $A_{FB}$ and FL as a function of $q^{2}$, as measured at LHCb with
$B\rightarrow K^{*}\mu^{+}\mu^{-}$ muon . The SM predictions are given by the
cyan (light) band, and this prediction integrated in the $q^{2}$ bins is
indicated by the purple (dark) regions.
Figure 2: The mass distributions of $B\rightarrow K^{*}\mu^{+}\mu^{-}$ in six
$q^{2}$ bins. The solid line shows a fit with a double-Gaussian signal
component (thin-green line) and an exponential background component (dashed-
red line).
Figure 3: Definition of the angles $\phi$, $\theta_{K}$ and $\theta_{L}$ in
the decay $B\rightarrow K^{*}e^{+}e^{-}$.
## 3 Analysis formalism
$B\rightarrow K^{*}e^{+}e^{-}$ can be uniquely described by four variables:
$q^{2}$ and three angular variables, $\theta_{L}$, $\theta_{K}$ and $\phi$,
(the definitions of which can be seen in Fig. 3). Following the formalism as
described in krug , the differential decay distribution can be written in
terms of these variables as:
$\displaystyle\frac{d\Gamma}{dq^{2}d\cos\Theta_{l}d\cos\Theta_{K}d\phi}=$
$\displaystyle\frac{9}{32\pi}[I_{1}\left(\cos\Theta_{K}\right)+I_{2}\left(\cos\Theta_{K}\right)\cos
2\Theta_{l}+$ $\displaystyle
I_{3}\left(\cos\Theta_{K}\right)\sin^{2}\Theta_{l}\cos
2\phi+I_{4}\left(\cos\Theta_{K}\right)\sin 2\Theta_{l}\cos\phi+$
$\displaystyle
I_{5}\left(\cos\Theta_{K}\right)\sin\Theta_{l}\cos\phi+I_{6}\left(\cos\Theta_{K}\right)\cos\Theta_{l}+$
$\displaystyle
I_{7}\left(\cos\Theta_{K}\right)\sin\Theta_{l}\sin\phi+I_{8}\left(\cos\Theta_{K}\right)\sin
2\Theta_{l}\sin\phi\ +$ $\displaystyle
I_{9}\left(\cos\Theta_{K}\right)\sin^{2}\Theta_{l}\sin 2\phi]$ (1)
When measuring this rate at LHCb, the 3D angular acceptance,
$\varepsilon\left(\cos\Theta_{l},\cos\Theta_{K},\phi\right)$ must also be
taken into account. It is assumed to be factorisable as the products of
$\varepsilon_{1}$, the acceptance as a function of $\phi$, and
$\varepsilon_{D}$, the acceptance as a function of $\cos\Theta_{K}$ and
$\cos\Theta_{L}$. Furthermore, assuming that $\varepsilon_{1}$ is an even
function, Equation 3 can be simplified by performing the $\phi$ transformation
that if $\phi$ $>$0, then $\phi$=$\phi$+$\pi$. A similar transformation can be
performed for $\cos\Theta_{L}$. Equation 3 can then be written as:
$\displaystyle\frac{d\Gamma}{dq^{2}d\cos\Theta_{l}d\cos\Theta_{K}d\phi}=$
$\displaystyle\frac{9}{32\pi}[I_{1}\left(\cos\Theta_{K}\right)+I_{2}\left(\cos\Theta_{K}\right)\cos
2\Theta_{l}+$ $\displaystyle
I_{3}\left(\cos\Theta_{K}\right)\sin^{2}\Theta_{l}\cos
2\phi+I_{9}\left(\cos\Theta_{K}\right)\sin^{2}\Theta_{l}\sin 2\phi]$
$\displaystyle\times\varepsilon_{D}\left(\cos\Theta_{l},\cos\Theta_{K}\right)$
(2)
In order to minimize theoretical uncertainties, it is desirable to measure
ratios of the amplitudes. Neglecting the lepton mass, the remaining I terms in
equation 3 can be written in terms of three such parameters,
$\mathrm{F_{L},A_{T}^{(2)},A_{Im}}$:
$\begin{split}\mathrm{F_{L}}&=\frac{\mathrm{\left|A_{0}\right|^{2}}}{\mathrm{\left|A_{0}\right|^{2}}+\left|A_{\bot}\right|^{2}+\left|A_{\|}\right|^{2}}\\\
\mathrm{A_{T}^{(2)}}&=\frac{\mathrm{\left|A_{\bot}\right|^{2}-\left|A_{\|}\right|^{2}}}{\mathrm{\left|A_{\bot}\right|^{2}+\left|A_{\|}\right|^{2}}}\\\
\mathrm{A_{Im}}&=\frac{\mathrm{\Im(A^{*}_{\bot L}A_{\bot L})-\Im(A^{*}_{\bot
R}A_{\bot
R})}}{\mathrm{\left|A_{0}\right|^{2}}+\left|A_{\bot}\right|^{2}+\left|A_{\|}\right|^{2}}\end{split}$
(3)
When expressed in terms of the helicity amplitudes, for small real values of
$\frac{A_{R}}{A_{L}}$, one obtains
$A_{T}^{(2)}\approx-2\frac{A_{right}}{A_{left}}$.
## 4 $B\rightarrow$ K∗e+e- Monte Carlo studies
Although work is ongoing on the analysis of the $B\rightarrow$ K∗e+e- data,
and yield predictions from Monte Carlo (MC) have been validated using the
control channel $B\rightarrow$ K∗J$/\Psi$ with J$/\Psi\rightarrow$(e+e-),
there is not yet, at the time of this conference, enough data to perform the
analysis or test the fitting procedure. Toy MC studies have therefore been
carried out for this purpose schune . 190k signal events were generated using
EvtGen, and separated into files containing 250 events: the predicted yields
from MC studies with 2fb-1 at a centre of mass energy of 14 TeV, excluding
effects from LHCb’s high level trigger. By performing the fit on each file, it
is shown that with 200-250 signal events and a signal to background ratio of
the order of 1, a precision of 0.2 is attainable on $\mathrm{A_{T}^{2}}$,
equivalent to an accuracy on the fraction of wrongly polarised photons of 0.1.
An example of one of the fits for one toy MC study can be seen in Fig. 4. The
analysis also demonstrates that the measurements are not sensitive to the
knowledge of the angular acceptance, and hence shall not be systematics
limited.
Figure 4: Example of the fit of $\cos\Theta_{L}$, $\cos\Theta_{K}$ and $\phi$
for one toy MC study containing 250 signal events and no background events.
## References
* (1) for example E. Lunghi and J. Matias, J. High Enerfy Physics, 04, (2007) 058
* (2) Y. Grossman and D. Pirjol, J. High Enerfy Physics, 06, (2009) 029
* (3) LHCB-CONF-2011-039
* (4) C. Bobeth, G. Hiller and D. van Dyk, J. High Enerfy Physics, 07, (2010) 098
* (5) CF. Kruger and J. Matias, Physics Rev, D71, (2005) 094009
* (6) J. LeFrançois, M.H. Schune, LHCb-2009-008
|
arxiv-papers
| 2012-01-24T15:03:16 |
2024-09-04T02:49:26.629697
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michelle Nicol (for the LHCb Collaboration)",
"submitter": "Michelle Nicol L",
"url": "https://arxiv.org/abs/1201.5000"
}
|
1201.5101
|
# Slicing the Torus: Obscuring Structures in Quasars
Martin Elvis1 1 Harvard-Smithsonian Center for Astrophysics, 60 Garden St.
Cambridge MA02138 USA elvis@cfa.harvard.edu
###### Abstract
Quasars and Active Galactic Nuclei (AGNs) are often obscured by dust and gas.
It is normally assumed that the obscuration occurs in an oblate ”obscuring
torus”, that begins at the radius at which the most refractive dust can remain
solid. The most famous form of this torus is a donut-shaped region of
molecular gas with a large scale-height. While this model is elegant and
accounts for many phenomena at once, it does not hold up to detailed tests.
Instead the obscuration in AGNs must occur on a wide range of scales and be
due to a minimum of three physically distinct absorbers. Slicing the ”torus”
into these three regions will allow interesting physics of the AGN to be
extracted.
## 1 The Quasar Standard Model
There is a ”standard model” for quasars, which was put in place within a
decade of the discovery of quasars [1], i.e. by 1973. This standard model
consists of three elements: (1) a supermassive (106-109 M⊙) black hole (SMBH,
[2]), surrounded by (2) an accretion disk [3], with (3) a relativistic jet [4,
5] emerging perpendicular to the disk and originating at just a few
Schwarzchild radii away from the black hole. The elements of this model
successfully account for, in turn: (1) the total power output of the quasar,
from the gravitational energy released by infall to near the event horizon of
the black hole; (2) the maximum temperature of 50,000 K - 100,000 K of the
ultraviolet (UV) continuum that dominates the luminosity, from the
thermalization of the gravitational energy release in the accretion disk; and
(3) the phenomenology of apparent superluminal motion, rapid variability and
polarization of the radio emission in those quasars where a jet is pointing
almost directly at us (the blazars). This is a pretty good list of successes,
and they have held up well against decades of tests.
However, unlike the predictive power of the contemporaneous particle physics
standard model [6], the quasar standard model predicts little of the rich
phenomenology of quasars: (a) the various ’types’ of active galaxies [7, 8];
(b) the maximally hot dust found in quasars, but not in starburst galaxies [9,
10, 11], (c) the strong X-ray emission [12, 13], and (d) all of the many
atomic emission and absorption features seen in the spectra of quasars [14],
not even the broad ($\sim$1% $c$) emission lines that led to their initial
recognition as exceptional objects [15].
As a response to several of these gaps, a fourth element of the standard model
has been commonly accepted since about 1985: a flattened, but large scale-
height, obscuring torus.
## 2 The Obscuring Donut Torus
A flattened obscuring torus coaligned with the accretion disk is an
appealingly simple addition to the standard model. It is able to explain both
the variety of types of AGNs, the existence of maximally hot dust, and the bi-
conical morphology often shown by the narrow emission lines of AGNs. This
paradigm is known as the Unified Scheme [16, 17, 18, 19, 20]. The Unified
Scheme posits obscuration by an optically and geometrically thick ”torus”,
lying between the inner, broad ($\sim$104 km s-1), and outer ($\sim$103 km
s-1), narrow, line emitting regions.
The torus is almost universally taken to be a large scale-height (H/R$\sim$1),
cold structure, rich in molecular gas and dust, that is co-planar with the
accretion disk [18]. As such it resembles a donut with a hole in the center,
notably in the famous illustration in Urry & Padovani [20]. This specific form
of non-spherical obscuring region has been called the ”donut torus” [21]. The
”strong” form of the Unified Scheme asserts that our orientation relative to
the one jet/disk/torus axis explains all of the variety of AGN types.
The strong Unified Scheme cannot be 100% correct as, e.g. the incidence of
X-ray obscuration is clearly more common at low luminosities [22, 23, 24, 25],
requiring some modification [26].
Nontheless this scheme does explain [16, 27]: (1) the distinction between type
1 (with broad emission lines) and type 2 (no broad emission lines) AGNs, due
to optical dust obscuration between the two emission regions; (2) the presence
of heavy X-ray obscuration in type 2 AGNs, due mainly to gas; (3) the
biconical geometry of the outer narrow line region, due to geometric
collimation of the continuum; (4) the finding of polarized broad lines in
otherwise purely narrow-lined type 2 AGNs [27, 28, 29], due to scattering off
warm electrons above the torus; and (5) the relative space densities of type 1
and type 2 AGNs. All these achievements apply equally to radio-loud AGNs [17],
and can be successfully extended to connect the luminosity functions of
blazars and radio galaxies [30].
This is a long list of accomplishments, and there can be no doubt that
flattened obscuring regions are important in AGNs. However, the elegant
reduction of these effects to a single region cannot be sustained for both
theoretical and observational reasons.
### 2.1 Problems with the Donut Torus
There are four theoretical challenges to the donut torus picture of AGN
obscuration. While none of them is individually inescapable, together they are
a significant challenge. They are:
1. 1.
The large, H/R$\sim$1, scale height in cold material: Clearly a cold
($\sim$100 K) medium does not have thermal velocities of greater than the
$\sim$1000 km s-1 of the narrow emission lines, as would be required to reach
H/R$\sim$1 interior to the region emitting those lines. (This assumes that the
lines have widths comparable to virial or Keplerian values.) The alternative
is that the material is highly clumped and has many clouds on highly inclined
orbits. A clumpy torus explains the observed AGN spectral energy distributions
(SEDs) better than a continuous medium [31]. A clumpy torus is observationally
supported by the emission seen outside of the bicones in NGC 4151, both in
optical emission lines (the ”rogue clouds” of [32]) and soft X-rays [33].
However these clouds must then collide with one another and collapse on some
unclear, but probably short, timescale. A non-static, though possibly steady-
state, model is needed (e.g.[34, 35]).
2. 2.
The energy and photon deficit problem for the broad emission line region:
Netzer (1987) noted that the observed UV continuum could not be extrapolated
in a simple, accretion disk like, way (e.g. [36]) and still provide sufficient
photons to ionize the gas producing the broad emission lines. More recent
inferred continuum shapes suggest even fewer photons in the extreme UV [37,
38]. Nor does the continuum provide sufficient energy input to power the broad
emission lines [39]. Netzer’s solution was to locate the broad emission line
region directly above the accretion disk. In this way the broad emission line
gas sees a more powerful continuum than an observer at a random angle. This
explanation works as, above the disk, neither the geometrical cosine $\theta$
nor limb darkening [40] diminish the continuum. However, if the torus is co-
planar with the disk, and covers the direct view of the disk over 80% of side-
on viewing directions, then we can only see the broad emission lines almost
perpendicular to the disk, so we also see the same continuum as that gas. The
continuum may not be a simple extrapolation of an $\alpha$-disk spectrum, but
could have a second, short-wavelength, peak, e.g. due to blurred Lyman-$\beta$
and HeII emitted by dense clouds at small radii Lawrence [41], which may solve
these problems.
3. 3.
The impedence of feedback: As is now well known, the mass of the stellar bulge
and the mass of the SMBH in a galaxy are closely tied together (see, e.g.
[42]). Assuming that the two masses are causally linked (but see [43]), some
feedback mechanism must keep the two growing in synchrony. There are three
ways the power output of a quasar can interact with the host galaxy
interstellar medium (ISM): (1) radiation; (2) broad slow winds; (3) narrow
relativistic jets. While relativistic jets from their central galaxies are
clearly important in clusters of galaxies [44], this mechanism is unlikely to
be important in either less massive systems with lower density ISMs, or in the
10 times more numerous radio-quiet quasars. However, both radiation and slow
winds will be inhibited by the presence of an exterior donut torus that blocks
80% of the sky as seen from the inner nucleus. This large covering factor
increases the efficiency needed to deposit the $\sim$5% of the total radiative
quasar power needed to unbind the host galaxy ISM [45, 46] to $\sim$25% of the
escaping radiation. More efficient scenarios have been proposed however [47].
4. 4.
The wide range of obscuring X-ray column density: The donut torus was invoked
to explain AGNs such as NGC 1068, in which the direct central radiation was
completely blocked by a Compton thick obscurer [$\tau_{Compton}$ =1
corresponds to NH=2$\times$1024cm-2]. However, there are many intermediate
type AGNs, with a wide range of lower X-ray column densities, and optical
reddening values. Values of NH from $\sim$1021cm-2 to $>$1025cm-2 are often
observed. This is roughly the difference between a thin sheet of paper and a
brick, so that a single physical cause is not required, and may be hard to
produce. Usually, in the donut torus scheme, these intermediate column
densities are attributed to viewing the donut torus at a grazing angle [48].
In the case of NGC 4151 this may be correct [32]. However, these intermediate
column densities are rather common [49]. The extended ”atmosphere” of the
donut torus must then cover $\sim$35% of the sky seen from the inner nucleus,
which is a substantial change to the picture. A clumpy wind can produce the
observed NH distribution [31].
One recurring observational problem is that the broad line widths seem to be
dominated by orbital rotation [50, 51, 52]. That requires that most broad
emission line regions are seen at a large angle to the rotation axis, but the
donut torus would prevent this.
These problems can be eased if one or more of the assumptions of the strong
Unified Scheme are relaxed: if the ratio of type 2:type 1 AGNs is smaller than
had been thought, if the torus is not co-planar, if the X-ray and optical (or
equivalently the gas and dust) obscurers are not tightly connected, or if the
obscuring region can be sliced into several layers. I will discuss this last
possibility next.
## 3 Slicing the Torus
In the years since the donut torus was proposed there has been much detailed
observational work on each of the features that the scheme was invented to
explain. One consequence is accumulating evidence that obscuration in AGNs
occurs on both smaller and larger scales than that of the hot dust limited
radius of the donut torus alone. Here I slice the obscuration into three
scales (Figure 1).
Figure 1: Scales of obscuration in quasars and AGNs. The horizontal scale is
in log(Rg).
### 3.1 Smaller Scale Obscuration
Large amplitude changes in absorbing X-ray column density have been seen on
briefer and briefer time intervals. There are now numerous examples of large
changes ($\Delta$N${}_{H}>$1023-24cm-2) in a few days or hours (e.g. [53,
54]). The most dramatic example is NGC 1365, in which the X-ray source
underwent a total eclipse by a Compton-thick cloud within two days, and
emerged within another two days [55]. For any velocity comparable to the
Keplerian velocity for the eclipsing clouds, these events all imply that they
lie at about the same distance as the broad emission line gas
($r<$10${}^{3-4}~{}R_{S}$) and have comparably high densities
($n_{e}>$109-10cm-3). The obvious conclusion is that the eclipsers are
discrete clouds of broad emission line gas. Further investigations seem to be
revealing unexpected features in these clouds with surprising implications
[56, 57].
For our purpose here, these eclipses show that obscuration, at least by gas,
occurs well within the dust sublimation radius. The observed low dust-to-gas
ratios, from comparisons of optical reddening with X-ray absorption [58, 59],
may be explained by a mix of inner, dust-free, and outer, dusty, absorbers
[60].
As the UV continuum source has $\sim$10 times the radius and so $\sim$100
times the area of the X-ray continuum, it is less likely that similar large
amplitude eclipses could be seen at UV wavelengths, which makes it hard to
tell if the clouds are dusty, as suggested by Czerny & Hryniewicz [61]. (In
principle transiting exoplanet techniques could look for these partial
eclipses.)
### 3.2 Larger Scale Obscuration
Since 1980 there have been many papers demonstrating a clear connection
between the obscuration of the AGN and the inclination angle of the host
galaxy [62, 16, 63, 64, 65, 66, 67, 68]. Evidently there is some obscuring
region dominated by the host galaxy dynamics, and so outside the SMBH sphere
of influence, though the radial scale of this obscurer is uncertain. Moreover,
as the radio jets of these galaxies lie at random angles to the host galaxy
disk plane, within a very wide cone [69, 70], the obscurer, the jet and the
accretion disk cannot all share the same axis.
Hubble imaging of type 1 and type 2 hosts also clearly shows that there are
many more dust lanes crossing the nuclei of type 2 AGNs than of type 1s on a
several 100 pc scale [22]. This obscuration lies beyond the region producing
the infrared emission from these AGNs. As these type 2 hosts are of later
morphological type than the type 1s, they undermine the strong Unified Scheme.
A different argument against the donut torus, coming from properties on
similarly large scales of up to $\sim$1 kpc , is based on the bi-conical
regions marked out by the narrow emission lines. A series of papers using
Hubble STIS long slit spectroscopy [71, 72] has found that these bi-cones are
not made up of pre-existing ISM illuminated by the AGN, as in the donut torus
model, but by outflowing hollow cones with wide opening angles. Hence the bi-
cone shapes are often not formed by a collimated incident radiation pattern,
but are matter-bounded, i.e. limited by the shape of the wind coming from the
nucleus. Moreover, the wide angles derived for the wind bi-cones ($\sim$50∘
[73]) imply that any torus covers a more limited solid angle than in the donut
torus model. (These bi-cones also show feedback in action, [74]).
Larger scale obscuration may be due to matter streaming toward the accretion
disk, as observed in H2 emission [75, 76], and predicted in some galaxy merger
models [77]. Improved imaging in molecular lines in the mm- and sub-mm bands,
primarily of CO, have begun to image obscuring structures at the $\sim$100 pc
radius scale. In the prototypical type 2 AGN, NGC 1068 (for which the donut
torus was largely originally created [27]), a Compton thick warped disk with
an inner radius of $\sim$70 pc curls up to just cross our line of sight to the
nucleus [78]. This molecular disk appears to be disconnected from any smaller
structure stretching down to the dust sublimation radius. ALMA will
revolutionize this field.
### 3.3 Hot Dust Scale Obscuration
Nonetheless, a substantial fraction of the AGN nuclear power is absorbed and
re-radiated by hot dust in most AGNs [79, 80]. The peak of the AGN dust
emission lies in the 10 $\mu$m to 20 $\mu$m range, implying temperatures of
order 200 K, and reaches shortward to $\sim$1 ${\mu}$, implying T$\sim$1800 K,
comparable to the maximum dust sublimation temperature. For the simple black
body case [24] this region spans a factor of $\sim$1000 in radius, with most
of the emitting area, and presumably most of the mass, at the larger radii.
Indeed, direct mid-IR imaging at 12$\mu$m with VLT [81, 77, 81], shows that
the mid-IR emission becomes tightly correlated with the AGN continuum only
within $\sim$600 Rsub, comparable with the expected scale. More extended
12$\mu$m emission dilutes the correlation and so presumably arises from star
formation.
The dust emitting radius [$R_{dust}(T)$] can be expressed in terms of the
Eddington ratio of the accretion rate onto the black hole
(L/0.1$L_{Edd}=\lambda_{0.1}$), and the Schwarzchild radius (Rg):
$R_{dust}(T)=1.5\times
10^{6}\lambda_{0.1}~{}L_{44}^{0.5}~{}T_{200K}^{-2.8}R_{g}$ [24]111Following
[82], for $\tau$=3, and a typical dust grain size $a=0.05\mu m$. Here $L_{44}$
is the AGN UV continuum luminosity in units of 1044 erg s-1, and $T_{200K}$ is
the dust temperature in units of 200 K.. Radii of about 106Rg apply to the 200
K dust, while the hottest dust, at T$\sim$1800 K, has R(hot)$\sim$3000 Rg.
It is instructive to compare these dust emitting radii with the size of the
SMBH sphere of influence, $R_{BHinfl}=G~{}M_{BH}/\sigma^{2}$ [42], so that
$R_{BHinfl}=10^{6}\sigma_{200}~{}R_{g}$. The main AGN dust emission thus spans
the region in which the host galaxy potential takes over from the black hole
potential (Figure 1). We might expect interesting things to happen as this
boundary is crossed. A stable H/R$\sim$1 torus is unlikely to be continuous
across this region.
The most discussed alternative to explain this dust emission is a dusty wind
off the accretion disk. The wind may be driven by magneto-centrifugal forces
[83, 84, 21], by external irradiation of the disk making it unbound [35], or
by star formation in the gravitationally unstable part of the disk [85]. A
steady state accretion model [34] can also sustain a large scale height torus,
but only at locally super-Eddington accretion rates.
A warped disk [79] can have a large covering factor without needing a large
scale height, and so is an alternative to a planar torus. This form has gained
new relevance as the means by which a black hole is fed and spun up has been
investigated in more detail. Individual accretion events with random angular
momentum vectors will naturally produce a low spin black hole [86]. Lawrence &
Elvis [24] point out that a warped disk will result, and will produce a type 1
or type 2 appearance depending on orientation, similar to that from a donut
torus, but with observable differences. Small angle warps are seen in some AGN
megamasers on parsec scales [87], and are implied in the Galactic Center [88].
A simple warped disk, with no rotation of the line of nodes, and random
accretion directions, predicts a 1:1 type 2:type 1 ratio [24]. These authors
then show that this ratio is consistent with observations, if LINERS are
excluded, for AGN samples selected by reasonably isotropic indicators (radio,
[OIII], mid-IR fluxes). The exception is samples selected in X-rays, which
show a higher fraction of X-ray obscured objects. This suggests an extra,
dust-free, absorber of X-rays, probably at smaller radii. These absorbers
could be the broad emission line clouds noted above (§3.1).
A 1:1 ratio, though, also eases several of the objections to the donut torus.
The resulting wider opening angles ease the photon and energy budget deficits,
allow wide angle biconical winds to escape, and so greater feedback, and
require a more physically reasonable smaller scale height torus. The solution
is still open.
Another prediction of the warped disk model is that the radio jets and the bi-
cones will not generally be aligned. Several anecdotal examples suggest that
this is the case [24], but a general survey is needed. The canonical Hubble
snapshot survey [70] needs to be repeated to greater depth, perhaps with AO on
8-meter class telescopes. Infrared interferometers are now beginning to
resolve this region as well as, or better than, Hubble [89, 90, 91], so we may
be able to see warped AGN dust disks directly in a few years. Eventually,
imaging reverberation mapping will let us image the broad line region motions
too, in all three spatial and velocity components [92].
## 4 Conclusions
A complete physical model for quasars is developing, but has some way to go:
The atomic emission and absorption features all have promising explanations in
terms of winds or failed winds subject to radiation pressure [57]. Although a
great deal is know about the behavior of the X-ray source, physically it is
normally ascribed to a ”hot corona” which is not understood. I anticipate that
accurate measurements of the temperature of the X-ray emission by NuSTAR [93]
will begin to unpick this knot.
The origin of the AGN powered hot dust and of the various types of AGN is
becoming more refined, but also, perhaps, more interesting and tractable. The
view of AGN obscuration as being due to a single donut torus does not capture
the complexity now realized to exist within AGNs. While acknowledging that the
illustration from Urry & Padovani is graphic and immediate, I would urge that
it should not be our default picture, as this form tends to become imprinted,
and is then hard to get beyond.
In the emerging, more complex, picture - as in the donut torus model - the
various types of AGNs continue to be explained by obscuration. However, at
least three separate obscuring regions must exist in quasars and AGNs: (1) an
inner, probably dust-free, region producing rapid X-ray eclipses, most likely
due to broad emission line clouds; (2) an intermediate scale dusty region that
reprocesses much of the AGN luminosity into thermal dust emission, including
the hottest dust. This region spans the boundary between the black hole- and
host-dominated gravitational potentials, and is perhaps in the form of a
warped disk; (3) outer regions on 0.1-1 kpc scales, connected to the host disk
and/or dust lanes.
By separating out the effects of these different obscurers we are starting to
learn much more about the inner structure of AGNs, and their feeding on sub-
kiloparsec scales .
The author would like to the thank his many collaborators in this area over
the years, especially Andy Lawrence and Guido Risaliti, Bozena Czerny, and
Gordon Richards, and is grateful to the scientific organizing committee for
the opportunity to present this work at AHAR 2011.
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|
arxiv-papers
| 2012-01-24T19:59:10 |
2024-09-04T02:49:26.636460
|
{
"license": "Public Domain",
"authors": "Martin Elvis",
"submitter": "Martin Elvis",
"url": "https://arxiv.org/abs/1201.5101"
}
|
1201.5108
|
# Event-by-event generation of electromagnetic fields in heavy-ion collisions
Wei-Tian Deng deng@fias.uni-frankfurt.de Frankfurt Institute for Advanced
Studies, D-60438 Frankfurt am Main, Germany Xu-Guang Huang xhuang@itp.uni-
frankfurt.de Frankfurt Institute for Advanced Studies, D-60438 Frankfurt am
Main, Germany Institut für Theoretische Physik, J. W. Goethe-Universität,
D-60438 Frankfurt am Main, Germany
###### Abstract
We compute the electromagnetic fields generated in heavy-ion collisions by
using the HIJING model. Although after averaging over many events only the
magnetic field perpendicular to the reaction plane is sizable, we find very
strong electric and magnetic fields both parallel and perpendicular to the
reaction plane on the event-by-event basis. We study the time evolution and
the spatial distribution of these fields. Especially, the electromagnetic
response of the quark-gluon plasma can give non-trivial evolution of the
electromagnetic fields. The implications of the strong electromagnetic fields
on the hadronic observables are also discussed.
###### pacs:
25.75.-q, 25.75.Ag
## I Introduction
Relativistic heavy-ion collisions provide us the methods to create and explore
strongly interacting matter at high energy densities where the deconfined
quark-gluon plasma (QGP) is expected to form. The properties of matter
governed by quantum chromodynamics (QCD) have been studied at the Relativistic
Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and at the
Large Hadron Collider (LHC) at CERN. Measurements performed at RHIC in Au + Au
collisions at center-of-mass energy $\sqrt{s}=200$ GeV per nucleon pair and at
LHC in Pb + Pb collisions at center-of-mass energy $\sqrt{s}=2.76$ TeV per
nucleon pair have revealed several unusual properties of this hot, dense,
matter (_e.g._ , its very low shear viscosity arXiv:0804.4015 ;
arXiv:1011.2783 , and its high opacity for energetic jets Wang:2003mm ;
Vitev:2002pf ; Eskola:2004cr ; Turbide:2005fk ).
Due to the fast, oppositely directed, motion of two colliding ions, off-
central heavy-ion collisions can create strong transient magnetic fields
Rafelski:1975rf . As estimated by Kharzeev, McLerran, and Warringa
arXiv:0711.0950 , the magnetic fields generated in off-central Au + Au
collision at RHIC can reach $eB\sim m_{\pi}^{2}\sim 10^{18}$ G, which is
$10^{13}$ times larger than the strongest man-made steady magnetic field in
the laboratory. The magnetic field generated at LHC energy can be $10$ times
larger than that at RHIC arXiv:0907.1396 . Thus, heavy-ion collisions provide
a unique terrestrial environment to study QCD in strong magnetic fields. It
has been shown that a strong magnetic field can convert topological charge
fluctuations in the QCD vacuum into global electric charge separation with
respect to the reaction plane arXiv:0711.0950 ; arXiv:0808.3382 ;
arXiv:0911.3715 . This so-called chiral magnetic effect may serve as a sign of
the local P and CP violation of QCD. Experimentally, the STAR arXiv:0909.1717
; arXiv:0909.1739 , PHENIX Ajitanand:2010rc , and ALICE collaboration:2011sma
Collaborations have reported the measurements of the two-particle correlations
of charged particles with respect to the reaction plane, which are
qualitatively consistent with the chiral magnetic effect, although there are
still some debates arXiv:1011.6053 ; arXiv:0912.5050 ; arXiv:1008.4919 ;
arXiv:1101.1701 ; arXiv:0911.1482 .
Besides the chiral magnetic effect, there can be other effects caused by the
strong magnetic fields including the catalysis of chiral symmetry breaking
hep-ph/9405262 , the possible splitting of chiral and deconfinement phase
transitions arXiv:1004.2712 , the spontaneous electromagnetic
superconductivity of QCD vacuum arXiv:1008.1055 ; arXiv:1101.0117 , the
possible enhancement of elliptic flow of charged particles arXiv:1102.3819 ;
arXiv:1108.4394 , the energy loss due to the synchrotron radiation of quarks
arXiv:1006.3051 , the emergence of anisotropic viscosities arXiv:1108.4394 ;
arXiv:0910.3633 ; arXiv:1108.0602 , the induction of the electric quadrupole
moment of the QGP arXiv:1103.1307 , _etc_.
The key quantity of all these effects are the strength of the magnetic fields.
Most of the previous works estimated the magnetic-field strength based on the
averaging over many events arXiv:0711.0950 ; arXiv:0907.1396 ; arXiv:1003.2436
; arXiv:1103.4239 ; arXiv:1107.3192 , thus, due to the mirror symmetry of the
collision geometry, only the $y$-component of the magnetic field remains
sizable, $e\langle B_{y}\rangle\sim m_{\pi}^{2}$, while other components are
$\langle B_{x}\rangle=\langle B_{z}\rangle=0$. Hereafter, we use the angle
bracket to denote event average. We choose the $z$ axis along the beam
direction of the projectile, $x$ axis along the impact parameter ${\bf b}$
from the target to the projectile, and $y$ axis perpendicular to the reaction
plane, as illustrated in Fig. 1.
Figure 1: The geometrical illustration of the off-central collisions with
impact parameter $b$. Here “T” for target and “P” for projectile.
However, in many cases, the final hadronic signals are measured on the event-
by-event basis. Thus, it is important to study how the magnetic fields are
generated on the event-by-event basis. Such a study was recently initiated by
Bzdak and Skokov arXiv:1111.1949 . They showed that the $x$-component of the
magnetic field can be as strong as the $y$ component on the event-by-event
basis, due to the fluctuation of the proton positions in the colliding nuclei.
Besides, they also found that the event-by-event generated electric field can
be comparable to the magnetic field.
The aim of our work is to give a detailed study of the space-time structure of
the event-by-event generated electromagnetic fields in the heavy-ion
collisions. We perform our calculation by using the heavy ion jet interaction
generator (HIJING) model Wang:1991hta ; Deng:2010mv ; Deng:2010xg . HIJING is
a Monte-Carlo event generator for hadron productions in high energy p + p, p +
A, and A + A collisions. It is essentially a two-component model, which
describes the production of hard parton jets and the soft interaction between
nucleon remnants. In HIJING, the hard jets production is controlled by
perturbative QCD, and the interaction of nucleon remnants via soft gluon
exchanges is described by the string model Sjostrand:1987su .
This paper is organized as follows. We give a general setup of our calculation
in Sec. II. In Sec. III, we present our numerical results. We discuss the
influence of the non-trivial electromagnetic response of the QGP on the time
evolution of the electromagnetic fields in Sec. IV. We conclude with
discussions and summary in Sec. V. We use natural unit $\hbar=c=1$.
## II General Setup
We use the Liénard-Wiechert potentials to calculate the electric and magnetic
fields at a position ${\mathbf{r}}$ and time $t$,
$\displaystyle e{\bf E}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}\frac{{\bf R}_{n}-R_{n}{\bf
v}_{n}}{(R_{n}-{\bf R}_{n}\cdot{\bf v}_{n})^{3}}(1-v_{n}^{2}),$ (1)
$\displaystyle e{\bf B}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}\frac{{\bf v}_{n}\times{\bf
R}_{n}}{(R_{n}-{\bf R}_{n}\cdot{\bf v}_{n})^{3}}(1-v_{n}^{2}),$ (2)
where $Z_{n}$ is the charge number of the $n$th particle, ${\bf
R}_{n}={\mathbf{r}}-{\mathbf{r}}_{n}$ is the relative position of the field
point ${\mathbf{r}}$ to the source point ${\mathbf{r}}_{n}$, and
${\mathbf{r}}_{n}$ is the location of the $n$th particle with velocity ${\bf
v}_{n}$ at the retarded time $t_{n}=t-|{\mathbf{r}}-{\mathbf{r}}_{n}|$. The
summations run over all charged particles in the system. Although there are
singularities at $R_{n}=0$ in Eqs. (1)-(2), in practical calculation of ${\bf
E}$ and ${\bf B}$ at given $(t,{\mathbf{r}})$, the events causing such
singularities rarely appear. So, we omit such events in our numerical code. In
non-relativistic limit, $v_{n}\ll 1$, Eq. (1) reduces to the Coulomb’s law and
Eq. (2) reduces to the Biot-Savart law for a set of moving charges,
$\displaystyle e{\bf E}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}\frac{{\bf R}_{n}}{R^{3}_{n}},$
(3) $\displaystyle e{\bf B}(t,{\mathbf{r}})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi}\sum_{n}Z_{n}\frac{{\bf v}_{n}\times{\bf
R}_{n}}{R^{3}_{n}}.$ (4)
To calculate the electromagnetic fields at moment $t$, we need to know the
full phase space information of all charged particles before $t$. In the
HIJING model, the position of each nucleon before collision is sampled
according to the Woods-Saxon distribution. The energy for each nucleon is set
to be $\sqrt{s}/2$ in the center-of-mass frame. Assuming that the nucleons
have no transverse momenta before collision, the value of the velocity of each
nucleon is given by $v_{z}^{2}=1-(2m_{N}/\sqrt{s})^{2}$, where $m_{N}$ is the
mass of the nucleon. At RHIC and LHC, $v_{z}$ is very large, so the nuclei are
Lorentz contracted to pancake shapes.
We set the initial time $t=0$ as the moment when the two nuclei completely
overlap. The collision time for each nucleon is given according to its initial
longitudinal position $z_{N}^{L}=z_{N}\cdot 2m_{N}/\sqrt{s}$ and velocity
$v_{z}$, where $z_{N}$ is the initial longitudinal position of nucleon in the
rest frame of the nucleus. The probability of two nucleon colliding at a given
impact parameter $b$ is determined by the Glauber model. In this paper, we
call nucleons without any interaction spectators and those that suffer at
least once elastic or inelastic collision participants before their first
collision. The participants will exchange their momenta and energies, and
become remnants after collision. Differently from the spectators and the
participants, the remnants can have finite transverse momenta. In our
calculation, we find that although the spectators and participants are the
main sources of fields at $t\leq 0$, remnants can give important contributions
at $t>0$. In the HIJING model, we neglect the back reaction of the
electromagnetic field on the motions of the the charged particles. This is a
good approximation before the collision happens because the electromagnetic
field is weak at that time. We will discuss the feed back effect of the
electromagnetic fields on QGP in Sec. IV by using a magnetohydrodynamic
treatment.
After collision, many partons may be produced and the hot, dense, QGP may
form. As the QGP is nearly neutral, we neglect the contributions from the
produced partons to the generation of the electromagnetic field. However, if
the electric conductivity of the QGP is large, the QGP can have significant
response to the change of the external electromagnetic field. This can become
substantial for the time evolution of the fields in the QGP. We will discuss
this point in detail in Sec. IV.
## III Numerical Results
### III.1 Impact parameter dependence
We first show the impact parameter dependence of the electromagnetic fields at
${\mathbf{r}}={\bf 0}$ and $t=0$. The left panel of Fig. 2 is the results for
Au + Au collision at RHIC energy $\sqrt{s}=200$ GeV; the right panel of Fig. 2
is for Pb + Pb collision at LHC energy $\sqrt{s}=2.76$ TeV.
Figure 2: (Color online) The electromagnetic fields at $t=0$ and
${\mathbf{r}}={\bf 0}$ as functions of the impact parameter $b$.
As seen from Eq. (2), $\langle B_{x}(t,{\bf 0})\rangle=0$, while $\langle
B_{y}(t,{\bf 0})\rangle<0$ when $b>0$. Also, from Eqs. (1)-(2), we find that
there are always $|E_{y}(0,{\bf 0})|\approx|B_{x}(0,{\bf 0})|$ and
$|B_{y}(0,{\bf 0})|\geq|E_{x}(0,{\bf 0})|$ when $v_{z}$ is large [See Eqs.
(6)-(7)]. These facts are reflected in Fig. 2. Although the $x$-component of
the magnetic field as well as the $x$\- and $y$-components of the electric
field vanish after averaging over many events, their magnitudes in each event
can be huge due to the fluctuations of the proton positions in the nuclei.
Thus, following Bzdak and Skokov arXiv:1111.1949 , we plot the averaged
absolute values $\langle|E_{x,y}|\rangle$ and $\langle|B_{x,y}|\rangle$ at
${\mathbf{r}}={\bf 0}$ and $t=0$. Similar with the findings in Ref.
arXiv:1111.1949 , we find that $\langle|B_{x}|\rangle$,
$\langle|E_{x}|\rangle$, and $\langle|E_{y}|\rangle$ are comparable to
$\langle|B_{y}|\rangle$, and the following equalities hold approximately,
$\langle|E_{x}|\rangle\approx\langle|E_{y}|\rangle\approx\langle|B_{x}|\rangle$.
But our results at RHIC energy are about three times smaller than that
obtained in Ref. arXiv:1111.1949 . We checked that this is because the
thickness of the nuclei in our calculation is finite while the authors of Ref.
arXiv:1111.1949 assumed that the nuclei are infinitely thin. We can also
observe that, at small $b$ region, contrary to $\langle B_{y}\rangle$ which is
proportional to $b$, the fields caused by fluctuation are not sensitive to
$b$.
### III.2 Collision energy dependence
Figure 3: (Color online) The collision energy dependence of the
electromagnetic fields at ${\mathbf{r}}={\bf 0}$ and $t=0$.
We see from Fig. 2 that the magnitudes of all the fields at LHC energy is
around $14$ times bigger than that at RHIC energy. To study the collision
energy dependence more carefully, we calculate the fields at $t=0$ and
${\mathbf{r}}={\bf 0}$ for different $\sqrt{s}$. To high precision, the linear
dependence of the fields on the collision energy is obtained, as shown in Fig.
3. Thus, the following scaling law holds for event-by-event generated
electromagnetic fields as well as for event-averaged magnetic fields,
$\displaystyle e\cdot{\rm Field}\propto\sqrt{s}f(b/R_{A}),$ (5)
where $R_{A}$ is the radius of the nucleus and $f(b/R_{A})$ is a universal
function which has the shapes as shown in Fig. 2 for
$\langle|B_{x,y}|\rangle$, $\langle|E_{x,y}|\rangle$, and $\langle
B_{y}\rangle$.
Actually, a more general form of Eq. (5) can be derived from Eqs. (1)-(2). As
the fields at $t=0$ are mainly caused by spectators and participants whose
velocity $v_{n}=v_{z}=\sqrt{1-(2m_{N}/\sqrt{s})^{2}}\approx 1$, the electric
and magnetic fields at $t=0$ in the transverse plane can be expressed as
$\displaystyle e{\bf E}_{\perp}(0,{\bf r})$ $\displaystyle\approx$
$\displaystyle\frac{e^{2}}{4\pi}\frac{\sqrt{s}}{2m_{N}}\sum_{n}\frac{{\bf
R}_{n\perp}}{R_{n\perp}^{3}},$ (6) $\displaystyle e{\bf B}_{\perp}(0,{\bf r})$
$\displaystyle\approx$
$\displaystyle\frac{e^{2}}{4\pi}\frac{\sqrt{s}}{2m_{N}}\sum_{n}\frac{{\bf
e}_{nz}\times{\bf R}_{n\perp}}{R_{n\perp}^{3}},$ (7)
where ${\bf e}_{nz}$ is the unit vector in $\pm z$ direction depending on
whether the $n$th proton is in the target or in the projectile, ${\bf
R}_{n\perp}$ is the transverse position of the $n$th proton which is
independent of $\sqrt{s}$, and $R_{n\perp}=|{\bf R}_{n\perp}|$.
### III.3 Spatial distribution
Figure 4: (Color online) The spatial distributions of the electromagnetic
fields in the transverse plane at $t=0$ for $b=0$ (upper panels) and $b=10$ fm
(lower panels) at RHIC energy. The unit is $m_{\pi}^{2}$. The dashed circles
indicate the two colliding nuclei.
The spatial distributions of the magnetic and electric fields are evidently
inhomogeneous. We show in Fig. 4 the contour plots of $\langle
B_{x,y,z}\rangle$, $\langle E_{x,y,z}\rangle$, $\langle|B_{x,y,z}|\rangle$,
and $\langle|E_{x,y,z}|\rangle$ at $t=0$ in the transverse plane at RHIC
energy. The upper two panels are for $b=0$ and the lower two panels are for
$b=10$ fm. The spatial distribution of the transverse fields for LHC energy is
merely the same as Fig. 4 but the fields have $2760/200\approx 14$ times
larger magnitudes everywhere according to Eqs. (6)-(7). The spatial
distribution of the fields in the reaction plane was studied in Ref.
arXiv:1103.4239 .
First, as we expected, the longitudinal fields $\langle B_{z}\rangle$,
$\langle E_{z}\rangle$, $\langle|B_{z}|\rangle$, and $\langle|E_{z}|\rangle$
are much smaller than the transverse fields. Second, the event-averaged fields
$\langle B_{x,y}\rangle$ and $\langle E_{x,y}\rangle$ distribute similarly
with the fields generated by two uniformly charged, oppositely moving, discs.
Third, the spatial distribution of the magnetic fields is very different from
that of the electric fields on the event-by-event basis. For central
collisions, both $\langle|B_{x}|\rangle$ and $\langle|B_{y}|\rangle$
distribute circularly and concentrate at ${\mathbf{r}}={\bf 0}$, while
$\langle|E_{x}|\rangle$ and $\langle|E_{y}|\rangle$ peak around $x=\pm R_{A}$
and $y=\pm R_{A}$ with $R_{A}$ the radius of the nucleus. We notice that for
off-central collision, the $y$-component of the electric field varies steeply
along $y$-direction, reflecting the fact that at $t=0$ a large amount of net
charge stays temporally in the “almond”-shaped overlapping region.
### III.4 Probability distribution over events
Figure 5: (Color online) The probability densities $P(B_{x},B_{y})$ and
$P(E_{x},E_{y})$ for different impact parameters for Au + Au collisions at
$\sqrt{s}=200$ GeV.
Although we used the event-averaged absolute values $\langle|B_{x,y}|\rangle$
and $\langle|E_{x,y}|\rangle$ to characterize the event-by-event fluctuations
of the electromagnetic fields, it would have more practical relevance to see
the probability distribution of the magnetic field, defined as
$\displaystyle P(B_{x},B_{y})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\frac{d^{2}N}{dB_{x}dB_{y}},$ (8)
where $N$ is the number of events. Similarly, we can define $P(E_{x},E_{y})$.
After simulating $10^{6}$ events, we obtain $P(B_{x},B_{y})$ and
$P(E_{x},E_{y})$ at $t=0$ and ${\mathbf{r}}={\bf 0}$ for Au + Au collisions at
$\sqrt{s}=200$ GeV, as shown in Fig. 5. As expected, the probability
distribution of the magnetic (electric) field peaks at ${\bf B}={\bf 0}$
(${\bf E}={\bf 0}$) for central collision, while the probability distribution
for magnetic field is shifted to finite $B_{y}$ for off-central collision.
This is more clearly shown in Fig. 6, where we depict the one-dimensional
probability density $P(B_{x})\equiv\int dB_{y}P(B_{x},B_{y})$ (other
probability densities are analogously defined).
The probability distributions for Pb + Pb collisions at $\sqrt{s}=2.76$ TeV
have analogous shapes with Fig. 5 but much more spread, as clearly shown in
the lower panels of Fig. 6. This is because the strength of the field
generated at LHC can be obtained approximately from that at RHIC by a
$\sqrt{s_{\rm LHC}/s_{\rm RHIC}}$-scaling according to Eqs. (6)-(7). Thus,
after normalization, the probability distributions at LHC energy are related
to that at RHIC energy by
$\displaystyle P_{\rm LHC}(B_{x},B_{y})\approx\frac{s_{\rm RHIC}}{s_{\rm
LHC}}P_{\rm RHIC}\left(\sqrt{\frac{s_{\rm RHIC}}{s_{\rm
LHC}}}B_{x},\sqrt{\frac{s_{\rm RHIC}}{s_{\rm LHC}}}B_{y}\right),$ (9)
$\displaystyle P_{\rm LHC}(E_{x},E_{y})\approx\frac{s_{\rm RHIC}}{s_{\rm
LHC}}P_{\rm RHIC}\left(\sqrt{\frac{s_{\rm RHIC}}{s_{\rm
LHC}}}E_{x},\sqrt{\frac{s_{\rm RHIC}}{s_{\rm LHC}}}E_{y}\right).$ (10)
Figure 6: (Color online) The probability densities $P(B_{x,y})$ and
$P(E_{x,y})$ for central collision $b=0$ and off-central collision $b=10$ fm
for Au + Au collision (upper panels) at $\sqrt{s}=200$ GeV and for Pb + Pb
collision at $\sqrt{s}=2.76$ TeV (lower panels).
### III.5 Early-stage time evolution
Figure 7: (Color online) The time evolution of electromagnetic fields at
${\mathbf{r}}=0$ with impact parameter $b=0$, and $b=10$ for Au + Au
collisions at $\sqrt{s}=200$ GeV and Pb + Pb collisions at $\sqrt{s}=2.76$
TeV. After collision, the remnants can essentially slow down the decay of the
transverse fields, and enhance the longitudinal fields.
In Fig. 7, we show our results of the early-stage time evolution of the
electromagnetic fields at ${\mathbf{r}}={\bf 0}$ in both central collisions
and off-central collisions with $b=10$ fm, for Au + Au collision at
$\sqrt{s}=200$ GeV and for Pb + Pb collision at $\sqrt{s}=2.76$ TeV. We take
into account the contributions from charged particles in spectators,
participants, and remnants. Around $t=0$, we checked that the contributions
from the remnants are negligibly small, while the contributions from
participants can be as large as that from spectators. However, at a latter
time when the spectators have already moved far away from the collision
region, the contributions from the remnants become important because they move
much slower than the spectators. These remnants can essentially slow down the
decay of the transverse fields, as seen from Fig. 7. Another evident effect of
the remnants is the substantial enhancements of the longitudinal magnetic and
electric fields caused by the position fluctuation of the remnants, which have
non-zero transverse momenta. Particularly, although $\langle|B_{z}|\rangle$ is
at least one order smaller than $\langle|B_{x,y}|\rangle$ for $t\lesssim 1$
fm/c, $\langle|E_{z}|\rangle$ can evolve to the same amount of
$\langle|E_{x,y}|\rangle$ in a very short time after collision and then they
decay very slowly.
For central collision, all the fields are generated due to the position
fluctuations of the charged particles. As seen from Fig. 7, these fluctuations
lead to sizable $\langle|E_{x}|\rangle=\langle|E_{y}|\rangle$ and
$\langle|B_{x}|\rangle=\langle|B_{y}|\rangle$ around $t=0$, but they drop very
fast. Note that the fields drop faster for larger collision energy $\sqrt{s}$.
For off-central collisions, the $y$-component of the magnetic field are much
larger than other fields at $t=0$. But at a latter time when the spectators
move far away, the contributions of remnants dominate, and lead to
$\langle|B_{y}|\rangle\approx\langle|B_{x}|\rangle>\langle B_{y}\rangle$.
A common feature of all the fluctuation-caused transverse fields is that they
all increase very fast before collision (due to the fast approaching of the
nuclei), then they drop steeply after $t=0$ (due to the high-speed leaving of
the spectators away from the collision center), and then decay very slowly
(due to that contribution from the slowly moving remnants take over that from
the spectators). After the early-stage evolution, the produced QGP may get
enough time to respond to the electromagnetic fields, which may substantially
modify the picture of evolution. We discuss this point in the next section.
## IV Response of the QGP to electromagnetic fields
In the calculations above, we have neglected the electromagnetic response of
the matter produced in the collision (_i.e._ , we assumed the produced matter
is ideally electrically insulating). However, if the produced matter, after a
short early-stage evolution, is in the QGP phase, the electric conductivity
$\sigma$ is not negligible. At high temperature, the perturbative QCD gives
that $\sigma\approx 6T/e^{2}$ Arnold:2003zc , and the lattice calculations
give that $\sigma\approx 7C_{\rm EM}T$ Gupta:2003zh , or $\sigma\approx
0.4C_{\rm EM}T$ Aarts:2007wj ; Ding:2010ga , or $\sigma\approx(1/3)C_{\rm
EM}T$-$C_{\rm EM}T$ Ding:2010ga ; Francis:2011bt for temperature of several
$T_{c}$, where $C_{\rm EM}\equiv\sum_{f}e_{f}^{2}$, $f=u,d,s$, and $e_{f}$ is
the charge of quark with flavor $f$. Thus it is expected that the QGP can have
non-trivial electromagnetic response. Such electromagnetic response can
substantially influence the time evolution of the electromagnetic fields in
the QGP.
To have an estimation of the electromagnetic response of QGP, we use the
following Maxwell’s equations,
$\displaystyle\displaystyle\nabla\times{\bf E}=-\frac{\partial{\bf
B}}{\partial t},$ (11)
$\displaystyle\displaystyle\frac{1}{\mu}\nabla\times{\bf
B}=\epsilon\frac{\partial{\bf E}}{\partial t}+{\bf J},$ (12)
where $\mu$ and $\epsilon$ are the permeability and permittivity of the QGP,
respectively, and are assumed as constants. ${\bf J}$ is the electric current
determined by the Ohm’s law,
$\displaystyle{\bf J}=\sigma\left({\bf E}+{\bf v}\times{\bf B}\right),$ (13)
where ${\bf v}$ is the flow velocity of QGP. Using Eq. (13), we can rewrite
the Maxwell’s equations as
$\displaystyle\displaystyle\frac{\partial{\bf B}}{\partial
t}=\nabla\times({\bf v}\times{\bf B})+\frac{1}{\sigma\mu}\left(\nabla^{2}{\bf
B}-\mu\epsilon\frac{\partial^{2}{\bf B}}{\partial t^{2}}\right),$ (14)
$\displaystyle\displaystyle\frac{\partial{\bf E}}{\partial
t}+\frac{\partial{\bf v}}{\partial t}\times{\bf B}={\bf
v}\times(\nabla\times{\bf E})+\frac{1}{\sigma\mu}\left(\nabla^{2}{\bf
E}-\mu\epsilon\frac{\partial^{2}{\bf E}}{\partial t^{2}}\right),$ (15)
where we have used the Gauss’s laws $\nabla\cdot{\bf B}=0$ and
$\nabla\cdot{\bf E}=\rho=0$ with the assumption that the net electric charge
density of the QGP is zero. Equation (14) is the induction equation, which
plays a central role in describing the dynamo mechanism of stellar magnetic
field generation. The first terms on the right-hand sides of Eqs. (14)-(15)
are the convection terms, while the last terms are the diffusion terms. The
ratio of these two types of terms are characterized by the magnetic Reynolds
number $R_{m}$,
$\displaystyle R_{m}\equiv LU\sigma\mu,$ (16)
where $L$ is the characteristic length or time scale of the QGP, $U$ is the
characteristic velocity of the flow.
Because the theoretical result of $\sigma$ is quite uncertain, the value of
$R_{m}$ is also uncertain. For example, by assuming $\mu=\epsilon=1$, setting
the characteristic length scale $L\sim 10$ fm, and the typical velocity $U\sim
0.5$, we can estimate $R_{m}$ at $T=350$ MeV as $R_{m}\sim 0.2$ if we use
$\sigma\approx 0.4C_{\rm EM}T$ in Refs. Aarts:2007wj ; Ding:2010ga , or
$R_{m}\sim 4$ if we use $\sigma\approx 7C_{\rm EM}T$ in Ref. Gupta:2003zh , or
$R_{m}\sim 600$ if we use $\sigma\approx 6T/e^{2}$ in Ref. Arnold:2003zc .
If $R_{m}\ll 1$, we can neglect the convection terms in Eq. (14) and Eq. (15).
Tuchin studied this case arXiv:1006.3051 (with additional condition $U\ll 1$
so that the second-order time-derivative terms in the diffusion terms are
neglected), and concluded that the magnetic field can be considered as
approximately stationary during the QGP lifetime111To reach this conclusion,
Tuchin used $\sigma=6T^{2}/T_{c}$ to estimate the magnetic diffusion time
$\tau=(L/2)^{2}\sigma/4$ and found that, for $L=10$ fm and $T=2T_{c}\approx
400$MeV, $\tau\approx 150$ fm is much longer than the lifetime of the QGP.
However, if, for example, $\sigma\approx 0.4C_{\rm EM}T$ is used, the magnetic
diffusion time is $\tau\approx 0.3$ fm, which is much shorter than what Tuchin
obtained..
If $R_{m}\gg 1$, we can neglect the diffusion terms in Eqs. (14)-(15), _i.e._
, take the ideally conducting limit,
$\displaystyle\displaystyle\frac{\partial{\bf B}}{\partial
t}=\nabla\times({\bf v}\times{\bf B}),$ (17) $\displaystyle{\bf E}=-{\bf
v}\times{\bf B}.$ (18)
It is well-known that Eq. (17) leads to the frozen-in theorem for ideally
conducting plasma (_i.e._ , the magnetic lines are frozen in the plasma
elements or more precisely the magnetic flux through a closed loop defined by
plasma elements keeps constant Jackson ).
We now use Eqs. (17)-(18) to estimate how the electromagnetic field evolves in
a QGP with $R_{m}\gg 1$. To this purpose, we have to know the evolution of
${\bf v}$ first. By assuming that the bulk evolution of the QGP is governed by
strong dynamics, we can neglect the influence of the electromagnetic field on
the evolution of the velocity ${\bf v}$. We assume the Bjorken picture for the
longitudinal expansion,
$\displaystyle v_{z}=\frac{z}{t}.$ (19)
Because the early transverse expansion is slow, following Ref.
Ollitrault:2007du , we adopt a linearized ideal hydrodynamic equation to
describe the transverse flow velocity ${\bf v}_{\perp}$,
$\displaystyle\frac{\partial}{\partial t}{\bf
v}_{\perp}=-\frac{1}{\varepsilon+P}\nabla_{\perp}P=-c_{s}^{2}\nabla_{\perp}\ln{\mathfrak{s}},$
(20)
where we used $\varepsilon+P=T{\mathfrak{s}}$, $\mathfrak{s}$ is the entropy
density, and $c_{s}=\sqrt{\partial P/\partial\varepsilon}$ is the speed of
sound. For simplicity, we choose an initial Gaussian transverse entropy
density profile as in Ollitrault:2007du ,
$\displaystyle{\mathfrak{s}}(x,y)={\mathfrak{s}}_{0}\exp{\left(-\frac{x^{2}}{2a^{2}_{x}}-\frac{y^{2}}{2a^{2}_{y}}\right)},$
(21)
where $a_{x,y}$ are the root-mean-square widths of the transverse
distribution. They are of order of the nuclei radii if the impact parameter is
not large. For example, for Au + Au collisions at RHIC, $a_{x}\sim a_{y}\sim
3$ fm for $b=0$, and $a_{x}\sim 2$ fm, $a_{y}\sim 3$ fm for $b=10$ fm. One can
then easily solve Eq. (20) and obtain,
$\displaystyle v_{x}$ $\displaystyle=$
$\displaystyle\frac{c_{s}^{2}}{a^{2}_{x}}xt,$ (22) $\displaystyle v_{y}$
$\displaystyle=$ $\displaystyle\frac{c_{s}^{2}}{a^{2}_{y}}yt.$ (23)
Substituting the velocity fields above into Eq. (17), we obtain a linear
differential equation for ${\bf B}(t)$. For a given initial condition ${\bf
B}^{0}({\mathbf{r}})={\bf B}(t=t_{0},{\mathbf{r}})$ where $t_{0}$ is the
formation time of the QGP, it can be solved analytically,
$\displaystyle B_{x}(t,x,y,z)$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})}B_{x}^{0}\left(xe^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})},ye^{-\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})},z\frac{t_{0}}{t}\right),$
(24) $\displaystyle B_{y}(t,x,y,z)$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})}B_{y}^{0}\left(xe^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})},ye^{-\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})},z\frac{t_{0}}{t}\right).$
(25)
Because $B_{z}$ is always much smaller than $B_{x}$ and $B_{y}$, we are not
interested in it. The electric fields can be obtained from Eq. (18) once we
have ${\bf B}(t,{\mathbf{r}})$.
To reveal the physical content in Eqs. (24)-(25), we notice that, by
integrating Eq. (19), Eq. (22), and Eq. (23), a fluid cell located at
$(x_{0},y_{0},z_{0})$ at time $t_{0}$ will flow to the coordinate $(x,y,z)$ at
time $t$ with
$\displaystyle x$ $\displaystyle=$ $\displaystyle
x_{0}\exp{\left[\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})\right]},$ (26)
$\displaystyle y$ $\displaystyle=$ $\displaystyle
y_{0}\exp{\left[\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})\right]},$ (27)
$\displaystyle z$ $\displaystyle=$ $\displaystyle z_{0}\frac{t}{t_{0}}.$ (28)
Thus,we can rewrite Eqs. (24)-(25) as
$\displaystyle B_{x}(t,x,y,z)$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})}B_{x}\left(t_{0},x_{0},y_{0},z_{0}\right),$
(29) $\displaystyle B_{y}(t,x,y,z)$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})}B_{y}\left(t_{0},x_{0},y_{0},z_{0}\right).$
(30)
As the areas of the cross sections of the QGP expand according to
$t\,\exp{\left(\frac{c_{s}^{2}}{2a_{y}^{2}}t^{2}\right)}$ in the $y$-$z$ plane
and $t\,\exp{\left(\frac{c_{s}^{2}}{2a_{x}^{2}}t^{2}\right)}$ in the $x$-$z$
plane, Eqs. (29)-(30) mean that the magnetic line flows with the fluid cell
and is diluted due to the expansion of the QGP. These are just the
manifestations of the frozen-in theorem, We also note that Eqs. (24)-(25) can
be written in explicit scaling forms,
$\displaystyle
t\,e^{\frac{c_{s}^{2}}{2a_{y}^{2}}t^{2}}B_{x}\left(t,xe^{\frac{c_{s}^{2}}{2a_{x}^{2}}t^{2}},ye^{\frac{c_{s}^{2}}{2a_{y}^{2}}t^{2}},z\,t\right)$
$\displaystyle=$ $\displaystyle
t_{0}\,e^{\frac{c_{s}^{2}}{2a_{y}^{2}}t_{0}^{2}}B_{x}\left(t_{0},xe^{\frac{c_{s}^{2}}{2a_{x}^{2}}t_{0}^{2}},ye^{\frac{c_{s}^{2}}{2a_{y}^{2}}t_{0}^{2}},z\,t_{0}\right),$
(31) $\displaystyle
t\,e^{\frac{c_{s}^{2}}{2a_{x}^{2}}t^{2}}B_{y}\left(t,xe^{\frac{c_{s}^{2}}{2a_{x}^{2}}t^{2}},ye^{\frac{c_{s}^{2}}{2a_{y}^{2}}t^{2}},z\,t\right)$
$\displaystyle=$ $\displaystyle
t_{0}\,e^{\frac{c_{s}^{2}}{2a_{x}^{2}}t_{0}^{2}}B_{y}\left(t_{0},xe^{\frac{c_{s}^{2}}{2a_{x}^{2}}t_{0}^{2}},ye^{\frac{c_{s}^{2}}{2a_{y}^{2}}t_{0}^{2}},z\,t_{0}\right).$
(32)
From Eqs. (24)-(25), we see that the evolution of ${\bf B}$ is strongly
influenced by its initial spatial distribution. However, the time evolution of
the magnetic fields at the center of the collision region, ${\mathbf{r}}={\bf
0}$, takes very simple forms,
$\displaystyle B_{x}(t,{\bf 0})$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{y}^{2}}(t^{2}-t_{0}^{2})}B_{x}^{0}({\bf
0}),$ (33) $\displaystyle B_{y}(t,{\bf 0})$ $\displaystyle=$
$\displaystyle\frac{t_{0}}{t}e^{-\frac{c_{s}^{2}}{2a_{x}^{2}}(t^{2}-t_{0}^{2})}B_{y}^{0}({\bf
0}).$ (34)
Setting $a_{x}\sim a_{y}\sim 3$ fm and $c_{s}^{2}\sim 1/3$, we see from Eqs.
(33)-(34) that for $t\lesssim 5$ fm the magnetic fields decay inversely
proportional to time.
## V Summary and Discussions
In summary, we have utilized the HIJING model to investigate the generation
and evolution of the electromagnetic fields in heavy-ion collisions. The cases
of Au + Au collision at $\sqrt{s}=200$ GeV and Pb + Pb collision at
$\sqrt{s}=2.76$ TeV are considered in detail. Although after averaging over
many events only the component $B_{y}$ remains, the event-by-event fluctuation
of the positions of charged particles can induce components
$B_{x},E_{x},E_{y}$ as large as $B_{y}$. They can reach the order of several
$m_{\pi}^{2}/e$. The spatial structure of the electromagnetic field is studied
and a very inhomogeneous distribution is found. We study also the time
evolution of the fields including the early-stage and the QGP-stage
evolutions. We find that the remnants can give substantial contribution to the
fields during the early-stage evolutions. The non-trivial electromagnetic
response of the QGP, which is sensitive to the electric conductivity, gives
non-trivial time dependence of the fields in it, see Sec. IV. We check both in
numerical calculation (Fig. 3) and analytical derivations (Eqs. (6)-(7)) that
the electric and magnetic fields at $t=0$ have approximately linear dependence
on the collision energy $\sqrt{s}$.
The strong event-by-event fluctuation of the electromagnetic field may lead to
important implications for observables which are sensitive to the
electromagnetic field. We point out two examples here.
(1) From Fig. 2 we see that although the electric fields $E_{x}$ and $E_{y}$
at ${\mathbf{r}}={\bf 0}$ can be very strong, they are roughly equal. Then one
expects that the strong electric fields should not have a significant
contribution for the correlation observable
$\langle\cos(\phi_{1}+\phi_{2}-2\Psi_{RP})\rangle$ which is sensitive to the
chiral magnetic effect Voloshin:2004vk , where $\phi_{1,2}$ are the azimuthal
angles of the final-state charged particles, and $\Psi_{RP}$ is the azimuthal
angle of the reaction plane. On the other hand, from Fig. 4 we see that in the
overlapping region for peripheral collisions, the electric field perpendicular
to the reaction plane has larger gradient than that parallel to the reaction
plane. Thus, a strong, out-of-plane electric field develops away from the
origin ${\mathbf{r}}={\bf 0}$ in the overlapping region. Note that the
direction of this electric field points outside of the reaction plane. If the
electric conductivity of the matter produced in the collision is large, this
out-of-plane electric field can drive positive (negative) charges to move
outward (toward) the reaction plane, and thus induce an electric quadrupole
moment. Such an electric quadrupole moment, as argued in Ref. arXiv:1103.1307
, can lead to an elliptic flow imbalance between $\pi^{+}$ and $\pi^{-}$.
Thus, it will be interesting to study how strong the electric quadrupole
moment can be induced by this out-of-plane electric field. Note that such
electric quardrupole configuration does not contribute to the correlation
$\langle\cos(\phi_{1}+\phi_{2}-2\Psi_{RP})\rangle$.
(2) It is known that quarks produced in off-central heavy-ion collision can be
possibly polarized due to the spin-orbital coupling of QCD Liang:2004ph ;
Gao:2007bc ; Huang:2011ru . The strong magnetic field can cause significant
polarization of quarks as well through the interaction between the quark
magnetic moment and the magnetic field. As estimated by Tuchin arXiv:1006.3051
, a magnetic field of order $m_{\pi}^{2}/e$ can almost immediately polarize
light quarks. Such polarization, contrary to the polarization due to spin-
orbital coupling, will depend on the charges of quarks, and build a spin-
charge correlation for quarks, _i.e._ , the positively charged quarks are
polarized along the magnetic field while the negatively charged quarks are
polarized opposite to the magnetic field. If we expect that the strong
interaction in the QGP and the hadronization processes do not wash out this
spin-charge correlations, we should observe similar spin-charge correlation
for final-state charged hadrons.
Acknowledgments: We thank A. Bzdak, V. Skokov, H. Warringa, and Z. Xu for
helpful discussions and comments. This work is supported by the Helmholtz
International Center for FAIR within the framework of the LOEWE program
(Landesoffensive zur Entwicklung Wissenschaftlich- Ökonomischer Exzellenz)
launched by the State of Hesse. The calculations are partly performed at the
Center for Scientific Computing of J. W. Goethe University. Some of the
figures are plotted using LevelScheme toolkitCaprio:2005dm for Mathematica.
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|
arxiv-papers
| 2012-01-24T20:30:34 |
2024-09-04T02:49:26.644126
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei-Tian Deng, Xu-Guang Huang",
"submitter": "Xu-Guang Huang",
"url": "https://arxiv.org/abs/1201.5108"
}
|
1201.5131
|
# Self Assembled Clusters of Spheres Related to Spherical Codes
Carolyn L. Phillips Applied Physics Program, University of Michigan, Ann
Arbor, Michigan, 48109, USA Eric Jankowski Department of Chemical
Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA Michelle
Marval Department of Materials Science and Engineering, University of
Michigan, Ann Arbor, Michigan, 48109, USA Sharon C. Glotzer corresponding
author _E-mail address:_ sglotzer@umich.edu Department of Chemical
Engineering, University of Michigan, Ann Arbor, Michigan, 48109, USA
Department of Materials Science and Engineering, University of Michigan, Ann
Arbor, Michigan, 48109, USA Applied Physics Program, University of Michigan,
Ann Arbor, Michigan, 48109, USA
###### Abstract
We consider the thermodynamically driven self-assembly of spheres onto the
surface of a central sphere. This assembly process forms self-limiting, or
terminal, anisotropic clusters ($N$-clusters) with well defined structures. We
use Brownian dynamics to model the assembly of $N$-clusters varying in size
from two to twelve outer spheres, and free energy calculations to predict the
expected cluster sizes and shapes as a function of temperature and inner
particle diameter. We show that the arrangements of outer spheres at finite
temperatures are related to spherical codes, an ideal mathematical sequence of
points corresponding to densest possible sphere packings. We demonstrate that
temperature and the ratio of the diameters of the inner and outer spheres
dictate cluster morphology and dynamics. We find that some $N$-clusters
exhibit collective particle rearrangements, and these collective modes are
unique to a given cluster size $N$. We present a surprising result for the
equilibrium structure of a $5$-cluster, which prefers an asymmetric square
pyramid arrangement over a more symmetric arrangement. Our results suggest a
promising way to assemble anisotropic building blocks from constituent
colloidal spheres.
## I Introduction
Anisotropic particles are compelling building blocks for self-assembled
materials because their directional interactions can be exploited to create
complicated and useful patterns Glotzer and Solomon (2007); Sacanna and Pine
(2011); Jones _et al._ (2010); Fejer _et al._ (2010, 2011); Williamson _et
al._ (2011); Jankowski and Glotzer (2011). One way to create anisotropic
building blocks is to self-assemble them from simpler particles, where the
building block represents a free-energy minimizing structure. Recently a
number of papers have been published synthesizing and simulating compound
building blocks that are clusters of spheresWales and Doye (1997); Sciortino
_et al._ (2010); Mossa _et al._ (2004); Williamson _et al._ (2011); Chen
_et al._ (2007a, b); Manoharan _et al._ (2003); Lauga and Brenner (2004); Cho
_et al._ (2005a, b, 2008); Hong _et al._ (2006). Colloidal spheres are
attractive candidates for assembly because they can be made from a wide
variety of polymers and metals, and their interaction potentials can be tuned
with organic ligands, solvents, and salts.
Here we consider a class of self-limiting, or “terminal”, colloidal clusters
created by self-assembling a small population of one type of particle, the
“halo” particle (HP), around a second type of particle, the central particle
(CP). The clusters are terminal because the only attractive interaction is
between the HP and CP, which are dilute in the fluid of HP, and therefore
steric restrictions among co-adsorbed HPs inhibit further growth. The
resulting clusters have structures determined by the interactions among the
adsorbed HPs, which self-organize around the CP to minimize their free energy.
Figure 1: (Color) A terminal $N$-cluster with an octahedral structure ($N$ =
6) is self-assembled from a bath of HP and a CP. This cluster has applications
as an anisotropic building block, could be used to manufacture a “patchy
particle” by imparting patches on the CP at the contact points, or could be
locked into a nanocolloidal cage structure.
Arrangements of HPs on the surface of a CP have been studied extensively by
mathematicians in the context of optimal arrangements of points on a
sphereTóth (1964); nja ; Melnyk _et al._ (1977); Whyte (1952). The solutions
provide a library of anisotropic clusters that can in principle be created
with properly designed interactions among the constituent particles. In this
work we study hard sphere HPs that are attractive only to dilute CPs and not
to other HPs, thereby producing clusters of HPs around a single CP. The
arrangements of these HPs bear comparison to a particular set of solutions,
the _spherical codes_ , for certain ratios of particle diameters. We
investigate the self-assembly of these clusters as a function of temperature,
where entropy controls the equilibrium structure of the cluster, and in a
semi-open system, where HP are free to bind and unbind from the CP surface. We
also consider the effect of temperature on the cluster structures and dynamics
at deviations from the perfectly dense packings that correspond to solutions
of the spherical code.
This paper is organized as follows. In Section II we briefly review sphere
surface extremal point problems. In Section III, we introduce the methods we
use to study the terminal $N$-clusters, including Brownian dynamics
simulations, free energy calculations, and metrics for cluster structure and
mobility. In Section IV, we report the results of our simulations, free energy
calculations, and analyses. We find that terminal $N$-clusters self assemble
across a range of diameters and temperatures and the structure of these
clusters resemble spherical code solutions. These findings are supported by
free energy calculations, which predict cluster sizes and distributions. Using
Brownian dynamics and free energy calculations, we explain the surprising
observation of a dominant low-symmetry $N$ = 5 cluster, a deviation from the
spherical code prediction. We calculate changes in cluster structure across a
range of diameter ratios and investigate the dynamics for different cluster
sizes, including collective modes. We find that the dynamics for clusters of
different sizes are different. In Section V, we discuss several ways this work
can be extended to create more types of anisotropic particles via tuning of
the particle interactions, constructing additional shells of particles, and
creating structurally reconfigurable particles. In Section VI we conclude with
a summary of our findings.
## II Sphere Surface Extremal Points and Spherical Codes
Figure 2: (Color) The arrangement of points (pink) that correspond to each
spherical code solution for $1\leq N\leq 12$. The point group of each
arrangement is shown to the upper right of each arrangement, and the densest
packing diameter ratio $D_{c}/D_{h}=\Lambda_{N}$ is shown to the lower right.
For $N=5$, the triangular bipyramid configuration is shown. Other $N=5$
configurations are shown and discussed in Figures 10-9 .
The problem of finding extremal points obtained by optimally distributing
points on the surface of a sphere to minimize a function $f$ has been well
studied in the field of mathematicsWhyte (1952); Edmundson (1992); Melnyk _et
al._ (1977). The problem is typically posed as follows:
_Given N points on the surface of a sphere of radius R, what arrangement of
the N points minimizes a function f_?
If $f=k\sum^{N}_{i\neq j}r_{i,j}^{-n}$, where $r_{i,j}$ is the Euclidean
distance between the points $i$ and $j$, and $n=1$, minimizing $f$ corresponds
to the Thomson problem, whose solution describes the distribution of identical
point charges on the surface of a sphere. As $n\rightarrow\infty$, the problem
corresponds to the _spherical code_ , (also known as the Fejes Tóth, or Tammes
problem), whose solution maximizes the minimum distance between any two sets
of pointsTóth (1964); nja ; Melnyk _et al._ (1977); Whyte (1952). Other
possible choices for $f$ include minimizing the maximum distance of any point
to its closest neighbor, also known as the _sphere covering problem_ , and
maximizing the volume of the convex hull of the points. For each of these
problems solutions are exactly known for some values of $N$, while various
numerical searches have suggested best solutions for other $N$. For the
functions mentioned, tables of putative solutions up to at least $N=130$ can
be found in Ref. nja .
Fig. 2 depicts the spherical code solutions for $1\leq N\leq 12$. The
arrangement of points for ${N}=4$ corresponds to the vertices of a regular
tetrahedron, ${N}=6$ an octahedron, ${N}=8$ a square anti-prism, and ${N}=12$
an icosahedron. The point arrangement of ${N}=11$ is equal to the ${N}=12$
solution minus a single point, or an icosahedron with one truncated pentagonal
face. For each ${N}$, the point group – the group of isometries that keeps one
point fixed – of the arrangementMelnyk _et al._ (1977) is shown in the upper
right corner. Each optimal arrangement of ${N}$ points on the surface of the
sphere is unique except for ${N}=5$ which has a continuum of solutions ranging
from a triangular bipyramid (point group $D_{3h}$, shown in Fig. 2 and Fig.
10b) to a square pyramid (point group $C_{4v}$, shown in Fig. 10a). All
solutions in the continuum have two points at opposite poles of the central
sphere and differ by the positions of the three remaining points on the
equator. The square pyramid arrangement is equal to the $N$ = 6 solution minus
a single point. We discuss these structures in detail in Section IV.4.
If the $N$ points represent sphere centers, the spherical code solution
corresponds to the densest packing of $N$ hard halo spheres that all “kiss” a
central sphere. For any packing of spheres around a central sphere, we define
$\Lambda$ to be the ratio of the central sphere diameter, $D_{c}$, to the halo
sphere diameter, $D_{h}$. We denote the minimal possible diameter ratio for
$N$ spheres, which corresponds to the spherical code solution, as
$\Lambda_{N}$. In Fig. 2, $\Lambda_{N}$ of each arrangement is shown to four
significant digits in the bottom right corner. Notably,
$\Lambda_{N=5}=\Lambda_{N=6}$ and $\Lambda_{N=11}=\Lambda_{N=12}$. In one of
mathematics’ most famous debates, Isaac Newton and David Gregory argued
whether the kissing number of unit spheres ($\Lambda=1$) is 12 or 13. Had it
been known that a central unit sphere can only be kissed by 13 spheres if
their radii is $r\leq 0.9165$, or $\Lambda_{13}=1.0911$nja , this would have
settled the question. Isaac Newton’s conjecture that the kissing number is 12
was not proven until 1953Aste and Weaire (2000).
We also note that the spherical code solutions for $N$ = 3-12, except $N$ = 5,
are rigid or jammed. They contain no “rattlers”, defined as spheres not in
isostatic contact with other spheresCohn _et al._ (2011); Tarnai and Zs.
(1983), and cannot be deformed other than global isometriesCohn _et al._
(2011); Tarnai and Zs. (1983).
## III Methods
To predict and compare the terminal $N$-clusters of halo particles bonded to
central particles we use computational tools that sample equilibrium
statistical mechanical ensembles. In particular, Brownian dynamics simulations
of model particles are used to perform computer experiments of self-assembly
and the results of these simulations are compared against cluster
probabilities calculated from a free energy analysis based upon numerical
partition function calculationsJankowski and Glotzer (2011). We also calculate
detailed structural and dynamic quantities for each cluster.
### III.1 Hard Sphere and Sticky Sphere Model
In a semi-open system, the spherical code solutions of Section II correspond
to perfectly hard spheres adsorbed on a perfectly sticky sphere.
Mathematically, perfectly hard spheres are points interacting via a function
that steps from infinity to zero (Fig. 3a) and perfectly sticky spheres are
points interacting via the same function plus an infinitely narrow square well
function (Fig. 3b). In this work we use radially-shifted Weeks-Chandler-
Andersen (WCA) and Morse models of hard and sticky spheres, respectively,
which allow for computational efficiency as well as direct comparison with
their ideal mathematical counterparts (Fig 3). They also capture, in a general
sense, the repulsive and attractive interactions of the constituent particles
we have in mind. As nanoparticle synthesis continues to mature, the types of
interactions that can be used to guide the self-assembly of small particles
can be precisely tuned over wide ranges of length and energy scales, and the
models used in simulations can be suitably adjusted.
Figure 3: (a) A mathematically ideal hard particle interaction is shown in
solid black compared to the hard particle interaction (in dashed blue) given
by the WCA potential (Eqn. 4). (b) A sticky sphere with a kissing contact
potential when $\delta\rightarrow 0$ is compared to a model sticky sphere (in
dashed blue) given by the Morse potential (Eqn. 8).
The radially-shifted WCA potential is given byWeeks _et al._ (1971)
$\displaystyle U_{WCA}=\left\\{\begin{array}[]{l
l}4\epsilon\left(\left(\frac{\sigma}{r-\alpha}\right)^{12}-\left(\frac{\sigma}{r-\alpha}\right)^{6}\right)&+\epsilon\\\
&r<r_{cutoff}\\\ 0&r\geq r_{cutoff}\\\ \end{array}.\right.$ (4)
The shifting parameter $\alpha$ is defined as $\alpha=\sigma_{h}-\sigma$,
where $\sigma_{h}$ is the WCA “diameter” of the HP, and
$r_{cutoff}=2^{1/6}\sigma+\alpha$. The interaction between two HPs can be made
arbitrarily hard relative to their size by increasing $\sigma_{h}$. The cost
of increasing $\sigma_{h}$ is that the dimensionless time $\tau$ that elapses
over each time step is reduced as $\tau\propto 1/\sigma_{h}$. We choose
$\sigma_{h}=3\sigma$ for its computational efficiency and its relatively
“hard” modeling of spheres. The energy parameter $\epsilon$ also determines
the “hardness” of the HP, as a larger energetic penalty to overlapping
corresponds to a “harder” potential. The cost of increasing $\epsilon$ is that
a smaller simulation time step is needed to model a steeper function. The
energy parameter is set to $\epsilon=(0.1/T)$, where $T$ is the temperature of
the simulation, so that the hardness of the HP is independent of temperature.
Using Eqn. 4 to model hard HPs is effective because of the large potential
energy penalty associated with two HPs approaching closer than $3\sigma$.
However, due to the soft nature of the potential, spheres with sufficient
kinetic energy can, in principle, approach as close as $2\sigma$. It is
therefore useful to determine the effective hard particle diameter of HP
modeled by Eqn. 4. We use the Barker-Henderson equationBarker and Henderson
(1967) to calculate the effective diameter
$D_{h,e}=\int_{0}^{\sigma_{BH}}(1-e^{-\beta u(r)})dr\simeq 3.0786\sigma$ (5)
where $\beta=1/k_{B}T$, $u(r)=U_{WCA}$ and the potential is zero at
$\sigma_{BH}=2^{1/6}\sigma+\alpha$. For the purpose of assessing the error in
our calculations based on the effective diameter, we can characterize two HPs
as contacting when the interaction energy between them is in the range
$0<U_{WCA}<10k_{B}T$, which corresponds to $3.0\sigma<D_{h,e}<3.1225\sigma$.
The radially-shifted Morse potentialMorse (1929) used to model the “kissing”
contact potential between the HP and CP is given by
$\displaystyle U_{M}=\left\\{\begin{array}[]{l
l}E_{0}\left(e^{-2\beta(r-r_{0})}-2e^{-\beta(r-r_{0})}\right)&\quad
r<r_{cutoff}\\\ 0&\quad r\geq r_{cutoff}\\\ \end{array}\right.$ (8)
where $E_{0}=5$ determines the depth of the energy well, $\beta=5.0/\sigma$
determines the width of the energy well, and $r_{0}$ determines the radial
displacement of the bottom of the energy well. The Morse potential interaction
range is truncated at $r_{cutoff}=2.5\sigma+(r_{0}-\sigma)$.
An effective CP diameter can be calculated by defining an HP and CP as
_bonded_ when the distance between the two particle centers is at the minimum
of Eqn. 8. More properly, an HP and CP are bonded when they remain
positionally correlated because the HP remains within a given displacement of
the CP. We use the minima of Eqn. 8 and the effective diameter of Eqn. 5 to
define an effective CP diameter $D_{c,e}=2r_{0}-D_{h,e}$. The ratio of the CP
to HP diameter is therefore $\Lambda^{m}$ = $D_{c,e}/D_{h,e}$, (_m_ for
_molecular dynamics_). We choose to keep the hardness of the HP-HP interaction
constant for all the MD simulations by holding $D_{h,e}$ (i.e. $\sigma_{h}$)
constant while varying $r_{0}$ to change $D_{c,e}$.
The non-infinitesimal potential well width of Eqn. 8 permits the bond between
the CP and HPs to stretch a small amount while remaining bonded. At some
$\Lambda^{m}$ ratios, this stretching, though small, may be enough to
accommodate an additional HP bond to the CP. It is therefore useful to define
the CP _bond-stretched effective diameter_ $D_{bs,c,e}=2\bar{R}_{hc}-D_{h,e}$,
where $\bar{R}_{hc}$ is the average center-to-center distance between a bonded
HP and CP measured in a simulation. The bond-stretched diameter ratio is
defined as $\Lambda^{bs}=D_{bs,c,e}/D_{h,e}$ ($bs$ for _bond-stretched_).
$\Lambda^{bs}$ is always greater than $\Lambda^{m}$. When the cluster is
loosely packed, the difference between the two measures converges to zero.
### III.2 Brownian Dynamics
To model mixtures of halo particles and central particles assembling in a
thermal bath we perform Brownian dynamics (BD) simulations, implemented in
HOOMD-blueHOO (2010). The natural units of this system are: the effective
diameter of the HP, $D_{h,e}=3.0786\sigma$; the mass of a HP, $m$; and the
depth of the HP-CP energy well, $E_{0}$. The volume fraction, $\phi$, is
defined as the ratio of the total volume of the HPs and CPs to the simulation
box volume, the dimensionless time is $t^{*}=t/(D_{h,e}\sqrt{m/E_{0}})$ , and
the dimensionless temperature is $T^{*}=k_{B}T/E_{0}$. We use periodic
boundary conditions. Each particle is subjected to conservative, random, and
drag forces, and its motion is governed by the Langevin equation discussed
further in Zhang _et al._ (2003); Iacovella _et al._ (2005); Iacovella and
Glotzer (2008). We use a value for the drag coefficient $\gamma=0.726$
$m/t^{*}$. The same drag coefficient is applied to HPs in both the free and
bound state. The conserved forces between particles are per Eqns. 4 and 8
above.
### III.3 Free Energy Calculations
The relative probability of finding a particular cluster of $N$ HPs bound to a
CP can be predicted using free energy calculations detailed in references
McGinty (1971); Meng _et al._ (2010); Jankowski and Glotzer (2011). For a
given $\Lambda$, the partition function is defined by the appropriately
weighted sum over all possible configurations of $N$ HPs bound to a CP for
$N=1$ to $\infty$. The contributions of the distinguishable microstates to the
partition function are calculated numerically.
The partition function is calculated assuming ideal hard spheres and sticky
spheres (Fig. 3). Given HPs of diameter $D_{h}$ and a CP of diameter $D_{c}$,
the interaction potential between ideal HPs is defined as,
$\displaystyle U_{H-H}(r)=\left\\{\begin{array}[]{l l}\infty&\quad r<D_{h}\\\
0&\quad r\geq D_{h}\\\ \end{array}\right.$ (11)
and the interaction potential between an ideal HP and an ideal CP is defined
as
$\displaystyle U_{H-C}(r)=\left\\{\begin{array}[]{l l}\infty&\quad
r<(D_{c}+D_{h})/2\\\ -E_{0}&\quad r=(D_{c}+D_{h})/2\\\ 0&\quad
r>(D_{c}+D_{h})/2\\\ \end{array}\right.$ (15)
We define $\Lambda^{f}=D_{c}/D_{h}$ (_f_ for _free energy calculation_). As in
section III.1, to vary $\Lambda^{f}$, $D_{h}$ is held constant (set to
$D_{h,e}$ from Eqn. 5) and $D_{c}$ is changed.
If $\Lambda_{N=M}\leq\Lambda^{f}<\Lambda_{N=M+1}$, then configurations of
${M}$ HPs bonded to the CP minimize the potential energy and configurations
with more than $M$ HPs have infinite potential energy (zero probability).
Configurations with fewer than $M$ HPs bonded to the CP increase the entropy
of the cluster. When $k_{B}T\approx E_{0}$, the free energy can be minimized
by clusters with fewer than $M$ HPs, because the entropy gained by the
remaining HPs on the CP balances the increase in potential energy. In the
grand canonical ensemble, at a fixed $\Lambda^{f}$, the probability of
observing a particular cluster $s$ is given by the Boltzmann distribution:
$P_{s}=e^{-\beta F_{s}}=\frac{\Omega_{s}e^{-\beta(U_{s}-\mu N)}}{\mathcal{Z}}$
(16)
where $\mathcal{Z}=\sum_{s}\Omega_{s}\exp(-\beta(U_{s}-\mu N))$ is the
partition function and $U_{s}-\mu N=NE_{0}$. Without loss of generality, we
treat $\mu$ =0. (A non-zero $\mu$ will only induce a uniform temperature shift
in our final results.)
In practice, calculating $\mathcal{Z}$ exactly is difficult, but by assuming
that only a small number of clusters contribute to $\mathcal{Z}$McGinty
(1971); Meng _et al._ (2010); Jankowski and Glotzer (2011), the relative
probabilities of these clusters can be determined. As in McGinty (1971); Meng
_et al._ (2010); Jankowski and Glotzer (2011), the degeneracy $\Omega_{s}$ can
be written as a product of three independent terms, the translational,
$Z_{t}$, rotational, $Z_{r}$, and vibrational $Z_{v}$ partition functions. The
translational partition function is approximately equal for all the clusters
because they are all small compared to the accessible volume, and thus
contributes equally to the $\Omega_{s}$ of each cluster.
To calculate the rotational and vibrational partition functions for an
$N$-cluster, we first assume an equilibrium configuration defined by $N$ HPs
in a spherical code configuration at a radial displacement of ($D_{c}$ \+
$D_{h}$)/2 from the CP. The rotational partition function is then calculated
as $Z_{r}=c_{r}\frac{\sqrt{I}}{\kappa}$, where $c_{r}$ is a temperature-
dependent constant that is the same for all the clusters, $I$ is the
determinant of the moment of inertia tensor, and $\kappa$ is the symmetry
number of the spherical code configuration under rotation. Each sphere is
given a unit mass. The vibrational partition function is proportional to the
product of the vibrational freedom, or freedom to rattle, of each sphere in
the cluster. The vibrational freedom of each HP can be measured as the
fractional area of the surface of the CP it has access to, subject to the
restrictions imposed by its neighboring spheres. We approximate the
vibrational area available to a given HP in a particular configuration by
using a Monte Carlo numerical approach whereby new positions for the HP are
randomly generated and accepted if the HP does not overlap another HP. The
accessible vibrational area is proportional to the total number of accepted
positions that are part of a contiguous area that includes the HP’s original
position divided by the total number of random trials. If
$\Lambda^{f}=\Lambda_{N}$, when the diameter ratio matches the spherical code
ratio, then most, if not all, of the spheres in an $N$-cluster are jammed and
have no vibrational freedom.
The free energy calculation is approximate, as it does not consider the
contribution of collective modes of HP motion to $\mathcal{Z}$, which, in
certain systems can help stabilize one configuration over another Haji-Akbari
_et al._ (2011). As we show in Section IV.2, each cluster has a small
$\Lambda$ range, $\Lambda>\Lambda_{N}$ where collective modes are not present,
and only local rattling is observed. Outside this range, we expect some error
in the calculation of relative probabilities to accumulate. The benefit of
this free energy approximation is demonstrated by both its favorable
comparison to predictions made by BD simulations and by its ability to rapidly
predict the entire phase diagram. Applying a more computationally intensive
method to perform an exact free energy comparison would be an interesting
topic for future study.
### III.4 Structure and mobility measures of a cluster
The HPs in an $N$-cluster for $\Lambda=\Lambda_{N}$ are confined to a unique
$N$ spherical code solution and cannot rearrange or even rattle for $N$ = 3 to
12, excepting $N$ = 5. For $\Lambda>>\Lambda_{N}$, the HPs are free to
randomly arrange on the surface of the CP. We aim to understand the structure
and dynamics of the $N$-cluster between these extremes. We perform BD
simulations of pre-assembled clusters wherein the HPs are restricted to the
surface of the CPs, a constraint imposed during the integration of the
equations of motion. This allows the dynamics of HP rearrangement to be
isolated from the dynamics of assembly and disassembly and prevents any
stretching of bonds from influencing the structures observed. Similar to
section III.1 the CP diameter is defined as $D_{c,e}=2r_{0}-D_{h,e}$, where
$r_{0}$ is the fixed distance between the CP and HP centers and $D_{h,e}$ is
the same as defined in Eqn. 5. The CP to HP diameter ratio in the constrained
system is thus $\Lambda^{c}=D_{c,e}/D_{h,e}$ ($c$ for _constrained_) and
varied by changing $r_{0}$. $\Lambda_{c}$ is initialized such that the HP can
be sparsely randomly distributed on the surface (i.e.
$\Lambda^{c}>>\Lambda_{N}$), and then slowly decreased over the course of a
simulation to a target $\Lambda^{c}$.
The angular displacement between two HP bound to the same CP, or
$\theta=\angle ACB\;$ is defined by the centers of two HPs $A$ and $B$ and the
CP $C$. To characterize the structure of the cluster, the distribution of
angular displacements between pairs of HPs, $n(\theta)$ for $\theta=[0,\pi]$
are measured for a fixed $N$ and $\Lambda^{c}$ over all HP pairs every
$10^{4}$ time steps during a simulation with $10^{9}$ total time steps. The
value of $n(\theta)$ for a given $N$ and $\Lambda^{c}$ represents the
likelihood of finding an HP at an angle $\theta$ relative to a given HP, and
$\int n(\theta)d\theta=N-1$. $n(\theta)$ is analogous to a pair correlation
function.
To characterize the dynamics we calculate the time scale over which $\theta$
is no longer correlated with itself. We define the mobility parameter $\tau$
from
$C(\theta(t),\theta(t+\delta t))=e^{-\tau t}$ (17)
where $C(\theta(t),\theta(t+\delta t))$ is the normalized angular
autocorrelation function and $t$ is time. In this work, $\tau$ has units of
1/10,000 time steps. The more mobile an HP is on the surface of the CP, the
more rapidly its angular displacement with respect to other HP decorrelates.
When the rate of decay of angular correlations is zero, all the HP in the
cluster are fully caged. We only calculate $\tau$ for clusters that display
more than one distinct peak in their $n(\theta)$ distributions so that
position swapping can be distinguished from local rattling. The lower bound on
the $\tau$ measurement is $1.5\cdot 10^{-4}$ because below this the HP
position swaps occur too infrequently over a $10^{9}$ time step simulation for
accurate values of $\tau$ to be measured. We calculate $\tau$ as a function of
$N$ and $\Delta\Lambda^{c}=\Lambda^{c}-\Lambda_{N}$, or the difference between
the diameter ratio of the $N$-cluster and the $N$ spherical code solution
ratio. Because the HPs in the simulation are not perfectly hard and are
constrained to the CP surface, it is possible for meaningful measurements to
be made when $\Delta\Lambda^{c}<0$.
### III.5 The calculation of $\Lambda$
In this paper, to elucidate different properties of $N$-clusters, several
calculation methods are used, necessitating four different ways to determine
$\Lambda=D_{c}/D_{h}$, the ratio of the central particle diameter, $D_{c}$, to
halo particle diameter, $D_{h}$. Each way was chosen to best represent the
effective diameters of the HP and CP in the particular method. These $\Lambda
s$ are comparable to each other and to the spherical code solution ratios,
$\Lambda_{N}$ of Fig. 2. We indicate the calculate type by the superscript $x$
of $\Lambda^{x}$, where $x\in\\{m,bs,c,f\\}$, where the $m$, Brownian
(molecular) dynamics; $bs$, bond-stretched; $c$, constrained; and $f$, free
energy calculation, as defined above.
## IV Results
Figure 4: (Color) Top: The $N$ clusters that self-assemble as a function of
$\Lambda^{m}$ and temperature is shown. The average $N$ of the self-assembled
cluster at $T^{*}=0.02$ is shown as a black line. The maximum and minimum N in
the simulation is shaded grey. The average $N$ of the self-assembled cluster
at $T^{*}=0.1$ is a red solid line. The maximum and minimum N in the
simulation is shaded pink. Example clusters self-assembled in simulation are
shown. Bottom: Accounting for bond-stretching and the effective diameter of
the HP, the lowest ratio where a cluster of size $N$ observed in the quasi-
statically decreasing simulation (blue triangles) and for the self-assembled
simulations (black circles) are compared to the spherical code predictions
(pink star). Error bars for the quasi-static simulation ratios are generated
from the contact range of two HP. Figure 5: (Color) The distributions of
cluster sizes as a function of temperature and $\Lambda$ as given by the free
energy calculation and the BD simulations are compared. Bottom left corner:
phase diagram of the free energy prediction of the most probable cluster size.
Lower right and upper left corners: in-page slices of the probability of
finding each cluster size $P_{N}$ as predicted by the free energy calculation
and BD simulation at the high and low temperature. Upper right corner: the
three most common clusters found in the BD simulation at the high temperature
and $\Lambda^{m}=0.46$.
### IV.1 Self-assembly and free energy of $N$-clusters
Using Brownian dynamics we simulate the self-assembly of clusters as a
function $\Lambda^{m}$ and at two different temperatures to investigate the
effect of thermal noise on the distribution of stable terminal $N$-clusters.
We compare these results to the known spherical code solutions and to free
energy calculations.
Brownian dynamics simulations of self-assembly are initialized by placing
$1000$ CPs on a cubic lattice, spaced so as to behave as independent systems.
The lattice is embedded in a bath of HPs at a total volume fraction of
$\phi=0.24-0.27$. The bath contains a minimum of four times as many HPs per CP
as the maximum cluster size observed for that $\Lambda^{m}$. We perform a
total of 760 simulations of 20$\times 10^{6}$ time steps, with time step size
$\Delta t^{*}=0.00363$ at low ($T^{*}=0.02$) and high ($T^{*}=0.1$)
temperatures. With this set of simulations, we calculate the cluster size
distribution as a function of $\Lambda^{m}$.
At the low temperature we observe that the cluster sizes are highly
monodisperse as a function of $\Lambda^{m}$. In Fig. 4, the mean cluster size
assembled at the low temperature, $T^{*}=0.02$ for $0.01<\Lambda^{m}<1$ is
shown as a black solid line. Grey shading indicates the range of cluster sizes
observed at a particular $\Lambda^{m}$. Over this range of $\Lambda^{m}$
clusters are uniform in size except when $\Lambda^{m}$ is near a value where
there is a transition from one mean cluster size to another. At these
transitions, we observe a narrowly distributed mixture of cluster sizes; e.g.,
at $\Lambda^{m}=0.71$ for the $T^{*}=0.02$ curve, we find equal numbers of
clusters containing $8$ or $9$ HPs.
In comparison to the low temperature data, the clusters at high temperature
are both smaller on average, and have a broader distribution of sizes as a
function of $\Lambda^{m}$. In Fig. 4, we show the distribution of clusters
assembled at high temperature, $T^{*}=0.1$ (red). The region shaded pink
represents the range of cluster sizes measured at a given $\Lambda^{m}$ at
$T^{*}=0.1$. At $\Lambda^{m}=0.71$ for the $T^{*}=0.1$ curve, we now observe
clusters of 5, 6 7, and 8 HPs. We also observe that the $N$ = 5 and $N$ = 11
clusters are not stable at any $\Lambda^{m}$ at low temperature but are
present in the broader distribution of clusters at high temperature.
To test the stability of the self-assembled clusters at low temperature, we
perform a simulation wherein the diameters of the CPs in large pre-assembled
clusters are slowly decreased. A single system with $\Lambda^{m}$ = 0.9489 is
equilibrated for 20$\times 10^{6}$ time steps at $T^{*}=0.02$, at which time
every CP is bonded to 12 HPs. Subsequently $D_{c,e}$ is decreased at a rate of
$4.833\times 10^{-8}\sigma/\Delta t$ until $\Lambda^{m}$ = 0.0101. As
discussed in reference Phillips and Glotzer , this decrease in the diameter is
slow enough that the system remains quasi-static, i.e. the system is
approximately in equilibrium. For this system, as $\Lambda^{m}$ approaches a
transition ratio, 1-3 HPs detach from a given CP and re-enter the bath, until
only two HP are bonded to each CP. In effect, this quasi-statically decreased
$D_{c,e}$ simulation disassembles the clusters as a function of $\Lambda^{m}$.
If bond-stretching is taken into account, we find that at a low temperature
($T^{*}=0.02$) the $N$-clusters self-assemble at the $\Lambda$ ratio predicted
by the spherical code solutions. In tightly packed clusters, bond stretching
makes $\Lambda^{bs}>\Lambda^{m}$. In the bottom plot of Fig. 4, the lowest
$\Lambda^{bs}$ at which a cluster of size $N$ is observed for the self-
assembled (black circles) and quasi-statically decreased (blue triangles)
simulation data is shown and compared to the spherical code $\Lambda_{N}$
ratio (pink stars). Blue error bars indicate the $\Lambda^{bs}$ ranges from
quasi-statically decreased simulations, generated by assuming that the true
diameter of an HP is the limits of the contact range defined in section III.1.
Good correspondence between the predicted and measured ratios is observed when
bond-stretching and the appropriate effective diameters of the particles is
accounted for.
We calculate the free energies of all clusters from $N=2$ to $N=12$ over a
temperature range of $0.02\leq T^{*}\leq 0.2$ and diameter ratio range of
$0.05\leq\Lambda^{f}\leq 1.09$. In the bottom left plot of Fig. 5, we report a
“phase diagram” of the most probable cluster at each combination of $T^{*}$
and $\Lambda^{f}$. The plots in the bottom right and top left of Fig. 5 show
data from an in-page slice of the phase diagram at the low and high
temperature, $T^{*}=0.02$ and $T^{*}=0.1$, and directly compare it to cluster
distributions from the BD simulation data of Fig. 4. For example, the three
clusters and probabilities depicted in the upper right of Fig. 5 are from a
single high temperature BD simulation with $\Lambda^{m}$ = 0.46.
We see that the free energy calculations support the findings of the BD
simulations. At high temperature and at a given $\Lambda$, both show a
decrease in cluster size relative to the low temperature data, as well as a
broadening in the distribution of cluster sizes. Discrepancies in peak height
and shape between the two predictions in Fig. 5 are likely due to the soft
sphere approximation, not accounting for the change in the contact energy or
effective diameter due to bond-stretching, and also to neglecting the
collective vibrational modes in the free energy calculations. However, the
free energy calculation shows that most of the features of the BD simulations
at higher temperatures can be attributed to the offsetting of the increase in
potential energy (i.e. fewer bonded HPs) by the commensurate increase in
vibrational freedom of the remaining bonded HPs.
Consistent with the BD simulation data, the free energy calculation also
predicts that $N$ = 5 clusters are not stable at low ($T^{*}<0.06$)
temperatures. Spherical code solutions indicate that the densest $N$ = 5
clusters occur at the same $\Lambda$ as the densest $N$ = 6 cluster. Thus, at
low temperature, when the free energy is dominated by the potential energy
term, the $N$ = 6 cluster is always stable over an $N$ = 5 cluster. However,
the free energy calculation predicts that at $T^{*}>0.06$ there is a $\Lambda$
range where an $N$ = 5 cluster is the most probable cluster. This $\Lambda$
range is observable in the high temperature BD simulation data. The
stabilization of the $N$ = 5 cluster over the $N$ = 6 cluster at higher
$T^{*}$ arises from the non-negligible contribution of the vibrational
partition function, the only term in the partition function that significantly
differs between the two clusters. The $N$ = 11 cluster is similarly predicted
to be unstable at low temperatures but stable over the $N$ = 12 cluster at a
higher temperature. The free energy calculation also predicts an entropic
stabilization of the $N$ = 7 and 9 clusters over the $N$ = 8 and 10 clusters,
respectively, at higher $T^{*}$ and “triple points”, at which the
probabilities of three clusters (e.g. 4, 5, and 6; or 7, 8, and 9) are equal.
### IV.2 Structure of $N$-clusters
Figure 6: (Color) The distribution of angular displacements $n(\theta)$ for
each cluster. The $n(\theta)$ shows a structural fingerprint particular to
each cluster. Figure 7: Cluster mobility as a function of the
$\Delta\Lambda^{c}=\Lambda^{c}-\Lambda_{N}$. Note that for the $N$ = 6 and $N$
= 12 clusters, the HPs do not become measurably mobile for
$\Delta\Lambda^{c}>>$ 0\. At the other extreme, $N$ = 5 and $N$ = 10 are
mobile for $\Delta\Lambda^{c}<$ 0\. Each data point is extracted from a linear
fitting with a goodness of fit $>$ 0.95.
We next consider how the structure of each $N$-cluster changes as
$\Lambda>\Lambda_{N}$. Clusters that have a large range of $\Lambda$ over
which their structure is ordered and stable are desirable targets for
synthesis. We investigate the structure and dynamics of the clusters by
modeling HPs constrained to the surface of a CP, as described in Section III.4
for different $N$ and $\Lambda^{c}$.
In simulation, we observe that a cluster of size $N$ generally exhibits three
different dynamics over different ranges of $\Lambda^{c}$. In the first range,
each HP remains locally caged. Each HP of the $N$-cluster can be assigned to
one point of the $N$ spherical code solution of Fig. 2 and that mapping
remains invariant under the dynamics of the cluster. Like an atom in a crystal
lattice, each HP rattles about its point. In the second range, each HP can
almost always be assigned to one point of the spherical code solution, however
the mapping does not remain invariant under the dynamics of the cluster. The
HPs sporadically rearrange but are still generally found rattling about the
points of the spherical code solution. In the third range, the HPs cannot be
assigned to points of the spherical code solution and move freely on the
surface. Between the second and third range, we suspect that there is no
distinct measurable boundary, but simply an increasing likeliness of a cluster
being in “transitional” states. Below we show how the two measures introduced
in section III.4 capture the signature features of these ranges.
In Figure 6, the distribution of angular displacements, $n(\theta)$, is shown
for $2\leq N\leq 12$ HPs constrained to the surface of a CP at $T^{*}=0.02$.
For each $N$, $n(\theta)$, is shown for three different $\Lambda^{c}$,
corresponding to $\Lambda_{N}$, $\Lambda_{N+1}$ and a midpoint between the
two. These three distributions are shown in order of increasing $\Lambda^{c}$
from left to right. Note that for $N=2$, $\Lambda_{N=2}$ is zero, as it is
always possible to add a second HP to a CP with one bound HP, regardless of CP
size. In this case, we arbitrarily choose the minimum $\Lambda^{c}$ =
$\Lambda_{N+1}/2$.
For each cluster we observe a unique $n(\theta)$ structure fingerprint that
softens as $\Lambda^{c}$ increases. For $N\leq 4$, each HP has one equidistant
ring of neighbors, resulting in $n(\theta)$ having a single peak that broadens
as $\Lambda^{c}$ increases. For $N>4$, each $n(\theta)$ has multiple peaks.
For $N>4$ except $N=5$ and $N=10$, the first peak at $\Lambda_{N}$ is narrow
and not connected to other peaks, indicating HPs are locally caged at their
spherical code pointsWeeks and Weitz (2002). The width of a peak is
proportional to the rattling of a HP within its local cage. As $\Lambda^{c}$
increases, the peaks broaden and eventually become connected. This broadening
and overlapping is associated with the degradation of the well-defined
structure by increased rattling and sporadic rearranging. In no case did we
find any evidence of new structures emerging. We note that for the clusters
$N=5$ and $N=10$ the peaks are not distinct at the smallest $\Lambda^{c}$
considered. For all $N$, if $\Lambda^{c}>>\Lambda_{N}$, then the HPs sample
uniformly random arrangements on the CP surface and the $n(\theta)$
distribution is a cosine function of $\theta$, truncated to zero when $\theta$
is less than the angular diameter of the HP (e.g. in Fig. 6, $N$ = 2,
$n(\theta)$ is a truncated cosine function for each $\Lambda^{c}$). In Fig. 6
for $N>2$, insofar as the distributions are far from converged to a cosine
function, we observe structure derived from the underlying spherical code
solution over the entire range of $\Lambda$ considered
### IV.3 Mobility of $N$-clusters
We next consider the dynamics of the HPs on the CP surface. As described in
section IIID, we can measure how rapidly the angular displacements of the HPs
decorrelate at a given $\Lambda^{c}$. We call this measure the mobility
parameter, $\tau$. When $\tau=0$, the HPs are in the first dynamical range;
that is, each HP is fully caged and the angular displacement between any two
HPs does not decorrelate. When $\tau>0$, but small, the HPs are in the second
dynamical range. In Figure 7, the $\tau$ of different clusters are calculated
as a function of increasing $\Lambda^{c}$ relative to the ratio at which the
cluster is predicted to assemble, $\Delta\Lambda^{c}=\Lambda^{c}-\Lambda_{N}$.
We observe that the size of the first dynamical range varies widely among
clusters. Noticeably, the $N=6$ and $N=12$ clusters are not measurably mobile
until $\Delta\Lambda^{c}$ is large. Note that in Fig. 6, the midpoint
$n(\theta)$ data of both $N$ = 6 and $N$ = 12 still have distinct separated
peaks. In contrast, the $N=11$ cluster becomes mobile at a much lower
$\Delta\Lambda^{c}$ than $N$ = 12 cluster, despite having nearly the same
spherical code solution and $\Lambda_{N=11}=\Lambda_{N=12}$. Comparing the $N$
= 11 and $N$ = 12 distributions in Fig. 6, at $\Lambda^{c}$ = 0.9021 the two
clusters have nearly identical $n(\theta)$ distributions. At
$\Lambda^{c}=0.9489$, $N=11$ is measurably mobile ($\tau=1.5\cdot 10^{-4})$)
but still has distinct peaks in $n(\theta)$ that are only slightly softer than
that of $N=12$. By $\Lambda^{c}$ = 1.095 the $n(\theta)$ peaks are noticeably
softened for the $N$ = 11 cluster relative to that of the $N$ = 12 cluster and
the peaks are connected. At this ratio, the mobility of the $N=11$ cluster is
$\tau=0.19$ while the $N=12$ cluster is just measurably mobile ($\tau=4\cdot
10^{-4}$). In comparison, the $N$ = 10 cluster is mobile even at the ratio at
which it first self-assembles. The rapid increase of the mobility as a
function of increasing $\Lambda^{c}$ in Fig. 7 is consistent with the
connected peaks and the rapid softening of the peak structure for $N$ = 10 in
Fig. 6.
The fact that HP mobilities for a particular cluster depend upon the cluster’s
structure is not surprising. However, it is not obvious that clusters of
different sizes should have such variation in the widths of the first
dynamical range indicated in Fig. 7. There is little correlation, for example,
between the mobility of the $N$ cluster and the range of $\Lambda$ over which
the $N$ cluster is stable in Fig. 4. We find that the HPs in the $N=5$ and
$N=10$ clusters are never fully caged, while the HPs for $N=6$ and $N=12$ are
fully caged for a large range of $\Lambda^{c}$. The mobilities for $N=7$, 8,
9, and 11 lie between these extremes. Note that $N=6$ and $N=12$ clusters have
highly symmetrical spherical code point arrangements with octahedral and
icosahedral structures, respectively. Their HP centers define the vertices of
Platonic solids with equilateral triangle faces. For $N$ = 7, 8, 9, 10, and
11, the convex polyhedra defined by the centers of the HP have pentagonal ($N$
= 11), square ($N$ = 8 and 10) or nearly square ($N$ = 7, 9, and 10) faces. We
hypothesize that these non-triangular “defects” in the spherical code
solutions are responsible for the increased mobility of these clusters by
providing locations where the barrier to rearrangement is low. However, for
such small systems, the rearrangements of HPs in a mobile cluster is more
appropriately viewed as a rearrangement of the entire cluster rather than a
localized rearrangement.
Figure 8: (Color) The rearrangements of clusters $N$ = 5-12. $N$ = 12 has two
rearrangements.
As discussed above, in the second dynamical range, the HPs in an $N$ cluster
can almost always be mapped to the points of the $N$ spherical code solution,
but that mapping does not remain invariant. We examine the $N$ = 5-12 clusters
at the $\Lambda^{c}$ at which each cluster is first observed to be measurably
mobile per Fig. 7 to understand how the HPs in a cluster rearrange. We find
(Fig. 8) that the $N$ = 5-11 clusters each have a single unique (discounting
reflections or rotations) rearrangement, which permutes the HPs over the
spherical code points. The $N$ = 12 cluster has two unique rearrangements.
Short movies of these rearrangements can be found online in the supplemental
material. For the $N$ = 5 and $N$ = 11 clusters, which have structures
equivalent to the $N+1$ spherical code minus a single point, a rearrangement
consists of a single HP “hopping” a gap to the available $N+1$ point. (The
reason the $N$ = 5 is found in this particular configuration is discussed in
the next section.) The clusters $N$ = 6, 10, and 12 exhibit a permutation
whereby a ring of HPs rotate relative to the cluster in a manner resembling a
twist of a _Rubik’s Cube™_. The clusters $N$ = 7, 8, 9, and 12 exhibit a
permutation whereby the cluster “buckles” into a new permutation of the
spherical code points. We find that the addition of the rearranging action for
$N$ = 5-12 is sufficient to make each cluster ergodic. That is, every possible
assignment of each HP to the spherical code points can be explored by the
cluster with no inaccessible microstates. This ergodicity is shown, using
group theory, in the supplemental materials.
For clusters $N$ = 6, 7, 8, 9, 10, and 12, the rearranging action is a
collective motion of particles in the cluster. Although this entropic
contribution is not considered by the free energy calculation in Section
III.3, the free energy calculations compare well with the BD simulations,
demonstrating that local rattling is more important than collective modes for
some ranges of $\Lambda$.
### IV.4 Breaking the degeneracy for $N$ = 5
#### IV.4.1 BD simulations
The $N=5$ spherical code has a continuum of solutions ranging from the
vertices of a square pyramid to a triangular bipyramid. For dense $N=5$
clusters at non-zero temperature, we seek the relative likelihood of the
cluster adopting particular configurations from the solution continuum. For
this, we construct an order parameter that can distinguish between different
configurations in our BD simulations.
All $N$ = 5 spherical code solutions have two points at opposite poles of the
central sphere and differ by the positions of the three remaining points on
the equator. The order parameter is constructed by, first, dividing the five
HPs into “pole” HPs and “equator” HPs. The neighbor distances, or distance
between each HP and the four other HPs is measured. HPs that do not have one
neighbor distance that $>$ 1.2 times the distance of the other three neighbor
distances are “equator” HPs. Second, the “equator” HP that is closest to other
“equator” HPs or has the minimum summed neighbor distances is selected and and
its center is labeled $A$.
Finally, an angle measurement is constructed in the plane of the equator as
follows. The centers of the pair of “pole” HPs are labeled $P_{1}$ and $P2$.
The points $A$, $P_{1}$, and $P_{2}$ define a plane $S_{1}$. The centers of
the two remaining HP are labeled $E_{1}$ and $E_{2}$. The line through $E_{1}$
and $E_{2}$ intersects $S_{1}$ at $E_{S}$ and $\hat{n}$ is the normal vector
to $S_{1}$. A plane $S_{2}$ orthogonal to $S_{1}$ is constructed from the
point $E_{S}$, $A$, and $A+\hat{n}$. The coordinates are translated and
rotated so that $A$ and $E_{S}$ are both on the $y$-axis of $S_{2}$ and $A$
has $x$-$y$ coordinates (0, $r_{0}$), where $r_{0}$ is the distance between
the center of an HP and the CP. The origin corresponds to the center of the
CP. The points $E_{1}$ and $E_{2}$ are projected to the plane $S_{2}$ and the
angles ($<\pi/2$) to the $x$-axis of $S_{2}$ is measured. The order parameter
$\chi$ is defined as this angle, sampled twice per configuration. Each angle
pair uniquely specifies a configuration in the solution continuum. A perfect
square pyramid configuration corresponds to two measurements of $\chi=0$ and a
perfect triangular bipyramid configuration corresponds to two measurements of
$\chi=\pi/6$ ($\approx 0.524$) radians. Fig. 9a illustrates how the order
parameter was constructed, and shows a sampling of the HP positions in the
$S_{2}$ plane from a simulation at $\Lambda^{c}=0.4$. The red circles
correspond to the triangular bipyramid positions.
We performed BD simulations of clusters of 5 HP at $\Lambda^{c}=0.4142$ and
$0.400$ with $T^{*}=0.02$. Two histograms are shown of the sampled $\chi$ at
the two ratios, 0.4142 and 0.4 in Fig. 9a and 9b, respectively. The figures
show that the degenerate continuum of $N=5$ spherical code solutions is broken
by the introduction of thermal noise. Surprisingly, we find that the square
pyramid is the preferred structure, even over the more symmetrical triangular
bipyramid. As the cluster is packed tighter, an even stronger preference for
the square pyramid configuration over other configurations emerges.
Figure 9: (Color) (a) The order parameter $\chi$ is constructed by measuring
the angle of the particles on the equator. Scattered points from a simulation
overlay an image of an SP configuration. Red circles indicate the sphere
centers of a TBP configuration. In (b) and (c) the distribution of $\chi$
sampled in from a BD simulation is shown as a function of the diameter ratio
$\Lambda^{c}$ = 0.4142 and 0.4 respectively.
#### IV.4.2 Free Energy
Figure 10: (Color) (Color) (a) The square pyramid (SP) and (b) the triangular
bipyramid (TBP) $N$ = 5 spherical codes. The jammed and unjammed kissing
spheres in each configuration are colored dark grey and pink, respectively.
The path that the unjammed spheres can follow is traced on the central sphere.
For (b) the central sphere is transparent so the full path around the equator
can be seen. In the graph at the bottom, at the low temperature, $T^{*}=0.02$,
the preference for the SP (black solid) over the TBP (red solid) is evident as
the HP diameter approach the limiting packing diameter. This preference (black
and red dashed lines) is even stronger at high temperature, $T^{*}=0.1$.
To understand the preference for the square pyramid configuration in the BD
simulation we use a free energy calculation, which elucidates the role of
entropy in breaking the degeneracy. The more symmetrical triangular bipyramid
configuration is used as the reference configuration.
The square pyramid and triangular bipyramid HP clusters are shown in Fig.
10(a) and Fig. 10(b). In Fig. 10(c), using a free energy calculation, the
probability of observing the square pyramid relative to the triangular
bipyramid is shown at two temperatures as the HP diameter, $D_{h}$, approaches
the diameter of spheres corresponding to the densest possible packing,
$D_{h,N=5}$. For dense clusters, we observe the square pyramid is always the
most likely configuration at nonzero temperature. We find this preference is
because the square pyramid has the most vibrational freedom. In Fig. 10(a) and
Fig. 10(b), the locally unjammed HP in a cluster at $\Lambda=\Lambda_{N=5}$
are colored pink and the HP that are locally jammed are colored grey. In the
degenerate continuum of $N=5$ spherical code solutions, only the square
pyramid has only one locally jammed HP, and thus, the highest vibrational
freedom.
In Fig. 7, the $N=5$ cluster becomes decreasingly mobile as the cluster is
packed tighter, i.e. on decreasing $\Delta\Lambda^{c}$. Unlike clusters for
other values of $N$, as $\tau\rightarrow 0$, the HPs in the $N=5$ cluster
become locally caged for entropic, rather than energetic, reasons.
## V Discussion
There are a number of ways the sticky sphere assembly method described above
can be extended to create interesting new species of anisotropic particles.
For example, we can now ponder a more general question. Given a desired
arrangement of points, what HP-CP interactions and HP-HP interactions will
result in self-assembly of the arrangement? The analogous mathematical
question was posed by L.L. Whyte in 1952, _“What spherical arrangements [of
points] possess extremal properties of any kind?”_ Whyte (1952) Ideally, we
seek HP-HP interactions and HP-CP interactions that self-assemble repeatable
and desirable patterns of HPs on the CP.
Cohn and KumarCohn and Kumar (2007) show that all potential energy functions
of distance that are completely monotonic, such as inverse power laws, share a
subset of universally optimal solution configurations. If the function is
strictly completely monotonic, then the universally optimal solution is also
unique. For points on the surface of a sphere, the only known universally
optimal solutions are Cohn and Kumar (2007); Ballinger _et al._ (2009) $N$ =
1-4, 6, and 12; that is, a single point, antipodal points, points forming an
equilateral triangle on the equator, and tetrahedral, octahedral, and
icosahedral arrangement of points. For our purposes, this means certain
desired point arrangements (e.g. a ring of 12 points distributed around the
equator of a sphere such as modeled in reference Zhang and Glotzer (2004)) are
likely to be inherently difficult to achieve from HP-HP interactions.
Restricting themselves to isotropic pair potentials and identical particles,
Cohn and KumarCohn and Kumar (2009) constructed separate decreasing convex
potential energy functions that have cubic (N=8) and dodecahedral (N=20)
configurations as their minimum. Thus references Cohn and Kumar (2007);
Ballinger _et al._ (2009); Cohn and Kumar (2009) imply that to assemble
certain clusters, it will be necessary to use more complicated HPs with
carefully constructed potentials, including non-completely monotonic or
anisotropic interactions.
Using an alternative approach, complex clusters may also be possible by simply
adding stages to the assembly process. For example, if, after the terminal
$N$-cluster of Fig. 1 is created, the bath of HPs is replaced by a bath of new
HPs coated with the same complementary material as the CP, a second shell of
spheres can be added to the first. The structure of this shell will also
depend on the entropy and energy of the cluster at a given temperature. If the
HPs in the second shell preferentially sit in the interstices of the first
shell, the polyhedron they form will be the _dual_ of the polyhedron of the
first shell. This can make new types of point arrangements possible. For
example, the dual of the octahedron is the cube. A cubic arrangement of eight
points on the surface of a sphere is not found as a minimum among most common
spherical surface functionsnja . A second shell of HPs that preferentially
assemble the dual of the first shell of HPs may be a physically more viable
method of assembling a cubic arrangement of spheres without requiring the
elaborately constructed HP-HP interaction potential of Cohn and KumarCohn and
Kumar (2009).
The results presented here may also be used to guide the synthesis of
reconfigurable $N=5$ clusters. As shown in section IV.4 above, a small change
in the packing fraction of the $N=5$ cluster introduces a significant change
in the structure of the cluster. Thus, changing the effective diameter of the
central particle by a modest amount induces a switch between a relatively
isotropic disordered cluster and an anisotropic square pyramidal cluster.
There may also be a correspondence between other mathematical sequences of
points distributed on a sphere and terminal cluster assembly problems. For
example in reference Chen _et al._ (2007a, b), clusters of cones and spheres
were shown to form unique and precisely packed clusters arising from free
energy minimization subject to a convexity constraint. The authors find a
packing sequence identical to that obtained from minimizing the second moment
of the mass distribution of a cluster of particles constrained to a convex
hull. We observe that the packing sequence produced in Chen _et al._ (2007a)
also bears a strong resemblance to the distribution of points on the surface
of a sphere that maximizes the convex hull nja .The authors further showed
that sequence successfully describes the polyhedral structures formed by
colloidal spheres self-assembling on an evaporating dropletManoharan _et al._
(2003); Lauga and Brenner (2004). That work serves another example of the
correspondence between mathematical solutions of extremal points on the
surface of a sphere and cluster structures obtainable in experiments.
Another interesting variant to consider is a shaped central particle, as was
done in reference Vernizzi and Olvera de la Cruz (2007) for the Thomson
problem. Considering the packing of HP around a shaped CP may lead to novel
clusters and is a generally unexplored problem.
## VI Conclusion
In this paper we have demonstrated that hard and sticky spheres can self-
assemble into terminal $N$-clusters with interesting and, in some cases,
unexpected, anisotropies. These clusters have predictable preferred structures
that depend on temperature and sphere diameter ratio. We find that some
clusters exhibit collective particle rearrangements, and these collective
modes are unique to a given cluster size.
If assembled directly from a bath at low temperature, certain cluster sizes
(e.g. $N=4,6,12$) form robustly, while other clusters occur only over small
ranges with relatively mobile structures (e.g. $N$ = 7,9,10) and still others
cannot be formed at all (e.g. $N$ = 5,11). A “multi-step” process that
assembles the clusters from a bath at a higher temperature, removes the bath,
and lowers the temperature may enable these hard-to-form clusters to be formed
robustly as well. It may even be possible to adjust the effective diameter of
the HP or CP as a step in the assembly process. Our free energy calculations
and Brownian (molecular) dynamics predictions of cluster structure provide a
guide for designing such a process for optimal yield of a desired cluster size
with a well-ordered structure. Clusters fabricated in this way may find use as
building blocks for subsequent self-assembly, as templates for manufacturing
precisely placed circular patches on the surface of a spherical particle,
creating nanocolloidal cages, or fabricating reconfigurable particles.
### VI.1 Acknowledgements
We acknowledge Oleg Gang and Alexei Tkachenko for discussions of related
problems. We acknowledge Daphne Klotsa for her helpful comments on the
manuscript. CLP, MM and SCG were supported by the U.S. Department of Energy,
Office of Basic Energy Sciences, Division of Materials Sciences and
Engineering, under award DE-FG02-02ER46000, the U.S. Department of Energy
Computational Science Graduate Fellowship. EJ received support from the James
S. McDonnell Foundation 21st Century Science Research Award/Studying Complex
Systems, grant no. 220020139 and from a National Defense Science and
Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. SCG is also supported by
the DOD/DDRE under the National Security Science & Engineering Faculty
Fellowship award No. N00244-09-1-0062. Any opinions, findings, and conclusions
or recommendations expressed in this publication are those of the author(s)
and do not necessarily reflect the views of the DOD/DDRE. This work was also
supported by the Non-Equilibrium Energy Research Center (NERC), an Energy
Frontier Research Center funded by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences under Award Number DE-SC0000989.
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|
arxiv-papers
| 2012-01-24T21:30:43 |
2024-09-04T02:49:26.652502
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carolyn L. Phillips, Eric Jankowski, Michelle Marval, and Sharon C.\n Glotzer",
"submitter": "Eric Jankowski",
"url": "https://arxiv.org/abs/1201.5131"
}
|
1201.5252
|
# Zitterbewegung of electrons in quantum wells and dots in presence of an in-
plane magnetic field
Tutul Biswas and Tarun Kanti Ghosh
Department of Physics, Indian Institute of Technology-Kanpur, Kanpur-208 016,
India
###### Abstract
We study the effect of an in-plane magnetic field on the $zitterbewegung$ (ZB)
of electrons in a semiconductor quantum well (QW) and in a quantum dot (QD)
with the Rashba and Dresselhaus spin-orbit interactions. We obtain a general
expression of the time-evolution of the position vector and current of the
electron in a semiconductor quantum well. The amplitude of the oscillatory
motion is directly related to the Berry connection in momentum space. We find
that in presence of the magnetic field the ZB in a quantum well does not
vanish when the strengths of the Rashba and Dresselhaus spin-orbit
interactions are equal. The in-plane magnetic field helps to sustain the ZB in
quantum wells even at low value of $k_{0}d$ (where $d$ is the width of the
Gaussian wavepacket and $k_{0}$ is the initial wave vector). The trembling
motion of an electron in a semiconductor quantum well with high Lande g-factor
(e.g. InSb) sustains over a long time, even at low value of $k_{0}d$. Further,
we study the ZB of an electron in quantum dots within the two sub-band model
numerically. The trembling motion persists in time even when the magnetic
field is absent as well as when the strengths of the SOI are equal. The ZB in
quantum dots is due to the superposition of oscillatory motions corresponding
to all possible differences of the energy eigenvalues of the system. This is
an another example of multi-frequency ZB phenomenon.
###### pacs:
75.70Tj,03.65.-w,73.21.La,73.21.Fg
## I Introduction
In recent years there is a growing interest in the field of spin based
electronic devices. There has been a lot of study in this field after the
proposal of the spin field effect transistor by Datta and Das datta . The
charge carriers carry spin in addition to their charges. The ultimate goal of
this field is to control the spin degree of freedom of the charge carriers to
produce and detect spin-polarized current in semiconductor nanostructures
fabian . One can develop device technology wolf and quantum information
processing david in future on the basis of the manipulation of the spin
degree of freedom. The coupling between intrinsic spin of an electron with its
orbital angular momentum constitutes the intrinsic spin-orbit interaction
(SOI) in low-dimensional semiconducting systems. Particularly, there are two
kind of SOI present in low-dimensional semiconductor structures. One is the
Rashba rashba spin-orbit interaction (RSOI) and another is the Dresselhaus
dress spin-orbit interaction (DSOI). The RSOI arises mainly from the
inversion asymmetry of the confining potential in semiconductor
heterojunctions. The strength of RSOI is proportional to the electric field
externally applied which can be tuned by an external bias nitta ; mats or
internally generated due to the crystal potentials. On the other hand DSOI is
present in bulk materials and semiconductor heterostructures which lack bulk
inversion symmetry. The form of DSOI term strongly depends on the growth
direction of semiconductor quantum well (QW) car ; chen ; andr . The strength
of DSOI depends on properties of the material and the crystal structure.
In 1930, Erwin Schrodinger Schro predicted that a free particle described by
the relativistic Dirac equation will perform an oscillatory motion which is
known as the $zitterbewegung$ (ZB). The free relativistic Dirac particle has
two energy branches: $\epsilon({\bf
p})=\pm\sqrt{{{{m}_{0}}^{2}}c^{4}+{c^{2}}{p^{2}}}$. It is well understood that
this oscillatory motion results from the interference between these two energy
branches huang . The large oscillation frequency $\omega_{{}_{c}}\simeq
10^{21}$ Hz and the small oscillation amplitude $\lambda_{c}\simeq 10^{-13}$ m
is not accessible to the modern experimental techniques. Most of the studies
on the ZB of electrons used plane waves to describe the electrons. It was
pointed out by Lock lock that plane wave is not a localized state and
therefore, rapid oscillations on the average position of a plane wave has some
limitations. He also demonstrated that when an electron is described by a wave
packet, the ZB oscillation has a transient character.
In 2005, Zawadzki et al. zawadki studied ZB in narrow gap semiconductor (NGS)
by using the analogy between $\bf k\cdot\bf p$ theory of the energy bands in
NGS and the free Dirac relativistic equation for electrons. They found much
more favorable amplitude and frequency of the oscillation than those in a
vacuum for a free electron. At the same time, Schliemann et al. john ; loss
has studied the ZB of an electron in III-V zincblende semiconductor QWs in the
presence of the SOI thoroughly. The above studies initiated intense
theoretical research on the ZB of electrons in various condensed matter
systems review ; demi ; rusin ; rusin1 ; maksinova ; winkler ; lamata ; nature
.
The proposal of an experimental scheme for observing ZB in ultra-cold clark
atomic gases is also given. The trembling motion was proposed for photons in a
two-dimensional photonic crystal and for Ramsey interferometry photonic . It
was reported in Ref. sonic that an acoustic analog of ZB in a macroscopic
two-dimensional sonic crystal was observed. A general theory of ZB of a multi-
band Hamiltonian is studied by David and Cserti cserti . Recently, Vaseghi
vaseghi et al. studied the effect of an external perpendicular magnetic field
on the ZB for both quantum wire and quantum dot within the two-sub-band model.
The dimensionless parameter $p_{0}=k_{0}d$ dictates whether the motion will be
oscillatory or not. It was shown in Ref. [14] that the motion will be
oscillatory when $p_{0}\gg 1$. It was also shown that the ZB vanishes when the
strengths of the RSOI and DSOI are equal. This is due to an additional
conserved quantity.
In this work we study effect of the in-plane magnetic field on the ZB of
electrons in a semiconductor QW and in a QD with the Rashba and Dresselhaus
SOI. We obtain an analytical expression of the time-evolution of the position
vector of an electron in a QW by using the Schrodinger picture. The advantage
of using the Schordinger picture is to see the direct relation between the
amplitude of the oscillatory motion and the Berry connection in momentum
space. We find that in presence of the magnetic field the ZB in a quantum well
does not vanish even when the strengths of the Rashba and Dresselhaus spin-
orbit interactions are equal. The ZB in a QW sustains even at very low value
of $k_{0}d$ due to the presence of the magnetic field. The trembling motion of
an electron in a semiconductor QW with high Lande g-factor (e.g. InSb)
sustains over a long time, even at very low value of $k_{0}d$. The time-period
of the oscillation decreases as the magnetic field strength is increased.
Next, we study the ZB of an electron in a QD within the two sub-band model
numerically. The trembling motion does not fade away in time even when the
magnetic field is absent as well as when the strength of the SOIs is equal.
This paper is organized as follows. In section II, we review the Hamiltonian
and eigenstates of a 2DEG with both type of SOI in the presence of an in-plane
magnetic field. In section III, we derive time-evolution of the electron’
position vector in the Schordinger picture. The numerical results and
discussions are given in section V. Section VI contains the calculation and
result of ZB of electrons in a GaAs/AlGaAs QD. The conclusion is presented in
Section VII.
## II Two-dimensional electron gas with spin-orbit interactions
We consider a 2DEG with the Rashba and Dresselhaus spin-orbit interactions in
presence of an in-plane magnetic field
${\bf{B}}=\hat{x}{B}_{x}+\hat{y}{B}_{y}$. The single-particle Hamiltonian of
this system is given by
$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{{\bf
p}^{2}}{2m^{\ast}}+\frac{\alpha}{\hbar}\big{(}{\sigma}_{x}{p}_{y}-{\sigma}_{y}{p}_{x}\big{)}+\frac{\beta}{\hbar}\big{(}{\sigma}_{x}{p}_{x}-{\sigma}_{y}{p}_{y}\big{)}$
(1) $\displaystyle+$
$\displaystyle\frac{g^{\ast}}{2}{\mu_{{}_{B}}}\big{(}B_{x}\sigma_{x}+B_{y}\sigma_{y}\big{)},$
where ${\bf p}$ and $m^{\ast}$ is the momentum and the effective mass of an
electron, respectively. Here $\alpha$ and $\beta$ are the strengths of the
Rashba and Dresselhaus spin-orbit interaction. The last term in the
Hamiltonian is the Zeeman term due to the application of the in-plane magnetic
field $\bf B$, $g^{\ast}$ is the effective Lande-g factor and $\mu_{{}_{B}}$
is the Bohr magneton. The energy eigenvalues and the corresponding eigenstates
of the system chang ; rod are, respectively, given by
$E_{\pm}({\bf
k})=\frac{\hbar^{2}\big{(}k_{x}^{2}+k_{y}^{2}\big{)}}{2m^{\ast}}\pm\sqrt{C^{2}+D^{2}},$
(2)
where $C=\alpha k_{y}+\beta k_{x}+g^{\ast}\mu_{{}_{B}}B_{x}/2$, $D=\alpha
k_{x}+\beta k_{y}-g^{\ast}\mu_{{}_{B}}B_{y}/2$, and the spin eigenstates are
$\displaystyle|k,+\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
-ie^{-i\theta_{k}}\end{pmatrix},\hskip
7.22743pt|k,-\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}-ie^{-i\theta_{k}}\\\
1\end{pmatrix},$ (3)
with $\theta_{k}=\tan^{-1}(C/D)$.
In absence of the magnetic field, a new conserved quantity
$\Sigma=\sigma_{x}-\sigma_{y}$ exists for $\alpha=\beta$ john_prl ; john_prb .
In this situation, the spin eigenstates $|k,\pm\rangle$ become independent of
the wave vector $k$ and therefore the ZB does not exist due to the absence of
spin randomization john ; loss . The situation is completely different when
the in-plane magnetic field is present. It is clear from Eq. (3) that the spin
states are always function of $k$ even at $\alpha=\beta$ as long as ${\bf
B}\neq 0$. Therefore, spin randomization occurs and we would expect to see the
ZB which is shown in the next section.
### II.1 Time-evolution of the wave packet and the Zitterbewegung
We shall use the Schordinger picture to analyze the time-evolution of the
electron position vector. We represent the initial wave function of an
electron by a Gaussian wave packet with initial spin polarization along the
$z$ axis as given by
$\displaystyle\psi\big{(}{\bf r},0\big{)}=\frac{1}{2\pi}\int\,d^{2}ka({\bf
k},0)e^{i\bf k\cdot r}\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$ (6)
where $a({\bf k},0)=d/(\sqrt{\pi})e^{-\frac{1}{2}d^{2}\big{(}{\bf
k-k}_{0}\big{)}^{2}}$. Here, $d$ and ${\bf k}_{0}$ is the initial width and
the initial wave vector of the wave packet, respectively. The time-evolution
of the initial wave packet in the Schordinger representation can be obtained
in the usual manner as $\psi({\bf r},t)=U(t)\psi({\bf r},0)$, where
$U(t)=e^{-iHt/\hbar}$ is the time-evolution operator. After doing some
straightforward algebra we obtain,
$\displaystyle\psi({\bf r},t)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\ d^{2}ka({\bf k},0)e^{i\bf k\cdot
r}e^{-i\frac{\hbar k^{2}}{2m^{\ast}}t}$ (11) $\displaystyle\times$
$\displaystyle\Big{\\{}\cos[\omega({\bf k})t]\left(\begin{array}[]{c}1\\\
0\end{array}\right)-e^{i\theta_{k}}\sin[\omega({\bf
k})t]\left(\begin{array}[]{c}0\\\ 1\end{array}\right)\Big{\\}},$
where $\omega({\bf k})=\big{[}E_{+}({\bf k})-E_{-}({\bf k})\big{]}/(2\hbar)$.
Equation (11) is nothing but the Fourier transformation of the following
function:
$\phi({\bf k},t)=a({\bf k},0)e^{-i\frac{\hbar
k^{2}}{2m^{\ast}}t}\left(\begin{array}[]{c}\cos[\omega({\bf k})t]\\\
-e^{i\theta_{k}}\sin[\omega({\bf k})t]\end{array}\right).$
The expectation value of position operator is then simply given by
$\displaystyle\langle{\bf r}(t)\rangle=i\int d^{2}k\phi^{\dagger}({\bf
k},t){\bf\nabla}_{\bf k}\phi({\bf k},t).$ (13)
Now it is straightforward to show that
$\displaystyle\langle{\bf r}(t)\rangle=\langle{\bf r}(0)\rangle$
$\displaystyle+$ $\displaystyle\frac{\hbar{\bf
k}(0)}{m^{\ast}}t-\frac{1}{2}\int d^{2}k\left|a({\bf
k,0})\right|^{2}\Big{(}{\bf\nabla}_{\bf k}\theta_{k}\Big{)}$ (14)
$\displaystyle\times$ $\displaystyle\Big{\\{}1-\cos[\omega({\bf
k})t]\Big{\\}}.$
The last oscillatory term $\cos[\omega({\bf k})t]$ in Eq. (14) indicates the
ZB. Two important observations can be made by analyzing Eq. (14). First, the
amplitude of the ZB is directly proportional to the Berry connection $\langle
k,\pm|i\frac{\partial}{\partial{\bf k}}|k,\pm\rangle={\bf\nabla}_{\bf
k}\theta_{k}$. Second, for $\alpha=\beta$ the amplitude (or the Berry
connection) of the oscillation does not vanish when the in-plane magnetic
field is present. But it vanishes if the magnetic field is absent.
After substituting the expressions for $a({\bf k},0)$, ${\bf\nabla}_{\bf
k}\theta_{k}$ in Eq. (14) and considering the initial velocity of the wave
packet is along $y$-direction (i.e. $k_{0x}=0,k_{0y}=k_{0}$) one can easily
obtain the following expressions
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{d}{2\pi}e^{-d^{2}k_{0}^{2}}\int^{2\pi}_{0}d\phi\int_{0}^{\infty}dqe^{-q^{2}+2dqk_{0y}\sin\phi}$
(15) $\displaystyle\times$
$\displaystyle\frac{q^{2}(1-\eta^{2})\sin\phi+q(\varepsilon_{x}+\eta\varepsilon_{y})}{Q^{2}}$
$\displaystyle\times$ $\displaystyle\Big{[}1-\cos\Big{(}\frac{2\alpha}{\hbar
d}tQ\Big{)}\Big{]},$
and
$\displaystyle\langle y(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{\hbar
k_{0y}}{m^{\ast}}t$ (16) $\displaystyle+$
$\displaystyle\frac{d}{2\pi}e^{-d^{2}k_{0}^{2}}\int^{2\pi}_{0}d\phi\int_{0}^{\infty}dqe^{-q^{2}+2dqk_{0y}\sin\phi}$
$\displaystyle\times$
$\displaystyle\frac{q^{2}(\eta^{2}-1)\cos\phi+q(\eta\varepsilon_{x}+\varepsilon_{y})}{Q^{2}}$
$\displaystyle\times$ $\displaystyle\Big{[}1-\cos\Big{(}\frac{2\alpha}{\hbar
d}tQ\Big{)}\Big{]},$
where $Q^{2}=(q\cos\phi+\eta q\sin\phi-\varepsilon_{y})^{2}+(q\sin\phi+\eta
q\cos\phi+\varepsilon_{x})^{2}$, $\eta=\beta/\alpha$, $q=kd$,
$\varepsilon_{x}=g^{\ast}\mu_{{}_{B}}B_{x}d/(2\alpha)$ and
$\varepsilon_{y}=g^{\ast}\mu_{{}_{B}}B_{y}d/(2\alpha)$.
From Eqs. (15) and (16) one can infer that ZB is absent in the limit
$d\rightarrow 0$. On the other hand, as $d\rightarrow\infty$ the Gaussian
approaches to a delta function i.e. $a({\bf k},0)=\delta({\bf k}-{\bf k}_{0})$
and in this limit we find an analytic expression for the ZB as given by
$\displaystyle\langle x(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{(1-\eta^{2})k_{0y}+\frac{g^{\ast}\mu_{{}_{B}}}{2\alpha}(B_{x}+\eta
B_{y})}{(k_{0y}+\frac{g^{\ast}\mu_{{}_{B}}B_{x}}{2\alpha})^{2}+(\eta
k_{0y}-\frac{g^{\ast}\mu_{{}_{B}}B_{y}}{2\alpha})^{2}}$ (17)
$\displaystyle\times$
$\displaystyle\Big{\\{}1-\cos[\omega(k_{0y})t]\Big{\\}},$
and
$\displaystyle\langle y(t)\rangle$ $\displaystyle=$ $\displaystyle\frac{\hbar
k_{0y}}{m^{\ast}}t$ (18) $\displaystyle+$
$\displaystyle\frac{1}{2}\frac{\frac{g^{\ast}\mu_{{}_{B}}}{2\alpha}(\eta
B_{x}+B_{y})}{(k_{0y}+\frac{g^{\ast}\mu_{{}_{B}}B_{x}}{2\alpha})^{2}+(\eta
k_{0y}-\frac{g^{\ast}\mu_{{}_{B}}B_{y}}{2\alpha})^{2}}$ $\displaystyle\times$
$\displaystyle\Big{\\{}1-\cos[\omega(k_{0y})t]\Big{\\}}.$
We also calculate the expectation values of the velocity of electron in $x$
and $y$ direction. They are given by
$\displaystyle\langle v_{x}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{\partial\langle x\rangle}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{v_{{}_{R}}}{2\pi}e^{-d^{2}k_{0y}^{2}}\int^{2\pi}_{0}d\phi\int_{0}^{\infty}dqe^{-q^{2}+2dqk_{0y}\sin\phi}$
$\displaystyle\times$
$\displaystyle\frac{q^{2}(1-\eta^{2})\sin\phi+q(\varepsilon_{x}+\eta\varepsilon_{y})}{Q}\sin\Big{(}\frac{2\alpha
Q}{\hbar d}t\Big{)},$
and
$\displaystyle\langle v_{y}(t)\rangle$ $\displaystyle=$
$\displaystyle\frac{\hbar k_{0y}}{m^{\ast}}$ $\displaystyle+$
$\displaystyle\frac{v_{{}_{R}}}{2\pi}e^{-d^{2}k_{0y}^{2}}\int^{2\pi}_{0}d\phi\int_{0}^{\infty}dqe^{-q^{2}+2dqk_{0y}\sin\phi}$
$\displaystyle\times$
$\displaystyle\frac{q^{2}(\eta^{2}-1)\cos\phi+q(\eta\varepsilon_{x}+\varepsilon_{y})}{Q}\sin\Big{(}\frac{2\alpha
Q}{\hbar d}t\Big{)}.$
Here, $v_{{}_{R}}=\Omega d=2\alpha/\hbar$ is the velocity corresponding to the
RSOI. The corresponding currents are simply given by $\langle
j_{x}(t)\rangle=e\langle v_{x}(t)\rangle$ and $\langle
j_{y}(t)\rangle=e\langle v_{y}(t)\rangle$, where $e$ is the electronic charge.
Eqs. (16) and (II.1) tell us that the ZB along $y$ direction vanishes when in-
plane magnetic field and the DSOI are absent simultaneously john because in
this case $\int_{0}^{2\pi}e^{2dqk_{0y}\cos{\phi}}\sin{\phi}d\phi=0$. But when
there is a finite in-plane magnetic field present the ZB in $y$ direction does
not vanish even at $\beta=0$. So this is an effect of the in-plane magnetic
field on ZB.
### II.2 Numerical Results and Discussion
In this sub-section we evaluate time-evolution of the observables position
vector and current density for different values of the parameters like
magnetic field, $k_{0}d$, $\beta$ etc.
GaAs/AlGaAs QW: We consider GaAs/AlGaAs quantum well for which the effective
Lande $g$-factor is $g^{\ast}=-0.44$. The value of the Rashba coefficient is
taken to be $\alpha=1.0\times 10^{-11}$ eV-m. We set the condition $dk_{0}=5$
in all the cases. To investigate the time dependence of the expectation values
of position and current of electron, we numerically evaluate Eqs. (15), (16),
(II.1) and (II.1). Here it should be mentioned that Eqs. (16) and (18) contain
two parts: the first depends linearly on time and the second one is
oscillatory in time (responsible for ZB). The magnitude of the first term is
very large compared to that of the second one. So if we plot this as a
function of time we will get a straight line due to the dominating first term.
We consider three cases corresponding to three different values of the
parameter $\beta$ namely $\beta=0$, $\beta=0.5\alpha$ and $\beta=\alpha$. For
each case we plot $\langle x(t)\rangle/d$, $\langle j_{x}(t)\rangle/ev_{R}$
and $\langle j_{y}(t)-j_{0}\rangle/ev_{R}$ as a function of $\Omega t$ for
different values of magnetic field in Figs. [1-3]. Here we define the quantity
$j_{0}$ as $j_{0}=e\hbar k_{0y}/m^{\ast}$. From Fig. [1(a)], one can see that
the amplitude of the ZB increases as the magnetic field increases from its
zero value. It is also noticeable that the ZB pattern is more oscillatory with
increasing magnetic field. The current shows (in Figs. [1(b),1(c)]) the
similar behavior as the position but it oscillates about zero. From Figs. [2]
and [3] we can see that as we increase the value of $\beta$, the amplitude of
ZB decreases. One important point is to be noted here that there is a definite
phase difference between the currents in $x$ and $y$ direction when
$\beta\neq\alpha$ as evident from Figs. [1] and [2]. But the situation is
different when $\beta=\alpha$. It can be seen from Fig. [3(b), 3(c)] the
currents are oscillating in the same phase and this can be easily understood
by analyzing Eqs. [II.1, II.1].
Figure 1: Here $\langle x(t)\rangle/d$, $\langle
j_{x}(t)\rangle/{ev_{{}_{R}}}$ and $\langle
j_{y}(t)-j_{0}\rangle/{ev_{{}_{R}}}$ are plotted as a function of $\Omega t$.
In the all the cases we set $\beta=0$ for GaAs/AlGaAs QW. Here, solid line:
($B_{x}=0$, $B_{y}=0$), dotted line: ($B_{x}=1/\sqrt{2}$ T, $B_{y}=1/\sqrt{2}$
T), dashed line: ($B_{x}=6$ T, $B_{y}=8$ T). Figure 2: Here $\langle
x(t)\rangle/d$, $\langle j_{x}(t)\rangle/{ev_{{}_{R}}}$ and $\langle
j_{y}(t)-j_{0}\rangle/{ev_{{}_{R}}}$ are plotted as a function of $\Omega t$.
In all the cases we set $\beta=0.5\alpha$ for GaAs/AlGaAs QW. Here, solid
line: ($B_{x}=0$, $B_{y}=0$), dotted line: ($B_{x}=1/\sqrt{2}$ T,
$B_{y}=1/\sqrt{2}$ T), dashed line: ($B_{x}=6$ T, $B_{y}=8$T).
In Ref. [14] it was shown that the ZB vanishes if $\alpha=\beta$ in the
absence of any external magnetic field. But Fig. [3] shows that the ZB is
still present when $\alpha=\beta$. This is due to the application of the
external in-plane magnetic field.
Figure 3: Here $\langle x(t)\rangle/d$, $\langle
j_{x}(t)\rangle/{ev_{{}_{R}}}$ and $\langle
j_{y}(t)-j_{0}\rangle/{ev_{{}_{R}}}$ are plotted as a function of $\Omega t$.
In all the cases we set $\beta=0.5\alpha$ for GaAs/AlGaAs QW. Here, solid
line: ($B_{x}=1/\sqrt{2}$ T, $B_{y}=1/\sqrt{2}$ T), dotted line: ($B_{x}=3$ T,
$B_{y}=4$ T), dashed line: ($B_{x}=6$ T, $B_{y}=8$ T). Figure 4: $\langle
x(t)\rangle/d$ is plotted as a function of $\Omega t$ for InSb QW. Here, we
set $\beta=0.5\alpha$.
InSb QW: Here, we consider InSb QW for which the the effective Lande
$g$-factor is very high (e.g $g^{\ast}=-50$) as compared to GaAs/AlGaAs. The
RSOI strength is taken to be $\alpha=0.9\times 10^{-11}$ eV-m. We set here
$\beta=0.5\alpha$ and $dk_{0}=5$. In Fig. [4] $\langle x(t)\rangle/d$ is
plotted with respect to $\Omega t$ for different values of the magnetic field.
But the situation is different from the GaAs/AlGaAs QW case. Although the
amplitude decreases but the number of oscillations contained in ZB within the
same time range is quite large as we increase the magnetic field. Since the
magnitude of $g^{\ast}$ is large the coupling between electron’s spin and
magnetic field is strong.
One can obtain more oscillation in ZB by increasing the magnitude of the
parameter $dk_{0}$ and we have shown that when $dk_{0}>>1$ the pattern is
completely oscillatory as evident from Eq.(10). In all these cases it is
observed that the ZB is transient in nature i.e it’s amplitude decreases with
time and it is a direct consequence of Lock’s lock prediction that the ZB of
electron will not be persistent in time if it is represented by a wave packet.
## III Zitterbewegung in a Quantum dot
In this section we would like to study ZB of electrons in a semiconductor QD.
We consider a 2DEG confined by an isotropic harmonic oscillator potential
$V(x,y)=(1/2)m\omega^{2}\big{(}x^{2}+y^{2}\big{)}$. In this context our
Hamiltonian reads as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{p^{2}}{2m}+V(x,y)+\frac{\alpha}{\hbar}\big{(}\sigma_{x}p_{y}-\sigma_{y}p_{x}\big{)}$
(21) $\displaystyle+$
$\displaystyle\frac{\beta}{\hbar}\big{(}\sigma_{x}p_{x}-\sigma_{y}p_{y}\big{)}+\frac{g^{\ast}}{2}{\mu_{{}_{B}}}\big{(}B_{x}\sigma_{x}+B_{y}\sigma_{y}\big{)}.$
We introduce the conventional harmonic oscillator creation and annhilation
operators as $a_{x}=(x/l+ilp_{x}/\hbar)/\sqrt{2}$,
$a_{x}^{\dagger}=(x/l-ilp_{x}/\hbar)/\sqrt{2}$,
$a_{y}=(y/l+ilp_{y}/\hbar)/\sqrt{2}$, and
$a_{y}^{\dagger}=(y/l-ilp_{y}/\hbar)/\sqrt{2}$, where
$l=\sqrt{\hbar/(m\omega)}$ is the harmonic oscillator length.
This Hamiltonian can be re-written as
$\displaystyle H$ $\displaystyle=$
$\displaystyle\hbar\omega\Big{(}a_{x}^{\dagger}a_{x}+a_{y}^{\dagger}a_{y}+1\Big{)}$
(22) $\displaystyle+$
$\displaystyle\frac{i\alpha}{\sqrt{2}l}\Big{\\{}\big{(}a_{x}-a_{x}^{\dagger}\big{)}\sigma_{y}-\big{(}a_{y}-a_{y}^{\dagger}\big{)}\sigma_{x}\Big{\\}}$
$\displaystyle+$
$\displaystyle\frac{i\beta}{\sqrt{2}l}\Big{\\{}\big{(}a_{y}-a_{y}^{\dagger}\big{)}\sigma_{y}-\big{(}a_{x}-a_{x}^{\dagger}\big{)}\sigma_{x}\Big{\\}}$
$\displaystyle+$
$\displaystyle\frac{g^{\ast}}{2}\mu_{{}_{B}}\big{(}B_{x}\sigma_{x}+B_{y}\sigma_{y}\big{)}.$
We consider only two lowest occupied energy states (ground state and first
excited state) of a two-dimensional harmonic oscillator potential. This
approximation is known as the “two sub-band model”. Within this approximation
the Hilbert space spanned by the following six basis vectors:
$|0,0,\uparrow\rangle$, $|0,0,\downarrow\rangle$, $|1,0,\uparrow\rangle$,
$|1,0,\downarrow\rangle$, $|0,1,\uparrow\rangle$, $|0,1,\downarrow\rangle$.
Here, $\uparrow$ and $\downarrow$ represent the $z$-component of the
electron’s spin vector.
Within six basis vectors one can write the Hamiltonian in a matrix form as
$H=\begin{pmatrix}H_{11}&H_{12}&0&H_{14}&0&H_{16}\\\
H_{21}&H_{22}&H_{23}&0&H_{25}&0\\\ 0&H_{32}&H_{33}&H_{34}&0&0\\\
H_{41}&0&H_{43}&H_{44}&0&0\\\ 0&H_{52}&0&0&H_{55}&H_{56}\\\
H_{61}&0&0&0&H_{65}&H_{66}\end{pmatrix}$
The matrix elements are as follows:
$H_{11}=H_{22}=\hbar\omega=\varepsilon_{0}$,
$H_{33}=H_{44}=H_{55}=H_{66}=2\hbar\omega=\varepsilon_{1}$,
$H_{12}=H_{21}^{\ast}=H_{34}=H_{56}=H_{43}^{\ast}=H_{65}^{\ast}=g^{\ast}\mu_{{}_{B}}\big{(}B_{x}-iB_{y}\big{)}/2$,
$H_{14}=H_{41}^{*}=-H_{23}^{*}=-H_{32}=(\alpha-i\beta)/(l\sqrt{2})$,
$H_{25}=H_{52}^{*}=-H_{61}=H_{25}^{\ast}=-H_{16}^{*}=-(\beta+i\alpha)/(l\sqrt{2})$.
Here $\varepsilon_{0}=\hbar\omega$ is the zero-point energy and
$\varepsilon_{1}=2\hbar\omega$ is the 1st excited state energy of the two-
dimensional harmonic oscillator.
We want to determine the expectation value of the time-dependent position
operator in this system. Let us consider at $t=0$ the system is in the ground
state and the spin is oriented along the positive $z$-direction. This initial
state is given by $|\psi(0)\rangle=|0,0,\uparrow\rangle$, so the expectation
value of the position operator is given by
$\displaystyle\langle x_{{}_{H}}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle\psi(0)|x_{{}_{H}}(t)|\psi(0)\rangle$ (23)
$\displaystyle=$
$\displaystyle\langle\psi(0)|e^{iHt/\hbar}x(0)e^{-iHt/\hbar}|\psi(0)\rangle$
$\displaystyle=$ $\displaystyle\langle
0|VV^{\dagger}e^{iHt/\hbar}VV^{\dagger}xVV^{\dagger}e^{-iHt/\hbar}VV^{\dagger}|0\rangle$
$\displaystyle=$ $\displaystyle\langle
0|VV^{\dagger}e^{iHt/\hbar}VXV^{\dagger}e^{-iHt/\hbar}VV^{\dagger}|0\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{6}\Big{(}\sum_{j=1}^{6}V_{j1}H_{ij}e^{i\frac{(\lambda_{i}-\lambda_{j})t}{\hbar}}\Big{)}V_{i1}^{\dagger},$
where $V$ is the diagonalization matrix which diagonalizes the Hamiltonian $H$
and $VV^{\dagger}=I$. Also, $\omega_{ij}=(\lambda_{i}-\lambda_{j})/\hbar$ is
the beating frequency.
The $V$ matrix also diagonalizes $U(t)$ and becomes
$U_{diag}(t)=VU(t)V^{\dagger}=e^{-i\lambda_{i}t/\hbar}I_{ij}$ and
$X=VxV^{\dagger}$. The position operator is given by
$x=\lambda\big{(}a+a^{\dagger}\big{)}/\sqrt{2}$. Within the above mentioned
basis this can be written in a matrix form as
$x=\frac{\lambda}{\sqrt{2}}\begin{pmatrix}0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\end{pmatrix}$
Figure 5: Plots of $\langle x(t)\rangle/\lambda$ vs $\omega t$ for GaAs/AlGaAs
QD for different values of magnetic fields. Here, Fig. (a): ($B_{x}=0.1$ T,
$B_{y}=0.1$ T), Fig. (b): ($B_{x}=1/\sqrt{(}2)$ T, $B_{y}=1/\sqrt{(}2)$ T),
Fig. (c): ($B_{x}=6$ T, $B_{y}=8$ T). We consider only RSOI i.e. $\beta=0$.
### III.1 Numerical Results and Discussion
The ZB of a GaAs/AlGaAs quantum dot in the presence of the in-plane magnetic
field with SOI is investigated here. The value of the Rashba strength is taken
as $\alpha=1.0\times 10^{-11}$ eV-m and the zero-point energy of the harmonic
oscillator potential is fixed to the value $\varepsilon_{0}=5$ meV. We find
the expectation values of the position coordinate as a function of time which
are plotted in Figs. [5] and [6]. In Fig. [5] we fix $\beta=0$ and vary the
magnetic field strengths. The magnetic field is kept constant and $\beta$ is
varied in Fig. [6]. The ZB in quantum dots is similar to the beating effect in
the classical wave mechanics with different frequencies. The oscillatory
motion is due to the superposition of individual oscillatory motions with
frequencies corresponding to the all possible energy eigenvalue differences of
the Hamiltonian. In this case, there are six non-degenerate eigenvalues and we
have six values of the energy differences. The number of beating frequencies
is six.
Figure 6: Plots of $\langle x(t)\rangle/\lambda$ vs $\omega t$ for GaAs/AlGaAs
QD for different values of $\beta$. The value of magnetic field is kept fixed
to $(B_{x},B_{y})=(0.1,0.1)T$. Figures (a),(b) and (c) correspond to
$\beta=0$, $\beta=0.5\alpha$ and $\beta=\alpha$ respectively.
## IV Conclusion
In this work we have investigated the effect of an external in-plane magnetic
field on the ZB of an electron in semiconductor QW and QD with Rashba and
Dresselhaus spin-orbit interactions. For QW a general expression of the
expectation values of position coordinate and current due to ZB within the
Gaussian wave packet is obtained. For QW case, the oscillatory quantum motion
of electron which is represented by a wave packet shows transient behavior and
this signature is a proof of Lock’s argument. Another important point is that
ZB does not vanish even at $\alpha=\beta$ when a finite in-plane magnetic
field is present. The $y$-component of current also performs ZB motion with
finite magnetic field. We study the same problem for high Lande g-factor QW
like InSb in comparison with low Lande g-factor QW like GaAs-AlGaAs. We have
also studied the problem of ZB in a GaAs/AlGaAs QD numerically. The ZB in
GaAs/AlGaAs QD is persistent in time. The ZB in quantum dots shows beating-
like pattern and it is similar to the the beating effect in the classical wave
mechanics.
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arxiv-papers
| 2012-01-25T12:40:48 |
2024-09-04T02:49:26.665612
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tutul Biswas and Tarun Kanti Ghosh",
"submitter": "Tutul Biswas",
"url": "https://arxiv.org/abs/1201.5252"
}
|
1201.5292
|
-titleConference Title, to be filled 11institutetext: IPHC Strasbourg.
# B-tagging at CMS
Cristina Ferro 11 Cristina.Ferro@cern.ch
###### Abstract
The identification of $b$ jets is a crucial issue to study and characterize
various channels like top quark events and many new physics scenarios.
Different b-tagging techniques are defined in CMS which benefit from the long
life time, high mass and large momentum fraction of the b-hadron produced in
b-quark jet. Efficient algorithms have been developed based on the measure of
b-hadron secondary vertex or on tracks with a large impact parameter. Data
collected in $pp$ collisions at $\sqrt{s}$=7TeV in 2011 are used to estimate
both the b-tagging efficiency and the mistag rate from light flavor jets.
## 1 Introduction
The b-tagging algorithms in CMS mainly rely on the long life time, high mass
and large momentum fraction of b hadrons produced in b-quark jets, as well as
on the presence of soft leptons from semi-leptonic b decaysref:BTV_11_001 .
Due to the high instantaneous luminosity during the 2011 data taking, the
number of collision taking place in the same bunch crossing (pileup events) is
of the order of 5 to 11 on average.
Figure 1: Number of tracks associated to a jet without any selection cut.
Figure 2: Number of tracks associated to a jet after selection cuts.
The presence of pileup increases the track multiplicity in the events, as we
can see in Fig.(1). This is why a special selection of the tracks was applied
in order to remove the tracks originating from pileupref:BTV_11_002 . In
Fig.(2), the number of tracks passing the selection cuts shows a smaller
pileup dependence.
## 2 The b-tagging observables
The b-tagging algorithms and their study are based on the measure of three
main variables: the impact parameter significance of the tracks, the position
of the secondary vertex, and the transverse momentum of the muon relative to
the jet direction. In the following a brief description of these variable is
presented.
### 2.1 The impact parameter significance
Figure 3: Geometric meaning of the impact parameter significance.
The impact parameter (IP) is defined as the distance between the track and the
primary interaction vertex (PV) at the point of closest approach. The IP is
positive (negative) if the track is produced downstream (upstream) with
respect to the PV along the jet direction (Fig.(3)). The IP is calculated in 3
dimensions thanks to the good x-y-z resolution provided by the pixel detector.
An important features of the IP is that it is Lorentz invariant and due to the
b-hadron lifetime the typical IP scale is set by c$\tau\sim$480 $\mu$m. In
practice, the impact parameter significance IP/$\sigma$(IP) is used in order
to take into account resolution effects.Thanks to the long lifetime of the
b-hadrons the IP from b-jets is expected to be mainly positive, while for the
light jets it is almost symmetric with respect to zero (Fig.4).
### 2.2 The secondary vertex
Thanks to the high resolution of the CMS traking system, it is possible to
directly reconstruct the secondary vertex, the point where the b hadron decays
(Fig.(3)). The vertex reconstruction is performed using the adaptive vertex
fitter. The resulting list of vertices is then subject to a cleaning
procedure, rejecting SV candidates that share 65$\%$ or more of their tracks
with the PV.
### 2.3 The transverse momentum of the muon
Semileptonic decays of b hadrons give rise to b jets that contain a muon with
a branching ratio of about 11$\%$, or 20$\%$ when
b$\rightarrow$c$\rightarrow$l cascade decays are included. This is why the
reconstructed muons inside a jet are used to study the performance of the
lifetime-based tagging algorithms. The muons are seeded from the CMS muon
chambers, and are then linked to tracks found in the tracking system to form
global muons. The CMS muon system is able to measure muons with high
acceptance resolution and efficiency.
## 3 B-tagging algorithms
Severla b-tagging algorithms are used in CMS ref:BTV_11_001 , ref:BTV_11_002 .
The output of each algorithm is a $discriminator$ value on which the user can
cut on to select different regions in the efficiency versus purity phase
space. In Fig.(4)these discriminators are presented.
Figure 4: Discriminators for: $Top$ $left$ Track Counting High Efficiency
(IP/$\sigma$(IP)), $center$ Track Counting High purity (IP/$\sigma$(IP)),
$right$ JetProbability, $Bottom$ $left$ JetBProbability, $center$ Simple
Secondary Vertex High efficiency, $right$ Simple Secondary Vertex High purity.
* •
The track counting algorithm identifies a b-jet if there are at least N tracks
with a significance of the impact parameter above a given threshold. The
tracks are ordered in decreasing IP/$\sigma$(IP) and the discriminator is the
impact parameter significance of the Nth track . To get an high b-jet
efficiency we can use the IP/$\sigma$(IP) of the second track (TCHE), to
select b-jets with high purity the third track is the better choice (TCHP).
* •
The Jet Probability algorithm relies on the IP/$\sigma$(IP) measurement of all
tracks in a jet. One can use the negative tail of the IP/$\sigma$(IP)
distribution to extract the probability density function (PDF) for tracks not
coming from b/c-jets. By integrating on the PDF, we can compute the
$probability$ for tracks to originate from the PV. Then combining the
probability of the tracks we can assign to the jet a probability to come from
the PV. The JetBprobability is then defined in a similar way but giving more
weight to the four most displaced tracks.
* •
Soft-Lepton tagging algorithms rely on the properties of muons or electrons
from semileptonic b-decay. Due to the large b-quark mass, the momentum of the
muon transverse to the jet axis, $p_{T}^{rel}$ , is larger for muons from
b-hadron decays than for muons in light flavor jets.
* •
Secondary Vertex tagging algorithms rely on the reconstruction of at least one
secondary vertex. The significance of the 3D flight distance is used as a
discriminating variable. Two variants based on the number of tracks at SV are
considered: N$\geq$2 for $high$ $efficiency$ (SSVHE), and Ntr$\geq$3 for
$high$ $purity$ (SSVHP) [2].
The $combined$ $secondary$ $vertex$ algorithm includes this information and
provides discrimination even when no secondary vertices are found. The mass of
reconstructed charged particles at the secondary vertex is used to measure the
b-tagged sample purity.
## 4 Performance of the taggers
Figure 5: Performance of all b-taggers obtained on the simulated QCD events.
The performance are shown as udsg jets tagging efficiency versus b-jets
tagging efficiency.
Figure 6: Light flavor mistag efficiency versus b-tagging efficiency for
different pileup scenario, for the TCHE tagger.
Varying the cuts on the discriminator, we obtain different efficiency of the
taggers. We establish standard operating points as, $loose$ (L), $medium$ (M),
and $tight$ (T), being the value at which the tagging of udsg jets is
estimated from MC to be 10$\%$, 1$\%$, or 0.1$\%$, respectively, for jet
transverse momentum of about 80 GeV. In Fig.(5) the performance for different
taggers are shown. In Fig.(6) the effects of the pileup on the performance of
the TCHE tagger is presented. Thanks to the good selection on tracks the
performance of the taggers are not compromised by the pileup events.
## 5 Physics results
Many measurements have been obtained using the b-tagging algorithms at
$\sqrt{s}$ = 7 TeV. Some of them used the b-tagging algorithms already at
trigger levelref:TOP-11-007 . Indeed, at trigger level, the b-quark candidates
can be selected if they have at least one or two tracks with a 3D impact
parameter significance above a given threshold. The motivation for applying
b-tagging in the trigger is a reduction of the trigger rates, while keeping
the signal efficiency high at the same time. The typical rate reduction is a
factor of 5-10. In the following a list of the main 2011 physics results
obtained thanks to the b-tagging algorithms is presented:
* •
B-PHYSICS:
* –
Inclusive production cross section of b-jetsref:4 .
* •
EW PHYSICS:
* –
Measurement of associated charm production in W final stateref:8 .
* •
Top-PHYSICS:
* –
Cross-section measurement of top pair production in various final states:
dileptonic ref:9 ,ref:10 , ref:11 , ref:14 , lepton+jets ref:12 , all hadronic
ref:TOP-11-007 .
* –
Single top in t channel ref:13 .
* –
Top mass measurement ref:14 .
* •
New PHYSICS:
* –
Search for supersymmetry in events with b-jets and missing transverse momentum
ref:15 .
* –
Search for supersymmetry in all hadronic events ref:16 .
* –
Search for an Heavy Bottom-like quark ref:17 .
* –
Search for an Heavy Top-like quark ref:18 .
* –
Search for pair production of a fourth-generation t’ quark in the lepton-plus-
jets channel ref:19 .
* –
Inclusive search for a fourth generation of quarks ref:20 .
## References
* (1) CMS PAS BTV-11-001, CMS Collaboration, Performance of b-jet identification in CMS.
* (2) CMS PAS BTV-11-002, CMS Collaboration, Status of b-tagging tools for 2011 data analysis.
* (3) CMS PAS TOP-11-007, CMS Collaboration, Measurement of the t$\bar{t}$ production cross section in the fully hadronic decay channel in pp collisions at $\sqrt{s}=7$TeV.
* (4) J. High Energy Phys. 03 (2011) 090, CMS Collaboration, Inclusive b-hadron production cross section with muons in pp collision at $\sqrt{s}=7$TeV.
* (5) CMS PAS EWK-11-013, CMS Collaboration, Measurement of associated charm production in W final state I pp collision at $\sqrt{s}=7$ TeV.
* (6) Phys. Lett. B 695 (2011) 424-443, CMS Collaboration, First measurement of the cross section for top-quark pair production in pp collisions at $\sqrt{s}=7$ TeV.
* (7) CMS PAS TOP-11-005, CMS Collaboration, Measurement of the t $\bar{t}$ production cross section in the dilepton channel in pp collisions at $sqrt{s}=7$TeV.
* (8) CMS PAS TOP-11-006, CMS Collaboration, First measurement of the t$\bar{t}$ production cross section in the dilepton channel with tau leptons in the final state in pp collisions at $\sqrt{s}=7$ TeV.
* (9) CERN-PH-EP-2011-085, CMS Collaboration, Measurement of the t$\bar{t}$ Pair Production Cross Section at $\sqrt{s}=7$ TeV using b-quark Jet Identification Techniques in Lepton + Jet Events.
* (10) Phys.Rev.Lett.107(2011)091802, CMS Collaboration, Measurement of the t-channel single top quark production cross section in pp collision at $\sqrt{s}=7$ TeV.
* (11) J.High Energy Phys.07(2011)049, CMS Collaboration, Measurement of the t$\bar{t}$ production cross section and the top quark mass In the dilepton channel in pp collision at $\sqrt{s}=7$ TeV.
* (12) J.High Energy Phys.07(2011)091802, CMS Collaboration, Search for supersymmetry in events with b-jets and missing transverse momentum at LHC.
* (13) CMS PAS SUS-11-006, CMS Collaboration, Search for supersymmetry in all hadronic events with b-jet.
* (14) CMS PAS EXO-11-036, CMS Collaboration, Search for an Heavy Bottom-like quark in 1.14 $fb^{-}1$ of pp collision at $\sqrt{s}=7$ TeV.
* (15) CMS PAS EXO-11-050, CMS Collaboration, Search for an Heavy Top-like quark in the dilepton Final state in pp collision at $\sqrt{s}=7$ TeV.
* (16) CMS PAS EXO-11-051, CMS Collaboration, Search for pair production of a fourth-generation $t^{`}$ quark in the lepton-plus-jets channel with the CMS experiment.
* (17) CMS PAS EXO-11-054, CMS Collaboration, Inclusive search for a fourth generation of quarks with the CMS experiment.
|
arxiv-papers
| 2012-01-25T14:59:34 |
2024-09-04T02:49:26.674260
|
{
"license": "Public Domain",
"authors": "Cristina Ferro",
"submitter": "Ferro Cristina",
"url": "https://arxiv.org/abs/1201.5292"
}
|
1201.5418
|
# A Transformation-based Implementation
for CLP with Qualification and Proximity ††thanks: This work has been
partially supported by the Spanish projects STAMP (TIN2008-06622-C03-01),
PROMETIDOS–CM (S2009TIC-1465) and GPD–UCM (UCM–BSCH–GR58/08-910502).
R. CABALLERO M. RODRÍGUEZ-ARTALEJO and C. A. ROMERO-DÍAZ
Departamento de Sistemas Informáticos y Computación Universidad Complutense
Facultad de Informática 28040 Madrid Spain
{rafa,mario}@sip.ucm.es, cromdia@fdi.ucm.es
(19 March 2011, 7 January 2012; 12 January 2012)
###### Abstract
To appear in Theory and Practice of Logic Programming (TPLP)
Uncertainty in logic programming has been widely investigated in the last
decades, leading to multiple extensions of the classical LP paradigm. However,
few of these are designed as extensions of the well-established and powerful
CLP scheme for Constraint Logic Programming. In a previous work we have
proposed the SQCLP (proximity-based qualified constraint logic programming)
scheme as a quite expressive extension of CLP with support for qualification
values and proximity relations as generalizations of uncertainty values and
similarity relations, respectively. In this paper we provide a transformation
technique for transforming SQCLP programs and goals into semantically
equivalent CLP programs and goals, and a practical Prolog-based implementation
of some particularly useful instances of the SQCLP scheme. We also illustrate,
by showing some simple—and working—examples, how the prototype can be
effectively used as a tool for solving problems where qualification values and
proximity relations play a key role. Intended use of SQCLP includes flexible
information retrieval applications.
###### keywords:
Constraint Logic Programming, Program Transformation, Qualification Domains
and Values, Similarity and Proximity Relations, Flexible Information
Retrieval.
## 1 Introduction
Many extensions of LP (logic programming) to deal with uncertain knowledge and
uncertainty have been proposed in the last decades. These extensions have been
proposed from different and somewhat unrelated perspectives, leading to
multiple approaches in the way of using uncertain knowledge and understanding
uncertainty.
A recent work by us [RR10] focuses on the declarative semantics of a new
proposal for an extension of the CLP scheme supporting qualification values
and proximity relations. More specifically, this work defines a new generic
scheme SQCLP (proximity-based qualified constraint logic programming) whose
instances $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ are
parameterized by a proximity relation $\mathcal{S}$, a qualification domain
$\mathcal{D}$ and a constraint domain $\mathcal{C}$. The current paper is
intended as a continuation of [RR10] with the aim of providing a semantically
correct program transformation technique that allows us to implement a sound
and complete implementation of some useful instances of SQCLP on top of
existing CLP systems like SICStus Prolog [sicstus] or SWI-Prolog [swipl]. In
the introductory section of [RR10] we have already summarized some related
approaches of SQCLP with a special emphasis on their declarative semantics and
their main semantic differences with SQCLP. In the next paragraphs we present
a similar overview but, this time, putting the emphasis on the goal resolution
procedures and system implementation techniques, when available.
Within the extensions of LP using annotations in program clauses we can find
the seminal proposal of quantitative logic programming by [VE86] that inspired
later works such as the GAP (generalized annotated programs) framework by
[KS92] and our former scheme QLP (qualified logic programming). In the
proposal of van Emden, one can find a primitive goal solving procedure based
on and/or trees (these are similar to the alpha-beta trees used in game
theory), used to prune the search space when proving some specific ground atom
for some certainty value in the real interval $[0,1]$. In the case of GAP, the
goal solving procedure uses constrained SLD resolution in conjunction with
a—costly—computation of so-called reductants between variants of program
clauses. In contrast, QLP goal solving uses a more efficient resolution
procedure called SLD($\mathcal{D}$) resolution, implemented by means of real
domain constraints, used to compute the qualification value of the head atom
based on the attenuation factor of the program clause and the previously
computed qualification values of the body atoms. Admittedly, the gain in
efficiency of SLD($\mathcal{D}$) w.r.t. GAP’s goal solving procedure is
possible because QLP focuses on a more specialized class of annotated
programs. While in all these three approaches there are some results of
soundness and completeness, the results for the QLP scheme are the stronger
ones (again, thanks to its also more focused scope w.r.t. GAP).
From a different viewpoint, extensions of LP supporting uncertainty can be
roughly classified into two major lines: approaches based on fuzzy logic
[Zad65, Haj98, Ger01] and approaches based on similarity relations.
Historically, Fuzzy LP languages were motivated by expert knowledge
representation applications. Early Fuzzy LP languages implementing the
resolution principle introduced in [Lee72] include Prolog-Elf [IK85], Fril
Prolog [BMP95] and F-Prolog [LL90]. More recent approaches such as the Fuzzy
LP languages in [Voj01, GMV04] and Multi-Adjoint LP (MALP for short) in the
sense of [MOV01a] use clause annotations and a fuzzy interpretation of the
connectives and aggregation operators occurring in program clauses and goals.
The Fuzzy Prolog system proposed in [GMV04] is implemented by means of real
constrains on top of a CLP($\mathcal{R}$) system, using a syntactic expansion
of the source code during the Prolog compilation. A complete procedural
semantics for MALP using reductants has been presented in [MOV01b]. A method
for translating a MALP like program into standard Prolog has been described in
[JMP09].
The second line of research mentioned in the previous paragraph was motivated
by applications in the field of flexible query answering. Classical LP is
extended to Similarity-based LP (SLP for short), leading to languages which
keep the classical syntax of LP clauses but use a similarity relation over a
set of symbols $S$ to allow “flexible” unification of syntactically different
symbols with a certain approximation degree. Similarity relations over a given
set $S$ have been defined in [Zad71, Ses02] and related literature as fuzzy
relations represented by mappings $\mathcal{S}:S\times S\to[0,1]$ which
satisfy reflexivity, symmetry and transitivity axioms analogous to those
required for classical equivalence relations. Resolution with flexible
unification can be used as a sound and complete goal solving procedure for SLP
languages as shown e.g. in [AF02, Ses02]. SLP languages include Likelog [AF99,
Arc02] and more recently SiLog [LSS04], which has been implemented by means of
an extended Prolog interpreter and proposed as a useful tool for web knowledge
discovery.
In the last years, the SLP approach has been extended in various ways. The
SQLP (similarity-based qualified logic programming) scheme proposed in [CRR08]
extended SLP by allowing program clause annotations in QLP style and
generalizing similarity relations to mappings $\mathcal{S}:S\times S\to D$
taking values in a qualification domain not necessarily identical to the real
interval $[0,1]$. As implementation technique for SQLP, [CRR08] proposed a
semantically correct program transformation into QLP, whose goal solving
procedure has been described above. Other related works on transformation-
based implementations of SLP languages include [Ses01, MOV04]. More recently,
the SLP approach has been generalized to work with proximity relations in the
sense of [DP80] represented by mappings $\mathcal{S}:S\times S\to[0,1]$ which
satisfy reflexivity and symmetry axioms but do not always satisfy
transitivity. SLP like languages using proximity relations include
Bousi$\sim$Prolog [JR09] and the SQCLP scheme [RR10]. Two prototype
implementations of Bousi$\sim$Prolog are available: a low-level implementation
[JR09b] based on an adaptation of the classical WAM (called Similarity WAM)
implemented in Java and able to execute a Prolog program in the context of a
similarity relation defined on the first order alphabet induced by that
program; and a high-level implementation [JRG09] done on top of SWI-Prolog by
means of a program transformation from Bousi$\sim$Prolog programs into a so-
called Translated BPL code than can be executed according to the weak SLD
resolution principle by a meta-interpreter.
Let us now refer to approaches related to constraint solving and CLP. An
analogy of proximity relations in the context of partial constraint
satisfaction can be found in [FW92], where several metrics are proposed to
measure the proximity between the solution sets of two different constraint
satisfaction problems. Moreover, some extensions of LP supporting uncertain
reasoning use constraint solving as implementation technique, as discussed in
the previous paragraphs. However, we are only aware of three approaches which
have been conceived as extensions of the classical CLP scheme proposed for the
first time in [JL87]. These three approaches are: [Rie98phd] that extends the
formulation of CLP by [HS88] with quantitative LP in the sense of [VE86] and
adapts van Emden’s idea of and/or trees to obtain a goal resolution procedure;
[BMR01] that proposes a semiring-based approach to CLP, where constraints are
solved in a soft way with levels of consistency represented by values of the
semiring, and is implemented with clp(FD,S) for a particular class of
semirings which enable to use local consistency algorithms, as described in
[GC98]; and the SQCLP scheme proposed in our previous work [RR10], which was
designed as a common extension of SQLP and CLP.
As we have already said at the beginning of this introduction, this paper
deals with transformation-based implementations of the SQCLP scheme. Our main
results include: a) a transformation technique for transforming SQCLP programs
into semantically equivalent CLP programs via two specific program
transformations named elimS and elimD; and b) and a practical Prolog-based
implementation which relies on the aforementioned program transformations and
supports several useful SQCLP instances. As far as we know, no previous work
has dealt with the implementation of extended LP languages for uncertain
reasoning which are able to support clause annotations, proximity relations
and CLP style programming. In particular, our previous paper [CRR08] only
presented a transformation analogous to elimS for a programming scheme less
expressive than SQCLP, which supported neither non-transitive proximity
relations nor CLP programming. Moreover, the transformation-based
implementation reported in [CRR08] was not implemented in a system.
The reader is assumed to be familiar with the semantic foundations of LP
[Llo87, Apt90] and CLP [JL87, JMM+98]. The rest of the paper is structured as
follows: Section 2 gives an abridged presentation of the SQCLP scheme and its
declarative semantics, followed by an abstract discussion of goal solving
intended to serve as a theoretical guideline for practical implementations.
Section 3 briefly discusses two specializations of SQCLP, namely QCLP and CLP,
which are used as the targets of the program transformations elimS and elimD,
respectively. Section 4 presents these two program transformations along with
mathematical results which prove their semantic correctness, relying on the
declarative semantics of the SQCLP, QCLP and CLP schemes. Section
LABEL:sec:practical presents a Prolog-based prototype system that relies on
the transformations proposed in the previous section and implements several
useful SQCLP instances. Finally, Section LABEL:sec:conclusions summarizes
conclusions and points to some lines of planned future research.
## 2 The Scheme SQCLP and its Declarative Semantics
In this section we first recall the essentials of the SQCLP scheme and its
declarative semantics, which were developed in detail in previous works [RR10,
RR10TR]. Next we present an abstract discussion of goal solving intended to
serve as a theoretical guideline for practical implementations of SQCLP
instances.
### 2.1 Constraint Domains
As in the CLP scheme, we will work with constraint domains related to
signatures. We assume an universal programming signature $\Gamma=\langle
DC,DP\rangle$ where $DC=\bigcup_{n\in\mathbb{N}}DC^{n}$ and
$DP=\bigcup_{n\in\mathbb{N}}DP^{n}$ are countably infinite and mutually
disjoint sets of free function symbols (called data constructors in the
sequel) and defined predicate symbols, respectively, ranked by arities. We
will use domain specific signatures $\Sigma=\langle DC,DP,PP\rangle$ extending
$\Gamma$ with a disjoint set $PP=\bigcup_{n\in\mathbb{N}}PP^{n}$ of primitive
predicate symbols, also ranked by arities. The idea is that primitive
predicates come along with constraint domains, while defined predicates are
specified in user programs. Each $PP^{n}$ may be any countable set of $n$-ary
predicate symbols.
Constraint domains $\mathcal{C}$, sets of constraints $\Pi$ and their
solutions, as well as terms, atoms and substitutions over a given
$\mathcal{C}$ are well known notions underlying the CLP scheme. In this paper
we assume a relational formalization of constraint domains as mathematical
structures $\mathcal{C}$ providing a carrier set $C_{\mathcal{C}}$ (consisting
of ground terms built from data constructors and a given set $B_{\mathcal{C}}$
of $\mathcal{C}$-specific basic values) and an interpretation of various
$\mathcal{C}$-specific primitive predicates. For the examples in this paper we
will use a constraint domain $\mathcal{R}$ which allows to work with
arithmetic constraints over the real numbers, and is defined to include:
* •
The set of basic values $B_{\mathcal{R}}=\mathbb{R}$. Note that
$C_{\mathcal{R}}$ includes ground terms built from real values and data
constructors, in addition to real numbers.
* •
Primitive predicates for encoding the usual arithmetic operations over
$\mathbb{R}$. For instance, the addition operation $+$ over $\mathbb{R}$ is
encoded by a ternary primitive predicate $op_{+}$ such that, for any
$t_{1},t_{2}\in C_{\mathcal{R}}$, $op_{+}(t_{1},t_{2},t)$ is true in
$\mathcal{R}$ iff $t_{1},t_{2},t\in\mathbb{R}$ and $t_{1}+t_{2}=t$. In
particular, $op_{+}(t_{1},t_{2},t)$ is false in $\mathcal{R}$ if either
$t_{1}$ or $t_{2}$ includes data constructors. The primitive predicates
encoding other arithmetic operations such as $\times$ and $-$ are defined
analogously.
* •
Primitive predicates for encoding the usual inequality relations over
$\mathbb{R}$. For instance, the ordering $\leq$ over $\mathbb{R}$ is encoded
by a binary primitive predicate $cp_{\leq}$ such that, for any $t_{1},t_{2}\in
C_{\mathcal{R}}$, $cp_{\leq}(t_{1},t_{2})$ is true in $\mathcal{R}$ iff
$t_{1},t_{2}\in\mathbb{R}$ and $t_{1}\leq t_{2}$. In particular,
$cp_{\leq}(t_{1},t_{2})$ is false in $\mathcal{R}$ if either $t_{1}$ or
$t_{2}$ includes data constructors. The primitive predicates encoding the
other inequality relations, namely $>$, $\geq$ and $>$, are defined
analogously.
We assume the following classification of atomic $\mathcal{C}$-constraints:
defined atomic constraints $p(\overline{t}_{n})$, where $p$ is a program-
defined predicate symbol; primitive constraints $r(\overline{t}_{n})$ where
$r$ is a $\mathcal{C}$-specific primitive predicate symbol; and equations
$t==s$.
We use $\mbox{Con}_{\mathcal{C}}$ as a notation for the set of all
$\mathcal{C}$-constraints and $\kappa$ as a notation for an atomic primitive
constraint. Constraints are interpreted by means of $\mathcal{C}$-valuations
$\eta\in\mbox{Val}_{\mathcal{C}}$, which are ground substitutions. The set
$\mbox{Sol}_{\mathcal{C}}(\Pi)$ of solutions of
$\Pi\subseteq\mbox{Con}_{\mathcal{C}}$ includes all the valuations $\eta$ such
that $\Pi\eta$ is true when interpreted in $\mathcal{C}$.
$\Pi\subseteq\mbox{Con}_{\mathcal{C}}$ is called satisfiable if
$\mbox{Sol}_{\mathcal{C}}(\Pi)\neq\emptyset$ and unsatisfiable otherwise.
$\pi\in\mbox{Con}_{\mathcal{C}}$ is entailed by
$\Pi\subseteq\mbox{Con}_{\mathcal{C}}$ (noted
$\Pi~{}{\models_{\mathcal{C}}}~{}\pi$) iff
$\mbox{Sol}_{\mathcal{C}}(\Pi)\subseteq\mbox{Sol}_{\mathcal{C}}(\pi)$.
### 2.2 Qualification Domains
Qualification domains were inspired by [VE86] and firstly introduced in [RR08]
with the aim of providing elements, called qualification values, which can be
attached to computed answers. They are defined as structures
$\mathcal{D}=\langle
D,\trianglelefteqslant,\text{{b}},\text{{t}},\circ\rangle$ verifying the
following requirements:
1. 1.
$\langle D,\trianglelefteqslant,\text{{b}},\text{{t}}\rangle$ is a lattice
with extreme points b (called infimum or bottom element) and t (called maximum
or top element) w.r.t. the partial ordering $\trianglelefteqslant$ (called
qualification ordering). For given elements $d,e\in D$, we write $d\sqcap e$
for the greatest lower bound ($glb$) of $d$ and $e$, and $d\sqcup e$ for the
least upper bound ($lub$) of $d$ and $e$. We also write $d\vartriangleleft e$
as abbreviation for $d\trianglelefteqslant e\land d\neq e$.
2. 2.
$\circ:D\times D\rightarrow D$, called attenuation operation, verifies the
following axioms:
1. (a)
$\circ$ is associative, commutative and monotonic w.r.t.
$\trianglelefteqslant$.
2. (b)
$\forall d\in D:d\circ\text{{t}}=d$ and $d\circ\text{{b}}=\text{{b}}$.
3. (c)
$\forall d,e\in D:d\circ e\trianglelefteqslant e$.
4. (d)
$\forall d,e_{1},e_{2}\in D:d\circ(e_{1}\sqcap e_{2})=(d\circ
e_{1})\sqcap(d\circ e_{2})$.
For any $S=\\{e_{1},e_{2},\ldots,e_{n}\\}\subseteq D$, the $glb$ (also called
infimum of $S$) exists and can be computed as $\bigsqcap S=e_{1}\sqcap
e_{2}\sqcap\cdots\sqcap e_{n}$ (which reduces to t in the case $n=0$). The
dual claim concerning $lub$s is also true. As an easy consequence of the
axioms, one gets the identity $d\circ\bigsqcap S=\bigsqcap\\{d\circ e\mid e\in
S\\}$.
Some of the axioms postulated for the attenuation operator—associativity,
commutativity and monotonicity—are also required for t-norms in fuzzy logic,
usually defined as binary operations over the real number interval $[0,1]$.
More generally, there are formal relationships between qualification domains
and some other existing proposals of lattice-based structures for uncertain
reasoning, such as the lower bound constraint frames proposed in [Ger01], the
multi-adjoint lattices for fuzzy LP languages proposed in [MOV01a, MOV01b] and
the semirings for soft constraint solving proposed in [BMR01, GC98]. However,
qualification domains are a class of mathematical structures that differs from
all these approaches. Their base lattices do not need to be complete and the
axioms concerning the attenuation operator require additional properties
w.r.t. t-norms. Some differences w.r.t. multi-adjoint algebras and the
semirings from [BMR01] have been discussed in more detail in [CRR08] and
[RR10], respectively.
Many useful qualification domains are such that $\forall d,e\in
D\setminus\\{\text{{b}}\\}:d\circ e\neq\text{{b}}$. In the sequel, any
qualification domain $\mathcal{D}$ that verifies this property will be called
stable. More technical details, explanations and examples concerning
qualification domains can be found in [RR10TR]. Examples include three basic
qualification domains which are stable, namely: the qualification domain
$\mathcal{B}$ of classical boolean values, the qualification domain
$\mathcal{U}$ of uncertainty values, the qualification domain $\mathcal{W}$ of
weight values. Moreover, Theorem 2.1 of [RR10TR] shows that the ordinary
cartesian product $\mathcal{D}_{1}\\!\times\mathcal{D}_{2}$ of two
qualification domains is again a qualification domain, while the strict
cartesian product $\mathcal{D}_{1}\\!\otimes\mathcal{D}_{2}$ of two stable
qualification domains is a stable qualification domain.
### 2.3 Expressing a Qualification Domain in a Constraint Domain
The SQCLP scheme depends crucially on the ability to encode qualification
domains into constraint domains, in the sense defined below:
###### Definition 2.1 (Expressing $\mathcal{D}$ in $\mathcal{C}$)
A qualification domain $\mathcal{D}$ is expressible in a constraint domain
$\mathcal{C}$ if there is an injective mapping
$\imath:D\setminus\\{\text{{b}}\\}\to C$ (thought as an embedding of
$D\setminus\\{\text{{b}}\\}$ into $C$) and moreover:
1. 1.
There is a $\mathcal{C}$-constraint $\mathsf{qVal}(X)$ with free variable $X$
such that $\mbox{Sol}_{\mathcal{C}}(\mathsf{qVal}(X))$ is the set of all
$\eta\in\mbox{Val}_{\mathcal{C}}$ verifying $\eta(X)\in ran(\imath)$.
_Informal explanation:_ For each qualification value $x\in
D\setminus\\{\text{{b}}\\}$ we think of $\imath(x)\in C$ as the representation
of $x$ in $\mathcal{C}$. Therefore, $ran(\imath)$ is the set of those elements
of $C$ which can be used to represent qualification values, and
$\mathsf{qVal}(X)$ constraints the value of $X$ to be some of these
representations.
2. 2.
There is a $\mathcal{C}$-constraint $\mathsf{qBound}(X,Y,Z)$ with free
variables $X$, $Y$ and $Z$ encoding “$x\trianglelefteqslant y\circ z$” in the
following sense: any $\eta\in\mbox{Val}_{\mathcal{C}}$ such that
$\eta(X)=\iota(x)$, $\eta(Y)=\iota(y)$ and $\eta(Z)=\iota(z)$ verifies
$\eta\in\mbox{Sol}_{\mathcal{C}}(\mathsf{qBound}(X,Y,Z))$ iff
$x\trianglelefteqslant y\circ z$.
_Informal explanation:_ $\mathsf{qBound}(X,Y,Z)$ constraints the values of
$X,Y,Z$ to be the representations of three qualification values $x,y,z\in
D\setminus\\{\text{{b}}\\}$ such that $x\trianglelefteqslant y\circ z$.
In addition, if $\mathsf{qVal}(X)$ and $\mathsf{qBound}(X,Y,Z)$ can be chosen
as existential constraints of the form $\exists X_{1}\ldots\exists
X_{n}(B_{1}\land\ldots\land B_{m})$—where $B_{j}~{}(1\leq j\leq m)$ are
atomic—we say that $\mathcal{D}$ is existentially expressible in
$\mathcal{C}$.
It can be proved that $\mathcal{B}$, $\mathcal{U}$, $\mathcal{W}$ and and any
qualification domain built from these with the help of the strict cartesian
product $\otimes$ are existentially expressible in any constraint domain
$\mathcal{C}$ that includes the basic values and computational features of
$\mathcal{R}$. The example below illustrates the existential representation of
three typical qualification domains in $\mathcal{R}$:
###### Example 2.1
1. 1.
$\mathcal{U}$ can be existentially expressed in $\mathcal{R}$ as follows:
$D_{\mathcal{U}}\setminus\\{\text{{b}}\\}=D_{\mathcal{U}}\setminus\\{0\\}=(0,1]\subseteq\mathbb{R}\subseteq
C_{\mathcal{R}}$; therefore $\imath$ can be taken as the identity embedding
mapping from $(0,1]$ into $\mathbb{R}$. Moreover, $\mathsf{qVal}(X)$ can be
built as the existential $\mathcal{R}$-constraint $cp_{<}(0,X)\land
cp_{\leq}(X,1)$ and $\mathsf{qBound}(X,Y,Z)$ can be built as the existential
$\mathcal{R}$-constraint $\exists X^{\prime}(op_{\times}(Y,Z,X^{\prime})\land
cp_{\leq}(X,X^{\prime}))$.
2. 2.
$\mathcal{W}$ can be existentially expressed in $\mathcal{R}$ as follows:
$D_{\mathcal{W}}\setminus\\{\text{{b}}\\}=D_{\mathcal{W}}\setminus\\{\infty\\}=[0,\infty)\subseteq\mathbb{R}\subseteq
C_{\mathcal{R}}$; therefore $\imath$ can be taken as the identity embedding
mapping from $[0,\infty)$ into $\mathbb{R}$. Moreover, $\mathsf{qVal}(X)$ can
be built as the existential $\mathcal{R}$-constraint $cp_{\geq}(X,0)$ and
$\mathsf{qBound}(X,Y,Z)$ can be built as the existential
$\mathcal{R}$-constraint $\exists X^{\prime}(op_{+}(Y,Z,X^{\prime})\land
cp_{\geq}(X,X^{\prime}))$.
3. 3.
$\mathcal{U}{\otimes}\mathcal{W}$ can be existentially expressed in
$\mathcal{R}$ as follows:
$D_{\mathcal{U}{\otimes}\mathcal{W}}\setminus\\{\text{{b}}\\}=(0,1]\times[0,\infty)\subseteq\mathbb{R}\times\mathbb{R}$;
therefore
$\imath:D_{\mathcal{U}{\otimes}\mathcal{W}}\setminus\\{\text{{b}}\\}\to
D_{\mathcal{R}}$ can bee defined as $\imath(x,y)$ = pair$(x,y)$, using a
binary constructor pair $\in DC^{2}$ to represent the ordered pair $(x,y)$ as
an element of $D_{\mathcal{R}}$. Moreover, taking into account the two
previous items of the example:
* •
$\mathsf{qVal}(X)$ can be built as $\exists X_{1}\exists X_{2}(X==$
pair$(X_{1},X_{2})\land cp_{<}(0,X_{1})\land cp_{\leq}(X_{1},1)\land
cp_{\geq}(X_{2},0))$.
* •
$\mathsf{qBound}(X,Y,Z)$ can be built as $\exists X_{1}\exists
X^{\prime}_{1}\exists X_{2}\exists X^{\prime}_{2}\exists Y_{1}\exists
Y_{2}\exists Z_{1}\exists Z_{2}(X==$ pair$(X_{1},X_{2})\land Y==$
pair$(Y_{1},Y_{2})\land Z==$ pair$(Z_{1},Z_{2})\land
op_{\times}(Y_{1},Z_{1},X^{\prime}_{1})\land
cp_{\leq}(X_{1},X^{\prime}_{1})\land op_{+}(Y_{2},Z_{2},X^{\prime}_{2})\land
cp_{\geq}(X_{2},X^{\prime}_{2}))$.
### 2.4 Programs and Declarative Semantics
Instances $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ of the SQCLP
scheme are parameterized by so-called admissible triples
$\langle\mathcal{S},\mathcal{D},\mathcal{C}\rangle$ consisting of a constraint
domain $\mathcal{C}$, a qualification domain $\mathcal{D}$ and a proximity
relation $\mathcal{S}:S\times S\to D$—where $D$ is the carrier set of
$\mathcal{D}$ and $S$ is the set of all variables, basic values and signature
symbols available in $\mathcal{C}$—satisfying the following properties:
* •
$\forall x\in S:\mathcal{S}(x,x)=\text{{t}}$ (reflexivity).
* •
$\forall x,y\in S:\mathcal{S}(x,y)=\mathcal{S}(y,x)$ (symmetry).
* •
$\mathcal{S}$ restricted to $\mathcal{V}\\!ar$ behaves as the identity — i.e.
$\mathcal{S}(X,X)=\text{{t}}$ for all $X\in\mathcal{V}\\!ar$ and
$\mathcal{S}(X,Y)=\text{{b}}$ for all $X,Y\in\mathcal{V}\\!ar$ such that
$X\neq Y$.
* •
For any $x,y\in S$, $\mathcal{S}(x,y)\neq\text{{b}}$ can happen only if:
* –
$x=y$ are identical.
* –
$x$ and $y$ are both: basic values; data constructor symbols with the same
arity; or defined predicate symbols with the same arity.
In particular, $\mathcal{S}(p,p^{\prime})\neq\text{{b}}$ cannot happen if $p$
and $p^{\prime}$ are syntactically different primitive predicate symbols.
A proximity relation $\mathcal{S}$ is called similarity iff it satisfies the
additional property $\forall x,y,z\in
S:\mathcal{S}(x,z)\trianglerighteqslant\mathcal{S}(x,y)\sqcap\mathcal{S}(y,z)$
(transitivity). A given proximity relation $\mathcal{S}$ can be extended to
work over terms, atoms and other syntactic objects in an obvious way. The
definition for the case of terms is as follows:
1. 1.
For any term $t$, $\mathcal{S}(t,t)=\text{{t}}$.
2. 2.
For $X\in\mathcal{V}\\!ar$ and for any term $t$ different from $X$,
$\mathcal{S}(X,t)=\mathcal{S}(t,X)=\text{{b}}$.
3. 3.
For any two data constructor symbols $c$ and $c^{\prime}$ with different
arities, $\mathcal{S}(c(\overline{t}_{n}),$
$c^{\prime}(\overline{t^{\prime}}_{m}))=\text{{b}}$.
4. 4.
For any two data constructor symbols $c$ and $c^{\prime}$ with the same arity,
$\mathcal{S}(c(\overline{t}_{n}),$
$c^{\prime}(\overline{t^{\prime}}_{n}))=\mathcal{S}(c,c^{\prime})\sqcap\mathcal{S}(t_{1},t^{\prime}_{1})\sqcap\cdots\sqcap\mathcal{S}(t_{n},t^{\prime}_{n})$.
For the case of finite substitutions $\sigma$ and $\theta$ whose domain is a
subset of a finite set of variables $\\{X_{1},\ldots,X_{m}\\}$,
$\mathcal{S}(\sigma,\theta)$ can be naturally defined as
$\mathcal{S}(X_{1}\sigma,X_{1}\theta)\sqcap\ldots\sqcap\mathcal{S}(X_{m}\sigma,X_{m}\theta)$.
A $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program is a set
$\mathcal{P}$ of _qualified program rules_ (also called _qualified clauses_)
$C:A\xleftarrow{\alpha}B_{1}\sharp w_{1},\ldots,B_{m}\sharp w_{m}$, where $A$
is a defined atom, $\alpha\in D\setminus\\{\text{{b}}\\}$ is called the
attenuation factor of the clause and each $B_{j}\sharp w_{j}~{}(1\leq j\leq
m)$ is an atom $B_{j}$ annotated with a so-called threshold value
$w_{j}\in(D\setminus\\{\text{{b}}\\})\uplus\\{?\\}$. The intended meaning of
$C$ is as follows: if for all $1\leq j\leq m$ one has $B_{j}\sharp e_{j}$
(meaning that $B_{j}$ holds with qualification value $e_{j}$) for some
$e_{j}\trianglerighteqslant^{?}w_{j}$, then $A\sharp d$ (meaning that $A$
holds with qualification value $d$) can be inferred for any $d\in
D\setminus\\{\text{{b}}\\}$ such that
$d\trianglelefteqslant\alpha\circ\bigsqcap_{j=1}^{m}e_{j}$. By convention,
$e_{j}\trianglerighteqslant^{?}w_{j}$ means $e_{j}\trianglerighteqslant w_{j}$
if $w_{j}~{}{\neq}~{}?$ and is identically true otherwise. In practice
threshold values equal to ‘?’ and attenuation values equal to t can be
omitted.
| % Book representation: book( ID, Title, Author, Lang, Genre, VocLvl, Pages
).
---|---
1 | library([ book(1, ‘Tintin’, ‘Hergé’, french, comic, easy, 65),
2 | book(2, ‘Dune’, ‘F.P. Herbert’, english, sciFi, medium, 345),
3 | book(3, ‘Kritik der reinen Vernunft’, ‘I. Kant’, german, philosophy, difficult, 1011),
4 | book(4, ‘Beim Hauten der Zwiebel’, ‘G. Grass’, german, biography, medium, 432) ])
| % Auxiliary predicate for computing list membership:
5 | member(B, [B$\mid$_])
6 | member(B, [_$\mid$T]) $\leftarrow$ member(B, T)
| % Predicates for getting the explicit attributes of a given book:
7 | getId(book(ID, _Title, _Author, _Lang, _Genre, _VocLvl, _Pages), ID)
8 | getTitle(book(_ID, Title, _Author, _Lang, _Genre, _VocLvl, _Pages), Title)
9 | getAuthor(book(_ID, _Title, Author, _Lang, _Genre, _VocLvl, _Pages), Author)
10 | getLanguage(book(_ID, _Title, _Author, Lang, _Genre, _VocLvl, _Pages), Lang)
11 | getGenre(book(_ID, _Title, _Author, _Lang, Genre, _VocLvl, _Pages), Genre)
12 | getVocLvl(book(_ID, _Title, _Author, _Lang, _Genre, VocLvl, _Pages), VocLvl)
13 | getPages(book(_ID, _Title, _Author, _Lang, _Genre, _VocLvl, Pages), Pages)
| % Function for guessing the reader level of a given book:
14 | guessRdrLvl(B, basic) $\leftarrow$ getVocLvl(B, easy), getPages(B, N), N $<$ 50
15 | guessRdrLvl(B, intermediate) $\xleftarrow{0.8}$ getVocLvl(B, easy), getPages(B, N), N $\geq$ 50
16 | guessRdrLvl(B, basic) $\xleftarrow{0.9}$ getGenre(B, children)
17 | guessRdrLvl(B, proficiency) $\xleftarrow{0.9}$ getVocLvl(B, difficult), getPages(B, N), N $\geq$ 200
18 | guessRdrLvl(B, upper) $\xleftarrow{0.8}$ getVocLvl(B, difficult), getPages(B, N), N $<$ 200
19 | guessRdrLvl(B, intermediate) $\xleftarrow{0.8}$ getVocLvl(B, medium)
20 | guessRdrLvl(B, upper) $\xleftarrow{0.7}$ getVocLvl(B, medium)
| % Function for answering a particular kind of user queries:
21 | search(Lang, Genre, Level, Id) $\leftarrow$ library(L)#1.0, member(B, L)#1.0,
22 | getLanguage(B, Lang), getGenre(B, Genre),
23 | guessRdrLvl(B, Level), getId(B, Id)#1.0
| % Proximity relation $\mathcal{S}_{s}$:
24 | $\mathcal{S}_{s}$(sciFi, fantasy) = $\mathcal{S}_{s}$(fantasy, sciFi) = 0.9
25 | $\mathcal{S}_{s}$(adventure, fantasy) = $\mathcal{S}_{s}$(fantasy, adventure) = 0.7
26 | $\mathcal{S}_{s}$(essay, philosophy) = $\mathcal{S}_{s}$(philosophy, essay) = 0.8
27 | $\mathcal{S}_{s}$(essay, biography) = $\mathcal{S}_{s}$(biography, essay) = 0.7
Figure 1: $\mbox{SQCLP}(\mathcal{S}_{s},\,\mathcal{U},\mathcal{R})$-program
$\mathcal{P}_{\\!s}$ (Library with books in different languages)
Figure 1 shows a simple
$\mbox{SQCLP}(\mathcal{S}_{s},\,\mathcal{U},\mathcal{R})$-program
$\mathcal{P}_{\\!s}$ which illustrates the expressivity of the SQCLP scheme to
deal with problems involving flexible information retrieval. Predicate search
can be used to answer queries asking for books in the library matching some
desired language, genre and reader level. Predicate guessRdrLvl takes
advantage of attenuation factors to encode heuristic rules to compute reader
levels on the basis of vocabulary level and other book features. The other
predicates compute book features in the natural way, and the proximity
relation $\mathcal{S}_{s}$ allows flexibility in any unification (i.e. solving
of equality constraints) arising during the invocation of the program
predicates.
The declarative semantics of a given
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program $\mathcal{P}$
relies on qualified constrained atoms (briefly qc-atoms) of the form $A\sharp
d\Leftarrow\Pi$, intended to assert that the validity of atom $A$ with
qualification degree $d\in D$ is entailed by the constraint set $\Pi$. A qc-
atom is called defined, primitive or equational according to the syntactic
form of $A$; and it is called observable iff $d\in D\setminus\\{\text{{b}}\\}$
and $\Pi$ is satisfiable.
Program interpretations are defined as sets of observable qc-atoms which obey
a natural closure condition. The results proved in [RR10] show two equivalent
ways to characterize declarative semantics: using a fix-point approach and a
proof-theoretical approach. For the purposes of the present paper it suffices
to consider the proof-theoretical approach that relies on a formal inference
system called Proximity-based Qualified Constrained Horn Logic—in symbols,
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$—intended to infer
observable qc-atoms from $\mathcal{P}$ and consisting of the three inference
rules displayed in Figure 2. Rule SQEA depends on a relation $\approx_{d,\Pi}$
between terms that is defined in the following way: $t\approx_{d,\Pi}s$ iff
there exist two terms $\hat{t}$ and $\hat{s}$ such that
$\Pi~{}{\models_{\mathcal{C}}}~{}t==\hat{t}$,
$\Pi~{}{\models_{\mathcal{C}}}~{}s==\hat{s}$ and $\text{{b}}\neq
d\trianglelefteqslant\mathcal{S}(\hat{t},\hat{s})$. Recall that the notation
$\Pi~{}{\models_{\mathcal{C}}}~{}\pi$ makes sense for any
$\mathcal{C}$-constraint $\pi$ and is a shorthand for
$\mbox{Sol}_{\mathcal{C}}(\Pi)\subseteq\mbox{Sol}_{\mathcal{C}}(\pi)$, as
explained in Subsection 2.1. The relation $\approx_{d,\Pi}$ allows to deduce
equations from $\Pi$ in a flexible way, i.e. taking the proximity relation
$\mathcal{S}$ into account. In the sequel we will use $t\approx_{d}s$ as a
shorthand for $t\approx_{d,\emptyset}s$, which holds iff $\text{{b}}\neq
d\trianglelefteqslant\mathcal{S}(t,s)$.
SQDA $\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{j}\theta\sharp
e_{j}\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{p^{\prime}(\overline{t^{\prime}}_{n})\sharp d\Leftarrow\Pi}$
---
if $(p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m})\in\mathcal{P}$, $\theta$ subst.,
$\mathcal{S}(p^{\prime},p)=d_{0}\neq\text{{b}}$,
$e_{j}\trianglerighteqslant^{?}w_{j}~{}(1\leq j\leq m)$ and
$d\trianglelefteqslant\bigsqcap_{i=0}^{n}d_{i}\sqcap\alpha\circ\bigsqcap_{j=1}^{m}e_{j}$.
SQEA $\displaystyle\frac{}{\quad(t==s)\sharp d\Leftarrow\Pi\quad}$ if
$t\approx_{d,\Pi}s$. SQPA $\displaystyle\frac{}{\quad\kappa\sharp
d\Leftarrow\Pi\quad}$ if $\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$.
Figure 2: Proximity-based Qualified Constrained Horn Logic
We write $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ \varphi$ to indicate
that $\varphi$ can be deduced from $\mathcal{P}$ in
$\mbox{SQCHL}(\mathcal{S},$ $\mathcal{D},\mathcal{C})$, and $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!k}\ \varphi$ in the case
that the deduction can be performed with exactly $k$ SQDA inference steps. As
usual in formal inference systems,
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proofs can be represented
as proof trees whose nodes correspond to qc-atoms, each node being inferred
from its children by means of some
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ inference step.
The following theorem, proved in [RR10TR], characterizes least program models
in the scheme SQCLP. This result allows to use
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$-derivability as a logical
criterion for proving the semantic correctness of program transformations, as
we will do in Section 4.
###### Theorem 2.1 (Logical characterization of least program models in
SQCHL)
For any $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$, its least model can be characterized as:
$\mathcal{M}_{\mathcal{P}}=\\{\varphi\mid\varphi\mbox{ is an observable
defined qc-atom and }\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ \varphi\\}\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}$
### 2.5 Goals and Goal Solving
Goals for a given $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$ have the form
$G~{}:~{}A_{1}\sharp W_{1},~{}\ldots,~{}A_{m}\sharp W_{m}\talloblong
W_{1}\trianglerighteqslant^{?}\\!\beta_{1},~{}\ldots,~{}W_{m}\trianglerighteqslant^{?}\\!\beta_{m}$
abbreviated as $(A_{i}\sharp
W_{i},~{}W_{i}\trianglerighteqslant^{?}\\!\beta_{i})_{i=1\ldots m}$. The
$A_{i}\sharp W_{i}$ are called annotated atoms. If all atoms
$A_{i},\,i=1\ldots m,$ are equations $t_{i}==s_{i}$, the goal $G$ is called a
unification problem. The pairwise different variables
$W_{i}\in\mathcal{W}\\!ar$ are called qualification variables; they are taken
from a set $\mathcal{W}\\!ar$ assumed to be disjoint from the set
$\mathcal{V}\\!ar$ of data variables used in terms. The conditions
$W_{i}\trianglerighteqslant^{?}\\!\beta_{i}$ (with
$\beta_{i}\in(D\setminus\\{\text{{b}}\\})\uplus\\{?\\}$) are called threshold
conditions and their intended meaning (relying on the notations ‘?’ and
‘$\trianglerighteqslant^{?}$’) is as already explained when introducing
program clauses in Subsection 2.4. In the sequel, $\mathrm{war}(o)$ will
denote the set of all qualification variables occurring in the syntactic
object $o$. In particular, for a goal $G$ as displayed above,
$\mathrm{war}(G)$ denotes the set $\\{W_{i}\mid 1\leq i\leq m\\}$. In the case
$m=1$ the goal is called atomic. The following definition relies on
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$-derivability to provide a
natural declarative notion of goal solution:
###### Definition 2.2 (Possible Answers and Goal Solutions)
Assume a given $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$ and a goal $G$ for $\mathcal{P}$ with the syntax displayed
above. Then:
1. 1.
A possible answer for $G$ is any triple $ans=\langle\sigma,\mu,\Pi\rangle$
such that $\sigma$ is a $\mathcal{C}$-substitution, $W\\!\mu\in
D\setminus\\{\text{{b}}\\}$ for all $W\in\mathrm{dom}(\mu)$, and $\Pi$ is a
satisfiable and finite set of atomic $\mathcal{C}$-constraints. The
qualification value $\lambda_{ans}=\bigsqcap_{i=1}^{m}W_{i}\mu$ is called the
qualification level of $ans$.
2. 2.
A possible answer $\langle\sigma,\mu,\Pi\rangle$ is called a solution for $G$
iff the conditions $W_{i}\mu=d_{i}\trianglerighteqslant^{?}\\!\beta_{i}$ and
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
A_{i}\sigma\sharp W_{i}\mu\Leftarrow\Pi$ hold for all $i=1\ldots m$. Note that
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
A_{i}\sigma\sharp W_{i}\mu\Leftarrow\Pi$ amounts to
$t_{i}\sigma\approx_{W_{i}\mu,\Pi}s_{i}\sigma$ in the case that $A_{i}$ is an
equation $t_{i}==s_{i}$. The set of all solutions for $G$ w.r.t. $\mathcal{P}$
is noted $\mbox{Sol}_{\mathcal{P}}(G)$.
3. 3.
A solution $\langle\eta,\rho,\Pi\rangle$ for $G$ is called ground iff
$\Pi=\emptyset$ and $\eta\in\mbox{Val}_{\mathcal{C}}$ is a variable valuation
such that $A_{i}\eta$ is a ground atom for all $i=1\ldots m$. The set of all
ground solutions for $G$ w.r.t. $\mathcal{P}$ is noted
$\mbox{GSol}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$.
4. 4.
A ground solution
$gsol=\langle\eta,\rho,\emptyset\rangle\in\mbox{GSol}_{\mathcal{P}}(G)$ is
subsumed by a possible answer $ans=\langle\sigma,\mu,\Pi\rangle$ iff
$W_{i}\mu\trianglerighteqslant W_{i}\rho$ for $i=1\ldots m$ (which implies
$\lambda_{ans}\trianglerighteqslant\lambda_{gsol}$) and there is some
$\nu\in\mbox{Sol}_{\mathcal{C}}(\Pi)$ such that $X\eta=X\sigma\nu$ holds for
each variable $X\in\mathrm{var}(G)$.
5. 5.
A ground solution
$gsol=\langle\eta,\rho,\emptyset\rangle\in\mbox{GSol}_{\mathcal{P}}(G)$ is
subsumed by a possible answer $ans=\langle\sigma,\mu,\Pi\rangle$ in the
flexible sense iff $\lambda_{ans}\trianglerighteqslant\lambda_{gsol}$ and
there is some $\nu\in\mbox{Sol}_{\mathcal{C}}(\Pi)$ such that
$\mathcal{S}(X\eta,X\sigma\nu)\trianglerighteqslant\lambda_{gsol}$ holds for
each variable $X\in\mathrm{var}(G)$.
A possible goal $G_{s}$ for the library program displayed in Figure 1 is
$G_{s}:$ search(german, essay, intermediate, ID)#W $\talloblong$ W $\geq$ 0.65
and one solution for $G_{s}$ is $\langle\\{\textit{ID}\mapsto
4\\},\\{\textit{W}\mapsto 0.7\\},\emptyset\rangle$. In this simple case, the
constraint set $\Pi$ within the solution is empty.
The following example will be used to discuss some implementation issues in
Subsection LABEL:sec:practical:SQCLP:eqsimrel.
###### Example 2.2
Assume the admissible triple
$\langle\mathcal{S},\mathcal{U},\mathcal{R}\rangle$ where the proximity
relation $\mathcal{S}$ is such that: $\mathcal{S}(a,b)=\mathcal{S}(b,a)=0.9$,
$\mathcal{S}(a,c)=\mathcal{S}(c,a)=0.9$, and
$\mathcal{S}(b,c)=\mathcal{S}(c,b)=0.4$. Let $\mathcal{P}$ be the empty
program. Then, the goal $G$:
$(X==Y)\sharp W_{1},\ (X==b)\sharp W_{2},\ (Y==c)\sharp W_{3}\talloblong
W_{1}\geq 0.8,\ W_{2}\geq 0.8,\ W_{3}\geq 0.8$
is a unification problem. Its valid solutions in the sense of Definition 2.2
include $\mbox{sol}_{i}=\langle\sigma_{i},\mu_{i},\emptyset\rangle$
($i=1,2,3$), where:
$\begin{array}[]{l@{\hspace{7.5mm}}l}\sigma_{1}=\\{X\mapsto a,\ Y\mapsto
a\\}\hfil\hskip 21.33955pt&\mu_{1}=\\{W_{1}\mapsto 1,\ W_{2}\mapsto 0.9,\
W_{3}\mapsto 0.9\\}\\\ \sigma_{2}=\\{X\mapsto b,\ Y\mapsto a\\}\hfil\hskip
21.33955pt&\mu_{2}=\\{W_{1}\mapsto 0.9,\ W_{2}\mapsto 1,\ W_{3}\mapsto
0.9\\}\\\ \sigma_{3}=\\{X\mapsto a,\ Y\mapsto c\\}\hfil\hskip
21.33955pt&\mu_{3}=\\{W_{1}\mapsto 0.9,\ W_{2}\mapsto 0.9,\ W_{3}\mapsto
1\\}\\\ \end{array}$
as well as some less interesting solutions assigning lower qualification
values to the variables $W_{i}$ ($i=1,2,3$). In this simple example, all the
solutions are ground, but this is not always the case in general. Note that
sol2 is subsumed by sol1 in the flexible sense because:
* •
$\nu=\varepsilon\in\mbox{Sol}_{\mathcal{C}}(\emptyset)$ satisfies
$\mathcal{S}(X\sigma_{2},X\sigma_{1}\varepsilon)=\mathcal{S}(b,a)=0.9\trianglerighteqslant
0.9$ and also
$\mathcal{S}(Y\sigma_{2},Y\sigma_{1}\varepsilon)=\mathcal{S}(a,a)=1\trianglerighteqslant
0.9$.
* •
The qualification level of both sol2 and sol1 is $0.9$, thus trivially,
$0.9\trianglerighteqslant 0.9$.
Moreover, sol3 is also subsumed by sol1 in the flexible sense, because:
* •
$\nu=\varepsilon\in\mbox{Sol}_{\mathcal{C}}(\emptyset)$ satisfies
$\mathcal{S}(X\sigma_{3},X\sigma_{1}\varepsilon)=\mathcal{S}(a,a)=1\trianglerighteqslant
0.9$ and also
$\mathcal{S}(Y\sigma_{3},Y\sigma_{1}\varepsilon)=\mathcal{S}(c,a)=0.9\trianglerighteqslant
0.9$.
* •
The qualification level of both sol3 and sol1 is $0.9$, thus trivially,
$0.9\trianglerighteqslant 0.9$.
In fact, it is easy to check that any of the three ground solutions sol1, sol2
and sol3 subsumes the other two in the flexible sense.
In practice, users of SQCLP languages will rely on some available goal solving
system for computing goal solutions. The following definition provides an
abstract specification of semantically correct goal solving systems which will
serve as a theoretical guideline for the implementation presented in Section
LABEL:sec:practical:
###### Definition 2.3 (Correct Abstract Goal Solving Systems for SQCLP)
An abstract goal solving system for
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ is any device
$\mathcal{CA}$ that takes a program $\mathcal{P}$ and a goal $G$ as input and
yields a set $\mathcal{CA}_{\mathcal{P}}(G)$ of possible answers
$\langle\sigma,\mu,\Pi\rangle$ (called computed answers) as output. Moreover:
1. 1.
$\mathcal{CA}$ is called sound iff every computed answer is a solution, i.e.
$\mathcal{CA}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$.
2. 2.
$\mathcal{CA}$ is called weakly complete iff for every ground solution
$gsol\in\mbox{GSol}_{\mathcal{P}}(G)$ there is some computed answer
$ans\in\mathcal{CA}_{\mathcal{P}}(G)$ such that $ans$ subsumes $gsol$.
3. 3.
$\mathcal{CA}$ is called weakly complete in the flexible sense iff for every
ground solution $gsol\in\mbox{GSol}_{\mathcal{P}}(G)$ there is some computed
answer $ans\in\mathcal{CA}_{\mathcal{P}}(G)$ such that $ans$ subsumes $gsol$
in the flexible sense.
4. 4.
$\mathcal{CA}$ is called correct iff it is both sound and weakly complete.
5. 5.
$\mathcal{CA}$ is called correct in the flexible sense iff it is both sound
and weakly complete in the flexible sense.
Extensions of the well-known SLD-resolution procedure [Llo87, Apt90] can be
used as a basis to obtain correct goal solving systems for extended LP
languages. In particular, constraint SLD-resolution provides a correct goal
solving system for instances of the CLP scheme, as proved e.g. in
[JMM+98]111In fact, constraint SLD-resolution is complete in a stronger sense
than weak completeness. As proved in [JMM+98], every solution - even if it is
not ground - is subsumed in a suitable sense by a finite set of computed
solutions.. Several extensions of the SLD-resolution, tailored to different LP
languages supporting uncertain reasoning, have already been mentioned in
Section 1.
Rather than developing an extension of SLD resolution tailored to the SQCLP
scheme, our aim in this paper is to to investigate goal solving systems based
on a semantically correct program transformation from SQCLP into CLP. Sections
4 and LABEL:sec:practical present the transformation technique and its
implementation on top of a CLP Prolog system, respectively. As we will explain
in Subsection LABEL:sec:practical:SQCLP, weak completeness as specified in
Definition 2.3(2) is very hard to achieve in a practical implementation, while
flexible weak completeness in the sense of Definition 2.3(3) is a satisfactory
notion for extended LP languages which use proximity relations. For instance,
similarity-based SLD resolution as presented in [Ses02] is complete in a
flexible sense. Therefore, the Prolog-based prototype system presented in
Section LABEL:sec:practical aims at soundness and weak completeness in the
flexible sense, as specified in Definition 2.3(3). The definition and lemma
below can be used as an abstract guideline for converting a correct goal
solving system $\mathcal{CA}$ into another goal solving system $\mathcal{FCA}$
which is correct in the flexible sense and may be easier to implement, because
it yields smaller sets of computed answers.
###### Definition 2.4 (Flexible Restrictions of an Abstract Goal Solving
System)
Let $\mathcal{CA}$ and $\mathcal{FCA}$ be two abstract goal solving systems
for $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$. We say that
$\mathcal{FCA}$ is a flexible restriction of $\mathcal{CA}$ iff the two
following conditions hold for any choice of a program $\mathcal{P}$ and a goal
$G:(A_{i}\sharp
W_{i},~{}W_{i}\trianglerighteqslant^{?}\\!\beta_{i})_{i=1\ldots m}$:
1. 1.
$\mathcal{FCA}_{\mathcal{P}}(G)\subseteq\mathcal{CA}_{\mathcal{P}}(G)$.
Informally, $\mathcal{FCA}$ is restricted to compute some of the answers
computed by $\mathcal{CA}$.
2. 2.
For each $ans=\langle\sigma,\mu,\Pi\rangle\in\mathcal{CA}_{\mathcal{P}}(G)$
there is some
$\widehat{ans}=\langle\hat{\sigma},\hat{\mu},\Pi\rangle\in\mathcal{FCA}_{\mathcal{P}}(G)$
such that $\lambda_{\widehat{ans}}\trianglerighteqslant\lambda_{ans}$ and
$\mathcal{S}(X\sigma,X\hat{\sigma})\trianglerighteqslant\lambda_{ans}$ holds
for each variable $X\in\mathrm{var}(G)$. Informally, each answer computed by
$\mathcal{CA}$ is close (w.r.t. $\mathcal{S}$) to some of the answers computed
by $\mathcal{FCA}$.
###### Lemma 2.1 (Flexible Correctness of Flexible Restrictions)
Let $\mathcal{CA}$ be a correct abstract goal solving system for
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$. Then any flexible
restriction $\mathcal{FCA}$ of $\mathcal{CA}$ is correct in the flexible
sense.
* Proof
* By assumption, $\mathcal{CA}$ is sound and weakly complete. We must prove soundness and weak completeness in the flexible sense for $\mathcal{FCA}$. Let a $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program $\mathcal{P}$ and a goal $G:(A_{i}\sharp W_{i},~{}W_{i}\trianglerighteqslant^{?}\\!\beta_{i})_{i=1\ldots m}$ for $\mathcal{P}$ be given.
— _Soundness._
$\mathcal{FCA}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$ trivially
follows from
$\mathcal{FCA}_{\mathcal{P}}(G)\subseteq\mathcal{CA}_{\mathcal{P}}(G)$ (true
because $\mathcal{FCA}$ refines $\mathcal{CA}$) and
$\mathcal{CA}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$ (true
because $\mathcal{CA}$ is sound).
— _Weak completeness in the flexible sense._ In order to check the conditions
stated in Definition 2.3(3), let
$gsol=\langle\eta,\rho,\emptyset\rangle\in\mbox{GSol}_{\mathcal{P}}(G)$ be
given. Since $\mathcal{CA}$ is weakly complete, there is some
$ans=\langle\sigma,\mu,\Pi\rangle\in\mathcal{CA}_{\mathcal{P}}(G)$ that
subsumes $gsol$ and hence:
1. [$(f)$]
2. $(a)$
$W_{i}\mu\trianglerighteqslant W_{i}\rho$ for $i=1\ldots m$, which implies
$\lambda_{ans}\trianglerighteqslant\lambda_{gsol}$.
3. $(b)$
There is some $\nu\in\mbox{Sol}_{\mathcal{C}}(\Pi)$ such that
$X\eta=X\sigma\nu$ holds for all $X\in\mathrm{var}(G)$.
Since $\mathcal{FCA}$ is a flexible refinement of $\mathcal{CA}$, there is
some
$\widehat{ans}=\langle\hat{\sigma},\hat{\mu},\Pi\rangle\in\mathcal{FCA}_{\mathcal{P}}(G)$
that is close to $ans$ and thus verifies:
1. [$(f)$]
2. $(c)$
$\lambda_{\widehat{ans}}\trianglerighteqslant\lambda_{ans}$.
3. $(d)$
$\mathcal{S}(X\sigma,X\hat{\sigma})\trianglerighteqslant\lambda_{ans}$ holds
for all $X\in\mathrm{var}(G)$.
Now we can claim:
1. [$(f)$]
2. $(e)$
$\lambda_{\widehat{ans}}\trianglerighteqslant\lambda_{gsol}$ — follows from
$(c)$ and $(a)$.
3. $(f)$
$\mathcal{S}(X\sigma\nu,X\hat{\sigma}\nu)\trianglerighteqslant\lambda_{gsol}$
holds for all $X\in\mathrm{var}(G)$ — follows from $(d)$ and $(a)$.
4. $(g)$
$\mathcal{S}(X\eta,X\hat{\sigma}\nu)\trianglerighteqslant\lambda_{gsol}$ holds
for all $X\in\mathrm{var}(G)$ — follows from $(f)$ and $(b)$.
Since $\nu\in\mbox{Sol}_{\mathcal{C}}(\Pi)$, (e) and (g) guarantee that
$\widehat{ans}$ subsumes $gsol$ in the flexible sense. This finishes the
proof.
Let us finish this section with a remark concerning unification. Both our
implementation and SLD-based goal solving systems for SLP languages—we view
[AF02, Ses02] as representative proposals of this kind; others have been cited
in Section 1—must share the ability to solve unification problems modulo a
given proximity relation $\mathcal{S}:S\times S\to[0,1]$ over signature
symbols, that is assumed to be transitive in [Ses02] and some other related
works, but not in Bousi$\sim$Prolog [JR09, JR09b] and our own approach. The
lack of transitivity makes a crucial difference. The unification algorithms
modulo $\mathcal{S}$ known for the case that $\mathcal{S}$ is a similarity
relation fail to be complete in the flexible sense if $\mathcal{S}$ is a non-
transitive proximity relation. More details on this issue are given in
Subsection LABEL:sec:practical:SQCLP when discussing the implementation of
unification modulo $\mathcal{S}$ in our prototype system for SQCLP
programming.
## 3 The Schemes QCLP & CLP as Specializations of SQCLP
As discussed in the concluding section of [RR10], several specializations of
the SQCLP scheme can be obtained by partial instantiation of its parameters.
In particular, QCLP and CLP can be defined as schemes with instances:
$\begin{array}[]{r@{\hspace{1mm}}c@{\hspace{1mm}}l}\mbox{QCLP}(\mathcal{D},\mathcal{C})\hskip
2.84526pt&~{}{=_{\mathrm{def}}{}}\hfil\hskip
2.84526pt&\mbox{SQCLP}(\mathcal{S}_{\mathrm{id}},\mathcal{D},\mathcal{C})\\\
\mbox{CLP}(\mathcal{C})\hskip 2.84526pt&~{}{=_{\mathrm{def}}{}}\hfil\hskip
2.84526pt&\mbox{SQCLP}(\mathcal{S}_{\mathrm{id}},\mathcal{B},\mathcal{C})=\mbox{QCLP}(\mathcal{B},\mathcal{C})\\\
\end{array}$
with $\mathcal{S}_{\mathrm{id}}$ the identity proximity relation and
$\mathcal{B}$ the qualification domain including just the two classical
boolean values. As explained in the introduction, QCLP and CLP are the targets
of two program transformations to be developed in Section 4. In this brief
section we provide an explicit description of the syntax and semantics of
these two schemes, derived from their behaviour as specializations of SQCLP.
### 3.1 Presentation of the QCLP Scheme
As already explained, the instances of QCLP can be defined by the equation
QCLP($\mathcal{D}$,$\mathcal{C}$) =
SQCLP($\mathcal{S}_{\mathrm{id}}$,$\mathcal{D}$,$\mathcal{C}$). Due to the
admissibility of the parameter triple
$\langle\mathcal{S}_{\mathrm{id}},\mathcal{D},\mathcal{C}\rangle$, the
qualification domain $\mathcal{D}$ must be (existentially) expressible in the
constraint domain $\mathcal{C}$. Technically, the QCLP scheme can be seen as a
common extension of the classical CLP scheme for Constraint Logic Programming
[JL87, JMM+98] and the QLP scheme for Qualified Logic Programming originally
introduced in [RR08]. Intuitively, QCLP programming behaves like SQCLP
programming, except that proximity information other than the identity is not
available for proving equalities.
Program clauses and observable qc-atoms in QCLP are defined in the same way as
in SQCLP. The library program $\mathcal{P}_{\\!s}$ in Figure 1 becomes a
$\mbox{QCLP}(\,\mathcal{U},\mathcal{R})$-program $\mathcal{P}^{\prime}_{\\!s}$
just by replacing $\mathcal{S}_{\mathrm{id}}$ for $\mathcal{S}$. Of course,
$\mathcal{P}^{\prime}_{\\!s}$ does not support flexible unification as it was
the case with $\mathcal{P}_{\\!s}$.
As explained in Subsection 2.4, the proof system consisting of the three
displayed in Figure 2 characterizes the declarative semantics of a given
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program $\mathcal{P}$. In
the particular case $\mathcal{S}=\mathcal{S}_{\mathrm{id}}$, the inference
rules specialize to those displayed in Figure 3, yielding a formal proof
system called _Qualified Constrained Horn Logic_ —in symbols,
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$—which characterizes the declarative
semantics of a given $\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$. Note that rule SQEA depends on a relation $\approx_{\Pi}$
between terms that is defined to behave the same as the specialization of
$\approx_{d,\Pi}$ to the case $\mathcal{S}=\mathcal{S}_{\mathrm{id}}$. It is
easily checked that $t\approx_{\Pi}s$ does not depend on $d$ and holds iff
$\Pi~{}{\models_{\mathcal{C}}}~{}t==s$. Both $\approx_{d,\Pi}$ and
$\approx_{\Pi}$ allow to use the constraints within $\Pi$ when deducing
equations. However,
$c(\overline{t}_{n})\approx_{\Pi}c^{\prime}(\overline{s}_{n})$ never holds in
the case that $c$ and $c^{\prime}$ are not syntactically identical.
QDA $\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{j}\theta\sharp
e_{j}\Leftarrow\Pi~{})_{j=1\ldots m}~{}}{p(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi}$
---
if $(p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m})\in\mathcal{P}$, $\theta$ subst.,
$e_{j}\trianglerighteqslant^{?}w_{j}~{}(1\leq j\leq m)$ and
$d\trianglelefteqslant\bigsqcap_{i=1}^{n}d_{i}\sqcap\alpha\circ\bigsqcap_{j=1}^{m}e_{j}$.
QEA $\displaystyle\frac{}{\quad(t==s)\sharp d\Leftarrow\Pi\quad}$ if
$t\approx_{\Pi}s$. QPA $\displaystyle\frac{}{\quad\kappa\sharp
d\Leftarrow\Pi\quad}$ if $\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$.
Figure 3: Qualified Constrained Horn Logic
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proof trees and the
notations related to them can be naturally specialized to
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$. In particular, we will use the
notation $\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \varphi$
(resp. $\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!k}\ \varphi$)
to indicate that the qc-atom $\varphi$ can be inferred in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ from the program $\mathcal{P}$ (resp.
it can be inferred by using exactly $k$ QDA inference steps). Clearly, Theorem
2.1 specializes to QCHL yielding the following result that is stated here for
convenience:
###### Theorem 3.1 (Logical characterization of least program models in QCHL)
For any $\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program $\mathcal{P}$, its
least model can be characterized as:
$\mathcal{M}_{\mathcal{P}}=\\{\varphi\mid\varphi\mbox{ is an observable
defined qc-atom and }\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
\varphi\\}\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}$
Concerning goals and their solutions, their specialization to the particular
case $\mathcal{S}=\mathcal{S}_{\mathrm{id}}$ leaves the syntax of goals $G$
unaffected and leads to the following definition, almost identical to
Definition 2.2:
###### Definition 3.1 (Possible Answers and Goal Solutions in QCLP)
Assume a given $\mbox{QCLP}(\mathcal{S},\mathcal{D}){\mathcal{C}}$-program
$\mathcal{P}$ and a goal $G:(~{}A_{i}\sharp
W_{i},W_{i}\trianglerighteqslant^{?}\\!\beta_{i}~{})_{i=1\ldots m}$. Then:
1. 1.
Possible answers $ans=\langle\sigma,\mu,\Pi\rangle$ for $G$ and their
qualification levels are defined as in SQCLP (see Definition 2.2(1)).
2. 2.
A solution for $G$ is any possible answer $\langle\sigma,\mu,\Pi\rangle$ that
verifies the conditions in Definition 2.2(2), except that the requirement
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
A_{i}\sigma\sharp W_{i}\mu\Leftarrow\Pi$ used in Definition 2.2 for SQCLP
becomes now $\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
A_{i}\sigma\sharp W_{i}\mu\Leftarrow\Pi$ for QCLP. The set of all solutions
for $G$ is noted $\mbox{Sol}_{\mathcal{P}}(G)$.
3. 3.
The subset $\mbox{GSol}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$
of all ground solutions is defined exactly as in Definition 2.2(3).
4. 4.
The subsumption relation between a ground solution
$\langle\eta,\rho,\emptyset\rangle\in\mbox{GSol}_{\mathcal{P}}(G)$ and an
arbitrary solution $\langle\sigma,\mu,\Pi\rangle$ is defined exactly as in
Definition 2.2(4). Subsumption in the flexible sense cannot be considered in
QCLP due to the absence of a proximity relation.
Finally, the notion of correct abstract goal solving system for SQCLP given in
Definition 2.3 specializes to QCLP with only one minor modification: weak
completeness in the flexible sense cannot be considered here, due to the
absence of a proximity relation. Therefore, we state the following definition:
###### Definition 3.2 (Correct Abstract Goal Solving Systems for QCLP)
An abstract goal solving system for $\mbox{QCLP}(\mathcal{D},\mathcal{C})$ is
any device $\mathcal{CA}$ that takes a program $\mathcal{P}$ and a goal $G$ as
input and yields a set $\mathcal{CA}_{\mathcal{P}}(G)$ of possible answers
$\langle\sigma,\mu,\Pi\rangle$ (called computed answers) as output. Moreover:
1. 1.
$\mathcal{CA}$ is called sound iff every computed answer is a solution, i.e.
$\mathcal{CA}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$.
2. 2.
$\mathcal{CA}$ is called weakly complete iff for every ground solution
$gsol\in\mbox{GSol}_{\mathcal{P}}(G)$ there is some computed answer
$ans\in\mathcal{CA}_{\mathcal{P}}(G)$ such that $ans$ subsumes $gsol$.
3. 3.
$\mathcal{CA}$ is called correct iff it is both sound and weakly complete.
### 3.2 Presentation of the CLP Scheme
As already explained, the instances of CLP can be defined by the equation
CLP($\mathcal{C}$) =
SQCLP($\mathcal{S}_{\mathrm{id}},\mathcal{B},\mathcal{C}$), or equivalently,
CLP($\mathcal{C}$) = QCLP($\mathcal{B},\mathcal{C}$). Due to the fixed choice
$\mathcal{D}=\mathcal{B}$, the only qualification value $d\in
D\setminus\\{\text{{b}}\\}$ available for use as attenuation factor or
threshold value is $d=\text{{t}}$. Therefore, CLP can only include threshold
values equal to ‘?’ and attenuation values equal to the top element
$\text{{t}}=true$ of $\mathcal{B}$. As explained in Section 2, such trivial
threshold and attenuation values can be omitted, and CLP clauses can be
written with the simplified syntax $A\leftarrow B_{1},\ldots,B_{m}$.
Since $\text{{t}}=true$ is the only non-trivial qualification value available
in CLP, qc-atoms $A\sharp d\Leftarrow\Pi$ are always of the form $A\sharp
true\Leftarrow\Pi$ and can be written as $A\Leftarrow\Pi$. Moreover, all the
side conditions for the inference rule QDA in Figure 3 become trivial when
specialized to the case $\mathcal{D}=\mathcal{B}$. Therefore, the
specialization of $\mbox{QCHL}(\mathcal{D},\mathcal{C})$ to the case
$\mathcal{D}=\mathcal{B}$ leads to the formal proof system called _Constrained
Horn Logic_ —in symbols, $\mbox{CHL}(\mathcal{C})$—consisting of the three
inference rules displayed in Figure 4, which characterizes the declarative
semantics of a given $\mbox{CLP}(\mathcal{C})$-program $\mathcal{P}$.
DA
$\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta)\Leftarrow\Pi~{})_{i=1\ldots
n}\quad(~{}B_{j}\theta\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{p(\overline{t^{\prime}}_{n})\Leftarrow\Pi}$
---
if $(p(\overline{t}_{n})\leftarrow B_{1},\ldots,B_{m})\in\mathcal{P}$ and
$\theta$ subst.
EA $\displaystyle\frac{}{\quad(t==s)\Leftarrow\Pi\quad}$ if $t\approx_{\Pi}s$.
PA $\displaystyle\frac{}{\quad\kappa\Leftarrow\Pi\quad}$ if
$\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$.
Figure 4: Constrained Horn Logic
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof trees and the notations related
to them can be naturally specialized to $\mbox{CHL}(\mathcal{C})$. In
particular, we will use the notation $\mathcal{P}\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ \varphi$ (resp. $\mathcal{P}\
{\vdash}_{\\!\mathcal{C}}^{\\!k}\ \varphi$) to indicate that the qc-atom
$\varphi$ can be inferred in $\mbox{CHL}(\mathcal{C})$ from the program
$\mathcal{P}$ (resp. it can be inferred by using exactly $k$ DA inference
steps). Clearly, Theorem 3.1 specializes to CHL yielding the following result
that is stated here for convenience:
###### Theorem 3.2 (Logical characterization of least program models in CHL)
For any $\mbox{CLP}(\mathcal{C})$-program $\mathcal{P}$, its least model can
be characterized as:
$\mathcal{M}_{\mathcal{P}}=\\{\varphi\mid\varphi\mbox{ is an observable
defined qc-atom and }\mathcal{P}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
\varphi\\}\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}$
Concerning goals and their solutions, their specialization to the scheme CLP
leads to the following definition:
###### Definition 3.3 (Goals and their Solutions in CLP)
Assume a given $\mbox{CLP}(\mathcal{C})$-program $\mathcal{P}$. Then:
1. 1.
Goals for $\mathcal{P}$ have the form $G:\,A_{1},\ldots,A_{m}$, abbreviated as
$(A_{i})_{i=1\ldots m}$, where $A_{i}~{}(1\leq i\leq m)$ are atoms.
2. 2.
A possible answer for a goal $G$ is any pair $ans=\langle\sigma,\Pi\rangle$
such that $\sigma$ is a $\mathcal{C}$-substitution and $\Pi$ is a satisfiable
and finite set of atomic $\mathcal{C}$-constraints.
3. 3.
A possible answer $\langle\sigma,\Pi\rangle$ is called a solution for $G$ iff
$\mathcal{P}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\ A_{i}\sigma\Leftarrow\Pi$ holds
for all $i=1\ldots m$. The set of all solutions for $G$ is noted
$\mbox{Sol}_{\mathcal{P}}(G)$.
4. 4.
A solution $\langle\eta,\Pi\rangle$ for $G$ is called ground iff
$\Pi=\emptyset$ and $\eta\in\mbox{Val}_{\mathcal{C}}$ is a variable valuation
such that $A_{i}\eta$ is a ground atom for all $i=1\ldots m$. The set of all
ground solutions for $G$ is noted $\mbox{GSol}_{\mathcal{P}}(G)$. Obviously,
$\mbox{GSol}_{\mathcal{P}}(G)\subseteq\mbox{Sol}_{\mathcal{P}}(G)$.
5. 5.
A ground solution
$\langle\eta,\emptyset\rangle\in\mbox{GSol}_{\mathcal{P}}(G)$ is subsumed by
$\langle\sigma,\Pi\rangle$ iff there is some
$\nu\in\mbox{Sol}_{\mathcal{C}}(\Pi)$ s.t. $\eta=_{\mathrm{var}(G)}\sigma\nu$.
The notion of correct abstract goal solving system for SQCLP given in
Definition 3.2 specializes to CLP with a minor change, namely: computed
answers are pairs $\langle\sigma,\Pi\rangle$. Formally, the definition for CLP
is as follows:
###### Definition 3.4 (Correct Abstract Goal Solving Systems for CLP)
A goal solving system for $\mbox{CLP}(\mathcal{C})$ is any device
$\mathcal{CA}$ that takes a program $\mathcal{P}$ and a goal $G$ as input and
yields a set $\mathcal{CA}_{\mathcal{P}}(G)$ of possible answers
$\langle\sigma,\Pi\rangle$ (called computed answers) as output. Moreover,
soundness, weak completeness and weak correctness of $\mathcal{CA}$ are
defined exactly as in Definition 3.2.
We close this Subsection with a technical lemma that will be useful for
proving some results in Subsection 4.2:
###### Lemma 3.1
Assume an existential $\mathcal{C}$-constraint $\pi(\overline{X}_{n})=\exists
Y_{1}\ldots\exists Y_{k}(B_{1}\land\ldots\land B_{m})$ with free variables
$\overline{X}_{n}$ and a given $\mbox{CLP}(\mathcal{C})$-program $\mathcal{P}$
including the clause $C:\,p(\overline{X}_{n})\leftarrow B_{1},\ldots,B_{m}$,
where $p\in DP^{n}$ does not occur at the head of any other clause of
$\mathcal{P}$. Then, for any n-tuple $\overline{t}_{n}$ of $\mathcal{C}$-terms
and any finite and satisfiable $\Pi\subseteq\mbox{Con}_{\mathcal{C}}$, one
has:
1. 1.
$\mathcal{P}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(p(\overline{t}_{n})\Leftarrow\Pi)\Longrightarrow\Pi\models_{\mathcal{C}}\pi(\overline{t}_{n})$,
where $\pi(\overline{t}_{n})$ stands for the result of applying the
substitution $\\{\overline{X}_{n}\mapsto\overline{t}_{n}\\}$ to
$\pi(\overline{X}_{n})$.
2. 2.
The opposite implication
$\Pi\models_{\mathcal{C}}\pi(\overline{t}_{n})\Longrightarrow\mathcal{P}\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ (p(\overline{t}_{n})\Leftarrow\Pi)$ holds if
$\overline{t}_{n}$ is a ground term tuple. Note that for ground
$\overline{t}_{n}$ the constraint entailment
$\Pi\models_{\mathcal{C}}\pi(\overline{t}_{n})$ simply means that
$\pi(\overline{t}_{n})$ is true in $\mathcal{C}$.
* Proof
* We prove each item separately:
1. 1.
Assume $\mathcal{P}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(p(\overline{t}_{n})\Leftarrow\Pi)$. Note that $C$ is the only clause for $p$
in $\mathcal{P}$ and that each atom $B_{j}$ in $C$’s body is an atomic
constraint. Therefore, the $\mbox{CHL}(\mathcal{C})$ proof must use a DA step
based on an instance $C\theta$ of clause $C$ such that
$\Pi\models_{\mathcal{C}}t_{i}==X_{i}\theta$ holds for all $1\leq i\leq n$ and
$\Pi\models B_{j}\theta$ holds for all $1\leq j\leq m$. These conditions and
the syntactic form of $\pi(\overline{X}_{n})$ obviously imply
$\Pi\models_{\mathcal{C}}\pi(\overline{t}_{n})$.
2. 2.
Assume now $\Pi\models_{\mathcal{C}}\pi(\overline{t}_{n})$ and
$\overline{t}_{n}$ ground. Then $\pi(\overline{t}_{n})$ is true in
$\mathcal{C}$, and due to the syntactic form of $\pi(\overline{X}_{n})$, there
must be some substitution $\theta$ such that $X_{i}\theta=t_{i}$ (syntactic
identity) for all $1\leq i\leq n$ and $B_{j}\theta$ is ground and true in
$\mathcal{C}$ for all $1\leq j\leq m$. Trivially,
$\Pi\models_{\mathcal{C}}t_{i}==X_{i}\theta$ holds for all $1\leq i\leq n$ and
$\Pi\models_{\mathcal{C}}B_{j}\theta$ also holds for all $1\leq j\leq m$.
Then, it is obvious that $\mathcal{P}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(p(\overline{t}_{n})\Leftarrow\Pi)$ can be proved by using a DA step based on
the instance $C\theta$ of clause $C$.
We remark that the second item of the previous lemma can fail if
$\overline{t}_{n}$ is not ground. This can be checked by presenting a
counterexample based on the constraint domain $\mathcal{R}$, using the syntax
for $\mathcal{R}$-constraints explained in [RR10TR]. Consider the existential
$\mathcal{R}$-constraint $\pi(X)=\exists Y(op_{+}(Y,Y,X))$, and a
$\mbox{CLP}(\mathcal{R})$-program $\mathcal{P}$ including the clause
$C:\,p(X)\leftarrow op_{+}(Y,Y,X)$ and no other occurrence of the defined
predicate symbol $p$. Consider also $\Pi=\\{cp_{\geq}(X,0.0)\\}$ and $t=X$.
Then $\Pi\models_{\mathcal{R}}\pi(X)$ is obviously true, because any real
number $x\geq 0.0$ satisfies $\exists Y(op_{+}(Y,Y,x))$ in $\mathcal{R}$.
However, there is no $\mathcal{R}$-term $s$ such that
$\Pi\models_{\mathcal{R}}op_{+}(s,s,X)$, and therefore there is no instance
$C\theta$ of clause $C$ that can be used to prove $\mathcal{P}\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ (p(X)\Leftarrow\Pi)$ by applying a DA step.
## 4 Implementation by Program Transformation
The purpose of this section is to introduce a program transformation that
transforms $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ programs and
goals into semantically equivalent $\mbox{CLP}(\mathcal{C})$ programs and
goals. This transformation is performed as the composition of the two
following specific transformations:
1. 1.
elimS — Eliminates the proximity relation $\mathcal{S}$ of arbitrary
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ programs and goals,
producing equivalent $\mbox{QCLP}(\mathcal{D},\mathcal{C})$ programs and
goals.
2. 2.
elimD — Eliminates the qualification domain $\mathcal{D}$ of arbitrary
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$ programs and goals, producing
equivalent $\mbox{CLP}(\mathcal{C})$ programs and goals.
Thus, given a $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$—resp. $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-goal
$G$—, the composition of the two transformations will produce an equivalent
$\mbox{CLP}(\mathcal{C})$-program
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(\mathcal{P}))$—resp.
$\mbox{CLP}(\mathcal{C})$-goal
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(G))$—.
###### Example 4.1 (Running example:
$\mbox{SQCLP}(\mathcal{S}_{r},\,\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-program
$\mathcal{P}_{r}$)
As a running example for this section, consider the
$\mbox{SQCLP}(\mathcal{S}_{r},\,\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-program
$\mathcal{P}_{r}$ as follows:
$R_{1}$ | famous(sha) $\xleftarrow{(0.9,1)}$
---|---
$R_{2}$ | wrote(sha, kle) $\xleftarrow{(1,1)}$
$R_{3}$ | wrote(sha, hamlet) $\xleftarrow{(1,1)}$
$R_{4}$ | good_work(G) $\xleftarrow{(0.75,3)}$ famous(A)#(0.5,100), authored(A, G)
$S_{1}$ | $\mathcal{S}_{r}$(wrote, authored) = $\mathcal{S}_{r}$(authored, wrote) = (0.9,0)
$S_{2}$ | $\mathcal{S}_{r}$(kle, kli) = $\mathcal{S}_{r}$(kli, kle) = (0.8,2)
where the constants $shakespeare$, $king\\_lear$ and $king\\_liar$ have been
respectively replaced, for clarity purposes in the subsequent examples, by
$sha$, $kle$ and $kli$.
In addition, consider the
$\mbox{SQCLP}(\mathcal{S}_{r},\,\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-goal
$G_{r}$ as follows:
good_work(X)#W $\talloblong$ W $\trianglerighteqslant^{?}$ (0.5,10)
We will illustrate the two transformation by showing, in subsequent examples,
the program clauses of $\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r})$ and
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r}))$
and the goals $\mathrm{elim}_{\mathcal{S}}(G_{r})$ and
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(G_{r}))$.
In the following subsections we explain both transformations in detail and we
show that they can be used to specify abstract goal solving systems for SQCLP.
### 4.1 Transforming SQCLP into QCLP
In this subsection we assume that the triple
$\langle\mathcal{S},\mathcal{D},\mathcal{C}\rangle$ is admissible. In the
sequel we say that a defined predicate symbol $p\in DP^{n}$ is affected by a
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program $\mathcal{P}$ iff
$\mathcal{S}(p,p^{\prime})\neq\text{{b}}$ for some $p^{\prime}\\!$ occurring
in $\mathcal{P}$. We also say that an atom $A$ is relevant for $\mathcal{P}$
iff some of the three following cases hold: a) $A$ is an equation $t==s$; b)
$A$ is a primitive atom $\kappa$; or c) $A$ is a defined atom
$p(\overline{t}_{n})$ such that $p$ is affected by $\mathcal{P}$.
As a first step towards the definition of the first program transformation
elimS, we define a set $EQ_{\mathcal{S}}$ of
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$ program clauses that emulates the
behaviour of equations in $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$.
The following definition assumes that the binary predicate symbol $\sim\ \in
DP^{2}$ (used in infix notation) and the nullary predicate symbols
$\mbox{pay}_{\lambda}\in DP^{0}$ are not affected by $\mathcal{P}$.
###### Definition 4.1
We define $EQ_{\mathcal{S}}$ as the following
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program:
$\begin{array}[]{l@{\hspace{0mm}}c@{\hspace{0mm}}l}EQ_{\mathcal{S}}&~{}{=_{\mathrm{def}}{}}&\\{~{}X\sim
Y\xleftarrow{\text{{t}}}(X==Y)\sharp?~{}\\}\\\ &&~{}\bigcup~{}\\{~{}u\sim
u^{\prime}\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?\mid
u,u^{\prime}\in B_{\mathcal{C}}\mbox{ and
}\mathcal{S}(u,u^{\prime})=\lambda\neq\text{{b}}~{}\\}\\\
&&~{}\bigcup~{}\\{~{}c(\overline{X}_{n})\sim
c^{\prime}(\overline{Y}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?,\
((X_{i}\sim Y_{i})\sharp?)_{i=1\ldots n}\mid c,c^{\prime}\in DC^{n}\\\
&&\qquad\mbox{and }\mathcal{S}(c,c^{\prime})=\lambda\neq\text{{b}}~{}\\}\\\
&&~{}\bigcup~{}\\{~{}\mbox{pay}_{\lambda}\xleftarrow{\lambda}\ \mid\exists
x,y\in S\mbox{ such that }\mathcal{S}(x,y)=\lambda\neq\text{{b}}~{}\\}.\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}\\\
\end{array}$
The following lemma shows the relation between the semantics of equations in
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ and the behaviour of the
binary predicate symbol ‘$\sim$’ defined by $EQ_{\mathcal{S}}$ in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$.
###### Lemma 4.1
Consider any two arbitrary terms $t$ and $s$; $EQ_{\mathcal{S}}$ defined as in
Definition 4.1; and a satisfiable finite set $\Pi$ of
$\mathcal{C}$-constraints. Then, for every $d\in D\setminus\\{\text{{b}}\\}$:
$t\approx_{d,\Pi}s\Longleftrightarrow EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
* Proof
* We separately prove each implication.
[$\Longrightarrow$] Assume $t\approx_{d,\Pi}s$. Then, there are two terms
$\hat{t}$, $\hat{s}$ such that:
$(1)~{}t\approx_{\Pi}\hat{t}\qquad(2)~{}s\approx_{\Pi}\hat{s}\qquad(3)~{}\hat{t}\approx_{d}\hat{s}$
We use structural induction on the form of the term $\hat{t}$.
* –
$\hat{t}=Z$, $Z\in\mathcal{V}\\!ar$. From (3) we have $\hat{s}=Z$. Then (1)
and (2) become $t\approx_{\Pi}Z$ and $s\approx_{\Pi}Z$, therefore
$t\approx_{\Pi}s$. Now $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
can be proved with a proof tree rooted by a QDA step of the form:
$\displaystyle\frac{~{}(t==X\theta)\sharp\text{{t}}\Leftarrow\Pi\qquad(s==Y\theta)\sharp\text{{t}}\Leftarrow\Pi\qquad(X==Y)\theta\sharp\text{{t}}\Leftarrow\Pi~{}}{(t\sim
s)\sharp d\Leftarrow\Pi}$
using the clause $X\sim Y\xleftarrow{\text{{t}}}(X==Y)\sharp?\in
EQ_{\mathcal{S}}$ instantiated by the substitution $\theta=\\{X\\!\mapsto t,\
Y\\!\mapsto s\\}$. Therefore the three premises can be derived from
$EQ_{\mathcal{S}}$ with QEA steps since $t\approx_{\Pi}t$, $s\approx_{\Pi}s$
and $t\approx_{\Pi}s$, respectively. Checking the side conditions of all
inference steps is straightforward.
* –
$\hat{t}=u$, $u\in B_{\mathcal{C}}$. From (3) we have $\hat{s}=u^{\prime}$ for
some $u^{\prime}\in B_{\mathcal{C}}$ such that
$d\trianglelefteqslant\lambda=\mathcal{S}(u,u^{\prime})$. Then (1) and (2)
become $t\approx_{\Pi}u$ and $s\approx_{\Pi}u^{\prime}$, which allow to build
a proof of $EQ_{\mathcal{S}}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
(t\sim s)\sharp d\Leftarrow\Pi$ by means of a QDA step using the clause $u\sim
u^{\prime}\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?$.
* –
$\hat{t}=c$, $c\in DC^{0}$. From (3) we have $\hat{s}=c^{\prime}$ for some
$c^{\prime}\in DC^{0}$ such that
$d\trianglelefteqslant\lambda=\mathcal{S}(c,c^{\prime})$. Then (1) and (2)
become $t\approx_{\Pi}c$ and $s\approx_{\Pi}c^{\prime}$, which allow us to
build a proof of $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
by means of a QDA step using the clause $c\sim
c^{\prime}\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?$.
* –
$\hat{t}=c(\overline{t}_{n})$, $c\in DC^{n}$ with $n>0$. In this case, and
because of (3), we can assume $\hat{s}=c^{\prime}(\overline{s}_{n})$ for some
$c^{\prime}\in DC^{n}$ satisfying $d\trianglelefteqslant
d_{0}~{}{=_{\mathrm{def}}}~{}\mathcal{S}(c,c^{\prime})$ and
$d\trianglelefteqslant d_{i}~{}{=_{\mathrm{def}}}~{}\mathcal{S}(t_{i},s_{i})$
for $i=1\dots n$. Then $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
with a proof tree rooted by a QDA step of the form:
$\displaystyle\frac{~{}\begin{array}[]{l@{\hspace{1cm}}l}(t==c(\overline{t}_{n}))\sharp\text{{t}}\Leftarrow\Pi\hfil\hskip
28.45274pt&\mbox{pay}_{d_{0}}\sharp d_{0}\Leftarrow\Pi\\\
(s==c^{\prime}(\overline{s}_{n}))\sharp\text{{t}}\Leftarrow\Pi\hfil\hskip
28.45274pt&(~{}(t_{i}\sim s_{i})\sharp d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\
\end{array}~{}}{(t\sim s)\sharp d\Leftarrow\Pi}$
using the $EQ_{\mathcal{S}}$ clause $C:c(\overline{X}_{n})\sim
c^{\prime}(\overline{Y}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{d_{0}}\sharp?,((X_{i}\sim
Y_{i})\sharp?)_{i=1\ldots n}$ instantiated by the substitution
$\theta=\\{X_{1}\mapsto t_{1},\ Y_{1}\mapsto s_{1},\ \dots,\ X_{n}\mapsto
t_{n},\ Y_{n}\mapsto s_{n}\\}$. Note that $C$ has attenuation factor t and
threshold values $?$ at the body. Therefore, the side conditions of the QDA
step boil down to $d\trianglelefteqslant d_{i}~{}(1\leq i\leq n)$ which are
true by assumption. It remains to prove that each premise of the QDA step can
be derived from $EQ_{\mathcal{S}}$ in QCHL($\mathcal{D},\mathcal{C}$):
* *
$EQ_{\mathcal{S}}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
(t==c(\overline{t}_{n}))\sharp\text{{t}}\Leftarrow\Pi$ and $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
(s==c^{\prime}(\overline{s}_{n}))\sharp\text{{t}}\Leftarrow\Pi$ are trivial
consequences of $t\approx_{\Pi}c(\overline{t}_{n})$ and
$s\approx_{\Pi}c^{\prime}(\overline{s}_{n})$, respectively. In both cases, the
QCHL($\mathcal{D}$,$\mathcal{C}$) proofs consist of one single QEA step.
* *
$EQ_{\mathcal{S}}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
\mbox{pay}_{d_{0}}\sharp d_{0}\Leftarrow\Pi$ can be proved using the clause
$\mbox{pay}_{d_{0}}\\!\xleftarrow{d_{0}}\ \in EQ_{\mathcal{S}}$ in one single
QDA step.
* *
$EQ_{\mathcal{S}}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t_{i}\sim
s_{i})\sharp d_{i}\Leftarrow\Pi$ for ${i=1\ldots n}$. For each $i$, we observe
that $t_{i}\approx_{d_{i},\Pi}s_{i}$ holds because of $\hat{t}_{i}=t_{i}$,
$\hat{s}_{i}=s_{i}$ which satisfy $t_{i}\approx_{\Pi}\hat{t}_{i}$,
$s_{i}\approx_{\Pi}\hat{s}_{i}$ and $\hat{t}_{i}\approx_{d_{i}}\hat{s}_{i}$.
Since $\hat{t}_{i}=t_{i}$ is a subterm of $\hat{t}=c(\overline{t}_{n})$, the
inductive hypothesis can be applied.
[$\Longleftarrow$] Let $T$ be a $\mbox{QCHL}(\mathcal{D},\mathcal{C})$-proof
tree witnessing $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$.
We prove $t\approx_{d,\Pi}s$ reasoning by induction on the number $n=\|T\|$ of
nodes in $T$ that represent conclusions of QDA inference steps. Note that all
the program clauses belonging to $EQ_{\mathcal{S}}$ define either the binary
predicate symbol ‘$\sim$’ or the nullary predicates $\mbox{pay}_{\lambda}$.
Basis ($n=1$).
In this case we have for the QDA inference step that there can be used three
possible $EQ_{\mathcal{S}}$ clauses:
1. 1.
The program clause is $X\sim Y\xleftarrow{\text{{t}}}(X==Y)\sharp?$. Then the
QDA inference step must be of the form:
$\displaystyle\frac{~{}(t==t^{\prime})\sharp
d_{1}\Leftarrow\Pi\quad(s==s^{\prime})\sharp
d_{2}\Leftarrow\Pi\quad(t^{\prime}==s^{\prime})\sharp
e_{1}\Leftarrow\Pi~{}}{(t\sim s)\sharp d\Leftarrow\Pi}$
with $d\trianglelefteqslant d_{1}\sqcap d_{2}\sqcap e_{1}$. The proof of the
three premises must use the QEA inference rule. Because of the conditions of
this inference rule we have $t\approx_{\Pi}t^{\prime}$,
$s\approx_{\Pi}s^{\prime}$ and $t^{\prime}\approx_{\Pi}s^{\prime}$. Therefore
$t\approx_{\Pi}s$ is clear. Then $t\approx_{d,\Pi}s$ holds by taking
$\hat{t}=\hat{s}=t$ because, trivially, $t\approx_{\Pi}\hat{t}$,
$s\approx_{\Pi}\hat{s}$ and $\hat{t}\approx_{d}\hat{s}$.
2. 2.
The program clause is $u\sim
u^{\prime}\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?$ with
$u,u^{\prime}\in B_{\mathcal{C}}$ such that
$\mathcal{S}(u,u^{\prime})=\lambda\neq\text{{b}}$. The QDA inference step must
be of the form:
$\displaystyle\frac{~{}(t==u)\sharp
d_{1}\Leftarrow\Pi\quad(s==u^{\prime})\sharp
d_{2}\Leftarrow\Pi\quad\mbox{pay}_{\lambda}\sharp
e_{1}\Leftarrow\Pi~{}}{(t\sim s)\sharp d\Leftarrow\Pi}$
with $d\trianglelefteqslant d_{1}\sqcap d_{2}\sqcap e_{1}$. Due to the forms
of the QEA inference rule and the $EQ_{\mathcal{S}}$ clause
$\mbox{pay}_{\lambda}\xleftarrow{\lambda}$, we can assume without loss of
generality that $d_{1}=d_{2}=\text{{t}}$ and $e_{1}=\lambda$. Therefore
$d\trianglelefteqslant\lambda$. Moreover, the
QCHL($\mathcal{D}$,$\mathcal{C}$) proofs of the first two premises must use
QEA inferences. Consequently we have $t\approx_{\Pi}u$ and
$s\approx_{\Pi}u^{\prime}$. These facts and $u\approx_{d}u^{\prime}$ imply
$t\approx_{d,\Pi}s$.
3. 3.
The program clause is $c\sim
c^{\prime}\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?$ with
$c,c^{\prime}\in DC^{0}$ such that
$\mathcal{S}(c,c^{\prime})=\lambda\neq\text{{b}}$. The QDA inference step must
be of the form:
$\displaystyle\frac{~{}(t==c)\sharp
d_{1}\Leftarrow\Pi\quad(s==c^{\prime})\sharp
d_{2}\Leftarrow\Pi\quad\mbox{pay}_{\lambda}\sharp
e_{1}\Leftarrow\Pi~{}}{(t\sim s)\sharp d\Leftarrow\Pi}$
with $d\trianglelefteqslant d_{1}\sqcap d_{2}\sqcap e_{1}$. Due to the forms
of the QEA inference rule and the $EQ_{\mathcal{S}}$ clause
$\mbox{pay}_{\lambda}\xleftarrow{\lambda}$, we can assume without loss of
generality that $d_{1}=d_{2}=\text{{t}}$ and $e_{1}=\lambda$. Therefore
$d\trianglelefteqslant\lambda$. Moreover, the
QCHL($\mathcal{D}$,$\mathcal{C}$) proofs of the first two premises must use
QEA inferences. Consequently we have $t\approx_{\Pi}c$ and
$s\approx_{\Pi}c^{\prime}$. These facts and $c\approx_{d}c^{\prime}$ imply
$t\approx_{d,\Pi}s$.
Inductive step ($n>1$).
In this case $t$ and $s$ must be of the form $t=c(\overline{t}_{n})$ and
$s=c^{\prime}(\overline{s}_{n})$. The $EQ_{\mathcal{S}}$ clause used in the
QDA inference step at the root must be of the form:
$c(\overline{X}_{n})\sim
c^{\prime}(\overline{Y}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{d_{0}}\sharp?,\
((X_{i}\sim Y_{i})\sharp?)_{i=1\ldots n}$
with $\mathcal{S}(c,c^{\prime})=d_{0}\neq\text{{b}}$. The inference step at
the root will be:
$\displaystyle\frac{~{}\begin{array}[]{l@{\hspace{1cm}}l}(t==c(\overline{t}_{n}))\sharp
d_{1}\Leftarrow\Pi\hfil\hskip 28.45274pt&pay_{d_{0}}\sharp
e_{0}\Leftarrow\Pi\\\ (s==c^{\prime}(\overline{s}_{n}))\sharp
d_{2}\Leftarrow\Pi\hfil\hskip 28.45274pt&(~{}(t_{i}\sim s_{i})\sharp
e_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\ \end{array}~{}}{(t\sim s)\sharp
d\Leftarrow\Pi}$
with $d\trianglelefteqslant d_{1}\sqcap d_{2}\sqcap\bigsqcap_{i=0}^{n}e_{i}$.
Due to the forms of the $EQ_{\mathcal{S}}$ clause
$\mbox{pay}_{d_{0}}\\!\xleftarrow{d_{0}}$ and the QEA inference rule there is
no loss of generality in assuming $d_{1}=d_{2}=\text{{t}}$ and $e_{0}=d_{0}$,
therefore we have $d\trianglelefteqslant d_{0}\sqcap\bigsqcap_{i=1}^{n}e_{i}$.
By the inductive hypothesis $t_{i}\approx_{e_{i},\Pi}s_{i}~{}(1\leq i\leq n)$,
i.e. there are constructor terms $\hat{t}_{i}$, $\hat{s}_{i}$ such that
$t_{i}\approx_{\Pi}\hat{t_{i}}$, $s_{i}\approx_{\Pi}\hat{s}_{i}$ and
$\hat{t}_{i}\approx_{e_{i}}\hat{s}_{i}$ for $i=1\ldots n$. Thus, we can build
$\hat{t}=c(\hat{t}_{1},\ldots,\hat{t}_{n})$ and
$\hat{s}=c^{\prime}(\hat{s}_{1},\ldots,\hat{s}_{n})$ having
$t\approx_{d,\Pi}s$ because:
* –
$t\approx_{\Pi}\hat{t}$, i.e.
$c(\overline{t}_{n})\approx_{\Pi}c(\overline{\hat{t}}_{n})$, by decomposition
since $t_{i}\approx_{\Pi}\hat{t}_{i}$.
* –
$s\approx_{\Pi}\hat{s}$, i.e.
$c^{\prime}(\overline{s}_{n})\approx_{\Pi}c^{\prime}(\overline{\hat{s}}_{n})$,
again by decomposition since $s_{i}\approx_{\Pi}\hat{s}_{i}$.
* –
$\hat{t}\approx_{d}\hat{s}$, since $d\trianglelefteqslant
d_{0}\sqcap\bigsqcap_{i=1}^{n}e_{i}\trianglelefteqslant\mathcal{S}(c,c^{\prime})\sqcap\bigsqcap_{i=1}^{n}\mathcal{S}(\hat{t}_{i},\hat{s}_{i})=\mathcal{S}(\hat{t},\hat{s})\enspace.\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}$
We are now ready to define elimS acting over programs and goals.
###### Definition 4.2
Assume a $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$ and a $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-goal
$G$ for $\mathcal{P}$ whose atoms are all relevant for $\mathcal{P}$. Then we
define:
1. 1.
For each atom $A$, let $A_{\sim}$ be $t\sim s$ if $A:t==s$; otherwise let
$A_{\sim}$ be $A$.
2. 2.
For each clause
$C:(p(\overline{t}_{n})\xleftarrow{\alpha}\overline{B})\in\mathcal{P}$ let
$\hat{\mathcal{C}}_{\mathcal{S}}$ be the set of
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$ clauses consisting of:
* —
The clause
$\hat{C}:(\widehat{p}_{C}(\overline{t}_{n})\xleftarrow{\alpha}\overline{B}_{\sim})$,
where $\widehat{p}_{C}\in DP^{n}$ is not affected by $\mathcal{P}$ (chosen in
a different way for each $C$) and $\overline{B}_{\sim}$ is obtained from
$\overline{B}$ by replacing each atom $A$ occurring in $\overline{B}$ by
$A_{\sim}$.
* —
A clause
$p^{\prime}(\overline{X}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{\lambda}\sharp?,\
((X_{i}\sim t_{i})\sharp?)_{i=1\ldots n},\
\widehat{p}_{C}(\overline{t}_{n})\sharp?$ for each $p^{\prime}\in DP^{n}$ such
that $\mathcal{S}(p,p^{\prime})=\lambda\neq\text{{b}}$. Here,
$\overline{X}_{n}$ must be chosen as $n$ pairwise different variables not
occurring in the clause $C$.
3. 3.
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ is the
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program
$EQ_{\mathcal{S}}\cup\hat{\mathcal{P}}_{\mathcal{S}}$ where
$\hat{\mathcal{P}}_{\mathcal{S}}~{}{=_{\mathrm{def}}}~{}\bigcup_{C\in\mathcal{P}}\hat{\mathcal{C}}_{\mathcal{S}}$.
4. 4.
$\mathrm{elim}_{\mathcal{S}}(G)$ is the
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$-goal $G_{\sim}$ obtained from $G$ by
replacing each atom $A$ occurring in $G$ by $A_{\sim}$.
The following example illustrates the transformation elimS.
###### Example 4.2 (Running example:
$\mbox{QCLP}(\mathcal{U}\\!\otimes\\!\mathcal{W},\,\mathcal{R})$-program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r})$)
Consider the
$\mbox{SQCLP}(\mathcal{S}_{r},\,\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-program
$\mathcal{P}_{r}$ and the goal $G_{r}$ for $\mathcal{P}_{r}$ as presented in
Example 4.1. The transformed
$\mbox{QCLP}(\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r})$ is as follows:
$\hat{R}_{1}$ | f̂amous${}_{R_{1}}$(sha) $\xleftarrow{(0.9,1)}$
---|---
$R_{1.1}$ | famous(X) $\leftarrow$ pay${}_{\text{{t}}}$, X$\sim$sha, f̂amous${}_{R_{1}}$(sha)
$\hat{R}_{2}$ | ŵrote${}_{R_{2}}$(sha, kle) $\xleftarrow{(1,1)}$
$R_{2.1}$ | wrote(X, Y) $\leftarrow$ pay${}_{\text{{t}}}$, X$\sim$sha, Y$\sim$kle, ŵrote${}_{R_{2}}$(sha, kle)
$R_{2.2}$ | authored(X, Y) $\leftarrow$ pay(0.9,0), X$\sim$sha, Y$\sim$kle, ŵrote${}_{R_{2}}$(sha, kle)
$\hat{R}_{3}$ | ŵrote${}_{R_{3}}$(sha, hamlet) $\xleftarrow{(1,1)}$
$R_{3.1}$ | wrote(X, Y) $\leftarrow$ pay${}_{\text{{t}}}$, X$\sim$sha, Y$\sim$hamlet, ŵrote${}_{R_{3}}$(sha, hamlet)
$R_{3.2}$ | authored(X, Y) $\leftarrow$ pay(0.9,0), X$\sim$sha, Y$\sim$hamlet, ŵrote${}_{R_{3}}$(sha, hamlet)
$\hat{R}_{4}$ | ĝood_work${}_{R_{4}}$(G) $\xleftarrow{(0.75,3)}$ famous(A)#(0.5,100), authored(A, G)
$R_{4.1}$ | good_work(X) $\leftarrow$ pay${}_{\text{{t}}}$, X$\sim$G, ĝood_work${}_{R_{4}}$(G)
% Program clauses for $\sim$: | % Program clauses for pay:
---|---
X $\sim$Y $\leftarrow$ X==Y | pay${}_{\text{{t}}}$ $\leftarrow$
kle $\sim$ kli $\leftarrow$ pay(0.8,2) | pay(0.9,0) $\xleftarrow{(0.9,0)}$
$[\ldots]$ | pay(0.8,2) $\xleftarrow{(0.8,2)}$
Finally, the goal $\mathrm{elim}_{\mathcal{S}}(G_{r})$ for
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r})$ is as follows:
good_work(X)#W $\talloblong$ W $\trianglerighteqslant^{?}$ (0.5,10)
The next theorem proves the semantic correctness of the program
transformation.
###### Theorem 4.1
Consider a $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$, an atom $A$ relevant for $\mathcal{P}$, a qualification value
$d\in D\setminus\\{\text{{b}}\\}$ and a satisfiable finite set of
$\mathcal{C}$-constraints $\Pi$. Then, the following two statements are
equivalent:
1. 1.
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp
d\Leftarrow\Pi$
2. 2.
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A_{\sim}\sharp d\Leftarrow\Pi$
where $A_{\sim}$ is understood as in Definition 4.2(1).
* Proof
* We separately prove each implication.
[1. $\Rightarrow$ 2.] (the transformation is complete). Assume that $T$ is a
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proof tree witnessing
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp
d\Leftarrow\Pi$. We want to show the existence of a
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree $T^{\prime}$ witnessing
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A_{\sim}\sharp d\Leftarrow\Pi$.
We reason by complete induction on $\|T\|$. There are three possible cases
according to the syntactic form of the atom $A$. In each case we argue how to
build the desired proof tree $T^{\prime}$.
— $A$ is a primitive atom $\kappa$. In this case $A_{\sim}$ is also $\kappa$
and $T$ contains only one SQPA inference node. Because of the inference rules
SQPA and QPA, both $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ \kappa\sharp
d\Leftarrow\Pi$ and $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \kappa\sharp d\Leftarrow\Pi$ are
equivalent to $\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$, therefore $T^{\prime}$
trivially contains just one QPA inference node.
— $A$ is an equation $t==s$. In this case $A_{\sim}$ is $t\sim s$ and $T$
contains just one SQEA inference node. We know $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ (t==s)\sharp
d\Leftarrow\Pi$ is equivalent to $t\approx_{d,\Pi}s$ because of the inference
rule SQEA. From this equivalence follows $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
due to Lemma 4.1 and hence $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
by construction of $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$. In this case,
$T^{\prime}$ will be a proof tree rooted by a QDA inference step.
— $A$ is a defined atom $p^{\prime}(\overline{t^{\prime}}_{n})$ with
$p^{\prime}\in DP^{n}$. In this case $A_{\sim}$ is
$p^{\prime}(\overline{t^{\prime}}_{n})$ and the root inference of $T$ must be
a SQDA inference step of the form:
$\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{j}\theta\sharp
e_{j}\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{~{}p^{\prime}(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi~{}}~{}(\clubsuit)$
with $C:(p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m})\in\mathcal{P}$, $\theta$ substitution,
$\mathcal{S}(p^{\prime},p)=d_{0}\neq\text{{b}}$,
$e_{j}\trianglerighteqslant^{?}w_{j}~{}(1\leq j\leq m)$,
$d\trianglelefteqslant d_{i}~{}(0\leq i\leq n)$ and
$d\trianglelefteqslant\alpha\circ e_{j}~{}(1\leq j\leq m)$—which means
$d\trianglelefteqslant\alpha$ in the case $m=0$. We can assume that the first
$n$ premises at ($\clubsuit$) are proved in
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ w.r.t. $\mathcal{P}$ by
proof trees $T_{1i}~{}(1\leq i\leq n)$ satisfying $\|T_{1i}\|<\|T\|~{}(1\leq
i\leq n)$, and the last $m$ premises at ($\clubsuit$) are proved in
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$ w.r.t. $\mathcal{P}$ by
proof trees $T_{2j}~{}(1\leq j\leq m)$ satisfying $\|T_{2j}\|<\|T\|~{}(1\leq
j\leq m)$.
By Definition 4.2, we know that the transformed program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ contains two clauses of the
following form:
$\begin{array}[]{c@{\hspace{1mm}}cl}\hat{C}\hfil\hskip
2.84526pt&:&\hat{p}_{C}(\overline{t}_{n})\xleftarrow{\alpha}B_{\sim}^{1}\sharp
w_{1},~{}\ldots,~{}B_{\sim}^{m}\sharp w_{m}\\\ \hat{C}_{p^{\prime}}\hfil\hskip
2.84526pt&:&p^{\prime}(\overline{X}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{d_{0}}\sharp?,~{}(~{}(X_{i}\sim
t_{i})\sharp?~{})_{i=1\ldots n},~{}\hat{p}_{C}(\overline{t}_{n})\sharp?\\\
\end{array}$
where $X_{i}~{}(1\leq i\leq n)$ are fresh variables not occurring in $C$ and
$B_{\sim}^{j}~{}(1\leq j\leq m)$ is the result of replacing ‘$\sim$’ for ‘==’
if $B_{j}$ is equation; and $B_{j}$ itself otherwise. Given that the $n$
variables $X_{i}$ do not occur in $C$, we can assume that
$\sigma~{}{=_{\mathrm{def}}}~{}\theta^{\prime}\uplus\theta$ with
$\theta^{\prime}~{}{=_{\mathrm{def}}}~{}\\{X_{1}\mapsto
t^{\prime}_{1},~{}\ldots,~{}X_{n}\mapsto t^{\prime}_{n}\\}$ is a well-defined
substitution. We claim that $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A_{\sim}\sharp d\Leftarrow\Pi$
can be proved with a proof tree $T^{\prime}$ rooted by the QDA inference step
($\spadesuit$.1), which uses the clause $\hat{C}_{p^{\prime}}$ instantiated by
$\sigma$ and having $d_{n+1}=d$.
$\displaystyle\frac{~{}\begin{array}[]{l}(~{}(t^{\prime}_{i}==X_{i}\sigma)\sharp\text{{t}}\Leftarrow\Pi~{})_{i=1\ldots
n}\\\ \mbox{pay}_{d_{0}}\sigma\sharp d_{0}\Leftarrow\Pi\\\ (~{}(X_{i}\sim
t_{i})\sigma\sharp d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\
\hat{p}_{C}(\overline{t}_{n})\sigma\sharp d_{n+1}\Leftarrow\Pi\\\
\end{array}~{}}{~{}p^{\prime}(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi~{}}~{}(\spadesuit.1)\quad\displaystyle\frac{~{}\begin{array}[]{l}(~{}(t^{\prime}_{i}==X_{i}\theta^{\prime})\sharp\text{{t}}\Leftarrow\Pi~{})_{i=1\ldots
n}\\\ \mbox{pay}_{d_{0}}\sharp d_{0}\Leftarrow\Pi\\\
(~{}(X_{i}\theta^{\prime}\sim t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\
\hat{p}_{C}(\overline{t}_{n}\theta)\sharp d_{n+1}\Leftarrow\Pi\\\
\end{array}~{}}{~{}p^{\prime}(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi~{}}~{}(\spadesuit.2)$
By construction of $\sigma$, ($\spadesuit$.1) can be rewritten as
($\spadesuit$.2), and in order to build the rest of $T^{\prime}$, we show
that each premise of ($\spadesuit$.2) admits a proof in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ w.r.t. the transformed program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$:
* –
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
(t^{\prime}_{i}==X_{i}\theta^{\prime})\sharp\text{{t}}\Leftarrow\Pi$ for
$i=1\ldots n$. Straightforward using a single QEA inference step since
$X_{i}\theta^{\prime}=t^{\prime}_{i}$ and
$t^{\prime}_{i}\approx_{\Pi}t^{\prime}_{i}$ is trivially true.
* –
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \mbox{pay}_{d_{0}}\sharp
d_{0}\Leftarrow\Pi$. Immediate using the clause
$(\mbox{pay}_{d_{0}}\xleftarrow{d_{0}})\in\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$
with a single QDA inference step.
* –
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (X_{i}\theta^{\prime}\sim
t_{i}\theta)\sharp d_{i}\Leftarrow\Pi$ for $i=1\ldots n$. From the first $n$
premises of ($\clubsuit$) we know $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
(t^{\prime}_{i}==t_{i}\theta)\sharp d_{i}\Leftarrow\Pi$ with a proof tree
$T_{1i}$ satisfying $\|T_{1i}\|<\|T\|$ for $i=1\ldots n$. Therefore, for
$i=1\ldots n$, $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t^{\prime}_{i}\sim
t_{i}\theta)\sharp d_{i}\Leftarrow\Pi$ with some
QCHL($\mathcal{D}$,$\mathcal{C}$) proof tree $T^{\prime}_{1i}$ by inductive
hypothesis. Since $(X_{i}\theta^{\prime}\sim t_{i}\theta)=(t^{\prime}_{i}\sim
t_{i}\theta)$ for $i=1\ldots n$, we are done.
* –
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
\hat{p}_{C}(\overline{t}_{n}\theta)\sharp d\Leftarrow\Pi$. This is proved by a
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree with a QDA inference step
node at its root of the following form:
$\displaystyle\frac{~{}(~{}(t_{i}\theta==t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{\sim}^{j}\theta\sharp
e_{j}\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{~{}\hat{p}_{C}(\overline{t}_{n}\theta)\sharp
d\Leftarrow\Pi~{}}~{}(\heartsuit)$
which uses the program clause $\hat{C}$ instantiated by the substitution
$\theta$. Once more, we have to check that the premises can be derived in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ from the transformed program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ and that the side conditions of
($\heartsuit$) are satisfied:
* *
The first $n$ premises can be trivially proved using QEA inference steps.
* *
The last $m$ premises can be proved w.r.t.
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ with some
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof trees $T^{\prime}_{2j}~{}(1\leq
j\leq m)$ by the inductive hypothesis, since we have premises
$(~{}B_{j}\theta\sharp e_{j}\Leftarrow\Pi~{})_{j=1\ldots m}$ at ($\clubsuit$)
that can be proved in $\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$
w.r.t. $\mathcal{P}$ with proof trees $T_{2j}$ of size
$\|T_{2j}\|<\|T\|~{}(1\leq j\leq m)$.
* *
The side conditions—namely: $e_{j}\trianglerighteqslant^{?}w_{j}~{}(1\leq
j\leq m)$, $d\trianglelefteqslant d_{i}~{}(1\leq i\leq n)$ and
$d\trianglelefteqslant\alpha\circ e_{j}~{}(1\leq j\leq m)$—trivially hold
because they are also satisfied by ($\clubsuit$).
Finally, we complete the construction of $T^{\prime}$ by checking that
($\spadesuit$.2) satisfies the side conditions of the inference rule QDA:
* –
All threshold values at the body of $\hat{C}_{p^{\prime}}$ are ‘?’, therefore
the first group of side conditions becomes
$d_{i}\trianglerighteqslant^{?}\,\,?~{}(0\leq i\leq n+1)$, which are trivially
true.
* –
The second side condition reduces to $d\trianglelefteqslant\text{{t}}$, which
is also trivially true.
* –
The third, and last, side condition is $d\trianglelefteqslant\text{{t}}\circ
d_{i}~{}(0\leq i\leq n+1)$, or equivalently $d\trianglelefteqslant
d_{i}~{}(0\leq i\leq n+1)$. In fact, $d\trianglelefteqslant d_{i}~{}(0\leq
i\leq n)$ holds due to the side conditions in ($\clubsuit$), and
$d\trianglelefteqslant d_{n+1}$ holds because $d_{n+1}=d$ by construction of
($\spadesuit$.1) and ($\spadesuit$.2).
[2. $\Rightarrow$ 1.] (the transformation is sound). Assume that $T^{\prime}$
is a $\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree witnessing
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A_{\sim}\sharp d\Leftarrow\Pi$.
We want to show the existence of a
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proof tree $T$ witnessing
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp
d\Leftarrow\Pi$. We reason by complete induction of $\|T^{\prime}\|$. There
are three possible cases according to the syntactic form of the atom
$A_{\sim}$. In each case we argue how to build the desired proof tree $T$.
— $A_{\sim}$ is a primitive atom $\kappa$. In this case $A$ is also $\kappa$
and $T^{\prime}$ contains only one QPA inference node. Both
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \kappa\sharp d\Leftarrow\Pi$ and
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
\kappa\sharp d\Leftarrow\Pi$ are equivalent to
$\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$ because of the inference rules QPA
and SQPA, therefore $T$ trivially contains just one SQPA inference node.
— $A_{\sim}$ is of the form $t\sim s$. In this case $A$ is $t==s$ and
$T^{\prime}$ is rooted by a QDA inference step. From
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim s)\sharp d\Leftarrow\Pi$
and by construction of $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ we have
$EQ_{\mathcal{S}}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t\sim
s)\sharp d\Leftarrow\Pi$. By Lemma 4.1 we get $t\approx_{d,\Pi}s$ and, by the
definition of the SQEA inference step, we can build $T$ as a proof tree with
only one SQEA inference node proving $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ (t==s)\sharp
d\Leftarrow\Pi$.
— $A_{\sim}$ is a defined atom $p^{\prime}(\overline{t}_{n})$ with
$p^{\prime}\in DP^{n}$ and $p^{\prime}\neq\,\,\sim$. In this case $A=A_{\sim}$
and the step at the root of $T^{\prime}$ must be a QDA inference step using a
clause $C^{\prime}\in\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ with head
predicate $p^{\prime}$ and a substitution $\theta$. Because of Definition 4.2
and the fact that $p^{\prime}$ is relevant for $\mathcal{P}$, there must be
some clause
$C:(p(\overline{t}_{n})\xleftarrow{\alpha}\overline{B})\in\mathcal{P}$ such
that $\mathcal{S}(p,p^{\prime})=d_{0}\neq\text{{b}}$, and $C^{\prime}$ must be
of the form:
$C^{\prime}:p^{\prime}(\overline{X}_{n})\xleftarrow{\text{{t}}}\mbox{pay}_{d_{0}}\sharp?,~{}((X_{i}\sim
t_{i})\sharp?)_{i=1\ldots n},~{}\hat{p}_{C}(\overline{t}_{n})\sharp?$
where the variables $\overline{X}_{n}$ do not occur in $C$. Thus the QDA
inference step at the root of $T^{\prime}$ must be of the form:
$\displaystyle\frac{~{}\begin{array}[]{l}(~{}(t^{\prime}_{i}==X_{i}\theta)\sharp
d_{1i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\ \mbox{pay}_{d_{0}}\theta\sharp
e_{10}\Leftarrow\Pi\\\ (~{}(X_{i}\sim t_{i})\theta\sharp
e_{1i}\Leftarrow\Pi~{})_{i=1\ldots n}\\\
\hat{p}_{C}(\overline{t}_{n})\theta\sharp e_{1(n+1)}\Leftarrow\Pi\\\
\end{array}~{}}{~{}p^{\prime}(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi~{}}~{}(\spadesuit)$
and the proof of the last premise must use the only clause for $\hat{p}_{C}$
introduced in $\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ according to
Definition 4.2, i.e.:
$\hat{C}:\hat{p}_{C}(\overline{t}_{n})\xleftarrow{\alpha}B_{\sim}^{1}\sharp
w_{1},~{}\ldots,~{}B_{\sim}^{m}\sharp w_{m}\enspace.$
Therefore, the proof of this premise must be of the form:
$\displaystyle\frac{~{}(~{}(t_{i}\theta==t_{i}\theta^{\prime})\sharp
d_{2i}\Leftarrow\Pi~{})_{i=1\ldots
n}\quad(~{}B_{\sim}^{j}\theta^{\prime}\sharp
e_{2j}\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{~{}\hat{p}_{C}(\overline{t}_{n})\theta\sharp
e_{1(n+1)}\Leftarrow\Pi~{}}~{}(\heartsuit)$
for some substitution $\theta^{\prime}$ not affecting $\overline{X}_{n}$. We
can assume that the last $m$ premises in ($\heartsuit$) are proved in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ w.r.t.
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ by proof trees $T^{\prime}_{j}$
satisfying $\|T^{\prime}_{j}\|<\|T^{\prime}\|~{}(1\leq j\leq m)$. Then we use
the substitution $\theta^{\prime}$ and clause $C$ to build a
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proof tree $T$ with a SQDA
inference step at the root of the form:
$\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta^{\prime})\sharp
e_{1i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{j}\theta^{\prime}\sharp
e_{2j}\Leftarrow\Pi~{})_{j=1\ldots
m}~{}}{~{}p^{\prime}(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi~{}}~{}(\clubsuit)$
Next we check that the premises of this inference step admit proofs in
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},$ $\mathcal{C})$ and that $(\clubsuit)$
satisfies the side conditions of a valid SQDA inference step.
* –
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
(t^{\prime}_{i}==t_{i}\theta^{\prime})\sharp e_{1i}\Leftarrow\Pi$ for
$i=1\ldots n$.
* *
From the premises $((X_{i}\sim t_{i})\theta\sharp
e_{1i}\Leftarrow\Pi)_{i=1\ldots n}$ of $(\spadesuit)$ and by construction of
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ we know $EQ_{\mathcal{S}}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (X_{i}\sim t_{i})\theta\sharp
e_{1i}\Leftarrow\Pi~{}(1\leq i\leq n)$. Therefore by Lemma 4.1 we have
$X_{i}\theta\approx_{e_{1i},\Pi}t_{i}\theta$ for $i=1\dots n$.
* *
Consider now the premises $((t^{\prime}_{i}==X_{i}\theta)\sharp
d_{1i}\Leftarrow\Pi)_{i=1\ldots n}$ of $(\spadesuit)$. Their proofs must rely
on QEA inference steps, and therefore $t^{\prime}_{i}\approx_{\Pi}X_{i}\theta$
holds for $i=1\dots n$.
* *
Analogously, from the proofs of the premises
$((t_{i}\theta==t_{i}\theta^{\prime})\sharp d_{2i}\Leftarrow\Pi)_{i=1\ldots
n}$ we have $t_{i}\theta\approx_{\Pi}t_{i}\theta^{\prime}$ (or equivalently
$t_{i}\theta^{\prime}\approx_{\Pi}t_{i}\theta$) for $i=1\ldots n$.
From the previous points we have $X_{i}\theta\approx_{e_{1i},\Pi}t_{i}\theta$,
$t^{\prime}_{i}\approx_{\Pi}X_{i}\theta$ and
$t_{i}\theta^{\prime}\approx_{\Pi}t_{i}\theta$, which by Lemma 2.7(1) of
[RR10TR] imply
$t^{\prime}_{i}\approx_{e_{1i},\Pi}t_{i}\theta^{\prime}~{}(1\leq i\leq n)$.
Therefore the premises $((t^{\prime}_{i}==t_{i}\theta^{\prime})\sharp
e_{1i}\Leftarrow\Pi)_{i=1\ldots n}$ can be proven in
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ using a SQEA inference
step.
* –
$\mathcal{P}\ {\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
B_{j}\theta^{\prime}\sharp e_{2j}\Leftarrow\Pi$ for $j=1\ldots m$. We know
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ B_{\sim}^{j}\theta^{\prime}\sharp
e_{2j}\Leftarrow\Pi$ with a proof tree $T^{\prime}_{j}$ satisfying
$\|T^{\prime}_{j}\|<\|T^{\prime}\|~{}(1\leq j\leq m)$ because of
($\heartsuit$). Therefore we have, by inductive hypothesis, $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\
B_{j}\theta^{\prime}\sharp e_{2j}\Leftarrow\Pi$ for some
$\mbox{SQCHL}(\mathcal{S},\mathcal{D},\mathcal{C})$ proof tree $T_{j}~{}(1\leq
j\leq m)$.
* –
$\mathcal{S}(p,p^{\prime})=d_{0}\neq\text{{b}}$. As seen above.
* –
$e_{2j}\trianglerighteqslant^{?}w_{j}$ for $j=1\ldots m$. This is a side
condition of the QDA step in $(\heartsuit)$.
* –
$d\trianglelefteqslant e_{1i}$ for $i=1\ldots n$. Straightforward from the
side conditions of $(\spadesuit)$, which include
$d\trianglelefteqslant\text{{t}}\circ e_{1i}$ for $(0\leq i\leq n+1)$.
* –
$d\trianglelefteqslant\alpha\circ e_{2j}$ for $j=1\ldots m$. This follows from
the side conditions of $(\spadesuit)$ and $(\heartsuit)$, since we have
$d\trianglelefteqslant\text{{t}}\circ e_{1i}$ for $i=0\ldots n+1$ (in
particular $d\trianglelefteqslant e_{1(n+1)}$) and
$e_{1(n+1)}\trianglelefteqslant\alpha\circ e_{2j}$ for $j=1\ldots m$.
Finally, the next theorem extends the previous result to goals.
###### Theorem 4.2
Let $G$ be a goal for a
$\mbox{SQCLP}(\mathcal{S},\mathcal{D},\mathcal{C})$-program $\mathcal{P}$
whose atoms are all relevant for $\mathcal{P}$. Assume
$\mathcal{P}^{\prime}=\mathrm{elim}_{\mathcal{S}}(\mathcal{P})$ and
$G^{\prime}=\mathrm{elim}_{\mathcal{S}}(G)$. Then,
$\mbox{Sol}_{\mathcal{P}}(G)=\mbox{Sol}_{\mathcal{P}^{\prime}}(G^{\prime})$.
* Proof
* According to the definition of goals in Section 2, and Definition 4.2, $G$ and $G^{\prime}$ must be of the form $(A_{i}\sharp W_{i},W_{i}\,{\trianglerighteqslant}^{?}\beta_{i})_{i=1\ldots m}$ and $(A_{\sim}^{i}\sharp W_{i},W_{i}\,{\trianglerighteqslant}^{?}\beta_{i})_{i=1\ldots m}$, respectively. By Definitions 2.2 and 3.1, both $\mbox{Sol}_{\mathcal{P}}(G)$ and $\mbox{Sol}_{\mathcal{P}^{\prime}}(G^{\prime})$ are sets of triples $\langle\sigma,\mu,\Pi\rangle$ where $\sigma$ is a $\mathcal{C}$-substitution, $\mu:\mathrm{war}(G)\to D_{\mathcal{D}}\setminus\\{\text{{b}}\\}$ (note that $\mathrm{war}(G)=\mathrm{war}(G^{\prime})$) and $\Pi$ is a satisfiable finite set of $\mathcal{C}$-constraints. Moreover:
1. 1.
$\langle\sigma,\mu,\Pi\rangle\in\mbox{Sol}_{\mathcal{P}}(G)$ iff
$W_{i}\mu=d_{i}\trianglerighteqslant^{?}\\!\beta_{i}$ and $\mathcal{P}\
{\vdash}_{\\!\mathcal{S},\mathcal{D},\mathcal{C}}^{\\!}\ A_{i}\sigma\sharp
W_{i}\mu\Leftarrow\Pi$ $(1\leq i\leq m)$.
2. 2.
$\langle\sigma,\mu,\Pi\rangle\in\mbox{Sol}_{\mathcal{P}^{\prime}}(G^{\prime})$
iff $W_{i}\mu=d_{i}\trianglerighteqslant^{?}\\!\beta_{i}$ and
$\mathcal{P}^{\prime}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
A_{\sim}^{i}\sigma\sharp W_{i}\mu\Leftarrow\Pi$ $(1\leq i\leq m)$.
Because of Theorem 4.1, conditions (1) and (2) are equivalent.
### 4.2 Transforming QCLP into CLP
The results presented in this subsection are dependant on the assumption that
the qualification domain $\mathcal{D}$ is existentially expressible in the
constraint domain $\mathcal{C}$ via an injective mapping
$\imath:D_{\mathcal{D}}\setminus\\{\text{{b}}\\}\to C_{\mathcal{C}}$ and two
existential $\mathcal{C}$-constraints of the following form:
* $\mathsf{qVal}(X)=\exists U_{1}\ldots\exists U_{k}(B_{1}\land\ldots\land B_{m})$
* $\mathsf{qBound}(X,Y,Z)=\exists V_{1}\ldots\exists V_{l}(C_{1}\land\ldots\land C_{q})$
The intuition behind $\mathsf{qVal}(X)$ and $\mathsf{qBound}(X,Y,Z)$ has been
explained in Definition 2.1. Roughly, they are intended to represent
qualification values from $\mathcal{D}$ and the behaviour of $\mathcal{D}$’s
attenuation operator $\circ$ by means of $\mathcal{C}$-constraints. Moreover,
the assumption that $\mathsf{qVal}(X)$ and $\mathsf{qBound}(X,Y,Z)$ have the
existential form displayed above allows to build CLP clauses for two predicate
symbols ${\mathit{q}Val}\in DP^{1}$ and ${\mathit{q}Bound}\in DP^{3}$ which
will capture the behaviour of the two corresponding constraints in the sense
of Lemma 3.1. More precisely, we consider the
$\mbox{CLP}(\mathcal{C})$-program $E_{\mathcal{D}}$ consisting of the
following two clauses:
* ${\mathit{q}Val}(X)\leftarrow B_{1},\ \ldots,\ B_{m}$
* ${\mathit{q}Bound}(X,Y,Z)\leftarrow C_{1},\ \ldots,\ C_{q}$
The next example shows the CLP clauses in $E_{\mathcal{D}}$ for
$\mathcal{C}=\mathcal{R}$ and three different choices of a qualification
domain $\mathcal{D}$ that is existentially expressible in $\mathcal{R}$,
namely: $\mathcal{U}$, $\mathcal{W}$ and $\mathcal{U}{\otimes}\mathcal{W}$. In
each case, the CLP clauses in $E_{\mathcal{D}}$ are obtained straightforwardly
from the $\mathcal{R}$ constraints $\mathsf{qVal}(X)$ and
$\mathsf{qBound}(X,Y,Z)$ shown in Example 2.1.
###### Example 4.3
1. 1.
$E_{\mathcal{U}}$ consists of the following two clauses:
${\mathit{q}Val}(X)\leftarrow{\mathit{c}p}_{<}(0,X),\
{\mathit{c}p}_{\leq}(X,1)$
---
${\mathit{q}Bound}(X,Y,Z)\leftarrow{\mathit{o}p}_{\times}(Y,Z,X^{\prime}),\
{\mathit{c}p}_{\leq}(X,X^{\prime})$
1. 2.
$E_{\mathcal{W}}$ consists of the following two clauses:
${\mathit{q}Val}(X)\leftarrow{\mathit{c}p}_{\geq}(X,0)$
---
${\mathit{q}Bound}(X,Y,Z)\leftarrow{\mathit{o}p}_{+}(Y,Z,X^{\prime}),\
{\mathit{c}p}_{\geq}(X,X^{\prime})$
1. 3.
$E_{\mathcal{U}{\otimes}\mathcal{W}}$ consists of the following two clauses:
${\mathit{q}Val}(X)\leftarrow X=={\mathsf{p}air}(X_{1},X_{2}),\
{\mathit{c}p}_{<}(0,X_{1}),\ {\mathit{c}p}_{\leq}(X_{1},1),\
{\mathit{c}p}_{\geq}(X_{2},0)$
---
${\mathit{q}Bound}(X,Y,Z)\leftarrow X=={\mathsf{p}air}(X_{1},X_{2}),\
Y=={\mathsf{p}air}(Y_{1},Y_{2}),\ Z=={\mathsf{p}air}(Z_{1},Z_{2}),$
$\qquad{\mathit{o}p}_{\times}(Y_{1},Z_{1},X^{\prime}_{1}),\
{\mathit{c}p}_{\leq}(X_{1},X^{\prime}_{1}),\
{\mathit{o}p}_{+}(Y_{2},Z_{2},X^{\prime}_{2}),\
{\mathit{c}p}_{\geq}(X_{2},X^{\prime}_{2})\hbox
to0.0pt{\quad\leavevmode\hbox{\begin{picture}(6.5,6.5)\put(0.0,0.0){\framebox(6.5,6.5)[]{}}\end{picture}}\hss}$
In general, the CLP clauses in $E_{\mathcal{D}}$ along with other techniques
explained in the rest of this subsection will be used to present semantically
correct transformations from $\mbox{QCLP}(\mathcal{D},\mathcal{C})$ into
$\mbox{CLP}(\mathcal{C})$, working both for programs and goals. All our
results will work under the assumption that ${\mathit{q}Val}\in DP^{1}$ and
${\mathit{q}Bound}\in DP^{3}$ are chosen as fresh predicate symbols not
occurring in the $\mbox{QCLP}(\mathcal{D},\mathcal{C})$ programs and goals to
be transformed. The next technical lemma ensures that the predicates qVal and
qBound correctly represent the behaviour of the constraints $\mathsf{qVal}(X)$
and $\mathsf{qBound}(X,Y,Z)$.
###### Lemma 4.2
For any satisfiable finite set $\Pi$ of $\mathcal{C}$-constraints one has:
1. 1.
For any ground term $t\in C_{\mathcal{C}}$:
$t\in\mbox{ran}(\imath)\iff\mathsf{qVal}(t)\mbox{ true in }\mathcal{C}\iff
E_{\mathcal{D}}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
{\mathit{q}Val}(t)\Leftarrow\Pi$
2. 2.
For any ground terms $r=\imath(x)$, $s=\imath(y)$, $t=\imath(z)$ with
$x,y,z\in D_{\mathcal{D}}\setminus\\{\text{{b}}\\}$:
$x\trianglelefteqslant y\circ z\iff\mathsf{qBound}(r,s,t)\mbox{ true in
}\mathcal{C}\iff E_{\mathcal{D}}\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
{\mathit{q}Bound}(r,s,t)\Leftarrow\Pi$
The two items above are also valid if $E_{\mathcal{D}}$ is replaced by any
$\mbox{CLP}(\mathcal{C})$-program including the two clauses in
$E_{\mathcal{D}}$ and having no additional occurrences of qVal and qBound at
the head of clauses.
###### Proof 4.1.
Immediate consequence of Lemma 3.1 and Definition 2.1.
Transforming Atoms
---
TEA | $({t==s})^{\mathcal{T}}\\!=(t==s,~{}\imath(\text{{t}}))$.
TPA | $({\kappa})^{\mathcal{T}}\\!=(\kappa,~{}\imath(\text{{t}}))$ with $\kappa$ primitive atom.
TDA | $({p(\overline{t}_{n})})^{\mathcal{T}}\\!=(p^{\prime}(\overline{t}_{n},W),~{}W)$ with $p\in DP^{n}$ and $W$ a fresh CLP variable.
Transforming qc-Atoms
TQCA | $\displaystyle\frac{A^{\mathcal{T}}=(A^{\prime},w)}{\quad(A\sharp d\Leftarrow\Pi)^{\mathcal{T}}=(A^{\prime}\Leftarrow\Pi,~{}\\{\mathsf{qVal}(w),\ \mathsf{qBound}(\imath(d),\imath(\text{{t}}),w)\\})\quad}$
Transforming Program Clauses
TPC | $\displaystyle\frac{(~{}B_{j}^{\mathcal{T}}=(B_{j}^{\prime},w^{\prime}_{j})~{})_{j=1\dots m}}{\quad C^{\mathcal{T}}=p^{\prime}(\overline{t}_{n},W)~{}\leftarrow~{}qV\\!al(W),\ \left(\begin{array}[]{l}qV\\!al(w_{j}^{\prime}),\ \ulcorner w^{\prime}_{j}\trianglerighteqslant^{?}\\!\imath(w_{j})\urcorner,\\\ qBound(W,\imath(\alpha),w^{\prime}_{j}),\ B^{\prime}_{j}\end{array}\right)_{j=1\ldots m}\quad}$
where $C:p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m}$, $W$ is a fresh CLP variable and
$\ulcorner w^{\prime}_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner$ is
omitted if $w_{j}=\ ?$, otherwise abbreviates
$qBound(\imath(w_{j}),\imath(\text{{t}}),w^{\prime}_{j})$.
Transforming Goals
TG | $\displaystyle\frac{(~{}B_{j}^{\mathcal{T}}=(B_{j}^{\prime},w^{\prime}_{j})~{})_{j=1\dots m}}{\quad\mathrm{elim}_{\mathcal{D}}(G)=\left(\begin{array}[]{l}qV\\!al(W_{j}),\ \ulcorner W_{j}\trianglerighteqslant^{?}\imath(\beta_{j})\urcorner,\\\ qV\\!al(w^{\prime}_{j}),\ qBound(W_{j},\imath(\text{{t}}),w^{\prime}_{j}),\ B_{j}^{\prime}\end{array}\right)_{j=1\ldots m}\quad}$
where $G:(B_{j}\sharp W_{j},W_{j}\trianglerighteqslant^{?}\beta_{j})_{j=1\dots
m}$ and $\ulcorner W_{j}\trianglerighteqslant^{?}\imath(\beta_{j})\urcorner$
as in TPC above.
Figure 5: Transformation rules
Now we are ready to define the transformations from
$\mbox{QCLP}(\mathcal{D},\mathcal{C})$ into $\mbox{CLP}(\mathcal{C})$.
###### Definition 3.
Assume that $\mathcal{D}$ is existentially expressible in $\mathcal{C}$, and
let $\mathsf{qVal}(X)$, $\mathsf{qBound}(X,Y,Z)$ and $E_{\mathcal{D}}$ be as
explained above. Assume also a $\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$ and a $\mbox{QCLP}(\mathcal{D},\mathcal{C})$-goal $G$ for
$\mathcal{P}$ without occurrences of the defined predicate symbols $qV\\!al$
and $qBound$. Then:
1. 1.
$\mathcal{P}$ is transformed into the $\mbox{CLP}(\mathcal{C})$-program
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})$ consisting of the two clauses in
$E_{\mathcal{D}}$ and the transformed $C^{\mathcal{T}}\\!$ of each clause
$C\in\mathcal{P}$, built as specified in Figure 5. The transformation rules of
this figure translate each $n$-ary predicate symbol $p\in DP^{n}$ into a
different $(n+1)$-ary predicate symbol $p^{\prime}\in DP^{n+1}$.
2. 2.
$G$ is transformed into the $\mbox{CLP}(\mathcal{C})$-goal
$\mathrm{elim}_{\mathcal{D}}(G)$ built as specified in Figure 5. Note that the
qualification variables $\overline{W}_{\\!n}$ occurring in $G$ become normal
CLP variables in the transformed goal.
The first three rules in Figure 5 are used for transforming atoms. For
convenience, the transformation of an atom produces a pair where the first
value is the transformed atom and the second one is either a new variable or
the representation of t. In the first two cases, namely TEA and TPA, the
transformation behaves as the identity and no new variables are introduced.
The third case, namely TDA, corresponds to the transformation of a defined
atom. In this case, a new CLP variable $W$—intended to represent the
qualification value associated to the atom—is added as its last argument. The
rule TQCA transforms qc-atoms of the form $A\sharp d\Leftarrow\Pi$ by means of
the transformation of $A$ using one of the three aforementioned transformation
rules. This transformation returns a pair $(A^{\prime},w)$ in which, as shown
above, $w$ can be either a new variable or the representation of t. Since $w$
can be a new variable $W$, the constraint $\mathsf{qVal}(w)$ is introduced to
ensure that it represents a qualification value. Finally, the constraint
$\mathsf{qBound}(\imath(d),\imath(\text{{t}}),w)$ encodes
“$d\trianglelefteqslant\text{{t}}\circ w$,” or equivalently
“$d\trianglerighteqslant w.$” The rule TPC is employed for transforming
program clauses $C:p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m}$ where each $w_{i}$ is either a qualification
value or $?$ indicating that proving the atom with any qualification value
different from b is acceptable. The rule introduces a new variable $W$
together with a constraint ${\mathit{q}Val}(W)$. The variable represents the
qualification value associated to the computation of user defined atoms
involving $p$ (renamed as $p^{\prime}$ in the transformed program). The
premises $(B_{j}^{\mathcal{T}}=(B_{j}^{\prime},w^{\prime}_{j}))_{j=1\dots m}$
transform the atoms in the body of the clause using in each case either TEA,
TPA or TDA. Therefore, each $w^{\prime}_{j}$ obtained in this way represents a
qualification value encoded as a constraint value. Moreover, the qualification
value encoded by $w^{\prime}_{j}$ must be greater or equal than the
corresponding qualification value $w_{j}$ that occurs in the program clause.
These two requirements are represented as ${\mathit{q}Val}(w_{j}^{\prime}),\
\ulcorner w^{\prime}_{j}\trianglerighteqslant^{?}\\!\imath(w_{j})\urcorner$ in
the transformed clause. The predicate call
${\mathit{q}Bound}(W,\imath(\alpha),w^{\prime}_{j})$ ensures that the value in
$W$ must be less than or equal to “$\alpha\circ w^{\prime}_{j}$” for every
$j$. For each $j=1\dots m$ all the atoms associated to the transformation of
$B_{j}$ precede the transformed atom $B^{\prime}_{j}$. In a Prolog-based
implementation, this helps to prune the search space as soon as possible
during the computations. The ideas behind rule TG are similar. A goal
$G:(B_{j}\sharp W_{j},W_{j}\trianglerighteqslant^{?}\beta_{j})_{j=1\dots m}$
is transformed by introducing atoms in charge of checking that: each $W_{j}$
is a valid qualification value; each $W_{j}$ is indeed less than or equal to
the representation of $\beta_{j}$ in CLP; each value $w_{j}$—obtained during
the transformation of the atoms $B_{j}$—corresponds to an actual qualification
value; and finally, that each $W_{j}$ is satisfactory—i.e. less or equal
to—w.r.t. its corresponding $w_{j}$ before effectively introducing the
transformed atoms $B^{\prime}_{j}$. The following example illustrates the
transformation elimD.
###### Example 4 (Running example: $\mbox{CLP}(\mathcal{R})$-program
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r}))$).
Consider the
$\mbox{QCLP}(\mathcal{U}{\otimes}\mathcal{W},\mathcal{R})$-program
$\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r})$ and the goal
$\mathrm{elim}_{\mathcal{S}}(G_{r})$ for the same program as presented in
Example 4.2. The transformed $\mbox{CLP}(\mathcal{R})$-program
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r}))$ is
as follows:
$\hat{R}_{1}$ | f̂amous${}_{R_{1}}$(sha, W) $\leftarrow$ qVal(W), qBound(W, t, (0.9,1))
---|---
$R_{1.1}$ | famous(X, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay${}_{\text{{t}}}$(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, sha, W2),
| qVal(W3), qBound(W, t, W3), f̂amous${}_{R_{1}}$(sha, W3)
$\hat{R}_{2}$ | ŵrote${}_{R_{2}}$(sha, kle, W) $\leftarrow$ qVal(W), qBound(W, t, (1,1))
---|---
$R_{2.1}$ | wrote(X, Y, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay${}_{\text{{t}}}$(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, sha, W2),
| qVal(W3), qBound(W, t, W3), $\sim$(Y, kle, W3),
| qVal(W4), qBound(W, t, W4), ŵrote${}_{R_{2}}$(sha, kle, W4)
$R_{2.2}$ | authored(X, Y, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay(0.9,0)(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, sha, W2),
| qVal(W3), qBound(W, t, W3), $\sim$(Y, kle, W3),
| qVal(W4), qBound(W, t, W4), ŵrote${}_{R_{2}}$(sha, kle, W4)
$\hat{R}_{3}$ | ŵrote${}_{R_{3}}$(sha, hamlet, W) $\leftarrow$ qVal(W), qBound(W, t, (1,1))
$R_{3.1}$ | wrote(X, Y, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay${}_{\text{{t}}}$(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, sha, W2),
| qVal(W3), qBound(W, t, W3), $\sim$(Y, hamlet, W3),
| qVal(W4), qBound(W, t, W4), ŵrote${}_{R_{3}}$(sha, hamlet, W4)
$R_{3.2}$ | authored(X, Y, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay(0.9,0)(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, sha, W2),
| qVal(W3), qBound(W, t, W3), $\sim$(Y, hamlet, W3),
| qVal(W4), qBound(W, t, W4), ŵrote${}_{R_{3}}$(sha, hamlet, W4)
$\hat{R}_{4}$ | ĝood_work${}_{R_{4}}$(G, W) $\leftarrow$ qVal(W),
| qVal(W1), qBound((0.5,100), t, W1), qBound(W, (0.75,3), W1), famous(A, W1),
| qVal(W2), qBound(W, (0.75,3), W2), authored(A, G, W2)
$R_{4.1}$ | good_work(X, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1), pay${}_{\text{{t}}}$(W1),
| qVal(W2), qBound(W, t, W2), $\sim$(X, G, W2),
| qVal(W3), qBound(W, t, W3), ĝood_work${}_{R_{4}}$(G, W3)
| % Program clauses for $\sim$:
| $\sim$(X, Y, W) $\leftarrow$ qVal(W), qVal(t), qBound(W, t, t), X==Y
| $\sim$(kle, kli, W) $\leftarrow$ qVal(W), qVal(W1), qBound(W, t, W1),
pay(0.8,2)(W1)
| $[\ldots]$
| % Program clauses for pay:
| pay${}_{\text{{t}}}$(W) $\leftarrow$ qVal(W), qBound(W, t, t)
| pay(0.9,0)(W) $\leftarrow$ qVal(W), qBound(W, t, (0.9,0))
| pay(0.8,2)(W) $\leftarrow$ qVal(W), qBound(W, t, (0.8,2))
| % Program clauses for qVal & qBound:
| qVal((X1,X2)) $\leftarrow$ X1 $>$ 0, X1 $\leq$ 1, X2 $\geq$ 0
| qBound((W1,W2), (Y1,Y2), (Z1,Z2)) $\leftarrow$ W1 $\leq$ Y1 $\times$ Z1, W2
$\geq$ Y2 $+$ Z2
Finally, the goal
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(G_{r}))$ for
$\mathrm{elim}_{\mathcal{D}}(\mathrm{elim}_{\mathcal{S}}(\mathcal{P}_{r}))$ is
as follows:
qVal(W), qBound((0.5,10), t, W), qVal(W’), qBound(W, t, W’), good_work(X,
W’)
Note that, in order to improve the clarity of the program clauses of this
example, the qualification value $(1,\\!0)$—top value in
$\mathcal{U}{\otimes}\mathcal{W}$—has been replaced by t.
The next theorem proves the semantic correctness of the program
transformation.
###### Theorem 5.
Let $A$ be an atom such that $qV\\!al$ and $qBound$ do not occur in $A$.
Assume $d\in D\setminus\\{\text{{b}}\\}$ such that $(A\sharp
d\Leftarrow\Pi)^{\mathcal{T}}=(A^{\prime}\Leftarrow\Pi,\Omega)$. Then, the two
following statements are equivalent:
1. 1.
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp
d\Leftarrow\Pi$
2. 2.
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
A^{\prime}\\!\rho\Leftarrow\Pi$ for some
$\rho\in\mbox{Sol}_{\mathcal{C}}(\Omega)$ such that
$\mathrm{dom}(\rho)=\mathrm{var}(\Omega)$.
###### Proof 4.2.
We separately prove each implication.
[1. $\Rightarrow$ 2.] (the transformation is complete). We assume that $T$ is
a $\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree witnessing $\mathcal{P}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp d\Leftarrow\Pi$. We want
to show the existence of a $\mbox{CLP}(\mathcal{C})$ proof tree $T^{\prime}$
witnessing $\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ A^{\prime}\\!\rho\Leftarrow\Pi$ for some
$\rho\in\mbox{Sol}_{\mathcal{C}}(\Omega)$ such that
$\mathrm{dom}(\rho)=\mathrm{var}(\Omega)$. We reason by complete induction on
$\|T\|$. There are three possible cases, according to the the syntactic form
of the atom $A$. In each case we argue how to build the desired proof tree
$T^{\prime}$.
— $A$ is a primitive atom $\kappa$. In this case TQCA and TPA compute
$A^{\prime}=\kappa$ and $\Omega=\\{\mathsf{qVal}(\imath(\text{{t}})),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),\imath(\text{{t}}))\\}$. Now,
from $\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \kappa\sharp
d\Leftarrow\Pi$ follows $\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$ due to the
QPA inference, and therefore taking $\rho=\varepsilon$ we can prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
\kappa\varepsilon\Leftarrow\Pi$ with a proof tree $T^{\prime}$ containing
only one PA node. Moreover, $\varepsilon\in\mbox{Sol}_{\mathcal{C}}(\Omega)$
is trivially true because the two constraints belonging to $\Omega$ are
obviously true in $\mathcal{C}$.
— $A$ is an equation $t==s$. In this case TQCA and TEA compute
$A^{\prime}=(t==s)$ and $\Omega=\\{\mathsf{qVal}(\imath(\text{{t}})),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),\imath(\text{{t}}))\\}$. Now,
from $\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t==s)\sharp
d\Leftarrow\Pi$ follows $t\approx_{\Pi}s$ due to the QEA inference, and
therefore taking $\rho=\varepsilon$ we can prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(t==s)\varepsilon\Leftarrow\Pi$ with a proof tree $T^{\prime}$ containing
only one EA node. Moreover, $\varepsilon\in\mbox{Sol}_{\mathcal{C}}(\Omega)$
is trivially true because the two constraints belonging to $\Omega$ are
obviously true in $\mathcal{C}$.
— $A$ is a defined atom $p(\overline{t^{\prime}}_{n})$ with $p\in DP^{n}$. In
this case TQCA and TDA compute
$A^{\prime}=p^{\prime}(\overline{t^{\prime}}_{n},W)$ and
$\Omega=\\{\mathsf{qVal}(W),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),W)\\}$ where $W$ is a fresh CLP
variable. On the other hand, $T$ must be rooted by a QDA step of the form:
$\displaystyle\frac{~{}(~{}(t^{\prime}_{i}==t_{i}\theta)\sharp
d_{i}\Leftarrow\Pi~{})_{i=1\ldots n}\quad(~{}B_{j}\theta\sharp
e_{j}\Leftarrow\Pi~{})_{j=1\ldots m}~{}}{p(\overline{t^{\prime}}_{n})\sharp
d\Leftarrow\Pi}\quad(\clubsuit)$
using a clause $C:(p(\overline{t}_{n})\xleftarrow{\alpha}B_{1}\sharp
w_{1},\ldots,B_{m}\sharp w_{m})\in\mathcal{P}$ instantiated by a substitution
$\theta$ and such that the side conditions
$e_{j}\trianglerighteqslant^{?}w_{j}~{}(1\leq j\leq m)$,
$d\trianglelefteqslant d_{i}~{}(1\leq i\leq n)$ and
$d\trianglelefteqslant\alpha\circ e_{j}~{}(1\leq j\leq m)$ are fulfilled.
For $j=1\ldots m$ we can assume
$B_{j}^{\mathcal{T}}=(B^{\prime}_{j},w^{\prime}_{j})$ and thus
$(B_{j}\theta\sharp
e_{j}\Leftarrow\Pi)^{\mathcal{T}}=(B^{\prime}_{j}\theta\Leftarrow\Pi,\Omega_{j})$
where $\Omega_{j}=\\{\mathsf{qVal}(w^{\prime}_{j}),\
\mathsf{qBound}(\imath(e_{j}),\imath(\text{{t}}),w^{\prime}_{j})\\}$. The
proof trees $T_{j}$ of the last $m$ premises of $(\clubsuit)$ will have less
than $\|T\|$ nodes, and hence the induction hypothesis can be applied to each
$(B_{j}\theta\sharp e_{j}\Leftarrow\Pi)$ with $1\leq j\leq m$, obtaining
CHL($\mathcal{C}$) proof trees $T^{\prime}_{j}$ proving
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
B^{\prime}_{j}\theta\rho_{j}\Leftarrow\Pi$ for some
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$ with
$\mathrm{dom}(\rho_{j})=\mathrm{var}(\Omega_{j})$.
Consider $\rho=\\{W\mapsto\imath(d)\\}$ and
$C^{\mathcal{T}}\in\mathrm{elim}_{\mathcal{D}}(\mathcal{P})$ of the form:
$C^{\mathcal{T}}:~{}p^{\prime}(\overline{t}_{n},W^{\prime})~{}\leftarrow~{}qV\\!al(W^{\prime}),\
\left(\begin{array}[]{l}qV\\!al(w_{j}^{\prime}),~{}\ulcorner
w^{\prime}_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner,\\\
qBound(W^{\prime},\imath(\alpha),w^{\prime}_{j}),~{}B^{\prime}_{j}\\\
\end{array}\right)_{j=1\ldots m.}\\\ $
Obviously, $\rho\in\mbox{Sol}_{\mathcal{C}}(\Omega)$ and
$\mathrm{dom}(\rho)=\mathrm{var}(\Omega)$. To finish the proof we must prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
A^{\prime}\\!\rho\Leftarrow\Pi$. We claim that this can be done with a
CHL($\mathcal{C}$) proof tree $T^{\prime}$ whose root inference is a DA step
of the form:
$\displaystyle\frac{~{}\begin{array}[]{l}~{}~{}~{}~{}(~{}(t^{\prime}_{i}\rho==t_{i}\theta^{\prime})\Leftarrow\Pi~{})_{i=1\ldots
n}\\\ ~{}~{}~{}~{}(W\rho==W^{\prime}\theta^{\prime})\Leftarrow\Pi\\\
~{}~{}~{}~{}qV\\!al(W^{\prime})\theta^{\prime}\Leftarrow\Pi\\\
\left(\begin{array}[]{l}qV\\!al(w^{\prime}_{j})\theta^{\prime}\Leftarrow\Pi\\\
\ulcorner
w^{\prime}_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner\theta^{\prime}\Leftarrow\Pi\\\
qBound(W^{\prime},\imath(\alpha),w^{\prime}_{j})\theta^{\prime}\Leftarrow\Pi\\\
B^{\prime}_{j}\theta^{\prime}\Leftarrow\Pi\end{array}\right)_{j=1\ldots m}\\\
\end{array}~{}}{p^{\prime}(\overline{t^{\prime}}_{n},W)\rho\Leftarrow\Pi}~{}(\spadesuit)$
using $C^{\mathcal{T}}$ instantiated by the substitution
$\theta^{\prime}=\theta\uplus\rho_{1}\uplus\dots\uplus\rho_{m}\uplus\\{W^{\prime}\mapsto\imath(d)\\}$.
We check that the premises of ($\spadesuit$) can be derived from
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})$ in CHL($\mathcal{C}$):
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(t^{\prime}_{i}\rho==t_{i}\theta^{\prime})\Leftarrow\Pi$ for $i=1\ldots n$. By
construction of $\rho$ and $\theta^{\prime}$, these are equivalent to prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(t^{\prime}_{i}==t_{i}\theta)\Leftarrow\Pi$ for $i=1\ldots n$ and these hold
with CHL($\mathcal{C}$) proof trees of only one EA node because of
$t^{\prime}_{i}\approx_{\Pi}t_{i}\theta$, which is a consequence of the first
$n$ premises of ($\clubsuit$).
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(W\rho==W^{\prime}\theta^{\prime})\Leftarrow\Pi$. By construction of $\rho$
and $\theta^{\prime}$, this is equivalent to prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
(\imath(d)==\imath(d))\Leftarrow\Pi$ which results trivial.
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
qV\\!al(W^{\prime})\theta^{\prime}\Leftarrow\Pi$. By construction of
$\theta^{\prime}$, this is equivalent to prove
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
qV\\!al(\imath(d))\Leftarrow\Pi$. We trivially have that
$\imath(d)\in\mbox{ran}(\imath)$. Then, by Lemma 4.2, this premise holds.
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
qV\\!al(w^{\prime}_{j})\theta^{\prime}\Leftarrow\Pi$ for $j=1\ldots m$. By
construction of $\theta^{\prime}$ and Lemma 4.2 we must prove, for any fixed
$j$, that $\mathsf{qVal}(w^{\prime}_{j}\rho_{j})$ is true in $\mathcal{C}$. As
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$ we know
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\mathsf{qVal}(w^{\prime}_{j}))$,
therefore $\mathsf{qVal}(w^{\prime}_{j}\rho_{j})$ is trivially true in
$\mathcal{C}$.
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
\ulcorner
w^{\prime}_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner\theta^{\prime}\Leftarrow\Pi$
for $j=1\ldots m$. We reason for any fixed $j$. If $w_{j}=\ ?$ this results
trivial. Otherwise, it amounts to
$\mathsf{qBound}(\imath(w_{j}),\imath(\text{{t}}),w^{\prime}_{j}\rho_{j})$
being true in $\mathcal{C}$, by construction of $\theta^{\prime}$ and Lemma
4.2. As seen before, $\mathsf{qVal}(w^{\prime}_{j}\rho_{j})$ is true in
$\mathcal{C}$, therefore $w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$ for
some $e^{\prime}_{j}\in D\setminus\\{\text{{b}}\\}$. From the side conditions
of ($\clubsuit$) we have $w_{j}\trianglelefteqslant e_{j}$. On the other hand,
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$ and, in particular,
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\mathsf{qBound}(\imath(e_{j}),\imath(\text{{t}}),w^{\prime}_{j}))$.
This, together with $w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$, means
$e_{j}\trianglelefteqslant e^{\prime}_{j}$, which with
$w_{j}\trianglelefteqslant e_{j}$ implies $w_{j}\trianglelefteqslant
e^{\prime}_{j}$, i.e.
$\mathsf{qBound}(\imath(w_{j}),\imath(\text{{t}}),w^{\prime}_{j}\rho_{j})$ is
true in $\mathcal{C}$.
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
qBound(W^{\prime},\imath(\alpha),w^{\prime}_{j})\theta^{\prime}\Leftarrow\Pi$
for $j=1\ldots m$. We reason for any fixed $j$. By construction of
$\theta^{\prime}$ and Lemma 4.2, we must prove that
$\mathsf{qBound}(\imath(d),\imath(\alpha),w^{\prime}_{j}\rho_{j})$ is true in
$\mathcal{C}$. As seen before, $\mathsf{qVal}(w^{\prime}_{j}\rho_{j})$ is true
in $\mathcal{C}$, therefore $w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$
for some $e^{\prime}_{j}\in D\setminus\\{\text{{b}}\\}$. From the side
conditions of ($\clubsuit$) we have $d\trianglelefteqslant\alpha\circ e_{j}$.
On the other hand, $\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$ and, in
particular,
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\mathsf{qBound}(\imath(e_{j}),\imath(\text{{t}}),w^{\prime}_{j}))$.
This, together with $w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$, means
$e_{j}\trianglelefteqslant e^{\prime}_{j}$. Now,
$d\trianglelefteqslant\alpha\circ e_{j}$ and $e_{j}\trianglelefteqslant
e^{\prime}_{j}$ implies $d\trianglelefteqslant\alpha\circ e^{\prime}_{j}$,
i.e. $\mathsf{qBound}(\imath(d),\imath(\alpha),w^{\prime}_{j}\rho_{j})$ is
true in $\mathcal{C}$.
* •
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
B^{\prime}_{j}\theta^{\prime}\Leftarrow\Pi$ for $j=1\ldots m$. In this case,
it is easy to see that
$B^{\prime}_{j}\theta^{\prime}=B^{\prime}_{j}\theta\rho_{j}$ by construction
of $\theta^{\prime}$ and because of the program transformation rules. On the
other hand, proof trees $T^{\prime}_{j}$ proving
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
B^{\prime}_{j}\theta\rho_{j}\Leftarrow\Pi$ can be obtained by inductive
hypothesis as seen before.
[2. $\Rightarrow$ 1.] (the transformation is sound). We assume that
$T^{\prime}$ is a a CHL($\mathcal{C}$) proof tree witnessing
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
A^{\prime}\rho\Leftarrow\Pi$ for some
$\rho\in\mbox{Sol}_{\mathcal{C}}(\Omega)$ such that
$\mathrm{dom}(\rho)=\mathrm{var}(\Omega)$. We want to to show the existence of
a $\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree $T$ witnessing
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp
d\Leftarrow\Pi$. We reason by complete induction on $\|T^{\prime}\|$. There
are three possible cases according to the the syntactic form of the atom
$A^{\prime}$. In each case we argue how to build the desired proof tree $T$.
— $A^{\prime}$ is a primitive atom $\kappa$. In this case due to TQCA and TPA
we can assume $A=\kappa$ and $\Omega=\\{\mathsf{qVal}(\imath(\text{{t}})),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),\imath(\text{{t}}))\\}$. Note
that $\mathrm{dom}(\rho)=\mathrm{var}(\Omega)=\emptyset$ implies
$\rho=\varepsilon$. Now, from $\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ \kappa\varepsilon\Leftarrow\Pi$ follows
$\Pi~{}{\models_{\mathcal{C}}}~{}\kappa$ due to the PA inference, and
therefore we can prove $\mathcal{P}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ \kappa\sharp d\Leftarrow\Pi$ with
a proof tree $T$ containing only one QPA node.
— $A^{\prime}$ is an equation $t==s$. In this case due to TQCA and TEA we can
assume $A=(t==s)$ and $\Omega=\\{\mathsf{qVal}(\imath(\text{{t}})),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),\imath(\text{{t}}))\\}$. Note
that $\mathrm{dom}(\rho)=\mathrm{var}(\Omega)=\emptyset$ implies
$\rho=\varepsilon$. Now, from $\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\
{\vdash}_{\\!\mathcal{C}}^{\\!}\ (t==s)\varepsilon\Leftarrow\Pi$ follows
$t\approx_{\Pi}s$ due to the EA inference, and therefore we can prove
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ (t==s)\sharp
d\Leftarrow\Pi$ with a proof tree $T$ containing only one QEA node.
— $A^{\prime}$ is a defined atom $p^{\prime}(\overline{t^{\prime}}_{n},W)$
with $p^{\prime}\in DP^{n+1}$. In this case due to TQCA and TDA we can assume
$A=p(\overline{t^{\prime}}_{n})$ and $\Omega=\\{\mathsf{qVal}(W),\
\mathsf{qBound}(\imath(d),\imath(\text{{t}}),W)\\}$. On the other hand,
$T^{\prime}$ must be rooted by a DA step ($\spadesuit$) using a clause
$C^{\mathcal{T}}\in\mathrm{elim}_{\mathcal{D}}(\mathcal{P})$ instantiated by a
substitution $\theta^{\prime}$. We can assume that ($\spadesuit$),
$C^{\mathcal{T}}$ and the corresponding clause $C\in\mathcal{P}$ have the form
already displayed in [1. $\Rightarrow$ 2.].
By construction of $C^{\mathcal{T}}$, we can assume
$B_{j}^{\mathcal{T}}=(B^{\prime}_{j},\ w^{\prime}_{j})$. Let
$\theta=\theta^{\prime}{\upharpoonright}\mathrm{var}(C)$ and
$\rho_{j}=\theta^{\prime}{\upharpoonright}\mathrm{var}(w^{\prime}_{j})~{}(1\geq
j\geq m)$. Then, due to the premises
$qV\\!al(w^{\prime}_{j})\theta^{\prime}\Leftarrow\Pi$ of ($\spadesuit$) and
Lemma 4.2 we can assume $e^{\prime}_{j}\in D\setminus\\{\text{{b}}\\}~{}(1\leq
j\leq m)$ such that $w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$.
To finish the proof, we must prove $\mathcal{P}\
{\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ A\sharp d\Leftarrow\Pi$. We claim
that this can be done with a $\mbox{QCHL}(\mathcal{D},\mathcal{C})$ proof tree
$T$ whose root inference is a QDA step of the form of ($\clubsuit$), as
displayed in [1. $\Rightarrow$ 2.], using clause $C$ instantiated by $\theta$.
In the premises of this inference we choose $d_{i}=\text{{t}}~{}(1\leq i\leq
n)$ and $e_{j}=e^{\prime}_{j}~{}(1\leq j\leq m)$. Next we check that these
premises can be derived from $\mathcal{P}$ in
$\mbox{QCHL}(\mathcal{D},\mathcal{C})$ and that the side conditions are
fulfilled:
* •
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\
(t^{\prime}_{i}==t_{i}\theta)\sharp d_{i}\Leftarrow\Pi$ for $i=1\ldots n$.
This amounts to $t^{\prime}_{i}\approx_{\Pi}t_{i}\theta$ which follows from
the first $n$ premises of ($\spadesuit$) given that
$t^{\prime}_{i}\rho=t^{\prime}_{i}$ and $t_{i}\theta^{\prime}=t_{i}\theta$.
* •
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ B_{j}\theta\sharp
e_{j}\Leftarrow\Pi$ for $j=1\ldots m$. From
$B_{j}^{\mathcal{T}}=(B_{j}^{\prime},w^{\prime}_{j})$ and due to rule TQCA, we
have $((B_{j}\theta)\sharp
e_{j}\Leftarrow\Pi)^{\mathcal{T}}=(B_{j}\theta\Leftarrow\Pi,\Omega_{j})$ where
$\Omega_{j}=\\{\mathsf{qVal}(w^{\prime}_{j}),\
\mathsf{qBound}(\imath(e_{j}),\imath(\text{{t}}),$ $w^{\prime}_{j})\\}$. From
the premises of ($\spadesuit$) and the fact that
$B^{\prime}_{j}\theta^{\prime}=B^{\prime}_{j}\theta\rho_{j}$ we know that
$\mathrm{elim}_{\mathcal{D}}(\mathcal{P})\ {\vdash}_{\\!\mathcal{C}}^{\\!}\
B^{\prime}_{j}\theta\rho_{j}\Leftarrow\Pi$ with a CHL($\mathcal{C}$) proof
tree $T^{\prime}_{j}$ such that $\|T^{\prime}_{j}\|<\|T^{\prime}\|$. Therefore
$\mathcal{P}\ {\vdash}_{\\!\mathcal{D},\mathcal{C}}^{\\!}\ B_{j}\theta\sharp
e_{j}\Leftarrow\Pi$ follows by inductive hypothesis provided that
$\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$. In fact, due to the form of
$\Omega_{j}$, $\rho_{j}\in\mbox{Sol}_{\mathcal{C}}(\Omega_{j})$ holds iff
$w^{\prime}_{j}\rho_{j}=\imath(e^{\prime}_{j})$ for some $e^{\prime}_{j}$ such
that $e_{j}\trianglelefteqslant e^{\prime}_{j}$, which is the case because of
the choice of $e_{j}$.
* •
$e_{j}\trianglerighteqslant^{?}w_{j}$ for $j=1\ldots m$. Trivial in the case
that $w_{j}=\ ?$. Otherwise they are equivalent to $w_{j}\trianglelefteqslant
e^{\prime}_{j}$ which follow from premises $\ulcorner
w^{\prime}_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner\theta^{\prime}\Leftarrow\Pi$
(i.e. $\ulcorner
w^{\prime}_{j}\rho_{j}\trianglerighteqslant^{?}\imath(w_{j})\urcorner\Leftarrow\Pi$)
of ($\spadesuit$) and Lemma 4.2.
* •
$d\trianglelefteqslant d_{i}$ for $i=1\ldots n$. Trivially hold due to the
choice of $d_{i}=\text{{t}}$.
* •
$d\trianglelefteqslant\alpha\circ e_{j}$ for $j=1\ldots m$. Note that
$\rho\in\mbox{Sol}_{\mathcal{C}}(\Omega)$ implies the existence of
$d^{\prime}\in D\setminus\\{\text{{b}}\\}$ such that
$\imath(d^{\prime})=W\rho$ and $d\trianglelefteqslant d^{\prime}$. On the
other hand, $e_{j}=e^{\prime}_{j}$ by choice. It suffices to prove
$d^{\prime}\trianglelefteqslant\alpha\circ e^{\prime}_{j}$ for $j=1\ldots m$.
Premises of ($\spadesuit$) and Lemma 4.2 imply that
$\mathsf{qBound}(W^{\prime}\theta^{\prime},\imath(\alpha),w^{\prime}_{j}\theta^{\prime})$
is true in $\mathcal{C}$. Moreover,
$W^{\prime}\theta^{\prime}=W\rho=\imath(d^{\prime})$ because of another
premise of ($\spadesuit$) and
$w^{\prime}_{j}\theta^{\prime}=\imath(e^{\prime}_{j})$ as explained above.
Therefore
$\mathsf{qBound}(W^{\prime}\theta^{\prime},\imath(\alpha),w^{\prime}_{j}\theta^{\prime})$
amounts to
$\mathsf{qBound}(\imath(d^{\prime}),\imath(\alpha),\imath(e^{\prime}_{j}))$
which guarantees $d^{\prime}\trianglelefteqslant\alpha\circ
e^{\prime}_{j}~{}(1\leq j\leq m)$.
The goal transformation correctness is established by the next theorem, which
relies on the previous result.
###### Theorem 6.
Let $G$ be a goal for a $\mbox{QCLP}(\mathcal{D},\mathcal{C})$-program
$\mathcal{P}$ such that $qV\\!al$ and $qBound$ do not occur in $G$. Let
$\mathcal{P}^{\prime}=\mathrm{elim}_{\mathcal{D}}(\mathcal{P})$ and
$G^{\prime}=\mathrm{elim}_{\mathcal{D}}(G)$. Assume a
$\mathcal{C}$-substitution $\sigma$, a mapping $\mu:\mathrm{war}(G)\to
D_{\mathcal{D}}\setminus\\{\text{{b}}\\}$ and a satisfiable finite set of
$\mathcal{C}$-constraints $\Pi$. Then, the following two statements are
equivalent:
1. 1.
$\langle\sigma,\mu,\Pi\rangle\in\mbox{Sol}_{\mathcal{P}}(G)$.
2. 2.
$\langle\theta,\Pi\rangle\in\mbox{Sol}_{\mathcal{P}^{\prime}}(G^{\prime})$ for
some $\theta$ that verifies the following requirements:
1. (a)
$\theta=_{\mathrm{var}(G)}\sigma$,
2. (b)
$\theta=_{\mathrm{war}(G)}\mu\imath$ and
3. (c)
$W\theta\in\mbox{ran}(\imath)$ for each
$W\in\mathrm{var}(G^{\prime})\setminus(\mathrm{var}(G)\cup\mathrm{war}(G))$.
|
arxiv-papers
| 2012-01-25T23:47:26 |
2024-09-04T02:49:26.684962
|
{
"license": "Public Domain",
"authors": "R. Caballero, M. Rodriguez-Artalejo and C. A. Romero-Diaz",
"submitter": "Rafael Caballero",
"url": "https://arxiv.org/abs/1201.5418"
}
|
1201.5424
|
# Almost All of Kepler’s Multiple Planet Candidates are Planets
Jack J. Lissauer11affiliation: NASA Ames Research Center, Moffett Field, CA
94035, USA , Geoffrey W. Marcy22affiliation: Astronomy Department, University
of California, Berkeley, CA 94720, USA , Jason F. Rowe11affiliation: NASA Ames
Research Center, Moffett Field, CA 94035, USA 33affiliation: SETI
Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA , Stephen T.
Bryson11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA ,
Elisabeth Adams44affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138, USA , Lars A. Buchhave55affiliation: Niels
Bohr Institute, University of Copenhagen, DK-2100, Copenhagen, Denmark
66affiliation: Centre for Star and Planet Formation, Natural History Museum of
Denmark, University of Copenhagen, DK-1350 Copenhagen, Denmark , David R.
Ciardi77affiliation: Exoplanet Science Institute/Caltech, Pasadena, CA 91125,
USA , William D. Cochran88affiliation: Department of Astronomy, University of
Texas, Austin, TX 78712, USA , Daniel C. Fabrycky99affiliation: Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA
1010affiliationmark: , Eric B. Ford1212affiliation: University of Florida,
211 Bryant Space Science Center, Gainesville, FL 32611, USA , Francois
Fressin44affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , John Geary44affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138,
USA , Ronald L. Gilliland1313affiliation: Space Telescope Science Institute,
Baltimore, MD 21218, USA , Matthew J. Holman44affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , Steve B.
Howell11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA ,
Jon M. Jenkins11affiliation: NASA Ames Research Center, Moffett Field, CA
94035, USA 33affiliation: SETI Institute/NASA Ames Research Center, Moffett
Field, CA 94035, USA , Karen Kinemuchi11affiliation: NASA Ames Research
Center, Moffett Field, CA 94035, USA 1414affiliation: Bay Area Environmental
Institute, CA , David G. Koch11affiliation: NASA Ames Research Center, Moffett
Field, CA 94035, USA , Robert C. Morehead1212affiliation: University of
Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA , Darin
Ragozzine44affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , Shawn E. Seader11affiliation: NASA Ames
Research Center, Moffett Field, CA 94035, USA 33affiliation: SETI
Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA , Peter G.
Tanenbaum11affiliation: NASA Ames Research Center, Moffett Field, CA 94035,
USA 33affiliation: SETI Institute/NASA Ames Research Center, Moffett Field, CA
94035, USA , Guillermo Torres44affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , Joseph D.
Twicken11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA
33affiliation: SETI Institute/NASA Ames Research Center, Moffett Field, CA
94035, USA Jack.Lissauer@nasa.gov
###### Abstract
We present a statistical analysis that demonstrates that the overwhelming
majority of Kepler candidate multiple transiting systems (multis) indeed
represent true, physically-associated transiting planets. Binary stars provide
the primary source of false positives among Kepler planet candidates, implying
that false positives should be nearly randomly-distributed among Kepler
targets. In contrast, true transiting planets would appear clustered around a
smaller number of Kepler targets if detectable planets tend to come in systems
and/or if the orbital planes of planets encircling the same star are
correlated. There are more than one hundred times as many Kepler planet
candidates in multi-candidate systems as would be predicted from a random
distribution of candidates, implying that the vast majority are true planets.
Most of these multis are multiple planet systems orbiting the Kepler target
star, but there are likely cases where (a) the planetary system orbits a
fainter star, and the planets are thus significantly larger than has been
estimated, or (b) the planets orbit different stars within a binary/multiple
star system. We use the low overall false positive rate among Kepler multis,
together with analysis of Kepler spacecraft and ground-based data, to validate
the closely-packed Kepler-33 planetary system, which orbits a star that has
evolved somewhat off of the main sequence. Kepler-33 hosts five transiting
planets with periods ranging from 5.67 to 41 days.
###### Subject headings:
planetary systems; stars: individual (Kepler-33/KOI-707/KIC 9458613); methods:
statistical
††slugcomment: ApJ in press1111affiliationtext: Hubble Fellow
## 1\. Introduction
Roughly one-third of Kepler’s planet candidates announced by Borucki et al.
(2011) are associated with targets that have more than one candidate planet.
False positives (FPs) plague ground-based transit searches, but the exquisite
quality of Kepler photometry, combined with the ability to measure small
deviations in center of light during transits (Jenkins et al., 2010; Batalha
et al., 2010), have been used to cleanse the sample prior to presentation in
Borucki et al. (2011). Accounting for candidates on each one’s individual
merit, Morton & Johnson (2011) estimated the fidelity of Kepler’s planet
candidates (fraction of the candidates expected to be actual planets) to be
above $90\%$. Yet the fidelity of multiple planet candidates is likely to be
higher than that for singles (Latham et al., 2011; Lissauer et al., 2011a). We
show herein that the vast majority of Kepler’s multiple planet candidates are
true multiple planet systems.
The majority of Kepler planet discoveries announced to date have been
confirmed dynamically, using radial velocity variations of the target star or
departures of observed transit times from a linear ephemeris (transit timing
variations, TTVs). However, three of the first sixteen Kepler planets do not
have dynamical evidence supporting the discovery, but rather these planets
have been “validated” by showing that the probability that the observed signal
is produced by a planet around the host star is at least 100 times as large as
that of a false positive (Torres et al., 2011; Fressin et al., 2011; Lissauer
et al., 2011b). For these three planets, the validation process was based on
the results of the BLENDER code, which performs detailed comparisons between
the observed Kepler lightcurve and the lightcurves predicted for an ensemble
of theoretically conceivable FPs to reject FP models that are not consistent
with the observations. BLENDER then sums the a priori probability for the
allowed false positives and compares that to the a priori probability for the
planet model. BLENDER validates a planet when the resulting odds ratio very
strongly favors the planet model. Although all three of these planets orbit
stars that also have dynamically-confirmed planets, planetary multiplicity was
not used quantitatively to aid in their validation.
Morton & Johnson (2011) do a more cursory analysis, but one that is far
easier, enabling them to perform it for all of the candidate planets
(“candidates”, henceforth) that are listed in the Borucki et al. (2011) paper.
The study by Morton & Johnson (2011) assumes a flat 20% planet occurrence rate
as a prior for their Bayesian analysis (i.e., the average number of planets
per star is assumed to be 0.2), and they do not consider planets orbiting
background stars to be FPs. Morton & Johnson (2011) use the depth and period
of the transit, together with the magnitude, stellar properties and galactic
latitude of the target star (henceforth “target”) to provide an upper bound on
the FP prior; for high SNR transits, this upper bound on the FP prior can be
much higher than the one computed using detailed lightcurve matching by
BLENDER to rule out many FP scenarios (Torres et al., 2011; Fressin et al.,
2011). Indeed, Morton & Johnson (2011) do not examine Kepler lightcurves, but
rather assume for their calculations that various false-positive vetting
procedures using Kepler photometry have been done, such as elimination of
candidates with V-shaped lightcurves or where the centroid of light received
on the Kepler focal plane is in a different position during a transit than it
is outside of transit, and that the planet candidates show no evidence for
being FPs. However, these cuts were not applied to all of the candidates
listed in Borucki et al. (2011). Some candidates were not vetted as
extensively as assumed by Morton & Johnson (2011), and other candidates did
not pass one of the cleanness tests but still remain viable, if somewhat more
suspect, planetary candidates (e.g., grazing planetary transits, as well as
grazing stellar eclipses, produce V-shaped lightcurves), and thus remain on
Borucki et al. (2011)’s candidate list.
We develop herein a quantitative method to estimate the true planet fraction
of the overall sample of Kepler candidate multiple planets that is based upon
the observation that there are far more multiples than would be expected for a
random distribution of candidates among Kepler targets (Lissauer et al.,
2011a). This “overabundance” of multiples is illustrated by the fact that
fewer than 1% of targets have a candidate on Borucki et al. (2011)’s candidate
list, whereas more than 15% of targets with at least one candidate have
multiple candidates, and more than 25% of targets with at least two candidates
have a third candidate. Such a result is expected if planets often appear
within systems of planets, especially if such systems tend to be flat in the
distribution of inclinations. High multiplicity, low inclination, planetary
systems are consistent with Kepler observations (Lissauer et al., 2011a).
We describe an approach to validation of specific multiple planet system
candidates that is intermediate between those of Morton & Johnson (2011) and
BLENDER, and we apply these techniques to validate the very flat and
dynamically-packed KOI-707 system111Targets with planetary candidates are
given an integer KOI (Kepler Object of Interest) number, and individual
candidates are denoted by the target KOI number followed by a decimal point
and a two digit number specifying the order in which they were identified. For
example, KOI-707.02 is the second planetary candidate to have been identified
by transit-like signatures in the lightcurve of the target star KOI-707.. In
this approach, applied herein exclusively to KOI-707, we examine the target
and candidates individually, supplementing the Kepler Input Catalog (KIC,
Brown et al., 2011) with spectroscopic observations of the target and high-
resolution imaging, including Keck guider camera exposures and speckle imaging
(Howell et al., 2011) and/or adaptive optics (Troy et al., 2000; Hayward et
al., 2001), to search for other stars in the target aperture. We check the
lightcurve for a shape consistent with a transit. We verify that the centroid
motion is nil or consistent with a transit around the target star and not a
transit/ eclipse of any other stars observed nearby. But we do not perform
detailed matching of observed lightcurves to FP scenarios. We then use
multiplicity to effectively increase the planet priors and allow for
validation in a Bayesian sense. Note that for candidate giant ($\sim$ 1
Jupiter radius) planets, one would need to consider the possibility of the
transit signal being produced by a late M star partially eclipsing the target
star (perhaps with dilution); for smaller candidates, the possibility of
transits of fainter stars by larger planets must be taken into account.
The probability of a candidate in a multiple system being a false positive is
contrasted to that of a single candidate in Section 2. Other factors involved
in validating transiting planet candidates are discussed in Section 3. We
apply our method to the KOI-707 candidate multi-planet system in Section 4. We
conclude in Section 5 with a summary of our results and a discussion of their
implications.
## 2\. Statistical Validation of Multi-Planet Candidates
Kepler has found far more multiple candidate systems than would be the case if
candidates were randomly distributed among target stars (Lissauer et al.,
2011a). False positives are expected to be nearly randomly distributed among
Kepler targets, whereas true transiting planets could be clustered if planets
whose sizes and periods are adequate for transits to be detected often come in
multiples, as is the case for planets detected by radial velocity variations
(Wright et al., 2009), and/or if planetary systems tend to be flat, so
geometry leads to higher transit probabilities for other planets if one planet
is seen to transit (Ragozzine & Holman, 2010). We quantify the excess of
multiples and its implications for the overall fidelity of the sample of
multiple candidates in this section.
We use the following notation and input parameters: The number of planet
candidates is $n_{c}=1199$, the number of targets with two or more candidates,
i.e., the number of candidate multiple planet systems, is $n_{m}=170$, and the
number of targets from which the sample is drawn is $n_{t}$ = 160,171. These
numerical values are taken from Lissauer et al. (2011a), who removed from the
1235 candidates listed in Borucki et al. (2011) those objects that were
estimated to have radii greater than twice that of Jupiter, those for which
only one transit had been observed, and those identified as FPs by Howard et
al. (2011). We denote the number of actual planets among the candidates as
$n_{p}$, the number of false positives as $n_{f}$, and the fidelity of the
sample (fraction of candidates that are planets) as $P$, so
$n_{c}=n_{p}+n_{f}$ and $P=n_{p}/n_{c}$.
For our calculations in Equations (1–9), we make the following two
assumptions:
i) FPs are randomly distributed among the targets;
ii) there is no correlation between the probability of a target to host one or
more detectable planets and to display FPs.
In contrast, we do not assume that planets are randomly distributed among the
targets. The previous verifications of the two-planet Kepler-10 system
(Batalha et al., 2011; Fressin et al., 2011), the three-planet Kepler-9
(Holman et al., 2010; Torres et al., 2011) and Kepler-18 systems (Cochran et
al., 2011), and especially the six-planet Kepler-11 system (Lissauer et al.,
2011b), provide strong evidence of the non-random distribution of transiting
planets among Kepler target stars.
The above two assumptions, together with observed values of $n_{c}$, $n_{m}$
and $n_{t}$ and an assumed lower bound on $P$, allow us to estimate lower
bounds on the fidelity of candidates in various classes of candidate multi-
planet systems using simple algebra. For example, assumption (1) implies that
the expected number of targets with $k$ false positives, $E(k)$, is given by a
Poisson distribution of $n_{t}$ members whose mean is given by the average
number of false positives per target, $\lambda=(1-P)n_{c}/n_{t}$. This
expectation value is given by the following formula:
$\displaystyle E(k)=\frac{\lambda^{k}e^{-\lambda}}{k!}n_{t}$ $\displaystyle=$
$\displaystyle\frac{((1-P)\frac{n_{c}}{n_{t}})^{k}e^{-(1-P)\frac{n_{c}}{n_{t}}}}{k!}n_{t}$
$\displaystyle\approx$
$\displaystyle\frac{((1-P)\frac{n_{c}}{n_{t}})^{k}}{k!}n_{t},$
where the approximation in Equation (1) is valid for $\lambda\ll 1$.
We next compute the expected number of Kepler targets with multiple candidates
at least one of which is a FP. In some cases, we make approximations that
yield conservative estimates of the expected numbers of true planets (i.e.,
overestimate the expected number of FPs). Our results, presented in Equations
(2–7), are given first as general formulae, then as a function of the fraction
of candidates that are true planets, $P$, for the observed values of 160,171
targets, 1199 candidates, and 170 multiple candidate systems stated above, and
finally these value if $P$ = 0.5 or 0.9222We choose to present numerical
results for $P=0.9$ because we consider it to be a reasonable estimate of the
actual planet fidelity rate, and $P=0.5$ to show that even for an overall FP
rate far higher than studies suggest, the expected number of FPs among the
multis is remarkably low.. These final numbers are to be compared to the
observation that there are 115 candidate 2-planet systems, with a total of 230
planet candidates, and 55 candidate systems of 3 or more transiting planets,
with a total of 178 planet candidates.
Equation (1) yields estimates of the number of targets with two or three false
positives:
$\displaystyle 2~{}{\rm FPs:}~{}\frac{((1-P)n_{c})^{2}}{2n_{t}}$
$\displaystyle=$ $\displaystyle 4.49(1-P)^{2}$ $\displaystyle=$ $\displaystyle
1.12~{}{\rm or}~{}0.045;$ $\displaystyle 3~{}{\rm
FPs:}~{}\frac{((1-P)n_{c})^{3}}{6n_{t}^{2}}$ $\displaystyle=$ $\displaystyle
0.011(1-P)^{3}$ $\displaystyle=$ $\displaystyle 0.0014~{}{\rm or}~{}1.1\times
10^{-5}.$
The number of targets expected to have four or more false positives is very
small and can be neglected for our purposes. Equation (1) also yields the
expected number of targets with a single FP under the same assumptions, 597.3
or 119.8.
The assumed lack of correlation between the propensity of a target to have
false positives and true planets implies that the probability that a given
target hosts both a planet and one or more false positive is equal to the
product of these individual probabilities. The probability of a target having
a detected transiting planet is given by $Pn_{c}/n_{t}$, and the probability
of it showing a false positive is given by Equation (1). Thus, the estimated
numbers of targets with at least one planet as well as one or more false
positives are given by:
$\displaystyle 1~{}{\rm planet}~{}+1~{}{\rm FP:}$
$\displaystyle\frac{n_{c}P}{n_{t}}\times\frac{n_{c}(1-P)}{n_{t}}\times n_{t}$
$\displaystyle=$ $\displaystyle P(1-P)\frac{n_{c}^{2}}{n_{t}}$
$\displaystyle=$ $\displaystyle 8.98P(1-P)=2.25~{}{\rm or}~{}0.81;$
$\displaystyle 1~{}{\rm planet}~{}+2~{}{\rm FPs:}$ $\displaystyle
P(1-P)^{2}\frac{n_{c}^{3}}{2n_{t}^{2}}$ $\displaystyle=$ $\displaystyle
0.034P(1-P)^{2}$ $\displaystyle=$ $\displaystyle 0.0042~{}{\rm or}~{}3\times
10^{-4}.$
The number of targets expected to have both a planet and three or more false
positives is very small and can be neglected for our purposes. Somewhat lower
estimates of the number of multi- candidate systems with one planet and at
least one FP than given by Equations (4) and (5) can be derived by noting that
the observed multiplicity implies fewer (than $n_{c}$) targets with a nonzero
number of planet candidates.
To derive an estimate of the expected number of targets with multiple true
planets as well as at least one FP, replace the term in Equations (4) and (5)
that represents the probability of a given target having a true planet,
$Pn_{c}/n_{t}$, with a term representing the probability of a given target
having a multi-planet system, $n_{m}/n_{t}$. (Assuming all multis are multi-
planet systems to estimate of this term is conservative in the sense that it
may lead to an overestimate in the number of false positives in multi-
candidate systems.)
$\displaystyle{\rm 2~{}or~{}more~{}planets}~{}+1~{}{\rm FP:}$
$\displaystyle\frac{n_{m}}{n_{t}}\times\frac{n_{c}(1-P)}{n_{t}}\times n_{t}$
$\displaystyle=$ $\displaystyle(1-P)\frac{n_{m}n_{c}}{n_{t}}=1.27(1-P)$
$\displaystyle=$ $\displaystyle 0.64~{}{\rm or}~{}0.13;$
$\displaystyle{\rm 2~{}or~{}more~{}planets}~{}+2~{}{\rm FPs:}$
$\displaystyle\frac{n_{m}}{n_{t}}\times(1-P)^{2}\times\frac{n_{c}^{2}}{2n_{t}}$
$\displaystyle=$ $\displaystyle 0.0048(1-P)^{2}$ $\displaystyle=$
$\displaystyle 0.0012~{}{\rm or}~{}5\times 10^{-5}.$
The number of targets expected to have both multiple planets and three or more
false positives is extremely small and can be neglected for our purposes.
### 2.1. Numerical Estimates for Kepler Candidates
Adding up the above estimates of FPs of different types yields expected totals
of 4.5 or 0.9 FPs (for assumed FP rates of 50% and 10%, respectively) in
doubles (compared to a total of 230 candidates) and 0.65 or 0.13 FPs in
systems with three or more candidates (compared to a total of 178 candidates).
These numbers suggest that validation of doubles at the 99% level by this
method alone is marginal. Triples and higher multiplicity candidates are more
strongly validatable, enough so that the one dynamically-unstable candidate
triple system noted by Lissauer et al. (2011a), KOI-284, which has a pair of
candidates with a period ratio $1.0383$, should be viewed as somewhat
surprising (but see Section 2.4).
It is of interest to compare the (low) false positive rates for multis
estimated in the previous paragraph with the presumed FP rates for singles
upon which their calculations are based. For 50% FPs in singles, our estimates
above correspond to 2% FPs in doubles (compare the first numbers at the end of
equations 2 and 4 with the 230 candidates in Kepler candidate 2-planet
systems) and 0.4% FPs for targets with three or more candidates (compare the
first numbers at the end equations 3 and 5–7 with the 178 candidates in Kepler
candidate 3 or more planet systems); for 10% FPs in singles, we estimate 0.4%
FPs in doubles (compare the final numbers at the end of equations 2 and 4 with
the 230 candidates in Kepler candidate 2-planet systems) and 0.1% for targets
with three or more candidates (compare the final numbers at the end equations
3 and 5–7 with the 178 candidates in Kepler candidate 3 or more planet
systems). These values are consistent with a “multiplicity boost” in the
Bayesian prior for planets of $\sim$ 25 for a system with one additional
candidate and of $\sim$ 100 for a system with more than one additional
candidate. The probability that a target hosts a planet candidate, 1/150, can
be compared to the probability of a target that hosts one candidate hosts
another, 1/6, and that an additional candidate has been found for higher
multiplicity systems, 1/3. These latter values, together with the assumption
that FPs and true planets are uncorrelated, lead to an estimate of the
multiplicity boost for candidate doubles of 30, slightly larger than the value
of 25 computed above, and to a somewhat reduced estimate of $\sim$ 50 for
candidates in systems of higher multiplicity.
The multiplicity boost introduced in the previous paragraph can be used to
quantify the increased probability that a set of transit-like signatures
represent a real transiting planet if one or more additional set of transit-
like signatures are detected for the same Kepler target; i.e., to compute an
estimate the probability of a particular candidate in a multiple candidate
system being a true planet from an estimate of that candidate being a planet
that does not account for multiplicity. We denote the probability of
planethood of the candidate in question computed without factoring in
multiplicity by $P_{1}$ and the probability of it being a planet accounting
for multiplicity by $P_{2}$ if it is in a two candidate system and by $P_{3+}$
if it is a member of a three or more candidate system. In our formulation,
multiplicity effectively increases the Bayesian estimate of the planet prior
but leaves the FP prior unchanged. Thus the probability of the planet
interpretation accounting for multiplicity is related to that neglecting
multiplicity according to the following approximate formulae:
$\displaystyle P_{2}$ $\displaystyle\approx$
$\displaystyle\frac{25P_{1}}{25P_{1}+(1-P_{1})}$ $\displaystyle~{}($
$\displaystyle\approx 1-\frac{1-P_{1}}{25}~{}{\rm for}~{}1-P_{1}\ll 1,~{}{\rm
and}~{}\approx 25P_{1}~{}{\rm for}~{}P_{1}\ll 1)$ $\displaystyle P_{3+}$
$\displaystyle\approx$ $\displaystyle\frac{50P_{1}}{50P_{1}+(1-P_{1})}$
$\displaystyle($ $\displaystyle\approx 1-\frac{1-P_{1}}{50}~{}{\rm
for}~{}1-P_{1}\ll 1,~{}{\rm and}~{}\approx 50P_{1}~{}{\rm for}~{}P_{1}\ll 1),$
where in each case we use the smaller value computed in the previous
paragraph. Rough estimates for the planet probabilities of individual Kepler
candidate multis can be obtained by applying Expressions (8) and (9) to the
estimates of planet probability by Morton & Johnson (2011); the Morton &
Johnson (2011) planet probabilities are given by $P_{1}$ in the above formulae
since their numbers do not account for multiplicity. Note that this
multiplicity boost incorporates self-consistently both the coplanarity boost
discussed in Cochran et al. (2011) and the systems boost (that planets tend to
come in systems) without need to disentangle these effects.
The above calculations of the multiplicity boost do not assume that any of the
candidates of the target under consideration have been confirmed or validated
as planets. Does validating one or more candidate as a true planet(s)
substantially increase the probability that other candidates of the same
target are also planets? For realistic estimates of FP rates (e.g., 10%), the
equations above show that most FPs in multis are likely to be combinations of
1 FP with one or more true planets, so validating one candidate does not
substantially increase confidence in the others from a statistical viewpoint.
Nonetheless, validating one candidate does eliminate some of the non-
statistical issues raised in the following three subsections. In particular,
if that validated candidate is shown to not only be a planet but also to orbit
the Kepler target star, then the probability that the other candidates are
physically-associated with the target increases.
### 2.2. Caveats
To examine the validity of the three assumptions stated above, we divide FPs
into two classes, (I) chance-alignment blends such as background eclipsing
binaries (BGEBs) and (II) physical triple stars (PTs), where grazing eclipses
of the target star by a stellar companion are considered together with PTs for
this purpose. We note that it is a matter of choice how to count cases where
the transit signal is produced by a planet transiting a star other than the
Kepler target. Larger planets transiting stars that are not physically
associated with the star providing most of the target’s flux can be considered
to fall within the chance-alignment blends/BGEB class of FPs (as they must be
for Kepler’s statistical census of planets, because background stars are not
in the denominator of the ratio taken to calculate the fraction of stars with
planets, so their planets must not be included in the numerator), or
alternatively not be regarded as FPs when assessing individual candidates and
the reliability of the sample. Planets transiting physical companions to the
star providing most of the flux can either be viewed as members of the PT
class of FPs or not classified as FPs; how these planets are categorized for
Kepler’s statistical census of planets is also a matter of choice.
Assumptions (i) and (ii) lead to the following caveats:
(i): The expected number of BGEBs (chance-alignment blends) that can cause FPs
varies with the magnitude and galactic latitude of the target star, whereas
those of PTs do not (except to the extent that the distribution of stellar
types among Kepler targets depends somewhat on magnitude). The variations in
expected BGEBs act to increase the expected total number of systems with 2 or
more FPs compared to the value given by the above formulae. When looking at
individual candidate systems, the FP probability of those around faint targets
and at low galactic latitude is larger, whereas the planetary interpretation
becomes more likely for brighter targets and those at higher galactic
latitude. However, the distributions in galactic latitude of the planet
candidates and of the multiple candidate systems both track that of all Kepler
targets, whereas the distribution of expected BGEB FPs is quite different
(Figure 1); this suggests that the fraction of candidates (both singles and
multis) that are produced by BGEBs is small.
Figure 1.— Distributions in galactic latitude of all 163,986 Kepler Quarter 2
exoplanet targets (solid turquoise curve), the 961 targets with planet
candidates meeting our criteria (solid dark red curve) and the 170 multiple
candidate systems (solid black curve). The dashed curve shows the distribution
of expected BGEB FPs, which we compute by weighting each target by the sky
density of stars with magnitudes between 15.0 and 20.0 according to the
Besancon model of the galaxy (Robin et al. 2003, 2004). All curves are
normalized to give a total of unity. Error bars represent the statistical
uncertainty (1$\sigma$ binomial variation) of each of the planet candidate
curves at quartile points in their distributions. The basic shape of the three
solid curves is governed by the orientation of the Kepler field on the sky,
where the amount of area covered increases from a point at galactic latitude
$5^{\circ}$ to a region spanning $16^{\circ}$ in longitude at $13^{\circ}$ in
galactic latitude and then decreases to a point at $21^{\circ}$ latitude; the
second most important factor is a weighting towards the galactic plane because
of the higher spatial density of target stars in that portion of the field.
The dashed curve is more heavily weighted towards the galactic plane because,
to first approximation, it is weighted quadratically in the spatial density of
stars (more precisely, the product of the spatial densities of target stars
and background stars), whereas the other curves are only weighted linearly.
The spatial density of background stars drops by a factor of 17 from the
lowest latitude Kepler observes to the highest latitude. The similarity of
curves for targets, planets and multis, and the difference of these from the
curve estimating BGEB FPs, suggests that the fraction of candidates that are
produced by BGEBs (as well as that resulting from planets transiting
background stars) is small.
The statistical analysis that we presented above does not discriminate between
multiple candidates resulting from multiple planets orbiting a background star
and a single candidate resulting from a single planet orbiting a background
star. However, the correlations and lack thereof between the curves shown in
Figure 1 also imply that the fraction of KOIs caused by single or multiple
planets transiting a background star is small. In some cases, it is especially
unlikely that a multiple candidate system is produced by several planets
orbiting a background star, either because its largest member would be greater
than planetary in size, or because the requirement of dynamical stability
would imply implausibly small densities for the larger planets orbiting the
fainter star. Additionally, as the fraction of giant planets among the multis
is smaller than it is for the singles (Latham et al., 2011), the probability
of a multi being a system of larger planets orbiting a fainter star is less
than that of a single candidate orbiting a star fainter than the target.
A second class of multiple FPs can be produced from a single physical system,
either a triple star with two sets of eclipses or a background star with
multiple transiting planets. Triple star systems are unstable for small period
ratios and produce large eclipse timing variations for moderate period ratios,
so if large timing variations are not detected, we do not need to consider
triple star system FPs (nor grazing eclipsing binary stars with a transiting
planet) unless period ratios are large.
Quadruple star systems consisting of two pairs of eclipsing binaries (e.g.,
Graczyk et al., 2011) could produce a small excess probability of a pair of
FPs. Such systems, as well as line of sight pairs of physically-unrelated EBs,
also might be more likely to mimic planetary-depth transits because of mutual
dilution of the fraction of light in the Kepler aperture.
We note also that FPs are inherently more likely among both small (Earth-size
and smaller) and large (Jupiter-size and larger) candidates than among
intermediate (Neptune-size) candidates, independent of multiplicity. Most
small candidates produce low-amplitude transits that reduce our ability to
remove false positives from the candidate list on the basis of the shape of
the lightcurve or from differences in the centroid of the location of the dip
relative to that of the target. Small candidates, including multis, are more
likely to be planets orbiting stars fainter than the target simply because the
Neptune-size planets that could produce such signals are far more common
according to Kepler observations than are giant planets (Borucki et al.,
2011), and this difference in abundance is even larger for the multis than for
the singles (Latham et al., 2011). At the other extreme of the planet size
distribution, giant candidates appear to be more contaminated by stellar false
positives (Demory & Seager, 2011) of the type which plague ground-based
surveys than do Neptune-size candidates.
(ii): Planets and FPs of both types are more easily detectable for quieter
targets; planets and PTs are more detectable for brighter targets, but BGEBs
that are bright enough to mimic observable transits of the target star are
more likely for fainter targets. A grazing eclipsing binary with a
circumstellar or circumbinary transiting planet analogous to Kepler-16 (Doyle
et al., 2011) would violate our assumption of a lack of correlation between
planets and FPs, but such a configuration requires a large period ratio for
stability and an even larger period ratio for the planet to have modest TTVs.
The assumption of no correlation between planets and FPs would be violated if
the orientations of orbital planes of PTs with one star having a planet are
not random. Alignments of planetary orbital planes for binary stars for which
both stars possess planets also violate this assumption, although members of
this class are true planets that are all physically associated (albeit less
directly) with the target star and thus would for many purposes not be
classified as FPs.
The numbers quoted in Equations (2–9) do not incorporate the caveats noted in
this subsection. Correlations between the propensity of a target to have one
or more FPs and to host one or more identified true planets (ii) are likely to
be at most a factor of a few. False positives of the BGEB class are likely to
be concentrated among faint target stars that are located at low galactic
latitude (Morton & Johnson, 2011); this concentration increases the expected
number of targets with two (or more) FPs, but also points to a substantial
portion of Kepler’s target list whose members have a priori FP likelihood that
is lower than or comparable to those given by the numbers above.
### 2.3. Period Ratios: Resonances & Stability
The distribution of candidate period ratios exhibits spikes near strong mean
motion resonances, which implies that candidates with such period ratios are
more likely to be true planets than those candidates not in or very near such
resonances (Lissauer et al., 2011a). This clustering can be used to increase
confidence in candidates with periods near resonances; such a “near-resonance
boost” can be quantified using the measured values of the normalized distance
from resonance computed from the distribution of planet period ratios in
Lissauer et al. (2011a). Candidate resonant planets can, however, arise from
period aliases in analysis of light curves; such aliases are not of the form
of the astrophysical FPs considered in this paper, and we note that special
care to search for them is an important aspect of validating candidate
resonant planets (see the discussion of KOI-730 and the Note Added in Proof in
Lissauer et al. (2011a)).
Resonant period ratios increase confidence in various candidates, and as
mentioned at the beginning of Section 2.1, one of the candidate systems,
KOI-284, is highly suspect because two of the candidates have periods that
differ from one another by $<4\%$, which would rapidly lead to a dynamical
instability on a very short time scale (but see the discussion of this system
in Section 2.4). Do non-resonant periods reduce the confidence in individual
multis and do stable systems produce an increase in our confidence? The
answers to both parts of this question are yes, but in most cases these
effects are minor. While resonances are overpopulated, a substantial majority
of multis are not in nor near such resonances (Lissauer et al., 2011a; Wright
et al., 2011). And while drawing planetary periods randomly from the observed
ensemble of transiting planet periods gives many more nominally unstable
systems than are observed, the fraction of unstable configurations from such a
random draw is still small. Nonetheless, in high-multiplicity densely-packed
systems, wherein most configurations with the same number of planets randomly
spread over the same range in period would likely be unstable, the boost in
confidence from stability can be significant. Note that special period ratio
boosts and stability boosts are in addition to the standard multiplicity
boost, i.e., they can be applied together as two multiplicative factors in the
planet prior for appropriate systems.
### 2.4. KOI-284 & Multis Composed of Planets Transiting Differing Stars in a
Binary
A very rough estimate of the expected number of multiple candidate systems
that orbit different members of a binary/multiple star system (rather than a
single system of planets orbiting the same star) can be made by assuming that
half of Kepler’s targets are binaries, that each star within a binary is as
likely to host both single and multiple transiting planets as are single
stars, and that the probability of one star within a binary hosting transiting
planets is uncorrelated with that of its binary companion. These assumptions
imply $n_{c}$ planets spread around $3n_{t}/2$ stars, and thus
$(2/3)(n_{c}/n_{t})$ planets per star and $2n_{c}^{2}/(9n_{t})\approx 2$ cases
in which multis are composed of planets transiting differing members of a
stellar binary. Positive correlations between the propensity of stars in a
binary to have transiting planets (resulting from the abundance of planets
and/or alignment of orbital planes) would increase this number, but the fact
that a significant majority of binaries are composed of stars of substantially
differing surface brightness combined with the paucity of giant planets
transiting low luminosity stars should reduce the number of multis comprised
of two planetary systems because of the difficulty of detecting planets around
secondary stars.
As mentioned above, KOI-284, with three planetary candidates having periods of
6.18, 6.42, 18.0 days and nominal sizes of $\sim$ 2 R⊕ (Earth radii), is the
only one of 170 multi-candidate system identified in Borucki et al. (2011)
that would be clearly dynamically unstable if all of the planets orbited the
same star (Lissauer et al., 2011a)333The other multi that was unstable
according to the integrations reported in Lissauer et al. (2011a) was the 4
candidate system KOI-191. This instability resulted from the strong
perturbations of giant planet candidate KOI-191.01 on the nearby outermost
candidate KOI-191.04. As the period ratio of these two candidates was 1.258,
Lissauer et al. (2011a) speculated that the system was librating in a 5:4 mean
motion resonance and that this resonance protected the planets from close
approaches. Subsequent data analysis shows that the period of KOI-191.04 was
underestimated by a factor of two. The updated value of the period of this
candidate is 38.6516 $\pm$ 0.0012 days. The revised period places the planets
much farther apart, and the system is stable with the updated parameters..
Analysis of a McDonald observatory spectrum of this bright (Kepler magnitude
$Kp=11.8$) target yields a temperature of $\sim$ 6250 K and log$g=4.5$, which
suggest a F7V star that is twice as luminous as the Sun. Speckle images of
KOI-284 obtained on 24 June 2010 revealed two stars differing in brightness by
less than one magnitude and separated from one another by slightly less than
1′′ (Howell et al., 2011). Spectroscopic observations of each individual star
obtained at Keck in 2011 show the two stars to have nearly identical spectra
and have a difference in radial velocity of 0.94 $\pm$ 0.1 km/s. The nearly
identical velocities are consistent with their being gravitationally bound.
Their separation of $0.9^{\prime\prime}$ at a distance of roughly 500 pc
implies a projected separation of 450 AU; the relative orbital velocities for
two 1 $M_{\odot}$ stars on a circular orbit with semimajor axis $a$ = 450 AU
is 2 km/s. These observations suggest that the three candidates may well all
be transiting planets in the same stellar system, with one of the stars
hosting one of the six day period planets and the other two planets orbiting
the other star in the binary. Thus, this system would not call into question
the statistical arguments presented in Section 2.1. A detailed analysis of
both Kepler and ground-based data to test this hypothesis and elucidate
particulars is currently underway (Bryson et al., 2012).
## 3\. Transit Characteristics: Planets vs. False Positives
The numbers derived in Section 2 suggest that the sample of multi-planet
candidates identified by Borucki et al. (2011) is likely to include only a
small fraction of false positives, of order 1%. But because of the
uncertainties associated with the caveats discussed in Section 2, as well as
any factors that we may have omitted, we do not consider it appropriate to
view all of Kepler’s multi-planet candidates to be validated planets at the
99% level. Nonetheless, it is possible to identify candidates with various
favorable characteristics as detailed below, and to examine the available data
for systems individually for signs of potential inconsistencies with the
planet hypothesis. See Haswell (2011) for a more detailed discussion of
transit characteristics.
The shape and duration of a candidate transit can be used to distinguish
events produced by planets from false positives produced by known
astrophysical sources. The BLENDER code (Torres et al., 2011) compares the
observed lightcurve to synthetic lightcurves of both a planet transiting the
target star and a vast array of possible astrophysical false positives
involving binary stars and larger planets transiting fainter stars. If the
planet hypothesis does not provide an acceptable fit to the data, the
candidate is not viable; candidates for which the planet fit is marginal are
called into question, and such candidates are not considered further herein.
Astrophysical false positives are considered credible if they provide an
acceptable fit (3$\sigma$) to the data. Then the a priori likelihoods of the
planet hypothesis and that of a false positive are compared, and a probability
that the apparent transit is caused by a blend is computed. The process can be
very involved, especially for candidates with orbital periods long enough that
astrophysical false positives could plausibly be produced by objects with
eccentric (not tidally-damped) orbits (Fressin et al., 2011; Lissauer et al.,
2011b).
True planetary transits of the star that is the dominant source of light in
the aperture are wavelength-independent (neglecting the small effects of
stellar limb-darkening and contamination by flux from other stars in the
aperture), whereas false positives are not (except in the unlikely case that
the effective temperatures of the contributing stars are similar). By
providing infrared time series spanning times of transit, the Warm Spitzer
Mission can assist in the validation of many transiting planet systems.
Several candidates in multiple systems have been observed, and they all
exhibit achromatic transit depths, consistent with planetary signals (Desert
et al., 2012).
The durations of transit-like signatures can be used to assess the probability
of the planetary interpretation, especially when more than one planet
candidate is present for a given target. Transit and eclipse durations depend
on several factors, including the masses and radii of the two objects, the
impact parameter, $b$, which is defined as the minimum projected separation
between the center of the smaller body (planet in the case of a planetary
transit) and that of the larger body (star) measured in units of the radius of
the larger body, and the period and eccentricity of the orbit. Central
transits (those for which the center of the transiting body passes over the
center of the transited body) by a planet on a circular orbit have durations
that vary as the cube root of the orbital period (Kepler, 1619) and inversely
to the cube root of the stellar density (Seager & Mallén-Ornelas, 2003); these
relationships hold for FPs as long as the bright star is substantially larger
and more massive than the other body. When the system’s mass is dominated by
the primary but the radius of the secondary cannot be ignored, then the
duration, $T_{\rm dur}$, (1st contact – 4th contact, which corresponds to the
values reported in Borucki et al. (2011) and in Table 3 of this paper), of a
central transit for a circular orbit satisfies the following proportionality
relationship:
$T_{\rm dur}\propto\frac{R_{\star}+R_{p}}{R_{\star}}P_{\rm orb}^{1/3}.$ (10)
If the transit duration is measured over the time interval when the center of
the planet covers the disk of the star, the first factor on the right-hand
side of Expression (10) is no longer relevant and the proportionality is
simply to $P_{\rm orb}^{1/3}$. Note that the orbital period, $P_{\rm orb}$, is
very accurately measured by Kepler, as is the ratio of the planet’s radius,
$R_{p}$, to that of the star, $R_{\star}$, provided dilution by nearby stars
is small.
Planetary transits with an impact parameter smaller than half the star’s
radius have a duration that exceeds $3^{1/2}/2\approx 86.67\%$ that of a
central transit. Such transits account for almost half of all transiting
planets, and over half of the transits observed to a specified SNR threshold.
The fraction of transits with durations greater than half that of a central
transit is $3^{1/2}/2$; again specifying a SNR threshold increases the
observed fraction. Orbital eccentricities also influence the durations of
transits and FPs; in most cases, durations are reduced because speeds are
highest near periastron (Kepler, 1609), where geometric considerations imply
that transit probabilities are larger.
## 4\. Validation of the 5-planet system Kepler-33
There are two primary purposes of this work: demonstrating the very high
credibility of the ensemble of Kepler multi-planet candidates and providing a
framework to validate particular systems of candidates as planets. We now turn
to a specific system that is both intrinsically interesting because it is the
closest analog to Kepler-11 observed to date (Lissauer et al., 2011a) and has
characteristics very favorable for validation by multiplicity.
KOI-707 (KIC 9458613) is a $Kp=14$ star for which four planet candidates,
ranging in period from 13 to 41 days, were announced by Borucki et al. (2011);
a fifth candidate planet, with shorter orbital period and smaller size, has
subsequently been identified. The lightcurve of this target is shown in Figure
2.
Figure 2.— Kepler photometric time series data for KOI-707 (KIC 9458613;
Kepler-33) in 29.426 minute intervals (standard Kepler long cadence
observations). Calibrated data from the spacecraft with each quarter
normalized to its median are shown in the top panel. The bottom panel displays
the lightcurve after detrending with a polynomial filter (Rowe et al., 2010).
The entire 8 Quarters of Kepler data analyzed herein are displayed. Note the
difference in vertical scales between the two panels. The midpoint times of
the five sets of periodic transits (including those that were not observed
because they occurred during data gaps) are indicated by triangles of
differing colors. Close-ups of folded lightcurves near transit times of each
of the planets are shown in Figures 7 and 8.
We performed an analysis of pixel-level Kepler spacecraft data that shows that
the transit signals come from a location on the sky that is coincident with
that of the target star within small uncertainty, and we obtained high-
resolution images of the target that did not show any nearby stars capable of
producing the transit signals; these studies are presented in Section 4.1. Our
analyses of the transit lightcurves and spectra taken of the target star are
presented in Section 4.2. In Section 4.3, we review the characteristics of
this system that allow for its validation by multiplicity. We discuss the
properties of the Kepler-33 planetary system in Section 4.4.
### 4.1. Location of Transit Signature on the Plane of the Sky
A seeing-limited image taken at Lick shows a star $7^{\prime\prime}$ to the NW
that is 4.5 magnitudes fainter than KOI-707, but no stars closer to the
target. On 12 June 2011, KOI-707 was imaged in the 880 nm filter
($\Delta\lambda$ = 50 nm) of our speckle camera at the WIYN 3.5-m telescope
located on Kitt Peak. This star showed no companions within a 2.8 $\times$ 2.8
arcsec region down to 3.1 magnitudes fainter than the target star; stars
within $0.05^{\prime\prime}$ of the target star would not have been detected.
Details of this speckle imaging data collection and reduction precesses are
fully described in Howell et al. (2011).
We check to see whether the transit signal is due to a source other than
KOI-707 using the difference image technique described in Torres et al.
(2011). This method fits the measured Kepler pixel response function (PRF) to
a difference image formed from the average in-transit and average out-of-
transit pixel images in order to calculate the position of the difference
image centroid. This difference image centroid position is compared with the
position of the PRF-fit centroid of the average out-of-transit image, as well
as to the predicted deviations in centroid locations due to scene crowding.
The results for all 5 planet candidates are presented in Table 1 and shown in
Figure 3.
Figure 3.— Centroid fitting results are shown for all 5 planet candidates of
KOI-707. The origin is the center of light received outside of transit times.
The red asterisk is the expected centroid of light during transit assuming the
transit is on the target star; it is displaced from the origin because some of
the light entering the aperture comes from known background stars whose
brightness would not be affected by transits of the target star. (The numbers
following the red asterisk are the KIC number and Kepler magnitude of
KOI-707.) The fitted centroids of the transit signatures observed in Quarters
1 through 8 are shown as the thin black crosses, where the arms of these
crosses show the uncertainty in RA and Dec. Some quarters as missing for some
candidates due to failure of the PRF fit. The robust average centroid across
quarters is shown by the thick black cross, with the black solid circle giving
the 3$\sigma$ uncertainty in the offset distance. The robust average of each
modeled quarterly offsets is shown as a thick gray cross, with the gray dashed
circle showing the average model 3$\sigma$ offset distance uncertainty. When
the circle showing the observed 3$\sigma$ uncertainty in the difference image
has significant intersection with the dashed circle showing the modeled
3$\sigma$ uncertainty in the position of the target star, the data are
consistent with the observed transit signal being on KOI-707.
The uncertainty in the quarterly centroid locations displayed in Figure 3 is
based upon propagating pixel-level uncertainty and does not include possible
PRF fit bias. Sources of PRF fit bias include scene crowding, because the fit
is of a single PRF that assumes a single star, and PRF error. The centroid
offset is the difference between the positions of the centroids of the
difference image and out-of-transit images, so common biases such as PRF error
should approximately cancel. There is, however, a quarter-dependent residual
bias. Statistical analysis across many uncrowded Kepler targets indicates that
the residual bias due to PRF error is zero-mean, so we average the quarterly
centroid positions. Specifically, we compute a robust $\chi^{2}$ minimizing
best-fit position to the quarterly position data. The uncertainties of the
average are estimated by taking the uncertainties of the fit and performing a
bootstrap analysis. This average for each planet candidate is shown on Figure
3.
Bias due to crowding (stars close enough to the target that they contribute
light observed in the aperture), however, does not cancel because, to the
extent that variations in other field stars are not correlated with transits,
field stars do not contribute to the difference image. This introduces a
systematic bias component that remains in the average over quarters. To
investigate the possibility that the observed offsets are due to crowding
bias, we model the local scene using stars from the KIC and the measured PRF,
induced the transit on KOI-707 with the ephemeris of each planet candidate,
and performed the same PRF fit analysis on the model difference and out-of-
transit images as that performed on the pixel data. If the resulting modeled
average offset matches the observed average offset, then the observed offset
can be explained by crowding bias.
The locations of the transit signatures (difference image centroids) of each
of KOI-707’s five planet candidates lie within $1^{\prime\prime}$ of the
predicted position, and given the uncertainties listed in Table 1, none of the
signatures are significantly offset and all are fully consistent with transits
on the target star. Furthermore, the lack of near neighbor stars seen in the
speckle image rules out some of the possible false positive scenarios that
could produce a centroid shift of the magnitude suggested by the marginally
significant offset observed.
### 4.2. Lightcurves, Spectra, Derived Properties
We obtained a spectrum of the target star KOI-707/Kepler-33 using the HiRES
spectrometer at Keck I 10-m telescope. Classification of this spectrum using
SME (Spectroscopy Made Easy, (Valenti & Piskunov, 1996; Valenti & Fischer,
2005)) yields the stellar parameter values shown in Table 2 (the final two
columns in this table show results from constrained fits described at the end
of this subsection). The mean stellar density based on spectroscopic
information was estimated by matching the determined distributions of $T_{\rm
eff},\log g$ and [Fe/H] to the Yonsei-Yale evolution tracks. This allowed the
determination of stellar mass and radius and mean stellar density posterior
distribution based on spectroscopy. The spectrum of KOI-707 is consistent with
two sets of stellar evolution tracks (Figure 4), leading to double-peaked
probability distributions and asymmetric uncertainties in several stellar
parameters. We plot the probability distributions of the fundamental
parameters of stellar mass and age in Figure 5 and those of the stellar
density, which is a key input to transit fitting, in Figure 6, and we include
in Table 2 various measures of uncertainty beyond the simple standard
deviation for the values of various derived stellar parameters that are either
critical to the validation and properties of KOI-707 and/or distributed highly
asymmetrically. Note that both solutions are somewhat above the main sequence,
implying an evolved star.
Figure 4.— Plotted are the Yonsei-Yale evolutionary tracks for 1.1 M⊙ (red),
1.2 M⊙ (green), 1.3 M⊙ (blue), 1.4 M⊙ (cyan) and 1.5 M⊙ (magenta) stars with
[Fe/H] = 0.15. Ages, in Gyr, are marked along the tracks. Note the uneven
rates of evolution along these tracks. The gray dotted tracks are spaced by
0.01 M⊙ in stellar mass and 0.02 Gyr in time to illustrate the distribution of
field stars on the $\log g-T_{\rm eff}$ plane; the apparent
gaps/discontinuities in these tracks are caused by rapid changes at various
stages of stellar evolution. The three boxes mark the 1, 2 and 3 $\sigma$
confidence regions of the spectroscopic SME analysis of the spectrum of
KOI-707 taken at Keck. There are degeneracies in determining a unique stellar
mass and radius for specified values of $T_{\rm eff}$ and $\log g$. For
example, a 4.4 Gyr, 1.3 M⊙ model and a 5.5 Gyr, 1.2 M⊙ model have nearly
identical $T_{\rm eff}$ and $\log g$. The low-mass/old peaks in the
probability distribution functions shown in Figure 5 result from the pile-up
of stellar evolution tracks as core H is exhausted in the transition regime
from radiative cores of 1.1 M⊙ and smaller stars to convective cores in 1.2 M⊙
and larger stars. The high-mass/young peaks result from the overlap between
tracks for the early shell H burning phase of stars with convective cores with
tracks for the late core H burning phase of stars that are $\sim
0.12~{}M_{\odot}$ more massive. Figure 5.— The relative probability
distribution of estimates for the mass and age of the star Kepler-33
(KOI-707), computed using the SME analysis of the star’s spectrum and a prior
based on the frequency distribution of stars, as described in the text. Both
curves are double-peaked because two solutions, both recently evolved off the
main sequence, are consistent with the data. Integration under the curves
implies that the younger, more massive, solution is a bit more probable than a
somewhat older and fainter star. Figure 6.— The relative probability
distribution of estimates for the density of the star Kepler-33 (KOI-707). The
top curve was computed using an MCMC analysis of the transit photometry data,
whereas the results shown in the bottom panel incorporated the SME analysis of
the star’s spectrum and a prior based on the frequency distribution of stars.
The spectral solution is highly asymmetric because two groups of stars are
consistent with the data (see Figure 4). Note the difference in horizontal
scales between the two panels.
As an independent check of the values of the SME parameters, we also derived
values by matching the spectrum to synthetic spectra (Torres et al., 2002;
Buchhave et al., 2010), and in addition we employ a new fitting scheme
(Stellar Parameter Classification, SPC) that is currently under development
and being readied for publication, allowing us to extract precise stellar
parameters from the spectra. The SPC analysis yielded the following
parameters: $T_{\rm eff}=5849\pm 50$ K, $\log g=4.07\pm 0.10$, [m/H]=$0.12\pm
0.08,v\sin i=3.4\pm 0.5$ km/s, RV = $13.779\pm 0.020$ km/s. The SME and SPC
parameters are fully consistent, and the nominal value of the key parameter
$\log g$ from SPC is slightly ($<1\sigma$) larger than the SME value. In
contrast, the derived stellar radius from SME is $1.82\pm 0.16~{}R_{\odot}$,
substantially larger (more than $3\sigma$ formal error, see Table 2) than the
radius estimate of 1.29 R⊙ based upon the KIC that was used by used by Borucki
et al. (2011).
We measured the radial velocity of KOI-707 relative to the barycenter of our
Solar System. The method involves measuring the Doppler shift of the stellar
spectrum obtained with the Keck I telescope and HiRES spectrometer on BJD =
2455782.9441 relative to the solar spectrum by a $\chi^{2}$ fit of the two
spectra. To set the zero point of the wavelength scale accurately, we measure
the displacement of telluric lines from the Earth’s atmosphere, serving as the
“iodine cell”. The resulting radial velocities are accurate to 0.1 km/s, based
on measurements of 2000 stars (Chubak & Marcy, 2012) and comparison with IAU
standard stars. For KOI-707, the radial velocity is $v_{\star}$ = 14.09 km/s,
indicating motion of the star away from the Solar System at a speed typical of
the velocity dispersion of stars in the Galactic disk. Orbital periods of the
planets quoted in Table 3 are as perceived from the barycenter of our Solar
System. As the Kepler-33 system is receding from us at $4.7\times 10^{-5}c$,
where $c$ represents the speed of light, the actual orbital periods in the
rest frame of the Kepler-33 system are 0.999953 times as large as the
tabulated values.
We used Q1-Q8 long-cadence Kepler aperture photometry (i.e., only pixel-level
corrections such as bias and dark smear are applied to the data and there is
no co-detrending). At this level of data reduction, there are still
instrumental artifacts on timescales similar to the planetary orbits. To
detrend the data (remove long-period variations), a second-order polynomial
was fitted to segments of sequential Kepler photometry data outside of
transits (i.e., in-transit observations were given zero weight), and then this
polynomial was subtracted from the data. A segment is defined as a length of
data that is uninterrupted for 5 or more consecutive long cadences ($\sim$2.5
hours). Each segment is then combined to produce a final time series that is
then normalized by the median.
We adopt a simple model to fit the data. We fit for the mean stellar density
($\rho_{\star}$), and each planet’s orbital period ($P_{\rm orb}$), transit
epoch ($T_{0}$), scaled planetary radius ($R_{p}/R_{\star}$) and impact
parameter ($b$). The planets are assumed to be on circular orbits; planet-
planet interactions are implicitly accounted for by allowing the center of
transit time to vary in order to best-fit transit shape (minimization of
scatter on residuals). The transit was described with the analytic formulae of
Mandel & Agol (2002), and we adopted a non-linear limb darkening law with
coefficients fixed based on the Claret & Bloemen (2011) tabulation for the
Kepler band with stellar parameters from SME analysis. A best fit model was
computed using a Levenberg-Marquardt algorithm to minimize the $\chi^{2}$
statistic and initial uncertainties were estimated via the construction of a
co-variance matrix.
The best fit model and co-variances were used to seed a hybrid-MCMC (Markov
Chain Monte-Carlo) algorithm to determine posterior distributions of our model
parameters. The model is considered hybrid, as we randomly use a Gibbs-sampler
and a buffer of previous chain parameters to produce proposals to jump to a
new location in the parameter space. The addition of the buffer allows for a
calculation of vectorized jumps that allow for efficient sampling of highly
correlated parameter space.
A direct comparison of the mean stellar density determined by the circular
transit-model (0.236 $\pm$ 0.080) and the mean stellar density determined from
spectroscopy (0.301 $\pm$ 0.067) differ by less than 1$\sigma$. This shows
that we can produce a self-consistent circular model in which all five planets
orbit the same star and produce stellar parameters that agree with
spectroscopy.
Close-ups of the lightcurves of each of the planets near planetary transit are
shown in Figure 7. Transit durations of all five KOIs vary in proportion to
the $P_{\rm orb}^{1/3}$ to within 5%, and to within 3% when planet sizes are
factored in as specified in Expression (10). The remarkable similarity of
these transit durations when appropriately scaled to orbital periods is
apparent in Figure 8, which shows the same data as Figure 7 but with the time
coordinate scaled by $P_{\rm orb}^{1/3}$. Such similarity would be the
consequence of an ensemble of planets transiting the same star on low
eccentricity orbits with similar impact parameters. The impact parameters
given in Table 3 are indeed quite similar for all of the planets. But the
inclinations implied to produce these parameters require a very fortuitous
arrangement in which tilt to the line of sight is just enough larger in the
inner planets than the outer ones. In contrast, if the system is viewed nearly
edge on, i.e., $i\approx 90^{\circ}$, then both a wider range of impact
parameters would be allowed (because transit durations are less sensitive to
the precise value of $b$ when $b\ll 1$) and similar values of $b$ imply a very
flat system rather than a fortuitous ensemble of tilts.
Figure 7.— Detrended Kepler flux from Kepler-33 shown phased at the period of
each transit signal and zoomed to a 30-hour region around mid-transit. Black
dots represent individual Kepler long cadence observations. The folded light
curves corresponding to the model fits are shown in colors that correspond to
these used in Figure 2. In each panel, the best-fit model for the other 4
planets was removed before plotting. Each panel has an identical vertical
scale, to show the relative depths, and identical horizontal scale, to show
the relative durations. Figure 8.— The folded lightcurves near the transit of
each planet in the Kepler-33 system, as shown in Figure 7, but in this case
the time coordinate in each panel has been scaled to $P_{\rm orb}^{1/3}$ of
the planet in question. Vertical lines mark the beginning and end times for
each transit. For planets on circular orbits with the same impact parameter,
durations normalized in this manner should be the same for all planets. The
remarkable similarity in normalized durations of these planets is very strong
evidence that they all orbit the same star.
We are thus motivated to perform a second set of fits constrained by the
assumptions of circular orbits for all planets and $b=0$ for the planet with
the longest normalized (according to Expression 10) transit duration,
KOI-707.01. Two such fits were performed, one for the low mass stellar
solution (where we imposed the constraint $M_{\star}<1.21~{}M_{\odot}$) and
the other for the high mass ($M_{\star}>1.21~{}M_{\odot}$) solution. We show
in the far right columns of Tables 2 and 3 both the high and low stellar mass
“Flat” fits; no uncertainties are quoted for these fits because the $b=0$
assumption constrains results so tightly that computed uncertainties are
unrealistically low.
The fitting procedure that we used to compute the planetary parameters listed
in Table 3 did not account for the small (1 – 2%) contribution of the second
star in the aperture to the flux measured by Kepler. This dilution implies
that the planet radii quoted are overestimated by 0.5 – 1%, a difference far
smaller than the uncertainties in their actual values.
### 4.3. Validation of the Kepler-33 Planetary System
The five planetary candidates of KOI-707 are extremely likely to represent a
true 5 planet system orbiting the target star. There are several lines of
evidence that support this conclusion. First, the analysis presented in
Section 2 implies that the overwhelming majority of Kepler multiple-planet
candidates are true planets and bound within the same physical system.
Second, the planetary system is already closely-packed dynamically for the
sizes of planets around an undiluted target star (see Section 4.4). Were the
planetary system to orbit another star that is not seen because it is
significantly fainter than the target, the larger planets required to match
the observations would necessitate implausibly low densities for the planetary
system to be stable.
Third, the spatial location analysis presented in Section 4.1, combined with
the galactic latitude distribution of targets and planet candidates shown in
Figure 1 and the relatively high galactic latitude of KOI-707 (15∘), together
imply that it is unlikely that the signals are due to a source not physically
associated with the target star.
Fourth, the long durations of the transits relative to the planets’ orbital
periods imply that they are produced by light being blocked from a star whose
density is significantly less than those of cool dwarfs, and/or they are
produced by apocentric transits of objects with substantial orbital
eccentricity. Substantial orbital eccentricities would make the system
unstable if the planets orbited the same star. It is highly implausible that
two stars in an unresolved stellar binary would be so similarly coeval to have
nearly equal densities characteristic of a somewhat evolved star.
Fifth, the remarkable similarity of the transit durations when appropriately
scaled to orbital periods (Expression (10) and Figure 8) can be understood as
the consequence of an ensemble of planets transiting the same star with
similar impact parameters, but would otherwise be an unlikely coincidence.
The above characteristics of KOI-707.01–05, combined with the overall high
validity of Kepler multis, imply that these candidates have a well above 99%
probability of composing a true multi-planet system, hence we consider these
candidates to be validated planets and name the system Kepler-33.
### 4.4. Characteristics of the Kepler-33 Planetary System
Numerical integrations to test stability of the system were performed using
masses of the planets from the nominal mass-radius relationship based upon
planets within our Solar System presented in Lissauer et al. (2011a). This
relationship is given by:
$M_{p}=\Big{(}\frac{R_{p}}{R_{\oplus}}\Big{)}^{2.06}M_{\oplus},$ (11)
where R⊕ and M⊕ are the radius and mass of the Earth, respectively. In
previous work, Lissauer et al. (2011a) integrated the KOI-707 system using
parameters based on the data presented by Borucki et al. (2011) and assuming
planar initially circular orbits; the model system survived intact for the
entire $3\times 10^{8}$ years simulated. All of our integrations also assume
that the system is planar. For all planets having zero initial eccentricity,
all three models of the system listed in Tables 2 and 3 survived intact for
the entire $10^{7}$ years simulated.
Of the three Kepler-33 planetary system models that we integrated, the one
based directly on the spectroscopic stellar parameters had the largest
planetary sizes and thus largest estimated planetary masses, with the
fractional differences in estimated planetary masses from one model to another
being larger than those for stellar mass. Thus, the spectroscopic parameters
yielded the largest interplanetary perturbations, and presumably the largest
TTV signatures. We examined distributions of TTV signals over 2 year periods
during the first 2 Myr of this simulation. These theoretical TTVs for the
inner planet were smaller than 30 seconds, and those for Kepler-33c less than
90 seconds, both values below current measurement capabilities. The outer
three planets exhibited far larger TTVs, 9 – 18 minutes for Kepler-33d, 23 –
38 minutes for Kepler-33e and 23 – 35 minutes for Kepler-33f, roughly
consistent with variations detected. While this model used relatively high
mass estimates, the integration began with circular orbits, which produce
smaller TTVs, so the results do not yield a preference between the three
models of planetary sizes listed in Table 3. A detailed analysis of planetary
TTVs is beyond the scope of this paper, but we note that it may well soon be
possible to derive estimates of the masses of some or all of the outer three
planets using observed TTVs.
The properties of the 5 planet Kepler-33 system are analogous to those of the
6 planet Kepler-11 system (Lissauer et al., 2011b). Both systems include 4
planets with orbital periods between 13 and 47 days that are comparable in
radius to, or somewhat smaller in size than, Neptune, as well as a smaller
planet closer to the star. Both systems are thus very closely-packed in a
dynamical sense. The Kepler-11 system is slightly more dynamically close-
packed than is the Kepler-33 system, both in the sense that the ratios of the
orbital periods of neighboring planets are a bit larger for Kepler-33’s
planets (Lissauer et al., 2011a) and as evidenced from smaller TTVs detected
in the Kepler-33 system.
In some aspects, the Kepler-33 system can be viewed as a less extreme cousin
of Kepler-11, with 5 known transiting planets rather than 6 and spacing that
not quite as close, yet dynamically much tighter than most systems (Lissauer
et al., 2011a). However, the new system’s inner planet, Kepler-33b, is smaller
and closer to its star than is its counterpart, Kepler-11b. The high density
of Kepler-11b relative to its brethren, coupled with its smaller size,
indicates a composition richer in heavy elements (Lissauer et al., 2011b).
While the mass and thus the density of Kepler-33b is not known, it is unlikely
that a planet this small ($1.5-1.8~{}R_{\oplus}$) and close (0.07 AU) to its
bright star (3 L⊙) could have retained a substantial H/He atmosphere. This
lends support to the suggestion by Lissauer et al. (2011b) that planets such
as Kepler-11b may have once had atmospheres of light gases that subsequently
escaped as a result of stellar irradiation. The slightly shorter normalized
duration of the small inner planet Kepler-33b suggests that this planet has
somewhat larger inclination and/or eccentricity than do the outer four
planets. We note that Mercury, the innermost and smallest planet in our Solar
System, also has the largest eccentricity and inclination of the 8 planets
known to orbit our Sun.
Stability of the Kepler-33 planetary system excludes large eccentricities of
the four outer planets, although interplanetary perturbations must cause
eccentricities to be non-zero. The remarkable similarity in the normalized
transit durations of the five planets orbiting Kepler-33 would be very
unlikely if the eccentricities and inclinations of planetary orbits are
significant (Figure 9). Moreover, the consequent similarity in computed impact
parameters despite the planets’ differing orbital distances requires that
either the system is viewed near edge-on, with impact parameters $b\ll 1$, or
a very fortuitous coincidence producing an anticorrelation between
inclinations and orbital periods. We view such a coincidence as unlikely, and
therefore prefer the planetary parameters closer to those listed in final two
columns of Tables 2 and 3 (the “Flat” fits) than to those listed in the second
column of Tables 2 and 3.
Figure 9.— The probability of observing all five planets in the Kepler-33
system to have period-normalized transit durations within 3.0% of one another
given that all five planets are seen to transit the host star is shown as a
function of mean mutual inclination. Points were derived using Monte Carlo
simulations that assumed parameters from the spectroscopic fit given in the
second column of Tables 2 and 3. Results with the flat fit parameters (not
shown) are nearly identical. Mutual inclinations for each realization are
drawn from a Rayleigh distribution with respect to random reference plane
(i.e., random observer). The black dots are for the $e=0$ case. The blue
triangles, yellow diamonds and red squares show results from simulations in
which the eccentricity assigned to each planet was drawn form a Rayleigh
distribution with mean value $<e>~{}=0.01$, $<e>~{}=0.03$ and $<e>~{}=0.05$,
respectively. The addition of even a small range of eccentricities greatly
reduces probability of highly correlated transit durations.
## 5\. Conclusions
The calculations presented in Section 2 show that the vast majority of
Kepler’s multi-planet candidates are indeed planets. The essence of our
argument is that fewer than 10-2 of Kepler targets have a planet candidate
listed in Borucki et al. (2011). If 10% or fewer of these candidates are FPs,
then fewer than 10-3 of Kepler targets have a false positive, and if FPs are
not correlated with other planet candidates, then fewer than 10-5 of Kepler’s
$\sim$ 160,000 targets, i.e., a total of at most 2 targets, are expected to
have a false positive as well as another transit-like signature that makes
them a candidate multi-planet system. This number can be compared to 170
candidate multi-planet systems with a total of 408 planet candidates announced
by Borucki et al. (2011). Accurate estimates of the false positive rate in
this population require detailed analysis of both Kepler and ground-based data
to account for the caveats expressed in Section 2.2. Nonetheless, we expect
that $\gtrsim 400$ of the 408 planet candidates in multis that were announced
by Borucki et al. (2011) are indeed exoplanets, although a small fraction of
these planets orbit stars other than their nominal Kepler target star. Note
that the smallest planet candidates are most prone to false positives, because
low SNR limits the restrictiveness in parameter space of tests of lightcurve
shape and location of the centroid of the transit signal on the sky plane, and
the abundance of astrophysical signals that are capable of producing lower
amplitude events is greater. Also note that stellar parameters, and thus
planetary sizes, are prone to large uncertainties for candidates that have no
high-SNR spectroscopic observations from the ground.
From a Bayesian statistical perspective, multiplicity effectively allows us to
increase the planet prior, and thus enables the validation of candidates as
being true planets at a specified high probability level without going to the
extreme lengths often required to provide a very small upper bound on the FP
prior. We use these arguments along with a detailed analysis of Kepler data to
validate the Kepler-11 analog KOI-707 system, which we name Kepler-33.
Kepler was competitively selected as the tenth Discovery mission. Funding for
this mission is provided by NASA’s Science Mission Directorate. The authors
thank the many people who gave so generously of their time to make the Kepler
mission a success, chief among them Bill Borucki, who has devoted decades to
developing and implementing Kepler. Useful comments were provided by Natalie
Batalha, Ruth Murray-Clay, Dimitar Sasselov, Jason Steffen, and especially by
Tim Brown. Kevin Zahnle and Mark Marley provided constructive comments on the
manuscript. D. C. F. acknowledges NASA support through Hubble Fellowship grant
#HF-51272.01-A, awarded by STScI, operated by AURA under contract NAS 5-26555.
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Table 1Centroid offsets for KOI-707 based on PRF fits. Planet | Transit Offset from KOI-707 | Transit Offset from Prediction
---|---|---
| (arcsec) | (Offset/$\sigma$) | (arcsec) | (Offset/$\sigma$)
.01 | $0.17\pm 0.29$ | $0.59$ | $0.39\pm 0.26$ | $1.50$
.02 | $0.30\pm 0.22$ | $1.35$ | $0.53\pm 0.22$ | $2.38$
.03 | $0.25\pm 0.39$ | $0.63$ | $0.65\pm 0.40$ | $1.64$
.04 | $0.37\pm 0.93$ | $0.39$ | $0.50\pm 0.67$ | $0.75$
.05 | $1.06\pm 0.42$ | $2.53$ | $0.71\pm 0.48$ | $1.47$
Table 2 Characteristics of the Star Kepler-33 Parameter | Median | $\sigma$ | $+1\sigma$ | $-1\sigma$ | $+3\sigma$ | $-3\sigma$ | Flat, $M_{\star}>1.2~{}M_{\odot}$ | Flat, $M_{\star}<1.2~{}M_{\odot}$ |
---|---|---|---|---|---|---|---|---|---
Spectroscopy | | | | | | | | |
$M_{\star}$ (M⊙) | 1.291 | 0.082 | 0.063 | -0.121 | 0.212 | -0.200 | 1.271 | 1.164 |
$R_{\star}$ (R⊙) | 1.82 | 0.16 | 0.14 | -0.18 | 0.49 | -0.42 | 1.663 | 1.615 |
$Z$ | 0.0250 | 0.0019 | | | | | 0.0248 | 0.0241 |
$T_{\rm eff}$ (K) | 5904 | 47 | | | | | 5899 | 5880 |
log($L_{\star}$/L⊙) | 0.556 | 0.080 | | | | | 0.476 | 0.446 |
Age (Gyr) | 4.27 | 0.87 | 0.74 | -1.03 | 2.68 | -1.65 | 4.31 | 5.92 |
log$g$ (cm2/s) | 4.027 | 0.056 | | | | | 4.0994 | 4.087 |
$v_{\star}$ (km/s) | 14.09 | 0.1 | | | | | | |
$\rho_{\star}$ (g/cm3) | 0.300 | 0.066 | 0.049 | -0.079 | 0.230 | -0.147 | 0.3883 | 0.3885 |
Transit Model | | | | | | | | |
$\rho_{\star}$ (g/cm3) | 0.288 | 0.081 | 0.091 | -0.066 | 0.112 | -0.215 | 0.03897 | 0.3897 |
Table 3 Characteristics of the Kepler-33 Planets Parameter | Median | $\sigma$ | $+1\sigma$ | $-1\sigma$ | $+3\sigma$ | $-3\sigma$ | Flat, $M_{\star}>1.2~{}M_{\odot}$ | Flat, $M_{\star}<1.2~{}M_{\odot}$
---|---|---|---|---|---|---|---|---
Kepler-33b = KOI-707.05 | | | | | | | |
$T_{0}$ (BJD-2454900) | 64.8981 | 0.0075 | | | | | 64.8973 | 64.8973
$P_{\rm orb}$ (days) | 5.66793 | 0.00012 | | | | | 5.667939 | 5.667939
$b$ | 0.50 | 0.16 | 0.16 | -0.20 | 0.34 | -0.41 | 0.298 | 0.298
$R_{p}/R_{\star}$ | 0.00877 | 0.00046 | | | | | 0.00853 | 0.00853
$R_{p}$ (R⊕) | 1.74 | 0.18 | | | | | 1.549 | 1.504
$i$ (degrees) | 86.39 | 1.17 | 0.96 | -1.62 | 2.90 | -3.16 | 88.03 | 88.03
$a/R_{\star}$ | 7.87 | 0.87 | | | | | 8.714 | 8.714
$a$ (AU) | 0.0677 | 0.0014 | | | | | 0.06740 | 0.06544
Trdepth (ppm) | 86.8 | 6.8 | | | | | 87.7 | 87.7
Trdur (hr) | 4.88 | 0.16 | 0.16 | -0.15 | 0.45 | -0.54 | 4.80 | 4.80
Kepler-33c = KOI-707.04 | | | | | | | |
$T_{0}$ (BJD-2454900) | 76.6764 | 0.0042 | | | | | 76.6777 | 76.6777
$P_{\rm orb}$ (days) | 13.17562 | 0.00014 | | | | | 13.17563 | 13.17563
$b$ | 0.44 | 0.17 | 0.17 | -0.21 | 0.38 | -0.38 | 0.144 | 0.144
$R_{p}/R_{\star}$ | 0.01602 | 0.00057 | | | | | 0.01560 | 0.01560
$R_{p}$ (R⊕) | 3.20 | 0.30 | | | | | 2.833 | 2.751
$i$ (degrees) | 88.19 | 0.72 | 0.58 | -1.06 | 1.55 | -1.92 | 89.46 | 89.46
$a/R_{\star}$ | 13.8 | 1.5 | | | | | 15.292 | 15.292
$a$ (AU) | 0.1189 | 0.0025 | | | | | 0.11287 | 0.11484
Trdepth (ppm) | 297.7 | 9.1 | | | | | 299.8 | 299.8
Trdur (hr) | 6.700 | 0.104 | 0.096 | -0.107 | 0.325 | -0.295 | 6.616 | 6.616
Kepler-33d = KOI-707.01 | | | | | | | |
$T_{0}$ (BJD-2454900) | 122.6342 | 0.0018 | | | | | 122.6341 | 122.6341
$P_{\rm orb}$ (days) | 21.77596 | 0.00011 | | | | | 21.775984 | 21.7759
$b$ | 0.44 | 0.17 | 0.14 | -0.23 | 0.38 | -0.40 | 0 (assumed) | 0 (assumed)
$R_{p}/R_{\star}$ | 0.02667 | 0.00087 | | | | | 0.02589 | 0.02589
$R_{p}$ (R⊕) | 5.35 | 0.49 | | | | | 4.700 | 4.564
$i$ (degrees) | 88.71 | 0.51 | 0.61 | -0.48 | 1.08 | -1.37 | 90 (assumed) | 90 (assumed)
$a/R_{\star}$ | 19.3 | 2.1 | | | | | 21.38 | 21.38
$a$ (AU) | 0.1662 | 0.0035 | | | | | 0.16533 | 0.16053
Trdepth (ppm) | 831.4 | 10.5 | | | | | 831 | 831
Trdur (hr) | 8.011 | 0.097 | 0.065 | -0.103 | 0.345 | -0.192 | 7.987 | 7.987
Kepler-33e = KOI-707.03 | | | | | | | |
$T_{0}$ (BJD-2454900) | 68.8715 | 0.0048 | | | | | 68.8720 | 68.8720
$P_{\rm orb}$ (days) | 31.78440 | 0.00039 | | | | | 31.78438 | 31.78438
$b$ | 0.47 | 0.16 | 0.14 | -0.21 | 0.38 | -0.37 | 0.204 | 0.204
$R_{p}/R_{\star}$ | 0.02011 | 0.00072 | | | | | 0.01955 | 0.01955
$R_{p}$ (R⊕) | 4.02 | 0.38 | | | | | 3.550 | 3.446
$i$ (degrees) | 88.94 | 0.37 | 0.45 | -0.35 | 0.84 | -1.07 | 89.576 | 89.576
$a/R_{\star}$ | 24.9 | 2.8 | | | | | 27.51 | 27.51
$a$ (AU) | 0.2138 | 0.0045 | | | | | 0.2127 | 0.20656
Trdepth (ppm) | 467 | 12 | | | | | 469 | 469
Trdur (hr) | 8.90 | 0.13 | 0.12 | -0.13 | 0.40 | -0.38 | 8.819 | 8.819
Kepler-33f = KOI-707.02 | | | | | | | |
$T_{0}$ (BJD-2454900) | 105.5763 | 0.0040 | | | | | 105.5757 | 105.5757
$P_{\rm orb}$ (days) | 41.02902 | 0.00042 | | | | | 41.02903 | 41.02903
$b$ | 0.43 | 0.17 | 0.15 | -0.23 | 0.39 | -0.42 | 0.130 | 0.130
$R_{p}/R_{\star}$ | 0.02227 | 0.00076 | | | | | 0.02171 | 0.02171
$R_{p}$ (R⊕) | 4.46 | 0.41 | | | | | 3.941 | 3.827
$i$ (degrees) | 89.17 | 0.34 | 0.33 | -0.42 | 0.68 | -0.98 | 89.772 | 89.772
$a/R_{\star}$ | 29.5 | 3.3 | | | | | 32.61 | 32.61
$a$ (AU) | 0.2535 | 0.0054 | | | | | 0.2522 | 0.2449
Trdepth (ppm) | 579 | 12 | | | | | 581 | 581
Trdur (hr) | 9.87 | 0.13 | 0.11 | -0.15 | 0.43 | -0.36 | 9.732 | 9.732
|
arxiv-papers
| 2012-01-26T01:31:49 |
2024-09-04T02:49:26.701892
|
{
"license": "Public Domain",
"authors": "Jack J. Lissauer, Geoffrey W. Marcy, Jason F. Rowe, Stephen T. Bryson,\n Elisabeth Adams, Lars A. Buchhave, David R. Ciardi, William D. Cochran,\n Daniel C. Fabrycky, Eric B. Ford, Francois Fressin, John Geary, Ronald L.\n Gilliland, Matthew J. Holman, Steve B. Howell, Jon M. Jenkins, Karen\n Kinemuchi, David G. Koch, Robert C. Morehead, Darin Ragozzine, Shawn E.\n Seader, Peter G. Tanenbaum, Guillermo Torres, Joseph D. Twicken",
"submitter": "Jack Lissauer",
"url": "https://arxiv.org/abs/1201.5424"
}
|
1201.5490
|
# On $p$-adic interpolating Function Associated with Modified Dirichlet’s
type of twisted $q$-Euler Numbers and Polynomials with weight $\alpha$
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr and Hassan Jolany
School of Mathematics, Statistics and Computer Science, University of Tehran,
Iran hassan.jolany@khayam.ut.ac.ir
(Date: January 18, 2012)
###### Abstract.
The $q$-calculus theory is a novel theory that is based on finite difference
re-scaling. The rapid development of $q$-calculus has led to the discovery of
new generalizations of $q$-Euler polynomials involving $q$-integers. The
present paper deals with the modified Dirichlet’s type of twisted $q$-Euler
polynomials with weight $\alpha$. We apply the method of generating function
and $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$, which are
exploited to derive further classes of $q$-Euler numbers and polynomials. To
be more precise we summarize our results as follows, we obtain some
combinatorial relations between modified Dirichlet’s type of twisted $q$-Euler
numbers and polynomials with weight $\alpha$. Furthermore we derive witt’s
type formula and Distribution formula (Multiplication theorem) for modified
Dirichlet’s type of twisted $q$-Euler numbers and polynomials with weight
$\alpha$. In section three, by applying Mellin transformation we define
$q$-analogue of modified twisted $q$-$l$-functions of Dirichlet’s type and
also we deduce that it can be written as modified Dirichlet’s type of twisted
$q$-Euler polynomials with weight $\alpha$. Moreover we will find a link
between modified twisted Hurwitz-zeta function and $q$-analogue of modified
twisted $q$-$l$-functions of Dirichlet’s type which yields a deeper insight
into the effectiveness of this type of generalizations. In addition we
consider $q$-analogue of partial zeta function and we derive behavior of the
modified $q$-Euler $L$-function at $s=0$. In final section, we construct
$p$-adic twisted Euler $q$-$L$ function with weight $\alpha$ and interpolate
Dirichlet’s type of twisted $q$-Euler polynomials with weight $\alpha$ at
negative integers. Our new generating function possess a number of interesting
properties which we state in this paper.
###### Key words and phrases:
Euler numbers and polynomials, $q$-Euler numbers and polynomials, Modified
twisted $q$-Euler numbers and polynomials with weight $\alpha,$ Modified
Dirichlet’s type twisted $q$-Euler numbers and polynomials with weight
$\alpha$
###### 2000 Mathematics Subject Classification:
05A10, 11B65, 28B99, 11B68, 11B73.
## 1\. Introduction, Definitions and Notations
$p$-adic numbers and $L$-functions theory plays a vital and important role in
mathematics. $p$-adic numbers were invented by the German mathematician Kurt
Hensel [1], around the end of the nineteenth century. In spite of their being
already one hundred years old, these numbers are still today enveloped in an
aura of mystery within the scientific community. The $p$-adic integral was
used in mathematical physics, for instance, the functional equation of the
$q$-Zeta function, $q$-Stirling numbers and $q$-Mahler theory of integration
with respect to the ring $\mathbb{Z}_{p}$ together with Iwasawa’s $p$-adic
$q$-$L$ functions. A $p$-adic zeta function, or more generally a $p$-adic
$L$-function, is a function analogous to the Riemann zeta function, or more
general $L$-functions, but whose domain and target are $p$-adic (where $p$ is
a prime number). For example, the domain could be the $p$-adic integers
$\mathbb{Z}_{p}$, a profinite $p$-group, or a $p$-adic family of Galois
representations, and the image could be the $p$-adic numbers $\mathbb{Q}_{p}$
or its algebraic closure. For a prime number $p$ and for a Dirichlet character
defined modulo some integer, the $p$-adic $L$-function was constructed by
interpolating the values of complex analytic $L$-function at non-positive
integers. In this paper our main focus will be on $p$-adic interpolation of
modified Dirichlet’s type of twisted $q$-Euler polynomials with weight
$\alpha$. Actually interpolation is the process of defining a continuous
function that takes on specified values at specified points. During the
development of $p$-adic analysis, researches were made to derive a meromorphic
function, defined over the $p$-adic number field, which would interpolate the
same or at least similar values as the Dirichlet $L$-function at non-positive
integers. Finding the interpolation functions of special orthogonal numbers
and polynomials started by H. Tsumura [29], and P. T. Young [34], for the
Bernoulli and Euler polynomials. After Taekyun Kim and Yilmaz Simsek, studied
on $p$-adic interpolation functions of these numbers and polynomials. L. C.
Washington [30], constructed one-variable $p$-adic-$L$-function which
interpolates generalized classical Bernoulli numbers at negative integers.
Diamond [35], obtained formulas which express the values of $p$-adic
$L$-function at positive integers in terms of the $p$-adic log gamma function.
Next Fox in [32], introduced the two-variable $p$-adic $L$-functions and T.
Kim [21], constructed the two-variable $p$-adic $q$-$L$-function, which is
interpolation function of the generalized $q$-Bernoulli polynomials. P. T.
Young [34], gave $p$-adic integral representations for the two-variable
$p$-adic $L$-function introduced by Fox. T. Kim and S.-H. Rim [17], introduced
twisted $q$-Euler numbers and polynomials associated with basic twisted
$q$-$l$-functions by using $p$-adic $q$-invariant integral on $\mathbb{Z}_{p}$
in the fermionic sense. Also, Jang et al. [33], investigated the $p$-adic
analogue twisted $q$-$l$-function, which interpolates generalized twisted
$q$-Euler numbers attached to Dirichlet’s character. In this paper, we will
construct a $p$-adic interpolation function of modified Dirichlet’s type of
twisted $q$-Euler polynomials with weight $\alpha$.
Imagine that $p$ be a fixed odd prime number. Throughout this paper
$\mathbb{Z}_{p}$, $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ will be denote the
ring of $p$-adic rational integers, the field of $p$-adic rational numbers,
and the completion of algebraic closure of $\mathbb{Q}_{p}$, respectively. Let
$\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}:=\mathbb{N}\cup\left\\{0\right\\}$. In this paper, we
assume that $\alpha\in\mathbb{Q}$ and $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}<1$.
The $p$-adic absolute value $\left|.\right|_{p}$, is normally defined by
$\left|x\right|_{p}=\frac{1}{p^{r}}\text{,}$
where $x=p^{r}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and $r\in\mathbb{Q}$.
As well-known definition, Euler polynomials are defined by
$\frac{2}{e^{t}+1}e^{xt}=\sum_{n=0}^{\infty}\boldsymbol{E}_{n}\left(x\right)\frac{t^{n}}{n!}=e^{\boldsymbol{E}\left(x\right)t}\text{,}$
with the usual convention about replacing $\boldsymbol{E}^{n}\left(x\right)$
by $\boldsymbol{E}_{n}\left(x\right)$ (for more information, see [8, 9, 14,
15, 16])
.
A $p$-adic Banach space $B$ is a $\mathbb{Q}_{p}$-vector space with a lattice
$B^{0}$ ($\mathbb{Z}_{p}$-module) separated and complete for $p$-adic
topology, ie.,
$B^{0}\simeq\lim_{\overleftarrow{n\in\mathbb{N}}}B^{0}/p^{n}B^{0}\text{.}$
For all $x\in B$, there exists $n\in\mathbb{Z}$, such that $x\in p^{n}B^{0}$.
Define
$v_{B}\left(x\right)=\sup_{n\in\mathbb{N}\cup\left\\{+\infty\right\\}}\left\\{n:x\in
p^{n}B^{0}\right\\}\text{.}$
It satisfies the following properties:
$\displaystyle v_{B}\left(x+y\right)$ $\displaystyle\geq$
$\displaystyle\min\left(v_{B}\left(x\right),v_{B}\left(y\right)\right)\text{,}$
$\displaystyle v_{B}\left(\beta x\right)$ $\displaystyle=$ $\displaystyle
v_{p}\left(\beta\right)+v_{B}\left(x\right)\text{, if
}\beta\in\mathbb{Q}_{p}\text{.}$
Then, $\left\|x\right\|_{B}=p^{-v_{B}\left(x\right)}$ defines a norm on $B,$
such that $B$ is complete for $\left\|.\right\|_{B}$ and $B^{0}$ is the unit
ball.
A measure on $\mathbb{Z}_{p}$ with values in a $p$-adic Banach space $B$ is a
continuous linear map
$f\mapsto\int
f\left(x\right)\mu=\int_{\mathbb{Z}_{p}}f\left(x\right)\mu\left(x\right)$
from $C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$, (continuous function
on $\mathbb{Z}_{p}$) to $B$. We know that the set of locally constant
functions from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{p}$ is dense in
$C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ so.
Explicitly, for all $f\in C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$,
the locally constant functions
$f_{n}=\sum_{i=0}^{p^{n}-1}f\left(i\right)1_{i+p^{n}\mathbb{Z}_{p}}\rightarrow\text{
}f\text{ in }C^{0}\text{.}$
Now if $\mu\in\boldsymbol{D}_{0}\left(\mathbb{Z}_{p},\mathbb{Q}_{p}\right)$,
set
$\mu\left(i+p^{n}\mathbb{Z}_{p}\right)=\int_{\mathbb{Z}_{p}}1_{i+p^{n}\mathbb{Z}_{p}}\mu$.
Then $\int_{\mathbb{Z}_{p}}f\mu$ is given by the following “Riemann sums”
$\int_{\mathbb{Z}_{p}}f\mu=\lim_{n\rightarrow\infty}\sum_{i=0}^{p^{n}-1}f\left(i\right)\mu\left(i+p^{n}\mathbb{Z}_{p}\right)\text{.}$
T. Kim defined $\mu$ as follows:
$\mu_{-q}\left(a+p^{n}\mathbb{Z}_{p}\right)=\frac{\left(-q\right)^{a}}{\left[p^{n}\right]_{-q}}\text{,}$
so,
$\displaystyle I_{-q}\left(f\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{\left[p^{N}\right]_{-q}}\sum_{x=0}^{p^{N}-1}\left(-1\right)^{x}f\left(x\right)q^{x}\text{,
(for details, see \cite[cite]{[\@@bibref{}{kim
15}{}{}]},\cite[cite]{[\@@bibref{}{kim 16}{}{}]},\cite[cite]{[\@@bibref{}{kim
17}{}{}]}). }$
Where $\left[x\right]_{q}$ is a $q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{,}$
note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ cf. [2-35].
If we take $f_{1}\left(x\right)=f\left(x+1\right)$ in (1), then we easily see
that
(1.2)
$qI_{-q}\left(f_{1}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$
By expression (1.2), we readily see that,
(1.3)
$\left(-1\right)^{n-1}I_{-q}\left(f\right)+q^{n}I_{-q}\left(f_{n}\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{n-1-l}q^{l}f\left(l\right)\text{,}$
where $f_{n}(x)=f\left(x+n\right)$.
Recently, Rim et al. [26] defined the modified weighted $q$-Euler numbers
$\boldsymbol{E}_{n,q}^{\left(\alpha\right)}$ and the modified weighted
$q$-Euler polynomials
$\boldsymbol{E}_{n,q}^{\left(\alpha\right)}\left(x\right)$ by using $p$-adic
$q$-integral on $\mathbb{Z}_{p}$ in the form
$\boldsymbol{E}_{n,q}^{\left(\alpha\right)}=\int_{\mathbb{Z}_{p}}q^{-\xi}\left[\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)\text{,
for }n\in\mathbb{N}^{\ast}\text{ and }\alpha\in\mathbb{Z}\text{.}$
Let $C_{p^{n}}=\left\\{w\mid w^{p^{n}}=1\right\\}$ be the Cylic group of order
$p^{n}$, and let
$\mathbf{T}_{\mathbf{p}}=\lim_{n\rightarrow\infty}C_{p^{n}}=C_{p^{\infty}}=\underset{n\geq
0}{\cup}C_{p^{n}}\text{,}$
note that $\mathbf{T}_{\mathbf{p}}$ is locally constant space (for details,
see [17, 23, 24, 27-30, 33]).
In [23], let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$ and $w\in\mathbf{T}_{\mathbf{p}}$. If we
take $f(x)=\chi\left(x\right)w^{x}e^{tx}$, then we have
$f(x+d)=\chi\left(x\right)w^{x}w^{d}e^{tx}e^{td}$. From (1.3), we see that
(1.4)
$\int_{X}\chi\left(x\right)w^{x}e^{tx}d\mu_{-q}\left(x\right)=\frac{\left[2\right]_{q}\sum_{i=0}^{d-1}\left(-1\right)^{d-1-i}q^{i}\chi\left(i\right)w^{i}e^{ti}}{q^{d}w^{d}e^{td}+1}\text{.}$
In view of (1.4), it is considered by
(1.5)
$F_{w,\chi}^{q}(t)=\frac{\left[2\right]_{q}\sum_{i=0}^{d-1}\left(-1\right)^{d-1-i}q^{i}\chi\left(i\right)w^{i}e^{ti}}{q^{d}w^{d}e^{td}+1}=\sum_{n=0}^{\infty}E_{n,\chi,w}^{q}\frac{t^{n}}{n!}\text{,
}\left|t+\log\left(qw\right)\right|<\frac{\pi}{d}\text{.}$
Let us consider the modified twisted $q$-Euler polynomials with weight
$\alpha$ as follows:
(1.6)
$\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(x\right)=\int_{\mathbb{Z}_{p}}q^{-\xi}w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)\text{,
for }n\in\mathbb{N}^{\ast}\text{.}$
By (1.6), and applying combinatorial techniques we have,
$\displaystyle\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(x\right)$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}q^{\alpha\left(n-k\right)x}\boldsymbol{E}_{n-k,q}^{\left(\alpha,w\right)}\left[x\right]_{q^{\alpha}}^{k}$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\boldsymbol{E}_{k,q}^{\left(\alpha,w\right)}\left[x\right]_{q^{\alpha}}^{n-k}\text{,}$
where
$\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(0\right):=\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}$
are called modified twisted $q$-Euler numbers with weight $\alpha$.
By (1.6), we get generating function of modified twisted $q$-Euler polynomials
as follows:
$\displaystyle\tciFourier^{\left(\alpha\right)}\left(t,x,q,w\right)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(x\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}e^{t\left[x+m\right]_{q^{\alpha}}}\text{.}$
By using a complex contour integral representation and (1), we get modified
twisted Hurwitz-zeta function as follows:
$\displaystyle\widetilde{\boldsymbol{\zeta}}_{q}^{\left(\alpha,w\right)}\left(s,x\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\tciFourier^{\left(\alpha\right)}\left(-t,x,q,w\right)t^{s-1}dt$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\left(\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-t\left[x+m\right]_{q^{\alpha}}}\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}w^{m}}{\left[m+x\right]_{q^{\alpha}}^{s}}\text{.}$
By (1) and (1), we now establish a relation between
$\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(x\right)$ and
$\widetilde{\boldsymbol{\zeta}}_{q}^{\left(\alpha,w\right)}\left(s,x\right)$
as follows:
(1.10)
$\widetilde{\boldsymbol{\zeta}}_{q}^{\left(\alpha,w\right)}\left(-n,x\right)=\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(x\right)\text{.}$
In this paper, we construct the generating function of modified Dirichlet’s
type twisted $q$-Euler polynomials with weight $\alpha$ in the $p$-adic case.
Also, we give Witt’s formula for this type polynomials. Moreover, we obtain a
new $p$-adic $q$-Euler $L$-function with weight $\alpha$ associated with
Dirichlet’s character $\chi,$ as follows:
$l_{p,q}^{\left(\alpha,w\right)}\left(-n\mid\chi\right)=\widetilde{E}_{n,\chi_{n}}^{\left(\alpha,w\right)}-\frac{1}{\left[p^{-1}\right]_{q^{\alpha
F}}^{n}}\chi_{n}\left(p\right)\widetilde{E}_{n,\chi_{n}}^{\ast\left(\alpha,w\right)}$
where $n\in\mathbb{N}^{\ast}$.
## 2\. Properties of Modified Dirichlet’s type of twisted $q$-Euler numbers
and polynomials
In this section, by using fermionic $p$-adic $q$-integral equations on
$\mathbb{Z}_{p}$, some interesting identities and relations of the modified
Dirichlet’s type of twisted $q$-Euler numbers and polynomials with weight
$\alpha$, are given.
###### Definition 1.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$. For each $n\in\mathbb{N}^{\ast}$ and $w\in
T_{p}$. Modified Dirichlet’s type of twisted $q$-Euler polynomials with weight
$\alpha$ defined by means of the following generating function:
(2.1)
$\tciFourier^{\left(\alpha\right)}\left(t,x,q,w\mid\chi\right)=\sum_{n=0}^{\infty}\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\frac{t^{n}}{n!}$
where
(2.2)
$\tciFourier^{\left(\alpha\right)}\left(t,x,q,w\mid\chi\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\chi\left(m\right)e^{t\left[x+m\right]_{q^{\alpha}}}\text{.}$
From (2.1) and (2.2) we obtain,
$\sum_{n=0}^{\infty}\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\chi\left(m\right)\left[x+m\right]_{q^{\alpha}}^{n}\right)\frac{t^{n}}{n!}\text{.}$
Therefore, we state the following theorem:
###### Theorem 1.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$. For each $n\in\mathbb{N}^{\ast}$ and $w\in
T_{p}$ we have
(2.3)
$\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\chi\left(m\right)\left[x+m\right]_{q^{\alpha}}^{n}\text{.}$
By using (2.3), we can write
$\displaystyle\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\left(-1\right)^{l+md}w^{l+md}\chi\left(l+md\right)\left[x+l+md\right]_{q^{\alpha}}^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{d-1}\left(-1\right)^{l}w^{l}\chi\left(l\right)\sum_{m=0}^{\infty}\left(-1\right)^{m}\left(w^{d}\right)^{m}\sum_{k=0}^{n}\binom{n}{k}\left(-1\right)^{k}q^{\alpha
k\left(x+l+md\right)}$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{d-1}\left(-1\right)^{l}w^{l}\chi\left(l\right)\sum_{k=0}^{n}\binom{n}{k}\left(-1\right)^{k}q^{\alpha
k\left(x+l\right)}\sum_{m=0}^{\infty}\left(-1\right)^{m}\left(w^{d}\right)^{m}\left(q^{\alpha
kd}\right)^{m}$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{d-1}\left(-1\right)^{l}w^{l}\chi\left(l\right)\sum_{k=0}^{n}\frac{\binom{n}{k}\left(-1\right)^{k}q^{\alpha
k\left(x+l\right)}}{q^{\alpha kd}w^{d}+1}\text{.}$
So, we obtain the following corollary:
###### Corollary 1.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}.$ For each $n\in\mathbb{N}^{\ast}$ and $w\in
T_{p}$ we have
$\displaystyle\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\chi\left(m\right)\left[x+m\right]_{q^{\alpha}}^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left[\alpha\right]_{q}^{n}\left(1-q\right)^{n}}\sum_{l=0}^{d-1}\left(-1\right)^{l}w^{l}\chi\left(l\right)\sum_{k=0}^{n}\frac{\binom{n}{k}\left(-1\right)^{k}q^{\alpha
k\left(x+l\right)}}{q^{\alpha kd}w^{d}+1}\text{.}$
By applying
$f(\xi)=q^{-\xi}\chi\left(\xi\right)w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}$
into (1),
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\xi}\chi\left(\xi\right)w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q^{\alpha}\right)^{n}}\sum_{k=0}^{n}\binom{n}{k}\left(-1\right)^{k}q^{\alpha
kx}\int_{\mathbb{Z}_{p}}\chi\left(\xi\right)w^{\xi}q^{a\xi
k-\xi}d\mu_{-q}\left(\xi\right)\text{,}$
where from (1.3), we easily see that
(2.5) $\int_{\mathbb{Z}_{p}}\chi\left(\xi\right)w^{\xi}q^{\xi\alpha
k-\xi}d\mu_{-q}\left(\xi\right)=\frac{\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{\alpha
kl}w^{l}\chi\left(l\right)}{q^{\alpha kd}w^{d}+1}\text{.}$
By using (2) and (2.5) we obtain
(2.6)
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\xi}\chi\left(\xi\right)w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\left(1-q^{\alpha}\right)^{n}}\sum_{l=0}^{n}\binom{n}{k}\left(-1\right)^{k}q^{\alpha
kx}\frac{\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{\alpha
kl}w^{l}\chi\left(l\right)}{q^{\alpha kd}w^{d}+1}$ $\displaystyle=$
$\displaystyle\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\text{.}$
Last from equivalent, we obtain Witt’s type formula of modified Dirichlet’s
type of twisted$\ q$-Euler polynomials with weight $\alpha$ as follows:
###### Theorem 2.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$. For each $n\in\mathbb{N}^{\ast}$ and $w\in
T_{p}$ we obtain
(2.7)
$\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)=\int_{\mathbb{Z}_{p}}q^{-\xi}\chi\left(\xi\right)w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)\text{.}$
By (2.2), we obtain $functional$ $equation$ as follows:
$\tciFourier^{\left(\alpha\right)}\left(t,x,q,w\mid\chi\right)=e^{t\left[x\right]_{q^{\alpha}}}\tciFourier^{\left(\alpha\right)}\left(q^{x}t,q,w\mid\chi\right)\text{.}$
By using the definition of the generating function
$\tciFourier^{\left(\alpha\right)}\left(t,x,q,w\mid\chi\right)$ as follows:
$\sum_{n=0}^{\infty}\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\frac{t^{n}}{n!}=\left(\sum_{n=0}^{\infty}\left[x\right]_{q^{\alpha}}^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}q^{n\alpha
x}\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(\chi\right)\frac{t^{n}}{n!}\right)\text{,}$
by the Cauchy product in the above equation, we have
$\sum_{n=0}^{\infty}\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
lx}\widetilde{E}_{l,q}^{\left(\alpha,w\right)}\left(\chi\right)\left[x\right]_{q^{\alpha}}^{n-l}\right)\frac{t^{n}}{n!}\text{,}$
Therefore, by comparing the coefficients of $\frac{t^{n}}{n!}$ on the both
sides of the above equation, we can state following theorem:
###### Theorem 3.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$. For each $n\in\mathbb{N}^{\ast}$ and $w\in
T_{p}$ we have
(2.8)
$\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)=\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
xl}\widetilde{E}_{l,q}^{\left(\alpha,w\right)}\left(\chi\right)\left[x\right]_{q^{\alpha}}^{n-l}\text{.}$
So, by using $umbral$ $calculus$ convention in equality (2.8), we get
(2.9)
$\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)=\left(q^{\alpha
x}\widetilde{E}_{q}^{\left(\alpha,w\right)}\left(\chi\right)+\left[x\right]_{q^{\alpha}}\right)^{n}\text{,}$
where
$\left(\widetilde{E}_{q}^{\left(\alpha,w\right)}\left(\chi\right)\right)^{n}$
is replaced by $\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(\chi\right)$.
From (1.3) we arrive at the following theorem:
###### Theorem 4.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$, $w\in T_{p}$ and $m\in\mathbb{N}^{\ast}$ we
get
$w^{n}\widetilde{E}_{m,q}^{\left(\alpha,w\right)}\left(n\mid\chi\right)+\left(-1\right)^{n-1}\widetilde{E}_{m,q}^{\left(\alpha,w\right)}\left(\chi\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{n-1-l}\chi\left(l\right)w^{l}\left[l\right]_{q^{\alpha}}^{m}\text{.}$
So, from (1.3), and some combinatorial techniques we can write
(2.10)
$\displaystyle\int_{\mathbb{Z}_{p}}q^{-\xi}\chi\left(\xi\right)w^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d\right]_{q^{\alpha}}^{n}}{\left[d\right]_{-q}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)w^{a}\int_{\mathbb{Z}_{p}}q^{-d\xi}w^{d\xi}\left[\frac{x+a}{d}+\xi\right]_{q^{d\alpha}}^{n}d\mu_{\left(-q\right)^{d}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d\right]_{q^{\alpha}}^{n}}{\left[d\right]_{-q}}\sum_{a=0}^{d-1}\left(-1\right)^{a}w^{a}\chi\left(a\right)\boldsymbol{E}_{n,q^{d}}^{\left(\alpha,w^{d}\right)}\left(\frac{x+a}{d}\right)\text{.}$
Therefore, by (2.10), we obtain the following theorem:
###### Theorem 5.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$, $w\in T_{p}$ and $n\in\mathbb{N}^{\ast}$ we
have
$\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)=\frac{\left[d\right]_{q^{\alpha}}^{n}}{\left[d\right]_{-q}}\sum_{a=0}^{d-1}\left(-1\right)^{a}w^{a}\chi\left(a\right)\boldsymbol{E}_{n,q^{d}}^{\left(\alpha,w^{d}\right)}\left(\frac{x+a}{d}\right)\text{.}$
## 3\. Modified Dirichlet’s type of twisted $q$-Euler $L$-function with
weight $\alpha$
In this section, our goal is to consider interpolation function of the
generating functions of modified Dirichlet’s type of twisted $q$-Euler
polynomials with weight $\alpha$. For $s\in\mathbb{C}$, $w\in T_{p}$ and
$\chi$ be a Dirichlet’s character with conductor $d(=odd)\in\mathbb{N}$, by
applying the Mellin transformation in equation (2.2), we obtain
$\displaystyle\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,s\mid\chi\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(s\right)}\mathop{\displaystyle\oint}t^{s-1}\tciFourier^{\left(\alpha\right)}\left(-t,x,q,w\mid\chi\right)dt$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}w^{m}\chi\left(m\right)\left(\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-t\left[m+x\right]_{q^{\alpha}}}dt\right)\text{,}$
so, from above equality, we have
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,s\mid\chi\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)w^{m}}{\left[m+x\right]_{q^{\alpha}}^{s}}\text{.}$
Consequently, we are in position to define modified Dirichlet’s type of
twisted $q$-Euler $L$-function as follows:
###### Definition 2.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$ and $w\in T_{p}$ we have
(3.1)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,s\mid\chi\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)w^{m}}{\left[m+x\right]_{q^{\alpha}}^{s}}\text{,}$
for all $s\in\mathbb{C}$. We note that
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,s\mid\chi\right)$
is analytic function in the whole complex $s$-plane.
By substituting $s=-n$ into (3.1) we easily get
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,-n\mid\chi\right)=\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\text{,}$
which led to stating following theorem:
###### Theorem 6.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$, $w\in T_{p}$ and $n\in\mathbb{N}^{\ast}$,
we define
(3.2)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(x,-n\mid\chi\right)=\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(x\mid\chi\right)\text{.}$
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(1,s\mid\chi\right)=\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$
which is the modified Dirichlet’s type of twisted $q$-Euler $L$-function with
weight $\alpha$. Now, by applying combinatorial techniques we can write,
(3.3)
$\displaystyle\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)w^{m}}{\left[m\right]_{q^{\alpha}}^{s}}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m=1}^{\infty}\sum_{a=0}^{d-1}\frac{\left(-1\right)^{a+dm}\chi\left(a+dm\right)w^{a+dm}}{\left[a+dm\right]_{q^{\alpha}}^{s}}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left[2\right]_{q^{d}}}\left[d\right]_{q^{\alpha}}^{-s}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)w^{a}\left[\left[2\right]_{q^{d}}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\left(w^{d}\right)^{m}}{\left[\left(\frac{a}{d}+m\right)\right]_{q^{d\alpha}}^{s}}\right]$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left[2\right]_{q^{d}}}\left[d\right]_{q^{\alpha}}^{-s}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)w^{a}\widetilde{\zeta}_{q^{d}}^{\left(\alpha,w^{d}\right)}\left(s,\frac{a}{d}\right)\text{.}$
So, by previous calculation we can state following theorem:
###### Theorem 7.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$ and $w\in T_{p}$ we have
(3.4)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{d}}}\left[d\right]_{q^{\alpha}}^{-s}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)w^{a}\widetilde{\boldsymbol{\zeta}}_{q^{d}}^{\left(\alpha,w^{d}\right)}\left(s,\frac{a}{d}\right)\text{.}$
We now consider the partial-zeta function
$\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(s,a,w\mid F\right)$
as follows:
(3.5) $\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(s,a,w\mid
F\right)=\left[2\right]_{q}\sum_{\underset{m>0}{m\equiv
a\left(\mathop{\mathrm{m}od}F\right)}}\frac{\left(-1\right)^{m}w^{m}}{\left[m\right]_{q^{\alpha}}^{s}}\text{.}$
If $F\equiv 1\left(\mathop{\mathrm{m}od}2\right)$, then we have
(3.6)
$\displaystyle\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(s,a,w\mid
F\right)$ $\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{\underset{m>0}{m\equiv
a\left(\mathop{\mathrm{m}od}F\right)}}\frac{\left(-1\right)^{m}w^{m}}{\left[m\right]_{q^{\alpha}}^{s}}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{m>0}\frac{\left(-1\right)^{mF+a}w^{mF+a}}{\left[mF+a\right]_{q^{\alpha}}^{s}}$
$\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left[2\right]_{q^{F}}}\frac{\left(-1\right)^{a}w^{a}}{\left[F\right]_{q^{\alpha}}^{s}}\left[\left[2\right]_{q^{F}}\sum_{m>0}\frac{\left(-1\right)^{m}\left(w^{F}\right)^{m}}{\left[m+\frac{a}{F}\right]_{q^{\alpha
F}}^{s}}\right]$ $\displaystyle=$
$\displaystyle\frac{\left[2\right]_{q}}{\left[2\right]_{q^{F}}}\frac{\left(-1\right)^{a}w^{a}}{\left[F\right]_{q^{\alpha}}^{s}}\widetilde{\boldsymbol{\zeta}}_{q^{F}}^{\left(\alpha,w^{F}\right)}\left(s,\frac{a}{F}\right)$
By expressions (3.2) and (3.6) we get the following theorem:
###### Theorem 8.
Let $F\equiv 1\left(\mathop{\mathrm{m}od}2\right)$, $w\in T_{p}$ , $q$,
$s\in\mathbb{C}$, $\left|q\right|<1$ and $n\in\mathbb{N}^{\ast}$ we have
(3.7) $\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(-n,a,w\mid
F\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{F}}}\left(-1\right)^{a}w^{a}\left[F\right]_{q^{\alpha}}^{n}\boldsymbol{E}_{n,q^{F}}^{\left(\alpha,w^{F}\right)}\left(\frac{a}{F}\right)\text{.}$
By expressions (3.4) and (3.7), we obtain the following corollary:
###### Corollary 2.
Let $\chi$ be a Dirichlet’s character with conductor
$d\left(=odd\right)\in\mathbb{N}$, $w\in T_{p}$ and $F\equiv
1\left(\mathop{\mathrm{m}od}2\right)$ we have
(3.8)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)=\sum_{a=0}^{F-1}\chi\left(a\right)\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(s,a,w\mid
F\right)\text{.}$
By (1) and (3.7), we modify the $q$-analogue of the partial zeta function with
weight $\alpha$ as follows:
(3.9) $\widetilde{\boldsymbol{H}}_{q}^{\left(\alpha\right)}\left(s,a,w\mid
F\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{F}}}\left(-1\right)^{a}w^{a}\left[a\right]_{q^{\alpha}}^{-s}\sum_{l=0}^{\infty}\binom{-s}{l}q^{\alpha
al}\left(\frac{\left[F\right]_{q^{\alpha}}}{\left[a\right]_{q^{\alpha}}}\right)^{l}\boldsymbol{E}_{l,q^{F}}^{\left(\alpha,w^{F}\right)}\text{.}$
Let $f\left(\text{=odd}\right)$ and $a$ be the positive integer with $0\leq
a<F$. Then, (3.8) reduces to
(3.10)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{F}}}\sum_{a=0}^{F-1}\chi\left(a\right)\left(-1\right)^{a}w^{a}\left[a\right]_{q^{\alpha}}^{-s}\sum_{l=0}^{\infty}\binom{-s}{l}q^{\alpha
al}\left(\frac{\left[F\right]_{q^{\alpha}}}{\left[a\right]_{q^{\alpha}}}\right)^{l}\boldsymbol{E}_{l,q^{F}}^{\left(\alpha,w^{F}\right)}\text{.}$
By expression (3.10), we see that
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$
is an analytic function $s\in\mathbb{C}$, with except $s=0$. Furthermore, for
each $n\in\mathbb{Z}$, with $n\geq 0$, we get
(3.11)
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(-n\mid\chi\right)=\widetilde{E}_{n,q}^{\left(\alpha,w\right)}\left(\chi\right)\text{.}$
By using (3.9), (3.10) and (3.11) we derive behavior of the modified
Dirichlet’s type of twisted $q$-Euler $L$-function with weight $\alpha$ at
$s=0$ as follows:
###### Theorem 9.
The following likeable identity
$\widetilde{\boldsymbol{L}}_{q}^{\left(\alpha,w\right)}\left(0\mid\chi\right)=\frac{1+q}{1+w^{F}}\sum_{a=0}^{F-1}\left(-1\right)^{a}\chi\left(a\right)w^{a}\text{,}$
is true.
## 4\. Modified $\boldsymbol{p}$-Adic Twisted Interpolation $q$-$l$-Function
with weight $\alpha$
In this section, we construct modified $p$-adic twisted $q$-Euler
$l$-function, which interpolate modified Dirichlet’s type of twisted $q$-Euler
polynomials at negative integers. Firstly, Washington constructed $p$-adic
$l$-function which interpolates generalized classical Bernoulli numbers.
Here, we use some the following notations, which will be useful in reminder of
paper.
Let $\omega$ denote the $Kummer$ character by the conductor $f_{\omega}=p$.
For an arbitrary character $\chi$, we set $\chi_{n}=\chi\omega^{-n}$,
$n\in\mathbb{Z}$, in the sense of product of characters.
Let
$\displaystyle\left\langle a\right\rangle$ $\displaystyle=$
$\displaystyle\omega^{-1}\left(a\right)a=\frac{a}{\omega\left(a\right)}\text{,}$
$\displaystyle\left\langle a\right\rangle_{q}$ $\displaystyle=$
$\displaystyle\frac{\left[a\right]_{q}}{\omega\left(a\right)}\text{.}$
Thus, we note that $\left\langle a\right\rangle\equiv
1\left(\mathop{\mathrm{m}od}p\mathbb{Z}_{p}\right)$. Let
$A_{j}\left(x\right)=\sum_{n=0}^{\infty}a_{n,j}x^{n}\text{,
}a_{n,j}\in\mathbb{C}_{p}\text{, }j=0,1,2,...$
be a sequence of power series, each convergent on a fixed subset
$T=\left\\{s\in\mathbb{C}_{p}\mid\left|s\right|_{p}<p^{-\frac{2-p}{p-1}}\right\\}\text{,}$
of $\mathbb{C}_{p}$ such that
(1) $a_{n,j}\rightarrow a_{n,0}$ as $j\rightarrow\infty$ for any $n$;
(2) for each $s\in T$ and $\epsilon>0$, there exists an
$n_{0}=n_{0}\left(s,\epsilon\right)$ such that
$\left|\sum_{n\geq n_{0}}a_{n,j}s^{n}\right|_{p}<\epsilon\text{ for }\forall
j\text{.}$
So,
$\lim_{j\rightarrow\infty}A_{j}\left(s\right)=A_{0}\left(s\right)\text{, for
all }s\in T\text{.}$
This was constructed by Washington [30] to indicate that each functions
$\omega^{-s}\left(a\right)a^{s}$ and
$\sum_{l=0}^{\infty}\binom{s}{l}\left(\frac{F}{a}\right)^{l}B_{l}\text{,}$
where $F$ is multiple of $p$ and $f$ and $B_{l}$ is the $l$-th Bernoulli
numbers, is analytic on $T$ (for more information, see [30]).
Assume that $\chi$ be a Dirichlet’s character with conductor $f\in\mathbb{N}$
with $f\equiv 1(\mathop{\mathrm{m}od}2)$. Thus, we consider the modified
Dirichlet’s type of twisted $p$-adic $q$-Euler $l$-function with weight
$\alpha$, $l_{p,q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$, which
interpolate the modified Dirichlet’s type of twisted $q$-Euler polynomials
with weight $\alpha$ at negative integers.
For $f\in\mathbb{N}$ with $f\equiv 1\left(\mathop{\mathrm{m}od}2\right)$, let
us assume that $F$ is positive integral multiple of $p$ and $f=f_{\chi}$. We
are now ready to give definition of
$l_{p,q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$ as follows:
(4.1)
$l_{p,q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)=\sum_{a=0}^{F-1}\chi\left(a\right)\left(-1\right)^{a}w^{a}\left\langle
a\right\rangle_{q^{\alpha}}^{-s}\sum_{l=0}^{\infty}\binom{-s}{l}q^{\alpha
al}\left(\frac{\left[F\right]_{q^{\alpha}}}{\left[a\right]_{q^{\alpha}}}\right)^{l}\boldsymbol{E}_{l,q^{F}}^{\left(\alpha,w^{F}\right)}\text{.}$
By (4.1), we note that $l_{p,q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)$
is analytic for $s\in T$.
For $n\in\mathbb{N}$, we have
(4.2)
$\widetilde{E}_{n,\chi_{n}}^{\left(\alpha,w\right)}=\left[F\right]_{q^{\alpha}}^{n}\sum_{a=0}^{F-1}\left(-1\right)^{a}\chi_{n}\left(a\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{a}{F}\right)\text{.}$
If $\chi_{n}\left(p\right)\neq 0$, then $\left(p,f_{\chi_{n}}\right)=1$, and
thus the ratio $\frac{F}{p}$ is a multiple of $f_{\chi_{n}}$.
Let
$\lambda=\left\\{\frac{a}{p}\mid a\equiv
0\left(\mathop{\mathrm{m}od}p\right)\text{ for }a_{i}\in\mathbb{Z}\text{ with
}0\leq a_{i}<F\right\\}\text{.}$
Thus, we have
$\displaystyle\left[F\right]_{q^{\alpha}}^{n}\sum_{\underset{p\mid
a}{a=0}}^{F-1}\left(-1\right)^{a}\chi_{n}\left(a\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{a}{F}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\left[p^{-1}\right]_{q^{\alpha
F}}^{n}}\left[\frac{F}{p}\right]_{q^{\alpha}}^{n}\chi_{n}\left(p\right)\sum_{\underset{\eta\in\lambda}{a=0}}^{\frac{F}{p}}\left(-1\right)^{\eta}\chi_{n}\left(\eta\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{\eta}{F/p}\right)\text{.}$
By (4), we can define the second modified twisted generalized Euler numbers
attached to $\chi$ as follows:
(4.4)
$\widetilde{E}_{n,\chi_{n}}^{\ast\left(\alpha,w\right)}=\left[\frac{F}{p}\right]_{q^{\alpha}}^{n}\sum_{\underset{\eta\in\lambda}{a=0}}^{\frac{F}{p}}\left(-1\right)^{\eta}\chi_{n}\left(\eta\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{\eta}{F/p}\right)\text{.}$
By (4.2), (4) and (4.4), we readily get that
(4.5)
$\displaystyle\widetilde{E}_{n,\chi_{n}}^{\left(\alpha,w\right)}-\frac{1}{\left[p^{-1}\right]_{q^{\alpha
F}}^{n}}\chi_{n}\left(p\right)\widetilde{E}_{n,\chi_{n}}^{\ast\left(\alpha,w\right)}$
$\displaystyle=$
$\displaystyle\left[F\right]_{q^{\alpha}}^{n}\sum_{\underset{p\nshortmid
a}{a=0}}^{F-1}\left(-1\right)^{a}\chi_{n}\left(a\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{a}{F}\right)$
By (4.1) and (1), we readily see that
$\displaystyle l_{p,q}^{\left(\alpha,w\right)}\left(-n\mid\chi\right)$
$\displaystyle=$
$\displaystyle\left[F\right]_{q^{\alpha}}^{n}\sum_{\underset{p\nshortmid
a}{a=0}}^{F-1}\left(-1\right)^{a}\chi_{n}\left(a\right)\boldsymbol{E}_{n,q}^{\left(\alpha,w\right)}\left(\frac{a}{F}\right)$
$\displaystyle=$
$\displaystyle\widetilde{E}_{n,\chi_{n}}^{\left(\alpha,w\right)}-\frac{1}{\left[p^{-1}\right]_{q^{\alpha
F}}^{n}}\chi_{n}\left(p\right)\widetilde{E}_{n,\chi_{n}}^{\ast\left(\alpha,w\right)}\text{.}$
Consequently, we state the following Theorem:
###### Theorem 10.
Let $n\in\mathbb{N}$, the following equalities
$l_{p,q}^{\left(\alpha,w\right)}\left(s\mid\chi\right)=\sum_{a=1}^{F}\chi\left(a\right)\left(-1\right)^{a}w^{a}\left\langle
a\right\rangle_{q^{\alpha}}^{-s}\sum_{l=0}^{\infty}\binom{-s}{l}q^{\alpha
al}\left(\frac{\left[F\right]_{q^{\alpha}}}{\left[a\right]_{q^{\alpha}}}\right)^{l}\boldsymbol{E}_{l,q^{F}}^{\left(\alpha,w^{F}\right)}\text{,}$
and
$l_{p,q}^{\left(\alpha,w\right)}\left(-n\mid\chi\right)=\widetilde{E}_{n,\chi_{n}}^{\left(\alpha,w\right)}-\frac{1}{\left[p^{-1}\right]_{q^{\alpha
F}}^{n}}\chi_{n}\left(p\right)\widetilde{E}_{n,\chi_{n}}^{\ast\left(\alpha,w\right)}\text{,}$
are true.
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|
arxiv-papers
| 2012-01-26T11:48:17 |
2024-09-04T02:49:26.717816
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Hassan Jolany",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1201.5490"
}
|
1201.5600
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-006 LHCb-PAPER-2011-038 January 26, 2012
Searches for Majorana neutrinos in $B^{-}$ decays
The LHCb Collaboration†††Authors are listed on the following pages.
Searches for heavy Majorana neutrinos in $B^{-}$ decays in final states
containing hadrons plus a $\mu^{-}\mu^{-}$ pair have been performed using 0.41
fb-1 of data collected with the LHCb detector in proton-proton collisions at a
center-of-mass energy of 7 TeV. The $D^{+}\mu^{-}\mu^{-}$ and
$D^{*+}\mu^{-}\mu^{-}$ final states can arise from the presence of virtual
Majorana neutrinos of any mass. Other final states containing $\pi^{+}$,
$D_{s}^{+}$, or $D^{0}\pi^{+}$ can be mediated by an on-shell Majorana
neutrino. No signals are found and upper limits are set on Majorana neutrino
production as a function of mass, and also on the $B^{-}$ decay branching
fractions.
Keywords: LHC, Majorana, Neutrino, $B$ Decays
PACS: 14.40.Nd, 13.35.Hb, 14.60.Pq
Submitted to Physics Review D
The LHCb Collaboration
R. Aaij38, C. Abellan Beteta33,n, B. Adeva34, M. Adinolfi43, C. Adrover6, A.
Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M. Alexander48, G.
Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22, S. Amato2, Y. Amhis36, J.
Anderson37, R.B. Appleby51, O. Aquines Gutierrez10, F. Archilli18,35, L.
Arrabito55, A. Artamonov 32, M. Artuso53,35, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back45, D.S. Bailey51, V. Balagura28,35, W. Baldini16,
R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48, C. Bauer10,
Th. Bauer38, A. Bay36, I. Bediaga1, S. Belogurov28, K. Belous32, I. Belyaev28,
E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson47, J. Benton43, R.
Bernet37, M.-O. Bettler17, M. van Beuzekom38, A. Bien11, S. Bifani12, T.
Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35, F. Blanc36, C. Blanks50,
J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A. Bondar31, N. Bondar27, W.
Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V. Bowcock49, C. Bozzi16, T.
Brambach9, J. van den Brand39, J. Bressieux36, D. Brett51, M. Britsch10, T.
Britton53, N.H. Brook43, H. Brown49, A. Büchler-Germann37, I. Burducea26, A.
Bursche37, J. Buytaert35, S. Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo
Gomez33,n, A. Camboni33, P. Campana18,35, A. Carbone14, G. Carboni21,k, R.
Cardinale19,i,35, A. Cardini15, L. Carson50, K. Carvalho Akiba2, G. Casse49,
M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph. Charpentier35, N. Chiapolini37,
K. Ciba35, X. Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35,
H.V. Cliff44, J. Closier35, C. Coca26, V. Coco38, J. Cogan6, P. Collins35, A.
Comerma-Montells33, F. Constantin26, A. Contu52, A. Cook43, M. Coombes43, G.
Corti35, B. Couturier35, G.A. Cowan36, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, F. De Lorenzi12, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P.
Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil
Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A.
Dziurda23, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van
Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella16,e,14, E.
Fanchini20,j, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, M. Ferro-Luzzi35, S. Filippov30, C.
Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M.
Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D.
Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J.
Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R.
Gauld52, N. Gauvin36, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44,
V.V. Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, B.
Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53, G. Haefeli36,
C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11, R. Harji50, N.
Harnew52, J. Harrison51, P.F. Harrison45, T. Hartmann56, J. He7, V. Heijne38,
K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, K. Holubyev11, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, R.F. Koopman39, P. Koppenburg38, M. Korolev29, A.
Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M. Kreps45, G. Krocker11, P.
Krokovny31, F. Kruse9, K. Kruzelecki35, M. Kucharczyk20,23,35,j, T.
Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G. Lafferty51, A. Lai15,
D. Lambert47, R.W. Lambert39, E. Lanciotti35, G. Lanfranchi18, C.
Langenbruch11, T. Latham45, C. Lazzeroni42, R. Le Gac6, J. van Leerdam38,
J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefran$c$cois7, O. Leroy6, T.
Lesiak23, L. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11,
B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-
March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F. Machefert7, I.V.
Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D.
Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R.
Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín
Sánchez7, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C. Matteuzzi20,
M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, G. McGregor51, R.
McNulty12, M. Meissner11, M. Merk38, J. Merkel9, R. Messi21,k, S.
Miglioranzi35, D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S.
Monteil5, D. Moran12, P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K.
Müller37, R. Muresan26, B. Muryn24, B. Muster36, M. Musy33, J. Mylroie-
Smith49, P. Naik43, T. Nakada36, R. Nandakumar46, I. Nasteva1, M. Nedos9, M.
Needham47, N. Neufeld35, A.D. Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V.
Niess5, N. Nikitin29, T. Nikodem11, A. Nomerotski52,35, A. Novoselov32, A.
Oblakowska-Mucha24, V. Obraztsov32, S. Oggero38, S. Ogilvy48, O. Okhrimenko41,
R. Oldeman15,d,35, M. Orlandea26, J.M. Otalora Goicochea2, P. Owen50, B.K.
Pal53, J. Palacios37, A. Palano13,b, M. Palutan18, J. Panman35, A.
Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G. Passaleva17,
G.D. Patel49, M. Patel50, S.K. Paterson50, G.N. Patrick46, C. Patrignani19,i,
C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M.
Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A.
Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrella16,35, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B.
Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V. Pugatch41, A.
Puig Navarro33, W. Qian53, J.H. Rademacker43, B. Rakotomiaramanana36, M.S.
Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos Reis1,
S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa Romero5, P. Robbe7, E.
Rodrigues48,51, F. Rodrigues2, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, M. Sannino19,i, R. Santacesaria22, C. Santamarina
Rios34, R. Santinelli35, E. Santovetti21,k, M. Sapunov6, A. Sarti18,l, C.
Satriano22,m, A. Satta21, M. Savrie16,e, D. Savrina28, P. Schaack50, M.
Schiller39, S. Schleich9, M. Schlupp9, M. Schmelling10, B. Schmidt35, O.
Schneider36, A. Schopper35, M.-H. Schune7, R. Schwemmer35, B. Sciascia18, A.
Sciubba18,l, M. Seco34, A. Semennikov28, K. Senderowska24, I. Sepp50, N.
Serra37, J. Serrano6, P. Seyfert11, M. Shapkin32, I. Shapoval40,35, P.
Shatalov28, Y. Shcheglov27, T. Shears49, L. Shekhtman31, O. Shevchenko40, V.
Shevchenko28, A. Shires50, R. Silva Coutinho45, T. Skwarnicki53, N.A. Smith49,
E. Smith52,46, K. Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, B.
Souza De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S.
Stahl11, O. Steinkamp37, S. Stoica26, S. Stone53,35, B. Storaci38, M.
Straticiuc26, U. Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25,
P. Szczypka36, T. Szumlak24, S. T’Jampens4, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, P.
Urquijo53, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P.
Vazquez Regueiro34, S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, H. Voss10, S. Wandernoth11, J.
Wang53, D.R. Ward44, N.K. Watson42, A.D. Webber51, D. Websdale50, M.
Whitehead45, D. Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46,
M. Williams50, F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A.
Wotton44, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R. Young47,
O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang53, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A. Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated to 6
56Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Leptons constitute a crucially important sector of elementary particles. Half
of the leptons are neutrinos. Yet we do not know if they are Dirac or Majorana
particles, the latter case characterized by being their own antiparticles [1].
Since the observation of neutrino oscillations has indisputably established
that neutrinos have non-zero mass, it is possible to distinguish the two types
experimentally. Finding neutrinoless double $\beta$ decay has long been
advocated as a premier demonstration of the possible Majorana nature of
neutrinos [2]. The Feynman diagram is shown in Fig. 1. We also show the
fundamental quark and lepton level process. An impressive lower limit from
neutrinoless double $\beta$ decays in nuclei has already been obtained on the
half-life of $\cal{O}$$(10^{25})$ years [3] for coupling to $e^{-}$.
Figure 1: (a) Diagram of neutrinoless double $\beta$ decay when two neutrons
in a nucleus decay simultaneously. (b) The fundamental diagram for changing
lepton number by two units.
Similar processes can occur in $B^{-}$ decays. The diagram is shown in Fig.
2(a). In this reaction there is no restriction on the mass of the Majorana
neutrino as it acts as a virtual particle. In this paper, unlike in
neutrinoless double beta decays, a like-sign dimuon is considered rather than
two electrons. The only existing limit is from a recent Belle measurement [4]
using the $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ channel. We consider only
final states where the $c\overline{d}$ pair forms a final-state meson, either
a $D^{+}$ or a $D^{*+}$, so the processes we are looking for are
$B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$. In this paper mention of a specific
reaction also implies inclusion of the charge conjugate reaction.
Figure 2: Feynman diagrams for $B$ decays involving an intermediate heavy
neutrino ($N$). (a) $B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$, (b)
$B^{-}\rightarrow\pi^{+}(D_{s}^{+})\mu^{-}\mu^{-}$, and (c) $B^{-}\rightarrow
D^{0}\pi^{+}\mu^{-}\mu^{-}$.
There are other processes involving $b$-quark decays that produce a light
neutrino that can mix with a heavy neutrino, designated as $N$. The heavy
neutrino can decay as $N\rightarrow W^{+}\mu^{-}$. In Fig. 2(b) we show the
annihilation processes $B^{-}\rightarrow\pi^{+}(D_{s}^{+})\mu^{-}\mu^{-}$,
where the virtual $W^{+}$ materializes either as a $\pi^{+}$ or $D_{s}^{+}$.
These decays have been discussed in the literature [5, 6, *Zhang:2010um].
We note that it is also possible for the $B^{-}\rightarrow
D^{(*)+}\mu^{-}\mu^{-}$ decay modes shown in Fig. 2(a) to proceed by a Cabibbo
suppressed version of the process in Fig. 2(b) where the virtual $W^{+}$ forms
$D^{(*)+}$. Similarly, the decay modes shown in Fig. 2(b) could be produced
via Cabibbo suppressed versions of the process in Fig. 2(a). Here the
$\pi^{+}\mu^{-}\mu^{-}$ final state requires a $b\rightarrow u$ quark
transition while for the $D_{s}^{+}\mu^{-}\mu^{-}$ final state, one of the
virtual $W^{-}$ must couple to a $\overline{s}$ quark rather than a
$\overline{d}$.
The lifetimes of $N$ are not predicted. We assume here that they are long
enough that the natural decay width is narrower than our mass resolution which
varies between 2 and 15 MeV111In this paper we use units where the speed of
light, $c$, is set equal to one. depending on mass and decay mode. For
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$, we can access the Majorana mass
region between approximately 260 and 5000 MeV while for $B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$, the Majorana mass region is between 2100 and 5150
MeV. In the higher mass region, the $W^{+}$ may be more likely to form a
$D_{s}^{+}$ meson than a $\pi^{+}$. The
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ search was first performed by Mark-II
[8] and then by CLEO [9]. LHCb also performed a similar search using a smaller
0.04 fb-1 data sample [10] giving an upper limit of $5.8\times 10^{-8}$ at 95%
confidence level (CL). The decay of $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$
has never been investigated.
Finally, in Fig. 2(c) we show how prolific semileptonic decays of the $B^{-}$
can result in the $D^{0}\pi^{+}\mu^{-}\mu^{-}$ final state. This process has
never been probed [11]. We benefit from the higher value of the CKM coupling
$|V_{cb}|$ relative to $|V_{ub}|$ in the annihilation processes shown in Fig.
2(b). The accessible region for Majorana neutrino mass is between 260 and 3300
MeV. For all the modes considered in this paper, we search only for decays
with muons in the final state, though electrons, and $\tau$ leptons in cases
where sufficient energy is available, could also be produced. Searches have
also been carried out looking for like-sign dileptons in hadron collider
experiments [12, *Chatrchyan:2011wba, *Abulencia:2007rd, *Acosta:2004kx].
## 2 Data sample and signal
We use a data sample of 0.37 fb-1 collected with the LHCb detector [16] in the
first half of 2011 and an additional 0.04 fb-1 collected in 2010 at a center-
of-mass energy of 7 TeV.
The detector elements are placed along the beam line of the LHC starting with
the vertex detector, a silicon strip device that surrounds the proton-proton
interaction region having its first active layer positioned 8 mm from the beam
during collisions. It provides precise locations for primary $pp$ interaction
vertices, the locations of decays of long-lived particles, and contributes to
the measurement of track momenta. Further downstream, other devices used to
measure track momenta include a large area silicon strip detector located in
front of a 4 Tm dipole magnet, and a combination of silicon strip detectors
and straw-tube drift chambers placed behind. Two Ring Imaging Cherenkov (RICH)
detectors are used to identify charged hadrons. An electromagnetic calorimeter
is used for photon detection and electron identification, followed by a hadron
calorimeter, and a system that distinguishes muons from hadrons. The
calorimeters and the muon system provide first-level hardware triggering,
which is then followed by a software high level trigger.
Muons are triggered on at the hardware level using their penetration through
iron and detection in a series of tracking chambers. Projecting these tracks
through the magnet to the primary event vertex allows a determination of their
transverse momentum, $p_{\rm T}$. Events from the 2011 data used in this
analysis were triggered on the basis of a single muon having a $p_{\rm T}$
greater than 1480 MeV, or two muons with their product $p_{\rm T}$ greater
than 1.69 GeV2. To satisfy the higher level trigger, the muon candidates must
also be detached from the primary vertex.
Candidate $B^{-}$ decays are found using tracking information, and particle
identification information from the RICH and muon systems. The identification
of pions, kaons and muons is based on combining the information from the two
RICH detectors, the calorimeters and the muon system. The RICH detectors
measure the angles of emitted Cherenkov radiation with respect to each charged
track. For a given momentum particle this angle is known, so a likelihood for
each hypothesis is computed. Muon likelihoods are computed based on track hits
in each of the sequential muon chambers. In this analysis we do not reject
candidates based on sharing hits with other tracks. This eliminates a possible
bias that was present in our previous analysis [10]. Selection criteria are
applied on the difference of the logarithm of the likelihood between two
hypotheses. The efficiencies and the mis-identification rates are obtained
from data using $K_{S}$, $D^{*+}\rightarrow\pi^{+}D^{0}$, $D^{0}\rightarrow
K^{-}\pi^{+}$ and $J/\psi\rightarrow\mu^{+}\mu^{-}$ event samples that provide
almost pure pion, kaon, and muon sources.
Efficiencies and and rejection rates depend on the momentum of the final state
particles. For the RICH detector generally the pion or kaon efficiencies
exceed 90% and the rejection rates are of the order of 5% [17]. The muon
system provides efficiencies exceeding 98% with rejection rates on hadrons of
better than 99%, depending on selection criteria [18]. Tracks of good quality
are selected for further analysis. In order to ensure that tracks have good
vertex resolution we insist that they all have $p_{\rm T}>300$ MeV. For muons
this requirement varies from 650$-$800 MeV depending on the final state. All
tracks must be inconsistent with having been produced at the primary vertex
closest to the candidate $B^{-}$ meson’s decay point. The impact parameter
(IP) is the minimum distance of approach of the track with respect to the
primary vertex. Thus we form the IP $\chi^{2}$ by testing the hypothesis that
the IP is equal to zero, and require it to be large; the values depend on the
decay mode and range from 4 to 35.
Table 1: Charm and charmonium branching fractions Particle | Final state | Branching fraction (%)
---|---|---
$D^{0}$ | $K^{-}\pi^{+}$ | 3.89$\pm$ | 0.05 [3]
$D^{+}$ | $K^{-}\pi^{+}\pi^{+}$ | 9.14$\pm$ | 0.20 [3]
$D_{s}^{+}$ | $K^{-}K^{+}\pi^{+}$ | 5.50$\pm$ | 0.27 [19]
$D^{*+}$ | $\pi^{+}D^{0}$ | 67.7$\pm$ | 0.5 [3]
$\psi(2S)$ | $\pi^{+}\pi^{-}J/\psi$ | 32.6$\pm$ | 0.5 [3]
$J/\psi$ | $\mu^{+}\mu^{-}$ | 5.93$\pm$ | 0.06 [3]
## 3 Normalization channels
Values for branching fractions will be normalized to well measured channels
that have the same number of muons in the final state and equal track
multiplicities. The first such channel is $B^{-}\rightarrow J/\psi K^{-}$. Its
branching fraction is ${\cal{B}}(B^{-}\rightarrow J/\psi K^{-})=(1.014\pm
0.034)\times 10^{-3}$ [3]. We use the $J/\psi\rightarrow\mu^{+}\mu^{-}$ decay
mode. The product branching fraction of this normalization channel is
$(6.013\pm 0.021)\times 10^{-5}$, and is known to an accuracy of $\pm$2%. The
charm meson decay modes used in this paper are listed in Table 1, along with
their branching fractions and those of the charmonium decays in the
normalization channels.
To select the $J/\psi K^{-}$ normalization channel, the $p_{\rm T}$
requirement is increased to 1100 MeV for the $K^{-}$ and 750 MeV for the
muons. To select $B^{-}$ candidates we further require that the three tracks
form a vertex with a $\chi^{2}<7$, and that this $B^{-}$ candidate points to
the primary vertex at an angle not different from its momentum direction by
more than 4.47 mrad, and that the impact parameter $\chi^{2}$ of the $B^{-}$
is less than 12. The same requirements will be used for the
$\pi^{+}\mu^{-}\mu^{-}$ selection. The total efficiency for
$\mu^{+}\mu^{-}K^{-}$ is (0.99$\pm$0.01)%, where the $\mu^{+}\mu^{-}$ come
from $J/\psi$ decay.
The invariant mass of $K^{-}\mu^{+}\mu^{-}$ candidates is shown in Fig. 3(a).
In this analysis the $\mu^{+}\mu^{-}$ invariant mass is required to be within
50 MeV of the $J/\psi$ mass. We use a Crystal Ball function (CB) to describe
the signal [20], a Gaussian distribution for the partially reconstructed
background events, and a linear distribution for combinatorial background. The
CB function provides a convenient way to describe the shape of the
distribution, especially in the mass region below the peak where radiative
effects often produce an excess of events that falls away gradually, a so
called “radiative tail”. The CB function is
$f(m;\alpha,n,{m}_{0},\sigma)=\left\\{\begin{array}[]{l
l}\exp\left(-\frac{(m-m_{0})^{2}}{2\sigma^{2}}\right)&\quad\mbox{for
$\frac{m-m_{0}}{\sigma}>-\alpha$}\\\
A\cdot\left(b-\frac{m-m_{0}}{\sigma}\right)^{-n}&\quad\mbox{for
$\frac{m-m_{0}}{\sigma}\leq-\alpha$}\\\ \end{array}\right.$ (1)
where
$A=\left(\frac{n}{|\alpha|}\right)^{n}\cdot\exp\left(-\frac{|\alpha|^{2}}{2}\right)$
$b=\frac{n}{|\alpha|}-|\alpha|.$
The measured mass of each candidate is indicated as $m$, while $m_{0}$ and
$\sigma$ are the fitted peak value and resolution, and $n$ and $\alpha$ are
parameters used to model the radiative tail. We use the notation $\sigma$ in
the rest of this paper to denote resolution values found from CB fits.
Using an unbinned log-likelihood fit yields 47,224$\pm$222 $B^{-}\rightarrow
J/\psi K^{-}$ events. Within a $\pm 2\sigma$ signal window about the peak
mass, taken as the signal region, there are 44,283 of these events. The number
of signal events in this window is also determined using the total number of
events and subtracting the number given by the background fit. The difference
is 119 events, and this is taken as the systematic uncertainty of 0.3%. The
width of the signal peak is found to be 19.1$\pm$0.1 MeV. Monte Carlo
simulations are based on event generation using Pythia [21], followed by a
Geant-4 [22] based simulation of the LHCb detector [23]. The $J/\psi K^{-}$
mass resolution is 20% larger than that given by the LHCb simulation. All
simulated mass resolutions in this paper are increased by this factor.
Figure 3: Invariant mass of (a) candidate $J/\psi K^{-}$ decays, and (b)
candidate $J/\psi K^{-}\pi^{+}\pi^{-}$ decays. The data are shown as the
points with error bars. Both the partially reconstructed background and the
combinatorial background are shown, although the combinatorial background is
small and barely visible. The solid curve shows the total. In both cases the
candidate $\mu^{+}\mu^{-}$ is required to be within $\pm$50 MeV of the
$J/\psi$ mass, and in (b) the dimuon pair is constrained to have the $J/\psi$
mass.
For final states with five tracks, we change the normalization channel to
$B^{-}\rightarrow\psi(2S)K^{-}$, with
$\psi(2S)\rightarrow\pi^{+}\pi^{-}J/\psi$, and
$J/\psi\rightarrow\mu^{+}\mu^{-}$. The branching fraction for this channel is
${\cal{B}}(B^{-}\rightarrow\psi(2S)K^{-})=(6.48\pm 0.35)\times 10^{-4}$ [3].
Events are selected using a similar procedure as for $J/\psi K^{-}$ but adding
a $\pi^{+}\pi^{-}$ pair, that must have an invariant mass when combined with
the $J/\psi$ which is compatible with the $\psi(2S)$ mass, and that forms a
consistent vertex with the other $B^{-}$ decay candidate tracks. The total
efficiency for $\mu^{+}\mu^{-}\pi^{+}\pi^{-}K^{-}$ is (0.078$\pm$0.002)%,
without inclusion of the $\psi(2S)$ or $J/\psi$ branching fractions. The
$B^{-}$ candidate mass plot is shown in Fig. 3(b). Here the $\mu^{+}\mu^{-}$
pair is constrained to the $J/\psi$ mass. (In what follows, whenever the final
state contains a ground-state charm meson, its decay products are constrained
to their respective charm masses.)
The data are fitted with a CB function for signal, a Gaussian distribution for
partially reconstructed background and a linear function for combinatorial
background. There are 767$\pm$29 signal events in a $\pm 2\sigma$ window about
the peak mass. The difference between this value and a count of the number of
events in the signal region after subtracting the background implies a 0.7%
systematic uncertainty on the yield.
## 4 Analysis of $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ and
$D^{*+}\mu^{-}\mu^{-}$
Decay diagrams for $B^{-}\rightarrow D^{(*)+}\mu^{-}\mu^{-}$ are shown in Fig.
2(a). Since the neutrinos are virtual, the process can proceed for any value
of neutrino mass. It is also possible for these decays to occur via a Cabibbo
suppressed process similar to the ones shown in Fig. 2(b), where the virtual
$W^{+}$ materializes as a $c\overline{d}$ pair. If this occurred we would
expect the Cabibbo allowed $D_{s}^{+}\mu^{-}\mu^{-}$ final state to be about
an order of magnitude larger. The search for Majorana neutrinos in this
channel are discussed in Section 6. The $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$
and $D^{*+}\rightarrow\pi^{+}D^{0},~{}D^{0}\rightarrow K^{-}\pi^{+}$ channels
are used. The decay products of the $D^{+}$ and $D^{0}$ candidates are
required to have invariant masses within $\pm$25 MeV of the charm meson mass,
and for $D^{*+}$ candidate selection the mass difference
$m(\pi^{+}K^{-}\pi^{+})-m(K^{-}\pi^{+})$ is required to be within $\pm 3$ MeV
of the known $D^{*+}-D^{0}$ mass difference.
The $D^{(*)+}\mu^{-}\mu^{-}$ candidate mass spectra are shown in Fig. 4. No
signals are apparent. The $B^{-}$ mass resolution is 15.7$\pm$0.5 MeV for the
$D^{+}$ channel and 14.1$\pm$0.6 MeV for the $D^{*+}$ channel. The background
has two components, one from mis-reconstructed $B$ decays that tends to peak
close to the $B^{-}$ mass, called “peaking backgrounds”, and random track
combinations that are parameterized by a linear function. To predict the
combinatorial background in the signal region we fit the data in the sidebands
with a straight line. In the $D^{+}$ mode we observe six events in the signal
region, while there are five in the $D^{*+}$ mode. The combinatorial
background estimates are 6.9$\pm$1.1 and 5.9$\pm$1.0 events, respectively.
Peaking backgrounds are estimated from mis-identification probabilities,
determined from data, coupled with Monte Carlo simulation. For these two
channels peaking backgrounds are very small. The largest, due to
$B^{-}\rightarrow D^{+}\pi^{-}\pi^{-}$, is only 0.04 events.
The total efficiencies for $D^{+}\mu^{-}\mu^{-}$ and $D^{*+}\mu^{-}\mu^{-}$
are $(0.099\pm 0.007)$% and $(0.066\pm 0.005)$%, respectively; here the charm
branching fractions are not included. The systematic errors are listed in
Table 2 for this mode and other modes containing charm mesons that will be
discussed subsequently. Trigger efficiency uncertainties are evaluated from
differences in the 2010 and 2011 data samples. The largest systematic
uncertainties are due to the branching fractions of the normalization channels
and the trigger efficiencies. The uncertainty on the background is taken into
account directly when calculating the upper limits as explained below. Other
uncertainties arise from errors on the charmed meson branching fractions. For
these final states the uncertainty due to different final state track momenta
with respect to the normalization mode are very small, on the order of 0.2%.
Other channels have uncertainties due to varying efficiencies as a function of
Majorana mass, and these are entered in the row labeled “Efficiency modeling”.
The detector efficiency modeling takes into account the different acceptances
that could be caused by having different track momentum spectra. For example,
the track momenta depend on the Majorana neutrino mass for on-shell neutrinos.
These uncertainties are ascertained by simulating the detector response at
fixed Majorana masses and finding the average excursion from a simple fit to
the response and the individually simulated mass points. This same method is
used for other modes.
Figure 4: Invariant mass spectrum for (a) $B^{-}\rightarrow D^{+}\mu^{-}\mu^{-}$ candidates, and (b) $B^{-}\rightarrow D^{*+}\mu^{-}\mu^{-}$ candidates. The solid lines show the linear fits to the data in the mass sidebands. Table 2: Systematic uncertainties for $B^{-}\rightarrow DX\mu^{-}\mu^{-}$ modes. Source | Systematic uncertainty (%)
---|---
Common to all modes |
${\cal{B}}(B^{-}\rightarrow\psi(2S)K^{-})$ | 5.4
${\cal{B}}(\psi(2S)\rightarrow J/\psi\pi^{+}\pi^{-})$ | 1.5
${\cal{B}}(J/\psi\rightarrow\mu^{+}\mu^{-})$ | 1.0
Uncertainty in signal shape | 3.0
Yield of reference channel | 0.7
$\mu$ PID | 0.6
Mode specific | $D_{s}^{+}$ | $D^{0}\pi^{+}$ | $D^{+}$ | $D^{*+}$
Trigger | 4.9 | 9.3 | 5.5 | 4.8
Efficiency modeling | 10.0 | 6.7 | |
PID ($K/\pi$) | 1.0 | | |
Charm decay ${\cal{B}}$’s | 4.9 | 1.3 | 2.2 | 1.5
Total | 13.8 | 13.2 | 8.8 | 8.2
To set upper limits on the branching fraction the number of events $N_{\rm
obs}$ within $\pm 2\sigma$ of the $B^{-}$ mass are counted. The distributions
of the number of events ($N$) are Poisson with the mean value of ($S+B$),
where $S$ indicates the expectation value of signal and $B$ background. For a
given number of observed events in the signal region, the upper limit is
calculated using the probability for $N\leq N_{\rm obs}$:
$P(N\leq N_{\rm obs})=\sum_{N\leq N_{\rm obs}}\frac{(S+B)^{N}e^{-(S+B)}}{N!}.$
(2)
A limit at 95% CL for branching fraction calculations is set by having
$P(N\leq N_{\rm obs})=0.05.$ The systematic errors are taken into account by
varying the calculated $S$ and $B$, assuming Gaussian distributions.
The upper limits on the branching fractions at 95% CL are measured to be
$\displaystyle{\cal{B}}(B^{-}\rightarrow D^{+}\mu^{-}\mu^{-})$
$\displaystyle<$ $\displaystyle 6.9\times 10^{-7}~{}{\rm and}$
$\displaystyle{\cal{B}}(B^{-}\rightarrow D^{*+}\mu^{-}\mu^{-})$
$\displaystyle<$ $\displaystyle 2.4\times 10^{-6}~{}.$
The limit on the $D^{+}$ channel is more stringent than a previous limit from
Belle of $1\times 10^{-6}$ at 90% CL [4], and the limit on the $D^{*+}$
channel is the first such result.
## 5 Analysis of $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$
The selection of $\pi^{+}\mu^{-}\mu^{-}$ events uses the same criteria as
described for $J/\psi K^{-}$ in Section 3, except for like-sign rather than
opposite-sign dimuon charges and pion rather than kaon identification. The
invariant mass distribution of $\pi^{+}\mu^{-}\mu^{-}$ candidates is shown in
Fig. 5. The mass resolution for this final state is 20.3$\pm$0.2 MeV. An
interval of $\pm$2$\sigma$ centered on the $B^{-}$ mass is taken as the signal
region.
Figure 5: Invariant mass distribution of $\pi^{+}\mu^{-}\mu^{-}$. The
estimated backgrounds are also shown. The curve is the sum of the peaking
background and the combinatoric background.
There are 7 events in the signal region, but no signal above background is
apparent. The peaking background, estimated as 2.5 events, is due to
misidentified $B^{-}\rightarrow J/\psi K^{-}$ or $J/\psi\pi^{-}$ decays; the
shape is taken from simulation. The combinatorial background is determined to
be 5.3 events from a fit to the $\pi^{+}\mu^{-}\mu^{-}$ mass distribution
excluding the signal region. The total background in the signal region then is
$7.8\pm 1.3$ events.
Since the putative neutrinos considered here decay into $\pi^{+}\mu^{-}$, and
are assumed to have very narrow widths, more sensitivity is obtained by
examining this mass distribution, shown in Fig. 6, for events in the $B^{-}$
signal region.
Figure 6: Invariant mass distribution of $\pi^{+}\mu^{-}$ in the $\pm 2\sigma$
region of the $B^{-}$ mass with both peaking and combinatorial background
superimposed. The peaking background at 3100 MeV is due to misidentified
$B^{-}\rightarrow J/\psi X$ decays.There are two combinations per event.
There is no statistically significant signal at any mass. There are three
combinations in one mass bin near 2530 MeV; however two of the combinations
come from one event, while it is possible to only have one Majorana neutrino
per $B^{-}$ decay. Upper limits at 95% confidence level on the existence of a
massive Majorana neutrino are set at each $\pi^{+}\mu^{-}$ mass by searching a
signal region whose width is $\pm 3\sigma_{N}$, where $\sigma_{N}$ is the mass
resolution, at each possible Majorana neutrino mass, $M_{N}$. This is done in
very small steps in $\pi^{+}\mu^{-}$ mass and so produces a continuous curve.
If a mass combination is found anywhere in the $\pm 3\sigma_{N}$ interval it
is considered as part of the observed yield. To set upper limits the mass
resolution and the detection efficiency as a function of $\pi^{+}\mu^{-}$ mass
need to be known. Monte Carlo simulation of the mass resolution as a function
of the Majorana neutrino mass is shown in Fig. 7, along with resolutions of
other channels. The overall efficiencies for different values of $M_{N}$ are
shown in Fig. 8. A linear interpolation is used to obtain values between the
simulated points.
Figure 7: Majorana mass resolutions for the three $B^{-}$ decays as a function
of Majorana mass. Figure 8: Detection efficiencies for the three $B^{-}$
decays as a function of Majorana mass. Charm meson decay branching fractions
are not included.
Many systematic errors in the signal yield cancel in the ratio to the
normalization channel. The remaining systematic uncertainties are listed in
Table 3. The largest sources of error are the modeling of the detector
efficiency (5.3%) and the measured branching fractions
${\cal{B}}\left(B^{-}\rightarrow J/\psi K^{-}\right)$ (3.4%), and
${\cal{B}}\left(J/\psi\rightarrow\mu^{+}\mu^{-}\right)$ (1.0%).
Table 3: Systematic uncertainties for $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ measurement. Selection criteria | Systematic uncertainties (%)
---|---
$K/\pi$ PID | 1.0
$\mu$ PID | 0.6
Muon selection | 0.6
Trigger | 1.0
Yields of reference channel | 0.4
Efficiency modeling | 5.3
${\cal{B}}(B^{-}\rightarrow J/\psi K^{-}$) | 3.4
${\cal{B}}(J/\psi\rightarrow\mu^{+}\mu^{-}$) | 1.0
Total | 6.7
To set upper limits on the branching fraction, the number of events $N_{\rm
obs}$ at each $M_{N}$ value (within $\pm 3\sigma_{N}$) are counted, and the
procedure described in the last section applied. Estimated background levels
are taken from Fig. 6. Figure 9(a) shows the upper limit on
${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})$ as a function of $M_{N}$ at
95% CL. For most of the neutrino mass region, the limits on the branching
ratio are $<8\times 10^{-9}$. Assuming a phase space decay of the $B^{-}$ we
also determine
${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})<1.3\times 10^{-8}~{}{\rm
at~{}95\%~{}CL.}$
These limits improve on previous results from CLEO $({<1.4\times
10^{-6}}~{}{\rm at~{}90\%~{}CL})$ [9], and LHCb $({<5.8\times 10^{-8}}~{}{\rm
at~{}95\%~{}CL})$ [10].
Figure 9: Upper limits at 95% CL as a function of the putative Majorana
neutrino mass, (a) for ${\cal{B}}(B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-})$ as a
function of the $\pi^{+}\mu^{-}$ mass, (b) for ${\cal{B}}(B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$) as a function of the $D_{s}^{+}\mu^{-}$ mass, and
(c) for ${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})$ as a function
of the $\pi^{+}\mu^{-}$ mass.
## 6 Analysis of $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$
The process $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ is similar to
$B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$, with the difference being that the
heavy neutrino can decay into $D_{s}^{+}\mu^{-}$. Here we consider only
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decays. Our analysis follows a
similar procedure used for the $\pi^{+}\mu^{-}\mu^{-}$ channel. Candidate
$D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$ decays are selected by having an
invariant mass within $\pm$25 MeV of the $D_{s}^{+}$ mass. A Majorana neutrino
candidate decay is then looked for by by having the $D_{s}^{+}$ candidate
decay tracks form a vertex with an opposite-sign muon candidate. Then this
neutrino candidate must form a vertex with another muon of like-sign to the
first one consistent with a $B^{-}$ decay detached from the primary vertex.
The invariant mass spectrum of $D_{s}^{+}\mu^{-}\mu^{-}$ candidates is shown
in Fig. 10. The mass resolution is 15.5$\pm$0.3 MeV.
Figure 10: Invariant mass spectrum for $B^{-}\rightarrow
D_{s}^{+}\mu^{-}\mu^{-}$ candidates. The line shows the fit to the data
excluding the $B^{-}$ mass signal region.
There are 12 events within the $B^{-}$ candidate mass region; it appears that
there is a dip in the number of events here. An unbinned fit to the data in
the sidebands gives an estimate of 22 events. The fluctuation at the $B^{-}$
mass, therefore, is about two standard deviations. Peaking background
contributions at the level of current sensitivity are negligible
($\sim$3$\times 10^{-4}$); thus only combinatorial background is considered.
After selecting the events in the $B^{-}$ signal region, we plot the
$D_{s}^{+}\mu^{-}$ invariant mass distribution, which is shown in Fig. 11. A
background estimate is made using the sideband data in $B^{-}$ candidate mass
(see Fig. 10), by fitting to a 4th order polynomial. The background estimated
from the sidebands is also shown in the figure. The normalization is absolute
and in agreement with the data. The data in the signal region is consistent
with the background estimate. The systematic error due to the fitting
procedure is estimated using the difference between this fit and the one
obtained using a 6th order polynomial.
Figure 11: Invariant mass spectrum of $D_{s}^{+}\mu^{-}$ from
$B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ events in the signal region with
the background estimate superimposed (solid curve). There are two combinations
per event.
The overall efficiencies for different values of $M_{N}$ are shown in Fig. 8.
As done previously, during the scan over the accessible Majorana neutrino mass
region we use a $\pm$3$\sigma_{N}$ mass window around a given Majorana mass.
The resolution is plotted in Fig. 7 as a function of $M_{N}$. Systematic
uncertainties are listed in Table 2.
Again we provide upper limits as a function of the Majorana neutrino mass,
shown in Fig. 9(b), only taking into account combinatorial background in this
case as the peaking background is absent. For neutrino masses below 5 GeV, the
limits on the mass dependent branching fractions are mostly $<6\times
10^{-7}$. We also determine an upper limit on the total branching fraction.
Since the background estimate of 22 events exceeds the observed level of 12
events we use the $CL_{s}$ method for calculating the upper limit [24].
Assuming a phase space decay of the $B^{-}$ we find
${\cal{B}}(B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-})<5.8\times 10^{-7}~{}{\rm
at~{}95\%~{}CL.}$
## 7 Analysis of $B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$
A prolific source of neutrinos is semileptonic $B^{-}$ decay. Majorana
neutrinos could be produced via semileptonic decays as shown in Fig. 2(c).
Here the mass range probed is smaller than in the case of
$\pi^{+}\mu^{-}\mu^{-}$ due to the presence of the $D^{0}$ meson in the final
state. The sensitivity of the search in this channel is also limited by the
need to reconstruct the $D^{0}\rightarrow K^{-}\pi^{+}$ decay. We do not
explicitly veto $D^{*+}\rightarrow\pi^{+}D^{0}$ decays as this would introduce
an additional systematic uncertainty. The invariant mass distribution of
$D^{0}\pi^{+}\mu^{-}\mu^{-}$ is shown in Fig. 12. The mass resolution is
14.4$\pm$0.2 MeV.
Figure 12: Invariant mass distribution of $D^{0}\pi^{+}\mu^{-}\mu^{-}$. The
solid line shows a linear fit to the data in the sidebands of the $B^{-}$
signal region.
Peaking backgrounds are essentially absent; the largest source is
$B^{-}\rightarrow D^{0}\pi^{-}\pi^{-}\pi^{+}$ which contributes only 0.13
events in the signal region. The combinatorial background, determined by a
linear fit to the sidebands of the $B^{-}$ signal region, predicts 35.9
events, while the number observed is 33.
The $\pi^{+}\mu^{-}$ invariant mass for events within two standard deviations
of the $B^{-}$ mass is shown in Fig. 13. The background shape is estimated by
a 5th order polynomial fit to the sideband data (see Fig. 12) and also shown
on the figure. The systematic error on this background is estimated using a
7th order polynomial fit.
Figure 13: Invariant mass distribution of $\pi^{+}\mu^{-}$ for
$B^{-}\rightarrow D^{0}\mu^{-}\mu^{-}\pi^{+}$ in the signal region and with
estimated background distribution superimposed. There are two combinations per
event.
The $\pi^{+}\mu^{-}$ mass resolution is shown in Fig. 7. The $M_{N}$ dependent
efficiencies are shown in Fig. 8. They vary from 0.2% to 0.1% over most of the
mass range. Systematic errors are listed in Table 2. The largest sources of
error are the trigger, and the $M_{N}$ dependent efficiencies.
The upper limits for ${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})$
as a function of the $\pi^{+}\mu^{-}$ mass are shown in Fig. 9(c). For
Majorana neutrino masses $<$ 3.0 GeV, the upper limits are less than
$1.6\times 10^{-6}$ at 95% CL. The limit on the branching fraction assuming a
phase space decay is
${\cal{B}}(B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-})<1.5\times
10^{-6}~{}{\rm at~{}95\%~{}CL.}$
## 8 Conclusions
A search has been performed for Majorana neutrinos in the $B^{-}$ decay
channels, $D^{(*)+}\mu^{-}\mu^{-}$, $\pi^{+}\mu^{-}\mu^{-}$,
$D_{s}^{+}\mu^{-}\mu^{-}$, and $D^{0}\pi^{+}\mu^{-}\mu^{-}$ that has only
yielded upper limits. The $D^{(*)+}\mu^{-}\mu^{-}$ channels may proceed via
virtual Majorana neutrino exchange and thus are sensitive to all Majorana
neutrino masses. They also could occur via the same annihilation process as
the other modes, though this would be Cabibbo suppressed. The other channels
provide limits for neutrino masses between 260 and 5000 MeV. The bounds are
summarized in Table 4. These limits are the most restrictive to date.
Table 4: Summary of upper limits on branching fractions. Both the limits on the overall branching fraction assuming a phase space decay, and the range of limits on the branching fraction as a function of Majorana neutrino mass ($M_{N}$) are given. All limits are at 95% CL. Mode | ${\cal{B}}$ upper limit | Approx. limits as function of $M_{N}$
---|---|---
$D^{+}\mu^{-}\mu^{-}$ | $6.9\times 10^{-7}$ |
$D^{*+}\mu^{-}\mu^{-}$ | $2.4\times 10^{-6}$ |
$\pi^{+}\mu^{-}\mu^{-}$ | $1.3\times 10^{-8}$ | $(0.4-1.0)\times 10^{-8}$
$D_{s}^{+}\mu^{-}\mu^{-}$ | $5.8\times 10^{-7}$ | $(1.5-8.0)\times 10^{-7}$
$D^{0}\pi^{+}\mu^{-}\mu^{-}$ | $1.5\times 10^{-6}$ | $(0.3-1.5)\times 10^{-6}$
Our search has thus far ignored the possibility of a finite neutrino lifetime.
Figure 14 shows the relative detection efficiency as a function of Majorana
neutrino lifetime, for (a) $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ for a mass
of 3 GeV, (b) $B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV,
and (c) $B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$ for a mass of 2 GeV. All
sensitivity is lost for lifetimes longer than $10^{-10}$ s to $10^{-11}$ s,
depending on the decay mode. Note that for the $D^{(*)+}\mu^{-}\mu^{-}$ final
states the detection efficiency is independent of the neutrino lifetime, since
the neutrino acts a virtual particle.
Figure 14: Relative efficiencies as a function of Majorana neutrino lifetime
for (a) $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV, (b)
$B^{-}\rightarrow D_{s}^{+}\mu^{-}\mu^{-}$ for a mass of 3 GeV, and (c)
$B^{-}\rightarrow D^{0}\pi^{+}\mu^{-}\mu^{-}$ for a Majorana neutrino mass of
2 GeV. Where the error bars are not visible, they are smaller than radii of
the points.
Our upper limits in the $\pi^{+}\mu^{-}\mu^{-}$ final state can be used to
establish neutrino mass dependent upper limits on the coupling $|V_{\mu 4}|$
of a heavy Majorana neutrino to a muon and a virtual $W$. The matrix element
has been calculated in Ref. [5]. The results are shown in Fig. 15 as a
function of $M_{N}$.
Figure 15: Upper limits on $|V_{\mu 4}|^{2}$ at 95% CL as a function of the
Majorana neutrino mass from the $B^{-}\rightarrow\pi^{+}\mu^{-}\mu^{-}$
channel.
A model dependent calculation of ${\cal{B}}(B^{-}\rightarrow
D^{0}\pi^{+}\mu^{-}\mu^{-})$ can also be used to extract $|V_{\mu 4}|$ [11],
but the $\pi^{+}\mu^{-}\mu^{-}$ mode is more sensitive. For the
$D^{(*)+}\mu^{-}\mu^{-}$ channels upper limits cannot be extracted until there
is a theoretical calculation of the hadronic form-factor similar to those
available for neutrinoless double $\beta$ decay.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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|
arxiv-papers
| 2012-01-26T18:43:35 |
2024-09-04T02:49:26.727683
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F.\n Constantin, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, S. De Capua, M. De\n Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp,\n M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, S. Easo, U. Egede, V. Egorychev, S.\n Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch.\n Elsasser, D. Elsby, D. Esperante Pereira, A. Falabella, E. Fanchini, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M.\n Ferro-Luzzi, S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R.\n Forty, M. Frank, C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra\n Tico, L. Garrido, D. Gascon, C. Gaspar, R. Gauld, N. Gauvin, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.\n A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G. Haefeli,\n C. Haen, S. C. Haines, T. Hampson, S. Hansmann-Menzemer, R. Harji, N. Harnew,\n J. Harrison, P. F. Harrison, T. Hartmann, J. He, V. Heijne, K. Hennessy, P.\n Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K. Holubyev, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D. Hutchcroft, D.\n Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah\n Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D.\n Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M.\n Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji,\n Y. M. Kim, M. Knecht, R. Koopman, P. Koppenburg, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K. Kruzelecki, M.\n Kucharczyk, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\'evre,\n A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li Gioi, M.\n Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H.\n Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, M.\n Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, M. Musy, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Nedos, M.\n Needham, N. Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N.\n Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrella, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H. Rademacker, B.\n Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S. Redford, M. M. Reid,\n A. C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, D. A. Roa Romero, P.\n Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J. Rogers, S.\n Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino,\n J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, M.\n Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E. Santovetti,\n M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, P.\n Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N. A. Smith, E. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F.\n Soomro, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V. K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T.\n Szumlak, S. T'Jampens, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez\n Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, B. Viaud,\n I. Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L. Wiggers, G.\n Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi, M. Witek,\n W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z. Yang, R.\n Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y.\n Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1201.5600"
}
|
1201.5606
|
# Leading Relativistic Corrections to the Kompaneets Equation
Lowell S. Brown and Dean L. Preston Los Alamos National Laboratory
Los Alamos, New Mexico 87545
###### Abstract
We calculate the first relativistic corrections to the Kompaneets equation for
the evolution of the photon frequency distribution brought about by Compton
scattering. The Lorentz invariant Boltzmann equation for electron-photon
scattering is first specialized to isotropic electron and photon
distributions, the squared scattering amplitude and the energy-momentum
conserving delta function are each expanded to order $v^{4}/c^{4}$, averages
over the directions of the electron and photon momenta are then carried out,
and finally an integration over the photon energy yields our Fokker-Planck
equation. The Kompaneets equation, which involves only first- and second-order
derivatives with respect to the photon energy, results from the order
$v^{2}/c^{2}$ terms, while the first relativistic corrections of order
$v^{4}/c^{4}$ introduce third- and fourth-order derivatives. We emphasize that
our result holds when neither the electrons nor the photons are in thermal
equilibrium; two effective temperatures characterize a general, non-thermal
electron distribution. When the electrons are in thermal equilibrium our
relativistic Fokker-Planck equation is in complete agreement with the most
recent published results, but we both disagree with older work.
††preprint: LA-UR 12-00207
## I Introduction
The Kompaneets kom equation,
$\frac{\partial}{\partial t}\,f(t,\omega)=\frac{\sigma_{\rm\scriptscriptstyle
T}\
n_{e}}{m_{e}\,c}\,\frac{1}{\omega^{2}}\,\frac{\partial}{\partial\omega}\,\omega^{4}\,\Bigg{\\{}T\,\frac{\partial\,f(t,\omega)}{\partial\omega}+\hbar\Big{[}1+f(t,\omega)\Big{]}\,f(t,\omega)\Bigg{\\}}\,,$
(1)
describes the scattering of unpolarized, low energy photons of frequency
$\omega$ on a dilute distribution of non-relativistic electrons when all the
particles — photons and electrons — are distributed isotropically in their
momenta. The non-relativistic total photon-electron cross section is the
Thomson cross section $\sigma_{\rm\scriptscriptstyle T}$. The electron number
density and mass are denoted by $n_{e}$ and $m_{e}$. The photon phase space
distribution $f(t,\omega)$ is normalized such that the number $n_{\gamma}$ of
photons per unit volume is given by
$n_{\gamma}(t)=2\int\frac{(d^{3}{\bf k})}{(2\pi)^{3}}\,f(t,\omega)\,,$ (2)
in which the prefactor 2 counts the number of photon polarization states and
${\bf k}$ is the photon wave-number vector with $|{\bf k}|c=\omega$. If the
electrons are in thermal equilibrium described by a Maxwell-Boltzmann
distribution, then $T$ is the temperature (in energy units) $T_{e}$ of this
thermal distribution. However, the Kompaneets equation (1) holds for any
isotropic distribution of electron momenta with $T$ defined to be 2/3 of the
average energy in this distribution brown . For photons with a Planck
distribution,
$f(t,\omega)\to
f^{(0)}(\omega)=\frac{1}{\exp\\{\hbar\omega\,/\,T_{\gamma}\\}-1}\,.$ (3)
The terms in the curly braces in the Kompaneets equation (1) vanish when
$T_{\gamma}=T$. In particular, if $T=T_{\gamma}=T_{e}$, there is a time-
independent photon distribution in thermal equilibrium with the
electrons111Since Compton scattering preserves the photon number, the
collision term on the right-hand side of Eq. (1) also vanishes for a general
Bose-Einstein distribution of massless particles at temperature $T$,
$f(t,\omega)\to
f^{(\alpha)}(\omega)=[\exp\\{(\hbar\omega/T)-\alpha\\}-1]^{-1}$..
Our purpose here is to examine the first relativistic corrections to the
Kompaneets equation. These corrections have been previously computed by
Challinor and Lasenby (C&L) C&L for the case in which the electrons are in a
thermal distribution. Using the method of C&L, Itoh, Kohyama, and Nozawa itoh
carried out the expansion to a much higher order in $v/c$. Subsequently,
Sazonov and Sunyaev SS confirmed the previous work of Challinor and Lasenby.
Here we use a method that is quite different from that employed by C&L, a
method that does not require that the electrons be in thermal equilibrium.
Although the structure of our result is quite different from that found by
C&L, we agree with C&L in the number of higher-order derivatives with respect
to the photon frequency $\omega$ which must supplement the Kompaneets equation
to correctly account for the relativistic corrections. Such higher-order
derivative terms are missing from the ad-hoc treatments of Cooper coop and of
Prasad, Shestakov, Kershaw, and Zimmerman prasad . These authors assume
(incorrectly) that the relativistic corrections may be accounted for by simply
replacing the factor $\omega^{4}$ that stands just before the curly braces in
Eq. (1) by a function $\alpha(\omega,T)$ which is determined so as to give the
rate of change of the photon energy density including the first relativistic
corrections. We compute both the rate of energy exchange between the photons
and electrons and the Sunyaev–Zel’dovich effect ZS ; SZ1 ; SZ2 which follow
from the relativistically corrected Kompaneets equation. Including the first
relativistic corrections, our results entail two effective temperatures
$T_{\rm eff\,1}$ and $T_{\rm eff\,2}$ which are defined by energy moments of
the electron phase-space distribution. When the electron distribution is
restricted to a thermal, relativistic Maxwell-Boltzmann distribution at
temperature $T$, $T_{\rm eff\,1}=T_{\rm eff\,2}=T$ and we find, after some
algebra, that our results that have a completely different structure are, in
fact, in complete agreement with those of C&L. Moreover, the rate of energy
exchange that we compute (also written down by C&L) agrees with that found
earlier by Woodward wood .
Our presentation is organized as follows: After describing the general method
we use in Sec. II, we then outline the calculation in Sec. III using the
results of several Appendices. Finally, our results are shown in Sec. IV: Sec.
IV.1 presents our general result, Sec. IV.2 gives its restriction to the case
in which the photons are in thermal equilibrium at temperature $T_{\gamma}$,
Sec. IV.3 derives the rate of energy transport between photons at temperature
$T_{\gamma}$ and the electrons in a general distribution, and finally, in Sec.
IV.4 the Sunyaev-Zel’dovich effect for non-thermal electrons with the first
relativistic correction is briefly described.
## II Relativistic Boltzmann Equation for Isotropic Scattering
We start from the Lorentz invariant form of the Boltzmann equation for
electron-photon scattering:
$\displaystyle k\partial f(x,k)$ $\displaystyle=$
$\displaystyle\int\frac{(d^{3}{\bf
p}^{\prime})}{(2\pi)^{3}}\frac{1}{2E^{\prime}}\ \frac{(d^{3}{\bf
k}^{\prime})}{(2\pi)^{3}}\frac{1}{2\omega^{\prime}}\,\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\frac{1}{2E}\,(2\pi)^{4}\,\delta^{4}(p^{\prime}+k^{\prime}-p-k)$
$\displaystyle\Big{|}T(p^{\prime},k^{\prime};p,k)\Big{|}^{2}\,\Big{\\{}[1+f(x,k)]\,2\,g(x,p^{\prime})\,f(x,k^{\prime})-[1+f(x,k^{\prime})]\,2\,g(x,p)\,f(x,k)\Big{\\}}\,.$
Here we revert to units in which $\hbar=1=c$, but we shall return to
conventional units when we write the final result. The left-hand side of the
equation involves the relativistic scalar $k\partial=\omega(\partial/\partial
t)+{\bf k}\cdot\nabla$. We are assuming that the electrons and photons are not
polarized. Hence $|T|^{2}$ denotes the square of the Lorentz invariant
scattering amplitude that is summed over the initial and final electron and
photon spins. It is divided by the initial electron spin weight $g_{e}=2$ so
as to describe the average scattering from an initially unpolarized ensemble
of electrons. It is divided by the square of the photon spin weight
$g_{\gamma}^{2}=4$ because initially there is an unpolarized mixture and
finally the scattering is into the scalar density $f(x,k)$ that describes a
typical photon [with the factor $g_{\gamma}=2$ needed to provide the photon
number count in Eq. (2)]. The function $g(x,p)$ is the electron phase space
density. We choose our Lorentz metric to have signature $(-\,+\,+\,+)$ so that
$t=x^{0}=-x_{0}$ while for the spatial coordinates $x^{k}=x_{k}$.
We now specialize to the isotropic case of interest where $f(x,k)\to
f(t,\omega)$ and $g(x,p)\to g(t,E)$, with the electron number density given by
$n_{e}=2\int\frac{(d^{3}{\bf p})}{(2\pi)^{3}}\,g(t,E)\,.$ (5)
The integration variables $p$ and $p^{\prime}$ in Eq. (LABEL:boltzmann) are
dummy variables. We shall make the interchange $p\leftrightarrow p^{\prime}$
in the first set of terms in Eq. (LABEL:boltzmann) so as to have a common
factor of $g(t,E)$ for the two ‘scattering in to’ and ‘scattering out of’
terms. To keep a convenient form, we shall also use the detailed balance
relation
$|T(p^{\prime},k^{\prime};p,k)|^{2}=|T(p,k;p^{\prime},k^{\prime})|^{2}$ (6)
for this first term in Eq. (LABEL:boltzmann). Finally, we note that the ${\bf
p}^{\prime}$ integration is best performed using
$\frac{(d^{3}{\bf
p}^{\prime})}{(2\pi)^{3}}\frac{1}{2E^{\prime}}=\frac{(d^{3}{\bf
p}^{\prime})}{(2\pi)^{3}}\frac{1}{2\sqrt{{{\bf
p}^{\prime}}^{2}+m_{e}^{2}}}=\frac{(d^{4}p^{\prime})}{(2\pi)^{3}}\,\delta\left({p^{\prime}}^{2}+m_{e}^{2}\right)$
(7)
against the four-dimensional delta function which now replaces
$p^{\prime}=p+k-k^{\prime}\,,$ (8)
giving
${p^{\prime}}^{2}+m_{e}^{2}=2p\left(k-k^{\prime}\right)-2kk^{\prime}\,.$ (9)
In this fashion, we obtain
$\displaystyle\omega\frac{\partial}{\partial t}\,f(t,\omega)$ $\displaystyle=$
$\displaystyle\int\frac{(d^{3}{\bf
k}^{\prime})}{(2\pi)^{3}}\frac{1}{2\omega^{\prime}}\,\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\frac{1}{2E}\,2\,g(t,E)\,2\pi\,\delta\left(2p\left(k-k^{\prime}\right)-2kk^{\prime}\right)$
(10)
$\displaystyle\qquad\Big{\\{}\Big{|}T(p^{\prime},k;p,k^{\prime})\Big{|}^{2}\,\Big{\\{}[1+f(t,\omega)]\,f(t,\omega^{\prime})$
$\displaystyle\qquad\qquad\qquad-\Big{|}T(p^{\prime},k^{\prime};p,k)\Big{|}^{2}\,[1+f(t,\omega^{\prime})]\,f(t,\omega)\Big{\\}}\,,$
in which the four-momentum $p^{\prime}$ in
$\Big{|}T(p^{\prime},k;p,k^{\prime})\Big{|}^{2}$ is determined by Eq. (8).
The angular part of the integrations over ${\bf p}$ and ${\bf k}^{\prime}$
pick out the completely rotationally invariant part of the integrand. Thus,
with angular brackets denoting the average over all the orientations of the
vectors within it, we may make the replacement
$\displaystyle\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}\,\Big{|}T(p^{\prime},k^{\prime};p,k)\Big{|}^{2}$
$\displaystyle\to$
$\displaystyle\left\langle\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}\,\Big{|}T(p^{\prime},k^{\prime};p,k)\Big{|}^{2}\right\rangle$
(11) $\displaystyle\equiv$ $\displaystyle s(p;\omega^{\prime},\omega)\,.$
In view of these remarks, we may write Eq. (10) as
$\frac{\partial}{\partial
t}f(t,\omega)=\frac{1}{\omega^{2}}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{1}{2E}\,2g(p)\,F(t,\omega;p)\,,$ (12)
with
$\displaystyle F(t,\omega;p)$ $\displaystyle=$
$\displaystyle\frac{\omega}{2\pi}\int_{0}^{\infty}\\!\\!\omega^{\prime}\,d\omega^{\prime}\,\Big{\\{}s(p;\omega,\omega^{\prime})\,[1+f(t,\omega)]\,f(t,\omega^{\prime})-s(p;\omega^{\prime},\omega)\,[1+f(t,\omega^{\prime})]\,f(t,\omega)\Big{\\}}.$
For the evaluation of Eq. (LABEL:ccolll) it is convenient to separate the
weight that appears there into symmetric and antisymmetric parts:
$s(p;\omega^{\prime},\omega)=s^{\rm\scriptscriptstyle
S}(p;\omega^{\prime},\omega)+s^{\rm\scriptscriptstyle
A}(p;\omega^{\prime},\omega)\,,$ (14)
with
$s^{\rm\scriptscriptstyle
S}(p;\omega,\omega^{\prime})=+s^{\rm\scriptscriptstyle
S}(p;\omega^{\prime},\omega)\,,$ (15)
and
$s^{\rm\scriptscriptstyle
A}(p;\omega,\omega^{\prime})=-s^{\rm\scriptscriptstyle
A}(p;\omega^{\prime},\omega)\,.$ (16)
With this decomposition, Eq. (LABEL:ccolll) becomes
$\displaystyle F(t,\omega;p)$ $\displaystyle=$
$\displaystyle\frac{\omega}{2\pi}\int_{0}^{\infty}\omega^{\prime}\,d\omega^{\prime}\,\Big{[}s^{\rm\scriptscriptstyle
S}(p;\omega^{\prime},\omega)-s^{\rm\scriptscriptstyle
A}(p;\omega^{\prime},\omega)\Big{]}\,f(t,\omega^{\prime})$ (17)
$\displaystyle\qquad-\frac{\omega}{2\pi}\,f(t,\omega)\,\int_{0}^{\infty}\omega^{\prime}\,d\omega^{\prime}\,\Big{[}s^{\rm\scriptscriptstyle
S}(p;\omega^{\prime},\omega)+s^{\rm\scriptscriptstyle
A}(p;\omega^{\prime},\omega)\Big{]}$
$\displaystyle\qquad-\frac{\omega}{2\pi}\,2\,f(t,\omega)\,\int_{0}^{\infty}\omega^{\prime}\,d\omega^{\prime}\,s^{\rm\scriptscriptstyle
A}(p;\omega^{\prime},\omega)\,f(t,\omega^{\prime})\,.$
## III Expansions and Angular Averages
It proves convenient to use the angles $\alpha$ , $\alpha^{\prime}$ between
${\bf p}$ and ${\bf k}$ , ${\bf k}^{\prime}$, and the angle $\theta$ between
${\bf k}$ and ${\bf k}^{\prime}$. We also use the velocity
$v=\frac{|{\bf p}|}{E}=\frac{|{\bf p}|}{E(|{\bf p}|)}<1\,.$ (18)
The delta function in Eq. (11) now becomes
$\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}=\frac{1}{2E}\
\delta\Big{(}\omega\left(1-v\cos\alpha\right)-\omega^{\prime}\left(1-v\cos\alpha^{\prime}\right)-(\omega\omega^{\prime}/E)\left(1-\cos\theta\right)\Big{)}\,,$
(19)
and the squared scattering amplitude (49) now appears as
$\displaystyle|T(p^{\prime},k^{\prime};p,k)|^{2}$ $\displaystyle=$
$\displaystyle 6\pi\,m_{e}^{2}\,\sigma_{\rm\scriptscriptstyle
T}\,\Bigg{\\{}2+\frac{1-\cos\theta}{(1-v\cos\alpha)(1-v\cos\alpha^{\prime})}\left[\frac{\omega\omega^{\prime}}{E^{2}}\,(1-\cos\theta)-2\left(1-v^{2}\right)\right]$
(20)
$\displaystyle\qquad\qquad+\left(1-v^{2}\right)^{2}\,\frac{(1-\cos\theta)^{2}}{(1-v\cos\alpha)^{2}(1-v\cos\alpha^{\prime})^{2}}\Bigg{\\}}\,.$
As we shall see, the Kompaneets equation results from expanding Eqs. (19) and
(20) in powers of $v$, with this equation resulting from the order $v^{2}$
terms. The leading relativistic corrections that concern us require that the
expansion be carried out to order $v^{4}$. We note that since in the
applications that we envisage, $\omega,\,\omega^{\prime}\sim T\sim
p^{2}/m_{e}\sim v^{2}\,E$, $\omega/E$ or $\omega^{\prime}/E$ should be counted
as being of order $v^{2}$.
The needed expansion of the delta function (19) in powers of $v$ reads
$\displaystyle
2E\,\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}$
$\displaystyle=$ $\displaystyle\delta\Big{(}\omega-\omega^{\prime}\Big{)}$
(21) $\displaystyle-$
$\displaystyle\Bigg{[}v\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)\delta^{\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{]}$
$\displaystyle+$
$\displaystyle\Bigg{[}\frac{1}{2}\,v^{2}\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)^{2}\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{\omega\omega^{\prime}}{E}\left(1-\cos\theta\right)\delta^{\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{]}$
$\displaystyle-$
$\displaystyle\Bigg{[}\frac{1}{3!}\,v^{3}\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)^{3}\delta^{\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle-\,v\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)\frac{\omega\omega^{\prime}}{E}\left(1-\cos\theta\right)\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{]}$
$\displaystyle+$
$\displaystyle\Bigg{[}\frac{1}{4!}\,v^{4}\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)^{4}\delta^{\prime\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle-\frac{1}{2}\,v^{2}\left(\omega\cos\alpha-\omega^{\prime}\cos\alpha^{\prime}\right)^{2}\frac{\omega\omega^{\prime}}{E}\left(1-\cos\theta\right)\delta^{\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle\qquad+\frac{1}{2}\,\left(\frac{\omega\omega^{\prime}}{E}\right)^{2}\left(1-\cos\theta\right)^{2}\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{]}+\,{\cal
O}(v^{5})\,.$
The term of order $v^{0}$, $\delta(\omega-\omega^{\prime})$, makes no
contribution, since for it the two parts of the collision integral, the
‘scattering in to’ and the ‘scattering out of’, cancel. Hence we need expand
the squared amplitude (20) only to order $v^{3}$ to obtain results good to
order $v^{4}$:
$\displaystyle|T(p^{\prime},k^{\prime};p,k)|^{2}$ $\displaystyle\simeq$
$\displaystyle 6\pi\,m_{e}^{2}\,\sigma_{\rm\scriptscriptstyle
T}\Bigg{\\{}v^{0}\,\Bigg{[}\left(1+\cos^{2}\theta\right)\Bigg{]}$ (22)
$\displaystyle-$ $\displaystyle
2\,v\,\left(1-\cos\theta\right)\,\Bigg{[}\left(\cos\alpha+\cos\alpha^{\prime}\right)\,\cos\theta\Bigg{]}$
$\displaystyle+$ $\displaystyle
v^{2}\,\left(1-\cos\theta\right)\,\Bigg{[}\left(\cos^{2}\alpha+\cos^{2}\alpha^{\prime}+2\cos\alpha\cos\alpha^{\prime}\right)$
$\displaystyle\qquad+\left(2-3\cos^{2}\alpha-3\cos^{2}\alpha^{\prime}-4\cos\alpha\cos\alpha^{\prime}\right)\cos\theta\Bigg{]}$
$\displaystyle+$ $\displaystyle
v^{3}\,\left(1-\cos\theta\right)\,\Bigg{[}\Big{(}-2\cos\alpha-2\cos\alpha^{\prime}+2\cos^{3}\alpha+2\cos^{3}\alpha^{\prime}$
$\displaystyle\qquad\qquad\qquad+4\cos^{2}\alpha\cos\alpha^{\prime}+4\cos\alpha\cos^{2}\alpha^{\prime}\Big{)}$
$\displaystyle\qquad-\Big{(}-4\cos\alpha-4\cos\alpha^{\prime}+4\cos^{3}\alpha+4\cos^{3}\alpha^{\prime}$
$\displaystyle\qquad\qquad\qquad+6\cos^{2}\alpha\cos\alpha^{\prime}+6\cos\alpha\cos^{2}\alpha^{\prime}\Big{)}\cos\theta\Bigg{]}\Bigg{\\}}\,.$
We now multiply the expressions (21) and (22) together and retain the
resulting terms up to those of order $v^{4}$. We then use the result of
angular averaging over the directions of ${\bf p}$ detailed in Appendix B and
subsequently average over the direction between ${\bf k}$ and ${\bf
k}^{\prime}$, the average over $\cos\theta$. To present the results in a
compact form, we separately record the order $v^{2}$ result, the one that
gives the Kompaneets equation,
$\displaystyle s_{2}(p;\omega^{\prime},\omega)$ $\displaystyle=$
$\displaystyle\langle|T(p^{\prime},k^{\prime};p,k)|^{2}\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}\rangle\Big{|}_{2}$
(23) $\displaystyle=$ $\displaystyle
4\,\pi\,\frac{m_{e}^{2}}{E}\,\sigma_{\rm\scriptscriptstyle
T}\,\Bigg{\\{}v^{2}\,\frac{\omega\omega^{\prime}}{3}\,\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}-\frac{\omega\,\omega^{\prime}}{E}\,\delta^{\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{\\}}\,,$
and the order $v^{4}$ result which gives the leading relativistic correction
to the Kompaneets equation
$\displaystyle s_{4}(p;\omega^{\prime},\omega)$ $\displaystyle=$
$\displaystyle\langle|T(p^{\prime},k^{\prime};p,k)|^{2}\,\delta\Big{(}2p\left(k-k^{\prime}\right)-2kk^{\prime}\Big{)}\rangle\Big{|}_{4}$
(24) $\displaystyle=$
$\displaystyle\frac{\pi}{15}\,\frac{m_{e}^{2}}{E}\,\sigma_{\rm\scriptscriptstyle
T}\,\Bigg{\\{}v^{4}\,\bigg{[}2\left(\omega-\omega^{\prime}\right)^{2}\omega\,\omega^{\prime}+\frac{14}{5}\,\omega^{2}\,\omega^{\prime\,2}\Bigg{]}\,\delta^{\prime\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle-10\,v^{2}\,\Bigg{[}\frac{2}{25}\,v^{2}\,\left(\omega-\omega^{\prime}\right)\omega\,\omega^{\prime}+\left(\left(\omega-\omega^{\prime}\right)^{2}+\frac{14}{5}\omega\,\omega^{\prime}\right)\,\frac{\omega\,\omega^{\prime}}{E}\Bigg{]}\,\delta^{\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle+\Bigg{[}-\frac{16}{5}\,v^{4}\,\omega\,\omega^{\prime}+4\,v^{2}\,\left(\omega-\omega^{\prime}\right)\frac{\omega\,\omega^{\prime}}{E}+42\left(\frac{\omega\,\omega^{\prime}}{E}\right)^{2}\Bigg{]}\,\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}$
$\displaystyle+\Bigg{[}12\,v^{2}\,\frac{\omega\,\omega^{\prime}}{E}\Bigg{]}\,\delta^{\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\Bigg{\\}}\,.$
There are no terms of odd order in $v$ in the somewhat lengthly algebra
required to obtain these formulae. They simplify with the aid of the delta
function identities presented in Appendix C:
$\displaystyle\frac{\omega\,\omega^{\prime}}{2\pi\,m_{e}^{2}}\,s^{\rm\scriptscriptstyle
S}_{2}(p;\omega^{\prime},\omega)$ $\displaystyle=$ $\displaystyle
2\,\sigma_{\rm\scriptscriptstyle
T}\,v^{2}\,\frac{(\omega\,\omega^{\prime})^{2}}{3\,E}\,\delta^{\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\,,$
(25)
$\displaystyle\frac{\omega\,\omega^{\prime}}{2\pi\,m_{e}^{2}}\,s^{\rm\scriptscriptstyle
A}_{2}(p;\omega^{\prime},\omega)$ $\displaystyle=$
$\displaystyle-2\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{(\omega\,\omega^{\prime})^{2}}{E^{2}}\,\delta^{\prime}\Big{(}\omega-\omega^{\prime}\Big{)}\,,$
(26)
and
$\displaystyle\frac{\omega\,\omega^{\prime}}{2\pi\,m_{e}^{2}}\,s^{\rm\scriptscriptstyle
S}_{4}(p;\omega^{\prime},\omega)$ $\displaystyle=$
$\displaystyle\sigma_{\rm\scriptscriptstyle
T}\,\frac{(\omega\,\omega^{\prime})^{2}}{15\,E}\Bigg{\\{}\frac{7}{5}\,v^{4}\,\omega\,\omega^{\prime}\,\delta^{\prime\prime\prime\prime}\left(\omega-\omega^{\prime}\right)$
(27)
$\displaystyle\qquad\qquad\qquad\quad+\Bigg{[}\frac{58}{5}\,v^{4}+21\,\frac{\omega\,\omega^{\prime}}{E^{2}}\Bigg{]}\delta^{\prime\prime}\left(\omega-\omega^{\prime}\right)\Bigg{\\}}\,,$
$\displaystyle\frac{\omega\,\omega^{\prime}}{2\pi\,m_{e}^{2}}\,s^{\rm\scriptscriptstyle
A}_{4}(p;\omega^{\prime},\omega)$ $\displaystyle=$
$\displaystyle-\sigma_{\rm\scriptscriptstyle
T}\,v^{2}\,\frac{(\omega\,\omega^{\prime})^{2}}{15\,E^{2}}\Bigg{\\{}14\,\omega\,\omega^{\prime}\,\delta^{\prime\prime\prime}\Big{(}\omega-\omega^{\prime}\Big{)}+28\,\delta^{\prime}\left(\omega-\omega^{\prime}\right)\Bigg{\\}}\,.$
(28)
## IV Results
It is now a straightforward although tedious matter to insert the forms above
into Eq. (17), perform the $\omega^{\prime}$ integrals, and place the results
into Eq. (12) to secure the Kompaneets equation and its leading relativistic
corrections.
### IV.1 General Result
$\displaystyle\omega^{2}\,\frac{\partial}{\partial t}\,f(t,\omega)\ $
$\displaystyle=$ $\displaystyle\sigma_{\rm\scriptscriptstyle
T}\,\frac{n_{e}}{m_{e}\,c}\,\frac{d}{d\omega}\,\omega^{4}\,\Bigg{\\{}T_{\rm
eff\,1}\,\frac{d\,f(t,\omega)}{d\omega}\,+\hbar\,\left[1+f(t,\omega)\right]\,f(t,\omega)\
$ $\displaystyle+$
$\displaystyle\frac{1}{m_{e}\,c^{2}}\left(\frac{5}{2}\,T^{2}_{\rm
eff\,2}+21\,T^{2}_{\rm
eff\,1}\right)\,\frac{d\,f(t,\omega)}{d\omega}+\frac{47}{2m_{e}\,c^{2}}\,T_{\rm
eff\,1}\,\hbar\,\left[1+f(t,\omega)\right]\,f(t,\omega)$ $\displaystyle-$
$\displaystyle\frac{7\,\hbar\omega^{2}}{10m_{e}\,c^{2}}\,\Bigg{[}6\,T_{\rm
eff\,1}\,\left(\frac{d\,f(t,\omega)}{d\omega}\right)^{2}-\hbar\frac{d\,f(t,\omega)}{d\omega}+T^{2}_{\rm
eff\,1}\frac{1}{\hbar}\,\frac{d^{3}f(t,\omega)}{d\omega^{3}}\Bigg{]}\Bigg{\\}}$
$\displaystyle-$ $\displaystyle\frac{21\,n_{e}\,\sigma_{\rm\scriptscriptstyle
T}}{5\,m^{2}_{e}\,c^{3}}\frac{d^{2}}{d\omega^{2}}\,\omega^{5}\Bigg{\\{}\Big{(}T^{2}_{\rm
eff\,2}+2\,T^{2}_{\rm eff\,1}\Big{)}\,\frac{d\,f(t,\omega)}{d\omega}+3T_{\rm
eff\,1}\,\hbar\left[1+f(t,\omega)\right]f(t,\omega)\bigg{\\}}$
$\displaystyle+$ $\displaystyle\frac{7\,n_{e}\,\sigma_{\rm\scriptscriptstyle
T}}{10m^{2}_{e}\,c^{3}}\frac{d^{3}}{d\omega^{3}}\,\omega^{6}\,\Bigg{\\{}\Big{(}T^{2}_{\rm
eff\,2}+T^{2}_{\rm eff\,1}\Big{)}\frac{d\,f(t,\omega)}{d\omega}+2T_{\rm
eff\,1}\,\hbar\left[1+f(t,\omega)\right]f(t,\omega)\Bigg{\\}}\,.$
Here we have reverted to conventional units and used the effective temperature
definitions (76) and (78) in Appendix D which, for convenience, we repeat
here:
$T_{\rm eff\,1}=\frac{1}{n_{e}}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{p^{2}\,c^{2}}{3E(p)}\,2\,g(t,E)\,,$ (30)
and
$T^{2}_{\rm eff\,2}=\frac{4}{15\,n_{e}}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\left(\frac{p^{2}}{2m_{e}}\right)^{2}\,2\,g(t,E)\,.$ (31)
We have chosen to order the terms in our result (LABEL:whee!) so as to have
successive parts involve overall higher derivatives. Since each part starts
out with at least one overall derivative, the result conserves photon number
as it must. Only the part with the single overall derivative $d/d\omega$
contributes to the rate of energy exchange between the photons and the
electrons. Similarly, the rate at which the second moment $(\hbar\omega)^{2}$
changes with time is affected only by the parts involving $d/d\omega$ and
$d^{2}/d\omega^{2}$ while all the parts contribute to the time rate of change
of the $(\hbar\omega)^{3}$ moment. We have kept some photon frequency
derivatives within the sequence of increasingly higher overall derivative so
that the sum of the terms in each of these groups vanishes in thermal
equilibrium. Each of these groupings in the result (LABEL:whee!) vanishes, in
fact, in the more general situation in which the photon distribution is of the
Planck form (3) but with the electron $g(E)$ constrained only to have $T_{\rm
eff\,1}=T_{\rm eff\,2}=T_{\gamma}$. With the electrons in thermal equilibrium,
$T_{\rm eff\,1}$ reduces without approximation to the electron temperature
$T_{e}$, and $T_{\rm eff\,2}\to T_{e}$ in first approximation, an
approximation that suffices since $T_{\rm eff\,2}$ only occurs in the
relativistic correction terms. In this case, each of these groupings of terms
vanishes for a Planck distribution when $T_{\gamma}=T_{e}$. Our expression
(LABEL:whee!) is in complete agreement with the work of Challinor and Lasenby
C&L in the limit in which the electrons are in thermal equilibrium at
temperature $T_{e}$. However, as mentioned previously, the structure of our
expression (LABEL:whee!) for electrons in thermal equilibrium — which is
equivalent to that of C&L — differs completely from the previous (incorrect)
results of Cooper coop and of Prasad, Shestakov, Kershaw, and Zimmerman
prasad .
### IV.2 Photons In Thermal Equilibrium
For photons in thermal equilibrium at a temperature $T_{\gamma}$ [the Planck
distribution (2) or, more generally, a Bose-Einstein distribution of photons]
the result reduces to
$\displaystyle\omega^{2}\,\frac{\partial}{\partial t}\,f(t,\omega)\ $
$\displaystyle=$
$\displaystyle-4\,\frac{d}{d\omega}\,\omega^{3}\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{n_{e}}{m_{e}\,c}\,\Bigg{\\{}\left(1-\frac{21}{20}\,\frac{\hbar^{2}\,\omega^{2}}{m_{e}c^{2}\,T_{\gamma}}\right)\,\Big{(}T_{\rm
eff\,1}-T_{\gamma}\Big{)}$ (32)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{5}{2\,m_{e}\,c^{2}}\,\Big{(}T^{2}_{\rm
eff\,2}-T_{\rm eff\,1}\,T_{\gamma}\Big{)}\,\Bigg{\\}}\,f_{0}(\omega)$
$\displaystyle+\,\frac{d^{2}}{d\omega^{2}}\,\omega^{4}\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{n_{e}}{m_{e}\,c}\,\Bigg{\\{}\left(1-\frac{7}{10}\,\frac{\hbar^{2}\,\omega^{2}}{m_{e}c^{2}\,T_{\gamma}}\right)\,\Big{(}T_{\rm
eff\,1}-T_{\gamma}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\,\,+\frac{47}{2\,m_{e}\,c^{2}}\Big{(}T^{2}_{\rm
eff\,2}-T_{\rm eff\,1}\,T_{\gamma}\Big{)}\Bigg{\\}}\,f_{0}(\omega)$
$\displaystyle-\,\frac{d^{3}}{d\omega^{3}}\,\omega^{5}\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{42\,n_{e}}{5\,m^{2}_{e}\,c^{3}}\,\Big{(}T^{2}_{\rm eff\,2}-\,T_{\rm
eff\,1}\,T_{\gamma}\Big{)}\,f_{0}(\omega)$
$\displaystyle+\,\frac{d^{4}}{d\omega^{4}}\,\omega^{6}\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{7\,n_{e}}{10\,m^{2}_{e}\,c^{3}}\,\Big{(}T^{2}_{\rm eff\,2}-T_{\rm
eff\,1}\,T_{\gamma}\Big{)}\,f_{0}(\omega)\,.$
### IV.3 Energy Transport
The energy transfer per unit volume to the photons is given by
$\dot{u}_{\gamma}=2\int\frac{(d^{3}{\bf
k})}{(2\pi)^{3}}\,\omega\,\frac{\partial}{\partial t}\,f(t,\omega)\,.$ (33)
From Eq. (32), we see that, including the first relativistic corrections, this
energy transfer between a photon distribution in equilibrium at temperature
$T_{\gamma}$ and an arbitrary isotropic distribution of election energies
involves
$\displaystyle 2\int\frac{(d^{3}{\bf k})}{(2\pi)^{3}}\,\omega\,f_{0}(\omega)$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\,\frac{1}{\pi^{2}}\,\int_{0}^{\infty}d\omega\,\omega^{3}\,\exp\left\\{-n\,\frac{\omega}{T_{\gamma}}\right\\}$
(34) $\displaystyle=$ $\displaystyle
T^{4}_{\gamma}\,\frac{3!}{\pi^{2}}\sum_{n=1}^{\infty}\,\frac{1}{n^{4}}=T^{4}_{\gamma}\,\frac{3!}{\pi^{2}}\,\zeta(4)=\frac{\pi^{2}}{15}\,T^{4}_{\gamma}=u_{\gamma}\,,$
and
$\displaystyle 2\int\frac{(d^{3}{\bf
k})}{(2\pi)^{3}}\,\omega^{3}\,f_{0}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\,\frac{1}{\pi^{2}}\,\int_{0}^{\infty}d\omega\,\omega^{5}\,\exp\left\\{-n\,\frac{\omega}{T_{\gamma}}\right\\}$
(35) $\displaystyle=$ $\displaystyle
T^{6}_{\gamma}\,\frac{5!}{\pi^{2}}\sum_{n=1}^{\infty}\,\frac{1}{n^{6}}=T^{6}_{\gamma}\,\frac{5!}{\pi^{2}}\,\zeta(6)=\frac{8\,\pi^{4}}{63}\,T^{6}_{\gamma}=\frac{120\,\pi^{2}}{63}\,T^{2}_{\gamma}\,u_{\gamma}\,.$
Hence, again reverting to conventional units,
$\displaystyle\dot{u}_{\gamma}=4\,\sigma_{\rm\scriptscriptstyle
T}\,\frac{n_{e}}{m_{e}\,c}\,\Bigg{\\{}\left(1-2\,\pi^{2}\,\frac{T_{\gamma}}{m_{e}\,c^{2}}\right)\,\Big{(}T_{\rm
eff\,1}-T_{\gamma}\Big{)}+\frac{5}{2\,m_{e}\,c^{2}}\,\Big{(}T^{2}_{\rm
eff\,2}-T_{\rm eff\,1}\,T_{\gamma}\Big{)}\,\Bigg{\\}}\,u_{\gamma}\,.$ (36)
In the electron thermal equilibrium limit in which $T_{\rm eff\,1}=T_{\rm
eff\,2}=T_{e}$, the energy transfer rate (36) agrees with the rate given by
Woodward wood which was later confirmed by Challinor and Lasenby C&L . The
result (36) holds, of course, for a general isotropic distribution of
electrons with the two effective temperatures $T_{\rm eff\,1}$ and $T_{\rm
eff\,2}$ defined by the integrals (30) and (31).
### IV.4 Sunyaev-Zel’dovich Effect For Non-thermal Electrons
Equation (32) can be used to generalize published results on the Sunyaev-
Zel’dovich effect, the distortion of the cosmic microwave background, a Planck
distribution at a very low temperature, by high energy electrons in hot
plasmas in galactic clusters. The distortion involves changing the time
derivative in the Boltzmann equation to the proper coordinate distance $\ell$
along the line of sight through the plasma cloud, $t\to\ell/c$. In doing so in
the relativistic corrected Kompaneets formula (32), we encounter two
dimensionless variables
$y_{1}=\sigma_{\rm\scriptscriptstyle T}\,\int d\ell\,n_{e}(\ell)\,\frac{T_{\rm
eff\,1}(\ell)}{m_{e}\,c^{2}}\,,$ (37)
and
$y_{2}=\sigma_{\rm\scriptscriptstyle T}\,\int
d\ell\,n_{e}(\ell)\,\frac{T^{2}_{\rm eff\,2}(\ell)}{m^{2}_{e}\,c^{4}}\,,$ (38)
that replace the conventional $y$ parameter222Note that even in the case of a
plasma in local thermodynamic equilibrium with an electron temperature
$T_{e}(\ell)$, the relativistic corrections involve a different $y$ parameter
($y=y_{2}$) defined by the electron number weighted average of
$T^{2}_{e}(\ell)$ rather than the first power $T_{e}(\ell)$ that appears in
$y=y_{1}$.. It is now convenient to define
$x=\frac{\hbar\omega}{T_{\gamma}}\,,$ (39)
and write the Planck distribution as $f_{0}(x)=[\exp\\{x\\}-1]^{-1}$. Since
the microwave background temperate $T_{\gamma}$ is so low, while the electrons
have average energies that are several keV, we may neglect the very small
ratios $T_{\gamma}/T_{\rm{eff}\,1}$ and $T_{\gamma}/T_{\rm{eff}\,2}$ as well
as $T_{\gamma}/m_{e}c^{2}$. With the omission of these terms, carrying out the
derivatives in Eq. (32) yields the spectral distortion
$\displaystyle\frac{\Delta f(x)}{f_{0}(x)}$ $\displaystyle=$ $\displaystyle
y_{1}\,x\,\left[1+f_{0}(x)\right]\,\Big{\\{}x-4+2\,xf_{0}(x)\Big{\\}}$ (40)
$\displaystyle+y_{2}\,x\,\left[1+f_{0}(x)\right]\,\Big{\\{}\frac{x}{10}\left(235-84\,x+7\,x^{2}\right)-10$
$\displaystyle\qquad\quad+\frac{1}{5}\,\left(235-252\,x+49\,x^{2}\right)x\,f_{0}(x)$
$\displaystyle\qquad\quad+\frac{126}{5}\,(x-2)\,x^{2}f_{0}^{2}(x)+\frac{84}{5}\,x^{3}f_{0}^{3}(x)\Big{\\}}\,.$
In the limit of small $x$, the Rayleigh-Jeans region, Eq. (40) simplifies to
$\frac{\Delta f(x)}{f_{0}(x)}=-2\,y_{1}+\frac{17}{5}\,y_{2}\,.$ (41)
In contrast to previous papers on the Sunyaev-Zel’dovich effect, Eqs. (40) and
(41) hold when the electrons are not in thermal equilibrium; when they are in
equilibrium Eqs. (40) and (41) agree with the expressions obtained by
Challinor and Lasenby333These authors, however, do not define the proper
parameters $y_{1}$ and $y_{2}$ that, as noted in the previous footnote, are
needed for the relativistic treatment, but rather use a $y=y_{1}$ and then
multiply this by an undefined electron temperature to obtain a $y_{2}$
parameter for the relativistic corrections. C&L .
## Appendix A Squared Amplitude Details
We first express the fully relativistic squared amplitude text as
$\displaystyle|T(p^{\prime},k^{\prime};p,k)|^{2}=6\pi\,m_{e}^{2}\,\sigma_{\rm\scriptscriptstyle
T}\Bigg{\\{}\left(\frac{\kappa^{\prime}}{\kappa}+\frac{\kappa}{\kappa^{\prime}}\right)+2\left(\frac{m^{2}_{e}}{\kappa}-\frac{m^{2}_{e}}{\kappa^{\prime}}\right)+\left(\frac{m^{2}_{e}}{\kappa}-\frac{m^{2}_{e}}{\kappa^{\prime}}\right)^{2}\Bigg{\\}}\,,$
(42)
in which $\sigma_{\rm\scriptscriptstyle T}=8\pi\,r_{0}^{2}/3$ is the Thomson
cross section with $r_{0}$ the classical electron radius and, with our space-
like metric
$\kappa=-pk=p^{0}k^{0}-{\bf p}\cdot{\bf k}\
\,,\qquad\qquad\kappa^{\prime}=-pk^{\prime}=p^{0}{k^{\prime}}^{0}-{\bf
p}\cdot{\bf k}^{\prime}\,.$ (43)
Note that in terms of these variables the delta function constraint (19) reads
$\kappa^{\prime}-\kappa=kk^{\prime}\,.$ (44)
It is convenient to use the variable
$\bar{\kappa}=\sqrt{\kappa\kappa^{\prime}}\,,$ (45)
so that the relation (44) may be written as
$\frac{\kappa^{\prime}}{\kappa}=1+\frac{kk^{\prime}}{\kappa}=1+\sqrt{\frac{\kappa^{\prime}}{\kappa}}\,\frac{kk^{\prime}}{\bar{\kappa}}\,.$
(46)
The proper solution of this quadratic equation, written in terms of
$\kappa^{\prime}/\kappa$, is
$\frac{\kappa^{\prime}}{\kappa}=\frac{1}{2}\left[\left(\frac{kk^{\prime}}{\bar{\kappa}}\right)^{2}+2+\frac{kk^{\prime}}{\bar{\kappa}}\sqrt{\left(\frac{kk^{\prime}}{\bar{\kappa}}\right)^{2}+4}\,\,\right]\,.$
(47)
Similarly,
$\frac{\kappa}{\kappa^{\prime}}=\frac{1}{2}\left[\left(\frac{kk^{\prime}}{\bar{\kappa}}\right)^{2}+2-\frac{kk^{\prime}}{\bar{\kappa}}\sqrt{\left(\frac{kk^{\prime}}{\bar{\kappa}}\right)^{2}+4}\,\,\right]\,.$
(48)
Making use of Eq. (44) and some algebra now presents
$\displaystyle|T(p^{\prime},k^{\prime};p,k)|^{2}$ $\displaystyle=$
$\displaystyle 6\pi\,m_{e}^{2}\,\sigma_{\rm\scriptscriptstyle
T}\,\left\\{\left(\frac{kk^{\prime}}{\bar{\kappa}}\right)^{2}+2+\,2\,m^{2}_{e}\,\frac{kk^{\prime}}{\bar{\kappa}^{2}}\,+\,m^{4}_{e}\,\left(\frac{kk^{\prime}}{\bar{\kappa}^{2}}\right)^{2}\right\\}\,.$
(49)
## Appendix B Angular Averages
The calculation outlined in the text involves the integration of
$\cos\alpha=\hat{\bf p}\cdot\hat{\bf
k}=\hat{p}^{\,l}\,\hat{k}^{\,l}\,,\qquad\qquad\cos\alpha^{\prime}=\hat{\bf
p}\cdot\hat{\bf k}^{\prime}=\hat{p}^{\,l}\,\hat{k}^{\prime\,l}\,,$ (50)
and of the products of the powers $\cos^{m}\alpha\,\cos^{n}\alpha^{\prime}$,
over the solid angle of the momentum ${\bf p}$ associated with the electron
distribution $g(t,E)$. This is equivalent to averaging over all orientations
of the unit vector $\hat{\bf p}$, and this averaging can be performed at any
stage of the computation — it may be performed before the actual integral over
${\bf p}$ is carried out. The averages may be expressed as contractions of
outer products $\hat{k}^{l}\,\cdots$ and $\hat{k}^{\prime\,m}\cdots$ with the
rotationally invariant tensors that result from the angular averages
$\langle\hat{p}^{k}\,\hat{p}^{l}\,\cdots\rangle$. For example,
$\langle\cos\alpha\,\cos\alpha^{\prime}\rangle=\hat{k}^{l}\,\hat{k}^{\prime\,m}\,\langle\hat{p}^{k}\,\hat{p}^{m}\rangle\,.$
(51)
This is the method that we shall employ.
Under the angular average
$\langle\hat{p}^{\,l}\rangle=0\,,$ (52)
and since ${\bf p}$ is a vector, not a pseudo-vector,
$\langle\hat{p}^{\,l}\,\hat{p}^{\,m}\,\hat{p}^{\,n}\rangle=0\,,$ (53)
because the only rotationally invariant, third rank tensor, is the pseudo-
tensor $\epsilon^{lmn}$. The lowest-order correlation is
$\langle\hat{p}^{\,l}\hat{p}^{\,m}\rangle=\frac{1}{3}\,\delta^{lm}\,,$ (54)
where $\delta^{kl}$ is the matrix element of the invariant unit matrix. The
overall coefficient is determined by the trace
$\langle\hat{p}^{\,l}\hat{p}^{\,l}\rangle=\langle 1\rangle=1\,.$ The final
average that we shall need is
$\langle\hat{p}^{\,k}\hat{p}^{\,l}\hat{p}^{\,m}\hat{p}^{\,n}\rangle=\frac{1}{15}\,\left[\delta^{kl}\,\delta^{mn}+\delta^{km}\,\delta^{ln}+\delta^{kn}\,\delta^{lm}\right]\,.$
(55)
Here the particular combination of the delta symbols on the right-hand side is
required to reproduce the complete symmetry of the left-hand side under any
permutation of the indices $k\,,\,l\,,\,m\,,\,n$. The overall coefficient
again follows from taking the trace over any index pair and comparing the
result with Eq. (54).
Therefore,
$\langle\cos\alpha\rangle=0=\langle\cos\alpha^{\prime}\rangle\,,$ (56)
and
$\langle\cos\alpha\cos\alpha^{\prime}\rangle=\frac{1}{3}\,\hat{\bf
k}\cdot\hat{\bf k}^{\prime}=\frac{1}{3}\,\cos\theta\,,$ (57)
from which follow
$\langle\cos^{2}\alpha\rangle=\frac{1}{3}=\langle\cos^{2}\alpha^{\prime}\rangle\,.$
(58)
Since the angular average of three momentum vectors vanishes,
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\langle\cos^{3}\alpha\rangle=\langle\cos^{2}\alpha\cos\alpha^{\prime}\rangle=\langle\cos\alpha\cos^{2}\alpha^{\prime}\rangle=\langle\cos^{3}\alpha\rangle\,.$
(59)
Next,
$\langle\cos^{3}\alpha\cos\alpha^{\prime}\rangle=\frac{1}{15}\left[\hat{\bf
k}^{2}\,\hat{\bf k}\cdot\hat{\bf k}^{\prime}+\hat{\bf k}^{2}\,\hat{\bf
k}\cdot\hat{\bf k}^{\prime}+\hat{\bf k}\cdot\hat{\bf k}^{\prime}\,\hat{\bf
k}^{2}\right]=\frac{1}{5}\,\cos\theta\,,$ (60)
and
$\langle\cos\alpha\cos^{3}\alpha^{\prime}\rangle=\frac{1}{5}\,\cos\theta\,.$
(61)
Finally
$\langle\cos^{2}\alpha\cos^{2}\alpha^{\prime}\rangle=\frac{1}{15}\left[\hat{\bf
k}^{2}\,\hat{\bf k}^{\prime\,2}+(\hat{\bf k}\cdot\hat{\bf
k}^{\prime})^{2}+(\hat{\bf k}\cdot\hat{\bf
k}^{\prime})^{2}\right]=\frac{1}{15}\left[1+2\cos^{2}\theta\right]\,,$ (62)
from which follow
$\langle\cos^{4}\alpha\rangle=\frac{1}{5}=\langle\cos^{4}\alpha^{\prime}\rangle\,.$
(63)
## Appendix C Delta Function Identities
The work in the text involves various derivatives of
$\delta(\omega-\omega^{\prime})=\delta(x)$ multiplied by various powers of
$\omega-\omega^{\prime}=x$. Simple manipulations can be performed to place a
derivative $d^{n}/dx^{n}$ to the left of a power $x^{m}$, leaving lower
derivatives and lower powers of $x$. Again, the lower derivatives can be
ordered to the left so that all the derivatives appear as total derivatives.
According to the rules of generalized functions, any resulting term of the
form $x^{l}\,\delta(x)$ gives a vanishing contribution. The simplest example
of this procedure is
$\displaystyle x\,\delta^{\prime}(x)$ $\displaystyle=$
$\displaystyle\frac{d}{dx}\,\left[x\ \delta(x)\right]-\delta\left(x\right)\,.$
(64)
As we have just noted, the first term may be discarded. Moreover, the delta
function $\delta(x)=\delta(\omega-\omega^{\prime})$ with no derivative can
also be omitted because it gives rise to equal contributions from the
‘scattering in to’ and ‘scattering out of’ terms in the Boltzmann equation
which cancel. We use the symbol $\,\,`\\!\\!=^{\prime}\,$ to denote the only
effective parts that remain after the manipulations described above have been
made. Thus, we write Eq. (64) as
$\displaystyle x\,\delta^{\prime}(x)\,\,`\\!\\!=^{\prime}\,0\,.\hbox{}\qquad$
(65)
More involved computations lead to the effective results
$\displaystyle
x\,\delta^{\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,-2\,\delta^{\prime}(x)\,,$
(66) $\displaystyle
x^{2}\,\delta^{\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,0\,,\hbox{}\qquad$ (67)
$\displaystyle
x\,\delta^{\prime\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,-3\,\delta^{\prime\prime}(x)\,,$
(68) $\displaystyle
x^{2}\,\delta^{\prime\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,6\,\delta^{\prime}(x)\,,$
(69) $\displaystyle
x^{2}\,\delta^{\prime\prime\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,+12\,\delta^{\prime\prime}(x)\,,$
(70)
and
$\displaystyle
x^{4}\delta^{\prime\prime\prime\prime}(x)\,\,`\\!\\!=^{\prime}\,0\,.\hbox{}\qquad$
(71)
## Appendix D Effective Temperatures Defined By Electron Distribution
Integrals
Here we shall explain the definitions of the two effective temperatures
$T_{\rm eff\,1}$ and $T_{\rm eff\,2}$ that reduce to the electron temperature
$T_{e}$ when the electron relativistic phase-space distribution $g(t,E)$ is
restricted to be a Maxwell-Boltzmann distribution.
For our system of free, relativistic electrons, the number density is given
for an arbitrary phase space distribution $g(t,E)$ by
$n_{e}=\int\frac{(d^{3}{\bf p})}{(2\pi)^{3}}\,2g(t,E)\,.$ (72)
For the case of thermal equilibrium,
$g(p)={\rm const.}\,\exp\left\\{-\frac{E(p)}{T}\right\\}\,,$ (73)
in which
$E(p)=\left[p^{2}+m_{e}^{2}\right]^{1/2}$ (74)
is the total relativistic energy of an electron with momentum $p$.
To obtain the definition of $T_{\rm eff\,1}$, we note that partial integration
gives444What follows is a proof that, for a dilute, non-interacting gas, the
familiar equation of state $p=n\,T$ holds even for relativistic particles.
$\displaystyle n_{e}$ $\displaystyle=$ $\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,2\,{\rm
const.}\,\exp\left\\{-\frac{E(p)}{T}\right\\}\,\frac{1}{3}\,\frac{\partial}{\partial{\bf
p}}\cdot{\bf p}$ (75) $\displaystyle=$ $\displaystyle-\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,{\rm const.}\,\frac{1}{3}\,{\bf
p}\cdot\frac{\partial}{\partial{\bf
p}}\,2\,\exp\left\\{-\frac{E(p)}{T}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{p^{2}}{3E(p)}\,2\,g(t,E)\,.$
Thus we define
$T_{\rm eff\,1}=\frac{1}{n_{e}}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{p^{2}}{3E(p)}\,2\,g(t,E)\,.$ (76)
We have just shown that in thermal equilibrium, $T_{\rm eff\,1}\to T$ is an
exact, relativistic result.
To obtain the definition of $T_{\rm eff\,2}$, we note that we are computing
only the first relativistic corrections to the Compton Fokker-Planck equation.
Hence, for an equilibrium distribution, we may approximate a relativistic
correction integral by
$\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\left(\frac{p^{2}}{2m}\right)^{2}\,2\,g(t,E)$
$\displaystyle\simeq$ $\displaystyle 2\,{\rm
const.}\,\exp\left\\{-m_{e}/T\right\\}\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\left(\frac{p^{2}}{2m_{e}}\right)^{2}\,\exp\left\\{-\frac{p^{2}}{2m_{e}T}\right\\}$
(77) $\displaystyle\simeq$ $\displaystyle\frac{15}{4}\,T^{2}\,n_{e}\,.$
Therefore, for an arbitrary distribution, we shall define
$T^{2}_{\rm eff\,2}=\frac{4}{15\,n_{e}}\,\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\left(\frac{p^{2}}{2m_{e}}\right)^{2}\,2\,g(t,E)\,.$ (78)
To leading order, $T_{\rm eff\,2}\to T$ when $g(t,E)$ becomes a thermal
distribution.
With these results in hand, we can now evaluate the the integrals needed in
the text. The following two integrals appear with the lowest-order functions
$s_{2}^{\rm\scriptscriptstyle S}$ and $s_{2}^{\rm\scriptscriptstyle A}$, and
thus they must be evaluated to both lowest and first non-leading orders:
$\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{v^{2}}{E^{2}}\,g(t,E)$ $\displaystyle=$
$\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{p^{2}}{E^{4}}\,g(t,E)$ (79) $\displaystyle\simeq$
$\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{1}{m_{e}^{3}}\left[\frac{p^{2}}{E}-\frac{6}{m_{e}}\,\left(\frac{p^{2}}{2m_{e}}\right)^{2}\right]\,g(t,E)$
$\displaystyle=$
$\displaystyle\frac{3}{2}\,\frac{n_{e}}{m_{e}^{3}}\,\,\left[\,T_{\rm
eff\,1}\,-\frac{15}{2}\,\frac{T^{2}_{\rm eff\,2}}{m_{e}}\right]\,,$
$\displaystyle\int\frac{(d^{3}{\bf p})}{(2\pi)^{3}}\,\frac{1}{E^{3}}\,g(t,E)$
$\displaystyle\simeq$ $\displaystyle\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{1}{m_{e}^{3}}\left[1-\frac{3}{2}\,\frac{p^{2}}{m_{e}^{2}}\right]\,g(t,E)$
(80) $\displaystyle\simeq$
$\displaystyle\frac{n_{e}}{2m_{e}^{3}}\left[1-\frac{9}{2}\,\frac{T_{\rm
eff\,1}}{m_{e}}\right]\,.$
On the other hand, for the higher order $s_{4}^{\rm\scriptscriptstyle S}$ and
$s_{4}^{\rm\scriptscriptstyle A}$ terms, we need only leading evaluations:
$\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{v^{4}}{E^{2}}\,g(t,E)\simeq\frac{15}{2}\,\frac{n_{e}}{m_{e}^{2}}\,\frac{T^{2}_{\rm
eff\,2}}{m_{e}^{2}}\,,$ (81) $\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{1}{E^{4}}\,g(t,E)\simeq\frac{n_{e}}{2m_{e}^{4}}\,,$
(82) $\int\frac{(d^{3}{\bf
p})}{(2\pi)^{3}}\,\frac{v^{2}}{E^{3}}\,g(t,E)\simeq\frac{3}{2}\,\frac{n_{e}}{m_{e}^{4}}\,T_{\rm
eff\,1}\,.$ (83)
## References
* (1) A. S. Kompaneets, Soviet Phys. JETP (Engl. Transl.) 4, 730 (1957).
* (2) L. S. Brown, Ann. Phys. (NY) 200, 190 (1990).
* (3) A. Challinor and A. Lasenby, Astrophys. J. 499, 1 (1998).
* (4) N. Itoh, Y. Kohyama, and S. Nozawa, Astrophys. J. 502, 7 (1998).
* (5) S. Y. Sazonov and R. A. Sunyaev, Astrophys. J. 543, 28 (2000).
* (6) G. Cooper, Phys. Rev. D 3, 2312 (1971).
* (7) M. K. Prasad, A. I. Shestakov, D. S. Kershaw, and G. B. Zimmerman, J. Quant. Spectrosc. Radiat. Transfer 40, 29 (1988).
* (8) P. Woodward, Phys. Rev. D 1, 2731 (1970).
* (9) Ya. B. Zel’dovich and R. A. Sunyaev, Astrophys. Space Sci. 4, 301 (1969).
* (10) R. A. Sunyaev and Ya. B. Zel’dovich, Comments Astrophys. Space Phys. 4, 173 (1972).
* (11) R. A. Sunyaev and Ya. B. Zel’dovich, Ann. Rev. Astron. Astrophys. 18, 537 (1980).
* (12) See, for example, M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley Pub. Co., Reading MA, 1995), Eq. (5.87).
|
arxiv-papers
| 2012-01-26T19:03:06 |
2024-09-04T02:49:26.737371
|
{
"license": "Public Domain",
"authors": "Lowell S. Brown and Dean L. Preston",
"submitter": "Lowell Brown",
"url": "https://arxiv.org/abs/1201.5606"
}
|
1201.5728
|
# Functional Programming and Security
Yusuf Moosa Motara
Department of Computer Science,
Rhodes University
South Africa
y.motara@ru.ac.za
(November 2011)
###### Abstract
This paper analyses the security contribution of typical functional-language
features by examining them in the light of accepted information security
principles. Imperative and functional code are compared to illustrate various
cases. In conclusion, there may be an excellent case for the use of functional
languages on the grounds of better security; however, empirical research
should be done to validate this possibility.
## Introduction
The functional paradigm dates from the 1930’s, when Alonzo Church created a
system called lambda calculus for understanding computation in terms of
functions[1]. At the time, as is the case at present, the dominant way in
which to understand computation was as a series of operations that changed (or
_mutated_) data. Lambda calculus provided a theoretical basis for
understanding computation as a series of variable transformations, called
_functions_ , instead. It is important to note that the transformation of a
variable is not equivalent to the _mutation_ of data; at the end of the
transformation, there is simply some new data, which is the transformation
result. In functional terms, we can say that the data is _immutable_ : once
created, it cannot be mutated. Furthermore, there is no logical reason for a
variable to be restricted to being data only; therefore, a variable may be a
function, and may be transformed and created and passed to a different
function in the same way that a variable would be. Both of these ideas — that
data is immutable, and that functions are data — are important aspects of
functional thought, and we can see most features of modern-day functional
languages as extensions of these ideas.
The hypothesis of this paper is that the functional paradigm, and the feature-
set common to most widely-known functional languages, leads to code that is by
default more secure than comparable mainstream imperative code. Tevis &
Hamilton suggested the functional paradigm briefly as a more secure
alternative in [2, pp. 4–5], and Tevis followed this up with a deeper look in
[3], where specific reference was made to Haskell. This paper expands on his
work, going into more depth and citing typically-functional features, as well
as making explicit comparisons to the imperative model. The integration of
some functional features into the imperative mainstream is discussed in [4],
where the authors attempt to create a version of Java that includes
functionally-pure methods, and discuss the security benefits of this approach
as they proceed.
## Security principles
To test the hypothesis, a good definition of “security” which is not specific
to the imperative paradigm is necessary. Whereas Tevis has used the chapters
of the “Secure Programming Cookbook in C and C++”[5] to derive security
principles, I have selected the principles provided in the second edition of
“Writing Secure Code”[6], by Howard & LeBlanc. The reason for this is that the
former is specifically geared towards implementation (since it is a cookbook)
in imperative languages, and provides implementation advice rather than
_principles_ for security. The latter, though it still contains a bias towards
imperative languages, describes a theoretical framework that underlies the
security recommendations that are provided. Furthermore, [6] has been accepted
by industry bodies such as OWASP[7] as being an acceptable baseline for
reasoning about code security, and is part of the recommended reading list for
Microsoft’s Security Development Lifecycle methodology[8].
Suitable comparison languages are also needed, to make the case with reference
to concrete examples. I have chosen to use C# 2111Note that these numbers
refer to _language_ versions, not Common Language Runtime (CLR) or .Net
versions! and F# 2 as imperative and functional languages respectively. The
functionality found in the latter is also found in most other functional
languages, such as Haskell, Scala, and OCaml. The former is widely-used, and
implements a set of functionality that is common to other mainstream
“enterprise”-style languages, such as Java. It is also a managed language,
just as F# is; this is necessary for a fair comparison, since comparing the
security benefits of unmanaged code _vs_ managed code would almost certainly
result in a “win” for the managed option[6, 535–6]. Though C# is currently at
version 3, I have chosen to use version 2 because version 3 brings with it
several functional-style changes[9], such as support for lambda-expressions
and expression-based programming in the form of LINQ222Language Integrated
Query. Other mainstream imperative languages, such as C++ or Java, do not have
these features. Therefore, to be generalisable to a wider set of readers who
are familiar with imperative code in an alternative language, I have chosen to
use an older version of C#. Another point that should be addressed is that F#
is not a _pure_ functional language, and one can use non-functional constructs
from within it. I will not use any such constructs in my examples.
For each of the principles that are covered, I have attempted to unpack the
meaning of the principle _as it applies to programming_. Some principles have
been left unchanged; however, others are different from a design or deployment
perspective, but have the same root cause as far as programming is
concerned333This conclusion seems to have been reached independently by [7],
which leaves out certain design- or deployment-focused principles.. These
latter principles have been rationalised as follows:
* •
_Fix Security Issues Correctly_ and _Learn from Mistakes_ both refer to the
idea of understanding the root cause of a defect, and fixing that. The former
principle emphasises determining the root cause, and the latter emphasises
process change to try and ensure that similar mistakes do not occur in future.
* •
_Backward Compatibility Will Always Give You Grief_ and _Employ Secure
Defaults_ both refer to changing, removing or deprecating features to provide
a more secure product.
* •
_Never Depend on Security Through Obscurity Alone_ is an advisory principle
rather than a rule or guideline. It is advisory since it takes no position on
“Security Through Obscurity” itself, but merely says that obscurity (if
employed) should not be the only security measure used. The positive
formulation of this would say that many defenses, possibly including security
through obscurity, should be used. This formulation is quite similar to the
principle of _Use Defense in Depth_ , and the two principles will therefore be
covered together.
### Plan on Failure
When a failure occurs in a particular method, the imperative programmer is
typically faced with two possibilities444In C#, there is also the option of
using an out parameter (though this option is rarely exercised in industry);
in C++, pointers or references are possibilities. None of this functionality
is available in Java. to return a null value, or to throw an exception. This
is particularly true in cases where failures are not expected: for example,
when a dependent system, which has historically always been available, goes
down. Throwing an exception is possible, but is typically a last-resort since
it has serious performance implications for mainstream imperative languages
like Java[10, §6.1][11] and C#[12].
The alternative is to return a null value, thus indicating that the desired
result of the function call is unavailable. However, this option has
consequences of its own that make it difficult to recommend. The first is that
the null value _implies_ that an error may have occurred, but may also mean
that the call succeeded and the correct value to be returned is null. There is
no way for the caller to determine which of these two semantics is the correct
one, other than to make more method calls. The second is that an imperative
language does not typically force a caller to handle the null value
immediately, and the value doesn’t carry a stack trace with it. An exception
only occurs once the value is dereferenced, which could be in a completely
different part of the program; this makes debugging very difficult.
In either case, the caller must remember to create an alternative line of
logic to deal with an error in the called function. Forgetting to do so leads
to a violation of this security principle.
In functional languages, the difference between success and failure is made
explicit by a _discriminated union_ (called Option in F# and Scala, and Maybe
in Haskell). A discriminated union differentiates between a fixed number of
cases, with each case optionally having a type associated with it. For
example, the definition of the Option discriminated union is:
let ’a Option =
| Some of ’a
| None
It is typically used as shown in the following pseudocode:
let getData x =
if dataIsOK x then
let result = (* create result here *)
Some result
else
None
match getData someInput with
| Some x -> // handle success case
| None -> // failure case
The caller of the getData function is forced to acknowledge that the call may
not be successful, in which case the None case will be invoked. In the event
that the call is successful, the Some class will be invoked instead. Note that
it is quite possible for the return value to be Some null; in this event, it
is clear that the call has succeeded, and the correct value happens to be
null. However, in idiomatic functional code, there is no reason for the null
value to appear at all. Haskell, for example, has no conception of null555The
closest that Haskell comes to null is by having the concept of undefined or
“bottom”, which differs significantly from null in terms of behaviour; for
example, any evaluation of undefined, whether in the context of comparison or
not, results in immediate program termination[13, §3.1].. Therefore, the
return value would likely be Some None, rather than Some null. Since null is
not used, there is never any danger of a null value being dereferenced.
Functional languages plan for failure explicitly; imperative languages imply
failure implicitly.
Note that I have used a typically functional feature known as _pattern
matching_ to evaluate a result based on the _structure_ of the result, rather
than the content of the result. It does not matter what the value that is
“wrapped” within the Some case is. All that matters is that, structurally, a
Some case is distinguished from a None case. This emphasis on the structure of
data, rather than the content of data (such as the current value of a
variable), is a line of thought that will be seen in other features as well.
### Fail to a Secure Mode
In [6, p. 65], Howard & LeBlanc provide the following pseudocode to describe a
failure of this principle:
DWORD dwRet = IsAccessAllowed(...);
if (dwRet == ERROR_ACCESS_DENIED) {
// Security check failed.
// Inform user that access is denied.
} else {
// Security check OK.
// Perform task.
}
They challenge the reader to find the problem with the above code, which seems
to be quite secure at first glance. In fact, the issue is in the comparison
dwRet == ERROR_ACCESS_DENIED. Secure code should only execute when dwRet is
equal to NO_ERROR; in the imperative code presented, any return code other
than ERROR_ACCESS_DENIED will result in the secure path being taken. This
approach of checking against known-good values and denying all other values is
called _whitelisting_ or _deny-by-default_ in other spheres of computer
science. Imperative languages don’t make whitelisting easy, however; in fact,
by making it easy to return from a function at any point, they encourage the
practise of checking for certain _invalid_ values early on, and returning an
error if any of those values is detected.
The same code could be expressed much more safely in a functional manner.
Given the following discriminated union:
type AccessLevel =
| Administrator of CredentialInfo
| PowerUser of int * CredentialInfo
| NormalUser of CredentialInfo
| Unauthorised
…the following code is easy to write:
let level = levelOf user
match level with
| Administrator _ -> (* secure code here *)
| PowerUser (n,_) when n > 9000 -> (* alternative secure path *)
| _ -> (* reject request *)
The most important feature of this code is that the secure paths —
Administrator-level access, and PowerUser access when the level is above 9000
— are whitelisted. All other paths are rejected by the default case. The above
code is also the easiest and most idiomatic functional code to write, which
means that whitelisting is the default position in a functional language. By
contrast, code which does not whitelist is more difficult to write:
let level = levelOf user
match level with
| NormalUser _ | Unauthorised -> (* reject *)
| Administrator _ -> (* secure code here *)
| PowerUser (n,_) when n > 9000 -> (* alternative secure path *)
| PowerUser _ -> (* reject *)
Not only is this more to write, and includes duplicate rejection code, the
language compiler will generally complain if the AccessLevel discriminated
union is changed and the non-whitelisting code is not updated. A programmer
quickly feels the maintenance burden and duplication burden of insecure code,
and is therefore encouraged to take the secure path by default.
### Assume External Systems Are Insecure
The same kind of whitelisting mechanism that is used to fail securely can be
used to parse, check, and potentially fail-securely with regard to incoming
data from an external system. There are two types of input that are considered
in this section: structured input, such as may be obtained from a third-party
web service, and unstructured input, such as console or text-box input.
The mechanism itself is a simple extension of the pattern matching concept:
the input is parsed by a function into a structured type, which is then
pattern matched against. In F#, this mechanism is called _active patterns_[14,
15]; in Haskell, it is implemented as a compiler extension to the reference
compiler and called _view patterns_[16]; in Scala, the feature is called
_extractor patterns_[17]; and in OCaml, the macro system provides a superset
of the required functionality.
Converting structured input into a pattern-matchable form is a problem that
arises only in languages that interoperate or integrate closely with with non-
functional code; languages that don’t interoperate easily are presented with
the problem of parsing unstructured input (such as bytes from the network)
into a structured form instead. To handle structured input, it is sufficient
to transform the input into a form that is suitable for pattern-matching.
type StructuredRecord = { Name : string; Age : int }
let (|Structured|_|) (x : ConventionalObject) =
if x = null then None
else Some { Name=x.Name; Age=x.Age }
let greet o =
match o with
| Structured { Name="Bruce" } -> "Welcome back, Mr Wayne."
| Structured { Age=a } when a < 25 -> "Wassup?"
| Structured { Name=n } -> sprintf "Hello %s!" n
| _ -> "Couldn’t convert object; failure case here."
In the above example, a conventional object (of type ConventionalObject) is
converted into a _record type_ by the active pattern666Strictly speaking, the
example shows a _partial_ active pattern; even simpler active patterns with
discriminated cases are possible (see [15]) Structured. If the object cannot
be converted – in this case, if it is null – the pattern match falls through
to the default case, where failure would be handled securely. The active
pattern could also have converted ConventionalObject to a tuple, discriminated
union, or any other pattern-matchable object.
For unstructured input, we can parse securely using a variation777Based on a
code sample presented in [14, p. 32–3]. on the above.
let (|Match|_|) pattern input =
let m = Regex.Match (input, pattern)
if m.Success then Some (List.tail [for x in m.Groups -> x.Value])
else None
let identify input =
match input with
| Match @"^Username=(\w+)$" [user] -> printfn "User is %s" user
| Match @"^Name=(\w+)\s+(\w+)$" [name;surname] -> printfn "Full name is %s %s" name surname
| _ -> (* handle failure *)
Once again, we specify the whitelisted cases, extract the data which matches
the whitelist, and any input that does not match is handled by the default
failure case.
### Minimise Your Attack Surface
The “attack surface” of an application is defined by example in [6, p. 57] as
including sockets, named pipes, RPC888Remote Procedure Call endpoints,
services, dynamic web pages, accounts, files, directories, and registry keys —
as well as any other possible vectors. Manadhata lays out a much more
sophisticated and systematic analysis of what the term “attack surface” means
for a programmer in his 2008 thesis[18, §3.2.7]:
> Given a system, $s$, and its environment, $E_{s}$, $s$’s attack surface is
> the triple $\langle{}M^{E_{s}},C^{E_{s}},I^{E_{s}}\rangle{}$, where
> $M^{E_{s}}$ is the set of entry and exit points, $C^{E_{s}}$ is the set of
> channels, and $I^{E_{s}}$ is the set of untrusted data items of $s$.
Entry points may be direct, which means that they accept input directly from
the environment, or indirect, which means that they accept input from a chain
of inputs which terminates in the environment. Exit points may similarly be
direct or indirect. A channel is a path by which entry and exit points receive
and send data respectively, such as a TCP socket or web request/response.
Lastly, an untrusted data item is “a persistent data item $d$ such that a
direct entry point of $s$ reads $d$ from the data store or a direct exit point
of $s$ writes $d$ to the data store”[18, p. 26], where a _persistent_ item is
visible to both system and user across different executions of $s$.
The most obvious entry and exit points are found in the public Application
Programming Interface (API) of a software component. In this regard, there
appear to be no significant differences between the size of a functional API
and the size of a non-functional API. For example, the C#
System.Collections.Generic.Dictionary[19] type API consists of 7 constructors,
9 public methods999Excluding 5 public methods inherited from System.Object but
not overloaded., and 5 properties for a total of 21 entry/exit points; by
contrast, the Microsoft.FSharp.Collections.Map[20] module contains 25
functions, which are its entry/exit points. There appears to be no attack
surface advantage that accrues to functional programming.
The situation changes once multi-core programming is considered. Recall the
triple $\langle{}M^{E_{s}},C^{E_{s}},I^{E_{s}}\rangle{}$: in a thread-using
imperative program, $C^{E_{s}}$ includes mutable thread-accessible memory,
$M^{E_{s}}$ includes all reads and writes to thread-shared variables, and
$I^{E_{s}}$ includes all variables that threads can potentially alter. These
variables need to be synchronised and protected to avoid deadlocks, livelocks,
and race conditions.
void Expirer() {
List<K> to_expire = new List<K>();
while (true) {
Thread.Sleep(expiry);
to_expire.Clear();
lock (storage) {
foreach (var kvp in storage) {
if (DateTime.Now - kvp.Value.Item2 >= expiry)
to_expire.Add(kvp.Key);
}
}
foreach (var k in to_expire)
lock (storage) { storage.Remove(k); }
}
}
The code sample above could be part of a very basic C# timed-expiry cache. It
shows the thread within the cache that expires items at the appropriate time.
Note that the storage dictionary must be held exclusively while the thread
does cache expiry. As a performance optimisation, the programmer releases
storage between removals of each expired item. This has the effect of allowing
an attacker to insert a key at the appropriate timing and have it almost
instantly removed by the system; whether this unintended consequence has any
effect on security depends on the system that the flawed cache is used in.
However, what is important to note is that an entry point into the system can
allow a value to be removed from an untrusted data store: the attack surface
of the multi-threaded program is increased.
let cache timeout =
let expiry = TimeSpan.FromMilliseconds (float timeout)
let exp = Map.filter (fun _ (_,dt) -> DateTime.Now-dt >= expiry)
let newValue k v = Map.add k (v, DateTime.Now)
MailboxProcessor.Start(fun inbox ->
let rec loop map =
async {
let! msg = inbox.TryReceive timeout
match msg with
| Some (Get (key, channel)) ->
match map |> Map.tryFind key with
| Some (v,dt) when DateTime.Now-dt < expiry ->
channel.Reply (Some v)
return! loop map
| _ ->
channel.Reply None
return! loop (Map.remove key map)
| Some (Set (key, value)) -> return! loop (newValue key value map)
| None -> return! loop (exp map)
}
loop Map.empty
)
Though the functional world has not yet settled on a dominant paradigm for
multi-core programming – for example, Haskell prefers software transactional
memory[21], and OCaml researchers are still investigating alternatives – one
of the oldest patterns for multi-core programming is found in Erlang[22], and
implemented in Scala[23] and F#: the actor model. In this pattern, messages
are sent to “actors”, who take some action based on the message content and,
if requested, return some reply to the sender. Messages and responses are
immutable, and the state within an actor is kept in an immutable data
structure; it is impossible for the flaw in the imperative cache system to
exist in the functional implementation. Immutability means that the attack
surface for a single-core program is the same as the attack surface for a
multi-core program; the same cannot be said for the imperative paradigm.
### Employ Secure Defaults / Backward Compatibility Will Always Give You Grief
This principle refers to removing uncommonly-used or insecure-by-default
features from a default installation, or deprecating/removing those features
altogether. A “feature” is some functionality that makes up part of a system;
in the imperative object-oriented world, creating a new feature may involve
creating new classes, determining the interactions between those classes, and
integrating the classes with existing code. At a minimum, a new feature
involves the alteration of existing methods to provide different
functionality.
Given this common-sense description of what a feature is, it follows that the
granularity of a system — the extent to which features can be decomposed and
regarded as separate units of functionality — depends largely on the coupling
of features to the surrounding code of the system. Features that are tightly-
coupled to the surrounding code are difficult to disable since removing them
requires a large number of changes to the surrounding code. Conversely,
features that are loosely-coupled require only a small amount of work to
disable.
Different methods have been proposed for measuring coupling. Green et al.[24,
p. 23–9] provide a good summary of the past 25 years of work on slice-based
methods101010These methods follow on from, and may be considered to be
superior to, the metrics proposed by Henry & Kafura in [25]; see, for example,
the discussion of flow _vs_ bandwidth in [26, p. 8] or [24, p. 23]. for
determining program coupling. Briefly, a _backward slice_ is the set of
program statements that affect the calculation of a variable, and a _forward
slice_ is the set of program statements that are affected by the calculation
of a variable. In layman’s terms, to calculate the coupling for a module one
starts at _output variable_ s — which may be a function return values, global
variables, printed variables, or reference parameters — and determines the
full backward slice for that variable, which provides all the calculations
that it is dependent on. It is then easy to see which dependent variables are
found in other modules and to determine the extent of the coupling.
While slicing can be generalisable and applicable to object-orientated
languages, its roots are found in traditional imperative languages such as C.
Chidamber & Kemerer proposed the _Coupling Between Objects_ (CBO) metric[27,
p. 485–7], defined as usage of methods or attributes of a class $A$ from a
class $B$. Briand et al. provide an overview of coupling metrics in [28], some
of which are more complex than Chidamber & Kemerer’s understanding, and some
of which are simpler. All coupling metrics agree on the CBO metric as a
baseline, and differ on which other aspects of the object-oriented approach
(such as inheritance) affect coupling.
It is clear, from the body of work on coupling, that this measure of
complexity is considered important, with “more than 30 different measures of
object-oriented coupling”[28, p. 3] found extant in 1999. It is taken as a
given that coupling can be mitigated, but not eliminated in non-trivial
projects. Many authors have stated that tightly-coupled modules are difficult
to untangle — and, consequently, tightly-coupled features are difficult to
disable. Furthermore, attempting to disable or remove a tightly-coupled
feature is an error-prone task.
The fundamental reason for the existence of coupling measures is implied by
slice-based analysis: the value of an output variable depends on the
calculation of dependent variables, and affects the calculation of other
variables in turn. This reveals, once again, the emphasis that the imperative
paradigm places on the importance of variable _value_ , as opposed to data
_structure_. In object-oriented terms, the problems that coupling points to
are a microcosm of this effect:
* •
two invocations of the same method on a class may return entirely different
results, depending on the current state of the class
* •
the state of a class may be altered by other functions (in C# and Java, this
is accomplished via “getters” and “setters” in programs that follow best
practice)
Coupling is not an active research area in the functional programming space.
As a consequence of immutability, pure functions always return the same value
when provided with the same input parameters; there is no “local state” that
can affect the calculation. This is known as _referential transparency_.
Actors are an exception to this norm: when using the actor model for multi-
core programming, it can be argued that an actor could contain some local
state. This state is immutable: more specifically, it can be said that there
is no state that changes, but there is a different immutable state that exists
for each request and which cannot be affected by anything other than the
actor. Inheritance, global mutable variables, mutable reference parameters,
and other aspects of the imperative or object-oriented world that lead to
coupling are largely unknown.
The immutability of data and its consequence for pure functions means that
features are, by default, loosely-coupled in a functional system. Disabling or
removing features to achieve secure defaults is correspondingly easier.
### Fix Security Issues Correctly / Learn From Mistakes
To fix security issues correctly or learn from mistakes, a programmer should
be able to understand the reasoning behind the program statements. Consider
the following imperative code:
public class LargeItem {
Ψ...
public Entity Mangle(Entity e, int n) {
if (!Reevaluate(e)) return null;
if (n > 0x33 && n < 0x55 && n != 0x44)
e.Discombobulate(0x55, n, 0x33);
return e;
}
...
}
A typical call to this code could look something like
large.Mangle(someEntity, input).Zombify();
What could go wrong with this code?
* •
The call to Mangle could return null, causing the call to the Zombify member
to throw a null-reference exception.
* •
Reevaluate could alter the value of the passed Entity, which could change the
result of the Discombobulate call.
* •
Mangle will always perform a call to Reevaluate, but will only sometimes
perform a call to Discombobulate. If the caller expects Discombobulate to be
called for every invocation, this could lead to unexpected behaviour.
* •
Reevaluate will succeed or fail based on the internal state of the LargeItem
instance, which may or may not be appropriate at the time of the call.
* •
The range-check for the first and third parameters of Discombobulate are
performed in Mangle, outside of the body of Discombobulate.
This (incomplete) list should be sufficient to show some of the issues that
could arise when determining why a problem has arisen at the call-site.
Diagnosing any of the behaviours listed requires analysis of all the values
that could affect the computation at the call-site, and one of the easiest
ways to do this is by attaching a debugger and examining values.
Given industry demands for productivity (and, quite possibly, programmer ego!)
there is an incentive to “resolve” bugs as quickly as possible. For example,
if the issue is a null-reference exception, the easiest fix is to write an if-
statement to check for the null value, and only call the Zombify member if
there isn’t a null return. The further down the call-tree the programmer goes,
the more dependent variables there are to consider, and the more difficult the
“right” solution is to determine — let alone fix correctly. Reasoning about
the code becomes increasingly difficult and dependent upon runtime values.
The corresponding functional code may look as follows:
let mangle n relatedInfo entity =
entity
|> reevaluate relatedInfo
|> Option.map (discombobulate n)
...
let mangled =
someEntity
|> mangle input related
mangled |> Option.map zombify
A few points are worth mentioning about this code, as compared to the
imperative version of the same logic:
* •
The state information by which the imperative Mangle would determine the
result of Reevaluate is explicitly encoded as a parameter in the functional
version. Therefore, all items that could affect the calculation of mangled can
be identified by visual inspection.
* •
The value of someEntity remains the same throughout. The transformed value may
be used explicitly by referencing the mangled symbol instead of the someEntity
symbol.
* •
No null-reference exception is possible (see “Plan On Failure”, p. Plan on
Failure).
* •
discombobulate cannot have any incidental side-effects (such as resetting a
boolean flag) since mutable data does not exist in the functional paradigm.
Whether it is called or not is only relevant in terms of the transformation of
someEntity.
* •
The range-check has been shifted into the body of discombobulate (not shown).
If it were _not_ shifted, the “pipelining” syntax of mangle would be
interrupted, and the code would be less idiomatic; I believe that pipelining
syntax visually encourages the programmer to shift conditionals towards their
correct place.
The most important difference, however, is that it is possible for a
programmer to reason about the program that is being written _without_
reference to runtime variable values, and with reference to the structure of
the data that is being transformed. This makes it easier to determine why a
particular issue has arrived, pinpoint the cause of the issue, and apply the
correct fix.
### Use Defense in Depth / Never Depend on Security Through Obscurity Alone
Defense in depth involves layering “defensive” code (such as authorisation
checks, resource-scoping code, and input/output-encoding) around the system,
with the goal of stopping an attacker who has circumvented $n$ layers at layer
$n+1$. Some features of the functional paradigm lend themselves to this
easily: for example, the fact that data is immutable makes it extraordinarily
difficult for an attacker to modify a “shared” resource (such as a path to
“authorised” plugin objects), since there are no mutable shared resources.
However, it is one thing to have features that enhance security, and quite
another to arrange for them to be layered as defenses that must be broken
through by a determined attacker. In an imperative object-oriented language,
one would trade readability for security with a construction such as
StaticMethods.EncodeOutput(
StaticMethods.CheckCredentials(
StaticMethods.CheckAuthorisation(
StaticMethods.ValidateToken(
...actual method call here...
))));
In C#, extension methods ease the syntax by removing the necessity of
referencing StaticMethods at every turn; in Java, the import static directive
has the same effect. Nevertheless, the code is cumbersome to write. At a
minimum, all of the defensive measures might be placed into a separate static
method, and the resulting defensive code might be as simple as
DefendOutput(...actual method call here...);
If a programmer fails to call DefendOutput, no checks are done. The
alternative approach suggests placing the checks within the relevant methods,
so that they done even if the caller doesn’t call a separate method such as
DefendOutput.
if (!StaticMethods.CheckCredentials(...)) return null;
if (!StaticMethods.CheckAuthorisation(...)) return null;
if (!StaticMethods.ValidateToken(...)) return null;
...
return StaticMethods.EncodeOutput(output);
This transfers the burden of ensuring that security checks are done from
client code to library code. Recognising this, various frameworks for
imperative languages, especially in the domain of web applications, have
created special syntax and attributes to make the process of applying security
checks easier.
Using a functional language, the situation becomes somewhat easier since
functions can be easily created via _composition_ , _chaining_ , and _partial
application_.
Composition and chaining refers to the creation of a new function by combining
existing functions. One of the applications of this is creating
“chokepoints”[6, p. 345–7], which are functions that encode input or output.
let encodeInput (s : string) = (* take a string as input, return a string as output *)
let someFunction x = (* manipulate x and output a string *)
let someOtherFunction x y z = (* manipulate x, y, and z, and output a string *)
let encodedFunction = someFunction >> encodeInput
let encodedOtherFunction a b c = someOtherFunction a b c |> encodeInput
In this example, encodedFunction is the result of composing someFunction and
encodeInput; any input to f will be processed by someFunction, then by
encodeInput, transparently to the caller. encodedOtherFunction chains
someOtherFunction to encodeInput — composition is not possible since
someOtherFunction and encodeInput take different numbers of parameters — and
is also transparent to the user. Exposing only the composed/chained functions
will ensure that all input and output is always encoded.
Partial application is a way to specify some of a function’s arguments in
order to specialise that function.
let isAuthorised user contextInfo functionality = true
let canDo user contextInfo = isAuthorised user contextInfo
// usage:
// canDo ViewReceipts
canDo is a specialisation of isAuthorised for a specific user and context. At
the beginning of every invocation of functionality (for example, at the start
of each web request), it is useful to create partially-applied functions which
“capture” the current context, and pass these functions onwards for use
elsewhere. The resulting functions are more general in nature by virtue of
being specialised: a caller of canDo does not need to know that user and
contextInfo exist, and only needs to know which functionality (such as
ViewReceipts) is required.
While there appears to be no good reason for chaining and composing functions
to not be used as part of the imperative paradigm, the fact is that small
functions which merely combine or specialise other functions are not idiomatic
in most imperative languages. I theorise that this is because some commonly-
found features of functional languages are not found in mainstream imperative
languages.
The first of these features is _referential transparency_ , which has already
been discussed briefly on p. Employ Secure Defaults / Backward Compatibility
Will Always Give You Grief. An implication of referential transparency is that
any symbol can be replaced by the calculations used to obtain that symbol,
with no change in the meaning of the program. Since variables in the
imperative paradigm are mutable, we cannot be sure that a symbol will remain
equivalent to the result of its initial assignment; nor can we be sure that
the variables which were used to calculate the symbol will not change between
the calculation of the symbol and our use thereof. Without this certainty,
partial application of a function requires explicitly capturing the _current_
value of a variable, which (in turn) may entail creating a custom data
structure – quite a lot of trouble to take for some eventual notational
convenience! Chaining and composing functions causes some difficulty as well,
since a call such as DoX(DoY(data)) makes debugging more difficult: if an
error occurs, is it as a result of DoX or DoY? A programmer examining the
issue would have to follow a variable into the DoY call, then into the DoX
call, which breaks the line-based debugging that is a feature of many
development environments. Composing more functions, DoX(DoY(DoZ(data))),
exacerbates this issue.
The second of these features is the granularity of units of functionality. In
a functional language, the standard unit of self-contained functionality is
the function. It is dependent on nothing except for its input parameters, and
may be called freely from any place that is able to reference it; there is no
“container” that must be created before it can be used. In an imperative non-
object-oriented language, the standard unit of functionality is the program:
all functions can modify program state, via references or global variables,
and it is telling that one of the criticisms of non-object-oriented imperative
languages is that they can easily lead to “spaghetti code” if not structured
correctly[29]. In object-oriented imperative code, the standard unit is the
object: to do anything useful, the programmer has to create an object, and
call methods on it. To compose or chain methods in different non-static
classes, a programmer must pass along references to each class; to simulate
partial application, a design pattern such as Command[30, p. 233–41] may need
to be used. Due to the granularity of units of functionality being the class,
rather than the function, combining functions is an unwieldy proposition.
Lastly, functional programming languages typically use a form of type
inference based on the Hindley-Milner algorithm[31, 32], which allows for
functions that support _parametric polymorphism_. This concept is best
explained by an example:
let double x = x * 2
let result = List.map double [0;1;2;3;4]
result will be the list [0;2;4;6;8]; the function double has been applied to
each element of the input list [0;1;2;3;4] by the List.map function. This
seems reasonable, until we consider that List.map doesn’t simply work on lists
of integers; it can work on lists of any type. Creating a function that works
with any types is something that has been difficult in statically-typed
imperative languages: Java has used boxing and type-erasure with compiler-time
checks to solve it, C upcasted to void* and used the sizeof operator, C++ has
developed the concept of templates, and C# chose reified generics as their
solution. F#, Haskell, Scala, and OCaml use Hindley-Milner type inference
instead. The type inference algorithm selects the most _general_ possible type
for a symbol, given the way in which the symbol is used. In the case of
List.map, the function signature is
(’a $\rightarrow$ ’b) $\rightarrow$ ’a list $\rightarrow$ ’b list
In other words, given a function which takes a ’a and returns a ’b, and a list
of ’a, a list of ’b will be returned; each of these parameters is generalised
(or _polymorphic_) to the maximum extent possible. This process of
generalisation occurs automatically.
By contrast, in most statically-typed object-oriented imperative languages,
type inference does not play a large role. Generalisation is explicit, along
the lines of class hierarchies and interfaces: to generalise a method, one
either passes in a superclass in the inheritance hierarchy, or extracts it
into a named interface and then uses that interface as a parameter to the
appropriate method. Explicit generalisation makes combining methods into a
manual exercise of matching types, which means that imperative programmers may
see it as a needless and laborious task.
### Don’t Mix Code and Data
The desire to mix code and data is one that arises when user control of a
system, or user interactivity, is desired. Lotus 1-2-3 version 2.0 is cited as
one of the first widely-used systems to incorporate interactivity in the form
of macros — data which was interpreted as code — and thus “perform custom
actions defined by the user”[6, p. 67]. A safer method of custom code
execution is sandboxing, with many advances made in that area in recent
times[33].
Nevertheless, the desire to create some form of user interactivity that is
less powerful (or more restricted) than native code execution, but more
powerful than pre-built parameterised commands, may exist. For example, a game
developer may wish to allow users to script certain behaviours within a game,
but not have any significant level of access to the API of the game. This can
be achieved by creating a domain-specific language (DSL), which is interpreted
and gives rise to the desired results.
A DSL may be categorised as either internal or external[34] depending on
whether the DSL is expressed in terms of the host language or not. Internal
DSLs are difficult to compare in any meaningful way since they necessitate a
comparison of host language syntaxes and the manner in which they relate to
the DSL that is being examined; such a comparison is fraught with
subjectivity, since one programmer’s verbosity is another programmer’s
explictness. External DSLs don’t suffer from this issue, and are more powerful
cognitive tools in any case since they do not force a user to think in the
syntax of a language that has not been built to match the problem domain.
In both imperative and functional languages, a parser for the DSL would be
generated based on a formal grammar. The parse results in an abstract syntax
tree, and here the functional and imperative worlds diverge. In an imperative
language, the nodes of the tree must be examined by manual inspection or by
creating classes for each important construct in the tree. In a functional
language, the tree is typically represented as a discriminated union, and
pattern-matching is used to decompose and process it; the usual security
benefits of whitelisting apply.
### Use Least Privilege
It is axiomatic that security should be embedded at the lowest possible level.
For example, if an operating system allows a programmer to lock a security-
relevant resource for exclusive use, it makes no sense to create application-
level code to repeatedly test whether the resource has been modified; another
application may circumvent such code, or merely use the operating system
functionality to go around it. This does not contradict the idea of defense in
depth, since in-depth security measures should be orthogonal.
The same low-level security measures that are available to imperative code are
also available to functional code: access-control lists, permissions, locks,
exclusive channels, and so on. Some of these may not be used as much by
idiomatic functional code — for example, the actor model allows for lock-free
multi-core programming, which makes mutexes irrelevant — but whether a given
measure is typically used or not has no impact on its availability for use.
This is as true for privilege-affecting security measures as it is for more
general security measures.
Implementing the principle of least privilege in an imperative language
involves data flow analysis as a first step[6, p. 61, 73–5]. _A priori_ , we
can understand the value of data-flow analysis as soon as we accept that
different resources should have different levels of access associated with
them. For example, anybody may be able to read a network-shared public file;
creating a trust relationship between two networked machines should
intuitively be a more privileged operation. Furthermore, the type and context
of access matters: reading a file from a desktop application and writing a
file from a web browser are two very different scenarios.
Schwartz et al. point out the difficulties inherent in conventional data flow
analysis by formalising existing literature around dynamic111111Static taint
analysis involves no runtime checks, and is not considered under in this
paper. taint analysis[35]. Taint analysis revolves around the idea that input
from an untrusted source is “tainted”, and should not be used until it has
been validated in some way. Howard & LeBlanc devote an entire chapter to this
idea[6, p. 341–62], underscoring its importance. This does not mean that data
flow analysis is impossible: it is an area of active research, and some of the
tools that have been developed for dynamic taint analysis are promising[36,
37].
In a functional language, the data flow is easier to perceive, and programmers
rarely need tools to explicate it. This is partly because of a feature that
has already been discussed: discriminated unions. Most dynamic taint analysis
tools work by tagging security-relevant data[35, 36] and examining them at
crucial points; there is no need for an additional tool to do this when the
ability to tag and differentiate data is built into the language.
Discriminated unions do not account for the majority of the data flow clarity
of functional languages, however. _Higher-order_ functions, of which we have
already encountered one (List.map), accounts for the rest. A higher-order
function is simply a function which uses a function as input or output. Using
higher-order functions allows a programmer to perform common operations with
the least code possible. For example, it is frequently the case that a single
value must be calculated on the basis of a sequence of values: any imperative
function which returns a scalar value and contains a loop would fall into this
category. In C#, such a function might look like
public double AveragePrice(double[] competitors) {
double average = 0.0;
for (int i = 0; i < competitors.Length; ++i) {
average = ((average * i) + competitors[i]) / (i + 1);
}
return average;
}
The corresponding F# could be
List.fold (fun (i,average) price ->
i+1., (average*i + price) / (i+1.)
) (0.,0.) >> snd
A few points arise as the two versions are compared.
* •
The imperative version gives little indication that it operates on a sequence
and uses each item in that sequence to arrive at a particular value. It
_could_ be implied by the method name, the scalar return value, and the array
of values passed as a parameter, but the method could also take into account
class state or some other data. Without looking at the body of the method, it
is impossible to know for sure. On the functional side, the mere fact that a
function called fold has been invoked tells the reader immediately that a
sequence of values is being reduced to a single value. Other common names in
functional programming, such as map or filter, provide similar recognizability
of the operation being applied: the data flow can be understood instantly by
noticing the name of the higher-order function.
* •
The core of imperative logic — the average calculation itself — is passed
through as a function in the functional version, and is visually next to the
invocation of fold. A reader wishing to know _how_ the reduction to a single
value occurs has the answer immediately available. In the imperative version,
this is located in the middle of the loop, and is not immediately obvious.
* •
The initial value to use during the calculation is declared at the top of the
imperative version, though the reader has no idea of its importance until the
return statement is seen or the calculation itself is examined. In the
functional version, the initial value (0.,0.) is provided next to the core
logic.
* •
There is no explicit indexing in the functional version. In the imperative
version, the i variable performs a dual task: indexing the competitors array,
and counting the number of items that have been seen thus far. Given a
different loop algorithm — for example, one that requires the variable to
start from 1 instead of 0 — it is easy for a programmer to make either a
calculation mistake or an indexing mistake, both of which could have security
implications. Since there is only one purpose for the i symbol in the
functional version, this mistake is more difficult to make.
The functional advantage in the case of least-privilege is a subtle one: it is
easier to understand data flows in a functional language, and correspondingly
easier to understand where to apply which level of privilege.
### Remember That Security Features != Secure Features
This principle states that security should be built in to features, rather
than being “tacked-on” at the end of a development cycle. On the program
level, building secure features requires understanding of the flow of data and
the reasons for that flow, rather than understanding each statement in
isolation. Alan Perlis referred to this in his 1982 epigram, “A programming
language is low level when its programs require attention to the
irrelevant”[38]. As the examples from the previous section show, this
understanding is easier to obtain when using a functional language; therefore,
secure features are correspondingly easier to develop, and insecure features
are easier to detect.
It can be said that building security in is a matter of choosing the secure
option whenever a choice is presented. In a functional language, the default
loose coupling of the functional paradigm make dependencies explicit, so that
any feature that is determined to be “insecure” can be removed or disabled
with minimal trouble. Choosing to use whitelists instead of blacklists is a
security-conscious decision, and it is one that the functional paradigm makes
easy. Iterating through objects rather than indexing through a collection is a
more security-conscious way of programming, as is explicitly indicating
failure. Importantly, an imperative language tends to regard these choices as
having _equal value_ ; a functional language values correctness (in the
mathematical sense), and its constructs typically reflect a push towards that.
## Caveats and possibilities
Languages differ, whether they are functional or imperative. Not all languages
are statically-typed, for example: Erlang is a functional language that is
dynamically-typed, just as Ruby and Python are dynamically-typed imperative
languages. Frameworks, libraries, syntax, and language philosophy (among other
concerns) make any comparison between imperative languages difficult, and the
same is true of functional languages.
Even the distinction between functional and imperative can become blurred:
since both F# and OCaml allow data to be mutable, are they really “functional”
languages? Javascript is considered to be an imperative language, but
CoffeeScript[39] (which compiles to Javascript) has functionally-inspired
features; which is it? To overcome this difficulty of categorisation, any
languages which support immutability by default, higher-order functions, and
referential transparency — all of which are associated with the functional
paradigm — have been considered as functional languages, whether statically-
or dynamically-typed.
I have compared statically-typed languages in this paper since I am most
familiar with them. Many of the points that have been made should be equally
applicable to dynamically-typed languages, and most of the examples should be
understandable and portable across language boundaries. The aim of this paper
has been to compare the functional _paradigm_ , as evidenced by features found
in typical functional languages, with the imperative paradigm. It has not been
intended as a comprehensive language comparison between C# and F#, nor should
it be mistaken for such.
### A failure of vision?
From the case presented, it seems clear that functional languages may have
certain security advantages over imperative languages. Empirical studies could
validate this idea; as a reference point, the 2011 study [40] examines
> …data from 362 projects of four different firms. These projects spanned a
> wide range of programming languages, application domain, process choices,
> and development sites spread over 15 countries and 5 continents.
The mean defect density121212Number of defects per thousand lines of code
(defects/KLOC) was $\frac{1}{10.06}\approx 0.1\pm 0.04$, with a maximum of
$\frac{1}{0.01}\approx 100$ and minimum $\frac{1}{240.05}\approx 0.004$[40, p.
265]. Although a breakdown of defect density by language is not provided, the
authors list Java, .Net, PHP, C, C++, and assembly language as the set of
languages that were used[40, p. 263]; all of these are imperative131313Data
was collected up to and including 2009. F# was only included as an official
part of the .Net ecosystem in 2010..
No comparable study for the functional paradigm exists. The most relevant
comparison point appears to be [41], which tracked defect density in the GHC
Haskell compiler over a number of versions. The density varied from 0.49 to
0.04, with a startling decline from 0.29 (GHC version 5.02.2) to 0.04 (GHC
version 5.04). Subsequent to version 5.04, the maximum defect density has been
0.08. The authors of [41], Sherriff et al., provide no explanation for this
sharp drop, nor do they explain their methodology for determining what a
“defect” is, other than saying that they used “detailed documentation and
defect logs” [41, p. 2]. Furthermore, given that the chosen project was the
reference Haskell compiler, it might be assumed that the programmers on the
project are expert Haskell programmers: this may bias the defect density in
the project’s favour.
It is clear that more empirical research must be done in this area before any
hard conclusions can be drawn. That research is all the more difficult to do
since functional languages are not as widely-used as imperative languages. It
is my hope that this paper will provide some incentive for security-conscious
programmers and organisations to seriously investigate the use of the
functional paradigm, and thus move the field towards a better understanding of
it and its relationship to the existing imperative model.
## References
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|
arxiv-papers
| 2012-01-27T09:23:34 |
2024-09-04T02:49:26.748478
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yusuf Moosa Motara",
"submitter": "Yusuf Motara",
"url": "https://arxiv.org/abs/1201.5728"
}
|
1201.5762
|
# Calibration System with Optical Fibers for Calorimeters at Future Linear
Collider Experiments
Jaroslav Zalesak
_for the CALICE Collaboration_ Manuscript received November 15, 2011.J.
Zalesak is with the Institute of Physics of the Academy of Science, Na
Slovance 2, CZ-18121 Prague, Czech Republic (phone: 420-26605-2707), (_e-mail:
zalesak@fzu.cz_)This work is supported by the Ministry of Education, Youth and
Sports of the Czech Republic under the projects AVO Z3407391, AVO Z10100502,
LC527 and INGO LA09042.
###### Abstract
We report on several versions of the calibration and monitoring system
dedicated to scintillator tile calorimeters built within the CALICE
collaboration and intended for future linear collider experiments. Whereas the
first, a 1 m3 analogue hadron calorimeter prototype, was already built and
tested in beam, second — technological prototype — is currently being
developed. Both prototypes are based on scintillating tiles that are
individually read out by new photodetectors, silicon photomultipliers (SiPMs).
Since the SiPM response shows a strong dependence on the temperature and bias
voltage and the SiPM saturates due to the limited number of pixels, it needs
to be monitored. The monitoring system has to have sufficient flexibility to
perform several different tasks. The self-calibration property of the SiPMs
can be used for the gain monitoring using a low intensity of the LED light. A
routine monitoring of all SiPMs during test beam operations is achieved with a
fixed-intensity light pulse. The full SiPM response function is cross-checked
by varying the light intensity from zero to the saturation level. We
concentrate especially on the last aspect — the high dynamic range.
###### Index Terms:
ILC, CALICE, Calorimetry, LED, Calibration, Optical fiber, Silicon
photomultiplier.
## I Introduction
The CALICE collaboration [1] is developing a hadronic calorimeter (HCAL) with
very high granularity for future linear colliders (ILC, CLIC). The
collaboration built 1 m3 physics prototype in 2005–6 [2] and currently CALICE
is building an engineering prototype [3].
The HCAL readout chain of these prototypes contains scintillator tiles with
embedded wavelength-shifting fibers and small SiPM (Silicon Photo Multiplier)
photo-detectors. The electrical signal is adjusted by a preamplifier and a
shaper and digitized by a 12-bit ADC.
The variation of characteristics of the complete chain (gain, saturation)
depends mainly on changes of the temperature and operation voltage of the
SiPM. For the correct offline reconstruction of the energy deposition, the
calibration runs have to be included into the data-taking process, since the
condition inside the detector can change.
The HCAL prototypes will be shortly described in section I-A and I-B. The
section I-C gives information about the calibration procedure we are using.
The calibration systems for these prototypes will be described in section II
with impact on the electronic principles of operation, together with a
performance results. The section III will describe a solution for multi-tiles-
illumination by a single optical fiber and also gives insight into the ongoing
development.
### I-A Physics Prototype
Figure 1: Configuration of the ECAL, HCAL and TCMT in the beam test at CERN.
The AHCAL (Analogue Hadron CALorimeter) 1 m3 physics prototype with 7608
active readout channels has been in test beams at the CERN SPS in 2006 and
2007 (Fig. 1). Beam tests at Fermilab followed in years 2008 and 2009 and now
the physics prototype continues to run with tungsten absorber at the CERN PS
in 2010 and at the SPS in 2011.
The physics prototype is made of 38 layers with scintillator tiles, interlaced
with 16 mm Fe absorber plates (or with the 10 mm W plates). The dimension of
one layer is 90 cm x 90 cm and it contains scintillator tiles with different
granularity (Fig. 2): 20 pcs of 12x12 cm2 tiles, 96 pcs of 6x6 cm2 tiles and
first 30 layers have 100 pcs of 3x3 cm2 tiles in the middle. Last 8 layers
have 6x6 cm2 tiles in the middle instead. The active element of the readout is
a compact photo-detector — a silicon photomultiplier SiPM (see section I-C).
with 1156 pixels. The AHCAL analogue board carries 1 ASIC for 18 SiPM
channels. A second board contains control and configuration electronics and
provides correct voltage for each SiPM. For the calibration a board called CMB
is used (see section II-A).
Figure 2: Layout of the scintillator tiles inside the cassette, active part of
one layer of the AHCAL physics prototype.
### I-B Engineering Prototype
Next version of the AHCAL is called the engineering prototype. The basic
electronics structural element of the engineering prototype (Fig. 3) is the
HCAL base unit (HBU). The engineering prototype is built to demonstrate the
feasibility of the electronics integration in the HCAL calorimeter for ILC,
using 3 cm x 3 cm scintillator tiles with SiPM photo-detectors, embedded
readout [4] and implementing a concept of power pulsing.
Figure 3: Wedge of the HCAL barrel (1/16 of the half-barrel) with one
integrated layer shown consisting several HBUs and driving and read-out cards.
Fig. 3 shows one half of the HCAL octagon. Each wedge consists of 48 layers.
Each layer is made of absorber (16 mm Fe or 10 mm W) and active layer of 2-3
rows with 6 HBUs connected together in a cassette, 2.2 m long.
The HBU unit (Fig. 4) is 36 cm x 36 cm large board, which has 144 scintillator
tiles with embedded SiPMs with 796 pixels on its back side. All 144 channels
are read-out by 4 ASICs. Details on the HBU are described in [4]. Two options
of optical calibrations are proposed for the HBU: distributed SMD LEDs [3] and
an external LED driver and optical fiber distribution (sections II and III).
Figure 4: The HBU unit with 4 ASICs which read-out 144 scintillator tile
channels placed on the back side.
### I-C SiPM and Calibration
SiPMs are photo detectors, that consist of hundreds (up to thousands) pixels,
each working as an avalanche photodiode in Geiger mode. Each pixel is coupled
via a resistor to a single output, creating an analogue signal with amplitude
equivalent to the number of fired pixels.
The light from the scintillator tile (Fig. 5(a)) is collected by a wavelength-
shifting fiber, that re-emits the light in green (where SiPM is most
sensitive) and guides the light toward the active area of the SiPM (1 mm2,
Fig. 5(b)). The other end of the fiber has a mirror.
Figure 5: (a) Photo of a 3 cm x 3 cm scintillator tile for the physics
prototype with the embedded WLS fiber and SiPM; (b) Detail of the SiPM chip.
The gain of the SiPM varies as a result of the manufacturing process. The
typical gain of the SiPM is 106 and is highly dependent on temperature T and
bias voltage. The typical gain varies by -1.7 %/K and +2.5 %/0.1V. The
temperature and voltage affects also the photodetection efficiency of SiPM,
which results in a response variation of -4.5 %/K and +7 %/0.1V in total.
The gain of the SiPM can be obtained from the Single Photo-electron Spectrum
(SPS) as a distance among the peaks (Fig. 6(a)). Low intensity light flashes
are required for this calibration measurement.
The SiPM needs correction for the saturation curve, as the number of fired
pixels is approaching the total number of pixels as can be seen in Fig. 6(b).
The saturation curve is also affected by the time structure of the photon
shower, as the pixels can recover and fire again on the next photon, creating
signal effectively higher than the total number of pixels in SiPM.
Figure 6: (a) Gain extraction from the single photo-electron spectrum; (b)
SiPM saturation effects.
## II Calibration LED Driver Cards and Principles
Since the SiPMs are sensitive to changes in temperature and operation voltage,
the LED system has been installed to monitor the stability of the readout
chain in time. The calibration system needs sufficient flexibility to perform
several different tasks. Gain calibration: we utilize the self-calibration
properties of the SiPMs to achieve the calibration of an ADC in terms of
pixels that is needed for non-linearity corrections and for a direct
monitoring of the SiPM gain (in Fig. 6(a)). Further we monitor all SiPMs
during test beam operations with a fixed intensity light pulse. Saturation: we
cross check the full SiPM response function by varying the light intensity
from zero to the saturation level (in Fig. 6(b)).
To satisfy such requirements we have developed several versions of calibration
and monitoring boards.
Our calibration boards use an UV-LED as a light source for calibration. The
UV-LEDs require a special driver in order to make them shine fast ($\leq$10
ns) with an amplitude covering several orders of magnitude in light intensity.
### II-A Calibration and Monitoring Board (CMB)
Figure 7: Layout of the CMB board: 12 LEDs on the left side and 12 PIN
photodiodes on right side.
The first system we developed was called the CMB, Calibration and Monitoring
Board [5]. It consists of 12 UV LEDs (Fig. 7) with a special fast driver
optimized for rectangular pulses. The pulses are 10 ns wide in order to match
real signals from hadron showers in the calorimeter as closely as possible. By
varying the control voltage, the LED intensity covers the full dynamic range
from zero to saturation (about 70 minimum ionizing particles). The LED
illuminates a fiber-bundle of 18 + 1. One fiber comes back to the CMB to
monitor the emitted light by a PIN photo diode with preamplifier on the board.
The other 18 fibers are rooted to each scintillator tile equipped with one
SiPM each.
Figure 8: (a) Operation principle of the CMB LED driver; (b) Oscillograph from
the LED pins during a pulse; green: anode, violet: cathode; Bottom part is a
zoom of the selected region from the upper part. Upper traces: 10 ns per div,
bottom traces: 1 ns per div.
The CMB is controlled by a slow control CAN-bus protocol. The output light is
adjustable both in amplitude and length (5–100 ns). The trigger is provided
externally by T-calib signal. The CMB provides readout from temperature
sensors from several places on the detector layer. The amplitude covers the
whole range of light intensities, from single photons up to the SiPM
saturation.
CMB utilizes a IC from IXIS company (IXLD02). In principle this driver works
as a H-bridge, where upper part is tightened to the supply voltage through a
protective resistor. Bottom part of the H-bridge (two FETs) are very fast
switches, which are closed in idle (Fig. 8(b), start of the oscillograph) — no
current passes the LED. Before the pulse, the LED is reverse biased
(controlled by the enable signal in Fig. 8(a)). The pulse is generated by
reversing the driver’s ”puls” signals. The LED is biased in the forward
direction (Fig. 8(b) bottom detail), forcing the LED to emit light. At the end
of the pulse, the LED is reverse biased again. The reverse bias is an
important step, because without discharging the LED tends to continue emitting
light. The time of rising and falling edges of the pulse is about 1 ns.
Figure 9: Design of a toroidal inductance (for low RFI 35 nH) on a PCB board
for the quasi resonant LED driver .
Figure 10: Quasi-resonant LED driver: (a) driver circuit; (b) operation
principle.
### II-B Quasi-resonant LED driver
To further improve the performance of the LED driver we abandoned the
rectangular pulses. The Quasi Resonant LED driver (QRLED) produces short 5 ns
long electrical pulses for LEDs of the sinusoidal shape. The QRLED is foreseen
as a source of the tunable LED light for calibration of SiPMs in the
engineering hadron calorimeter module. The short electrical pulses are created
in the toroidal inductance made directly on the PCB. This design resulted from
the requirement on the minimal height of the electronic circuitry in the
compact engineering module.
The driver uses embedded PCB toroidal inductors (Fig. 9), which help to reduce
the EMI (electromagnetic interference).
The LED driver works in principle similarly to the boost DC-DC converter (Fig.
10). In the idle mode, the V2 voltage is higher than V1 and the LED is reverse
biased. Before the pulse, the inductor (Fig 10(a): ”L”) is switched to the
ground, enabling the current to flow through the inductor and store energy.
When the switching transistor is switched off, the circuit is left in the
resonance configuration. The energy (current through ”L”) is smoothly
transferred to the capacitor (voltage on ”C”), back and so on. The resonance
is heavily dumped by the dump resistor (”RD”), therefore only the first
overshoot on ”C” is high enough to bias the LED directly.
The first overshoot (sine wave) can be tens of volts, forcing the LED to pass
through a current up to 1 A. The negative part of the sine wave helps to
reverse bias the LED in a very short time. The circuit is tuned such that the
following sine wave will not overcome the direct-bias threshold and the LED is
kept reverse biased as can be seen in Fig. 10(b).
### II-C External Calibration System — QMB6
Figure 11: QMB6 performance: (a) single photon spectrum measured at low light
intensity (red color represents pedestal, no light); (b) saturation curves for
all tiles illuminated by one UV-LED with pulse of 3.5 ns width.
For the second option of the SiPM calibration considered for the engineering
prototype, a dedicated LED calibration board [6] was developed at Institute of
Physics in Prague. It consists of 6 Quasi-Resonant LED drivers. The board is
controlled by the CANbus, or by a DIP switches in the Stand-alone mode.
Peak LED optical power was measured as 0.4 nJ in a single pulse with Thorlabs
power-meter PM100d with S130VC sensor and 3 mm UV LED. The beam test at DESY
[7] in 2010 showed that a single LED is able to saturate 12 tiles at once with
signal equivalent to 200 MIPs. The pulse length is fixed at 3.5 ns by the ”L”
and ”C” components (Fig. 10(a)) parameters.
Examples of the QMB6 performance could be seen in Fig. 11 displaying a single-
photon spectrum of one illuminated tile channel at low light intensity and
saturation curves for all measured channels, respectively.
The test in 4 T magnetic field [8] showed almost negligible influence to the
QMB6 operation ($<$1%).
Figure 12: 3D view of the PCB layout of the QMB1 Figure 13: Scheme of the QMB1
### II-D 1-ch Led Driver (QMB1)
Final 1-channel prototype (QMB1) is currently under development, in Fig. 12 we
can see three dimensional topology of designed QMB1 pcb. To fit the tile
spacing the board is only 3 cm wide and 14 cm in depth. Each board consists
only of one driver (for one UV or blue LED) but system is proposed as highly
modular as one board behaves as a master and several others as slaves. The
master board communicates with DAQ via CAN bus and it is also used as a
distributor of the LVDS trigger system as is pictured in scheme in Fig. 13.
Pulse length is fixed to 5 ns to produce optical pulse. The light amplitude is
smoothly variable (max $\sim$1 A through a LED). The goal is to illuminate 72
tiles by a single LED. This driver with optical fibers will be integrated to
the newly built calorimeter prototype whose design can be seen in Fig. 3.
## III Light fiber distribution system
The original concept, well proven by several years in operation in the beam
tests, consists of a single LED sending light to a bunch of 19 fibers carried
by an optical connector. 18 fibers are routed each to a single tile, 1 fiber
was routed back to a PIN diode for the feedback.
Figure 14: (a) principle of the light emitted by the notch; (b) first notch,
closest to the LED; (c) last notch, at the rear fiber end.
In the next step we simplified the distribution of UV flashes to scintillator
tiles. Instead of using one fiber for each tile, a series of cuts on a single
optical fiber illuminates a row of tiles below the fiber. The challenge is to
make the light flashes equal with the same amount of light for each tile.
The specially ”notched” plastic optical fiber has spots along its length. The
notch is a special cut (Fig. 14(a)) on the fiber’s surface, which reflects
light perpendicularly to the fiber path (Fig. 15). Depth of notches varies
from beginning to the end of the fiber to guarantee uniform light output from
each notch (Fig. 14(b,c)).
Figure 15: View of the optical fiber with notches. The fiber is illuminated
by a continuous light to visualize the notches.
Test of several hand-made prototype fibers with different number of notches
were performed. We achieved the spread of light better than 20% on the fiber
length of 210 cm with 72 notches [6]. Other options, 24 and 12-notched fibers
have been manufactured with $\pm$15% and $\pm$10% homogeneity, respectively.
An example of results on light output uniformity measurement with 24 notches
is displayed in Fig. 16.
Figure 16: Light output measurement as function of the position of a notch
along fiber
The optical fiber development continues with building a semi-automatic machine
(Fig. 17) for the notched fiber production. New optical coupling to the bunch
of 3 optical fibers is being developed. In the final prototype the light will
be guided from single LED by 3 fibers having 24 notches each as is displayed
in Fig. 18. Fibers will be routed on the component side of the HBU, where SMD
components are not mounted. The light from the fiber will be guided to the
tiles (sitting on the back side) through the 3 mm hole in the PCB of HBU.
Figure 17: Semi-automatic notch-fiber machine Figure 18: Foreseen
illumination of row with 72 tiles on 6 HBUs by 3 fibers with 24 notches each.
## Acknowledgment
The author gratefully thanks to his colleagues from the Institute of Physics
in Prague: Jaroslav Cvach, Milan Janata, Jiri Kvasnicka, Denis Lednicky, Ivo
Polak, Jan Smolik for very valuable contributions to the results presented in
this paper. Special thanks go to Mark Terwort and Mathias Reinecke (DESY) for
their help and realization the measurements needed for this work.
## References
* [1] CALICE home page:
https://twiki.cern.ch/twiki/bin/view/CALICE/CaliceCollaboration.
* [2] C. Adloff et al., _Construction and commissioning of the CALICE analog hadron calorimeter prototype_ , JINST 5 (2010) P05004, arXiv:1003.2662.
* [3] Mathias Reinecke, _Towards a full scale prototype of the CALICE tile hadron calorimeter_ , IEEE Nucl. Science Symp. (NSS), this proceedings, NP3.M-76, Valencia, 23th–29th Oct 2011.
* [4] Mark Terwort, _Concept and status of the CALICE analog hadron calorimeter engineering prototype_ , talk presented at Technology and Instrumentation in Particle Physics 2011 (TIPP11), Chicago, 21th Jun 2011, to be published in Physics Procedia.
* [5] I. Polak, _Development of Calibration system for AHCAL_ , talk presented at ECFA conference in Valencia 15th Nov 2007
www-hep2.fzu.cz/calice/files/ECFA_Valencia.Ivo_CMB_Devel_nov06.pdf
* [6] J. Cvach, M. Janata, J. Kvasnicka, D. Lednicky, I. Polak, J. Smolik, J. Zalesak, _Calibration prototype for the EUDET HCAL_ , EUDET-Report-2008-07, www.eudet.org/e26/e26/e27/e825/EUDET_report_07.pdf.
* [7] J. Cvach, J. Kvasnicka, I. Polak, J. Zalesak, _Beam test of the QMB6 calibration board and HBU0 prototype_ , EUDET-Memo-2010-21, www.eudet.org/e26/e28/e86887/e106794/EUDET-Memo-2010-21.pdf.
* [8] J. Cvach, J. Hladky, M. Janata, J. Kvasnicka, D. Lednicky, I. Polak, J. Zalesak, _Magnetic field tests of the QRLED driver_ , EUDET-Memo-2009-005, www.eudet.org/e26/e28/e42441/e62562/memo_2009-5.doc.
|
arxiv-papers
| 2012-01-27T12:29:06 |
2024-09-04T02:49:26.759931
|
{
"license": "Public Domain",
"authors": "Jaroslav Zalesak (for the CALICE Collaboration) (Institute of Physics\n of the ASCR, Prague, Czech Republic)",
"submitter": "Jaroslav Zalesak",
"url": "https://arxiv.org/abs/1201.5762"
}
|
1201.5960
|
Vol.0 (200x) No.0, 000–000
11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China; wangxin@nao.cas.cn
22institutetext: State Key Laboratory of Space Weather, Chinese Academy of
Sciences, Beijing 100080, China
Received 2012 April 12; accepted 2012 May 7
# A CME-driven shock analysis of the 14-Dec-2006 SEP event
Xin Wang 1122 Yihua Yan 11
###### Abstract
Observations of the interplanetary shock provide us with strong evidence of
particle acceleration to multi-MeV energies, even up to GeV energy, in a solar
flare or coronal mass ejection (CME). Diffusive shock acceleration is an
efficient mechanism for particle acceleration. For investigating the shock
structure, the energy injection and energy spectrum of a CME-driven shock, we
perform dynamical Monte Carlo simulation of the 14-Dec-2006 CME-driven shock
using an anisotropic scattering law. The simulated results of the shock fine
structure, particle injection, and energy spectrum are presented. We find that
our simulation results give a good fit to the observations from multiple
spacecraft.
###### keywords:
Acceleration of Particles, CME-driven Shock, Solar Energetic Particles,
Numerical Simulation
## 1 Introduction
It is widely accepted that there are two classes of solar energetic particle
(SEP) events, although recent observations indicate that the actual processes
may be much more complicated (Pick & Vilmer, 2008). The first class is normal
impulsive SEP events, which are connected with the large solar flare (Miller,
1997). The second class is gradual SEP events, which are responsible by
diffusive shock acceleration (DSA) associating with fast coronal mass
ejections (CMEs)(Cane, Reames, & von Rosenvinge, 1991; Yan et al., 2006). In
solar magnetic connection region, CME and flares are two type of
manifestations of the same magnetic energy release process (Wang et al., 1996;
Zhang et al., 2001; Zhang et al., 2007). Both CMEs and flares result in
particle acceleration that constitute an SEP event. But which manifestation
dominates the particle injection is still not clear (Li et al., 2009; Le et
al., 2011; Qin & Shalchi, 2009). Some numerical models suggest that mixed
particle acceleration by both flares and CME-driven shocks provide much better
fits to the in-situ observations. Since the particle injection process is
connected with the complicated nonlinear effects in the particle acceleration
processes and also there exists difference of injection mechanism in two type
of manifestations, so we just put forward to a pure shock numerical model to
calculate the particle injection in the CME-driven shock. We expect that it
would be helpful for understanding the particle injection problem in the SEP
events.
DSA theory was first introduced in the later of 1970’s ( Krymsky (1977);
Axford et al. (1977); Bell (1978); Blandford & Ostriker (1978). In the past
several decades, the accumulation of the increasingly observational data from
many spacecraft investigated the nonlinear diffusive shock acceleration
(NLDSA) mechanism, which is the most efficient accelerator in many
astrophysical and space physical environments (Bednarz & Ostrowski, 1999;
Malkov & Drury, 2001; Bykov et al., 2009; Bykov & Treumann, 2011; Lu, Xia, &
Wang, 2006; Zhang, Bi, & Hu, 2006). With the development of technology in
observational equipments, especially in spacecraft working in deep space,
there are a lot of models for modeling the various nonlinear interaction with
the diffusive shock acceleration. Several main approaches for studying the
nonlinear DSA includes: the two-fluid model (Drury & Völk, 1981; Drury,
Axford, & Summers, 1982); the numerical model (Berezhko & Völk, 2000; Kang &
Jones, 2007; Zirakashvili, 2007; Verkhoglyadova et al., 2010); the stationary
or dynamical Monte Carlo model (Ellison & Eichler, 1984; Knerr, Jokipii &
Ellison, 1996; Vladimirov, Ellison & Bykov, 2006); the semi-analytical model
(Malkov et al., 2000; Caprioli, Amato & Blasi, 2010) and etc. Among these
approaches, the Monte Carlo method addresses the nonlinear effects of DSA by
assuming that the entire particle population undergoes a random walk under a
certain scattering law (Ellison, Möbius & Paschmann, 1990; Knerr, Jokipii &
Ellison, 1996; Wang & Yan, 2011).
There are three important non-linear processes of DSA theory including the
particle injection, particle confinement, and shock robustness (Malkov &
Drury, 2001; Hu, 2009). Owing to the fact that walking processes of the
particles can be controlled self-consistently in the Monte Carlo method, the
Monte Carlo method has an advantage for simulating the particle injection. We
have already studied the energy translation processes of an Earth-bow shock
using the dynamical Monte Carlo method with a prescribed multiple anisotropic
scattering angular distributions (Wang & Yan, 2012). We find that the
acceleration efficiency increases as the dispersion of the scattering angular
distribution increases from an anisotropic case to an isotropic case. Here, we
will further investigate this important particle injection problem in the CME-
driven shock using the dynamical Monte Carlo method. There exist a few
different properties between the Earth-bow shock and the CME-driven shock:
Firstly, Earth-bow shock has a stationary downstream bulk flow but the CME-
driven shock has a dynamical downstream bulk flow; Secondly, the CME-driven
shock front has an opposite motion compared with the Earth-bow shock’s
evolution; Thirdly, the CME-driven shock has an extended plane shock front
structure near the Earth, but the Earth-bow shock front has a stationary bow
shock geometry. We predict those differences would produce a different non-
linear properties including the evolution of the shock fine structure, energy
injection rate and even the energy spectral shape. This paper will focus on
the understanding some of the non-linear properties of the planetary CME-
driven shock. The 14-Dec-2006 shock event was fortuitous as it provides us an
opportunity for applying the dynamical Monte Carlo package code, which was
developed on the Matlab platform (Wang & Yan, 2011).
This paper is structured as follows: In Section 2, we present the specific
observations for the 14-Dec-2006 CME-driven shock event; The detailed
description of the method is given in Section 3; We present the simulated
results and discussions in Section 4; Finally, Section 5 presents the summary
and some conclusions.
## 2 Observations
Figure 1: The plot shows the key parameters of the 14-Dec-06 shock event in
the Wind spacecraft, and the data come from
$http://cdaweb.gsfc.nasa.gov/cdaweb$.
Figure 2: The upper panel represents the proton density profile vs its
position at the end of the simulation. The middle panel denotes the solar wind
thermal velocity profile in the local frame vs the time. The lower panel
indicates the bulk flow speed profile vs its position at the end of the
simulation. The vertical lines in the upper and lower panels both show the
final FEB position at the end of simulation.
The unusual group of CME-driven shock events of solar cycle 23 was observed in
December 2006 at the solar active region 10930. Halo CMEs were observed by the
LASCO coronagraphs in association with the events of 13 and 14 December, with
speeds of 1774km/s and 1042km/s, respectively. Because the 14 December solar
event was better magnetically connected to the Earth, so it provided the best
opportunity for testing the nonlinear effect and efficiency of the diffusive
shock acceleration (DSA) mechanism. As shown in Figure 1, an overview of key
parameter observations from the Proton Monitor (PM) instruments on Wind/SWE
for the CME shock event of 14 December 2006 is given in detail. This event
originated on the western hemisphere of the Sun. It showed an abrupt
fluctuation in intensity of proton density and solar wind thermal speed during
the decay of the 13 December solar event. The initial particle increase
following the 14 December solar event was seen in the higher energy range, as
expected for velocity dispersion. There was also a higher background at the
lower energy associated with the 13 December solar event and the related shock
(von Rosenvinge et al., 2009). Simultaneously, many spikes were also detected
to be superposed on the radio continuum in the frequency range 2.6-3.8 GHz by
the digital spectrometers of NAOC. These spikes were found to have complex
structures associated with other radio burst signatures connecting with the
in-situ SEP event observations (Wang et al., 2008).
Both Wind and ACE were in orbit around the L1 Lagrangian point $\sim$ 1.5
million km upstream of the Earth. Similar intensity modulations were observed
at Wind and ACE. As has also been noted by Mulligan et al. (2008), the
variations of the particle intensity and smooth magnetic fields observed by
near-Earth spacecraft occurred in the duration of the interplanetary coronal
mass ejection (ICME) driving the shock on 14 December which was related to the
13 December solar event. Solar wind observations from ACE show evidence of the
presence of the ICME, which had the enhanced magnetic field and a smooth
rotated “magnetic cloud” in the upstream shock and the intervening sheath
region, respectively. According to the Wind magnetic cloud list, the axis
orientation of this “magnetic cloud” was $\theta$ = $27^{o}$, $\phi$ =
$85^{o}$. In addition, Liu et al. (2008) estimate that the “cloud” axis
direction was $\theta$ =$-57^{o}$, $\phi$ = $81^{o}$ in GSE coordinates. Thus,
both agree that the axis was closely aligned west to east but differ in
whether it was inclined north or south, most likely because different
intervals were considered in their analysis.
Table 1: The Parameters of the Simulated Shock
Simulated Parameters | Dimensionless value | Scaled value
---|---|---
Physical | Upstream bulk speed | $u_{u}$=-0.4467 | -600km/s
Downstream bulk speed | $u_{d}$=-0.7742 | -1042km/s
Relative inflow velocity | $\Delta u$=0.3275 | 442km/s
Inflow sonic Mach number | M=17.5 | …
Thermal speed | $\upsilon_{0}$=0.0342 | 46km/s
Scattering time | $\tau$=0.3333 | 0.052s
| Box size | $X_{max}$=300 | $10R_{e}$
Numerical | 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 | $3R_{e}$
Notes: The physical parameters are taken from the Wind spacecraft, and the
numerical parameters are decided by the 14-Dec-2006 CME-driven shock.
## 3 The method
### 3.1 Physical model
We consider a plane-parallel shock where the supersonic flow moves from the
Sun to the Earth (in the rest frame) along the x-axis direction. The shock was
observed by Wind, SOHO, and ACE spacecraft near the Earth in the location of
the first Lagrangian point $L1\sim$ 1.5 million km ($\sim$ 250$R_{e}$, where
$R_{e}$ is the radius of the Earth) upstream of the Earth on 14 December. All
trajectories of the spacecraft in the 348th day corresponding to the 14
December 2006 are shown in Figure 3. With the CME-driven shock propagating
from the Sun along x-axis to the Earth, its shock front were encountered by
Wind, SOHO, and ACE spacecraft located in $X_{GSE}$ between the 250$R_{e}$ and
180$R_{e}$ upstream to the Earth. These three spacecraft moved about 10$R_{e}$
distance in their obits on the 348th day. The distances of all these three
spacecraft from the Sun-Earth line were within 50Re along the $Y_{GSE}$ and
$Z_{GSE}$ directions. The 14-Dec-2006 shock event originated from the western
hemisphere of the Sun with an interplanetary “ magnetic cloud” axis
orientation of $\theta$ = $27^{o}$, $\phi$ = $85^{o}$. And the actual
trajectory of Wind spacecraft at that moment is just tangent to the Sun-Earth
line with an angle $\phi$ = $80^{o}$ as shown in the lower panel of the Figure
3. As far as the position of Wind spacecraft is concerned, the observed CME-
driven shock is just a parallel diffusive shock. So the observation of Wind
spacecraft provided an example of semi-parallel shock for applying our
dynamical Monte Carlo code to understand the particle injection problem of DSA
theory.
Figure 3: The diagrams show the realistic obits of the near-Earth spacecraft.
The obit data are taken from $http://cdaweb.gsfc.nasa.gov/cdaweb$.
The important physical parameters of this simulation include the upstream bulk
flow velocity ($u_{u}$), the downstream bulk flow velocity ($u_{d}$), the
relative bulk inflow velocity difference ($\Delta u$), the inflow sonic Mach
number ($|u_{d}|/c_{s}$), which is 17.5 (where $c_{s}\equiv(\gamma
kT/m)^{1/2}$, $c_{s}$ is the upstream sound speed), the upstream thermal
velocity [$v_{0}\equiv(kT/m)^{1/2}$], and the constant scattering time
($\tau$), which is 2/5 times of the scattering time ($\tau_{0}=0.13s$) used by
Knerr, Jokipii & Ellison (1996) in the Earth-bow shock simulation. Since there
are some differences in the shock geometry between the CME-driven shock (i.e.
at L1 point) and the Earth-bow shock, we chose the scattering time 0.4 times
smaller than that in the Earth-bow shock, which is equivalent to a 2.5 times
larger FEB distance than that in the Earth-bow shock. The specific physical
parameters and numerical parameters are listed in the Table 1 with their
dimensionless values and scaled values, respectively.
### 3.2 Mathematical model
According to the observed 14-Dec-2006 CME-driven shock, the schematic diagram
of the simulation box can be designed as one-dimensional parallel shock along
the x-axis direction. As shown in Figure 4, the initial particles with a
relative bulk flow speed difference ($\Delta u$) move from the right to the
left. The initial particles have a background Maxwellian thermal distribution
with an initial temperature ($T_{0}\equiv mv_{0}^{2}/k$) in the local frame.
To begin and maintain the shock simulation, particles are assumed to flow into
the simulation box from the pre-inflow box (PIB) at the right boundary. Then,
with the continuous particle flow moving forward one time step, only those
particles which move into the main simulation box are actually added to the
simulation. This process naturally leads to a flux-weighted inflow population.
At the left boundary of the box, a reflective wall acts to produce a CME-
driven shock moving from the left to the right. Considering the geometry of
the 14 December shock event, we just follow the parallel component of the CME-
driven shock observed in Wind spacecraft.
Figure 4: The schematic diagram of the simulation box. The left reflective
wall acts ICME to produce the ICME-driven shock prorogating from the left
boundary to the right boundary of the box.
Figure 4 also shows one typical particle and its local ($V_{L}$) or box frame
($V$) velocity in the upstream region and downstream region, respectively. The
majority of the incoming particles cross the shock front only once from the
upstream region to the downstream region and stay in the downstream region. A
small portion of the particles can effectively scatter off the resonant MHD
wave self-generated by the energetic particles and return to the upstream
region to re-cross the shock front for additional energy gains (Liu, Petrosian
& Mason, 2004). Thus an anisotropic energetic particle distribution, but not a
strict Maxwellian distribution, is produced in the diffusive regions. It is
this elastic interaction between individual particles and the collective
background that allows the Fermi acceleration to occur.
The position of the FEB could coincide with a location upstream of the shock
where particles are no longer able to scatter effectively and return to the
shock. A reasonable FEB farther out in front of the shock moves, companying
with the shock front, at the same shock evolutional velocity $V_{sh}$. This
constant FEB distance is acted to inform a precursor region which is showed by
the shadow area in the middle of the Figure 4. If one particle archives to the
highest energy, and exceeds the position of the FEB in front of the shock, it
will be taken as the escaped particle and removed from the simulation system.
According to the actual motion of Wind spacecraft in the duration, the
spacecraft moved about 10$R_{e}$ distance in their obits on the 348th day. To
simulate the shock formation and evolution, the total length of the simulation
box is set to be 10$R_{e}$, the length of the FEB is set to be $\sim$3
$R_{e}$. The scattering time is set to be $\tau=0.052s$ on the basis of the
Earth-bow shock model (Knerr, Jokipii & Ellison, 1996).
The important numerical parameters include the box size ($x_{max}$), the time
to evolve the whole system ($t_{max}$), the number of grid zones ($n_{x}$),
the initial number of particles per zone ($n_{0}$), and the size of the time
step ($dt$). Because of the similar character of the plasma flow near the
Earth, we take some numerical parameters as in the Earth-bow shock model
(Knerr, Jokipii & Ellison, 1996). Specifically, the total box length
$x_{max}=300$ is divided into $n_{x}=600$ grids, with each grid length being
$\Delta x=1/2$; the total time $t_{max}=2400$ is divided into $n_{t}=72000$
steps by $dt$, with each step being $dt=1/30$. All numerical parameters are
listed in Table 1. The physical parameters and the numerical parameters
constitute the whole simulation parameter list. As shown in Table 1, each
dimensionless value is corresponding to its scaled value. The scale factors
for distance, velocity, and time are $x_{scale}=10R_{e}/300$,
$v_{scale}=442kms^{-1}/0.3275$, and $t_{scale}=x_{scale}/v_{scale}$,
respectively.
The presented simulations apply the same steps like the Earth-bow shock model
(Knerr, Jokipii & Ellison, 1996) including three sub steps: (i) All the
particles moving with their velocities in the simulation box along the $x$
axis direction. (ii) Summing particle masses and velocities over each
background computational grid. (iii) Invoking the scattering angular
distribution law. The particle diffusive processes in the presented
simulations are dominated by the Gaussian scattering angular distributions.
The scattering rate is $R_{s}=dt/\tau$, which implies that only this fraction
of particles is able to scatter off the scattering center frozen in the
background fluid. The candidate does not change its route until it is selected
to scatter once again. So the particle’s mean free path is proportional to the
local thermal velocities in the local frame with
$\lambda=V_{L}\cdot\tau.$ (1)
For an individual proton, 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 scattering angles consist of two variables: $\delta\theta$ and
$\delta\phi$. Once a particle has a collision with the background 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 $f(\delta\theta,\delta\phi)$. Here, we will just present the
results of the CME-driven shock using the Gaussian scattering angular
distribution with a standard deviation value of $\sigma=\pi$.
## 4 Results
### 4.1 Data analysis
Since the individual particle energy can be examined at any given time in the
simulation, so the energy function over time can be obtained. At first, we can
calculate the necessary energy functions for further analysis. In this
simulation, we obtained the total energy function in the box, the loss energy
function escaped from the FEB, and the injected energy function which is the
energy summation of the injected energetic particles from the downstream
region at the local velocity of $V_{L}=U_{0}$ over time. Then, at the end of
the simulation, we obtained the final values of the total energy
$E_{tot}=3.5666$, the energy loss $E_{loss}=0.2010$, and the energy injection
$E_{inj}=0.5464$, respectively. The final energy injection rate $R_{inj}$,
which represents the acceleration efficiency, can be defined by the formula as
follows.
$R_{inj}=E_{inj}/E_{tot}$ (2)
The injection rate is so important for a CME-driven shock, because it is
connected with the facts that the shock how distribute itself energy to
accelerate cosmic ray (CR) and to “heat” the thermal background plasma. By a
series of simulations, we give a plausible injection rate with a value of
$R_{inj}=15.32\%$ for the 14-Dec-2006 shock. Under this condition, we obtain
the maximum energy particle with the dimensionless value of $VL_{max}=20.2609$
and the scaled value of $E_{max}=3.8684MeV$. In addition, because there exists
some energy losses in the simulation system, the shock fine structures do not
completely agree with the situ observations. Finally, according to the DSA
theory, the energy spectrum index can be calculated based on the simulated
compression ratio (i.e., $\Gamma=(r+2)/[2(r-1)]$). We calculated the total
energy spectral index with a value of $\Gamma_{tot}=0.8406$ and the vicious
subshock energy spectral index with a value of $\Gamma_{sub}=1.1074$,
respectively.
Figure 5: The extend energy spectra in two plots are calculated over the
entire simulation region and the only downstream region at the end of the
simulation, respectively. The solid extended curve with a “power-law” tail in
each plot represents the final shocked energy spectrum. The thick curve with a
narrow peak denotes the initial Maxwellian energy spectrum. Figure 6: Fluency
spectra of the protons measured in the two largest December 2006 SEP events by
multiple spacecraft. The energy range is from 5 to 100 MeV adapted from
Mewaldt et al. (2008).
Figure 5 shows the simulated energy spectra. The first plot shows the energy
spectrum with the “double-peak” structures averaged over the entire simulation
box at the end of the simulation. The second plot shows an energy spectrum
with a “power-law” tail averaged only over the downstream region at the end of
the simulation. In this plot, the thick solid curve with a narrow peak
represents the initial Maxwellian thermal energy spectrum in the shock frame.
As viewed from the first plot, the double peaks imply that there exist two
thermal particle distributions in the entire simulation box: the left peak
represents the “heated” downstream flow distribution and the right peak
represents the Maxwellian distribution in the unshocked upstream flow. Turn to
look at the second plot, we find the final extend energy spectrum at the left
of the panel shows several decade times wider than that in the initial energy
spectrum at the right of the panel. This means that there exist a large
temperature difference between the shocked downstream region and the unshocked
upstream region.
As shown in Figure 6, the spectra of protons in the two largest December 2006
SEP events by ACE, STEROEO, and SAMPEX instruments are reported. The particle
intensities started increasing at the beginning of the December 5 and the
other at the onset of the December 13 event. These two events both have
spectra that roll over in a similar fashion beyond $\sim$50MeV, as in the
20-Jan-2005 SEP event (Mewaldt et al., 2008; Wang, Zhao, & Zhou, 2009; Bartoli
et al., 2012; Wang et al., 2010). The fitting energy spectral shape of the
12-14 December 2006 events are showed in the lower panel and the spectral
index is marked as a value of $E^{-1.07}$ in the lower energy range. The
predicted subshock energy spectral index ($\Gamma_{sub}=1.1074$) from our
simulation is consistent with the observed energy spectral index in the lower
energy range. Owing to computer constraints on the size of the simulation
grid, this simulated energy spectrum is just in the range from keV to MeV. We
speculate that the second “roll over” on the higher energy spectrum could be
obtained if a larger simulation box size is used. This will be investigated in
a future simulation. There are two conditions suggesting that the “roll-over”
would be reproduced at high energy: (1) The FEB distance decides the maximum
diffusive length (i.e. $FEB\equiv\lambda_{max}=\tau\cdot p_{max}$). If we
enlarge the FEB distance and the total simulation box, we can obtain the
larger $P_{max}$ in the new simulation system. (2) In the Figure 6, we can see
the first power law $E^{-1.07}$ as the input function of the second power law
$E^{-2.45}$ at the high energy range. Simultaneously, in the Figure 5, we can
see the heated Maxwellian thermal distribution, which would be represented by
a similar power law $E^{-0.5}$ averaged over the respective energy range, as
the input function of the first power law $E^{-1.1074}$.
### 4.2 Shock structures
At the end of the simulation, the simulated shock with the specific parameter
values are given as follows: the shock position $X_{sh}=121.5$, the FEB
position $X_{f}=31.5$, the shock evolutional velocity $V_{sh}=-0.0744$, the
subshock velocity $V_{sub}=0.2103$, total compression ratio $r_{tot}=5.4034$,
subshock’s compression ratio $r_{sub}=3.4697$, total energy spectral index
$\Gamma_{tot}=0.8406$, subshock’s energy spectral index $\Gamma_{sub}=1.1074$,
particle injection rate $R_{inj}=15.32\%$, energy loss $E_{loss}=0.2010$, the
maximum energy particle’s local velocity $VL_{max}=20.2609$, and the maximum
energy particle $E_{max}=3.8684MeV$.
First, we present the entire shock evolution with the temperature profile of
the time sequences as shown in Figure 7. The supersonic continuous inflow with
an initial Maxwellian thermal velocity $v_{0}$ in each grid evolves from the
begin to the end of the simulation. Their kinetic energies are translated into
the random thermal energetic particles by the “heating” processes in the
downstream region resulting in a distinct enhancement temperature profiles in
the shock front with the time. The profile of the thermal temperature shows
the upstream averaged temperature of $T_{0}=2.5\times 10^{5}K$ and the
downstream averaged temperature of $T_{d}=9.0\times 10^{6}K$. This means that
the CME-driven shock can “heat” the background plasma efficiently and provide
the first-order Fermi acceleration mechanism by crossing the shock front for
accelerating the particles, which are injected from the “heated” downstream
region into the precursor region.
Figure 2 shows a group of profiles of the physical parameters in the
simulation. From top to bottom, the upper panel shows the proton density
profile vs its position. The proton density is presented by the scaled value.
The enhanced density flux apparently appears in the position of the shock
front. The intensity of the density in the downstream is about five times
larger than those in the unshocked upstream bulk flow. By comparing the proton
density profiles between the the Figure 2 and the Figure 1, the simulated bulk
flow has a lightly higher proton density intensity in the downstream bulk flow
than that in the observed downstream bulk flow. The middle panel in the Figure
2 denotes that the thermal velocity profile evolutes with the time. The
profile at time of T$\sim$600 (it is zero before the simulation time T$<$600,
take account of the injection from the PIB) has an initial Maxwellian thermal
velocity of $v_{0}=46km/s$ until it is shocked. After the profile is shocked
as shown in middle panel of the Figure 2, it reaches an average thermal
velocity with a value around $<v_{d}>=300km/s$ till the end of the simulation.
Also the thermal velocity profile shows a slightly larger enhancement than
that from the observation by Wind spacecraft. Although the simulated proton
density and the solar wind thermal velocity are slightly larger than those
from in-situ observations, we suggest that this is caused by the insufficiency
of particles in the simulation. We have demonstrated that it is the case using
a series of simulations with different initial number of particles per cell.
The lower panel indicates the profiles of the bulk flow speed vs its position
at the end of the simulation. The profile shows an upstream bulk flow speed
$U_{u}=-600km/s$ and the downstream bulk flow speed $U_{d}=-1042km/s$ which
are followed by the observations. The complex shock front fine structure will
be showed in Figure 8 at the end of the simulation. The final evolutional
positions of the FEB and the shock front are $X_{f}=31.5$ and $X_{sh}=121.5$
in the x-axis, respectively. The distance between these two locations is just
the size of the precursor region where the particle acceleration processes
occur. It is just this region slowed the incoming upstream bulk flow speed
$U_{u}$ down to the downstream bulk flow speed $U_{d}$. The bulk flow speed in
precursor region is between the two bulk flow speeds (i.e.
$U_{u}>U_{p}>U_{d}$). From our simulation, we can see that the particle
acceleration process and the “back pressure” due to the energetic particles
occurred mostly in precursor region which results in a non-linear shock
structure that is characterized by a bulk flow speed gradient. According to
the evolutional shock front position $X_{sh}$ with the time, we can calculate
the shock evolutional velocity $V_{sh}$ as follows.
$V_{sh}=\frac{|X_{max}-X_{sh}|}{T_{max}},$ (3)
where, the $X_{max}$ is the total length of the simulation box, and the
$T_{max}$ is the total simulation time. Then, we are able to calculate the
total shock compression ratio in the shock frame as follows.
$r_{tot}=\frac{\Delta U+|V_{sh}|}{|V_{sh}|},$ (4)
where $\Delta$U is the relative bulk flow speed between the upstream and
downstream, $V_{sh}$ is the shock velocity.
Figure 7: The mesh plot represents the evolutional bulk flow temperature
profiles in their positions with the time. The lower temperature represents
the upstream bulk flow. The higher temperature represents the downstream bulk
flow. The apparently distinguished boundary traces the shock front positions
with time.
Figure 8 shows the shock fine structure with the bulk flow speed near the
shock front at the end of the simulation. $V_{sub}=0.2103$ shows the bulk flow
speed of the subshock, $V_{d}\simeq 0$ shows the bulk flow speed of the
downstream region, $V_{sh}=-0.0744$ represents the value of the opposite shock
evolutional velocity, and $U_{0}=0.3275$ represents the incoming bulk flow
speed with a related bulk flow speed difference of $\Delta U$. All zones of
the precursor, subshock and downstream are divided by a vertical dashed line
and a solid line in the plot. These three zones constitute the total shock
fine structure in the simulated shock region. The smooth precursor has a long
scale in the range from the subshock’s position $X_{sub}$ to the FEB position
$X_{f}$, which is invisible, beyond the left boundary of the plot. This zone
is called the diffusive zone where the bulk flow speed will be slowed by the
“back pressure” of the accelerated particles. The subshock region with a
narrow scale of a three-grid-length has a deep drop of the bulk flow speed, in
which the bulk flow speed vary from the subshock velocity $V_{sub}$ to the
downstream velocity $V_{d}$. The scale of the three-grid-length is almost
identical to the averaged thermal mean free path over the downstream region.
The subshock velocity $V_{sub}$ is decided by the horizontal dot-dashed line
with a value of $V_{sub}=0.2103$. The downstream velocity $V_{d}$ is marked
with a horizontal dashed line at the end of the simulation, which should be
with an averaged value of $<V_{d}>=0$ over the entire simulation time in the
box frame. The nagative shock evolutional velocity marked with a horizontal
solid line shows an value of $V_{sh}=-0.0744$. We can calculate the subshock’s
compression ratio according to Rankie-Hongniout relationships in the shock
frame as follows.
$r_{sub}=\frac{V_{sub}+V_{sh}}{<V_{d}>+V_{sh}},$ (5)
where, we take the averaged value of the downstream velocity $<V_{d}>$ equal
to zero.
Figure 8: The bulk flow speed fine structure of the simulated shock at the end
of the simulation. The vertical dash line and vertical solid line split the
entire region into three sections: precursor, subshock and downstream region.
## 5 Summary and conclusions
In summary, we performed a dynamical Monte Carlo simulations on the
14-Dec-2006 CME-driven shock using an anisotropic scattering law. The specific
temperature profile, shock fine structures, particle injection function, as
functions of time, are presented. We examined the correlation between the
energy injection and the shock energy translation processes of the
interplanetary CME-driven shock. Simultaneously, we find the simulated CME-
driven shock energy spectrum provides a good fit to the observations from the
multiple spacecraft.
In conclusion, the dynamical Monte Carlo simulation of the 14-Dec-2006 CME-
driven shock demonstrates that the energy spectrum is affected by the specific
non-linear factor of the DSA. This paper focus on the energy injection, which
is one of important nonlinear effects of the DSA. By calculating the energy
injection rate of the CME-driven shock, we can understand how the CME-driven
shock distributes its shock energy to accelerate the energetic particles by
first-order Fermi acceleration mechanism as well as how it heats the solar
wind background bulk flow at a certain efficiency. We give an energy injection
rate of $R_{inj}=15.32\%$ in the 14-Dec-2006 CME-driven shock. We guess that
this predicted injection rate could satisfy the required energies of the
observed SEP events, which should be released from the CME-driven shock.
###### Acknowledgements.
The work was supported by the Chinese Academy Sciences NSFC grant 10921303,
and the National Basic Research Program of the MOST Grant (No. 2011CB 811401).
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|
arxiv-papers
| 2012-01-28T13:41:56 |
2024-09-04T02:49:26.769626
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xin Wang, Yihua Yan",
"submitter": "Xin Wang Mr.",
"url": "https://arxiv.org/abs/1201.5960"
}
|
1201.6158
|
11institutetext: SISSA/ISAS and INFN Trieste, Italy
# Neutrino Physics
Andrea Romanino
###### Abstract
These lectures aim at providing a pedagogical overview of neutrino physics. We
will mostly deal with standard neutrinos, the ones that are part of the
Standard Model of particle physics, and with their standard dynamics, which is
enough to understand in a coherent picture most of the rich data available.
After introducing the basic theoretical framework, we will illustrate the
experimental determination of the neutrino parameters and their theoretical
implications, in particular for the origin of neutrino masses.
## 0.1 Introduction
Neutrino physics has played a crucial role in particle physics since the birth
of the theory of weak interactions, but the advances in a field requiring the
detection of such an elusive particle, have been characterized by long time
scales until 1998. After about 70 years of slow (but steady) progress, the
findings of the Super-Kamiokande (SK) experiment in 1998 triggered an
impressive acceleration and a renewed interest in the field.
There are various reasons for the interest in neutrino physics. First of all,
after decades in which the interpretation of neutrino experiments testing
neutrino transitions has been plagued by the uncertain knowledge of the
initial fluxes, the SK experiment started an era in which the data
interpretation has been relatively clean, with measurements either relatively
independent of the flux uncertainties or based on quite a precise knowledge of
the fluxes.
On the theoretical side, the evidence of small but non-vanishing neutrino
masses represents one of the few clear indications of physics beyond the
standard model of particle physics (SM). The latter predicts in fact vanishing
neutrino masses, unless supplemented by additional degrees of freedom or
effective interactions (which can be anyway considered as strong hints of new
degrees of freedom living at a higher energy scale).
Moreover, neutrinos play a crucial role in cosmology, another field which is
witnessing a fast expansion. They could be responsible for the origin of the
baryon asymmetry of the universe, i.e. our very existence, they enter the
determination of the spectrum of the cosmic microwave background (CMB), the
determination of the large scale structures (LSS) in the universe, the
delicate chemical equilibriums determining the light element abundances during
big bang nucleosynthesis (BBN). Not to mention astrophysics, where they
represent a powerful probe of the dynamics of the Sun, of core-collapse
supernovae, of high energy sources, etc.
Last but not least, neutrinos play an important role in many particle physics
models and phenomena, and may allow to indirectly access energy scales
otherwise largely out of the reach of (natural and laboratory) particle
accelerators. They are for example important ingredients in grand unified
theories (GUTs), flavour models, lepton flavour violation, lepton violation,
neutrinoless double beta decay.
In this lectures we will discuss many of the issues mentioned above. After
some historical remarks, we will discuss neutrino phenomenology, in particular
what do we know, and how, about the neutrino parameters. We will then discuss
the theoretical impact of the knowledge on neutrino parameters, in particular
the implications on the origin of neutrino masses and of the pattern of
neutrino masses and mixings. Useful general references are [1, 2]. We will use
natural units.
## 0.2 The birth of neutrino physics
(a) (b) (c)
Figure 1: Neutrino indirect discoveries: (a) electron neutrino from the beta
decay spectrum; (b) muon neutrino from the decay of pions from cosmic rays;
(c) tau neutrino from $e^{+}e^{-}\to e^{\pm}\mu^{\mp}X$ anomalous events at
SPEAR.
The existence of neutrinos has been first postulated by Pauli [3] in 1930 as a
“desperate remedy” to save energy conservation in the beta decay of nuclei. In
the absence of neutrinos, the two body decay of a nucleus made of $Z$ protons
and $A$ nucleons, $(A,Z)\to(A,Z+1)+e^{-}$, would produce a monochromatic
spectrum for the emitted electron. On the other hand, a continuous spectrum,
typical of a three body decay, was observed (see Fig. 1a). Hence Pauli’s
postulation of the existence of what we now call an electron anti-neutrino,
$\overline{\nu}_{e}$, in the final state of the decay:
$(A,Z)\to(A,Z+1)+e^{-}+\overline{\nu}_{e}.$ (1)
Fermi took Pauli seriously and in 1934 provided a quantitative description of
the phenomenon in terms of an effective interaction that would become the
basis of the theory of weak interactions [4, 5]. His success lead to a wide
acceptance of the neutrino hypothesis. The experimental progress in neutrino
physics has been quite slow in most of the neutrino history, and it was only
in 1956 that Reines and Cowan [6, 7] were able to establish experimentally the
existence of the neutrino emitted by nuclear reactors by detecting them
through the inverse beta reaction
$\overline{\nu}_{e}+p\to n+e^{+}.$ (2)
The muon neutrino was also introduced to account for missing momentum. Some
cosmic ray tracks observed in balloon experiments were in fact exhibiting
90-degrees kinks in correspondence to what would be interpreted as the decay
of a charged pion into a muon and a muon neutrino:
$\pi^{+}\to\mu^{+}\nu_{\mu}$ (see Fig. 1b). While such tracks were observed in
the late 40’s, when the pion was discovered, it was only in 1964 that
Lederman, Schwartz, and Steinberger [8] detected muon neutrinos by producing
the first, prototypical, artificial neutrino beam at Brookhaven. The latter
was obtained by sending a beam of protons on target to produce pions, mostly
decaying into muons and neutrinos as above. This is still a widely used method
to produce neutrinos, in the laboratory and in Nature. The neutrinos were then
detected through interactions with matter that were found to produce muons,
not electrons, thus indicating that the neutrino detected was not the same as
the electron neutrino detected by Reines and Cowan. Moreover, the separate
conservation of individual (electron, muon) lepton numbers holds in such
processes, once the neutrinos are given the same lepton number as their
corresponding lepton. As a consequence, when decaying into an electron, a muon
should produce both a muon neutrino and an electron antineutrino: $\mu\to
e\,\nu_{\mu}\overline{\nu}_{e}$. The fact that muon decay is mostly a three
body decay is confirmed by the analysis of the electron spectrum. Another
reason why the two neutrinos emitted in the muon decay should be different is
that otherwise the muon decay operator would induce a $\mu\to e\gamma$ decay
rate much larger than what experimentally allowed.
Tau neutrinos were associated in 1975 to the tau lepton discovery and detected
only recently, in 2000. Tau leptons were discovered at the SPEAR $e^{+}e^{-}$
accelerator, which observed anomalous events $e^{+}e^{-}\to e^{\pm}\mu^{\mp}X$
[9] (see Fig. 1c). $X$ represents here one or more invisible particles whose
presence in the final state was inferred by the measurement of the missing
momentum. The charged particles in the final state are produced by the decay
(too fast to be observed) of a $\tau^{+}\tau^{-}$ pair produced in the
collision. Analogously to the muon case, the $\tau$ lepton should decay into a
lighter lepton and a couple of neutrinos, to conserve the individual lepton
numbers. The tau neutrinos was thus introduced. Its detection at the DONUT
experiment [10] at FNAL was achieved by producing a tau neutrino flux through
the $D_{s}$ decay into $\tau\,\nu_{\tau}$ and through the challenging
observation of the tau produced by the $\nu_{\tau}$ interaction in matter.
The obvious question is now whether the story is over or there exists other
neutrino species, besides the electron, muon and tau ones. An important
constraint on the existence of such additional neutrinos is given by the
measurement of the $Z$-boson width at LEP. The decay width depends on the
number of (kinematically accessible) decays channels. The measurement agrees
with the prediction one obtains taking into account the known charged
particles with $m<M_{Z}/2$ and the three ($2.98\pm 0.01$) known neutrinos. The
existence of additional neutrinos with $m<M_{Z}/2$ is therefore excluded. Note
that by neutrino here we mean a particle with the same interactions as the
three known neutrinos. “Sterile neutrinos”, hypothetical particles not having
any SM gauge interaction, would not contribute to the $Z$ width and are
therefore allowed by the $Z$-width constraint. The latter, moreover,
translates into a constraint on the number of fermion families. The known
fermions are organised in three families with identical gauge quantum numbers,
each including a neutrino. The existence of additional families incorporating
light ($m<M_{Z}/2$) neutrinos is therefore also excluded.
Neutrino oscillation have also a long history. They were postulated in 1957 by
Pontecorvo [11, 12] who, in analogy with $K^{0}$-$\overline{K}^{0}$
oscillations, considered neutrino-antineutrino oscillations. The possibility
of mixing among electron and muon neutrinos was then considered by Maki,
Nakagawa, and Sakata [13] in 1962, in analogy to the Gell-Mann hypothesis of
quark mixing. The first experimental evidence of neutrino oscillations came in
1968, when Davis, using and experimental technique suggested by Pontecorvo
[14, 15], observed a deficit of about 50% in the measured solar neutrino flux
[16, 17] with respect to what predicted by Bahcall [18] a few years before.
Such an evidence was however plagued by the uncertainties on the theoretical
prediction of the solar neutrino flux. As we will see, it was only relatively
recently that the SNO experiment was able to get rid of such uncertainties and
confirm both the deficit and Bahcall’s prediction.
## 0.3 Neutrino parameters
Before discussing what we know about them and how, let us define the neutrino
parameters. In order to put the discussion in context, we start by describing
the theoretical framework and by illustrating the difference between Dirac and
Majorana neutrino.
Most neutrino experiments are characterized by energies much lower than the
electroweak scale, $v=174\,\mathrm{GeV}$. At such scales, the electroweak
symmetry is badly broken and it is convenient to describe the dynamics of the
particles light enough to be produced as initial or final states by means of
an effective hamiltonian that does not involve heavy fields and obeys the
unbroken QED and QCD symmetries (U(1)${}_{\text{em}}$ and SU(3)c
respectively).
Neutrinos do not couple to photons (QED) nor gluons (QCD). Their interactions
are described by an effective lagrangian generated by $W$ and $Z$ exchanges at
the electroweak scale. Such a four fermion interaction was first introduced by
Fermi and its detailed form was spelled out later [19, 20, 21]:
$\mathcal{L}_{E\ll
M_{Z}}^{\text{eff}}=\mathcal{L}_{\text{QED+QCD}}+4\,\frac{G_{F}}{\sqrt{2}}\,j_{c}^{\mu}j_{c\mu}^{\dagger}+\text{N.C.}+\ldots,$
(3)
where “N.C.” denotes the neutral current term, which is not relevant in the
following considerations, and the charged current is given by
$j^{\mu}_{c}=\overline{\nu_{e_{i}}}\,\gamma^{\mu}P_{L}\,e_{i}+\overline{u_{i}}\,\gamma^{\mu}P_{L}\,d_{i}$
(4)
in terms of the three charged lepton Dirac fields $e_{i}$ ($e_{1}\equiv e$,
$e_{2}\equiv\mu$, $e_{3}\equiv\tau$), the three neutrino “flavour eigenstates”
$\nu_{e_{i}}$, $i=1,2,3$, the three up quarks $u_{i}$ ($u_{1}\equiv u$,
$u_{2}\equiv c$, $u_{3}\equiv t$), the three down quarks $d_{i}$ ($d_{1}\equiv
d$, $d_{2}\equiv s$, $d_{3}\equiv b$). $P_{L}=(1-\gamma_{5})/2$ is the
projector on left-chirality fields, $\psi_{L}\equiv P_{L}\psi$, and
$P_{R}=(1+\gamma_{5})/2$ is the projector on right-chirality fields,
$\psi_{R}\equiv P_{R}\psi$. In the massless limit, left-chirality fields are
associated to left-handed helicity particles and right-handed helicity
antiparticles. In the massive limit this is not the case.
Neutrinos are allowed to have non-vanishing masses $m_{1}$, $m_{2}$, $m_{3}$.
We denote by $\nu_{i}$, $i=1,2,3$, the neutrino mass eigenstates fields, which
by definition diagonalize the mass matrix. The flavour eigenstates,
$\nu_{e_{i}}$, $i=1,2,3$, diagonalize the charged current. The neutrino
flavour eigenstates can be expressed in terms of the mass eigenstates through
a $3\times 3$ unitary matrix $U$ called Pontecorvo Maki Nakagawa Sakata (PMNS)
matrix. The matrix is unitary because it has to preserve the canonical form of
the kinetic term of the neutrino fields. As in the case of the CKM matrix
describing quark mixing, not all the 9 parameters parameterizing the matrix
$U$ are physical. The physical parameters are three mixing angles and,
depending on the Dirac or Majorana nature of the neutrinos (see below), one or
three CP-violating phases. The standard parameterization of the PMNS matrix in
the case of Majorana neutrinos is
$U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}s^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}s^{i\delta}&c_{23}c_{13}\end{array}\right)\begin{pmatrix}1&0&0\\\
0&e^{i\alpha}&0\\\ 0&0&e^{i\beta}\end{pmatrix},$ (5)
where $s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$,
$\theta_{12},\theta_{23},\theta_{13}$ are three mixing angles, $\delta$ is a
CP-violating phases that is physical in both the Majorana and Dirac neutrino
cases, and $\alpha,\beta$ are two CP-violating phases that are physical only
in the case of Majorana neutrinos and are therefore sometimes called Majorana
phases. In the case of Dirac neutrinos, the standard parameterization of the
PMNS matrix is given by the first factor only in the RHS eq. (5).
From a qualitative point of view, Dirac and Majorana neutrinos differ as
follows. In the case of Dirac neutrinos, neutrino and antineutrino are two
different particles, each with two possible values of the helicity, for a
total of 4 degrees of freedom. Also, the neutrino and antineutrino fields are
independent and the neutrino field, as all Dirac fields, splits into two
independent components with definite chirality (value of $\gamma_{5}$):
$\nu=\nu_{L}+\nu_{R}$. A mass term for Dirac neutrinos does not break lepton
number.
In the case of Majorana neutrinos, the neutrino particle coincides with its
antiparticle111In the case of massless neutrinos, the two helicities do not
mix and can be associated to two massless fermions, the neutrino and the
antineutrino, with one degree of freedom each. In the case of massive
neutrinos, the two helicities are mixed by the mass term (they are both part
of the same irreducible representation of the Poincaré group) and represent a
single particle., which gives a total of 2 degrees of freedom. The neutrino
and antineutrino fields are not independent (they are related by a matrix
transformation) and only have one chirality: $\nu=\nu_{L}$. A mass term for a
single Majorana neutrino field necessarily breaks lepton number and any U(1)
charge associated to the neutrino.
In the massless limit, the distinction between Dirac and Majorana neutrinos is
irrelevant. Indeed, the $\nu_{R}$ component of the Dirac field, if it exists,
does not have in this case any interaction, gauge or Yukawa, nor it is mixed
by a mass term to the $\nu_{L}$ component. Therefore, it does not affect the
dynamics of the fields produced in the experiments. As a consequence, telling
Majorana from Dirac neutrinos in experiments in which the energy is much
larger than the neutrino mass (so that we approach the massless limit) is
difficult. In particular, the distinction between Majorana and Dirac neutrinos
is irrelevant in oscillation experiments and in most other neutrino
experiments except when lepton number violation plays a role, as in the case
of neutrinoless double beta decay (see Section 0.4.7).
From a pragmatic point of view, what above is what needed for the
comprehension of this and the next Section. In the next subsection we will
give additional theoretical details on the nature of neutrino masses that will
be mostly needed as a background to Section 0.5 only and can be omitted at a
first reading.
### 0.3.1 The neutrino mass term
In this interlude we would like to discuss in greater theoretical detail the
form of the neutrino mass term, the difference between Dirac and Majorana
neutrinos, and how the neutrino parameters arise. This requires a basic
knowledge of quantum field theory.
The charged fermions are described by Dirac spinors, four component complex
fields. From the point of view of Lorentz transformations, Dirac spinors are
not elementary, though. For example, the electron field, $e$, contains two
independent components that have different (inequivalent) Lorentz
transformations, characterized by their chirality: $e=e_{L}+e_{R}$. In order
to be able to write the most general Lorentz invariant mass term or
interaction, it is useful to list the fields with equivalent Lorentz
transformations. Each Standard Model charged fermion field, and each
conjugated field, decomposes into a left and a right component:
$u_{i}=u_{iL}+u_{iR}$, $\overline{u_{i}}=\overline{u_{iL}}+\overline{u_{iR}}$,
$d_{i}=d_{iL}+d_{iR}$, $\overline{d_{i}}=\overline{d_{iL}}+\overline{d_{iR}}$,
$e_{i}=e_{iL}+e_{iR}$, $\overline{e_{i}}=\overline{e_{iL}}+\overline{e_{iR}}$,
where $i=1,2,3$ is the family index and I have omitted the color index of
quarks. Only the left-handed component of the neutrino field has been observed
so far, therefore we do not include a right-handed component in the list for
the time being: $\nu_{i}=\nu_{iL}$, $\overline{\nu_{i}}=\overline{\nu_{iL}}$.
Note that the conjugated of a right-chirality field also has left-chirality.
The left-chirality fields are therefore $e_{iL}$, $u_{iL}$, $d_{iL}$,
$\overline{e_{i}}_{L}=\overline{e_{iR}}$,
$\overline{u_{i}}_{L}=\overline{u_{iR}}$,
$\overline{d_{i}}_{L}=\overline{d_{iR}}$, $\nu_{iL}$. The conjugated fields
have all right-chirality. The most general Lorentz invariant gauge
transformation can in principle mix all the left-chirality fields. Once the
gauge transformations have been defined, the most general mass term is given
by
$\frac{m_{ij}}{2}\psi_{iL}\psi_{jL}+\text{h.c.},$ (6)
where $\psi_{iL}$ denotes a generic left-chirality field (the 7 (per family)
fields listed above, in the case of the SM), a proper Lorentz invariant
contraction of the Lorentz indexes is understood, and the mass matrix $m_{ij}$
is symmetric and should be invariant under gauge transformation. It is then
easy to write the most general mass term for the fermions above. In the
effective theory we are considering, the relevant gauge symmetries are the QED
and QCD ones, U(1)${}_{\text{em}}$ and SU(3)c. Under SU(3)c transformations,
each left-chirality quark field $u_{iL}$, $d_{iL}$ transforms as a triplet and
each left-chirality antiquark field, $\overline{u_{iR}}$, $\overline{d_{iR}}$,
transforms as an anti-triplet (leptons are of course invariant). Under
U(1)${}_{\text{em}}$ transformations, each left chirality fermion transforms
according to its electric charge: $Q_{u_{L}}=2/3$, $Q_{d_{L}}=-1/3$,
$Q_{e_{L}}=-1$, $Q_{\nu_{L}}=0$, $Q_{\overline{u_{R}}}=-2/3$,
$Q_{\overline{d_{R}}}=1/3$, $Q_{\overline{e_{R}}}=1$. Note that the neutrino
field does not feel either QCD or QED interactions, hence its elusiveness. It
is then an easy exercise to write the most general mass term in the form in
eq. (6) and invariant under QED and QCD transformations:
$m^{U}_{ij}\overline{u_{iR}}u_{jL}+m^{D}_{ij}\overline{d_{iR}}d_{jL}+m^{U}_{ij}\overline{e_{iR}}e_{jL}+\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}+\text{h.c.}.$
(7)
A few comments are in order. The charged fermions mass terms couple two
independent fields (that can be combined into a Dirac field). Such mass terms
are called “Dirac” mass terms. The factor 1/2 is missing there because, for
example,
$(m^{U}_{ij}/2)\overline{u_{iR}}u_{jL}+(m^{U}_{ji}/2)u_{iL}\overline{u_{jR}}=m^{U}_{ij}\overline{u_{iR}}u_{jL}$,
where we have used the fact that the mass term, when written in the form in
eq. (6), is symmetric. The neutrino mass term, on the other hand, involves the
same set of fields. Such a mass term is called a “Majorana” mass term. Any
charge carried by the $\nu_{iL}$ (such as total lepton number, for example) is
violated by such a mass term. That is why the neutrinos are the only fermions
in the above list for which a Majorana mass term is allowed: they are the only
neutral fields (under the gauge symmetries we are considering). Note that the
observed smallness of neutrino masses, compared to all the other fermion
masses, is not explained at this level: the QED and QCD gauge symmetries allow
a mass term for both the charged fermions and the neutrinos. We will see in
Section 0.5 that a natural explanation for the smallness of neutrino masses
arises once the whole SM gauge symmetry
$G_{\text{SM}}=\text{SU(3)}_{c}\times\text{SU(2)}_{L}\times\text{U(1)}_{Y}$ is
considered.
If neutrinos have a right-chirality component $\nu_{R}$ (also uncharged under
QED and QCD interactions), the most general mass term for neutrinos becomes
$\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}\to\frac{m^{L}_{ij}}{2}\nu_{iL}\nu_{jL}+\frac{m^{R}_{ij}}{2}\nu_{iR}\nu_{jR}+m^{N}_{ij}\overline{\nu_{iR}}\nu_{jL}.$
(8)
The first two terms are Majorana, while the third one is a Dirac mass term.
The Dirac limit, in which lepton number is conserved if $\nu_{L}$ and
$\nu_{R}$ are given the same lepton number as $e_{L}$ and $e_{R}$, is obtained
for $m^{L}=m^{R}=0$. We will see that the right-chirality components of
neutrinos, if present, are allowed to get a mass term much heavier than the
electroweak scale. This may account for their absence from the effective
lagrangian we are considering and for the smallness of the light neutrino
masses. For the time being we stick to the economical and theoretically
appealing case in which only left-chirality neutrinos are present.
Fermion masses and mixings are obtained when writing the lagrangian in terms
of mass eigenstates. In order to do that, it suffices to determine new linear
combinations of the initial fields,
$\displaystyle d^{\prime}_{iL}$ $\displaystyle=U^{d_{L}}_{ij}d_{jL},$
$\displaystyle d^{\prime}_{iR}$ $\displaystyle=U^{d_{R}}_{ij}d_{jR},$
$\displaystyle u^{\prime}_{iL}$ $\displaystyle=U^{u_{L}}_{ij}u_{jL},$
$\displaystyle u^{\prime}_{iR}$ $\displaystyle=U^{u_{R}}_{ij}u_{jR},$ (9a)
$\displaystyle e^{\prime}_{iL}$ $\displaystyle=U^{e_{L}}_{ij}e_{jL},$
$\displaystyle e^{\prime}_{iR}$ $\displaystyle=U^{e_{R}}_{ij}e_{jR},$
$\displaystyle\nu^{\prime}_{iL}$ $\displaystyle=U^{\nu_{L}}_{ij}\nu_{jL},$
(9b)
such that: i) the kinetic term for the new field is still canonical and ii)
the mass terms can be written as
$\begin{split}m^{U}_{ij}\overline{u_{iR}}u_{jL}+m^{D}_{ij}\overline{d_{iR}}d_{jL}+m^{U}_{ij}\overline{e_{iR}}e_{jL}+\frac{m^{\nu}_{ij}}{2}\nu_{iL}\nu_{jL}+\text{h.c.}=\\\
m_{u_{i}}\overline{u^{\prime}_{iR}}u^{\prime}_{iL}+m_{d_{i}}\overline{d^{\prime}_{iR}}d^{\prime}_{iL}+m_{e_{i}}\overline{e^{\prime}_{iR}}e^{\prime}_{iL}+\frac{m_{i}}{2}\nu^{\prime}_{iL}\nu^{\prime}_{iL}+\text{h.c.}=\\\
m_{u_{i}}\overline{u^{\prime}_{i}}u^{\prime}_{i}+m_{d_{i}}\overline{d^{\prime}_{i}}d^{\prime}_{i}+m_{e_{i}}\overline{e^{\prime}_{i}}e^{\prime}_{i}+\left(\frac{m_{i}}{2}\nu^{\prime}_{iL}\nu^{\prime}_{iL}+\text{h.c.}\right).\end{split}$
(10)
Because of the first requirement, the mixing matrices $U$ must be unitary.
Because of the second one, they should satisfy
$m^{D}=U_{d_{R}}^{\dagger}m^{D}_{\text{diag}}U^{\phantom{\dagger}}_{d_{L}},\quad
m^{E}=U_{u_{R}}^{\dagger}m^{U}_{\text{diag}}U^{\phantom{\dagger}}_{u_{L}},\quad
m^{E}=U_{e_{R}}^{\dagger}m^{E}_{\text{diag}}U^{\phantom{\dagger}}_{e_{L}},\quad
m^{L}=U^{T}_{\nu}m^{\nu}_{\text{diag}}U^{\phantom{\dagger}}_{\nu},$ (11)
where the diagonal matrices have non-negative eigenvalues. It turns out that
given generic complex matrices $m^{U,D,E}$ and given a symmetric complex
matrix $m^{\nu}$, it is always possible to find unitary matrices
$U^{d_{L},d_{R},u_{L},u_{R},e_{L},e_{R},\nu_{L}}$ satisfying eqs. (11). We can
then express the whole lagrangian in terms of the primed fields with definite
mass, and drop the primes for convenience. The QED and QCD gauge lagrangian do
not change form, as the transformations in eqs. (0.3.1) conserve the gauge
quantum numbers. The only change arises in the charged current, which in terms
of the mass eigenstate fields, becomes
$j^{\mu}_{c}=V_{ij}\overline{u_{iL}}\gamma^{\mu}d_{jL}+U^{\dagger}_{ij}\overline{\nu_{iL}}\gamma^{\mu}e_{jL},$
(12)
where $V=U_{u_{L}}U^{\dagger}_{d_{L}}$ is the Cabibbo Kobayashi Maskawa (CKM)
quark mixing matrix and $U=U_{e_{L}}U^{\dagger}_{\nu_{L}}$ is the PMNS lepton
mixing matrix.
Not all the parameters in $V$, $U$ are physical. Let us consider the CKM
matrix first. It is possible to write
$V=\operatorname{Diag}(e^{i\gamma_{1}},e^{i\gamma_{2}},e^{i\gamma_{3}})V_{\text{standard}}\operatorname{Diag}(1,e^{i\alpha},e^{i\beta})$,
where $V_{\text{standard}}$ is in the form of the first factor in the RHS of
eq. (5). Moreover, it is possible to reabsorb the phases $\gamma_{i}$,
$i=1,2,3$, $\alpha,\beta$ through a redefinition of the left-handed fields
$u_{iL}\to u_{iL}e^{i\gamma_{i}}$, $d_{2L}\to d_{2L}e^{-i\alpha}$, $d_{3L}\to
d_{3L}e^{-i\beta}$. This allows to write $V$ in the standard form
$V_{\text{standard}}$. On the other hand, the field phase transformations used
to bring $V$ in the standard form, introduce phases in the mass terms in the
second line of eq. (10). In order to get rid of them once and for all, it is
possible to redefine the phases of the right-chirality fields $d_{iR},u_{iR}$.
This shows that the phases $\gamma_{i}$, $\alpha$, $\beta$ are not physical,
as they can be completely eliminated from the lagrangian. In the lepton sector
the story would be exactly the same if the neutrinos had a Dirac mass term
$m^{N}_{ij}\overline{\nu_{iR}}\nu_{iL}$ as the charged fermions. On the other
hand if, as we assume, the neutrino mass term is Majorana, the phases $\alpha$
and $\beta$ in
$U=\operatorname{Diag}(e^{i\gamma_{1}},e^{i\gamma_{2}},e^{i\gamma_{3}})U_{\text{standard}}\operatorname{Diag}(1,e^{i\alpha},e^{i\beta})$
end up being physical. Indeed, while it is possible to eliminate those phases
from $U$ by redefining $\nu_{2L}\to\nu_{2L}e^{-i\alpha}$,
$\nu_{3L}\to\nu_{3L}e^{-i\beta}$, this transformation would move those phases
in the neutrino Majorana mass term. As the latter does not involve an
independent field whose phase can be rotated to eliminate $\alpha,\beta$ once
and for all, the phases $\alpha,\beta$ turn out not to be unphysical. They are
called Majorana phases.
### 0.3.2 Physical lepton mass and mixing parameters (Majorana neutrinos)
Let us now go back to phenomenology. As we have seen, the physical mass and
mixing parameters in the lepton sector are the 6 charged lepton and neutrino
masses and the 6 mixing parameters
$m_{e},\;m_{\mu},\;m_{\tau},\quad
m_{1},\;m_{2},\;m_{3},\quad\theta_{23},\;\theta_{12},\;\theta_{13},\;\delta,\quad\alpha,\;\beta.$
(13)
The physical ranges of the above parameters are $m_{e,\mu,\tau,1,2,3}\geq 0$,
$0\leq\theta_{23,12,13}\leq\pi/2$, $0\leq\delta<2\pi$,
$0\leq\alpha,\beta<\pi$.
One important remark concerns the ordering of the neutrino mass eigenstates.
In the charged lepton sector (and in the quark sector), the mass eigenstates
are ordered with their masses: $m_{e_{1}}<m_{e_{2}}<m_{e_{3}}$. In the case of
neutrinos, the convention used is different. By definition, we call $\nu_{1}$
and $\nu_{2}$ the two neutrinos whose masses are closer in value, with
$m_{1}<m_{2}$. The third mass eigenstate has a larger separation in mass from
$\nu_{1}$ and $\nu_{2}$, but can be heavier or lighter. If $m_{3}>m_{1,2}$, we
say that the neutrinos have a “normal” hierarchy. If $m_{3}<m_{{1,2}}$, we say
they have an inverse hierarchy. Let us call
${\Delta m^{2}_{ij}}\equiv m^{2}_{j}-m^{2}_{i}.$ (14)
Then we have, by definition, ${\Delta m^{2}_{12}}>0$ and $0<{\Delta
m^{2}_{12}}<|{\Delta m^{2}_{23}}|$. Corresponding to the two possible
hierarchies, ${\Delta m^{2}_{23}}$ can have both signs: ${\Delta
m^{2}_{23}}>0$ in the case of normal hierarchy and ${\Delta m^{2}_{23}}<0$ in
the case of inverse hierarchy.
Neutrino oscillation phenomena do not depend on the absolute values of
neutrino masses but only on the squared mass differences in eq. (14). It is
then useful to use the following set of lepton mass and mixing parameters,
equivalent to the one in eq. (13):
$m_{e},\;m_{\mu},\;m_{\tau},\quad{\Delta m^{2}_{12}},\;|{\Delta
m^{2}_{23}}|,\;\operatorname{sign}({\Delta
m^{2}_{23}}),\;\theta_{23},\;\theta_{12},\;\theta_{13},\;\delta,\quad
m_{\text{lightest}},\;\alpha,\;\beta.$ (15)
The neutrino masses $m_{1}$, $m_{2}$, $m_{3}$ have been traded for the
equivalent set of parameters $m_{\text{lightest}}$, the lightest neutrino
mass, and ${\Delta m^{2}_{12}}$, $|{\Delta m^{2}_{23}}|$,
$\operatorname{sign}({\Delta m^{2}_{23}})$. The mass parameters ${\Delta
m^{2}_{12}}$, $|{\Delta m^{2}_{23}}|$, $\operatorname{sign}({\Delta
m^{2}_{23}})$, the mixing angles $\theta_{23}$, $\theta_{12}$, $\theta_{13}$
and the phase $\delta$ are accessible to neutrino oscillation experiments. The
absolute scale of neutrino masses, represented by $m_{\text{lightest}}$, and
the Majorana phases $\alpha,\beta$ (if physical) are not. The third squared
mass difference, ${\Delta m^{2}_{13}}$ is obviously not independent of the
first two, ${\Delta m^{2}_{13}}={\Delta m^{2}_{12}}+{\Delta m^{2}_{23}}$. The
experiment shows that ${\Delta m^{2}_{12}}\ll{\Delta m^{2}_{23}}$, so that
${\Delta m^{2}_{13}}\approx{\Delta m^{2}_{23}}$. We have then in first
approximation only two squared mass differences, which are sometimes called
“solar” and “atmospheric”: ${\Delta m^{2}_{\text{SUN}}}\equiv{\Delta
m^{2}_{12}}$, ${\Delta m^{2}_{\text{ATM}}}\equiv{\Delta
m^{2}_{23}}\approx{\Delta m^{2}_{13}}$. The name refers, as we will see, to
the neutrino source that was first used to measure those parameters. Analogous
names are sometimes used for the corresponding mixing angles:
$\theta_{\text{SUN}}\equiv\theta_{12}$,
$\theta_{\text{ATM}}\equiv\theta_{23}$.
The experimental situation is the following. The charged lepton masses are of
course well known. The solar and atmospheric squared mass differences,
together with the corresponding mixing angles, are also known. There are
bounds on $\theta_{13}$ and $m_{\text{lightest}}$. No information is available
at present on $\operatorname{sign}({\Delta m^{2}_{23}})$, $\alpha$, $\beta$.
Before discussing in detail the experimental determination of the neutrino
parameters, we summarize the most relevant information available at present:
$\begin{gathered}\begin{aligned} {\Delta m^{2}_{\text{ATM}}}&\sim 2.4\times
10^{-3}\,\mathrm{eV}^{2}&\theta_{23}&\sim 45^{\circ}&&\text{(ATM, K2K,
Minos)}\\\ {\Delta m^{2}_{\text{SUN}}}&\sim 0.76\times
10^{-4}\,\mathrm{eV}^{2}&\theta_{12}&\sim 35^{\circ}&&\text{(SUN, KamLAND)}\\\
&\qquad\theta_{13}<7^{\circ}\;\text{($2\sigma$)}&&&&\text{(CHOOZ, Minos + ATM,
SUN)}\end{aligned}\\\\[5.69054pt] \begin{aligned}
&|m_{ee}|=|U^{2}_{ei}m_{\nu_{i}}|<\mathcal{O}\left(1\right)\times
0.4\,\mathrm{eV}&\quad&\text{(Heidelberg-Moscow)}\\\
&(m^{\dagger}m)_{ee}=|U_{ei}|^{2}m^{2}_{\nu_{i}}<(2.2\,\mathrm{eV})^{2}&\quad&\text{(Mainz,
Troktsk)}\\\
&\sum_{i}m_{\nu_{i}}<\mathcal{O}\left(1\right)\,\mathrm{eV}\;\text{(priors)}&\quad&\text{(Cosmology)}.\end{aligned}\end{gathered}$
(16)
The experiments from which the information is obtained are also indicated.
“ATM” and “SUN” denote the atmospheric and solar neutrino experiments
respectively. $U_{ei}$ denotes the “1i” element of the PMNS matrix, which can
be expressed in terms of the parameters in eq. (15) through eq. (5). The bound
from cosmology is subject to uncertainties associated to the priors used in
the analysis.
## 0.4 The determination of the neutrino parameters
At present, most experimental information on the neutrino mass and mixing
parameters comes from experiments measuring neutrino transitions, which are by
now known to be due to neutrino oscillations. There are also beta decay
experiments aiming at a measurement of the absolute scale of neutrino masses,
$m_{\text{lightest}}$; neutrinoless double beta decay experiments, sensitive
to lepton number violation (Majorana vs Dirac neutrinos) and, in the case of
Majorana neutrinos, to both $m_{\text{lightest}}$ and the Majorana phases
$\alpha,\beta$; and experiments in astrophysics and cosmology, sensitive to
different neutrino properties.
### 0.4.1 The physics of neutrino oscillation experiments
Neutrino oscillations arise from the misalignment of the neutrino flavour
eigenstate fields, $\nu_{e,\mu\tau}$, coupled to the charged leptons in the
charged current interactions, and neutrino mass eigenstate fields,
$\nu_{1,2,3}$, eigenstates of the free hamiltonian and therefore associated to
the free propagation. Such a misalignment, as we have seen, is quantified by
the PMNS matrix: $\nu_{e_{i}}=U_{ih}\nu_{h}$,
$\overline{\nu}_{e_{i}}=U^{*}_{ih}\overline{\nu}_{h}$. The one-particle states
relations involve the conjugate matrix elements:
$|\nu_{e_{i}}\rangle=U^{*}_{ih}|\nu_{h}\rangle$,
$|\overline{\nu}_{e_{i}}\rangle=U_{ih}|\overline{\nu}_{h}\rangle$. Neutrinos
are produced by the charged current interactions of charged leptons, typically
electrons or muons. They are therefore in a flavour eigenstate, i.e. in a
coherent superposition of mass eigenstates. Suppose a neutrino
$|\nu_{e_{i}}\rangle=U^{*}_{ih}|\nu_{h}\rangle$ is produced by the interaction
with the lepton $e_{i}$ and it freely evolves. Let us compute the probability
that the neutrino is found after a time $t$ to be a $|\nu_{e_{j}}\rangle$
neutrino, for example by means of a charged current interaction with the
lepton $e_{j}$. The free evolution of the initial state gives
$e^{-iHt}|\nu_{e_{i}}\rangle=U^{*}_{ih}e^{-iE_{h}t}|\nu_{h}\rangle$, where
$E^{2}_{h}=(p^{2}+m^{2}_{h})$. The probability that the neutrino is found to
be a $|\nu_{e_{j}}\rangle$ neutrino is therefore
$P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})=\left|\langle\nu_{e_{j}}|e^{-iHt}|\nu_{e_{i}}\rangle\right|^{2}=|U_{jh}e^{-iE_{h}t}U^{\dagger}_{hi}|^{2}\approx|U_{jh}e^{-i\frac{m^{2}_{h}}{2E}t}U^{\dagger}_{hi}|^{2},$
(17)
where we have approximated $E_{h}\approx p+m^{2}_{h}/(2E)$, as $E\gg m_{\nu}$
in all neutrino oscillation experiments, and the time $t$ can be replaced by
the length travelled $L$. We have not specified the helicity of the neutrino,
as it is not necessary. This is because in all neutrino oscillation
experiments the neutrino energy is way larger than its mass, $E\gg m_{\nu}$.
Since the neutrino interaction only involves the left-chirality component, the
neutrino produced will be mostly in an left-handed helicity state, whether it
is Majorana or Dirac, and helicity flips, either at production, detection, or
during propagation, are largely negligible. Moreover, in the $E\gg m_{\nu}$
limit, the oscillation probabilities do not depend on Majorana phases. The
oscillation probabilities satisfy
$P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\overline{\nu}_{e_{j}}\to\overline{\nu}_{e_{i}})$,
because of CPT invariance. If CP is conserved,
$P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\overline{\nu}_{e_{i}}\to\overline{\nu}_{e_{j}})$,
and equivalently $P(\nu_{e_{i}}\to\nu_{e_{j}})=P(\nu_{e_{j}}\to\nu_{e_{i}})$
if $T$ is conserved. The total oscillation probability is of course one,
$\sum_{j}P(\nu_{e_{i}}\to\nu_{e_{j}})=1$.
It is instructive to consider the simplest case of two neutrino oscillations.
Let us then consider the electron and muon neutrinos only. Up to phases
redefinitions, their mixing can be described by a real orthogonal $2\times 2$
matrix, i.e. a rotation by an angle $\theta$, which gives a simple expression
for the oscillation probability:
$\begin{aligned} \nu_{e}&=\nu_{1}\cos\theta+\nu_{2}\sin\theta\\\
\nu_{\mu}&=-\nu_{1}\sin\theta+\nu_{2}\cos\theta\end{aligned}\quad\Rightarrow\quad
P(\nu_{e}\rightarrow\nu_{\mu})=\sin^{2}2\theta\sin^{2}\frac{\Delta
m^{2}L}{4E},$ (18)
where ${\Delta m^{2}}=m^{2}_{2}-m^{2}_{1}$ can be taken positive by
definition. In order to obtain predictions for the outcome of realistic
experiments, the oscillation probability has to be convoluted with the source
energy spectrum, the distribution in the position of the neutrino emission and
detection, the scattering cross sections, the experimental resolution and
efficiency.
Let us comment on the form of the two neutrino oscillation formula. The
oscillation amplitude $A=\sin^{2}2\theta$ is determined by the mixing angle
$\theta$ and does not allow to distinguish (in vacuum) the physically
inequivalent $\theta$ and $\pi/2-\theta$ values. The squared mass difference
determines the oscillation length $\lambda=4\pi E/{\Delta m^{2}}\approx
2.48\,\text{km}(E(\text{GeV})/{\Delta m^{2}}(\text{eV}^{2}))$, or equivalently
the oscillation phase $\phi={\Delta m^{2}}L/(4E)\approx 1.27({\Delta
m^{2}}(\text{eV}^{2})L(\text{km})/E(\text{GeV}))$. In order to determine both
the oscillation parameters, it is best to consider an experiment in which the
neutrinos travel a distance comparable to their oscillation length,
$L\sim\lambda$. In the $L\ll\lambda$ limit, in fact,
$P(\nu_{e}\rightarrow\nu_{\mu})\approx\sin^{2}2\theta\,(\Delta
m^{2}L/(4E))^{2}$ and even a detailed measurement of the oscillation
probability as a function of $E$ and $L$ would determine the product
$\sin^{2}2\theta\cdot\Delta m^{2}$ only. In this limit, oscillations have not
enough time to occur and the expression for the probability can be obtained in
perturbation theory (which represents a check of the correctness of the
formula). The oscillation probability is proportional to $(L/E)^{2}$ and the
neutrino flux decreases with the geometrical factor $1/L^{2}$, therefore the
number of neutrino oscillation events measured in a detector is approximately
independent of the distance $L$, within this limit. In the $L\gg\lambda$
limit, on the other hand, the oscillations are so fast that they average out
and
$P(\nu_{e}\rightarrow\nu_{\mu})\approx\sin^{2}2\theta/2=\sin^{2}\theta\cos^{2}\theta+\cos^{2}\theta\sin^{2}\theta$.
Only the mixing angle can be measured in this limit. The oscillation
probability is independent of $E$, $L$ and the number of oscillation events
decreases with $1/L^{2}$. In this “classical” limit, the oscillation
probability is the sum (over $i$) of the probabilities that the initial
neutrino $\nu_{e}$ is a $\nu_{i}$ times the probability that the neutrino
$\nu_{i}$ is observed to be a $\nu_{\mu}$.
Figure 2: Typical sensitivity plot of a neutrino experiment.
The situation is illustrated in Fig. 2, where the typical sensitivity plot of
a neutrino experiment is plotted. Assuming that the experiment is sensitive to
a given (averaged) oscillation probability, the sensitivity in the
$\sin^{2}2\theta$–${\Delta m^{2}}$ plane is shown. The two limits considered
above can be recognized in the lower and upper part of the plot respectively.
In order to measure both $\sin^{2}2\theta$ and ${\Delta m^{2}}$, a measurement
of the averaged probability is not enough. The $E$ or $L$ dependence has also
to be measured, better if in the $L\sim\lambda$, or $({\Delta
m^{2}}L/(4E))\sim 1$, regime.
The above derivation of the oscillation formula is simplistic for a number of
reasons. First of all, it holds in vacuum only. The coherent (or incoherent)
effect of matter in neutrino propagation can be very important, as we will
show below. Moreover, the neutrino coherence assumed in the derivation of the
oscillation formula can be lost for a number of reasons, besides the necessary
averages mentioned above. Because the wave packets associated to the different
mass eigenstates making up a given flavour eigenstate travel at slightly
different velocities, for example (this is relevant when the distance
travelled is very large). Or because the neutrino production process typically
involves at least another particle in the final state. The quantum mechanics
reduction to the neutrino subsystem also induces a loss of coherence. Finally,
the derivation assumed that the neutrino mass eigenstates are all in a pure
eigenstate with same definite momentum. It is sometime argued that it is more
appropriate to assume that they have the same energy instead. A proper
derivation would take into account the precise form of the density matrix
describing the initial state and its momentum distribution, as obtained from
the dynamics of the production process. In this context, using the fixed
momentum or fixed energy description just amounts to a change of variable in
the integration over the momentum distribution of the initial neutrino state.
Let us consider now the three neutrino case. The exact three neutrino formulas
are
$\displaystyle\begin{aligned}
P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})&=P(\overline{\nu}_{e_{j}}\rightarrow\overline{\nu}_{e_{i}})=P_{\text{CPC}}+P_{\text{CPV}}\\\
P(\overline{\nu}_{e_{i}}\rightarrow\overline{\nu}_{e_{j}})&=P(\nu_{e_{j}}\rightarrow\nu_{e_{i}})=P_{\text{CPC}}-P_{\text{CPV}}\end{aligned}$
$\displaystyle\begin{aligned}
P_{\text{CPC}}&=\delta_{ij}-4\operatorname{Re}(J^{ji}_{12})S^{2}_{12}-4\operatorname{Re}(J^{ji}_{23})S^{2}_{23}-4\operatorname{Re}(J^{ji}_{31})S^{2}_{31}\\\
P_{\text{CPV}}&=8\sigma_{ij}J_{\text{CP}}S_{12}S_{23}S_{31}\end{aligned}$ (19)
$\displaystyle S_{hk}=\sin\frac{{\Delta m^{2}_{hk}}L}{4E},\quad
J^{ji}_{hk}=U^{\phantom{\dagger}}_{jh}U^{\dagger}_{hi}U^{\phantom{\dagger}}_{ik}U^{\dagger}_{kj},\quad\operatorname{Im}(J^{ji}_{hk})=\sigma_{ji}\sigma_{hk}J_{\text{CP}},\quad\sigma_{ij}=\sum_{k}\epsilon_{ijk}=\pm
1,0,$
where $P_{\text{CPC}}$ and $P_{\text{CPV}}$ are the CP conserving and CP
violating parts of the oscillation probability respectively. Note again the
independence of the formulas above of Majorana phases and of the absolute
neutrino mass scale.
Despite the existence of three neutrinos, the results of neutrino oscillation
experiments are often shown in a two neutrino oscillation context and mainly
determine a single mixing angle and squared mass difference. This is because
the experimental values of the neutrino parameters are such that often, in
first approximation, the three neutrino oscillation formula reduces to a two
neutrino one. For example, the CHOOZ experiment, as we will see, measures the
probability of electron neutrino disappearance, $P(\nu_{e}\to\nu_{e})$, for
$L/E$ values such that the $S_{12}$ terms in eq. (19) are negligible (because
of the small ${\Delta m^{2}_{12}}$). For the same reason $S_{23}\approx
S_{12}$, so that we can approximate $P(\nu_{e}\rightarrow\nu_{e})\approx
1-\sin^{2}2\theta_{13}\sin^{2}({\Delta m^{2}_{23}}L/(4E))$, a two neutrino
oscillation formula with $\theta=\theta_{13}$ and ${\Delta m^{2}}={\Delta
m^{2}_{23}}$. This allows the CHOOZ experiment to set an upper bound on
$\theta_{13}$ [22] that makes the $\theta_{13}$ contribution negligible, in
first approximation, in the solar and atmospheric neutrino experiments. In
particular, in atmospheric neutrino experiments, $S^{2}_{12}\ll 1$,
$S^{2}_{23}\approx S^{2}_{13}$, $\theta_{13}\ll 1$, so that
$P(\nu_{\mu}\rightarrow\nu_{\tau})\approx\sin^{2}2\theta_{23}\sin^{2}({\Delta
m^{2}_{23}}L/(4E))$, $P(\nu_{e}\rightarrow\nu_{\mu,\tau})\ll 1$, and the
results can be interpreted in terms of $\nu_{\mu}\leftrightarrow\nu_{\tau}$
oscillations with $\theta=\theta_{23}$ and ${\Delta m^{2}}={\Delta
m^{2}_{23}}$. In solar neutrino experiments, detecting the disappearance of
the electron neutrinos produced by the Sun, the $S^{2}_{23}$ and $S^{2}_{13}$
terms are suppressed by $\theta_{13}$ and $P(\nu_{e}\rightarrow\nu_{e})\approx
1-\sin^{2}2\theta_{12}\sin^{2}({\Delta m^{2}_{12}}L/(4E))$, leading to a
determination of $\theta=\theta_{12}$ and ${\Delta m^{2}}={\Delta
m^{2}_{12}}$.
In the next subsections, we will discuss the experimental determination of
neutrino parameters by oscillation experiments.
### 0.4.2 Experimental determination of ${\Delta m^{2}_{23}}$ and
$\theta_{23}$
The experimental determination of ${\Delta m^{2}_{23}}$ and $\theta_{23}$ is
mostly due to the Super-Kamiokande (SK) measurement of atmospheric neutrinos,
and to the K2K, Minos, and Opera experiments. The result of a global fit is
shown in Fig. 3a [23]. Let us discuss the main ingredients entering the above
determination.
(a) (b)
Figure 3: Global fit of the ${\Delta m^{2}_{23}}$ and $\theta_{23}$ parameters
(a). Schematic representation of the distance travelled by atmospheric
neutrinos (b).
#### Atmospheric neutrinos
Atmospheric neutrinos arise from cosmic ray interactions in the atmosphere.
Charged pion produced by such interactions decay mostly through the decay
chain $\pi^{+}\to\mu^{+}\nu_{\mu}\to
e^{+}\nu_{e}\overline{\nu}_{\mu}\nu_{\mu}$ (analogously for negative pions, SK
does not tell neutrinos from antineutrinos), thus producing in first
approximation two muon neutrinos for each electron neutrino. The ratio of muon
to electron neutrinos reaching the Earth is actually slightly larger than two
because i) energetic muons have a long life-time and may not decay before
reaching the Earth and ii) Kaons are also produced by cosmic ray interactions.
Atmospheric neutrinos are detected by experiments placed underground (to
shield cosmic rays, but not neutrinos). The neutrinos travel a distance
ranging from $10\,\text{km}$ to more than $10^{4}\,\text{km}$, as shown in
Fig. 3b. Their energy ranges from 0.1 to 10 and more GeV. The oscillation
phase for ${\Delta m^{2}_{23}}$ oscillations is therefore typically ${\Delta
m^{2}_{23}}L/(4E)=10^{-2}$–$10^{2}$, centred around 1, the value we argued is
experimentally the best to reveal oscillation. As the neutrino flux produced
in the atmosphere is obtained by theoretical simulation characterized by
significant uncertainties, the measurement of the absolute muon or electron
neutrino flux does not allow to firmly establish the occurrence of neutrino
flavour transitions. On the other hand, the measurement of the muon to
electron neutrino ratio has a smaller theoretical uncertainty and is therefore
more reliable. Even more reliable is the variation of the muon and electron
fluxes (and their ratio) with the distance travelled, i.e. with the direction
(zenith angle) from which they reach the detector. The latter measurements by
SK provided in 1998 the first firm evidence of neutrino flavour transitions
and opened the modern era of neutrino physics.
Super-Kamiokande is a large water Cherenkov detector located in the Kamioka
mine, in Japan, $2.7\,\text{km}$ underground. It contains about 50 ktons of
water and is surrounded by about 13000 photomultipliers. In order to perform
the analysis above, a measurement of the neutrino flavour, direction, the
energy is needed. Let us see how such information is, at least partial,
obtained.
Neutrinos can be detected through their charged current interactions with the
nuclei: $\nu_{e_{i}}+N\to e_{i}+N^{\prime}$. The lepton $e_{i}=e,\mu$ produced
in the interaction is ultra-relativistic and produces a cone of Cherenkov
light while it travels through the water, which is detected by the
photomultipliers. When a lepton stops inside the detector the photomultipliers
detect a ring of Cherenkov light. The nature of the lepton can be told by the
shape of the ring: a muon produces a relatively clean ring, while the ring
produced by electrons is more fuzzy, as shown in Fig. 4. The position of the
ring allows to determine the lepton direction, which is correlated to the
neutrino direction if the neutrino energy is larger than about a GeV. If the
lepton is produced in the detector and stops inside the detector, its energy
can be measured by the amount of Cherenkov light collected by the
photomultipliers. The lepton energy is not strongly correlated to the neutrino
energy, but it cannot exceed it, which is enough to provide an handle on the
energy dependence of the neutrino flavour transition probability. The best
events are therefore the “fully contained multi-GeV” events. Neutrinos also
interact through neutral current interactions with nuclei such as
$\nu+N\to\nu+N+\pi^{0}\to\nu+N+\gamma\gamma$, which also produce a signal in
the detector. Tau leptons can also be produced if the neutrino is energetic
enough to exceed the kinematical threshold for production. Taus quickly decay
into hadrons, producing a signal similar to the neutral current one.
(a) (b)
Figure 4: Cherenkov rings produced by a muon (a) and an electron (b) in Super-
Kamiokande.
By now the statistics accumulated by Super-Kamiokande is impressive. Not only
it allows to establish neutrino transitions without any doubt, despite the
oscillation pattern is too smeared out by the poor neutrino-lepton energy
correlation to be observed explicitly, but it also allows to attribute the
transitions to $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. In
particular, no depletion of the electron neutrino flux with the distance
travelled has been observed, which is compatible with the CHOOZ bound [22] on
$\nu_{e}$ transitions. Also, oscillations into sterile neutrinos, hypothetical
additional neutrinos not feeling any SM gauge interaction, are ruled out or
bound to have a marginal role. The same holds for exotic disappearance
mechanisms such as neutrino decay, or Lorentz or CPT violation.
#### Accelerator experiments
The Super-Kamiokande results have been confirmed by a number of experiments
using neutrinos produced at accelerators. Opera is a sophisticated detector at
the Gran Sasso laboratory in Italy designed to explicitly detect $\nu_{\tau}$
appearance from a $\nu_{\mu}$ neutrino beam produced at CERN. Such an
appearance would confirm the indirect, but solid, interpretation of the SK
results in terms of $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. The tau
produced by the $\nu_{\tau}$ charged current interaction in the detector is
observed in emulsion films. Unfortunately, the expected statistics is not very
high, but a first candidate $\nu_{\tau}$ event has been recently reported
[24]. The K2K experiment in Japan used the SK detector to measure the
disappearance of $\nu_{\mu}$ from a pulsed beam produced at KEK. The initial
flux is measured by a detector placed near the neutrino source. The average
neutrino energy is slightly above $1\,\mathrm{GeV}$, and the distance
travelled by neutrinos is about $250\,\text{km}$, which gives an oscillation
phases of order one, as desired. The muon scattering angle in the detector can
be measured, together with its energy. The kinematics of the charged current
interaction then allows to reconstruct the neutrino energy. The experimental
results have been reported in [25]. Another important experiment is Minos, in
the Sudan mine, $735\,\text{km}$ north of Fermilab, where the (pulsed)
$\nu_{\mu}$ beam is produced. The average neutrino energy is higher than in
K2K, to give again an oscillation phase around one. As in the case of K2K, the
initial flux is measured by a near detector. Neutrino interactions in steel
are measured in this case by means of a magnetized tracking calorimeter. Minos
can observe $\nu_{\mu}$ charged current events (penetrating muons) and
therefore $\nu_{\mu}$ disappearance, which gives a determination of
$\theta_{23}$ and ${\Delta m^{2}_{23}}$ in agreement with the SK one [26]. It
can also see neutral current interactions of any neutrino (they produce a
diffuse hadron shower), which confirms that oscillations into sterile
neutrinos cannot account for the $\nu_{\mu}$ disappearance [27]. It can detect
$\nu_{e}$ charged current interactions (compact electromagnetic showers) and
therefore set a bound on $\nu_{\mu}\to\nu_{e}$ oscillations, which translates
into a bound on $\theta_{13}$ [28] compatible (although at present weaker)
with the CHOOZ one. The presence of a magnetic field allows Minos to tell
$\mu^{+}$ from $\mu^{-}$, which in turn allows to test CP-violation (although
with a poor sensitivity). The possibility to switch from a $\nu_{\mu}$ to a
$\overline{\nu}_{\mu}$ beam allows to test CPT violation. A mild, not very
significant, tension between the $\nu_{\mu}$ and $\overline{\nu}_{\mu}$
determinations of the oscillation parameters has been recently reported [29].
### 0.4.3 Experimental determination of ${\Delta m^{2}_{12}}$ and
$\theta_{12}$
The experimental determination of ${\Delta m^{2}_{12}}$ and $\theta_{12}$ is
mainly due to the SK, SNO, Borexino, KamLAND experiments. The result of a
global fit is shown in Fig. 5a [23]. Let us discuss the main ingredients
entering the above determination. In order to illustrate the physics of solar
neutrinos, it is necessary to discuss neutrino propagation in matter
(a) (b)
Figure 5: Global fit of the ${\Delta m^{2}_{12}}$ and $\theta_{12}$ parameters
(a). Dependence of the mixing angle in matter with the neutrino energy in the
case ${\Delta m^{2}}>0$ and for two values of the mixing angle in vacuum,
$\theta=0.6$ (dashed curve) and $\theta=0.06$ (solid curve) (b).
#### Matter effects in neutrino propagation
As neutrinos do not feel electromagnetic or strong interactions, they can
travel through ordinary matter without experiencing a single scattering
interaction. The mean free path of a neutrino in a medium as dense as the
Earth’s mantle is in fact $\lambda(E)\sim
10^{9}\,\text{km}\,(\text{GeV}/E)^{2}$ and even in the core of the Sun is
$\lambda(E)\sim 10^{10}\,\text{km}\,(10\,\mathrm{MeV}/E)^{2}$, both much
larger than the distance travelled in the medium. The energy normalization is
appropriate for atmospheric and solar neutrinos respectively. Only in
extraordinarily dense matter, such as a proto-neutron star core, neutrinos
have a mean free path, $\lambda(E)\sim
10\,\text{cm}\,(100\,\mathrm{MeV}/E)^{2}$, trapping them in a random walk
lasting about 10 seconds.
This does not mean, however, that matter does not affect neutrino propagation
in the Earth and in the Sun. While incoherent scattering is proportional to
the square of the weak interaction Fermi coupling suppressing the process,
forward coherent scattering [30], affecting the phase of the neutrino wave
function, is proportional to only one power of the Fermi coupling. Let us then
compare the rate of incoherent scattering, $dP_{\text{sc}}/dx$, where
$P_{\text{sc}}$ is the incoherent scattering probability, and the rate of
change of the neutrino phase due to coherent forward scattering,
$d\phi_{\text{co}}/dx$:
$\begin{aligned} &\text{incoherent:}&&dP_{\text{sc}}/dx\sim
G^{2}_{\text{F}}E^{2}n\\\ &\text{coherent:}&&d\phi_{\text{co}}/dx\sim
G_{\text{F}}n\end{aligned}\rightarrow\frac{dP_{\text{sc}}}{d\phi_{\text{co}}}\sim
G_{\text{F}}E^{2}\sim 10^{-5}\left(\frac{E}{\text{GeV}}\right)^{2},$ (20)
where $n$ is the matter number density and $E$ is the neutrino energy. We
therefore see that the coherent effect is largely dominant. While the effect
on the neutrino phase would be unobservable in the absence of neutrino
oscillations, the impact on oscillations may be significant, as the phases of
the three neutrino flavour eigenstates are affected in different ways.
Coherent scattering in the propagation can be accounted for by adding to the
free hamiltonian for the three neutrinos an effective “MSW” [30, 31, 32]
potential. In the flavour eigenstate basis, the Hamiltonian then reads
$H=\frac{1}{2E}U\begin{pmatrix}m^{2}_{1}&&\\\ &m^{2}_{2}&\\\
&&m^{2}_{3}\end{pmatrix}U^{\dagger}+\begin{pmatrix}V&&\\\ &0&\\\
&&0\end{pmatrix}+\text{universal terms,}$ (21)
where $U$ is the PMNS matrix and $V$ is the MSW potential. The three flavour
neutrinos feel different potentials in matter, because they have different
weak interactions. At the tree level, in neutral matter with no muon or tau
lepton number (or with $L_{\mu}=L_{\tau}$) and a negligible neutrino density,
such as the Earth or Sun matter, $V_{\mu}=V_{\tau}$ and
$V=V_{e}-V_{\mu}=\sqrt{2}G_{\text{F}}n_{e}$, where $n_{e}$ is the electron
neutrino number density. The difference between the electron neutrino
potential and the muon and tau one is due to the fact that the electron
neutrino can interact through charged current interactions with electrons,
while the muon and tau neutrinos cannot. The Hamiltonian in eq. (21)
determines neutrino propagation. In the antineutrino case, $U\to U^{*}$ and
$V\to-V$. Let us see what are the consequences of the presence of the MSW
potential are as far as neutrino propagation _in constant density_ is
concerned.
The case of constant density is relevant for the neutrino propagation in the
Earth, for example. The Earth has a density profile that is not constant but
can be in first approximated to be constant both in the mantle, where the mass
density is approximately $\rho_{m}\sim 3$–$5\,\text{g}/\text{cm}^{3}$, and in
the core, where $\rho_{c}\sim 10$–$15\,\text{g}/\text{cm}^{3}$. The effect of
the Earth in neutrino propagation is important for i) atmospheric neutrinos
(only through the subdominant $\nu_{e}\leftrightarrow\nu_{\mu,\tau}$
transitions, as $\nu_{\mu}\leftrightarrow\nu_{\tau}$ transitions are not
affected), ii) solar and supernova neutrinos, iii) terrestrial experiments (in
the case of a long baseline).
One of the most interesting consequences of the presence of the MSW term is
the possibility of a resonant enhancement of the oscillation amplitude. In
order to illustrate such an effect, let us consider a simple two neutrino
case, in which the Hamiltonian can be written as
$H=\begin{pmatrix}\displaystyle\sin^{2}\theta+\frac{2EV}{\Delta
m^{2}}&\sin\theta\cos\theta\\\
\sin\theta\cos\theta&\cos^{2}\theta\end{pmatrix}\frac{\Delta
m^{2}}{2E}+\text{universal terms}.$ (22)
It is then clear then a resonant enhancement of the mixing angle takes place
when the two diagonal elements coincide. In such a case, the mixing angle “in
matter” (i.e. obtained from the diagonalization of the matrix in eq. (22)),
$\theta_{m}$, becomes maximal, $\theta_{m}=45^{\circ}$, no matter how small is
the mixing angle in vacuum, and the squared mass difference in matter gets
correspondingly suppressed:
$\frac{2EV}{\Delta m^{2}}=\cos 2\theta\Rightarrow\left\\{\begin{aligned}
&(\sin 2\theta)_{m}=1,\;\\\ &(\Delta m^{2})_{m}=\Delta m^{2}\sin
2\theta\end{aligned}\right..$ (23)
Such a mixing enhancement takes place for an appropriate value of the neutrino
energy if $\theta<45^{\circ}$ and $V\cdot{\Delta m^{2}}>0$ or if
$\theta>45^{\circ}$ and $V\cdot{\Delta m^{2}}<0$. The dependence of the mixing
angle in matter with the neutrino energy is shown in Fig. 5b in the case
${\Delta m^{2}}>0$ and for two values of the mixing angle in vacuum,
$\theta=0.6$ (dashed curve) and $\theta=0.06$ (solid curve). We see that there
exists an energy for which the enhancement is maximal even for small mixing
angles, but the energy in which the angle is sizeable is correspondingly
small. It is also interesting to follow the dependence of the two Hamiltonian
neutrino eigenstates, $(\nu_{1,2})_{m}$, with the neutrino energy, in the case
$\theta\ll 1$, for example (still assuming $V\cdot{\Delta m^{2}}>0$, so that
the resonance does take place). When the neutrino energy is small,
$(\nu_{1,2})_{m}$ coincide with the mass eigenstates $\nu_{1,2}$. Since for
$\theta\ll 1$ the electron neutrino is close to $\nu_{1}$, we have
$\nu_{e}\approx(\nu_{1})_{m}$. In the opposite limit in which $(2EV/{\Delta
m^{2}})\gg 1$, the first diagonal element in eq. (22) becomes the heaviest and
dominates, which means that $\nu_{e}\approx(\nu_{2})_{m}$. By crossing the
resonance, the electron neutrino moves from the first to the second
Hamiltonian (propagation) eigenstate. The same effect takes place if the
neutrino energy is constant but the matter density, and therefore $V$, varies.
Which may play an important role for the neutrino propagation in matter with
varying density.
The precise relation between the mixing angle and squared mass difference in
vacuum and in matter is given, in the two neutrino case, by the following
formulas:
$\sin^{2}2\theta_{m}=\frac{\sin^{2}2\theta}{1+\displaystyle\left(\frac{2EV}{\Delta
m^{2}}\right)^{2}-2\cos 2\theta\frac{2EV}{\Delta m^{2}}},\quad(\Delta
m^{2})_{m}=\Delta m^{2}\left[1+\left(\frac{2EV}{\Delta m^{2}}\right)^{2}-2\cos
2\theta\frac{2EV}{\Delta m^{2}}\right]^{1/2}.$
We can define a resonant energy $E_{\text{res}}$ by
$\frac{2EV}{\Delta m^{2}}=\frac{E}{E_{\text{res}}}\cos 2\theta,\quad
E_{\text{res}}=\frac{\Delta m^{2}}{2V}\cos 2\theta\approx
8\,\mathrm{GeV}\left(\frac{\Delta m^{2}}{2\cdot
10^{-3}\,\mathrm{eV}^{2}}\frac{n_{e}}{1.65\,\text{gr/cm}^{3}}\right).$ (24)
Note also that
$\frac{(\sin^{2}2\theta)_{m}}{\sin^{2}2\theta}=\left[\frac{\Delta
m^{2}}{(\Delta m^{2})_{m}}\right]^{2}.$ (25)
The equations above show that matter effects are negligible when $E\ll
E_{\text{res}}$ or when $L\ll\lambda_{m}$, where $\lambda_{m}$ is the
oscillation length in matter. In the latter case, in fact, one can approximate
$\sin\phi\approx\phi$, where $\phi$ is the oscillation phase. Matter effects
then cancel because of eq. (25).
Let us now consider propagation in matter with varying density. Let us still
stick to the two neutrino case. The Hamiltonian is time dependent, as the MSW
potential varies during the propagation: $H(t)=H_{\text{free}}+V(t)$. The
exact solution for the evolution of the neutrino wave functions are non-
trivial and have usually to be obtained numerically. There is however one
important case in which the evolution is easy to follow: the adiabatic limit.
In such a limit, the variation of the Hamiltonian is much slower than the
variation of the oscillation phase. As a consequence, a neutrino which at a
certain time is in a given eigenstate of the full Hamiltonian $H(t)$ will
remain, in first approximation, in that eigenstate (which however will vary
together with $H(t)$). Such an adiabatic evolution takes place if
$\frac{d\theta_{m}}{dx}\ll\frac{(\Delta m^{2})_{m}}{2E}$ (26)
during the evolution, where $dx$ represents the variation in the neutrino
position.
An important consequence of the adiabatic evolution is the possibility of
large flavour swaps even for small mixing angles. This may happen if the
neutrino, while traveling, crosses a resonance because of the variation in the
matter density and therefore of $V$. The situation is illustrated in Fig. 6a.
There, the dependence of the two Hamiltonian eigenstates (in units of ${\Delta
m^{2}}/(2E)$) with $V$ ($2EV/{\Delta m^{2}}$) is shown for a small value of
the mixing angle, $\theta=0.06$. In the relevant case, solar neutrino
oscillations, the mixing angle will not be small, but this example better
shows how striking the effect can be. Consider the case in which an electron
neutrino is emitted in a medium with a density high enough that $2EV/{\Delta
m^{2}}\gg 1$ (the inner part of the Sun for example). As we have seen, in such
conditions, $\nu_{e}\approx(\nu_{2})_{m}$. Suppose now the density decreases
during the evolution until is vanishes and the evolution is adiabatic. Once
the neutrino is out of the medium, it will still be in the second eigenstate
of the Hamiltonian. Which in vacuum is
$\nu_{2}=\nu_{e}\sin\theta+\nu_{\mu}\cos\theta$. The probability that the
electron neutrino has become a muon neutrino is therefore
$P(\nu_{e}\to\nu_{\mu})\approx\cos^{2}\theta$. The transition probability
turns out to be close to one even if the mixing angle is small. Such an effect
cannot hold for arbitrarily small mixing angles of course (for $\theta=0$
there cannot be any effect), which means that the adiabatic approximation must
fail when $\theta$ is small enough. This can be seen from eq. (26). The
adiabatic condition is worse at the resonance, where eq. (26) becomes
$\gamma\equiv\frac{\Delta
m^{2}}{2E(V^{\prime}/V)_{\text{res}}}\frac{\sin^{2}2\theta}{\cos 2\theta}\gg
1,$ (27)
where $V^{\prime}$ is the derivative of the MSW potential with respect to the
position. If $\theta$ is small enough, the adiabatic condition at the
resonance is not fulfilled. If the adiabatic condition holds at production and
detection and it fails only in a small region around the resonance, the “level
crossing” probability is given in first approximation by the Landau-Zener
formula
$P(\nu_{1}\rightarrow\nu_{2})\equiv P_{c}\approx e^{-\gamma/2},$ (28)
where $\gamma$ is the adiabaticity parameter in eq. (27). The Landau-Zener
approximation fails in the extreme non-adiabatic regime, $\gamma\ll 1$.
(a) (b)
Figure 6: Dependence of the Hamiltonian eigenstates on the MSW potential for
$\theta=0.06$ (a). Contributions to the solar neutrino flux (b).
#### Solar neutrinos
The discussion of the neutrino evolution in varying density applies to solar
neutrinos. Solar neutrinos are electron neutrinos produced in the burning
process that produces the solar energy:
$4p+2e\to{}^{4}\text{He}+2\nu_{e}+26.7\,\mathrm{MeV}$. The process takes place
through different reactions. Correspondingly, we have different types of solar
neutrinos, characterized by different energy spectra. Among them, we have the
pp neutrinos, from the $pp\to de^{+}\nu_{e}$ reaction, that have by far the
largest flux (which is then well known because it can be derived from a
measurement of the total solar luminosity) but have quite a small energy,
$E<0.42\,\mathrm{MeV}$; the Be neutrinos, from
${}^{7}\text{Be}+e\to{}^{7}\text{Li}+\nu_{e}$, with a significant,
monochromatic flux with $E=0.863\,\mathrm{MeV}$; and B neutrinos, with a small
flux, but a more energetic spectrum, extending up to more than
$10\,\mathrm{MeV}$. The latter are the only ones that can be seen by the SK
and SNO experiments. The different contributions to the solar neutrino flux
are shown in Fig. 6b.
Several experiments have been devised to measure the solar neutrino flux,
starting from the historical Chlorine experiment in the Homestake mine in the
US, by Davis [17], which gave the first evidence of a neutrino deficit,
although with respect to an uncertain theoretical prediction. The latter was a
radiochemical experiment. The neutrino reaction
$\nu_{e}\,\mbox{}^{37}\text{Cl}\rightarrow e\,\mbox{}^{37}\text{Ar}$, with
energy threshold $E>0.814\,\mathrm{GeV}$, was detected by separating the few
tens of atoms of Argon produced by chemical methods and by counting them
through their beta decay back to the initial isotope. Analogous methods were
used in the Gallium experiments (SAGE [33], at the Baksan lake, Russia and
Gallex/GNO [34], at the Gran Sasso laboratories), exploiting the
$\nu_{e}{}^{71}\text{Ga}\to e\,{}^{71}\text{Ge}$ reaction, with threshold
$E>0.233\,\mathrm{MeV}$. Such experiments were not able to measure the time at
which the reaction happened nor the direction of the incoming neutrinos, but
they have the lowest energy thresholds, as shown in Fig. 6b. In particular,
the Gallium experiments are the only ones sensitive to pp neutrinos.
The Super-Kamiokande experiment detects solar neutrinos [35] through elastic
scattering with electrons in the water, $\nu_{e,x}\to\nu_{e,x}e$, where $x$
stands for $\mu$ or $\tau$, with an energy threshold $E>5.5\,\mathrm{MeV}$.
The electron and muon/tau neutrino cross sections are different, with the
latter smaller by a factor 6–7, because the charged current interactions do
not contribute.
The Sudbury Neutrino Observatory (SNO) experiment, near Sudbury, Canada, uses
heavy water, $D_{2}O$, and can detect neutrinos through three types of
processes. Elastic scattering (ES), $\nu_{e,x}e\to\nu_{e,x}e$, involves all
types of neutrinos. Electron and muon/tau neutrinos have different cross
sections, however, as in the SK case. If $\Phi_{e}$ and $\Phi_{\mu+\tau}$ are
the electron and muon/tau neutrino flux reaching the Earth, the ES measurement
determines $\Phi_{e}+0.155\Phi_{\mu+\tau}$. The neutrino direction can be
determined from the electron direction, which allows to tell the solar
neutrinos from the background by their direction. Charged current interactions
(CC), $\nu_{e}D\to ppe$, only involve electron neutrinos and therefore
determine $\Phi(\nu_{e})$. The neutrino energy spectrum can be reconstructed
from the electron one. Neutral current interactions (NC),
$\nu_{x}D\to\nu_{x}pn$, involve all types of neutrinos with equal cross
section. They therefore allow to determine the total neutrino flux
$\Phi_{e}+\Phi_{\mu+\tau}$. The SNO experiment has played for solar neutrino a
role similar to SK for atmospheric neutrinos, to the extent to which it
allowed to obtain a clear evidence of solar electron neutrino transitions,
independent of the theoretical uncertainties on the initial neutrino flux.
This is because the three reactions, ES, CC, and NC, measure three independent
linear combinations of the electron and muon/tau fluxes. It is then possible
to determine (and over-constrain) both fluxes, as shown in Fig. 7a [36]. In
particular, the total neutrino flux reaching the Earth (directly given by the
NC measurement), barring exotic phenomena, determines the initial electron
neutrino flux. From the experimental point of view, the use of heavy water is
necessary in order to obtain neutrino CC interactions (in water only
antineutrinos can interact with the proton in the Hydrogen and the neutrino
interaction with the neutrons in the Oxygen has a too high thresholds).
Chlorine was added in a second phase of the experiment to enhance the neutron
capture cross section, which, through the $\gamma$ produced, is an important
handle to detect the crucial NC processes [37]. Adding ${}^{3}\text{He}$
proportional chambers in the third phase of the experiment further improves
the NC detection, as it allows to see the single neutrons.
The Borexino experiment, at the Gran Sasso laboratories, also uses the elastic
scattering process, as SK and SNO, but is sensitive to lower energy neutrinos,
in particular to the ${}^{7}\text{Be}$ ones, as it uses a scintillator
detector. Such a measurement [38] is important as it constrains the electron
neutrino survival probability for values of the neutrino energy in which
matter effects in the sun are negligible. Moreover, such an experiment was
able to measure “geo-neutrinos” [39], $\overline{\nu}_{e}$ from natural
radioactivity with $E<3\,\mathrm{MeV}$.
#### KamLAND
The Kamioka Liquid-scintillator Anti-Neutrino Detector (KamLAND) experiment,
near the Super-Kamiokande experiment, also plays a crucial role in the
determination of the ${\Delta m^{2}_{12}}$ and $\theta_{12}$ parameters, as
the determination is not obtained by using solar neutrinos, but terrestrial
neutrinos ($\overline{\nu}_{e}$) emitted by several nuclear reactors in Japan.
The neutrino energy is of the order of a few MeV, the average distance
travelled is about 200 km, giving an order one oscillation phase for the
${\Delta m^{2}_{12}}$ oscillation frequency, ${\Delta m^{2}_{12}}L/(4E)\sim
1$. The detection is performed by means of the CC interaction
$\overline{\nu}_{e}p\to e^{+}n$ in the scintillator, with both the electron
and the delayed coincidence with the $\gamma$ signal from the neutron capture
used to observe it. The neutrino energy is directly related to the positron
energy, $E_{\nu_{e}}=E_{e^{+}}+m_{n}-m_{p}$, which allows to measure the
neutrino oscillation probability as a function of the energy and as a
consequence i) to obtain a good ${\Delta m^{2}_{12}}$ determination [40] and
ii) to observe the oscillation pattern, including an oscillation dip, in the
survival probability, as shown in Fig. 7b [40].
(a) (b)
Figure 7: Determination of the electron and muon/tau neutrino fluxes by SNO
(a). Electron antineutrino survival probability at KamLAND (b).
### 0.4.4 The unknown oscillation parameters
The mixing angle $\theta_{13}$ has not been measured yet, but both direct and
indirect bounds have been obtained from the CHOOZ and Minos experiments,
mentioned above, and from the analysis of subleading effects in the
atmospheric and solar neutrino experiments. The result of a global fit on
$\theta_{13}$ are shown in Fig. 8a [23].
The determination of $\theta_{13}$ is important for several reasons. It offers
an handle on the origin of the neutrino (and quark) masses and mixing angles.
In particular, it allows to discriminate among different flavour models. And
it is important for phenomenology, as it is crucial in the study of leptonic
CP-violation, supernova signals, and subleading effects, for example in
$\nu_{\mu}\leftrightarrow\nu_{\tau}$ transitions at the ${\Delta m^{2}_{23}}$
oscillation frequency. From the experimental point of view, a rich
experimental program is available. Several terrestrial experiments are running
or have been planned using different techniques: conventional beams obtained
from pion decays, so called “beta-beams”, obtained from the beta decay of
radio-active ions circulating in a storage ring with long straight sections,
and neutrino factory beams, obtained from the decay of muons also circulating
in a storage ring. A summary of the prospects on the $\theta_{13}$
determination are shown in Fig. 8b for different values of the experimental
parameters [41]. The figure uses the GLoBES package [42, 43]. References for
the single experiments are shown in Figure.
(a) (b)
Figure 8: Result of a global fit of $\theta_{13}$ (a). Future prospects on the
determination of $\theta_{13}$ (b).
Let us now discuss the determination of the sign of ${\Delta m^{2}_{23}}$. I
remind that this parameter determines the pattern of neutrino masses, enters
the analysis of supernova neutrino signals and of long baseline terrestrial
neutrino experiments, and determines the possibility to measure neutrinoless
double beta decay (see below). This parameter can be determined in the
presence of matter effects. Let us consider the three neutrino effective
Hamiltonian for propagation in matter and let us take the ${\Delta
m^{2}_{12}}=0$ limit for simplicity (${\Delta m^{2}_{12}}$ effects are
subleading in the experiments meant to measure $\operatorname{sign}({\Delta
m^{2}_{23}})$):
$H_{\text{eff}}=\frac{1}{2E}\left[U\begin{pmatrix}0&&\\\ &0&\\\ &&{\Delta
m^{2}_{23}}\end{pmatrix}U^{\dagger}\pm\begin{pmatrix}2EV&&\\\ &0&\\\
&&0\end{pmatrix}\right].$ (29)
The relative sign between the two terms on the RHS depends on whether neutrino
or antineutrino oscillations are considered. In the Earth, the MSW potential
is positive, $V>0$. We therefore easily see that if ${\Delta m^{2}_{23}}>0$
the resonant enhancement of oscillations can take place for neutrinos but not
antineutrinos, whereas if ${\Delta m^{2}_{23}}<0$ the enhancement takes place
for antineutrinos. This offers an handle to measure
$\operatorname{sign}({\Delta m^{2}_{23}})$. The neutrino energy should be of
the order of the resonant energy, say $10\,\mathrm{GeV}$, as determined by
$|{\Delta m^{2}_{23}}|$ and $V$. Moreover, a long baseline is needed, so that
the oscillation phase is not small. I remind in fact that for small
oscillation phases $\phi$, the oscillating factor in the probaility can be
approximated as $\sin\phi\approx\phi$ and matter effects cancel (see eq. (25)
and below). Finally, the effect shows up in the
$\nu_{e}\leftrightarrow\nu_{\mu,\tau}$ channel. This can be seen from eq. (29)
by observing that in the limit ${\Delta m^{2}_{12}}=0$ the $\theta_{12}$
rotation in the PMNS matrix is not physical and $U$ can be approximated with a
23 rotation by the angle $\theta_{23}$. In this limit, matter effects do not
affect the $\nu_{\mu}\leftrightarrow\nu_{\tau}$ oscillations. The prospects
for the measurement of the sign of ${\Delta m^{2}_{23}}$ are summarized in Fig
23 of [41].
Let us now discuss the possible determination of the CP-violating phase
$\delta$. We do not need to stress the importance of investigating whether CP-
violation is present not only in the quark sector, but also in the lepton
sector. On top of that, leptonic CP-violation could explain the origin of the
Baryon asymmetry in the universe through the leptogenesis mechanism. Neutrino
oscillation experiments offer the possibility to study leptonic CP-violation
associated to the CP phase $\delta$, which is certainly physical, whether
neutrinos are Dirac or Majorana. The Majorana phases $\alpha$, $\beta$, if
physical, cannot be accessed by oscillation experiments. The CP-violating
phase $\delta$ can determine a difference between the neutrino and
antineutrino oscillation probabilities:
$\displaystyle P(\nu_{e_{i}}\rightarrow\nu_{e_{j}})$
$\displaystyle=P(\overline{\nu}_{e_{j}}\rightarrow\overline{\nu}_{e_{i}})=P_{\text{CPC}}+P_{\text{CPV}}$
(30) $\displaystyle
P(\overline{\nu}_{e_{i}}\rightarrow\overline{\nu}_{e_{j}})$
$\displaystyle=P(\nu_{e_{j}}\rightarrow\nu_{e_{i}})=P_{\text{CPC}}-P_{\text{CPV}}$
(see also eq. (19)). At accelerators experiments aiming at a measurement of
such difference, due to the smallness of ${\Delta m^{2}_{12}}/|{\Delta
m^{2}_{23}}|$ and $\theta_{13}$, we can approximate
$\displaystyle\left.\begin{aligned}
P(\nu_{\mu}\leftrightarrow\nu_{\tau})_{\text{CPC}}&\approx\sin^{2}\theta_{23}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\\\
P(\nu_{e}\leftrightarrow\nu_{\mu})_{\text{CPC}}&\approx\sin^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\\\
P(\nu_{e}\leftrightarrow\nu_{\tau})_{\text{CPC}}&\approx\cos^{2}\theta_{23}\sin^{2}2\theta_{13}\sin^{2}\frac{{\Delta
m^{2}_{23}}L}{4E}\end{aligned}\right\\}+{\Delta m^{2}_{\text{SUN}}}\text{
corr.}$ (31) $\displaystyle\;\;P_{\text{CPV}}=\pm\cos\theta_{13}\sin
2\theta_{12}\sin 2\theta_{23}\sin 2\theta_{13}\sin\delta\,\sin\frac{{\Delta
m^{2}_{12}}L}{4E}\sin^{2}\frac{{\Delta m^{2}_{23}}L}{4E}.$ (32)
The formulas above show that CP-violation has a chance to show up in
$\nu_{e}\leftrightarrow\nu_{\mu}$ oscillations [44]. First of all, two out of
the three angles entering the CP-violating part of the probability in eq. (32)
are large (unlike the quark mixing angles). If the baseline of the experiment
is large enough, the term oscillating with the atmospheric frequency is also
of order one. If the phase $\delta$ is not too small, the CP-violating part of
the probability is then only suppressed by $\sin 2\theta_{13}$ and the solar
phase $\sin({\Delta m^{2}_{12}}L/(4E))$, which are not necessarily too small.
On top of that, the CP-conserving part of the
$\nu_{e}\leftrightarrow\nu_{\mu}$ probability is suppressed by two powers of
$\sin 2\theta_{13}$, whereas the CP-violating part is suppressed by only one
power. This means that the smaller $\sin 2\theta_{13}$, the larger is the
asymmetry between the probabilities in the neutrino and antineutrino channel,
$a_{\text{CP}}=\frac{P(\nu_{e}\rightarrow\nu_{\mu})-P(\overline{\nu}_{e}\rightarrow\overline{\nu}_{\mu})}{P(\nu_{e}\rightarrow\nu_{\mu})+P(\overline{\nu}_{e}\rightarrow\overline{\nu}_{\mu})}\propto\frac{1}{\sin
2\theta_{13}+\text{ corr.}}.$ (33)
A smallish $\theta_{13}$ is therefore not necessarily a curse for CP-violation
[44]. On the one hand, the total number of events decreases with
$\sin^{2}2\theta_{13}$, and therefore the statistical error on the measurement
of the asymmetry increases as $\delta a\sim 1/\sqrt{N}\propto 1/\sin
2\theta_{13}$, where $N$ is the average number of events. On the other hand,
the asymmetry signal also increases with $1/\sin 2\theta_{13}$. The
statistical significance of the measurement, $\delta a/a$, is therefore
approximately constant [45]. Such a behaviour cannot hold for an arbitrarily
small value of $\theta_{13}$, of course. This is indeed the case for two
reasons: i) the corrections in eq. (33) become dominant compared to the $\sin
2\theta_{13}$ term and ii) the number of expected events may become smaller
than one.
Experimentally, the measurement of CP-violation is complicated by the fake
sources of neutrino-antineutrino asymmetry. In particular, one has to consider
the CP-asymmetry of the source, which typically does not emit the same number
of neutrinos and antineutrinos, the CP-asymmetry of the Earth, made of matter
and not antimatter, through which the neutrinos travel, and the CP-asymmetry
of the target. A measurement of CP-violation therefore requires a good
knowledge of the initial neutrino fluxes, of the Earth (electron) density
profile, and of the neutrino cross sections. To cope with such difficulties,
it would be useful to have a measurement of the energy spectrum, two
baselines, and to measure more than a single oscillation channel. Neutrino
factories are especially suited for measuring CP-violation, as the neutrino
flux is very high and quite pure. The prospects for the measurement of the
phase $\delta$ are summarized in Fig 22 of [41].
### 0.4.5 Supernova neutrinos
Supernova neutrinos are emitted during the core collapse of type-II
supernovas. Their study can i) provide further informations on the neutrino
parameters, modulo the uncertainties on the spectrum and intensity of the
source, ii) probe the physics of the collapse, and iii) constrain exotic
neutrino transitions, such as oscillations into sterile neutrinos.
Type-II supernovas originate from the collapse of large stars. The burning
process produces heavier and heavier elements in their core. If the star is
large enough, the gravitational pressure becomes too large to be stood by the
core, and leads to the collapse of the atomic structures. The core, which
before collapse has a radius $R\sim 8000\,$km, a density $\rho\sim
10^{9}\,\text{g/cm}^{3}$ and a temperature $T\sim 0.7\,\mathrm{MeV}$, shrinks
to a proto-neutron star formed by nuclear matter with $R\sim 30\,$km,
$\rho\sim 3\cdot 10^{14}\,\text{g/cm}^{3}$, $T\sim 30\,\mathrm{MeV}$. In the
process, an impressive amount of energy, $E\sim 3\cdot 10^{53}\,$erg,
corresponding essentially to the gravitational energy released, is emitted.
Only about 0.01% of this energy goes into light, about 1% goes into kinetic
energy, and the remaining 99% is emitted through neutrinos. The neutrino
emission is not instantaneous, however. The matter density in the proto-
neutron star is so high that the neutrino mean free path is of the order of 10
cm. The time it takes to the neutrinos to diffuse out is then
$t_{\text{diff}}\sim 3R^{2}/\lambda\sim 10\,$sec. A handful of nupernova
neutrinos where detected when the supernova SN 1987A exploded in the
Magellanic Cloud, 50 kpc away, in 1987. The time distribution of the neutrino
events confirmed the qualitative success of the picture above.
The observation of the neutrino emission constrains the possibility of
invisible, or faster escape channels for the energy to be released. One such
example is neutrino oscillations into sterile neutrinos. If the oscillation
rate was large enough, sterile neutrinos, which do not interact with matter
and would not be trapped inside the core, would immediately escape, carrying
away the neutrino energy. A strong bound on a possible active-sterile mixing
angle follows, $\sin^{2}2\theta_{s}\lesssim 10^{-8}$, which can be evaded if
the sterile neutrino mass is small enough [46, 47, 48, 49, 50]. Such limits
are particularly interesting [51, 52] in the case of neutrinos from extra-
dimension [53, 54, 55, 56, 57, 58, 59, 60]. Other invisible channels
constrained by the observation of supernova neutrinos are the conversion into
axions or into KK gravitons in large extra dimension scenarios.
Supernovas in our galaxy are expected to explode with an uncertain, but not
very exciting rate of about one every 30 years or more. However, if such an
event took place, the present neutrino detectors would gather an impressive
amount of data. A supernova 10 kpc away would produce about 8000 neutrino
events in SK, 800 in SNO, and 330 in KamLAND, thus allowing a detailed study
of the flavour, energy, and time spectrum of the neutrinos reaching us. The
distortions of such spectra compared to the expectations in the absence of
oscillations (which have a significant degree of uncertainty) might provide
information on $\theta_{13}$ and the sign of ${\Delta m^{2}_{23}}$ [61, 62].
### 0.4.6 Anomalous anomalies
While the three neutrino oscillation picture consistently and precisely
explains an impressive amount of experimental data, the results of the LSND
experiment [63, 64] do not fit in the picture. Using a neutrino beam from pion
decay detected in a scintillator, such an experiment found an evidence of
$\overline{\nu}_{\mu}\to\overline{\nu}_{e}$ transitions that, if interpreted
in terms of neutrino oscillations, would require a squared mass difference
larger than the atmospheric one, ${\Delta m^{2}_{\text{LSND}}}>{\Delta
m^{2}_{\text{ATM}}}$. Such a third squared mass difference would require the
introduction of a fourth light neutrino. As we have seen in the introduction,
the number of light “active” neutrinos (i.e. with the gauge interactions of
standard neutrinos) is bound by the measurement of the $Z$ boson width to be
three. The forth neutrino should then be sterile. Such an interpretation poses
a number of problems. From the theoretical point of view, in order to account
for a light sterile neutrino one should explain while an explicit mass term
for it, not forbidden by the electroweak symmetry (unlike the one for active
neutrinos), would be absent or extremely small. This can be however accounted
for by an appropriate symmetry. Moreover, even if the presence of a fourth
neutrino, it is not easy to fit the observed anomaly [65, 66, 67] because of
the bounds from the Karmen [68] and Bugey [69] experiments. The LSND anomaly
is being tested by MiniBOONE, which uses about the same value of $L/E$, but
with $\mathcal{O}\left(10\right)$ larger values of $L$, $E$. MiniBOONE can run
both in a neutrino and antineutrino mode. The present situation is the
following. The neutrino run excludes the LSND signal at more than 90%
confidence level (it observes an anomaly, but at the wrong value of $L/E$, and
in the low energy region that is more sensitive to the backgrounds) [70]. The
antineutrino run, on the other hand, seems to find an excess compatible with
LSND [71].
### 0.4.7 Beyond oscillations
We now discuss the bounds and prospects of determination of the neutrino
parameters that cannot be probed with oscillation experiments,
$m_{\text{lightest}}$, and the Majorana phases $\alpha$, $\beta$ (assuming
they are physical).
As said, the determination of the squared mass differences ${\Delta
m^{2}_{23}}$ and ${\Delta m^{2}_{12}}$ does not determine the absolute value
of neutrino masses. On the other hand, the latter can be obtained from the
additional knowledge of $m_{\text{lightest}}$. We have indeed
$m^{2}_{1}=m^{2}_{\text{lightest}}$, $m^{2}_{2}={\Delta
m^{2}_{12}}+m^{2}_{\text{lightest}}$,
$m^{2}_{3}=m^{2}_{\text{lightest}}+{\Delta m^{2}_{12}}+{\Delta m^{2}_{23}}$ in
the case of normal hierarchy and $m^{2}_{3}=m^{2}_{\text{lightest}}$,
$m^{2}_{2}=m^{2}_{\text{lightest}}-{\Delta m^{2}_{23}}$,
$m^{2}_{1}=m^{2}_{\text{lightest}}-{\Delta m^{2}_{12}}-{\Delta m^{2}_{23}}$ in
the case of inverse hierarchy (in which case ${\Delta m^{2}_{23}}<0$). In
principle, $m_{\text{lightest}}$ can have any value. If
$m_{\text{lightest}}\ll({\Delta m^{2}_{12}})^{1/2}\sim 0.01\,\mathrm{eV}$, the
three neutrinos have masses $m_{3}\approx|{\Delta m^{2}_{23}}|^{1/2}\sim
0.05\,\mathrm{eV}$, $m_{2}\approx({\Delta m^{2}_{12}})^{1/2}\sim
0.01\,\mathrm{eV}$, $m_{1}=m_{\text{lightest}}\ll m_{2}$ in the normal
hierarchy case and $m_{1}\approx m_{2}\approx|{\Delta m^{2}_{23}}|^{1/2}\sim
0.05\,\mathrm{eV}$, $m_{3}=m_{\text{lightest}}\ll m_{1,2}$ in the inverse
hierarchy case (in which case it actually suffices to assume
$m_{\text{lightest}}\ll|{\Delta m^{2}_{23}}|^{1/2}\sim 0.05\,\mathrm{eV}$). If
$m_{\text{lightest}}\gg|{\Delta m^{2}_{23}}|^{1/2}\sim 0.05\,\mathrm{eV}$, the
three neutrinos are approximately degenerate, $m_{1}\approx m_{2}\approx
m_{3}\approx m_{\text{lightest}}$.
Beta decay experiments exploit the fact that a non vanishing neutrino mass
modifies the endpoint of the electron spectrum in beta decays
$(A,Z)\to(A,Z+1)+e^{-}+\overline{\nu}_{e}$, where $A$ and $Z$ are the mass and
atomic number of the decaying atom. This is a purely kinematical effect
illustrated in Fig. 9a (taken from [72]). The tritium decay
${}^{3}\text{H}\to{}^{3}\text{He}+e^{-}+\overline{\nu}_{e}$ is often used for
this purpose. The decay spectrum depends in general on the composition of
$\nu_{e}$ in terms of the three mass eigenstates and on their masses. Given
that the present sensitivities are larger than $|{\Delta m^{2}_{23}}|^{1/2}$,
the spectrum only depends on the combination $(m^{\dagger}m)_{ee}$, where $m$
is the light neutrino mass matrix is the flavour basis:
$\displaystyle\frac{dN}{dE}\propto\sum|U_{eh}|^{2}\Gamma(m^{2}_{h},E)\approx\Gamma(m^{2}_{\nu_{e}},E),$
(34) $\displaystyle
m^{2}_{\nu_{e}}\equiv(m^{\dagger}m)_{ee}=|U_{eh}|^{2}m^{2}_{h}=c^{2}_{13}(m^{2}_{1}c^{2}_{12}+m^{2}_{2}s^{2}_{12})+m^{2}_{3}s^{2}_{13}.$
(35)
The present bound from the Mainz [73] and Troitsk [74] experiments is in the
degenerate neutrino regime, in which $m_{\nu_{e}}\approx m_{\text{lightest}}$,
and give $m_{\text{lightest}}<2.3\,\mathrm{eV}$. The Katrin experiment [75]
promises to lower the sensitivity down to $0.2\,\mathrm{eV}$.
(a) (b)
Figure 9: Modification of the beta decay spectrum in the presence of a non-
vanishing neutrino mass (a). Different beta decay processes and their
microscopic mechanisms (b).
Another handle on the absolute value of neutrino masses is provided by
neutrinoless double beta ($0\nu 2\beta$) decay, $(A,Z)\to(A,Z+2)+2e^{-}$. As
mentioned earlier, $0\nu 2\beta$ signals lepton number violation and is
induced by a Majorana neutrino mass term (see Fig. 9b) at a rate
$\Gamma\propto|m_{ee}|^{2}\langle Q\rangle^{2}$, where $\langle Q\rangle$ is
the matrix element of the hadronic part of the operator inducing the decay and
$m_{ee}=U_{eh}^{2}m_{h}=c^{2}_{13}(m_{1}c^{2}_{12}+m_{2}s^{2}_{12}e^{2i\alpha})+m_{3}s^{2}_{13}e^{2i\beta^{\prime}}$
(36)
is the 11 element of the light neutrino mass matrix in the flavour basis, with
$\beta^{\prime}=\beta-\delta$. The $0\nu 2\beta$ rate therefore probes both
the absolute scale of neutrino masses and the Majorana phases $\alpha$,
$\beta$.
In order to measure the $0\nu 2\beta$ decay, a nucleus $(A,Z)$ for which the
beta decay, but not the double beta one, is kinematically forbidden, is
needed. It is then possible to discriminate the neutrinoless decay from the
standard two neutrino decay $(A,Z)\to(A,Z+2)+2e^{-}+2\nu_{e}$. Indeed, the
latter has a continuous spectrum for the sum of the energies of the two
electrons, with endpoint at the $Q$-value of the decay, while in the former
the energy of the electrons must coincide with the $Q$ value. If the energy
resolution of the electron energy measurement is precise enough, it is then
possible to exclude most of the events due to the standard two neutrino decay.
The determination, or bound, one obtains is however plagued by the
$\mathcal{O}\left(50\%\right)$ uncertainty associated to the matrix element
$\langle Q\rangle$. The Heidelberg-Moscow collaboration, using the
$\mbox{}^{76}\text{Ge}\rightarrow\mbox{}^{76}\text{Se}+2e^{-}$ decay, sets a
limit $|m_{ee}|<\mathcal{O}\left(1\right)\times 0.4\,\mathrm{eV}$ [76]. The
claim of a signal has also been reported by a subgroup of the collaboration
[77]. A rich experimental program is available in this field, with prospects
of lowering the bound down to a few$\times 10^{-2}\,\mathrm{eV}$.
Neutrinos play a role in cosmology through their effect on the Cosmic
Microwave Background (CMB) and the formation of Large Scale Structures in the
universe (LSS). The effect on CMB is due to the fact that the anisotropies in
the photon radiation at decoupling (which takes place at a temperature of
about $0.3\,\mathrm{eV}$) are sensitive to the total radiation density, and in
particular to the energy fraction in neutrinos. In turn, the latter is
determined by the mere sum of the three neutrino masses,
$m_{\text{cosmo}}=m_{1}+m_{2}+m_{3}$, whose knowledge is of course equivalent
to the knowledge of $m_{\text{lightest}}$. The effect on LSS is due to the
fact that the free streaming of relativistic non-interacting particles
smoothes the density fluctuations leading to the large scale structures
observed today. The length scale of the effect depends again on the neutrino
masses.
By fitting the available data on CMB and LSS, it is possible to find an upper
bound on $m_{\text{lightest}}$. However, the latter depends on a number of
assumptions (although plausible and consistent) on the cosmological model. It
is assumed, for example, that the structures are generated by gaussian
adiabatic fluctuations, that the spectral index is constant, that the particle
spectrum is the SM one, that the dark matter is cold and the dark energy is
accounted for by a non-vanishing cosmological constant. Moreover, we note that
the LSS constraint is more powerful but less reliable, as the effect of
neutrino masses is larger at smaller scales, where the numerical simulations
are more difficult. The bound one obtains at 99% confidence level is
$m_{\text{cosmo}}<2.6\,\mathrm{eV}$ when conservatively using the CMB data
only and $m_{\text{cosmo}}<0.5\,\mathrm{eV}$ if the LSS data is also taken
into account [78].
Besides CMB and LSS, neutrinos also affect Big Bang nucleosynthesis (BBN) and
possibly Baryogenesis. The present relative abundance of protons, neutrons,
and light elements is determined during BBN by standard inverse beta reactions
involving neutrinos at their decoupling temperature $T\sim\,\mathrm{MeV}$. The
Baryon asymmetry in the universe is quantified by the number density of
Baryons (minus the negligible density of anti-Baryons), usually normalized to
the photon density, $n_{B}/n_{\gamma}\approx 6\cdot 10^{-10}$. It is believed
that the asymmetry between Baryons and anti-Baryons, $n_{B}>0$, originated
dynamically during the evolution of the universe. On the other hand, the SM of
particle physics cannot account for such a dynamical origin. However, it has
been proposed that the Baryon asymmetry could originate from a lepton
asymmetry generated by the simplest dynamics underlying the origin of neutrino
masses, the see-saw mechanism [79] (see next Section). More specifically, the
idea is that a lepton asymmetry is formed by the CP-asymmetric, out of
equilibrium decay of heavy right-handed neutrinos (transformed into a Baryon
asymmetry by sphalerons) [80, 81, 82]. Although by far not the only one, this
is an economical and successful Baryogenesis mechanism that allows, under
hypotheses, to relate the single number characterizing the Baryon asymmetry to
the neutrino parameters.
The summary of theoretical expectations and bounds on $m_{\text{lightest}}$ is
shown in Fig. 10 [2]. In Fig. 10a the parameter $m_{\nu_{e}}$ probed by beta
decay experiments is plotted as a function of the lightest neutrino mass,
taking into account the present uncertainties on ${\Delta m^{2}_{12}}$ and
$|{\Delta m^{2}_{23}}|$, for the two signs of ${\Delta m^{2}_{23}}$. The bound
from the Mainz and Troitsk experiments are shown together with the expected
bound from Katrin. Fig. 10b shows an analogous plot for the parameter probed
by $0\nu 2\beta$ decay. The darker regions correspond to the uncertainty
associated to the unknown Majorana phases, with the oscillation parameters
fixed at their present central values. The lighter region account for the
additional uncertainty on the oscillation parameters. Finally, Fig. 10c shows
the situation for the parameter probed by cosmology.
(a) (b) (c)
Figure 10: Summary of bounds on $m_{\text{lightest}}$.
## 0.5 Theoretical implications
After having illustrated the phenomenology associated to neutrino masses and
mixings and the determination of the neutrino parameters, we conclude by
discussing the theoretical impact of the information that has been gathered so
far.
From the theoretical point of view, the relevant information emerging from the
data in eq. (16) can be summarized as follows:
$\begin{gathered}m_{\nu_{i}}\ll 174\,\mathrm{GeV}\\\\[2.84526pt]
\theta_{23}\sim 45^{\circ}\text{(= $45^{\circ}$?)}\qquad\theta_{12}\sim
30^{\circ}\text{--}35^{\circ}\neq
45^{\circ}\qquad\theta_{13}<7^{\circ}\\\\[1.42262pt] |{\Delta
m^{2}_{12}}/{\Delta m^{2}_{23}}|\approx 0.035\ll 1.\end{gathered}$ (37)
The most important theoretical guideline is the smallness of neutrino masses.
We then have the surprising fact that two out of three mixing angles turn out
to be large, unlike what found in the quark sector. In particular,
$\theta_{23}$ is compatible with being maximal. While the present uncertainty
is too large to draw conclusions, it would be interesting to know whether
$\theta_{23}$ is indeed maximal (i.e. $45^{\circ}$ up to small corrections) or
just large (i.e. $\mathcal{O}\left(1\right)$). A maximal angle would in fact
be an indication of a non-trivial flavour structure [83]. As for the solar
angle, we know that it is definitely not maximal, although compatible with the
so called tri-bimaximal prediction [84]. The squared mass difference hierarchy
implies that the ratio $m_{2}/m_{3}$ is about 0.2 or larger, not as small as
the typical charged fermion mass hierarchy. In the following, we will
concentrate on the first guideline, the smallness of neutrino masses, and its
implications for the origin of neutrino masses.
There is no doubt that neutrino masses are indeed very small compared to the
natural scale of fermion masses, the electroweak scale, $v=174\,\mathrm{GeV}$:
$m_{\nu}/v\lesssim 10^{-12}$. On the other hand, some of the charged fermion
masses are also quite small compared to $v$, the smallest being the electron
mass, suppressed by a factor $m_{e}/v\approx 0.3\cdot 10^{-5}$. Still, the
smallness of neutrino masses seems to be peculiar. Not only because twelve
orders of magnitude are more than five. Also because all the three families of
neutrinos are bound to be that light. On the contrary, the suppression of the
lightest charged fermion masses seems to be related to the hierarchy among
different families, the heaviest families being suppressed compared to the
electroweak scale at most by a couple of orders of magnitude. Moreover, there
is a compelling explanation for the peculiar smallness of neutrino masses, as
we now see.
### 0.5.1 The origin of neutrino masses
We have seen in Section 0.3.1 that the observed smallness of neutrino masses
is not explained at the level of the effective theory below the electroweak
scale: the QED and QCD gauge symmetries allow a mass term for both the charged
fermions and the neutrinos. Things are different when considering the full SM
gauge symmetry
$G_{\text{SM}}=\text{SU(3)}_{c}\times\text{SU(2)}_{L}\times\text{U(1)}_{Y}$.
It is well known that such a symmetry forbids any fermion mass term, both for
charged fermions and neutrinos. In order to see that, it suffices to show that
no gauge invariant mass term in the form eq. (6) can be written for the left
handed fermion fields. The latter transform under the $\text{SU(2)}_{L}$ gauge
symmetry either as doublets, $L_{i}=(\nu_{iL},e_{iL})^{T}$,
$Q_{i}=(u_{iL},d_{iL})^{T}$, or as singlets, $\overline{e_{iR}}$,
$\overline{u_{iR}}$, $\overline{d_{iR}}$. Their hypercharges are -1/2, 1/6, 1,
-2/3, 1/3 respectively. It is then easy to see that it is not possible to
combine any two such left handed fermions in a gauge invariant combination.
The fact that no fermion mass term is allowed in the limit in which the
electroweak symmetry is unbroken can be rephrased by saying that the SM is a
“chiral” theory. This property might be the very reason why the fermions we
observe have a mass so much smaller than, say, the Planck scale: they are
protected by the electroweak symmetry. As a consequence, a SM fermion mass has
to be proportional to at least one power of $v$. Here is the crucial
difference between charged fermions and neutrinos: while the charged fermion
mass term arises proportional to one power of $v$, the neutrino mass term
needs at least two powers of $v$. Let us see why this is the case and what are
the consequences.
In the SM the electroweak symmetry is broken by the vacuum expectation value
(vev) of the Higgs field. The Higgs is a complex scalar field that transforms
as a doublet under SU(2)L, $H=(h^{+},h^{0})$, and has hypercharge 1/2. The
Higgs potential is such that the value of the (neutral component of the) field
in the ground state does not vanish. Such a vev is denoted by $\langle
h^{0}\rangle$ and provides the value of the electroweak scale, $\langle
h^{0}\rangle=v$. The electroweak symmetry is broken because the ground state
value of the Higgs is not invariant under electroweak gauge transformations.
While mass terms for the SM fermions are not allowed, Yukawa interactions with
the Higgs are. For example, the electron can interact with the Higgs through
the gauge invariant interaction
$\lambda_{E}\overline{e_{R}}L\,H^{\dagger}=\lambda_{E}\overline{e_{R}}(e_{L}h^{0}+\nu_{L}h^{+})$,
where family indexes have been understood. Once the Higgs field is expressed
in terms of the displacement from the ground state value, $h^{0}=v+\delta
h^{0}$, a mass term is generated for the electron in the form
$m_{E}\overline{e_{R}}e_{L}$, with $m_{E}=\lambda_{E}v$.
The electron Yukawa interaction above has the property of being
“renormalizable”. For our purposes, this means that the coupling in front has
a non negative dimension in mass (it is in fact dimensionless). Note that a
lagrangian (density) has dimension 4 in energy. Therefore, renormalizable
operators have dimension 4 or less. Non-renormalizable terms are instead
characterized by coefficients proportional to inverse powers of a mass scale,
or cut-off, $\Lambda$. Such terms are thought not to be fundamental but to
arise as remnant, effective terms from a more fundamental (possibly
renormalizable) theory living at a scale related to $\Lambda$. They are indeed
inconsistent (at least perturbatively) at energies larger than $\Lambda$. A
dimension $4+n$ non-renormalizable operator is suppressed by $n$ powers of
$\Lambda$.
One can then wonder what is the most general form of the renormalizable Yukawa
interactions of the SM fermions with the Higgs, and which is the most general
fermion mass term that can be generated at the renormalizable level. The
answer is provided by the SM flavour lagrangian
$\displaystyle\mathcal{L}_{\text{SM}}^{\text{flavor}}$
$\displaystyle=\lambda^{E}_{ij}\overline{e_{iR}}L_{j}H^{\dagger}$
$\displaystyle+\lambda^{D}_{ij}\overline{d_{iR}}Q_{j}H^{\dagger}$
$\displaystyle+\lambda^{U}_{ij}\overline{u_{iR}}Q_{j}H$
$\displaystyle+\text{h.c.}$ (38) $\displaystyle=m^{E}_{ij}e^{c}_{i}e_{j}$
$\displaystyle+m^{D}_{ij}d^{c}_{i}d_{j}$
$\displaystyle+m^{U}_{ij}u^{c}_{i}u_{j}$ $\displaystyle+\text{h.c.}+\ldots,$
with
$m^{E}_{ij}=\lambda^{E}_{ij}v\quad m^{D}_{ij}=\lambda^{D}_{ij}v\quad
m^{U}_{ij}=\lambda^{U}_{ij}v\quad m^{\nu}_{ij}=0.$ (39)
We therefore see that, unlike charged fermion masses, neutrino masses do not
arise in the SM even after electroweak symmetry breaking, if one sticks to
renormalizable interactions. This is a good starting point to understand the
smallness of neutrino masses. Of course, we have in the end to account for the
fact that neutrino masses are not vanishing. In order to do that, some
ingredient has to be added to the SM as a renormalizable theory. While there
are certainly several possibilities, we can distinguish two main options.
Either the new ingredients live at a scale $\Lambda\gg v$ (the standard
example being the addition of heavy right-handed neutrinos giving rise to the
see-saw mechanism) or the new ingredients live at a scale $\Lambda\lesssim v$
(the standard example being Dirac neutrinos). Let us consider the two
possibilities in turn.
### 0.5.2 $\Lambda\gg v$
This case is particularly interesting. It can be described in a model
independent way by making use of a central theorem of effective field theory.
At the electroweak scale and below, the effect of whatever are the additional
heavy degrees of freedom to be added in order to account for neutrino masses,
can be described in terms of effective interactions involving only light
degrees of freedom and symmetries. In particular, we do not need to know the
specific form of the high energy theory in order to parameterize its effect at
the electroweak scale in a model independent way. Such effective interactions
are non-renormalizable, i.e. suppressed by powers of the scale $\Lambda$ at
which they arise. We therefore have
$\mathcal{L}_{E\ll\Lambda}^{\text{eff}}=\mathcal{L}_{\text{SM}}^{\text{ren}}+\mathcal{L}_{\text{SM}}^{\text{NR}},$
(40)
where $\mathcal{L}_{\text{SM}}^{\text{ren}}$ is the renormalizable SM
lagrangian and $\mathcal{L}_{\text{SM}}^{\text{NR}}$ accounts for the
effective interactions.
The effect at energies $E\ll\Lambda$ of dimension $4+n$ effective interactions
arising at the scale $\Lambda$ is suppressed by $(E/\Lambda)^{n}$, as it can
be inferred from simple dimensional analysis. As a consequence, the most
relevant effective operators are those with lowest dimension: $4+1$. It turns
out that in the SM it is possible to write only one such dimension 5 operator:
$\mathcal{L}^{\text{NR}}_{\text{SM}}=\frac{h_{ij}}{2\Lambda}(HL_{i})(HL_{j})+\text{higher
dimension},$ (41)
where SU(2)L invariant contractions are understood. Note that separating the
coefficient of the operator in eq. (41) in a dimensionless numerator $h_{ij}$
and a dimensionful denominator $\Lambda$ is purely conventional. When doing
that, we are implicitly identifying with $\Lambda$ the scale at which the
degrees of freedom generating the operator live and with $h_{ij}$ the
combination of dimensionless couplings, loop factors, etc. entering the
determination of the operator.
Once the electroweak symmetry is broken and the neutral Higgs component
acquires a vev, a neutrino mass term is generated,
$m^{\nu}_{ij}=h_{ij}v\times\frac{v}{\Lambda}.$ (42)
We therefore see that, unlike charged fermions, neutrino masses turn out in
this context to be proportional to two powers of the electroweak symmetry
breaking scale, and therefore to be suppressed by a factor $v/\Lambda$
compared to the charged fermion masses, as a consequence of their
(unspecified) origin at the high scale $\Lambda$. The smallness of neutrino
masses is then understood in terms of the heaviness of the scale $\Lambda$ at
which they originate (and at which lepton number if broken). It is also
possible to invert the relation in eq. (42) to obtain an estimate of the scale
$\Lambda$ in terms of the measured value of the neutrino masses:
$\Lambda\sim 0.5\times
10^{15}\,\mathrm{GeV}h\left(\frac{0.05\,\mathrm{eV}}{m_{\nu}}\right).$ (43)
As the coupling $h$ cannot be much larger than 1, $\Lambda$ has to be of the
order or smaller than $10^{15}\,\mathrm{GeV}$. Still, $\Lambda$ might be not
too far from the GUT scale. Neutrino masses open in this context an indirect
window on scales that could never be probed directly.
Let us summarize the results of the discussion so far. The smallness of
neutrino masses can be economically understood in a model-independent way in
terms of the heaviness of the scale at which lepton number is violated. What
makes them special is the fact that they are the only fermions in the SM for
which a mass does not arise (after EWSB) from a renormalizable interaction
with the Higgs field. They turn out to be Majorana.
Such an understanding is very appealing, but is based on the fact that, unlike
the other charged fermions, neutrinos were not given a right-handed component.
In the presence of a right-handed component $\nu_{R}$, it would be possible to
write a renormalizable neutrino Yukawa interaction with the Higgs,
$\lambda^{N}_{ij}\overline{\nu_{iR}}L_{j}\,H$, providing neutrino masses
proportional to the electroweak scale, just as for all the other fermions. The
real question might then appear to be: why neutrinos should not have a right-
handed components, as all the other fermions? What makes them special from
this point of view?
The answer is simple. In order to make the neutrino Yukawa interaction gauge
invariant, the right-handed neutrinos should be neutral under all SM
interactions222To be precise, they could also be in a triplet of SU(2)L.. Then
they would be the only fermions for which an explicit mass term in the form in
eq. (6) would be allowed:
$\frac{M_{ij}}{2}\overline{\nu_{iR}}\,\overline{\nu_{jR}}.$ (44)
Unlike all other fermions, their mass would not be bound to vanish in the
limit in which the electroweak symmetry is unbroken, would not be bound to be
proportional to powers of the electroweak scale. Therefore, right-handed
neutrinos would be the only fermions for which a mass much larger than $v$
would be allowed. In which case, according to the effective theory theorem
mentioned above, their effect at the electroweak scale and below, can be
described in terms of the effective interaction in eq. (41). Indeed,
neglecting the momentum in the right-handed neutrino propagator in the “see-
saw” [85, 86, 87, 88, 89] diagram in Fig. 11a, one obtains the effective
interaction in eq. (41), with
$\frac{h}{\Lambda}=-\lambda_{N}^{T}\frac{1}{M}\lambda_{N}\quad\Rightarrow\quad
m_{\nu}=-m_{\text{D}}^{T}\frac{1}{M}m_{\text{D}},\quad\text{where}\quad
m_{D}=\lambda_{N}v.$ (45)
This is the celebrated see-saw formula. In this context, it turns out to be
just an example, probably the simplest, of heavy physics generating the
operator in eq. (41).
Figure 11: The see-saw diagram.
It turns out that there are only three possible types of heavy degrees of
freedom that can be exchanged at the tree level to generate the operator in
eq. (41): the exchange of SM-singlet fermions (type I see-saw, or just see-
saw), of a hypercharge -1 SU(2)L scalar triplet [90, 91, 92, 93] (type II see-
saw), or of a zero hypercharge SU(2)L fermion triplet [94] (type III see-saw).
### 0.5.3 $\Lambda\lesssim v$
While the $M\gg v$ option is very appealing, as it provides a solid,
economical, compelling understanding of the smallness of neutrino masses, only
based on the hypothesis that the new ingredients needed to account for
neutrino masses are heavier than the electroweak scale, the possibility that
such new ingredients are lighter than $v$ cannot be excluded. In this case,
the general effective description we used in the previous case does not hold
and each case should be considered separately.
A paradigmatic example is provided by Dirac neutrinos. As discussed above,
such neutrinos have a right-handed component, but their mass terms, which
could in principle be as heavy as the Planck scale, is assumed to be much
smaller than the neutrino masses themselves. The latter then arise from the
Yukawa interaction $\lambda^{N}_{ij}\overline{\nu_{iR}}L_{j}\,H$, as for the
other fermions, and turn out to be in the Dirac form (as in eq. (8) with
$m^{L}=m^{R}=0$), with $m_{N}=\lambda_{N}v$.
Such a possibility requires the Majorana mass term for the right-handed
neutrinos in eq. (44) to be smaller than the Planck scale by almost 30 orders
of magnitude and the $\lambda_{N}$ entries to be all smaller than about
$10^{-11}$. Lepton number conservation can force $M_{ij}=0$. We would then
only have to cope with very small Yukawas or to account for their smallness
with appropriate mechanisms [95, 96, 97]. In any case, it is fair to say that
additional structure must be added to explain what the simplest option,
$\Lambda\gg v$, gives for free.
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|
arxiv-papers
| 2012-01-30T10:29:11 |
2024-09-04T02:49:26.784989
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrea Romanino (SISSA, Trieste and INFN, Trieste)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1201.6158"
}
|
1201.6164
|
11institutetext: Physics Department, University of Helsinki, and Helsinki
Institute of Physics, Finland
# Cosmological inflation
K. Enqvist
###### Abstract
The very basics of cosmological inflation are discussed. We derive the
equations of motion for the inflaton field, introduce the slow-roll
parameters, and present the computation of the inflationary perturbations and
their connection to the temperature fluctuations of the cosmic microwave
background.
## 0.1 Introduction
Cosmological inflation is, by definition, a period of superluminal expansion
in the very early universe[1]. In practice, the rate of expansion is usually
taken to be (quasi)exponential. Superluminal expansion does not contradict the
theory of relativity, which states that no _signal_ can propagate faster than
the speed of light. One cannot use the expansion of space to send any signal,
and as we will see, superluminal expansion is indeed a solution to the field
equations of general relativity.
Historically, inflation was introduced to solve certain fine-tuning issues in
the hot Big Bang scenario. However, its main attraction turned out to provide
a mechanism for the origin of the density perturbations required for structure
formation. Density perturbations leave their imprint on the cosmic microwave
background (CMB) and were first detected by the COBE satellite [2].
The simplest starting point for cosmology is to assume that the universe is
homogeneous and isotropic. It can be shown that the most general metric
describing such a Friedmann–Robertson–Walker (FRW) universe reads
$ds^{2}=dt^{2}-a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ k=\pm 1,0\leavevmode\nobreak\ ,$
(1)
where $k$ is the spatial curvature parameter. Here the convention is such that
today the scale factor $a(t_{\rm now})=1$. Within the FRW framework, matter is
homogeneous and continuous. Therefore, it can be described by the energy-
momentum tensor of a perfect fluid, given by
$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}\leavevmode\nobreak\ .$ (2)
Here $\rho$ is the energy density and $p$ the pressure of the fluid. In the
rest frame of the fluid, where the four-velocity is given by $u=(1,0)$, we
find $T_{00}=\rho,\leavevmode\nobreak\ T_{ii}=p$. One then substitutes the
metric (1) and the energy-momentum tensor (2) into the Einstein equations,
which yield evolution equations for $a(t)$ and the components of the energy-
momentum tensor:
$\displaystyle H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}\rho-\frac{k}{a^{2}}\leavevmode\nobreak\ ,$ (3)
$\displaystyle\dot{\rho}+3H(p+\rho)$ $\displaystyle=$ $\displaystyle
0\leavevmode\nobreak\ ,$ (4)
where $G$ is the gravitational constant and $H$ is the Hubble parameter. These
must be supplemented by an equation of state $p=w\rho$, whence it follows from
(4) that $\rho\propto a^{-3(w+1)}$. For cold dust, $w=0$; for relativistic
particles (radiation), $w=1/3$.
If $k=0$, the geometry is said to be flat. By virtue of (3), in such a case
the energy density is directly related to the Hubble rate:
$\rho=\frac{3H^{2}}{8\pi G}\equiv\rho_{c}\leavevmode\nobreak\ ,$ (5)
where $\rho_{c}$ is called the critical density. Usually the densities are
written in units of the critical density and are then denoted by
$\Omega=\rho/\rho_{c}\leavevmode\nobreak\ .$ (6)
From the Friedmann equation (3) we then find that the difference from the
critical density evolves as (with $a_{\rm now}=1$)
$\frac{(\Omega-1)_{\rm now}}{(\Omega-1)}=\frac{H^{2}a^{2}}{H^{2}_{\rm
now}}\leavevmode\nobreak\ .$ (7)
Let us assume $a(t)\propto t^{q}$ so that $H\propto q/t$. For (adiabatic)
radiation domination, $q=1/2$, while for matter domination $q=2/3$. Then we
find that at earlier times
$(\Omega-1)=\left(\frac{t}{t_{\rm now}}\right)^{2(1-q)}(\Omega-1)_{\rm
now}\leavevmode\nobreak\ .$ (8)
According to the WMAP observations [3], today $\Omega$ differs from 1 by at
most a few per cent. Hence (8) implies a considerable fine-tuning of the
initial value of $\Omega$. For example, let us consider the electroweak phase
transition at $t_{EW}\simeq 10^{-11}$ s and assume for simplicity radiation
domination before $t=380\ 000$ yrs followed by matter domination up to $t_{\rm
now}=13.7\times 10^{9}$ yrs. Then one finds that $\Omega-1$ should have been
smaller than today by a factor of $7.5\times 10^{-28}$. The existence of dark
energy does not change the conclusion in any qualitative way.
Such fine-tuning obviously calls for a dynamical explanation, a process that
would automatically yield the initial condition $\Omega=1$. Cosmological
inflation solves this problem by assuming that in the very early universe, the
energy of the vacuum $\rho_{\Lambda}$ was constant and much bigger than any
other energy form, including the curvature. A universe with constant vacuum
energy is called the de Sitter universe, where the Friedmann equation (3)
would read
$H^{2}=H_{0}^{2}-\frac{k}{a^{2}}\leavevmode\nobreak\ ,$ (9)
with $3H_{0}^{2}=8\pi G\rho_{\Lambda}$. The solution to (9) at late times
behaves as $a(t)=a(t_{0})\exp(H_{0}t)$. The curvature terms $k/a^{2}\to 0$
while $\Omega-1\propto\exp(-2H_{0}t)$. Eventually the vacuum energy should
decay, but if the lifetime is long enough, the initial condition problem is
solved as $\Omega$ is driven exponentially close to 1.
Vacuum energy solves also another initial-condition problem, the homogeneity
of the CMB. We see photons arriving at us from every direction with
temperatures that are almost exactly equal. They were created at the (somewhat
illogically named) recombination at redshift $z\simeq 1100$ when the ambient
plasma became transparent to photons. The problem is that at the recombination
time there were thousands of regions that apparently had not had time to be in
causal contact. This can be seen as follows. Photons travel along geodesics
$ds^{2}=0$, and taking the flat ($\Omega=1$) FRW space for simplicity, we thus
write (defining the $x$-axis to lie along the photon trajectory)
$ds^{2}=dt^{2}-a^{2}dx^{2}=0$. The physical distance a photon travels is then
given by
$d_{\gamma}(t)=a(t)\int
dx=a(t)\int_{0}^{t}dta^{-1}=\frac{t^{q}t^{1-q}}{1-q}\leavevmode\nobreak\ ,$
(10)
where we have again assumed that $a\propto t^{q}$. A causal volume at the
recombination time $t_{RC}=380\ 000$ yrs is $V\sim d_{RC}^{3}=(2t_{RC})^{3}$.
The volume of the visible universe today is roughly (neglecting dark energy)
$V_{U}\sim(t_{\rm now}/(1-\frac{2}{3}))^{3}$ and at recombination was smaller
by a factor of $z^{3}$ but still much larger than a single causal volume. Thus
one finds that at recombination our universe contained about $10^{5}$ acausal
regions.
Let us observe that in contrast to (10), in the de Sitter universe photons
travel the physical distance
$d_{\gamma}(t)=a(t)\int dx\simeq\frac{1}{H_{0}}e^{H_{0}t}\leavevmode\nobreak\
\leavevmode\nobreak\ ,t\to\infty,$ (11)
so that two observers separated by an initial distance $d_{i}$ will at late
times find themselves at a distance $d(t)=d_{i}e^{H_{0}t}$. They cannot
communicate if $d(t)>d_{\gamma}(t)$ or if $d_{i}>{1}/{H_{0}}\equiv d_{H}$.
Thus de Sitter space has an event horizon, a radial distance from the observer
beyond which no information can be gathered. The horizon appears because the
expansion is superluminal: photons cannot keep up with the exponential
expansion rate.
Assume now that the de Sitter expansion stops at $a_{\rm end}$ and, in the
sudden decay approximation, the vacuum energy $\rho_{\Lambda}$ is converted
instantaneously to radiation with
$\rho_{\Lambda}=\frac{3H_{0}^{2}}{8\pi G}\sim T_{0}^{4}\leavevmode\nobreak\ ,$
(12)
where $T_{0}$ is the temperature of the universe after the conversion. This
would then mean the beginning of the ‘normal’ Big Bang expansion with $a\sim
t^{1/2}$. However, the preceding superluminal expansion has not been without
consequences:
* •
two points $A$ and $B$ initially in causal contact
($d_{AB}(t_{0})<H_{0}^{-1}$) will lose causal contact because of the
exponential expansion when $d_{AB}(t)>H_{0}^{-1}$ ($A$ is said to have left
$B$’s horizon);
* •
once ‘normal’ hot Big Bang expansion begins, the horizon grows as $d_{H}\sim
t$ while $d_{AB}(t)\sim t^{q}$ with $q<1$ so that eventually
$d_{H}>d_{AB}(t)$; hence $A$ and $B$ will again come into causal contact ($A$
is said to re-enter $B$’s horizon).
Thus, according to inflation, while it seems that there are acausal regions in
the CMB sky, they actually have been in causal contact in the very beginning.
We should require that every point in the CMB sky as seen today had initially
been in causal contact; i.e., the size of the visible universe at the
beginning of inflation is $d_{U}<H_{0}^{-1}$. It has then been stretched by
expansion of the universe as
$d_{U}=\frac{a_{\rm now}}{a_{\rm
end}}e^{H_{0}\tau}H_{0}^{-1}=\frac{T_{0}}{T_{\rm now}}e^{N}H_{0}^{-1}$ (13)
where $\tau$ is the duration of inflation and $N=H_{0}\tau=\ln(a(t_{\rm
end})/a_{i})$ is the number of e-folds (here we neglect the details of the
expansion history). The actual observed size of the universe today is very
roughly $d_{\rm now}\sim 3/2H_{\rm now}$ (this does not account for dark
energy) so that requiring $d_{U}>d_{\rm now}$ translates into
$N>\ln\left[\frac{d_{\rm now}T_{0}}{d_{U}T_{\rm now}}\right]\sim 60$ (14)
for reference values $T_{0}\sim 10^{15}$ GeV, $T_{\rm now}\simeq 3$ K, $H_{\rm
now}\simeq 70$ km s-1/Mpc. Thus, if inflation lasted more than about 60
e-folds, both the fine-tuning of $\Omega$ and the homogeneity of CMB find a
natural solution. Note that because of the exponential expansion, during
inflation the universe becomes essentially empty.
## 0.2 The inflaton
### 0.2.1 Equation of state of a scalar field
If vacuum energy dominates, from the continuity equation (4) we find the
equation of state
$p\simeq-\rho\leavevmode\nobreak\ .$ (15)
Eventually the vacuum energy should decay and provide us with the beginning of
the hot Big Bang. The simplest way to achieve this is to assume the existence
of a singlet scalar field $\phi$, called the inflaton, with a potential
$V(\phi)$ which is of a specific type. In curved spacetime a scalar field
theory is given by the action
$S=\int d^{4}x\sqrt{-g}{\cal L[\phi]}\leavevmode\nobreak\ ,$ (16)
where $g={\rm det}\ g_{\mu\nu}=-a^{6}$ for the flat FRW metric (1), while the
Lagrangian reads
${\cal
L}=\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{a^{2}}(\nabla\phi)^{2}-V(\phi)\leavevmode\nobreak\
.$ (17)
The energy-momentum tensor for a general Lagrangian reads
$T^{\mu\nu}=\frac{\partial{\cal
L}}{\partial(\partial_{\nu}\phi)}\partial^{\mu}\phi-g^{\mu\nu}{\cal L}$ (18)
so that by treating the scalar field as a perfect fluid, we find from (2) that
$\rho=T_{00}=\dot{\phi}^{2}-\left[\frac{1}{2}\left(\dot{\phi}^{2}-\frac{1}{a^{2}}(\nabla\phi)^{2}\right)-V\right]=\frac{1}{2}\dot{\phi}^{2}+\frac{1}{a^{2}}(\nabla\phi)^{2}+V\leavevmode\nobreak\
.$ (19)
We define comoving pressure $p$ as
$\displaystyle a^{2}p$ $\displaystyle=$ $\displaystyle
T_{kk}=(\partial_{k}\phi)^{2}+a^{2}\left[\frac{1}{2}\left(\dot{\phi}^{2}-\frac{1}{a^{2}}(\nabla\phi)^{2}\right)-V\right]$
(20) $\displaystyle=$ $\displaystyle
a^{2}\left(\frac{1}{2}\dot{\phi}^{2}-V\right)-\frac{1}{6}(\nabla\phi)^{2}\leavevmode\nobreak\
,$
where the factor $1/6$ comes about because we assume an isotropic universe.
Note that the spatial gradients in $\rho$ and $p$ tend to die away with
expansion; this (to some extent) justifies the assumption of a flat metric in
(17). Thus we find that
$\frac{p}{\rho}\simeq\frac{\frac{1}{2}\dot{\phi}^{2}-V}{\frac{1}{2}\dot{\phi}^{2}+V}\simeq-1$
(21)
if $\dot{\phi}^{2}\ll V$. In other words, if the kinetic energy of the scalar
field is much smaller than the potential energy, one can have a de Sitter-like
period of exponential expansion.
The equation of motion for $\phi$ is the Euler-Lagrange equation:
$0=\frac{\partial\sqrt{-g}{\cal
L}}{\partial\phi}-\partial_{\mu}\frac{\partial\sqrt{-g}{\cal
L}}{\partial\partial_{\mu}\phi}=\ddot{\phi}+3H\dot{\phi}-\frac{1}{a^{2}}\nabla^{2}\phi+V^{\prime}\leavevmode\nobreak\
.$ (22)
If we require $\dot{\phi}^{2}\ll V$, then we should also require
$\dot{\phi}\ddot{\phi}\ll\dot{V}=V^{\prime}\dot{\phi}$ or $\ddot{\phi}\ll
V^{\prime}$. This is called _slow roll_.
### 0.2.2 Motion of the inflaton
Let us now assume that the inflaton gradients can be neglected and we may
focus on the homogeneous field $\phi=\phi(t)$. The equation of motion (22)
reads then
$\ddot{\phi}(t)+3H\dot{\phi}(t)+\Gamma\dot{\phi}(t)+V^{\prime}(\phi(t))=0$
(23)
where we have added by hand the decay width $\Gamma$. (In field theory decay
width is the imaginary part of self-energy: $\Gamma=2\ {\rm Im}\ E$.) Decay
can begin only when $\Gamma\gtrsim H(t)$. Let us assume that near $\phi\simeq
0$ we may expand the potential as
$V=V_{0}-a\phi-b\phi^{2}-\dots$ (24)
where $a$ and $b$ area some parameters. The potential is very flat if
$a,\leavevmode\nobreak\ b$ are very small. It then follows that the Hubble
rate is given by $H^{2}=H_{0}^{2}\simeq 8\pi GV_{0}/3$, where $H_{0}$ is
constant. Assuming that initially $\phi\simeq 0$, $\dot{\phi}\simeq 0$ we find
that field motion is slow with $\ddot{\phi}\ll|V^{\prime}|$, whence the
equation of motion can be written simply as
$3H_{0}\dot{\phi}\simeq-V^{\prime}\leavevmode\nobreak\ .$ (25)
Around the true vacuum $\phi=\phi_{*}$ we may write the equation of motion as
(assuming again that $\Gamma\ll H(t)$)
$\ddot{\phi}+3H\dot{\phi}+\frac{1}{2}m_{\phi}^{2}(\phi-\phi_{*})^{2}\simeq
0\leavevmode\nobreak\ ,$ (26)
where $m_{\phi}$ can be called the physical mass of the inflaton, which
typically is much bigger than $V^{\prime\prime}$ during inflation. Writing
$\xi=A(t)(\phi-\phi_{*})$ one finds for the amplitude $A(t)\propto a^{-3/2}$,
while averaging over one oscillation cycle the mean pressure is found[4] to be
$\langle p\rangle=0$. Hence a harmonically oscillating field behaves as cold
matter with $\rho\propto a^{-3}$ and $a\propto t^{2/3\leavevmode\nobreak\ }$.
The inflaton oscillations will continue until $H\simeq\Gamma$, whence the
inflaton starts to decay into some (relativistic) particles. The decay
products will eventually thermalize among themselves. The energy density
stored in the oscillations is transformed at $t=t_{\rm dec}$ into radiation so
that [see also (12)]
$\rho_{\rm thermal}=\frac{g_{*}\pi^{2}}{30}T_{\rm RH}^{4}=\rho_{\rm
end}\left(\frac{a_{0}}{a(t_{\rm dec})}\right)^{3}\leavevmode\nobreak\ ,$ (27)
where $\rho_{\rm end}\simeq V_{0}$ is the inflaton energy density at the end
of inflation and $T_{\rm RH}$ is the _reheat temperature_. The decay time is
given by the condition $t_{\rm dec}=\Gamma^{-1}=H(t_{\rm dec})$.
Thus we arrive at the following scenario:
* •
initially the inflaton is in the flat part of the potential, slowly rolling;
the universe is expanding (quasi)exponentially;
* •
as the field gathers speed, slow-roll conditions no longer hold and inflation
ends;
* •
inflaton begins to oscillate about the true minimum $\phi_{*}$; the universe
is expanding as if dominated by cold matter;
* •
when $H\simeq\Gamma$, the inflaton starts to decay and (re)heats the universe,
which then starts expanding in a radiation-dominated phase
The decay of the inflaton field can in principle proceed also in a non-
perturbative manner through so-called parametric resonance [5]. By coupling
the inflaton to other scalar fields, one may arrange for a situation where the
decay products are generated in bursts while the inflaton oscillates past the
minimum of the potential. This results in a much more efficient reheating than
the conventional perturbative decay.
### 0.2.3 Slow-roll parameters and the number of e-folds
It is convenient to define slow-roll parameters that characterize the inflaton
potential during inflation. The spectral index of the perturbations can also
be expressed in terms of the slow-roll parameters, as we will see. Since
$V\gg\dot{\phi}^{2}$, from the slow-roll equation of motion (25) we deduce
that
$1\gg\frac{1}{V}\left(\frac{V^{\prime}}{3H}\right)^{2}\simeq\frac{1}{V}\frac{(V^{\prime})^{2}}{8\pi
GV_{0}}\leavevmode\nobreak\ .$ (28)
Let us therefore define the slow-roll parameter $\epsilon$ as
$\epsilon\equiv\frac{M_{P}^{2}}{16\pi}\left(\frac{V^{\prime}}{V}\right)^{2}\equiv\frac{M^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\ll
1$ (29)
where $M=M_{P}/\sqrt{8\pi}$ defines the reduced Planck mass. Taking the
derivative $d(V^{\prime})^{2}/d\phi$ one finds that $V^{\prime\prime}/V\ll
16\pi/M_{P}^{2}$ so that one may define
$\eta\equiv\frac{M_{P}^{2}}{8\pi}\frac{V^{\prime\prime}}{V}\equiv
M^{2}\frac{V^{\prime\prime}}{V}\ll 1\leavevmode\nobreak\ .$ (30)
The slow-roll conditions become violated and inflation ends when
$\epsilon,\eta\simeq 1$.
The number of e-folds is then given by
$N=\ln\frac{a(t_{\rm end})}{a(_{i})}=\int_{t_{i}}^{t_{\rm
end}}dtH(t)\leavevmode\nobreak\ ,$ (31)
where $t_{\rm end}$ corresponds to $\epsilon(\phi),\leavevmode\nobreak\
\eta(\phi)\simeq 1$. Using the slow-roll equation (25) we may write (31) as
$N=-\frac{1}{M^{2}}\int_{\phi_{i}}^{\phi_{\rm
end}}d\phi\frac{V}{V^{\prime}}\leavevmode\nobreak\ .$ (32)
Comparing this with (29) we see that the required large number of e-folds can
be obtained if the slow-roll parameter $\epsilon$ is small enough.
## 0.3 Inflationary perturbations
### 0.3.1 Evolution of field perturbations
Like any quantum field, the inflaton is subject to fluctuations. Hence, we
should write the inflaton as
$\phi(x,t)=\phi_{0}(t)+\delta\phi(x,t)\leavevmode\nobreak\ ,$ (33)
where $\phi_{0}(t)$ is the homogeneous part which is treated here as the
background field that is the solution to the slow-roll equation (25), and
$\delta\phi(x,t)$ is the (small) perturbation. Since during inflation the
energy density $\rho\sim V$, field fluctuations source also perturbations of
energy density with $\delta\rho\propto\delta
V(\phi(x,t))=V^{\prime}(\phi_{0})\delta\phi(x,t)$. Eventually, such
perturbations can be observed in the microwave sky as temperature
fluctuations. Given the inflationary model, we are thus in a position to
calculate the spectrum of the CMB fluctuations. This requires two things: 1)
assumptions about the initial conditions for the field perturbation, and 2)
understanding the evolution of the perturbation.
Assuming that the perturbation $\delta\phi(x,t)$ is small, after substituting
(33) to the equation of motion (22) we find that to lowest order
$\delta\phi(x,t)$ obeys
$\delta\ddot{\phi}_{k}+3H(\phi_{0})\delta\dot{\phi}_{k}+\left(\frac{k^{2}}{a^{2}}+m_{\phi}^{2}\right)\delta\phi_{k}=0\leavevmode\nobreak\
,$ (34)
where we have moved to Fourier space.
Obviously, for massless fields and for long-wavelength fluctuations with $k\to
0$, one finds that $\delta\phi_{k}\to$ const. This means that once beyond the
horizon, the field perturbation freezes. Because it has lost causal contact
with the horizon patch where it originated, it can no longer be modified by
local physics. This is of course good news as it implies that the spectrum of
perturbations can be computed if we only knew its amplitude at the time it
crosses the horizon.
For a perturbation well within the horizon, $k/a\gg H$. Moreover, the slow-
roll condition requires that $m_{\phi}\ll H$. Hence at early times we may
write (34) as
$\delta\ddot{\phi}_{k}+3H(\phi_{0})\delta\dot{\phi}_{k}+\frac{k^{2}}{a^{2}}\delta\phi_{k}=0\leavevmode\nobreak\
.$ (35)
Hence we have an evolution equation for the perturbation, but what is the
initial condition for the Fourier mode $\delta\phi_{k}$? The sensible
assumption is that at very small distance scales, well inside the horizon,
curvature can be neglected and that locally space looks Minkowski. It is then
sufficient to consider quantum fluctuations in empty flat space111The concept
of vacuum in curved space is not unproblematic since the particle content for
observers in different frames can be different. This particular choice is
called the Bunch–Davies vacuum. and quantize the free inflaton field in the
usual manner with $\phi_{k}=w_{k}(t)a_{k}+w_{k}^{*}(t)a_{-k}^{\dagger}$ where
the amplitude is given by
$w_{k}=V^{-1/2}\sqrt{\frac{1}{2E_{k}}}\exp(-iE_{k}t)\leavevmode\nobreak\ ,$
(36)
while
$[a_{k},a_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}},\leavevmode\nobreak\
[a_{k},a_{k^{\prime}}]=0$. The vacuum is then defined through the usual
condition $a_{k}|0\rangle=0$ with $a_{k}^{\dagger}|0\rangle\propto|k\rangle$.
It is easy to see that in the vacuum the inflaton VEV is zero whereas the
variance is given by
$\langle 0|\phi^{2}|0\rangle=\sum_{k}|w_{k}|^{2}\leavevmode\nobreak\ .$ (37)
This suggests that the initial perturbation $\delta\phi_{k}$ at $t\to-\infty$
should be identified with the root-mean-square of the the variance (37). Thus,
we should find the solution of (35) with $\delta\phi_{k}=w_{k}$ and
$w_{k}(t\to-\infty)$ given by (36) in a box of size $L$. Here we will ignore
the variation of $H$ and assume simply that $H(\phi_{0}(t))\simeq H_{0}$; this
is consistent with the slow-roll assumption. It is straightforward to show
that the solution is given by
$w_{k}=\frac{1}{L^{3/2}}\frac{H}{(2k^{3})^{1/2}}\left(i+\frac{k}{aH}\right)\exp{\left(\frac{ik}{aH}\right)}$
(38)
which reduces to the flat space result222This can be seen by expanding
$k/(aH)$ about some reference time $t_{0}$ by writing
${k}/({aH})={k}/({aH})|_{t=t_{0}}+{d}[{k}/({aH})]/dt|_{t=t_{0}}(t-t_{0})$.
(36) in the limit $t\to-\infty$. The initial field perturbation (37) is
Gaussian, but later evolution can, depending on the model, also generate small
non-Gaussian features.
At horizon exit $t=t_{*}$, we have $k=aH$ so that the perturbation amplitude
is then given by
$|w_{k}|^{2}=\frac{H^{2}(t_{*})}{2L^{3}k^{3}}\leavevmode\nobreak\ .$ (39)
The power spectrum $P(k)$ associated with a fluctuation $\delta_{k}$ is
defined as
$P(k)=\frac{V}{(2\pi)^{3}}k^{2}d\Omega\langle|\delta_{k}|^{2}\rangle=\frac{Vk^{3}}{4\pi^{2}E_{k}}\langle|\delta_{k}|^{2}\rangle\leavevmode\nobreak\
,$ (40)
where $V$ is the volume of the box. Thus at the horizon exit we find the power
spectrum of the inflaton fluctuations as
$P(k,t_{*})=\frac{Vk^{3}}{4\pi^{2}E_{k}}\frac{H^{2}(t_{*})}{2L^{3}k^{3}}=\left(\frac{H}{2\pi}\right)^{2}_{k=aH}\leavevmode\nobreak\
.$ (41)
Note that the box-size dependence cancels. Equation (41) is a central result
for cosmological inflation.
Because we are assuming slow roll, the value of $H$ is almost constant in
time. It then follows that the amplitude of the perturbation (39) is almost
the same for any arbitrary scale $k$ at the horizon exit so that the observed
spectrum should be (almost) scale independent.
### 0.3.2 From field perturbations to temperature fluctuations
Inflaton perturbations generate density perturbations, or more generally,
perturbation in the energy-momentum tensor, which is the source for the
metric. Hence, there will also be metric perturbations. In the so-called
Newtonian gauge the perturbed metric reads, neglecting now vector and tensor
perturbations (gravitational waves),
$ds^{2}=(1+2\Phi)dt^{2}-a^{2}(t)(1+2\Psi)dx^{2}\leavevmode\nobreak\ ,$ (42)
where $\Psi$ and $\Phi$ are called the Bardeen potentials; for a perfect non-
viscous fluid $\Psi=\Phi$.
However, in general relativity one is free to change coordinates in any way
one wishes by performing a general coordinate transformation $g_{\mu\nu}\to
g^{\prime}_{\mu\nu}$. Therefore, one could ‘gauge away’ the density
perturbation by making a coordinate transformation such that
$\delta\rho(x,t)=0$ everywhere (this is called ‘constant density
hypersurface’). Obviously, we need some gauge-invariant description of the
perturbation in order to really be able to decide what is observable.
Without going into details, let us state that one can show[6] that the
comoving curvature perturbation
${\cal R}=-\frac{H}{\dot{\phi}}\delta\phi$ (43)
is both gauge invariant and also remains constant outside horizon. It is
related to the Bardeen potential by $\Psi=\frac{3}{5}{\cal R}$. In the
literature one often uses also the curvature perturbation $\zeta$, which
outside the horizon is simply given by $\zeta=-{\cal R}$.
CMB observations yield correlators such as
$\langle\frac{\delta T}{T}\frac{\delta T}{T}\rangle_{k}\equiv
P_{T}(k)\leavevmode\nobreak\ ,$ (44)
or the CMB temperature power spectrum. One can show that for large angular
distances ${\delta T}/{T}=-\frac{1}{3}\Psi$, which is called the Sachs–Wolfe
effect. For small angular distances (with multipoles $l\gtrsim 50$) one needs
also to consider hydrodynamical effects in the photon–baryon plasma, which
give rise to the well-known acoustic oscillations in the CMB power spectrum.
Here, let us focus on the Sachs–Wolfe effect alone. We need to compute the
power spectrum for the comoving curvature perturbation:
$P_{\cal R}(k)=\langle{\cal R}_{k}{\cal
R}_{k}\rangle=\frac{H^{2}}{\dot{\phi}^{2}}\langle\delta\phi_{k}\delta\phi_{k}\rangle=\frac{H^{2}}{\dot{\phi}^{2}}\left(\frac{H}{2\pi}\right)^{2}_{k=aH}.$
(45)
Using the slow-roll condition (25) and the definition of the slow-roll
parameter $\epsilon$ (29) one then finds
$P_{\cal
R}(k)=\frac{1}{24\pi^{2}}\frac{V}{M^{4}}\frac{1}{\epsilon}\leavevmode\nobreak\
.$ (46)
The amplitude of the perturbation, often called the COBE normalization, is
conventionally defined as as $\delta_{H}^{2}=4P_{\cal R}/25$ with the observed
value $\delta_{H}=1.9\times 10^{-5}$. This constrains the scale of the
inflaton potential to be $(V/\epsilon)^{1/4}=0.027M$. Thus, for a very low
scale model, say, with $V\simeq M_{W}^{4}$, one would require an extremely
flat potential with $\epsilon$ very close to zero.
The CMB spectrum is specified by its amplitude and by the spectral index $n$.
Purely phenomenologically we may thus write
$P_{\cal R}=Ak^{n-1}\leavevmode\nobreak\ ,$ (47)
where $n-1={d\ln P_{\cal R}}/{d\ln k}$. Thus, if the spectrum were exactly
scale independent, we would find $n=1$. Because of the slow roll, we expect
the deviation from scale independence to be small. Let us now compute the
spectral index. At the horizon exit $k=aH$ so that the differential $d\ln
k={da}/{a}+{dH}/{H}\simeq Hdt$ since during inflation $H\simeq$ const. From
the slow-roll condition (25) we find that $dt=-3Hd\phi/V^{\prime}$ so that
$\frac{d}{d\ln
k}=-\frac{V^{\prime}}{3H^{2}}\frac{d}{d\phi}=-M^{2}\frac{V^{\prime}}{V}\frac{d}{d\phi}\leavevmode\nobreak\
.$ (48)
Then ${d\epsilon}/{d\ln k}=4\epsilon^{2}-2\epsilon\eta$ and
$n-1=2\eta-6\epsilon\leavevmode\nobreak\ .$ (49)
Hence given the inflaton model, we are able to predict both the amplitude and
the spectral index of the CMB temperature fluctuation. WMAP7[3] yields
$1-n=0.037\pm 0.014$ (for CMB alone). Hence typically (but not necessarily)
$\epsilon\sim{\cal O}(0.01)$. Other typical values would then be the scale of
the potential during inflation $V^{1/4}\simeq 10^{16}$ GeV and the Hubble rate
during inflation $H\simeq 10^{14}$ GeV. However, all these number are very
much model dependent.
Scalar-field-driven inflation is a generic idea but as of yet, there is no
compelling, particle physics motivated theory of inflation. Instead, there
exists a vast number of different models. There are models with many inflaton
fields, models based on extra dimensions, models based on the Higgs field with
a non-minimal coupling to gravity, models where the the superluminal expansion
and the primordial perturbation are generated by different fields (for reviews
of the various inflaton models, see Ref. [7]). The present observations of the
CMB spectral index yield some interesting constraints on the models, but
perhaps the best hope for a decisive test of the various models could be
provided by the ESA Planck Surveyor Mission [8], which is expected to put a
stringent limit to the non-Gaussianity of the primordial perturbation, or
perhaps even observe it. Should that happen, many models would immediately be
ruled out.
## References
* [1] For a textbook, see, for example, David Lyth and Andrew Liddle, _Cosmological Inflation and Large-Scale Structure_ (Cambridge University Press, 2000).
* [2] C. L. Bennett et al., Astrophys. J. 464 (1996) L1 [arXiv:astro-ph/9601067].
* [3] WMAP results are available at the URL lambda.gsfc.nasa.gov.
* [4] M. S. Turner, Phys. Rev. D 28 (1983) 1243.
* [5] L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. Lett. 73 (1994) 3195 [arXiv:hep-th/9405187]; Y. Shtanov, J. H. Traschen, and R. H. Brandenberger, Phys. Rev. D 51 (1995) 5438 [arXiv:hep-ph/9407247]; J. H. Traschen and R. H. Brandenberger, Phys. Rev. D 42 (1990) 2491; L. Kofman, A. D. Linde, and A. A. Starobinsky, Phys. Rev. D 56 (1997) 3258 [arXiv:hep-ph/9704452].
* [6] For details, see Viatcheslav F. Mukhanov, H. A. Feldman, and Robert H. Brandenberger, Phys. Rep. 215 203-333 (1992).
* [7] David H. Lyth and Antonio Riotto, Phys. Rep. 314 1-146 (1999) [arXiv:: hep-ph/9807278]; Anupam Mazumdar and Jonathan Rocher, arXiv:1001.0993 [hep-ph].
* [8] See www.esa.int/SPECIALS/Planck/index.html
|
arxiv-papers
| 2012-01-30T10:51:17 |
2024-09-04T02:49:26.800867
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "K. Enqvist (Univ. Helsinki and Helsinki Inst. Phys.)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1201.6164"
}
|
1201.6166
|
# Conditional and Unique Coloring of Graphs
P.Venkata Subba Reddy and K. Viswanathan Iyer
Dept. of Computer Science and Engineering
National Institute of Technology
Tiruchirapalli 620 015, India
email : venkatpalagiri@gmail.com, kvi@nitt.edu Author for correspondence
###### Abstract
For integers $k>0$ and $0<r\leq\Delta$ (where $r\leq k$), a conditional
$(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of
$G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices
with at least $\min\\{r,d(v)\\}$ differently colored neighbors. The smallest
integer $k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called
the $r$th order conditional chromatic number, denoted by $\chi_{r}(G)$. For
different values of $r$ we first give results (exact values or bounds for
$\chi_{r}(G)$ depending on $r$) related to the conditional coloring of graphs.
Then we obtain $\chi_{r}(G)$ of certain parameterized graphs viz., windmill
graph, line graph of windmill graph, middle graph of friendship graph, middle
graph of a cycle, line graph of friendship graph, middle graph of complete
$k$-partite graph, middle graph of a bipartite graph and gear graph. Finally
we introduce _unique conditional colorability_ and give some related results.
Keywords: conditional coloring; conditional chromatic number; operations on
graphs; windmill graph; middle graph; gear graph.
MSC (2010) classification. 68R10, 05C15.
## 1 Introduction
Let $G=(V(G),E(G))$ be a simple, connected, undirected graph. For a vertex
$v\in V(G)$, the open neighborhood of $v$ in $G$ is defined by $N_{G}(v)$=
{$u\in V(G):(u,v)\in E(G)$}, and the degree of $v$ is denoted by
$d(v)$=$|N_{G}(v)|$. Let $\delta(G),\;\Delta(G)$ and $\omega(G)$ (or simply
$\delta,\;\Delta$ and $\omega$) denote, respectively the minimum degree, the
maximum degree and the clique number of $G$. For an integer $k>0$, a proper
$k$-coloring of a graph $G$ is a surjective mapping $c\colon
V(G)\to\\{1,\ldots,k\\}$ such that if $(u,v)\in E(G)$, then $c(u)\neq c(v)$.
The smallest $k$ such that $G$ has a proper $k$-coloring is the chromatic
number $\chi(G)$ of $G$. Given a set $S\subseteq V(G)$ we define
$c(S)=\\{c(u):u\in S$ }. For integers $k>0$ and $0<r\leq\Delta$ (where $r\leq
k$), a conditional $(k,r)$-coloring of $G$ is a surjective mapping $c\colon
V(G)\to\\{1,\ldots,k\\}$ such that both the following conditions (C1) and (C2)
hold:
> (C1) If $(u,v)\in E(G)$, then $c(u)\neq c(v)$.
> (C2) For any $v\in V(G)$, $|c(N_{G}(v))|\geq$ min {$d(v),r$ }.
The smallest integer $k$ such that $G$ has a conditional $(k,r)$-coloring is
called the $r$th order conditional chromatic number of $G$, denoted by
$\chi_{r}(G)$. It is proved in [6], that deciding whether $\chi_{r}(G)\leq k$
is $NP$-complete. For undefined notations/terminology we refer to standard
texts in graph theory such as [1, 3, 4, 8]).
## 2 Conditional colorability of some graphs
We start with the definitions of Cartesian product and join of two graphs. Let
$G=(V(G),E(G))$ and $H=(V(H),E(H))$ be two graphs. The Cartesian product
$G\Box H$ of $G$ and $H$ has the vertex set $V(G)\times V(H)$ and the edge set
$E(G\Box H)=\\{((x_{1},x_{2}),(y_{1},y_{2})):(x_{1},y_{1})\in
E(G)\;\text{and}\;x_{2}=y_{2},\;\text{{or}}\;(x_{2},y_{2})\in
E(H)\;\text{and}\;x_{1}=y_{1}\\}$. The join $G+H$ has the vertex set $V(G)\cup
V(H)$ and the edge set $E(G)\cup E(H)\cup CE$ where the cross-edge set
$CE=\\{(u_{g},u_{h}):u_{g}\in V(G)\;\text{and}\;u_{h}\in V(H)\\}$.
###### Theorem 2.1.
Let $G_{1}$ and $G_{2}$ be two graphs where $\chi(G_{1})=k_{1}$,
$\chi(G_{2})=k_{2}$ and w.l.o.g. let $k_{1}\leq k_{2}$. Then
$\chi_{r}(G_{1}+G_{2})$ = $\chi(G_{1}+G_{2})$ = $k_{1}+k_{2}$, where $r\leq
k_{1}+1$.
###### Proof.
In the graph $G_{1}+G_{2}$, $V(G_{2})\subset N_{G_{1}+G_{2}}(u)$ if $u\in
V(G_{1})$ or $V(G_{1})\subset N_{G_{1}+G_{2}}(u)$ if $u\in V(G_{2})$.
Therefore $c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, and in any proper
$k$-coloring of $G_{1}+G_{2}$, for all $u\in V(G_{1}+G_{2})$,
$|c(N_{G_{1}+G_{2}}(u))|\geq\min\\{d(u),r\\}$. This implies that every proper
$k$-coloring of $G_{1}+G_{2}$ is also a proper $(k,r)$-coloring – if not, we
get a contradiction: suppose that $\chi(G_{1}+G_{2})=k\neq k_{1}+k_{2}$; since
$c(V(G_{1}))\cap c(V(G_{2}))=\emptyset$, either $k_{1}\neq\chi(G_{1})$ or
$k_{2}\neq\chi(G_{2})$ which contradicts the given condition. ∎
###### Theorem 2.2.
Let $T_{1},T_{2}$ be two non trivial trees with $n_{1},n_{2}$ number of
vertices respectively and w.l.o.g. let $n_{1}\leq n_{2}$. Then
$\chi_{r}(T_{1}+T_{2})=2(r-1)$, where $4\leq r\leq\ n_{1}+1$.
###### Proof.
Every nontrivial tree has at least two vertices with degree one [4]. Therefore
there exist vertices $u,v$ where $u\in V(T_{1})$ and $v\in V(T_{2})$, such
that $d(u)=d(v)=1$. Therefore, $d_{u}(T_{1}+T_{2})=1+n_{2}$ and
$d_{v}(T_{1}+T_{2})=1+n_{1}$. If $\chi_{r}(T_{1}+T_{2})<2(r-1)$, then either
$|c(V(T_{1}))|<r-1$ or $|c(V(T_{2}))|<r-1$ or both because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$. Hence (C2) is violated at $u$ or $v$ or both.
Therefore, $\chi_{r}(T_{1}+T_{2})\geq 2(r-1)$. Since every tree is
$2$-colorable and $r\geq 4$, properly color $V(T_{1})$, $V(T_{2})$ in
$T_{1}+T_{2}$ using $r-1$ colors each such that $|c(V(T_{1}+T_{2}))|=2(r-1)$.
The resulting coloring is a conditional $(2(r-1),r)$-coloring of
$T_{1}+T_{2}$, as (C1) is satisfied because $c(V(T_{1}))\cap
c(V(T_{2}))=\emptyset$, and (C2) is satisfied because $c(V(T_{1}))\cap\
c(V(T_{2}))=\emptyset$ and for all $w\in V(T_{1}+T_{2})$,
$|c(N_{T_{1}+T_{2}}(w))|\geq(r-1)+1\geq$ min $\\{d(w),r\\}$. Hence the result.
∎
###### Theorem 2.3.
Given any two graphs $G_{1}$ and $G_{2}$, let $r_{1}$ and $r_{2}$ be such that
$r_{1}\geq\delta(G_{1})$ and $r_{2}\geq\delta(G_{2})$. Then $\chi_{r}(G_{1}\
\Box\ G_{2})\leq\chi_{r_{1}}(G_{1}).\chi_{r_{2}}(G_{2})$ where
$r\leq\delta(G_{1})+\delta(G_{2})$.
###### Proof.
Let $\chi_{r_{1}}(G_{1})=g_{1}$ and $\chi_{r_{2}}(G_{2})=g_{2}$. Let
$c_{G_{1}}$ (resp. $c_{G_{2}}$) be a proper $(g_{1},r_{1})$\- (resp.
$(g_{2},r_{2})$-) coloring of $G_{1}$ (resp. $G_{2}$). Then let $c_{G_{1}\Box
G_{2}}$ be a coloring of $G_{1}\Box G_{2}$ wherein we assign to any vertex
$(u_{1},u_{2})\in V(G_{1}\ \Box\ G_{2})$ the color denoted by the ordered pair
$(c_{g_{1}}(u_{1}),c_{g_{2}}(u_{2}))$. This coloring uses $g_{1}.g_{2}$ colors
and it defines a proper coloring of $G_{1}\ \Box\ G_{2}$. Therefore
$c_{G_{1}\Box G_{2}}$ satisfies (C1). Let $(u_{1},u_{2})\in V(G_{1}\ \Box\
G_{2})$ such that $u_{1}\in V(G_{1})$ and $u_{2}\in V(G_{2})$. Since
$c_{G_{1}}$ and $c_{G_{2}}$ satisfy (C2), by the definition of $G_{1}\Box
G_{2}$, a vertex $(u_{1},u_{2})$ has at least
$\min\\{r_{1},\delta(G_{1})\\}=\delta(G_{1})$ distinctly colored neighbors of
the form $(u^{\prime},u_{2})$ because $|c(N_{G_{1}}(u_{1}))|\geq\delta(G_{1})$
and at least $\min\\{r_{2},\delta(G_{2})\\}=\delta(G_{2})$ distinctly colored
neighbors of the form $(u_{1},u^{\prime\prime})$ because
$|c(N_{G_{2}}(u_{2}))|\geq\delta(G_{2})$. Therefore $|c(N_{G_{1}\ \Box\
G_{2}}((u_{1},u_{2}))|\geq\delta(G_{1})+\delta(G_{2})\geq r$. Hence
$c_{G_{1}\Box G_{2}}$ satisfies (C2) and the result follows. ∎
###### Theorem 2.4.
Let $G(V_{1},V_{2},E)$ be a bipartite graph, $S_{1}=\bigcap_{u\in
V_{1}}N_{G}(u)$, $S_{2}=\bigcap_{v\in V_{2}}N_{G}(v)$ and w.l.o.g. let
$|S_{1}|\leq|S_{2}|$. Then $\chi_{r}(G)=2r$ where $r\leq|S_{1}|$ .
###### Proof.
In any proper coloring of $G$, from the given conditions $|c(V_{1})|\geq r$
and $|c(V_{2})|\geq r$ as $G$ is bipartite. Since $r\leq|S_{1}|$ and
$c(S_{1})\cap c(S_{2})=\emptyset$ we have $\chi_{r}(G)\geq 2r$. But there
exists a proper $2r$-coloring of $G$ such that $|c(S_{1})|=|c(S_{2})|=r$
because every bipartite graph is bicolorable and $r\geq 2$. This coloring also
satisfies (C2) as $S_{1}\subseteq V_{2}$ and $S_{2}\subseteq V_{1}$. Thus
$\chi_{r}(G)\leq 2r$. Hence $\chi_{r}(G)=2r$. ∎
###### Theorem 2.5.
Let $L(T)$ be the line graph of complete $k$-ary tree $T$ with height $h\geq
2$. Then
$\chi_{r}(L(T))=\left\\{\begin{array}[]{l l}k+1,&\quad\mbox{if $r\leq k$.
{}}\\\ 2k+1,&\quad\mbox{if $r=\Delta$. {}}\\\ \end{array}\right.$
###### Proof.
Let $V(L(T))=\left\\{v_{1},v_{2},\dotsc,v_{e(h)}\right\\}$, where
$e(h)=\frac{k^{h+1}-1}{k-1}-1$. In $T$ we assume that the root is at level $0$
and for each $l$ ($1\leq l\leq h$), $v_{e(l-1)+1}$ to $v_{e(l)}$ represent the
edges between levels $l-1$ and $l$, numbered from ‘left’ to ‘right’. It can be
seen that $\Delta(L(T))=2k$ and $\omega(L(T))=k+1$. In the ordering
$v_{e(h)},\dotsc,v_{1}$ of the vertices of $L(T)$, for each $i$ ($1\leq i\leq
e(h)$), $v_{i}$ is a simplicial vertex in the subgraph induced by
$\\{v_{i},\dotsc,v_{1}\\}$. Hence the ordering is a p.e.o. and $L(T)$ is
chordal. As every chordal graph is perfect, $\chi(L(T))=\omega(L(T))=k+1$.
Since every vertex of $L(T)$ is in a $K_{k+1}$, we also have
$\chi_{r}(L(T))=k+1$ if $r\leq k$. Thus $\chi_{r}(L(T))=k+1$, if $r\leq k$.
From [5] we know $\chi_{r}(G)\geq\min\\{r,\Delta\\}+1$. Taking $G=L(T)$ we
have $\chi_{r}(L(T))\geq\min\\{r,\Delta\\}+1=2k+1$ if $r=\Delta$. Similar to
the greedy (vertex) coloring, color the vertices in the order
$v_{1},\dotsc,v_{e(h)}$ by assigning to each vertex the first available color
not already used for any of the lower indexed vertices within distance two. In
the assumed order, each vertex has at most $\Delta$ lower indexed vertices
within distance two; therefore $\chi_{\Delta}(L(T))\leq\Delta+1=2k+1$. Hence
$\chi_{r}(L(T))=2k+1$ if $r=\Delta$. ∎
## 3 Conditional colorability of some parameterized
graphs
For this section we need the following definitions. The middle graph $M(G)$
of $G$ is the graph whose vertex set corresponds to $V(G)\cup E(G)$; in $M(G)$
two vertices are adjacent iff $(i)$ they are adjacent edges of $G$ or $(ii)$
one is a vertex and the other is an edge incident with it [7]. The windmill
graph $Wd(k,n)$ consists of $n$ copies of $K_{k}$ and identifying one vertex
from each $K_{k}$ as the common center vertex. In particular $Wd(3,n)$ is
called the Friendship graph $F_{n}$ [2]. A complete $k$-partite graph
$K(n_{1},\ldots,n_{k})$ has vertex set $V=V_{1}\cup\ldots\cup V_{k}$ where
$V_{1},\ldots,V_{k}$ are mutually disjoint with $|V_{i}|=n_{i}$; each vertex
$v\in V_{i}$ is connected to all vertices of $V\setminus
V_{i},\;i=1,\ldots,k$. The $n$-gear $G_{n}$ consists of a cycle $C_{2n}$ on
$2n$ vertices where every other vertex on the cycle is adjacent to a
$(2n+1)^{\rm th}$ center vertex labeled $v_{0}$. The vertices in the $C_{2n}$
are labeled sequentially $v_{1},\ldots,v_{2n}$ such that for $1\leq i\leq
2n-1$, $v_{i}$ is adjacent to $v_{i+1}$, $v_{1}$ is adjacent to $v_{2n}$, and
every vertex in $V_{oddi}=\\{v_{i}|\;\text{i is odd and}\;1\leq i\leq 2n-1\\}$
is adjacent to $v_{0}$.
We begin with two lemmas followed by our propositions.
###### Lemma 3.1.
For a non-negative integer $r\leq\Delta$ let Vset-$d2r$ in a graph $G$ be a
set $S_{d2r}\subseteq V(G)$ with the following two properties:
> $(i)$ For all $u\in S_{d2r}$, $d(u)\leq r$.
> $(ii)$ For all $u_{1},u_{2}\in S_{d2r}$ either $(u_{1},u_{2})\in E(G)$ or
> there exists a $u_{3}\in S_{d2r}$ such that $u_{1},u_{2}\in N(u_{3})$ or
> both.
Then $\chi_{r}(G)\geq|S_{d2r}|$.
###### Proof.
Assume that $\chi_{r}(G)<|S_{d2r}|$. Then there exist at least two vertices
$u_{1},u_{2}\in S_{d2r}$ such that $c(u_{1})=c(u_{2})$. By the definition of
Vset-$d2r$ $(ii)$ holds; if $(u_{1},u_{2})\in E(G)$, (C1) is violated; if
$u_{3}\in S_{d2r}$ such that $u_{1},u_{2}\in N(u_{3})$,
$|c(N(u_{3}))|<min\\{r,d(u_{3})\\}=d(u_{3})$ and hence (C2) is violated at
$u_{3}$. Therefore $\chi_{r}(G)\geq|S_{d2r}|$. ∎
###### Lemma 3.2.
Given a graph $G$, let $c\colon V(G)\to\\{1,\ldots,k\\}$ be a coloring such
that for a given $r$, $c$ satisfies (C2). Let the condition (C3) be
> (C3) For each edge $uv$ in $G$ there exists a vertex $w$ such that $d(w)$
> $\leq r$ and $u,v\in N_{G}(w)$.
If $G$ satisfies (C3) also then $c$ satisfies (C1) and hence $c$ defines a
conditional $(k,r)$-coloring of $G$.
###### Proof.
The proof by contradiction is straightforward. ∎
###### Proposition 3.3.
For integers $k\geq 3,n\geq 2$ let $Wd(k,n)$ be a windmill graph. Then
$\chi_{r}(Wd(k,n))=\left\\{\begin{array}[]{l l}k,&\quad\mbox{if $2\leq r\leq
k-1$. {}}\\\ min\\{r,\Delta\\}+1,&\quad\mbox{if $r\geq k$. {}}\\\
\end{array}\right.$
###### Proof.
Every vertex $v$ of $Wd(k,n)$ is contained in a $K_{k}$; it can be seen that
$|c(N(v))|\geq k-1$ in any proper coloring $c$ of $Wd(k,n)$. Therefore if
$2\leq r\leq k-1$, conditional $(\chi(Wd(k,n)),r)$-coloring of $Wd(k,n)$
exists and we know that $\chi(Wd(k,n))=k$. From [5] we have $\chi_{r}(G)\geq
min\\{r,\Delta\\}+1$. Taking $G=Wd(k,n)$ we get $\chi_{r}(Wd(k,n))$ $\geq
min\\{r,\Delta\\}+1$. In $Wd(k,n)$ only the center vertex has a degree
$n(k-1)>k$. For a $k^{\prime}$ if $k^{\prime}>k$, then every $k$-colorable
graph is also $k^{\prime}$-colorable. Hence if $r\geq k$, then a proper
$(min\\{r,\Delta\\}+1)$-coloring of $Wd(k,n)$ exists, which is also a
conditional $(min\\{r,\Delta\\}+1,r)$-coloring. Therefore
$\chi_{r}(Wd(k,n))\leq min\\{r,\Delta\\}+1$. Hence
$\chi_{r}(Wd(k,n))=min\\{r,\Delta\\}+1$ if $r\geq k$. ∎
###### Proposition 3.4.
Let $L(Wd(k,n))$ be the line graph of windmill graph $Wd(k,n)$. Then
$\chi_{\Delta}(L(Wd(k,n)))=n(k-1)+\binom{k-1}{2}=z\;(say).$
###### Proof.
It follows that $|V(L(Wd(k,n)))|=n\binom{k}{2}=inx(k,n)$ (say). Let
$V(L(Wd(k,n)))=\\{v_{1},\ldots,v_{inx(k,n)}\\}$. We assume that in $Wd(k,n)$,
for all $1\leq i\leq n,v_{(i-1)(k-1)+1}$ to $v_{i(k-1)}$ and
$v_{n(k-1)+inx(k-1,i-1)+1}$ to $v_{n(k-1)+inx(k-1,i)}$ represent respectively
the edges of $i^{th}$ copy of $K_{k}$ incident with and not incident with the
center vertex. It can be seen that $L(Wd(k,n))$ has a clique
$\\{v_{1},\ldots,v_{n(k-1)}\\}$ and a Vset-$d2r$
$S_{d2r}=\\{v_{1},\ldots,v_{z}\\}$. By lemma 3.1, $\chi_{\Delta}(L(Wd(k,n)))$
$\geq|S_{d2r}|=z$. We now define the coloring assignment $c\colon
V(L(Wd(k,n)))\to\\{1,\ldots,z\\}$ as follows:
$c(v_{i})=\begin{cases}i,&\text{if $1\leq i\leq z.$}\\\
i\mod{\binom{k-1}{2}}+n(k-1)+1,&\text{otherwise.}\end{cases}$
In $c$ the if-case uses $z$ and the otherwise-case doesn’t use any extra
color. For all $1\leq i\leq n$
$|c(\\{v_{(i-1)(k-1)+1},\ldots,v_{i(k-1)}\\})$ $\cup$
$c(\\{v_{n(k-1)+inx(k-1,i-1)+1},\ldots,v_{n(k-1)+inx(k-1,i)}\\})|$
=
$|\\{v_{(i-1)(k-1)+1},\ldots,v_{i(k-1)}\\}\cup\\{v_{n(k-1)+inx(k-1,i-1)+1},\ldots,v_{n(k-1)+inx(k-1,i)}\\}|$
and
$|c\\{v_{1},\ldots,v_{n(k-1)}\\}|=n(k-1)$; hence (C2) is satisfied at all the
vertices. The graph $G=L(Wd(k,n))$ can be seen to satisfy (C3) in lemma 3.2.
By lemma 3.2, $\chi_{\Delta}(L(Wd(k,n)))\leq z$; hence
$\chi_{\Delta}(L(Wd(k,n)))=z$. ∎
###### Proposition 3.5.
Let $L(F_{n})$ be the line graph of $F_{n}$. Then
$\chi_{r}(L(F_{n}))=\begin{cases}2n,&\text{if $r<\Delta$.}\\\ 2n+1,&\text{if
$r=\Delta$.}\end{cases}$
###### Proof.
We have the following two cases:
Case 1: $r<\Delta:$ Let $r^{\prime}=\Delta-1$. Since $|V(L(F_{n}))|=3n$, let
$V(L(F_{n}))=\\{v_{1},\ldots,v_{3n}\\}$. We assume that for all $1\leq i\leq
n,v_{2i-1}$ and $v_{2i}$ represent the edges of $i^{\rm th}$ copy of $K_{3}$
incident with the center vertex and $v_{2n+i}$ represents the edge of $i^{\rm
th}$ copy of $K_{3}$ not incident with the center vertex of $F_{n}$. As
$\\{v_{1},\ldots,v_{2n}\\}$ is the maximum size clique of $L(F_{n})$,
$\omega(L(F_{n}))=2n$. Since $\chi_{r}(G)\geq\omega(G)$, taking $r=r^{\prime}$
and $G=L(F_{n})$ we have $\chi_{r^{\prime}}(L(F_{n}))\geq 2n$. We now define
the coloring assignment $c\colon V(L(F_{n}))\to\\{1,\ldots,2n\\}$ as follows:
$c(v_{i})=\begin{cases}i,&\text{if $1\leq i\leq 2n$.}\\\ 2n,&\text{if
$2n+1\leq i\leq 3n-1$.}\\\ 1,&\text{if $i=3n$.}\end{cases}$
Note that in $c$ the first case uses $2n$ colors and the remaining cases use
no new colors. It can be verified that $c$ defines a conditional
$(2n,r^{\prime})$-coloring of $L(F_{n})$. Thus
$\chi_{r^{\prime}}(L(F_{n}))\leq 2n$; hence $\chi_{r^{\prime}}(L(F_{n}))=2n$.
From [5] it follows that $\omega(G)\leq\chi_{r_{1}}(G)\leq\chi_{r_{2}}(G)$ if
$r_{1}\leq r_{2}$. Taking $G=L(F_{n}),r_{1}=r$ and $r_{2}=r^{\prime}$ it
follows that $\chi_{r}(L(F_{n}))=2n$.
Case 2: $r=\Delta:$ Since $F_{n}=Wd(3,n)$, by theorem 1 we have
$\chi_{\Delta}(L(Wd(3,n)))=\chi_{\Delta}(L(F_{n}))=2n+1$. ∎
###### Proposition 3.6.
Let $M(K(n_{1},\ldots,n_{k}))$ be the middle graph of $K(n_{1},\ldots,n_{k})$.
Then
$\chi_{\Delta}(M(K(n_{1},\ldots,n_{k})))=k+l.$
where $n=\sum_{i=1}^{k}n_{i}$ and $l=1/2\sum_{i=1}^{k}n_{i}(n-n_{i})$.
###### Proof.
We know that $K(n_{1},\ldots,n_{k})$ has $l$ edges,
$|V(M(K(n_{1},\ldots,n_{k})))|=l+n$. Let
$V(M(K(n_{1},\ldots,n_{k})))=\\{v_{1},\ldots,v_{l+n}\\}$ and $n_{0}=0$. We
assume that $v_{1}$ to $v_{l}$ represent the edges and for all $1\leq i\leq
k,v_{l+1+\sum_{j=0}^{i-1}n_{j}}$ to $v_{l+\sum_{j=0}^{i}n_{j}}$ represent the
$i^{\rm th}$ partition vertices of $K(n_{1},\ldots,n_{k})$. Let
$r=\Delta,V_{e}=\\{v_{1},\ldots,v_{l}\\}$ and
$V_{v}=\\{v_{l+1},\ldots,v_{l+n}\\}$. It can be easily seen that
$M(K(n_{1},\ldots,n_{k}))$ has a Vset-$d2r$ given by
$S_{d2r}=V_{e}\;\cup\;\\{v_{l+1},v_{l+n_{1}+1},v_{l+(n_{1}+n_{2})+1},v_{l+(n_{1}+n_{2}+n_{3})+1},\ldots,v_{l+(n_{1}+\ldots+n_{k-1})+1}\\}$.
Thus by lemma 3.1, $\chi_{r}(M(K(n_{1},\ldots,n_{k})))\geq|S_{d2r}|=k+l$. We
now define the coloring assignment $c\colon V(M(K(n_{1},\ldots,n_{k})))$
$\to\\{1,\ldots,k+l\\}$ as follows:
$c(v_{i})=\begin{cases}i,&\text{if $1\leq i\leq l.$}\\\ l+p,&\text{otherwise,
where $p$ is such that $1+\sum_{j=0}^{p-1}n_{j}\leq
i-l\leq\sum_{j=0}^{p}n_{j}$.}\end{cases}$
In $c$ the first case uses $l$ colors and the remaining case uses $k$ new
colors. We have $|c(V_{e})|=|V_{e}|$ and $c(V_{e})\cap c(V_{v})=\emptyset$;
for any two vertices $v_{i},v_{i^{\prime}}\in V_{v}$ if there exists no $p$
such that $l+1+\sum_{j=0}^{p-1}n_{j}\leq i,i^{\prime}\leq
l+\sum_{j=0}^{i}n_{j}$ then $c(v_{i})\neq c(v_{i^{\prime}})$; hence (C2) is
satisfied at all the vertices. Taking $G=M(K(n_{1},\ldots,n_{k}))$ it can be
seen that (C3) of lemma 3.2 is satisfied. By lemma 3.2,
$\chi_{\Delta}(M(K(n_{1},\ldots,n_{k})))\leq k+l$. Hence
$\chi_{\Delta}(M(K(n_{1},\ldots,n_{k})))=k+l$. ∎
###### Proposition 3.7.
For $n\geq 4$, let $M(C_{n})$ be the middle graph of $C_{n}$. Then
$\chi_{r}(M(C_{n}))=\left\\{\begin{array}[]{l l}3,&\quad\mbox{if $r=2$. {}}\\\
4,&\quad\mbox{if $r=3$. {}}\\\ \end{array}\right.$
###### Proof.
Let $V(M(C_{n}))=\\{v_{1},\ldots,v_{2n}\\}$. We assume that $v_{1}$ to $v_{n}$
and $v_{n+1}$ to $v_{2n}$ represent the vertices and edges of $C_{n}$
respectively where for all $i$ ($1\leq i\leq n$) $v_{n+i}$ is incident with
both $v_{i}$ and $v_{i\;mod\;n+1}$. We have the following cases:
Case 1: $r=2:$ Since $r<\Delta,\;\chi_{r}(M(C_{n}))\geq 3$. We define the
coloring assignment $c\colon V(M(C_{n}))\to\\{1,2,3\\}$ thus:
For even $n$
$c(v_{i})=\begin{cases}1,&\text{if $1\leq i\leq n.$}\\\ 2,&\text{if $n+1\leq
i\leq 2n$ and $i:$ odd.}\\\ 3,&\text{otherwise.}\end{cases}$
For odd $n$
$c(v_{i})=\begin{cases}1,&\text{if $i=1$ or both $n+1\leq i\leq 2n$ and $i:$
odd.}\\\ 2,&\text{if $i=2n$ or $2\leq i\leq n-1$.}\\\
3,&\text{otherwise.}\end{cases}$
In both the cases it can be verified that $c$ defines a conditional
$(3,r)$-coloring of $M(C_{n})$. Thus $\chi_{r}(M(C_{n}))\leq 3$; hence
$\chi_{r}(M(C_{n}))=3$.
Case 2: $r=3:$ Since $r<\Delta,\;\chi_{r}(M(C_{n}))\geq 4$. We define the
coloring assignment $c\colon V(M(C_{n}))\to\\{1,2,3,4\\}$ as follows:
$c(v_{i})=\begin{cases}1,&\text{if $n+1\leq i\leq 2n$ and $(i-n):$ even.}\\\
2,&\text{if $1\leq i\leq n$ and $i:$ odd.}\\\ 3,&\text{if $i=n+1$ or both
$4\leq i\leq n$ and $i:$ even.}\\\ 4,&\text{otherwise.}\end{cases}$
It can be verified that $c$ defines a conditional $(4,r)$-coloring of
$M(C_{n})$. Thus $\chi_{r}(M(C_{n}))\leq 4$; hence $\chi_{r}(M(C_{n}))=4$. ∎
###### Proposition 3.8.
Let $M(F_{n})$ be the middle graph of $F_{n}$. Then
$\chi_{r}(M(F_{n}))=\left\\{\begin{array}[]{l l}2n+1,&\quad\mbox{if $r\leq
2n$. {}}\\\ 2n+2,&\quad\mbox{if $r=2n+1$. {}}\\\ 2n+4,&\quad\mbox{if
$r=\Delta$. {}}\\\ \end{array}\right.$
###### Proof.
From the definition we have $|V(M(F_{n}))|=5n+1$. Let
$V(M(F_{n}))=\\{v_{1},\ldots,v_{5n+1}\\}$. We assume that for all $i$ ($1\leq
i\leq n)\;v_{2i-1}$ and $v_{2i}$ represent the edges of $i^{\rm th}$ copy of
$K_{3}$ incident with the center vertex, $v_{2(n+i)}$ and $v_{2(n+i)+1}$
represent the vertices of $i^{\rm th}$ copy of $K_{3}$ excluding the center
vertex,$v_{4n+i+1}$ represents the edge of $i^{\rm th}$ copy of $K_{3}$ not
incident with the center vertex and $v_{2n+1}$ represents the center vertex of
$F_{n}$. It is clear that $\Delta(M(F_{n}))=2n+2$. We have the following cases
:
Case 1: $r\leq 2n:$ Let $r^{\prime}=2n$. Since $S=\\{v_{1},\ldots,v_{2n+1}\\}$
is the maximum size clique of $M(F_{n}),\omega(M(F_{n}))=2n+1$. We know that
$\chi_{r}(G)\geq\omega(G)$, taking $G=M(F_{n})$ and $r=r^{\prime}$, we get
$\chi_{r^{\prime}}(M(F_{n}))\geq 2n+1$. Now we define the coloring assignment
$c\colon V(M(F_{n}))\to\\{1,\ldots,2n+1\\}$ as follows:
$c(v_{i})=\begin{cases}i,&\text{if $1\leq i\leq 2n+1$.}\\\ 3,&\text{if
$i=2n+2$.}\\\ 4,&\text{if $i=2n+3$.}\\\ 1,&\text{if $2n+4\leq i\leq 4n+1$ and
$(i-2n):$ even.}\\\ 2,&\text{if $2n+4\leq i\leq 4n+1$ and $(i-2n):$ odd.}\\\
2n+1,&\text{if $4n+2\leq i\leq 5n+1$.}\end{cases}$
It is clear that the total number of colors used in $c$ is $2n+1$. Since $S$
is a clique, $|c(S)|=|S|=2n+1$ and $r^{\prime}<2n+1$, for all $v\in S$ (C2) is
satisfied at $v$. For all $i$ ($1\leq i\leq n)$
$|c(\\{v_{2i-1},v_{2i},v_{2(n+i)},v_{2(n+i)+1},v_{4n+i+1}\\})|=|\\{v_{2i-1},v_{2i},v_{2(n+i)},v_{2(n+i)+1},v_{4n+i+1}\\}|$;
therefore the remaining vertices also satisfy (C2). If we take $G$ to be
$M(F_{n})$ it follows (C3) of lemma 3.2 is satisfied. By lemma 3.2,
$\chi_{r^{\prime}}(M(F_{n}))\leq 2n+1$; hence
$\chi_{r^{\prime}}(M(F_{n}))=2n+1$. From [3] we infer
$\omega(G)\leq\chi_{r_{1}}(G)\leq\chi_{r_{2}}(G)$ if $r_{1}\leq r_{2}$. Taking
$G=M(F_{n}),r_{1}=r$ and $r_{2}=r^{\prime}$, it follows that
$\chi_{r}(M(F_{n}))=2n+1$.
Case 2: $r=2n+1:$ Since $r<\Delta,\;\chi_{r}(M(F_{n}))\geq r+1$. Now we define
the coloring assignment $c^{\prime}\colon V(M(F_{n}))\to\\{1,\ldots,2n+2\\}$
as follows:
$c^{\prime}(v_{i})=\begin{cases}c(v_{i})\,\;\text{as defined in case
1},&\text{if $1\leq i\leq 4n+1$}\;\text{and}\\\
2n+2,&\text{otherwise.}\end{cases}$
In $c^{\prime}$ the if-case uses $2n+1$ and the otherwise-case uses one new
color. Since $|c^{\prime}(S)|=|S|$ and $c^{\prime}(S)\cap
c^{\prime}(\\{v_{4n+2},\ldots,v_{5n+1}\\})=\emptyset$, for all $v\in S$ (C2)
is satisfied at $v$. By extending the argument similar to case 1, we can
conclude that $c^{\prime}$ defines a conditional $(2n+2,r)$-coloring of
$M(F_{n})$. Thus $\chi_{r}(M(F_{n}))\leq 2n+2$; hence
$\chi_{r}(M(F_{n}))=2n+2$.
Case 3: $r=\Delta:\,M(F_{n})$ has a Vset-$d2r$
$S_{d2r}=\\{v_{1},\ldots,v_{2n+3},v_{4n+2}\\}$; by lemma 3.1,
$\chi_{r}(M(F_{n}))\geq|S_{d2r}|=2n+4$. We now define the coloring assignment
$c\colon V(M(F_{n}))\to\\{1,\ldots,2n+4\\}$ as follows:
$c(v_{i})=\begin{cases}i,&\text{if $1\leq i\leq 2n+3$.}\\\ 2n+3,&\text{if
$2n+4\leq i\leq 4n+1$ and $(i-2n):$ even.}\\\ 2n+4,&\text{if $i=4n+2$ or both
$2n+4\leq i\leq 4n+1$ and $(i-2n):$ odd.}\\\ 2n+2,&\text{if $4n+3\leq i\leq
5n+1$.}\end{cases}$
It is clear that in $c$ the total number of colors used is $2n+4$. For all $i$
($1\leq i\leq n)$ we have $|c(S)\cup
c(\\{v_{2(n+i)},v_{2(n+i)+1},v_{4n+i+1}\\})|=|S\cup\\{v_{2(n+i)},v_{2(n+i)+1},v_{4n+i+1}\\}|$;
therefore all the vertices satisfy (C2). With $G=M(F(n))$ we see that (C3) of
lemma 3.2 is satisfied. Thus by lemma 3.2, $\chi_{r}(M(F_{n}))\leq 2n+4$.
Hence $\chi_{r}(M(F_{n}))=2n+4$. ∎
###### Proposition 3.9.
Let $M(K(n_{1},n_{2}))$ be the middle graph of $K(n_{1},n_{2})$ and w.l.o.g.
assume $n_{1}\leq n_{2}$. Then
$\chi_{r}(M(K(n_{1},n_{2})))=\begin{cases}n_{2}+1,&\text{if $\,r\leq
n_{2}$.}\\\ n_{2}+2,&\text{if $\,r=n_{2}+1$.}\end{cases}$
###### Proof.
We have $|V(M(K(n_{1},n_{2})))|=n_{1}n_{2}+n$ where $n=n_{1}+n_{2}$. Let the
vertex set $V(M(K(n_{1},n_{2})))$ be $\\{v_{1},\ldots,v_{n+n_{1}n_{2}}\\}$. We
assume that $v_{1}$ to $v_{n_{1}}$ represent the vertices of the first
partition, $v_{n_{1}+1}$ to $v_{n}$ represent the vertices of the second
partition and $v_{n+1}$ to $v_{n+n_{1}n_{2}}$ represent the edges of
$K(n_{1},n_{2})$. We also assume that for all $i$ ($1\leq i\leq
n_{1})\;v_{n+(i-1)n_{2}+1}$ to $v_{n+in_{2}}$ represent the edges incident at
$v_{i}$ and for all $j$ ($1\leq j\leq n_{2})\;v_{n+(i-1)n_{2}+j}$ is incident
with $v_{i}$ and $v_{n_{1}+j}$. It is clear that
$\chi_{\Delta}(M(K(n_{1},n_{2})))=n$. We have the following cases:
Case 1: $r\leq n_{2}:$ Let $r^{\prime}=n_{2}$. Since
$\\{v_{1},v_{n+1},\ldots,v_{n+n_{2}}\\}$ is the maximum size clique of
$M(K(n_{1},n_{2}))$, $\omega(M(K(n_{1},n_{2})))=n_{2}+1$. We know,
$\chi_{r}(G)\geq\omega(G)$; taking $G=M(K(n_{1},n_{2}))$ and $r=r^{\prime}$,
we get $\chi_{r^{\prime}}(M(K(n_{1},n_{2})))\geq n_{2}+1$. We define the
coloring assignment $c\colon V(M(K(n_{1},n_{2})))$ $\to\\{1,\ldots,n_{2}+1\\}$
as follows:
$c(v_{i})=\begin{cases}n_{2}+1,&\text{if $1\leq i\leq n$.}\\\
1+(\lfloor(i-1-n)/n_{2}\rfloor+(i-n))\mod{n_{2}},&\text{otherwise.}\end{cases}$
In $c$ the if-case uses one and the otherwise-case uses $n_{2}$ new colors.
For all $i$ ($1\leq i\leq
n_{1})\;|c(N[v_{i}])|=|c(\\{v_{i},v_{n+(i-1)n_{2}+1},\ldots,v_{n+in_{2}}\\})|=|\\{v_{i},v_{n+(i-1)n_{2}+1},\ldots,v_{n+in_{2}}\\}|=|N[v_{i}]|$;
hence for all $v\in$
$V(M(K(n_{1},n_{2})))\setminus\\{v_{n_{1}+1},\ldots,v_{n}\\}$ (C2) is
satisfied at $v$. For all $j$ ($n_{1}+1\leq j\leq
n)\;|c(N(v_{j}))|=|c(\\{v_{j+in_{2}}:\forall i:1\leq i\leq
n_{1}\\})|=|\\{v_{j+in_{2}}:\forall i:1\leq i\leq n_{1}\\}|=|N(v_{j})|$; hence
(C2) is satisfied at all $v_{j}$. Now (C3) of lemma 3.2 is satisfied if we set
$G=M(K(n_{1},n_{2}))$. By lemma 3.2, $\chi_{r^{\prime}}(M(K(n_{1},n_{2})))\leq
n_{2}+1$; hence $\chi_{r^{\prime}}(M(K(n_{1},n_{2})))=n_{2}+1$. We know that
$\omega(G)\leq\chi_{r_{1}}(G)\leq\chi_{r_{2}}(G)$ if $r_{1}\leq r_{2}$. Taking
$G=M(K(n_{1},n_{2})),r_{1}=r$ and $r_{2}=r^{\prime}$, it follows that
$\chi_{r}(M(K(n_{1},n_{2})))=n_{2}+1$.
Case 2: $r=n_{2}+1:$ Since $r<\Delta,\;$ $\chi_{r}(M(K(n_{1},n_{2})))\geq
r+1$. We now define the coloring assignment $c^{\prime}\colon
V(M(K(n_{1},n_{2})))\to\\{1,\ldots,n_{2}+2\\}$ as follows:
$c^{\prime}(v_{i})=\begin{cases}n_{2}+2,&\text{if $1\leq i\leq n_{1}$.}\\\
c(v_{i})\,\text{as defined in case 1},&\text{otherwise.}\end{cases}$
In $c^{\prime}$ the if-case uses one and the otherwise-case uses $n_{2}+1$ new
colors. $c^{\prime}(\\{v_{1},\ldots,v_{n}\\})\cap
c^{\prime}(\\{v_{n+1},\ldots,v_{n+n_{1}n_{2}}\\})$ =
$\emptyset,\,c^{\prime}(\\{v_{1},\ldots,v_{n_{1}}\\})\cap
c^{\prime}(\\{v_{n_{1}+1},\ldots,v_{n}\\})=\emptyset$ and for all $i$ ($1\leq
i\leq n)\;|c^{\prime}(N(v_{i}))|=|N(v_{i})|$; hence (C2) is satisfied at all
the vertices. By setting $G=M(K(n_{1},n_{2}))$ we reason (C3) of lemma 3.2 is
satisfied. By lemma 3.2, $\chi_{r}(M(K(n_{1},n_{2})))\leq n_{2}+2$. Hence
$\chi_{r}(M(K(n_{1},n_{2})))=n_{2}+2$. ∎
###### Proposition 3.10.
If $G_{n}$ is the $n$-gear then for any $n\geq 3$
$\chi_{r}(G_{n})=\left\\{\begin{array}[]{l l}4,&\quad\mbox{if $r=2$. {}}\\\
\chi_{2}(C_{2n})+1,&\quad\mbox{if $r=3$.{}}\\\
min\\{r,\Delta\\}+1,&\quad\mbox{if $r\geq 4$. {}}\\\ \end{array}\right.$
###### Proof.
Let $k=\chi_{r}(G_{n})$. From [5] we know $\chi_{r}(G)\geq
min\\{r,\Delta\\}+1$. Taking $G=G_{n}$ we have $\chi_{r}(G_{n})\geq
min\\{r,\Delta\\}+1$. Let $k=\chi_{r}(G_{n})$.
Case 1: $r=2:$ We have $k\geq 3$. We assume that $k=3$. Let $c\colon
V(G_{n})\to\\{1,2,3\\}$ be a conditional $(3,2)$-coloring where $c(v_{i})=i$
for $i=1,2,3$. Then across $v_{1},v_{2},v_{3}$ (C1) must be true and in
particular (C2) must hold at $v_{2}$. To satisfy (C1) at $v_{3},c(v_{4})\neq
3$. We branch into two cases. If $c(v_{4})=1$, then by (C2) at $v_{4}$ we must
have $c(v_{5})\notin\\{1,3\\}$. Therefore we must have $c(v_{5})=2$. To
satisfy (C1), $c(v_{0})\notin\\{1,2,3\\}$. On the other hand if $c(v_{4})=2$
then to satisfy (C2) at $v_{4}$ we must have $c(v_{5})\notin\\{2,3\\}$. Hence
$c(v_{5})=1$. To satisfy (C2) at $v_{3}$ while preserving proper coloring,
$c(v_{0})\notin\\{1,2,3\\}$. Therefore $k\geq 4$. To show that $k=4$, it
suffices to construct a conditional $(4,2)$-coloring of $G_{n}$. Define
$c\colon V(G_{n})\to\\{1,2,3,4\\}$ as follows:
Case 1.1: $n\equiv 0\pmod{3}:$ Set $c(v_{0})=4$ and
$c(v_{i})=\left\\{\begin{array}[]{l l}1,&\quad\mbox{if \ $i\bmod 3=1$. {}}\\\
2,&\quad\mbox{if \ $i\bmod 3=2$.{}}\\\ 3,&\quad\mbox{if \ $i\bmod 3=0$. {}}\\\
\end{array}\right.$
Case 1.2: $n\equiv 2\pmod{3}:$ Modify $c$ in case (1.1) by the assignment
$c(v_{2n})=2$.
Case 1.3: $n\equiv 1\pmod{3}:$ Modify $c$ in case (1.1) by the assignments
$c(v_{2n-4})=2,c(v_{2n-3})=1,c(v_{2n-2})=3,c(v_{2n-1})=2$ and $c(v_{2n})=3$.
It can be verified that $c$ is a conditional $(4,2)$-coloring of $G_{n}$. Thus
$k\leq 4$, and hence $\chi_{2}(G_{n})=4$.
Case 2: $r=3:$ We have $k\geq 4$. By (C1) $c(v_{i})\neq c(v_{0})$ and by (C2)
at $v_{i}$, $c(v_{i+1})\neq c(v_{0})$ for all odd $i$ in the range $1\leq
i\leq 2n-1$. Hence $c(v_{i})\neq c(v_{0})$ for all $i\;(i\neq 0)$. Since
$d(v_{i})\leq 3$ for $1\leq i\leq 2n$ and $r=3$, a conditional
$(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$ gives a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$. In turn, a conditional
$(\chi_{2}(C_{2n}),2)$-coloring of $C_{2n}$ with an additional color to
$v_{0}$ gives a conditional $(\chi_{3}(G_{n}),3)$-coloring of $G_{n}$.
Case 3: $r\geq 4:$ Let $l=\min\\{r,\Delta(G_{n})\\}$. We know
$\chi_{r}(G_{n})\geq l+1$. Since $v_{0}$ is the only vertex with $d(v_{0})\geq
r$ and $\chi_{r}(C_{2n})\leq l$, conditional $(l,r)$-coloring of
$V(G_{n})\setminus\\{v_{0}\\}$ such that $|c(V_{oddi})|=l$, with $(l+1)$th
color assigned to $v_{0}$ results in a conditional $(l+1,r)$-coloring of
$V(G_{n})$. Thus $\chi_{r}(G_{n})\leq l+1$, and hence
$\chi_{r}(G_{n})=\min\\{r,\Delta\\}+1$. ∎
## 4 Unique conditional colorability
If $\chi(G)=k$ and every $k$-coloring of $G$ induces the same partition of
$V(G)$ then $G$ is called uniquely k-colorable. In a similar way we define
unique $(k,r)$-colorability of graphs.
Definition. If $\chi_{r}(G)=k$ and every conditional $(k,r)$-coloring of $G$
induces the same partition of $V(G)$, then $G$ will be called uniquely
$(k,r)$-_colorable_.
With the following propositions we explore further unique $(k,r)$-coloring.
###### Proposition 4.1.
If $G$ is uniquely $p$-colorable and $r\leq p-1$ then $\chi_{r}(G)=p$.
###### Proof.
Since $G$ is uniquely $p$-colorable, let $c\colon V(G)\to\\{1,2,\ldots,p\\}$
be the proper coloring of $G$ and w.l.o.g. for $1\leq i\leq p$ let the color
class $C_{i}$ be defined as $C_{i}=\\{v:c(v)=i\\}$. For all $u\in V(G)$ if
$u\in C_{i}$ then for all $j\in\\{1,2,\ldots,p\\}$ there exists a $v\in
C_{j}\;(j\neq i)$ – this implies that for every $u\in V(G)$, $d(u)\geq p-1$
and $|c(N(u))|=p-1$. Note that $c$ is also a conditional $(p,r)$-coloring of
$G$ because (C2) is also satisfied as for every $u\in V(G),|c(N(u))|=p-1\geq$
min $\\{r,d(u)\\}$. ∎
The definition of conditional $(k,r)$-coloring of $G$ and Proposition 4.1
together imply:
###### Corollary.
Every uniquely $p$-colorable graph $G$ is also uniquely $(p,p-1)$-colorable.
###### Proposition 4.2.
For every $k\geq 3$, there exists a uniquely $(3,2)$-colorable graph $G_{k}$
with $k+2$ vertices.
###### Proof.
We take $G_{1}$ to be $C_{3}$. Suppose that $k\geq 1$ and assume that $G_{k}$
has been obtained. From $G_{k}$, we construct $G_{k+1}$ by introducing a new
vertex $w$. The vertex and edge sets of $G_{k+1}$ are defined thus:
$\displaystyle V(G_{k+1})$
$\displaystyle=V(G_{k})\cup\\{w\\},\text{where}\;w\notin V(G_{k}).$
$\displaystyle E(G_{k+1})$
$\displaystyle=E(G_{k})\cup\\{(u,w),(v,w)\\},\text{where}\ (u,v)\in E(G_{k}).$
Evidently, $|V(G_{k})|=k+2$. We need to show that for any $k$, $G_{k}$ is
uniquely $(3,2)$-colorable. We prove the result by induction on $k$. Clearly
$G_{1}$ is uniquely $(3,2)$-colorable. For some $k>1$, assume $G_{k}$ is
uniquely $(3,2)$-colorable. Then $G_{k+1}$ is also uniquely $(3,2)$-colorable
because the new vertex $w\in G_{k+1}$ is assigned a third color different from
its two neighbors which are adjacent; by the inductive hypothesis the result
follows. ∎
###### Proposition 4.3.
Every path $P_{n}\;(n\geq 3)$ is uniquely $(3,2)$-colorable.
###### Proof.
From [5] we get $\chi_{2}(P_{n})=3$ . Define a conditional $(3,2)$-coloring
$c\colon V(P_{n})\to\\{1,2,3\\}$ by
$\displaystyle c^{-1}(1)=C_{1}$ $\displaystyle=\\{v_{i}:i\bmod 3=1\\},$
$\displaystyle c^{-1}(2)=C_{2}$ $\displaystyle=\\{v_{i}:i\bmod 3=2\\},$
$\displaystyle c^{-1}(3)=C_{3}$ $\displaystyle=\\{v_{i}:i\bmod 3=0\\},$
where, $C_{1}$,$C_{2}$ and $C_{3}$ are the color classes. In the conditional
$(3,2)$-coloring of $P_{n}$, for any two vertices $v_{i},v_{j}\;(i\neq j)$ in
$V(P_{n})$ if $|i-j|\bmod 3=0$ then $v_{i}$ and $v_{j}$ must be colored same;
otherwise either (C1) will be violated at $v_{min\\{i,j\\}+1}$ or (C2) will be
violated at $v_{max\\{i,j\\}-1}$. Since $c$ is the only coloring wherein for
all $v_{i},v_{j}\in V(P_{n})$, $c(v_{i})=c(v_{j})$, if $|i-j|\bmod 3=0$,
$P_{n}$ is uniquely $(3,2)$-colorable. ∎
###### Proposition 4.4.
If $T$ ($\neq P_{n}$) is a rooted tree with $n$ verices and $k=\chi_{r}(T)$
then $T$ is not uniquely $(k,r)$-colorable unless $k=n$.
###### Proof.
Let the root of $T$ be a vertex of degree $\Delta(T)$. Let $c\colon
V(T)\to\\{1,2,\ldots,k\\}$ be a conditional $(k,r)$-coloring of $T$. We show
that a new conditional $(k,r)$-coloring of $T$ can be obtained based on $c$ if
$k\neq n$. We know that every tree $T$ has at least two vertices say, $u,v$
with degree one. Let $p(v)$ denote the parent of $v$. We have the following
cases:
Case 1: $p(u)=p(v):$
Case 1.1: $c(u)=c(v):$ Since $r\geq 2$ assigning one of the colors from the
set $c(N_{T}(p(u)))\setminus\\{c(v)\\}$ to $u$ results in a new conditional
$(k,r)$-coloring.
Case 1.2: $c(u)\neq c(v):$
Case 1.2.1: There exists a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$ Interchanging the colors of $u$ and $v$ results in a different
induced partition of $V(T)$.
Case 1.2.2: There doesn’t exist a vertex $w\in V(T)$ such that $c(w)=c(u)$ or
$c(w)=c(v):$
Case 1.2.2.1: $r\geq\Delta:$ If $n\neq\Delta+1$ then there exists a vertex
$w^{\prime}\in N_{T}(p(u))$ such that $d(w^{\prime})\geq 2$; then
interchanging the colors of $u$ and $w^{\prime}$ results in a different
induced partition of $V(T)$ because the subtree rooted at $w^{\prime}$ doesn’t
contain any vertex colored $c(u)$. If $n=\Delta+1$ then $k=n$.
Case 1.2.2.2: $r<\Delta:$ There must exist at least two vertices
$w_{1},w_{2}\in N_{T}(p(u))$ such that $c(w_{1})=c(w_{2})$. Interchanging the
colors of $u$ and $w_{1}$ results in a different induced partition of $V(T)$
because the subtree rooted at $w_{1}$ doesn’t contain any vertex colored
$c(u)$.
Case 2: $p(u)\neq p(v):$
Case 2.1: $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or $d(p(v))<$ min
$\\{r,\Delta(T)\\}:$ Assigning to $u$ any color in the set $c(V(T))\setminus
c(N_{T}[p(u)])$ if $d(p(u))<$ min $\\{r,\Delta(T)\\}$ or to $v$ any color in
the set $c(V(T))\setminus c(N_{T}[p(v)])$ if $d(p(v))<$ min
$\\{r,\Delta(T)\\}$ gives a new conditional $(k,r)$-coloring.
Case 2.2: $d(p(u))>$ min $\\{r,\Delta(T)\\}$ or $d(p(v))>$ min
$\\{r,\Delta(T)\\}:$ If $d(p(u))>$ min $\\{r,\Delta(T)\\}$ there must exist a
vertex $u^{\prime}\in N_{T}(p(u))$ such that $c(u)=c(u^{\prime})$ or two
vertices $u_{1},u_{2}\in N_{T}(p(u))$ such that $c(u_{1})=c(u_{2})\neq c(u)$.
Assigning to $u$ any of the color in the set
$c(V(T))\setminus\\{c(u^{\prime}),c(p(u))\\}$ or interchanging the colors of
$u$ and $u_{1}$ and the colors $c(u)$, $c(u_{1})$ in the subtree rooted at
$u_{1}$ gives a new conditional $(k,r)$-coloring in the former and latter
cases respectively. The case $d(p(v))>$ min $\\{r,\Delta(T)\\}$ is similar.
Case 2.3: $d(p(u))=d(p(v))=$ min $\\{r,\Delta(T)\\}:$
Case 2.3.1: min $\\{r,\Delta(T)\\}>2:$ There exists a $w\in V(T)$ such that
$c(u)\neq c(w)$ and $p(u)=p(w)$. Making the color of $u$ as $c(w)$ and in the
subtree rooted at $w$, swapping the colors $c(u)$ and $c(w)$ gives a new
conditional $(k,r)$-coloring.
Case 2.3.2: min $\\{r,\Delta(T)\\}=2:$ Since $T\neq P_{n}$ so $\Delta(T)>2$
and $r=2$. There exists an ancestor of $u$ and $v$ with degree $\geq 2$
because the root of $T$ has maximum degree. Let $w$ with $d(w)\geq 2$ be the
closest ancestor of $u$ and $v$. Then there exists two children of $w$ namely
$w_{1}$ and $w_{2}$ which are ancestors of $u$, $v$ respectively. If $w$ is
the root of $T$ and $c(w_{1})\neq c(w_{2})$, interchanging the colors
$c(w_{1})$ and $c(w_{2})$ in the subtree rooted at $w_{1}$ gives a new
conditional $(k,r)$-coloring. If $w$ is not the root of $T$ and $c(w_{1})\neq
c(w_{2})$, interchanging the colors $c(w_{1})$ and $c(w_{2})$ in the subtree
rooted at $w$ gives a new conditional $(k,r)$-coloring. Otherwise (i.e., if
$c(w_{1})=c(w_{2})$), there exists a $w_{3}\in N_{T}(w)$ such that
$c(w_{3})\neq c(w_{1})$ and interchanging the colors $c(w_{1})$ and $c(w_{2})$
in the subtree rooted at $w_{1}$ gives a new conditional $(k,r)$-coloring. ∎
## References
* [1] R. Balakrishnan and K. Renganathan, A Textbook of Graph Theory, Springer, 2000.
* [2] J. A. Gallian, A dynamic survey of graph labeling, Elect. J. Combin. 16 (DS6) (2009).
* [3] M. C. Golumbic, Algorithmic graph theory and perfect graphs, Elsevier, North Holland, 2004.
* [4] F. Harary, Graph theory, Addison-Wesley, MA, 1969.
* [5] H. J. Lai, J. Lin, B. Montgomery, T. Shui and S. Fan, Conditional colorings of graphs, Discr. Math. 306 (2006), pp. 1997-2004.
* [6] X. Li, X. Yao, W. Zhou and H. Broersma, Complexity of conditional colorability of graphs, Appl. Math. Lett. 22 (2009), pp. 320-324.
* [7] D. Michalak, On middle and total graphs with coarseness number equal 1, Lect. Notes in Math. 1018: Graph Theory, Springer, (1983), 139-150.
* [8] D. B. West, Introduction to graph theory, Prentice-Hall, 2003.
|
arxiv-papers
| 2012-01-30T10:54:47 |
2024-09-04T02:49:26.808003
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "P. V. Subba Reddy and K. V. Iyer",
"submitter": "Iyer Viswanathan K.",
"url": "https://arxiv.org/abs/1201.6166"
}
|
1201.6268
|
-titleHadron Collider Physics symposium (HCP-2011), Paris, France, November 14-18 2011 11institutetext: Department of Physics, University of Oxford, Oxford, United Kingdom
# A search for time-integrated $C\\!P$ violation in $\boldmath D^{0}\to
h^{-}h^{+}$ decays
M. J. Charles 11 m.charles1@physics.ox.ac.uk on behalf of the LHCb
collaboration
###### Abstract
The preliminary results of a search for time-integrated $C\\!P$ violation in
$D^{0}\to h^{-}h^{+}$ ($h=K$, $\pi$) decays performed with 0.6 fb-1 of data
collected by LHCb in 2011 are presented. The flavour of the charm meson is
determined by the charge of the slow pion in the $D^{*+}\to D^{0}\pi^{+}$ and
$D^{*-}\to\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{-}$ decay
chains. The difference in $C\\!P$ asymmetry between $D^{0}\to K^{-}K^{+}$ and
$D^{0}\to\pi^{-}\pi^{+}$, $\Delta A_{C\\!P}\equiv
A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+})$, is measured to be
$\Delta A_{C\\!P}=\left[-0.82\pm 0.21(\mathrm{stat.})\pm
0.11(\mathrm{syst.})\right]\%$. This differs from the hypothesis of $C\\!P$
conservation by $3.5\sigma$.
## 1 Introduction
The time-integrated $C\\!P$ asymmetry, $A_{C\\!P}(f)$, for the final states
$f=K^{-}K^{+}$ and $f=\pi^{-}\pi^{+}$ has two contributions: an indirect
component (to a good approximation universal for $C\\!P$ eigenstates in the
Standard Model) and a direct component (in general final state dependent). In
the limit of U-spin symmetry, the direct component is equal and opposite in
sign for $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ bib:grossman_kagan_nir . In the
Standard Model, $C\\!P$ violation is expected to be small bib:cicerone ;
bib:lenz ; bib:grossman_kagan_nir . However, in the presence of physics beyond
the Standard Model the rate of $C\\!P$ violation could be enhanced
bib:grossman_kagan_nir ; bib:littlest_higgs . No prior evidence of $C\\!P$
violation in the charm sector has been found.
The most precise measurements to date of the time-integrated $C\\!P$
asymmetries in $D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$ were made by
the CDF, BABAR, and BELLE collaborations bib:cdf_paper ; bib:babar_paper2008 ;
bib:belle_paper2008 . LHCb has previously reported preliminary results based
on 37 $\rm pb^{-1}$ of data collected in 2010 bib:lhcb2010 . In the limit that
the efficiency for selected events is independent of the decay time, the
difference between the two time-integrated asymmetries, $\Delta
A_{C\\!P}\equiv A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+})$, is equal
to the difference in the direct $C\\!P$ asymmetry. However, if the dependence
of the efficiency on the decay time is different for the $K^{-}K^{+}$ and
$\pi^{-}\pi^{+}$ final states, a contribution from indirect $C\\!P$ violation
remains. The Heavy Flavor Averaging Group (HFAG) has combined time-integrated
and time-dependent measurements of $C\\!P$ asymmetries taking account of the
different decay time acceptance to obtain world-average values for the
indirect $C\\!P$ asymmetry of $a_{C\\!P}^{\mathrm{ind}}=(-0.03\pm 0.23)\%$ and
the difference in direct $C\\!P$ asymmetry between the final states of $\Delta
a_{C\\!P}^{\mathrm{dir}}=(-0.42\pm 0.27)\%$ bib:hfag .
In these proceedings, LHCb results bib:this_result_conf ;
bib:this_result_eprint for the measurement of the difference in integrated
$C\\!P$ asymmetries between $D^{0}\to K^{-}K^{+}$ and
$D^{0}\to\pi^{-}\pi^{+}$, performed with approximately 0.6 $\rm fb^{-1}$ of
data collected in 2011, are presented. The initial state ($D^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) is tagged by requiring a
$D^{*+}\to D^{0}\pi^{+}$ decay. The use of charge-conjugate modes is implied
throughout, except in the definition of asymmetries.
## 2 Formalism
The raw asymmetry for tagged $D^{0}$ decays to a final state $f$ is given by
$A_{\mathrm{raw}}(f)$, defined as:
$A_{\mathrm{raw}}(f)\equiv\frac{N(D^{*+}\to
D^{0}(f)\pi^{+})\,-\,N(D^{*-}\to\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\bar{f})\pi^{-})}{N(D^{*+}\to
D^{0}(f)\pi^{+})\,+\,N(D^{*-}\to\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}(\bar{f})\pi^{-})},$ (1)
where $N(X)$ refers to the number of reconstructed events of decay $X$ after
background subtraction.
The raw asymmetries may be written as a sum of various components, coming from
both physics and detector effects:
$\displaystyle A_{\mathrm{raw}}(f)$ $\displaystyle=$ $\displaystyle
A_{C\\!P}(f)\,+\,A_{\mathrm{D}}(f)\,+\,A_{\mathrm{D}}(\pi_{\mathrm{s}})\,+\,A_{\mathrm{P}}(D^{*+}).$
(2)
Here, $A_{C\\!P}(f)$ is the intrinsic physics $C\\!P$ asymmetry,
$A_{\mathrm{D}}(f)$ is the asymmetry for selecting the $D^{0}$ decay into the
final state $f$, $A_{\mathrm{D}}(\pi_{\mathrm{s}})$ is the asymmetry for
selecting the ‘slow pion’ from the $D^{*+}$ decay chain, and
$A_{\mathrm{P}}(D^{*+})$ is the production asymmetry for prompt $D^{*+}$
mesons. The asymmetries $A_{C\\!P}$, $A_{\mathrm{D}}$ and $A_{\mathrm{P}}$ are
defined in the same fashion as $A_{\mathrm{raw}}$.
For a two-body decay of a spin-0 particle to a self-conjugate final state
there can be no $D^{0}$ detection asymmetry, i.e.
$A_{\mathrm{D}}(K^{-}K^{+})=A_{\mathrm{D}}(\pi^{-}\pi^{+})=0.$ Moreover, to
first order $A_{\mathrm{D}}(\pi_{\mathrm{s}})$ and $A_{\mathrm{P}}(D^{*+})$
cancel out in the difference
$A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+}),$
leaving a quantity, defined as $\Delta A_{C\\!P}$, equal to the difference in
physics asymmetries:
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle\equiv$ $\displaystyle
A_{C\\!P}(K^{-}K^{+})\,-\,A_{C\\!P}(\pi^{-}\pi^{+}),$ (3) $\displaystyle=$
$\displaystyle
A_{\mathrm{raw}}(K^{-}K^{+})\,-\,A_{\mathrm{raw}}(\pi^{-}\pi^{+}).$ (4)
To minimize second order effects, related to the slightly different kinematic
properties of the two decay modes, the analysis is done in bins of the
relevant kinematic variables, as shown later in Secs. 4 and 5. The physics
asymmetry of each final state may be written at first order as bib:bigi_d2hh :
$\displaystyle A_{C\\!P}(f)$ $\displaystyle\approx$ $\displaystyle
a^{\mathrm{dir}}_{C\\!P}(f)\,+\,\frac{\langle
t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P},$ (5)
where $a^{\mathrm{dir}}_{C\\!P}(f)$ is the asymmetry coming from direct
$C\\!P$ violation for the decay, $\langle t\rangle$ is the average decay time
in the sample used, $\tau$ the true $D^{0}$ lifetime, and
$a^{\mathrm{ind}}_{C\\!P}$ is the asymmetry associated with $C\\!P$ violation
in the mixing. To a good approximation this latter quantity is universal
bib:grossman_kagan_nir , and so
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=$
$\displaystyle\left[a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})\,-\,a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})\right]\,+\,\frac{\Delta\langle
t\rangle}{\tau}a^{\mathrm{ind}}_{C\\!P},$ (6)
where $\Delta\langle t\rangle$ is the difference in average decay time of the
$D^{0}$ mesons in the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ samples. In the limit
that $\Delta\langle t\rangle$ vanishes, $\Delta A_{C\\!P}$ probes the
difference in direct $C\\!P$ violation between the two decays.
## 3 Dataset and selection
A description of the LHCb detector may be found in Ref. LHCb . The field
direction in the LHCb dipole is such that charged particles are deflected in
the horizontal plane. The current direction in the dipole was changed several
times during data taking; about 60% of the data was taken with one polarity
and 40% with the other.
Selections are applied to provide samples of $D^{*+}\to D^{0}\pi^{+}$
candidate decays, with $D^{0}\to K^{-}K^{+}$ and $\pi^{-}\pi^{+}$. A loose
$D^{0}$ selection including a mass window of full width 100 MeV$/c^{2}$ was
already applied during data taking, in the final stage of the high level
trigger (HLT). In the offline analysis only candidates that were accepted by
this trigger algorithm are considered. Both the offline and HLT selections
impose a variety of requirements on kinematics and decay time to isolate the
decays of interest, including requirements on the track fit quality, on the
$D^{0}$ and $D^{*+}$ vertex fit quality, on the transverse momentum of the
$D^{0}$ ($\mbox{$p_{\rm T}$}>2$ GeV$/c$), on the decay time $t$ of the $D^{0}$
($ct>100\,\,\upmu\rm m$), on the helicity angle of the $D^{0}$ decay, that the
$D^{0}$ points back to a primary vertex, and that the $D^{0}$ daughter tracks
do not. Fiducial requirements are imposed to ensure that the tagging soft pion
lies within the central region of the detector acceptance. In addition, the
offline analysis exploits the capabilities of the RICH system to distinguish
between pions and kaons when reconstructing the $D^{0}$.
Defining the mass difference as $\delta m\equiv
m(h^{+}h^{-}\pi^{+})-m(h^{+}h^{-})-m(\pi^{+})$, the mass and mass difference
spectra of selected candidates are shown in Figs. 1 and 2, respectively. The
difference in width between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ mass spectra
arises from the different opening angles of the two decays. The $D^{*+}$
signal yields are approximately $1.44\times 10^{6}$ in the tagged $K^{-}K^{+}$
sample, and $0.38\times 10^{6}$ in the tagged $\pi^{-}\pi^{+}$ sample. The
fractional difference in average decay time of $D^{0}$ candidates passing the
selection between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ samples is
$\Delta\langle t\rangle/\tau=(9.8\pm 0.9)\%$.
Figure 1: Fits to the $m(K^{-}K^{+})$ and $m(\pi^{-}\pi^{+})$ spectra of
$D^{*+}$ candidates passing the selection and satisfying $0<\delta m<15$
MeV$/c^{2}$. The vertical blue lines indicate the signal window of 1844–1884
MeV$/c^{2}$.
Figure 2: Fits to the mass difference spectra, where the $D^{0}$ is
reconstructed in the final states $K^{-}K^{+}$ (top) and $\pi^{-}\pi^{+}$
(bottom), with a $D^{0}$ mass lying in the window of 1844–1884 MeV$/c^{2}$.
## 4 Fit procedure
Fits are performed on the samples in order to determine
$A_{\mathrm{raw}}(K^{-}K^{+})$ and $A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. The
analysis is performed in 54 kinematic bins, divided by $p_{\rm T}$ and
pseudorapidity ($\eta$) of the $D^{*+}$ candidates, momentum of the tagging
soft pion, and whether the initial trajectory of the slow pion is towards the
left or right half of the detector. A binning is imposed for the reason that
the production and detection asymmetries can in general vary with $p_{\rm T}$
and $\eta$, and so can the detection efficiency of the two different $D^{0}$
decays, in particular through the effects of the particle identification
requirements.
The events are further partitioned in two ways. First, the data are divided
between the two dipole magnet settings. Second, the first 350 pb-1 of data are
processed separately from the remainder, with the division aligned with a
break in data taking due to a LHC technical stop. In total, therefore, 216
independent measurements are considered for each decay mode.
One-dimensional fits to the mass difference spectra are performed. The fits
include separate components for the signal and background lineshapes. The
signal is described as the sum of two Gaussian functions with a common mean,
convolved with an asymmetric function. The background is described by an
empirical function of the form
$\left[1-e^{-(\delta m-\delta m_{0})/c}\right],$
where $\delta m_{0}$ and $c$ are parameters describing the threshold and shape
of the function, respectively. In each case an unbinned maximum likelihood fit
is used. The $D^{*+}$ and $D^{*-}$ samples are fitted simultaneously and share
several shape parameters, though a charge-dependent offset in the central
value and overall scale factor in the mass resolution are allowed. The raw
asymmetry in the yields of the signal component is extracted directly from
this simultaneous fit. An example fit from one measurement bin is shown in
Fig. 3.
The one-dimensional fits used for the tagged data do not distinguish between
correctly reconstructed signal and backgrounds that peak in the mass
difference. Such backgrounds can arise from $D^{*+}$ decays in which the
correct slow pion is found but the $D^{0}$ is partially mis-reconstructed.
However, these backgrounds are suppressed by the use of tight particle
identification requirements and a narrow $D^{0}$ mass window. From studies of
the $D^{0}$ mass sidebands this contamination is found to be approximately 1%
of the signal yield.
Figure 3: Example fit used in the $\Delta A_{CP}$ analysis. The first
kinematic bin of the first run period with magnet up polarity is shown for the
$D^{0}\to K^{-}K^{+}$ final state.
## 5 Results and systematic uncertainties
A value of $\Delta A_{C\\!P}$ is determined in each measurement bin using the
results for $A_{\mathrm{raw}}(K^{-}K^{+})$ and
$A_{\mathrm{raw}}(\pi^{-}\pi^{+})$. The $\chi^{2}/ndf$ of these measurements
has a value of 211/215. A weighted average is performed to yield the result
$\Delta A_{C\\!P}=(-0.82\pm 0.21)\%$.
Numerous robustness checks are made, including monitoring the value of $\Delta
A_{C\\!P}$ as a function of time (Fig. 4), re-performing the measurement with
more restrictive RICH particle identification requirements, and using a
different $D^{*+}$ selection. Potential biases due to the inclusive hardware
trigger selection are investigated with the subsample of data in which one of
the signal final-state tracks is directly responsible for the hardware trigger
decision. In all cases good stability is observed.
Figure 4: Time-dependence of the measurement. The data are divided into 19
disjoint, contiguous, time-ordered blocks and the value of $\Delta A_{CP}$
measured in each block. The red dashed line shows the result for the combined
sample.
Systematic uncertainties are assigned by loosening the fiducial requirement on
the soft pion; by assessing the effect of potential effect peaking background
in toy Monte Carlo studies; by repeating the analysis with the asymmetry
extracted through sideband subtraction instead of a fit; with all candidates
but one (chosen at random) removed in events with multiple candidates; and
comparing with the result obtained with no kinematic binning. In each case the
full value of the change in result is taken as the systematic uncertainty.
These uncertainties are listed in Table 1. The sum in quadrature is $0.11\%$.
Table 1: Summary of absolute systematic uncertainties for $\Delta A_{C\\!P}$. Effect | Uncertainty
---|---
Fiducial requirement | 0.01%
Peaking background asymmetry | 0.04%
Fit procedure | 0.08%
Multiple candidates | 0.06%
Kinematic binning | 0.02%
Total | 0.11%
## 6 Conclusions
LHCb has measured the time-integrated difference in $C\\!P$ asymmetry between
$D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$ decays, $\Delta
A_{C\\!P}=\left(a^{\mathrm{dir}}_{C\\!P}(K^{-}K^{+})\,-\,a^{\mathrm{dir}}_{C\\!P}(\pi^{-}\pi^{+})\right)\,+\,0.098\,a^{\mathrm{ind}}_{C\\!P}$,
to be
$\Delta A_{C\\!P}=\left[-0.82\pm 0.21(\mathrm{stat.})\pm
0.11(\mathrm{syst.})\right]\%$
with 0.6 fb-1 of 2011 data, where the first uncertainty is statistical and the
second systematic. Combining the statistical and systematic uncertainties in
quadrature, the significance of the measured deviation from zero is
$3.5\sigma$. The result is consistent with the current HFAG world average
bib:hfag .
Subsequent to this result being presented, several papers on the theoretical
interpretation have been written, examining the possible size and uncertainty
in the SM contribution and considering possible new physics contributions. A
partial list may be found in Ref. bib:reaction:Isidori ; bib:reaction:Brod ;
bib:reaction:Wang ; bib:reaction:Rozanov ; bib:reaction:Pirtskhalava ;
bib:reaction:Hochberg ; bib:reaction:Cheng ; bib:reaction:Bhattacharya ;
bib:reaction:Chang . We thank our theory colleagues for their help in
understanding this result.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* (1) Y. Grossman, A. L. Kagan and Y. Nir, Phys. Rev. D 75 (2007) 036008.
* (2) S. Bianco, F. L. Fabbri, D. Benson and I. Bigi, Riv. Nuovo Cim. 26N7 (2003) 1.
* (3) M. Bobrowski, A. Lenz, J. Riedl and J. Rohrwild, JHEP 1003 (2010) 009.
* (4) I. I. Bigi, M. Blanke, A. J. Buras and S. Recksiegel, JHEP 0907 (2009) 097.
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* (6) BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 100 (2008) 061803.
* (7) Belle Collaboration, M. Staric et al., Phys. Lett. B 670 (2008) 190.
* (8) LHCb Collaboration, LHCb-CONF-2011-023.
* (9) Heavy Flavor Averaging Group, D. Asner et al., arXiv:1010.1589 [hep-ex]; http://www.slac.stanford.edu/xorg/hfag/
charm/EPS11/DCPV/direct_indirect_cpv.html.
* (10) LHCb Collaboration, LHCb-CONF-2011-061.
* (11) LHCb Collabroation, R. Aaij et al., arXiv:1112.0938 [hep-ex] (submitted to Phys. Rev. Lett.).
* (12) I. I. Bigi, A. Paul and S. Recksiegel, JHEP 1106 (2011) 089.
* (13) LHCb Collaboration, A. A. Alves et al., JINST 3 (2008) S08005.
* (14) G. Isidori, J. F. Kamenik, Z. Ligeti and G. Perez, arXiv:1111.4987 [hep-ph].
* (15) J. Brod, A. L. Kagan and J. Zupan, arXiv:1111.5000 [hep-ph].
* (16) K. Wang and G. Zhu, arXiv:1111.5196 [hep-ph].
* (17) A. N. Rozanov and M. I. Vysotsky, arXiv:1111.6949 [hep-ph].
* (18) D. Pirtskhalava and P. Uttayarat, arXiv:1112.5451 [hep-ph].
* (19) Y. Hochberg and Y. Nir, arXiv:1112.5268 [hep-ph].
* (20) H.-Y. Cheng and C.-W. Chiang, arXiv:1201.0785 [hep-ph].
* (21) B. Bhattacharya, M. Gronau and J. L. Rosner, arXiv:1201.2351 [hep-ph].
* (22) X. Chang, M.-K. Du, C. Liu, J.-S. Lu and S. Yang, arXiv:1201.2565 [hep-ph].
|
arxiv-papers
| 2012-01-30T16:13:35 |
2024-09-04T02:49:26.817276
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. J. Charles (for the LHCb Collaboration)",
"submitter": "Matthew Charles",
"url": "https://arxiv.org/abs/1201.6268"
}
|
1201.6274
|
# Unusual size effects on thermoelectricity in a strongly correlated oxide
J. Ravichandran jayakanth@berkeley.edu Applied Science and Technology
Graduate Group, University of California, Berkeley, CA 94720, USA Materials
Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA A. K. Yadav Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA Department of Materials Science and
Engineering, University of California, Berkeley, CA 94720, USA W. Siemons
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak
Ridge, TN 37831, USA M. A. McGuire Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA V. Wu
Department of Materials Science and Engineering, University of California,
Berkeley, CA 94720, USA A. Vailionis Geballe Laboratory for Advanced
Materials, Stanford University, Stanford, California 94305, USA A. Majumdar
ARPA-E, US Department of Energy, 1000 Independence Avenue, Washington, DC
20585, USA R. Ramesh Materials Sciences Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA Department of Materials Science and
Engineering, University of California, Berkeley, CA 94720, USA Department of
Physics, University of California, Berkeley, CA 94720, USA SETP, US
Department of Energy, 1000 Independence Avenue, Washington, DC 20585, USA
###### Abstract
We investigated size effects on thermoelectricity in thin films of a strongly
correlated layered cobaltate. At room temperature, the thermopower is
independent of thickness down to 6 nm. This unusual behavior is inconsistent
with the Fuchs-Sondheimer theory, which is used to describe conventional
metals and semiconductors, and is attributed to the strong electron
correlations in this material. Although the resistivity increases, as
expected, below a critical thickness of $\sim$ 30 nm. The temperature
dependent thermopower is similar for different thicknesses but resistivity
shows systematic changes with thickness. Our experiments highlight the
differences in thermoelectric behavior of strongly correlated and uncorrelated
systems when subjected to finite size effects. We use the atomic limit Hubbard
model at the high temperature limit to explain our observations. These
findings provide new insights on decoupling electrical conductivity and
thermopower in correlated systems.
###### pacs:
71.27.+a,79.10.-n,73.50.Lw,73.50.-h
## I Introduction
Among various energy conversion methods, thermoelectricity deals with direct
inter-conversion of thermal and electrical energy. The efficiency of a
thermoelectric heat engine is related to a material dependent figure of merit,
$Z$, given by $S^{2}\sigma/\kappa$, where $S$ is the thermopower or Seebeck
coefficient, $\sigma$ and $\kappa$ are the electrical and thermal
conductivities (lattice and electronic), respectively. In conventional
thermoelectric materials, electrical conductivity and thermopower are governed
by the density of states, chemical potential and the scattering mechanism. Due
to this coupling between thermopower and electrical conductivity, achieving
high $Z$ has been a challenging task. Hicks and DresselhausHicks and
Dresselhaus (1993a, b) proposed quantum confinement as a means to enhance the
thermoelectric power factor ($S^{2}\sigma$) in nanostructured materials.
NanostructuringKim et al. (2006); Poudel et al. (2008) showed no significant
enhancement in power factor, as the enhancement in thermopower was offset by
the decrease in electrical conductivity (both mobility and carrier density).
Nevertheless, nanostructuring is an effective means to reduce the lattice part
of the thermal conductivity without significantly affecting electrical
transport. Investigations exploring quantum confinement effects have primarily
centered around conventional semiconductors, which show band-like transport.
Reduced dimensions in materials can have profound influence on transport
properties due to effects such as quantization and changes in scattering
mechanism. Thin films are the commonly used to study two dimensional transport
behavior. There are several reports on thickness dependent transport
measurements on thin film materials showing band-like transport,Pichard et al.
(1980); Worden (1958); Cho and Kim (2005); Ganesan and Sivaramakrishnan
(2000); Rogacheva et al. (2003, 2002); Ohtomo and Hwang (2004) but few studies
focus on size effects on strongly correlated materials.Schultz et al. (2009)
Several of these investigations have centered around the effect of transverse
confinement on thermoelectric transport. Recent investigations have shown
large thermoelectric responses in complex oxides.Okuda et al. (2001); Terasaki
et al. (1997); Lee et al. (2006) Particularly, strongly correlated oxides such
as cobaltates show enhanced thermopower, which cannot be explained by band-
like transport. The transport behavior in these cobaltates has been explained
by the Hubbard model, with the incorporation of spin degeneracy.Koshibae and
Maekawa (2001); Wang et al. (2003) Size dependent thermoelectric measurements
on a strongly correlated cobaltate poses several interesting questions about
the role of quantum confinement in thermoelectric transport, the effect of
thickness on the mobility and spin degeneracy. In this article, we report
unusual size effects on thermoelectricity in a strongly correlated
thermoelectric oxide, Bi2Sr2Co2Oy (abbreviated as BSCO) and use the Hubbard
model to explain the physics behind the observations. We performed
thermoelectric transport measurements in thin films of BSCO both as a function
of thickness and temperature to elucidate size effects on this system and
discuss possible directions for correlated thermoelectrics.
## II Experimental Methods
Thin films of BSCO (3 – 170 nm thick) were grown on yttria stablized zirconia
(YSZ) substrates using pulsed laser deposition from a stoichiometric ceramic
target of BSCO. The growth was carried out at 700∘C, with an oxygen partial
pressure of 500 mTorr. All films were grown using a 248 nm KrF excimer laser
with a fluence of 2 J/cm2 at a repetition rate of 1 Hz. Films were
characterized by X-ray diffraction (XRD) for phase purity and crystallinity,
X-ray reflectivity (XRR) for thickness, atomic force microscopy (AFM) for
surface roughness and Rutherford backscattering (RBS) for chemical
composition. All transport measurements were carried out in the van der Pauw
geometry. Triangular ohmic metal contacts of side 1 mm (15 nm Ti/100 nm Pd)
were deposited on the corners of the films using electron beam evaporation.
Hall measurements at room temperature were carried out in air using a home-
built apparatus with a 1 Tesla electromagnet. Low temperature resistivity and
thermopower measurements were performed in a Quantum Design PPMS.
Figure 1: (a) The out-of-plane x-ray diffraction pattern for BSCO film on YSZ
is shown. The inset compares the rocking curve for the substrate and the film.
(b) Pole figure scan for (116) plane of BSCO. The four peaks at $\phi$ = 45∘,
135∘, 225∘ and 315∘ correspond to the substrate YSZ (111). The corresponding
2$\theta$ value for the scan is 29.75∘. (c) Pole figure scan for (11 14) plane
of BSCO. The four peaks at $\phi$ = 0∘, 90∘, 180∘ and 270∘ correspond to the
substrate YSZ (2 2 0). The corresponding 2$\theta$ value for the scan is
50.46∘. Figure 2: The thickness dependent thermopower and resistivity of thin
films of BSCO measured at room temperature. The thicknesses of the films range
from 6 – 170 nm. The resistivity of the films was calculated by dividing the
measured sheet resistance by the measured thickness. The actual resistivity
can be estimated only after considering the thickness of dead layer present in
these films, which we show in Fig. 3, but the overall trend remains the same.
## III Results and Discussion
The XRD measurements indicated that the films were single phase and oriented
with c-axis (axis perpendicular to the layers) along the out-of-plane
direction. The out-of-plane x-ray diffraction pattern for the BSCO film grown
on YSZ is shown in Fig. 1(a). All the peaks can be indexed to the (00$l$)
planes of BSCO and no secondary phase or other orientations were observed. The
inset of Fig. 1(a) shows the rocking curve for the film as compared to the
substrate. The films showed rocking curves with full-width-at-half-maximum of
$\sim$ 0.05–0.2∘ (compared to substrate’s 0.02∘). In order to establish the
in-plane epitaxial relationship between the film and substrate, we performed
phi-pole scans about (116) and (11 14) peaks of BSCO. The pole figures for the
two cases are shown in Fig. 1(b) and Fig. 1(c) respectively. It is evident
from the figures that the BSCO film doesn t have any preferential in-plane
epitaxial relationship with the substrate. Thus, we have excellent out-of-
plane texturing but no in-plane relationship with the substrate, leading to a
wire texture scenario. In Fig. 1(b), four peaks at $\phi$ = 45∘, 135∘, 225∘
and 315∘ correspond to the substrate YSZ (111). The corresponding 2$\theta$
value for the scan is 29.75∘. In Fig. 1(c), four peaks at $\phi$ = 0∘, 90∘,
180∘ and 270∘ correspond to the substrate YSZ (220). The corresponding
2$\theta$ value for the scan is 50.46∘.
The primary result of this work is summarized in Fig. 2 where thickness
dependent thermopower and resistivity for BSCO films at room temperature are
shown. The nominal resistivity remained constant till $\sim$30 nm and for
lower thicknesses the resistivity increased with decreasing thickness.
Surprisingly, the thermopower remained constant ($\sim$100–110 $\mu$V/K) over
the studied thickness range. In comparison, thermopower of Se doped Bi2Te3
decreases by 65% from the bulk value when the thickness is decreased to 50
nm.Cho and Kim (2005) Unlike BSCO, in a typical metalPichard et al. (1980);
Worden (1958) or a semiconductorCho and Kim (2005); Ganesan and
Sivaramakrishnan (2000), decreasing thickness results in decrease in
thermopower and increase in resistivity, due to surface scattering. In those
systems, the observed thickness dependent thermoelectric properties can be
explained by Fuchs-Sondheimer theorySondheimer (1952) quantitatively for
metalsPichard et al. (1980); Worden (1958) and qualitatively for
semiconductors.Cho and Kim (2005); Ganesan and Sivaramakrishnan (2000);
Rogacheva et al. (2003) Fuchs-Sondheimer theory uses the energy dependent
surface scattering as an additional scattering mechanism which becomes
dominant when the thickness of the films are comparable to the bulk mean free
path of the electrons. The thickness dependent thermopower and resistivity as
predicted by this theory are given below.
Figure 3: The sheet carrier density and Hall mobility as a function of
thickness at room temperature. (Top panel) The sheet carrier density data was
an excellent fit for a straight line with a small offset in thickness $\sim$ 4
nm. (Bottom panel) The thermopower and the actual resistivity as a function of
thickness, with the surface scatterings fits clearly showing the deviation for
the thermopower data. The surface scattering model used the mean free path
estimated from the resistivity data and typical values for the energy
scattering dependent scattering term as 0.1 and 0.2.Cho and Kim (2005) The
actual resistivity of the films is calculated accounting for the dead layer.
$\rho_{f}=\rho_{b}\left(1+\frac{3l_{b}}{8t}(1-p)\right)$ (1)
$S_{f}=S_{b}\left(1-\frac{3l_{b}}{8t}(1-p)\frac{U_{b}}{1+U_{b}}\right)$ (2)
where $\rho_{f}$ is film resistivity, $\rho_{b}$ is bulk resistivity, $S_{f}$
is film thermopower, $S_{b}$ is bulk thermopower, $l_{b}$ is bulk electron
mean free path, $t$ is the thickness, $p$ is the specularity parameter and
$U_{b}$ is the energy dependent scattering term. If the Fuchs-Sondheimer model
is applicable to our system, we expect a decrease in thermopower as thickness
decreases, contradictory to the observed constant thermopower. We use in-depth
transport measurements, to eliminate different scenarios under which this
conventional theory can be applicable to our case and to establish the role of
strong correlations in explaining our observations.
Figure 4: Temperature dependent thermopower for films of thickness 115, 88,
16 and 15 nm. The films showed very similar temperature dependence over the
measured temperature range. The inset shows the magnetic field dependence of
thermopower for a 88 nm film at 20 K with the field applied along the
temperature gradient. The calculated spin entropic contribution to thermopower
is shown as a red line.
First, it is essential to understand the contribution of carrier concentration
and mobility in increasing resistivity with decreasing thickness. Hall
measurements were used to measure the sheet carrier density and mobility (see
Fig. 3). The sheet carrier density scaled linearly with thickness and has a
small offset of $\sim$ 4 nm in the thickness axis. Thus, there is no thickness
dependent change in the volume carrier density but an insulating dead layer of
thickness $\sim$ 4 nm is present. We confirmed this by growing a film of
$\sim$ 3 nm thickness and found it to be insulating. The measured Hall
mobility showed a sharp decrease below 30 nm, as shown in Fig. 3. Hence, we
can conclude that the increase in resistivity shown in Fig. 2 is caused by the
presence of a dead layer and the mobility reduction caused by surface
scattering below $\sim$ 30 nm. We account for the presence of dead layer and
plot the revised resistivity in Fig. 3, which still shows very similar
thickness dependence as depicted in Fig. 2 and an excellent fit for the
surface scattering dominated resistivity shown in Eqn.1. On the other hand,
the calculated thermopower values (using Eqn.2) with the mean free path values
obtained from the resistivity fit and a typical energy dependence scattering
term of 0.1 and 0.2, show clear deviations from the experimentally measured
values below 50 and 30 nm respectively.
The presence of a competing mechanism such as quantum confinement,Hicks and
Dresselhaus (1993a)which compensates for the thermopower decrease due to
surface scattering, can validate the applicability of Fuchs-Sondheimer theory.
The lack of changes in the bulk carrier concentration is inconsistent with the
presence of quantum confinement but is not a definitive proof to rule out this
scenario completely. Typically, the surface scattering is not expected to show
a strong temperature dependence as it is a boundary dominated mechanism. On
the other hand, mechanisms like quantum confinement are expected to show a
strong temperature dependence, as the effect is stronger at lower
temperatures. Thus, if the temperature dependent thermopower does not show any
significant deviation from the bulk at the lower thicknesses, we can conclude
that Fuchs-Sondheimer theory is not applicable to our case and thermopower is
insensitive to surface scattering. The temperature dependent thermopower as
shown in Fig. 4 clearly depicts a very similar temperature dependence for
thicknesses of 115, 88, 16 and 15 nm films down to 80 K (for thinner films the
thermopower measurement became unreliable below this temperature due to high
resistance of the films). Thus, the temperature dependence of thermopower
remains bulk-like even in thin samples confirming that thermopower is robust
against surface scattering and the non-applicability of Fuchs-Sondheimer
theory in this system. It is important to note that the presence of wire-
texture in these films, suggests that grain boundary scattering should also be
considered for the electron scattering mechanisms. Typically, grain boundary
scattering doesn’t show any thickness dependence,Pichard et al. (1980); Worden
(1958) hence, will not change the conclusions derived here.
The magnetic field dependence of thermopower for an 88 nm film at 20 K is
shown in the inset of Fig. 4. The observed field dependent thermopower is in
excellent agreement with the spin entropic contribution to thermopower (Eqn.
3).
$Q(H,T)/Q(0,T)={ln[2cosh(u)]-utanh(u)}/ln(2)$ (3)
where u = $g\mu_{B}$H/2k${}_{\mathrm{B}}T$ and $g$ is the Landé $g$-factor
(here, $g$=2). This observation is consistent with the experiments on NaxCoO2
($g$=2.2),Wang et al. (2003) confirming the role of strong correlation and
spin entropy in thermoelectric properties of BSCO.
Figure 5: Temperature dependent resistivity for films of different thickness.
At the bulk limit, the films show the characteristic metal–insulator
transition with a transition temperature $\sim$ 100 K. As the thickness is
decreased the transition temperature shifts to higher temperature and below 23
nm, the films remained insulating till 300 K.
Finally, we studied the size effects on resistivity at low temperatures by
performing temperature dependent resistivity measurements (shown in Fig. 5) on
films with different thicknesses. Single crystalsFujii et al. (2002); Yamamoto
et al. (2002) and thin filmsWang et al. (2009a, b) of BSCO have shown a metal-
insulator transition with transition temperatures $\sim$80–140 K.
Interestingly, the transition temperature shifted to higher temperatures as we
decreased the thickness. This shift in transition temperature needs further
investigation for a clear understanding. Moreover, due to the presence of
unconventional Hall effectEng et al. (2006) in the cobaltates at low
temperatures, extensive Hall effect investigations are necessary to uncover
the exact origin of this shift.
As we have already established that BSCO is also a correlated system similar
to other cobaltates, it is essential to put these findings in perspective
within the framework of the Hubbard model.Koshibae et al. (2000); Uchida et
al. (2011); Koshibae and Maekawa (2001); Mukerjee and Moore (2007) The
transport coefficients predicted for NaxCoO2Mukerjee and Moore (2007) using
the atomic limit Hubbard model in the high temperature limit are given as:
$S=-\frac{k_{B}}{e}log\left(\frac{2(1-x)}{x}\right)$ (4)
$\rho=\frac{Vh^{2}}{8\pi^{2}e^{2}\eta a^{2}t^{2}\tau\beta x(1-x)}$ (5)
where $x$ is filling, kB is the Boltzmann constant, e is the electronic
charge, $\beta$ is 1/k${}_{\mathrm{B}}T$, $\eta$ is the lattice structure
dependent constant, $a$ is the lattice constant, $t$ is the bandwidth, $\tau$
is the relaxation time, $V$ is the unit cell volume and h is Planck’s
constant.
The thermopower relation shown in Eqn. 4 does not depend on the relaxation
time and hence is consistent with our conclusion that thermopower is robust to
changes in the scattering mechanism. Besides, the resistivity clearly shows an
inverse scaling with the scattering time, hence consistent with our
observations. Thus, it is evident that the Fuchs-Sondheimer theory doesn’t
account for the thermopower and resistivity measurements, but the simple
Hubbard based clearly explains the thickness and temperature dependence of
thermoelectric properties in this strongly correlated system. It is important
to note that Eqn.4 and Eqn. 5 change qualitatively, if the limits on the
energy scales such as thermal energy($k_{B}T$) and bandwidth ($t$) are
different from the assumed limit here ($t\ll k_{B}T$). These variations still
doesn’t change the overall conclusion that thermopower is independent of
scattering time. Our experiment elegantly establishes that the thermopower, at
the high temperature limit, is independent of scattering parameter. It is
important to comment on the relevance of the Eqn.4, in estimating the valence
state of Co. Using the average room temperature thermopower of 110 $\mu V/K$,
we estimate the x to be 0.36. Thus the estimated average cobalt valence in
this compound is 3.36, which is very close to the reported value of 3.3.Morita
et al. (2004) Further, the resistivity in the metallic regime can be explained
using Eqn. 5 but there is no clear insight on how the metal insulator
transition can be understood using the Hubbard model.
## IV Summary
In summary, we have studied the size effects on thermoelectricity in thin
films of a strongly correlated cobaltate system. The thermopower is
insensitive to surface scattering unlike resistivity, which increases with
decreasing thickness below $\sim$ 30 nm. These observations can be explained
by the atomic limit Hubbard model. Unlike conventional thermoelectric
materials, the insensitivity of thermopower to scattering mechanism in
strongly correlated systems simplifies the decoupling of thermopower and
electrical conductivity. Hence, the next step towards complete decoupling of
thermopower and electrical conductivity in a correlated system is only
dependent on understanding the limits of filling dependence of thermoelectric
properties. Since nanostructuring is a proven route to decrease the lattice
part of thermal conductivity without affecting electrical properties,
designing nanostructured correlated materials can lead to decoupling of all
the three thermoelectric parameters and hence, a pathway to high
thermoelectric efficiency.
###### Acknowledgements.
The work was supported by the Division of Materials Sciences and Engineering,
Office of Basic Energy Sciences, U.S Department of Energy. JR acknowledges
support from the Link foundation. The authors gratefully acknowledge the
assistance of Dr. K. M. Yu with the RBS measurements, target preparation by J.
Wu and useful discussions with Dr. S. Mukerjee, Dr. Jay Sau, Dr. B.
Kavaipatti, Dr. M. Trassin, Dr. C.-W Liang and Dr. Y.-H Chu.
## References
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* Hicks and Dresselhaus (1993b) L. Hicks and M. Dresselhaus, Physical Review B 47, 16631 (1993b).
* Kim et al. (2006) W. Kim, J. Zide, A. Gossard, D. Klenov, and S. Stemmer, Physical Review Letters 96, 045901 (2006).
* Poudel et al. (2008) B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan, D. Wang, A. Muto, D. Vashaee, et al., Science 320, 634 (2008).
* Pichard et al. (1980) C. Pichard, C. Tellier, and A. Tosser, Journal of Physics F: Metal Physics 10, 2009 (1980).
* Worden (1958) D. Worden, Journal Of Physics And Chemistry Of Solids 6, 89 (1958).
* Cho and Kim (2005) K. Cho and I. Kim, Materials Letters 59, 966 (2005).
* Ganesan and Sivaramakrishnan (2000) N. Ganesan and V. Sivaramakrishnan, Journal Of Physics D-Applied Physics 21, 784 (2000).
* Rogacheva et al. (2003) E. Rogacheva, O. Nashchekina, Y. Vekhov, M. Dresselhaus, and S. Cronin, Thin Solid Films 423, 115 (2003).
* Rogacheva et al. (2002) E. I. Rogacheva, T. V. Tavrina, S. N. Grigorov, O. N. Nashchekina, V. V. Volobuev, A. G. Fedorov, K. A. Nasedkin, and M. S. Dresselhaus, Journal of Electronic Materials 31, 298 (2002).
* Ohtomo and Hwang (2004) A. Ohtomo and H. Hwang, Applied Physics Letters 84, 1716 (2004).
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* Lee et al. (2006) M. Lee, L. Viciu, L. Li, Y. Wang, M. Foo, S. Watauchi, R. Pascal, R. Cava, and N. Ong, Nature Materials 5, 537 (2006).
* Koshibae and Maekawa (2001) W. Koshibae and S. Maekawa, Physical Review Letters 87, 236603 (2001).
* Sondheimer (1952) E. H. Sondheimer, Advances In Physics 1, 1 (1952).
* Fujii et al. (2002) T. Fujii, I. Terasaki, T. Watanabe, and A. Matsuda, Japanese Journal Of Applied Physics Part 2-Letters & Express Letters 41, 783 (2002).
* Yamamoto et al. (2002) T. Yamamoto, K. Uchinokura, and I. Tsukada, Physical Review B 65, 184434 (2002).
* Wang et al. (2009a) S. Wang, A. Venimadhav, S. Guo, K. Chen, Q. Li, A. Soukiassian, D. G. Schlom, M. B. Katz, X. Q. Pan, W. Wong-Ng, et al., Applied Physics Letters 94, 022110 (2009a).
* Wang et al. (2009b) S. Wang, Z. Zhang, L. He, M. Chen, W. Yu, and G. Fu, Applied Physics Letters 94, 162108 (2009b).
* Eng et al. (2006) H. Eng, P. Limelette, W. Prellier, C. Simon, and R. Frésard, Physical Review B 73, 33403 (2006).
* Koshibae et al. (2000) W. Koshibae, K. Tsutsui, and S. Maekawa, Physical Review B 62, 6869 (2000).
* Uchida et al. (2011) M. Uchida, K. Oishi, M. Matsuo, W. Koshibae, Y. Onose, M. Mori, J. Fujioka, S. Miyasaka, S. Maekawa, and Y. Tokura, Physical Review B 83, 165127 (2011).
* Mukerjee and Moore (2007) S. Mukerjee and J. Moore, Applied Physics Letters 90, 112107 (2007).
* Morita et al. (2004) Y. Morita, J. Poulsen, K. Sakai, T. Motohashi, T. Fujii, I. Terasaki, H. Yamauchi,and M. Karppinen, Journal of Solid State Chemistry 177, 3149 (2004).
|
arxiv-papers
| 2012-01-30T16:38:21 |
2024-09-04T02:49:26.822777
|
{
"license": "Public Domain",
"authors": "J. Ravichandran, A. K. Yadav, W. Siemons, M. A. McGuire, V. Wu, A.\n Vailionis, A. Majumdar, and R. Ramesh",
"submitter": "Jayakanth Ravichandran",
"url": "https://arxiv.org/abs/1201.6274"
}
|
1201.6315
|
# Thermodynamic Tree: The Space of Admissible Paths
Alexander N. Gorban Department of Mathematics, University of Leicester, UK
(ag153@le.ac.uk).
###### Abstract
Is a spontaneous transition from a state $x$ to a state $y$ allowed by
thermodynamics? Such a question arises often in chemical thermodynamics and
kinetics. We ask the more formal question: is there a continuous path between
these states, along which the conservation laws hold, the concentrations
remain non-negative and the relevant thermodynamic potential $G$ (Gibbs
energy, for example) monotonically decreases? The obvious necessary condition,
$G(x)\geq G(y)$, is not sufficient, and we construct the necessary and
sufficient conditions. For example, it is impossible to overstep the
equilibrium in 1-dimensional (1D) systems (with $n$ components and $n-1$
conservation laws). The system cannot come from a state $x$ to a state $y$ if
they are on the opposite sides of the equilibrium even if $G(x)>G(y)$. We find
the general multidimensional analogue of this 1D rule and constructively solve
the problem of the thermodynamically admissible transitions.
We study dynamical systems, which are given in a positively invariant convex
polyhedron $D$ and have a convex Lyapunov function $G$. An admissible path is
a continuous curve in $D$ along which $G$ does not increase. For $x,y\in D$,
$x\succsim y$ ($x$ precedes $y$) if there exists an admissible path from $x$
to $y$ and $x\sim y$ if $x\succsim y$ and $y\succsim x$. The tree of $G$ in
$D$ is a quotient space $D/\sim$. We provide an algorithm for the construction
of this tree. In this algorithm, the restriction of $G$ onto the 1-skeleton of
$D$ (the union of edges) is used. The problem of existence of admissible paths
between states is solved constructively. The regions attainable by the
admissible paths are described.
###### keywords:
Lyapunov function, convex polyhedron, attainability, tree of function,
entropy, free energy
###### AMS:
37A60, 52A41, 80A30, 90C25
## 1 Introduction
### 1.1 Motivation, ideas and a simple example
“Applied dynamical systems” are models of real systems. The available
information about the real systems is incomplete and uncertainties of various
types are encountered in the modeling. Often, we view them as errors: errors
in the model structure, errors in coefficients, in the state observation and
many others. Nevertheless, there is an order in this world of errors: some
information is more reliable, we trust in some structures more and even
respect them as laws. Some other data are less reliable. There is an hierarchy
of reliability, our knowledge and beliefs (described, for example by R.
Peierls [53] for model making in physics). Extracting as many consequences
from the more reliable data either without or before use of the less reliable
information is a task which arises naturally.
In our paper, we study dynamical systems with a strictly convex Lyapunov
function $G$ defined in a positively invariant convex polyhedron $D$. For
them, we analyze the admissible paths, along which $G$ decreases
monotonically, and find the states that are attainable from the given initial
state along the admissible paths. The main area of applications of these
systems is chemical kinetics and thermodynamics. The motivation of our
research comes from the hierarchy of reliability of the information in these
applications.
Let us discuss the motivation in more detail. In chemical kinetics, we can
rank the information in the following way. First of all, the list of reagents
and conservation laws should be known. Let the reagents be
$A_{1},A_{2},\ldots,A_{n}$. The non-negative real variable $N_{i}\geq 0$, the
amount of $A_{i}$ in the mixture, is defined for each reagent, and $N$ is the
vector of composition with coordinates $N_{i}$. The conservation laws are
presented by the linear balance equations:
$b_{i}(N)=\sum_{j=1}^{n}a_{i}^{j}N_{j}={\rm const}\;\;(i=1,\ldots,m)\,.$ (1)
We assume that the linear functions $b_{i}(N)$ ($i=1,\ldots,m$) are linearly
independent.
The list of the components together with the balance conditions (1) is the
first part of the information about the kinetic model. This determines the
space of states, the polyhedron $D$ defined by the balance equations (1) and
the positivity inequalities $N_{i}\geq 0$. This is the background of kinetic
models and any further development is less reliable. The polyhedron $D$ is
assumed to be bounded. This means that there exist such coefficients
$\lambda_{i}$ that the linear combination $\sum_{i}\lambda_{i}b_{i}(N)$ has
strictly positive coefficients: $\sum_{i}\lambda_{i}a_{i}^{j}>0$ for all
$j=1,\ldots,n$.
The thermodynamic functions provide us with the second level of information
about the kinetics. Thermodynamic potentials, such as the entropy, energy and
free energy are known much better than the reaction rates and, at the same
time, they give us some information about the dynamics. For example, the
entropy increases in isolated systems. The Gibbs free energy decreases in
closed isothermal systems under constant pressure, and the Helmholtz free
energy decreases under constant volume and temperature. Of course, knowledge
of the Lyapunov functions gives us some inequalities for vector fields of the
systems’ velocity but the values of these vector fields remain unknown. If
there are some external fluxes of energy or non-equilibrium substances then
the thermodynamic potentials are not Lyapunov functions and the systems do not
relax to the thermodynamic equilibrium. Nevertheless, the inequality of
positivity of the entropy production persists and this gives us useful
information even about the open systems. Some examples are given in [26, 28].
The next, third part of the information about kinetics is the reaction
mechanism. It is presented in the form of the stoichiometric equations of the
elementary reactions:
$\sum_{i}\alpha_{\rho i}A_{i}\to\sum_{i}\beta_{\rho i}A_{i}\,,$ (2)
where $\rho=1,\ldots,m$ is the reaction number and the stoichiometric
coefficients $\alpha_{\rho i},\beta_{\rho i}$ ($i=1,\ldots,n$) are nonnegative
integers.
A stoichiometric vector $\gamma_{\rho}$ of the reaction (2) is a
$n$-dimensional vector with coordinates $\gamma_{\rho i}=\beta_{\rho
i}-\alpha_{\rho i}$, that is, ‘gain minus loss’ in the $\rho$th elementary
reaction.
The concentration of $A_{i}$ is an intensive variable $c_{i}=N_{i}/V$, where
$V>0$ is the volume. The vector $c=N/V$ with coordinates $c_{i}$ is the vector
of concentrations. A non-negative intensive quantity, $r_{\rho}$, the reaction
rate, corresponds to each reaction (2). The kinetic equations in the absence
of external fluxes are
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{\rho}r_{\rho}\gamma_{\rho}\,.$ (3)
If the volume is not constant then equations for concentrations include
$\dot{V}$ and have different form.
For perfect systems and not so fast reactions the reaction rates are functions
of concentrations and temperature given by the mass action law and by the
generalized Arrhenius equation. A special relation between the kinetic
constants is given by the principle of detailed balance: For each value of
temperature $T$ there exists a positive equilibrium point where each reaction
(2) is equilibrated with its reverse reaction. This principle was introduced
for collisions by Boltzmann in 1872 [10]. Wegscheider introduced this
principle for chemical kinetics in 1901 [67]. Einstein in 1916 used it in the
background for his quantum theory of emission and absorption of radiation
[17]. Later, it was used by Onsager in his famous work [51]. For a recent
review see [30].
At the third level of reliability of information, we select the list of
components and the balance conditions, find the thermodynamic potential, guess
the reaction mechanism, accept the principle of detailed balance and believe
that we know the kinetic law of elementary reactions. However, we still do not
know the reaction rate constants.
Finally, at the fourth level of available information, we find the reaction
rate constants and can analyze and solve the kinetic equations (3) or their
extended version with the inclusion of external fluxes.
Of course, this ranking of the available information is conventional, to a
certain degree. For example, some reaction rate constants may be known even
better than the list of intermediate reagents. Nevertheless, this hierarchy of
the information availability, list of components – thermodynamic functions –
reaction mechanism – reaction rate constants, reflects the real process of
modelling and the stairs of available information about a reaction kinetic
system.
It seems very attractive to study the consequences of the information of each
level separately. These consequences can be also organized ‘stairwise’. We
have the hierarchy of questions: how to find the consequences for the dynamics
(i) from the list of components, (ii) from this list of components plus the
thermodynamic functions of the mixture, and (iii) from the additional
information about the reaction mechanism.
The answer to the first question is the description of the balance polyhedron
$D$. The balance equations (1) together with the positivity conditions
$N_{i}\geq 0$ should be supplemented by the description of all the faces. For
each face, some $N_{i}=0$ and we have to specify which $N_{i}$ have zero
value. The list of the corresponding indices $i$, for which $N_{i}=0$ on the
face, $I=\\{i_{1},\ldots,i_{k}\\}$, fully characterizes the face. This problem
of double description of the convex polyhedra [49, 14, 21] is well known in
linear programming.
The list of vertices [6] and edges with the corresponding indices is necessary
for the thermodynamic analysis. This is the 1-skeleton of $D$. Algorithms for
the construction of the 1-skeletons of balance polyhedra as functions of the
balance values were described in detail in 1980 [26]. The related problem of
double description for convex cones is very important for the pathway analysis
in systems biology [58, 22].
In this work, we use the 1-skeleton of $D$, but the main focus is on the
second step, i.e. on the consequences of the given thermodynamic potentials.
For closed systems under classical conditions, these potentials are the
Lyapunov functions for the kinetic equations. For example, for perfect systems
we assume the mass action law. If the equilibrium concentrations $c^{*}$ are
given, the system is closed and both temperature and volume are constant then
the function
$G=\sum_{i}c_{i}(\ln({c_{i}}/{c_{i}^{*}})-1)$ (4)
is the Lyapunov function; it should not increase in time. The function $G$ is
proportional to the free energy $F=RTG+{\rm const}$ (for detailed information
about the Lyapunov functions for kinetic equations under classical conditions
see the textbook [68] or the recent paper [33]).
If we know the Lyapunov function $G$ then we have the necessary conditions for
the possibility of transition from the vector of concentrations $c$ to
$c^{\prime}$ during the non-stationary reaction: $G(c)\geq G(c^{\prime})$
because the inequality $G(c(t_{0}))\geq G(c(t_{0}+t))$ holds for any time
$t\geq 0$.
The inequality $G(c)\geq G(c^{\prime})$ is necessary if we are to reach
$c^{\prime}$ from the initial state $c$ by a thermodynamically admissible
path, but it is not sufficient because in addition to this inequality there
are some other necessary conditions. The simplest and most famous of them is:
if $D$ is one-dimensional (a segment) then the equilibrium $c^{*}$ divides
this segment into two parts and both $c(t_{0})$ and $c(t_{0}+t)$ ($t>0$) are
always on the same side of the equilibrium.
Fig. 1: The balance simplex (a), the levels of the Lyapunov function (b) and
the thermodynamic tree (c) for the simple system of three components,
$A_{1},A_{2},A_{3}$. Algorithm for finding a vertex $v\succsim c$ (d).
In 1D systems the overstepping of the equilibrium is forbidden. It is
impossible to overstep a point in dimension one, but it is possible to
circumvent a point in higher dimensions. Nevertheless, in any dimension the
inequality $G(c)\geq G(c^{\prime})$ is not sufficient if we are to reach
$c^{\prime}$ from the initial state $c$ along an admissible path. Some
additional restrictions remain in the general case as well. A two-dimensional
example is presented in Fig. 1. Let us consider the mixture of three
components, $A_{1,2,3}$ with the only conservation law $c_{1}+c_{2}+c_{3}=b$
(we take for illustration $b=1$) and the equidistribution in equilibrium
$c_{1}^{*}=c_{2}^{*}=c_{3}^{*}=1/3$. The balance polyhedron is the triangle
(Fig. 1a). In Fig. 1b the level sets of
$G=\sum_{i=1}^{3}c_{i}(\ln(3c_{i})-1)$
are presented. This function achieves its minimum at equilibrium,
$G(c^{*})=-1$. On the edges, the function $G$ achieves its conditional
minimum, $g_{0}$, in the middles, and $g_{0}=\ln(3/2)-1$. $G$ reaches its
maximal value, $g_{\max}=\ln 3-1$, at the vertices.
If $G(c^{*})<g\leq g_{0}$ then the level set $G(c)=g$ is connected. If
$g_{0}<g\leq g_{\max}$ then the corresponding level set $G(c)=g$ consists of
three components (Fig. 1b). The critical value is $g=g_{0}$. The critical
level $G(c)=g_{0}$ consists of three arcs. Each arc connects two middles of
the edges and divides $D$ in two sets. One of them is convex and includes two
vertices, the other includes the remaining vertex.
A thermodynamically admissible path is a continuous curve along which $G$ does
not increase. Therefore, such a path cannot intersect these arcs ‘from
inside’, i.e. from values $G(c)\leq g_{0}$ to bigger values, $G(c)>g_{0}$. For
example, if an admissible path starts from the state with 100% of $A_{2}$,
then it cannot intersect the arc that separates the vertex with 100% $A_{1}$
from two other vertices. Therefore, any vertex cannot be reached from another
one and if we start from 100% of $A_{2}$ then the reaction cannot overcome the
threshold $\sim$77.3% of $A_{1}$, that is the maximum of $c_{1}$ on the
corresponding arc (Fig. 1b). This is an example of the 2D analogue of the 1D
prohibition of overstepping of equilibrium.
For $x,y\in D$, $x\succsim y$ ($x$ precedes $y$) if there exists a
thermodynamically admissible path from $x$ to $y$, and $x\sim y$ if $x\succsim
y$ and $y\succsim x$. The equivalence classes with respect to $x\sim y$ in $D$
are the connected components of the level sets $G(c)=g$. The quotient space
$\mathcal{T}=D/\sim$ is the space of these connected components. For the
canonical projection we use the standard notation $\pi:D\to\mathcal{T}$. This
is the tree of the connected components of the level sets of $G$. (Here “tree”
stands for a one dimensional continuum, a sort of dendrites [13], and not for
a tree in the sense of the graph theory.)
If $x\sim y$ then $G(x)=G(y)$. Therefore, we can define the function $G$ on
the tree: $G(\pi(c))=G(c)$. It is convenient to draw this tree on the plane
with the vertical coordinate $g=G(x)$ (Fig. 1c). The equilibrium $c^{*}$
corresponds to a root of this tree, $\pi(c^{*})$. If $G(c^{*})<g\leq g_{0}$
then the level set $G(c)=g$ corresponds to one point on the tree. The level
$G(c)=g_{0}$ corresponds to the branching point, and each connected component
of the level sets $G(c)=g$ with $g_{0}<g\leq g_{\max}$ corresponds to a
separate point on the tree. The terminal points (“leaves” with $g>g_{0}$) of
the tree correspond to the vertices of $D$.
An ordered segment $[x,y]$ or $[y,x]$ ($x\succsim y$) on the tree
$\mathcal{T}$ consists of such points $z$ that $x\succsim z\succsim y$. A
continuous curve $\varphi:[0,1]\to D$ is an admissible path if and only if its
image $\pi\circ\varphi:[0,1]\to\mathcal{T}$ is a path that goes monotonically
down in the coordinate $g$. Such a monotonic path in $\mathcal{T}$ from a
point $x$ to the root is just a segment $[x,\pi(c^{*})]$. On this segment,
each point $y$ is unambiguously characterized by $g=G(y)$. Therefore, if for
$c\in D$ we know the value $G(c)$ and a vertex $v\succsim c$, then we can
unambiguously describe the image of $c$ on the tree: $\pi(c)$ is the point on
the segment $[\pi(v),\pi(c^{*})]$ with the given value of $G$, $g=G(c)$.
We can find a vertex $v\succsim c$ by a chain of central projections: the
first step is the central projection of $c$ onto the border of $D$ with center
$c^{*}$. The result is the point $c^{\prime}$ on a face (in Fig. 1d this is
the point $c^{\prime}$ on an edge). The second step is the central projection
of the point $c^{\prime}$ onto the border of the face with the center at the
partial equilibrium ${c^{*}}^{\prime}$ (that is, the minimizer of $G$ on the
face) and so on (Fig. 1d). If the projection on a face is the partial
equilibrium then for any vertices $v$ of the face $v\succsim c$. In
particular, if the face is a vertex $v$ then $v\succsim c$. For a simple
example presented in Fig. 1d this is the vertex $A_{1}$.
In this paper, we extend these ideas and observations to any dynamical system,
which is given in a positively invariant convex polyhedron and has there a
strictly convex Lyapunov function. The class of chemical kinetic equations for
closed systems provides us standard and practically important examples of the
systems of this class.
### 1.2 A bit of history
It seems attractive to use an attainable region instead of the single
trajectory in situations with incomplete information or with information with
different levels of reliability. Such situations are typical in many areas of
science and engineering. For example, the theory for the continuous–time
Markov chain is presented in [2, 27] and for the discrete–time Markov chains
in [3].
Perhaps, the first celebrated example of this approach was developed in
biological kinetics. In 1936, A.N. Kolmogorov [40] studied the dynamics of
interacting populations of prey ($x$) and predator ($y$) in the general form:
$\dot{x}=xS(x,y),\;\;\dot{y}=yW(x,y)$
under monotonicity conditions: $\partial S(x,y)/\partial y<0$, $\partial
W(x,y)/\partial y<0$. The zero isoclines, given by equations $S(x,y)=0$ or
$W(x,y)=0$, are graphs of two functions $y(x)$. These isoclines divide the
phase space into compartments with curvilinear borders. The geometry of the
intersection of the zero isoclines, together with some monotonicity
conditions, contain important information about the system dynamics that we
can find [40] without exact knowledge of the kinetic equations. This approach
to population dynamics was applied to various problems [45, 7]. The impact of
this work on population dynamics was analyzed in the review [62].
In 1964, Horn proposed to analyze the attainable regions for chemical reactors
[36]. This approach became popular in chemical engineering. It was applied to
the optimization of steady flow reactors [23], to batch reactor optimization
without knowledge of detailed kinetics [19], and for optimization of the
reactor structure [34]. An analysis of attainable regions is recognized as a
special geometric approach to reactor optimization [18] and as a crucially
important part of the new paradigm of chemical engineering [35].
Many particular applications were developed, from polymerization [63] to
particle breakage in a ball mill [47] and hydraulic systems [28]. Mathematical
methods for the study of attainable regions vary from Pontryagin’s maximum
principle [46] to linear programming [38], the Shrink-Wrap algorithm [43], and
convex analysis. In 1979 it was demonstrated how to utilize the knowledge
about partial equilibria of elementary processes to construct the attainable
regions [24]. The attainable regions significantly depend on the reaction
mechanism and it is possible to use them for the discrimination of mechanisms
[29].
Thermodynamic data are more robust than the reaction mechanism. Hence, there
are two types of attainable regions. The first is the thermodynamic one, which
use the linear restrictions and the thermodynamic functions [25]. The second
is generated by thermodynamics and stoichiometric equations of elementary
steps (but without reaction rates) [24, 31]. R. Shinnar and other authors [61]
rediscovered this approach. There was even an open discussion about priority
[9].
Some particular classes of kinetic systems have rich families of the Lyapunov
functions. Krambeck [41] studied attainable regions for linear systems and the
$l_{1}$ Lyapunov norm instead of the entropy. Already simple examples
demonstrate that the sets of distributions which are accessible from a given
initial distribution by linear kinetic systems (Markov processes) with a given
equilibrium are, in general, non-convex polytopes [24, 27, 70]. The geometric
approach to attainability was developed for all the thermodynamic potentials
and for open systems as well [26]. Partial results for chemical kinetics and
some other engineering systems are summarized in [68, 28].
The tree of the level set components for differentiable functions was
introduces in the middle of the 20 century by Adelson-Velskii and Kronrod [1,
42] and Reeb [56]. Sometimes these trees are called the Reeb trees [20] but
from the historical point of view it may be better to call them the Adelson-
Velskii – Kronrod – Reeb (or AKR) trees. These trees were essentially used by
Kolmogorov and Arnold [4] in solution of the Hilbert’s superposition problem
(the ideas, their relations to dynamical systems and role in the Arnold’s
scientific life are discussed in his lecture [5]).
The general Reeb graph can be defined for any topological space $X$ and real
function $f$ on it. It is the quotient space of $X$ by the equivalence
relation “$\sim$” defined by $x\sim y$ holds if and only if $f(x)=f(y)$ and
$x$, $y$ are in the same connected component of $f^{-1}(f(x))$. Of course,
this “graph” is again not a discrete object from the graph theory but a
topological space. It has application in differential topology (Morse theory
[48]), in topological shape analysis and visualization [20, 39], in data
analysis [64] and in asymptotic analysis of fluid dynamics [44, 59]. The books
[20, 39] include many illustration of the Reeb graphs. The efficient mesh-
based methods for the computation of the graphs of level set components are
developed for general scalar fields on 2- and 3-dimensional manifolds [16].
Some time ago the tree of entropy in the balance polyhedra was rediscovered as
an adequate tool for representation of the attainable regions in chemical
thermodynamics [25, 26]. It was applied to analysis of various real systems
[37, 69]. Nevertheless, some of the mathematical backgrounds of this approach
were delayed in development and publications. Now, the thermodynamically
attainable regions are in extensive use in chemical engineering and beyond
[18, 19, 23, 28, 34, 35, 36, 37, 38, 41, 43, 46, 47, 60, 61, 63, 69]. In this
paper we aim to provide the complete mathematical background for the analysis
of the thermodynamically attainable regions. For this purpose, we construct
the trees of strictly convex functions in a convex polyhedron. This problem
allows a general meshless solution in higher dimensions because topological
and geometrical simplicity (the domain $D$ is a convex polyhedron and the
function $G$ is strictly convex in D). In this paper, we present this solution
in detail.
### 1.3 The problem of attainability and its solution
Let us formulate precisely the problem of attainability and its solution
before the exposition of all technical details and proofs. Our results are
applicable to any dynamical system that obeys a continuous strictly convex
Lyapunov function in a positively invariant convex polyhedron. The situations
with uncertainty, when the specific dynamical system is not given with an
appropriate accuracy but the Lyapunov function is known, give a natural area
of application of these results.
Here and below, $D$ is a convex polyhedron in $\mathbb{R}^{n}$, $D_{0}$
consists of the vertices of $D$, $D_{1}$ is the union of the closed edges of
$D$, that is, the 1-skeleton of $D$, and $\widetilde{D_{1}}$ is the graph
whose vertices correspond to the vertices of $D$ and edges correspond to the
edges of $D$, (the graph of the 1-skeleton) of $D$. We use the same notations
for vertices and edges of $D$ and $\widetilde{D_{1}}$.
Let a real continuous function $G$ be given in $D$. We assume that $G$ is
strictly convex in $D$ [57]. Let $x^{*}$ be the minimizer of $G$ in $D$ and
let $g^{*}=G(x^{*})$ be the corresponding minimal value.
The level set $S_{g}=\\{x\in D\,|\,G(x)=g\\}$ is closed and the sublevel set
$U_{g}=\\{x\in D\,|\,G(x)<g\\}$ is open in $D$ (i.e. it is the intersection of
an open subset of $\mathbb{R}^{n}$ with $D$).
Let us transform $\widetilde{D_{1}}$ into a labeled graph. Each vertex $v\in
D_{0}$ is labeled by the value $\gamma_{v}=G(v)$ and each edge $e=[v,w]\subset
D_{1}$ is labeled by the minimal value of $G$ on the segment $[v,w]\subset D$,
$g_{e}=\min_{[v,w]}G(x)$. The vertices and edges of $\widetilde{D_{1}}$ are
labeled by the same numbers as the correspondent vertices and edges of
$D_{1}$. By definition, the graph $\widetilde{D_{1}}\setminus U_{g}$ consists
of the vertices and edges of $\widetilde{D_{1}}$, whose labels $\gamma\geq g$.
The graph $\widetilde{D_{1}}\setminus U_{g}$ depends on $g$ but this is a
piecewise constant dependence. It changes only at $g=\gamma$, where $\gamma$
are some of the labels of the graph $\widetilde{D_{1}}$. Therefore, it is not
necessary to find this graph and to analyze connectivity in it for each value
$G(y)=g$.
###### Definition 1.
A continuous path $\varphi[0,1]\to D$ is admissible if the function
$G(\varphi(x))$ does not increase on $[0,1]$. For $x,y\in D$, $x\succsim y$
($x$ precedes $y$) if there exists an admissible path $\varphi[0,1]\to D$ with
$\varphi(0)=x$ and $\varphi(1)=y$; $x\sim y$ if $x\succsim y$ and $y\succsim
x$.
The relation “$\succsim$” is transitive. It is a preorder on $D$. The relation
“$\sim$” is an equivalence.
###### Definition 2.
The tree of $G$ in $D$ is the quotient space $\mathcal{T}=D/\sim$.
The equivalence classes of $\sim$ are the path-connected components of the
level sets $S_{g}$. For the natural projection of $D$ on $\mathcal{T}$ we use
the notation $\pi:D\to\mathcal{T}$. We denote by $\pi^{-1}(z)\subset D$ the
set of preimages of $z\in\mathcal{T}$. The preorder “$\succsim$” on $D$
transforms into a partial order on $\mathcal{T}$: $\pi(x)\succsim\pi(y)$ if
and only if $x\succsim y$. We call $\mathcal{T}$ also the thermodynamic tree
keeping in mind the thermodynamic applications. The “tree” $\mathcal{T}$ is a
1D continuum. We have to distinguish this continuum from trees in the graph-
theoretic sense which have the same graphical representation but are discrete
objects. In Sec. 3.2 (“Coordinates on the thermodynamic tree”) we describe the
tree structure of this continuum. It includes the root, the edges, the
branching points and leaves but the edges are represented as the real line
segments.
###### Definition 3.
Let $x,y\in\mathcal{T}$, $x\succsim y$. An ordered segment $[x,y]$ (or
$[y,x]$) consists of such points $z\in\mathcal{T}$ that $x\succsim z\succsim
y$.
In Sec. 3 we prove that any ordered segment $[x,y]$ ($x\neq y$) in
$\mathcal{T}$ is homeomorphic to $[0,1]$. A continuous curve $\varphi:[0,1]\to
D$ is an admissible path if and only if its image
$\pi\circ\varphi:[0,1]\to\mathcal{T}$ is monotonic in the partial order on
$\mathcal{T}$. Such a monotonic path in $\mathcal{T}$ from $x$ to $y$
($x\succsim y$) is just a path along a segment $[x,y]$. Each point $z$ on this
segment is unambiguously characterized by the value of $G(z)$.
We also use the notation $[x,y]$ for the usual closed segments in
$\mathbb{R}^{n}$ with ends $x,y$: $[x,y]=\\{\lambda
x+(1-\lambda)y\,|\,\lambda\in[0,1]\\}$. The degenerated segment $[x,x]$ is
just a point $\\{x\\}$. The segments without one end are $(x,y]$ and $[x,y)$
and $(x,y)$ is the segment in $\mathbb{R}^{n}$ without both ends.
The attainability problem: Let $x,y\in D$ and $G(x)\geq G(y)$. Is $y$
attainable from $x$ by an admissible path?
The solution of the attainability problem can be found in several steps:
1. 1.
Find two vertices of $D$, $v_{x}$ and $v_{y}$, that precede $x$ and $y$,
correspondingly. Such vertices always exist. There may be several such
vertices in $D$. We can use any of them.
2. 2.
Construct the graph $\widetilde{D_{1}}\setminus U_{G(y)}$ by deletion from
$\widetilde{D_{1}}$ all the elements with the labels $\gamma<G(y)$.
3. 3.
$y$ is attainable from $x$ by an admissible path if and only if $v_{x}$ and
$v_{y}$ are connected in the graph $\widetilde{D_{1}}\setminus U_{G(y)}$.
So, to check the existence of an admissible path from $x$ to $y$ we should
check the inequality $G(x)\geq G(y)$ (the necessary condition) then go up in
$G$ values and find the vertices, $v_{x}$ and $v_{y}$, that precede $x$ and
$y$, correspondingly (such vertices always exist). Then we should go down in
$G$ values to $G(y)$ and check whether the vertices $v_{x}$ and $v_{y}$ are
connected in the graph $\widetilde{D_{1}}\setminus U_{G(y)}$. The classical
problem of determining whether two vertices in a graph are connected may be
solved by many search algorithms [52, 50], for example, by the elementary
breadth–first or depth–first search algorithms.
The procedure “find a vertex $v_{x}\in D_{0}$ that precedes $x\in D$” can be
implemented as follows:
1. 1.
If $x=x^{*}$ then any vertex $v\in D_{0}$ precedes $x$.
2. 2.
If $x\neq x^{*}$ then consider the ray
$r_{x}=\\{x^{*}+\lambda(x-x^{*})\,|\,\lambda\geq 0\\}$. The intersection
$r_{x}\cap D$ is a closed segment $[x^{*},x^{\prime}]$. We call $x^{\prime}$
the central projection of $x$ onto the border of $D$ with the center $x^{*}$;
$x^{\prime}\succsim x$.
3. 3.
The central projection $x^{\prime}$ always belongs to an interior of a face
$D^{\prime}$ of $D$, $0\leq\dim D^{\prime}<\dim D$. If $\dim D^{\prime}>0$
then set $x:=x^{\prime}$, $D:=D^{\prime}$, $x^{*}:={\rm
argmin}\\{G(z)\,|\,z\in D^{\prime}\\}$ and go to step 1.
4. 4.
If $\dim D^{\prime}=0$ then it is a vertex $v\succsim x$ we are looking for.
The dimension of the face decreases at each step, hence, after not more than
$\dim D-1$ steps we will definitely obtain the desired vertex. A simple
example is presented in Fig. 1d.
The information about all connected components of $\widetilde{D_{1}}\setminus
U_{g}$ for all values of $g$ is summarized in the tree of $G$ in $D$,
$\mathcal{T}$ (Definition 2). The tree $\mathcal{T}$ can be described as
follows (Theorem 15): it is the space of pairs $(g,M)$, where
$g\in[\min_{D}G(x),\max_{D}G(x)]$ and $M$ is a connected component of
$\widetilde{D_{1}}\setminus U_{g}$, with the partial order relation:
$(g,M)\succsim(g^{\prime},M^{\prime})$ if $g\geq g^{\prime}$ and $M\subseteq
M^{\prime}$. For $x,y\in D$, $x\succsim y$ if and only if
$\pi(x)\succsim\pi(y)$.
The tree $\mathcal{T}$ may be constructed gradually, by descending from the
maximal value of $G$, $g=g_{\max}$ (Sec. 3.3). At $g=g_{\max}$, the graph
$\widetilde{D_{1}}\setminus U_{g}$ consists of the isolated vertices with the
labels $\gamma=g_{\max}$ (generically, this is one vertex). Going down in $g$,
we add to $\widetilde{D_{1}}\setminus U_{g}$ the elements, vertices and edges,
in descending order of their labels. After adding each element we record the
changes in the connected components of $\widetilde{D_{1}}\setminus U_{g}$.
For each point $z\in\mathcal{T}$, $z=(g,M)$, its preimage in $D$,
$\pi^{-1}(g,M)$, may be described by the equation $G(x)=g$ supplemented by a
set of linear inequalities. Computationally, these linear inequalities can be
produced by a convex hull operation from a finite set. This finite set is
described explicitly in Sec. 3.4.
For each point $z=(g,M)$ the set of all $z^{\prime}=(g^{\prime},M^{\prime})$
attainable by admissible paths from $z$ has a simple description,
$g^{\prime}\leq g$, $M^{\prime}\supseteq M$.
The tree of $G$ in $D$ provides a workbench for the analysis of various
questions about admissible paths. It allows us to reduce the $n$-dimensional
problems in $D$ to some auxiliary questions about such 1D or even discrete
objects as the tree $\mathcal{T}$ and the labeled graph $\widetilde{D_{1}}$.
For example, we use the thermodynamic tree to solve the following problem of
attainable sets: For a given $x\in D$ describe the set of all $y\precsim x$ by
a system of inequalities. For this purpose, we find the image of $x$ in
$\mathcal{T}$, $\pi(x)$, then define the set of all points attainable by
admissible paths from $\pi(x)$ in $\mathcal{T}$ and, finally, describe the
preimage of this set in $D$ by the system of inequalities (Sec. 3.4).
### 1.4 The structure of the paper
In Sec. 2, we present several auxiliary propositions from convex geometry. We
constructively describe the result of the cutting of a convex polyhedron $D$
by a convex set $U$: The description of the connected components of
$D\setminus U$ is reduced to the analysis of the 1D continuum $D_{1}\setminus
U$, where $D_{1}$ is the 1-skeleton of $D$.
In Sec. 3, we construct the tree of level set components of a strictly convex
function $G$ in the convex polyhedron $D$ and study the properties of this
tree. The main result of this section is the algorithm for construction of
this tree (Sec. 3.3). This construction is applied to the description of the
attainable sets in Sec. 3.4. These sections include some practical recipes and
it is possible to read them independently, immediately after Introduction.
Several examples of the thermodynamic trees for chemical systems are presented
in Sec. 4.
## 2 Cutting of a polyhedron $D$ by a convex set $U$
### 2.1 Connected components of $D\setminus U$ and of $D_{1}\setminus U$
Let $D$ be a convex polyhedron in $\mathbb{R}^{n}$. We use the notations:
${\rm Aff}(D)$ is the minimal linear manifold that includes $D$; $d=\dim{\rm
Aff}(D)=\dim D$ is the dimension of $D$; $ri(D)$ is the interior of $D$ in
${\rm Aff}(D)$; $r\partial(D)$ is the border of $D$ in ${\rm Aff}(D)$.
For $P,Q\subset\mathbb{R}^{n}$ the Minkowski sum is $P+Q=\\{x+y\,|\,x\in
P,y\in Q\\}$. The convex hull (conv) and the conic hull (cone) of a set
$V\subset\mathbb{R}^{n}$ are:
${\rm
conv}(V)=\left\\{\sum_{i=1}^{q}\lambda_{i}v_{i}\,\left|\,q>0,\,v_{1},\ldots,v_{q}\in
V,\,\lambda_{1},\ldots\lambda_{q}>0,\,\sum_{i=1}^{q}\lambda_{i}=1\right.\right\\}\,;$
${\rm cone}(V)=\left\\{\left.\sum_{i=1}^{q}\lambda_{i}v_{i}\,\right|\,q\geq
0,\,v_{1},\ldots,v_{q}\in V,\,\lambda_{1},\ldots\lambda_{q}>0,\,\right\\}\,.$
For a set $D\subset\mathbb{R}^{n}$ the following two statements are equivalent
(the Minkowski–Weyl theorem):
1. 1.
For some real (finite) matrix $A$ and real vector $b$,
$D=\\{x\in\mathbb{R}^{n}\,|Ax\leq b\\}$ ;
2. 2.
There are finite sets of vectors
$\\{v_{1},\ldots,v_{q}\\}\subset\mathbb{R}^{n}$ and $\\{r_{1},\ldots
r_{p}\\}\subset\mathbb{R}^{n}$ such that
$D={\rm conv}\\{v_{1},\ldots\,v_{q}\\}+{\rm cone}\\{r_{1},\ldots,r_{p}\\}\,.$
(5)
Every polyhedron has two representations, of type (1) and (2), known as
(halfspace) $H$-representation and (vertex) $V$-representation, respectively.
We systematically use both these representations. Most of the polyhedra in our
paper are bounded, therefore, for them only the convex envelope of vertices is
used in the $V$-representation (5).
The $k$-skeleton of $D$, $D_{k}$, is the union of the closed $k$-dimensional
faces of $D$:
$D_{0}\subset D_{1}\subset\ldots\subset D_{d}=D\,.$
$D_{0}$ consists of the vertices of $D$ and $D_{1}$ is a one-dimensional
continuum embedded in $\mathbb{R}^{n}$. We use the notation
$\widetilde{D_{1}}$ for the graph whose vertices correspond to the vertices of
$D$ and edges correspond to the edges of $D$, and call this graph the graph of
the 1-skeleton of $D$.
Let $U$ be a convex subset of $\mathbb{R}^{n}$ (it may be a non-closed set).
We use $U_{0}$ for the set of vertices of $D$ that belong to $U$, $U_{0}=U\cap
D_{0}$, and $U_{1}$ for the set of the edges of $D$ that have non-empty
intersection with $U$. By default, we consider the closed faces of $D$, hence,
the intersection of an edge with $U$ either includes some internal points of
the edge or consists from one of its ends. We use the same notation $U_{1}$
for the set of the corresponding edges of $\widetilde{D_{1}}$.
A set $W\subset P\subset\mathbb{R}^{n}$ is a path-connected component of $P$
if it is its maximal path-connected subset. In this section, we aim to
describe the path-connected components of $D\setminus U$. In particular, we
prove that these components include the same sets of vertices as the connected
components of the graph $\widetilde{D_{1}}\setminus U$. This graph is produced
from $\widetilde{D_{1}}$ by deletion of all the vertices that belong to
$U_{0}$ and all the edges that belong to $U_{1}$.
###### Lemma 4.
Let $x\in D\setminus U$. Then there exists such a vertex $v\in D_{0}$ that the
closed segment $[v,x]$ does not intersect $U$: $[v,x]\subset D\setminus U$.
###### Proof.
Let us assume the contrary: for every vertex $v\in D_{0}$ there exists such
$\lambda_{v}\in(0,1]$ that $x+\lambda_{v}(v-x)\in U$. The convex polyhedron
$D$ is the convex hull of its vertices. Therefore, $x=\sum_{v\in
D_{0}}\kappa_{v}v$ for some numbers $\kappa_{v}\geq 0$, $v\in D_{O}$,
$\sum_{v\in D_{0}}\kappa_{v}=1$.
Let
$\delta_{v}=\frac{\kappa_{v}}{\lambda_{v}\sum_{v^{\prime}\in
D_{0}}\frac{\kappa_{v^{\prime}}}{\lambda_{v^{\prime}}}}\,.$
It is easy to check that $\sum_{v\in D_{0}}\delta_{v}=1$ and
$x=\sum_{v\in D_{0}}\delta_{v}(x+\lambda_{v}(v-x))\,.$ (6)
According to (6), $x$ belongs to the convex hull of the finite set
$\\{x+\lambda_{v}(v-x)\,|\,v\in D_{0}\\}\subset U$. $U$ is convex, therefore,
$x\in U$ but this contradicts to the condition $x\notin U$. Therefore, our
assumption is wrong and there exists at least one $v\in D_{0}$ such that
$[v,x]\cap U=\emptyset$. ∎
So, if a point from the convex polyhedron $D$ does not belong to a convex set
$U$ then it may be connected to at least one vertex of $D$ by a segment that
does not intersect $U$. Let us demonstrate now that if two vertices of $D$ may
be connected in $D$ by a continuous path that does not intersect $U$ then
these vertices can be connected in $D_{1}$ by a path that is a sequence of
edges $D$, which do not intersect $U$.
###### Lemma 5.
Let $v,v^{\prime}\in D_{0}$, $v,v^{\prime}\notin U$. Suppose that
$\varphi:[0,1]\to(D\setminus U)$ is a continuous path, $\varphi(0)=v$ and
$\varphi(1)=v^{\prime}$. Then there exists such a sequence of vertices
$\\{v_{0},\ldots,v_{l}\\}\subset(D\setminus U)$ that any two successive
vertices, $v_{i},v_{i+1}$, are connected by an edge
$e_{i,i+1}\subset(D_{1}\setminus U)$.
###### Proof.
Let us, first, prove the statement: the vertices $v,v^{\prime}$ belong to one
path-connected component of $D\setminus U$ if and only if they belong to one
path-connected component of $D_{1}\setminus U$.
Let us iteratively transform the path $\varphi$. On the $k$th iteration we
construct a path that connects $v$ and $v^{\prime}$ in $D_{d-k}\setminus U$,
where $d=\dim D$ and $k=1,\ldots,d-1$. We start from a transformation of path
in a face of $D$.
Let $S\subset D_{j}$ be a closed $j$-dimensional face of $D$, $j\geq 2$ and
let $\psi:[0,1]\to(D_{j}\setminus U)$ be a continuous path, $\psi(0)=v$,
$\psi(1)=v^{\prime}$ and $\psi([0,1])\cap U=\emptyset$. We will transform
$\psi$ into a continuous path $\psi_{S}:[0,1]\to(D_{j}\setminus U)$ with the
following properties: (i) $\psi_{S}(0)=v$, $\psi_{S}(1)=v^{\prime}$, (ii)
$\psi_{S}([0,1])\cap U=\emptyset$, (iii) $\psi_{S}([0,1])\setminus
S\subseteq\psi([0,1])\setminus S$ and (iv) $\psi_{S}([0,1])\cap
ri(S)=\emptyset$. The properties (i) and (ii) are the same as for $\psi$, the
property (iii) means that all the points of $\psi_{S}([0,1])$ outside $S$
belong also to $\psi([0,1])$ (no new points appear outside $S$) and the
property (iv) means that there are no points of $\psi_{S}([0,1])$ in $ri(S)$.
To construct this $\psi_{S}$ we consider two cases:
1. 1.
$U\cap ri(S)\neq\emptyset$, i.e. there exists $y^{0}\in U\cap ri(S)$;
2. 2.
$U\cap ri(S)=\emptyset$.
In the first case, let us project any $\psi(\tau)\in ri(S)$ onto
$r\partial(S)$ from the center $y^{0}$. Let $y\in S$, $y\neq y^{0}$. There
exists such a $\lambda(y)\geq 1$ that $y^{0}+\lambda(y)(y-y^{0})\in
r\partial(S)$. This function $\lambda(y)$ is continuous in
$S\setminus\\{y^{0}\\}$. The function $\lambda(y)$ can be expressed through
the Minkowski gauge functional [32] defined for a set $K$ and a point $x$:
$\begin{split}p_{K}(x)=\inf\\{r>0\,|\,x\in
rK\\};\;\;\lambda(y)=\left(p_{(D-y_{0})}(y-y_{0})\right)^{-1}\,.\end{split}$
Let us define for any $y\in ri(S)$, $y\neq y^{0}$ a projection
$\pi_{S}(y)=y^{0}+\lambda(y)(y-y^{0})$. This projection is continuous in
$S\setminus\\{y^{0}\\}$, and $\pi_{S}(y)=y$ if $y\in r\partial(S)$. It can be
extended as a continuous function onto whole $D_{j}\setminus\\{y^{0}\\}$:
$\pi_{S}(y)=\left\\{\begin{array}[]{ll}\pi_{S}(y)&\mbox{ if }y\in
S\setminus\\{y^{0}\\}\,;\\\ y&\mbox{ if }y\in D_{j}\setminus
S\,.\end{array}\right.$
The center $y^{0}\in U$. Because of the convexity of $U$, if $y\notin U$ then
$y^{0}+\lambda(y-y^{0})\notin U$ for any $\lambda\geq 1$. Therefore, the path
$\psi_{S}(t)=\pi_{S}(\psi(t))$ does not intersect $U$ and satisfies all the
requirements (i)-(iv).
Let us consider the second case, $U\cap ri(S)=\emptyset$. There are the
moments of the first entrance of $\psi(t)$ in $S$ and the last going of this
path out of $S$:
$\tau_{1}=\min\\{\tau\,|\,\psi(\tau)\in
S\\},\;\;\tau_{2}=\max\\{\tau\,|\,\psi(\tau)\in S\\}\,,$
$0\leq\tau_{1}\leq\tau_{2}\leq 1$. Let $y^{1}=\psi(\tau_{1})$ and
$y^{2}=\psi(\tau_{2})$. If $y^{1}=y^{2}$ then we can just delete the loop
between $\tau_{1}$ and $\tau_{2}$ from the path $\psi(\tau)$ and get the path
that does not enter $ri(S)$. So, let $y^{1}\neq y^{2}$.
These points belong to $r\partial(S)$. Let $y^{S}\in ri(S)$ be an arbitrary
point in the relative interior of $S$ which does not belong to the segment
$[y^{1},y^{2}]$ ($\dim S\geq 2$). The segments $[y^{1},y^{S}]$ and
$[y^{2},y^{S}]$ do not intersect $U$ because the following reasons: $U\cap
S\subset r\partial(S)$ (may be empty), neither $y^{1}$ nor $y^{2}$ belong to
$U$, and all other points of the 3-vertex polygonal chain
$[y^{1},y^{S},y^{2}]$ belong to $ri(S)$.‘
Let $P(y^{1},y^{S},y^{2})$ be a plane that includes the chain
$[y^{1},y^{S},y^{2}]$. The intersection $S\cap P(y^{1},y^{S},y^{2})$ is a
convex polygon. The convex set $U\cap S\cap P(y^{1},y^{S},y^{2})$ belongs to
the border of this polygon. Therefore, it belongs to one side of it (Fig. 2)
(may be empty) because convexity of the polygon and of the set $U$. The couple
of points $y^{1},y^{2}$ cut the border of the polygon in two connected broken
lines. At least one of them does not intersect $U$ (Fig. 2). Let us substitute
$\psi$ on the interval $[\tau_{1},\tau_{2}]$ by this broken line. The new path
does not intersect $ri(S)$. Let us use for this new path the notation
$\psi_{S}(t)$. The path $\psi_{S}$ does not intersect $ri(S)$ and $U$, and all
the points on them outside $S$ are the points on the path $\psi$ for the same
values of the argument $\tau$.
Fig. 2: Intersection of a face $S$ with the plane $P(y^{1},y^{S},y^{2})$ when
$U\cap ri(S)=\emptyset$ (Lemma 5, case 2). In this intersection, $U\cap
S\subset r\partial(S)$ belongs to one side of the polygon (the bold segment).
The dashed lines with arrows represent the 3-vertex polygonal chain
$[y^{1},y^{S},y^{2}]$. There exists a path from $y^{1}$ to $y^{2}$ along the
boundary of the polygon. In Fig., this is the polygonal chain that follows the
solid lines with arrows.
So, for any closed face $S\subset D$ with $\dim S=j\geq 2$ and a continuous
path $\psi:[0,1]\to(D_{j}\setminus U)$ that connects the vertices
$v,v^{\prime}$ of $D$ ($\psi(0)=v$, $\psi(1)=v^{\prime}$) we construct a
continuous path $\psi_{S}:[0,1]\to(D_{j}\setminus U)$ that connects the same
vertices, does not intersect $ri(S)$ and takes no new values outside $S$,
$\psi_{S}([0,1])\setminus S\subseteq\psi([0,1])\setminus S$.
Let us order the faces $S\subseteq D$ with $\dim S\geq 2$ in such a way that
$\dim S_{i}\geq\dim S_{j}$ for $i<j$: $D=S_{0},S_{1},\ldots,S_{\ell}$. Let us
start from a given path $\varphi:[0,1]\to D\setminus U$ that connects the
vertices $v$ and $v^{\prime}$ and let us apply sequentially the described
procedure:
$\theta=(\ldots(((\varphi_{S_{0}})_{S_{1}})_{S_{2}})\ldots)_{S_{\ell}}\,.$
By the construction, this path $\theta$ does not intersect any relative
interior $ri(S_{k})$ ($k=0,1,\ldots,\ell$). Therefore, the image of $\theta$
belongs to $D_{1}$, $\theta:[0,1]\to(D_{1}\setminus U)$. It can be transformed
into a simple path in $D_{1}\setminus U$ by deletion of all loops (if they
exist). This simple path (without self-intersections) is just the sequence of
edges we are looking for. ∎
Lemmas 4, 5 allow us to describe the connected components of the
$d$-dimensional set $D\setminus U$ through the connected components of the
one-dimensional continuum $D_{1}\setminus U$.
###### Proposition 6.
Let $W_{1},\ldots,W_{q}$ be all the path-connected components of $D\setminus
U$. Then $W_{i}\cap D_{0}\neq\emptyset$ for all $i=1,\ldots,q$, the continuum
$D_{1}\setminus U$ has $q$ path-connected components and $W_{i}\cap D_{1}$ are
these components.
###### Proof.
Due to Lemma 4, each path-connected component of $D\setminus U$ includes at
least one vertex of $D$. According to Lemma 5, if two vertices of $D$ belong
to one path-connected component of $D\setminus U$ then they belong to one
path-connected component of $D_{1}\setminus U$. The reverse statement is
obvious, because $D_{1}\subset D$ and a continuous path in $D_{1}$ is a
continuous path in $D$. ∎
We can study connected components of a simpler, discrete object, the graph
$\widetilde{D_{1}}$. The path-connected components of $D\setminus U$
correspond to the connected components of the graph
$\widetilde{D_{1}}\setminus U$. (This graph is produced from
$\widetilde{D_{1}}$ by deletion all the vertices that belong to $U_{0}$ and
all the edges that belong to $U_{1}$).
###### Proposition 7.
Let $W_{1},\ldots,W_{q}$ be all the path-connected components of $D\setminus
U$. Then the graph $\widetilde{D_{1}}\setminus U$ has exactly $q$ connected
components and each set $W_{i}\cap D_{0}$ is the set of the vertices of $D$ of
one connected component of $\widetilde{D_{1}}\setminus U$.
###### Proof.
Indeed, every path between vertices in $D_{1}$ includes a path that connects
these vertices and is the sequence of edges. (To prove this statement we just
have to delete all loops in a given path.) Therefore, the vertices
$v_{1},v_{2}$ belong to one connected component of $\widetilde{D_{1}}\setminus
U$ if and only if they belong to one path-connected component of
$D_{1}\setminus U$. The rest of the proof follows from Proposition 6. ∎
We proved that the path-connected components of $D\setminus U$ are in one-to-
one correspondence with the components of the graph
$\widetilde{D_{1}}\setminus U$ (the correspondent components have the same
sets of vertices). In applications, we will meet the following problem. Let a
point $x\in D\setminus U$ be given. Find the path-connected component of
$D\setminus U$ which includes this point. There are two basic ways to find
this component. Assume that we know the connected components of
$\widetilde{D_{1}}\setminus U$. First, we can examine the segments $[x,v]$ for
all vertices $v$ of $D$. At least one of them does not intersect $U$ (Lemma
4). Let it be $[x,v_{0}]$. We can find the connected component
$\widetilde{D_{1}}\setminus U$ that contains $v_{0}$. The point $x$ belongs to
the correspondent path-connected component of $D\setminus U$. This approach
exploits the $V$-description of the polyhedron $D$. The work necessary for
this method is proportional to the number of vertices of $D$.
Another method is based on projection on the faces of $D$. Let $x\in ri(D)$.
We can take any point $y^{0}\in D\setminus U$ and find the unique
$\lambda_{1}>1$ such that $x^{1}=y_{0}+\lambda_{1}(x-y^{0})\in r\partial(D)$.
Let $x^{1}\in ri(S_{1})$, where $S_{1}$ is a face of $D$. If $S_{1}\cap
U=\emptyset$ then we can take any vertex $v_{0}\in S_{1}$ and find the
connected component $\widetilde{D_{1}}\setminus U$ that contains $v_{0}$. This
component gives us the answer. If $S_{1}\cap U\neq\emptyset$ then we can take
any $y^{1}\in S_{1}\cap U$ and find the unique $\lambda_{2}>1$ such that
$x^{2}=y^{1}+\lambda_{2}(x^{1}-y^{1})\in r\partial(S)$. This $x^{2}$ belongs
to the relative boundary of the face $S_{1}$. If $x^{2}$ is not a vertex then
it belongs to the relative interior of some face $S_{2}$, $\dim S_{2}>0$ and
we have to continue. At each iteration, the dimension of faces decreases.
After $d=\dim D$ iterations at most we will get the vertex $v$ we are looking
for (see also Fig. 1) and find the connected component of
$\widetilde{D_{1}}\setminus U$ which gives us the answer. Here we exploit the
$H$-description of $D$.
### 2.2 Description of the connected components of $D\setminus U$ by
inequalities
Let $W_{1},\ldots,W_{q}$ be the path-connected components of $D\setminus U$.
###### Proposition 8.
For any set of indices $I\subset\\{1,\ldots,q\\}$ the set
$K_{I}=U\bigcup\left(\bigcup_{i\in I}W_{i}\right)$
is convex.
###### Proof.
Let $y^{1},y^{2}\in K_{I}$. We have to prove that $[y^{1},y^{2}]\subset
K_{I}$. Five different situations are possible:
1. 1.
$y^{1},y^{2}\in U$;
2. 2.
$y^{1}\in U,\,y^{2}\in W_{i}$, $i\in I$;
3. 3.
$y^{1},y^{2}\in W_{i}$, $i\in I$, $[y^{1},y^{2}]\cap U=\emptyset$;
4. 4.
$y^{1},y^{2}\in W_{i}$, $i\in I$, $[y^{1},y^{2}]\cap U\neq\emptyset$;
5. 5.
$y^{1}\in W_{i},\,y^{2}\in W_{j}$, $i,j\in I$, $i\neq j$.
We will systematically use two simple facts: (i) the convexity of $U$ implies
that its intersection with any segment is a segment and (ii) if $x^{1}\in
W_{i}$ and $x^{2}\in D\setminus W_{i}$ then the segment $[x^{1},x^{2}]$
intersects $U$ because $W_{i}$ is a path-connected component of $U$.
In case 1, $[y^{1},y^{2}]\subset U\subset K$ because convexity $U$.
In case 2, there exists such a point $y^{3}\in(y^{1},y^{2})$ that
$[y^{1},y^{3})\subseteq U\cap[y^{1},y^{2}]\subseteq[y^{1},y^{3}]$. The segment
$(y^{3},y^{2}]$ cannot include any point $x\in D\setminus W_{i}$ because it
does not include any point from $U$. Therefore, in this case
$(y^{3},y^{2}]\subset W_{i}\subset K$ and $y^{3}\in K$ because it belongs
either to $U$ or to $W_{i}$.
In case 3, $[y^{1},y^{2}]\subset W_{i}\subset K$ because $W_{i}$ is a path-
connected component of $D\setminus U$ and $[y^{1},y^{2}]\cap U=\emptyset$.
In case 4, $[y^{1},y^{2}]\cap U$ is a segment $L$ with the ends $x^{1},x^{2}$.
It may be $[x^{1},x^{2}]$ ($y^{1}<x^{1}\leq x^{2}<y^{2}$), $(x^{1},x^{2}]$
($y^{1}\leq x^{1}<x^{2}<y^{2}$), $[x^{1},x^{2})$ ($y^{1}<x^{1}<x^{2}\leq
y^{2}$), or $(x^{1},x^{2})$ ($y^{1}\leq x^{1}<x^{2}\leq y^{2}$). This segment
cuts $[y^{1},y^{2}]$ in three segments: $[y^{1},y^{2}]=L_{1}\cup L\cup L_{2}$,
$L_{1}$ includes $y^{1}$ and $L_{2}$ includes $y^{2}$. Therefore,
$L_{1}\subset W_{i}$, $L\subset U$ and $L_{2}\subset W_{i}$ because $W_{i}$ is
a path-connected component of $D\setminus U$ and $U$ is convex. So,
$[y^{1},y^{2}]\subset K$.
In case 5, $[y^{1},y^{2}]\cap U$ is also a segment $L$ with the ends
$x^{1},x^{2}$. It may be $[x^{1},x^{2}]$ ($y^{1}<x^{1}\leq x^{2}<y^{2}$),
$(x^{1},x^{2}]$ ($y^{1}\leq x^{1}<x^{2}<y^{2}$), $[x^{1},x^{2})$
($y^{1}<x^{1}<x^{2}\leq y^{2}$), or $(x^{1},x^{2})$ ($y^{1}\leq
x^{1}<x^{2}\leq y^{2}$). This segment cuts $[y^{1},y^{2}]$ in three segments:
$[y^{1},y^{2}]=L_{1}\cup L\cup L_{2}$, $L_{1}$ includes $y^{1}$ and $L_{2}$
includes $y^{2}$. Therefore, $L_{1}\subset W_{i}$, $L\subset U$ and
$L_{2}\subset W_{j}$ because $W_{i,j}$ are path-connected components of
$D\setminus U$ and $U$ is convex. So, $[y^{1},y^{2}]\subset K$. ∎
Typically, the set $U$ is represented by a set of inequalities, for example,
$G(x)\leq g$. It may be useful to represent the path-connected components of
$D\setminus U$ by inequalities. For this purpose, let us first construct a
convex polyhedron $Q\subset U$ with the same number of path-connected
components in $D\setminus Q$, $V_{1},\ldots,V_{q}$ and with inclusons
$W_{i}\subset V_{i}$. We will construct $Q$ as a convex hull of a finite set.
Let us select the edges $e$ of $D$ which intersect $U$ but the intersection
$e\cap U$ does not include vertices of $D$. For every such edge we select one
point $x_{e}\in e\cap U$. The set of these points is $Q_{1}$. By definition,
$Q={\rm conv}(U_{0}\cup Q_{1})\,.$ (7)
$Q$ is convex, hence, we can apply all the previous results about the
components of $D\setminus U$ to the components of $D\setminus Q$.
###### Lemma 9.
The set $U_{0}\cup Q_{1}$ is the set of vertices of $Q$.
###### Proof.
A point $x\in U_{0}\cup Q_{1}$ is not a vertex of $Q={\rm conv}(U_{0}\cup
Q_{1})$ if and only if it is a convex combination of other points from this
set: there exist such $x_{1},\ldots,x_{k}\in U_{0}\cup Q_{1}$ and
$\lambda_{1},\ldots,\lambda_{k}>0$ that $x_{i}\neq x$ for all $i=1,\ldots,k$
and
$\sum_{i=1}^{k}\lambda_{i}=1\,,\;\;\sum_{i=1}^{k}\lambda_{i}x_{i}=x\,.$
If $x\in U_{0}$ then this is impossible because $x$ is a vertex of $D$ and
$U_{0}\cup Q_{1}\subset D$. If $x\in Q_{1}$ then it belongs to the relative
interior of an edge of $D$ and, hence, may be a convex combination of points
$D$ from this edge only. By construction, $U_{0}\cup Q_{1}$ may include only
one internal point from an edge and in this case does not include a vertex
from this edge. Therefore, all the points from $Q_{1}$ are vertices of $Q$. ∎
###### Lemma 10.
The set $D\setminus Q$ has $q$ path-connected components $V_{1},\ldots,V_{q}$
that may be enumerated in such a way that $W_{i}\subset V_{i}$ and
$W_{i}=V_{i}\setminus U$.
###### Proof.
To prove this statement about the path-connected components, let us mention
that $Q$ and $U$ include the same vertices of $D$, the set $U_{0}$, and cut
the same edges of $D$. Graphs $\widetilde{D_{1}}\setminus Q$ and
$\widetilde{D_{1}}\setminus U$ coincide. $Q\subset U$ because of the convexity
of $U$ and definition of $Q$. To finalize the proof, we can apply Proposition
7. ∎
###### Proposition 11.
Let $I$ be any set of indices from $\\{1,\ldots,q\\}$.
$Q\bigcup\left(\bigcup_{i\in I}V_{i}\right)={\rm conv}\left(U_{0}\bigcup
Q_{1}\bigcup\left(\bigcup_{i\in I}\left(D_{0}\bigcap
V_{i}\right)\right)\right)$ (8)
###### Proof.
On the left hand side of (8) we see the union of $Q$ with the connected
components $V_{i}$ ($i\in I$). On the right hand side there is a convex
envelope of a finite set. This finite set consists of the vertices of $Q$,
($U_{0}\cup Q_{1}$) and the vertices of $D$ that belong to $V_{i}$ ($i\in I$).
Let us denote by $R_{I}$ the right hand side of (8) and by $L_{I}$ the left
hand side of (8).
$L_{I}$ is convex due to Proposition 8 applied to $Q$ and $V_{i}$. The
inclusion $R_{I}\subseteq L_{I}$ is obvious because $L_{I}$ is convex and
$R_{I}$ is defined as a convex hull of a subset of $L_{I}$. To prove the
inverse inclusion, let us consider the path-connected components of
$D\setminus R_{I}$. Sets $V_{j}$ ($j\notin I$) are the path-connected
components of $D\setminus R_{I}$ because they are the path-connected
components of $D\setminus Q$, $Q\subset R_{I}$ and $R_{I}\cap V_{j}=\emptyset$
for $j\notin I$. There exist no other path-connected components of $Q\subset
R_{I}$ because all the vertices of $V_{i}$ ($i\in I$) belong to $R_{I}$ by
construction, hence, $D_{0}\setminus R_{I}\subset\cup_{j\notin I}V_{j}$. Due
to Lemma 4 every path-connected component of $D\subset R_{I}$ includes at
least one vertex of $D$. Therefore, $V_{j}$ ($j\notin I$) are all the path-
connected components of $D\setminus R_{I}$ and $D\setminus R_{I}=\cup_{j\notin
I}V_{j}$. Finally, $R_{I}=D\setminus\cup_{j\notin
I}V_{j}=Q\cup\left(\cup_{i\in I}V_{i}\right)=L_{I}$. ∎
According to Lemma 9, each path-connected component $W_{i}\subset D\setminus
U$ can be represented in the form $W_{i}=V_{i}\setminus U$, where $V_{i}$ is a
path-connected component of $D\setminus Q$. By construction, $Q\subset U$,
hence
$W_{i}=(Q\cup V_{i})\setminus U\,.$ (9)
If $U$ is given by a system of inequalities then representations (8) and (9)
give us the possibility to represent $W_{i}$ by inequalities. Indeed, the
convex envelope of a finite set in (8) may be represented by a system of
linear inequalities. If the sets $Q\cup V_{i}$ and $U$ in (9) are represented
by inequalities then the difference between them is also represented by the
system of inequalities.
The description of the path-connected component of $D\setminus U$ may be
constructed by the following steps:
1. 1.
Construct the graph of the 1-skeleton of $D$, this is $\widetilde{D_{1}}$;
2. 2.
Find the vertices of $D$ that belong to $U$, this is the set $U_{0}$;
3. 3.
Find the edges of $D$ that intersect $U$, this is the set $U_{1}$.
4. 4.
Delete from $\widetilde{D_{1}}$ all the vertices from $U_{0}$ and the edges
from $U_{1}$, this is the graph $\widetilde{D_{1}}\setminus U$;
5. 5.
Find all the connected components of $\widetilde{D_{1}}\setminus U$. Let the
sets of vertices of these connected components be $V_{01},\ldots,V_{0q}$;
6. 6.
Select the edges $e$ of $D$ which intersect $U$ but the intersection $e\cap U$
does not include vertices of $D$. For every such an edge select one point
$x_{e}\in e\cap U$. The set of these points is $Q_{1}$.
7. 7.
For every $i=1,\ldots,q$ describe the polyhedron $R_{i}={\rm conv}(U_{0}\cup
Q_{1}\cup V_{0i})$;
8. 8.
There exists $q$ path-connected components of $D\setminus U$:
$W_{i}=R_{i}\setminus U$.
Every step can be performed by known algorithms including algorithms for the
solution of the double description problem [49, 14, 21] and the convex hull
algorithms [55].
Fig. 3: Construction of the path-connected components $W_{i}$ of $D\setminus
U$ for the simple example. (a) The balance simplex $D$, the set $U$ and the
path-connected components $W_{i}$; (b) The polyhedron $Q={\rm conv}(U_{0}\cup
Q_{1})$ (7) ($U_{0}=\emptyset$, $Q_{1}$ consists of the middles of the edges)
and the connected components $V_{i}$ of $D\setminus Q$: $V_{i}=\\{c\in
D\,|\,c_{i}>1/2\\}$; (c) The set $R_{1}={\rm conv}(U_{0}\cup
Q_{1}\cup(D_{0}\cap V_{1}))$; (d) The connected components $W_{1}$ described
by the inequalities (as $R_{1}\setminus U$ (8)).
Let us use the simple system of three reagents, $A_{1,2,3}$ (Fig. 1) to
illustrate the main steps of the construction of the path-connected
components. The polyhedron $D$ is here the 2D simplex (Fig. 1a). The plane
${\rm Aff}D$ is given by the balance equation $c_{1}+c_{2}+c_{3}=1$. We select
$U=\\{c\,|\,G(c)\leq g_{0}\\}$ as an example of a convex set (Fig. 3a). It
includes no vertices of $D$, hence, $U_{0}=\emptyset$. $U$ intersects each
edge of $D$ in the middle point, hence, $U_{1}$ includes all the edges of $D$.
The graph $\widetilde{D_{1}}\setminus U$ consists of three isolated vertices.
Its connected components are these isolated vertices. $Q_{1}$ consists of
three points, the middles of the edges $(1/2,1/2,0)$, $(1/2,0,1/2)$ and
$(0,1/2,1/2)$ (in this example, the choice of these points is unambiguous,
Fig. 3a).
The polyhedron $Q$ is a convex hull of these three points, that is the
triangle given in ${\rm Aff}(D)$ by the system of three inequalities
$c_{1,2,3}\leq 1/2$ (Fig. 3b). The connected components of $D\setminus Q$ are
the triangles $V_{i}$ given in $D$ by the inequalities $c_{i}>1/2$. In the
whole $\mathbb{R}^{3}$, these sets are given by the systems of an equation and
inequalities:
$V_{i}=\\{c\,|\,c_{1,2,3}\geq 0,\,c_{1}+c_{2}+c_{3}=1,\,c_{i}>1/2\\}\,.$
The polyhedron $R_{i}$ is the convex hull of four points, the middles of the
edges and the $i$th vertex (Fig. 3c). In $D$, $R_{i}$ is given by two linear
inequalities, $c_{j}\leq 1/2,\,j\neq i$. In the whole $\mathbb{R}^{3}$, these
inequalities should be supplemented by the equation and inequalities that
describe $D$:
$R_{i}=\\{c\,|\,c_{1,2,3}\geq 0,\,c_{1}+c_{2}+c_{3}=1,\,c_{j}\leq 1/2\,(j\neq
i)\\}\,.$
The path-connected components of $D\setminus U$, $W_{i}$ are described as
$R_{i}\setminus U$ Fig. 3d): in $D$ we get $W_{i}=\\{c\,|\,c_{j}\leq
1/2\,(j\neq i),\,G(c)>g_{0}\\}$. In the whole $\mathbb{R}^{3}$,
$W_{i}=\\{c\,|\,c_{1,2,3}\geq 0,\,c_{1}+c_{2}+c_{3}=1,\,c_{j}\leq 1/2\,(j\neq
i),\,G(c)>g_{0}\\}\,.$
$V_{i}$ are convex sets in this simple example, therefore, it is possible to
simplify slightly the description of the components $W_{i}$ and to represent
them as $V_{i}\setminus U$:
$W_{i}=\\{c\,|\,c_{1,2,3}\geq
0,\,c_{1}+c_{2}+c_{3}=1,\,c_{i}>1/2,\,G(c)>g_{0}\\}\,$
(or $W_{i}=\\{c\,|\,c_{i}>1/2,\,G(c)>g_{0}\\}$ in $D$).
In the general case (more components and balance conditions), the connected
components $V_{i}$ may be non-convex, hence, description of these sets by the
systems of linear equations and inequalities may be impossible. Nevertheless,
there exists another version of the description of $W_{i}$ where a smaller
polyhedron is used instead of $R_{i}$
Let $V_{0i}$ be the set of vertices of a connected component of the graph
$\widetilde{D_{1}}\setminus U$. Let $E_{\rm out}(V_{0i})$ be the set of the
outer edges of $V_{0i}$ in $\widetilde{D_{1}}$ i.e., this is the set of edges
of $\widetilde{D_{1}}$ that connect vertices from $V_{0i}$ with vertices from
$D_{0}\setminus V_{0i}$. For each $e\in E_{\rm out}(V_{0i})$ the corresponding
edge $e\subset D_{1}$ intersects $U$ because $V_{0i}$ is the set of vertices
of a connected component of the graph $\widetilde{D_{1}}\setminus U$.
Let us select a point $x_{e}\in U\cap e$ for each $e\in E_{\rm out}(V_{0i})$
(we use the same notations for the edges from $\widetilde{D_{1}}$ and the
corresponding edges from ${D_{1}}$). Let us use the notation
$Q_{0i}=\\{x_{e}\,|\,e\in E_{\rm out}(V_{0i})\\}\,.$
###### Proposition 12.
The path-connected component $W_{i}$ of $D\setminus U$ allows the following
description:
$W_{i}={\rm conv}(Q_{0i}\cup V_{0i})\setminus U\,.$
###### Proof.
The set $Q_{0i}\subset U$ and $V_{0i}$ is the set of vertices of a connected
component of the graph $\widetilde{D_{1}}\setminus Q_{0i}$ by construction
because $Q_{0i}$ cuts all the outer edges of $V_{0i}$ in $D_{1}$. The rest of
the proof follows the proofs of Lemma 10 and Proposition 11. ∎
This proposition allows us to describe $W_{i}$ by the system of inequalities.
For this purpose, we have to use a convex hull algorithm and describe the
convex hull ${\rm conv}(Q_{0i}\cup V_{0i})$ by the system of linear
inequalities and then add the inequality that describes the set $\setminus U$.
In the simple system (Fig. 3), the connected components of the graph
$\widetilde{D_{1}}\setminus U$ are the isolated vertices. The set $Q_{0i}$ for
the vertex $A_{i}$ consists of two middles of its incident edges. In Fig. 3b,
the set ${\rm conv}(Q_{0i}\cup V_{0i})$ for $V_{0i}=\\{A_{1}\\}$ is the
triangle $V_{1}$.
## 3 Thermodynamic tree
### 3.1 Problem statement
Let a real continuous function $G$ be given in the convex bounded polyhedron
$D\subset\mathbb{R}^{n}$. We assume that $G$ is strictly convex in $D$, i.e.
the set (the epigraph of $G$)
${\rm epi}(G)=\\{(x,g)\,|\,x\in D,\,g\geq G(x)\\}\subset
D\times(-\infty,\infty)$
is convex and for any segment $[x,y]\subset D$ ($x\neq y$) $G$ is not constant
on $[x,y]$. A strictly convex function on a bounded convex set has a unique
minimizer. Let $x^{*}$ be the minimizer of $G$ in $D$ and let $g^{*}=G(x^{*})$
be the corresponding minimal value. The level set $S_{g}=\\{x\in
D\,|\,G(x)=g\\}$ is closed and the sublevel set $U_{g}=\\{x\in
D\,|\,G(x)<g\\}$ is open in $D$. The sets $S_{g}$ and $D\setminus U_{g}$ are
compact and $S_{g}\subset D\setminus U_{g}$.
Let $x,y\in D$. According to Corollary 16 proven in the next subsection, an
admissible path from $x$ to $y$ in $D$ exists if and only if $\pi(y)$ belongs
to the ordered segment $[\pi(x^{*}),\pi(x)]$. Therefore, to describe
constructively the relation $x\succsim y$ in $D$ we have to solve the
following problems:
1. 1.
How to construct the thermodynamic tree $\mathcal{T}$?
2. 2.
How to find an image $\pi(x)$ of a state $x\in D$ on the thermodynamic tree
$\mathcal{T}$?
3. 3.
How to describe by inequalities a preimage of an ordered segment of the
thermodynamic tree, $\pi^{-1}([w,z])\subset D$ ($w,z\in\mathcal{T}$,
$z\succsim w$)?
### 3.2 Coordinates on the thermodynamic tree
We get the following lemma directly from Definition 1. Let $x,y\in D$.
###### Lemma 13.
$x\sim y$ if and only if $G(x)=G(y)$ and $x$ and $y$ belong to the same path-
connected component of $S_{g}$ with $g=G(x)$.
The path-connected components of $D\setminus U_{g}$ can be enumerated by the
connected components of the graph $\widetilde{D_{1}}\setminus U_{g}$. The
following lemma allows us to apply this result to the path-connected
components of $S_{g}$.
###### Lemma 14.
Let $g>g^{*}$, $W_{1},\ldots,W_{q}$ be the path-connected components of
$D\setminus U_{g}$ and let $\sigma_{1},\ldots,\sigma_{p}$ be the path-
connected components of $S_{g}$. Then $q=p$ and $\sigma_{i}$ may be enumerated
in such a way that $\sigma_{i}$ is the border of $W_{i}$ in $D$.
###### Proof.
$G$ is continuous in $D$, hence, if $G(x)>g$ then there exists a vicinity of
$x$ in $D$ where $G(x)>g$. Therefore $G(y)=g$ for every boundary point $y$ of
$D\setminus U_{g}$ in $D$ and $S_{g}$ is the boundary of $D\setminus U_{g}$ in
$D$.
Let us define a projection $\theta_{g}:D\setminus U_{g}\to S_{g}$ by the
conditions: $\theta_{g}(x)\in[x,x^{*}]$ and $G(\theta_{g}(x))=g$. By
definition, the inequality $G(x)\geq g$ holds in $D\setminus U_{g}$. The
function $f_{x}(\lambda)=G((1-\lambda)x^{*}+\lambda x)$ is strictly
increasing, continuous and convex function of $\lambda\in[0,1]$,
$f_{x}(0)=g^{*}<g$, $f_{x}(1)=G(x)\geq g$. The function $f_{x}(\lambda)$
depends continuously on $x\in D\setminus U_{g}$ in the uniform metrics.
Therefore, the solution $\lambda_{x}$ to the equation $f_{x}(\lambda)=g$ on
$[0,1]$ exists (the intermediate value theorem), is unique, and continuously
depends on $x\in D\setminus U_{g}$. The projection $\theta_{g}$ is defined as
$\theta_{g}(x)=(1-\lambda_{x})x^{*}+\lambda_{x}x$.
The fixed points of the projection $\theta_{g}$ are elements of $S_{g}$. The
image of each path-connected component $W_{i}$ is a path-connected set. The
preimage of every path-connected component $\sigma_{i}$ is also a path-
connected set. Indeed, let $\theta_{g}(x)\in\sigma_{i}$ and
$\theta_{g}(y)\in\sigma_{i}$. There exists a continuous path from $x$ to $y$
in $D\setminus U_{g}$. It may be composed from three paths: (i) from $x$ to
$\theta_{g}(x)$ along the line segment $[x,\theta_{g}(x)]\subset[x,x^{*}]$
then a continuous path in $\sigma_{i}$ between $\theta_{g}(x)$ and
$\theta_{g}(y)$ (it exists because $\sigma_{i}$ is a path-connected component
of $S_{g}$ and it belongs to $D\setminus U_{g}$ because $S_{g}\subset
D\setminus U_{g}$) and, finally, from $\theta_{g}(y)$ to $y$ along the line
segment $[\theta_{g}(y),y]\subset[x^{*},y]$. Therefore, the image of a path-
connected component $W_{i}$ is a path-connected components of $S_{g}$ that may
be enumerated by the same index $i$, $\sigma_{i}$. This $\sigma_{i}$ is the
border of $W_{i}$ in $D$. ∎
The equivalence class of $x\in D$ is defined as $[x]=\\{y\in D\,|\,y\sim
x\\}$. Let $W(x)$ be a path-connected component of $D\setminus U_{g}$
($g=G(x)$) for which $\theta_{g}(W(x))=[x]$. Due to Lemma 14, such a component
exists and
$W(x)=\\{y\in D\,|\,y\succsim x\\}\,.$ (10)
Let us define a one-dimensional continuum $\mathcal{Y}$ that consists of the
pairs $(g,M)$, where $g^{*}\leq g\leq g_{\max}$ and $M$ is a set of vertices
of a connected component of $\widetilde{D_{1}}\setminus U_{g}$. For each
$(g,M)$ the fundamental system of neighborhoods consists of the sets
$V_{\rho}$ ($\rho>0$):
$V_{\rho}=\\{(g^{\prime},M^{\prime})\,|\,(g^{\prime},M^{\prime})\in\mathcal{Y},\,|g-g^{\prime}|<\rho,\,M^{\prime}\subseteq
M\\}\,.$ (11)
Let us define the partial order on $\mathcal{Y}$:
$(g,M)\succsim(g^{\prime},M^{\prime})\mbox{ if }g\geq g^{\prime}\mbox{ and
}M\subseteq M^{\prime}\,.$
Let us introduce the mapping $\omega:D\to\mathcal{Y}$:
$\omega(x)=(G(x),W(x)\cap D_{0})\,.$
###### Theorem 15.
There exists a homeomorphism between $\mathcal{Y}$ and $\mathcal{T}$ that
preserves the partial order and makes the following diagram commutative:
---
$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\omega}$$\textstyle{\mathcal{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Y}}$
###### Proof.
According to Lemmas 14, 10 and Proposition 7, $\omega$ maps the equivalent
points $x$ to the same pair $(g,M)$ and the non-equivalent points to different
pairs $(g,M)$. For any $x,y\in D$, $x\succsim y$ if and only if
$\omega(x)\succsim\omega(y)$.
The fundamental system of neighborhoods in $Y$ may be defined using this
partial order. Let us say that $(g,M)$ is compatible to
$(g^{\prime},M^{\prime})$ if $(g^{\prime},M^{\prime})\succsim(g,M)$ or
$(g,M)\succsim(g^{\prime},M^{\prime})\\}$. Then for $\rho>0$
$V_{\rho}=\\{(g^{\prime},M^{\prime})\in\mathcal{Y}\,|\,|\gamma-\gamma^{\prime}|<\rho\mbox{
and }(g^{\prime},M^{\prime})\mbox{ is compatible to }(g,M)\\}\,.$
For sufficiently small $\rho$ this definition coincides with (11).
So, by the definition of $\mathcal{T}$ as a quotient space $D/\sim$,
$\mathcal{Y}$ has the same partial order and topology as $\mathcal{T}$. The
isomorphism between $\mathcal{Y}$ and $\mathcal{T}$ establishes one-to-one
correspondence between the $\pi$-image of the equivalence class $[x]$,
$\pi([x])$, and the $\omega$-image of the same class, $\omega([x])$. ∎
$\mathcal{Y}$ can be considered as a coordinate system on $\mathcal{T}$. Each
point is presented as a pair $(g,M)$ where $g^{*}\leq g\leq g_{\max}$ and $M$
is a set of vertices of a connected component of $\widetilde{D_{1}}\setminus
U_{g}$. The map $\omega$ is the coordinate representation of the canonical
projection $\pi:D\to\mathcal{T}$. Now, let us use this coordinate system and
the proof of Theorem 15 to obtain the following corollary.
###### Corollary 16.
An admissible path from $x$ to $y$ in $D$ exists if and only if
$\pi(y)\in[\pi(x^{*}),\pi(x)]\,.$
###### Proof.
Let there exist an admissible path from $x$ to $y$ in $D$, $\varphi:[0,1]\to
D$. Then $\pi(x)\succsim\pi(y)$ in $\mathcal{T}$. Let $\pi(x)=(G(x),M)$ in
coordinates $\mathcal{Y}$. For any $v\in M$, $\pi(y)\in[\pi(x^{*}),\pi(v)]$
and $\pi(x)\in[\pi(x^{*}),\pi(v)]$.
Assume now that $\pi(y)\in[\pi(x^{*}),\pi(x)]$ and $\pi(x)=(G(x),M)$. Then the
admissible path from $x$ to $y$ in $D$ can be constructed as follows. Let
$v\in M$ be a vertex of $D$. $G(v)\geq G(x)$ for each $v\in M$. The straight
line segment $[x^{*},v]$ includes a point $x_{1}$ with $G(x_{1})=G(x)$ and
$y_{1}$ with $G(y_{1})=G(y)$. Coordinates of $\pi(x_{1})$ and $\pi(x)$ in
$\mathcal{Y}$ coincide as well as coordinates of $\pi(y_{1})$ and $\pi(y)$.
Therefore, $x\sim x_{1}$ and $y\sim y_{1}$. The admissible path from $x$ to
$y$ in $D$ can be constructed as a sequence of three paths: first, a
continuous path from $x$ to $x_{1}$ inside the path-connected component of
$S_{G(x)}$ (Lemma 13), then from $x_{1}$ to $y_{1}$ along a straight line and
after that a continuous path from $y_{1}$ to $y$ inside the path-connected
component of $S_{G(y)}$. ∎
To describe the space $\mathcal{T}$ in coordinate representation
$\mathcal{Y}$, it is necessary to find the connected components of the graph
$\widetilde{D_{1}}\setminus U_{g}$ for each $g$. First of all, this function,
$g\mapsto\mbox{ the set of connected components of }\widetilde{D_{1}}\setminus
U_{g}\,,$
is piecewise constant. Secondly, we do not need to solve at each point the
computationally heavy problem of the construction of the connected components
of the graph $\widetilde{D_{1}}\setminus U_{g}$ “from scratch”. The problem of
the parametric analysis of these components as functions of $g$ appears to be
much cheaper. Let us present a solution of this problem. At the same time,
this is a method for the construction of the thermodynamic tree in coordinates
$(g,M)$.
The coordinate system $\mathcal{Y}$ allows us to describe the tree structure
of the continuum $\mathcal{T}$. This structure includes a root,
$(g^{*},D_{0})$, edges, branching points and leaves.
Let $M$ be a connected component of $\widetilde{D_{1}}\setminus U_{g}$ for
some $g$, $g^{*}<g<g_{\max}$. If $M\subsetneqq D_{0}$ then the set of all
points $(g,M)\in\mathcal{T}$ has for a given $M$ the form
$(\underline{a}_{M},\overline{a}_{M}]\times M$,
$\underline{a}_{M}<\overline{a}_{M}$. We call this set an edge of
$\mathcal{T}$.
If $M$ includes all the vertices of $D$ ($M=D_{0}$) then the set of all points
$(g,M)\in\mathcal{T}$ has the form $[g^{*},\overline{a}_{D_{0}}]\times D_{0}$.
This may be either an edge (if $\overline{a}_{D_{0}}>g^{*}$) or just a root,
$\\{(g^{*},D_{0})\\}$, (this is possible in 1D systems).
Let us define the numbers
$\underline{a}_{M}=\inf\\{g\,|\,(g,M)\in\mathcal{T}\\}$. Let us introduce the
set of outer edges of $M$ in $\widetilde{D_{1}}$, $E_{\rm out}(M)$. This is
the set of edges of $\widetilde{D_{1}}$ that connect vertices from $M$ with
vertices from $D_{0}\setminus M$. We keep the same notation, $E_{\rm out}(M)$,
for the set of the corresponding edges of $D$.
$\underline{a}_{M}=\max_{e\in E_{out}(M)}\min\\{G(x)\,|\,x\in e\\}\,.$ (12)
This number, $\underline{a}_{M}$, is the “cutting value” of $G$ for $M$. It
cuts $M$ from the other vertices of $\widetilde{D_{1}}$ in the following
sense: if we delete from $\widetilde{D_{1}}$ all the edges $e$ with the label
values $<\underline{a}_{M}$ then $M$ will remain attached to some vertices
from $D_{0}\setminus M$. If we delete the edges with the label values
$\leq\underline{a}_{M}$ then $M$ becomes disconnected from $D_{0}\setminus M$.
There is the only connected component of $\widetilde{D_{1}}\setminus
U_{\underline{a}_{M}}$ that includes $M$, $M^{\prime}\supsetneqq M$. The pair
$(\underline{a}_{M},M^{\prime})\in\mathcal{T}$ is a branching point of
$\mathcal{T}$. The edge $(\underline{a}_{M},\overline{a}_{M}]\times M$
connects two vertices, the upper vertex $(\overline{a}_{M},M)$ and the lower
vertex, $(\underline{a}_{M},M^{\prime})$.
If $M$ consists of one vertex, $M=\\{v\\}$, then the point $(G(v),\\{v\\})$ is
a leaf of $\mathcal{T}$.
### 3.3 Construction of the thermodynamic tree
To construct the tree of $G$ in $D$ we need the graph $\widetilde{D_{1}}$ of
the 1-skeleton of the polyhedron $D$. Elements of $\widetilde{D_{1}}$ should
be labeled by the values of $G$. Each vertex $v$ is labeled by the value
$\gamma_{v}=G(v)$ and each edge $e=[v,w]$ is labeled by the minimal value of
$G$ on the segment $[v,w]\subset D$, $g_{e}=\min_{[v,w]}G(x)$. We need also
the minimal value $g^{*}=\min_{D}\\{G(x)\\}$ because the root of the tree is
$(g^{*},D_{0})$.
The strictly convex function $G$ achieves its local maxima in $D$ only in
vertices. The vertex $v$ is a (local) maximizer of $g$ if $g_{e}<\gamma_{v}$
for each edge $e$ that includes $v$. The leaves of the thermodynamic tree are
pairs $(\gamma_{v},\\{v\\})$ for the vertices that are the local maximizers of
$G$.
As a preliminary step of the construction, we arrange and enumerate the labels
of the elements of $\widetilde{D_{1}}$, the vertices and edges, in descending
order. Let there exist $l$ different label values:
$g_{\max}=a_{1}>a_{2}>\ldots>a_{l}$. Each $a_{k}$ is a value $\gamma_{v}=G(v)$
at a vertex $v\in D_{0}$ or the minimum of $G$ on an edge $e\subset D_{1}$ (or
both). Let $A_{i}$ be the set of vertices $v\in D_{0}$ with $\gamma_{v}=a_{i}$
and let $E_{i}$ be the set of edges of $D_{1}$ with $g_{e}=a_{i}$
($i=1,\ldots,l$).
Let us construct the connected components of the graph
$\widetilde{D_{1}}\setminus U_{g}$ starting from $a_{1}=g_{\max}$. The
function $G$ is strictly convex, hence, $a_{1}=\gamma_{v}$ for a set of
vertices $A_{1}\subset D_{0}$ but it is impossible that $a_{1}=g_{e}$ for an
edge $e$, hence, $E_{1}=\emptyset$.
The set of connected components of $\widetilde{D_{1}}\setminus U_{g}$ is the
same for all $g\in(a_{i+1},a_{i}]$. For an interval $(a_{2},a_{1}]$ the
connected components of $\widetilde{D_{1}}\setminus U_{g}$ are the one-element
sets $\\{v\\}$ for $v\in A_{1}$.
For $g\in[g^{*},a_{l}]$ the graph $\widetilde{D_{1}}\setminus U_{g}$ includes
all the vertices and edges of $\widetilde{D_{1}}$ and, hence, it is connected
for this segment. Let us take, formally, $a_{l+1}=g^{*}$.
Let $\mathcal{L}_{i}=\\{M^{i}_{1},\ldots,M_{k_{i}}^{i}\\}$ be the set of the
connected components of $\widetilde{D_{1}}\setminus U_{g}$ for
$g\in(a_{i},a_{i-1}]$ ($i=1,\ldots,l$). Each connected component is
represented by the set of its vertices $M^{i}_{j}$. Let us describe the
recursive procedure for construction of $\mathcal{L}_{i}$:
1. 1.
Let us take formally $\mathcal{L}_{0}=\emptyset$.
2. 2.
Assume that $\mathcal{L}_{i-1}$ is given and $i\leq l$. Let us find the set
$\mathcal{L}_{i}$ of connected components of $\widetilde{D_{1}}\setminus
U_{g}$ for $g=a_{i}$ (and, therefore, for $g\in(a_{i+1},a_{i}]$).
* •
Add the one-element sets $\\{v\\}$ for all $v\in A_{i}$ to the set
$\mathcal{L}_{i-1}=\\{M^{i-1}_{1},\ldots,M_{k_{i-1}}^{i-1}\\}\,.$
Denote this auxiliary set of sets as
$\widetilde{L}_{i,0}=\\{M_{1},\ldots,M_{q}\\}$, where $q=k_{i-1}+|A_{i}|$.
* •
Enumerate the edges from $E_{i}$ in an arbitrary order:
$e_{1},\ldots,e_{|E_{i}|}$. For each $k=0,\ldots,|E_{i}|$, create recursively
an auxiliary set of sets $\widetilde{L}_{i,k}$ by the union of some of
elements of $\widetilde{L}_{k-1}$: Let $\widetilde{L}_{i,k-1}$ be given and
$e_{k}$ connects the vertices $v$ and $v^{\prime}$. If $v$ and $v^{\prime}$
belong to the same element of $\widetilde{L}_{i,k-1}$ then
$\widetilde{L}_{i,k}=\widetilde{L}_{i,k-1}$. If $v$ and $v^{\prime}$ belong to
the different elements of $\widetilde{L}_{i,k-1}$, $M$ and $M^{\prime}$, then
$\widetilde{L}_{i,k}$ is produced from $\widetilde{L}_{i,k-1}$ by the union of
$M$ and $M^{\prime}$:
$\widetilde{L}_{i,k}=(\widetilde{L}_{i,k-1}\setminus\\{M\\}\setminus\\{M^{\prime}\\})\cup\\{M\cup
M^{\prime}\\}$
(we delete two elements, $M$ and $M^{\prime}$, from $\widetilde{L}_{i,k-1}$
and add a new element $M\cup M^{\prime}$).
The set $\mathcal{L}_{i}$ of connected components of
$\widetilde{D_{1}}\setminus U_{g}$ for $g=a_{i}$ is
$\mathcal{L}_{i}=\widetilde{L}_{i,|E_{i}|}$.
Generically, all the labels of the graph $\widetilde{D_{1}}$ vertices and
edges are different and the sets $E_{i}$ and $A_{i}$ include not more than one
element. Moreover, for each $i$ either $E_{i}$ or $A_{i}$ is generically empty
and the description of the recursive procedure may be simplified for the
generic case:
1. 1.
Let us take formally $\mathcal{L}_{0}=\emptyset$.
2. 2.
Assume that $\mathcal{L}_{i-1}$ is given and $i\leq l$.
* •
If $a_{i}$ is a label of a vertex $v$, $a_{i}=\gamma_{v}$, then add the one-
element set $\\{v\\}$ to the set $\mathcal{L}_{i-1}$:
$\mathcal{L}_{i}=\mathcal{L}_{i-1}\cup\\{\\{v\\}\\}$.
* •
Let $a_{i}$ be a label of an edge $e=[v,v^{\prime}]$. If $v$ and $v^{\prime}$
belong to the same element of $\mathcal{L}_{i-1}$ then
$\mathcal{L}_{i}=\mathcal{L}_{i-1}$. If $v$ and $v^{\prime}$ belong to the
different elements of $\mathcal{L}_{i-1}$, $M$ and $M^{\prime}$, then
$\mathcal{L}_{i}$ is produced from $\mathcal{L}_{i-1}$ by the union of $M$ and
$M^{\prime}$ (delete elements $M$ and $M^{\prime}$ and add an element $M\cup
M^{\prime}$).
The described procedure gives us the sets of connected components of
$\widetilde{D_{1}}\setminus U_{g}$ for all $g$ and, therefore, we get the tree
$\mathcal{T}$. The descent from the higher values of $G$ allows us to avoid
the solution of the computationally more expensive problem of the calculation
of the connected components of a graph at any level of $G$.
### 3.4 The problem of attainable sets
In this section, we demonstrate how to solve the problem of attainable sets.
For given $x\in D$ (an initial state) we describe the attainable set
$Att(x)=\\{y\in D\,|\,x\succsim y\\}$
by a system of inequalities. Let the tree $\mathcal{T}$ of $G$ in $D$ be given
and let all the pairs $(g,M)\in\mathcal{T}$ be described. We also use the
notation $Att(z)$ for sets attainable in $\mathcal{T}$ from $z\in\mathcal{T}$.
First of all, let us describe the preimage of a point $(g,M)\in\mathcal{T}$ in
$D$. It can be described by the equation $G(x)=g$ and a set of linear
inequalities. For each edge $e$ we select a minimizer of $G$ on $e$,
$x_{e}={\rm argmin}\\{G(x)\,|\,x\in e\\}$ (we use the same notations for the
elements of the graph $\widetilde{D_{1}}$ and of the continuum $D_{1}$). Let
$Q_{M}=\\{x_{e}\,|\,e\in E_{\rm out}(M)\\}\,.$
In particular, $\underline{a}_{M}=\max\\{G(x)\,|\,x\in Q_{M}\\}$.
The following Proposition is a direct consequence of Proposition 12.
###### Proposition 17.
The preimage of $(g,M)$ in $D$ is a set
$\pi^{-1}(g,M)=\\{x\in{\rm conv}(Q_{M}\cup M)\,|\,G(x)=g\\}\,.$ (13)
The sets $M$ and $Q_{M}$ in (13) do not depend on the specific value of $g$.
It is sufficient that the point $(g,M)\in\mathcal{T}$ exists.
Let us consider the second projection of $\mathcal{T}$, i.e., the set of all
connected components of the graph $\widetilde{D_{1}}\setminus U_{g}$ for all
$g$. For a connected component $M$, the lower chain of connected components is
a sequence $M=M_{1}\subsetneqq M_{2}\subsetneqq\ldots\subsetneqq M_{k}$.
(“Lower” here means the descent in the natural order in $\mathcal{T}$,
$\succsim$.) For a given initial element $M=M_{1}$ the maximal lower chain of
$M$ is the lower chain of $M$ that cannot be extended by adding new elements.
By construction of connected components, the maximal lower chain of $M$ is
unique for each initial element $M$. In the maximal lower chain
$\underline{a}_{M_{i}}=\overline{a}_{M_{i+1}}$.
For each set of values $H\subset(\underline{a}_{M},\overline{a}_{M}]$ the
preimage of the set $H\times M\subset\mathcal{T}$ is given by (13) as
$\pi^{-1}(H\times M)=\\{x\in{\rm conv}(Q_{M}\cup M)\,|\,G(x)\in H\\}\,.$ (14)
We describe the set $Att(x)$ for $x\in D$ by the following procedures: (i)
find the projection $\pi(x)$ of $x$ onto $\mathcal{T}$, (ii) find the
attainable set in $\mathcal{T}$ from $\pi(x)$, $Att(\pi(x))$, and (iii) find
the preimage of this set in $D$:
$Att(x)=\pi^{-1}(Att(\pi(x)))\,.$ (15)
The attainable set $Att(g,M)$ in $\mathcal{T}$ from $(g,M)\in\mathcal{T}$ is
constructed as a union of edges and its parts. Let $M=M_{1}\subsetneqq
M_{2}\subsetneqq\ldots\subsetneqq M_{k}=D_{0}$ be the maximal lower chain of
$M$. Then
$\begin{split}Att(g,M)=&(\underline{a}_{1},g]\times
M_{1}\cup(\underline{a}_{2},\underline{a}_{1}]\times M_{2}\cup\ldots\\\
&\cup(\underline{a}_{k-1},\underline{a}_{k-2}]\times
M_{k-1}\cup[\underline{a}_{k},\underline{a}_{k-1}]\times M_{k}\,,\end{split}$
(16)
where $\underline{a}_{i}=\underline{a}_{M_{i}}$.
To find the preimage of $Att(g,M)$ in $D$ we have to apply formula (14) to
each term of (16). In Sec. 1.3 we demonstrated how to find $\pi(x)$.
Therefore, each step of the solution of the problem of attainable set (15) is
presented.
## 4 Chemical thermodynamics: examples
Fig. 4: The balance polygon $D$ on the plane with coordinates $[S]$ and
$[ES]$ for the four–component enzyme–substrate system $S$, $E$, $ES$ $P$ with
two balance conditions, $b_{S}=[S]+[ES]+[P]={\rm const}$ and
$b_{E}=[E]+[ES]={\rm const}$. Fig. 5: The graph $\widetilde{D_{1}}(b)$ of the
one-skeleton of the balance polyhedron for the six-component system, ${\rm
H}_{2}$, ${\rm O}_{2}$, ${\rm H}$, ${\rm O}$, ${\rm H}_{2}{\rm O}$, ${\rm
OH}$, as a piece-wise constant function of $b=(b_{\rm H},b_{\rm O})$. For each
vertex the components are indicated which have non-zero concentrations at this
vertex.
### 4.1 Skeletons of the balance polyhedra
In chemical thermodynamics and kinetics, the variable $N_{i}$ is the amount of
the $i$th component in the system. The balance polyhedron $D$ is described by
the positivity conditions $N_{i}\geq 0$ and the balance conditions (1)
$b_{i}(N)={\rm const}$ ($i=1,\ldots,m$). Under the isochoric (the constant
volume) conditions, the concentrations $c_{i}$ also satisfy the balance
conditions and we can construct the balance polyhedron for concentrations.
Sometimes, the balance polyhedron is called the reaction simplex with some
abuse of language because it is not obligatory a simplex when the number $m$
of the independent balance conditions is greater than one.
The graph $\widetilde{D_{1}}$ depends on the values of the balance functionals
$b_{i}=b_{i}(N)=\sum_{j=1}^{n}a_{i}^{j}N_{j}$. For the positive vectors $N$,
the vectors $b$ with coordinates $b_{i}=b_{i}(N)$ form a convex polyhedral
cone in $\mathbb{R}^{m}$. Let us denote this cone by $\Lambda$.
$\widetilde{D_{1}}(b)$ is a piece-wise constant function on $\Lambda$. Sets
with various constant values of this function are cones. They form a partition
of $\Lambda$. Analysis of this partition and the corresponding values of
$\widetilde{D_{1}}$ can be done by the tools of linear programming [26]. Let
us represent several examples.
In the first example, the reaction system consists of four components: the
substrate $S$, the enzyme $E$, the enzyme-substrate complex $ES$ and the
product $P$. we consider the system under constant volume. We denote the
concentrations by $[S]$, $[E]$, $[ES]$ and $[P]$. There are two balance
conditions: $b_{S}=[S]+[ES]+[P]={\rm const}$ and $b_{E}=[E]+[ES]={\rm const}$.
Fig. 6: Transformations of the graph $\widetilde{D_{1}}(b)$ with changes of
the relation between $b_{\rm H}$ and $b_{\rm O}$: (a) transition from the
regular case $b_{\rm H}>2b_{\rm O}$ to the regular case $2b_{\rm O}>b_{\rm
H}>b_{\rm O}$ through the singular case $b_{\rm H}=2b_{\rm O}$, (b) transition
from the regular case $2b_{\rm O}>b_{\rm H}>b_{\rm O}$ to the regular case
$b_{\rm H}<b_{\rm O}$ through the singular case $b_{\rm H}=b_{\rm O}$. Fig. 7:
The graph $\widetilde{D_{1}}$ for the eight-component system, ${\rm H}_{2}$,
${\rm O}_{2}$, ${\rm H}$, ${\rm O}$, ${\rm H}_{2}{\rm O}$, ${\rm OH}$, ${\rm
H}_{2}{\rm O}_{2}$, ${\rm H}{\rm O}_{2}$ for the stoichiometric mixture,
$b_{\rm H}=2b_{\rm O}$. The vertices that correspond also to the six-component
mixture are distinguished by bold font.
For $b_{S}>b_{E}$ the polyhedron (here the polygon) $D$ is a trapezium (Fig.
4a). Each vertex corresponds to two components that have non-zero
concentrations in this vertex. For $b_{S}>b_{E}$ there are four such pairs,
$(ES,P)$, $(ES,S)$, $(E,P)$ and $(E,S)$. For two pairs there are no vertices:
for $(S,P)$ the value $b_{E}$ is zero and for $(ES,E)$ it should be
$b_{S}<b_{E}$. When $b_{S}=b_{E}$, two vertices, $(ES,P)$ and $(ES,S)$,
transform into one vertex with one non-zero component, $ES$, an the polygon
$D$ becomes a triangle (Fig. 4b). When $b_{S}<b_{E}$ then $D$ is also a
triangle and a vertex $ES$ transforms in this case into $(ES,E)$ (Fig. 4c).
For the second example, we select a system with six components and two balance
conditions: ${\rm H}_{2}$, ${\rm O}_{2}$, ${\rm H}$, ${\rm O}$, ${\rm
H}_{2}{\rm O}$, ${\rm OH}$;
$\begin{split}&b_{\rm H}=2N_{\rm H_{2}}+N_{\rm H}+2N_{\rm H_{2}O}+N_{\rm
OH}\,,\\\ &b_{\rm O}=2N_{\rm O_{2}}+N_{\rm O}+N_{\rm H_{2}O}+N_{\rm
OH}\,.\end{split}$
The cone $\Lambda$ is a positive quadrant on the plane with the coordinates
$b_{\rm H},b_{\rm O}$. The graph $\widetilde{D_{1}}(b)$ is constant in the
following cones in $\Lambda$ ($b_{\rm H},b_{\rm O}>0$): (a) $b_{\rm H}>2b_{\rm
O}$, (b) $b_{\rm H}=2b_{\rm O}$, (c) $2b_{\rm O}>b_{\rm H}>b_{\rm O}$, (d)
$b_{\rm H}=b_{\rm O}$ and (e) $b_{\rm H}<b_{\rm O}$ (Fig. 5).
The cases (a) $b_{\rm H}>2b_{\rm O}$, (c) $2b_{\rm O}>b_{\rm H}>b_{\rm O}$,
and (e) $b_{\rm H}<b_{\rm O}$ (Fig. 5) are regular: there are two independent
balance conditions and for each vertex there are exactly two components with
non-zero concentration. In case (a) ($b_{\rm H}>2b_{\rm O}$), if $b_{\rm H}\to
2b_{\rm O}$ then two regular vertices, ${\rm H}_{2},\,{\rm H}_{2}{\rm O}$ and
${\rm H},\,{\rm H}_{2}{\rm O}$, join in one vertex (case (b)) with only one
non-zero concentration, ${\rm H}_{2}{\rm O}$ (Fig. 6a). This vertex explodes
in three vertices ${\rm O},\,{\rm H}_{2}{\rm O}$; ${\rm O}_{2},\,{\rm
H}_{2}{\rm O}$ and ${\rm H}_{2}{\rm O},\,{\rm OH}$, when $b_{\rm H}$ becomes
smaller than $2b_{\rm O}$ (case (c), $2b_{\rm O}>b_{\rm H}>b_{\rm O}$) (Fig.
6a). Analogously, in the transition from the regular case (c) to the regular
case (e) through the singular case (d) ($b_{\rm H}=b_{\rm O}$) three vertices
join in one, ${\rm 0H}$ that explodes in two (Fig. 6b).
For the modeling of hydrogen combustion, the eight-component model is used
usually: ${\rm H}_{2}$, ${\rm O}_{2}$, ${\rm H}$, ${\rm O}$, ${\rm H}_{2}{\rm
O}$, ${\rm OH}$, ${\rm H}_{2}{\rm O}_{2}$, ${\rm H}{\rm O}_{2}$. In Fig. 7 the
graph $\widetilde{D_{1}}$ is presented for one particular relations between
$b_{\rm H}$ and $2b_{\rm O}$, $b_{\rm H}=2b_{\rm O}$. This is the so-called
“stoichiometric mixture” where proportion between $b_{\rm H}$ and $2b_{\rm O}$
is the same as in the “product”, ${\rm H}_{2}{\rm O}$.
### 4.2 Examples of the thermodynamic tree
Fig. 8: The thermodynamic tree for the four–component enzyme–substrate system
$S$, $E$, $ES$ $P$ (Fig. 4) with excess of substrate: $b_{S}>b_{E}$ (case
(a)). The vertices and edges are enumerated in order of $\gamma_{v}$ and
$g_{e}$ (starting from the greatest values). The order of these numbers is
represented in Fig. On the right, the graphs $\widetilde{D_{1}}\setminus
U_{g}$ are depicted. The solid bold line on the tree is the thermodynamically
admissible path from the initial state $E,S$ (enzyme plus substrate) to the
equilibrium. There are leaves at all levels $g=\gamma_{i}$. There are
branching points at $g=g_{1,2,3}$ and no vertices at $g=g_{4}$.
In this section, we present two example of the thermodynamic tree. First, let
us consider the trapezium (Fig. 4a). Let us select the order of numbers
$\gamma_{v}$ and $g_{e}$ according to Fig. 8. The vertices and edges are
enumerated in order of $\gamma_{v}$ and $g_{e}$ (starting from the greatest
values). The tree is presented in Fig. 8. On the right, the graphs
$\widetilde{D_{1}}\setminus U_{g}$ are depicted for all intervals
$(a_{i-1},a_{i}]$. For $(\gamma_{2},\gamma_{1}]$ it is just a vertex $v_{1}$.
For $(g_{1},\gamma_{2}]$ it consists of two disjoint vertices, $v_{1}$ and
$v_{2}$. For $(\gamma_{3},g_{1}]$ these two vertices are connected by an edge.
On the interval $(g_{2},\gamma_{3}]$ the graph $\widetilde{D_{1}}\setminus
U_{g}$ is an edge $(v_{1},v_{2})$ and an isolated vertex $v_{3}$. On
$(\gamma_{4},g_{2}]$ all three vertices $v_{1}$, $v_{2}$ and $v_{3}$ are
connected by edges. For $(g_{3},\gamma_{4}]$ the isolated vertex $v_{4}$ is
added to the graph $\widetilde{D_{1}}\setminus U_{g}$. For $g\leq g_{3}$ the
graph $\widetilde{D_{1}}\setminus U_{g}$ includes all the vertices and is
connected.
Fig. 9: The thermodynamic tree for the six–component ${\rm H}_{2}$–${\rm
O}_{2}$ system, ${\rm H}_{2}$, ${\rm O}_{2}$, ${\rm H}$, ${\rm O}$, ${\rm
H}_{2}{\rm O}$, ${\rm OH}$ with the stoichiometric hydrogen–oxygen ratio,
$b_{\rm H}=2b_{\rm O}$ (Fig. 5b). The order of numbers $\gamma_{i}$, $g_{j}$
is presented in Fig. On the right, the graph $\widetilde{D_{1}}\setminus
U_{g}$ is represented for $g=g_{10}$. For $g\leq g_{10}$, the graph
$\widetilde{D_{1}}\setminus U_{g}$ includes all the vertices and is connected.
The solid bold line on the tree is the thermodynamically admissible path from
the initial state ${\rm H}_{2},{\rm O}_{2}$ to the equilibrium. There are
leaves at all levels $g=\gamma_{i}$. There are branching points at
$g=g_{1-4,7,10}$ and no vertices at $g=g_{5,6,8,9}$.
For the second example (Fig. 9) we selected the six-component system (Fig. 5)
with the stoichiometric hydrogen–oxygen ratio, $b_{\rm H}=2b_{\rm O}$. The
selected order of numbers $\gamma_{i}$, $g_{j}$ is presented in Fig. 9.
## 5 Conclusion
We studied dynamical systems that obey a continuous strictly convex Lyapunov
function $G$ in a positively invariant convex polyhedron $D$. Convexity allows
us to transform $n$-dimensional problems about attainability and attainable
sets into an analysis of 1D continua and discrete objects.
We construct the tree (the Adelson-Velskii – Kronrod – Reeb tree [1, 42, 56])
of the function $G$ in $D$ and call this 1D continuum the thermodynamic tree.
The thermodynamic tree is a tool to solve the “attainability problem”: is
there a continuous path between two states, $x$ and $y$ along which the
conservation laws hold, the concentrations remain non-negative and the
relevant thermodynamic potential $G$ (Gibbs energy, for example) monotonically
decreases? This question arises often in non-equilibrium thermodynamics and
kinetics. The analysis of the admissible paths can be considered as a
dynamical analogue of the study of the steady states feasibility in chemical
and biochemical kinetics. In this recent study, the energy balance method, the
stoichiometric network theory, the entropy production analysis and the
advanced algorithms of convex geometry of polyhedral cones are used [8, 54].
The obvious inequality, $G(x)\geq G(y)$ is necessary but not sufficient
condition for existence of an admissible path from $x$ to $y$. In 1D systems,
the space of states is an interval and the thermodynamic tree has two leaves
(the ends of the interval) and one root (the equilibrium). In such a system, a
spontaneous transition from a state $x$ to a state $y$ is allowed by
thermodynamics if $G(x)\geq G(y)$ and $x$ and $y$ are on the same side of the
equilibrium, i.e. they belong to the same branch of the thermodynamic tree.
This is just a well known rule: “it is impossible to overstep the equilibrium
in one-dimensional systems”.
The construction of the thermodynamic tree gives us the multidimensional
analogue of this rule. Let $\pi:D\to\mathcal{T}$ be the natural projection of
the balance polyhedron $D$ on the thermodynamic tree $\mathcal{T}$. A
spontaneous transition from a state $x$ to a state $y$ is allowed by
thermodynamics if and only if $\pi(y)\in[\pi(x),\pi(N^{*})]$, where $N^{*}$ is
the equilibrium and $[\pi(x),\pi(N^{*})]$ is the ordered segment.
In this paper, we developed methods for solving the following problems:
1. 1.
How to construct the thermodynamic tree $\mathcal{T}$?
2. 2.
How to solve the attainability problem?
3. 3.
How to describe the set of all states attainable from a given initial state
$x$?
For this purpose, we analyzed the cutting of a convex polyhedron by a convex
set and developed the algorithm for construction of the tree of level set
components of a convex function in a convex polyhedron. In this algorithm, the
restriction of $G$ onto the 1-skeleton of $D$ is used. This finite family of
convex functions of one variable includes all necessary information for
analysis of the tree of the level set component of the convex function $G$ of
many variables.
In high dimensions, some steps of our analysis become computationally
expensive. The most expensive operations are the convex hull (description of
the convex hull of a finite set by linear inequalities) and the double
description operations (description of the faces of a polyhedron given by a
set of linear inequalities). Therefore, in high dimensions some of the problem
may be modified, for example, instead of the explicit description of the
convex hull it is possible to use the algorithm for solution of a problem:
does a point belong to this convex hull [55]. The computational aspects of the
discussed problems in higher dimensions deserve more attention and the proper
modifications of the problems should be elaborated. For example, two following
problems need to be solved efficiently:
* •
To find the maximal and the minimal value of any linear function $f$ in a
class of thermodynamic equivalence;
* •
To evaluate the maximum and the minimum of ${\mathrm{d}}G/{\mathrm{d}}t$ in
any class of thermodynamic equivalence:
$-\overline{\sigma}\leq{\mathrm{d}}G/{\mathrm{d}}t\leq-\underline{\sigma}\leq
0.$
For any $w\in\mathcal{T}$, the solution of the first problem allows us to find
an interval of values of any linear function of state in the corresponding
class of thermodynamic equivalence. We can use the results of Sec. 2.2 to
reformulate this problem as the convex programming problem.
The second problem gives us the possibility to consider dynamics of relaxation
on $\mathcal{T}$. On each interval on $\mathcal{T}$ we can write
$-\overline{\sigma}(g)\leq{\mathrm{d}}g/{\mathrm{d}}t\leq-\underline{\sigma}(g)\leq
0\,,$ (17)
where the functions $\overline{\sigma}(g),\underline{\sigma}(g)\geq 0$ depend
on the interval on $\mathcal{T}$.
This differential inequality (17) will be a tool for the study of the dynamics
of relaxation and may be considered as a reduced kinetic model that
substitutes dynamics on the $d$-dimensional balance polyhedron $D$ by dynamics
on the one-dimensional dendrite. The problem of the construction of the
reduced model (17) is closely related to the following problem [66]: “Can one
establish a lower bound on the entropy production, in terms of how much the
distribution function departs from thermodynamical equilibrium?” In 1982, C.
Cercignani [12] proposed a simple linear estimate for $\underline{\sigma}(g)$
for the Boltzmann equation (Cercignani’s conjecture). After that, these
estimates were studied and improved by many authors [15, 11, 65, 66] and now
the state of art achieved for the Boltzmann equation gives us some hints how
to create the relaxation model (17) on the thermodynamic tree for the general
kinetic systems. This may be the next step in the study of the thermodynamic
trees.
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|
arxiv-papers
| 2012-01-30T18:30:03 |
2024-09-04T02:49:26.830951
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1201.6315"
}
|
1201.6428
|
Chirikov criterion of resonance overlapping
for the model of molecular dynamics
Guzev M.A.
Institute for Applied Mathematics FEB RAS, Vladivostok
e-mail: guzev@iam.dvo.ru
The chaotic dynamics in a cell of particles’ chain interacting by means of
Lennard-Jones potential is considered. Chirikov criterion of resonance
overlapping is used as the condition of chaos. The asymptotic representation
for this function at low and high energies is obtained for the function
corresponding to the criterion. Keywords: molecular dynamics, Lennard-Jones
potential, Chirikov criterion, chaotic dynamics
Mathematical Subject Classification 2000: 37D45 n
## Introduction
It is well known [1] that investigation of the problem occurrence of chaotic
dynamics naturally led to the need to study simple models in which there is
chaos. Further study of the real physical processes in which there is chaos
showed that many simple models reflect its main, for example, the occurrence
of stochastic separatrix layer, formation of nonlinear resonances and etc.
One-dimensional chains of particles have also been the subject of research in
studying the problems of chaotic dynamics [1] because some problems for them
are reduced to the study of standard maps known for nonlinear dynamic systems.
On the other hand investigation of the phenomena of chaos chain of particles
is also important in connection with the active use of the particle method for
simulating the behavior of materials. Speaking about this method we consider
the particles as point masses, and not as discrete elements allowing to reduce
the equations of continuum mechanics to the difference system of ordinary
differential equations. One of the most well-developed variants of the
particle method is the method of molecular dynamics in the classical version
of which particles act as atoms and molecules. If interatomic potential is
known then the dynamics of molecular compounds can be modeled with high
accuracy.
Despite the good correlation between the results of computer simulation
observed experiment and material behavior there is a problem of understanding
their internal mechanisms in terms of nonlinear dynamics. For one-dimensional
system corresponding model problem is reduced to consideration of the particle
mass $m$ in the cell. The dynamics of particle is determined by the
Hamiltonian
$H=H(p,q,t)=\frac{p^{2}}{2m}+V(q),\qquad V(q)=U(q)+U(\xi(t)-q),$ (1)
in which $p$ is the particle momentum, and form of the function $\xi(t)$
depends on the collective behavior of particles in the system. Formula (1)
means that particle interacts with a stationary particle on the left side and
interacts with a particle moving in a law $q_{2}=\xi(t)$ on the right one. The
interaction between the particles is characterized by the potential $U(q)$.
The phenomenon of strong chaos for particles interacting by means of the
Lennard-Jones potential $U(q)$ was considered in [2]. Chirikov criterion of
resonance overlapping [3] was used as the condition of chaos. The function
$K_{n,n-1}$ corresponding to the criterion was calculated numerically with
respect to energy. In this article we obtain the asymptotic representation for
this function at low and high energies.
## 1 Chirikov criterion of resonance overlapping
Let us remind the model [2]. Interaction is determined by the Lennard-Jones
potential
$U(q)=U_{0}\left[\left(\dfrac{a}{q}\right)^{\alpha}-\dfrac{\alpha}{\beta}\left(\dfrac{a}{q}\right)^{\beta}\right]$
with $\alpha>\beta>0$. The perturbation $\xi(t)$ is given by the formula
$\xi(t)=2a\bigl{(}1+\varepsilon(t)\bigr{)},\;\varepsilon(t)=\alpha_{1}\cos\omega_{1}t$
with $|\alpha_{1}|\ll 1$. We introduce the dimensionless variables setting
$H\to H/U_{0}$, $q\to q/a-1$, $t\to t/t_{0}$, $\omega_{1}\to\omega_{1}t_{0}$
where $t_{0}=\sqrt{ma^{2}/U_{0}}$. Then Hamiltonian (1) is written in the form
$H(p,q,t)=\frac{p^{2}}{2}+V(q),\qquad V(q)=U(q+1)+U(1+\varepsilon(t)-q).$ (2)
Given the small amplitude $\varepsilon(t)$ we carry out the expansion of the
Hamiltonian (2) with respect to $\varepsilon(t)$:
$\begin{gathered}H(p,q,t)=H_{0}(p,q)-\varepsilon(t)H_{1}(q)+.\mkern-0.95mu.\mkern-0.95mu.,\\\
H_{0}(p,q)=\frac{p^{2}}{2}{+}W(q),\;H_{1}(q)=\dfrac{d\mkern
1.0muU\hfill}{dq\hfill}(1{-}q),\;W(q)=U(q{+}1){+}U(q{-}1).\end{gathered}$ (3)
Action-angle variables $(I,\varphi)$ are commonly used in studying the
dynamics of Hamiltonian systems [1]. The unperturbed action is conserved along
the phase trajectory and its value is determined by initial conditions
$(p_{0},q_{0})$. If you choose $p_{0}=0$ the correspondence between the
unperturbed action $I$ and the parameter $q_{0}$ is unique. Transformation to
the action-angle variables can be carried out numerically in accordance with
formulae
$\mkern-20.0muI(q_{0})=\frac{2}{\pi}\int\limits_{0}^{q_{0}}p(q_{0},q)dq,\quad\varphi(q,q_{0})=\omega\int\limits_{q_{0}}^{q}\frac{dq}{p(q_{0},q)},\quad\omega=\dfrac{\pi}{\displaystyle
2\int\limits_{0}^{q_{0}}\dfrac{dq}{p(q_{0},q)}},$ (4)
where $p(q_{0},q)=\sqrt{2\bigl{(}V(q_{0})-V(q)\bigr{)}}$. We express the
variables $(p,q)$ in the terms of $(I,\varphi)$. Then Hamiltonian $H(p,q,t)$
can be expanded in a Fourier series:
$\mkern-20.0mu\begin{aligned}
H(p,q,t)&=H_{0}(I){-}\frac{1}{2}\alpha_{1}\sum_{n}H_{n}(I)[\cos(n\varphi{+}\varphi_{0}{-}\omega_{1}t){+}\cos(n\varphi{+}\varphi_{0}{+}\omega_{1}t)],\\\
H_{n}&=\frac{1}{\pi}\int\limits_{0}^{2\pi}d\varphi U^{(1)}(1{-}q)\cos
n\varphi.\end{aligned}$ (5)
Let us consider the resonance condition for a three of numbers
$(n,I_{n},\omega_{1})$: $n\omega(I_{n})=\omega_{1}.$
To analyze the dynamics of the system in the vicinity of nonlinear resonance
we use the standard method [1]. In this case the only resonant harmonic in (5)
is fixed:
$H(p,q,t)=H_{0}(I)-\frac{1}{2}\alpha_{1}H_{n}(I)\cos(n\varphi{+}\varphi_{0}{-}\omega_{1}t).$
(6)
We introduce a new phase $\psi=n\varphi+\varphi_{0}-\omega_{1}t$ then the
equations of motion corresponding to Hamiltonian (6) have the form
$\displaystyle\dfrac{d\mkern 1.0muI\hfill}{dt\hfill}$
$\displaystyle=-\dfrac{\partial\mkern
1.0muH\hfill}{\partial\psi\hfill}=\frac{1}{2}\alpha_{1}H_{n}(I)\sin\psi,$ (7)
$\displaystyle\dfrac{d\mkern 1.0mu\psi\hfill}{dt\hfill}$
$\displaystyle=\dfrac{\partial\mkern 1.0muH\hfill}{\partial
I\hfill}=n\omega(I)-\omega_{1}-\frac{1}{2}\alpha_{1}\cos\psi\dfrac{d\mkern
1.0mu\hfill}{dI\hfill}H_{n}(I).$
As the equations (7) are examined in the vicinity of the resonance action
$I_{n1}$ it is assumed that the value $J=I{-}I_{n}$ is small. We expand
$H_{0}(I)$, $\omega(I)$ and neglect the terms $J^{2}$ correspondingly. Then
the equations (7) are written in the form:
$\dfrac{d\mkern
1.0muJ\hfill}{dt\hfill}=\frac{1}{2}\alpha_{1}H_{n}(I_{n})\sin\psi,\qquad\dfrac{d\mkern
1.0mu\psi\hfill}{dt\hfill}=nJ\dfrac{d\mkern
1.0mu\omega\hfill}{dI\hfill}(I_{n}).$
The obtained equations correspond to the so-called nonlinear pendulum
Hamiltonian [1]:
$\bar{H}(J,\psi)=n\frac{J^{2}}{2}\dfrac{d\mkern
1.0mu\omega\hfill}{dI\hfill}(I_{n})-\frac{1}{2}\alpha_{1}H_{n}(I_{n})\cos\psi.$
The width of the resonance $\Delta J$is calculated from the condition:
$\frac{(\Delta J)^{2}}{2}\dfrac{d\mkern
1.0mu\omega\hfill}{dI\hfill}(I_{n})=\frac{1}{2}\alpha_{1}H_{n}(I_{n});\qquad\Delta
J=\sqrt{\raise
4.30554pt\hbox{$\alpha_{1}H_{n}(I_{n})$}\mskip-5.0mu\bigg{/}\dfrac{d\mkern
1.0mu\omega\hfill}{dI\hfill}(I_{n})}.$
In terms of frequency $\omega$ we have
$\Delta\omega\approx\sqrt{\frac{1}{2}\alpha_{1}H_{n}(I_{n})\dfrac{d\mkern
1.0mu\omega\hfill}{dI\hfill}(I_{n})}.$ (8)
Consider the invariant curves for the resonant action $I_{n}$, $I_{n-1}$. In
order to form a chaotic region in the plane $(p,q)$ it is necessary to destroy
the curves corresponding to these values of the action. Let us assume that
$\Delta\Omega\;(\Delta I)$ is the corresponding width of the resonance and
$\delta\Omega=\omega(I_{n})-\omega(I_{(n-1)})$ ($\delta
I{=}I_{n}{-}I_{(n-1)}$) is the distance between resonances. The parameter $K$
characterizing the degree of resonance overlapping is equal to $K{=}\Delta
I\big{/}\delta I{\approx}\Delta\Omega\big{/}\delta\Omega$ [1]. In accordance
to Chirikov criterion [3] the overlap of resonance takes place on condition
that $|K|\geq 1$ :
$K_{n,n-1}(q_{0}(\omega_{1}))=\frac{\Delta\omega(I_{n})+\Delta\omega(I_{(n-1)})}{\omega(I_{n})-\omega(I_{(n-1)})}\sim
1.$ (9)
In [2] the function $K_{3,2}$ was calculated numerically at different
$|\alpha_{1}|\ll 1$ with respect to $q_{0}$ (energy). In this article we
construct an asymptotic representation for the functions
$K_{1,2}\bigl{(}q_{0}(\omega_{1})\bigr{)},\;K_{2,3}\bigl{(}q_{0}(\omega_{1})\bigr{)}$
at small and large energies.
## 2 Approximate action-angle variables at small energy
The criterion (9) includes the frequency and its derivative (8), and the value
of the Fourier coefficient (5). From the point of view of physics the
smallness of the energy values correspond to the fulfillment of the inequality
$E\ll 1$. This allows us to use an approximation for $W(q)$ (3) as polynomial
functions with respect to $q$ and construct approximate formulae for the
action-angle variables at $E\to 0$. Approximation for the potential $W$ is
obtained after expansion in the vicinity of the point $q=0$:
$W_{app}(q)=W_{0}+bq^{2}+cq^{4},\qquad b=U^{(2)}(1),\qquad
c=\frac{U^{(4)}(1)}{12}.$ (10)
Let us introduce the approximate action-angle variables assuming
$J(E)=\frac{1}{\pi}\int\limits_{Q_{1}}^{Q_{2}}p_{app}dq_{,}\qquad\psi=\Omega
t_{app},\qquad t_{app}=\int\limits_{Q_{1}}^{q}\frac{dq}{p_{app}}$
where $p_{app}=\sqrt{2\bigl{(}E-W_{app}(q)\bigr{)}}$, and the coordinates of
points $Q_{i}$ are determined from the equation $E=W_{app}(Q_{i})$. The
quantity $\Omega=\dfrac{\partial\mkern 1.0muE\hfill}{\partial
J\hfill}=2\dfrac{\pi}{T(E)}$ is the cyclic frequency where $T(E)$ is the
period of motion in the potential (10). Omitting simple calculations we have
the following presentation for $T(E)$:
$\begin{gathered}\frac{T}{2}=\int\limits_{Q_{1}}^{Q_{2}}\frac{dq}{\sqrt{2\bigl{(}E-W_{app}(q)\bigr{)}}}=\frac{1}{\sqrt{2c}}\int\limits_{-T_{1}}^{T_{1}}\frac{dl}{\sqrt{(T_{1}^{2}-l^{2})(A^{2}+l^{2})}},\\\
T_{1}=\sqrt{-b+\sqrt{b^{2}+\dfrac{4cH}{2b}}},\qquad H=E-W_{0},\qquad
A^{2}=\frac{b+\sqrt{b^{2}+4cH}}{2c}.\end{gathered}$
Substituting $t=B\cos\varphi$ we write this expression in the form :
$\frac{T}{2}=\sqrt{\frac{2}{c}}\frac{1}{\sqrt{T_{1}^{2}+A^{2}}}\int\limits_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^{2}\sin^{2}\varphi}}=\sqrt{\frac{2}{c}}\frac{1}{\sqrt{T_{1}^{2}+A^{2}}}{{\mathbf{K}}}(k),$
(11)
where ${{\mathbf{K}}}(k)$ is the complete elliptic integral. As a result
obtain
$\psi=\frac{\pi}{2}\frac{F(\Phi,k)}{{{\mathbf{K}}}(k)},\qquad q=T_{1}\cos\Phi$
(12)
where $F(\Phi,k)$ is the elliptic integral of the first order.
## 3 Asymptotic behavior of different quantities at small energy
We are interested in the asymptotic behavior of $T$ and the derivative
$\dfrac{\partial\mkern 1.0muT\hfill}{\partial H\hfill}{=}\dfrac{\partial\mkern
1.0muT\hfill}{\partial E\hfill}$ at $H\to 0$, then
$k=\frac{\sqrt{Hc}}{b}\bigl{(}1+\underline{O}(H)\bigr{)},\qquad\frac{1}{\sqrt{T_{1}^{2}+A^{2}}}=\sqrt{\frac{c}{b}}\left(1-\frac{Hb}{b^{2}}\right)+\underline{O}(H^{2}).$
(13)
The smallness of $k$ allows us to use the asymptotic formula for
${{\mathbf{K}}}(k)$:
${{\mathbf{K}}}(k)=\frac{\pi}{2}\left\\{1+\left(\frac{1}{2}\right)^{2}k^{2}+\left(\frac{1\cdot
3}{2\cdot 4}\right)^{2}k^{4}+.\mkern-0.95mu.\mkern-0.95mu.\right\\}.$ (14)
From here and (11) we have
$\frac{T}{2}=\frac{\pi}{\sqrt{2b}}\left(1-\frac{3Hc}{4b^{2}}\right)+\underline{O}(H^{2}).$
Hence we immediately obtain the leading order in the expression for the
frequency and its derivative which can be written as:
$\displaystyle\omega$
$\displaystyle=\sqrt{2b}+\underline{O}(H^{2})=\sqrt{2U^{(2)}(1)}+\underline{O}(H^{2}),$
(15) $\displaystyle\dfrac{\partial\mkern 1.0mu\omega\hfill}{\partial E\hfill}$
$\displaystyle=\dfrac{\partial\mkern 1.0mu\omega\hfill}{\partial
H\hfill}=\frac{3c}{\sqrt{2ba}}+\underline{O}(H)=\frac{U^{(4)}(1)}{4\sqrt{2U^{(2)}(1)}}+\underline{O}(H).$
Let us write the function $q(J,\psi)$ in the terms of variables $(J,\psi)$ .
The error does not exceed $\underline{O}(H)$. Because of $k\sim\sqrt{H}$ (13)
we use asymptotic formulae for $\mathbf{K}(k)$ (14) and ${\it
F}(\Phi,k)=\Phi+.\mkern-0.95mu.\mkern-0.95mu.$ It results in
$\psi=\Phi+\underline{O}(H)$ and $\Phi=\varphi+\underline{O}(H)$.
To calculate the Fourier coefficients (5) we expand the potential in the
vicinity of $q=0$:
$H_{n}=\frac{1}{\pi}\int\limits_{0}^{2\pi}d\varphi\cos
n\varphi\left[U^{(2)}(1)q+U^{(3)}(1)\frac{q^{2}}{2!}+U^{(4)}(1)\frac{q^{3}}{3!}+.\mkern-0.95mu.\mkern-0.95mu.\right].$
The function $q(J,\psi)$ is equal to $q=T_{1}\cos(\varphi{+}\underline{O}(H))$
where $T_{1}=\sqrt{H/b}+\underline{O}(H^{2})$. Then we have
$H_{2}\sim H\frac{U^{(3)}(1)}{4U^{(2)}(1)},\qquad H_{3}\sim\sqrt[]{H^{3}}.$
From (8), (15) we have inequalities
$|\Delta\omega(I_{1})|\sim\sqrt[4]{H}\gg|\Delta\omega(I_{2})|\sim\sqrt{H}\gg\break|\Delta\omega(I_{3})|\sim\sqrt[4]{H^{3}}$.
It allows us to obtain expression for functions $|K_{2,1}|,\;|K_{3,2}|$:
$\mkern-20.0mu\begin{aligned}
|K_{2,1}|&{\approx}\left|\frac{\Delta\omega(I_{1})}{2(\omega(I_{2})-\omega(I_{1}))}\right|{=}\left|\frac{\Delta\omega(I_{1})}{\omega_{1}}\right|{\sim}\frac{1}{\omega_{1}}\sqrt{\frac{1}{2}\alpha_{1}\sqrt{HU^{(2)}(1)}\frac{U^{(4)}(1)}{8U^{(2)}(1)}},\\\
|K_{3,2}|&{\approx}\left|\frac{\Delta\omega(I_{2})}{2(\omega(I_{3})-\omega(I_{2}))}\right|{=}3\left|\frac{\Delta\omega(I_{2})}{\omega_{1}}\right|{\sim}\frac{3}{\omega_{1}}\sqrt{\frac{1}{2}\alpha_{1}H\frac{U^{(3)}(1)}{32U^{(2)}(1)}\frac{U^{(4)}(1)}{U^{(2)}(1)}}.\end{aligned}$
(16)
## 4 Approximate action-angle variables at high energy and asymptotic
behavior of different quantities
From the point of view of physics high values of energy $E$ correspond to the
fulfillment of the inequality $E/{U_{0}}{\gg}1$. Since the potential
$W(q)=U(1{+}q)+U(1{-}q)$ is symmetric function with respect to $q\to-q$ it is
enough to consider it at the interval $(0,1)$. It is clear that for large
values of $E$ turning point coordinate $q_{1}\sim 1$. Then the leading
contribution in $V(q)$ is determined by $U(1-q)$. We introduce an approximate
potential
$W_{big}(q)=\frac{1}{(1-q)^{\alpha}}$ (17)
and action-angle variables $J_{big},\phi$
$J_{big}(E)=\frac{2}{\pi}\int\limits_{Q_{1}}^{1}p_{big}dq,\qquad\phi=\Omega_{big}t_{big},\qquad\Omega_{big}=\dfrac{\partial\mkern
1.0muE\hfill}{\partial J_{big}\hfill},\qquad
t_{big}=\int\limits_{Q_{1}}^{q}\frac{dq}{p_{big}},$ (18)
where function $p_{big}=\sqrt{2\bigl{(}E-W_{big}(q)\bigr{)}}$, and the
coordinate $Q_{1}$ is defined by the equation
$E=W_{big}(Q_{1}).$ (19)
From a mathematical point of view, values $I$, $J_{big}$ determine the area in
phase space for functions $p$, $p_{big}$ respectively. Area difference can be
estimated by the sum of functions
$|I(E)-J_{big}(E)|\leq\frac{2}{\pi}[|Q_{1}-q_{1}|\max|p|+|\int\limits_{Q_{1}}^{1}[p-p_{big}]dq|].$
(20)
Leading order of $Q_{1}{-}q_{1}$ is determined from (17), (19) and is equal to
$Q_{1}{-}q_{1}=\dfrac{Q_{1}^{\alpha{-}\beta{+}1}}{\beta}\bigl{(}1+\underline{O}(Q_{1}^{\alpha{-}\beta})\bigr{)}$.
Since $\max|p|\leq\sqrt{2E}$ then the first term in (20) has the following
order $(Q_{1}-q_{1})\sqrt{E}\sim\dfrac{1}{E^{1/2+(1-\beta)/\alpha}}$. The
integral term in (20) is equal to
$\left|\int\limits_{\mskip
4.0muQ_{1}}^{1}[p-p_{big}]dq\right|=\left|\int\limits_{\mskip
4.0muQ_{1}}^{1}\frac{W_{big}-W}{p+p_{big}}dq\right|\sim\left|\int\limits_{\mskip
4.0muQ_{1}}^{1}\frac{1}{p_{big}}dq\right|\sim
Q_{1}^{1-\beta+\alpha/2}\sim\frac{1}{E^{1/2+(1-\beta)/\alpha}}.$
Hence
$I(E)-J_{big}(E)\sim\frac{1}{E^{1/2+(1-\beta)/\alpha}}$ (21)
on condition that $\alpha>2(\beta-1)$ which provides a decrease in the right
side of (21). Differentiation (21) with respect to $E$ results in relations
for the frequencies corresponding to the action and their derivatives:
$\omega
E)-\Omega_{big}(E)\sim\frac{1}{E^{3/2+(1-\beta)/\alpha}},\qquad\dfrac{\partial\mkern
1.0mu\omega\hfill}{\partial E\hfill}-\dfrac{\partial\mkern
1.0mu\Omega_{big}\hfill}{\partial
E\hfill}\sim\frac{1}{E^{5/2+(1-\beta)/\alpha}}.$ (22)
The difference between the phases $\varphi$ and $\phi$ is written in the form:
$|\varphi-\phi|\leq|\omega-\Omega_{big}|t_{big}+\omega|t-t_{big}|\sim\frac{t_{big}}{E^{3/2+(1-\beta)/\alpha}}+\omega|\omega-\Omega_{big}|.$
For high energy frequency $\omega\sim t_{big}\sim\sqrt{E}$ then from here and
(22) we have
$\varphi-\phi\sim\frac{1}{E^{1+(1-\beta)/\alpha}}\to 0.$ (23)
Let us transform from phase to phase in the integral (5) with accuracy (23)
and use the equality $q(\phi+\pi,J_{big})=Q_{2}-q(\phi,J_{big})$. Since
$2=Q_{2}+Q_{1}$ we rewrite (5) in the following form:
$H_{n}=\frac{1}{2}\int\limits_{Q_{1}}^{Q_{2}}dq\left[U^{(1)}(2-q)+U^{(1)}(Q_{1}+q)(-1)^{n}\right]\cos\frac{\pi
n}{Q_{2}-Q_{1}}q.$
A leading term with respect to $E$ is determined by means of integration:
$H_{n}=\frac{1}{2}\left.\left[-U(L-q)+U(Q_{1}+q)(-1)^{n}\right]\cos\frac{\pi
n}{Q_{2}-Q_{1}}q\right|_{Q_{1}}^{Q_{2}}-\\\
-\frac{1}{2}\frac{\pi}{Q_{2}-Q_{1}}\int\limits_{Q_{1}}^{Q_{2}}dq\left[U(L-q)-U(Q_{1}+q)(-1)^{n}\right]\sin\frac{\pi
n}{Q_{2}-Q_{1}}q.$ (24)
The outside terms have a singular behavior with respect to $E$ since
$\displaystyle\frac{(-1)^{n}}{2}\left[-U(Q_{1})-U(2Q_{1})\right]=$
$\displaystyle-\frac{(-1)^{n}}{2}\frac{1}{Q_{1}^{\alpha}}[1+1/2^{\alpha}]+\underline{O}\left(\frac{1}{Q_{1}^{\beta}}\right)=$
$\displaystyle-E\frac{(-1)^{n}}{2}[1+1/2^{\alpha}]+\underline{O}\left(\frac{1}{Q_{1}^{\beta}}\right).$
The integral in (24) is smaller than these terms then the Fourier coefficient
is equal to
$H_{n}=-E\frac{(-1)^{n}}{2}[1+1/2^{\alpha}]+.\mkern-0.95mu.\mkern-0.95mu.$
(25)
From (9), (25) we obtain expression for functions $K_{n,n-1}$ :
$K_{n,n-1}=\sqrt{\frac{\alpha_{1}}{2}(1+1/2^{\alpha})}(n-1/2).$
The parameter $\alpha=12$ for the Lennard-Jones potential then we
obtain$K_{2,1}=1.5\sqrt{\alpha_{1}/2}$, $K_{3,2}=2.5\sqrt{\alpha_{1}/2}$.
## 5 Comparison with numerical results for $K_{2,1}(q_{0})$
The function $K_{2,1}(q_{0})$ was defined numerically (exact) and calculated
(asympt) in accordance with formula (16). Its behavior depending on the
initial position of the resonance trajectory is presented in Fig.1 (exact).
From here it is seen that formula (16) is a good approximation for
$K_{2,1}(q_{0})$ as for small as high energies.
Fig. 1. Graph of the function $K_{2,1}(q_{0})$ defined numerically (exact) and
calculated (asympt) in accordance with formula (16).
## 6 Acknoledgments
This work was supported by Russian Foundation for Basic Research (Project
11-01-12057-ofi-i-2011)
## References
1. 1.
Zaslavsky G. M., Sagdeev R. Z., Usikov D. A., Chernikov A. A. Weak chaos and
quasi-regular patterns. Cambridge University Press, 1991. 254p.
2. 2.
Guzev M. A., Koshel K. V., Izrailsky Yu. G. The effect of global chaos in a
chain of particles// Nonlinear Dynamics. 2010. V. 52, No. 5, pp. 291–305 (in
Russian)(http://nd.ics.org.ru/doc/r/pdf/1676/0)
3. 3.
Chirikov B.V. A universal instability of many-dimensional oscillator systems//
Phys. Rep., 1979. V. 52, No. 5, pp. 264–379.
|
arxiv-papers
| 2012-01-31T03:27:25 |
2024-09-04T02:49:26.848838
|
{
"license": "Public Domain",
"authors": "M.A. Guzev",
"submitter": "Mikhail Guzev",
"url": "https://arxiv.org/abs/1201.6428"
}
|
1201.6478
|
# Solutions to the non-autonomous ABS lattice equations: Casoratians and
bilinearization
Ying Shi***Corresponding author. E-mail: shiying0707@shu.edu.cn, Da-jun
Zhang†††E-mail: djzhang@staff.shu.edu.cn, Song-lin Zhao
Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China
###### Abstract
In the paper non-autonomous H1, H2, H3δ and Q1δ equations in the ABS list are
bilinearized. Their solutions are derived in Casoratian form. We also list out
some Casoratian shift formulae which are used to verify Casoratian solutions.
Key words: non-autonomous ABS list, Casoratian, bilinear, soliton solutions
PACS: 02.30.Ik, 05.45.Yv
## 1 Introduction
The “discrete integrable systems” has been a popular topic and is still
drawing more and more attention. Particularly in the recent ten years it
received much progress. The property of multidimensional consistency[1, 2, 3]
provides an approach to investigate integrability for the discrete systems
defined on an elementary quadrilateral:
$Q(u_{n,m},u_{n+1,m},u_{n,m+1},u_{n+1,m+1};p,q)=0,$ (1.1)
where $p,q$ are spacing parameters. The ABS list[3] contains all quadrilateral
lattice equations of the above form which are consistent-around-the-cube(CAC).
In the ABS list the spacing parameters $p,q$ can either constants or functions
$p_{n}$ and $q_{m}$, which corresponds to autonomous case or non-autonomous
case, respectively. In general an autonomous system means a
differential/difference model with constant coefficients while a non-
autonomous one means the model has coefficients varying with independent
variables but the model can not be transformed back to an autonomous one.
Obviously, the ABS lattice equations themselves are automatically non-
autonomous in the sense of taking $p=p_{n},q=q_{m}$, and this non-autonomous
case still keeps the CAC property.
Integrable non-autonomous systems have its own importance. It is known that
most of discrete Painlevé equations are non-autonomous ordinary difference
equations. In addition, for integrable non-autonomous forms of partial
difference equations, their reductions usually lead to integrable non-
autonomous mappings, which quite often are discrete Painlevé equations. To get
a non-autonomous version of a partial difference equation, taking (1.1) as an
example, one can replace constant lattice parameters $(p,q)$ by
$(p_{n,m},q_{n,m})$, but the integrability should be kept. Many criterions,
such as singularity confinement, conservation laws and algebraic entropy, have
been used to check integrability for the non-autonomous systems, for both
ordinary and partial difference cases[4, 5, 6, 7]. Besides, it is also
possible to deautonomise a discrete bilinear system if it contains spacing
parameters. With suitable deautonomisation the obtained non-autonomous
bilinear systems admit $N$-soliton solutions expressed through deformed
discrete exponential functions, (see [8, 9] as examples).
Recently, many solving approaches have been developed to find solutions for
autonomous lattice equations in the ABS list[10, 11, 12, 13, 14, 15, 16, 17,
18]. In [13] the H1, H2, H3δ and Q1δ equations in the ABS list were
bilinearized and their solutions were derived in Casoratian form. In the
present paper we will repeat the treatment of [13] to get bilinear forms as
well as solutions in Casoratian form for some non-autonomous ABS lattice
equations. As we have mentioned before, the ABS lattice equations with spacing
parameters $(p_{n},q_{m})$ are automatically non-autonomous and still CAC.
Their integrable aspects are also double checked by singularity confinement
and algebraic entropy approaches[7].
The paper is organized as follows. Section 2 contains some basic notations for
discrete systems and Casoratians and a list of non-autonomous ABS lattice
equations. In Section 3 the non-autonomous H1, H2, H3δ and Q1δ equations are
bilinearized and their solutions are derived in Casoratian form. The Appendix
contains a collection of Casoratian formulae of non-autonomous case.
## 2 Preliminaries
Conventionally, we use tilde/hat notations to express the shifts in $n$/$m$
directions, for example,
$u=u_{n,m},~{}~{}\widetilde{u}=u_{n+1,m},~{}~{}\underaccent{\tilde}{u}=u_{n-1,m},~{}~{}\widehat{u}=u_{n,m+1},~{}~{}\underaccent{\hat}{u}=u_{n,m-1},~{}~{}\widehat{\widetilde{u}}=u_{n+1,m+1}.$
By these notations the lattice equation (1.1) is rewritten as
$\mathcal{Q}(u,\widetilde{u},\widehat{u},\widehat{\widetilde{u}};p,q)=0.$
(2.1)
The non-autonomous ABS list is as follows[3, 7]:
$\displaystyle\mbox{H1}:~{}~{}$
$\displaystyle(u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})+q_{m}-p_{n}=0,$
(2.2a) $\displaystyle\mbox{H2}:~{}~{}$
$\displaystyle(u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})+(q_{m}-p_{n})(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}})+q_{m}^{2}-p_{n}^{2}=0,$
(2.2b) $\displaystyle\mbox{H3}_{\delta}:~{}~{}$ $\displaystyle
p_{n}(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-q_{m}(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})+\delta(p_{n}^{2}-q_{m}^{2})=0,$
(2.2c) $\displaystyle\mbox{Q1}_{\delta}:~{}~{}$ $\displaystyle
p_{n}(u-\widehat{u})(\widetilde{u}-\widehat{\widetilde{u}})-q_{m}(u-\widetilde{u})(\widehat{u}-\widehat{\widetilde{u}})+\delta^{2}p_{n}q_{m}(p_{n}-q_{m})=0,$
(2.2d) $\displaystyle\mbox{Q2}:~{}~{}$ $\displaystyle
p_{n}(u-\widehat{u})(\widetilde{u}-\widehat{\widetilde{u}})-q_{m}(u-\widetilde{u})(\widehat{u}-\widehat{\widetilde{u}})+p_{n}q_{m}(p_{n}-q_{m})(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}})$
$\displaystyle~{}~{}-p_{n}q_{m}(p_{n}-q_{m})(p_{n}^{2}-p_{n}q_{m}+q_{m}^{2})=0,$
(2.2e) $\displaystyle\mbox{Q3}_{\delta}:~{}~{}$
$\displaystyle\sin(p_{n}+q_{m})(u\widehat{\widetilde{u}}+\widetilde{u}\widehat{u})-\sin
p_{n}(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-\sin
q_{m}(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})$
$\displaystyle~{}~{}+\delta^{2}\sin p_{n}\sin q_{m}\sin(p_{n}+q_{m})=0,$
(2.2f) $\displaystyle\mbox{Q4}:~{}~{}$
$\displaystyle\text{sn}(p_{n}+q_{m})(u\widehat{\widetilde{u}}+\widetilde{u}\widehat{u})-\text{sn}p_{n}(u\widetilde{u}+\widehat{u}\widehat{\widetilde{u}})-\text{sn}q_{m}(u\widehat{u}+\widetilde{u}\widehat{\widetilde{u}})$
$\displaystyle~{}~{}+\text{sn}p_{n}\text{sn}q_{m}\text{sn}(p_{n}+q_{m})(1+k^{2}u\widetilde{u}\widehat{u}\widehat{\widetilde{u}})=0,$
(2.2g)
where $\delta$ is a constant, $p_{n}=p(n)$ and $q_{m}=q(m)$ are arbitrary non-
zero functions of discrete variables $n$ and $m$, respectively.
Here the forms of Q3δ and Q4 are in accordance with the autonomous version via
the parametrization introduced by Hietarinta [19]. We omit A1δ and A2 from the
above list because of the equivalence between A1δ and Q1δ by
$u\to(-1)^{n+m}u$, as well as A2 and Q3δ=0 by $u\to u^{(-1)^{n+m}}$.
The discrete version of Wronskian is Casoratian, which is a determinant of the
Casorati matrix:
$f=|\psi(n,m,l_{1}),\psi(n,m,l_{2}),\cdots,\psi(n,m,l_{N})|=|l_{1},l_{2},\cdots,l_{N}|,$
(2.3a) where the basic column vector is
$\psi(n,m,l)=(\psi_{1}(n,m,l),\psi_{2}(n,m,l),\cdots,\psi_{N}(n,m,l))^{T},$
(2.3b)
and the shifts are in $l$ direction. Using the standard short-hand
notations[20], we list the following often-used $N$th-order Casoratians
$|\widehat{N-1}|=|0,1,\cdots,N-1|,~{}~{}|\widehat{N-2},N|=|0,1,\cdots,N-2,N|,~{}~{}|-1,\widetilde{N-1}|=|-1,1,2,\cdots,N-1|.$
As in [13], since in (2.3) there are three direction variables, say $n,m$ and
$l$, one can introduce the operators $E^{\nu}$ $(\nu=1,2,3)$ by
$E^{1}\psi=\widetilde{\psi}=\psi(n+1,m,l),~{}~{}E^{2}\psi=\widehat{\psi}=\psi(n,m+1,l),~{}~{}E^{3}\psi=\bar{\psi}=\psi(n,m,l+1),$
(2.4)
then define a Casoratian w.r.t $E^{\nu}$-shift,
$|\widehat{N-1}|_{[\nu]}=|\psi,E^{\nu}\psi,(E^{\nu})^{2}\psi,\cdots,(E^{\nu})^{N-1}\psi|,~{}~{}(\nu=1,2,3).$
(2.5)
For these Casoratians we have
###### Proposition 1.
The Casoratians
$|\widehat{N-1}|_{{}_{[1]}}=|\widehat{N-1}|_{{}_{[2]}}=|\widehat{N-1}|_{{}_{[3]}},$
(2.6)
if their column vector $\psi(n,m,l)$ satisfies the relations
$\alpha_{n}\psi=\overline{\psi}-\widetilde{\psi},~{}~{}\beta_{m}\psi=\overline{\psi}-\widehat{\psi},$
(2.7)
where $\alpha_{n}$ and $\beta_{m}$ are arbitrary functions of discrete
variables $n$ and $m$, respectively, i.e. $\alpha_{n}=\alpha(n)$,
$\beta_{m}=\beta(m)$.
###### Proof.
By the definition of $E^{\nu}$ in (2.4) the relations (2.7) can be rewritten
as
$E^{3}\psi=(E^{1}+\alpha_{n})\psi,~{}~{}E^{3}\psi=(E^{2}+\beta_{m})\psi,$
(2.8)
from which one has
$\displaystyle(E^{3})^{k}=(E^{1}+\alpha_{n})^{k}=(E^{1})^{k}+\sum_{j=1}^{k}\sum_{\begin{subarray}{c}l_{j}=0\\\
l_{j}\leq
l_{j+1}\end{subarray}}^{k-j}\prod_{i=1}^{j}\alpha_{n+l_{i}}(E^{1})^{k-j},~{}~{}k=1,2,\cdots,N-1,$
(2.9a)
$\displaystyle(E^{3})^{k}=(E^{2}+\beta_{m})^{k}=(E^{2})^{k}+\sum_{j=1}^{k}\sum_{\begin{subarray}{c}l_{j}=0\\\
l_{j}\leq
l_{j+1}\end{subarray}}^{k-j}\prod_{i=1}^{j}\beta_{m+l_{i}}(E^{2})^{k-j},~{}~{}k=1,2,\cdots,N-1.$
(2.9b)
Then, substituting them into (2.5) one can easily obtain (2.6). ∎
This Proposition will bring more flexibility for Casoratian verifications.
Besides, for later convenience, we give the following Laplace expansion
property[20]:
###### Proposition 2.
Suppose that $\mathbf{B}$ is a $N\times(N-2)$ matrix, and a,b,c,d are $N$th-
order column vectors, then
$|\mathbf{B},\mathbf{a},\mathbf{b}||\mathbf{B},\mathbf{c},\mathbf{d}|-|\mathbf{B},\mathbf{a},\mathbf{c}||\mathbf{B},\mathbf{b},\mathbf{d}|+|\mathbf{B},\mathbf{a},\mathbf{d}||\mathbf{B},\mathbf{b},\mathbf{c}|=0.$
(2.10)
## 3 Bilinearization and Casoratian solutions
In the following we derive bilinear forms and Casoratian solutions for the
non-autonomous H1, H2, H3δ and Q1δ models in the non-autonomous ABS list
(2.2). Singularity confinement might provide a possible transformation to
connect discrete integrable systems with their bilinear forms, (see [21, 22]
as examples), but here we will roughly use the same transformations as for the
autonomous lattice equations[13]. It then turns out that these non-autonomous
lattice equations can share the same bilinear forms with those autonomous ones
except changing the spacing parameters accordingly.
To get Casoratian solutions one needs to use deformed discrete exponential
functions and develop corresponding Caosratian shift formulae, which we have
listed in Appendix.
### 3.1 Non-autonomous H1 equation
We note that the non-autonomous H1 has been solved in [8] through bilinear
approach and the Casoratian solutions were given, but here we give a more
generalized result.
With the parametrization
$p_{n}=c-a_{n}^{2},~{}~{}q_{m}=c-b_{m}^{2},~{}~{}(c~{}\mathrm{is~{}an~{}arbitrary~{}constant}),$
(3.1)
and through the transformation
$u_{n,m}=\frac{g_{n,m}}{f_{n,m}}-\sum^{n-1}_{i=n_{{}_{0}}}a_{i}-\sum^{m-1}_{j=m_{{}_{0}}}b_{j}-\gamma,~{}~{}(\gamma~{}\mathrm{is~{}an~{}arbitrary~{}constant}),$
(3.2)
the non-autonomous H1 (2.2a) is bilinearized by
$\displaystyle\mathcal{H}_{1}\equiv(\widehat{g}\widetilde{f}-\widetilde{g}\widehat{f})+(a_{n}-b_{m})(\widehat{f}\widetilde{f}-f\widehat{\widetilde{f}})=0,$
(3.3a)
$\displaystyle\mathcal{H}_{2}\equiv(g\widehat{\widetilde{f}}-\widehat{\widetilde{g}}f)+(a_{n}+b_{m})(f\widehat{\widetilde{f}}-\widehat{f}\widetilde{f})=0,$
(3.3b)
where in the transformation (3.2) $n_{0},m_{0}$ are arbitrary integers. The
connection between (2.2a) and (3.3) is
$-\bigl{[}\mathcal{H}_{1}+(a_{n}-b_{m})f\widehat{\widetilde{f}}\bigr{]}\bigl{[}\mathcal{H}_{2}+(a_{n}+b_{m})\widehat{f}\widetilde{f}\bigr{]}/(f\widehat{f}\widetilde{f}\widehat{\widetilde{f}})+(a_{n}^{2}-b_{m}^{2})\equiv\mathrm{H1},$
which is the same relation as the autonomous one[13].
Solutions to the bilinear equations (3.3) can be given by
###### Proposition 3.
The Casoratians
$f(\psi)=|\widehat{N-1}|_{{}_{[3]}},~{}~{}g(\psi)=|\widehat{N-2},N|_{{}_{[3]}},$
(3.4)
solve the non-autonomous bilinear equations (3.3), if the column vector
$\psi(n,m,l)$ symmetrically‡‡‡ Here the symmetric property between pairs
$(n,a_{n})$ and $(m,b_{m})$ means, for example, once we have (3.5), at the
same time we have
$b_{m-1}\underaccent{\hat}{\psi}=\psi-\overline{\underaccent{\hat}{\psi}},$
and
$\psi=A_{[n]}\omega,~{}~{}a_{n}\widetilde{\omega}=\omega+\widetilde{\overline{\omega}}.$
, in terms of the pairs $(n,a_{n})$ and $(m,b_{m})$, satisfies the shift
relations
$\displaystyle
a_{n-1}\underaccent{\tilde}{\psi}=\psi-\overline{\underaccent{\tilde}{\psi}},$
(3.5a)
$\displaystyle\psi=A_{[m]}\phi,~{}~{}b_{m}\widehat{\phi}=\phi+\widehat{\overline{\phi}},$
(3.5b)
where $\phi(n,m,l)$ is an auxiliary vector, the $N\times N$ transform matrix
$A_{[m]}$ is invertible, and the subscript $[m]$ specially means $A_{[m]}$
only depends on $m$ but is independent of $(n,l)$.
###### Proof.
We prove $\mathcal{H}_{1}$ in its down-tilde-hat version
$\mathcal{\underaccent{\hat}{\underaccent{\tilde}{H}}}_{1}$:
$\mathcal{\underaccent{\hat}{\underaccent{\tilde}{H}}}_{1}\equiv(\underaccent{\tilde}{g}\underaccent{\hat}{f}-\underaccent{\hat}{g}\underaccent{\tilde}{f})+(a_{n-1}-b_{m-1})(\underaccent{\tilde}{f}\underaccent{\hat}{f}-\underaccent{\hat}{\underaccent{\tilde}{f}}f).$
(3.6)
Using the formulae given in appendix A with $c=0$. In (3.6)
$f=|\widehat{N-1}|_{{}_{[3]}}$, $\underaccent{\hat}{\underaccent{\tilde}{f}}$,
$\underaccent{\hat}{f}$,
$\underaccent{\tilde}{g}+a_{n-1}\underaccent{\tilde}{f}$,
$\underaccent{\tilde}{f}$ and
$\underaccent{\hat}{g}+b_{m-1}\underaccent{\hat}{f}$ are (A.8c), (A.6e),
(A.9a), (A.6b) and (A.9e), respectively, we have
$\begin{array}[]{rl}\mathcal{\underaccent{\hat}{\underaccent{\tilde}{H}}}_{1}\equiv&-(a_{n-1}-b_{m-1})\underaccent{\hat}{\underaccent{\tilde}{f}}f+\underaccent{\hat}{f}(\underaccent{\tilde}{g}+a_{n-1}\underaccent{\tilde}{f})-\underaccent{\tilde}{f}(\underaccent{\hat}{g}+b_{m-1}\underaccent{\hat}{f})\\\
=&-a_{n-1}^{-N+2}~{}b_{m-1}^{-N+2}~{}[|\widehat{N-3},~{}\psi(N-2),~{}\psi(N-1)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\underaccent{\hat}{\psi}(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\\\
&-|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\\\
&+|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}}]\\\
=&0,\end{array}$
where we have made use of Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},\textbf{b},\textbf{c},\textbf{d})=(\psi(N-2),\psi(N-1),\underaccent{\hat}{\psi}(N-2),\underaccent{\tilde}{\psi}(N-2))$.
Next, we prove the down-tilde version of $\mathcal{H}_{2}$, which is
$\mathcal{\underaccent{\tilde}{H}}_{2}\equiv(\widehat{f}\underaccent{\tilde}{g}-\widehat{g}\underaccent{\tilde}{f})+(a_{n-1}+b_{m})(\widehat{f}\underaccent{\tilde}{f}-f\widehat{\underaccent{\tilde}{f}}).$
(3.7)
In (3.7) we take $f=|\widehat{N-1}|_{{}_{[3]}}$, and for
$\widehat{\underaccent{\tilde}{f}}$, $\underaccent{\tilde}{f}$,
$\widehat{g}-b_{m}\widehat{f}$, $\widehat{f}$ and
$\underaccent{\tilde}{g}+a_{n-1}\underaccent{\tilde}{f}$ we use (A.8b),
(A.6b), (A.9b), (A.6g) and (A.9a), with $c=0$, respectively. Then we have
$\begin{array}[]{rl}\mathcal{\underaccent{\tilde}{H}}_{2}\equiv&-(a_{n-1}+b_{m})f\widehat{\underaccent{\tilde}{f}}-\underaccent{\tilde}{f}(\widehat{g}-b_{m}\widehat{f})+\widehat{f}(\underaccent{\tilde}{g}+a_{n-1}\underaccent{\tilde}{f})\\\
=&-a_{n-1}^{-N+2}~{}b_{m}^{-N+2}|A_{[m+1]}A_{[m]}^{-1}|\cdot[|\widehat{N-3}~{}~{}\psi(N-2)~{}~{}\psi(N-1)|_{{}_{[3]}}\\\
&~{}\times|\widehat{N-3},~{}\underaccent{\tilde}{\psi}(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&-|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&+|\widehat{N-3},~{}\psi(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}]\\\
=&0,\end{array}$
by using Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},\textbf{b},\textbf{c},\textbf{d})=(\psi(N-2),~{}\psi(N-1),~{}\underaccent{\tilde}{\psi}(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2))$.
∎
For the explicit forms of $\psi$ together with the transformation matrices we
can take either (A.3) with $c=0$ or (A.4) with $c=0$.
### 3.2 Non-autonomous H2 equation
By the parametrization (3.1) with $c=0$, i.e.,
$p_{n}=-a_{n}^{2},~{}~{}q_{m}=-b_{m}^{2}$, we first rewrite the non-autonomous
H2 (2.2b) into
$\mathrm{H2}\equiv(u-\widehat{\widetilde{u}})(\widetilde{u}-\widehat{u})+(a_{n}^{2}-b_{m}^{2})(u+\widetilde{u}+\widehat{u}+\widehat{\widetilde{u}}-(a_{n}^{2}+b_{m}^{2}))=0.$
(3.8)
Then, taking the transformation
$u_{n,m}=U_{n,m}^{2}-2U_{n,m}\frac{g}{f}+\frac{h+s}{f},$ (3.9a) with
$U_{n,m}=\sum^{n-1}_{i=n_{{}_{0}}}a_{i}+\sum^{m-1}_{j=m_{{}_{0}}}b_{j}+\gamma,~{}~{}~{}~{}(\gamma~{}\text{is
an arbitrary constant}),$ (3.9b) and $s-h=\alpha
f,~{}~{}~{}~{}(\alpha~{}\text{is some constant}),$ (3.9c)
one can bilinearize (3.8) by
$\displaystyle\mathcal{H}_{1}\equiv(\widehat{g}\widetilde{f}-\widetilde{g}\widehat{f})+(a_{n}-b_{m})(\widehat{f}\widetilde{f}-f\widehat{\widetilde{f}})=0,$
(3.10a)
$\displaystyle\mathcal{H}_{2}\equiv(g\widehat{\widetilde{f}}-\widehat{\widetilde{g}}f)+(a_{n}+b_{m})(f\widehat{\widetilde{f}}-\widehat{f}\widetilde{f})=0,$
(3.10b)
$\displaystyle\mathcal{H}_{3}\equiv-(a_{n}+b_{m})\widehat{f}\widetilde{g}+a_{n}\widehat{\widetilde{f}}g+b_{m}f\widehat{\widetilde{g}}+\widehat{\widetilde{f}}h-f\widehat{\widetilde{h}}=0,$
(3.10c)
$\displaystyle\mathcal{H}_{4}\equiv-(a_{n}-b_{m})f\widehat{\widetilde{g}}+a_{n}\widetilde{f}\widehat{g}-b_{m}\widehat{f}\widetilde{g}+\widetilde{f}\widehat{h}-\widehat{f}\widetilde{h}=0,$
(3.10d) $\displaystyle\mathcal{H}_{5}\equiv
b_{m}(\widehat{f}g-f\widehat{g})+f\widehat{h}+\widehat{f}s-g\widehat{g}=0,$
(3.10e)
where the connection is
$\mathrm{H2}=\sum_{i=1}^{5}\mathcal{H}_{i}P_{i}/(f\widetilde{f}\widehat{f}\widehat{\widetilde{f}}),$
and
$\displaystyle
P_{1}=-4(a_{n}+b_{m})[(\widetilde{U}\widehat{\widetilde{U}}-a_{n}^{2}+b_{m}^{2})\widetilde{f}\widehat{f}-\widehat{\widetilde{U}}\widehat{f}\widetilde{g}-(a_{n}-b_{m})f\widehat{\widetilde{g}}],$
$\displaystyle
P_{2}=-4[(a_{n}-b_{m})(\widetilde{U}\widehat{\widetilde{U}}-a_{n}^{2}+b_{m}^{2})\widetilde{f}\widehat{f}+(\widetilde{U}\widehat{\widetilde{U}}-a_{n}^{2}+b_{m}^{2})\widetilde{f}\widehat{g}-\widetilde{U}\widehat{\widetilde{U}}\widehat{f}\widetilde{g}-(a_{n}-b_{m})\widetilde{U}f\widehat{\widetilde{g}}],$
$\displaystyle
P_{3}=4[(a_{n}-b_{m})U\widetilde{f}\widehat{f}+\widehat{U}\widetilde{f}\widehat{g}-\widetilde{U}\widehat{f}\widetilde{g}-\widetilde{f}\widehat{h}+\widehat{f}\widetilde{h}],$
$\displaystyle
P_{4}=4[(a_{n}+b_{m})(\widehat{U}f\widehat{\widetilde{f}}-\widehat{f}\widetilde{g})+\widetilde{U}(\widehat{\widetilde{f}}g-f\widehat{\widetilde{g}})],$
$\displaystyle
P_{5}=4(a_{n}^{2}-b_{m}^{2})\widetilde{f}\widehat{\widetilde{f}},$
where $U=U_{n,m}$ is defined in (3.9b). This is as same as the autonomous
case[13].
For solutions we have
###### Proposition 4.
The Casoratians
$f=|\widehat{N-1}|_{{}_{[3]}},~{}g=|\widehat{N-2},~{}N|_{{}_{[3]}},~{}h=|\widehat{N-3},~{}N-1,~{}N|_{{}_{[3]}},~{}s=|\widehat{N-2},~{}N+1|_{{}_{[3]}},$
(3.12)
solve the non-autonomous bilinear equations (3.10), where the basic column
vector $\psi$ satisfies the same conditions in Proposition 3.
###### Proof.
We skip the proof for $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ as they are just
the bilinear H1. We prove $\mathcal{H}_{5}$ and shifted $\mathcal{H}_{3}$ and
$\mathcal{H}_{4}$ in the following forms
$\displaystyle\mathcal{\underaccent{\tilde}{H}}_{3}\equiv-(a_{n-1}+b_{m})\underaccent{\tilde}{\h
f}g+a_{n-1}\widehat{f}\underaccent{\tilde}{g}+b_{m}\underaccent{\tilde}{f}\widehat{g}+\widehat{f}\underaccent{\tilde}{h}-\underaccent{\tilde}{f}\widehat{h},$
(3.13a)
$\displaystyle\underaccent{\hat}{\underaccent{\tilde}{\mathcal{H}}}_{4}\equiv-(a_{n-1}-b_{m-1})\underaccent{\hat}{\underaccent{\tilde}{f}}g+\underaccent{\hat}{f}(a_{n-1}\underaccent{\tilde}{g}+\underaccent{\tilde}{h})-\underaccent{\tilde}{f}(b_{m-1}\underaccent{\hat}{g}+\underaccent{\hat}{h}).$
(3.13b)
We still use the formulae given in appendix A with $c=0$.
For (3.13a), $g=|\widehat{N-2},~{}N|_{{}_{[3]}}$, and
$\widehat{\underaccent{\tilde}{f}}$, $\underaccent{\tilde}{f}$,
$b_{m}\widehat{g}-\widehat{h}$, $\widehat{f}$ and
$a_{n-1}\underaccent{\tilde}{g}+\underaccent{\tilde}{h}$ are (A.8b), (A.6b),
(A.9d), (A.6g) and (A.9c) respectively. Then we have
$\begin{array}[]{rl}\mathcal{\underaccent{\tilde}{H}}_{3}\equiv&-(a_{n-1}+b_{m})g\widehat{\underaccent{\tilde}{f}}+\underaccent{\tilde}{f}(b_{m}\widehat{g}-\widehat{h})+\widehat{f}(a_{n-1}\underaccent{\tilde}{g}+\underaccent{\tilde}{h})\\\
=&-a_{n-1}^{-N+2}~{}b_{m}^{-N+2}|A_{[m+1]}A_{[m]}^{-1}|\cdot[|\widehat{N-3},\psi(N-2),\psi(N)|_{{}_{[3]}}\\\
&~{}\times|\widehat{N-3},\underaccent{\tilde}{\psi}(N-2),\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&-|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&+|\widehat{N-3},~{}\psi(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}]\\\
=&0,\end{array}$
where we have made use of Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},~{}\textbf{b},~{}\textbf{c},~{}\textbf{d})=(\psi(N-2),~{}\psi(N),~{}\underaccent{\tilde}{\psi}(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2))$.
For (3.13b), $g=|\widehat{N-2},~{}N|_{{}_{[3]}}$, and
$\underaccent{\hat}{\underaccent{\tilde}{f}}$, $\underaccent{\hat}{f}$,
$a_{n-1}\underaccent{\tilde}{g}+\underaccent{\tilde}{h}$,
$\underaccent{\tilde}{f}$ and
$b_{m-1}\underaccent{\hat}{g}+\underaccent{\hat}{h}$ are (A.6a), (A.6e),
(A.9c), (A.6b) and (A.9f) respectively. Then we have
$\begin{array}[]{rl}\mathcal{\underaccent{\tilde}{H}}_{4}\equiv&-(a_{n-1}-b_{m-1})\underaccent{\hat}{\underaccent{\tilde}{f}}g+\underaccent{\hat}{f}(a_{n-1}\underaccent{\tilde}{g}+\underaccent{\tilde}{h})-\underaccent{\tilde}{f}(b_{m-1}\underaccent{\hat}{g}+\underaccent{\hat}{h})\\\
=&-a_{n-1}^{-N+2}~{}b_{m-1}^{-N+2}[|\widehat{N-3},~{}\psi(N-2),~{}\psi(N)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\underaccent{\hat}{\psi}(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\\\
&-|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\\\
&+|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}}]\\\
=&0,\end{array}$
where we have made use of Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},~{}\textbf{b},~{}\textbf{c},~{}\textbf{d})=(\psi(N-2),~{}\psi(N),~{}\underaccent{\hat}{\psi}(N-2),~{}\underaccent{\tilde}{\psi}(N-2))$.
For (3.10e), we have
$f=|\widehat{N-1}|_{{}_{[3]}},~{}g=|\widehat{N-2},N|_{{}_{[3]}},~{}s=|\widehat{N-3},N-1,N|_{{}_{[3]}}$,
and $\widehat{h}-b_{m}\widehat{g}$, $\widehat{g}-b_{m}\widehat{f}$ and
$\widehat{f}$ are provided by (A.9d), (A.9b) and (A.6g). Now we obtain
$\begin{array}[]{rl}\mathcal{H}_{5}\equiv&f(\widehat{h}-b_{m}\widehat{g})-g(\widehat{g}-b_{m}\widehat{f})+\widehat{f}s\\\
=&b_{m}^{-N+2}|A_{[m+1]}A_{[m]}^{-1}|\cdot[|\widehat{N-3},\psi(N-2),\psi(N-1)|_{{}_{[3]}}\cdot|\widehat{N-3},\psi(N),\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&-|\widehat{N-3},~{}\psi(N-2),~{}\psi(N)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\\\
&+|\widehat{N-3},~{}\psi(N-2),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}}\cdot|\widehat{N-3},~{}\psi(N-1),~{}\psi(N)|_{{}_{[3]}}]\\\
=&0,\end{array}$
where we have made use of Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},~{}\textbf{b},~{}\textbf{c},~{}\textbf{d})=(\psi(N-2),~{}\psi(N-1),~{}\psi(N),~{}\accentset{\circ}{E}^{2}\psi(N-2))$.
∎
For the explicit forms of $\psi$ together with the transformation matrices we
can take either (A.3) with $c=0$ or (A.4) with $c=0$.
### 3.3 Non-autonomous H3 equation
With the parametrization
$p_{n}=\frac{1+\alpha_{n}^{2}}{2\alpha_{n}},~{}~{}q_{m}=\frac{1+\beta_{m}^{2}}{2\beta_{m}},~{}~{}\alpha_{n}^{2}=-\frac{a_{n}-c}{a_{n}+c},~{}~{}\beta_{m}^{2}=-\frac{b_{m}-c}{b_{m}+c},$
the non-autonomous H3 equation (2.2c) admits two different sets of bilinear
forms. One is
$\displaystyle\mathcal{B}_{1}\equiv
2cf\widetilde{f}+(a_{n}-c)\widetilde{\overline{f}}\underaccent{\bar}{f}-(a_{n}+c)\overline{f}\widetilde{\underaccent{\bar}{f}}=0,$
(3.14a) $\displaystyle\mathcal{B}_{2}\equiv
2cf\widehat{f}+(b_{m}-c)\widehat{\overline{f}}\underaccent{\bar}{f}-(b_{m}+c)\overline{f}\widehat{\underaccent{\bar}{f}}=0,$
(3.14b)
and the other is
$\displaystyle\mathcal{B}_{1}^{\prime}\equiv(b_{m}+c)\overline{f}\widehat{\widetilde{f}}+(a_{n}-c)\widehat{\widetilde{\overline{f}}}f-(a_{n}+b_{m})\widetilde{f}\widehat{\overline{f}}=0,$
(3.15a)
$\displaystyle\mathcal{B}_{2}^{\prime}\equiv(a_{n}+c)f\underaccent{\bar}{\th
f}+(b_{m}-c)\widehat{\widetilde{f}}\underaccent{\bar}{f}-(a_{n}+b_{m})\widetilde{f}\widehat{\underaccent{\bar}{f}}=0,$
(3.15b)
$\displaystyle\mathcal{B}_{3}^{\prime}\equiv(a_{n}-c)(b_{m}+c)\widehat{\underaccent{\bar}{f}}\widetilde{\overline{f}}-(b_{m}-c)(a_{n}+c)\widetilde{\underaccent{\bar}{f}}\widehat{\overline{f}}-2c(a_{n}-b_{m})f\widehat{\widetilde{f}}=0.$
(3.15c)
Both of them share same transformation
$u_{n,m}=A~{}V_{n,m}\frac{\overline{f}_{n,m}}{f_{n,m}}+B~{}V_{n,m}^{-1}\frac{\underaccent{\bar}{f}_{n,m}}{f_{n,m}},~{}~{}AB=-\frac{1}{4}\delta,$
(3.16a) where
$V_{n,m}=\prod^{n-1}_{i=n_{0}}\alpha_{i}\prod^{m-1}_{j=m_{0}}\beta_{j}.$
(3.16b)
The connections are respectively
$\displaystyle\mathrm{H3}=\frac{-\delta^{2}B^{-2}V_{n,m}^{2}(a_{n}-c)(b_{m}-c)P_{1}+4\delta
P_{2}+16B^{2}V_{n,m}^{-2}(a_{n}+c)(b_{m}+c)P_{3}}{32(a_{n}^{2}-c^{2})(b_{m}^{2}-c^{2})f\widetilde{f}\widehat{f}\widehat{\widetilde{f}}},$
with
$\displaystyle
P_{1}=\widehat{\widetilde{\overline{f}}}[(b_{m}-c)\widehat{\overline{f}}\mathcal{B}_{1}-(a_{n}-c)\widetilde{\overline{f}}\mathcal{B}_{2}]-\overline{f}[(b_{m}+c)\widetilde{\overline{f}}\widehat{\mathcal{B}_{1}}-(a_{n}+c)\widehat{\overline{f}}\widetilde{\mathcal{B}_{2}}],$
$\displaystyle
P_{2}=2c[(b_{m}+c)(b_{m}-c)(\widehat{f}\widehat{\widetilde{f}}\mathcal{B}_{1}+f\widetilde{f}\widehat{\mathcal{B}_{1}})-(a_{n}+c)(a_{n}-c)(\widetilde{f}\widehat{\widetilde{f}}\mathcal{B}_{2}+f\widehat{f}\widetilde{\mathcal{B}_{2}})],$
$\displaystyle
P_{3}=\widehat{\widetilde{\underaccent{\bar}{f}}}[(b_{m}+c)\widehat{\underaccent{\bar}{f}}\mathcal{B}_{1}-(a_{n}+c)\widetilde{\underaccent{\bar}{f}}\mathcal{B}_{2}]-\underaccent{\bar}{f}[(b_{m}-c)\widetilde{\underaccent{\bar}{f}}\widehat{\mathcal{B}_{1}}-(a_{n}-c)\widehat{\underaccent{\bar}{f}}\widetilde{\mathcal{B}_{2}}],$
and
$\displaystyle\mathrm{H}3\equiv\frac{c}{f\widetilde{f}\widehat{f}\widehat{\widetilde{f}}}[A^{2}V_{n,m}^{2}\frac{\widetilde{\overline{f}}\widehat{f}\mathcal{B}_{1}^{\prime}-\widehat{\overline{f}}\widetilde{f}\mathcal{B}_{2}^{\prime}}{(a_{n}+c)(b_{m}+c)}+B^{2}V_{n,m}^{-2}\frac{\widehat{\underaccent{\bar}{f}}\widetilde{f}\underline{\mathcal{B}}_{1}^{\prime}-\widetilde{\underaccent{\bar}{f}}\widehat{f}\underline{\mathcal{B}}_{2}^{\prime}}{(a_{n}-c)(b_{m}-c)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+AB(\frac{\widehat{\underaccent{\bar}{f}}\widetilde{f}\mathcal{B}_{2}^{\prime}+\widetilde{\overline{f}}\widehat{f}\underline{\mathcal{B}}_{2}^{\prime}}{(a_{n}+c)(b_{m}-c)}-\frac{\widetilde{\underaccent{\bar}{f}}\widehat{f}\mathcal{B}_{1}^{\prime}+\widehat{\overline{f}}\widetilde{f}\underline{\mathcal{B}}_{1}^{\prime}}{(a_{n}-c)(b_{m}+c)}+\frac{2(a_{n}+b_{m})\widetilde{f}\widehat{f}\mathcal{B}_{3}^{\prime}}{(a_{n}^{2}-c^{2})(b_{m}^{2}-c^{2})})],$
which are similar to the one in the autonomous case[13]. For solutions we have
###### Proposition 5.
The Casoratians
$f=|\widehat{N-1}|_{{}_{[\nu]}},~{}~{}\nu=1,~{}2,~{}3,$ (3.18)
solve non-autonomous bilinear equations (3.14) and (3.15), if the basic column
vector $\psi(n,m,l)$ is symmetric in terms of pairs $(n,a_{n})$ and
$(m,b_{m})$, and together with auxiliary vectors $\omega(n,m,l)$ and
$\zeta(n,m,l)$, satisfies the following shift relations
$\displaystyle(c-a_{n})\underaccent{\bar}{\psi}=\psi-\widetilde{\underaccent{\bar}{\psi}},$
(3.19a)
$\displaystyle\psi=A_{[n]}\omega,~{}~{}(a_{n}+c)\widetilde{\omega}=\omega+\widetilde{\overline{\omega}},$
(3.19b)
$\displaystyle\psi=B_{[l]}\zeta,~{}~{}(c+b_{m})\overline{\zeta}=\zeta+{\widehat{\overline{\zeta}}},$
(3.19c)
where $A_{[n]}$ and $B_{[l]}$ are $N\times N$ transform matrices, and the
matrix product $B_{[l]}B_{[l+1]}^{-1}$ is independent of $l$.
###### Proof.
We prove (3.14a) in its down-tilde version
$\mathcal{\underaccent{\tilde}{B}}_{1}\equiv
2c\underaccent{\tilde}{f}f+(a_{n-1}-c)\overline{f}\underaccent{\bar}{\underaccent{\tilde}{f}}-(a_{n-1}+c)\overline{\underaccent{\tilde}{f}}\underaccent{\bar}{f}.$
(3.20)
In (3.20), for $f$, $\underaccent{\tilde}{f}$, $\overline{f}$,
$\underaccent{\bar}{\underaccent{\tilde}{f}}$, $\underaccent{\tilde}{\b f}$
and $\underaccent{\bar}{f}$, we make use of (A.7b), (A.8i), (A.7h), (A.8h),
(A.8l) and (A.7c) respectively, and get
$\displaystyle\mathcal{\underaccent{\tilde}{B}}_{1}\equiv$
$\displaystyle\prod_{j=0}^{N-3}(a_{n-1}-b_{m+j})^{-1}\prod_{j=0}^{N-3}(c-b_{m+j})^{-1}\prod_{j=0}^{N-3}(c+b_{m+j})^{-1}|B_{[l+1]}B_{[l]}^{-1}|$
$\displaystyle\times[-|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[2]}}\cdot|\widehat{N-3},~{}\underaccent{\bar}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle+|\widehat{N-3},~{}\psi(N-2),~{}\underaccent{\bar}{\psi}(N-2)|_{{}_{[2]}}\cdot|\widehat{N-3},~{}\underaccent{\tilde}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle-|\widehat{N-3},~{}\psi(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}\cdot|\widehat{N-3},~{}\underaccent{\tilde}{\psi}(N-2),~{}\underaccent{\bar}{\psi}(N-2)|_{{}_{[2]}}]$
$\displaystyle=$ $\displaystyle 0,$
with the help of Proposition 2 in which $\textbf{B}=(\widehat{N-3})$,
$(\textbf{a},\textbf{b},\textbf{c},\textbf{d})=(\psi(N-2),\underaccent{\tilde}{\psi}(N-2),\underaccent{\bar}{\psi}(N-2),\accentset{\circ}{E}^{3}\psi(N-2))$.
Here to get the coefficient $|B_{[l+1]}B_{[l]}^{-1}|$ we request
$B_{[l]}B_{[l+1]}^{-1}$ to be independent of $l$, i.e.
$B_{[l]}B_{[l-1]}^{-1}=B_{[l+1]}B_{[l]}^{-1}$.
$\mathcal{B}_{2}$ holds thanks to $\mathcal{B}_{1}$ and the _n-m_ symmetric
property of $\psi(n,m,l)$.
We prove $\mathcal{B}_{1}^{\prime}$ in its down-tilde shifted version, i.e.
$\mathcal{\underaccent{\tilde}{B}}_{1}^{\prime}\equiv(b_{m}+c)\underaccent{\tilde}{\b
f}\widehat{f}+(a_{n}-c){\widehat{\overline{f}}}\underaccent{\tilde}{f}-(a_{n-1}+b_{m}){\widehat{\overline{\underaccent{\tilde}{f}}}}f.$
(3.21)
In (3.21), we take $f=|\widehat{N-1}|_{{}_{[3]}}$, and
$\underaccent{\tilde}{\b f}$, $\widehat{f}$, ${\widehat{\overline{f}}}$,
$\underaccent{\tilde}{f}$ and ${\widehat{\overline{\underaccent{\tilde}{f}}}}$
as (A.8e), (A.6c), (A.6h), (A.6a) and (A.8f), respectively. Then we have
$\displaystyle\mathcal{\underaccent{\tilde}{B}}_{1}^{\prime}\equiv$
$\displaystyle(a_{n-1}-c)^{-N+2}(b_{m}+c)^{-N+2}|A_{[m+1]}A_{[m]}^{-1}|$
$\displaystyle\times[-|\psi(0),~{}\widetilde{N-2},~{}\psi(N-1)|_{{}_{[3]}}\cdot|\widetilde{N-2},~{}\underaccent{\tilde}{\psi}(N-1),~{}\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}}$
$\displaystyle+|\psi(0),~{}\widetilde{N-2},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[3]}}\cdot|\widetilde{N-2},~{}\psi(N-1),~{}\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}}$
$\displaystyle-|\psi(0),~{}\widetilde{N-2},~{}\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}}\cdot|\widetilde{N-2},~{}\psi(N-1),~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[3]}}]$
$\displaystyle=$ $\displaystyle 0,$
where we have made use of Proposition 2, in which
$\textbf{B}=(\widetilde{N-2})$,
$(\textbf{a},\textbf{b},\textbf{c},\textbf{d})=(\psi(0),\psi(N-1),\underaccent{\tilde}{\psi}(N-1),\accentset{\circ}{E}^{2}\psi(N-1))$.
$\mathcal{B}_{2}^{\prime}$ holds thanks to $\mathcal{B}_{1}^{\prime}$ and the
_n-m_ symmetric property of $\psi(n,m,l)$.
By a down-hat shift, $\mathcal{B}_{3}^{\prime}$ reads
$\mathcal{\underaccent{\hat}{B}}_{3}^{\prime}\equiv(a_{n}-c)(b_{m-1}+c)\underaccent{\bar}{f}\underaccent{\hat}{\tb
f}-(a_{n}+c)(b_{m-1}-c)\overline{f}\widetilde{\underaccent{\bar}{\underaccent{\hat}{f}}}-2c(a_{n}-b_{m-1})\underaccent{\hat}{f}\widetilde{f}.$
(3.23)
In (3.23), for $\underaccent{\bar}{f}$, $\underaccent{\hat}{\tb f}$,
$\overline{f}$, $\underaccent{\bar}{\underaccent{\hat}{\t f}}$,
$\underaccent{\hat}{f}$ and $\widetilde{f}$ we use (A.7d), (A.8n), (A.7e),
(A.8k), (A.7j) and (A.8j) respectively. Then we have
$\displaystyle\mathcal{\underaccent{\hat}{B}}_{3}^{\prime}\equiv$
$\displaystyle\prod_{i=1}^{N-2}(b_{m-1}-a_{n+i})^{-1}\prod_{i=1}^{N-2}(c-a_{n+i})^{-1}\prod_{i=1}^{N-2}(c+a_{n+i})^{-1}|B_{[l+1]}B_{[l]}^{-1}|$
$\displaystyle\times[|\widetilde{N-2},~{}\psi(0),~{}\underaccent{\bar}{\psi}(N-1)|_{{}_{[1]}}\cdot|\widetilde{N-2},~{}\underaccent{\hat}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}}$
$\displaystyle-|\widetilde{N-2},~{}\psi(0),~{}\underaccent{\hat}{\psi}(N-1)|_{{}_{[1]}}\cdot|\widetilde{N-2},~{}\underaccent{\bar}{\psi}(N-1),~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}}$
$\displaystyle+|\widetilde{N-2},~{}\psi(0),~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}}\cdot|\widetilde{N-2},~{}\underaccent{\bar}{\psi}(N-1),~{}\underaccent{\hat}{\psi}(N-1)|_{{}_{[1]}}]$
$\displaystyle=$ $\displaystyle 0,$
where we have made use of Proposition 2 in which
$\textbf{B}=(\widetilde{N-2})$,
$(\textbf{a},\textbf{b},\textbf{c},\textbf{d})=(\psi(0),\underaccent{\bar}{\psi}(N-1),\underaccent{\hat}{\psi}(N-1),\accentset{\circ}{E}^{3}\psi(N-1))$.
∎
For the explicit forms of $\psi$ together with the transformation matrices we
can take either (A.3) or (A.4).
### 3.4 Non-autonomous Q1 equation
The non-autonomous Q1 equation can have two different bilinearizations which
are also similar to their autonomous cases[13]. First, using the
parametrization
$p_{n}=\frac{rc^{2}}{a_{n}^{2}-c^{2}},~{}~{}~{}~{}q_{m}=\frac{rc^{2}}{b_{m}^{2}-c^{2}},$
(3.25a) where $c$ and $r$ are constants, and through the transformation
$u_{n,m}=AV_{n,m}\frac{\overline{\overline{f}}_{n,m}}{f_{n,m}}+BV_{n,m}^{-1}\frac{\underaccent{\bar}{\db{f}}_{n,m}}{f_{n,m}},~{}~{}AB=\frac{r^{2}\delta^{2}}{16},$
(3.25b) where $V_{n,m}$ is (3.16b) and
$\alpha_{n}=\frac{a_{n}-c}{a_{n}+c},~{}~{}\beta_{m}=\frac{b_{m}-c}{b_{m}+c}$,
the non-autonomous Q1 equation (2.2d) can be transformed to (3.14), i.e., one
of bilinear form for the non-autonomous H3 equation. So its solutions
consequently follow the Proposition 5. In this case, the connection is
$\mathrm{Q}1\equiv(\alpha_{n}\beta_{m}U_{n,m}^{2}A^{2}\overline{P}_{1}+\frac{r^{2}}{16}\alpha_{n}^{-1}\beta_{m}^{-1}(a_{n}+c)^{-2}(b_{m}+c)^{-2}\delta^{2}P_{2}+\alpha_{n}^{-1}\beta_{m}^{-1}U_{n,m}^{-2}B^{2}\underline{P}_{1})/f\widetilde{f}\widehat{f}\widehat{\widetilde{f}},$
where
$\begin{array}[]{rl}P_{1}=&Y\widetilde{Y}-X\widehat{X},~{}~{}~{}~{}X=\mathcal{B}_{1}-2cf\widetilde{f},~{}~{}~{}~{}Y=\mathcal{B}_{2}-2cf\widehat{f},\\\
P_{2}=&(a_{n}^{2}-c^{2})(b_{m}+c)^{2}\Bigl{(}\overline{X}\underaccent{\bar}{{\h
X}}-4c^{2}\overline{f}\,\widetilde{\overline{f}}\underaccent{\bar}{\h
f}\underaccent{\bar}{\th
f}\,\Bigr{)}+(a_{n}^{2}-c^{2})(b_{m}-c)^{2}\Bigl{(}\underaccent{\bar}{X}{\widehat{\overline{X}}}-4c^{2}\underaccent{\bar}{f}\,\underaccent{\bar}{\t
f}\,{\widehat{\overline{f}}}\,\widehat{\widetilde{\overline{f}}}\,\Bigr{)}\\\
&-4c^{2}(b_{m}^{2}-c^{2})\Bigl{(}X\widehat{X}-4c^{2}f\widetilde{f}\,\widehat{f}\,\widehat{\widetilde{f}}\,\Bigr{)}-(b_{m}^{2}-c^{2})(a_{n}+c)^{2}\Bigl{(}\overline{Y}\underaccent{\bar}{\t
Y}-4c^{2}\overline{f}\,{\widehat{\overline{f}}}\,\underaccent{\bar}{\t
f}\,\underaccent{\bar}{\th f}\,\Bigr{)}\\\
&-(b_{m}^{2}-c^{2})(a_{n}-c)^{2}\Bigl{(}\underaccent{\bar}{Y}\widetilde{\overline{Y}}-4c^{2}\underaccent{\bar}{f}\underaccent{\bar}{\h
f}\,\widetilde{\overline{f}}\,\widehat{\widetilde{\overline{f}}}\,\Bigr{)}+4c^{2}(a_{n}^{2}-c^{2})\Bigl{(}Y\widetilde{Y}-4c^{2}f\widehat{f}\,\widetilde{f}\,\widehat{\widetilde{f}}\,\Bigr{)}.\end{array}$
The second bilinear form employs the transformation
$u_{n,m}=W_{n,m}-(\frac{c^{2}}{r}-\delta^{2}r)\frac{g_{n,m}}{f_{n,m}},$
(3.26a) where $c,r$ are constants,
$W_{n,m}=\sum_{i=n_{0}}^{n-1}\alpha_{i}+\sum_{j=m_{0}}^{m-1}\beta_{j},~{}\alpha_{n}=p_{n}a_{n},~{}\beta_{m}=q_{m}b_{m},~{}p_{n}=\frac{c^{2}/r-\delta^{2}r}{a_{n}^{2}-\delta^{2}},~{}q_{m}=\frac{c^{2}/r-\delta^{2}r}{b_{m}^{2}-\delta^{2}},$
(3.26b)
and the bilinear form reads
$\displaystyle\mathcal{Q}_{1}$ $\displaystyle\equiv$
$\displaystyle(b_{m}-\delta){\widehat{\widetilde{\overline{f}}}}f+(a_{n}+\delta)\widehat{\widetilde{f}}\overline{f}-(a_{n}+b_{m}){\widetilde{\overline{f}}}\widehat{f}=0,$
(3.27a) $\displaystyle\mathcal{Q}_{2}$ $\displaystyle\equiv$
$\displaystyle(a_{n}-b_{m}){\widehat{\widetilde{\overline{f}}}}f+(b_{m}+\delta){\widetilde{\overline{f}}}\widehat{f}-(a_{n}+\delta)\widetilde{f}{{\widehat{\overline{f}}}}=0,$
(3.27b) $\displaystyle\mathcal{Q}_{3}$ $\displaystyle\equiv$
$\displaystyle\widetilde{f}{{\widehat{\overline{f}}}}-{\widetilde{\overline{f}}}\widehat{f}+(b_{m}-\delta){{\widehat{\overline{f}}}}\widetilde{g}-(a_{n}-\delta){\widetilde{\overline{f}}}\widehat{g}+(a_{n}-b_{m})\overline{f}\widehat{\widetilde{g}}=0,$
(3.27c) $\displaystyle\mathcal{Q}_{4}$ $\displaystyle\equiv$
$\displaystyle(a_{n}-b_{m})({\widehat{\widetilde{\overline{f}}}}g-\overline{f}\widehat{\widetilde{g}})+(a_{n}+b_{m})({\widetilde{\overline{f}}}\widehat{g}-{{\widehat{\overline{f}}}}\widetilde{g})=0.$
(3.27d)
The connection is
$\mathrm{Q}1=\frac{(c^{2}/r-\delta^{2}r)^{3}}{(a_{n}^{2}-\delta^{2})(b_{m}^{2}-\delta^{2})(a_{n}-b_{m})(a_{n}+\delta)\overline{f}f\widetilde{f}\widehat{f}\widehat{\widetilde{f}}}\quad\sum^{4}_{i=1}\mathcal{Q}_{i}P_{i},$
where
$\begin{array}[]{rl}P_{1}~{}=&(a_{n}-b_{m})~{}[-(a_{n}-b_{m})\widetilde{f}\widehat{f}g+(a_{n}+b_{m})f(\widehat{f}\widetilde{g}-\widetilde{f}\widehat{g})\\\
&\phantom{(a_{n}-b_{m})}-(a_{n}^{2}-\delta^{2})\widetilde{f}\widehat{g}g+(b_{m}^{2}-\delta^{2})\widehat{f}\widetilde{g}g+(a_{n}^{2}-b_{m}^{2})f\widetilde{g}\widehat{g}~{}],\\\
P_{2}~{}=&(a_{n}+b_{m})~{}[(a_{n}-b_{m})\widetilde{f}\widehat{f}g+(b_{m}-\delta)f(\widetilde{f}\widehat{g}-\widehat{f}\widetilde{g})+(a_{n}-b_{m})(b_{m}-\delta)\widetilde{g}(\widehat{f}g-f\widehat{g})],\\\
P_{3}~{}=&(a_{n}+b_{m})(a_{n}+\delta)~{}[(a_{n}-b_{m})\widetilde{f}\widehat{f}g+(b_{m}-\delta)\widetilde{f}f\widehat{g}-(a_{n}-\delta)\widehat{f}f\widetilde{g}~{}],\\\
P_{4}~{}=&(a_{n}+\delta)f[-(a_{n}-b_{m})\widetilde{f}\widehat{f}+(a_{n}-\delta)(b_{m}-\delta)(\widetilde{f}\widehat{g}-\widehat{f}\widetilde{g})].\end{array}$
For solutions to (3.27) we have
###### Proposition 6.
The Casoratians
$f=|\widehat{N-1}|_{{}_{[3]}},~{}g=|-1,\widetilde{N-1}|_{{}_{[3]}},$ (3.28)
solve the non-autonomous bilinear equations (3.27), if their basic column
vector $\psi(n,m,l)$ has symmetric property and satisfies the shift relations
(3.19) with $c=\delta$, as well as
$\psi=A_{[n]}A_{[m]}\sigma,~{}(a_{n}+\delta)\widetilde{\sigma}=\sigma+\widetilde{\overline{\sigma}},~{}(b_{m}+\delta)\widehat{\sigma}=\sigma+\widehat{\overline{\sigma}},$
(3.29)
where the matrices $A_{[n]}$, $A_{[m]}$ and their shifts are in an Abelian
group.
###### Proof.
The down-hat-bar shifted version of (3.27a) is
$\mathcal{\underaccent{\bar}{\underaccent{\hat}{Q}}}_{1}\equiv(b_{m-1}-\delta)\widetilde{f}\underaccent{\bar}{\underaccent{\hat}{f}}+(a_{n}+\delta)\widetilde{\underaccent{\bar}{f}}\underaccent{\hat}{f}-(a_{n}+b_{m-1})\widetilde{\underaccent{\hat}{f}}\underaccent{\bar}{f},$
(3.30)
and by using (A.6d), (A.6f), (A.8d), (A.6e), (A.8a) and
$\underaccent{\bar}{f}=|-1~{}~{}\widehat{N-3}~{}~{}\psi(N-2)|_{[3]}$, with
$c=\delta$ we can prove it is true. In fact, it is the same as the down hat-
bar version of $\mathcal{B}_{2}^{\prime}$.
By a down-tilde shift (3.27b) is written as
$\mathcal{\underaccent{\tilde}{Q}}_{2}\equiv(a_{n-1}-b_{m}){\widehat{\overline{f}}}\underaccent{\tilde}{f}+(b_{m}+\delta){\overline{f}}\widehat{\underaccent{\tilde}{f}}-(a_{n-1}+\delta)f{\widehat{\overline{\underaccent{\tilde}{f}}}}.$
(3.31)
Thanks to Proposition 1, here we use $f=|\widehat{N-1}|_{{}_{[2]}}$. Then with
the help of (A.7i), (A.7a), (A.7g), (A.8g) and (A.8m), with $c=\delta$, we can
rewrite (3.31) as
$\begin{array}[]{rl}\mathcal{\underaccent{\tilde}{Q}}_{2}~{}\equiv&\prod_{j=1}^{N-2}(\delta+b_{m+j})^{-1}\prod_{j=1}^{N-2}(a_{n-1}-b_{m+j})^{-1}|B_{[l+1]}B_{[l]}^{-1}|~{}[|\widetilde{N-1},~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}}\\\
&\times|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}-|\widehat{N-2},~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}}\cdot|\widetilde{N-1},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}\\\
&-|\widehat{N-1}|_{{}_{[2]}}\cdot|\widetilde{N-2},~{}\underaccent{\tilde}{\psi}(N-1),~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}}]\\\
&=0,\end{array}$
where we have made use of Proposition 2.
By a down-tilde-hat shift (3.27c) is written as
$\mathcal{\underaccent{\hat}{\underaccent{\tilde}{Q}}}_{3}\equiv-\underaccent{\hat}{\b
f}[\underaccent{\tilde}{f}+(a_{n-1}-\delta)\underaccent{\tilde}{g}]+\underaccent{\tilde}{\b
f}[\underaccent{\hat}{f}+(b_{m-1}-\delta)\underaccent{\hat}{g}]+(a_{n-1}-b_{m-1})\underaccent{\hat}{\dt{\b
f}}g.$ (3.32)
With $c=\delta$, substituting (A.10b),(A.10d), (A.10f), (A.10e), (A.10a),
(A.10c) and (A.10m) into the right-hand side of (3.32), we obtain
$\mathcal{\underaccent{\tilde}{ \dh
Q}}_{3}~{}\equiv(a_{n-1}-\delta)^{2}Y_{1}-(b_{m-1}-\delta)^{2}Y_{2}+(a_{n-1}-\delta)^{2}(b_{m-1}-\delta)^{2}Y_{3},$
where
$\displaystyle Y_{\mu}=$ $\displaystyle
f|E_{\mu}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}+\underaccent{\bar}{f}|E_{\mu}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}-g|E_{\mu}{\psi}(-1),\widehat{N-2}|_{{}_{[3]}},~{}~{}\mu=1,~{}2,$
$\displaystyle Y_{3}=$
$\displaystyle|\underaccent{\tilde}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}|\underaccent{\hat}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}-|\underaccent{\hat}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}|\underaccent{\tilde}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}$
$\displaystyle+g|\underaccent{\hat}{\psi}(-1),\underaccent{\tilde}{\psi}(-1),\widetilde{N-2}|_{{}_{[3]}},$
which are zeros in the light of Proposition 2 [13]. Thus, we have proved
$\mathcal{Q}_{3}=0$. To prove $\mathcal{Q}_{4}=0$, we go to prove
${\mathcal{Q}}_{4}^{{}^{\prime}}=\mathcal{Q}_{3}+\mathcal{Q}_{4}=0$, i.e.,
$\mathcal{Q}_{4}^{{}^{\prime}}\equiv-\widehat{\overline{f}}[\widetilde{f}-(a_{n}+\delta)\widetilde{g}]-\widetilde{\overline{f}}[\widehat{f}-(b_{m}+\delta)\widehat{g}]+(a_{n}-b_{m})\widehat{\widetilde{\overline{f}}}g=0,$
(3.34)
In the light of (A.10k), (A.10g), (A.10i), (A.10h), (A.10j), (A.10l) and
(A.10n), with $c=\delta$ we can rewrite $\mathcal{Q}_{4}^{{}^{\prime}}$ as
$\mathcal{Q}_{4}^{{}^{\prime}}~{}\equiv(a_{n}+\delta)^{2}Z_{1}-(b_{m}+\delta)^{2}Z_{2}-(a_{n}+\delta)^{2}(b_{m}+\delta)^{2}Z_{3},$
where
$\displaystyle Z_{\mu}=$ $\displaystyle
f|\accentset{\circ}{E}^{\mu}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}+\underaccent{\bar}{f}|\accentset{\circ}{E}^{\mu}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}-g|\accentset{\circ}{E}^{\mu}{\psi}(-1),\widehat{N-2}|_{{}_{[3]}},~{}~{}\mu=1,2,$
$\displaystyle Z_{3}=$
$\displaystyle|\accentset{\circ}{E}^{1}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}|\accentset{\circ}{E}^{2}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}-|\accentset{\circ}{E}^{2}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}$
$\displaystyle\times|\accentset{\circ}{E}^{1}{\psi}(-1),\widetilde{N-1}|_{{}_{[3]}}+g|\accentset{\circ}{E}^{2}{\psi}(-1),\accentset{\circ}{E}^{1}{\psi}(-1),\widetilde{N-2}|_{{}_{[3]}},$
which are also zeros[13] in the light of Proposition 2. Thus we finish the
proof. ∎
For the explicit forms of $\psi$ together with the transformation matrices we
can take either (A.3) with $c=\delta$ or (A.4) with $c=\delta$.
## 4 Conclusions
We have derived bilinear forms and Casoratian solutions for the non-autonomous
H1, H2, H3δ and Q1δ models in the non-autonomous ABS list. The transformations
that we used to fulfill bilinearization are quite similar to those used in the
autonomous cases[13]. Besides, the bilinear forms and Casoratian structures of
solutions are also similar to the autonomous cases. In addition, in Appendix
we listed Caosratian shift formulae for non-autonomous case.
It would be interesting to have a look at the non-autonomous deformation in
terms of the bilinearizations, bilinear equations and Casoratian vectors. With
comparison we can sum up the following deformations from autonomous case to
non-autonomous case:
$\begin{array}[]{llcl}\mathrm{spacing~{}paramaters:}&(a,b)&\to&(a_{n},b_{m}),\\\
\mathrm{linear~{}function:}&an+bm&\to&\sum^{n-1}_{i=n_{0}}a_{i}+\sum^{m-1}_{j=m_{0}}b_{j},\\\
\mathrm{discre~{}exponential~{}function:}&\left(\frac{a+k}{a-k}\right)^{n}\left(\frac{b+k}{b-k}\right)^{m}&\to&\prod_{i=n_{0}}^{n-1}(\frac{a_{i}+k}{a_{i}-k})\prod_{j=m_{0}}^{m-1}(\frac{b_{j}+k}{b_{j}-k})\end{array}$
We note that many discrete bilinear equations can be deautonomised using these
deformations (see [8]). Besides, actually, here we have seen that these
deformations can well keep the correspondence of autonomous and non-autonomous
ABS lattice equations.
### Acknowledgements
This project is supported by the NSF of China (11071157), Shanghai Leading
Academic Discipline Project (No.J50101) and Postgraduate Innovation Foundation
of Shanghai University (No. SHUCX111027).
## Appendix A Casoratian shift formulae
We derive shift formulae for the Casoratians
$f=|\widehat{N-1}|_{{}_{[\nu]}},~{}g=|\widehat{N-2},N|_{{}_{[\nu]}},~{}h=|\widehat{N-2},~{}s=|\widehat{N-3},N-1,N|_{{}_{[\nu]}},N+1|_{{}_{[\nu]}},$
(A.1)
for $\nu=1,2,3,$, where the basic column vector $\psi(n,m,l)$ satisfies the
following relation
$\displaystyle(a_{n-1}-c)\underaccent{\tilde}{\psi}=\psi-\underaccent{\tilde}{\b\psi},~{}~{}(b_{m-1}-c)\underaccent{\hat}{\psi}=\psi-\underaccent{\hat}{\b\psi},$
(A.2a) and its auxiliary vectors $\omega(n,m,l)$, $\phi(n,m,l)$,
$\zeta(n,m,l)$ and $\sigma(n,m,l)$ satisfy
$\displaystyle\psi=A_{[n]}\omega,~{}~{}(a_{n}+c)\widetilde{\omega}=\omega+\widetilde{\overline{\omega}},$
(A.2b)
$\displaystyle\psi=A_{[m]}\phi,~{}~{}(a_{n}+c)\widehat{\phi}=\phi+\widehat{\overline{\phi}},$
(A.2c)
$\displaystyle\psi=B_{[l]}\zeta,~{}~{}(c+b_{m})\overline{\zeta}=\zeta+\widehat{\overline{\zeta}},$
(A.2d)
$\displaystyle\psi=A_{[n]}A_{[m]}\sigma,~{}~{}(a_{n}+c)\widetilde{\sigma}=\sigma+\widetilde{\overline{\sigma}},~{}~{}(b_{m}+c)\widehat{\sigma}=\sigma+\widehat{\overline{\sigma}},$
(A.2e)
where $c$ is a constant, $A_{[n]}$, $A_{[m]}$ and $B_{[l]}$ only depends on
$n$, $m$ and $l$, respectively, $A_{[n]},A_{[m]}$ and their shifts are in an
Abelian group, and the matrix product $B_{[l+1]}B_{[l]}^{-1}$ is independent
of $l$.
We give two explicit forms for $\psi$ satisfying the above criterion (A.2).
One is
$\displaystyle\psi(n,m,l)$ $\displaystyle=$
$\displaystyle\psi^{+}(n,m,l)+\psi^{-}(n,m,l),$ (A.3a)
$\displaystyle\psi^{\pm}(n,m,l)$ $\displaystyle=$
$\displaystyle(\psi^{\pm}_{1}(n,m,l),\psi^{\pm}_{2}(n,m,l),\cdots,\psi^{\pm}_{N}(n,m,l))^{T},$
(A.3b) with $\psi_{r}^{\pm}(n,m,l)=\rho_{r}^{\pm}(c\pm
k_{r})^{l}\prod_{i=n_{{}_{0}}}^{n-1}(a_{i}\pm
k_{r})\prod_{j=m_{{}_{0}}}^{m-1}(b_{j}\pm k_{r}),~{}~{}r=1,2,\cdots,N,$ (A.3c)
where $\rho_{r}^{\pm}$ and $k_{r}$ are constants. The available transform
matrices are $\displaystyle
A_{[n]}=\mbox{Diag}(A_{{[n]}_{1}}(k_{1},n),\cdots,A_{{[n]}_{N}}(k_{N},n)),~{}~{}$
$\displaystyle
A_{{[n]}_{r}}(k_{r},n)=\prod_{i=n_{0}}^{n-1}(a_{i}^{2}-k_{r}^{2}),$ (A.3d)
$\displaystyle
A_{[m]}=\mbox{Diag}(A_{{[m]}_{1}}(k_{1},m),\cdots,A_{{[m]}_{N}}(k_{N},m)),~{}~{}$
$\displaystyle
A_{{[m]}_{r}}(k_{r},m)=\prod_{j=m_{0}}^{m-1}(b_{j}^{2}-k_{r}^{2}),$ (A.3e)
$\displaystyle
B_{[l]}=\mbox{Diag}(B_{{[l]}_{1}}(k_{1},l),\cdots,B_{{[l]}_{N}}(k_{N},l)),~{}~{}$
$\displaystyle B_{{[l]}_{r}}(k_{r},l)=(c^{2}-k_{r}^{2})^{l},$ (A.3f)
and the auxiliary vectors $\omega,\phi,\zeta,\sigma$ are correspondingly
defined by these transform matrices and $\psi$ through (A.2b)-(A.2e). Another
explicit form of $\psi$ is
$\displaystyle\psi(n,m,l)$ $\displaystyle=$
$\displaystyle\mathcal{A}_{+}\psi^{+}(n,m,l)+\mathcal{A}_{-}\psi^{-}(n,m,l),$
(A.4a) $\displaystyle\psi^{\pm}(n,m,l)$ $\displaystyle=$
$\displaystyle(\psi^{\pm}_{1}(n,m,l),\psi^{\pm}_{2}(n,m,l),\cdots,\psi^{\pm}_{N}(n,m,l))^{T},$
(A.4b) with
$\psi^{\pm}_{r}(n,m,l)=\frac{1}{(r-1)!}\partial_{k_{1}}^{r-1}[\rho_{1}^{\pm}(c\pm
k_{1})^{l}\prod_{i=n_{{}_{0}}}^{n-1}(a_{i}\pm
k_{1})\prod_{j=m_{{}_{0}}}^{m-1}(b_{j}\pm k_{1})],~{}~{}r=1,2,\cdots,N,$
(A.4c) where $\mathcal{A}_{\pm}$ are two arbitrary non-singular lower
triangular Toeplitz matrix(see [23]). The available transform matrices of this
case are $\displaystyle A_{[n]}=(a_{s,i}(k_{1}))_{N\times
N},~{}a_{s,i}(k_{1})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial_{k_{1}}^{s-i}}{(s-i)!}\displaystyle\prod_{i=n_{0}}^{n-1}(a_{i}^{2}-k_{1}^{2}),&\hbox{$s\geq
i$,}\\\ 0,&\hbox{$s<i$,}\end{array}\right.~{}s,i=1,\cdots,N,$ (A.4f)
$\displaystyle A_{[m]}=(a_{s,j}(k_{1}))_{N\times N},~{}a_{s,j}(k_{1})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial_{k_{1}}^{s-j}}{(s-j)!}\displaystyle\prod_{j=m_{0}}^{m-1}(b_{j}^{2}-k_{1}^{2}),&\hbox{$s\geq
j$,}\\\
0,&\hbox{$s<j$,}\end{array}\right.~{}s,j=1,\cdots,N,~{}~{}~{}~{}~{}~{}$ (A.4i)
$\displaystyle B_{[l]}=(b_{s,j}(k_{1}))_{N\times N},~{}b_{s,j}(k_{1})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial_{k_{1}}^{s-j}}{(s-j)!}(c^{2}-k_{1}^{2})^{l},&\hbox{$s\geq
j$,}\\\ 0,&\hbox{$s<j$,}\end{array}\right.~{}s,j=1,\cdots,N.$ (A.4l)
The auxiliary vectors $\omega,\phi,\zeta,\sigma$ are also correspondingly
defined by these transform matrices and $\psi$ through (A.2b)-(A.2e). Since
$A_{[n]},A_{[m]},B_{[l]}$ are non-singular lower triangular Toeplitz matrices
which compose an Abelian group (see [23]), the commutative property holds
automatically and the matrix product $B_{[l+1]}B_{[l]}^{-1}$ is independent of
$l$ (see [18]).
In the following we list some Casoratian shift formulae. For convenience, we
define operators $\accentset{\circ}{E}^{\nu}$, $\nu=1,2,3$, as follows
$\accentset{\circ}{E}^{1}\psi=A_{[n]}A_{[n+1]}^{-1}E^{1}\psi,~{}~{}\accentset{\circ}{E}^{2}\psi=A_{[m]}A_{[m+1]}^{-1}E^{2}\psi,~{}~{}\accentset{\circ}{E}^{3}\psi=B_{[l]}B_{[l+1]}^{-1}E^{3}\psi.$
(A.5)
These Casoratian shift formulae are
$\displaystyle(a_{n-1}-c)^{N-1}\underaccent{\tilde}{f}_{{}_{[3]}}=|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[3]}},$
(A.6a)
$\displaystyle(a_{n-1}-c)^{N-2}\underaccent{\tilde}{f}_{{}_{[3]}}=-|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}},$
(A.6b)
$\displaystyle(b_{m}+c)^{N-1}\widehat{f}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}},$
(A.6c)
$\displaystyle(a_{n}+c)^{N-2}\widetilde{f}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{1}\psi(N-2)|_{{}_{[3]}},$
(A.6d)
$\displaystyle(b_{m-1}-c)^{N-2}\underaccent{\hat}{f}_{{}_{[3]}}=-|\widehat{N-2},~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}},$
(A.6e) $\displaystyle(b_{m-1}-c)^{N-1}\underaccent{\bar}{\dh
f}_{{}_{[3]}}=|-1,~{}\widehat{N-3},~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}},$
(A.6f)
$\displaystyle(b_{m}+c)^{N-2}\widehat{f}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}},~{}$
(A.6g)
$\displaystyle(b_{m}+c)^{N-2}\widehat{\overline{f}}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widetilde{N-1},~{}\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}};$
(A.6h)
$\displaystyle\prod_{j=0}^{N-2}(a_{n-1}-b_{m+j})\underaccent{\tilde}{f}_{{}_{[2]}}=|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}},$
(A.7a)
$\displaystyle\prod_{j=0}^{N-3}(a_{n-1}-b_{m+j})\underaccent{\tilde}{f}_{{}_{[2]}}=-|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[2]}},$
(A.7b)
$\displaystyle\prod_{j=0}^{N-3}(c-b_{m+j})\underaccent{\bar}{f}_{{}_{[2]}}=-|\widehat{N-2},~{}\underaccent{\bar}{\psi}(N-2)|_{{}_{[2]}},$
(A.7c)
$\displaystyle\prod_{i=0}^{N-2}(c-a_{n+i})\underaccent{\bar}{f}_{{}_{[1]}}=|\widehat{N-2},~{}\underaccent{\bar}{\psi}(N-1)|_{{}_{[1]}},$
(A.7d)
$\displaystyle\prod_{i=0}^{N-2}(c+a_{n+i})\overline{f}_{{}_{[1]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}},$
(A.7e)
$\displaystyle\prod_{j=0}^{N-3}(c+a_{n+i})\overline{f}_{{}_{[1]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[1]}},$
(A.7f)
$\displaystyle\prod_{j=0}^{N-2}(c+b_{m+j})\overline{f}_{{}_{[2]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}},$
(A.7g)
$\displaystyle\prod_{j=0}^{N-3}(c+b_{m+j})\overline{f}_{{}_{[2]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widehat{N-2},~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}},~{}$
(A.7h)
$\displaystyle\prod_{j=1}^{N-2}(c+b_{m+j}){\widehat{\overline{f}}}_{{}_{[2]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widetilde{N-1},~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}},~{}$
(A.7i)
$\displaystyle\prod_{i=0}^{N-2}(b_{m-1}-a_{n+i})\underaccent{\hat}{f}_{{}_{[1]}}=|\widehat{N-2},~{}\underaccent{\hat}{\psi}(N-1)|_{{}_{[1]}};$
(A.7j)
$\displaystyle\\!\\!(b_{m-1}+a_{n})(a_{n}+c)^{N-2}(b_{m-1}-c)^{N-2}\widetilde{\underaccent{\hat}{f}}_{{}_{[3]}}\\!\\!=\\!\\!|A_{[n+1]}A_{[n]}^{-1}\\!|\cdot|\widehat{N-3},\underaccent{\hat}{\psi}(N-2),\accentset{\circ}{E}^{1}\psi(N-2)|_{{}_{[3]}},$
(A.8a)
$\displaystyle(a_{n-1}+b_{m})(b_{m}+c)^{N-2}(a_{n-1}-c)^{N-2}\underaccent{\tilde}{\h
f}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widehat{N-3},\underaccent{\tilde}{\psi}(N-2),\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}},$
(A.8b)
$\displaystyle(a_{n-1}-b_{m-1})(a_{n-1}-c)^{N-2}(b_{m-1}-c)^{N-2}\underaccent{\hat}{\underaccent{\tilde}{f}}_{{}_{[3]}}=|\widehat{N-3},~{}\underaccent{\hat}{\psi}(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}},$
(A.8c)
$\displaystyle(a_{n}+c)^{N-1}\widetilde{\underaccent{\bar}{f}}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}|\cdot|-1,~{}\widehat{N-3},~{}\accentset{\circ}{E}^{1}\psi(N-2)|_{{}_{[3]}},$
(A.8d) $\displaystyle(a_{n-1}-c)^{N-2}\underaccent{\tilde}{\b
f}_{{}_{[3]}}=-|\widetilde{N-1},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[3]}},$
(A.8e)
$\displaystyle\\!(a_{n-1}+b_{m})\\!(b_{m}+c)^{N-2}\\!(a_{n-1}-c)^{N-2}\\!\underaccent{\tilde}{\hb
f}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}\\!|\cdot|\widetilde{N-2},\\!\underaccent{\tilde}{\psi}(N-1),\accentset{\circ}{E}^{2}\psi(N-1)|_{{}_{[3]}}\\!,$
(A.8f) $\displaystyle\prod_{j=1}^{N-2}(a_{n-1}-b_{m+j})\underaccent{\tilde}{\h
f}_{{}_{[2]}}=-|\widetilde{N-1},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}},$
(A.8g)
$\displaystyle(a_{n-1}-c)\prod_{j=0}^{N-3}(c-b_{m+j})\prod_{j=0}^{N-3}(a_{n-1}-b_{m+j})\underaccent{\bar}{\dt
f}_{{}_{[2]}}=|\widehat{N-3},~{}\underaccent{\bar}{\psi}(N-2),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[2]}},$
(A.8h) $\displaystyle
2c\prod_{j=0}^{N-3}(c+b_{m+j})\prod_{j=0}^{N-3}(c-b_{m+j})f_{{}_{[2]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widehat{N-3},~{}\underaccent{\bar}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}},$
(A.8i) $\displaystyle
2c\prod_{i=1}^{N-2}(c+a_{n+i})\prod_{i=1}^{N-2}(c-a_{n+i})\widetilde{f}_{{}_{[1]}}=|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widetilde{N-2},~{}\underaccent{\bar}{\psi}(N-1),~{}\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}},$
(A.8j)
$\displaystyle(b_{m-1}-c)\prod_{i=1}^{N-2}(b_{m-1}-a_{n+i})\prod_{i=1}^{N-2}(c-a_{n+i})\widetilde{\underaccent{\bar}{\underaccent{\hat}{f}}}_{{}_{[1]}}=-|\widetilde{N-2},~{}\underaccent{\hat}{\psi}(N-1),~{}\underaccent{\bar}{\psi}(N-1)|_{{}_{[1]}},$
(A.8k)
$\displaystyle\\!\\!(a_{n-1}+c)\\!\\!\prod_{j=0}^{N-3}(c+b_{m+j}\\!\\!)\prod_{j=0}^{N-3}(a_{n-1}-b_{m+j})\overline{\underaccent{\tilde}{f}}_{{}_{[2]}}\\!\\!=|B_{[l+1]}B_{[l]}^{-1}\\!\\!|\cdot|\widehat{N-3},\underaccent{\tilde}{\psi}(N-2),\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}},$
(A.8l)
$\displaystyle(a_{n-1}+c)\\!\\!\prod_{j=1}^{N-2}\\!\\!(c+b_{m+j})\\!\\!\prod_{j=1}^{N-2}\\!\\!(a_{n-1}-b_{m+j}){\widehat{\overline{\underaccent{\tilde}{f}}}}_{{}_{[2]}}\\!\\!=|B_{[l+1]}B_{[l]}^{-1}\\!|\cdot|\widetilde{N-2},\underaccent{\tilde}{\psi}(N-1),\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[2]}},$
(A.8m)
$\displaystyle(b_{m-1}+c)\\!\\!\prod_{i=1}^{N-2}\\!(c+a_{n+i})\\!\\!\prod_{i=1}^{N-2}(b_{m-1}-a_{n+i})\widetilde{\overline{\underaccent{\hat}{f}}}_{{}_{[1]}}\\!\\!=\\!|B_{[l+1]}B_{[l]}^{-1}|\cdot|\widetilde{N-2},\underaccent{\hat}{\psi}(N-1),\accentset{\circ}{E}^{3}\psi(N-1)|_{{}_{[1]}};$
(A.8n)
$\displaystyle(a_{n-1}-c)^{N-2}[\underaccent{\tilde}{g}_{{}_{[3]}}+(a_{n-1}-c)\underaccent{\tilde}{f}_{{}_{[3]}}]=-|\widehat{N-3},~{}\psi(N-1),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}},$
(A.9a)
$\displaystyle(b_{m}+c)^{N-2}[\widehat{g}_{{}_{[3]}}-(b_{m}+c)\widehat{f}_{{}_{[3]}}]=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widehat{N-3},\psi(N-1),\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}},$
(A.9b)
$\displaystyle(a_{n-1}-c)^{N-2}[\underaccent{\tilde}{h}_{{}_{[3]}}+(a_{n-1}-c)\underaccent{\tilde}{g}_{{}_{[3]}}]=-|\widehat{N-3},~{}\psi(N),~{}\underaccent{\tilde}{\psi}(N-2)|_{{}_{[3]}},$
(A.9c)
$\displaystyle(b_{m}+c)^{N-2}[\widehat{h}_{{}_{[3]}}-(b_{m}+c)\widehat{g}_{{}_{[3]}}]=|A_{[m+1]}A_{[m]}^{-1}|\cdot|\widehat{N-3},~{}\psi(N),~{}\accentset{\circ}{E}^{2}\psi(N-2)|_{{}_{[3]}},$
(A.9d)
$\displaystyle(b_{m-1}-c)^{N-2}[\underaccent{\hat}{g}_{{}_{[3]}}+(b_{m-1}-c)\underaccent{\hat}{f}_{{}_{[3]}}]=-|\widehat{N-3},~{}\psi(N-1),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}},$
(A.9e)
$\displaystyle(b_{m-1}-c)^{N-2}[(b_{m-1}-c)\underaccent{\hat}{g}_{{}_{[3]}}+\underaccent{\hat}{h}_{{}_{[3]}}]=-|\widehat{N-3},~{}\psi(N),~{}\underaccent{\hat}{\psi}(N-2)|_{{}_{[3]}};$
(A.9f)
$\displaystyle\underaccent{\hat}{f}_{{}_{[3]}}=\underaccent{\bar}{f}_{{}_{[3]}}-(b_{m-1}-c)|\underaccent{\hat}{\psi}(-1)~{}~{}\widehat{N-2}|_{{}_{[3]}},$
(A.10a) $\displaystyle\underaccent{\hat}{\b
f}_{{}_{[3]}}=f_{{}_{[3]}}-(b_{m-1}-c)g_{{}_{[3]}}+(b_{m-1}-c)^{2}|\underaccent{\hat}{\psi}(-1)~{}~{}\widetilde{N-1}|_{{}_{[3]}},~{}$
(A.10b)
$\displaystyle\underaccent{\hat}{g}_{{}_{[3]}}=|\underaccent{\hat}{\psi}(-1)~{}~{}\widehat{N-2}|_{{}_{[3]}}-(b_{m-1}-c)|\underaccent{\hat}{\psi}(-1),~{}\psi(-1),~{}\widetilde{N-2}|_{{}_{[3]}},$
(A.10c)
$\displaystyle\underaccent{\tilde}{f}_{{}_{[3]}}=\underaccent{\bar}{f}_{{}_{[3]}}-(a_{n-1}-c)|\underaccent{\tilde}{\psi}(-1),~{}\widehat{N-2}|_{{}_{[3]}},$
(A.10d) $\displaystyle\underaccent{\tilde}{\b
f}_{{}_{[3]}}=f_{{}_{[3]}}-(a_{n-1}-c)g+(a_{n-1}-c)^{2}|\underaccent{\tilde}{\psi}(-1),~{}\widetilde{N-1}|_{{}_{[3]}},$
(A.10e)
$\displaystyle\underaccent{\tilde}{g}_{{}_{[3]}}=|\underaccent{\tilde}{\psi}(-1),~{}\widehat{N-2}|_{{}_{[3]}}-(a_{n-1}-c)|\underaccent{\tilde}{\psi}(-1),~{}\psi(-1),~{}\widetilde{N-2}|_{{}_{[3]}},$
(A.10f)
$\displaystyle(-1)^{N}\widetilde{f}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}|[\underaccent{\bar}{f}_{{}_{[3]}}-(a_{n}+c)|\accentset{\circ}{E}^{1}\psi(-1),~{}\widehat{N-2}|_{{}_{[3]}}],$
(A.10g)
$\displaystyle(-1)^{N}\widetilde{\overline{f}}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}|[f_{{}_{[3]}}+(a_{n}+c)g-(a_{n}+c)^{2}|\accentset{\circ}{E}^{1}\psi(-1),~{}\widetilde{N-1}|_{{}_{[3]}}],$
(A.10h)
$\displaystyle(-1)^{N}\widetilde{g}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}|[\\!-|\accentset{\circ}{E}^{1}\psi(-1),~{}\widehat{N-2}|_{{}_{[3]}}-(a_{n}+c)\\!\\!|\accentset{\circ}{E}^{1}\psi(-1),~{}\psi(-1),~{}\widetilde{N-2}|_{{}_{[3]}}],$
(A.10i)
$\displaystyle(-1)^{N}\widehat{f}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot[\underaccent{\bar}{f}_{{}_{[3]}}-(b_{m}+c)|\accentset{\circ}{F}^{2}\psi(-1),~{}\widehat{N-2}|_{{}_{[3]}}],$
(A.10j)
$\displaystyle(-1)^{N}\widehat{\overline{f}}_{{}_{[3]}}=|A_{[m+1]}A_{[m]}^{-1}|\cdot[f_{{}_{[3]}}+(b_{m}+c)g-(b_{m}+c)^{2}|\accentset{\circ}{E}^{2}\psi(-1),~{}\widetilde{N-1}|_{{}_{[3]}}],$
(A.10k)
$\displaystyle\\!\\!(-1)^{N}\widehat{g}_{{}_{[3]}}=\\!|A_{[m+1]}A_{[m]}^{-1}|\cdot[-|\accentset{\circ}{E}^{2}\psi(-1),\widehat{N-2}|_{{}_{[3]}}-\\!(b_{m}+c)|\accentset{\circ}{E}^{2}\psi(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}],$
(A.10l)
$\displaystyle\\!\\!(a_{n-1}\\!-b_{m-1})\\!\underaccent{\hat}{\underaccent{\tilde}{\b
f}}_{{}_{[3]}}\\!=\\!\\!(a_{n-1}-b_{m-1})\underaccent{\bar}{f}_{{}_{[3]}}\\!-\\!\\!(a_{n-1}\\!-c)^{2}|\underaccent{\tilde}{\psi}(-1),\widehat{N-2}|_{{}_{[3]}}\\!\\!+\\!\\!(b_{m-1}\\!-c)^{2}|\underaccent{\hat}{\psi}(-1),\widehat{N-2}|_{{}_{[3]}}\\!\\!$
$\displaystyle+\\!\\!(a_{n-1}\\!-c)^{2}(b_{m-1}\\!-c)\\!|\underaccent{\tilde}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}\\!\\!-(a_{n-1}\\!-c)(b_{m-1\\!}-c)^{2}|\underaccent{\hat}{\psi}(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}\\!\\!$
$\displaystyle+(a_{n-1}-c)^{2}(b_{m-1}-c)^{2}|\underaccent{\hat}{\psi}(-1),\underaccent{\tilde}{\psi}(-1),\widetilde{N-2}|_{{}_{[3]}},$
(A.10m)
$\displaystyle(a_{n}-b_{m})\widehat{\widetilde{\overline{f}}}_{{}_{[3]}}=|A_{[n+1]}A_{[n]}^{-1}||A_{[m+1]}A_{[m]}^{-1}|[(a_{n}-b_{m})\underaccent{\bar}{f}_{{}_{[3]}}-(a_{n}+c)^{2}|\accentset{\circ}{E}^{1}\psi(-1),\widehat{N-2}|_{{}_{[3]}}$
$\displaystyle~{}~{}+(b_{m}+c)^{2}|\accentset{\circ}{E}^{2}\psi(-1),\widehat{N-2}|_{{}_{[3]}}-(a_{n}+c)^{2}(b_{m}+c)|\accentset{\circ}{E}^{1}\psi(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}$
$\displaystyle~{}~{}+(a_{n}\\!+c)(b_{m}\\!+c)^{2}|\accentset{\circ}{E}^{2}\psi(-1),\psi(-1),\widetilde{N-2}|_{{}_{[3]}}$
$\displaystyle~{}~{}-(a_{n}\\!+c)^{2}(b_{m}+c)^{2}|\accentset{\circ}{E}^{2}\psi(-1),\accentset{\circ}{E}^{1}\psi(-1),\widetilde{N-2}|_{{}_{[3]}}].$
(A.10n)
In fact, these formulae can be proved in a similar way as in [13, 18, 17]. We
need to use the following relations which are derived from (A.2):
$\displaystyle(a_{n-1}-b_{m})\underaccent{\tilde}{\psi}=\psi-\underaccent{\tilde}{\h\psi},~{}~{}(b_{m-1}-a_{n})\underaccent{\hat}{\psi}=\psi-\underaccent{\hat}{\t\psi},$
(A.11a)
$\displaystyle(a_{n}+c)\widetilde{\psi}=A_{[n+1]}A_{[n]}^{-1}\psi+\widetilde{\overline{\psi}},~{}~{}(b_{m-1}+a_{n})\underaccent{\hat}{\t\psi}=\widetilde{\psi}+A_{[n+1]}A_{[n]}^{-1}\underaccent{\hat}{\psi},$
(A.11b)
$\displaystyle(c+b_{m})\overline{\psi}=B_{[l+1]}B_{[l]}^{-1}\psi+\widehat{\overline{\psi}},~{}(a_{n-1}+c)\overline{\underaccent{\tilde}{\psi}}=\overline{\psi}+B_{[l+1]}B_{[l]}^{-1}\underaccent{\tilde}{\psi},~{}2c\psi=\overline{\psi}+B_{[l]}B_{[l-1]}^{-1}\underaccent{\bar}{\psi},$
(A.11c)
$\displaystyle(b_{m}+c)\widehat{\psi}=A_{[m+1]}A_{[m]}^{-1}\psi+\widehat{\overline{\psi}},~{}~{}(a_{n-1}+b_{m})\widehat{\underaccent{\tilde}{\psi}}=\widehat{\psi}+A_{[m+1]}A_{[m]}^{-1}\underaccent{\tilde}{\psi}.$
(A.11d)
In the following as examples we only prove (A.7a) and (A.8l). Let us prove
(A.7a). For $\underaccent{\tilde}{f}_{{}_{[2]}}$, using the relation (A.11a)
we first have
$\displaystyle(a_{n-1}-b_{m})\underaccent{\tilde}{f}_{{}_{[2]}}$
$\displaystyle=|(a_{n-1}-b_{m})\underaccent{\tilde}{\psi}(0),~{}\underaccent{\tilde}{\psi}(1),~{}\cdots,~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}$
$\displaystyle=|\psi(0),~{}\underaccent{\tilde}{\psi}(1),~{}\cdots,~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}.$
Then, for the second column,
$\displaystyle(a_{n-1}-b_{m+1})(a_{n-1}-b_{m})\underaccent{\tilde}{f}_{{}_{[2]}}$
$\displaystyle=|\psi(0),~{}(a_{n-1}-b_{m+1})\underaccent{\tilde}{\psi}(1),~{}\cdots,~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}$
$\displaystyle=|\psi(0),~{}\psi(1),~{}\cdots,~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}}.$
Repeating this procedure we reach to
$\prod_{j=0}^{N-2}(a_{n-1}-b_{m+j})\underaccent{\tilde}{f}_{{}_{[2]}}=|\widehat{N-2},~{}\underaccent{\tilde}{\psi}(N-1)|_{{}_{[2]}},$
i.e., (A.7a). Next, let us prove (A.8i). Based on $\overline{\eqref{1-6}}$ and
using (A.11c), we first have
$\displaystyle(c+b_{m})\prod_{j=0}^{N-3}(c-b_{m+j})f_{{}_{[2]}}$
$\displaystyle=-|(c+b_{m})\overline{\psi}(0),~{}\overline{\psi}(1),~{}\cdots,~{}\overline{\psi}(N-2),~{}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle=-|B_{[l+1]}B_{[l]}^{-1}\psi(0),~{}\overline{\psi}(1),~{}\cdots,~{}\overline{\psi}(N-2),~{}\psi(N-2)|_{{}_{[2]}}.$
For the second column, we have
$\displaystyle(c+b_{m+1})(c+b_{m})\prod_{j=0}^{N-3}(c-b_{m+j})f_{{}_{[2]}}$
$\displaystyle=-|B_{[l+1]}B_{[l]}^{-1}\psi(0)$
$\displaystyle,~{}(c+b_{m+1})\overline{\psi}(1),~{}\cdots,\overline{\psi}(N-2),~{}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle=-|B_{[l+1]}B_{[l]}^{-1}\psi(0)$
$\displaystyle,~{}B_{[l+1]}B_{[l]}^{-1}\psi(1),~{}\cdots,~{}\overline{\psi}(N-2),~{}\psi(N-2)|_{{}_{[2]}}.$
Repeating this procedure, we reach to
$\displaystyle\prod_{j=0}^{N-3}(c+b_{m+j})\prod_{j=0}^{N-3}(c-b_{m+j})f_{{}_{[2]}}$
$\displaystyle~{}~{}~{}=-|B_{[l+1]}B_{[l]}^{-1}||\widehat{N-3},B_{[l]}B_{[l+1]}^{-1}\overline{\psi}(N-2),B_{[l]}B_{[l+1]}^{-1}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle~{}~{}~{}=|B_{[l+1]}B_{[l]}^{-1}||\widehat{N-3},B_{[l]}B_{[l+1]}^{-1}\psi(N-2),\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}},$
where we have rewritten $B_{[l]}B_{[l+1]}^{-1}\overline{\psi}(N-2)$ by
$\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}$. Then, still using (A.11c) and
the fact that $B_{[l+1]}B_{[l]}^{-1}$ is independent of $l$, we have
$\displaystyle
2c\prod_{j=0}^{N-3}(c+b_{m+j})\prod_{j=0}^{N-3}(c-b_{m+j})\underaccent{\tilde}{\b
f}_{{}_{[2]}}$
$\displaystyle~{}~{}~{}=|B_{[l+1]}B_{[l]}^{-1}||\widehat{N-3},B_{[l]}B_{[l+1]}^{-1}2c\,\psi(N-2),\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle~{}~{}~{}=|B_{[l+1]}B_{[l]}^{-1}||\widehat{N-3},B_{[l]}B_{[l+1]}^{-1}B_{[l]}B_{[l-1]}^{-1}\underaccent{\bar}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}}$
$\displaystyle~{}~{}~{}=|B_{[l+1]}B_{[l]}^{-1}||\widehat{N-3},\underaccent{\bar}{\psi}(N-2),~{}\accentset{\circ}{E}^{3}\psi(N-2)|_{{}_{[2]}},$
which is (A.8i).
## References
* [1] Nijhoff F W and Walker A J 2001 The discrete and continuous Painlevé vi@ hierarchy and the Garnier systems Glasgow Math. J. 43A 109-23.
* [2] Bobenko A I and Suris Y B 2002 Integrable systems on quad-graphs Int. Math. Res. Notices 11 573-611.
* [3] Adler V E, Bobenko A I and Suris Y B 2003 Classification of integrable equations on quad-graphs. The consistency approach Commun. Math. Phys. 233 513-43.
* [4] Grammaticos B, Ramani A and Papageorgiou V 1991 Do integrable mappings have the Painlevé property? Phys. Rev. Lett. 67 1825-28.
* [5] Papageorgiou V, Grammaticos B and Ramani A 1993 Integrable lattices and convergence acceleration algorithms Phys. Lett. A 179 111-15.
* [6] Sahadevan R, Rasinb O G and Hydon P E 2007 Integrability conditions for nonautonomous quad-graph equations J. Math. Anal. Appl. 331 712-26.
* [7] Grammaticos B and Ramani A 2010 Singularity confinement property for the (Non-Autonomous) Adler-Bobenko-Suris integrable lattice equations Lett. Math. Phys. 92 33-45.
* [8] Kajiwara K and Ohta Y 2008 Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation J. Phys. Soc. Jpn. 77 054004.
* [9] Kajiwara K and Ohta Y 2009 Bilinearization and Casorati determinant solutions to non-autonomous 1+1 dimensional discrete soliton equations MI Preprint Series 2009-6 1-16.
* [10] Atkinson J, Hietarinta J and Nijhoff F W 2007 Seed and soliton solutions for Adler’s lattice equation J. Phys. A: Math. Theor. 40 F1-F8.
* [11] Atkinson J, Hietarinta J and Nijhoff F W 2008 Soliton solutions for Q3 J. Phys. A: Math. Theor. 41 142001 (11pp).
* [12] Nijhoff F W, Atkinson J and Hietarinta J 2009 Soliton solutions for ABS lattice equations: I Cauchy matrix approach, J. Phys. A: Math. Theor. 42 No.404005(34pp).
* [13] Hietarinta J and Zhang D J Soliton solutions for ABS lattice equations: II Casoratians and bilinearization J. Phys. A: Math. Theor. 42 No.404006(30pp).
* [14] Atkinson J and Nijhoff F W A constructive approach to the soliton solutions of integrable quadrilateral lattice equations, Commun. Math. Phys. 299 283-304.
* [15] Nijhoff F W and Atkinson J 2010 Elliptic _N_ -soliton solutions of ABS lattice equations Int. Math. Res. Notices 20 3837-3895.
* [16] Butler S and Joshi N 2010 An inverse scattering transform for the lattice potential KdV equation Inverse Problems 26 115012 (28pp).
* [17] Zhang D J and Hietarinta J Generalized solutions for the H1 model in ABS list of lattice equations in Nonlinear and Modern Mathematical Physics (July 15-21, 2009, Beijing) Editors Ma W X, Hu X B and Liu Q P AIP Conf. Proc. Vol. 1212 Amer. Inst. Phys. Melville NY 2010 154-161.
* [18] Shi Y and Zhang D J 2011 Rational solutions of the H3 and Q1 models in the ABS lattice list SIGMA 7 046 11pages.
* [19] Hietarinta J 2005 Searching for CAC-maps J. Nonl. Math. Phys. 12 223-230.
* [20] Freeman N C and Nimmo J J C 1983 Soliton solutions of the KdV and KP equations: the Wronskian technique Phys. Lett. A 95 1-3.
* [21] Maruno K, Kajiwara K, Nakao S and Oikawa M 1997 Bilinearization of discrete soliton equations and singularity confinement Phys. Lett. A 229 173-82.
* [22] Kajiwara K, Maruno K and Oikawa M 2000 Bilinearization of discrete soliton equations through the singularity confinement test Chaos, Solitons and Fractals 11 33-39.
* [23] Zhang D J, Notes on solutions in Wronskian form to soliton equations: KdV-type, nlin.SI/0603008.
|
arxiv-papers
| 2012-01-31T09:07:30 |
2024-09-04T02:49:26.856565
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ying Shi, Da-jun Zhang and Song-lin Zhao",
"submitter": "Shi Ying",
"url": "https://arxiv.org/abs/1201.6478"
}
|
1201.6554
|
# Distinct Quantum States Can Be Compatible with a Single State of
Reality111This article was published by the American Physical Society under
the terms of the Creative Commons Attribution 3.0 License. Further
distribution of this work must maintain attribution to the authors and the
published article’s title, journal citation (PRL 109, 150404 (2012)), and DOI
(10.1103/PhysRevLett.109.150404).
Peter G. Lewis 1@physics.org Controlled Quantum Dynamics Theory Group,
Imperial College London, London SW7 2AZ, United Kingdom David Jennings
Controlled Quantum Dynamics Theory Group, Imperial College London, London SW7
2AZ, United Kingdom Jonathan Barrett Department of Mathematics, Royal
Holloway, University of London, Egham Hill, Egham TW20 0EX, United Kingdom
Terry Rudolph Controlled Quantum Dynamics Theory Group, Imperial College
London, London SW7 2AZ, United Kingdom
###### Abstract
Perhaps the quantum state represents information about reality, and not
reality directly. Wave function collapse is then possibly no more mysterious
than a Bayesian update of a probability distribution given new data. We
consider models for quantum systems with measurement outcomes determined by an
underlying physical state of the system but where several quantum states are
consistent with a single underlying state—i.e., probability distributions for
distinct quantum states overlap. Significantly, we demonstrate by example that
additional assumptions are always necessary to rule out such a model.
Broadly speaking, physicists today view the quantum state in one of two ways:
in correspondence with the real physical state of affairs or as representing
only an agent’s knowledge or information about some aspect of the physical
situation. The latter ‘epistemic’ viewpoint is primarily motivated by the
obvious parallel between the quantum process of instantaneous wavefunction
collapse, and the classical procedure of instantaneous updating of a
probability distribution, both of which occur upon the acquisition of
information regarding the outcome of a measurement process. The epistemic view
has a long history of illustrious advocates Einstein et al. (1935); Popper
(1967); BALLENTINE (1970); Peierls (1979); Jaynes (1980); Zeilinger A. (1999);
Caves et al. (2002); Spekkens (2007). In recent times a research programme has
arisen aimed at not only philosophically justifying the epistemic view, but
potentially deriving quantum theory from more primitive considerations about
information and/or Bayesian reasoning Brukner and Zeilinger (2001); Fuchs
(2010); Spekkens (2007); Leifer (2006); Leifer and Spekkens (2011, 2011);
Barnum and Wilce (2009); Hardy (2001); Barrett (2007).
At least two versions of the epistemic view can be distinguished. One is
operational: the quantum state represents information about which outcome will
occur if a measurement is performed on the system. Measurement itself is
treated as a primitive. The other is that the quantum state represents
information about some underlying physical state of the system, where this
underlying state need not be described by quantum theory.
In his 1935 letter to Schrödinger, in an attempt to explain what he really
meant in the EPR paper, Einstein writes (Einstein, ; Howard, 2006, p. 26):
> But then for the same [real] state of [the system] there are two (in general
> arbitrarily many) equally justified $\Psi$, which contradicts the hypothesis
> of a one-to-one or complete description of the real states.
Einstein came to this conclusion, which he termed ‘incompleteness’ of the
quantum state, using an argument based on what we now call (following
Schrödinger Schrödinger (1935)) _steering_ Harrigan and Spekkens (2010). Note
that the question he raises here is _not_ whether there are multiple states of
reality associated with a single wavefunction (one possible type of
incompleteness), but rather whether there are multiple wavefunctions
associated with a single real state. A natural way to understand this is as an
expression of the second kind of epistemic view above—that a quantum state
represents an agent’s information about an underlying reality, but is not part
of that reality itself.
Unfortunately, as shown later by Bell Bell (1966) Einstein’s specific argument
for incompleteness was based on a false premise (locality). However, a
question that stands on its own and is the topic of this article remains: is
it even mathematically possible to find an embedding of quantum theory in some
deeper theory where the quantum states are not always uniquely determined
given the underlying physical state? Following Harrigan and Spekkens (2010),
we refer to such a possibility as a $\psi$-_epistemic_ interpretation of the
quantum state.
Recently, a no-go theorem was proven Pusey _et al._ (2012) showing that a
$\psi$-epistemic interpretation is impossible. It is important to note that a
key assumption of the argument is _preparation independence_ —situations where
quantum theory assigns independent product states are presumed to be
completely describable by independently combining the two purportedly deeper
descriptions for each system. Here, we will show via explicit constructions
that without this assumption, $\psi$-epistemic models can be constructed with
all quantum predictions retained. Hence, we show that not only is the
‘preparation independence’ assumption of that particular no-go theorem
necessary, but also any similar no-go theorem will require non-trivial
assumptions beyond those required for a well-formed ontological model.
One of the most compelling motivations for exploring the $\psi$-epistemic view
is the amazing range of phenomena, normally considered uniquely quantum, that
can be derived by imposing only a simple principle restricting an agent’s
knowledge about a presumed underlying reality Spekkens (2007); Hardy (1999);
Bartlett _et al._ (2011). It is clear that the primary reason these theories
do manage to reproduce so many quantum-like phenomena is that the states of
knowledge (probability distributions) overlap on a non-trivial subset of the
underlying space of ‘hidden’ states, even when the agent’s knowledge is
maximal (the equivalent of a quantum pure state).
These toy theories do not, however, reproduce _all_ quantum states and
measurement statistics. About such theories which do reproduce all of quantum
theory much less is known—the main examples and constraints are outlined in
Harrigan and Rudolph (2007). In Hardy (2004) Hardy showed that the underlying
mathematical space of real states on which such theories are defined—the
‘ontic state’ space—must have cardinality at least that of the integers.
Spekkens Spekkens (2005) showed the models must be _preparation contextual_ in
addition to measurement contextual Kochen and Specker (1967). The manifold
dimension of the ontic state space was shown by Montina Montina (2008) to be
necessarily exponential assuming that the dynamics of the ontic states is
Markovian; he then went on to show the intriguing result that it is possible
to reduce the manifold dimension by one Montina (2011, 2011). This latter
construction results in probability distributions that intersect in the ontic
state space, however they do so only on a set of measure 0. The question of
economic representation of finite data in such models Hardy (2001); Galvão
(2009); Dakić _et al._ (2008); Wehner _et al._ (2008) has also received some
attention.
In another pair of recent works Colbeck and Renner (2011a, b), Colbeck and
Renner have argued that, given additional assumptions of experimenters’ free
choice in choosing measurement settings and no superluminal signaling of the
choice at the ontic level, the ontic states must be in one-to-one
correspondence with the quantum states. The models of the present Letter are
explicitly constructed for single systems only and, if applied to systems
composed of multiple separated subsystems, would involve superluminal
influences of measurement choices upon ontic variables. Hence, our results are
consistent with those of Colbeck and Renner (2011a, b).
Formal statement of the problem.–Following Harrigan and Spekkens (2010), an
_ontological model_ for a quantum system defines a measure space $\Lambda$,
the elements $\lambda$ of which are termed _ontic states_. These should be
thought of as the underlying physical states that a system can be in at a
given time. A pure quantum state $|\psi\rangle$ corresponds to an equivalence
class of experimental preparations, and is represented within an ontological
model as a probability distribution $\mu_{\psi}(\lambda)$ over $\Lambda$. The
distribution $\mu_{\psi}$ is called an epistemic state. It represents the
information that an agent has about the ontic state of a system, given that it
was prepared in a particular way. Dynamics correspond to trajectories of ontic
states through $\Lambda$, and these map to dynamical changes in the
probability distributions over $\Lambda$. This setting includes the standard
Hilbert space description through the trivial assignment of $\Lambda$ as the
complex projective space $CP^{d-1}$ (the boundary of the quantum state space),
and $\mu_{\psi}(\lambda)$ being a delta-function distribution centred at
$\lambda=|\psi\rangle\langle\psi|$ Beltrametti and Bugajski (1995). It is for
this reason the term ‘ontic’ is used—as opposed to ‘hidden’, for example.
Throughout, we shall consider only projective measurements on a finite
$d$-dimensional quantum system, described as a string of rank-1 projectors
$\Phi=\\{|\phi_{0}\rangle\langle\phi_{0}|,\cdots,|\phi_{d-1}\rangle\langle\phi_{d-1}|\\}$,
where $\sum_{k}|\phi_{k}\rangle\langle\phi_{k}|=\mathbbm{1}$. If a measurement
is performed, the probabilities for different outcomes are defined by the
ontic state $\lambda$ of the system. Hence each measurement is associated with
a set of $d$ _response functions_
$\\{\xi_{\phi_{k}}:\Lambda\rightarrow[0,1]\\}$, where
$\xi_{\phi_{k}}(\lambda)$ is the probability of obtaining the outcome
$|\phi_{k}\rangle\langle\phi_{k}|$ when the ontic state of the system is
$\lambda$. The response functions are positive semi-definite and normalized so
that $\sum_{k}\xi_{\phi_{k}}(\lambda)=1,\forall\lambda$. If the
$\xi_{\phi_{k}}(\lambda)$ take values only in $\\{0,1\\}$ the model is
_deterministic_. By the Kochen-Specker theorem Kochen and Specker (1967), a
deterministic ontological model for a quantum system must be measurement
contextual Harrigan and Rudolph (2007) if the system has dimension $d\geq 3$.
This means that the response function $\xi_{\phi_{k}}$ depends on the complete
set of co-measured projectors in $\Phi$. We do not indicate this dependence
for notational simplicity.
An ontological model is successful in explaining quantum measurement
statistics for measurement $\Phi$ and preparation $|\psi\rangle$ if and only
if it is the case that $\forall k$:
$\displaystyle\int_{\Lambda}\xi_{\phi_{k}}(\lambda)\mathrm{d}\mu_{\psi}$
$\displaystyle=\int_{\Lambda}\mu_{\psi}(\lambda)\xi_{\phi_{k}}(\lambda)\mathrm{d}\lambda=|\langle\phi_{k}|\psi\rangle|^{2}.$
(1)
$\Lambda$$\Lambda$$\mbox{Supp}\,(\mu_{\psi})$$\mbox{Supp}\,(\mu_{\phi})$$\mathcal{E}$
Figure 1: Schematic of two ontic state spaces in different ontological models.
The supports of epistemic states associated with four quantum states are
shown. In (a) each ontic state is in the support of the epistemic state for at
most one $\psi$: the model is $\psi$-ontic. In (b) those ontic states in the
highlighted ‘epistemic region’ $\mathcal{E}$ on the right do not uniquely
identify a quantum state, and could result from either of the associated
preparation procedures: the model is $\psi$-epistemic.
With these basic ingredients in place, an ontological model is
$\psi$-epistemic if at least two distinct quantum states $|\psi_{1}\rangle$
and $|\psi_{2}\rangle$ are described by distributions $\mu_{\psi_{1}}$ and
$\mu_{\psi_{2}}$ such that the intersection of their supports has non-zero
measure, shown schematically in Fig. 1. An ontological model is _$\psi$
-ontic_ otherwise Harrigan and Spekkens (2010); Spekkens (2005). The idea here
is that in a $\psi$-ontic model, the quantum state is uniquely determined by
the ontic state $\lambda$, since for any $\lambda$, there is only one
$|\psi\rangle$ such that $\lambda$ is contained in the support of
$\mu_{\psi}$. In this case, although the quantum state $|\psi\rangle$ plays an
epistemic role in defining a distribution $\mu_{\psi}$, the quantum state is
also a function of the ontic state, hence can justifiably be thought of as a
physical property of the system. Informally, the whole of the quantum state
$|\psi\rangle$ is ‘written into’ the real state of affairs.
In a $\psi$-epistemic model, on the other hand, there are at least some
circumstances in which two different quantum states describe systems in the
same ontic state $\lambda$. In this case, it is defensible to claim that the
quantum state ‘merely’ represents an agent’s information. A stronger
definition of the term $\psi$-epistemic would require that for _any_ non-
orthogonal states, $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$, the
distributions $\mu_{\psi_{1}}$ and $\mu_{\psi_{2}}$ overlap. In this Letter,
we stick with the weaker definition above.
_The original Bell model.–_ As a counterexample to ‘von-Neumann’s silly
assumption’ Bell Bell (1966) described a simple ontological model capable of
describing a quantum system of dimension $d$. The ontic state space consists
of pairs $(|\lambda\rangle,x)$, where $|\lambda\rangle$ is an element of
$CP^{d-1}$ and $x$ is an element of the interval $[0,1]$. For the moment
consider $d=2$. The ontic state space is then isomorphic to
$\Lambda=S^{2}\times[0,1]$, where $S^{2}$ is the two-dimensional Bloch sphere.
The quantum state $|\psi\rangle$ corresponds to a distribution
$\mu_{\psi}(\hat{\lambda},x)=\delta(\hat{\lambda}-\hat{\psi})$, where the
convention here, and in what follows, is to denote the unit Bloch vector
associated to $|\psi\rangle$ as $\hat{\psi}$. This choice of distribution is
uniform over the subset $\\{(\hat{\psi},x):0\leq x\leq 1\\}$. The model is
deterministic, with response functions given by
$\displaystyle\xi_{\phi_{k}}(\hat{\lambda},x)$ $\displaystyle=$
$\displaystyle\Theta\left[(|\langle\lambda|\phi_{0}\rangle|^{2}-x)(-1)^{k}\right],$
(2)
where $\Theta$ is the Heaviside step function. For a projective measurement
outcome $|\phi\rangle\langle\phi|$ and a quantum state $|\psi\rangle$, the
Born rule is satisfied since
$\displaystyle\int\mathrm{d}\hat{\lambda}\mathrm{d}x\,\,\mu_{\psi}(\hat{\lambda},x)\xi_{\phi}(\hat{\lambda},x)=|\langle\psi|\phi\rangle|^{2}.$
(3)
Note—as Bell did—that there are many possible choices for the response
functions that would work equally well. The only requirement is that the
support over the subset $\\{(\hat{\lambda},x):0\leq x\leq 1\\}$ has measure
equal to the probability occurring in the Born rule; how this support is
distributed is entirely arbitrary.
_A $\psi$-epistemic modification of the Bell model for a qubit.–_The Bell
model is $\psi$-ontic, since no two epistemic states, corresponding to
distinct quantum states, overlap. A different ontological model for qubit
systems was described by Kochen and Specker Kochen and Specker (1967), which
is $\psi$-epistemic. To date, however, no one has extended Kochen and
Specker’s model to systems of higher dimension than two Rudolph (2006). This
section shows how the Bell model can be modified in order to obtain a
$\psi$-epistemic model for qubits. Later, this model is extended to obtain a
$\psi$-epistemic model for quantum systems of arbitrary finite dimension.
A simple visualization of the ontic state space $\Lambda$ as an annulus with
$\hat{\lambda}$ specifying the direction, and $x$ the radial distance is
depicted in Fig. 2(a). Let $\hat{z}$ correspond to the north pole of the Bloch
sphere, and let $\hat{\lambda}\cdot\hat{z}=\cos(\theta_{\lambda})$, where
$\theta_{\lambda}$ is the polar angle of the Bloch vector $\hat{\lambda}$.
Label the upper ($\theta_{\lambda}<\pi/2$) hemisphere ${\cal R}_{0}$ and the
lower ($\theta_{\lambda}>\pi/2$) hemisphere ${\cal R}_{1}$. Given a projective
measurement
$\Phi=\\{|\phi_{0}\rangle\langle\phi_{0}|,|\phi_{1}\rangle\langle\phi_{1}|\\}$,
assume that the outcomes are labelled such that
$|\langle\phi_{0}|z\rangle|^{2}\geq|\langle\phi_{1}|z\rangle|^{2}$.
$\hat{\phi}_{0}$$\hat{\psi}$$\mbox{Supp}\,(\xi_{\phi_{0}})$$\mbox{Supp}\,(\xi_{\phi_{1}})$$\hat{\phi}_{1}$
(a)
$\hat{\psi}$${\cal E}_{0}$${\cal E}_{1}$${\cal R}_{0}$${\cal R}_{1}$ (b)
Figure 2: (a) Unmodified Bell model ontic state space along with supports of a
response function for outcome $\hat{\phi_{0}}$ (green), $\hat{\phi_{1}}$
(blue) and epistemic state $\mu_{\psi}$ (red line). The inner circle is the
surface of the Bloch sphere, the dashed line indicates it also corresponds to
$x=0$. The outer solid circle corresponds to $x=1$. (b) Modified Bell model,
showing the subsets ${\cal E}_{0}$ and ${\cal E}_{1}$. Any probability weight
that $\mu_{\psi}$ gives to ontic states within ${\cal E}_{0}$ can be
redistributed over ${\cal E}_{0}$.
With response functions defined as in Eq. (2), the ontic states in the set
${\cal E}_{0}=\left\\{(\hat{\lambda},x):\hat{\lambda}\in{\cal R}_{0}\mbox{ and
}0\leq x<(1-\sin\theta_{\lambda})/2\right\\}$ (4)
all result in the $\hat{\phi}_{0}$ outcome for any measurement $\Phi$.
Similarly, those in the set
${\cal E}_{1}=\left\\{(\hat{\lambda},x):\hat{\lambda}\in{\cal R}_{1}\mbox{ and
}(1+\sin\theta_{\lambda})/2<x\leq 1\right\\}$ (5)
all result in the $\hat{\phi}_{1}$ outcome. These sets are illustrated in Fig.
2(b).
In order to construct a $\psi$-epistemic model, note that if an epistemic
state $\mu_{\psi}$ assigns non-zero probability to the subset ${\cal E}_{0}$,
then this much probability weight can be redistributed over ${\cal E}_{0}$
without changing the Born rule statistics. This is because ontic states in
${\cal E}_{0}$ behave identically to one another, as far as predictions for
quantum measurement outcomes go. Similarly ${\cal E}_{1}$. Hence define a
modified Bell model such that for $\hat{\psi}\in{\cal R}_{0}$,
$\mu_{\psi}(\hat{\lambda},x)=\delta(\hat{\lambda}-\hat{\psi})\Theta\left(x-\frac{1}{2}(1-\sin\theta_{\psi})\right)\\\
+\frac{1}{2}(1-\sin\theta_{\psi})\mu_{{\cal E}_{0}}(\hat{\lambda},x)$ (6)
where $\theta_{\psi}$ is the polar angle of $\hat{\psi}$, and $\mu_{{\cal
E}_{0}}$ is essentially arbitrary but can be taken to be the uniform
distribution over ${\cal E}_{0}$. A similar expression defines $\mu_{\psi}$
for states with $\hat{\psi}\in{\cal R}_{1}$.
The modified Bell model still reproduces the Born rule, but now the model is
$\psi$-epistemic. Any two quantum states in the same hemisphere are described
by distributions that overlap, either in ${\cal E}_{0}$ or in ${\cal E}_{1}$.
The preparation of an ontic state from either of these regions does not reveal
a unique quantum state, but only reveals in which hemisphere the quantum state
resides.
_$\psi$ -epistemic modification in higher dimensions.–_Here, we modify the
Bell model to produce a $\psi$-epistemic model for arbitrary finite dimension.
The ontic state space for the $d$-dimensional Bell model is
$\Lambda=CP^{d-1}\times[0,1]$. The distribution corresponding to a quantum
state $|\psi\rangle$ is given by
$\mu_{\psi}(|\lambda\rangle,x)=\delta(|\lambda\rangle-|\psi\rangle)$. Response
functions for a measurement $\Phi$ are defined such that $\xi_{\phi_{k}}$ has
support of length $|\langle\phi_{k}|\lambda\rangle|^{2}$ on the line segment
$\\{(|\lambda\rangle,x):0\leq x\leq 1\\}$. Up to this constraint the response
functions are arbitrary. This model is $\psi$-ontic since the delta functions
do not overlap for distinct $|\psi\rangle$.
In order to construct a $\psi$-epistemic model, fix an arbitrary preferred
state $|0\rangle$. For each measurement $\Phi$, assume that the outcomes are
ordered such that
$|\langle\phi_{0}|0\rangle|^{2}\geq|\langle\phi_{1}|0\rangle|^{2}\geq\cdots\geq|\langle\phi_{d-1}|0\rangle|^{2}$.
Fix the response functions so that
$\xi_{\phi_{k}}(|\lambda\rangle,x)=1\quad\mathrm{if}\quad\sum_{i=0}^{k-1}|\langle\lambda|\phi_{i}\rangle|^{2}\leq
x<\sum_{i=0}^{k}|\langle\lambda|\phi_{i}\rangle|^{2}$ (7)
and
$\xi_{\phi_{k}}(|\lambda\rangle,x)=0\quad\mathrm{otherwise}.$ (8)
In Equation (7), $\sum_{i=1}^{k-1}|\langle\lambda|\phi_{i}\rangle|^{2}$ is
taken to be $0$ when $k=0$. For completeness, let us specify that for $x=1$,
$\xi_{\phi_{k}}(|\lambda\rangle,x)=1$ iff $k=d-1$.
Now the aim is to define a subset ${\cal E}_{0}$ of $\Lambda$, such that ontic
states in ${\cal E}_{0}$ predict the same outcomes for all measurements. To
this end, note that $|\langle\phi_{0}|0\rangle|^{2}\geq 1/d$. Given this, it
is easy to show that if $|\langle\lambda|0\rangle|^{2}>\frac{d-1}{d}$, then
$|\langle\phi_{0}|\lambda\rangle|^{2}>0$. For arbitrary $|\chi\rangle$, let
$z(|\chi\rangle)=\inf_{|\phi\rangle:|\langle\phi|0\rangle|^{2}\geq
1/d}|\langle\phi|\chi\rangle|^{2},$ (9)
where an explicit expression is easily found but not needed. Define
${\cal
E}_{0}=\left\\{(|\lambda\rangle,x):|\langle\lambda|0\rangle|^{2}>\frac{d-1}{d}\mbox{
and }0\leq x<z(|\lambda\rangle)\right\\}.$ (10)
Any ontic state $\lambda\in{\cal E}_{0}$ has the property that whatever
measurement is performed, the outcome is $|\phi_{0}\rangle\langle\phi_{0}|$.
The epistemic states can therefore be modified as above to produce a
$\psi$-epistemic model. Informally, the idea is the same: any probability that
$\mu_{\psi}$ assigns to ontic states within the set ${\cal E}_{0}$ can be
redistributed over the whole of ${\cal E}_{0}$ without changing Born rule
statistics. More specifically, when $|\langle\psi|0\rangle|^{2}\leq(d-1)/d$,
let
$\mu_{\psi}(|\lambda\rangle,x)=\delta(|\lambda\rangle-|\psi\rangle),$ (11)
and when $|\langle\psi|0\rangle|^{2}>(d-1)/d$, let
$\mu_{\psi}(|\lambda\rangle,x)=\delta(|\lambda\rangle-|\psi\rangle)\Theta(x-z(|\psi\rangle))+z(|\psi\rangle)\mu_{{\cal
E}_{0}}(|\lambda\rangle,x),$ (12)
where, as above, $\mu_{{\cal E}_{0}}$ is arbitrary but could be taken to be
the uniform distribution over ${\cal E}_{0}$.
This model is clearly very contrived. A degree of symmetry could be restored
by postulating a preferred basis $|0\rangle,\ldots,|d-1\rangle$, instead of a
preferred state. If $|\langle\lambda|j\rangle|^{2}>(d-1)/d$ for some $j$, then
relabel measurement outcomes such that
$|\langle\phi_{0}|j\rangle|^{2}\geq|\langle\phi_{1}|j\rangle|^{2}\geq\cdots\geq|\langle\phi_{d-1}|j\rangle|^{2}$.
In this case $|\langle\phi_{0}|\lambda\rangle|^{2}>0$, and sets ${\cal E}_{j}$
can be defined in analogy with ${\cal E}_{0}$ above. For any state
$|\psi\rangle$ with $|\langle\psi|j\rangle|^{2}>(d-1)/d$, any probability
assigned to ontic states within ${\cal E}_{j}$ can be redistributed over
${\cal E}_{j}$. This results in a model that is ‘more epistemic’ than the one
above. It is an open question whether more natural models can be found.
_Discussion.–_ Our results are particularly pertinent given the recent no-go
theorem of Pusey _et al._ (2012). The theorem shows that it is not possible
to construct $\psi$-epistemic models of quantum theory, given an assumption
that independent preparations produce uncorrelated ontic states. The present
Letter shows that $\psi$-epistemic models are possible if that assumption is
given up. None of these models is intuitive or motivated by physical
principles or considerations. The primary motivation for exploring the
possibility of $\psi$-epistemic models is to understand the formal limitations
of reproducing quantum theory from a deeper theory.
Such models may also play a useful role in other areas of quantum information.
The remarkable protocols of Toner and Bacon Toner and Bacon (2003) for
recovering the bipartite measurement statistics of a singlet state and
classical simulation of teleportation—using only one and two bits
(respectively) of classical communication—make use of shared randomness in a
way that lets the problem be recast in terms of correlated single qubit
ontological models that are $\psi$-epistemic. A related result has been
obtained by Montina Montina (2011), who has also put bounds on the classical
simulation cost for an arbitrary number of qubits.
There remain many open questions. In the models presented, it is not the case
that $\mu_{\psi}$ has non-zero overlap with $\mu_{\phi}$ for any pair of non-
orthogonal quantum states $|\psi\rangle$ and $|\phi\rangle$. Hence the models
do not satisfy the stronger definition of $\psi$-epistemic suggested above. It
would be interesting to establish whether such models exist. (Scott Aaronson
has recently combined some of the ideas that were presented in a preprint
version of this article with those of George Lowther to answer this question
in the affirmative; see Aaronson and Lowther (2006).)
The authors gratefully acknowledge support from the Engineering and Physical
Sciences Research Council (P.G.L., J.B., and T.R.), the Leverhulme Foundation
(T.R.) and the Royal Comission for the Exhibition of 1851 (D.J.).
## References
* Einstein et al. (1935) A. Einstein, B. Podolsky, and N. Rosen, Physical Review 47, 777 (1935).
* Popper (1967) K. R. Popper, in _Quantum Theory and Reality_ , edited by M. Bunge (Springer, 1967), chap. 1.
* BALLENTINE (1970) L. E. Ballentine, Rev. Mod. Phys. 42, 358 (1970).
* Peierls (1979) R. E. Peierls, _Surprises in Theoretical Physics_ (Princeton Universtiy Press, 1979), p. 32.
* Jaynes (1980) E. T. Jaynes, _Foundations of Radiation Theory and Quantum Electrodynamics_ (Plenum, 1980).
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* Spekkens (2007) R. W. Spekkens, Phys. Rev. A, 75, 032110 (2007).
* Brukner and Zeilinger (2001) Č. Brukner and A. Zeilinger, Phys. Rev. A, 63, 022113 (2001).
* Fuchs (2010) C. A. Fuchs, “QBism, the Perimeter of Quantum Bayesianism,” (2010), arXiv:1003.5209 [quant-ph].
* Leifer (2006) M. S. Leifer, Phys. Rev. A, 74, 042310 (2006), arXiv:quant-ph/0606022.
* Leifer and Spekkens (2011) M. S. Leifer and R. W. Spekkens, “Formulating quantum theory as a causally neutral theory of Bayesian inference,” (2011a), arXiv:1107.5849 [quant-ph].
* Leifer and Spekkens (2011) M. S. Leifer and R. W. Spekkens, “A Bayesian approach to compatibility, improvement, and pooling of quantum states,” (2011b), arXiv:1110.1085 [quant-ph].
* Barnum and Wilce (2009) H. Barnum and A. Wilce, “Information processing in convex operational theories,” (2009), arXiv:0908.2352 [quant-ph] .
* Hardy (2001) L. Hardy, “Quantum theory from five reasonable axioms,” (2001), arXiv:quant-ph/0101012.
* Barrett (2007) J. Barrett, Phys. Rev. A, 75, 032304 (2007).
* (17) A. Einstein, Unpublished letter to E. Schrödinger, 19th June 1935, EA 22-047.
* Howard (2006) D. Howard, “Revising the Einstein-Bohr dialogue,” online $<$http://www.nd.edu/$\sim$dhoward1/Revisiting the Einstein-Bohr Dialogue.pdf$>$ (2006).
* Schrödinger (1935) E. Schrödinger, Proc. Cambridge Phil. Soc., 31 (1935).
* Harrigan and Spekkens (2010) N. Harrigan and R. Spekkens, Foundations of Physics, 40, 125 (2010), ISSN 0015-9018, dOI: 10.1007/s10701-009-9347-0.
* Pusey _et al._ (2012) M. F. Pusey, J. Barrett, and T. Rudolph, Nat Phys advance online publication (2012), 10.1038/nphys2309.
* Hardy (1999) L. Hardy, “Disentangling nonlocality and teleportation,” (1999), arXiv:quant-ph/9906123.
* Bartlett _et al._ (2011) S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” (2011), arXiv:1111.5057 [quant-ph].
* Harrigan and Rudolph (2007) N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” (2007), arXiv:0709.4266 [quant-ph].
* Hardy (2004) L. Hardy, Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 35, 267 (2004), ISSN 1355-2198.
* Spekkens (2005) R. W. Spekkens, Phys. Rev. A, 71, 052108 (2005).
* Kochen and Specker (1967) S. Kochen and E. P. Specker, Journal of Mathematics and Mechanics, 17, 59 (1967).
* Montina (2008) A. Montina, Phys. Rev. A, 77, 022104 (2008).
* Montina (2011) A. Montina, Physics Letters A, 375, 1385 (2011a), ISSN 0375-9601.
* Montina (2011) A. Montina, Phys. Rev. A, 83, 032107 (2011b).
* Galvão (2009) E. F. Galvão, Phys. Rev. A, 80, 022106 (2009).
* Dakić _et al._ (2008) B. Dakić, M. Šuvakov, T. Paterek, and Č. Brukner, Phys. Rev. Lett., 101, 190402 (2008).
* Wehner _et al._ (2008) S. Wehner, M. Christandl, and A. C. Doherty, Phys. Rev. A, 78, 062112 (2008).
* Colbeck and Renner (2011a) R. Colbeck and R. Renner, Nat. Commun. 2, 411 (2011a).
* Colbeck and Renner (2011b) R. Colbeck and R. Renner, “Is a system’s wave function in one-to-one correspondence with its elements of reality?” (2011b), arXiv:1111.6597 [quant-ph] .
* Beltrametti and Bugajski (1995) E. G. Beltrametti and S. Bugajski, Journal of Physics A: Mathematical and General, 28, 3329 (1995).
* Bell (1966) J. S. Bell, Rev. Mod. Phys., 38, 447 (1966).
* Rudolph (2006) T. Rudolph, “Ontological Models for Quantum Mechanics and the Kochen-Specker theorem,” (2006), arXiv:quant-ph/0608120v1.
* Toner and Bacon (2003) B. F. Toner and D. Bacon, Phys. Rev. Lett., 91, 187904 (2003).
* Montina (2011) A. Montina, Phys. Rev. A, 84, 042307 (2011c).
* Werner (1989) R. F. Werner, Phys. Rev. A, 40, 4277 (1989).
* Aaronson and Lowther (2006) S. Aaronson and G. Lowther, http://mathoverflow.net/questions/95537/psi-epistemic-theories-in-3-or-more-dimensions (2012).
|
arxiv-papers
| 2012-01-31T14:41:40 |
2024-09-04T02:49:26.870538
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Peter G. Lewis, David Jennings, Jonathan Barrett and Terry Rudolph",
"submitter": "Peter Lewis",
"url": "https://arxiv.org/abs/1201.6554"
}
|
1201.6580
|
# Stackable and queueable permutations
Peter G. Doyle
(Version 1.0 dated 30 January 2012
No Copyright††thanks: The authors hereby waive all copyright and related or
neighboring rights to this work, and dedicate it to the public domain. This
applies worldwide. )
###### Abstract
There is a natural bijection between permutations obtainable using a stack
(those avoiding the pattern $312$) and permutations obtainable using a queue
(those avoiding $321$). This bijection is equivalent to one described by
Simion and Schmidt in 1985. We argue that this bijection might well have been
found back in 1968 by readers of volume 1 of Knuth’s _The Art of Computer
Programming_ , if Knuth had not assigned difficulty ratings to his exercises.
## 1 Warm-up
Let’s warm up by playing a solitaire game called _Double-Ended Knuth_ ; we
will call it _DEK_ for short, pronounced ‘deek’ so as to be indistinguishable
from ‘deque’ and ‘Dyck’. DEK is a bare-bones relative of familiar solitaire
games like Klondike. In DEK, we use a one-suit deck consisting of only the
thirteen hearts (say). We shuffle the deck thoroughly, and place the deck face
down on the table. The goal is to end with the cards in a pile face up,
running in order from ace to king. In addition to _the deck_ and _the pile_
(intially empty), we maintain a line of cards (initially empty), called _the
deque_ , spread out face up on the board. At any point, if the next card
needed for the pile is available as the top card of the deck or at either end
of the deque, we may move it up to the pile; otherwise, our only option is to
move the top card of the deck to either end of the deque.
Exercise. Play enough games of DEK to decide if you like it. This game is
definitely winnable, but if you are impatient, you might want to get rid of
the face cards and play with a 10-card deck.
## 2 Introduction
In volume 1 of _The Art of Computer Programming_ [4], Knuth shows that the
number of permutations of $(1,2,\ldots,n)$ that can be obtained using a stack
is the Catalan number $C_{n}=\frac{1}{2n+1}{{2n+1}\choose{n+1}}$. In an
exercise, he identifies the stackable permutations as those permutations
$(p_{1},p_{2},\ldots,p_{n})$ for which there are no indices $i<j<k$ such that
$p_{j}<p_{k}<p_{i}$. Nowadays it is common to describe these permutations as
_$312$ -avoiding_.
In volume 3 of _The Art of Computer Programming_ [5], Knuth shows that the
number of $321$-avoiding permutations (those that contain no decreasing
subsequence of length three) is also $C_{n}$. Knuth’s argument here is
roundabout: Starting with a $321$-avoiding permutation, you carry out the
Robinson-Schensted algorithm to get a pair of two-line Young tableaux of the
same shape, then fit the pair together to make a single Young tableau of shape
$(n,n)$. Any tableau of shape $(n,n)$ arises uniquely in this way, and the
number of such tableaux is $C_{n}$.
There is a more straight-forward argument, yielding a natural bijection
between $312$-avoiding and $321$-avoiding permutations. This bijection is
equivalent to one found in 1985 by Simion and Schmidt [9]. (See Claesson and
Kitaev [2].) We argue that this bijection might well have been found back in
1968 by readers of Knuth’s volume 1, if Knuth had not taken care to grade all
his exercises as to the expected difficulty of solving them.
## 3 One example
Maybe one example will allow you to skip the rest of this. Here are two
corresponding permutations, the first obtained using a stack and the second
obtained using a queue. Figure 1 shows the corresponding stack height and
queue length record.
Figure 1: Stack height and queue length: A peakless weak Dyck path
$\begin{array}[]{l|r|r}\mbox{input}&\mbox{stack}&\mbox{output}\\\
\hline\cr&&1,2,3,4,5,6,7\\\ &1&2,3,4,5,6,7\\\ 2&1&3,4,5,6,7\\\
2,1&&3,4,5,6,7\\\ 2,1&3&4,5,6,7\\\ 2,1&4,3&5,6,7\\\ 2,1,5&4,3&6,7\\\
2,1,5&6,4,3&7\\\ 2,1,5,7&6,4,3&\\\ 2,1,5,7,6&4,3&\\\ 2,1,5,7,6,4&3&\\\
2,1,5,7,6,4,3&&\end{array}$
$\begin{array}[]{l|r|r}\mbox{output}&\mbox{queue}&\mbox{input}\\\
\hline\cr&&1,2,3,4,5,6,7\\\ &1&2,3,4,5,6,7\\\ 2&1&3,4,5,6,7\\\
2,1&&3,4,5,6,7\\\ 2,1&3&4,5,6,7\\\ 2,1&3,4&5,6,7\\\ 2,1,5&3,4&6,7\\\
2,1,5&3,4,6&7\\\ 2,1,5,7&3,4,6&\\\ 2,1,5,7,3&4,6&\\\ 2,1,5,7,3,4&6&\\\
2,1,5,7,3,4,6&&\\\ \end{array}$
## 4 An exercise rated “M28”
Toward the beginning of volume 1 of _The Art of Computer Programming_ [4],
Knuth poses the following exercise. The rating “M28” identifies this exercise
as mathematically oriented, and of average to moderate difficulty, likely
requiring somewhere between fifteen to twenty minutes (difficulty “20”) and
over two hours (difficulty “30”) to solve. (The scale is ‘logarithmic’.)
Exercise 2.2.1-5 [M28] _Show that it is possible to obtain the permutation
$p_{1}\,p_{2}\,\ldots\,p_{n}$ from $1\,2\,\ldots\,n$ using a stack if and only
if there are no indices $i<j<k$ such that $p_{j}<p_{k}<p_{i}$. _
Solution. This exercise refers to previous exercises, which make clear that we
are to produce the permutation by a sequence of push and pop operations, where
the objects pushed are the numbers $1,2,\ldots,n$, in that order, and the
output is the sequence of objects popped. For example, by pushing and popping
in strict alternation, we get the identity permutation $(1,2,\ldots,n)$. By
doing $n$ pushes followed by $n$ pops, we get the reversing permutation
$(n,\ldots,2,1)$. Let us call any permutation obtainable in this way
_stackable_.
Rephrased in the language of pattern-avoiding permutations, the problem is to
show that a permutation is stackable if and only if it avoids the pattern
$312$.
This condition is necessary because to obtain a permutation containing a
pattern of form $312$, when we pop $3$, we would have to have $1$ and $2$ on
the stack, with $1$ above $2$, which is impossible.
To see that the condition is sufficient, suppose that we attempt to produce a
given output permutation in the obvious way—the only way possible—namely,
working through the desired output permutation in order, pushing to get the
next desired output onto the top of the stack if it is not already on the
stack, and hoping that it is at the top of the stack if it is already on the
stack. The only thing that can go wrong is if at some point when we wish to
take something (call it $1$) from the stack, there is something larger (call
it $2$) blocking it; if $1$ and $2$ are on the stack, it is because we have
already popped something larger still (call it $3$); so $3$ has been called
first, now $1$ is called, with $2$ to be called later.
If we associate to a stackable permutation the record of stack height as a
function of time, we get a Dyck path of length $2n$, meaning a path in the
integer lattice $\mathbf{Z}^{2}$ from $(0,0)$ to $(2n,0)$, made up of steps
$(1,1)$ and $(1,-1)$, that never goes below the $x$-axis. Any such Dyck path
arises from one and only one obtainable permutation. The number of Dyck paths
of length $2n$ is the Catalan number
$C_{n}=\frac{1}{2n+1}{{2n+1}\choose{n+1}}$. (Pick a sequence consisting of
$n+1$ $1$’s and $n$ $-1$’s; take the unique rotation for which all partial
sums are non-negative; discard the initial $1$; convert to a Dyck path.) Thus
the number of stackable permutations is $C_{n}$.
Notice that the answer to this problem does not change if, in addition to
pushing and popping, we have the option of transferring an object directly
from the input to the output. Now there is more than one sequence of
operations that produces a given permutation, because we have the option of
transferring an object directly from input to output, or pushing it and then
immediately popping it. To avoid this ambiguity, we may assume that we never
pop immediately after pushing, say because we wish to minimize the number of
operations. Now in the stack record of an obtainable permutation, all of the
peaks (consisting of a step $(1,1)$ followed immediately by a step $(1,-1)$)
will have been replaced by horizontal steps $(1,0)$; the total number of steps
in the path will now be shorter than $2n$ by the number of peaks removed. The
$C_{n}$ possible stack records will now consist of ‘peakless weak Dyck paths’.
## 5 An exercise rated “00”
Immediately following the exercise we’ve just discussed comes this throw-away
exercise, with difficulty rating “00”.
Exercise 2.2.1-6 [00] _Consider the problem of exercise 2, with a queue
substituted for a stack. What permutations of $1\,2\,\ldots\,n$ can be
obtained with the use of a queue? _
Solution. The set-up of exercise 2 is the same as that of exercise 5, the
preceeding exercise, which we’ve just discussed. If we interpret the problem
ungenerously, as Knuth’s “00” rating indicates that he expects us to do, the
only permutation that can be obtained is the identity permutation.
Let’s vary the problem by allowing direct transfers from input to output.
Surely this would be a reasonable alternative interpretation of permutations
that ‘can be obtained with the use of a queue’. While allowing direct
transfers does not change which permutations can be obtained with the use of a
stack, here we find that we can obtain all and only those permutations that
arise by interleaving two increasing subsequences. Let us call permutations
that can be obtained using a queue in this way _queueable_.
A permutation is queueable just if it is $321$-avoiding. The reason is that a
permutation can be obtained by interleaving two increasing subsequences just
if has no decreasing subsequence of length three. This is a simple consequence
of the Robinson-Schensted correspondence, but we do not need anything that
fancy. Any permutation obtained by interleaving two increasing subsequences
must avoid $321$, because if not, one of the two subsequences would contain
the pattern $21$, making it non-increasing. The argument in the other
direction is just like the argument above that a $312$-avoiding permutation is
stackable. Here we want to show that a $321$-avoiding permutation is
queueable. Once again, we work through the desired output permutation, hoping
that the next desired output is at the head of the queue if it is already in
the queue, and moving items to the queue to uncover it if it is still in the
input stream. The only thing that can go wrong is if when we want to take
something (call it $2$) from the queue, there is something larger (call it
$3$) blocking it; if $2$ is in the queue, it is because we had to make way to
retrieve something smaller (call it $1$) from the input stream. So in the
input $3$ was ahead of $2$, and $2$ was ahead of $1$.
Here, as in the case of a stack when we allowed direct transfers from input to
output, a given queueable permutation can arise in multiple ways, because we
can move an object to the queue early or late. To avoid this ambiguity, let us
assume that we never move an object to the queue unless and until we have to,
as in the procedure we’ve just described. In contrast to what we saw with
stacks, the preferred way of obtaining a permutation is not necessarily
quicker than competing ways, since moving an object to the queue early does
not increase the number of operations. But the preferred way does minimize the
aggregate time spent by objects on the queue, so if we imagine that a storage
fee is charged for the queue, the preferred way minimizes this fee. Note that
minimizing total time in storage works to pick out the preferred way of using
a stack, as well as a queue.
In picking out a preferred way to obtain a queueable permutation, we
implicitly pick out a canonical way to decompose it into (at most) two
increasing subsequences. One subsequence, consisting of those objects that are
transferred directly, consists of the ‘record-setting’ objects, namely, those
that are larger than any predecessor. This subsequence is of course
increasing, no matter what permutation we start with. A permutation is
queueable just if the complemetary subsequence is also increasing.
Now recall that in the case of stackable permutations, where we allow direct
transfers, the stack height record is a weak Dyck path; any weak Dyck path
determines a unique stackable permutation; and there is a bijection between
stackable permutations and peakless weak Dyck paths. The same holds here,
except that now instead of stack height we look at queue length: The queue
length record is a weak Dyck path; any weak Dyck path determines a unique
queueable permutation; and there is a bijection between queueable permuations
and peakless weak Dyck paths, associating to any queueable permutation the
queue length record of the preferred way of obtaining it. The main difference
is in the way you would go about reducing a general weak Dyck path to the
unique peakless weak Dyck path realizing the same queueable permutation.
Because of the correspondence to peakless weak Dyck paths, we get that the
number of queueable permutations is $C_{n}$, just as for stackable
permutations. Furthermore, we get a natural bijection between stackable and
queueable permutations by pairing up those that share the same peakless weak
Dyck path. This bijection is equivalent to that found by Simion and Schmidt
[9]. Claesson and Kitaev [2] call this bijection (or something equivalent to
it) the _Knuth-Richards bijection_ for the following reason. If we turn a
peakless weak Dyck path into a standard Dyck path in the obvious way, the
correspondence between queueable permutations and standard Dyck paths that we
get is equivalent to that described by Richards [7]. Turning a stackable
permuation into a Dyck path following Knuth, and turning the Dyck path into a
queueable permutation following Richards, we get a bijection between stackable
and queueable permutations.
In describing the bijection we have found as being mediated either by peakless
weak Dyck paths or standard Dyck paths, we are not doing it justice. We can
describe it more directly by saying that, if you set out to produce a given
stackable permutation using a stack, but by mistake you use a queue instead,
you get the corresponding queueable permutation, and vice versa.
Here’s another way to think of it. Suppose that we have at our disposal not a
stack or a queue, but a _set_ , from which we are able to recover objects in
any order. Now we can realize any permutation. Again, let us imagine that
there is a cost for using the set, so we use it as little as possible. To any
permutation there corresponds a unique peakless weak Dyck path, namely the set
size record, only now the correspondence is many-to-one. Start with an
arbitrary permutation $\sigma$, find the corresponding peakless weak Dyck
path, and let $\mathrm{stackit}(\sigma)$ be the corresponding stackable
permutation. The function $\mathrm{stackit}$ is an idempotent map
($\mathrm{stackit}\circ\mathrm{stackit}=\mathrm{stackit}$), projecting the set
of all permutations onto the set of stackable permutations. The mapping
$\mathrm{stackit}$ tells what you get if you try to produce a specified
permutation by means of a set, but unbeknownst to you the set is not a set but
a stack, and you always get the top element no matter which one you ask for.
Similarly, we get a idempotent function $\mathrm{queueit}$ telling what you
get if what you thought was a set was really a queue. Restricting
$\mathrm{queueit}$ to stackable permutations gives a bijection to queueable
permutations; its inverse is the restriction of $\mathrm{stackit}$ to
queueable permutations.
One more thing: It is plausible to say that using a queue, we can obtain any
permuation, because after all, we can retrieve any object from the queue by
succesively transferring cards from the output back to the input until we find
the object we are looking for. So here is another interpretation of the
problem that would justify a difficulty rating of “00”.
## 6 Conclusion
The drawback of labelling exercises according to the expected difficulty is
that it does not allow for ambiguous exercises, where the level of difficulty
may depend on how the exercise is interpreted. Assigning exercises is not like
writing a computer program, where ambiguity is to be avoided at all costs. The
problems that the real world poses for us are almost always ambiguous. And
even if unambiguous, how difficult they may be is seldom known in advance.
## 7 Problems
1. 1.
What permutations can be obtained using two stacks?
2. 2.
What permutations can be obtained using two queues?
3. 3.
What permutations can be obtained using one stack and one queue?
4. 4.
What is the probability of winning at DEK if you play optimally?
## References
* [1] Miklós Bóna. A survey of stack-sorting disciplines. Electron. J. Combin., 9(2):Article 1, 16, 2002/03. Permutation patterns (Otago, 2003).
* [2] Anders Claesson and Sergey Kitaev. Classification of bijections between 321- and 132-avoiding permutations. Sém. Lothar. Combin., 60:Article B60d, 2008, arXiv:0805.1325v1 [math.CO].
* [3] S. Even and A. Itai. Queues, stacks, and graphs. In Theory of machines and computations (Proc. Internat. Sympos., Technion, Haifa, 1971), pages 71–86. Academic Press, 1971.
* [4] Donald E. Knuth. The art of computer programming. Volume 1: Fundamental algorithms. Addison-Wesley, first edition, 1968.
* [5] Donald E. Knuth. The art of computer programming. Volume 3: Sorting and searching. Addison-Wesley, first edition, 1973.
* [6] Vaughan R. Pratt. Computing permutations with double-ended queues, parallel stacks and parallel queues. In Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973), pages 268–277. Assoc. Comput. Mach., New York, 1973\.
* [7] Dana Richards. Ballot sequences and restricted permutations. Ars Combin., 25:83–86, 1988.
* [8] Pierre Rosenstiehl and Robert E. Tarjan. Gauss codes, planar Hamiltonian graphs, and stack-sortable permutations. J. Algorithms, 5(3):375–390, 1984.
* [9] Rodica Simion and Frank W. Schmidt. Restricted permutations. European J. Combin., 6(4):383–406, 1985.
* [10] Robert Tarjan. Sorting using networks of queues and stacks. J. Assoc. Comput. Mach., 19:341–346, 1972.
* [11] Walter Unger. On the $k$-colouring of circle-graphs. In STACS 88, volume 294 of Lecture Notes in Comput. Sci., pages 61–72. Springer, 1988.
* [12] Walter Unger. The complexity of colouring circle graphs. In STACS 92, volume 577 of Lecture Notes in Comput. Sci., pages 389–400. Springer, 1992.
|
arxiv-papers
| 2012-01-31T15:44:30 |
2024-09-04T02:49:26.879131
|
{
"license": "Public Domain",
"authors": "Peter G. Doyle",
"submitter": "Peter G. Doyle",
"url": "https://arxiv.org/abs/1201.6580"
}
|
1201.6633
|
# A new class of generalized Bernoulli polynomials and Euler polynomials
N. I. Mahmudov
Eastern Mediterranean University
Gazimagusa, TRNC, Mersiin 10, Turkey
Email: nazim.mahmudov@emu.edu.tr
###### Abstract
The main purpose of this paper is to introduce and investigate a new class of
generalized Bernoulli polynomials and Euler polynomials based on the
$q$-integers. The $q$-analogues of well-known formulas are derived. The
$q$-analogue of the Srivastava–Pintér addition theorem is obtained. We give
new identities involving $q$-Bernstein polynomials.
## 1 Introduction
Throughout this paper, we always make use of the following notation:
$\mathbb{N}$ denotes the set of natural numbers, $\mathbb{N}_{0}$ denotes the
set of nonnegative integers, $\mathbb{R}$ denotes the set of real numbers,
$\mathbb{C}$ denotes the set of complex numbers.
The $q$-shifted factorial is defined by
$\left(a;q\right)_{0}=1,\ \ \
\left(a;q\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left(1-q^{j}a\right),\
\ \ n\in\mathbb{N},\ \ \
\left(a;q\right)_{\infty}={\displaystyle\prod\limits_{j=0}^{\infty}}\left(1-q^{j}a\right),\
\ \ \ \left|q\right|<1,\ \ a\in\mathbb{C}.$
The $q$-numbers and $q$-numbers factorial is defined by
$\left[a\right]_{q}=\frac{1-q^{a}}{1-q}\ \ \ \left(q\neq 1\right);\ \ \
\left[0\right]_{q}!=1;\ \ \ \
\left[n\right]_{q}!=\left[1\right]_{q}\left[2\right]_{q}...\left[n\right]_{q}\
\ \ \ \ n\in\mathbb{N},\ \ a\in\mathbb{C}$
respectively. The $q$-polynomial coefficient is defined by
$\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{n-k}\left(q;q\right)_{k}}.$
The $q$-analogue of the function $\left(x+y\right)^{n}$ is defined by
$\left(x+y\right)_{q}^{n}:={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\frac{1}{2}k\left(k-1\right)}x^{n-k}y^{k},\ \ \
n\in\mathbb{N}_{0}.$
The $q$-binomial formula is known as
$\left(1-a\right)_{q}^{n}=\left(a;q\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left(1-q^{j}a\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\frac{1}{2}k\left(k-1\right)}\left(-1\right)^{k}a^{k}.$
In the standard approach to the $q$-calculus two exponential functions are
used:
$\displaystyle e_{q}\left(z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\frac{1}{\left(1-\left(1-q\right)q^{k}z\right)},\
\ \ 0<\left|q\right|<1,\ \left|z\right|<\frac{1}{\left|1-q\right|},\ \ \ \ \ \
\ $ $\displaystyle E_{q}\left(z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right)}z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\left(1+\left(1-q\right)q^{k}z\right),\
\ \ \ \ \ \ 0<\left|q\right|<1,\ z\in\mathbb{C}.\ $
From this form we easily see that $e_{q}\left(z\right)E_{q}\left(-z\right)=1$.
Moreover,
$D_{q}e_{q}\left(z\right)=e_{q}\left(z\right),\ \ \ \
D_{q}E_{q}\left(z\right)=E_{q}\left(qz\right),$
where $D_{q}$ is defined by
$D_{q}f\left(z\right):=\frac{f\left(qz\right)-f\left(z\right)}{qz-z},\ \ \ \
0<\left|q\right|<1,\ 0\neq z\in\mathbb{C}.$
The above $q$-standard notation can be found in [1].
Over 70 years ago, Carlitz extended the classical Bernoulli and Euler numbes
and polynomials and introduced the $q$-Bernoulli and the $q$-Euler numbers and
polynomials (see [2], [3] and [4] ). There are numerous recent investigations
on this subject by, among many other authors, Cenki et al. ([12], [13], [14]),
Choi et al. ([15] and [16]), Kim et al. ([17]-[24]), Ozden and Simsek [25],
Ryoo et al. [28], Simsek ([29], [30] and [31]), and Luo and Srivastava [11],
Srivastava et al. [32].
We first give here the definitions of the $q$-Bernoulli and the $q$-Euler
polynomials of higher order as follows.
###### Definition 1
Let $q,\alpha\in\mathbb{C},\ 0<\left|q\right|<1.$ The $q$-Bernoulli numbers
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}$ and polynomials
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ in $x,y$ of order
$\alpha$ are defined by means of the generating function functions:
$\displaystyle\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<2\pi,$
$\displaystyle\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<2\pi.$
###### Definition 2
Let $q,\alpha\in\mathbb{C},\ 0<\left|q\right|<1.$ The $q$-Euler numbers
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}$ and polynomials
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ in $x,y$ of order
$\alpha$ are defined by means of the generating functions:
$\displaystyle\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<\pi,$
$\displaystyle\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<\pi.$
It is obvious that
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}$
$\displaystyle=\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,0\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=B_{n}^{\left(\alpha\right)}\left(x+y\right),\
\ \ \lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}=B_{n}^{\left(\alpha\right)},$
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}$
$\displaystyle=\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,0\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=E_{n}^{\left(\alpha\right)}\left(x+y\right),\
\ \ \lim_{q\rightarrow
1^{-}}\mathfrak{E}_{n,q}^{\left(\alpha\right)}=E_{n}^{\left(\alpha\right)},$
$\displaystyle\lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle=B_{n}^{\left(\alpha\right)}\left(x\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,y\right)=B_{n}^{\left(\alpha\right)}\left(y\right),\
\ \ $ $\displaystyle\lim_{q\rightarrow
1^{-}}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle=E_{n}^{\left(\alpha\right)}\left(x\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)=E_{n}^{\left(\alpha\right)}\left(y\right).$
Here $B_{n}^{\left(\alpha\right)}\left(x\right)$ and
$E_{n}^{\left(\alpha\right)}\left(x\right)$ denote the classical Bernoulli and
Euler polynomials of order $\alpha$ which are defined by
$\left(\frac{t}{e^{t}-1}\right)^{\alpha}e^{tx}=\sum_{n=0}^{\infty}B_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{\left[n\right]_{q}!}\
\ \ \ \ \text{and\ \ \ \
}\left(\frac{2}{e^{t}+1}\right)^{\alpha}e^{tx}=\sum_{n=0}^{\infty}E_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
In fact Definitions 1 and 2 define two different type
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ and
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$ of the
$q$-Bernoulli polynomials and two different type
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ and
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$ of the $q$-Euler
polynomials. Both polynomials
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ and
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$
($\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ and
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$) coincide with the
classical highe order Bernoulli polynomilas (Euler polynomilas) in the
limiting case $q\rightarrow 1^{-}$.
For the $q$-Bernoulli numbers $\mathfrak{B}_{n,q}$, the $q$-Euler numbers
$\mathfrak{E}_{n,q}$ of order $n$, we have
$\mathfrak{B}_{n,q}=\mathfrak{B}_{n,q}\left(0,0\right)=\mathfrak{B}_{n,q}^{\left(1\right)}\left(0,0\right),\
\ \ \ \
\mathfrak{E}_{n,q}=\mathfrak{E}_{n,q}\left(0,0\right)=\mathfrak{E}_{n,q}^{\left(1\right)}\left(0,0\right),$
respectively. Note that the $q$-Bernoulli numbers $\mathfrak{B}_{n,q}$ are
defined and studied in [26].
The aim of the present paper is to obtain some results for the above defined
$q$-Bernoulli and $q$-Euler polynomials. In this paper the $q$-analogues of
well-known results, for example, Srivastava and Pintér [10], Cheon [5], etc.,
will be given. Also the formulas involving the $q$-Stirling numbers of the
second kind, $q$-Bernoulli polynomials and Phillips $q$-Bernstein polynomials
are derived.
## 2 Preliminaries and Lemmas
In this section we shall provide some basic formulas for the $q$-Bernoulli and
$q$-Euler polynomials in order to obtain the main results of this paper in the
next section. The following result is $q$-analogue of the addition theorem for
the classical Bernoulli and Euler polynomials.
###### Lemma 3
_(Addition Theorems)_ For all $x,y\in\mathbb{C}$ we have
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x+y\right)_{q}^{n-k},\
\ \
\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\
\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(x+y\right)_{q}^{n-k},$ (5)
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right)y^{n-k}=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(0,y\right)x^{n-k},$
(10) $\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(x,0\right)y^{n-k}=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(0,y\right)x^{n-k}.$
(15)
In particular, setting $x=0$ and $y=0$ in (10) and (15), we get the following
formulas for $q$-Bernoulli and $q$-Euler polynomials, respectively.
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}x^{n-k},\ \ \
\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,y\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-\not
k\right)\left(n-k-1\right)/2}\mathfrak{B}_{k,q}^{\left(\alpha\right)}y^{n-k},$
(20) $\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}^{\left(\alpha\right)}x^{n-k},\ \ \
\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-\not
k\right)\left(n-k-1\right)/2}\mathfrak{E}_{k,q}^{\left(\alpha\right)}y^{n-k}.$
(25)
Setting $y=1$ and $x=1$ in (10) and (15), we get
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,1\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right),\
\ \
\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(1,y\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(0,y\right),$
(30) $\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,1\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(x,0\right),\
\ \
\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(1,y\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(0,y\right).$
(35)
Clearly (30) and (35) are $q$-analogues of
$B_{n}^{\left(\alpha\right)}\left(x+1\right)=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)B_{k}^{\left(\alpha\right)}\left(x\right),\ \ \
E_{n}^{\left(\alpha\right)}\left(x+1\right)=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)E_{k}^{\left(\alpha\right)}\left(x\right),$
respectively.
###### Lemma 4
We have
$\displaystyle
D_{q,x}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\left[n\right]_{q}\mathfrak{B}_{n-1,q}^{\left(\alpha\right)}\left(x,y\right),\
\ \
D_{q,y}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=\left[n\right]_{q}\mathfrak{B}_{n-1,q}^{\left(\alpha\right)}\left(x,qy\right),$
$\displaystyle
D_{q,x}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\left[n\right]_{q}\mathfrak{E}_{n-1,q}^{\left(\alpha\right)}\left(x,y\right),\
\ \
D_{q,y}\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=\left[n\right]_{q}\
\mathfrak{E}_{n-1,q}^{\left(\alpha\right)}\left(x,qy\right).$
###### Lemma 5
_(Difference Equations)_ We have
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(1,y\right)-\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$
$\displaystyle=\left[n\right]_{q}\mathfrak{B}_{n-1,q}^{\left(\alpha-1\right)}\left(0,y\right),$
(36)
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(1,y\right)+\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$
$\displaystyle=2\mathfrak{E}_{n,q}^{\left(\alpha-1\right)}\left(0,y\right),$
(37)
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)-\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,-1\right)$
$\displaystyle=\left[n\right]_{q}\mathfrak{B}_{n-1,q}^{\left(\alpha-1\right)}\left(x,-1\right),$
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)+\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,-1\right)$
$\displaystyle=2\mathfrak{E}_{n,q}^{\left(\alpha-1\right)}\left(x,-1\right).$
From (36) and (20), (37) and (25) we obtain the following formulas.
###### Lemma 6
We have
$\displaystyle\mathfrak{B}_{n-1,q}^{\left(\alpha-1\right)}\left(0,y\right)$
$\displaystyle=\frac{1}{\left[n+1\right]_{q}}\sum_{k=0}^{n}\left[\begin{array}[c]{c}n+1\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(0,y\right),$
(40) $\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha-1\right)}\left(0,y\right)$
$\displaystyle=\frac{1}{2}\left[\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(0,y\right)+\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(0,y\right)\right].$
(43)
Putting $\alpha=1$ in (40) and (43), and noting that
$\mathfrak{B}_{n,q}^{\left(0\right)}\left(0,y\right)=\mathfrak{E}_{n,q}^{\left(0\right)}\left(0,y\right)=q^{n\left(n-1\right)/2}y^{n},$
we arrive at the following expansions:
$\displaystyle y^{n}$
$\displaystyle=\frac{1}{q^{n\left(n-1\right)/2}\left[n+1\right]_{q}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n+1\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}\left(0,y\right),$ $\displaystyle
y^{n}$
$\displaystyle=\frac{1}{2q^{n\left(n-1\right)/2}}\left[{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}\left(0,y\right)+\mathfrak{E}_{n,q}\left(0,y\right)\right],$
which are $q$-analoques of the following familiar expansions
$y^{n}=\frac{1}{n+1}{\displaystyle\sum\limits_{k=0}^{n}}\left(\begin{array}[c]{c}n+1\\\
k\end{array}\right)B_{k}\left(y\right),\ \ \
y^{n}=\frac{1}{2}\left[{\displaystyle\sum\limits_{k=0}^{n}}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)E_{k}\left(y\right)+E_{n}\left(y\right)\right],$ (44)
respectively.
###### Lemma 7
_(Recurrence Relationships)_ The polynomials
$\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ and
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$ satisfy the
following difference relationships:
$\displaystyle{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(x,0\right)-{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(x,-1\right)$
$\displaystyle=\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}m^{j+1}\mathfrak{B}_{j,q}^{\left(\alpha-1\right)}\left(x,-1\right),$
(51)
$\displaystyle\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(\frac{1}{m},y\right)-{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(0,y\right)$
$\displaystyle=\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j-1}\mathfrak{B}_{j,q}^{\left(\alpha-1\right)}\left(0,y\right),$
(56)
$\displaystyle{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}^{\left(\alpha\right)}\left(x,0\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}^{\left(\alpha\right)}\left(x,-1\right)$
$\displaystyle=2{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}^{\left(\alpha-1\right)}\left(x,-1\right),$
(63)
$\displaystyle\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(\frac{1}{m},y\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{E}_{j,q}^{\left(\alpha\right)}\left(0,y\right)$
$\displaystyle=2{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{E}_{j,q}^{\left(\alpha-1\right)}\left(0,y\right).$
(68)
## 3 Explicit relationship between the $q$-Bernoulli and $q$-Euler
polynomials
In this section we shall investigate some explicit relationships between the
$q$-Bernoulli and $q$-Euler polynomials. Here some $q$-analogues of known
results will be given. We also obtain new formulas and their some special
cases below. These formulas are some extensions of the formulas of Srivastava
and Á. Pintér, Cheon and others.
We present natural $q$-extensions of th main results of the papers [10], [8],
see Theorems 8 and 13.
###### Theorem 8
For $n\in\mathbb{N}_{0}$, the following relationship
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\frac{1}{2m^{n}}\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\left[m^{k}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(x,-1\right)\right.$
$\displaystyle\left.+\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}m^{j+1}\mathfrak{B}_{j,q}^{\left(\alpha-1\right)}\left(x,-1\right)\right]\mathfrak{E}_{n-k,q}\left(0,my\right),$
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\frac{1}{2m^{n}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}m^{k}\left[\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(0,y\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(0,y\right)\right.$
$\displaystyle\left.+\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-1-j}\mathfrak{B}_{j,q}^{\left(\alpha-1\right)}\left(0,y\right)\right]\mathfrak{E}_{n-k,q}\left(mx,0\right)$
holds true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.
Proof. Using the following identity
$\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)=\frac{2}{e_{q}\left(\frac{t}{m}\right)+1}\cdot
E_{q}\left(\frac{t}{m}my\right)\cdot\frac{e_{q}\left(\frac{t}{m}\right)+1}{2}\cdot\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}e_{q}\left(tx\right)$
we have
$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{E}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\frac{t^{n}}{m^{n}\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle+\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{E}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=:I_{1}+I_{2}.$
It is clear that
$I_{2}=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{E}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\frac{t^{n}}{\left[n\right]_{q}!}=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}m^{k-n}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right)\mathfrak{E}_{n-k,q}\left(0,my\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
On the other hand
$\displaystyle I_{1}$
$\displaystyle=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\frac{t^{n}}{\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}{\displaystyle\sum\limits_{j=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}m^{-n}\mathfrak{E}_{j,q}\left(0,my\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right){\displaystyle\sum\limits_{j=0}^{n-k}}\left[\begin{array}[c]{c}n-k\\\
j\end{array}\right]_{q}m^{k-n}\mathfrak{E}_{j,q}\left(0,my\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}m^{-n}{\displaystyle\sum\limits_{j=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}\mathfrak{E}_{j,q}\left(0,my\right){\displaystyle\sum\limits_{k=0}^{n-j}}\left[\begin{array}[c]{c}n-j\\\
k\end{array}\right]_{q}m^{k}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
Therefore
${\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!}=\frac{1}{2}{\displaystyle\sum\limits_{n=0}^{\infty}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}m^{k-n}\left[\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(x,0\right)+m^{-k}{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{B}_{j,q}^{\left(\alpha\right)}\left(x,0\right)\right]\mathfrak{E}_{n-k,q}\left(0,my\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
It remains to use the formula (51).
Next we discuss some special cases of Theorem 8.
###### Corollary 9
For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship
$\displaystyle\mathfrak{B}_{n,q}\left(x,y\right)$
$\displaystyle=\frac{1}{2m^{n}}\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\left[m^{k}\mathfrak{B}_{k,q}\left(x,0\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}m^{j}\mathfrak{B}_{j,q}\left(x,-1\right)\right.$
$\displaystyle+\left.\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}m^{j+1}\left(x-1\right)_{q}^{j}\right]\mathfrak{E}_{n-k,q}\left(0,my\right),$
$\displaystyle\mathfrak{B}_{n,q}\left(x,y\right)$
$\displaystyle=\frac{1}{2m^{n}}{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
j\end{array}\right]_{q}m^{k}\left[m^{k}\mathfrak{B}_{k,q}\left(0,y\right)+{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{B}_{j,q}\left(0,y\right)\right.$
$\displaystyle\left.+\left[k\right]_{q}{\displaystyle\sum\limits_{j=0}^{k-1}}\left[\begin{array}[c]{c}k-1\\\
j\end{array}\right]_{q}q^{\frac{1}{2}j\left(j-1\right)}\left(\frac{1}{m}-1\right)_{q}^{k-1-j}y^{j}\right]\mathfrak{E}_{n-k,q}\left(mx,0\right)$
holds true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.
###### Corollary 10
[8] For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship
holds true.
$\displaystyle B_{n}\left(x+y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\left(B_{k}\left(y\right)+\frac{k}{2}y^{k-1}\right)E_{n-k}\left(x\right),$
$\displaystyle B_{n}\left(x+y\right)$
$\displaystyle=\frac{1}{2m^{n}}\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\left[m^{k}B_{k}\left(x\right)+m^{k}B_{k}\left(x-1+\frac{1}{m}\right)+km\left(1+m\left(x-1\right)\right)^{k-1}\right]E_{n-k,q}\left(my\right).$
###### Corollary 11
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\mathfrak{B}_{n,q}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\left(\mathfrak{B}_{k,q}\left(0,y\right)+q^{\frac{1}{2}\left(k-1\right)\left(k-2\right)}\frac{\left[k\right]_{q}}{2}y^{k-1}\right)\mathfrak{E}_{n-k,q}\left(x,0\right).$
(69)
###### Corollary 12
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\displaystyle\mathfrak{B}_{n,q}\left(x,0\right)$
$\displaystyle={\displaystyle\sum\limits_{\begin{subarray}{c}k=0\\\
\left(k\neq 1\right)\end{subarray}}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]\mathfrak{B}_{k,q}\mathfrak{E}_{n-k,q}\left(x,0\right)+\left(\mathfrak{B}_{1,q}+\frac{1}{2}\right)\mathfrak{E}_{n-1,q}\left(x,0\right),$
(72) $\displaystyle\mathfrak{B}_{n,q}\left(0,y\right)$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\ \left(k\neq
1\right)\end{subarray}}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{B}_{k,q}\mathfrak{E}_{n-k,q}\left(0,y\right)+\left(\mathfrak{B}_{1,q}+\frac{1}{2}\right)\mathfrak{E}_{n-1,q}\left(0,y\right).$
(75)
The formulas (69)-(75) are $q$-extension of the Cheon’s main result [5].
Notice that $\mathfrak{B}_{1,q}=-\frac{1}{\left[2\right]_{q}},$ see [26], and
the extra term becomes zeo for $q\rightarrow 1^{-}.$
###### Theorem 13
For $n\in\mathbb{N}_{0}$, the following relationship
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\frac{1}{m^{n-1}\left[k+1\right]_{q}}\left[2{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k+1-j}\mathfrak{E}_{j,q}^{\left(\alpha-1\right)}\left(0,y\right)\right.$
$\displaystyle-\left.{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k+1-j}\mathfrak{E}_{j,q}^{\left(\alpha\right)}\left(0,y\right)-\mathfrak{E}_{k+1,q}^{\left(\alpha\right)}\left(0,y\right)\right]\mathfrak{B}_{n-k,q}\left(mx,0\right),$
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n}\left[k+1\right]_{q}}\left[2{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}^{\left(\alpha-1\right)}\left(x,-1\right)\right.$
$\displaystyle-\left.{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}^{\left(\alpha\right)}\left(x,-1\right)-m^{k+1}\mathfrak{E}_{k+1,q}^{\left(\alpha\right)}\left(x,0\right)\right]\mathfrak{B}_{n-k,q}\left(0,my\right)$
holds true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.
Proof. The proof is based on the following identities
$\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)=\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}E_{q}\left(ty\right)\cdot\frac{e_{q}\left(\frac{t}{m}\right)-1}{t}\cdot\frac{t}{e_{q}\left(\frac{t}{m}\right)-1}e_{q}\left(\frac{t}{m}mx\right),$
$\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)=\left(\frac{2}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)\cdot\frac{e_{q}\left(\frac{t}{m}\right)-1}{t}\cdot\frac{t}{e_{q}\left(\frac{t}{m}\right)-1}E_{q}\left(\frac{t}{m}my\right)$
and similar to that of Theorem 8.
Next we discuss some special cases of Theorem 13.
###### Corollary 14
For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship
$\displaystyle\mathfrak{E}_{n,q}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{m^{-n}}{\left[k+1\right]_{q}}\left[2{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}m^{j}\left(x-1\right)_{q}^{j}\right.$
$\displaystyle\left.-{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}m^{j}\mathfrak{E}_{j,q}\left(x,-1\right)-m^{k+1}\mathfrak{E}_{k+1,q}\left(x,0\right)\right]\mathfrak{B}_{n-k,q}\left(0,my\right)$
holds true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.
###### Corollary 15
[8] For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship
holds true.
$\displaystyle E_{n}\left(x+y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\frac{2}{k+1}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\left(y^{k+1}-E_{k+1}\left(y\right)\right)B_{n-k}\left(x\right),$
$\displaystyle E_{n}\left(x+y\right)$
$\displaystyle=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\frac{m^{k-n+1}}{k+1}\left[2\left(x+\frac{1-m}{m}\right)^{k+1}-E_{k+1}\left(x+\frac{1-m}{m}\right)-E_{k+1}\left(x\right)\right]B_{n-k}\left(my\right).$
###### Corollary 16
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\mathfrak{E}_{n,q}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{2}{\left[k+1\right]_{q}}\left(q^{\frac{1}{2}k\left(k+1\right)}y^{k+1}-\mathfrak{E}_{k+1,q}\left(0,y\right)\right)\mathfrak{B}_{n-k,q}\left(x,0\right).$
###### Corollary 17
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\displaystyle\mathfrak{E}_{n,q}\left(x,0\right)$
$\displaystyle=-{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]\
\frac{2}{\left[k+1\right]_{q}}\mathfrak{E}_{k+1,q}\mathfrak{B}_{n-k,q}\left(x,0\right),$
$\displaystyle\mathfrak{E}_{n,q}\left(0,y\right)$
$\displaystyle=-\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{2}{\left[k+1\right]_{q}}\mathfrak{E}_{k+1,q}\mathfrak{B}_{n-k,q}\left(0,y\right).$
These formulas are $q$-analogues of the formula of Srivastava and Á. Pintér
[10].
## 4 $q$-Stirling Numbers and $q$-Bernoulli Polynomials
In this section, we aim to derive several formulas involving the $q$-Bernoulli
polynomials, the $q$-Euler polynomials of order $\alpha,$ the $q$-Stirling
numbers of the second kind and $q$-Bernstein polynomials.
###### Theorem 18
Each of the following relationships holds true for the Stirling numbers
$S_{2}(n,k)$ of the second kind:
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\sum_{j=0}^{n}\left(\begin{array}[c]{c}mx\\\
j\end{array}\right)j!{\displaystyle\sum\limits_{k=0}^{n-j}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}m^{j-n}\mathfrak{B}_{k,q}^{\left(\alpha\right)}\left(0,y\right)S_{2}\left(n-k,j\right),$
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\sum_{j=0}^{n}\left(\begin{array}[c]{c}mx\\\
j\end{array}\right)j!{\displaystyle\sum\limits_{k=0}^{n-j}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}m^{j-n}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(0,y\right)S_{2}\left(n-k,j\right).$
The familiar $q$-Stirling numbers $S(n,k)$ of the second kind are defined by
$\frac{\left(e_{q}\left(t\right)-1\right)^{k}}{\left[k\right]_{q}!}=\sum_{m=0}^{\infty}S_{2,q}\left(m,k\right)\frac{t^{m}}{\left[m\right]_{q}!},$
where $k\in\mathbb{N}.$ Next we give relationship between $q$-Bernstein basis
defined by Phillips [27] and $q$-Bernoulli polynomials
$b_{n,k}\left(q;x\right):=x^{k}\left(1-x\right)_{q}^{n-k}.$
###### Theorem 19
We have
$b_{n,k}\left(q;x\right)=x^{k}\sum_{m=0}^{n}\left[\begin{array}[c]{c}n\\\
m\end{array}\right]_{q}S_{2,q}\left(m,k\right)\mathfrak{B}_{n-m,q}^{\left(k\right)}\left(1,-x\right).$
(76)
Proof. The proof follows from the following identities.
$\displaystyle\frac{x^{k}t^{k}}{\left[k\right]_{q}!}e_{q}\left(t\right)E_{q}\left(-xt\right)$
$\displaystyle=\frac{x^{k}t^{k}}{\left[k\right]_{q}!}\sum_{n=0}^{\infty}\frac{\left(1-x\right)_{q}^{n}t^{n}}{\left[n\right]_{q}!}=\sum_{n=k}^{\infty}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{x^{k}\left(1-x\right)_{q}^{n-k}t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=\sum_{n=k}^{\infty}b_{n,k}\left(q;x\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
and
$\displaystyle\frac{x^{k}t^{k}}{\left[k\right]_{q}!}e_{q}\left(t\right)E_{q}\left(-xt\right)$
$\displaystyle=\frac{x^{k}\left(e_{q}\left(t\right)-1\right)^{k}}{\left[k\right]_{q}!}\frac{t^{k}}{\left(e_{q}\left(t\right)-1\right)^{k}}e_{q}\left(t\right)E_{q}\left(-xt\right)$
$\displaystyle=x^{k}\sum_{m=0}^{\infty}S_{2,q}\left(m,k\right)\frac{t^{m}}{\left[m\right]_{q}!}\sum_{n=0}^{\infty}\mathfrak{B}_{n,q}^{\left(k\right)}\left(1,-x\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=x^{k}\sum_{n=0}^{\infty}\left(\sum_{m=0}^{n}\left[\begin{array}[c]{c}n\\\
m\end{array}\right]_{q}S_{2,q}\left(m,k\right)\mathfrak{B}_{n-m,q}^{\left(k\right)}\left(1,-x\right)\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
Finally, in their limit case when $q\rightarrow 1^{-}$, these last result (76)
would reduce to the following formula for the classical Bernoulli polynomials
$B_{n}^{\left(k\right)}\left(x\right)$ and the Bernstein basis
$b_{n,k}\left(x\right)=x^{k}\left(1-x\right)^{n-k}:$
$b_{n,k}\left(x\right)=x^{k}\sum_{m=0}^{n}\left[\begin{array}[c]{c}n\\\
m\end{array}\right]_{q}S_{2}\left(m,k\right)\mathfrak{B}_{n-m}^{\left(k\right)}\left(1-x\right).$
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|
arxiv-papers
| 2012-01-31T17:49:55 |
2024-09-04T02:49:26.887046
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nazim I. Mahmudov",
"submitter": "Nazim Mahmudov Idris",
"url": "https://arxiv.org/abs/1201.6633"
}
|
1202.0007
|
# Direct Detection of Dark Matter Debris Flows
Michael Kuhlen Theoretical Astrophysics Center, University of California,
Berkeley, CA 94720 Mariangela Lisanti PCTS, Princeton University, Princeton,
NJ 08544 David N. Spergel Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08540
###### Abstract
Tidal stripping of dark matter from subhalos falling into the Milky Way
produces narrow, cold tidal streams as well as more spatially extended “debris
flows” in the form of shells, sheets, and plumes. Here we focus on the debris
flow in the Via Lactea II simulation, and show that this incompletely phase-
mixed material exhibits distinctive high velocity behavior. Unlike tidal
streams, which may not necessarily intersect the Earth’s location, debris flow
is spatially uniform at 8 kpc and thus guaranteed to be present in the dark
matter flux incident on direct detection experiments. At Earth-frame speeds
greater than 450 km/s, debris flow comprises more than half of the dark matter
at the Sun’s location, and up to 80% at even higher speeds. Therefore, debris
flow is most important for experiments that are particularly sensitive to the
high speed tail of the dark matter distribution, such as searches for light or
inelastic dark matter or experiments with directional sensitivity. We show
that debris flow yields a distinctive recoil energy spectrum and a broadening
of the distribution of incidence direction.
## I Introduction
The Galactic dark matter halo forms through a process of hierarchical
structure formation, with smaller halos merging together to form a larger
host. This process of halo accretion occurs over the span of billions of
years. Dark matter that merged early on virializes and is smoothly distributed
at present. However, more recent mergers can leave relic structures in the
Milky Way, observed as features in the spatial and velocity distribution of
Galactic halo stars. Understanding the origin of these features is critical
for piecing together the formation history of our Galaxy, and can provide
clues for distinctive signatures to search for with dark matter experiments.
The dark matter in the solar neighborhood is commonly assumed to be smoothly
distributed in space and to have a Maxwellian velocity distribution
Drukier:1986tm ; Lewin:1995rx . High resolution numerical simulations of the
hierarchical formation of Milky-Way-like dark matter halos, however, predict
large of phase-space substructure throughout the halo Diemand:2008in ;
Zemp:2008gw ; Springel:2008cc ; Vogelsberger:2008qb . This is in agreement
with collisionless dynamics and Liouville’s theorem, which imply that the
initial cold dark matter three-dimensional phase-space manifold111In Cold Dark
Matter theory, the thermal velocity dispersions are close to zero, yielding a
very thin three-dimensional sheet in phase space as an initial configuration.
evolves in a continuous manner by folding and stretching, but never tearing.
Gravitationally bound subhalos, such as those thought to host the Milky Way
dwarf satellite galaxies, are examples of spatial phase-space substructure.
The current census of Milky Way dwarf satellite galaxies stands at 22, but
many more are likely to be discovered with future surveys Tollerud:2008ze .
Additionally, the simulations predict the presence of many thousands of dark
subclumps within the Milky Way’s virial volume, too small to host a luminous
stellar component, but potentially interesting dark matter annihilation
sources (e.g., Kuhlen:2008aw, ; Kuhlen:2009kx, ). The substructure abundance
relative to the smooth host halo mass distribution is found to decrease
towards the Galactic Center, a natural consequence of the stronger disruptive
tidal forces and shorter dynamical times closer to the center of the
potential. Current estimates based on the results from the highest resolution
numerical simulations find that spatial substructure is unlikely to
significantly modify the local dark matter density at 8 kpc
Kamionkowski:2008vw ; Vogelsberger:2008qb ; Kamionkowski:2010mi . Barring
drastic changes in the properties of very low-mass subhalos,222Calculations of
the contribution of local substructure to $\bar{\rho}$(8 kpc) must extrapolate
subhalo scaling relations for many orders of magnitude below the simulations’
resolution limit. the assumption of a smooth dark matter distribution at 8 kpc
appears to be a good one.
The situation is quite different for velocity substructure. The same tidal
disruption processes that render the local dark matter distribution spatially
smooth are sources of velocity substructure. Indeed, the speed distributions
measured in high resolution numerical simulations exhibit deviations from the
standard Maxwellian assumption, especially at large speeds Kuhlen:2009vh ;
Vogelsberger:2008qb ; Lisanti:2010qx . As we show below, the vast majority of
high-speed dark matter particles in the solar neighborhood have been recently
accreted and are partially phase-mixed, having not yet come into equilibrium
with the rest of the halo. We can further distinguish between velocity
structure that is spatially localized, such as tidal streams, and that which
is spatially homogenized, which we designate as “debris flow” Lisanti:2011as .
Both streams and debris flow arise from the disruption of satellites that fall
into the Milky Way, but differ in the relative amount of phase-mixing that
they have undergone.
Tidal streams consist of material that has been stripped from an infalling
satellite, and that has not yet had the time to spatially mix. It is
dynamically cold (meaning its internal velocity dispersion is much less than
the Milky Way halo’s), and it is still spatially confined to a narrow stream
with a one-dimensional morphology. There are several known examples of stellar
tidal streams in the Milky Way halo. One of the most dramatic examples is the
Sagittarius stream Ivezic:2000ua ; Yanny:2000ty , which is clearly associated
with the on-going tidal disruption of the Sagittarius dwarf galaxy
Johnston:1995vd . The SDSS “Field of Streams” Belokurov:2006ms ;
Belokurov:2006kc contains a number of additional tidal stream candidates such
as the Monoceros and Orphan streams. The existence of stellar streams
associated with disrupting dwarf galaxies implies that dark matter streams
should have formed by a similar mechanism. The presence of such a stream in
the local neighborhood could significantly affect the predictions of event
rates and recoil spectra at direct detection experiments Alves:2010pt ;
Lang:2010cd ; Gondolo:2005hh ; Kuhlen:2009vh . While dark matter streams have
been identified in N-body simulations Zemp:2008gw ; Vogelsberger:2008qb ;
Kuhlen:2009vh ; Maciejewski:2010gz ; Elahi:2011dy , the probability that a
single stream dominates the local dark matter density is less than 1%
Vogelsberger:2008qb .
In this paper, we instead focus on debris flow, which represents a more
ubiquitous type of velocity substructure. The term “debris flow” refers to the
sum total of all material stripped from infalling subhalos that has not
completely phase-mixed. As such, it comprises dynamically cold and narrow
tidal streams from recently infalling subhalos, older tidal streams that have
been wrapped a number of times, as well as material that was lost from halos
in the form of sheets and plumes in the violent gravitational shocks
experienced at pericenter passages Choi:2008xi . Rather than considering
multiple debris flows, each associated with an individually disrupting
subhalo, we instead view debris flow as a single feature of the velocity
distribution.
The salient difference between an individual tidal stream and debris flow is
that the former is dynamically cold and has a one-dimensional morphology,
while the latter is dynamically hot and is spatially ubiquitous in the central
regions of the Milky Way halo. For a collisionless system, any diffusion in
configuration space must be accompanied by a decrease in the width of the
velocity distribution in order to preserve phase-space density. Therefore, one
might expect debris flow to be colder, not hotter, than tidal streams.
However, because debris flow is the superposition of tidal debris from many
disrupting satellites, its velocity dispersion is due to the relative velocity
of material stripped from distinct subhalos as well as from the intersections
of a single halo’s tidal stream with itself. While the individual fine-grained
components of the tidal debris must be very cold, in aggregate they appear
dynamically hot. As such, debris flow is velocity structure that is
intermediate between the fully equilibrated host halo and dynamically cold and
narrow tidal streams.
In Lisanti:2011as , we used the Via Lactea-II simulation Diemand:2008in ;
Kuhlen:2008qj to study one sub-component of the debris flow, namely the
portion that was bound to halos at the time of reionization ($z\sim 9$).
Already this component exhibited an interesting speed distribution strongly
peaked at $\sim 340$ km/s in the Galactic frame, quite unlike that of the
underlying relaxed host halo. Here we extend this analysis by following a
larger sample of subhalos throughout their entire accretion history. This
allows us to get a better understanding of the origin and make-up of the
debris flow.
Owing to its spatial homogeneity, this debris flow is guaranteed to be present
in the solar neighborhood, and it is therefore very important to understand
its implications for direct detection experiments, which are sensitive to the
local distribution of dark matter velocities Lewin:1995rx . These experiments
consist of shielded detectors that measure the recoil energies of target
nuclei scattering off dark matter particles passing through the Earth
Gaitskell:2004gd . The expected recoil spectrum is different for dark matter
that is in velocity substructure rather than in the equilibrated component of
the halo. We will show in § IV that debris flow results in a distinctive
recoil spectrum, with more high energy events than is typically expected from
the canonical Maxwellian velocity distribution. These differences may be
important in ameliorating the current tension between experiments, some of
which have been observing anomalous signals Bernabei:2008yi ; Bernabei:2010mq
; Aalseth:2010vx ; Aalseth:2011wp ; Angloher:2011uu while others have not
Ahmed:2012vq ; Aprile:2011hi ; Ahmed:2009zw ; Akerib:2005kh ; Edelweiss:2011cy
; Angle:2007uj ; Angle:2009xb ; Angloher:2004tr ; Alner:2007ja ;
Lebedenko:2008gb ; Akimov:2011tj ; Ahmed:2010wy ; Collar:2011kf ; Angle:2011th
.
This paper is organized as follows. In § II, we describe in detail our
procedure for identifying debris flow particles in the Via Lactea II
simulation, and show that debris flow dominates the local dark matter
distribution at high speeds. In § III, we present the speed distribution of
the debris flow and discuss its origin. In § IV, we go on to explore the
implications of debris flow for direct detection experiments. This section
also includes a simple model that accurately captures the phenomenology, and
which can be used to model debris flow effects without resorting to high
resolution numerical simulations. Finally, in § V,we present a brief
discussion of the results and a conclusion.
## II Identification of Debris Flow Particles
We use the Via Lactea-II (VL2) N-body simulation Diemand:2008in ;
Kuhlen:2008qj to study the formation of debris flow in the Milky Way. VL2 is
one of the highest resolution cosmological dark-matter-only simulations of the
formation of a galactic halo. It resolves the virial volume of a Milky-Way-
sized halo with about 1 billion particles of mass $4.1\times 10^{3}\text{
M}_{\odot}$ embedded in a cubic volume of 40 Mpc per side. The simulation is
initialized at redshift 104.3 assuming a WMAP3 $\Lambda$CDM cosmology
Spergel:2006hy , and evolved to the present. VL2 resolves large amounts of
substructure in the Galactic halo, including subhalos and dark matter streams
Zemp:2008gw ; Kuhlen:2009vh . Throughout the evolution, 400 full outputs,
equally spaced in time, were written to disk.
The 6DFOF halo-finding algorithm Diemand:2006ey was used to identify the
tightly bound centers of all (sub-)halos at 27 of the outputs (roughly every
$\sim 680$ Myr). These centers were used to construct spherical density
profiles, from which we calculated subhalo properties like $V_{\rm max}$,
$R_{\rm Vmax}$, and the tidal radius (or $R_{200}$ for isolated halos) and
corresponding mass. The subhalos are linked through time in two sets of
evolutionary tracks. The first one ($T_{0}$) starts with the 20 048 subhalos
that have an identifiable remnant at $z=0$ and reached $V_{\rm max}>4$ km/s at
some time, and traces their most massive progenitor halo backwards through
time from $z=0$. The second ($T_{4.56}$) starts with the 20 000 most massive
halos at $z=4.56$ and traces their descendant halos forward through time. The
overlap between the two tracks consists of 11 870 subhalos, and 7 433 subhalos
in $T_{4.56}$ do not have a $z=0$ remnant. Both sets of tracks (as well as
additional data) are available at the Via Lactea Project webpage VLwebsite .
For the 20 000 subhalos in $T_{4.56}$, we additionally traced the 6DFOF-linked
central particles through the intermediate outputs, so we have orbital
information (positions and velocities) at all 400 outputs (every $\sim$ 34
Myr).
A dark matter particle is labeled as “debris” if it was bound to some halo at
$z>0$ and is no longer bound to any halo but the host today. Operationally, we
restrict our analysis to the 4 232 452 particles located between 7.5 and 9.5
kpc at $z=0$, and determine for each of these particles to which (sub-)halos
it is bound at every one of the 27 coarse outputs. For every particle, we
determine whether it is debris, and, if so, at what redshift it was stripped
off its birth halo. Because our halo finding procedure does not “un-bind”
particles, we consider a particle to be bound to a halo if it lies within the
halo’s tidal radius. This may slightly overestimate the amount of debris,
because a small fraction of particles are assigned halo membership even though
they are just passing through, but we have explicitly checked that this is not
a large effect for a small subsample of halos, and do not expect this slight
overestimate to affect our conclusions.
Figure 1: Fractional density of debris particles above some minimum speed,
$v_{\text{min}}$, in the Earth’s rest frame (in June). The solid line is for
debris particles with a $z=0$ remnant halo (from tracks $T_{0}$), and the
dotted line for high redshift debris from halos that are completely disrupted
by $z=0$.
We construct debris catalogs from both $T_{0}$ and the subset of subhalos in
$T_{4.56}$ that do not have a $z=0$ remnant. Note that this constitutes a
marked improvement over our earlier study Lisanti:2011as , in which we
considered only particles that were bound around the time of reionization
$z\sim 9$. Figure 1 shows the fractional contribution ($N_{\rm debris}/N_{\rm
tot}$) above a given Earth rest-frame speed. In total ($v_{\rm min}=0$ km/s),
about 90% of all particles at 8 kpc are debris, with 70% having been stripped
from halos that were completely disrupted prior to $z=0$, and 20% from halos
with surviving remnants. At higher $v_{\rm min}$, the relative contributions
are reversed: debris from surviving subhalos exceeds debris from fully
disrupted subhalos at 400 km/s, and makes up more than half of all the
material at $v_{\rm min}>450$ km/s. Debris from surviving subhalos contributes
as much as 85% of the local material at $v_{\rm min}=650$ km/s. The $T_{0}$
debris curve is well fit by a Gauss error function,
$\epsilon(E_{R})\,\simeq\,0.22+0.34\,\left[{\rm erf}\\!\left(\frac{v_{\rm
min}-465\,{\rm km/s}}{185\,{\rm km/s}}\right)+1\right].$ (1)
## III Formation Process of Debris Flows
Figure 2: Top : Normalized speed distributions for debris from subhalos that
are still present at $z=0$ (solid line), from subhalos present at $z=4.56$ but
not at $z=0$ (dotted line), for all particles in the Milky Way (black dashed
line), and for non-debris particles (gray dashed line). The comparison is made
for particles in the radial shell $7.5<r<9.5$ kpc. Bottom: Histogram of speed
distribution for the debris flow (solid black), as well as the distributions
of particles from a sample of subhalos that contribute the most to the debris
flow (colored dashed: 19765:purple, 19624:green, 17928:blue, 17689:red,
18506:yellow). The left panel shows the distributions in the Galactic frame,
while the right panel is in the Earth frame (assuming $t_{\text{max}}$ = June
2).
The left panel of Figure 2 shows the Galactic rest-frame speed distribution of
the debris particles in the radial shell $7.5<r<9.5$ kpc, compared to the
distribution of all particles, as well as non-debris particles, in the same
radial shell. Note that these distributions are separately normalized in order
to highlight the difference in their shapes, but as a result their heights do
not reflect the relative contributions of each component (see Figure 1 for
that information). The total distribution (black dashed) exhibits the well-
known Diemand:2004kx ; Hansen:2005yj ; Vogelsberger:2008qb ; Kuhlen:2009vh ;
Fairbairn:2008gz ; Catena:2011kv departures from the shape of a Maxwellian
distribution, consisting of a deficit near the peak and an excess at high
speeds. The speed distribution for non-debris particles (grey dashed) is
similar to the distribution for debris from fully disrupted subhalos (dotted),
indicating that the $T_{4.56}$ debris has equilibrated with the host halo. In
contrast, the debris from surviving subhalos has an intriguing high-speed
behavior, with a distribution (solid) peaked at $\sim 350$ km/s. This is
consistent with the results of Lisanti:2011as , which considered only a subset
of particles contributing to the debris flow. For the remainder of this paper
we consider the debris from fully disrupted subhalos to be part of the
background halo, and henceforth the term “debris flow” will refer to debris
from subhalos with a surviving $z=0$ remnant only.
On the right side of Figure 2, we show the corresponding distributions shifted
into the Earth’s frame. These distributions are obtained by applying a
Galilean boost of
$\vec{v}_{e}(t)=\vec{v}_{\text{LSR}}+\vec{v}_{\text{pec}}+\vec{v}_{\oplus}(t),$
(2)
where $\vec{v}_{\text{LSR}}=(0,220,0)\text{ km/s}$ is the velocity of the
local standard of rest (LSR) Majewski:2008pz ,
$\vec{v}_{\text{pec}}=(10,5.23,7.17)\text{ km/s}$ is the Sun’s peculiar
velocity with respect to the LSR Dehnen:1997cq , and $\vec{v}_{\oplus}$ is the
velocity of the Earth in the Sun’s rest frame, as specified in Savage:2008er ;
Gelmini:2000dm . These velocities are taken in the coordinate system where
$\hat{x}$ points towards the Galactic center, $\hat{y}$ points in the
direction of Galactic rotation, and $\hat{z}$ points towards the Galactic
north pole. These coordinates are associated with the
($v_{r},v_{\theta},v_{\phi}$) coordinates of the VL2 particles, for an
arbitrary assignment of the Galactic plane. The right side of Fig. 2 shows
that the transformation into the Earth frame smooths out some of the peaks in
the debris flow distribution observed in the Galactic rest-frame.333This
smoothing arises because, in the transformation from Galactic to Earth frame,
particles with different Galactic frame speed can end up with the same Earth
frame speed, depending on their direction with respect to the Earth. The
debris flow distribution, however, maintains a significantly different shape
and is shifted towards higher speeds.
Subhalo ID | Mass ($z=0$) | $R_{\rm gc}(z=0)$ | Infall Mass | $z_{\rm infall}$ | $N_{\rm peri}$ | min($D_{\rm peri}$) | $f_{\rm debris}$
---|---|---|---|---|---|---|---
| [$\text{ M}_{\odot}$] | [${\rm kpc}$] | [$\text{ M}_{\odot}$] | | | [${\rm prop.kpc}$] |
19765 | $9.8\times 10^{6}$ | 20.9 | $4.1\times 10^{9}$ | 1.9 | 12 | 4.1 | $1.2\times 10^{-1}$
19624 | $5.8\times 10^{8}$ | 21.8 | $2.7\times 10^{10}$ | 1.6 | 6 | 6.6 | $9.3\times 10^{-2}$
17928 | $5.7\times 10^{7}$ | 42.3 | $5.8\times 10^{9}$ | 2.9 | 15 | 5.9 | $4.5\times 10^{-2}$
17689 | $1.2\times 10^{7}$ | 44.6 | $7.9\times 10^{9}$ | 2.9 | 15 | 3.7 | $3.2\times 10^{-2}$
18506 | $4.3\times 10^{6}$ | 34.1 | $1.1\times 10^{9}$ | 3.6 | 21 | 2.4 | $2.8\times 10^{-2}$
18646 | $2.9\times 10^{8}$ | 41.0 | $2.5\times 10^{9}$ | 1.3 | 4 | 44 | $1.3\times 10^{-3}$
Table 1: The top five subhalos contributing the most mass to the debris flow,
plus one fairly massive halo that contributes only very little. The table
lists each subhalo’s ID, its $z=0$ mass and Galacto-centric radius ($R_{\rm
gc}$), its mass at infall (first crossing of the host’s $R_{\rm vir}$) and the
redshift this occurred ($z_{\rm infall}$), the number of pericenter passages
($N_{\rm peri}$) and minimum pericenter distance ($D_{\rm peri}$) of its
orbit, as well as the mass fraction of debris ($f_{\rm debris}$) it
contributes. Figure 3: Mass ratio ($M/M_{\rm infall}$; blue dots), Galacto-
centric distance (blue lines) and relative speed (magenta lines) as a function
of time for the five subhalos contributing the most mass to the debris flow,
and one additional halo (bottom right panel) contributing only very little,
$f_{\rm debris}=9.8\times 10^{-4}$. See Table 1 for more information about
these subhalos. Subhalo masses have only been determined at coarsely spaced
outputs (every $\sim 680$ Myr), but the orbits of the subhalos’ most strongly
bound central particles (i.e. 6DFOF-linked) have been traced in the
intermediate outputs (every $\sim 34$ Myr). The dotted line indicates the
virial radius of the host halo.
The presence of several discrete sub-peaks in the debris distribution (e.g.,
at 330, 380, 420, and 460 km/s in the Galactic frame distribution) hints at
the importance of a few individual halos. This is confirmed in the lower left
panel of Figure 2, in which we show the debris flow distribution on a
logarithmic scale and over-plot the corresponding distributions (normalized to
total debris flow) for the five subhalos contributing the most mass to the
debris flow. Although no one subhalo dominates the peak at $\sim 350$ km/s,
individual features are easily identified as being associated with one of
these halos: the broad shoulder at $\sim 100$ km/s as well as the peak at 460
km/s, for example, are contributed by halo 19624, and the peaks at 330, 380,
and 420 km/s come from halo 19765. In total, these five subhalos make up 31.8%
of the debris flow. Figure 3 shows their mass loss and orbital information
(Galacto-centric distance and speed) as a function of time, and their
properties are summarized in Table 1. The number of pericenter passages
undergone by these subhalos is strikingly high. With the exception of 19624
($N_{\rm peri}=6$), they all have experienced more than 10 pericenter
passages, and subhalo 18506 had more than 20. For comparison, the mean number
of pericenter passages for all $T_{4.56}$ subhalos with at least one
pericenter passage is only 4.3. The top five debris-contributing subhalos all
have multiple deep pericenter passages, reaching considerably below 10 kpc,
which enables them to contribute to the local debris flow at 8 kpc.
The high number and depth of their pericenter passages is reflected in a large
amount of mass loss. From first infall until $z=0$, these five subhalos have
lost between 97.9% and 99.7% of their infall mass, and it is this material
that makes up their contribution to the debris flow. Note that their mass loss
is strongest in the earlier parts of their orbits Kravtsov:2004cm ;
Diemand:2007qr . As a contrast, we show in the bottom right panel of Figure 2
the orbital information for a sixth halo, which is representative of the
population of subhalos that only contributes weakly to the debris. This
subhalo has undergone a smaller number of pericenter passages, none of which
reach closer than 44 kpc from the center, and it has lost less than 90% of its
mass.
The speed of $\sim 350$ km/s at which the main debris flow peak occurs is
easily explained by energy conservation, as it simply reflects the speed of
the debris particles’ parent subhalos orbiting in the Galactic potential. The
five representative subhalos discussed above have infall redshifts between 3.6
and 1.6, and initial apocenter distances ranging from 74 to 176 kpc, with a
mean of $\langle D_{\rm apo}^{i}\rangle=96$ kpc. As the subhalos orbit in the
host halo, they lose mass from tidal stripping and their orbits shrink due to
the influence of dynamical friction and in response to the steadily growing
mass of the host halo interior to their orbits. This shrinking continues until
they become so light that dynamical friction ceases to be efficient ($M_{\rm
sub}/M_{\rm host}\lesssim 10^{-2}$, BoylanKolchin:2007ku ) and the host halo’s
mass accretion halts Diemand:2007qr . The apocenters of our five subhalos
shrink by a factor of $D_{\rm apo}^{f}/D_{\rm apo}^{i}=0.47$ to 0.78, with a
mean final apocenter of $\langle D_{\rm apo}^{f}\rangle=59$ kpc. At this
distance, they have a mean speed of $\langle v_{\rm apo}\rangle=54$ km/s. The
difference in the late time ($z<1$) host halo potential between 59 kpc and 8.5
kpc is $6.7\times 10^{4}$ (km/s)2, and hence conservation of energy implies a
mean speed at 8.5 kpc of 370 km/s, which is in very good agreement with the
peak speed of the debris flow. These results also correspond well with the
energy-infall relation recently elaborated on by Rocha:2011aa .
Figure 4: Scatter plots of infall mass vs. number of pericenter passages
(left) and closest pericenter approach vs. infall redshift (right). The size
and color of the symbols indicate the fractional contribution that a subhalo
makes to the debris flow. Only subhalos with surviving remnants at $z=0$ are
plotted.
In Figure 4, we extend our analysis to the full set of subhalos contributing
to the debris flow. In the left panel, we show a scatter plot of the subhalos’
infall mass versus their number of pericenter passages ($N_{\rm peri}$). The
size and color of the symbols represent the fraction of the debris flow that a
given subhalo contributes. In the right panel, we plot the distance of the
deepest pericenter approach (min($D_{\rm peri}$)) against the subhalo’s infall
redshift. These two plots clearly demonstrate that the trends observed for the
top five contributing subhalos continue to hold for the entire population. The
amount of material contributed to the debris flow tends to increase with
increasing infall mass, with larger $N_{\rm peri}$, and with decreasing
min($D_{\rm peri}$) of the subhalo. The largest fraction of debris flow is
contributed by subhalos accreting between $z=1.5$ and 4, and about 40% of the
debris is contributed by halos brought in with the last major merging event at
$z\sim 1.7$.
Infall Mass | $f_{\rm debris}$
---|---
$>10^{10}\text{ M}_{\odot}$ | 0.12
$10^{9}-10^{10}\text{ M}_{\odot}$ | 0.42
$10^{8}-10^{9}\text{ M}_{\odot}$ | 0.21
$10^{7}-10^{8}\text{ M}_{\odot}$ | 0.16
$10^{6}-10^{7}\text{ M}_{\odot}$ | 0.061
$<10^{6}\text{ M}_{\odot}$ | 0.027
Table 2: The fraction of the debris flow contributed by halos in the given
mass range.
To some degree, the larger fractional contribution of more massive individual
halos is simply a result of their having more material to lose. In principle,
it would be possible for the far greater number of low infall mass subhalos to
contribute more to the debris flow in aggregate than the more massive ones.
This turns out not to be the case: the majority ($>53$%) of debris flow is
contributed by subhalos with infall masses greater than $10^{9}\text{
M}_{\odot}$, and halos with infall masses below $10^{7}\text{ M}_{\odot}$ only
contribute $<10$% (see Table 2). This result gives us confidence that our
results are not highly sensitive to the numerical resolution of the VL2
simulation.
## IV Direct Detection Phenomenology
Direct detection experiments are sensitive to the scattering of dark matter
particles off a target nucleus. The recoil spectrum measured by these
experiments depends on the distribution of dark matter speeds and is thus
sensitive to the presence of local velocity substructure in the form of
streams or debris flow. The implications of tidal streams for direct detection
experiments have been explored by Freese:2003tt ; Gelmini:2004gm ;
Kuhlen:2009vh ; Lang:2010cd ; Alves:2010pt , and more recently by
Natarajan:2011gz in light of the CoGeNT anomaly Aalseth:2010vx ;
Aalseth:2011wp . In this section, we will derive a semi-analytic model for the
recoil spectrum of debris flows. The phenomenological model presented here is
a function of a single parameter and can easily be used to find the expected
spectrum of events from scattering off the debris flow.
For a direct detection experiment with target nucleus of mass $m_{N}$, the
differential scattering rate per unit detector mass is Lewin:1995rx
$\frac{dR}{dE_{R}}=\frac{\rho_{0}}{m_{N}m_{\text{dm}}}\sigma(E_{R})g(v_{\text{min}}),$
(3)
where $\rho_{0}$ ($\approx 0.3$ GeV/cm3) is the local dark matter density,
$m_{N}$ is the nuclear mass, $E_{R}$ is the nuclear recoil energy,
$\sigma(E_{R})$ is the energy-dependent scattering cross section, and
$g(v_{\text{min}})$ is a function of the detector’s threshold speed. The
differential scattering rate is sensitive to the distribution of dark matter
speeds in the Earth frame $f(v)$, and thus depends on whether the dark matter
is virialized or in a stream or debris flow. The relevant quantity is
$g(v_{\text{min}})=\int_{v_{\text{min}}}\frac{f(v)}{v}dv,$ (4)
where the threshold speed $v_{\text{min}}$ is given by
$\sqrt{m_{N}E_{R}/2\mu^{2}}$ for elastic scattering. If the scattering is
dominated by a Maxwellian distribution $f(v)\propto v^{2}e^{-v^{2}/v_{0}^{2}}$
in the Galactic frame, the expected recoil spectrum is exponentially falling
Lewin:1995rx . If, in contrast, the local dark matter is dominated by a
stream, then the scattering rate is constant up to a recoil energy
corresponding to $|\vec{v}_{\rm stream}-\vec{v}_{e}|$ Freese:2003tt , where
$\vec{v}_{e}$ is given in Eq. 2.
The particles in the debris flow have speeds characterized by the distribution
function
$f(v)=\frac{1}{N}\frac{dN}{dv}=\frac{1}{N}\frac{dN}{d\cos\theta_{e}}\frac{d\cos\theta_{e}}{dv}$
(5)
in the Earth frame, where N is the total number of debris particles and
$\theta_{e}$ is the angle between the velocities of the flow particles in the
Galactic frame and the direction of Earth’s motion. This angle is related to
the Earth-frame velocities through
$v^{2}=v_{\text{flow}}^{2}+v_{e}(t)^{2}-2v_{\text{flow}}v_{e}(t)\cos\theta_{e},$
(6)
where $v_{\text{flow}}$ is the speed of the debris flow in the Galactic frame.
Figure 5: Tangential vs radial velocity components (km/s) of dark matter
within a radial shell $7.5<r<9.5$ kpc in the Galactic frame. On the left, the
distribution for dark matter debris and in the middle, the distribution for
all VL2 particles in this radial shell. The right panel shows the distribution
of debris particles (blue triangles) and all VL2 particles without the debris
contribution (red circles) as a function of $\cos\theta_{e}$, where
$\theta_{e}$ is the angle between the velocities of the particles in the
Galactic frame and the direction of Earth’s motion.
A complete expression for f(v) depends on how the debris particles are
distributed as a function of $\cos\theta_{e}$. Figure 5 shows the tangential
and radial Galactic-frame velocity distributions for the debris (left) and for
all VL2 particles (middle) in a 7.5–9.5 kpc radial shall. The right panel
shows the distribution of debris particles as a function of $\cos\theta_{e}$.
The results show that the debris flow is nearly uniformly distributed
(isotropic) in $\cos\theta_{e}$, with $dN/d\cos\theta_{e}=N/2$.
To proceed, we make two simplifying assumptions. First, we neglect any
dispersion in the Galacto-centric speed of the debris flow and treat its
distribution as a delta function centered on $v_{\rm flow}$. Secondly, we
assume that the debris flow is isotropic. The true distribution of the debris
flow’s radial and tangential velocity components exhibits non-zero dispersion
and a small tangential bias, but, as we show below, our simplified model
nevertheless captures the main features of the recoil spectrum.
With these assumptions, the Earth-frame speed distribution function of the
debris flow is given by
$f_{\text{flow}}(v)=\begin{cases}\frac{1}{2}\frac{v}{v_{\text{flow}}v_{e}(t)}&{\rm
if}\;(v_{\rm flow}-v_{e})<v<(v_{\rm flow}+v_{e}),\\\ 0&{\rm
otherwise}.\end{cases}$ (7)
Substituting this into (4) and integrating, we find that the recoil spectrum
for the debris flow is proportional to
$g(v_{\text{min}})=\begin{cases}\frac{1}{v_{\text{flow}}}&{\rm
if}\;v_{\text{min}}<(v_{\text{flow}}-v_{e}),\\\
\frac{v_{\text{flow}}+v_{e}-v_{\text{min}}}{2v_{\text{flow}}v_{e}}&{\rm
if}\;(v_{\text{flow}}-v_{e})<v_{\text{min}}<(v_{\text{flow}}+v_{e}),\\\ 0&{\rm
if}\;v_{\text{min}}>(v_{\text{flow}}+v_{e}).\end{cases}$ (8)
The only input parameter in this equation is the speed of the debris flow in
the Galactic frame. The left panel of Figure 6 shows the semi-analytic model
for $g(v_{\text{min}})$ for $v_{\text{flow}}=340$ km/s (dashed red). Overlaid
on the same plot is the distribution obtained directly from the VL2 simulation
for the debris flow (solid black) and all other particles (solid gray) from
7.5–9.5 kpc. The semi-analytic model captures the main features of the debris
flow distribution remarkably well, even though it ignores the velocity
dispersion and small tangential bias of the flow.
Because debris flow particles have relatively large speeds compared to the
virialized component of the halo, they mainly contribute to nuclear recoils
with large energies. To illustrate this, let us consider the recoil energy
spectrum for a 10 GeV elastically scattering dark matter particle. The
scattering rate for this low-mass dark matter candidate is given by Eq. 3,
with an energy-dependent cross section Jungman:1995df
$\sigma(E_{R})=\frac{m_{N}\sigma_{N}}{2\mu^{2}}\frac{(f_{p}Z+f_{n}(A-Z))^{2}}{f_{p}^{2}}|F_{H}(E_{R})|^{2},$
(9)
where the detector target has charge $Z$ and atomic number $A$, $\mu$ is the
reduced mass of the dark matter-nucleus system, $\sigma_{N}$ is the cross
section for the dark matter-nucleus interaction at zero momentum transfer
(10${}^{-41}\text{ cm}^{2}$ for this example), and $f_{p,n}$ are the couplings
to the proton and neutron, respectively. We will take $f_{p}=f_{n}=1$ for the
rest of this section. The Helm form factor $F_{H}(E_{R})$ accounts for the
loss of coherence at large momentum transfer Helm:1956zz .
Figure 6: Left: $g(v_{\text{min}})$ for the debris flow (black) in a 7.5–9.5
kpc spherical shell in VL2. The gray line represents the same distribution for
all other particles in the same VL2 shell. The dashed red line is the
prediction of the semi-analytic model described in the text for
$v_{\text{flow}}=340$ km/s. These distributions are shown for
$t_{\text{max}}=\text{ June 2}$. Right: The modulated amplitude for a 10 GeV
dark matter elastically scattering off a Germanium target with cross section
$10^{-41}\text{cm}^{2}$. The gray line is the distribution for a Maxwellian
distribution with $v_{0}=220$ and $v_{\text{esc}}=550\text{ km/s}$, while the
dashed lines show the spectrum for debris flows with $v_{\text{flow}}=$ 340
(red), 400 (blue), and 460 (green) km/s. The solid red line is the
distribution for scattering off of both a Maxwellian and 340 km/s debris flow,
with relative density given by VL2. The modulated amplitude is half the
difference in rate at peak (June) and minimum (December).
The right panel of Figure 6 shows the recoil energy spectrum of the modulated
amplitude (half the difference between the maximum rate in June and the
minimum in December) for this dark matter candidate scattering off a Germanium
target. The gray line is the spectrum assuming a Maxwellian distribution with
$v_{0}=220\text{ km/s}$ and $v_{\text{esc}}=550\text{ km/s}$, while the dashed
lines are the distributions for a debris flow with Galacto-centric speeds of
340, 400, and 460 km/s (red, blue, green, respectively). The total scattering
rate will have contributions from both the virialized and unvirialized
components of the halo. Therefore, the total rate is a sum of the rates from
individual halo components - i.e., from the Maxwellian component,
$R_{\text{MB}}$, and the debris flow component, $R_{\text{debris}}$:
$\frac{dR_{\text{total}}}{dE_{R}}=(1-\epsilon(0))\frac{dR_{\text{MB}}}{dE_{R}}+\epsilon(0)\frac{dR_{\text{debris}}}{dE_{R}}.$
(10)
The relative contribution from either component depends on the relative
density fraction $\epsilon(0)=N_{\text{debris}}/N_{\text{tot}}$ between the
debris flow and the total number of halo particles, which in the VL2
simulation is 0.22 (see Figure 1). The solid red line in the right panel of
Figure 6 shows the recoil spectrum of the modulated amplitude when scattering
occurs off of both a Maxwellian and 340 km/s debris flow, with
$\epsilon(0)=0.22$. Clearly, the presence of the debris flow leads to more
significant modulation at recoil energies where a Maxwellian distribution
would give little contribution.
It is clear, then, that the density and speed of the debris flow can have
important implications for the expected distribution of events in direct
detection experiments. If the dark matter has a large scattering threshold,
such as in light elastic dark matter or inelastic dark matter
TuckerSmith:2001hy , it may be particularly sensitive to the presence of the
debris flow. Both of these scenarios have received attention recently, in
light of conflicting results from current experiments. The tightest limit for
spin-independent scattering interactions is currently set by XENON100
Aprile:2011hi , and improves upon bounds from CDMS Ahmed:2009zw ;
Akerib:2005kh , EDELWEISS Edelweiss:2011cy , XENON10 Angle:2007uj ;
Angle:2009xb , CRESST Angloher:2004tr , and ZEPLIN Alner:2007ja ;
Lebedenko:2008gb ; Akimov:2011tj . Despite the null results from these
experiments, the DAMA collaboration reports a $9\sigma$ annual modulation
signal Bernabei:2008yi ; Bernabei:2010mq and the CoGeNT experiment reports a
2.8$\sigma$ modulation Aalseth:2010vx . The CRESST experiment has also claimed
an excess of events that cannot be explained with background estimates
Angloher:2011uu . The three anomalies can be made consistent with each other
if the dark matter is light ${\cal O}(10\text{ GeV})$ and scatters off a non-
Maxwellian distribution Frandsen:2011gi . Debris flows may also be able to
explain the observed modulation in CoGeNT at unexpectedly large energies
Fox:2011px . We caution, however, that a non-Maxwellian velocity distribution
by itself does not appear to be sufficient to reconcile these signals with the
exclusion limits from the CDMS low-threshold analyses Ahmed:2010wy ;
Ahmed:2012vq ; Collar:2011kf , XENON10 S2 analysis Angle:2011th , and XENON100
results Aprile:2011hi (see Frandsen:2011gi ).
Figure 7: Mollweide projections of the distributions of incidence direction of
debris particles (left), all particles (middle), and a $10^{7}$ particle
realization of a Maxwellian halo (right). The coordinate system is chosen such
that the Galactic disk normal is aligned with the simulation’s
$\hat{y}$-direction, and the direction anti-parallel to the Earth’s motion is
in the center of the projection. From top to bottom, the rows show the
distributions for particles with Earth-frame speeds $<200\text{ km
s${}^{-1}$}$, $200-350\text{ km s${}^{-1}$}$, $350-500\text{ km s${}^{-1}$}$,
and $>500\text{ km s${}^{-1}$}$.
The presence of velocity substructure could be even more important for
experiments that are sensitive to the direction of the scattering dark matter
particles Kuhlen:2009vh , rather than just their speed. Indeed, directionally
sensitive detectors, such as DRIFT Burgos:2007zz , DMTPC Sciolla:2009fb ,
MIMAC Santos:2011kf , and NEWAGE Miuchi:2010hn (see Ahlen:2009ev for a
summary of the current state of experimental efforts), require large recoil
energies in order to follow the recoil tracks, which are typically only a few
millimeters in length. These experiments are thus quite likely to be impacted
by the non-Maxwellian velocity structure arising from debris flows.
To investigate the expected directional signatures of the debris flow in more
detail, we present in Figure 7 Mollweide projections of the distribution of
incidence directions for debris particles (left), for all particles (center),
and for a Maxwellian halo (right). The coordinate system is chosen such that
the direction anti-parallel to the Earth’s motion corresponds to the center of
the maps. We show the distributions split into four distinct Earth-frame speed
bins, in order to demonstrate trends with recoil energy. The incidence
directions of the debris flow particles are distributed more broadly and less
uniformly than for the Maxwellian halo, and they exhibit remarkable ring-like
structures, most pronounced in the 350 km/s $<v<$ 500 km/s bin.444The detailed
morphology and strength of these features depends somewhat on the orientation
of the Galactic disk plane, which is not specified in the purely dark matter
VL2 simulation. Such features arise because the debris flow is peaked at one
Galacto-centric speed ($\sim 350$ km/s), but is nearly isotropic in direction.
Debris flow particles that happen to be traveling in the direction anti-
parallel to Earth are boosted out of the 350–500 km/s bin, and far fewer lower
speed particles are boosted into this bin. The result is a hole in the center
of the distribution. Similar effects occur in the other speed bins.
Experimentally, there is no way to determine whether a given recoil event
originated with a debris flow or a relaxed halo particle, and so it perhaps
makes more sense to look at the combined distribution for all particles, as
shown in the middle column of Figure 7. Now, the ring-like features are washed
out by the dominating relaxed halo component, but in the two highest speed
(i.e. recoil energy) bins a pronounced asymmetry is still visible. Comparing
to the equivalent Maxwellian distributions (in the right column), it is
apparent that the debris flow has two effects at high speeds: (i) it broadens
the distribution of incidence directions and (ii) the peak of the distribution
(the hotspot direction) can be shifted away from the direction anti-parallel
to the Earth’s motion. The latter effect is due to anisotropy in the direction
of debris flow particles.
## V Conclusions
This work presents a detailed analysis of the properties of dark matter debris
flow in the VL2 simulation. Debris flow is an example of spatially-uniform
velocity substructure that consists of overlapping sheets, streams, plumes and
shells created by dark matter that is tidally stripped from subhalos falling
into the Milky Way. Subhalos that contribute dominantly to debris flow
typically have large infall mass ($\gtrsim 10^{9}$ M⊙) and make numerous
($\gtrsim 10$) pericenter passages, with a minimum pericenter distance within
8 kpc.
Debris flow is distinct from dark matter streams. Although both arise from
tidal disruption of satellites, streams are dynamically colder than debris
flow and have not had time to spatially mix. Streams consist of particles that
are spatially confined and coherent in velocity space. In contrast, debris
flow is spatially-mixed over a large volume, yet retains distinctive velocity
behavior because it is not completely virialized. In VL2, the debris flow has
a speed peaked at a magnitude of $\sim 340$ km/s.
Debris flow is ubiquitous in the solar neighborhood; approximately 20% of all
dark matter particles in VL2 between 7.5–9.5 kpc are identified as debris
flow. This fraction increases to 50% for particles with speeds greater than
450 km/s, and rises to 80% at 600 km/s. The prevalence of debris flow makes it
highly relevant for direct detection experiments. In particular, if the dark
matter has a large minimum scattering threshold, then direct detection
experiments will be sensitive to its presence. The recoil spectrum is
different from that expected for a standard Maxwellian distribution, with more
scattering events at large nuclear recoil energies. Our simple parametrization
for the debris flow recoil spectrum allows one to analytically determine the
deviations from a Maxwellian expectation for a debris flow of given speed and
density.
Although the primary focus of this work has been to study debris flow in the
context of dark matter, we conclude with some preliminary thoughts on the
relevance of debris flow to the stellar halo. The dense cores of subhalos were
the site of star formation billions of years ago and these stars are tidally-
stripped, along with dark matter, as the subhalos fall into the Milky Way. As
a result, the distribution of stars in the halo is not smooth, and exhibits
phase-space features that are correlated with accretion events in the Galaxy
Johnston:1996sb ; Johnston:1997fv ; Bullock:2005pi ; Johnston:2008fh ;
Harding:2000zt ; Helmi:1999ks . A tidal stream is an example of such a
feature, and evidence for streams has been found using deep photometric wide-
field surveys, such as SDSS Ivezic:2000ua ; Yanny:2000ty , the Spaghetti
Survey Morrison:2000gp and the Two Micron All Sky Survey 2MASS (see
Helmi:2008eq for a review).
The presence of stellar streams strongly suggests that debris flow should also
be present and potentially detectable. Ideally, a search for spatially-uniform
velocity substructure requires complete kinematic information of stars. The
upcoming GAIA satellite Perryman:2001sp will obtain the largest and most
accurate sample of proper motions in the solar neighborhood to date and will
therefore be an integral step in mapping out the velocity domain. In the
meantime, a study using the position and radial velocity measurements of
metal-poor main sequence turnoff stars in 137 SEGUE lines of sight has found
evidence for velocity substructure Schlaufman:2009jv . This study identified
10 high-confidence and 21 lower-confidence555They expect 3 false positives in
this subset. detections, referred to as ECHOS (Elements of Cold HalO
Substructure), within 17.5 kpc of the sun. Each detection consists of ${\cal
O}(20)$ stars uniformly distributed along a large patch of sky with a radial
velocity distribution that differs from the expected background. In addition,
the ECHOS are chemically distinct from the kinematically smooth stellar halo
background, strongly suggesting a separate origin Schlaufman:2011kf . Because
the detections are spread out over large areas of the sky, they do not exhibit
a stream-like morphology. Indeed, the morphology more closely resembles that
of debris flow, and it will be useful to explore whether the ECHOS can be
explained as the tidal debris of many infalling satellites.
Discovery of debris flow in the stellar halo would provide critical
information for dark matter searches, suggesting that there are more high
speed particles (with speeds in excess of the most probable speed) in the
solar neighborhood than expected for a Maxwellian standard halo model. This
would alter the expectation for the nuclear recoil spectrum and modulation
fraction in direct detection experiments, as well as the angular distribution
of events in directional experiments. In light of a detection, the results
from both the dark matter and stellar searches will shed light on the matter
distribution in our Galaxy and its tumultuous merger history.
## Acknowledgements
We thank Jesse Thaler, David Weinberg, and Neil Weiner for useful discussions,
and Jim Cline and Wei Xue for catching an error in our debris flow toy model.
ML acknowledges support from the Simons Postdoctoral Fellows Program. This
work was supported in part by the U.S. National Science Foundation, grant NSF-
PHY-0705682, the LHC Theory Initiative, Jonathan Bagger, PI, and NSF grants
OIA-1124453 (PI P. Madau) and OIA-1124403 (PI A. Szalay).
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|
arxiv-papers
| 2012-01-31T21:00:02 |
2024-09-04T02:49:26.899609
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Michael Kuhlen, Mariangela Lisanti, David N. Spergel",
"submitter": "Michael Kuhlen",
"url": "https://arxiv.org/abs/1202.0007"
}
|
1202.0023
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ Interval edge-colorings of
Cartesian products of graphs I Petros A. Petrosyan, Hrant H. Khachatrian and
Hovhannes G. Tananyan
# Interval edge-colorings of Cartesian products of graphs I
Petros A. Petrosyan, Hrant H. Khachatrian[MCSD], Hovhannes G. Tananyan email:
pet_petros@ipia.sci.amemail: hrant@egern.netemail: HTananyan@yahoo.com
Department of Informatics and Applied Mathematics,
Yerevan State University, 0025, Armenia Institute for Informatics and
Automation Problems,
National Academy of Sciences, 0014, Armenia Department of Applied Mathematics
and Informatics,
Russian-Armenian State University, 0051, Armenia
###### Abstract
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval
$t$-coloring if all colors are used, and the colors of edges incident to each
vertex of $G$ are distinct and form an interval of integers. A graph $G$ is
interval colorable if $G$ has an interval $t$-coloring for some positive
integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs.
For a graph $G\in\mathfrak{N}$, the least and the greatest values of $t$ for
which $G$ has an interval $t$-coloring are denoted by $w(G)$ and $W(G)$,
respectively. In this paper we first show that if $G$ is an $r$-regular graph
and $G\in\mathfrak{N}$, then $W(G\square P_{m})\geq W(G)+W(P_{m})+(m-1)r$
($m\in\mathbb{N}$) and $W(G\square C_{2n})\geq W(G)+W(C_{2n})+nr$ ($n\geq 2$).
Next, we investigate interval edge-colorings of grids, cylinders and tori. In
particular, we prove that if $G\square H$ is planar and both factors have at
least $3$ vertices, then $G\square H\in\mathfrak{N}$ and $w(G\square H)\leq
6$. Finally, we confirm the first author’s conjecture on the $n$-dimensional
cube $Q_{n}$ and show that $Q_{n}$ has an interval $t$-coloring if and only if
$n\leq t\leq\frac{n\left(n+1\right)}{2}$.
Keywords: edge-coloring, interval coloring, grid, cylinder, torus,
$n$-dimensional cube
AMS Subject Classification: 05C15, 05C76
## 1 Introduction
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval
$t$-coloring if all colors are used, and the colors of edges incident to each
vertex of $G$ are distinct and form an interval of integers. A graph $G$ is
interval colorable if $G$ has an interval $t$-coloring for some positive
integer $t$. Let $\mathfrak{N}$ be the set of all interval colorable graphs
[1, 14]. For a graph $G\in\mathfrak{N}$, the least and the greatest values of
$t$ for which $G$ has an interval $t$-coloring are denoted by $w(G)$ and
$W(G)$, respectively. The concept of interval edge-coloring was introduced by
Asratian and Kamalian [1]. In [1], they proved the following:
###### Theorem 1
If $G$ is a regular graph, then
(1)
$G\in\mathfrak{N}$ if and only if $\chi^{\prime}(G)=\Delta(G)$.
(2)
If $G\in\mathfrak{N}$ and $w(G)\leq t\leq W(G)$, then $G$ has an interval
$t$-coloring.
In [2], Asratian and Kamalian investigated interval edge-colorings of
connected graphs. In particular, they obtained the following two results.
###### Theorem 2
If $G$ is a connected graph and $G\in\mathfrak{N}$, then
$W(G)\leq\left(\mathrm{diam}(G)+1\right)\left(\Delta(G)-1\right)+1$.
###### Theorem 3
If $G$ is a connected bipartite graph and $G\in\mathfrak{N}$, then
$W(G)\leq\mathrm{diam}(G)\left(\Delta(G)-1\right)+1$.
Recently, Kamalian and Petrosyan [16] showed that these upper bounds cannot be
significantly improved.
In [13], Kamalian investigated interval colorings of complete bipartite graphs
and trees. In particular, he proved the following:
###### Theorem 4
For any $r,s\in\mathbb{N}$, the complete bipartite graph $K_{r,s}$ is interval
colorable, and
(1)
$w\left(K_{r,s}\right)=r+s-\gcd(r,s)$,
(2)
$W\left(K_{r,s}\right)=r+s-1$,
(3)
if $w\left(K_{r,s}\right)\leq t\leq W\left(K_{r,s}\right)$, then $K_{r,s}$ has
an interval $t$-coloring.
In [21], Petrosyan investigated interval colorings of complete graphs and
$n$-dimensional cubes. In particular, he obtained the following two results.
###### Theorem 5
If $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative, then
$W\left(K_{2n}\right)\geq 4n-2-p-q$.
###### Theorem 6
$W\left(Q_{n}\right)\geq\frac{n(n+1)}{2}$ for any $n\in\mathbb{N}$.
The $NP$-completeness of the problem of the existence of an interval edge-
coloring of an arbitrary bipartite graph was shown in [24]. A similar result
for regular graphs was obtained in [1, 2]. In [19, 22, 23], interval edge-
colorings of various products of graphs were investigated. Some interesting
results on interval colorings were also obtained in [3, 4, 6, 7, 8, 9, 10, 14,
15, 16, 17, 18, 19, 20]. Surveys on this topic can be found in some books [3,
12, 19].
In this paper we focus only on interval edge-colorings of Cartesian products
of graphs.
## 2 Notations, definitions and auxiliary results
Throughout this paper all graphs are finite, undirected, and have no loops or
multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of
$G$, respectively. The degree of a vertex $v$ in $G$ is denoted by $d_{G}(v)$,
the maximum degree of $G$ by $\Delta(G)$, and the chromatic index of $G$ by
$\chi^{\prime}(G)$. If $G$ is a connected graph, then the distance between two
vertices $u$ and $v$ in $G$, we denote by $d(u,v)$, and the diameter of $G$ by
$\mathrm{diam}(G)$. We use the standard notations $P_{n}$, $C_{n}$, $K_{n}$
and $Q_{n}$ for the simple path, simple cycle, complete graph on vertices and
the $n$-dimensional cube, respectively. A partial edge-coloring of a graph $G$
is a coloring of some of the edges of $G$ such that no two adjacent edges
receive the same color. If $\alpha$ is a partial edge-coloring of $G$ and
$v\in V(G)$, then $S\left(v,\alpha\right)$ denotes the set of colors appearing
on colored edges incident to $v$.
Let $[t]$ denote the set of the first $t$ natural numbers. Let $\left\lfloor
a\right\rfloor$ ($\left\lceil a\right\rceil$) denote the largest (least)
integer less (greater) than or equal to $a$. For two positive integers $a$ and
$b$ with $a\leq b$, the set $\left\\{a,\ldots,b\right\\}$ is denoted by
$\left[a,b\right]$. The terms and concepts that we do not define can be found
in [25].
Let $G$ and $H$ be graphs. The Cartesian product $G\square H$ is defined as
follows:
$V(G\square H)=V(G)\times V(H)$,
$E(G\square
H)=\\{(u_{1},v_{1})(u_{2},v_{2})\colon\,u_{1}=u_{2}~{}and~{}v_{1}v_{2}\in
E(H)~{}or~{}v_{1}=v_{2}~{}and~{}u_{1}u_{2}\in E(G)\\}$.
Clearly, if $G$ and $H$ are connected graphs, then $G\square H$ is connected,
too. Moreover, $\Delta(G\square H)=\Delta(G)+\Delta(H)$ and
$\mathrm{diam}(G\square H)=\mathrm{diam}(G)+\mathrm{diam}(H)$. The
$k$-dimensional grid $G(n_{1},n_{2},\ldots,n_{k})$, $n_{i}\in\mathbb{N}$ is
the Cartesian product of paths $P_{n_{1}}\square P_{n_{2}}\square\cdots\square
P_{n_{k}}$. The cylinder $C(n_{1},n_{2})$ is the Cartesian product
$P_{n_{1}}\square C_{n_{2}}$, and the torus $T(n_{1},n_{2})$ is the Cartesian
product of cycles $C_{n_{1}}\square C_{n_{2}}$.
We also need the following two lemmas.
###### Lemma 7
If $\alpha$ is an edge-coloring of a connected graph $G$ with colors
$1,\ldots,t$ such that the edges incident to each vertex $v\in V(G)$ are
colored by distinct and consecutive colors, and $\min_{e\in
E(G)}\\{\alpha(e)\\}=1$, $\max_{e\in E(G)}\\{\alpha(e)\\}=t$, then $\alpha$ is
an interval $t$-coloring of $G$.
* Proof.
For the proof of the lemma, it suffices to show that all colors are used in
the coloring $\alpha$ of $G$.
Let $u$ and $w$ be vertices such that $1\in S(u,\alpha)$ and $t\in
S(w,\alpha)$. Also, let $P=v_{1},\ldots,v_{k}$, where $u=v_{1}$ and $v_{k}=w$
be a $u,w$-path in $G$. If $k=1$, then $t\in S(u,\alpha)$ and all colors
appear on edges incident to $u$. Assume that $k\geq 2$. The sets
$S(v_{i},\alpha)$ for $v_{i}\in V(P)$ are intervals, and for $2\leq i\leq k$,
intervals $S(v_{i-1},\alpha)$ and $S(v_{i},\alpha)$ share a color. Thus, the
sets $S(v_{1},\alpha),\ldots,S(v_{k},\alpha)$ cover $[1,t]$. $\square$
The next lemma was proved by Behzad and Mahmoodian in [5].
###### Lemma 8
If both $G$ and $H$ have at least $3$ vertices, then the Cartesian product
$G\square H$ is planar if and only if $G\square H=G(m,n)$ or $G\square
H=C(m,n)$.
## 3 The Cartesian product of regular graphs
Interval edge-colorings of Cartesian products of graphs were first
investigated by Giaro and Kubale in [7], where they proved the following:
###### Theorem 9
If $G\in\mathfrak{N}$, then $G\square P_{m}\in\mathfrak{N}$ $(m\in\mathbb{N})$
and $G\square C_{2n}\in\mathfrak{N}$ $(n\geq 2)$.
It is well-known that $P_{m},C_{2n}\in\mathfrak{N}$ and
$W\left(P_{m}\right)=m-1$, $W\left(C_{2n}\right)=n+1$ for $m\in\mathbb{N}$ and
$n\geq 2$. Later, Giaro and Kubale [9, 19] proved a more general result.
###### Theorem 10
If $G,H\in\mathfrak{N}$, then $G\square H\in\mathfrak{N}$. Moreover,
$w(G\square H)\leq w(G)+w(H)$ and $W(G\square H)\geq W(G)+W(H)$.
Let us note that if $G\in\mathfrak{N}$ and $H=P_{m}$ or $H=C_{2n}$, then, by
Theorem 10, we obtain $w(G\square H)\leq w(G)+2$ and $W(G\square P_{m})\geq
W(G)+m-1$, $W(G\square C_{2n})\geq W(G)+n+1$. Now we improve the lower bound
in Theorem 10 for $W\left(G\square P_{m}\right)$ and $W\left(G\square
C_{2n}\right)$ when $G$ is a regular graph and $G\in\mathfrak{N}$. More
precisely, we show that the following two theorems hold.
###### Theorem 11
If $G$ is an $r$-regular graph and $G\in\mathfrak{N}$, then $G\square
P_{m}\in\mathfrak{N}$ $(m\in\mathbb{N})$ and $W\left(G\square P_{m}\right)\geq
W(G)+W\left(P_{m}\right)+(m-1)r$.
* Proof.
For the proof, we construct an edge-coloring of the graph $G\square P_{m}$
that satisfies the specified conditions.
Let $V(G)=\left\\{v_{1},\ldots,v_{n}\right\\}$ and $V\left(G\square
P_{m}\right)=\bigcup_{i=1}^{m}V^{i}$, where
$V^{i}=\left\\{v_{j}^{(i)}\colon\,1\leq j\leq n\right\\}$. Also, let
$E\left(G\square
P_{m}\right)=\bigcup_{i=1}^{m}E^{i}\cup\bigcup_{j=1}^{n}E_{j}$, where
$E^{i}=\left\\{v_{j}^{(i)}v_{k}^{(i)}\colon\,v_{j}v_{k}\in E(G)\right\\}$ and
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq m-1\right\\}$.
For $1\leq i\leq m$, define a subgraph $G^{i}$ of the graph $G\square P_{m}$
as follows: $G^{i}=\left(V^{i},E^{i}\right)$. Clearly, $G^{i}$ is isomorphic
to $G$ for $1\leq i\leq m$. Since $G\in\mathfrak{N}$, there exists an interval
$W(G)$-coloring $\alpha$ of $G$. Now we define an edge-coloring $\beta$ of the
subgraphs $G^{1},\ldots,G^{m}$.
For $1\leq i\leq m$ and for every edge $v_{j}^{(i)}v_{k}^{(i)}\in E(G^{i})$,
let
$\beta\left(v_{j}^{(i)}v_{k}^{(i)}\right)=\alpha(v_{j}v_{k})+(i-1)(r+1)$.
It is easy to see that the color of each edge of the subgraph $G^{i}$ is
obtained by shifting the color of the associated edge of $G$ by $(i-1)(r+1)$,
thus the set $S\left(v_{j}^{(i)},\beta\right)$ is an interval for each vertex
$v_{j}^{(i)}\in V(G^{i})$, where $1\leq i\leq m$, $1\leq j\leq n$. Now we
define an edge-coloring $\gamma$ of the graph $G\square P_{m}$.
For every $e\in E\left(G\square P_{m}\right)$, let
$\gamma(e)=\left\\{\begin{tabular}[]{ll}$\beta(e)$,&if $e\in E(G^{i}),$\\\
$\max S\left(v_{j}^{(i)},\beta\right)+1$,&if $e=v_{j}^{(i)}v_{j}^{(i+1)}\in
E_{j}$,\\\ \end{tabular}\right.$
where $1\leq i\leq m,1\leq j\leq n$.
Let us prove that $\gamma$ is an interval
$\left(W(G)+W\left(P_{m}\right)+(m-1)r\right)$-coloring of the graph $G\square
P_{m}$ for $m\in\mathbb{N}$.
First we show that the set $S\left(v_{j}^{(i)},\gamma\right)$ is an interval
for each vertex $v_{j}^{(i)}\in V\left(G\square P_{m}\right)$, where $1\leq
i\leq m,1\leq j\leq n$.
Case 1: $i=1$, $1\leq j\leq n$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq n$, we have
$\displaystyle S\left(v_{j}^{(1)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right),\ldots,\max
S\left(v_{j},\alpha\right)\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+1\right\\}$ $\displaystyle=$ $\displaystyle[\min
S\left(v_{j},\alpha\right),\max S\left(v_{j},\alpha\right)+1].$
Case 2: $2\leq i\leq m-1$, $1\leq j\leq n$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq n$, we have
$\displaystyle S\left(v_{j}^{(i)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(i-1)(r+1),\ldots,\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)\right\\}$ $\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(i-2)(r+1)+1\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(i-1)(r+1),\ldots,\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+1\right\\}$
$\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(i-2)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(i-1)(r+1)-1,\ldots,\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle[\min S\left(v_{j},\alpha\right)+(i-1)(r+1)-1,\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+1].$
Case 3: $i=m$, $1\leq j\leq n$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq n$, we have
$\displaystyle S\left(v_{j}^{(m)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(m-1)(r+1),\ldots,\max
S\left(v_{j},\alpha\right)+(m-1)(r+1)\right\\}$ $\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(m-2)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(m-1)(r+1)-1,\ldots,\max
S\left(v_{j},\alpha\right)+(m-1)(r+1)\right\\}$ $\displaystyle=$
$\displaystyle[\min S\left(v_{j},\alpha\right)+(m-1)(r+1)-1,\max
S\left(v_{j},\alpha\right)+(m-1)(r+1)].$
Next we prove that in the coloring $\gamma$ all colors are used. Clearly,
there exists an edge $v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\in E(G^{1})$ such that
$\gamma\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=1$, since in the coloring
$\alpha$ there exists an edge $v_{j_{0}}v_{k_{0}}$ with
$\alpha\left(v_{j_{0}}v_{k_{0}}\right)=1$ and
$\gamma\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=\beta\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=\alpha\left(v_{j_{0}}v_{k_{0}}\right)$.
Similarly, there exists an edge $v_{j_{1}}^{(m)}v_{k_{1}}^{(m)}\in E(G^{m})$
such that
$\gamma\left(v_{j_{1}}^{(m)}v_{k_{1}}^{(m)}\right)=W(G)+(m-1)(r+1)=W(G)+W\left(P_{m}\right)+(m-1)r$,
since in the coloring $\alpha$ there exists an edge $v_{j_{1}}v_{k_{1}}$ with
$\alpha\left(v_{j_{1}}v_{k_{1}}\right)=W(G)$ and
$\gamma\left(v_{j_{1}}^{(m)}v_{k_{1}}^{(m)}\right)=\beta\left(v_{j_{1}}^{(m)}v_{k_{1}}^{(m)}\right)=\alpha\left(v_{j_{1}}v_{k_{1}}\right)+(m-1)(r+1)$.
Now, by Lemma 7, we have that for each
$t\in[W(G)+W\left(P_{m}\right)+(m-1)r]$, there is an edge $e\in
E\left(G\square P_{m}\right)$ with $\gamma(e)=t$.
This shows that $\gamma$ is an interval
$\left(W(G)+W\left(P_{m}\right)+(m-1)r\right)$-coloring of the graph $G\square
P_{m}$ for $m\in\mathbb{N}$. Thus, $G\square P_{m}\in\mathfrak{N}$ and
$W\left(G\square P_{m}\right)\geq W(G)+W\left(P_{m}\right)+(m-1)r$. $\square$
###### Corollary 12
If $G$ is an $r$-regular graph and $G\in\mathfrak{N}$, then $G\square
Q_{n}\in\mathfrak{N}$ $(n\in\mathbb{N})$ and
$W\left(G\square Q_{n}\right)\geq W(G)+\frac{n(n+2r+1)}{2}$.
###### Theorem 13
If $G$ is an $r$-regular graph and $G\in\mathfrak{N}$, then $G\square
C_{2n}\in\mathfrak{N}$ $(n\geq 2)$ and $W\left(G\square C_{2n}\right)\geq
W(G)+W\left(C_{2n}\right)+nr$.
* Proof.
For the proof, we construct an edge-coloring of the graph $G\square C_{2n}$
that satisfies the specified conditions.
Let $V(G)=\left\\{v_{1},\ldots,v_{p}\right\\}$ and $V\left(G\square
C_{2n}\right)=\bigcup_{i=1}^{2n}V^{i}$, where
$V^{i}=\left\\{v_{j}^{(i)}\colon\,1\leq j\leq p\right\\}$. Also, let
$E\left(G\square
C_{2n}\right)=\bigcup_{i=1}^{2n}E^{i}\cup\bigcup_{j=1}^{p}E_{j}$, where
$E^{i}=\left\\{v_{j}^{(i)}v_{k}^{(i)}\colon\,v_{j}v_{k}\in E(G)\right\\}$ and
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq
2n-1\right\\}\cup\left\\{v_{j}^{(1)}v_{j}^{(2n)}\right\\}$.
For $1\leq i\leq 2n$, define a subgraph $G^{i}$ of the graph $G\square C_{2n}$
as follows: $G^{i}=\left(V^{i},E^{i}\right)$. Clearly, $G^{i}$ is isomorphic
to $G$ for $1\leq i\leq 2n$. Since $G\in\mathfrak{N}$, there exists an
interval $W(G)$-coloring $\alpha$ of $G$. Now we define an edge-coloring
$\beta$ of the subgraphs $G^{1},\ldots,G^{2n}$.
For $1\leq i\leq 2n$ and for every edge $v_{j}^{(i)}v_{k}^{(i)}\in E(G^{i})$,
let
$\beta\left(v_{j}^{(i)}v_{k}^{(i)}\right)=\left\\{\begin{tabular}[]{ll}$\alpha\left(v_{j}v_{k}\right)$,&if
$i=1$,\\\ $\alpha\left(v_{j}v_{k}\right)+(i-1)(r+1)+1$,&if $2\leq i\leq
n+1$,\\\ $\alpha\left(v_{j}v_{k}\right)+(2n+1-i)(r+1)$,&if $n+2\leq i\leq
2n$.\\\ \end{tabular}\right.$
It is easy to see that the color of each edge of the subgraph $G^{i}$ is
obtained by shifting the color of the associated edge of $G$ by $(i-1)(r+1)+1$
for $2\leq i\leq n+1$, and by $(2n-i+1)(r+1)$ for $n+2\leq i\leq 2n$, thus the
set $S\left(v_{j}^{(i)},\beta\right)$ is an interval for each vertex
$v_{j}^{(i)}\in V(G^{i})$, where $1\leq i\leq 2n$, $1\leq j\leq p$. Now we
define an edge-coloring $\gamma$ of the graph $G\square C_{2n}$.
For every $e\in E\left(G\square C_{2n}\right)$, let
$\gamma(e)=\left\\{\begin{tabular}[]{ll}$\beta(e)$,&if $e\in E(G^{i}),$\\\
$\max S\left(v_{j}^{(1)},\beta\right)+1$,&if $e=v_{j}^{(1)}v_{j}^{(2n)}\in
E_{j}$,\\\ $\max S\left(v_{j}^{(1)},\beta\right)+2$,&if
$e=v_{j}^{(1)}v_{j}^{(2)}\in E_{j}$,\\\ $\max
S\left(v_{j}^{(i)},\beta\right)+1$,&if $e=v_{j}^{(i)}v_{j}^{(i+1)}\in
E_{j},2\leq i\leq n$,\\\ $\max S\left(v_{j}^{(i)},\beta\right)+1$,&if
$e=v_{j}^{(i-1)}v_{j}^{(i)}\in E_{j},n+2\leq i\leq 2n$,\\\
\end{tabular}\right.$
where $1\leq i\leq 2n,1\leq j\leq p$.
Let us prove that $\gamma$ is an interval
$\left(W(G)+W\left(C_{2n}\right)+nr\right)$-coloring of the graph $G\square
C_{2n}$ for $n\geq 2$.
First we show that the set $S\left(v_{j}^{(i)},\gamma\right)$ is an interval
for each vertex $v_{j}^{(i)}\in V\left(G\square C_{2n}\right)$, where $1\leq
i\leq 2n,1\leq j\leq p$.
Case 1: $i=1$, $1\leq j\leq p$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq p$, we have
$\displaystyle S\left(v_{j}^{(1)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right),\ldots,\max
S\left(v_{j},\alpha\right)\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+2\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+1\right\\}$ $\displaystyle=$ $\displaystyle[\min
S\left(v_{j},\alpha\right),\max S\left(v_{j},\alpha\right)+2].$
Case 2: $2\leq i\leq n$, $1\leq j\leq p$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq p$, we have
$\displaystyle S\left(v_{j}^{(i)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+(i-1)(r+1)+1,\ldots,\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+1\right\\}$
$\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(i-2)(r+1)+2\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+2\right\\}$ $\displaystyle=$
$\displaystyle[\min S\left(v_{j},\alpha\right)+(i-1)(r+1),\max
S\left(v_{j},\alpha\right)+(i-1)(r+1)+2].$
Case 3: $i=n+1$, $1\leq j\leq p$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq p$, we have
$\displaystyle S\left(v_{j}^{(n+1)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min S\left(v_{j},\alpha\right)+n(r+1)+1,\ldots,\max
S\left(v_{j},\alpha\right)+n(r+1)+1\right\\}$ $\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(n-1)(r+1)+2\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+(n-1)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle[\min S\left(v_{j},\alpha\right)+n(r+1)-1,\max
S\left(v_{j},\alpha\right)+n(r+1)+1].$
Case 4: $n+2\leq i\leq 2n$, $1\leq j\leq p$.
By the definition of $\gamma$ and taking into account that $\max
S\left(v_{j},\alpha\right)-\min S\left(v_{j},\alpha\right)=r-1$ for $1\leq
j\leq p$, we have
$\displaystyle S\left(v_{j}^{(i)},\gamma\right)$ $\displaystyle=$
$\displaystyle\left\\{\min
S\left(v_{j},\alpha\right)+(2n+1-i)(r+1),\ldots,\max
S\left(v_{j},\alpha\right)+(2n+1-i)(r+1)\right\\}$
$\displaystyle\cup\left\\{\max
S\left(v_{j},\alpha\right)+(2n+1-i)(r+1)+1\right\\}\cup\left\\{\max
S\left(v_{j},\alpha\right)+(2n-i)(r+1)+1\right\\}$ $\displaystyle=$
$\displaystyle[\min S\left(v_{j},\alpha\right)+(2n-i+1)(r+1)-1,\max
S\left(v_{j},\alpha\right)+(2n-i+1)(r+1)+1].$
Next we prove that in the coloring $\gamma$ all colors are used. Clearly,
there exists an edge $v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\in E(G^{1})$ such that
$\gamma\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=1$, since in the coloring
$\alpha$ there exists an edge $v_{j_{0}}v_{k_{0}}$ with
$\alpha\left(v_{j_{0}}v_{k_{0}}\right)=1$ and
$\gamma\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=\beta\left(v_{j_{0}}^{(1)}v_{k_{0}}^{(1)}\right)=\alpha\left(v_{j_{0}}v_{k_{0}}\right)$.
Similarly, there exists an edge $v_{j_{1}}^{(n+1)}v_{k_{1}}^{(n+1)}\in
E(G^{n+1})$ such that
$\gamma\left(v_{j_{1}}^{(n+1)}v_{k_{1}}^{(n+1)}\right)=W(G)+n(r+1)+1=W(G)+W\left(C_{2n}\right)+nr$,
since in the coloring $\alpha$ there exists an edge $v_{j_{1}}v_{k_{1}}$ with
$\alpha\left(v_{j_{1}}v_{k_{1}}\right)=W(G)$ and
$\gamma\left(v_{j_{1}}^{(n+1)}v_{k_{1}}^{(n+1)}\right)=\beta\left(v_{j_{1}}^{(n+1)}v_{k_{1}}^{(n+1)}\right)=\alpha\left(v_{j_{1}}v_{k_{1}}\right)+n(r+1)+1$.
Now, by Lemma 7, we have that for each $t\in[W(G)+W\left(C_{2n}\right)+nr]$,
there is an edge $e\in E\left(G\square C_{2n}\right)$ with $\gamma(e)=t$.
This shows that $\gamma$ is an interval
$\left(W(G)+W\left(C_{2n}\right)+nr\right)$-coloring of the graph $G\square
C_{2n}$ for $n\geq 2$. Thus, $G\square C_{2n}\in\mathfrak{N}$ and
$W\left(G\square C_{2n}\right)\geq W(G)+W\left(C_{2n}\right)+nr$. $\square$
From Theorems 5 and 13, we have:
###### Corollary 14
If $n=p2^{q}$, where $p$ is odd and $q$ is nonnegative, then
$W\left(K_{2n}\square C_{2n}\right)\geq 2n^{2}+4n-1-p-q$.
Note that the lower bound in Corollary 14 is close to the upper bound for
$W\left(K_{2n}\square C_{2n}\right)$, since $\Delta\left(K_{2n}\square
C_{2n}\right)=2n+1$ and $\mathrm{diam}\left(K_{2n}\square C_{2n}\right)=n+1$,
by Theorem 2, we have $W\left(K_{2n}\square C_{2n}\right)\leq 2n^{2}+4n+1$.
## 4 Grids, cylinders and tori
Interval edge-colorings of grids, cylinders and tori were first considered by
Giaro and Kubale in [7], where they proved the following:
###### Theorem 15
If $G=G(n_{1},n_{2},\ldots,n_{k})$ or $G=C(m,2n)$, $m\in\mathbb{N},n\geq 2$,
or $G=T(2m,2n)$, $m,n\geq 2$, then $G\in\mathfrak{N}$ and $w(G)=\Delta(G)$.
For the greatest possible number of colors in interval colorings of cylinders
and tori, Petrosyan and Karapetyan [20] proved the following theorems:
###### Theorem 16
For any $m\in\mathbb{N},n\geq 2$, we have $W(C(m,2n))\geq 3m+n-2$.
###### Theorem 17
For any $m,n\geq 2$, we have $W(T(2m,2n))\geq\max\\{3m+n,3n+m\\}$.
First we consider grids. It is easy to see that $W\left(G(2,n)\right)=2n-1$
for any $n\in\mathbb{N}$. Now we provide a lower bound for
$W\left(G(m,n)\right)$ when $m,n\geq 2$.
###### Theorem 18
For any $m,n\geq 2$, we have $W(G(m,n))\geq 2(m+n-3)$.
* Proof.
For the proof, we are going to construct an edge-coloring of the graph
$G(m,n)$ that satisfies the specified conditions.
Let $V(G(m,n))=\left\\{v_{j}^{(i)}\colon\,1\leq i\leq m,1\leq j\leq
n\right\\}$ and $E(G(m,n))=\bigcup_{i=1}^{m}E^{i}\cup\bigcup_{j=1}^{n}E_{j}$,
where
$E^{i}=\left\\{v_{j}^{(i)}v_{j+1}^{(i)}\colon\,1\leq j\leq n-1\right\\}$ and
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq m-1\right\\}$.
Define an edge-coloring $\alpha$ of $G(m,n)$ as follows:
(1)
for $i=1,\ldots,m-1$, $j=1,\ldots,n-1$, let
$\alpha\left(v_{j}^{(i)}v_{j}^{(i+1)}\right)=2(i+j)-3$;
(2)
for $i=1,\ldots,m-1$, let
$\alpha\left(v_{n}^{(i)}v_{n}^{(i+1)}\right)=2(n+i)-5$;
(3)
for $j=1,\ldots,n-1$, let
$\alpha\left(v_{j}^{(1)}v_{j+1}^{(1)}\right)=2j$;
(4)
for $i=2,\ldots,m$, $j=1,\ldots,n-1$, let
$\alpha\left(v_{j}^{(i)}v_{j+1}^{(i)}\right)=2(i+j)-4$.
It is easy to see that $\alpha$ is an interval $(2(m+n-3))$-coloring of
$G(m,n)$ when $m,n\geq 2$. $\square$
Note that the lower bound in Theorem 18 is not far from the upper bound for
$W\left(G(m,n)\right)$, since $G(m,n)$ is bipartite,
$2\leq\Delta\left(G(m,n)\right)\leq 4$ and
$\mathrm{diam}\left(G(m,n)\right)=m+n-2$, by Theorem 3, we have
$W\left(G(m,n)\right)\leq 3(m+n-2)+1$.
From Theorems 10 and 18, we have:
###### Corollary 19
If $n_{1}\geq\cdots\geq n_{2k}\geq 2$ $(k\in\mathbb{N})$, then
$W(G(n_{1},n_{2},\ldots,n_{2k}))\geq 2\sum_{i=1}^{2k}n_{i}-6k$,
and if $n_{1}\geq\cdots\geq n_{2k+1}\geq 2$ $(k\in\mathbb{N})$, then
$W(G(n_{1},n_{2},\ldots,n_{2k+1}))\geq 2\sum_{i=1}^{2k}n_{i}+n_{2k+1}-6k-1$.
Next we consider cylinders. In [18], Khchoyan proved the following:
###### Theorem 20
For any $n\geq 3$, we have
(1)
$C(2,n)\in\mathfrak{N}$,
(2)
$w\left(C(2,n)\right)=3$,
(3)
$W\left(C(2,n)\right)=n+2$,
(4)
if $w\left(C(2,n)\right)\leq t\leq W\left(C(2,n)\right)$, then $C(2,n)$ has an
interval $t$-coloring.
Now we prove some general results on cylinders.
###### Theorem 21
For any $m\geq 3,n\in\mathbb{N}$, we have $C(m,2n+1)\in\mathfrak{N}$ and
$w\left(C(m,2n+1)\right)=\left\\{\begin{tabular}[]{ll}$4$,&if $m$ is even,\\\
$6$,&if $m$ is odd.\\\ \end{tabular}\right.$
* Proof.
Let $V(C(m,2n+1))=\left\\{v_{j}^{(i)}\colon\,1\leq i\leq m,1\leq j\leq
2n+1\right\\}$ and
$E(C(m,2n+1))=\bigcup_{i=1}^{m}{E}^{i}\cup\bigcup_{j=1}^{2n+1}{E}_{j}$ , where
$E^{i}=\left\\{v_{j}^{(i)}v_{j+1}^{(i)}\colon\,1\leq j\leq
2n\right\\}\cup\left\\{v_{1}^{(i)}v_{2n+1}^{(i)}\right\\}$,
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq m-1\right\\}$.
First we show that if $m$ is even, then $C(m,2n+1)$ has an interval
$4$-coloring.
For $1\leq i\leq\frac{m}{2}$, define a subgraph $C^{i}$ of the graph
$C(m,2n+1)$ as follows:
$C^{i}=\left(V^{2i-1}\cup V^{2i},E^{2i-1}\cup
E^{2i}\cup\left\\{v_{j}^{(2i-1)}v_{j}^{(2i)}\colon\,1\leq j\leq
2n+1\right\\}\right)$.
Clearly, $C^{i}$ is isomorphic to $C(2,2n+1)$ for $1\leq i\leq\frac{m}{2}$. By
Theorem 20, $C(2,2n+1)\in\mathfrak{N}$ and there exists an interval
$3$-coloring $\alpha$ of $C(2,2n+1)$. Now we define an edge-coloring $\beta$
of $C(m,2n+1)$. First we color the edges of $C^{i}$ according to $\alpha$ for
$1\leq i\leq\frac{m}{2}$. Then we color the edges
$v_{j}^{(2i)}v_{j}^{(2i+1)}\in E_{j}$ with color $4$ for $1\leq
i\leq\frac{m}{2}-1,1\leq j\leq 2n+1$. It is easy to see that $\beta$ is an
interval $4$-coloring of $C(m,2n+1)$. This shows that
$C(m,2n+1)\in\mathfrak{N}$ and $w(C(m,2n+1))\leq 4$. On the other hand,
$w(C(m,2n+1))\geq\Delta(C(m,2n+1))=4$; thus $w(C(m,2n+1))=4$ for even $m$.
Now assume that $m$ is odd.
First we show that $C(3,2n+1)$ has an interval $6$-coloring.
Define an edge-coloring $\gamma$ of $C(3,2n+1)$ as follows:
(1)
$\gamma\left(v_{1}^{(1)}v_{1}^{(2)}\right)=6$ and for
$j=2,\ldots,2\left\lfloor\frac{n+1}{2}\right\rfloor$, let
$\gamma\left(v_{j}^{(1)}v_{j}^{(2)}\right)=4$;
(2)
$\gamma\left(v_{2\left\lfloor\frac{n+1}{2}\right\rfloor+1}^{(1)}v_{2\left\lfloor\frac{n+1}{2}\right\rfloor+1}^{(2)}\right)=2$
and for $j=2\left\lfloor\frac{n+1}{2}\right\rfloor+2,\ldots,2n+1$, let
$\gamma\left(v_{j}^{(1)}v_{j}^{(2)}\right)=3$;
(3)
$\gamma\left(v_{1}^{(2)}v_{1}^{(3)}\right)=3$ and for
$j=2,\ldots,2\left\lfloor\frac{n+1}{2}\right\rfloor$, let
$\gamma\left(v_{j}^{(2)}v_{j}^{(3)}\right)=2$;
(4)
for $j=2\left\lfloor\frac{n+1}{2}\right\rfloor+1,\ldots,2n+1$, let
$\gamma\left(v_{j}^{(2)}v_{j}^{(3)}\right)=1$;
(5)
$j=1,\ldots,\left\lfloor\frac{n+1}{2}\right\rfloor$, let
$\gamma\left(v_{2j-1}^{(1)}v_{2j}^{(1)}\right)=\gamma\left(v_{2j-1}^{(2)}v_{2j}^{(2)}\right)=5$
and
$\gamma\left(v_{2j}^{(1)}v_{2j+1}^{(1)}\right)=\gamma\left(v_{2j}^{(2)}v_{2j+1}^{(2)}\right)=3$;
(6)
for $j=\left\lfloor\frac{n+1}{2}\right\rfloor+1,\ldots,n$, let
$\gamma\left(v_{2j-1}^{(1)}v_{2j}^{(1)}\right)=\gamma\left(v_{2j-1}^{(2)}v_{2j}^{(2)}\right)=4$
and
$\gamma\left(v_{1}^{(1)}v_{2n+1}^{(1)}\right)=\gamma\left(v_{1}^{(2)}v_{2n+1}^{(2)}\right)=4$;
(7)
for $j=\left\lfloor\frac{n+1}{2}\right\rfloor+1,\ldots,n$, let
$\gamma\left(v_{2j}^{(1)}v_{2j+1}^{(1)}\right)=\gamma\left(v_{2j}^{(2)}v_{2j+1}^{(2)}\right)=2$;
(8)
for $j=1,\ldots,\left\lfloor\frac{n+1}{2}\right\rfloor$, let
$\gamma\left(v_{2j-1}^{(3)}v_{2j}^{(3)}\right)=1$ and
$\gamma\left(v_{2j}^{(3)}v_{2j+1}^{(3)}\right)=3$;
(9)
for $j=\left\lfloor\frac{n+1}{2}\right\rfloor+1,\ldots,n$, let
$\gamma\left(v_{2j-1}^{(3)}v_{2j}^{(3)}\right)=2$ and
$\gamma\left(v_{1}^{(3)}v_{2n+1}^{(3)}\right)=2$;
(10)
for $j=\left\lfloor\frac{n+1}{2}\right\rfloor+1,\ldots,n$, let
$\gamma\left(v_{2j}^{(3)}v_{2j+1}^{(3)}\right)=3$.
It is not difficult to see that $\gamma$ is an interval $6$-coloring of
$C(3,2n+1)$ for which $S(v_{j}^{(3)},\gamma)=[1,3]$ when $1\leq j\leq 2n+1$.
Next we define an edge-coloring $\phi$ of $C(m,2n+1)$ as follows: first we
color the edges of the subgraph $C(3,2n+1)$ of $C(m,2n+1)$ according to
$\gamma$. Secondly, we color the edges of the remaining subgraph $C(m-3,2n+1)$
of $C(m,2n+1)$ according to $\beta$, and finally, we color the edges
$v_{j}^{(3)}v_{j}^{(4)}\in E_{j}$ with color $4$ for $1\leq j\leq 2n+1$. It is
easy to see that $\phi$ is an interval $6$-coloring of $C(m,2n+1)$. This shows
that $C(m,2n+1)\in\mathfrak{N}$ and $w(C(m,2n+1))\leq 6$.
Now we prove that $w(C(m,2n+1))\geq 6$ for odd $m$.
Let $\psi$ be an interval $w(C(m,2n+1))$-coloring of $C(m,2n+1)$ and
$w(C(m,2n+1))\leq 5$. Consider the set $S\left(v_{j}^{(i)},\psi\right)$ for
$1\leq i\leq m,1\leq j\leq 2n+1$. It is easy to see that if
$d\left(v_{j}^{(i)}\right)=3$, then $1\leq\min
S\left(v_{j}^{(i)},\psi\right)\leq 3$, and if $d\left(v_{j}^{(i)}\right)=4$,
then $1\leq\min S\left(v_{j}^{(i)},\psi\right)\leq 2$. Hence, $3\in
S\left(v_{j}^{(i)},\psi\right)$ for $1\leq i\leq m,1\leq j\leq 2n+1$, but this
implies that the edges with color $3$ form a perfect matching in $C(m,2n+1)$,
which contradicts the fact that $C(m,2n+1)$ does not have one. Thus,
$w(C(m,2n+1))=6$ for odd $m$. $\square$
Before we derive lower bounds for $W(C(2m,2n))$ and $W(C(2m,2n+1))$, let us
note that Lemma 8, Theorems 15 and 21 imply the following:
###### Corollary 22
If $G\square H$ is planar and both factors have at least $3$ vertices, then
$G\square H\in\mathfrak{N}$ and $w(G\square H)\leq 6$.
###### Theorem 23
If $m\in\mathbb{N},n\geq 2$, then $W(C(2m,2n))\geq 4m+2n-2$, and if
$m,n\in\mathbb{N}$, then $W(C(2m,2n+1))\geq 4m+2n-1$.
* Proof.
For the proof of the theorem, it suffices to construct edge-colorings that
satisfies the specified conditions. Let
$V(C(2m,2n))=\left\\{v_{j}^{(i)}\colon\,1\leq i\leq 2m,1\leq j\leq
2n\right\\}$ and $V(C(2m,2n+1))=\left\\{u_{j}^{(i)}\colon\,1\leq i\leq
2m,1\leq j\leq 2n+1\right\\}$. Also, let
$E(C(2m,2n))=\bigcup_{i=1}^{2m}E^{i}\cup\bigcup_{j=1}^{2n}E_{j}$ and
$E(C(2m,2n+1))=\bigcup_{i=1}^{2m}\overline{E}^{i}\cup\bigcup_{j=1}^{2n+1}\overline{E}_{j}$
, where
$E^{i}=\left\\{v_{j}^{(i)}v_{j+1}^{(i)}\colon\,1\leq j\leq
2n-1,\right\\}\cup\left\\{v_{1}^{(i)}v_{2n}^{(i)}\right\\}$,
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq 2m-1\right\\}$ and
$\overline{E}^{i}=\left\\{u_{j}^{(i)}u_{j+1}^{(i)}\colon\,1\leq j\leq
2n,\right\\}\cup\left\\{u_{1}^{(i)}u_{2n+1}^{(i)}\right\\}$,
$\overline{E}_{j}=\left\\{u_{j}^{(i)}u_{j}^{(i+1)}\colon\,1\leq i\leq
2m-1\right\\}$.
First we construct an interval $(4m+2n-2)$-coloring of $C(2m,2n)$ when
$m\in\mathbb{N},n\geq 2$.
Define an edge-coloring $\alpha$ of $C(2m,2n)$ as follows:
(1)
for $i=1,\ldots,m$, $j=1,\ldots,n$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j+1}^{(2i-1)}\right)=\alpha\left(v_{j}^{(2i)}v_{j+1}^{(2i)}\right)=4i+2j-4$;
(2)
for $i=1,\ldots,m$, $j=n+1,\ldots,2n-1$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j+1}^{(2i-1)}\right)=\alpha\left(v_{j}^{(2i)}v_{j+1}^{(2i)}\right)=4i-2j+4n-1$;
(3)
for $i=1,\ldots,m$, let
$\alpha\left(v_{1}^{(2i-1)}v_{2n}^{(2i-1)}\right)=\alpha\left(v_{1}^{(2i)}v_{2n}^{(2i)}\right)=4i-1$;
(4)
for $i=1,\ldots,m$, $j=1,\ldots,n$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j}^{(2i)}\right)=4i+2j-5$;
(5)
for $i=1,\ldots,m$, $j=n+1,\ldots,2n$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j}^{(2i)}\right)=4i-2j+4n$;
(6)
for $i=1,\ldots,m-1$, $j=2,\ldots,n+1$, let
$\alpha\left(v_{j}^{(2i)}v_{j}^{(2i+1)}\right)=4i+2j-3$;
(7)
for $i=1,\ldots,m-1$, $j=n+2,\ldots,2n$, let
$\alpha\left(v_{j}^{(2i)}v_{j}^{(2i+1)}\right)=4i-2j+4n+2$;
(8)
for $i=1,\ldots,m-1$, let
$\alpha\left(v_{1}^{(2i)}v_{1}^{(2i+1)}\right)=4i$.
Next we construct an interval $(4m+2n-1)$-coloring of $C(2m,2n+1)$ when
$m,n\in\mathbb{N}$.
Define an edge-coloring $\beta$ of $C(2m,2n+1)$ as follows:
(1)
for $i=1,\ldots,m$, $j=1,\ldots,n+1$, let
$\beta\left(u_{j}^{(2i-1)}u_{j+1}^{(2i-1)}\right)=\beta\left(u_{j}^{(2i)}u_{j+1}^{(2i)}\right)=4i+2j-4$;
(2)
for $i=1,\ldots,m$, $j=n+2,\ldots,2n$, let
$\beta\left(u_{j}^{(2i-1)}u_{j+1}^{(2i-1)}\right)=\beta\left(u_{j}^{(2i)}u_{j+1}^{(2i)}\right)=4i-2j+4n+1$;
(3)
for $i=1,\ldots,m$, let
$\beta\left(u_{1}^{(2i-1)}u_{2n+1}^{(2i-1)}\right)=\beta\left(u_{1}^{(2i)}u_{2n+1}^{(2i)}\right)=4i-1$;
(4)
for $i=1,\ldots,m$, $j=1,\ldots,n+2$, let
$\beta\left(u_{j}^{(2i-1)}u_{j}^{(2i)}\right)=4i+2j-5$;
(5)
for $i=1,\ldots,m$, $j=n+3,\ldots,2n+1$, let
$\beta\left(u_{j}^{(2i-1)}u_{j}^{(2i)}\right)=4i-2j+4n+2$;
(6)
for $i=1,\ldots,m-1$, $j=2,\ldots,n+1$, let
$\beta\left(u_{j}^{(2i)}u_{j}^{(2i+1)}\right)=4i+2j-3$;
(7)
for $i=1,\ldots,m-1$, $j=n+2,\ldots,2n+1$, let
$\beta\left(u_{j}^{(2i)}u_{j}^{(2i+1)}\right)=4i-2j+4n+4$;
(8)
for $i=1,\ldots,m-1$, let
$\beta\left(u_{1}^{(2i)}u_{1}^{(2i+1)}\right)=4i$.
It is straightforward to check that $\alpha$ is an interval
$(4m+2n-2)$-coloring of $C(2m,2n)$ when $m\in\mathbb{N},n\geq 2$, and $\beta$
is an interval $(4m+2n-1)$-coloring of $C(2m,2n+1)$ when $m,n\in\mathbb{N}$.
$\square$
Note that the lower bound in Theorem 23 is not so far from the upper bound for
$W\left(C(m,n)\right)$. Indeed, since $C(2m,2n)$ is bipartite,
$3\leq\Delta\left(C(2m,2n)\right)\leq 4$ and
$\mathrm{diam}\left(C(2m,2n)\right)=2m+n-1$, by Theorem 3, we have
$W\left(C(2m,2n)\right)\leq 3(2m+n-1)+1$. Similarly, since
$3\leq\Delta\left(C(2m,2n+1)\right)\leq 4$ and
$\mathrm{diam}\left(C(2m,2n+1)\right)=2m+n-1$, by Theorem 2, we have
$W\left(C(2m,2n+1)\right)\leq 3(2m+n)+1$. Next we would like to compare
obtained lower bounds for $W(C(m,n))$. If $m$ is even and $m<n$, then the
lower bound in Theorem 23 is better than in Theorem 16, if $m$ is even and
$m>n$, then the lower bound in Theorem 16 is better than in Theorem 23, and if
$m$ is even and $m=n$, then we obtain the same lower bound in both theorems.
In the following we consider tori. In [22], Petrosyan proved that the torus
$T(m,n)\in\mathfrak{N}$ if and only if $mn$ is even. Since $T(m,n)$ is
$4$-regular, by Theorem 1, we obtain that $w(T(m,n))=4$ when $mn$ is even. Now
we derive a new lower bound for $W(T(m,n))$ when $mn$ is even.
###### Theorem 24
For any $m,n\geq 2$, we have $W(T(2m,2n))\geq\max\\{3m+n+2,3n+m+2\\}$, and for
any $m\geq 2$, $n\in\mathbb{N}$, we have
$W\left(T(2m,2n+1)\right)\geq\left\\{\begin{tabular}[]{ll}$2m+2n+2$,&if $m$ is
odd,\\\ $2m+2n+3$,&if $m$ is even.\\\ \end{tabular}\right.$
* Proof.
First note that the lower bound for $W(T(2m,2n))$ $(m,n\geq 2)$ follows from
Theorem 13. For the proof of a second part of the theorem, it suffices to
construct an edge-coloring of $T(2m,2n+1)$ that satisfies the specified
conditions.
Let $V(T(2m,2n+1))=\left\\{v_{j}^{(i)}\colon\,1\leq i\leq 2m,1\leq j\leq
2n+1\right\\}$ and
$E(T(2m,2n+1))=\bigcup_{i=1}^{2m}E^{i}\cup\bigcup_{j=1}^{2n+1}E_{j}$, where
$E^{i}=\left\\{v_{j}^{(i)}v_{j+1}^{(i)}\colon\,1\leq j\leq
2n\right\\}\cup\left\\{v_{1}^{(i)}v_{2n+1}^{(i)}\right\\}$,
$E_{j}=\left\\{v_{j}^{(i)}v_{j}^{(i+1)}\colon\,1\leq i\leq
2m-1\right\\}\cup\left\\{v_{j}^{(1)}v_{j}^{(2m)}\right\\}$.
Define an edge-coloring $\alpha$ of $T(2m,2n+1)$ as follows:
(1)
for $j=1,\ldots,n+1$, let
$\alpha\left(v_{j}^{(1)}v_{j+1}^{(1)}\right)=\alpha\left(v_{j}^{(2m)}v_{j+1}^{(2m)}\right)=2j$;
(2)
for $j=n+2,\ldots,2n$, let
$\alpha\left(v_{j}^{(1)}v_{j+1}^{(1)}\right)=\alpha\left(v_{j}^{(2m)}v_{j+1}^{(2m)}\right)=2(2n+1-j)+3$
and
$\alpha\left(v_{1}^{(1)}v_{2n+1}^{(1)}\right)=\alpha\left(v_{1}^{(2m)}v_{2n+1}^{(2m)}\right)=3$;
(3)
for $j=1,\ldots,n+2$, let
$\alpha\left(v_{j}^{(1)}v_{j}^{(2m)}\right)=2j-1$;
(4)
for $j=n+3,\ldots,2n+1$, let
$\alpha\left(v_{j}^{(1)}v_{j}^{(2m)}\right)=2(2n+3-j)$;
(5)
for $i=1,\ldots,\left\lfloor\frac{m}{2}\right\rfloor$, $j=1,\ldots,n+1$, let
$\alpha\left(v_{j}^{(2i)}v_{j+1}^{(2i)}\right)=\alpha\left(v_{j}^{(2i+1)}v_{j+1}^{(2i+1)}\right)=\alpha\left(v_{j}^{(2m-2i)}v_{j+1}^{(2m-2i)}\right)=\alpha\left(v_{j}^{(2m-2i+1)}v_{j+1}^{(2m-2i+1)}\right)=4i+2j$;
(6)
for $i=1,\ldots,\left\lfloor\frac{m}{2}\right\rfloor$, $j=n+2,\ldots,2n$, let
$\alpha\left(v_{j}^{(2i)}v_{j+1}^{(2i)}\right)=\alpha\left(v_{j}^{(2i+1)}v_{j+1}^{(2i+1)}\right)=\alpha\left(v_{j}^{(2m-2i)}v_{j+1}^{(2m-2i)}\right)=\alpha\left(v_{j}^{(2m-2i+1)}v_{j+1}^{(2m-2i+1)}\right)=4i+2(2n+1-j)+3$
and
$\alpha\left(v_{1}^{(2i)}v_{2n+1}^{(2i)}\right)=\alpha\left(v_{1}^{(2i+1)}v_{2n+1}^{(2i+1)}\right)=\alpha\left(v_{1}^{(2m-2i)}v_{2n+1}^{(2m-2i)}\right)=\alpha\left(v_{1}^{(2m-2i+1)}v_{2n+1}^{(2m-2i+1)}\right)=4i+3$;
(7)
for $i=1,\ldots,\left\lceil\frac{m}{2}\right\rceil$, $j=2,\ldots,n+1$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j}^{(2i)}\right)=\alpha\left(v_{j}^{(2m-2i+1)}v_{j}^{(2m-2i+2)}\right)=4i+2j-3$;
(8)
for $i=1,\ldots,\left\lceil\frac{m}{2}\right\rceil$, $j=n+2,\ldots,2n+1$, let
$\alpha\left(v_{j}^{(2i-1)}v_{j}^{(2i)}\right)=\alpha\left(v_{j}^{(2m-2i+1)}v_{j}^{(2m-2i+2)}\right)=4(n+1+i)-2j$;
(9)
for $i=1,\ldots,\left\lceil\frac{m}{2}\right\rceil$, let
$\alpha\left(v_{1}^{(2i-1)}v_{1}^{(2i)}\right)=\alpha\left(v_{1}^{(2m-2i+1)}v_{1}^{(2m-2i+2)}\right)=4i$;
(10)
for $i=1,\ldots,\left\lfloor\frac{m}{2}\right\rfloor$, $j=1,\ldots,n+2$, let
$\alpha\left(v_{j}^{(2i)}v_{j}^{(2i+1)}\right)=\alpha\left(v_{j}^{(2m-2i)}v_{j}^{(2m-2i+1)}\right)=4i+2j-1$;
(11)
for $i=1,\ldots,\left\lfloor\frac{m}{2}\right\rfloor$, $j=n+3,\ldots,2n+1$,
let
$\alpha\left(v_{j}^{(2i)}v_{j}^{(2i+1)}\right)=\alpha\left(v_{j}^{(2m-2i)}v_{j}^{(2m-2i+1)}\right)=4i+2(2n+3-j)$.
It is not difficult to check that $\alpha$ is an interval $(2m+2n+3)$-coloring
of $T(2m,2n+1)$ when $m$ is even, and $\alpha$ is an interval
$(2m+2n+2)$-coloring of $T(2m,2n+1)$ when $m$ is odd. $\square$
From Theorems 1, 15 and 24, we have:
###### Corollary 25
If $G=T(2m,2n)$ $(m,n\geq 2)$ and $4\leq t\leq\max\\{3m+n+2,3n+m+2\\}$, then
$G$ has an interval $t$-coloring. Also, If $H=T(2m,2n+1)$ $(m\geq
2,n\in\mathbb{N})$, $m$ is odd and $4\leq t\leq 2m+2n+2$, then $H$ has an
interval $t$-coloring, and if $H=T(2m,2n+1)$ $(m\geq 2,n\in\mathbb{N})$, $m$
is even and $4\leq t\leq 2m+2n+3$, then $H$ has an interval $t$-coloring.
Let us note that the lower bound in Theorem 24 is not so far from the upper
bound for $W\left(T(m,n)\right)$. Indeed, since $T(2m,2n)$ is bipartite,
$\Delta\left(T(2m,2n)\right)=4$ and $\mathrm{diam}\left(C(2m,2n)\right)=m+n$,
by Theorem 3, we have $W\left(T(2m,2n)\right)\leq 3(m+n)+1$. Similarly, since
$\Delta\left(T(2m,2n+1)\right)=4$ and
$\mathrm{diam}\left(T(2m,2n+1)\right)=m+n$, by Theorem 2, we have
$W\left(T(2m,2n+1)\right)\leq 3(m+n+1)+1$.
## 5 $N$-dimensional cubes
It is well-known that the $n$-dimensional cube $Q_{n}$ is the Cartesian
product of $n$ copies of $K_{2}$. In [21], Petrosyan investigated interval
colorings of $n$-dimensional cubes and proved that $w\left(Q_{n}\right)=n$ and
$W\left(Q_{n}\right)\geq\frac{n(n+1)}{2}$ for any $n\in\mathbb{N}$. In the
same paper he also conjectured that $W\left(Q_{n}\right)=\frac{n(n+1)}{2}$ for
any $n\in\mathbb{N}$. Here, we prove this conjecture.
Let $e,e^{\prime}\in E(Q_{n})$ and $e=u_{1}u_{2}$, $e^{\prime}=v_{1}v_{2}$.
The distance between two edges $e$ and $e^{\prime}$ in $Q_{n}$, we define as
follows:
$d(e,e^{\prime})=\min_{1\leq i\leq 2,1\leq j\leq 2}d\left(u_{i},v_{j}\right)$.
Let $\alpha$ be an interval $t$-coloring of $Q_{n}$.
For any $e,e^{\prime}\in E(Q_{n})$, define
$\mathrm{sp}_{\alpha}\left(e,e^{\prime}\right)$ as follows:
$\mathrm{sp}_{\alpha}\left(e,e^{\prime}\right)=\left|\alpha(e)-\alpha(e^{\prime})\right|$.
For any $k,0\leq k\leq n-1$, define $\mathrm{sp}_{\alpha,k}$ as follows:
$\mathrm{sp}_{\alpha,k}=\max\left\\{\mathrm{sp}_{\alpha}\left(e,e^{\prime}\right)\colon\,e,e^{\prime}\in
E(Q_{n})~{}and~{}d(e,e^{\prime})=k\right\\}$.
First we need the following simple lemma.
###### Lemma 26
For any pair of vertices $u,v\in V(Q_{n})$ with $d(u,v)=k$, there exist
$v_{1},v_{2},\ldots,v_{k}$ ($v_{i}\neq v_{j}$ when $i\neq j$) vertices such
that $d(u,v_{i})=k-1$ and $vv_{i}\in E(Q_{n})$ for $i=1,\ldots,k$.
Now let $\alpha$ be an interval $W(Q_{n})$-coloring of $Q_{n}$. Clearly,
$\mathrm{sp}_{\alpha,0}=n-1$.
###### Theorem 27
If $1\leq k\leq n-1$, then
$\mathrm{sp}_{\alpha,k}\leq\mathrm{sp}_{\alpha,k-1}+n-k$.
* Proof.
Let $e,e^{\prime}\in E(Q_{n})$ be any two edges of $Q_{n}$ with
$d(e,e^{\prime})=k$. Without loss of generality, we may assume that
$\alpha(e)\geq\alpha(e^{\prime})$. Since $d(e,e^{\prime})=k$, there exist $u$
and $v$ vertices such that $u\in e$ and $v\in e^{\prime}$ and $d(u,v)=k$. By
Lemma 26, we have that there are $v_{1},v_{2},\ldots,v_{k}$ ($v_{i}\neq v_{j}$
when $i\neq j$) vertices such that $d(u,v_{i})=k-1$ and $vv_{i}\in E(Q_{n})$
for $i=1,\ldots,k$. Since $Q_{n}$ is $n$-regular, we have
$\min_{1\leq i\leq k}\alpha(v_{i}v)\leq\alpha(e^{\prime})+n-k$. (*)
Let $\alpha(e^{\prime\prime})=\min_{1\leq i\leq k}\alpha(v_{i}v)$. By (*), we
obtain
$\alpha(e^{\prime})\geq\alpha(e^{\prime\prime})-(n-k)$ and
$d(e,e^{\prime\prime})=k-1$.
Thus,
$\mathrm{sp}_{\alpha}\left(e,e^{\prime}\right)=\left|\alpha(e)-\alpha(e^{\prime})\right|\leq\left|\alpha(e)-\alpha(e^{\prime\prime})+n-k\right|\leq\left|\alpha(e)-\alpha(e^{\prime\prime})\right|+n-k\leq\mathrm{sp}_{\alpha,k-1}+n-k$.
Since $e$ and $e^{\prime}$ were arbitrary edges with $d(e,e^{\prime})=k$, we
obtain $\mathrm{sp}_{\alpha,k}\leq\mathrm{sp}_{\alpha,k-1}+n-k$. $\square$
###### Corollary 28
$\mathrm{sp}_{\alpha,n-1}\leq\frac{n\left(n+1\right)}{2}-1$.
* Proof.
By Theorem 27, we have
$\mathrm{sp}_{\alpha,n-1}\leq\mathrm{sp}_{\alpha,0}+n-1+n-2+\ldots+1=\frac{n\left(n+1\right)}{2}-1$.
$\square$
###### Corollary 29
$W\left(Q_{n}\right)\leq\frac{n\left(n+1\right)}{2}$ for any $n\in\mathbb{N}$.
* Proof.
Clearly, for any $e,e^{\prime}\in E(Q_{n})$, we have $d(e,e^{\prime})\leq
n-1$. Thus, by Corollary 28, we get
$W\left(Q_{n}\right)\leq\frac{n\left(n+1\right)}{2}$. $\square$
By Theorem 6 and Corollary 29, we obtain
$W\left(Q_{n}\right)=\frac{n\left(n+1\right)}{2}$ for any $n\in\mathbb{N}$.
Moreover, by Theorem 1, we have that $Q_{n}$ has an interval $t$-coloring if
and only if $n\leq t\leq\frac{n\left(n+1\right)}{2}$.
## References
* [1] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian).
* [2] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43.
* [3] A.S. Asratian, T.M.J. Denley, R. Haggkvist, Bipartite Graphs and their Applications, Cambridge University Press, Cambridge, 1998.
* [4] M.A. Axenovich, On interval colorings of planar graphs, Congr. Numer. 159 (2002) 77-94.
* [5] M. Behzad, E.S. Mahmoodian, On topological invariants of the product of graphs, Canad. Math. Bull. 12 (1969) 157-166.
* [6] Y. Feng, Q. Huang, Consecutive edge-coloring of the generalized $\theta$-graph, Discrete Appl. Math. 155 (2007) 2321-2327.
* [7] K. Giaro, M. Kubale, Consecutive edge-colorings of complete and incomplete Cartesian products of graphs, Congr, Numer. 128 (1997) 143-149.
* [8] K. Giaro, M. Kubale, M. Malafiejski, Consecutive colorings of the edges of general graphs, Discrete Math. 236 (2001) 131-143.
* [9] K. Giaro, M. Kubale, Compact scheduling of zero-one time operations in multi-stage systems, Discrete Appl. Math. 145 (2004) 95-103.
* [10] D. Hanson, C.O.M. Loten, B. Toft, On interval colorings of bi-regular bipartite graphs, Ars Combin. 50 (1998) 23-32.
* [11] W. Imrich, S. Klavzar, Product graphs: Structure and Recognition, John Wiley & Sons, New York, 2000.
* [12] T.R. Jensen, B. Toft, Graph Coloring Problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995.
* [13] R.R. Kamalian, Interval colorings of complete bipartite graphs and trees, preprint, Comp. Cen. of Acad. Sci. of Armenian SSR, Erevan, 1989 (in Russian).
* [14] R.R. Kamalian, Interval edge colorings of graphs, Doctoral Thesis, Novosibirsk, 1990.
* [15] R.R. Kamalian, A.N. Mirumian, Interval edge colorings of bipartite graphs of some class, Dokl. NAN RA, 97 (1997) 3-5 (in Russian).
* [16] R.R. Kamalian, P.A. Petrosyan, A note on upper bounds for the maximum span in interval edge-colorings of graphs, Discrete Math. 312 (2012) 1393-1399.
* [17] R.R. Kamalian, P.A. Petrosyan, A note on interval edge-colorings of graphs, Graphs and Combin. (2012) under review.
* [18] A. Khchoyan, Interval edge-colorings of subcubic graphs and multigraphs, Yerevan State University, BS thesis, 2010, 30p.
* [19] M. Kubale, Graph Colorings, American Mathematical Society, 2004.
* [20] P.A. Petrosyan, G.H. Karapetyan, Lower bounds for the greatest possible number of colors in interval edge colorings of bipartite cylinders and bipartite tori, Proceedings of the CSIT Conference (2007) 86-88.
* [21] P.A. Petrosyan, Interval edge-colorings of complete graphs and $n$-dimensional cubes, Discrete Math. 310 (2010) 1580-1587.
* [22] P.A. Petrosyan, Interval edge colorings of some products of graphs, Discuss. Math. Graph Theory 31(2) (2011) 357-373.
* [23] P.A. Petrosyan, H.H. Khachatrian, L.E. Yepremyan, H.G. Tananyan, Interval edge-colorings of graph products, Proceedings of the CSIT Conference (2011) 89-92.
* [24] S.V. Sevast’janov, Interval colorability of the edges of a bipartite graph, Metody Diskret. Analiza 50 (1990) 61-72 (in Russian).
* [25] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 2001.
|
arxiv-papers
| 2012-01-31T21:14:21 |
2024-09-04T02:49:26.910116
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Petros A. Petrosyan, Hrant H. Khachatrian and Hovhannes G. Tananyan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1202.0023"
}
|
1202.0219
|
# A new class of generalized Genocchi polynomials
N. I. Mahmudov
Eastern Mediterranean University
Gazimagusa, TRNC, Mersiin 10, Turkey
Email: nazim.mahmudov@emu.edu.tr
###### Abstract
The main purpose of this paper is to introduce and investigate a new class of
generalized Genocchi polynomials based on the $q$-integers. The $q$-analogues
of well-known formulas are derived. The $q$-analogue of the Srivastava–Pintér
addition theorem is obtained.
## 1 Introduction
Throughout this paper, we always make use of the following notation:
$\mathbb{N}$ denotes the set of natural numbers, $\mathbb{N}_{0}$ denotes the
set of nonnegative integers, $\mathbb{R}$ denotes the set of real numbers,
$\mathbb{C}$ denotes the set of complex numbers.
The $q$-shifted factorial is defined by
$\left(a;q\right)_{0}=1,\ \ \
\left(a;q\right)_{n}={\displaystyle\prod\limits_{j=0}^{n-1}}\left(1-q^{j}a\right),\
\ \ n\in\mathbb{N},\ \ \
\left(a;q\right)_{\infty}={\displaystyle\prod\limits_{j=0}^{\infty}}\left(1-q^{j}a\right),\
\ \ \ \left|q\right|<1,\ \ a\in\mathbb{C}.$
The $q$-numbers and $q$-numbers factorial is defined by
$\left[a\right]_{q}=\frac{1-q^{a}}{1-q}\ \ \ \left(q\neq 1\right);\ \ \
\left[0\right]_{q}!=1;\ \ \ \
\left[n\right]_{q}!=\left[1\right]_{q}\left[2\right]_{q}...\left[n\right]_{q}\
\ \ \ \ n\in\mathbb{N},\ \ a\in\mathbb{C}$
respectively. The $q$-polynomail coefficient is defined by
$\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}=\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{n-k}\left(q;q\right)_{k}}.$
The $q$-analogue of the function $\left(x+y\right)^{n}$ is defined by
$\left(x+y\right)_{q}^{n}:={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\frac{1}{2}k\left(k-1\right)}x^{n-k}y^{k},\ \ \
n\in\mathbb{N}_{0}.$
In the standard approach to the $q$-calculus two exponential function are
used:
$\displaystyle e_{q}\left(z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\frac{1}{\left(1-\left(1-q\right)q^{k}z\right)},\
\ \ 0<\left|q\right|<1,\ \left|z\right|<\frac{1}{\left|1-q\right|},\ \ \ \ \ \
\ $ $\displaystyle E_{q}\left(z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right)}z^{n}}{\left[n\right]_{q}!}=\prod_{k=0}^{\infty}\left(1+\left(1-q\right)q^{k}z\right),\
\ \ \ \ \ \ 0<\left|q\right|<1,\ z\in\mathbb{C}.\ $
From this form we easily see that $e_{q}\left(z\right)E_{q}\left(-z\right)=1$.
Moreover,
$D_{q}e_{q}\left(z\right)=e_{q}\left(z\right),\ \ \ \
D_{q}E_{q}\left(z\right)=E_{q}\left(qz\right),$
where $D_{q}$ is defined by
$D_{q}f\left(z\right):=\frac{f\left(qz\right)-f\left(z\right)}{qz-z}.$
Carlitz has introduced the $q$-Bernoulli numbers and polynomials in [1].
Srivastava and Pinter proved some relations and theorems between the Bernoulli
polynomials and Euler polynomials in [22]. They also gave some generalizations
of these polynomials. In [9]-[17], Kim et al. investigated some properties of
the $q$-Euler polynomials and Genocchi polynomials. They gave some recurrence
relations. In [2], Cenkci et al. gave the $q$-extension of Genocchi numbers in
a different manner. In [13], Kim gave a new concept for the $q$-Genocchi
numbers and polynomials. In [20], Simsek et al. investigated the $q$-Genocchi
zeta function and $l$-function by using generating functions and Mellin
transformation.
###### Definition 1
The $q$-Bernoulli numbers $\mathfrak{B}_{n,q}^{\left(\alpha\right)}$ and
polynomials $\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ in
$x,y$ of order $\alpha$ are defined by means of the generating function
functions:
$\displaystyle\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<2\pi,$
$\displaystyle\left(\frac{t}{e_{q}\left(t\right)-1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<2\pi.$
###### Definition 2
The $q$-Genocchi numbers $\mathfrak{G}_{n,q}^{\left(\alpha\right)}$ and
polynomials $\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ in
$x,y$ are defined by means of the generating functions:
$\displaystyle\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<\pi,$
$\displaystyle\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)$
$\displaystyle=\sum_{n=0}^{\infty}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!},\
\ \ \left|t\right|<\pi.$
It is obvious that
$\displaystyle\mathfrak{B}_{n,q}^{\left(\alpha\right)}$
$\displaystyle=\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(0,0\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=B_{n}^{\left(\alpha\right)}\left(x+y\right),\
\ \ \lim_{q\rightarrow
1^{-}}\mathfrak{B}_{n,q}^{\left(\alpha\right)}=B_{n}^{\left(\alpha\right)},$
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}$
$\displaystyle=\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(0,0\right),\ \ \
\lim_{q\rightarrow
1^{-}}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=G_{n}^{\left(\alpha\right)}\left(x+y\right),\
\ \ \lim_{q\rightarrow
1^{-}}\mathfrak{G}_{n,q}^{\left(\alpha\right)}=G_{n}^{\left(\alpha\right)}.$
Here $B_{n}^{\left(\alpha\right)}\left(x\right)$ and
$E_{n}^{\left(\alpha\right)}\left(x\right)$ denote the classical Bernoulli and
Genocchi polynomials of order $\alpha$ are defined by
$\left(\frac{t}{e^{t}-1}\right)^{\alpha}e^{tx}=\sum_{n=0}^{\infty}B_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{\left[n\right]_{q}!}\
\ \ \ \ \text{and\ \ \ \
}\left(\frac{2}{e^{t}+1}\right)^{\alpha}e^{tx}=\sum_{n=0}^{\infty}G_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
The aim of the present paper is to obtain some results for the $q$-Genocchi
polynomials. The $q$-analogues of well-known results, for example, Srivastava
and Pintér [pinter], Cheon [cheon], etc., can be derived from these
$q$-identities. The formulas involving the $q$-Stirling numbers of the second
kind, $q$-Bernoulli polynomials and $q$-Bernstein polynomials are also given.
Furthermore some special cases are also considered.
The following elementary properties of the $q$-Genocchi polynomials
$\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ of order $\alpha$
are readily derived from Definition. We choose to omit the details involved.
Property 1. _Special values of the_ $q$_-Genocchi polynomials of order_
$\alpha$_:_
$\mathfrak{E}_{n,q}^{\left(0\right)}\left(x,0\right)=x^{n},\ \ \
\mathfrak{E}_{n,q}^{\left(0\right)}\left(0,y\right)=q^{\frac{1}{2}n\left(n-1\right)}y^{n}.$
Property 2._Summation formulas for the_ $q$_-Genocchi polynomials of order_
$\alpha$_:_
$\displaystyle\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{k,q}^{\left(\alpha\right)}\left(x+y\right)_{q}^{n-k},\
\ \
\mathfrak{E}_{n,q}^{\left(\alpha\right)}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{n-k,q}^{\left(\alpha-1\right)}\mathfrak{E}_{k,q}\left(x,y\right),$
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(x,0\right)y^{n-k}=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(0,y\right)x^{n-k},$
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{G}_{k,q}^{\left(\alpha\right)}x^{n-k},\ \ \
\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(0,y\right)=\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}q^{\left(n-k\right)\left(n-k-1\right)/2}\mathfrak{G}_{k,q}^{\left(\alpha\right)}y^{n-k}.$
Property 3._Difference equations:_
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(1,y\right)+\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(0,y\right)$
$\displaystyle=2\left[n\right]_{q}\mathfrak{G}_{n-1,q}^{\left(\alpha-1\right)}\left(0,y\right),$
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,0\right)+\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,-1\right)$
$\displaystyle=2\left[n\right]_{q}\mathfrak{G}_{n-1,q}^{\left(\alpha-1\right)}\left(x,-1\right).$
Property 4. _Differential relations:_
$D_{q,x}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=\left[n\right]_{q}\mathfrak{G}_{n-1,q}^{\left(\alpha\right)}\left(x,y\right),\
\ \
D_{q,y}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)=\left[n\right]_{q}\
\mathfrak{G}_{n-1,q}^{\left(\alpha\right)}\left(x,qy\right).$
Property 5. _Addition theorem of the argument:_
$\mathfrak{E}_{n,q}^{\left(\alpha+\beta\right)}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\mathfrak{E}_{n-k,q}^{\left(\alpha\right)}\left(x,0\right)\mathfrak{E}_{k,q}^{\left(\beta\right)}\left(0,y\right).$
Property 6. _Recurrence Relationships:_
$\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(\frac{1}{m},y\right)+\sum_{k=0}^{n}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{n-k}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(0,y\right)=2\left[n\right]_{q}\sum_{k=0}^{n-1}\left[\begin{array}[c]{c}n-1\\\
k\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{n-1-k}\mathfrak{G}_{k,q}^{\left(\alpha-1\right)}\left(0,y\right).$
## 2 Explicit relationship between the $q$-Genocchi and the $q$-Bernoulli
polynomials
In this section we prove an interesting relationship between the $q$-Genocchi
polynomials $\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ of
order $\alpha$ and the $q$-Bernoulli polynomials. Here some $q$-analogues of
known results will be given. We also obtain new formulas and their some
special cases below.
###### Theorem 3
For $n\in\mathbb{N}_{0}$, the following relationship
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\frac{1}{m^{n-k-1}\left[k+1\right]_{q}}\left[2\left[k+1\right]_{q}{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\frac{1}{m^{k-j}}\mathfrak{G}_{j,q}^{\left(\alpha-1\right)}\left(x,-1\right)\right.$
$\displaystyle-\left.{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\frac{1}{m^{k+1-j}}\mathfrak{G}_{j,q}^{\left(\alpha\right)}\left(x,-1\right)-\mathfrak{G}_{k+1,q}^{\left(\alpha\right)}\left(x,0\right)\right]\mathfrak{B}_{n-k,q}\left(0,my\right).$
holds true between the $q$-Genocchi and the $q$-Bernoulli polynomials..
Proof. Using the following identity
$\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)=\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)\cdot\frac{e_{q}\left(\frac{t}{m}\right)-1}{t}\cdot\frac{t}{e_{q}\left(\frac{t}{m}\right)-1}\cdot
E_{q}\left(\frac{t}{m}my\right)$
we have
$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)\frac{t^{n}}{\left[n\right]_{q}!}$
$\displaystyle=\frac{m}{t}{\displaystyle\sum\limits_{n=0}^{\infty}}\left({\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-k}}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(x,0\right)-\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\right)\frac{t^{n}}{\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}$
$\displaystyle={\displaystyle\sum\limits_{n=1}^{\infty}}\left({\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-1-k}}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(x,0\right)-m\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,0\right)\right)\frac{t^{n-1}}{\left[n\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}\left({\displaystyle\sum\limits_{k=0}^{n+1}}\left[\begin{array}[c]{c}n+1\\\
k\end{array}\right]_{q}m^{k}\mathfrak{G}_{k,q}^{\left(\alpha\right)}\left(x,0\right)-m^{n+1}\mathfrak{G}_{n+1,q}^{\left(\alpha\right)}\left(x,0\right)\right)\frac{t^{n}}{m^{n}\left[n+1\right]_{q}!}{\displaystyle\sum\limits_{n=0}^{\infty}}\mathfrak{B}_{n,q}\left(0,my\right)\frac{t^{n}}{m^{n}\left[n\right]_{q}!}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}{\displaystyle\sum\limits_{k=0}^{n}}\frac{1}{m^{n}\left[k+1\right]_{q}}\left({\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}m^{j}\mathfrak{G}_{j,q}^{\left(\alpha\right)}\left(x,0\right)-m^{k+1}\mathfrak{G}_{k+1,q}^{\left(\alpha\right)}\left(x,0\right)\right)\mathfrak{B}_{n-k,q}\left(0,my\right)\frac{t^{n}}{\left[n\right]_{q}!}.$
It remains to use Poperty 6.
Since $\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$ is not
symmetric with respect to $x$ and $y$ we can prove a different from of the
above theorem. It should be stressed out that Theorems 3 and 4 coincide in the
limiting case when $q\rightarrow 1^{-}.$
###### Theorem 4
For $n\in\mathbb{N}_{0}$, the following relationship
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-k-1}\left[k+1\right]_{q}}\left[2\left[k+1\right]_{q}\sum_{j=0}^{k}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{G}_{j,q}^{\left(\alpha-1\right)}\left(0,y\right)\right.$
$\displaystyle-\left.\sum_{j=0}^{k+1}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k+1-j}\mathfrak{G}_{j,q}^{\left(\alpha\right)}\left(0,y\right)-\mathfrak{G}_{k+1,q}\left(0,y\right)\right]\mathfrak{B}_{n-k,q}\left(mx,0\right)$
holds true between the $q$-Genocchi and the $q$-Bernoulli polynomials.
Proof. The proof is based on the following identity
$\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}e_{q}\left(tx\right)E_{q}\left(ty\right)=\left(\frac{2t}{e_{q}\left(t\right)+1}\right)^{\alpha}E_{q}\left(ty\right)\cdot\frac{e_{q}\left(\frac{t}{m}\right)-1}{t}\cdot\frac{t}{e_{q}\left(\frac{t}{m}\right)-1}\cdot
e_{q}\left(\frac{t}{m}mx\right).$
Next we discuss some special cases of Theorems 3 and 4. By noting that
$\mathfrak{G}_{j,q}^{\left(0\right)}\left(0,y\right)=q^{\frac{1}{2}j\left(j-1\right)}y^{j},\
\ \ \ \
\mathfrak{G}_{j,q}^{\left(0\right)}\left(x,-1\right)=\left(x-1\right)_{q}^{j}$
we deduce from Theorems 3 and 4 Corollary 5 below.
###### Corollary 5
For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship
$\displaystyle\mathfrak{G}_{n,q}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-k-1}\left[k+1\right]_{q}}\left[2\left[k+1\right]_{q}\sum_{j=0}^{k}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}q^{\frac{1}{2}j\left(j-1\right)}y^{j}\right.$
$\displaystyle-\left.\sum_{j=0}^{k+1}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k+1-j}\mathfrak{G}_{j,q}\left(0,y\right)-\mathfrak{G}_{k+1,q}\left(0,y\right)\right]\mathfrak{B}_{n-k,q}\left(mx,0\right),$
$\displaystyle\mathfrak{G}_{n,q}\left(x,y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-k-1}\left[k+1\right]_{q}}\left[2\left[k+1\right]_{q}{\displaystyle\sum\limits_{j=0}^{k}}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\frac{1}{m^{k-j}}\left(x-1\right)_{q}^{j}\right.$
$\displaystyle-\left.{\displaystyle\sum\limits_{j=0}^{k+1}}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\frac{1}{m^{k+1-j}}\mathfrak{G}_{j,q}\left(x,-1\right)-\mathfrak{G}_{k+1,q}\left(x,0\right)\right]\mathfrak{B}_{n-k,q}\left(0,my\right).$
holds true between the $q$-Bernoulli polynomials and $q$-Euler polynomials.
###### Corollary 6
For $n\in\mathbb{N}_{0}$, $m\in\mathbb{N}$ the following relationship holds
true.
$\displaystyle G_{n}\left(x+y\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\frac{2}{k+1}\left(\left(k+1\right)y^{k}-G_{k+1,q}\left(y\right)\right)B_{n-k}\left(x\right),$
(3) $\displaystyle G_{n}\left(x+y\right)$
$\displaystyle=\sum_{k=0}^{n}\left(\begin{array}[c]{c}n\\\
k\end{array}\right)\frac{1}{m^{n-k-1}\left(k+1\right)}\left[2\left(k+1\right)G_{k}\left(y+\frac{1}{m}-1\right)-G_{k+1}\left(y+\frac{1}{m}-1\right)-G_{k+1}\left(y\right)\right]B_{n-k,q}\left(mx\right)$
(6)
between the classical Genocchi polynomials and the classical Bernoulli
polynomials.
Note that the formula (6) is new for the classical polynomials.
In terms of the $q$-Genocchi numbers
$\mathfrak{G}_{k,q}^{\left(\alpha\right)}$, by setting $y=0$ in Theorem 3, we
obtain the following explicit relationship between the $q$-Genocchi
polynomials $\mathfrak{G}_{k,q}^{\left(\alpha\right)}$ of order $\alpha$ and
the $q$-Bernoulli polynomials.
###### Corollary 7
The following relationship holds true:
$\displaystyle\mathfrak{G}_{n,q}^{\left(\alpha\right)}\left(x,0\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{1}{m^{n-k-1}\left[k+1\right]_{q}}\left[2\left[k+1\right]_{q}\sum_{j=0}^{k}\left[\begin{array}[c]{c}k\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k-j}\mathfrak{G}_{j,q}^{\left(\alpha-1\right)}\right.$
$\displaystyle-\left.\sum_{j=0}^{k+1}\left[\begin{array}[c]{c}k+1\\\
j\end{array}\right]_{q}\left(\frac{1}{m}-1\right)_{q}^{k+1-j}\mathfrak{G}_{j,q}^{\left(\alpha\right)}-\mathfrak{G}_{k+1,q}^{\left(\alpha\right)}\right]\mathfrak{B}_{n-k,q}\left(mx,0\right).$
###### Corollary 8
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\mathfrak{G}_{n,q}\left(x,y\right)={\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{2}{\left[k+1\right]_{q}}\left[\left[k+1\right]_{q}q^{\frac{1}{2}k\left(k-1\right)}y^{k}-\mathfrak{G}_{k+1,q}\left(0,y\right)\right]\mathfrak{B}_{n-k,q}\left(x,0\right).$
###### Corollary 9
For $n\in\mathbb{N}_{0}$ the following relationship holds true.
$\displaystyle\mathfrak{G}_{n,q}\left(x,0\right)$
$\displaystyle=-{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{2}{\left[k+1\right]_{q}}\mathfrak{G}_{k+1,q}\mathfrak{B}_{n-k,q}\left(x,0\right),$
$\displaystyle\mathfrak{G}_{n,q}$
$\displaystyle=-{\displaystyle\sum\limits_{k=0}^{n}}\left[\begin{array}[c]{c}n\\\
k\end{array}\right]_{q}\frac{2}{\left[k+1\right]_{q}}\mathfrak{G}_{k+1,q}\mathfrak{B}_{n-k,q}.$
## References
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* [4] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
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* [6] L. C. Jang and T. Kim, $q$-Genocchi numbers and polynomials associated with fermionic p-adic invariant integrals on Zp, Abstr. Appl. Anal. 2008 (2008), Art. ID 232187, 8 pp. doi:10.1155/2008/232187.
* [7] L. C. Jang, T. Kim, D. H. Lee, and D. W. Park, An application of polylogarithms in the analogue of Genocchi numbers, NNTDM 7 (2000), 66{70.
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|
arxiv-papers
| 2012-02-01T17:06:58 |
2024-09-04T02:49:26.924800
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nazim I. Mahmudov",
"submitter": "Nazim Mahmudov Idris",
"url": "https://arxiv.org/abs/1202.0219"
}
|
1202.0222
|
Subvarieties of hypercomplex manifolds with holonomy in $SL(n,{\mathbb{H}})$
Andrey Soldatenkov111Andrey Soldatenkov is partially supported by AG
Laboratory NRU-HSE, RF government grant, ag. 11.G34.31.0023, Misha
Verbitsky222Misha Verbitsky is partially supported by RFBR grant
10-01-93113-NCNIL-a, Simons-IUM fellowship, RFBR grant 09-01-00242-a, Science
Foundation of the SU-HSE award No. 10-09-0015 and AG Laboratory NRU-HSE, RF
government grant, ag. 11.G34.31.0023.
Abstract A hypercomplex manifold $M$ is a manifold with a triple $I,J,K$ of
complex structure operators satisfying quaternionic relations. For each
quaternion $L=aI+bJ+cK$, $L^{2}=-1$, $L$ is also a complex structure operator
on $M$, called an induced complex structure. We are studying compact complex
subvarieties of $(M,L)$, when $L$ is a generic induced complex structure.
Under additional assumptions (Obata holonomy contained in
$SL(n,{\mathbb{H}})$, existence of an HKT metric), we prove that $(M,L)$
contains no divisors, and all complex subvarieties of codimension 2 are
trianalytic (that is, also hypercomplex).
###### Contents
1. 1 Introduction
1. 1.1 Hypercomplex manifolds: an introduction
2. 1.2 Trianalytic subvarieties
2. 2 Introduction to the geometry of $SL(n,{\mathbb{H}})$-manifolds
1. 2.1 The quaternionic Dolbeault complex on $SL(n,{\mathbb{H}})$-manifolds
2. 2.2 Calibrations on $SL(n,{\mathbb{H}})$-manifolds
3. 2.3 Holomorphic Lagrangian subvarieties in $SL(n,{\mathbb{H}})$-manifolds
3. 3 Subvarieties in $SL(n,{\mathbb{H}})$-manifolds
## 1 Introduction
### 1.1 Hypercomplex manifolds: an introduction
Definition 1.1: 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 $I,J,K$, it is called hyperkähler
([Bes], [Bo]).
The term “hypercomplex manifold” is due to C. P. Boyer, [Bo], who classified
compact hypercomplex manifolds of quaternionic dimension 1, though the notion
was considered as early as in 1955, by M. Obata ([Ob]).
The first interesting non-hyperkähler examples of hypercomplex manifolds were
found by physicists in [SSTV], and independently by D. Joyce in [J]. In the
same paper, Joyce classified all homogeneous hypercomplex structures on simply
connected compact manifolds, using the Wang’s classification of homogeneous
spaces ([Wa]).
Next, we recall the definition of an HKT-metric.
Definition 1.2: Let $(M,I,J,K)$ be a hypercomplex manifold and $g$ a
quaternionic Hermitian metric. Consider the Hermitian forms:
$\omega_{I}(X,Y)=g(IX,Y),\quad\omega_{J}(X,Y)=g(JX,Y),\quad\omega_{K}(X,Y)=g(KX,Y).$
If any two of these forms are closed, the manifold is hyperkähler. Define
$\Omega_{I}=\omega_{J}+\sqrt{-1}\omega_{K}$. It is easy to check that
$\Omega_{I}\in\Lambda^{2,0}_{I}M$. The metric $g$ is called HKT (“hyperkähler
with torsion”) if $\partial\Omega_{I}=0$, where
$\partial\colon\Lambda^{p,q}_{I}M\to\Lambda^{p+1,q}_{I}M$ is the $(1,0)$-part
of the de Rham differential. In this case, the form $\Omega_{I}$ is called an
HKT-form, and $(M,I,J,K,g)$ an HKT-manifold.
HKT-metrics were introduced by P. S. Howe and G. Papadopoulos [HP] (see also
[GP]) and were much studied since then. Existence of an HKT metric puts a
significant constraint on a global geometry of a hypercomplex manifold ([FG],
[BDV]).
Since the advent of string theory, hypercomplex manifolds became an important
object in physics, because the corresponding $\sigma$-models exhibit
interesting supersymmetries ([GHR]). After A. Strominger’s paper [St],
supersymmetric $\sigma$-models associated with non-Kähler target spaces became
a popular object of study. Strominger proposed to use the antisymmetric
torsion connections on the target spaces. In mathematics, such structures were
studied by J. Bismut [Bi] in connection with the local index formula. In the
hypercomplex setting, Bismut connections were studied by P. S. Howe and G.
Papadopoulos in 1990-ies in a series of papers starting with [HP]. This
research lead them to the discovery of HKT metrics.
Since [GP], HKT-metrics became an important ingredient in the mathematical
study of hypercomplex geometry. The HKT metrics share much in common with the
Kähler structures. Like Kähler metrics, they are locally defined by a
potential ([BS]), but can be used to obtain Hodge-theoretic restrictions on
the cohomology ([V3]).
In the present paper, we use the HKT-geometry to study complex subvarieties in
hypercomplex manifolds.
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. Therefore, for any $a,b,c\in{\mathbb{R}}$, with
$a^{2}+b^{2}+c^{2}=1$, the almost complex structure $L=aI+bJ+cK$ is in fact
integrable. By Newlander-Nirenberg theorem, $L$ defines a complex structure on
$M$. We denote by $(M,L)$ the complex manifold corresponding to this complex
structure.
Definition 1.3: A complex structure $L=aI+bJ+cK$, with $a^{2}+b^{2}+c^{2}=1$,
is called induced by quaternions, and the corresponding family, parametrized
by ${\mathbb{C}}P^{1}\cong S^{2}$ — the twistor family.
Let $(V,I,J,K)$ be a quaternionic vector space of real dimension $4n$. The
group $GL(n,{\mathbb{H}})$ consists of linear transformations of $V$ that
preserve the complex structures $I,J$ and $K$. Consider the Hodge
decomposition $V\otimes_{\mathbb{R}}{\mathbb{C}}=V^{1,0}_{I}\oplus
V^{0,1}_{I}$, where $V^{1,0}_{I}$ and $V^{0,1}_{I}$ are eigenspaces of $I$
with eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$ respectively. Let
$\Lambda^{2n,0}_{I}V$ be the top exterior power of $V^{1,0}_{I}$. Recall that
$SL(n,{\mathbb{H}})$ is a subgroup in $GL(n,{\mathbb{H}})$ consisting of those
elements that act trivially on $\Lambda^{2n,0}_{I}V$.
Let $(M,I,J,K)$ be a hypercomplex manifold and $\nabla$ the corresponding
Obata connection. Denote by $\operatorname{Hol}(\nabla)$ the holonomy group of
$\nabla$. Since the Obata connection preserves the quaternionic structure, we
have $\operatorname{Hol}(\nabla)\subset GL(n,{\mathbb{H}})$.
Definition 1.4: If the holonomy group $\operatorname{Hol}(\nabla)$ of the
Obata connection on a hypercomplex manifold $M$ is contained in
$SL(n,{\mathbb{H}})$, we call $M$ an $SL(n,{\mathbb{H}})$-manifold.
Remark 1.5: It is easy to see that an $SL(n,{\mathbb{H}})$-manifold has a
trivial canonical bundle (in fact, the canonical bundle of such a manifold has
a canonical flat connection with trivial monodromy). The converse is also
true, for compact manifolds admitting an HKT-metric, as follows from the Hodge
theory of HKT-manifolds ([V8]).
Example 1.6: Let $G$ be a connected, simply connected nilpotent Lie group,
and $\Gamma\subset G$ a discrete, co-compact subgroup. The quotient
$N:=\Gamma\backslash G$ is called a nilmanifold. Suppose that $I,J,K$ are
left-invariant complex structures on $G$ that satisfy quaternionic relations.
Then the hypercomplex structure descends to $N$ and we call $N$ a hypercomplex
nilmanifold. It was shown in [BDV] that any hypercomplex nilmanifold is in
fact an $SL(n,{\mathbb{H}})$-manifold.
Example 1.7: One more example of an $SL(n,{\mathbb{H}})$-manifold, due to A.
Swann, is a torus fibrations over a hyperkähler base ([Sw]). Let $(X,I,J,K)$
be a hyperkähler manifold. A 2-form $\alpha\in\Lambda^{2}X$ is called anti-
self-dual if it is of type (1,1) with respect to any induced complex
structure. If $\alpha$ represents an integral cohomology class, then it
defines a principal $U(1)$-bundle over $X$. Given $4k$ such forms,
$\alpha_{1},\ldots,\alpha_{4k}$, we obtain a principal $T^{4k}$-bundle
$\pi\colon M\to X$. This bundle admits an instanton connection $A$, given by
1-forms $\theta_{i}\in\Lambda^{1}M$, such that
$d\theta_{i}=\pi^{*}(\alpha_{i})$. The hypercomplex structure on $M$ is
defined as follows: on horizontal subspaces of $A$ the quaternionic action is
lifted from $X$, and on vertical subspaces it is given by a flat hypercomplex
structure of $4k$-dimensional torus. From this construction it is easy to see
that $M$ is an $SL(n,{\mathbb{H}})$-manifold.
The Hopf manifold $H=({\mathbb{H}}^{n}\backslash 0)/\langle A\rangle$ equipped
with a standard hypercomplex structure is not an
$SL(n,{\mathbb{H}})$-manifold. Indeed, the holonomy of the Obata connection on
$H$ is ${\mathbb{Z}}$ acting on $TM$ as a matrix $A$ with all eigenvalues
$|\alpha_{i}|>1$.
It follows from the adjunction formula that none of the homogeneous
hypercomplex manifolds constructed by Joyce in [J] has holonomy in
$SL(n,{\mathbb{H}})$. Indeed, such a manifold is fibered over a homogeneous
Fano manifold with toric fibers, hence its canonical bundle is non-trivial.
However, an $SL(n,{\mathbb{H}})$-manifold has trivial (even flat) canonical
bundle. It was shown in [Sol] that the manifold $SU(3)$ with its Joyce
hypercomplex structure has holonomy $GL(2,{\mathbb{H}})$; a similar conjecture
is stated, but not proven, for all homogeneous hypercomplex manifolds.
### 1.2 Trianalytic subvarieties
Definition 1.8: Let $M$ be a hypercomplex manifold. A subset $Z\subset M$ is
called trianalytic if it is complex analytic in $(M,L)$ for all induced
complex structures $L$.
Geometry of trianalytic subvarieties was studied at some length in [V2]. It
was shown that any trianalytic subvariety can be desingularized by taking a
normalization, and this desingularization is smooth and hypercomplex.
The following theorem was proved in [V6] (see also [V1]).
Theorem 1.9: Let $M$ be a hyperkähler manifold, not necessarily compact. Then
there exists a countable subset $S\subset{\mathbb{C}}P^{1}$ of induced complex
structures, such that for all compact complex subvarieties $Z\subset(M,L)$,
$L\notin S$, the subset $Z\subset M$ is trianalytic.
Remark 1.10: We call an induced complex structure $L$ generic, if
$L\in{\mathbb{C}}P^{1}\backslash S$. If $L$ is a generic induced complex
structure on a hyperkähler manifold $M$, then $(M,L)$ has no compact complex
subvarieties except trianalytic subvarieties. Since trianalytic subvarieties
are hypercomplex in their smooth points, their complex codimension is even.
Therefore, such $(M,L)$ has no compact odd-dimensional subvarieties. This
implies that $(M,L)$ is not algebraic.
It is interesting to note that this result is manifestly false for a general
hypercomplex manifold. For example, consider a Hopf surface
$H:=({\mathbb{H}}\backslash 0)/(x\sim 2x)$. For each induced complex structure
$L=aI+bJ+cK$, the manifold $(H,L)$ is fibered over ${\mathbb{C}}P^{1}$ with
fibers elliptic curves, isomorphic to $({\mathbb{C}}\backslash 0)/(x\sim 2x)$.
Therefore, $(H,L)$ contains divisors for each induced complex structure $L$.
However, for $SL(n,{\mathbb{H}})$-manifolds we still retain some control over
subvarieties. In [GV], the results of [V1] and [V6] were interpreted in terms
of calibrations on hyperkähler manifolds (2.2). It turns out that some of the
calibrations constructed in hyperkähler geometry survive in a more general
hypercomplex setting (2.2). This is used to obtain a weaker version of 1.2:
Theorem 1.11: Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold admitting
an HKT-metric. Then there exists a countable subset
$S\subset{\mathbb{C}}P^{1}$, such that for any induced complex structure
$L\in{\mathbb{C}}P^{1}\backslash S$, the manifold $(M,L)$ has no compact
divisors, and all compact complex subvarieties $Z\subset(M,L)$ of complex
codimension 2 are trianalytic.
Proof: See the paragraph after the proof of 3.
Without an HKT assumption, one can prove non-existence of holomorphic
Lagrangian subvarieties (for a definition of holomorphic Lagrangian
subvarieties in $SL(n,{\mathbb{H}})$-manifolds, please see 2.3).
Theorem 1.12: Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold. Then there
exists a countable subset $S\subset{\mathbb{C}}P^{1}$, such that for any
induced complex structure $L\in{\mathbb{C}}P^{1}\backslash S$, the manifold
$(M,L)$ has no compact holomorphic Lagrangian subvarieties.
Proof: For any holomorphic Lagrangian subvariety $X\subset(M,I)$, one has
$TX\cap J(TX)=0$, because $TX\subset TM$ is a Lagrangian subspace, for any
quaternionic Hermitian metric. Therefore, 1.2 is immediately implied by 3 (see
also 2.2).
In the following section we recall some facts about
$SL(n,{\mathbb{H}})$-manifolds and calibrations. We prove 1.2 in Section 3.
## 2 Introduction to the geometry of $SL(n,{\mathbb{H}})$-manifolds
This section is an introduction to HKT geometry of
$SL(n,{\mathbb{H}})$-manifolds and their calibrations. We follow [GV] and
[V4].
### 2.1 The quaternionic Dolbeault complex on $SL(n,{\mathbb{H}})$-manifolds
In this subsection, we recall the definition of a quaternionic Dolbeault
algebra of a hypercomplex manifold. We follow [V4], though this complex is
essentially due to [CS].
Let $(M,I,J,K)$ be a hypercomplex manifold, $\dim_{\mathbb{R}}M=4n$. There is
a natural multiplicative action of $SU(2)\subset{\mathbb{H}}^{*}$ on
$\Lambda^{*}(M)$, associated with the hypercomplex structure.
It is well-known that any irreducible complex representation of $SU(2)$ is a
symmetric power $S^{i}(W_{1})$, where $W_{1}$ is a fundamental 2-dimensional
representation. We say that a representation $U$ has weight $i$ if it is
isomorphic to $S^{i}(W_{1})$. It follows from the Clebsch-Gordan formula that
the weight is multiplicative in the following sense: if $i\leqslant j$, then
$W_{i}\otimes W_{j}=\bigoplus_{k=0}^{i}W_{i+j-2k},$
where $W_{i}=S^{i}(W_{1})$ denotes the irreducible representation of weight
$i$.
Let $V^{i}\subset\Lambda^{i}(M)$ be a sum of all irreducible
subrepresentations $W\subset\Lambda^{i}(M)$ of weight $<i$. Since the weight
is multiplicative, $V^{*}=\bigoplus_{i}V^{i}$ is an ideal in $\Lambda^{*}(M)$.
It is easy to see that the de Rham differential $d$ increases the weight by
one at most: $dV^{i}\subset V^{i+1}$. So $V^{*}\subset\Lambda^{*}(M)$ is a
differential ideal in the de Rham DG-algebra $(\Lambda^{*}(M),d)$.
Definition 2.1: 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).
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 ([V3]):
Proposition 2.2: 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
${\cal
R}_{p,q}:\;\Lambda^{p+q,0}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M).$
(2.1)
Consider the projection
$\Pi_{+}^{p,q}\colon\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p,q}_{I,+}(M)$
and let
$R:\;\Lambda^{p,q}_{I}(M){\>\longrightarrow\>}\Lambda^{p+q,0}_{I}(M)$
denote the composition ${\cal R}_{p,q}^{-1}\circ\Pi_{+}^{p,q}$.
Now, let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold,
$\dim_{\mathbb{R}}M=4n$. Let $\Phi_{I}$ be a nowhere degenerate holomorphic
section of $\Lambda^{2n,0}_{I}(M)$. Assume that $\Phi_{I}$ is real, that is,
$J(\Phi_{I})=\overline{\Phi}_{I}$. Existence of such a form is equivalent to
$\operatorname{Hol}(\nabla)\subset SL(n,{\mathbb{H}})$, where $\nabla$ is the
Obata connection (see [V5]). 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
${\cal V}_{p,q}(\eta)\wedge\alpha=\eta\wedge
R(\alpha)\wedge\overline{\Phi}_{I},$ (2.2)
for any test form $\alpha\in\Lambda^{n-p,n-q}_{I}(M)$.
The following proposition establishes some important properties of ${\cal
V}_{p,q}$ (for the proof, see [V4], Proposition 4.2, or [AV1], Theorem 3.6):
Proposition 2.3: Let $(M,I,J,K)$ 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 weakly positive if and only if
$\eta$ is weakly 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 $n$.
### 2.2 Calibrations on $SL(n,{\mathbb{H}})$-manifolds
In this subsection, we recall the construction of the sequence of calibrations
on $SL(n,{\mathbb{H}})$-manifolds, following [GV]. These calibrations will
play the central role in the proof of the main theorem.
Definition 2.4: ([HL]) Let $(V,g)$ be a Euclidean space. For any $p$-form
$\eta\in\Lambda^{p}(V^{*})$, let $\operatorname{\sf comass}(\eta)$ be the
maximum of
$\frac{\eta(v_{1},v_{2},\ldots,v_{p})}{|v_{1}||v_{2}|\ldots|v_{p}|}$, for all
$p$-tuples $(v_{1},...,v_{p})$ of vectors in $V$.
Definition 2.5: ([HL]) A precalibration on a Riemannian manifold is a
differential form $\eta$ with $\operatorname{\sf comass}(\eta)\leqslant 1$
everywhere. A calibration is a precalibration which is closed.
Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold, with $\Phi_{I}$ a
holomorphic volume form on $(M,I)$ preserved by the Obata connection. We will
assume that $\Phi_{I}$ is real, that is $J(\Phi_{I})=\overline{\Phi}_{I}$. A
number of interesting calibrations can be constructed in this situation. The
following theorem was proved in [GV] (Theorem 5.4):
Theorem 2.6: Let $(M,I,J,K)$ 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 $V_{n+i,n+i}^{J}:=(\Phi_{I})_{J}^{n,n}\wedge\omega_{J}^{i}$ 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
$\widetilde{\omega}_{K}+\sqrt{-1}\>\widetilde{\omega}_{I}$.
Note, that $V_{n+i,n+i}^{J}\in\Lambda^{n+i,n+i}_{J}M$, but using the same
construction we can obtain a similar calibration
$V_{n+i,n+i}^{L}\in\Lambda^{n+i,n+i}_{L}M$ for any induced complex structure
$L$.
We will need the following characterization of the form $V_{n+i,n+i}^{I}$ (for
the proof, see [GV], Remark 3.8 and Proposition 3.9):
Proposition 2.7: Let $V_{n+i,n+i}^{I}\in\Lambda^{n+i,n+i}(M,I)$ be a
calibration from 2.2. Then $V_{n+i,n+i}^{I}$ is proportional to
$\mathcal{V}_{i,i}(\Omega_{I}^{i})$ and to
$\Pi_{+}^{n+i,n+i}(\omega_{I}^{n+i})$ with some positive coefficients that do
not depend on the complex structure $I$ (here $\Omega_{I}$ is an HKT form). In
particular, the form $V_{n+i,n+i}^{I}$ is of maximal weight and for any
$\alpha\in\Lambda^{n-i,n-i}$ we have
$V_{n+i,n+i}^{I}\wedge\alpha=a_{i}\Omega_{I}^{i}\wedge
R(\alpha)\wedge\overline{\Phi}_{I},$ (2.3)
where $a_{i}$ are some positive functions on $M$.
Remark 2.8: Note that the calibrations $V_{n+i,n+i}^{I}$ are constructed in
the case when the metric is HKT. However, this assumption is not necessary for
$i=0$. Since by 2.1 the form ${\cal V}_{0,0}(1)$ is always closed, 2.2 is true
for $i=0$ even if the metric is not HKT. This remark makes it possible to
prove 1.2 without the HKT assumption.
Remark 2.9: In general, the form $V_{n+i,n+i}^{J}$ is not parallel with
respect to the Obata connection. Otherwise, since $\Phi_{I}$ is parallel,
$\omega_{J}$ would also be parallel. Then the manifold $(M,I,J,K,g)$ would
necessarily be hyperkähler. In fact, $V_{n+i,n+i}^{J}$ is not parallel with
respect to any torsion-free connection on $M$ (see [GP], Claim 6.6).
### 2.3 Holomorphic Lagrangian subvarieties in $SL(n,{\mathbb{H}})$-manifolds
Let $(M,I,J,K)$ be a $SL(n,{\mathbb{H}})$-manifold, and
$\Phi_{J}\in\Lambda^{2n,0}(M,J)$ a section of the canonical bundle of $(M,J)$
parallel with respect to the Obata connection. Since $I$ and $J$ anticommute,
$I(\Phi_{J})$ is a section of $\Lambda^{0,2n}(M,J)$, hence satisfies
$I(\Phi_{J})=\alpha\overline{\Phi}_{J}$, for $\alpha$ a complex number such
that $|\alpha|=1$. Rescaling $\Phi_{J}$, we can always assume that
$I(\Phi_{J})=\overline{\Phi}_{J}$. Denote by
$\widetilde{V}_{n,n}:=\frac{1}{n!}(\operatorname{Re}\Phi_{J})^{n,n}_{I}$ the
$(n,n)$-part of $\Phi_{J}$, taken with respect to $I$. In [GV] it was shown
that $\widetilde{V}_{n,n}$ is a calibration for any quaternionic Hermitian
metric which satisfies $|\Phi_{J}|=1$. The corresponding calibrated
subvarieties were described ([GV, Proposition 5.1]) as follows.
Theorem 2.10: Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold, $X\subset
M$ a subvariety, and $\widetilde{V}_{n,n}\in\Lambda^{n,n}(M,I)$ the
calibration defined above. Consider a quaternionic Hermitian metric $h$ on
$(M,I,J,K)$, and let $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ be a
(2,0)-form constructed from $h$ as in 1.1. Then the following conditions are
equivalent.
(i)
$\widetilde{V}_{n,n}$ calibrates $X$.
(i)
$X\subset(M,I)$ is a complex subvariety which is Lagrangian with respect to
$\Omega$.
Proof: [GV, Proposition 5.1].
Definition 2.11: Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold, and
$X\subset(M,I)$ a complex subvariety. We say that $X$ is holomorphic
Lagrangian if it is calibrated by $\widetilde{V}_{n,n}$.
Remark 2.12: It is remarkable that one is able to define holomorphic
Lagrangian subvarieties in the absence of a holomorphic symplectic form. More
precisely, the property of being holomorphic Lagrangian is independent from
the choice of a quaternionic Hermitian structure which determines the
$(2,0)$-form $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$.
## 3 Subvarieties in $SL(n,{\mathbb{H}})$-manifolds
In this subsection, we prove the main result of this paper (3), which is used
to prove 1.2.
Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold equipped with an HKT-
metric $g$. In the previous subsection we have constructed a sequence of
closed positive forms $V_{n+i,n+i}^{I}\in\Lambda^{n+i,n+i}_{I}M$,
$i=0,1,...,n$. We will use these forms to prove 1.2.
The proof of 1.2 is based on an observation, which is essentially linear-
algebraic. Let $(U,I,J,K)$ be a quaternionic vector space of real dimension
$4n$, $\Phi_{I}\in\Lambda^{2n,0}_{I}(U^{*})$ a complex volume form and
$V_{n+i,n+i}^{I}$ the element of $\Lambda^{n+i,n+i}_{I}(U^{*})$ constructed in
2.2. Consider an $I$-invariant subspace $W\subset U$, of complex dimension
$n+i$. Note that $\dim_{\mathbb{C}}(W\cap J(W))\geqslant 2i$. Let
$\xi_{W}\in\Lambda^{n+i,n+i}_{I}U$ be a volume polyvector of $W$ (it is well
defined up to a scalar multiplier). Consider a function $\psi\colon
SU(2)\to{\mathbb{R}}$ mapping $g\in SU(2)$ to $\langle
V_{n+i,n+i}^{I},g(\xi_{W})\rangle$. Since $W$ is $I$-invariant, $\psi$ is
constant on the $U(1)$-subgroup of $SU(2)$ associated with the complex
structure $I$. This allows one to consider $\psi$ as a function on
$SU(2)/U(1)={\mathbb{C}}P^{1}$.
Proposition 3.1: In the above assumptions, let $\dim_{\mathbb{C}}(W\cap
J(W))=2k$. Then
(i)
If $k=i$, then $\psi$ considered as a function on ${\mathbb{C}}P^{1}$ has
strict extremum at the point corresponding to the complex structure $I$.
(ii)
If $k>i$, then $\psi$ is identically zero.
Proof: Let us fix a quaternionic-hermitian metric in $U$, such that
$\Phi_{I}\wedge\overline{\Phi_{I}}$ is its volume form. Denote by
$\eta_{W}\in\Lambda^{n-i,n-i}(U^{*})$ the form dual to $\xi_{W}$, that is
$\eta_{W}=*(\xi_{W}^{\sharp})$, where $\sharp$ denotes the duality with
respect to the metric and $*$ is the Hodge star operator.
Then we have $\psi(g)=\langle
V_{n+i,n+i}^{I},g(\xi_{W})\rangle=*(V_{n+i,n+i}^{I}\wedge g(\eta_{W}))$ and by
(2.3) we obtain
$V_{n+i,n+i}^{I}\wedge g(\eta_{W})=a_{i}\Omega_{I}^{i}\wedge
R(g(\eta_{W}))\wedge\overline{\Phi_{I}}.$
We can choose an orthonormal basis in $U^{1,0}$ of the form
$\langle e_{1},J\overline{e_{1}},\ldots,e_{n},J\overline{e_{n}}\rangle,$
such that
$W^{1,0}=\langle
e_{1},J\overline{e_{1}},\ldots,e_{k},J\overline{e_{k}},e_{k+1},e_{k+2},\ldots,e_{n+i-k}\rangle,$
If $k>i$ then for any $g\in SU(2)$ we see that $R(g(\eta_{W}))$ has to belong
to the $(2n-2i)$-th exterior power of the subspace in $(U^{*})^{1,0}$ spanned
by
$e_{k+1}^{*},J\overline{e_{k+1}^{*}},\ldots,e_{n}^{*},J\overline{e_{n}^{*}}$.
But this exterior power vanishes, so $R(g(\eta_{W}))=0$, which proves the
second part of the proposition.
If $k=i$ then
$\eta_{W}=J\overline{e_{i+1}^{*}}\wedge\ldots\wedge
J\overline{e_{n}^{*}}\wedge J{e_{i+1}^{*}}\wedge\ldots\wedge J{e_{n}^{*}}$
and
$R(\eta_{W})=e_{i+1}^{*}\wedge J\overline{e_{i+1}^{*}}\wedge\ldots\wedge
e_{n}^{*}\wedge J\overline{e_{n}^{*}}.$
Since $\Omega_{I}=\sum e_{j}^{*}\wedge J\overline{e_{j}^{*}}$, we see that in
this case $\Omega_{I}^{i}\wedge R(\eta_{W})$ does not vanish, that is
$\psi(1)\neq 0$. We claim that the function $\psi$ is non-constant in this
case. Otherwise, the function would be equal to its average over $SU(2)$,
which equals $\langle{\mathrm{A}v}_{SU(2)}V_{n+i,n+i}^{I},\xi_{W}\rangle$. But
in the last expression ${\mathrm{A}v}_{SU(2)}V_{n+i,n+i}^{I}=0$ because
$V_{n+i,n+i}^{I}$ is of the maximal weight and belongs to non-trivial
irreducible representation of $SU(2)$. Therefore, the average of $\psi$ is
zero.
Now, consider the action of $U(1)$ on the two-dimensional sphere by rotations
around the axis that passes through the two points corresponding to the
complex structures $I$ and $-I$. We claim that the function $\psi$ is
invariant under this action. This follows from the definition of $\psi$:
observe that $V_{n+i,n+i}^{I}$ and $\xi_{W}$ are $I$-invariant. Therefore,
$g\mapsto\langle V_{n+i,n+i}^{I},g(\xi_{W})\rangle$ is invariant (as a
function on $SU(2)$) under the adjoint action of the $U(1)$-subgroup
corresponding to $I$.
Since $\psi$ is an analytic non-constant function on the sphere, and it is
invariant under the $U(1)$-action considered above, it must have strict
extremum at $I$. This proves the proposition.
Let $(M,I,J,K)$ be an $SL(n,{\mathbb{H}})$-manifold equipped with an HKT-
metric, and $[Z]\in H_{2n+2i}(M,{\mathbb{Z}})$ an integer homology class.
Consider a function
$\varphi_{Z}:\;{\mathbb{C}}P^{1}{\>\longrightarrow\>}{\mathbb{R}}$ associating
to each $L\in{\mathbb{C}}P^{1}$ a number $\int_{Z}V^{L}_{n+i,n+i}$, where
$V^{L}_{n+i,n+i}\in\Lambda^{n+i,n+i}(M,L)$ is the corresponding calibration
form.
Note that if $L=aI+bJ+cK$ with $a^{2}+b^{2}+c^{2}=1$, then
$\omega_{L}=a\omega_{I}+b\omega_{J}+c\omega_{K}$. Since $V^{L}_{n+i,n+i}$ is
proportional to $\Pi_{+}^{n+i,n+i}(\omega_{L}^{n+i})$ with the coefficient
that does not depend on $L$ (see 2.2), we see that the function $\varphi_{Z}$
is a restriction to $S^{2}$ of a homogeneous polynomial in $\mathbb{R}^{3}$.
Such a function can have only a finite number of strict extrema (this follows
from the fact that real algebraic variety can have only finitely many
connected components, see [Wh]). Let $S\subset{\mathbb{C}}P^{1}$ be the set of
all strict extrema of $\varphi_{Z}$ for all integer homology classes. Since
for each fixed $[Z]\in H_{2n+2i}(M,{\mathbb{Z}})$ the set of strict extrema of
$\varphi_{Z}$ is finite, the set $S$ is countable.
Theorem 3.2: For each $L\in{\mathbb{C}}P^{1}\backslash S$, and any compact
complex subvariety $Z\subset(M,L)$ of complex dimension $n+i$, one has
$\dim_{\mathbb{C}}(TZ\cap J(TZ))>2i$.
Proof: Fix a volume form $dv$ on $Z$ and assume that $\dim_{\mathbb{C}}(TZ\cap
J(TZ))=2i$. Then
$\varphi_{Z}(L_{1})=\int_{Z}\langle V^{L_{1}}_{n+i,n+i},\xi_{TZ}\rangle dv.$
Fix an arbitrary smooth point $x\in Z$. Note that for any $g\in SU(2)$ we have
$\langle V^{L}_{n+i,n+i},g(\xi_{T_{x}Z})\rangle=\langle
V^{{\mathrm{A}d}_{g}L}_{n+i,n+i},\xi_{T_{x}Z}\rangle$. Thus, 3 implies that
the function $L_{1}\mapsto\langle V^{L_{1}}_{n+i,n+i},\xi_{T_{x}Z}\rangle$ has
strict extremum at $L_{1}=L$. Thus, $\varphi_{Z}$ also has strict extremum at
this point, which contradicts our assumption that
$L\in{\mathbb{C}}P^{1}\backslash S$.
Proof of 1.2: Let $L\in{\mathbb{C}}P^{1}\backslash S$ and $Z$ be a divisor in
$(M,L)$, that is a compact $L$-complex subvariety of complex dimension $2n-1$.
Then 3 implies that $\dim_{\mathbb{C}}(TZ\cap J(TZ))>2n-2$. Since $TZ\cap
J(TZ)$ is ${\mathbb{H}}$-invariant, the last inequality would imply that the
dimension equals $2n$, which is impossible. So there exist no divisors in
$(M,L)$.
Analogously, if $\dim_{\mathbb{C}}Z=2n-2$, then we have
$\dim_{\mathbb{C}}(TZ\cap J(TZ))>2n-4$. This implies that
$\dim_{\mathbb{C}}(TZ\cap J(TZ))=2n-2$, that is, $TZ$ is
${\mathbb{H}}$-invariant and $Z$ trianalytic. This completes the proof of the
theorem.
Remark 3.3: We should note that the existence of an HKT-metric was essential
for the proof of the main theorem. It still remains unclear if this condition
could be removed.
On the other hand, the condition that the holonomy of the Obata connection is
contained in $SL(n,{\mathbb{H}})$ is known to be necessary. There exist
examples of HKT-manifolds with odd-dimensional complex subvarieties for each
induced complex structure. An HKT-structure on compact Lie groups due to D.
Joyce ([J]) gives such an example: it is well-known (see e.g. [V7]) that these
manifolds admit a toric fibration over a rational base, hence they always
contain divisors.
Remark 3.4: Let $T$ be a compact hyperkähler torus, and $L$ a generic induced
complex structure. Then all complex subvarieties of $T$ are again tori ([KV]).
We conjecture that something similar would happen for nilmanifolds, and for
flat hypercomplex manifolds.
Question 3.5: Let $M$ be a compact $SL(n,{\mathbb{H}})$-manifold with flat
Obata connection, and $L$ a generic induced complex structure. Is it true that
all the complex subvarieties of $(M,L)$ are also flat?
Question 3.6: Let $M$ be a hypercomplex nilmanifold (1.1), and $L$ a generic
induced complex structure. Is it true that all complex subvarieties of $(M,L)$
are also nilmanifolds?
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* [V8] M. Verbitsky, Hypercomplex manifolds with trivial canonical bundle and their holonomy, arXiv:math/0406537, “Moscow Seminar on Mathematical Physics, II”, American Mathematical Society Translations, 2, 221 (2007).
* [Wa] Wang, Hsien-Chung Closed manifolds with homogeneous complex structure, Amer. J. Math. 76, (1954), 1-32.
* [Wh] Whitney, H., Elementary structure of real algebraic varieties, Ann. Math., 66 (1957), 545–556.
Andrey Soldatenkov
Laboratory of Algebraic Geometry,
National Research University Higher School of Economics,
7 Vavilova Str., Moscow, Russia, 117312
Misha Verbitsky
Laboratory of Algebraic Geometry,
National Research University Higher School of Economics,
7 Vavilova Str., Moscow, Russia, 117312
verbit@maths.gla.ac.uk, verbit@mccme.ru
|
arxiv-papers
| 2012-02-01T17:12:22 |
2024-09-04T02:49:26.931669
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andrey Soldatenkov, Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1202.0222"
}
|
1202.0295
|
# Quantitative nanostructure characterization using atomic pair distribution
functions obtained from laboratory electron microscopes
Milinda Abeykoon Condensed Matter Physics and Materials Science Department,
Brookhaven National Laboratory Christos D. Malliakas Department of
Chemistry, Northwestern University Pavol Juhás Department of Applied Physics
and Applied Mathematics, Columbia University Emil S. Božin Condensed Matter
Physics and Materials Science Department, Brookhaven National Laboratory
Mercouri G. Kanatzidis Department of Chemistry, Northwestern University
Chemistry Department, Argonne National Laboratory Simon J. L. Billinge
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory Department of Applied Physics and Applied Mathematics, Columbia
University
###### Abstract
Quantitatively reliable atomic pair distribution functions (PDFs) have been
obtained from nanomaterials in a straightforward way from a standard
laboratory transmission electron microscope (TEM). The approach looks very
promising for making electron-derived PDFs (ePDFs) a routine step in the
characterization of nanomaterials because of the ubiquity of such TEMs in
chemistry and materials laboratories. No special attachments such as energy
filters were required on the microscope. The methodology for obtaining the
ePDFs is described as well as some opportunities and limitations of the
method.
## I Introduction
One of the great challenges of nanoscience is to obtain the quantitative
structures of nanoparticles billi;s07 ; jadzi;s07 . The atomic pair
distribution function (PDF) method has recently emerged as a powerful tool for
doing this egami;b;utbp03 ; juhas;n06 ; billi;jssc08 ; petko;cm06 ;
page;chpl04 ; kodam;aca06 ; neder;pssc07 ; masad;prb07 ; petko;jpcc07 , but
obtaining the required high quality diffraction data to high momentum transfer
with good statistics generally requires synchrotron x-ray or spallation
neutron data from a national user facility. Here we show that data of
sufficient quality for quantitative analysis of nanoparticle structure using
the PDF can be obtained from transmission electron microscopes (TEM) available
at many research institutions. Quantitative structural models were applied to
PDFs of several nanoparticle systems showing that electron PDFs can be modeled
with the powerful emerging modeling tools for studying PDFs in general
farro;jpcm07 ; neder;b;dsadss08 ; tucke;jpcm07 ; cerve;jac10 . This approach
complements medium and high resolution imaging methods for studying
nanoparticles in the TEM. The ease of data collection and ubiquity of TEMs
will make this an important tool in the characterization of nanostructured
materials.
A challenge when using electrons as a probe is that they scatter strongly
cowle;mic04 ; cowle;b;edt92 and not according to the weak scattering
kinematical scattering equations on which the PDF analysis is based debye;ap15
; warre;b;xd90 . This would appear to rule out electrons as a source of
diffraction data for PDFs except in the cases of very dilute, such as gas-
phase schoo;nl05 , samples. However, kinematical, or nearly kinematical,
scattering is obtained from electrons when sample volumes are sufficiently
small that multiple scattering events are not of high probability before the
electrons exit the sample (typically a few nm of thickness), or when the
scattering from the samples is highly incoherent, for example the scattering
from amorphous materials and away from zone axes in a crystal
weiri;b;ecnafsdonm06 . In these latter cases there is still significant
multiple scattering, but it is sufficiently incoherent that it can be treated
as a background and subtracted and the resulting coherent signal can be
treated kinematically. This has been discussed in detail in a number of
publications cocka;armr07 ; ansti;u88 ; ankel;zna05 ; karle;pnas77 . This is
used in the rapidly growing field of electron crystallography
weiri;b;ecnafsdonm06 , and has been demonstrated in previous work of electron
diffraction (ED) from glasses and amorphous materials moss;prl69 ; cocka;aca88
; cocka;armr07 ; hirot;mcp03 ; noren;jac99 , though little has been done in
the way of quantitative modeling in those studies. In these respects, the
study of small nanoparticles is particularly favorable. The samples are
inherently thin, limited to the diameter of the nanoparticles when they are
dispersed as a sub-mono-layer on a holey carbon support, and the structure is
typically less coherent than from crystals because of the finite size effects
that significantly broaden Bragg peaks and the often lower symmetries of
nanoparticle structures due to surface and bulk relaxations. In fact
fortuitously, the scattering is most kinematical precisely for the small
nanoparticles ($<10$ nm) that are most beneficially studied using PDF methods
egami;b;utbp03 ; juhas;n06 ; billi;jssc08 ; petko;cm06 ; page;chpl04 ;
kodam;aca06 ; neder;pssc07 ; masad;prb07 ; petko;jpcc07 .
Here we show how to obtain PDFs from a normal transmission electron microscope
(TEM) found in many research labs. We find that the resulting electron PDFs
(ePDFs) can be modeled to extract quantitative structural information about
the local structure using PDF refinement programs such as PDFgui farro;jpcm07
. This opens the door to broader application of PDF methods for nanostructure
characterization since TEM is already a routine part of the nanoparticle
characterization process wang;jpcb00 ; won;jap06 . With this development, as
well as obtaining low and high resolution TEM _images_ of nanoparticles,
quantitative structural information, similar to that normally obtained from a
Rietveld refinement rietv;jac69 ; young;b;trm93 in bulk materials, is also
available from nanoparticles with little additional effort. This approach also
complements high resolution TEM by getting an average signal from a large
number of nanoparticles rather than giving information from a small part of
the sample that may not be representative. The fact that the real-space images
and the diffraction data suitable for structural analysis can be obtained at
the same time and from the same region of material is also a large advantage,
resulting in more complete information for the characterization of the sample.
In some cases, the small quantity of material required for ePDF, compared to
x-ray and neutron PDF measurements (xPDFs and nPDFs, respectively), may also
be a major advantage, as well as the ability to study thin films.
### Theoretical Background
The Fourier transform of X-ray or neutron powder diffraction data yields the
PDF, $G(r)$ according to farro;aca09
$\displaystyle G(r)=(2/\pi)\int_{Q_{\min}}^{Q_{\max}}Q[S(Q)-1]\sin(Qr)dQ,$ (1)
where the structure function, $S(Q)$, is the properly normalized powder
diffraction intensity and $Q$, for elastic scattering, is the magnitude of the
scattering vector, $Q=4\pi\sin(\theta)/\lambda$.warre;b;xd90 ; egami;b;utbp03
The PDF is also related to the atomic structure through
$\displaystyle G(r)=\frac{1}{Nr}\sum_{ij}\frac{f_{i}(0)f_{j}(0)}{\langle
f(0)\rangle^{2}}\delta(r-r_{ij})-4\pi r\rho_{o},$ (2)
where the sum goes over all pairs of atoms _i_ and _j_ separated by $r_{ij}$
in the model. The form factor of atom _i_ is $f_{i}(Q)$ and $\langle
f(Q)\rangle$ is the average over all atoms in the sample. In equation 2, the
scattering factors are evaluated at $Q=0$, which in the case of x-rays is the
atomic number of the atom. The double sums are taken over all atoms in the
sample. For a multicomponent system, $S(Q)$ can be written in terms of the
concentrations, $c_{i}$, of the atomswarre;b;xd90 ; egami;b;utbp03
$S(Q)=1+\frac{I(Q)-\sum c_{i}|f_{i}(Q)|^{2}}{\big{|}\sum
c_{i}f_{i}(Q)\big{|}^{2}}.$ (3)
In the case of electrons as a probe, the equations are the same, providing the
scattering can be treated kinematically cowle;b;edt92 ; however, the form-
factor must be that appropriate for electrons, $f_{e}(Q)$, which is the
Fourier transform of the electronic potential distribution of an atom. Note
that in the electron diffraction literature, it is common to use
$s=2\sin(\theta)/\lambda=Q/2\pi$ instead of $Q$ for the independent variable
in the scattering. The electron form factor, $f_{e}(Q)$, is different to, but
closely related to, the x-ray form factor of the same atom, $f_{x}(Q)$, which
is the Fourier transform of the electron density. A useful relationship
between $f_{e}(Q)$ and $f_{x}(Q)$ is cowle;b;edt92
$f_{e}(Q)=\frac{m_{e}e^{2}}{2\hbar^{2}}\left(\frac{Z-f_{x}(Q)}{Q^{2}}\right),$
(4)
where $m_{e}$ and $e$ are the mass and charge of the electron, respectively,
$\hbar$ is Plank’s constant, and $Z$ the atomic number. This equation does not
give a definite value for $f_{e}(Q)$ at $Q=0$, but $f_{e}$(0) can be
calculated by extrapolation or by using
$f_{e}(0)=4\pi^{2}\frac{me^{2}}{3\hbar^{2}}(Z\langle r_{e}^{2}\rangle),$ (5)
where $\langle r_{e}^{2}\rangle$ is the mean square radius of the electronic
shell of the atom.cowle;b;edt92 Figure 1 shows a comparison between x-ray and
electron form factors, $f_{x}$(Q) and $f_{e}$(Q) of Au.
In the case of single crystal ED, a rule of thumb is that when the crystal
thickness is greater than $\sim 300$-400 Å, data reduction must be done based
on the dynamical diffraction theory which assumes the presence of coherent
multiple scattering components of electrons cowle;b;edt92 . Depending on the
energy of the electrons, this thickness limit may even fall below these
numbers in the presence of heavy elements cowle;b;edt92 , and in the case of
electron powder diffraction, the average thickness of crystallites in the
specimen should also be less than a few hundred Ångströms to avoid dynamical
scattering effects. cowle;b;dp95 Coherent multiple scattering changes the
relative intensities of Bragg peaks from the kinematical structure factor
values, redistributes intensity to the weaker peaks at higher values of $Q$
cocka;aca88 and can allow symmetry disallowed peaks to appear in the pattern.
If $\Gamma$ is the elastic mean free path of the electron, it has been shown
that a PDF determined from a polycrystalline Pt sample does not affect the
positions of the PDF peaks for D/$\Gamma\leq 5$, where D is the particle size,
but it does affect the determination of coordination numbers ansti;u88 . Here
we show that model fits may be good, even in the presence of significant
multiple scattering, while refined thermal factors are underestimated, though
it is desirable to optimize experimental conditions such as to minimize
multiple scattering.
Incoherent multiple scattering can be observed in ED patterns in the form of
increased background cowle;b;edt92 which does not affect the relative
intensities of the Bragg peaks. This is why, in the case of a less coherent
structure, dynamical scattering effects are less important.
In this study, all the specimens used were nanosized samples: thin films,
discrete nanoparticles, or agglomerates of nanoparticles. In this case, the
hope was that multiple scattering would not introduce undue aberrations into
the kinematical diffraction pattern and a reliable PDF will result. We found
this to be largely true with an exception we discuss below.
## II Experimental
Nanocrystalline thin film, or dispersed nanoparticulate samples, were
distributed on a holey carbon grid and ED data taken with a short camera
length, to give the widest $Q$-range, and a relatively large beam-size (2-5
$\mu$m diameter) on the sample, to obtain the best possible powder average. To
improve the powder average different regions of the sample were illuminated by
translating the sample under the beam. In other respects, the TEM was used in
a standard configuration using a CCD detector and no energy filtering and
operated at 200 keV (wavelength, $\lambda=0.025079$ Å).
All selected area ED experiments were carried out at room temperature on a
Hitachi H8100 200 KeV transmission electron microscope equipped with a Gatan
Orius SC600 CCD Camera (24mm x 24 mm active area). Typical exposure time per
frame was around 0.3s. Formvar coated 300 mesh copper grids (Electron
Microscopy Sciences) stabilized with an evaporated carbon film were used to
support the metallic films and nanoparticles. Deposition of gold on the carbon
coated side of the TEM grid was performed with a Denton Vacuum DeskIII
sputterer and gave a uniform film. The thickness of the film was measured in
real time during the sputtering process with the aid of a thickness monitor
(Maxtek, Inc TM-350). Part of the grid was masked during the deposition and
this masked area was used to extract the diffraction intensity of the support.
No differences were found in the diffraction intensity data of the background
(carbon and polymer films) between different grids. Deposition on the side of
the grid that was coated with the polymer gave Au nanoparticles with a wide
range of sizes up to $\sim 100$ nm. NaCl nanoparticles were deposited on a TEM
grid by a radio-frequency thermal evaporation method.
For comparison, x-ray measurements on Au nanoparticles were carried out in the
rapid acquisition mode (RAPDF) chupa;jac03 using a Perkin Elmer amorphous
silicon 2D detector at X7B beamline of National Synchrotron Light Source
(NSLS) at Brookhaven National Laboratory (BNL). Nanoparticles in ethanol
solution were loaded in a 1 mm diameter kapton tube sealed at both ends, and
mounted perpendicular to the x-ray beam. The data were collected at room
temperature using the x-ray energy of $\sim$38 keV ($\lambda=0.3196$ Å). The
data were collected in a multiple 4 s exposures for a total collection time of
5 min.
To calibrate the conversion from detector coordinates to scattering angle, it
is necessary to measure the ED pattern from a standard of known lattice
parameters. The software for reducing the data to 1D, Fit2D hamme;hpr96 uses
this to optimize the effective sample-detector distance, find the center of
the Scherrer rings on the detector, and correct for aberrations such as any
deviation from orthogonality of the detector and the scattered beam. Typical
standards used by the program are Al2O3, CeO2, LaB6, NaCl and Si. However, for
the ED experiment it is necessary to have a nano-sample standard to obtain a
good powder average. For this gold nanoparticles of diameter $\sim$100 nm were
used and a literature value of 4.0782 Å for the lattice parameter. The
effective sample-detector distance depends on the settings of the magnetic
lenses used in the microscope. We assumed that the energy of the electrons,
200 keV, is well known (resulting in $\lambda=0.025079$ Å), though for the
most accurate results the electron wavelength should be calibrated using
standard methods.Once these calibration quantities are known, they are fixed
and the same values are used to convert the sample data. From this perspective
it is essential that the sample is measured under identical conditions as the
standard, including camera length and focus. We found that even scanning
around a sample to find a different viewing area resulted in a small variation
in the position on the detector of the center of the resulting diffraction
pattern. It was thus necessary to run a separate calibration run on each
diffraction pattern to determine the center of the rings, while keeping the
camera-length from the Au calibration.
## III Data Analysis
The 2D ED images were read and integrated into 1D powder diffraction patterns,
after masking the missing beam stop region. The data have to be further
processed to obtain the PDF. Corrections were applied to the raw data to
account for experimental effects and properly normalized and divided by
($\langle f_{e}(Q)\rangle^{2}$) egami;b;utbp03 , resulting in the total
scattering structure function, $S(Q)$. The kernel of the Fourier transform is
the reduced structure function, $F(Q)=Q[S(Q)-1]$. We used a home-written
program, PDFgetE, to carry out these steps. The PDF is then straightforwardly
obtained as the Fourier transform of $F(Q)$ according to Eq. 1, which is also
carried out in PDFgetE. Once the PDFs are obtained, they can be modeled using
existing PDF modeling programs. Here we used PDFgui farro;jpcm07 .
## IV Results
A low resolution TEM image of the 2.7 nm thick Au film is shown in Fig. 2(a).
The film is uniform and featureless in the image, but a region at the edge of
the film was selected for imaging so that the edge of the film gives a visual
cue to its presence. An ED pattern from a position away from the edge of the
film is shown in Fig. 2(b). We can see a series of concentric circles due to
the Scherrer powder diffraction rings in transmission geometry. The resulting
1D ED pattern, obtained by integrating around the rings in the 2D pattern is
shown in Fig. 2(c). Broad diffuse features are observed consistent with the
nanocrystallinity of the sample. Weak features are clearly evident up to
$Q=12$ Å-1 (Fig. 2(c) inset), but less apparent beyond that point.
The $F(Q)$ from the same data after correction is shown in Fig. 3(a) and the
resulting ePDFs in Fig. 3(c), with the calculated PDF from a model of the gold
fcc structure plotted on top in red. For comparison, in Fig. 3(b) and (d) we
show the x-ray derived $F(Q)$ and xPDF, respectively. Unfortunately this is
not a direct comparison between identical samples. We were not able to collect
x-ray data from the same film as the ePDF as it was too thin to get a
sufficient signal in the x-ray measurement.
The structure functions of the electron and x-ray data (Fig. 3(a) and (b),
respectively) are clearly highly similar. Features in the $eF(Q)$ are broader
than the x-ray case but the features are all recognizable and have the correct
relative intensities. Likewise, the e- and xPDFs (Fig. 3(c) and (d),
respectively) are highly similar, with the features in the ePDF of the
nanoscale film being broader. The quality of the fits is comparable for both
the ePDF and xPDF curves, with the ePDF giving a slightly lower (better)
agreement factor. The refined parameters are presented in Table 1. The breadth
of the ePDF peaks are accommodated in the model by giving gold very large
atomic displacement parameters (ADPs), twice as large as those in the x-ray
measured gold nanoparticles that are already large. This indicates the
presence of significant atomic scale disorder in the film and is not coming
from the ePDF measurement itself. This is discussed in greater detail below.
These results clearly demonstrate that quantitatively reliable ePDFs can be
obtained from nanocrystalline materials in a standard laboratory TEM. The
counting statistics from the electron data compare favorably to those from the
x-ray measurements (Fig. 3(a) and (b)), despite the much shorter measurement
time, suggesting that ePDF determination could become a useful general
characterization tool during nanoparticle synthesis. Two effects are clearly
evident in the $Q$-space data: low $Q$-space resolution and the rapid
diminishing of the amplitude of scattered features with increased $Q$. The
latter is likely to reflect real differences in the samples, with the range of
structural coherence being lower in the gold film than the gold nanoparticles
used in the x-ray experiment. The lower $Q$-space resolution could be either a
sample or a measurement effect but this cannot be disentangled without having
a well characterized, kinematically scattering, nanoparticle standard for ED,
which doesn’t currently exist. The sputtered gold film has an fcc gold
structure, like the bulk, but with significantly more disorder and a nanometer
range for the structural coherence.
The ED data were taken with a standard CCD camera and no filtering of
inelastically scattered electrons. This is the most straightforward protocol
for data collection as it is the standard setup in most laboratory TEMs. It is
expected to result in lower quality PDFs than those measured with energy
filtered electrons because of the higher backgrounds due to inelastically
scattered electrons cocka;aca88 . ED data collected with an image plate
detector are also expected to be higher quality due to the low intrinsic
detector noise and better dynamic range of that detector technology. Thus, the
resulting PDF shown in Fig. 3(c) represents the baseline of what is possible
without specialized instrumentation. The resulting $F(Q)$ shows excellent
signal to noise up to the maximum accessible $Q$-range of 17 Å-1, as evident
in Fig. 3(a).
To explore the size limits for Au NPs to scatter kinematically, we collected
data from larger, 100 nm Au nanoparticles, and the results are also given in
Table 1 and Figures 4 and 5. Comparing the integrated 1D diffraction patterns
of the large NPs and the thin Au film, Figs. 4(c) and 2(c), respectively, we
see similar features, but in the case of the NPs, the amplitudes of the
scattered intensities extend to much higher $Q$ values, as if there is a much
smaller Debye-Waller factor for the data. This can be clearly observed by
comparing the e$F(Q)$ of the large nanoparticles in Fig. 5(a) with those from
x-ray diffraction data, x$F(Q)$ in Fig. 5(b). The enhancement in the high-$Q$
features is large and is almost certainly due to significant coherent multiple
scattering in this sample. The resulting ePDF from the NPs has peaks that are
correspondingly sharp compared to the thin film gold and the xPDFs of gold
NPs. Regardless of the presence of significant multiple scattering, a model
was refined against the ePDF of the 100 nm Au nanoparticles to see the extent
that the refined structural model parameters are affected. The structure
refinement gave fits that were slightly worse but comparable in quality to the
xPDF fits (see Table 1), $R_{w}=0.24$. The refined values were similar also,
except for much smaller atomic displacement parameters (ADPs), due to the
artificially sharpened PDF peaks. It is somewhat remarkable that, in this
case, the dynamical scattering produces features in the $F(Q)$ with
approximately the correct relative amplitude, but extending to much
higher-$Q$. Not only are the PDF peaks in the right position ansti;u88 , but
have the right relative amplitudes. Gold may be a special case because the
structure factors are all either ones or zero’s.
This clearly shows that for a strong scatterer such as Au, 100 nm
nanoparticles already give significant dynamical effects. The resulting PDFs
give useful semi-quantitative and qualitative information but the refined
thermal parameters are not reliable. Indeed, the effect of the multiple
scattering to increase the real-space resolution by boosting the intensities
of the high-$Q$ peaks makes the PDF peaks sharper with the result that bond-
lengths can be extracted with greater precision from the ePDF data in this
case. When accurate PDF peak positions rather than quantitative peak
intensities are desired, this could be a significant advantage of the ePDF
method, for example, when looking for small peak splittings, or resolving peak
overlaps, to aid in structure solution.
A less trivial structure factor is obtained from binary compounds, such as the
NaCl studied here. The TEM image of the sample in Fig. 6(a) shows that it
consists of nanoscale crystallites, some of which have a cubic habit and
others that have no particular morphology. The corresponding ED pattern in
Fig. 6(b) shows clear and fairly uniform rings, with some spottiness from an
imperfect powder average. Fig. 6(c) shows the integrated ED pattern. The
$F(Q)$ and the resulting ePDF obtained from this data set is shown in Figs.
7(a) and (c), respectively. For comparison, an x$F(Q)$ and an xPDF obtained
from a bulk crystalline NaCl sample is also shown in Fig. 7(b) and (d).
The rock-salt structure model fits to the PDFs are shown in Fig. 7(c) and (d)
and the results are presented in Table 2. The e- and xPDFs are qualitatively
highly similar, with all features in the xPDF easily recognizable in the ePDF.
Notably, the relative intensities of adjacent peaks are similar between the e-
and xPDFs. Peaks in the ePDF die out in amplitude with increasing $r$ more
quickly, due to the broader features in the ED pattern. The overall quality of
the fit to the ePDF is worse than the xPDF of bulk NaCl. Refined lattice
constants agree well within the experimental uncertainty. The ePDF refined
thermal parameters are much smaller than those obtained from the x-ray data.
This is unlikely to be a real effect as both the x-ray and electron data were
measured at room temperature, and it is rather implausible that the
nanoparticulate samples have _less_ static structural disorder than bulk NaCl.
We therefore assume that this is the effect of multiple scattering in the
data, similar to that observed for large Au NPs. Clearly, ADPs refined from
ePDFs present a lower bound on actual sample ADPs. They are accurate in the
case where multiple scattering is negligible, but underestimate the thermal
motions and static disorder in the presence of multiple scattering.
## V Discussion and Conclusions
The Au and NaCl examples establish that quantitatively, or semi-
quantitatively, reliable PDFs can be obtained from nanomaterials using
electron diffraction data obtained on a standard laboratory TEM, without the
use of filtering. Because of the ease and speed of collecting such data and
the ubiquity of such instruments in chemistry and materials laboratories, if
the barriers to data processing could be overcome making the whole process
straightforward, this could become a broadly applicable standard and useful
characterization method for nanoparticles and thin films.
This work also explores the experimental parameters for obtaining good data
for reliable ePDFs from nanomaterials. Principally, samples should be thin
enough or, for the case of nanoparticles, have a sufficiently small diameter.
What this diameter is depends on the average atomic number of the sample. For
Au, 100 nm diameter NPs gave significant coherent multiple scattering, 2.7 nm
thick films did not. For all materials we expect that 10 nm and smaller
particles will scatter kinematically; and these are precisely in the size-
range of nanoparticles that benefit the most from a PDF analysis masad;prb07 .
Obtaining a good powder average is also a very important part of powder
diffraction regardless of the probing technique, XRD, ND or ED. This can be
easily achieved by using a large sample volume in ND and spinning the sample
in XRD. However, in ED, both of these methods become difficult due to the
limitations of the configuration and careful sample preparation in this regard
is very helpful. Again, for the particular application in nanoparticle
structure characterization, the small size of the particles means that better
powder averages can be obtained even from small sample volumes. However, the
quality of the powder average should be checked by visual inspection of the ED
images from the CCD, which is readily done as is evident in the figures in
this paper. The powder average can be improved by increasing the beam-spot
size on the sample and also by taking multiple images from different regions
of the sample and averaging them. The maximum Qmax attainable is determined by
the operational energy, camera length, dimensions of the detector and the
diameter of the microscope, but in general should be maximized. Our electron
microscope configuration equipped with a CCD camera limited Qmax to
$\sim(17-18)$ Å. The advantage of using a higher Qmax is the better real space
resolution that results in the ePDF. However, standard microscope
configurations naturally give sufficiently high $Q_{\max}$ values for most
applications. Thus there seems to be no impediment to the use of ED from
standard laboratory electron microscopes for quantitative nanoparticle
structural characterization using the PDF.
## VI Acknowledgements
We would like to acknowledge helpful discussions with Michael Thorpe. We also
thank Jon Hanson for allowing access to the X7B beamline at NSLS, which is
supported by DOE-BES under contract No DE-AC02-98CH10886. Work in the Billinge
group was supported by DOE-BES through account DE-AC02-98CH10886. Work in the
Kanatzidis group was supported by NSF through grant DMR-1104965.
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Figure 1: A comparison between normalized (to f(0)) x-ray and electron form
factors, $f_{x}$(Q) and $f_{e}$(Q) of Au. Figure 2: (a) A TEM image of the 2.7
nm thick Au film used for ED. (b) A false-color 2D ED pattern collected on
this sample using 200 keV electrons. Lighter colors indicate higher intensity.
The black bar across the middle of the image is the shadow of the beam-stop.
(c) 1D Au electron powder diffraction pattern obtained by integrating around
the rings in 2 (b). The inset shows the high $Q$ region of the ED pattern on
an expanded $y$-scale. The dotted lines are guides to the eye. Figure 3: (a)
Reduced structure function, $F(Q)$, of Au obtained from the integrated ED
pattern in Fig. 2(c), (b) An $F(Q)$ of Au nanoparticles calculated from an XRD
pattern collected at X7B at the NSLS. (c) Au bulk structure model fit to the
resulting ePDF from 3(a). (d) Au bulk structure model fit to the resulting
xPDF from 3(b). Observed and calculated PDFs are presented with blue circles
and a solid red line respectively. The difference between observed and
calculated is offset below (green solid lines). In both cases used Qmax=15.25
Å.
Figure 4: (a) A TEM image of $\sim$ 100 nm Au nanoparticles used for ED. Black
dots are the nanoparticles on the grid and the large white areas are the holes
in the grid. (b) A background subtracted ED image, collected from the same
region of the sample using 200 keV electrons. (c) The 1D ED pattern obtained
by integrating around the rings in 4(b). The inset shows a magnified region of
the integrated ED pattern as indicated by the dotted lines. This ED pattern
clearly suffers from multiple scattering due to the thickness of the sample.
.
Figure 5: (a) The reduced structure function, $F(Q)$, of Au calculated from
the integrated ED pattern in Fig. 4(c), (b) An F(Q) generated from an
integrated Au NP XRD pattern from a 2D data set collected at X7B at the NSLS.
(c) Au bulk structure model fit to the resulting ePDF from 5 (a). (d) Au bulk
structure model fit to the resulting xPDF from 5 (b). Observed and calculated
PDFs are presented with blue circles and a solid red lines respectively. The
difference between observed and calculated is offset below (green solid
lines). Figure 6: (a) A TEM image of the NaCl film used for ED. (b) A false-
color 2D ED image collected on this sample using 200 keV electrons. Lighter
colors indicate higher intensity. The black bar across the middle of the image
is the shadow of the beam-stop. (c) 1D ED pattern obtained by integrating
around the rings in 6 (b). The inset shows the high $Q$ region of the ED
pattern on an expanded $y$-scale. The dotted lines are guides to the eye.
Figure 7: (a) Reduced structure function, F(Q), of NaCl obtained from the
integrated ED pattern in Fig. 6(c). (b) An F(Q) of NaCl calculated from an
x-ray data set. (c) NaCl bulk structure model fit to the resulting ePDF from 7
(a). (d) NaCl bulk structure model fit to the resulting xPDF from 7 (b).
Observed and calculated PDFs are presented with blue circles and solid red
lines respectively. The difference between observed and calculated is offset
below (green solid lines). In both cases used Qmax=13.6 Å.
Table 1: Refined parameters for 2.7 nm thick nanoparticulate Au film, $\sim
100$ nm diameter nanoparticles (NP) from ePDFs and from a gold nanoparticle
sample from xPDFs. The structure model is the fcc bulk gold structure, space-
group Fm-3m. It was not possible to measure the nanoparticle size from the
ePDFs as we were not able to calibrate the intrinsic $Q$-space resolution of
the ED measurement allowing us to separate the instrumental resolution and
particle size effects in the ePDFs.
| ePDF (film) | ePDF (NP) | xPDF
---|---|---|---
Qmax (Å-1) | 15.25 | 15.25 | 15.25
Fit range (Å) | 1-20 | 1-20 | 1-20
Cell parameter (Å) | 4.075(3) | 4.076(2) | 4.058(1)
Uiso (Å2) | 0.033(4) | 0.006 (3) | 0.014(1)
Diameter (Å) | $\sim$27∗ | $\sim$1000∗∗ | 24.51(9)
Q-damp (Å-1) | 0.095(5) | 0.095(5) | 0.047(2)
Rw (%) | 17 | 24 | 20
∗film thickness measured during deposition ∗∗NP diameter estimated directly
from the TEM image
Table 2: Refined parameters for nanoparticulate NaCl from ePDFs and for a bulk powder of NaCl from the xPDF. The structure model is the fcc rock-salt structure, space-group Fm-3m. It was not possible to measure the nanoparticle size from the ePDFs as we were not able to calibrate the intrinsic $Q$-space resolution of the ED measurement allowing us to separate the instrumental resolution and particle size effects in the ePDFs. | ePDF | xPDF
---|---|---
Qmax (Å-1) | 13.6 | 13.6
Fit range (Å) | (0.2-30) | (0.2-30)
Cell parameter (Å) | 5.62(2) | 5.63(1)
$U_{iso}$ \- Na (Å2) | 0.007(5) | 0.027(1)
$U_{iso}$ \- Cl (Å2) | 0.004(4) | 0.016(1)
Q-damp (Å-1) | 0.095(5) | 0.06(1)
Rw % | 33 | 6
|
arxiv-papers
| 2012-02-01T21:16:21 |
2024-09-04T02:49:26.940582
|
{
"license": "Public Domain",
"authors": "Milinda Abeykoon, Christos D. Malliakas, Pavol Juhas, Emil S. Bozin,\n Mercouri G. Kanatzidis, Simon J. L. Billinge",
"submitter": "A. M. Milinda Abeykoon Ph.D.",
"url": "https://arxiv.org/abs/1202.0295"
}
|
1202.0350
|
# Chaotic mixing and fractals in a geophysical jet current
M.V. Budyansky, S.V. Prants Laboratory of Nonlinear Dynamical Systems,
V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of
Sciences, 690041 Vladivostok, Russia
###### Abstract
We model Lagrangian lateral mixing and transport of passive scalars in
meandering oceanic jet currents by two-dimensional advection equations with a
kinematic stream function with a time-dependent amplitude of a meander
imposed. The advection in such a model is known to be chaotic in a wide range
of the meander’s characteristics. We study chaotic transport in a stochastic
layer and show that it is anomalous. The geometry of mixing is examined and
shown to be fractal-like. The scattering characteristics (trapping time of
advected particles and the number of their rotations around elliptical points)
are found to have a hierarchical fractal structure as functions of initial
particle’s positions. A correspondence between the evolution of material lines
in the flow and elements of the fractal is established.
###### keywords:
Chaotic advection, meandering jet, fractals
###### PACS:
47.52.+j, 47.53.+n , 92.10.Ty
## 1 Introduction
Major western boundary currents in the ocean are meandering jets separating
water masses with different physical and biogeochemical characteristics. The
prominent examples are the Gulf Stream in the Atlantic Ocean and the Kuroshio
in the Pacific Ocean. These and similar "heat engines" define the climate in
large regions of the planet. Similar jets in the stratosphere play important
role in transport and distribution of chemical substances. From the
hydrodynamic point of view, they may be considered as jet flows with running
waves of different wave lengths and phase velocities imposed. The simplest
kinematic model of such a flow is a two-dimensional jet of an ideal fluid with
a given velocity profile that is perturbed by an amplitude-modulated wave
traveling from the west to the east. The problem of transport and mixing of
passive scalars in meandering jets has been considered by many authors in the
context of atmospheric and oceanic physics [1, 2, 3, 4, 5, 6, 7, 8].
The typical phase portrait (Fig. 1) consists of two chains of circulations
with a zigzag-like jet between them and resembles the phase portrait of a
particle in the field of two running waves. In the frame moving with the
velocity of one of the waves, the problem is topologically equivalent to the
motion of a periodically perturbed nonlinear physical pendulum that is known
to demonstrate chaotic oscillations [9, 10]. Different aspects of chaotic
mixing of passive particles in meandering jets in the atmosphere and the ocean
have been studied in the papers [1, 2, 3, 4, 5, 6, 7, 8].
In the paper we focus on topological and statistical aspects of the chaotic
transport and mixing in a specific kinematic model of an eastward meandering
jet which has been introduced in Refs. [2, 3] some years ago. We are motivated
by the desire to get a more deep insight into the evolution of material lines
in the flow and to establish a connection between dynamical, topological and
statistical characteristics of the flow. The equations of motion of passive
particles advected by a planar incompressible flow is known to have a
Hamiltonian form
$\displaystyle\dot{x}$ $\displaystyle=u(x,y,\,t)=-\frac{\partial\Psi}{\partial
y},$ (1) $\displaystyle\dot{y}$
$\displaystyle=v(x,y,\,t)=\frac{\partial\Psi}{\partial x},$
with the streamfunction $\Psi$ playing the role of a Hamiltonian and the
coordinates $(x,y)$ of a particle being canonically conjugated variables.
Thus, nonstationary two-dynamical advection is equivalent to a Hamiltonian
system with one and half degrees of freedom whose phase space coincides with
its configuration space. This property is very useful in visualizing geometric
invariant sets in real and numerical experiments.
## 2 Model flow
To be specific we consider a two-dimensional Bickley jet with the velocity
profile $u_{0}\operatorname{sech}^{2}y$ whose argument is modulated by a zonal
running wave [3]. The streamfunction in the fixed frame reference is the
following:
$\psi^{\prime}(x^{\prime},y^{\prime},\tau)=-\psi_{0}\tanh{\left(\frac{y^{\prime}-a\cos{k(x^{\prime}-c\tau)}}{\lambda\sqrt{1+k^{2}a^{2}\sin^{2}{k(x^{\prime}-c\tau)}}}\right)},$
(2)
where $a$, $k$ and $c$ are amplitude, wavenumber and phase velocity
respectively, $\lambda$ is a measure of the jet’s width. After introducing the
following notations:
$\displaystyle x$ $\displaystyle=k(x^{\prime}-c\tau),$ $\displaystyle y$
$\displaystyle=ky^{\prime},$ $\displaystyle t$
$\displaystyle=\psi_{0}k^{2}\tau,$ (3) $\displaystyle x^{\prime}$
$\displaystyle=\frac{x}{k}+c\tau,$ $\displaystyle y^{\prime}$
$\displaystyle=\frac{y}{k},$ $\displaystyle\tau$
$\displaystyle=\frac{t}{\psi_{0}k^{2}}.$
and
$A=ak,\qquad L=\lambda k,\qquad C=\frac{c}{\psi_{0}k},$ (4)
we get the advection equations (1) in the frame moving with the phase velocity
$c$:
$\begin{aligned}
\dot{x}&=\frac{1}{L\sqrt{1+A^{2}\sin^{2}x}\cosh^{2}\theta}-C,\\\
\dot{y}&=-\frac{A\sin x(1+A^{2}-Ay\cos
x)}{L\left(1+A^{2}\sin^{2}x\right)^{3/2}\cosh^{2}\theta},\end{aligned}\qquad\theta=\frac{y-A\cos
x}{L\sqrt{1+A^{2}\sin^{2}x}}.$ (5)
Figure 1: Streamlines of the unperturbed system (5) in the frame moving with
the meander’s phase velocity $c$.
The respective streamfunction
$\psi(x,y)=-\tanh{\left(\frac{y-A\cos x}{L\sqrt{1+A^{2}\sin^{2}x}}\right)}+Cy$
(6)
has three normalized control parameters: $L$, $A$ and $C$ are the jet’s width,
meander’s amplitude and its phase velocity. The scaling chosen results in
translational invariance of the phase portrait along the $x$-axis with the
period $2\pi$.
The detailed analysis of stationary points and bifurcations of Eqs. (5) has
been done in Ref. [11]. Stationary points may exist only under the condition
$LC\leqslant 1$. There are four stationary points, two of them are always
stable and the other ones are stable under the condition
$AL\operatorname{Arcosh}\sqrt{1/LC}>1$. If the additional condition
$C<1/L\cosh^{2}(1/AL)$ is fulfilled, there are additional four unstable saddle
points. Resuming one gets:
1. 1.
$C>C_{\text{cr1}}=1/L$, there are no stationary points.
2. 2.
$C_{\text{cr1}}>C>C_{\text{cr2}}=1/L\cosh^{2}(1/AL)$ and $C>C_{\text{cr3}}$,
there are two centers and two saddles with two separatrices connecting the
saddles. There are two separatrices each of which passes through its own
saddle. The jet between the separatrices is westward.
3. 3.
$C_{\text{cr1}}>C>C_{\text{cr2}}$ and $C<C_{\text{cr3}}$, the jet is eastward
with the same stationary points as in the case 2.
4. 4.
$C_{\text{cr2}}>C>C_{\text{cr3}}$, there are eight stationary points and two
separatrices. There are two separatrices, each of which connects two saddle
points. The jet is westward.
5. 5.
$C_{\text{cr2}}>C$ and $C<C_{\text{cr3}}$, the jet is eastward with the same
stationary points as in the case 4.
Therefore, from the point of view of existence and stability of stationary
states, there are three possibilities: 1) there are no stationary points; 2)
there are four stationary points, two centers and two saddles; 3) there are
eight stationary points, four centers and four saddles. A bifurcation between
the first and second regimes consists in arising two pairs ‘‘saddle-center’’.
A bifurcation between the second and third regimes consists in arising two
saddles and a center between them instead of one saddle (a fork-type
bifurcation).
There is one more bifurcation that does not change the number and stability of
stationary points but changes the topology of the flow. The values of the
streamfunction on the separatrices are equal on modulo but of opposite signs.
There is a critical value of the phase velocity $C=C_{\text{cr3}}$ under which
the separatrices coincide and the respective streamfunction is equal to zero.
If $C>C=C_{\text{cr3}}$, a free flow between the separatrices is westward,
whereas with $C<C=C_{\text{cr3}}$ it is eastward. It is difficult to find
$C=C_{\text{cr3}}$ analytically but it may be shown [11] that
$C_{\text{cr3}}>C_{\text{cr2}}$, if
$\frac{2(1+A^{2})}{AL\sinh(2/AL)}<1.$ (7)
Otherwise $C_{\text{cr3}}<C_{\text{cr2}}$.
The respective phase portraits are typical with Hamiltonian systems with
running waves in shear flows [6, 12]. In dependence on the values of the phase
velocity $C$ one can get different topologies: a homoclinic connection, a
heteroclinic connection and a separatrix reconnection. Being motivated by
eastward jet currents in the ocean and atmosphere, we deal in this paper with
the case 3 (see the phase portrait in Fig. 1).
## 3 Chaotic mixing and transport
Streamlines with the streamfunction (6), that is time-independent in the
moving frame, are shown in Fig. 1. The plot demonstrates three different
regions of the flow: a central eastward jet (J), north and south circulations
(C) and peripheral westward currents (P). The centers of the circulations are
at critical lines to be defined by the condition $u(y_{c})=c_{x}$,
$v(y_{c})=0$, and they are divided by two separatrices connecting saddle
points. No exchange between the north and south circulations is possible in
the unperturbed system.
Even the simplest periodic perturbation of the meander’s amplitude,
$A=A_{0}+\varepsilon\cos(\omega t+\varphi)$, results in spitting stable and
unstable manifolds of the saddle points. There arise stochastic layers with
chaotic mixing and transport of water masses fluxes of which depend on the
width of a layer, a number of overlapping resonances and other factors. As an
example, we plot in Fig. 2 the Poincaré section of the perturbed flow. The
control parameters throughout the paper are chosen as follows: $A=0.785$,
$C=0.1168$, $L=0.628$, $\varepsilon=0.0785$, $\omega=0.2536$ and
$\varphi=\pi/2$.
Figure 2: General view of the Poincaré section of the system with a
periodically modulated meander’s amplitude. Figure 3: (a) Poincaré sections in
the northern part of the first eastern frame, (b) stickiness to the island’s
border.
Due to the zonal and meridian symmetries it is enough to consider mixing in
the northern part of the first eastern frame only $0\leqslant x\leqslant
2\pi$. In Fig. 3a we plot the respective Poincaré section. The vortex core,
that survives under the perturbation, is immersed into a stochastic sea where
one can see 6 small islands belonging to the same resonance. Particles,
belonging to these islands, rotate around the elliptic point of the vortex
core. Zoom of the section nearby two of the islands is shown in Fig. 3b. The
feature we want to pay attention to is a stickiness to the boundaries of the
vortex core and of the islands that is visualized by increasing the density of
points near the respective boundaries. It is a typical phenomenon with
Hamiltonian systems [10] which influences essentially the transport of passive
particles.
Figure 4: (a) Probability distribution functions (PDFs) of lengths of
eastward (triangles) and westward (squares) flights obtained with $10$ chaotic
trajectories for $5\cdot 10^{6}$ steps. The parameters are:
$\varepsilon=0.0785$ and $\omega=0.2536$. (b) For short eastward flights,
$12\leqslant x_{f}\leqslant 90$, the PDF decays exponentially. (c) The tails
of both the PDFs decay as a power law with the slope $\nu=1.94\pm 0.09$
obtained with westward (squares) long flights $|x_{f}|>25$.
Without perturbation, the transport properties are very simple: particles
either rotate in circulations C or move eastward in the jet J and westward in
the peripheral currents P. Under a perturbation, the motion in the stochastic
layers become extremely sensitive to small variations in initial conditions,
and one is forced to use an statistical approach to describe transport. A
commonly used statistical measure of transport is the variance
$\sigma^{2}(t)=<x^{2}>$, where the averaging is supposed to be done over an
ensemble of particles. Long time behavior of $\sigma^{2}(t)$ and of the
probability density function of particle’s displacement $P(x,t)$ may be
anomalous with typical chaotic Hamiltonian systems [13]. When
$\sigma^{2}(t)\sim t$ and $P(x,t)$ is a Gaussian distribution, the advection
is known to be normal with a well-defined diffusion coefficient
$D=\lim_{t\to\infty}\sigma^{2}/2t$. When $\sigma^{2}(t)\sim t^{\gamma}$, with
$\gamma\neq 1$, $D$ is either zero or infinite [13], and one gets either
subdiffusion ($\gamma<1$) or super-diffusion ($\gamma>1$). If the trajectories
are dominated by sticking regions nearby boundaries of islands, where
particles spend a long time, subdiffusion results. Superdiffusion occurs when
particles in the jet travel long distances between sticking events. The
respective length and time PDFs are expected to be non Gaussian.
We will call ‘‘a flight’’ any event between two successive changes of signs of
the particle’s zonal velocity. In this terminology a sticking consists of a
number of flights with approximately equal flight times. The Poincaré section
of the flow with the perturbation strength $\varepsilon=0.0785$ and frequency
$\omega=0.2536$ is shown in Fig. 3a. The flight PDFs are computed with $10$
particles (initially placed in the first east frame inside a stochastic layer)
up to the time $t=5\cdot 10^{6}$. The PDF of the lengths of flights is shown
in Fig. 4a for both the directions. The asymmetry between the eastward
(‘‘positive’’) and westward (‘‘negative’’) flights is evident. Both the PDFs
can be roughly split into three distinctive regions. The very short flights
with small values of $|x_{f}|$ ($<2\pi$) are supposed to be dominated by
sticking to the boundaries of the vortex core and oscillatory islands. The PDF
for eastward flights with the lengths in the range $12\leqslant x_{f}\leqslant
90$ decays exponentially (Fig. 4b). The tails of both the PDFs are close to a
power-law decay $P(x_{f})\sim|x_{f}|^{-\nu}$. In Fig. 4c we estimate the
exponent for long westward flights, $|x_{f}|>25$, with corresponding error by
least-square fitting of the straight line to the log-log plot of the data to
be $\nu=1.94\pm 0.09$.
Figure 5: (a) The northern border between the circulation (C) and the
peripheral current (P), (b) stickiness to the border of a ballistic island.
Figure 6: Examples of ballistic trajectories: the blue and regular trajectory,
which is the upper one in (a) and the lower one in (b), is inside a ballistic
island, the green and weakly chaotic trajectory, which is the lower one in (a)
and the upper one in (b), is just outside the island. (a) $x-y$ plane, (b)
dependence of the zonal position on time. Stickiness and flight events are
evident with the green chaotic trajectory.
Besides resonant islands with particles moving around the elliptic point in
the same frame (Fig. 3), we have found so-called ballistic islands which were
situated both in the chaotic sea and in the peripheral jets (Fig. 5a).
Ballistic modes [10, 14, 15] correspond to the stable periodic motion of
particles from one frame to another. Particles, belonging to a ballistic
island, move with a constant zonal velocity from one frame to another. When
mapping their positions at the moments $t=2k\pi/\omega$ $(k=1,2,\dots)$ onto
the first frame, we see a chain of islands which are visible in Fig. 5a. In
Fig. 5b a zoom of the ballistic island is shown. Stickiness to the island’s
border is evident. In Fig. 6 we demonstrate two ballistic trajectories
corresponding to the ballistic island shown in Fig. 5b. If a particle at $t=0$
is placed inside the island, it travels to the west in a regular way (the
upper blue trajectory in Fig. 6a). Its zonal position $x$ grows linearly with
time (the lower blue trajectory in Fig. 6b). The lower green trajectory in
Fig. 6a corresponds to a particle placed initially nearby the border of the
same ballistic island from outside. Intermittent flight and sticking events
are evident in Fig. 6b (the upper green curve in that figure).
## 4 Fractal geometry of mixing
Figure 7: Fractal set of initial positions $y_{0}$ of particles that reach the
lines $x=0,\pm 2\pi$ after $n/2$ turns around the elliptic points. $T$ is a
time particles need to reach the lines $x=0,\pm 2\pi$. Indices $e$ and $w$
mean particles moving in the eastward and westward directions, respectively.
Figure 8: Fragments of the evolution of a material line in the first eastern
frame. The fragments of the fractal in Fig. 7 with $n_{e}=1,2,3$ are marked by
the respective letters.
Poincaré sections provide good impression about the structure of the phase
space but not about geometry of mixing. In this section we consider the
evolution of a material line consisting of a large number of particles
distributed initially on a straight line that transverses the stochastic layer
at $x=0$. A typical stochastic layer consists of an infinite number of
unstable periodic and chaotic orbits with islands of regular motion to be
imbeded. All the unstable invariant sets are known to possess stable and
unstable manifolds. When time progresses particle’s trajectories nearby a
stable manifold of an invariant set tend to approach the set whereas the
trajectories close to an unstable manifold go away from the set. Because of
such a very complicated heteroclinic structure, we expect a diversity of
particle’s trajectories. Some of them are trapped forever in the first eastern
frame $0\leqslant x\leqslant 2\pi$ rotating around the elliptic point along
heteroclinic orbits. Other ones quit the frame through the lines $x=0$ or
$x=2\pi$, and then either are trapped there or move to the neighbor frames
(including the first one), and so on to infinity.
To get a more deep insight into the geometry of chaotic mixing we follow the
methodology of our works [16, 17] and compute the time $T$, particles spend in
the neighbor circulation zones $-2\pi\leqslant x\leqslant 2\pi$ before
reaching the critical lines $x=0,x=\pm 2\pi$, and the number of times $n/2$
they wind crossing the lines $x=\pm\pi$. In the upper panel in Fig. 7 the
functions $n(y_{0})$ and $T(y_{0})$ are shown. The upper parts of each
function (with $n>0$ and $T>0$) represent the results for the particles with
initial positive zonal velocities which they have simply due to their
locations on the material line at $x=0$. These particles enter the eastern
frame and may change the direction of their motion many times before leaving
the frame through the lines $x=0$ or $x=2\pi$. The time moments of those
events we fix for all the particles with $1.9\leqslant y_{0}\leqslant 2.045$.
The lower "negative" parts of the functions $n(y_{0})$ and $T(y_{0})$
represent the results for the particles with initial negative zonal velocities
($y_{0}\geqslant 2.045$) which move initially to the first western frame
($-2\pi\leqslant x\leqslant 0$). In fact, $T_{e}(y_{0})$ and $T_{w}(y_{0})$
are the time moments when a particle with the initial position $y_{0}$ quits
the eastern or western frames, respectively. Both the functions consist of a
number of smooth U-like segments intermittent with poorly resolved ones.
Border points of each U-like segments separate particles belonging to stable
and unstable manifolds of the heteroclinic structure. The corresponding
initial $y$-positions is a set (of zero measure) of particles to be trapped
forever in the respective frame. A fractal-like structure of chaotic advection
in both the frames is shown in the upper panel in Fig. 7, and its fragments
for the first levels are shown in the middle panel for the eastern and the
western fragments separately. Particles with even values of $n$ quit one the
frames through the border $x=0$, those with odd $n$ – through the border
$x=2\pi$ for the eastern frame and $x=-2\pi$ for the western one.
Let us consider in detail the fractal-like structure in the eastern frame
keeping in mind that the results are similar with any other frame. The
$n_{e}(y_{0})$-dependence is a complicated hierarchy of sequences of segments
of the material line. Following to the authors of the paper [18], we call as
an epistrophe a sequence of segments of the $(n+1)$-level, converging to the
ends of a segment of a sequence of the $n$-th level, whose length decrease in
accordance with a law. At $n_{e}=1$ we see in Fig. 7 an epistrophe with
segment’s length A, B, C, D and so on decreasing as $l_{m}=l_{0}q^{m}$ with
$q\approx 0.46$. Letters a and b in Fig. 7 denote the first segments of the
epistrophes at the level $n_{e}=2$, whereas d and c —the first segments of the
epistrophes at the level $n_{e}=3$. The respective laws for all those
epistrophes are not exponential.
In Fig. 8 we demonstrate fragments of the evolution of the material line in
the first eastern frame at the moments indicated in the figure. Letters on the
line mark the corresponding segments of the $n_{e}(y_{0})$ and $T_{e}(y_{0})$
functions in Fig. 7. As an example, let us explain formation of the epistrophe
ABCD at the level $n_{e}=1$. With the period of perturbation
$T_{0}=2\pi/\omega\simeq 8\pi$, a portion from the north end of the material
line leaves the frame through its eastern border. Look at the segments A and B
at $t=15\pi$ and $t=23\pi$. They quit the first frame as a fold through the
period $T_{0}\simeq 8\pi$. The other segments – C, and D (not shown in Fig. 8)
do the same job. The epistrophe’s segments at the odd levels ($n=2k-1>1$) quit
the frame with the period of perturbation $T_{0}$ one by one being folded (c
and d segments). The folds of the segments of the $(2k-1)$-level are exterior
with respects to the folds of the segments of the $(2k+1)$-level. The
following empirical law is valid: $T_{2k-1}-T_{2k+1}\simeq 2T_{0}$, where
$T_{2k-1}$ is a time when the first segments of the epistrophes at the level
$(2k-1)$ (A with $n_{e}=1$) reach the line $x=2\pi$, and $T_{2k+1}$ the
respective time for the first segments of the epistrophes at the level $2k+1$
(c and d segments with $n_{e}=3$).
Segments of the epistrophes of the even levels ($n=2k$) leave the frame with
the period $T_{0}$ as well but through the border $x=0$ moving to the west. We
show the evolution of some of them at the moments $t=31\pi$ and $t=35\pi$ in
Fig. 8. Thus, the material line evolves by stretching and folding, and folds
quit the frame in both directions with the period of perturbation.
## 5 Conclusion
We have treated the problem of mixing and transport of passive particles in a
kinematic model of a meandering oceanic jet current from the point of view of
dynamical system’s theory. A careful simulation of the Hamiltonian equations
of advection has shown a complicated character of mixing under a time-
dependent perturbation of the meander’s parameters. Both the oscillatory and
ballistic resonant islands and sticking of trajectories to their boundaries
have been found. The transport has benn shown to be anomalous. The geometry of
mixing has been shown to be fractal-like. The trapping time of advected
particles and the number of their rotations around elliptical points have been
found to have a hierarchical fractal structure as functions of initial
particle’s positions. A correspondence between the evolution of material lines
in the flow and elements of the fractal has been established.
## 6 Acknowledgments
This work was supported by the Russian Foundation for Basic Research (Project
no.06-05-96032), by the Program ‘‘Mathematical Methods in Nonlinear Dynamics’’
of the Russian Academy of Sciences and by the Program for Basic Research of
the Far Eastern Division of the Russian Academy of Sciences.
## References
* [1] J. Sommeria, S.D. Meyers, Y.L. Swinney, Laboratory model of a planetary eastward jet, Nature, 337 (1989) 58–62.
* [2] A.S. Bower, A simple kinematic mechanism for mixing fluid parcels across a meandering jet, J. Phys. Oceanogr., 21 (1989) 173-180.
* [3] R.M. Samelson, Fluid exchange across a meandering jet, J. Phys. Oceanogr., 22 (1992) 431–440.
* [4] T.H. Solomon, E.R. Weeks, H.L. Swinney, Observation of anomalous diffusion and Levy flights in a two-dimensional rotating flow, Phys. Rev. Lett., 71 (1993) 3975–3978.
* [5] D. Del-Castillo-Negrete, P.J. Morrison, Chaotic transport by Rossby waves in shear flow, Phys. Fluids A, 5 (1993) 948–965.
* [6] J.B. Weiss, E. Knobloch, Mass transport by modulated traveling waves, Phys. Rev.A., 42 (1989) 2579–2589.
* [7] K. Ngan, T. Shepherd, Chaotic mixing and transport in Rossby-wave critical layers, J. Fluid Mech., 334 (1997) 315–351.
* [8] J.Q. Duan, S. Wiggins, Fluid exchange across a meandering jet with quasi-periodic time variability, J. Phys. Oceanogr., 26 (1996) 1176-1188.
* [9] B.V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979) 263–379.
* [10] G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford, Oxford University Press, 2005.
* [11] M.Yu. Uleysky, M.V. Budyansky, S.V. Prants, Chaotic advection in a meandering jet current, Nonlinear Dynamics, 1 (2006) is.2 [in Russian].
* [12] J.E. Howard, S.M Hohs, Stochasticity and reconnection in Hamiltonian systems, Phys. Rev.A, 29 (1984) 418–421.
* [13] M.F. Shlesinger, G.M. Zaslavsky, J. Klafter, Strange kinetics, Nature, 363 (1993) 31–38.
* [14] V. Rom-Kedar and G. Zaslavsky, Chaos, 9 (1999) 697–705.
* [15] A. Iomin, D. Gangardt, and S. Fishman, Phys. Rev. E. 57,(1998) 4054–4062.
* [16] M. Budyansky, M. Uleysky, S. Prants, Hamiltonian fractals and chaotic scattering by a topographical vortex and an alternating current, Physica D, 195 (2004) 369-378.
* [17] M.V. Budyansky, M.Yu. Uleysky, S.V. Prants, Chaotic scattering, transport, and fractals in a simple hydrodynamic flow, Zh. Eksp. Teor. Fiz., 126 (2004) 1167-1179 [JETP, 99 (2004) 1018-1027].
* [18] K.A. Mitchell, J.P. Handley, B. Tighe, J.B. Delos and S.K. Knudson, Geometry and topology of escape, Chaos, 13 (2003) 880-891.
|
arxiv-papers
| 2012-02-02T03:36:21 |
2024-09-04T02:49:26.950863
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. V. Budyansky, S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1202.0350"
}
|
1202.0352
|
# Mechanism of destruction of transport barriers in geophysical jets with
Rossby waves
M.Yu. Uleysky, M.V. Budyansky, and S.V. Prants Pacific Oceanological
Institute
of the Russian Academy of Sciences, 43 Baltiiskaya st., 690041 Vladivostok,
Russia
###### Abstract
The mechanism of destruction of a central transport barrier in a dynamical
model of a geophysical zonal jet current in the ocean or the atmosphere with
two propagating Rossby waves is studied. We develop a method for computing a
central invariant curve which is an indicator of existence of the barrier.
Breakdown of this curve under a variation of the Rossby wave amplitudes and
onset of chaotic cross-jet transport happen due to specific resonances
producing stochastic layers in the central jet. The main result is that there
are resonances breaking the transport barrier at unexpectedly small values of
the amplitudes that may have serious impact on mixing and transport in the
ocean and the atmosphere. The effect can be found in laboratory experiments
with azimuthal jets and Rossby waves in rotating tanks under specific values
of the wave numbers that are predicted in the theory.
###### pacs:
05.45.-a,05.60.Cd,47.52.+j
Transport and mixing of water (air) masses and their characteristics play a
crucial role in the ocean and atmosphere dynamics. In the Lagrangian approach
a particle with the position $\vec{r}$ is advected by an Eulerian velocity
field $\vec{v}(\vec{r},t)$
$\frac{d\vec{r}}{dt}=\vec{v}(\vec{r},t).$ (1)
It is known that a simple deterministic velocity field may produce practically
unpredictable particle trajectories, the phenomenon known as chaotic advection
Aref ; Ottino ; PK06 .
We study theoretically and numerically horizontal cross-jet transport in
geophysical zonal flows. To list a few we mention the Gulf Stream in the
Atlantic, the Kuroshio in the Pacific, and the polar night Antarctic jet in
the atmosphere, which are the jet currents separating water (air) masses with
different physical properties. Transport of particles across a geophysical jet
is of crucial importance and may cause, for example, depletion of ozone in the
atmosphere and heating and freshing of waters in the ocean. The velocity
fields of real flows are not, of course, regular, but if the Eulerian
correlation time is large as compared to the Lagrangian one, the problem may
be treated in the framework of chaotic advection concept.
The equations of motion of a passive particle with coordinates $x$ and $y$
advected by a two-dimensional incompressible flow with a stream function
$\Psi$ are known to have a Hamiltonian form Aref
$\frac{dx}{dt}=u(x,y,t)=-\frac{\partial\Psi}{\partial
y},\quad\frac{dy}{dt}=v(x,y,t)=\frac{\partial\Psi}{\partial x},$ (2)
with the phase space being the position space for advected particles. Chaotic
mixing and transport in jet flows have been extensively studied with kinematic
models, where the velocity field is a given function of $x$, $y$ and $t$
imitating real flows (see PK06 ; S92 ; RWig06 ; UBP07 and references
therein), and with dynamical models conserving the potential vorticity (see
PK06 ; P91 ; DM93 ; Rypina and references therein). The problem has been
studied as well in laboratory where azimuthal jets with Rossby waves have been
produced in rotating tanks SMS89 ; SHS93 . It has been found both numerically
and experimentally that fluid is effectively mixed along the jet, but in
common opinion a large gradient of the potential vorticity in the central part
prevents transport across the jet. A technique, based on computing the finite-
scale Lyapunov exponent, has been found useful in Ref. BLR01 to detect the
presence of cross-jet barriers in kinematic models. A comparison of properties
of cross-jet transport in kinematic and dynamical models of atmospheric zonal
jets has been done recently in Ref. P.H.Haynes . Up to now, the transport
barrier has been shown numerically Rypina ; RWig06 to be broken only with so
large values of the wave amplitudes that are beyond of the validity of linear
models and can be hardly observed in real flows.
The aim of the paper is to prove that cross-jet transport under appropriate
conditions is possible at comparatively small values of the wave amplitudes
and, therefore, may occur in geophysical jets. We develop a general method to
detect a core of the transport barrier and find a mechanism of its destruction
using the dynamical model of a zonal jet flow with two propagating Rossby
waves. The method comprises the identification of a central invariant curve
(CIC), which is an indicator of existence of the barrier, finding certain
resonant conditions for its destruction at given values of the wave numbers,
and detection of cross-jet transport.
Motion of two-dimensional incompressible fluid in the rotating frame is
governed by the equation for conserving potential vorticity
$(\partial/\partial t+\vec{v}\cdot\vec{\nabla})\Pi=0$. In the quasigeostrophic
approximation P87 , one gets $\Pi=\nabla^{2}\Psi+\beta y$, where $\beta$ is
the Coriolis parameter. The $x$ axis is chosen along the zonal flow, from the
west to the east and $y$ — along the gradient from the south to the north.
Barotropic perturbations of zonal flows produce Rossby waves which have an
essential impact on transport and mixing in the ocean and the atmosphere P87 .
The stream function is sought in the form
$\Psi=\Psi_{0}+\Psi_{\text{int}}=\Psi_{0}(y)+\sum_{j}\Phi_{j}(y)e^{ik_{j}(x-c_{j}t)},$
(3)
where $\Psi_{0}$ describes a zonal flow and $\Psi_{\text{int}}$ is its
perturbation which is supposed to be a superposition of zonal running Rossby
waves. After substituting (3) in the equation for the potential vorticity and
a linearization, one gets the Rayleigh-Kuo equation Kuo
$(u_{0}-c_{j})\Bigl{(}\frac{d^{2}\Phi_{j}}{dy^{2}}-k_{j}^{2}\Phi_{j}\Bigr{)}+\Bigl{(}\beta-\frac{d^{2}u_{0}}{dy^{2}}\Bigr{)}\Phi_{j}=0,$
(4)
where the zonal velocity $u_{0}=-d\Psi_{0}/dy$ has a single extremum at $y=0$.
If one takes the following zonal-velocity profile (the Bickley jet DM93 ):
$u_{0}(y)=U_{0}\operatorname{sech}^{2}{\frac{y}{D}},$ (5)
then Eq. (4) admits two neutrally stable solutions
$\Phi_{j}(y)=A_{j}U_{0}D\operatorname{sech}^{2}{\frac{y}{D}},\quad j=1,2,$ (6)
where $U_{0}$ is the maximal velocity in the flow, $D$ is a measure of its
width, and $A_{j}$ are the wave amplitudes. It is easy to check that (5) and
(6) are compatible with (4) if there is the following condition for the phase
velocities:
$c_{1,2}=\frac{U_{0}}{3}(1\pm\alpha),\quad\alpha\equiv\sqrt{1-\beta^{\ast}},\quad\beta^{\ast}\equiv\frac{3D^{2}\beta}{2U_{0}},$
(7)
which are connected with the wave numbers by the dispersion relation
$c_{1,2}=U_{0}D^{2}k_{1,2}^{2}/6$. Two neutrally stable Rossby waves exist if
$\beta D^{2}/U_{0}<2/3$.
Thus, the stream function of the zonal flow with two Rossby waves, satisfying
the conservation of the potential vorticity, has the form
$\Psi(x,y,t)=-U_{0}D\Bigl{(}\tanh{\frac{y}{D}}-\operatorname{sech}^{2}{\frac{y}{D}}\times\\\
\times\Bigl{[}A_{1}\cos{k_{1}(x-c_{1}t)}+A_{2}\cos{k_{2}(x-c_{2}t)}\Bigr{]}\Bigr{)}.$
(8)
One of the task of this paper is to present results in the form allowing a
comparison with laboratory experiments SMS89 ; SHS93 in which an azimuthal
jet at the radius $R$ with Rossby waves with the wave numbers $n_{1}$ and
$n_{2}$ has been produced:
$k_{1,2}=\frac{n_{1,2}}{R},\quad
c_{1,2}=\frac{U_{0}D^{2}}{6R^{2}}n_{1,2}^{2}.$ (9)
Let it be $n_{1}>n_{2}$, and the wave with $n_{1}$ is called the first one.
Let the wave numbers be represented as $n_{1}=mN_{1}$ and $n_{2}=mN_{2}$,
where $m\neq 1$ is the greatest common divisor and $N_{1}/N_{2}$ is an
irreducible fraction. Introducing new coordinates $x^{\prime}$, $y^{\prime}$,
and $t^{\prime}$
$x=\frac{(x^{\prime}+C_{2}t^{\prime})R}{m},\quad y=Dy^{\prime},\quad
t=\frac{R}{mU_{0}}t^{\prime},$ (10)
we rewrite the stream function (8) in the frame moving with the phase velocity
of the first wave
$\Psi^{\prime}(x^{\prime},y^{\prime},t^{\prime})=-\tanh{y^{\prime}}+A_{1}\operatorname{sech}^{2}{y^{\prime}}\cos(N_{1}x^{\prime})+\\\
+A_{2}\operatorname{sech}^{2}{y^{\prime}}\cos(N_{2}x^{\prime}+\omega_{2}t^{\prime})+C_{2}y^{\prime},$
(11)
where
$\omega_{2}\equiv\frac{2N_{2}(N_{1}^{2}-N_{2}^{2})}{3(N_{1}^{2}+N_{2}^{2})},\quad
C_{2}\equiv\frac{2N_{1}^{2}}{3(N_{1}^{2}+N_{2}^{2})}.$ (12)
Thus, we get the stream function (11) with the control parameters $N_{1}$ and
$N_{2}$ which are specified by the four experimental parameters: $U_{0}$,
$\beta$, $D$, and $R$. One can now study cross-jet transport with any
combination of the wave numbers $n_{1}$ and $n_{2}$ that can be realized in a
laboratory experiment by adjusting the radius $R$, the jet width $D$, the
maximal velocity $U_{0}$, and the Coriolis-like parameter $\beta$ SMS89 ;
SHS93 .
The advection equations (2) with the stream function (11) have the form
$\begin{gathered}\begin{aligned}
\frac{dx}{dt}=&-C_{2}+\operatorname{sech}^{2}{y}[1+2A_{1}\tanh{y}\cos{(N_{1}x)}+\\\
&+2A_{2}\tanh{y}\cos{(N_{2}x+\omega_{2}t)}],\end{aligned}\\\
\frac{dy}{dt}=-\operatorname{sech}^{2}{y}[A_{1}N_{1}\sin{(N_{1}x)}+A_{2}N_{2}\sin{(N_{2}x+\omega_{2}t)}],\end{gathered}$
(13)
where we omitted the primes over $x$, $y$, and $t$. If the amplitude of the
second wave is zero, $A_{2}=0$, then the set (13) is integrable. The phase
portrait of the steady flow with $A_{1}=0.2416$, $N_{1}=5$ and $N_{2}=1$ is
shown in Fig. 1a in the frame moving with the phase velocity of the first
wave. The eastward jet is situated between two chains with five vortices. The
southern and northern peripheral currents are westward in the moving frame. In
a steady flow all the particles follow streamlines. At $A_{2}>0$, chaos may
arise in a typical way: a stochastic layer appears at the place of the broken
separatrices (Fig. 1b and c).
Figure 1: (a) $A_{2}=0$. Phase portrait of the steady jet flow with $N_{1}=5$
and $N_{2}=1$ in the moving frame. (b) $A_{2}=0.09$. CIC (the bold curve) is a
barrier to transport across the jet. (c) $A_{2}=0.095$. Destruction of CIC and
onset of cross-jet transport.
At odd values of $N_{1}$ and $N_{2}$, Eqs. (13) have the two symmetries
$\hat{S}:\left\\{\begin{aligned} \tilde{x}&=\pi+x,\\\
\tilde{y}&=-y,\end{aligned}\right.\quad\hat{I}_{0}:\left\\{\begin{aligned}
\tilde{x}&=-x,\\\ \tilde{y}&=y,\end{aligned}\right.$ (14)
which are involutions, i. e., $\hat{S}^{2}=1$ and $\hat{I}_{0}^{2}=1$. Solving
the equation $\hat{I}_{0}(x_{j},y_{j})=\hat{S}(x_{j},y_{j})$, $j=1,2$, one
gets indicator points Aizawa : ($x_{1}=\pi/2$, $y_{1}=0$) and ($x_{2}=3\pi/2$,
$y_{2}=0$). Iterating them, we construct a CIC BUP09 in the central part of
the jet which is the last transport barrier in the sense that the CIC breaks
down and is replaced by a stochastic layer with variation of the wave
amplitudes. We illustrate this in Fig. 1. At $A_{2}=0.09$, the CIC together
with neighboring invariant curves forms a narrow transport barrier (Fig. 1b).
We define a CIC as a curve which is invariant under the operator $\hat{S}$ and
the evolution operator over the period $2\pi/\omega_{2}$. The CIC separates
the northern and southern parts of the flow. At $A_{2}=0.095$, the CIC breaks
down, and cross-jet transport becomes possible (Fig. 1c).
It is reasonable to suppose that destruction of CIC is caused by a ballistic
resonance between the maximal frequency of the particle motion in the central
jet and the perturbation frequency $\omega_{2}$. The first one is estimated
from Eq. (13) to be $f_{1}\simeq-C_{2}+1$, and the second one is given by
(12). So, the approximate condition of the ballistic resonance is
$\frac{f_{1}}{\omega_{2}}=\frac{N_{1}^{2}+3N_{2}^{2}}{2N_{2}(N_{1}^{2}-N_{2}^{2})}.$
(15)
At small amplitudes, this ratio gives an approximate estimate for the CIC
winding number $w$ BUP09 . Equating the right-hand side of Eq. (15) to a
rational number, one finds those values of the wave numbers $N_{1}$ and
$N_{2}$ for which the CIC is strongly influenced by the corresponding
resonance, and, therefore, cross-jet transport becomes possible.
Figure 2: Diagrams of (a) the winding number $w$ and (b) the maximal deviation
of iterations of the indicator point along the y-axis, $|y_{\rm max}|$, in the
space of the Rossby wave amplitudes $A_{1}$ and $A_{2}$. White zone: regime
with cross-jet transport.
In order to reveal a scenario for CIC destruction we plot in Fig. 2 the
dependencies of $w$ and the maximal deviation of iterations of the indicator
point along the y-axis, $|y_{\rm max}|$, on $A_{1}$ and $A_{2}$ for the pair
($N_{1}=5$, $N_{2}=1$). The bifurcation curves with the winding numbers
corresponding to certain resonances are shown in Fig. 2a. The even $2:1$
($w=0.5$) and odd $7:3$ ($w=0.4285$) ones produce two deep and wide spikes in
the plots. White color codes those values of the amplitudes $A_{1,2}$ at which
a CIC is broken. Comparing Figs. 2a and b, we see that the zone with broken
CIC in Fig. 2a correspond to the values $|y_{\rm max}|\simeq 1.5$ in Fig. 2b,
i.e., iterations of points, situated initially in the jet core, cover the
region of the size of order $\simeq 3$ jet’s half-width. It means breakdown of
central transport barrier at those values of $A_{1,2}$ at which the CIC is
broken. Figure 2 demonstrates clearly that destruction of the transport
barrier may happen at comparatively small values of the wave amplitudes
$(A_{1,2}<1)$. Our model is essentially a linear one, and the Rayleigh-Kuo
equation is valid to first order in the wave amplitudes.
To illustrate the mechanism of destruction of CIC we study the topology of the
phase space near the islands of the resonance $7:3$ (see the spike with
$w=0.4285\dots$ in Fig. 2). Let us fix $A_{1}=0.2418$ and gradually increase
$A_{2}$ away from zero. In the range $0<A_{2}<0.088$ the smooth CIC and
neighboring invariant curves form a transport barrier (Fig. 3a). At
$A_{2}\simeq 0.088$, invariant manifolds of hyperbolic orbits of the resonance
$7:3$ cross each other, the CIC breaks down, and there appears at its place a
narrow stochastic layer locked between remained invariant curves (Fig. 3b).
When $A_{2}$ increases further islands of the resonance $7:3$ diverge, and a
meandering CIC appears again between them (see Fig. 3c at $A_{2}=0.09$). At
$A_{2}>0.095$, CIC and surrounding invariant curves are destroyed, and cross-
jet transport becomes possible in a wide range of the $y$ coordinate (Fig. 3).
Thus, existence of a CIC is a sufficient but not necessary condition for
existence of a transport barrier. Animation of metamorphoses of topology of
the transport barrier and its destruction at a fixed value of one of the wave
amplitudes and variation of the other one can be found in Ref. animation .
Figure 3: Mechanism of CIC destruction and onset of cross-jet transport. (a)
$A_{2}=0.087$. Smooth CIC and neighboring invariant curves form a transport
barrier. (b) $A_{2}=0.088$. A narrow stochastic layer (shadowed strip) appears
at the place of the broken CIC. (c) $A_{2}=0.09$. CIC appears again as a
meandering curve. (d) $A_{2}=0.1$. Breakdown of CIC and onset of cross-jet
transport. Insets show magnification of the phase space region nearby the
resonance $7:3$.
In conclusion we discuss briefly a possibility for checking main results of
our work in laboratory experiments on chaotic advection in rotating fluid
SMS89 ; SHS93 imitating nonlinear geostrophical geophysical flows in the
ocean and the atmosphere. An azimuthal jet with Rossby waves was produced by
the action of the Coriolis force on radially pumped fluid in a rotating tank
with a slope imitating the $\beta$-effect on the rotating Earth. The measured
velocity field was rather well approximated by the model stream function (8)
SHS93 . Rapid mixing on either side of the jet was observed for a
quasiperiodic flow, but no significant transport was observed across the jet.
In our opinion the reason is that the experiments have been carried out under
conditions that are far away from the resonances which are capable of
destroying the central transport barrier at the values of the Rossby wave
numbers realized in the experiment. The results obtained in this paper allow
to specify those values of the control parameters of the flow, the Rossby wave
numbers, for which there exist specific resonances capable of destroying
transport barriers at comparatively small values of the wave amplitudes. Our
recommendation to observe cross-jet transport in laboratory is to produce an
azimuthal jet and Rossby waves with odd wave numbers whose values differ
significantly from each other, say ($N_{1}=5,N_{2}=1$) or ($N_{1}=7,N_{2}=3$).
The work was supported partially by the Program “Fundamental Problems of
Nonlinear Dynamics” of the Russian Academy of Sciences and by the Russian
Foundation for Basic Research (project no. 09-05-98520).
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* (12) G. Boffetta, G. Lacorata, G. Redaelli, and A. Vulpiani, Physica D. 159, 58 (2001).
* (13) P. H. Haynes, D. A. Poet, and E. F. Shuckburgh, J. Atmos. Sci. 64, 3640 (2007).
* (14) J. Pedlosky, Geophysical fluid dynamics (New-York: Springer-Verlag, 1987).
* (15) H.L. Kuo, J. Meteorol. 6, 105 (1949).
* (16) S. Shinohara and Y. Aizawa, Progr. Theor. Phys. 100, 219 (1998).
* (17) M. V. Budyansky, M. Yu. Uleysky, and S. V. Prants, Phys. Rev. E 79, 056215 (2009).
* (18) M. Yu. Uleysky, M. V. Budyansky, and S. V. Prants, http://dynalab.poi.dvo.ru/papers/mult1.avi and http://dynalab.poi.dvo.ru/papers/mult2.avi
|
arxiv-papers
| 2012-02-02T04:02:20 |
2024-09-04T02:49:26.958353
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Yu. Uleysky, M. V. Budyansky, S. V. Prants",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1202.0352"
}
|
1202.0360
|
# Lagrangian study of transport and mixing in a mesoscale eddy street
S.V. Prants prants@poi.dvo.ru [ M.V. Budyansky V.I. Ponomarev M.Yu. Uleysky
Pacific Oceanological Institute of the Russian Academy of Sciences,
43 Baltiiskaya st., 690041 Vladivostok, Russia
###### Abstract
We use dynamical systems approach and Lagrangian tools to study surface
transport and mixing of water masses in a selected coastal region of the Japan
Sea with moving mesoscale eddies associated with the Primorskoye Current.
Lagrangian trajectories are computed for a large number of particles in an
interpolated velocity field generated by a numerical regional multi-layer
eddy-resolving circulation model. We compute finite-time Lyapunov exponents
for a comparatively long period of time by the method developed and plot the
Lyapunov synoptic map quantifying surface transport and mixing in that region.
This map uncovers the striking flow structures along the coast with a
mesoscale eddy street and repelling material lines. We propose new Lagrangian
diagnostic tools — the time of exit of particles off a selected box, the
number of changes of the sign of zonal and meridional velocities — to study
transport and mixing by a pair of strongly interacting eddies often visible at
sea-surface temperature satellite images in that region. We develop a
technique to track evolution of clusters of particles, streaklines and
material lines. The Lagrangian tools used allow us to reveal mesoscale eddies
and their structure, to track different phases of the coastal flow, to find
inhomogeneous character of transport and mixing on mesoscales and
submesoscales and to quantify mixing by the values of exit times and the
number of times particles wind around the eddy’s center.
###### keywords:
Lagrangian transport , mesoscale eddy dynamics , Lyapunov exponents
url]www.dynalab.poi.dvo.ru
## 1 Introduction
Large-scale coherent structures, such as major currents and eddies, are key
ingredients organizing ocean flows. They are easily seen on surface
temperature and sea-surface height satellite data. In numerical models,
contours of potential vorticity can be used to reveal coherent structures.
However, those and other Eulerian means give snapshots at fixed times and
contain little information about transport and mixing of water masses which
are inherent Lagrangian notions. Lagrangian approach to study horizontal
transport and mixing in the ocean is rapidly becoming an effective method to
demonstrate inhomogeneous character of that mixing and existence of coherent
structures (Haller and Poje, 1998; Samelson and Wiggins, 2006; Koshel and
Prants, 2006; Haller, 2000; Haller and Yuan, 2000; Shadden et al., 2005; Ide
et al., 2002; Mancho et al., 2006; Jones and Winkler, 2002; Samelson, 1992;
Artale et al., 1997; Koshel et al., 2008; Izrailsky et al., 2008). There are
interesting papers on that approach studying transport and mixing in different
basins in the World Ocean with numerically generated velocity fields (Beron-
Vera et al., 2008; d’Ovidio et al., 2004; Mancho et al., 2008; Olascoaga et
al., 2006; García-Olivares et al., 2007; Miller et al., 1997) and velocity
fields derived from satellite altimeter measurements (d’Ovidio et al., 2009;
Waugh et al., 2006; Abraham and Bowen, 2002; Lehahn et al., 2007) or using
high-frequency radars (Lekien et al., 2005; Lipphardt et al., 2006; Gildor et
al., 2009). This approach does not aim at studying individual trajectories of
fluid particles but at searching for and identifying spatial structures
organizing the whole flow and known in theory of dynamical systems as
invariant manifolds.
Motion of a fluid particle is the trajectory of a dynamical system with given
initial conditions governed by the velocity field computed either by solving
the corresponding master equations or as the output of a numerical ocean model
or derived from a measurement. The phase space of that dynamical system is the
real space where many important phase-space objects known in the theory of
dynamical systems, such as stationary points, Kolmogorov-Arnold-Moser tori,
stable and unstable manifolds, periodic and chaotic orbits, etc., can be found
and studied.
In this work, we use velocity data from the Japan Sea circulation model
(Shapiro, 2000) to study and characterize surface transport and mixing in the
region comprising the Primorskoye (Liman) Current which is known by its rich
mesoscale activity. The instabilities of this current in the warm period
generate mesoscale coastal eddies (Ponomarev and Trusenkova, 2000) which
propagate downstream until the entrance to the Peter the Great Bay (the
latitude of Vladivostok, Russia). These eddies are generated in the model as a
kind of a mesoscale eddy street. The mesoscale and submesoscale dynamics over
the shelf and steep continental slope includes jet currents, streamers and
eddies being controlled by synoptic scale wind forcing and sea baroclinicity.
According to satellite data, the anticyclonic mesoscale eddies of relatively
small scale of the order $(10\div 50)$ km have been observed in the
northwestern marginal area directly over the steep continental slope. Near the
sea surface they often form pairs of strongly interacting eddies with spiral
patterns visible at satellite images. The anticyclonic mesoscale eddies of
larger scale have been clearly seen in the southern marginal area of the slope
and shelf of the Peter the Great Bay. The mesoscale dynamics over the
continental slope could be associated with the coastal Kelvin waves
propagating downstream with the Primorskoye Current to the southwest and
catched by the wide shelf of the Peter the Great Bay.
Many practical transport problems focus on tracking the evolution of small
fluid patches (clusters of particles) near the coast that becomes important in
the case of oil and other pollutant spills or harmful algal blooms. It is of
crucial interest to know the fate of the particles inside the patch. As to
basic research problems in physical oceanography, it is important to know
typical transport pathways along which coastal waters move to the open sea and
the open-sea waters move to the coast.
The aim of this paper is to study surface transport and mixing along the
mesoscale eddy street by computing Lagrangian trajectories for a large number
of particles advected by the Japan Sea circulation model known as the MHI
model (Marine Hydrophysical Institute, Sevastopol, Ukraine) (Shapiro, 2000).
Our paper is organized as follows. In section 2, we review briefly the
dynamical systems method to study transport and mixing in the ocean. Section 3
introduces the Japan Sea circulation model. Section 4 contains our main
results. (i) We compute finite-time Lyapunov exponents (FTLE), the time of
exit of a large number of particles off a selected sea region and streaklines.
It enables us to reveal mesoscale eddies, hyperbolic and non-hyperbolic
regions in the sea and to quantify mixing by values of the FTLE. (ii) We track
the evolution of material lines, crossing the eddies, and demonstrate
inhomogeneous character of transport and mixing on mesoscales and
submesoscales that can be quantified by values of exit times and the number of
times particles change their zonal velocity while moving in the preselected
box. (iii) We show that the evolution of neighbour patches of particles,
chosen on a ridge of largest values of the FTLE field and nearby, illustrates
strongly different transport paths of particles in hyperbolic sea regions. We
summarize our results in section 5. In Appendixes A and B, we illustrate the
complex pattern of chaotic mixing and transport by an idealized example of the
vortex flow and describe the method of computing the FTLEs, respectively.
## 2 Dynamical systems method to study transport and mixing in the ocean
In Lagrangian approach, a fluid particle is advected by the two-dimensional
Eulerian velocity field
$\frac{dx}{dt}=u(x,y,t),\quad\frac{dy}{dt}=v(x,y,t),$ (1)
where $(x,y)$ is the location of the particle, $u$ and $v$ are the zonal and
meridional components of its velocity at the location $(x,y)$. The motion is
considered on a plane because of a comparatively small size of the region we
will analyze in this paper. Even if the velocity field is fully deterministic,
the Lagrangian trajectories may be very complicated and practically
unpredictable. It means that a distance between two initially nearby particles
grows exponentially in time
$\|\delta{\mathbf{r}}(t)\|=\|\delta{\mathbf{r}}(0)\|\,e^{\lambda t},$ (2)
where $\lambda$ is a positive number, known as the Lyapunov exponent, which
characterizes asymptotically (at $t\to\infty$) the average rate of the
particle dispersion, and $\|\cdot\|$ is a norm of the vector
$\mathbf{r}=(x,y)$. It immediately follows from (2) that we are unable to
forecast the fate of the particles beyond the so-called predictability horizon
$T_{p}\simeq\frac{1}{\lambda}\ln\frac{\|\Delta\|}{\|\Delta(0)\|},$ (3)
where $\|\Delta\|$ is the confidence interval of the particle location and
$\|\Delta(0)\|$ is a practically inevitable inaccuracy in specifying the
initial location. The deterministic dynamical system (1) with a positive
maximal Lyapunov exponent for almost all vectors $\delta\mathbf{r}(0)$ (in the
sense of nonzero measure) is called chaotic. It should be stressed that the
dependence of the predictability horizon $T_{p}$ on the lack of our knowledge
of exact location is logarithmic, i. e., it is much weaker than on the measure
of dynamical instability quantified by $\lambda$. Simply speaking, with any
reasonable degree of accuracy on specifying initial conditions there is a time
interval beyond which the forecast is impossible, and that time may be rather
small for chaotic systems.
In the last two decades, dynamical systems methods have been applied to study
transport and mixing processes in the ocean (Haller and Poje, 1998; Samelson
and Wiggins, 2006; Koshel and Prants, 2006; Haller, 2000; Haller and Yuan,
2000; Shadden et al., 2005; Ide et al., 2002; Mancho et al., 2006; Jones and
Winkler, 2002; Miller et al., 1997; Samelson, 1992; Artale et al., 1997).
Since the phase plane of the two-dimensional dynamical system (1) is the
physical space for fluid particles, many abstract mathematical objects from
dynamical systems theory are material surfaces, points and curves in fluid
flows. Stagnation point in a steady flow is the fluid particle with zero
velocity. Besides “trivial” elliptic stagnation points, the motion around
which is stable, there are hyperbolic (saddle) stagnation points which
organize fluid motion in their neighbourhood in a specific way. There are two
opposite directions (for each saddle point) along which nearby trajectories
approach the point at an exponential rate and two other directions along which
nearby trajectories move away from it at an exponential rate.
We recall briefly some important notions from dynamical systems theory that
will be used in the present paper. Invariant manifold in a two-dimensional
flow is a material line, i. e., it is composed of the same fluid particles in
course of time. To introduce the notion of stable and unstable manifolds it is
instructive to consider a steady flow around a hyperbolic point which is the
fixed point the fluid motion around which is unstable. So, a fluid particle
approaches the hyperbolic point along its stable (unstable) invariant
manifolds when $t\to+(-)\infty$. Even in simple time-periodic flows, the
stable and unstable manifolds may intersect each other transversally creating
so-called homoclinic and heteroclinic tangles where the fluid motion is so
complicated that it may be strictly called chaotic, the phenomenon known as
chaotic advection (Aref, 1984; Ottino, 1989; Samelson, 1992; Pierrehumbert and
Yang, 1993; Koshel and Prants, 2006; Samelson and Wiggins, 2006). Close fluid
particles in the tangles rapidly diverge providing very effective mechanism
for mixing. Stable and unstable manifolds are important organizing structures
in the flow because they attract and repel fluid particles (not belonging to
them) at an exponential rate and partition the flow into regions with
different types of motion. Thus, they are transport barriers.
Stable and unstable manifolds are useful tools in studying realistic flows
modeling the ocean. In aperiodic flows it is possible to identify
aperiodically moving hyperbolic points with stable and unstable effective
manifolds (Haller and Poje, 1998). Unlike the manifolds in steady and periodic
flows, defined in the infinite time limit, the “effective” manifolds of
aperiodic hyperbolic trajectories have a finite lifetime. The point is that
they may play the same role in organizing oceanic flows as do invariant
manifolds in simpler flows. The effective manifolds in course of their life
undergo stretching and folding at progressively small scales and intersect
each other in the homoclinic points in the vicinity of which fluid particles
move chaotically. Trajectories of initially close fluid particles diverge
rapidly in these regions, and particles from other regions appear there. It is
the mechanism for effective transport and mixing of water masses in the ocean.
Moreover, stable and unstable effective manifolds constitute Lagrangian
transport barriers between different regions because they are material
invariant curves that cannot be crossed by purely advective processes.
Motion in any preselected region in a circulation basin may be considered as a
scattering problem in the sense that fluid particles come into the region from
outside and leave it sooner or later. Passive particles are advected by an
incoming flow into a mixing region, where their motion may be chaotic, and
then most of them are washed away from that region. It is known in theory of
chaotic scattering that there exists an abstract chaotic invariant set in a
bounded region of phase space consisting of an infinite number of hyperbolic
particle trajectories that never leave the mixing region (Ott, 2002; Tel et
al., 2005; Budyansky et al., 2004a, b). If a particle belongs to the set at an
initial moment, then it remains in the mixing region forever (in theory). Most
of particles sooner or later leave the mixing region, but their behavior is
strongly influenced by the presence of the chaotic invariant set. Each
trajectory in the set and therefore, the whole set possesses stable and
unstable manifolds. Theoretically, these manifolds have infinite spatial
extent, and the tracer, belonging to the stable manifold, is advected by the
incoming flow into the mixing region and remains there forever. The
corresponding initial conditions make up a set of zero measure. However, the
particles that are initially close to those in the stable manifold follow them
for a long time, eventually deviate from them and leave the mixing region
along the unstable manifold. In Appendix A, we illustrate these abstract
mathematical concepts with the simple model of the flow with a fixed point
eddy embedded in a background steady flow with the periodic tidal component
(Budyansky et al., 2004a, b) and explain how they can be used to characterize
inhomogeneous mixing in realistic eddy flows.
## 3 Numerical Japan sea circulation model
The Japan Sea is a deep marginal sea with shallow straits connected with the
East China Sea, Okhotsk Sea and North Pacific. The Japan Sea has three deep
basins, named as Tsushima and Yamato Basins in the southern sea area and the
deepest and largest Japan Basin in the northern sea area. The paper is focused
on simulation of the mesoscale dynamical processes over the shelf and
continental slope of the Japan Basin situated in the northwestern area of the
Japan Sea. The typical large scale circulation over the Northwestern Japan Sea
includes, two cyclonic gyres, the cold Primorskoye Current streamed
southwestward along the continental slope of the Japan Basin and the warm
northern current along the slope of Japanese Islands. The sea domain in
numerical experiments is characterized by thin shelf along the continental
coast of the Russian North Primorye Region (Primorsky Krai), wide shelf of the
Peter the Great Bay in the South Primorye Region and steep continental slope
in the whole sea area adjacent to the northwest Japan Sea coast. The
southwestern cyclonic gyre over southern and central areas of the Japan Basin
is simulated in the model domain as a large scale circulation.
The MHI ocean circulation model developed by N.B. Shapiro and E.N. Mikhaylova
at the Marine Hydrophysical Institute, Sevastopol, Ukraine (Shapiro, 2000) is
a set of 3D primitive equations under the hydrostatic and Boussinesq
approaches in Z coordinate system with a free surface boundary condition. It
belongs to a class of layered models, in which the sea consists of a number of
quasi-isopycnal layers. Interfacial surfaces between layers can freely move up
and down and layers can deform, physically vanish (outcrop) and restore. The
MHI model has been applied in Refs. (Ponomarev and Trusenkova, 2000;
Trusenkova et al., 2005) for simulation of the Japan Sea large scale
circulation, as well as to simulate mesoscale dynamics in the northwestern
Japan Sea area adjacent to the Primorsky Krai coast. Equations of the MHI
model, vertically integrated within layers, are formulated at the beta-plane,
with the $x$-axis directed from the west to east and the $y$-axis directed
from the south to north. A specific feature of the MHI model is that the
density (buoyancy) of any layer is allowed to vary with space and time.
The present study is focused on simulation of mesoscale dynamics over the
continental slope and shelf in the closed sea area of the cyclonic gyre
occupying the southern and central area of the Japan Basin. The sea domain is
$39^{\circ}$N – $44^{\circ}$N, $129^{\circ}$E – $138^{\circ}$E with the
horizontal grid steps $2.5$ km along latitude and $2.5$ km along longitude.
The total number of the grid points is $210\times 280$. We set $9$ quasi-
isopycnal layers including the upper mixed one. The bottom topography is
adopted from navigation maps. Islands can not be resolved in coastline for the
chosen mesh but they are represented in topography. The area of the cyclonic
gyre is suggested to be closed, and we use no slip boundary conditions for
current velocity at the sea domain contour including sea coast, northern,
eastern, and southern boundaries.
Figure 1: (a) The velocity field in the whole basin of the Japan Sea in the
6th layer of the MHI model with the horizontal resolution $1/16^{\circ}$ and
with initial temperature and salinity distributions estimated from the CTD
data of the oceanographic observations in summer 1999 (Talley et al., 2001).
(b) Depths of the main pycnocline (the interface between the 5th and 6th
layers of the model in August).
The initial conditions for realistic summer isopycnal interfaces, temperature
and salinity distribution in the model layers have been taken from
oceanographic survey 1999 (Talley et al., 2001). The MHI model has been
integrated with the time step of 4 min for a one year. The coefficients of
quasi-isopycnal biharmonic viscosity, harmonic viscosity, and diffusion used
in the momentum and heat/salt transfer equations have been varied
correspondingly from $10^{17}$ m4/s, $10^{7}$ m2/s and $0.4\times 10^{7}$ m2/s
in the model spin up (60 days) to $10^{16}$ m4/s, $10^{6}$ m2/s and $0.4\times
10^{6}$ m2/s during other months of the warm period of a year until mid
November. During the winter convection, the coefficients increase like in the
spin up process. The quasi-isopycnal harmonic viscosity is applied only near
the domain boundary in a warm period of a year and in the whole area in
winter.
We simulate the nonlinear mesoscale eddy dynamics over the shelf, continental
slope, and Japan Basin taking into account realistic bottom topography and
daily mean external atmospheric forcing. The near-surface daily atmospheric
conditions have been set from the NCEP/NCAR Reanalysis. It includes short wave
radiation flux, wind stress, wind speed, air temperature and precipitation.
The numerical experiments with minimized coefficients of the horizontal and
vertical viscosity show the intensive mesoscale dynamics, particularly,
mesoscale variability of anticyclonic/cyclonic eddies and streamers over the
shelf and continental slope. The anticyclonic eddies, generated over the shelf
break and continental slope, move usually southwestward along the slope like
the topographic Kelvin waves with prevailing phase velocity of about $6$–$8$
cm/s. The spatial scale of the anticyclonic eddies increases usually near the
Peter the Great Bay shelf where it exceeds significantly the baroclinic Rossby
deformation radius.
The current system and mesoscale dynamics over the continental slope and the
Peter the Great Bay shelf change substantially from summer to winter. The
strong northeastward boundary jet current is formed near the western coast of
the Peter the Great Bay from late October to November when the monsoon is
already changed from the summer type to the winter one. We have simulated
current velocity fields in the surface mixed layer for the warm period with
daily resolution in time. This warm period is associated with strong
baroclinic eddy activity in the Northwestern Japan Sea.
Figure 1 demonstrates the velocity field in the whole basin of the Japan Sea
in the 6th layer of the model (Ponomarev and Trusenkova, 2000; Trusenkova et
al., 2005) with the horizontal resolution $1/16^{\circ}$ and with initial
temperature and salinity distributions estimated from the CTD data of the
oceanographic observations in summer 1999 (Talley et al., 2001). An annual run
of atmospheric conditions corresponds to the end of the 20th century. The
mesoscale eddies are generated due to the baroclinic instability in the main
pycnocline and manifest themselves in the upper layers as well. The velocity
field in Fig. 1 demonstrates clearly the mesoscale eddies along the coast of
the Prymorsky Krai region between $43^{\circ}$N and $46^{\circ}$N.
Figure 2: Satellite image of the water surface temperature in the selected
region of the Japan Sea in the infrared range (NOAA AVHRR data, 15. 09. 1997).
Dark and white colors correspond to low and high temperatures, respectively.
Figure 3: The NOAA AVHRR infrared images of the vortex pairs in the mesoscale
eddy street associated with the Primorskoye Current. The upper panel:
Landsat-5 (TM) data (29. 09. 2007) with the resolution of 120 m. Coordinates:
$43^{\circ}38^{\prime}$N-$45^{\circ}35^{\prime}$N,
$135^{\circ}10^{\prime}$E-$138^{\circ}14^{\prime}$E. The lower panel:
Landsat-7 (ETM+) data (14. 09. 2008) with the resolution of 60 m. Coordinates:
$42^{\circ}10^{\prime}$N-$44^{\circ}10^{\prime}$N,
$133^{\circ}7^{\prime}$E-$136^{\circ}10^{\prime}$E.
## 4 Lagrangian results
### 4.1 Finite-time Lyapunov exponents and repelling material lines
In our analysis we focus on the region between latitudes $41^{\circ}$N and
$44^{\circ}$N and between longitudes $130^{\circ}$E and $136^{\circ}$E
comprising the Primorskoye Current flowing to the southwest along the
continental slope of the Primorsky Krai (Russia). Satellite data demonstrate
anticyclonic mesoscale eddies of relatively small scale which are observed in
this area directly over the steep continental slope. The satellite image of
the surface temperature in the infrared range in the part of this region is
shown in Fig. 2. Dark and white colors in the figure correspond to low and
high temperatures, respectively. The street of the anticyclonic eddies with
the scale of 50 km is visible along the coast of Primorsky Krai over the strip
shelf and steep continental slope of the Japan Basin. Centers of the eddies
are situated directly over the shelf break, 200 m depth. They often form pairs
of strongly interacting eddies visible at sea-surface temperature satellite
images. We demonstrate in Fig. 3 such vortex pairs in the NOAA AVHRR Landsat-5
(TM) and Landsat-7 (ETM+) infrared images in two different years with the
resolution of 120 and 60 m, respectively (http://glovis.usgs.gov/). Each image
shows the pair of two interconnected spirals with a stagnation point between
them. The cores of the eddies are at the distance $25\div 30$ km from the
coast of the Primorsky Krai. The images also show the upwelling phenomena
prevailing near the coast and cold/warm streamers injecting the cold/warm
water from the coastal upwelling/offshore areas into the anticyclonic eddies.
We will study in this section such a vortex pair by the Lagrangian methods
using upper-layer current velocity fields from numerical experiments with the
nonstationary hydrodynamic MHI sea circulation model.
The mesoscale dynamics in the Primorskoye Current over the shelf and steep
continental slope, where the centers of the anticyclonic eddies are situated
directly over the strip shelf break, is associated with the effect of coastal
Kelvin waves propagating southwestward along the shelf break. The integration
of both the MHI model and diagnostic model (Fyman and Ponomarev, 2008), using
observed temperature and salinity profiles from oceanographic R/V surveys,
shows the anticyclonic eddies with the scale of 50 km and the centers over the
shelf break moving downstream along the shelf and continental slope. The
anticyclonic eddies of the similar scale are simulated in the wide shelf of
the Peter the Great Bay. A snapshot of the vorticity field,
$\operatorname{rot}\mathbf{v}$, on one of the days in the first month of
integration of the MHI model (Fig. 4a) demonstrates the complex pattern of
mixing in that region with a number of anticyclonic eddies of different sizes
with negative vorticity.
Figure 4: (a) Snapshot of the vorticity field plotted vs initial particle’s
positions. Color modulates the values of the vorticity
$\operatorname{rot}\mathbf{v}$. (b) Lyapunov synoptic map shows the maximal
FTLE, $\lambda$, vs initial particle’s position. $\lambda$ is in units ${\rm
days}^{-1}$. Integration time is 50 days.
In theory of dynamical systems, the Lyapunov exponents, $\lambda$’s, are known
to be quantitative criteria of chaotic motion in the asymptotic limit. In
practice, one computes Lyapunov exponents for a finite time. The finite-time
Lyapunov exponent (FTLE) is the finite-time average of the maximal separation
rate for a pair of neighbouring advected particles. The FTLE at position
$\mathbf{r}$ at time $\tau$ is given by
$\lambda(\mathbf{r}(t))\equiv\frac{1}{\tau}\ln\sigma(G(t)),$ (4)
where $\tau$ is an integration time, and $\sigma(G(t))$ denotes the largest
singular value of the evolution matrix $G(t)$ which governs evolution of small
displacements in linearized advection equations (see Appendix B for the
derivation of formula (4)).
The FTLE is not an instantaneous separation rate, but rather measures the
integrated separation between trajectories. In real oceanic flows,
instantaneous streamlines can quickly diverge from actual particle’s
trajectories. The FTLEs adequately describe actual transport and mixing in the
ocean because they are derived directly from particle’s trajectories. They are
especially useful in oceanography because they are mathematical analogues of
drifter launching in the ocean and characterize quantitatively dispersion of
water masses. Computing the FTLE field in a selected geographic region, we get
the map that contains information about mixing properties in the region for a
given period of time. Comparing the maps in different seasons, we get an
information about variability in the region. Moreover, the Lyapunov maps
enable to reveal Lagrangian coherent structures hidden in the velocity field
including stable and unstable manifolds of finite-time hyperbolic
trajectories, large-scale transport barriers and eddies. It is interesting
that the FTLE is, in fact, an Eulerian quantity, but in the same time it is a
Lagrangian one because it is derived from particle’s trajectories.
A uniform grid of $1000\times 1000$ particles is advected by the numerically
generated velocity field. After $50$ days (starting on September, 15), the
FTLEs are computed using Eq. (4). Spatial distribution of the FTLEs, plotted
against initial positions in Fig. 4b, may be called a Lyapunov synoptic map.
This map shows that there is a large range of positive $\lambda$ values up to
$0.3$ days-1 which corresponds to Lyapunov mixing times ($e$-folding times)
down to 3 days. We would like to stress that the plot in Fig. 4b shows values
of $\lambda$ against initial particle’s positions accumulated for a rather
long time, 50 days.
The Lyapunov synoptic map in Fig. 4b reveals a number of structures. There are
mesoscale anticyclonic eddies, forming the street along the continental slope
of the Primorsky Krai, which is easily visible on the Lyapunov map and on the
vorticity map (Fig. 4a). The eddy’s cores are characterized by low values of
the Lyapunov exponents. The particles inside the cores tend to stay therein
for a comparatively long time. There are filaments that wind up around the
eddy’s centers in spirals which reveal transport pathways of an ejection of
water.
Moreover, there are very long filaments, ridges of $\lambda$, corresponding to
the largest Lyapunov exponents, which are not associated with any eddies. The
ridges with largest values of $\lambda$, sandwiched between the regions with
smaller $\lambda$’s values, mean that fluid particles placed initially on one
of those ridges will experience in the future strong hyperbolic behavior, i.
e., they will diverge from each other at an exponential rate for the computed
period of time. The ridges reveal stretching directions of the velocity field.
In the language of dynamical systems theory, they approximate stable manifolds
in a selected area.
Figure 5: Evolution of the coherent patches 1 (red) and 2 (green) and the
patch 3 (blue), which is strongly deformed, for 18 days. Spatial scale in the
panel at $t=18$ is different from the other ones. The size of each patch is 6
km along the latitude and 3 km along the longitude with $250\times 250$
particles in each one.
To demonstrate inhomogeneity of mixing in the region under consideration, we
compute the evolution of three fluid patches. The patch 3 was chosen at the
ridge of the Lyapunov map (it is the left rectangular among the three ones
marked in Fig. 4b) with the centroid located initially at
($x_{0}=133^{\circ}54^{\prime}$E and $y_{0}=42^{\circ}34^{\prime}$N), whereas
the patches 1 ($x_{0}=134^{\circ}2^{\prime}$E and
$y_{0}=42^{\circ}26^{\prime}$N) and 2 ($x_{0}=134^{\circ}2^{\prime}$E and
$y_{0}=42^{\circ}36^{\prime}$N) were chosen nearby on the both sides of the
ridge. Figure 5 compares their evolution for 18 days. The patches 1 and 2
remain coherent for this time but their fate is different: the patch 1 travels
to the south-west, whereas the patch 2 hits the coast to the north from its
initial location. Any ridge with largest values of $\lambda$ serves, in fact,
as a transport barrier for waters on both its sides.
The behavior of the patch 3 in Fig. 5 is different. It undergoes strong
stretching and folding to be elongated for 18 days over more than 600 km,
almost two order of magnitude greater than the patches 1 and 2 do. This is
because the patch 3 was initialized at the ridge of the Lyapunov map which
approximately corresponds to a stable manifold. As with any patch placed near
a stable manifold, the particles inside it align along the associated unstable
manifold in course of time. It takes 5-6 days for the patch 3 to reach the
unstable manifold. After that time, the patch undergoes rapid stretching and
folding.
### 4.2 Lagrangian diagnostic tools for revealing eddy’s structure
In this section we focus on Lagrangian study of transport and mixing of
passive particles by the pair of strongly interacting anticyclonic eddies in
the vortex street associated with the Primorskoye current. The satellite
images of the surface temperature (Fig. 3) often demonstrate such pairs in
different years. The snapshots of the vorticity field generated by the Japan
Sea MHI model (Fig. 4a) and the synoptic Lyapunov map (Fig. 4b) give the
evidence of the vortex pairs in the same region. In addition to the FTLE
field, we propose new Lagrangian diagnostic tools for revealing eddy’s
structure and eddy induced transport and mixing. To be concrete we consider
the most prominent pair of eddies in this region marked by the two straight
lines Fig. 4a. They are also easily visible in the Lyapunov synoptic map in
Fig. 4b.
It is worth to recall that the FTLE is an integrated quantity characterizing
the divergency of nearby advected particles for a comparatively long period of
time, 50 days in our case. So, the FTLE synoptic map is a field of this
quantity in geographic coordinates which are initial positions of the passive
particles. Figure 6b is a zoom of the Lyapunov synoptic map in Fig. 4b clearly
demonstrating the complex pattern of mixing of passive particles by the vortex
pair selected. The values of $\lambda$ form a spiral-like structure for each
of the eddies in the vortex pair. The very pair is surrounded by the ridges of
the largest values of the FTLE revealing stretching directions of the velocity
field which are stable manifolds of the hyperbolic trajectories of the vortex
pair. The spiral-like structure, in turn, is the pair of spiral bands, one
with largest values of $\lambda$ and the other one with small FTLE values. The
band with largest FTLE values is a collection of initial particle positions
which leave the corresponding eddy region in course of time, whereas the band
with minimal FTLE values marks those particles that do not quit the eddy for
50 days. Thus, the shadowed spirals in Fig. 6b provide the Lagrangian
information on the transport pathways along which advected particles quit the
corresponding eddy in course of time. Using the velocity fields and the
Lagrangian eddy patterns we can estimate the general speed of the anticyclonic
eddies moving southwestward along the shelf break as equal to $2\div 3$ cm/s,
while the current velocity in the upper mixed layer and on the sea surface is
about $10\div 20$ cm/s. It corresponds to the mean speed of the Primorskoye
Current in the upper layer of $200\div 400$ m in the main pycnocline.
The FTLE map has a fine grained structure with a number of details. To get
more smoothed picture of mixing it is sometimes useful to compute the so-
called exit-time map. We distribute initially $10^{6}$ particles in the box
shown in Fig. 4a and compute how long it takes for a particle with given
initial position to leave the box. The corresponding map, which is more
contrast than the Lyapunov one, is shown in Fig. 6a. It clearly demonstrates
that the northern eddy has a prominent central homogeneous spot (the eddy’s
core) with large values of the exit times. The southern eddy is represented by
the spiral beginning in the eddy’s core.
There is the quantity that may provide a representative picture of eddy-
induced advection, the number of times particles wind around the eddy’s
center. It is difficult to compute that number exactly for a large number of
particles but it may be approximated by the number of times a given particle
changes the sign of its zonal ($n_{x}$) and meridional ($n_{y}$) velocities
unless it reaches one of the borders of the box shown in Fig. 4a. We plot in
Figs. 6c and d geographic maps of those quantities where they are coded by the
color. Both the maps demonstrate clearly the spiral-like structure of the
vortex pair.
To illustrate Lagrangian motion of particles in the vortex pair and compare
that motion in each of the eddies, we track the evolution of material lines
crossing the cores of the eddies. We take two straight material lines in Fig.
4a with $30000$ particles in each one (the same lines are shown in Fig. 6a)
one of which crosses the core of the northern eddy and the other — the core of
the southern eddy. Their evolution for 14 days is shown in Fig. 7. The dotted
fragments in this figure appear because of insufficiently large initial
density of points. The particles in the middle fragments of each line begin to
rotate anticyclonically (the panel at $t=2$), form quickly the vortex pair
(the panel at $t=6$) and move downstream along with the eddy’s cores winding
around their centers. The outer fragments of the lines elongate to the north
and south along the unstable manifolds of the eddy street (the panel at
$t=10$). At $t=14$, the northern eddy catches the southern one up and then
both the eddies move together.
Figure 6: (a) Exit-time map shows how long in days ($T$) it takes for a
particle with a given initial position to quit the box shown in Fig. 4a. (b)
Lyapunov synoptic map of the region shows the values of $\lambda$ vs initial
particle’s position. (c) and (d) Maps of the number of times advected
particles change the sign of their zonal and meridional velocities, $n_{x}$
and $n_{y}$, respectively, unless they quit the box. Figure 7: Evolution of
the two material lines (Fig. 4a) crossing the cores of the northern and
southern eddies in the vortex pair. The red (blue) color refers to the
northern (southern) eddy. Figure 8: Evolution of the same two material lines
as in Fig. 7. The left (blue) and right (red) columns refer to the southern
and northern eddies, respectively. (a) Time of exit off the box $T$ in days vs
initial particle’s latitude position $y_{0}$. (b) Number of times, $n_{y}$,
the particles change the sign of their meridional velocity before they leave
the box vs $y_{0}$. (c) The corresponding maximal finite-time Lyapunov
exponent vs $y_{0}$.
To give a detailed description of the structure of each eddy in the vortex
pair we apply the method of particle’s scattering elaborated in Ref.
(Budyansky et al., 2004a). This method is explained briefly in Appendix A of
the present paper. We take the same lines as shown in Figs. 4a but compute now
dependencies of time of particle’s exit off the selected box, $T$, the number
of times particles change the sign of their meridional velocity, $n_{y}$, and
the maximal FTLE on initial particle’s latitude $y_{0}$ (see Figs. 8a, b and
c, respectively).
The left column in Fig. 8a with the latitudes $y_{0}<43^{\circ}07^{\prime}$N
represents the data for the southern eddy and the right column with the
latitudes $y_{0}>43^{\circ}07^{\prime}$N — the northern eddy. Their comparison
allows to see the difference between the two eddies in the vortex pair. It is
evident from Fig. 8a that particles prefer to quit the southern eddy,
including its core, more or less periodically by portions. Each portion is
represented by a $\cup$-like segment of the $T(y_{0})$ function which consists
of a large number of particles with approximately the same time of exit and
the same number of changes of the sign of their meridional velocity before
leaving the selected box. It is seen in the plot $n_{y}(y_{0})$ in Fig. 8b
that the particles belonging to a given $\cup$-like segment have the same
values of $n_{y}$. In difference from the southern eddy, particles quit the
northern eddy’s core practically at the same time. In other words, the
particles quit the southern eddy by portions along spiral-like transport
pathways, whereas the periphery of the northern eddy exchanges water with the
surrounding but its core moves coherently as a whole for a long time.
Comparison of the left and right plots $n_{y}(y_{0})$ in Fig. 8b gives an
additional information on particle advection by the two eddies. The particles
in the core of the northern eddy change the sign of their meridional velocity
$4\div 8$ times before leaving the selected box (the right column), whereas
the core of the southern eddy is more inhomogeneous (the left column). It is
confirmed by comparing the plots $\lambda(y_{0})$ for both the eddies in Fig.
8c.
The scattering plots enable to identify in the most unambiguous manner the
eddy’s core and its periphery. As an example let us consider the plots for the
northern eddy shown in the right column in Fig. 8. The eddy’s core is
represented in the $T(y_{0})$ plot in Fig. 8a by the smooth segment of the
length $\simeq 14$ km surrounded by the inhomogeneous structures which should
be attributed to the eddy’s periphery. The $\cup$-like segments of the
function $T(y_{0})$ beyond the eddy’s core represent the water masses evolving
more or less coherently and quitting the eddy by portions. This process is
manifested in Fig. 6 as spirals of the northern eddy which are less prominent
than those for the southern one. The motion of particles in the eddy’s
periphery is erratic due to numerous intersections of effective stable and
unstable manifolds.
Figure 8b gives the picture of circulation on submesoscales. We show in
Appendix A by means of the example with a simple vortex-current system that
both the trapping time for particles in the mixing zone and the number of
their full turns around the vortex have a hierarchical fractal structure as
functions of initial particle’s position. Both these functions (see Figs. 10a
and b) are singular on a Cantor set of initial conditions. Due to periodicity
of that idealized flow it became possible to explain in detail transport and
mixing of passive particles.
Numerically generated velocity fields in the ocean are, of course, aperiodic.
Moreover, we deal with moving eddies which are patches of nonzero vorticity.
So, we could not expect the simplified picture of scattering of material lines
by a moving eddy resembling that is shown in Appendix A. Nevertheless, we find
a kind of resemblance of the plots in Figs. 8a and b with the corresponding
plots in Figs. 10a and b. The function $T(y_{0})$ is a graph with smooth
$\cup$-like segments intermitted with badly resolved ones. There is no, of
course, singularities at the ends of smooth segments because of lacking of
periodicity of the velocity field and transient character of effective stable
and unstable manifolds. The plot $n_{y}(y_{0})$ in Fig. 8b is a hierarchy of
epistrophes, but it is organized in a much more complicated way as compared to
the ones in Refs. (Budyansky et al., 2004a) due to the same reasons. There is
no evident fractal structure in both the plots but there is a kind of self-
similarity and strong difference in behavior of neighbour particles whose exit
times $T$ and the number $n_{y}$ may differ by an order of magnitude for a
rather short period of time. Particles in each smooth segment in Fig. 8a have
approximately the same number of changes of the velocity sign. So, each such
segment represents a coherent portion of the whole material line which is
washed away off the selected box simultaneously. The segments in Fig. 8a,
which are not smooth, consist of a number of portions with their own fate and
very different values of $T$ and $n_{y}$.
Figure 9: Snapshots of the streakline obtained by injection of a dye for 45
days. The injection point ($x_{0}=135^{\circ}7^{\prime}$E,
$y_{0}=43^{\circ}23^{\prime}$N) was chosen at the periphery of one of the
eddies present there at the initialization time.
Important oceanographic information can be gained by computing so-called
streaklines in the active regions of the flow. Streakline at the time moment
$t$, passing through a point $(x,y)$, is a curve composed of all the fluid
particles which passed through that point before the moment $t$. Injecting a
dye into a point on the flow plane, one can visualize the corresponding
streakline. Recalling the recent catastrophic oil spill at the bottom in the
Gulf of Mexico, it is evident that tracking the evolution of streaklines in
numerically generated or measured velocity fields may provide useful
information about possible oil transport.
Figure 9 illustrates complex form of the streakline in the region under
consideration. The injection point ($x_{0}=135^{\circ}7^{\prime}$E,
$y_{0}=43^{\circ}23^{\prime}$N) was chosen to be at the periphery of one of
the anticyclonic eddies present there at the start of the injection. That’s
why the injected particles begin to encircle the eddy anticyclonically (see
the panels on 5th and 10th days). The initial portion of the injected
particles moves downstream together with that eddy (the panel at $t=14$). The
dotted fragments of the streakline appear because of insufficiently large
initial density of points. The particles, released during the phase without
eddies near the injection point, begin to move downstream (the panels at
$t=10$ and $t=14$). Moving with the Primorskoye Current, they successfully
catch up the downstream eddies, and the streakline begins to draw up all the
anticyclonic eddies present in the region (see the panels at $t=31$, $t=40$
and $t=45$). The resulting streakline at $t=45$ gives an approximate image of
the effective unstable manifold of the whole region.
## 5 Conclusion
We have demonstrated in this paper that Lagrangian tools and methods of
dynamical systems theory can help to gain new information on surface transport
and mixing on both mesoscales and submesoscales in the ocean. We have focused
on the selected region of the Japan Sea comprising the coastal Primorskoye
Current with a street of anticyclonic mesoscale eddies. Computing Lagrangian
trajectories for a large number of particles advected by the MHI numerical
model, we have studied eddy-induced surface transport and mixing in that
region. We have developed the method to compute the FTLEs for any velocity
field and plotted the Lyapunov synoptic map with a high resolution which can
be used to quantify mixing processes. The Landsat satellite infrared images,
high resolution numerical experiments with the MHI circulation model and
Lagrangian modelling of the mesoscale and submesoscale dynamics in the
Primorskoye Current system have shown strongly interacting mesoscale
anticyclonic eddies generated over the shelf break and steep continental slope
in the northwestern area of the Japan Sea.
The main attention has been paid to Lagrangian study of transport and mixing
by a vortex pair of strongly interacting eddies which often occur in that
region in summer and autumn periods. We proposed new Lagrangian diagnostic
tools, the time of exit of particles off a selected box, the number of changes
of the sign of zonal and meridional velocities, and computed synoptic maps for
these quantities. Along with the Lyapunov map, they have been shown to be able
to reveal mesoscale eddies, meso- and submesoscale filaments, repelling
material lines, hyperbolic and non-hyperbolic regions in the sea. In
particular, we have found that the eddies have a prominent spiral-like
structure resembling the spiral patterns at satellite images in that region.
Based on the theory of chaotic scattering, we developed the technique to track
evolution of clusters of particles, streaklines and material lines and applied
it to study in detail transport and mixing induced by the vortex pair. The so-
called scattering functions, dependencies of the exit times and the number of
times particles wind around the eddy’s center on initial particle positions,
give us important oceanographic information on eddy’s structure and eddy-
induced transport and mixing. In particular, they allow to identify in the
most unambiguous manner the eddy’s cores and periphery and to discover that
the eddies release the water to the surrounding by portions.
Lagrangian approach to transport and mixing problems seems to be perspective
because the results of computation of finite-time and finite-scale Lyapunov
exponents, exit times and winding numbers, tracking the evolution of
streaklines, material lines and patches of fluid particles will be more and
more realistic along with improvement of high-resolution numerical models of
the ocean circulation. The numerical results obtained within that approach can
be readily compared to the results of surface-drifter and subsurface-float
experiments which are rapidly becoming a common experimental technique in
oceanography (Garraffo et al., 2001; Molcard et al., 2003; Özgökmen et al.,
2003; Poje et al., 2002; Molcard et al., 2006). This approach can be used as
well for making predictions about possible oil and other pollutant transport,
biological productivity and other applications. An interesting application of
Lagrangian approach to study marine ecosystem dynamics has been found recently
in Ref. (Tew Kai et al., 2009) where it has been demonstrated that
frigatebirds may trace precisely Lagrangian coherent structures in the
Mozambique Channel which are ridges in the field of the finite-size Lyapunov
exponent.
## Acknowledgments
We would like to thank V. Dubina for providing us the satellite images in Fig.
3 and anonymous reviewers for valuable comments. The work was supported
partially by the Program “Fundamental Problems of Nonlinear Dynamics” of the
Russian Academy of Sciences, by the Russian Foundation for Basic Research
(project no. 09-05-98520) and by the Prezidium of the Far-Eastern Branch of
the RAS.
## Appendix A Geometry of chaotic scattering of particles in a simple vortex
model
It is instructive to demonstrate typical underlying structures that govern
complicated mixing and transport by ocean eddies and introduce some special
geometric concepts with an oversimplifed model of the oceanic flow with a
topographic eddy embedded in a background steady flow with the periodic tidal
component (Budyansky et al., 2004a, b).
Passive particles, advected by a steady flow with the periodic component
directed along the $y$-axis, enter the mixing region where a fixed point eddy
with the singular point at $x=y=0$ is located and then wash out to an outflow
region. Let us put a large number of particles outside the mixing region on a
line segment, crossing the current, with the fixed value of their
$y_{0}$-coordinates and different values of $x_{0\,i}$. We compute the time
$T$, when the particles reach the line $y=6$, and the number of particle’s
rotations $n$ around the point eddy before reaching this line. It is a problem
of chaotic scattering (Ott, 2002).
Figure 10: (a) Fractal dependence of the trapping time $T$ on initial
particle’s positions $x_{0}$ with the inset showing a 20-fold magnification of
one of the singularity zones. (b) Number of particle’s rotations $n$ around
the point eddy before reaching a fixed line. Mechanism of generating the
fractal with magnification of a small segment corresponding to the inset in
the panel (a).
Figure 10 demonstrates the typical scattering function $T(x_{0})$ with an
uncountable number of singularities which are unresolved in principle. The
inset in Fig. 10a shows a zoom by the factor of 20 of one of these singularity
zones. Successive magnifications confirm a self-similarity of the function
with increasing values of the trapping times $T$. To give an insight into a
mechanism of generating the fractal, we plot in Fig. 10b segments of the
initial string with a large number of particles which are trapped in the
mixing region after $n$ rotations around the fixed point eddy. After each
rotation, a portion of particles is washed out in the downstream region with
$y\geq 6$. This process resembles the mechanism of generating the famous
Cantor set but is more complicated. It is well known (see, for example, (Ott,
2002)) that in constructing the middle third Cantor set one takes the closed
interval $[0,\,1]$ and removes open middle third intervals from each of the
intervals remaining at each stage of the process. It is seen from Fig. 10b
that, starting with the given interval of initial points, different portions
of tracers are washed out by the flow from the intervals remaining after each
rotation around the vortex. Continuing in this way ad infinitum, we get a
Cantor set of remaining initial points.
Due to periodicity of that idealized flow it became possible to explain in
detail transport and mixing of passive particles (Budyansky et al., 2004a).
The point is that there exists an invariant chaotic set (saddle set) $\Lambda$
(Ott, 2002; Tel et al., 2005). This set is defined as the set of all
trajectories (except for Kolmogorov–Arnold–Moser tori and cantori) that never
leave the mixing region. It consists of an infinite number of unstable
periodic and aperiodic (chaotic) trajectories. The particles belonging to
$\Lambda$ remain on it forever. However, their measure is zero.
Each trajectory, belonging to $\Lambda$, and therefore the whole set has
stable, $\Lambda_{s}$, and unstable, $\Lambda_{u}$, manifolds which are
infinite material curves. The stable manifold of the chaotic set is defined as
the invariant set of trajectories approaching those in $\Lambda$ as
$t\to\infty$. The unstable manifold $\Lambda_{u}$ is defined as the stable
manifold corresponding to time-reversed dynamics. Following trajectories in
$\Lambda_{s}$, particles, advected by the incoming flow, enter the mixing
region and remain there forever. Particles that are initially close to those
in $\Lambda_{s}$ follow the corresponding trajectories, then deviate from them
and eventually leave the mixing region along the unstable manifold
$\Lambda_{u}$. So, if one chooses a material line in the incoming flow,
crossing the stable manifold $\Lambda_{s}$, it is expected that some advected
particles will stay in the mixing zone forever giving rise to singularities in
the $T(y_{0})$ function. It means that this function has a self-similar
(fractal) structure with smooth $\cup$-like segments with singularities at
their ends intermitted with wildly oscillating fragments with are unresolvable
under magnification in principle.
Particles, which are chosen to be close to $\Lambda_{s}$ in the incoming flow,
enter along $\Lambda_{s}$ into the mixing zone where they remain for a long
time and eventually exit the region along the unstable manifold. The set of
segments with equal number of turns around the vortex, $n$, were called
“epistrophes” in Ref. (Budyansky et al., 2004a) following to Ref. (Mitchell et
al., 2003). These epistrophes were shown to make up a hierarchy (Fig. 10b).
Each epistrophe converges to a limit point on the corresponding material-line
segment. The endpoints of each segment of the $n$th-level epistrophe are the
limit points of an $n+1$th-level epistrophe. The lengths of segments in an
epistrophe decreases in geometric progression. The common ratio of all the
progressions $q$ is related to the maximal Lyapunov exponent for the saddle
points as follows: $\lambda=-\ln q/2\pi$. The hierarchy of epistrophes in Fig.
10b determines transport of particles, and its fractal properties are
generated by the infinite sequences of intersections of the advected material
line with stable and unstable manifolds of the chaotic invariant set
$\Lambda$. The similar fractal-like picture of chaotic transport and mixing
has been found in more realistic models for a periodically meandering jet
current like the Gulf Stream and the Kuroshio (Koshel and Prants, 2006; Prants
et al., 2006; Uleysky et al., 2007).
## Appendix B Finite-time Lyapunov exponents
In this Appendix we derive the formula (4) for the finite-time Lyapunov
exponents (FTLE) used in this paper to compute the Lyapunov synoptic map for
the Japan sea region selected (Fig. 4b). The general problem of particle’s
advection by a flow in an abstract $n$-dimensional space is described by the
$n$-dimensional system of nonlinear ordinary differential equations in the
vector form
$\begin{gathered}\mathbf{\dot{x}}=\mathbf{f}(\mathbf{x},t),\quad\mathbf{x}=(x_{1},\dotsc,x_{n}),\\\
\mathbf{f}(\mathbf{x},t)=(f_{1}(x_{1},\dotsc,x_{n},t),\dots,f_{n}(x_{1},\dotsc,x_{n},t)).\end{gathered}$
(5)
The Lyapunov exponent at an arbitrary point $\mathbf{x_{0}}$ is given by
$\Lambda(\mathbf{x_{0}})=\lim_{t\to\infty}\lim_{\|\delta\mathbf{x}(0)\|\to
0}\frac{\ln(\|\delta\mathbf{x}(t)\|/\|\delta\mathbf{x}(0)\|)}{t},$ (6)
where $\delta\mathbf{x}(t)=\mathbf{x_{1}}(t)-\mathbf{x_{0}}(t)$,
$\mathbf{x_{0}}(t)$ and $\mathbf{x_{1}}(t)$ are solutions of the set (5),
$\mathbf{x_{0}}(0)=\mathbf{x_{0}}$. The limit exists, is the same for almost
all the choices of $\delta\mathbf{x}(0)$ and has a clear geometrical sense:
trajectories of two nearby particles diverge in time exponentially (in
average) with the exponent given by the Lyapunov exponent.
Due to smallness of $\delta\mathbf{x}$ one can linearize the set (5) in a
vicinity of some trajectory $\mathbf{x_{0}}(t)$ and obtain the system of time-
dependent linear equations (Greene and Kim, 1987)
$\begin{pmatrix}\delta\dot{x}_{1}\\\ {1}\\\
\delta\dot{x}_{n}\end{pmatrix}=J(t)\begin{pmatrix}\delta x_{1}\\\ {1}\\\
\delta x_{n}\end{pmatrix},$ (7)
where $J(t)$ is the Jacobian matrix of the system (5) along the trajectory
$\mathbf{x_{0}}(t)$
$J(t)=\begin{pmatrix}\dfrac{\partial f_{1}(\mathbf{x_{0}}(t),t)}{\partial
x_{1}}&\dots&\dfrac{\partial f_{1}(\mathbf{x_{0}}(t),t)}{\partial x_{n}}\\\
{3}\\\ \dfrac{\partial f_{n}(\mathbf{x_{0}}(t),t)}{\partial
x_{1}}&\dots&\dfrac{\partial f_{n}(\mathbf{x_{0}}(t),t)}{\partial
x_{n}}\end{pmatrix}.$ (8)
Solution of the linear system (7) can be found with the help of the evolution
matrix $G(t,t_{0})$
$\begin{pmatrix}\delta x_{1}(t)\\\ {1}\\\ \delta
x_{n}(t)\end{pmatrix}=G(t,t_{0})\begin{pmatrix}\delta x_{1}(t_{0})\\\ {1}\\\
\delta x_{n}(t_{0})\end{pmatrix}.$ (9)
The evolution matrix obeys the differential equation which can be obtained
after substituting (9) into (7)
$\dot{G}=JG,$ (10)
with the initial condition $G(t_{0},t_{0})=I$, where $I$ is the unit matrix.
Any evolution matrix has the important property
$G(t,t_{0})=G(t,t_{1})G(t_{1},t_{0}).$ (11)
One can write the singular-value decomposition of the evolution matrix as
follows:
$G(t,t_{0})=U(t,t_{0})D(t,t_{0})V^{T}(t,t_{0}),$ (12)
where $U$, $V$ are orthogonal and
$D=\operatorname{diag}(\sigma_{1},\dots,\sigma_{n})$ is diagonal. The
quantities $\sigma_{1},\dots,\sigma_{n}$ are called singular values of the
matrix $G$. The Lyapunov exponents are defined via singular values of the
evolution matrix as follows:
$\Lambda_{i}=\lim_{t\to\infty}\frac{\ln\sigma_{i}(t,t_{0})}{t-t_{0}}.$ (13)
Quantities
$\lambda_{i}(t,t_{0})=\frac{\ln\sigma_{i}(t,t_{0})}{t-t_{0}}$
are called finite-time Lyapunov exponents (Okushima, 2003). Thus, the FTLE is
the ratio of the logarithm of the maximal possible stretching in a given
direction to a time interval $t-t_{0}$.
Let us consider now our specific problem of particle’s advection on a plane
with $2\times 2$ evolution matrix and the singular-value decomposition
$G=UDV^{T}\Rightarrow\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\begin{pmatrix}\cos\phi_{2}&-\sin\phi_{2}\\\
\sin\phi_{2}&\cos\phi_{2}\end{pmatrix}\times\\\ \begin{pmatrix}\sigma_{1}&0\\\
0&\sigma_{2}\end{pmatrix}\begin{pmatrix}\cos\phi_{1}&-\sin\phi_{1}\\\
\sin\phi_{1}&\cos\phi_{1}\end{pmatrix}.$ (14)
Solution of these four algebraic equations has the form
$\begin{gathered}\sigma_{1}=\frac{\sqrt{(a+d)^{2}+(c-b)^{2}}+\sqrt{(a-d)^{2}+(b+c)^{2}}}{2},\\\
\sigma_{2}=\frac{\sqrt{(a+d)^{2}+(c-b)^{2}}-\sqrt{(a-d)^{2}+(b+c)^{2}}}{2},\\\
\phi_{1}=\frac{\operatorname{arctan2}{(c-b,\,a+d)}-\operatorname{arctan2}{(c+b,\,a-d)}}{2},\\\
\phi_{2}=\frac{\operatorname{arctan2}{(c-b,\,a+d)}+\operatorname{arctan2}{(c+b,\,a-d})}{2},\end{gathered}$
(15)
where function $\operatorname{arctan2}$ is defined as
$\operatorname{arctan2}{(y,x)}=\left\\{\begin{aligned} &\arctan{(y/x)},&x\geq
0,\\\ &\arctan{(y/x)}+\pi,&x<0.\end{aligned}\right.$ (16)
Equation (10) can not be numerically integrated over a large time because the
elements of the corresponding evolution matrix grow exponentially if one of
the Lyapunov exponents is positive. However, we can divide a large time
interval on subintervals with the duration which is less or order of the
Lyapunov time, $t_{\lambda}=1/\lambda$, and represent the whole evolution
matrix as a product of evolution matrices computed on these subintervals using
the property (11). We compute this product and the corresponding singular
values using the GNU Multiple Precision Arithmetic Library (http://gmplib.org)
in order to preserve the absolute precision of our representation of the
evolution matrix.
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|
arxiv-papers
| 2012-02-02T04:53:38 |
2024-09-04T02:49:26.966257
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.V. Prants, M.V. Budyansky, V.I. Ponomarev, M.Yu. Uleysky",
"submitter": "Michael Uleysky",
"url": "https://arxiv.org/abs/1202.0360"
}
|
1202.0487
|
# Paramagnetic spin correlations in colossal magnetoresistive La0.7Ca0.3MnO3
Joel S. Helton1,∗ Matthew B. Stone2 Dmitry A. Shulyatev3 Yakov M. Mukovskii3
Jeffrey W. Lynn1,† 1NIST Center for Neutron Research, National Institute of
Standards and Technology, Gaithersburg, Maryland 20899, USA 2Quantum
Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
37831, USA 3National University of Science and Technology “MISiS,” Moscow
119991, Russia
###### Abstract
Neutron spectroscopy measurements reveal dynamic spin correlations throughout
the Brillouin zone in the colossal magnetoresistive material La0.7Ca0.3MnO3 at
265 K ($\approx$1.03 $T_{C}$). The long-wavelength behavior is consistent with
spin diffusion, yet an additional and unexpected component of the scattering
is also observed in low-energy constant-$E$ measurements, which takes the form
of ridges of strong quasielastic scattering running along ($H$ 0 0) and
equivalent directions. Well-defined $Q$-space correlations are observed in
constant-$E$ scans at energies up to at least 28 meV, suggesting robust short-
range spin correlations in the paramagnetic phase.
###### pacs:
75.47.Gk, 75.40.Gb, 75.50.Cc, 78.70.Nx
Hole-doped perovskite manganites of the form La1-xCaxMnO3 (LCMO) feature a
ferromagnetic metallic ground state for 0.2 $<$ $x$ $<$ 0.5, with the highest
$T_{C}$ for the combined ferromagnetic and metal-insulator transition at an
optimal doping of $x$ $\approx$ 3/8.Wollan and Koehler (1955); Schiffer et al.
(1995); Cheong and Chen (1998) The colossal magnetoresistance (CMR) observed
at this transitionRamirez (1997) cannot be explained solely by Zener double-
exchange;Millis et al. (1995) rather, the physics underlying the CMR effect in
LCMO likely arises from strong coupling between the spin, charge, lattice, and
orbital degrees of freedomRöder et al. (1996); Varma (1996) and nanoscale
inhomogeneities between competing phases.Dagotto (2003, 2005) For $x$ = 0.3
LCMO, short-range static and dynamic polaron correlations are observed above
$T_{C}$Lynn et al. (2007); Bridges et al. (2010) with an ordering wave vector
of (0.25 0.25 0) signifying CE-typeGoodenough (1955) charge- and orbital-
ordered regions. The spin dynamics in the ferromagnetic phase are likewise
unconventional.Lynn (2000); Zhang et al. (2007) The spin wave dispersion
softens near the zone boundary, which can be fit to a phenomenological model
of first- and fourth-nearest-neighbor ferromagnetic Heisenberg interactions,
and displays anomalous spin wave damping.Ye et al. (2007); Dai et al. (2000)
The spin wave stiffness coefficient renormalizes but does not fully collapse
as $T_{C}$ is approached from below;Lynn et al. (1996) this contrasts with
higher-bandwidth manganite materials such as Pr1-xSrxMnO3 where the stiffness
fully collapses at $T_{C}$ as expected for a second-order ferromagnetic phase
transition.Fernandez-Baca et al. (1998) Above $T$ $\approx$ 200 K a spin
diffusive quasielastic component develops in the low-$q$ spectral weightLynn
et al. (1996); Dai et al. (2001) arising from a short-ranged localization of
electrons on the Mn3+/Mn4+ lattice. This quasielastic component displays a
strong field dependenceLynn et al. (1997) and dominates the spin fluctuation
spectrum near $T_{C}$ with a temperature dependence that closely matches those
of both lattice polarons and the bulk resistivity.Adams et al. (2000) It is
the development of this spin diffusive component, rather than the thermal
population of spin waves, that truncates the ferromagnetic metallic ground
state in a weakly first-order phase transition.Adams et al. (2004)
In this work we report neutron spectroscopy measurements on a single crystal
sample of La0.7Ca0.3MnO3. Spin correlations at 265 K, slightly above $T_{C}$,
are explored primarily through the $Q$ dependence of scattering at constant
energy transfer. The long-wavelength spin dynamics can be described by spin
diffusion with a short, almost temperature-independent correlation length of
$\approx$12 Å.Lynn et al. (1996) For $q$ transfers approaching the Brillouin
zone edge an additional anisotropic scattering component is observed at low
energies, in the form of ridges of strong quasielastic scattering intensity
running along ($H$ 0 0) and symmetry equivalent directions. To the best of our
knowledge, a component of this sort has not been reported in the paramagnetic
phase of any other isotropic ferromagnet. A close connection between the
low-$q$ quasielastic scattering and the colossal magnetoresistance has already
been established,Adams et al. (2000) supporting the possibility that the novel
high-$q$ paramagnetic scattering component presented here might also play a
role in the CMR physics. In the zone boundary ($H$ $K$ 0.5) plane spin
correlations are observed up to energies of at least 28 meV; this scattering
is qualitatively similar to the low-energy correlations in the ($H$ $K$ 0)
plane.
A 1.5 g single crystal sample of La0.7Ca0.3MnO3, previously used for
measurements of polaron correlations,Lynn et al. (2007) was grown by the
floating zone technique.Shulyatev et al. (2002) This is the highest
composition of LCMO for which single crystal samples have been successfully
grown, with $T_{C}$ = 257 K. The crystal structure is orthorhombic perovskite,
but given the presence of multiple crystallographic domains we employ the
cubic notation with $a$ = 3.87 Å. The experiment was carried out at the ARCS
time-of-flight chopper spectrometer at the Spallation Neutron Source, Oak
Ridge National Laboratory with an incident neutron energy of 50 meV at
temperatures of 100 K and 265 K. The 100 K spin wave dispersion in the ($\xi$
0 0) direction can be fit to $\hbar\omega$ =
2$S$[$|J_{1}|$(1-cos(2$\pi\xi$))+$|J_{4}|$(1-cos(4$\pi\xi$))] where the first-
and fourth-nearest neighbor ferromagnetic interactions are given by $S|J_{1}|$
= 6.18 $\pm$ 0.17 meV and $J_{4}/J_{1}$ = 0.19 $\pm$ 0.02, roughly in
agreement with the interaction values previously reportedYe et al. (2007) for
an LCMO sample with $T_{C}$ = 238 K.
Figure 1: (Color online) Intensity plot of $S(Q,\,\omega)$ measured at 265 K,
with the energy integrated between 3 meV $\leq$ $\hbar\omega$ $\leq$ 5 meV.
(a) Scattering in the ($H$ $K$ 0) plane. (b) Scattering in the (0 $K$ $L$)
plane. $Q$ perpendicular to the scattering plane has been integrated over
$\pm$ 0.16 Å-1. The black, red, purple, and orange dashed lines represent the
scan directions displayed in Fig. 2. (c) Integrated intensity of transverse
scans through scattering ridges, with the energy integrated between 2 meV
$\leq$ $\hbar\omega$ $\leq$ 6 meV. The red line is the Mn form factor squared.
Uncertainties throughout this paper are statistical and refer to one standard
deviation.
Figure 1 displays an intensity plot of $S(Q,\omega)$ at $T$ = 265 K in both
the ($H$ $K$ 0) and (0 $K$ $L$) scattering planes with the energy transfer
integrated between 3 meV $\leq$ $\hbar\omega$ $\leq$ 5 meV. This temperature
is in the paramagnetic phase of LCMO, at approximately 1.03 $T_{C}$. The most
prominent features of these data are circular rings of strong scattering
surrounding the Bragg positions. These rings can be attributed to spin
diffusive scatteringLynn et al. (1996) with an increase of the quasielastic
linewidth as $q$ moves away from the zone center.Chatterji et al. (2005);
Daoud-Aladine et al. (2006) For data with energy transfers from 5 meV to at
least 22 meV, the $q$ position corresponding to the maximum intensity of the
ring increases with the energy value of the constant-$E$ scan consistent with
the $\omega$ $\propto$ $q^{2.5}$ expectation of dynamical scaling theory;Lynn
(1984); Endoh and Hirota (1997); Halperin and Hohenberg (1969) in particular
the ring radii, as measured in scans along the ($\xi$ $\xi$ 0) direction, are
consistent with a quasielastic half width at half maximum (HWHM) that varies
with $q$ as $\Gamma(q)$ = $\Lambda q^{2.5}$ where $\Lambda$ = 18.9 $\pm$ 0.5
meV Å2.5. Closer to the zone boundary, the isotropy of the spin dynamics
breaks down and ridges of scattering are present which connect the rings along
($H$ 0 0) and equivalent directions. These ridges of scattering are strongest
in Brillouin zones at low $Q$ and the intensity falls off at higher $Q$ in a
manner roughly consistent with the Mn form factor squared, as displayed in
Fig. 1(c). These data reflect the integrated intensity of transverse scans
through ridges centered at positions equivalent to (0.5 0 0), (1 0.5 0), (1.5
0 0), (1 1 0.5), (1.5 1 0), and (2 0.5 0); all of the equivalent positions
within the range of the instrument were averaged. Given the strong coupling
between the magnetic and lattice degrees of freedom in CMR manganites it is
possible that these ridges of scattering also have a structural component;
with the current statistics we can only note that the $Q$ dependence suggests
that this scattering is primarily magnetic in origin.
Figure 2: (Color online) (a),(b) Differing intensities for scans in the (0
$\xi$ 0) and ($\xi$ $\xi$ 0) directions away from the (0 -1 0) reflection at
265 K. The blue data points and horizontal line correspond to the background
given by a scan in the ($\xi$ $\xi$ 0) direction at 100 K. (c) Scans through
the (0.5 1 0) position in both directions transverse to $\vec{q}$, measured at
265 K in the $\hbar\omega$ $\approx$ 3 meV data. (d) Transverse scan through
(0 0.4 0) in the elastic data, measured at 265 K. The lines in panels (c) and
(d) are Gaussian fits with a HWHM of 0.16 r.l.u. (0.26 Å-1). For panels
(a)-(c) the data have been integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$
directions transverse to the scan. For panel (d) the data have been integrated
over $\pm$ 0.13 Å-1 along $H$ and over $\pm$ 0.08 Å-1 along $K$.
These ridges of additional scattering are further explored in Fig. 2, where
scans in the (0 $\xi$ 0) and ($\xi$ $\xi$ 0) directions away from the (0 -1 0)
position are shown for energies centered at 3 meV [Fig. 2(a)] and 5 meV [Fig.
2(b)]. These scans are roughly isotropic through the ring, but when $q$
$\approx$ 0.55 Å-1 ($\approx$0.34 r.l.u. in the (0 $\xi$ 0) direction) the
data in the two directions diverge with the scattering in the ($\xi$ $\xi$ 0)
direction falling much faster. The width of these ridges of scattering in
directions transverse to $\vec{q}$ is about 0.26 Å-1 HWHM as shown in Fig.
2(c). Elastic or quasielastic scattering centered at $\vec{q}$ = (0.5 0 0) has
been observed in the CMR bilayer manganite La1.2Sr1.8Mn2O7Chatterji et al.
(2006) and the nearly half-doped manganite Pr0.55(Ca0.8Sr0.2)0.45MnO3Ye et al.
(2005) signifying short-range antiferromagnetic correlations. In
La0.7Ca0.3MnO3, the anomalous scattering near the (0.5 0 0) position instead
arises from a breakdown in dynamical scaling theory in which the energy width
of the quasielastic scattering ceases to be isotropic as $q$ approaches the
zone boundary, with smaller quasielastic widths for $\vec{q}$ values along the
($\xi$ 0 0) direction.
Figure 3: (Color online) (a) Energy dependence of the 265 K paramagnetic
scattering in constant-$Q$ scans. $Q$ values have been chosen so that
$\vec{q}$ away from (0 1 0) is along either the ($\xi$ 0 0) or ($\xi$ -$\xi$
0) directions, and with the magnitude of $q$ either 0.325 Å-1 or 0.812 Å-1.
The data have been folded across the $K$ = 0 plane and integrated over $\pm$
0.13 Å-1 in all $\vec{q}$ directions. (b) The energy dependence of the ridge
intensity, displayed as the difference in scattering for measurements at
$\vec{Q}$ = (0 0.6 0) and (0.283 0.717 0) (both with $q$ = 0.649 Å-1). The
gray line is a guide to the eye.
The intensity of this scattering anisotropy is energy-dependent, as shown in
the constant-$Q$ energy scans of Fig. 3(a). For smaller $q$ values where the
data are well described by simple spin diffusion, such as the $q$ = 0.325 Å-1
data shown in the figure, the energy scans do not depend on the orientation of
$\vec{q}$. In the higher-$q$ data, such as the $q$ = 0.812 Å-1 data shown in
the figure, the intensity of the ridge of extra scattering is demonstrated by
the difference in scattering for the $\vec{q}~{}||~{}(\xi~{}0~{}0)$ and
$\vec{q}$ $||$ ($\xi$ -$\xi$ 0) data. Figure 3(b) shows this difference in
scattering intensity for $q$ = 0.649 Å-1; this $\vec{q}$ position is shifted
slightly from the center of the ridge as the zone edge position will feature a
nuclear superlattice reflection in the elastic data. This anomalous scattering
near the zone edge is quasielastic in nature, having a maximum at the elastic
position and an energy HWHM of approximately 2.5 meV. This anisotropy ceases
to be measurable for $\hbar\omega$ $\gtrsim$ 15 meV; this is roughly the
energy transfer where rings of scattering surrounding adjacent Brillouin zone
centers begin to overlap. While this intensity is energy-dependent, the
transverse width of the ridges is not. A transverse scan through the ridge at
(0 0.4 0) is displayed in Fig. 2(d) for elastic scattering data (-1 meV $\leq$
$\hbar\omega$ $\leq$ 1 meV), showing a width (HWHM of 0.26 Å-1) equal to that
observed in the inelastic data.
Figure 4: (Color online) (a) Intensity plot of $S(Q,\,\omega)$ measured at 100
K, with the energy integrated between 24 meV $\leq$ $\hbar\omega$ $\leq$ 27
meV. $Q$ perpendicular to the scattering plane has been integrated over $\pm$
0.16 Å-1. The black and red dashed lines represent the scan directions in the
other panels. (b)-(c) Differing intensities for scans in the (0 $\xi$ 0) and
($\xi$ $\xi$ 0) directions away from the (1 0 0) reflection at 100 K. The data
have been integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$ directions
transverse to the scan.
These spin correlations above $T_{C}$ can be compared to the propagating spin
waves observed below $T_{C}$. Figure 4 displays data measured at 100 K. For
relatively low energies, such as $\hbar\omega$ $\approx$ 9 meV as shown in
Fig. 4(b), the spin waves are isotropic. At 100 K we do not observe any low-
energy anisotropy in $S(\vec{Q},\,\omega)$, in contrast to the anomalous
quasielastic ridges of scattering along the ($H$ 0 0) direction observed in
the paramagnetic phase. This is consistent with previous reportsLynn et al.
(1996) of the low-$q$ quasielastic scattering developing only at temperatures
approaching $T_{C}$. As expected, the widths of these spin waves in a
constant-$E$ scan (a HWHM of about 0.06 Å-1) are determined by the
instrumental resolution and are far narrower than the $Q$-space peaks arising
in the data above $T_{C}$. At higher energies, such as $\hbar\omega$ $\approx$
25.5 meV shown in Figs. 4(a) and 4(c), the peak in $Q$ space approaches the
zone boundary in the ($H$ 0 0) direction and thus the isotropy breaks down.
The constant-$E$ correlations in the ($H$ $K$ 0) plane develop into square
shapes which merge near the zone boundaries, consistent with the calculated
spin wave scattering.
Figure 5: (Color online) Scattering intensities with the energy integrated
between 24 meV $\leq$ $\hbar\omega$ $\leq$ 28 meV and $|L|$ = 0.5, measured at
265 K. (a) Intensity plot of the ($H$ $K$ 0.5) scattering plane. $Q$
perpendicular to the scattering plane has been integrated over $\pm$ 0.16 Å-1.
The black, red, and orange dashed lines represent the scan directions in the
other panels. (b) Differing intensities for scans in the ($\xi$ 0 0) and
($\xi$ $\xi$ 0) directions away from the (1 -1 0.5) reflection. (c) Scan in
the ($\xi$ 0 0) direction through the (1 -0.5 0.5) position. The Gaussian fit
has a HWHM of 0.16 r.l.u. (0.26 Å-1). The data in panels b and c have been
integrated over $\pm$ 0.13 Å-1 in both $\vec{q}$ directions transverse to the
scan.
The presence of persistent spin correlations at 265 K can be further explored
through the spin dynamics in the ($H$ $K$ 0.5) scattering plane. For low
energy transfers, the spin correlations in this plane will present as
scattering centered at integer positions, arising from the ridges of
scattering displayed in Fig. 1. For higher energy transfers, such as 24 meV
$\leq$ $\hbar\omega$ $\leq$ 28 meV for the data shown in Fig. 5, the magnetic
scattering displays a more complex structure. To increase statistics the
measured data have been folded across the $L$ = 0 plane, so that data with $L$
= -0.5 and $L$ = 0.5 have been averaged together. The intensity plot of
correlations in the ($H$ $K$ 0.5) plane, shown as Fig. 5(a), is qualitatively
quite similar to the correlations in the ($H$ $K$ 0) plane at lower energies.
A similar anisotropy between scans along the ($\xi$ 0 0) and ($\xi$ $\xi$ 0)
directions is observed, as shown in Fig. 5(b); these ridges have a transverse
width, shown in Fig. 5(c), consistent with the low-energy data. It should be
noted that the ($H$ $K$ 0.5) scattering plane lies along a Brillouin zone
edge, so that all correlations in this plane are in the high-$q$ regime where
simple spin diffusion theory should not be applicable. In particular, the
correlations yield peaks in $q$ that are far narrower than would be expected
from a purely diffusive model. A breakdown in scaling theory in which the
width of peaks in constant-$E$ scans falls far below the theoretical value at
high $q$ was also observed in paramagnetic iron and nickel.Lynn (1984) These
correlations are also qualitatively similar to the ferromagnetic phase
$Q$-space spin wave correlations displayed in LCMO. The bilayer manganite
La1.2Sr1.8MnO7 was likewise reportedChatterji et al. (2005) to display
$Q$-space spin correlations in the paramagnetic phase that qualitatively
resembled those of ferromagnetic spin waves. Given the dispersion relation
displayed by propagating spin waves in LCMO below $T_{C}$, the $Q$-space
correlations in the ($H$ $K$ 0) plane at an energy of $\omega_{0}$ will have
the same structure as correlations in the ($H$ $K$ 0.5) plane at an energy of
$\omega_{0}$+$\Delta\omega$ where $\Delta\omega$ = $4S|J_{1}|$ (such that
$\Delta\omega$ $\approx$ 25 meV at 100 K). Finding the correlations of Fig. 5
in data where $\hbar\omega$ $\approx$ 26 meV suggests a significant
renormalization of $\Delta\omega$ from the 100 K data; this is reminiscent of
the previously reported renormalization of the spin wave stiffness,Lynn et al.
(1996, 1997) where $D(T_{C})$ $\approx$ $D(T=0)$/2.
It is clear that the spin correlations in LCMO above $T_{C}$ result in well-
defined peaks in constant-$E$ scans that qualitatively resemble the
correlations from the spin wave excitations below $T_{C}$; similar results
have been previously reported in a bilayer manganite.Chatterji et al. (2005)
Despite these well-defined peaks in constant-$E$ scans, no clear peaks at
finite energy are observed in constant-$Q$ scans. For $\vec{q}$ positions away
from the ($H$ 0 0) direction, the positions of these peaks are well described
by spin diffusive dynamical scaling theory. The data at higher-$q$ values
deviate from the expectations of dynamical scaling theory primarily through
peaks in the $Q$-space correlations that are far narrower than the simple spin
diffusive model would predict. An additional component of the paramagnetic
scattering is also observed as ridges of unexpectedly strong quasielastic
scattering at low energy transfers and $\vec{q}$ positions parallel to
$\vec{a}^{\ast}$ or a symmetry equivalent direction. Intrinsic inhomogeneity
with small scale regions of competing phases is a common signature in strongly
correlated electron materials, including stripe order in high-$T_{C}$
superconducting cupratesTranquada et al. (1995) and polar nanoregions in
relaxor ferroelectrics.Blinc et al. (1999) Phase separation of this sort is
well known in CMR manganites, with considerable evidence for lattice
polaronsLynn et al. (2007) and spin polarons;De Teresa et al. (1997) LCMO
samples with smaller doping levels also display evidence of ferromagnetic
droplets.Hennion et al. (1998) The physics of colossal magnetoresistance in
LCMO has been modeled as a percolationMayr et al. (2001) or Griffiths
phaseSalamon et al. (2002) effect arising from the separation of various
competing phases.
The effects of hole doping and applied field on the spin fluctuation spectrum
of LCMO near $T_{C}$ suggest that the small-$q$ spin diffusive portion of the
scattering arises from the short length-scale hopping of electrons on the
Mn3+/Mn4+ lattice;Lynn et al. (1996) this diffusive scattering coexists with
spin waves near $T_{C}$ and drives the ferromagnetic phase transition. It is
also known that the temperature dependence of the low-$q$ quasielastic
scattering is quite similar to the temperature dependences of the bulk
resistivity and the polaron correlations, suggesting a close connection
between paramagnetic scattering and colossal magnetoresistance. The new
high-$q$ quasielastic scattering in the paramagnetic phase described in this
work represents spin correlations with wavelengths approaching atomic length
scales, comparable in size to the small polarons generated by localized
electrons. We hope that further measurements on the short-range spin
correlations near $T_{C}$ will shed new light on the physics of colossal
magnetoresistance.
J.S.H. acknowledges support from the NRC/NIST Postdoctoral Associateship
Program. This research at Oak Ridge National Laboratory’s Spallation Neutron
Source was sponsored by the Scientific User Facilities Division, Office of
Basic Energy Sciences, U. S. Department of Energy.
$\ast$ email: joel.helton@nist.gov
${\dagger}$ email: jeffrey.lynn@nist.gov
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|
arxiv-papers
| 2012-02-02T17:04:21 |
2024-09-04T02:49:26.980268
|
{
"license": "Public Domain",
"authors": "Joel S. Helton, Matthew B. Stone, Dmitry A. Shulyatev, Yakov M.\n Mukovskii, and Jeffrey W. Lynn",
"submitter": "Joel Helton",
"url": "https://arxiv.org/abs/1202.0487"
}
|
1202.0501
|
# Global modeling of transcriptional responses in interaction networks
Leo Lahti 1111to whom correspondence should be addressed, Juha E.A. Knuuttila
2 and Samuel Kaski 1,∗
1Aalto University School of Science and Technology, Helsinki Institute for
Information Technology HIIT and Adaptive Informatics Research Centre,
Department of Information and Computer Science, P.O. Box 15400, FI-00076
Aalto, Finland. 2Neuroscience Center, University of Helsinki, P.O. Box 54,
FI-00014, Finland
###### Abstract
Motivation: Cell-biological processes are regulated through a complex network
of interactions between genes and their products. The processes, their
activating conditions, and the associated transcriptional responses are often
unknown. Organism-wide modeling of network activation can reveal unique and
shared mechanisms between physiological conditions, and potentially as yet
unknown processes. Results: We introduce a novel approach for organism-wide
discovery and analysis of transcriptional responses in interaction networks.
The method searches for local, connected regions in a network that exhibit
coordinated transcriptional response in a subset of conditions. Known
interactions between genes are used to limit the search space and to guide the
analysis. Validation on a human pathway network reveals physiologically
coherent responses, functional relatedness between physiological conditions,
and coordinated, context-specific regulation of the genes. Availability:
Implementation is freely available in R and Matlab at
http://netpro.r-forge.r-project.org/ Contact: leo.lahti@iki.fi,
samuel.kaski@tkk.fi
NOTE: The final version of this manuscript has been published in
Bioinformatics 26(21):2713-2720, 2010.
## 1 Introduction
Coordinated activation and inactivation of genes through molecular
interactions determines cell function. Changes in cell-biological conditions
induce changes in the expression levels of co-regulated genes in order to
produce specific physiological responses. A huge body of information
concerning cell-biological processes is available in public repositories,
including gene ontologies [2], pathway models [44], regulatory information
[29], and protein interactions [20]. Less is known about the contexts in which
these processes are activated [39], and how individual processes are reflected
in gene expression [34]. Although gene expression measurements provide only an
indirect view to physiological processes, their wide availability provides a
unique resource for investigating gene co-regulation on a genome- and
organism-wide scale. This allows the detection of transcriptional responses
that are shared by multiple conditions, suggesting shared physiological
mechanism with potential biomedical implications, as demonstrated by the
Connectivity map [25] where a number of chemical perturbations on a cancer
cell line were used to reveal shared transcriptional responses between
disparate conditions to enhance screening of therapeutic targets.
Figure 1: Organism-wide analysis of transcriptional responses in a human
pathway interaction network reveals physiologically coherent activation
patterns and condition-specific regulation. One of the subnetworks and its
condition-specific responses, as detected by the NetResponse algorithm is
shown. The expression of each gene is visualized with respect to its mean
level of expression across all samples.
Transcriptional responses have been modeled using so-called gene expression
signatures [16]. A signature describes a co-expression state of the genes,
associated with particular conditions. Well-characterized signatures have
proven to be accurate biomarkers in clinical trials, and hence reliable
indicators of cell’s physiological state. Disease-associated signatures are
often coherent across tissues [7] or platforms [16]. Commercial signatures are
available for routine clinical practice [37], and other applications have been
suggested recently [7]. The established signatures are typically designed to
provide optimal classification performance between two particular conditions.
The problem with the classification-based signatures is that their
associations to the underlying physiological processes are not well understood
[30]. Our goal is to enhance the understanding by deriving transcriptional
signatures that are explicitly connected to well-characterized processes
through the network.
We introduce and validate a novel approach for organism-wide discovery and
analysis of transcriptional response patterns in interaction networks. Our
algorithm has been designed to detect and model local regions in a network,
each of which exhibits similar transcriptional response in a subset of
conditions. The algorithm is independent of predefined classifications for
genes or conditions. This extends the previous network-based approaches that
detect differentially expressed subnetworks between two predefined conditions
[17, 43]. Organism-wide analysis can reveal unique and shared mechanisms
between disparate conditions [24], and potentially as yet unknown processes
[35]. The proposed NetResponse algorithm provides an efficient model-based
tool for simultaneous feature selection and class discovery that utilizes
known interactions between genes to guide the analysis. Related approaches
include cMonkey [40] and a modified version of SAMBA biclustering [51]. These
are application-oriented tools that rely on additional, organism-specific
information, and their implementation is currently not available for most
organisms, including human. We provide a general-purpose algorithm whose
applicability is not limited to particular organisms.
NetResponse makes it possible to perform data-driven identification of
functionally coherent network components and their condition-specific
responses. This is useful since the commonly used alternatives, predefined
gene sets or pathways, are collections of intertwined processes rather than
coherent functional entities [35]. This has complicated their use in gene
expression analysis, and methods have consequently been suggested for
identifying the ’key condition-responsive genes’ of predefined gene sets [27],
or for decomposing predefined pathways into smaller functional modules
represented by gene expression signatures [4]. Our network-based search
procedure detects the coordinately regulated gene sets in a data-driven
manner. Gene expression provides functional information of the network that is
missing in purely graph-oriented approaches for studying cell-biological
networks [1]. The network brings in prior information of gene function and
connects the responses more closely to known processes. This would be missing
from purely gene expression-based methods such as biclustering [32], subspace
clustering, or other feature selection approaches [26, 42]. A key difference
to previous network-based clustering methods, including MATISSE [53] and
related approaches [13, 47] is that they assume a single correlated reponse
between all genes in a module. NetResponse additionally models condition-
specific responses of the network. This allows a more expressive definition of
a functional module, or a signature.
We validate the algorithm by modeling condition-specific transcriptional
responses in a human pathway interaction network across an organism-wide
collection of physiological conditions. The results highlight functional
relatedness between tissues, providing a global view on cell-biological
network activation patterns.
## 2 Methods
### 2.1 The NetResponse Algorithm
We introduce a new approach for global detection and characterization of
transcriptional responses in genome-scale interaction networks. NetResponse
searches for local, connected subnetworks where joint modeling of gene
expression reveals coordinated transcriptional response in particular
conditions (Fig. 1). More generally, it is a new algorithm for simultaneous
feature selection (for genes) and class discovery (for conditions) that
utilizes known interactions between genes to limit the search space and to
guide the analysis.
#### Gene expression signatures.
Subnetworks are the functional units of the interaction network in our model;
transcriptional responses are described in terms of subnetwork activation.
Given a physiological state, the underlying assumption is that gene expression
in subnetwork $n$ is regulated at particular levels to ensure proper
functioning of the relevant processes. This can involve simultaneous
activation and repression of the genes: sufficient amounts of mRNA for key
proteins has to be available while interfering genes may need to be silenced.
This regulation is reflected in a unique expression signature
$\boldsymbol{s}^{(n)}$, a vector describing the associated expression levels
of the subnetwork genes. The level of regulation varies from gene to gene;
expression of some genes is regulated at precise levels whereas other genes
fluctuate more freely. Given the physiological state, we assume that the
distribution of observed gene expression is Gaussian,
$\boldsymbol{x}^{(n)}\sim N(\boldsymbol{s}^{(n)},\Sigma^{(n)})$.
#### Modeling condition-specific transcriptional responses.
Each subnetwork is potentially associated with alternative transcriptional
states, activated in different conditions and corresponding to unique
combinations of processes. Since individual processes and their
transcriptional responses are in general unknown [27], detection of condition-
specific responses provides an efficient proxy for identifying functionally
distinct states of the network. Our task is to detect and characterize these
signatures. We assume that in a specific observation (measurement condition),
the subnetwork $n$ can be in any one of $R^{(n)}$ latent physiological states
indexed by $r$. Each state is associated with a unique expression signature
$\boldsymbol{s}_{r}^{(n)}$ over the subnetwork genes. Associations between the
observations and the underlying physiological states are unknown, and treated
as latent variables. This leads to a mixture model for gene expression in the
subnetwork $n$:
$\boldsymbol{x}^{(n)}\sim\sum_{r=1}^{R^{(n)}}w_{r}^{(n)}p(\boldsymbol{x}^{(n)}|\boldsymbol{s}_{r}^{(n)},\Sigma_{r}^{(n)}),$
(1)
where each component distribution $p$ is assumed to be Gaussian. In practice,
we assume a diagonal covariance matrix $\Sigma_{r}^{(n)}$.
A particular transcriptional response is characterized by the triple
$\\{\boldsymbol{s}_{r}^{(n)},\Sigma_{r}^{(n)},w_{r}^{(n)}\\}$. This defines
the shape, fluctuations, and frequency of the associated gene expression
signature in subnetwork $n$. The feasibility of the Gaussian modeling
assumption is supported by the previous observations of [22], where predefined
gene sets were used to investigate differences in gene expression between two
predefined sample groups. In our model, the subnetworks, transcriptional
responses and the activating conditions are learned from data. In one-channel
data such as Affymetrix arrays used in this study, the centroids
$\boldsymbol{s}_{r}^{(n)}$ describe absolute expression signals of the
preprocessed array data. Relative differences can be investigated by comparing
the detected responses. The model is applicable also on two-channel expression
data when a common reference sample is used for all arrays since the relative
differences are not altered by the choice of comparison baseline when the same
baseline is used for all samples.
Now the model has been specified assuming the subnetworks are given. In
practice they are learned from the data. In order to do this we make two
assumptions. First, we rely on the prior information in the global interaction
network, and assume that co-regulated gene groups are connected components in
this network. Second, we assume that the subnetworks are independent. This
allows a well-defined algorithm, and the subnetworks are then interpretable as
independent components of transcriptional regulation. In practice the
algorithm, described below, is an agglomerative approximation for searching
for locally independent subnetworks.
Figure 2: The agglomerative subnetwork detection procedure. Initially, each
gene is assigned in its own singleton subnetwork. Agglomeration proceeds by at
each step merging the two neighboring subnetworks that benefit most from joint
modeling of their transcriptional responses. This continues until no
improvement is obtained by merging the subnetworks.
### 2.2 Implementation
Efficient implementation is crucial for scalability. For fast computation, we
use an agglomerative procedure where interacting genes are gradually merged
into larger subnetworks (Fig. 2). Joint modeling of dependent genes reveals
coordinated responses and improves the likelihood of the data when compared to
independent models, giving the first criterion for merging the subnetworks.
However, increasing subnetwork size tends to increase model complexity and the
possibility of overfitting since the number of samples remains constant while
the dimensionality (subnetwork size) increases. To compensate for this effect,
we use a Bayesian information criterion [8] to penalize increasing model
complexity and to determine optimal subnetwork size.
The cost function for a subnetwork $G$ is $C(G)=-2L+qLog(N)$, where $L$ is the
(marginal) log-likelihood of the data, given the mixture model in Eq. 1, $q$
is the the number of parameters, and $N$ denotes sample size. NetResponse
searches for a joint model for the network genes that maximizes the likelihood
of observed gene expression, but avoids increasing model complexity through
penalizing an increasing number of model parameters. An optimal model is
searched for by at each step merging the subnetwork pair that produces the
maximal gain in the cost function. More formally, the algorithm merges at each
step the subnetwork pair $G_{i},G_{j}$ that minimizes the cost
$\Delta\mathcal{C}=-2(L_{i,j}-(L_{i}+L_{j}))+(q_{i,j}-(q_{i}+q_{j}))Log(N)$.
The agglomerative scheme is as follows:
Initialize: Learn univariate Gaussian mixture for the expression values of
each gene, and bivariate joint models for all potential gene pairs with a
direct link. Assign each gene into its own singleton subnetwork.
Merge: Merge the neighboring subnetworks $G_{i}$, $G_{j}$ that have a direct
link in the network and minimize the difference $\mathcal{C}$. Compute new
joint models between the newly merged subnetwork and its neighbors.
Terminate: Continue merging until no improvement is obtained by merging the
subnetworks ($\Delta\mathcal{C}\geq 0$).
The number $R^{(n)}$ of distinct transcriptional responses of the subnetwork
is unknown, and is estimated with an infinite mixture model. Learning several
multivariate Gaussian mixtures between the neighboring subnetworks at each
step is a computationally demanding task, in particular when the number of
mixture components is unknown. The Gaussian mixtures, including the number of
mixture components, are learned with an efficient variational Dirichlet
process implementation [23]. The likelihood $L$ in the model is approximated
by the lower bound of the variational approximation. The Gaussian mixture
detects a particular type of dependency between the genes. In contrast to
MATISSE [53] and other studies that use correlation or other methods to
measure global co-variation, the mixture model detects coordinated responses
that can be activated only in a few conditions. Condition-specific joint
regulation indicates functional dependency between the genes but it may have a
minor contribution to the overall correlation between gene expression
profiles. In principle, we could also model the dependencies in gene
fluctuations within each individual response with covariances of the Gaussian
components. However, this would heavily increase model complexity, and
therefore we leave dependencies in gene-specific fluctuations within each
response unmodeled, and focus on modeling differences between the responses.
NetResponse provides a full generative model for gene expression, where each
subnetwork is described with an independent joint mixture model. The maximum
subnetwork size is limited to 20 genes to avoid numerical instabilities in
computation. The infinite Gaussian mixture can automatically adapt model
complexity to the sample size. We model subnetworks of 1-20 genes across 353
samples; similar dimensionality per sample size has previously been used with
variational mixture models [15].
### 2.3 Data
#### Pathway interaction network.
We investigate the pathway interaction network based on the KEGG database of
metabolic pathways [19] provided by the SPIA package [52] of BioConductor
(www.bioconductor.org). This implements the pathway impact analysis method
originally proposed in [6], which is to our knowledge currently the only
freely available pathway analysis tool that considers pathway topology. SPIA
provides the data in a readily suitable form for our analysis. Other pathway
data sets, commonly provided in the BioPAX format, are not readily available
in a suitable pairwise interaction form. Directionality and types of the
interactions were not considered. Genes with no expression measurements were
removed from the analysis. We investigate the largest connected component of
the network with 1800 unique genes, identified by Entrez GeneIDs.
#### Gene expression data.
We analyzed a collection of normal human tissue samples from ten post-mortem
donors [41], containing gene expression measurements from 65 normal
physiological conditions. To ensure sample quality, RNA degradation was
minimized in the original study by flash freezing all samples within 8.5 h
postmortem. Only the samples passing Affymetrix quality measures were
included. Each condition has 3-9 biological replicates measured on the
Affymetrix HG-U133plus2.0 platform. The reproducibility of our findings is
investigated in an independent human gene expression atlas [49], measured on
the Affymetrix HG-U133A platform, where two biological replicates are
available for each measured condition. In the comparisons we use the 25
conditions available in both data sets (adrenal gland cortex, amygdala, bone
marrow, cerebellum, dorsal root ganglia, hypothalamus, liver, lung, lymph
nodes, occipital lobe, ovary, parietal lobe, pituitary gland, prostate gland,
salivary gland, skeletal muscle, spinal cord, subthalamic nucleus, temporal
lobe, testes, thalamus, thyroid gland, tonsil, trachea, and trigeminal
ganglia). Both data sets were preprocessed with RMA [18]. Certain genes have
multiple probesets, and a standard approach to summarize information across
multiple probesets is to use alternative probeset definitions based on probe-
genome remapping [5]. This would provide a single expression measure for each
gene. However, since the HG-U133A array represents a subset of probesets on
the HG-U133Plus2.0 array, the redefined probesets are not technically
identical between the compared data sets. To minimize technical bias in the
comparisons, we use probesets that are available on both platforms. Therefore,
we rely on manufacturer annotations of the probesets and use an alternative
approach (used e.g. by [38]), where one of the available probesets is selected
at random to represent each unique gene. Random selection is used to avoid
selection bias. When available, the ’xxxxxx_at’ probesets were used because
they are more specific by design than the other probe set types
(www.affymetrix.com).
### 2.4 Validation
The NetResponse algorithm is validated with an application on the pathway
interaction network of 1800 genes [52] across 353 gene expression samples from
65 physiological conditions in normal human body [41]. NetResponse is compared
to alternative approaches in terms of physiological coherence and
reproducibility of the findings.
#### Comparison methods.
NetResponse is designed for organism-wide modeling of transcriptional
responses in genome-scale interaction networks. Simultaneous detection of the
subnetworks and their condition-specific responses is a key feature of the
model. A straightforward alternative would be a two-step approach where the
subnetworks and their condition-specific responses are detected in separate
steps, although this can be unoptimal for detecting condition-specific
responses. Various methods are available for detecting subnetworks based on
network and gene expression data [13, 47] in the two-step approach. We use
MATISSE, a state-of-the-art algorithm described in [53]. MATISSE finds
connected subgraphs in the network such that each subgraph consists of highly
correlated genes. The output is a list of genes for each detected subnetwork.
Since MATISSE only clusters the genes, we model transcriptional responses of
the detected subnetworks in a separate step by using a similar mixture model
to the NetResponse algorithm. This combination is also new, and called
MATISSE+ in this paper. The second comparison method is the SAMBA biclustering
algorithm [50]. The output is a list of associated genes and conditions for
each identified bicluster. SAMBA detects gene sets with condition-specific
responses but, unlike NetResponse and MATISSE+, the algorithm does not utilize
the network. Influence of the prior network is additionally investigated by
randomly shuffling the gene expression vectors, while keeping the network and
the within-gene associations intact. Comparisons between the original and
shuffled data help to assess relative influence of the prior network on the
results. Comparisons to randomly shuffled genes in SAMBA are not included
since SAMBA does not use the network.
#### Reproducibility in validation data.
Reproducibility of the findings is investigated in an independent validation
data set in terms of significance and correlation (for details, see Section
2.3). Each comparison method implies a grouping for the physiological
conditions in each subnetwork, corresponding to the detected responses. It is
expected that physiologically relevant differences between the groups are
reproducible in other data sets. We tested this by estimating differential
expression between the corresponding conditions in the validation data for
each pairwise comparison of the predicted groups using a standard test for
gene set analysis (GlobalTest; [10]). To ensure that the responses are also
qualitatively similar in the validation data, we measured Pearson correlation
between the detected responses and those observed in the corresponding
conditions in validation data. The responses were characterized by the
centroids provided by the model in NetResponse and MATISSE+. For SAMBA we used
the mean expression level of each gene within each group of conditions since
SAMBA groups the conditions but does not characterize the responses. In
validation data, the mean expression level of each gene is used to
characterize the response within each group of conditions. Probesets were
available for 75% of the genes in the detected subnetworks in the validation
data; transcriptional responses with less than three probesets in the
validation data were not considered. Validation data contained corresponding
samples for $>79\%$ of the predicted responses in NetResponse, MATISSE+, and
SAMBA (Supplementary Table 1).
## 3 Results
The validation results reported below demonstrate that the NetResponse
algorithm is readily applicable for modeling transcriptional responses in
interaction networks on an organism-wide scale. While biomedical implications
of the findings require further investigation, NetResponse detects a number of
physiologically coherent and reproducible transcriptional responses in the
network, and highlights functional relatedness between physiological
conditions. It also outperformed the comparison methods in terms of
reproducibility of the findings.
### 3.1 Application to human pathway network
In total, NetResponse identified 106 subnetworks with 3-20 genes
(Supplementary data file). For each subnetwork, typically (median) 3 distinct
transcriptional responses were detected across the 65 physiological conditions
(Supplementary Fig. 1). One of the subnetworks with four distinct responses is
illustrated in Fig. 1. Each respose is associated with a subset of conditions.
Statistically significant differences between the corresponding conditions
were observed also in the independent validation data ($p<0.01$; GlobalTest).
Three of the four responses were also qualitatively similar (correlation
$>0.8$; Supplementary Fig. 2). The first response is associated with immune-
system related conditions such as spleen and tonsil. Responses 2-3 are
associated with neuronal conditions such as subthalamic or nodose nucleus, or
with central nervous system, for example accumbens and cerebellum. The fourth
group manifests a ’baseline’ signature that fluctuates around the mean
expression level of the genes. Testis and pituitary gland are examples of
conditions in this group. While most physiological conditions are strongly
associated with a particular response, samples from amygdala, bone marrow,
cerebral cortex, heart atrium, and temporal lobe manifested multiple
responses. In general, it is not well known how individual pathways are
manifested at gene expression level. While alternative responses reveal
condition-specific regulation, detection of physiologically coherent and
reproducible responses may indicate shared mechanisms between physiological
conditions. Although the responses may reflect previously unknown processes,
it is likely that some of them reflect the activation patterns of known
pathways. Overlapping pathways can provide a starting point for
interpretation. The subnetwork of Fig. 1 overlaps with various known pathways,
most remarkably with the MAPK pathway with 10 genes (detailed gene-pathway
associations are provided in the Supplementary data file; see subnetwork 12).
MAPK is a general signal transduction system that participates in a complex,
cross-regulated signaling network that is sensitive to cellular stimuli [54].
Association of MAPK to cell growth and proliferation could potentially explain
the differences between neuronal and other conditions. Six subnetwork genes
participate in the p53 pathway, which is a known regulator of the MAPK
signaling pathway. In addition, p53 is known to interact with a number of
other pathways, both as an upstream regulator, and a downstream target [55].
Both MAPK and p53 are associated with processes including cell growth,
differentiation, and apoptosis, and exhibit diverse cellular responses to
varying conditions. Condition-specific regulation can potentially explain the
detection of alternative transcriptional states of the subnetwork.
The detected responses characterize absolute expression signals in our
preprocessed one-channel array data. Systematic differences in the expression
levels of the individual genes are normalized out in the visualization by
showing the relative expression of each gene with respect to its mean
expression level across all samples. Note that the choice of a common baseline
does not affect the relative differences between the samples.
Figure 3: Associations between 65 physiological conditions (rows) and the
detected transcriptional responses of the pathway interaction network of Fig.
1. The shade indicates the probability of a particular transcriptional
response in each condition (black: $P=0$; white: $P=1$). Hierarchical
clustering based on the signature co-occurrence probabilities between each
pair of physiological conditions highlights their relatedness.
#### Condition-selective network activation.
Associations between the physiological conditions and the detected
transcriptional responses are shown in Fig. 3. Some responses are shared by
many conditions, while others are more specific to particular contexts such as
immune system, muscle, or the brain. Related physiological conditions often
exhibit similar network activation patterns, which is seen by grouping the
conditions according to co-occurrence probabilities of shared transcriptional
response. This is known as tissue-selectivity of gene expression [28].
#### Probabilistic tissue connectome.
Relatedness of physiological conditions can be measured in terms of shared
transcriptional responses (Supplementary Fig. 3). This is an alternative
formulation of the tissue connectome map suggested by [12] to highlight
functional connectivity between tissues based on the number of shared
differentially expressed genes at different thresholds. We use shared network
responses instead of shared gene count. The use of co-regulated gene groups is
expected to be more robust to noise than the use of individual genes. As the
overall measure of connectivity between physiological conditions, we use the
mean of signature co-occurrence probabilities over the subnetworks, given the
model in Eq. 1. The analysis reveals functional relatedness between the
conditions. In particular, two subcategories of the central nervous system
appear distinct from the other conditions. Closer investigation of the
observed responses would reveal how the conditions are related at
transcriptional level (Supplementary data file).
Figure 4: Comparison between the alternative approaches. Detected responses:
Fraction of genes participating in the detected transcriptional responses.
Reproducibility (significance): Fraction of responses that are reproducible in
the validation data in terms of differential expression between the associated
conditions ($p<0.05$; GlobalTest). Reproducibility (correlation): Median
correlation between the gene expression levels of the detected responses and
the corresponding conditions in the validation data.
### 3.2 Comparison to alternative approaches
NetResponse was compared to the alternative approaches in terms of
physiological coherence and reproducibility of the findings (Fig. 4;
Supplementary Table 1). NetResponse detected the largest amount of responses;
68% of the network genes were associated with a response, compared to 45% in
MATISSE+ and SAMBA. At the same time, NetResponse outperformed the comparison
methods in terms of reproducibility of the findings.
#### Physiological coherence.
The association between the responses and physiological conditions was
measured by normalized mutual information (NMI; [3]) between the sample-
response assignments and sample class labels within each subnetwork. The NMI
varies from 0 (no association) to 1 (deterministic association). The
transcriptional responses detected by NetResponse, MATISSE+, and SAMBA show
statistically significant associations to particular physiological conditions
with a significantly higher average NMI (0.46-0.50) than expected based on
randomly labeled data (0.26-0.32; $p<10^{-4}$; Wilcoxon test; Supplementary
Table 1). The highest average NMI (0.50) was obtained by NetResponse but
differences between NetResponse, MATISSE+, and SAMBA are not significant.
NetResponse is significantly physiologically more coherent also when compared
to results obtained with shuffled gene expression (NMI 0.22; $p<10^{-12}$).
The observations confirm the potential physiological relevance of the findings
in NetResponse, MATISSE+, and SAMBA.
#### Reproducibility.
The majority of the detected responses were reproducible both in terms of
significance and correlation (Supplementary Fig. 4) as described in Section
2.4. Of the predicted differences between groups of physiological conditions,
80% were significant in validation data with $p<0.05$ (GlobalTest), compared
to 72% and 63% in MATISSE+ and SAMBA, respectively, or 43% obtained for
randomly shuffled data with NetResponse (Fig. 4). The changes were also
qualitatively similar; in NetResponse the median correlation between the
detected responses and corresponding conditions in the validation data is
0.76, which is significantly higher ($p<0.01$; Wilcoxon test) than in the
comparison methods (MATISSE+: 0.64; SAMBA: 0.68), or in randomly shuffled
NetResponse data (0.14). NetResponse detected responses for a larger fraction
of the genes (68%) than the other methods. This seems an intrinsic property of
the algorithm since it detected responses for a similar fraction of the genes
also in the network with randomly shuffled genes (72%). However, only the
findings from the real data were reproducible.
## Discussion
Cell-biological networks may cover thousands of genes, but any change in the
physiological context typically affects only a small part of the network.
While gene function and interactions are often subject to condition-specific
regulation [28], they are typically studied only in particular experimental
conditions. Organism-wide analysis could reveal highly specialized functions
that are activated only in one or a few conditions. Detection of shared
responses between the conditions can reveal previously unknown functional
connections and help to formulate novel hypotheses of gene function in
previously unexplored contexts. We provide a well-defined algorithm for such
analysis.
The results support the validity of the model. NetResponse detected the
largest number of responses without compromising physiological coherence or
reproducibility of the findings compared to the alternatives. The most highly
reproducible results were obtained by NetResponse. Further analysis is needed
to establish the physiological role of the findings.
NetResponse is readily applicable for modeling condition-specific responses in
cell-biological networks, including pathways, protein interactions, and
regulatory networks. The network connects the responses to well-characterized
processes, and provides readily interpretable results that are less biased
towards known biological phenomena than methods based on predefined gene sets
that are routinely used in gene expression studies to bring in prior
information of gene function and to increase statistical power. However, these
are often collections of intertwined processes rather than coherent functional
entities. For example, pathways from KEGG may contain hundreds of genes while
only a small part of a pathway may be affected by changes in physiological
conditions [35]. This has complicated the use of predefined gene sets in gene
expression studies. [6] demonstrated that taking into account aspects of
pathway topology, such as gene and interaction types can improve the
estimation of pathway activity. While their SPIA algorithm measures the
activity of known pathways between two predefined conditions, our algorithm
searches for potentially unknown functional modules, and detects their
association to multiple conditions simultaneously. This is useful since
biomedical pathways are human-made descriptions of cellular processes, often
consisting of smaller, partially independent modules [4, 14]. Our data-driven
search procedure can rigorously identify functionally coherent network modules
where the interacting genes show coordinated responses. Joint modeling
increases statistical power which is useful since gene expression, and many
interaction data types such as protein-protein interactions, have high noise
levels. The probabilistic formulation accounts for biological and measurement
noise in a principled manner. Certain types of interaction data such as
transcription factor binding or protein interactions are directly based on
measurements. This can potentially help to discover as yet unknown processes
that are not described in the pathway databases [35]. False negative
interactions form a limitation for the current model because joint responses
of co-regulated genes can be modeled only when they form a connected
subnetwork.
The need for principled methods for analyzing large-scale collections of gene
expression data is increasing with their availability. Versatile gene
expression atlases contain valuable information about shared and unique
mechanisms between disparate conditions which is not available in smaller and
more specific experiments [24, 45]. For example, [25] demonstrated that large-
scale screening of cell lines under diverse conditions can enhance the finding
of therapeutic targets. Our model is directly applicable in similar
exploratory tasks, providing tools for organism-wide analysis of
transcriptional activity in normal human tissues [41, 49], cancer, and other
diseases [21, 31] in a genome- and organism-wide scale. Similar collections
are available for several model organisms including mouse [49], yeast [11],
and plants [46]. A key advantage of our approach compared to methods that
perform targeted comparisons between predefined conditions [17, 43] is that it
allows systematic organism-wide investigation when the responses and the
associated conditions are unknown. The motivation is similar to SAMBA and
other biclustering approaches that detect groups of genes that show
coordinated respose in a subset of conditions [32], but the network ties the
findings more tightly to cell-biological processes in our model. This can
focus the analysis and improve interpretability. Since the nonparametric
mixture model adjusts model complexity with sample size, our algorithm is
potentially applicable also in smaller and more targeted data sets. For
example, it could potentially advance disease subtype discovery by revealing
differential network activation in subsets of patients.
Many large-scale collections are continuously updated with new measurements.
Our algorithm provides no integration technique for new experiments yet; on-
line extensions that could directly integrate data from new experiments
provide an interesting topic for further study. Another potential extension
would be a fully-Bayesian treatment that would provide confidence intervals,
removing the need to assess significance of the results in a separate step.
While our model provides a model-based criterion for detecting the responses
without prior knowledge of the activating conditions, the statistical
significance of the findings has to be verified in further experiments. The
majority of the responses in our experiments could be verified in an
independent data set. Other potential extensions include adding more structure
to address the directionality, relevance and probabilities of the
interactions. Not all cell-biological processes have clear manifestations at
transcriptome level. Hence information of transcript and interaction types, as
in SPIA, could potentially help to improve the sensitivity of our approach. We
could also seek to loosen the constraints imposed by the prior network.
However, such extensions would come with an increased computational cost. The
simple and efficient implementation is a key advantage.
NetResponse is closely related to subspace clustering methods such as
agglomerative independent variable component analysis (AIVGA; [15]). However,
AIVGA and other model-based feature selection techniques [26, 42] consider all
potential connections between the features, which leads to a more limited
scalability. Finding a global optimum in our model would require exhaustive
combinatorial search over all potential subnetworks. Since the complexity
depends on the topology of the network, finding a general formulation for the
model complexity is problematic. The number of potential solutions grows
faster than exponentially with the number of features (genes) and links
between them, making exhaustive search in genome-scale interaction networks
infeasible. Approximative solutions are needed, and are often sufficient in
practice. A combination of techniques is used to achieve an efficient
algorithm compared to the model complexity. First, we focus the analysis on
those parts of the data that are supported by known interactions. This
increases modeling power and considerably limits the search space. Second, the
agglomerative scheme finds an approximative solution where at each step the
subnetwork pair that leads to the highest improvement in cost function is
merged. This finds a solution relatively fast compared to the complexity of
the task. Note that the order in which the subnetworks become merged may
affect the solution. Finally, the variational implementation considerably
speeds up mixture modeling [23]. The running time of our application was 248
min on a standard desktop computer (Intel 2.83GHz; Supplementary Fig. 5).
Investigation of a human pathway interaction network revealed condition-
specific regulation in the network, that is, groups of interacting genes whose
joint response differs between physiological conditions. This highlights the
condition-dependent nature of network activation, and emphasizes an important
shortcoming in the current gene set-based testing methods [36]: simply
measuring gene set ’activation’ is often not sufficient; it is also crucial to
characterize how the expression changes, and in which conditions. Organism-
wide modeling can provide quantitative information about these connections.
## Conclusions
We have introduced and validated a general-purpose algorithm for global
identification and characterization of transcriptional responses in genome-
scale interaction networks across diverse physiological conditions. An
organism-wide analysis of a human pathway interaction network validates the
model, and provides a global view on cell-biological network activation. The
results reveal shared and unique mechanisms between physiological conditions,
and potentially help to formulate novel hypotheses of gene function in
previously unexplored contexts.
## Funding
This work was supported by the Academy of Finland [207467] and the IST
Programme of the European Community, under the PASCAL 2 Network of Excellence
[ICT-216886]. LL and SK belong to the Finnish CoE on Adaptive Informatics
Research Centre of Academy of Finland, and to the Helsinki Institute for
Information Technology HIIT.
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## Supplementary Figures
A |
---|---
B |
C |
Figure 5: Supplementary Fig 1. Histograms of model statistics: A Number of
transcriptional responses in the subnetworks detected by NetResponse. B
Subnetwork size. C Number of physiolocial conditions associated with each
response.
Figure 6: Supplementary Fig 2 Reproducibility of transcriptional responses of
the subnetwork of Figure 1 (main text) in independent validation data.
Correlation: Qualitatively similar responses are observed in the validation
data (Pearson correlation $>0.8$), except for the fourth response (correlation
-0.27). Differential expression with respect to the mean level of each gene is
used in the comparisons. This removes overoptimistic bias in the correlations
caused by the systematic differences in the expression levels of the genes.
Significance: Each response is associated with a subset of conditions. The
differences between the corresponding conditions are statistically significant
($p<0.01$; GlobalTest) in the validation data for each pairwise comparison
between the predicted four groups of conditions. The investigated gene
expression atlas (Roth et al., 2006) and the validation data (Su et al., 2004)
have been measured on different array platforms (HG-U133Plus2 and HG-U133A,
respectively). Gene expression levels are here shown for the 17 (out of 20)
probesets that are available on both platforms.
Figure 7: Supplementary Fig 3 Tissue connectome based on the detected transcriptional responses of the human pathway interaction network. For each pair of tissues the overall probability of shared transcriptional response across the network is shown (black: $P=0$; white: $P=1$; see main text for details). This gives a probabilistic measure of tissue similarity based on network activation. The rows and columns are ordered with hierarchical clustering to highlight the relatedness between physiological conditions. A |
---|---
B |
Figure 8: Supplementary Fig 4 Reproducibility of the detected transcriptional
responses in the independent Su et al., 2004 validation data in terms of
significance and correlation. A Significance of differential expression
between each pair of associated conditions for predicted responses in the
validation data. 80% of the predicted differences between the conditions were
verified in the validation data with $p<0.05$ (GlobalTest). We tested only the
responses where corresponding conditions were available in the validation data
(81% of the responses). B Correlation between the detected responses in the
investigated data set and the corresponding conditions in the validation data.
Differential expression with respect to the mean level of each gene was used
in the comparisons. This removes the potential bias in the correlations caused
by the systematic differences in the expression levels of the genes.
Figure 9: Supplementary Fig 5 Running time for data sets of different sizes on the pathway network described in the main text. The running time for the GSE3526 data set investigated in the main text was 248 minutes (i.e. 4.1 hours). Computation time increases superlinearly with sample size from 33 minutes with 20 samples to 64 hours with 1977 samples. Model fitting in the algorithm can be parallelized, which will make the model scalable to larger data sets in standard multi-core desktop computers. The running time depends also on the size and connectivity of the network. Our investigated network represents a standard pathway network used in current organism-wide studies. The network has a median of 5 and a maximum of 105 direct interaction partners per gene. This reduces the search space considerably compared to models that would consider all potential interactions between the 1800 network genes. To investigate time consumption we have selected random subsets of various sizes (20, 50, 100, 200, and 353 samples) from the GSE3526 data, described in the main text and having 353 arrays in total. The data sets with 500 and more (1000, 1500, 1973) samples were obtained by picking random subsets from the GSE2109 data set, which has 1973 arrays in total (downloaded 30.5.2008 from http://www.ncbi.nlm.nih.gov/geo/). Both data sets were preprocessed as described in Section 2.3 in the main text. | NetRespose | NetResponse | MATISSE+ | MATISSE+ | SAMBA
---|---|---|---|---|---
| | (shuffled) | | (shuffled) |
Reproducibility (corr.) | 0.76 | 0.14 | 0.64 | 0.60 | 0.68
Reproducibility (signif.) | 0.80 | 0.43 | 0.72 | 0.71 | 0.63
Fraction of responses | 0.81 | 0.34 | 0.79 | 0.80 | 0.89
with validation data | | | | |
Physiol. coh. (NMI) | 0.50 | 0.22 | 0.49 | 0.41 | 0.46
Physiol. coh. (signif.) | $<10^{-4}$ | $0.65$ | $<10^{-4}$ | $<10^{-2}$ | $<10^{-4}$
Fraction of data | 0.68 | 0.72 | 0.45 | 0.40 | 0.45
assigned to subnetworks | | | | |
Table 1: Supplementary Table 1 Comparison statistics. Reproducibility
(correlation): Median correlation between the detected responses and the
corresponding conditions in the validation data. Reproducibility
(significance): Fraction of transcriptional responses that were reproducible
in the validation data (GlobalTest $p<0.05$). The results are shown for the
responses where corresponding conditions in the validation data were
available. Significance of differential expression was calculated for each
pairwise comparison between the associated conditions of the predicted
responses in the validation data. Transcriptional responses with validation
data: Fraction of transcriptional responses for which corresponding samples
were available for testing in the validation data. Physiological coherence
(NMI): Normalized mutual information between the detected transcriptional
responses and sample labels (physiological conditions). A higher NMI indicates
stronger association between the detected responses and physiological
conditions. The differences between NetResponse, MATISSE+, and SAMBA are not
significant. Physiological coherence (significance): Significance of the
physiological coherence (NMI) compared to the expectation based on randomly
labeled samples (Wilcoxon test p-value). Fraction of data assigned to
subnetworks: Fraction of genes participating in the detected responses.
|
arxiv-papers
| 2012-02-02T17:40:14 |
2024-09-04T02:49:26.988307
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Leo Lahti, Juha E. A. Knuuttila, Samuel Kaski",
"submitter": "Leo Lahti",
"url": "https://arxiv.org/abs/1202.0501"
}
|
1202.0667
|
# Additive colorings of planar graphs
Jarosław Grytczuk
Faculty of Mathematics and Information Science,
Warsaw University of Technology, 00-661 Warszawa, Poland
e-mail: grytczuk@mini.pw.edu.pl Tomasz Bartnicki, Sebastian Czerwiński
Faculty of Mathematics, Computer Science, and Econometrics,
University of Zielona Góra, 65-516 Zielona Góra, Poland
e-mail: t.bartnicki@wmie.uz.zgora.pl, s.czerwinski@wmie.uz.zgora.pl Bartłomiej
Bosek, Grzegorz Matecki, Wiktor Żelazny
Faculty of Mathematics and Computer Science,
Jagiellonian University, 30-348 Kraków, Poland
e-mail: bosek@tcs.uj.edu.pl, matecki@tcs.uj.edu.pl, zelazny@tcs.uj.edu.pl
###### Abstract
An _additive coloring_ of a graph $G$ is an assignment of positive integers
$\\{1,2,\ldots,k\\}$ to the vertices of $G$ such that for every two adjacent
vertices the sums of numbers assigned to their neighbors are different. The
minimum number $k$ for which there exists an additive coloring of $G$ is
denoted by $\eta(G)$. We prove that $\eta(G)\leqslant 468$ for every planar
graph $G$. This improves a previous bound $\eta(G)\leqslant 5544$ due to
Norin. The proof uses Combinatorial Nullstellensatz and coloring number of
planar hypergrahs. We also demonstrate that $\eta(G)\leqslant 36$ for
$3$-colorable planar graphs, and $\eta(G)\leqslant 4$ for every planar graph
of girth at least $13$. In a group theoretic version of the problem we show
that for each $r\geqslant 2$ there is an $r$-chromatic graph $G_{r}$ with no
additive coloring by elements of any Abelian group of order $r$.
## 1 Introduction
Let $G$ be a simple graph, and let $k$ be a positive integer. By a _coloring_
of $G$ we mean any function $f$ from the set of vertices $V(G)$ to the set
$\\{1,2,\ldots,k\\}$. Given a coloring $f$, consider the induced function
$S=S(f)$ on the set $V(G)$ defined by the formula
$S(v)=\sum_{x\in N(v)}f(x),$
where $N(v)$ denotes the set of neighbors of the vertex $v$ in $G$. The
initial coloring $f$ is called an _additive coloring_ of $G$ if $S(u)\neq
S(v)$ for every pair of adjacent vertices $u$ and $v$. The minimum number $k$
for which there exists an additive coloring of $G$ is denoted by $\eta(G)$.
The notion of additive coloring was introduced in [4] as a vertex version of
the 1-2-3-conjecture of Karoński, Łuczak, and Thomason [7]. In the original
problem the numbers are assigned to the edges of a graph, and prospective
color of a vertex $v$ is derived as the sum of numbers assigned to the edges
incident to $v$. It is conjectured that for every connected graph (except
$K_{2}$) one can produce a proper vertex coloring in this way using only three
numbers— 1, 2, and 3. Currently best bound is $5$, as proved by Kalkowski,
Karoński, and Pfender [6].
In the related additive coloring problem no finite bound is possible since for
cliques we have $\eta(K_{n})=n$. We conjecture however, that perhaps
$\eta(G)\leqslant\chi(G)$ for every graph $G$, where $\chi(G)$ denotes the
usual chromatic number. This conjecture is widely open as it is not known
whether $\eta(G)$ is bounded for bipartite graphs. In [4] we proved that
$\eta(G)\leqslant 3$ for planar bipartite graphs, and also that
$\eta(G)\leqslant 100280245065$ for general planar graphs. The later bound was
improved to $5544$ by Norin (personal communication). We present this proof in
section 2 for completeness.
In this note we obtain a further improvement of this bound. Our main result
asserts that $\eta(G)\leqslant 468$ for every planar graph $G$. The proof uses
Combinatorial Nullstellensatz of Alon [1], and the coloring number of
hyperhraphs represented by planar bipartite graphs. For planar graphs of girth
at least $13$ we get a much better bound by $4$, using a decomposition theorem
from [3].
## 2 Coloring number of graphs and hypergraphs
We start with presenting an unpublished result of Norin. Recall that the
coloring number $\mathop{\mathrm{c}ol}(G)$ of a graph $G$ is the least integer
$k$ such that there exists a linear ordering of the vertices
$v_{1},\ldots,v_{n}$ such that the number of _backward_ neighbors of $v_{i}$
(those contained in the set $\\{v_{1},\ldots,v_{i-1}\\}$) is at most $k-1$,
for every $i=1,2,\ldots,n$. It is well known that
$\mathop{\mathrm{c}ol}(G)\leqslant 6$ for every planar graph $G$.
###### Theorem 1
_(S. Norin)_ Let $G$ be a graph with chromatic number $\chi(G)=r$ and coloring
number $\mathop{\mathrm{c}ol}(G)=k$. Let $n_{1},\ldots,n_{r}$ be $r$ pairwise
coprime integers, with $n_{i}\geqslant k$ for all $i=1,2,\ldots,k$. Then
$\eta(G)\leqslant n_{1}\times\ldots\times n_{r}$. In particular,
$\eta(G)\leqslant 5544$ for every planar graph $G$ (by taking $n_{1}=7$,
$n_{2}=8$, $n_{3}=9$, and $n_{4}=11$).
Proof. Fix a proper coloring $c$ of a graph $G$ using colors
$\\{1,2,\ldots,r\\}$. Also, fix a linear ordering of the vertices realizing
$\mathop{\mathrm{c}ol}(G)=k$. Let $n_{1},\ldots,n_{r}$ be any positive
integers such that $\gcd(n_{i},n_{j})=1$ for every pair $i\neq j$, with
$n_{i}\geqslant k$ for all $i=1,2,\ldots,r$. Suppose now that each vertex $v$
is assigned with certain weight $n(v)\in\mathbb{Z}_{n_{j}}$, with $j=c(v)$.
Denote by $S_{i}(v)$ the sum of weights of all the neighbors of $v$ in color
$i$. More formally,
$S_{i}(v)=\mathop{\displaystyle\sum}\limits_{x\in N(v)\cap c^{-1}(i)}n(x),$
where summation is in the group $\mathbb{Z}_{n_{i}}$. Finally, let
$S(v)=(S_{1}(v),\ldots,S_{r}(v))$.
Since no neighbor of $v$ is colored with $c(v)$, we have $S_{j}(v)=0$ for
$j=c(v)$. Our aim is to modify weights $n(v)$ greedily so that
$S_{c(v)}(u)\neq 0$ for every backward neighbor $u$ of $v$. This will imply
that $S(u)\neq S(v)$ for every pair of adjacent vertices $u$ and $v$.
Suppose we have achieved this property for all vertices up to $v_{i-1}$ by
choosing appropriate weights $n(v_{1}),\ldots,n(v_{i-1})$. Now we have to find
a weight for the vertex $v_{i}$. Let $j=c(v_{i})$. For every backward neighbor
$u$ of $v_{i}$ there is only one value of $n(v_{i})$ making
$S_{j}(u)=0(\mathop{\mathrm{m}od}n_{j})$. Since there are at most $k-1$
backward neighbors of $v_{i}$, there are only $k-1$ forbidden values for
$n(v_{i})$. Since $n_{j}>k-1$, there is a free element of $\mathbb{Z}_{n_{j}}$
for the weight $n(v_{i})$.
To get an additive coloring of graph $G$ we assign to every vertex $v$, an
element $f(v)=(f_{1}(v),\ldots,f_{r}(v))$ of the group
$\mathbb{Z}_{n_{1}}\times\ldots\times\mathbb{Z}_{n_{r}}$, defined by
$f_{j}(v)=n(v)$ if $j=c(v)$, and $f_{j}(v)=0$, otherwise. This completes the
proof, as the group $\mathbb{Z}_{n_{1}}\times\ldots\times\mathbb{Z}_{n_{r}}$
is isomorphic to $\mathbb{Z}_{N}$, where $N=n_{1}\times\ldots\times n_{r}$.
The notion of coloring number can be generalized in a natural way for
hypergraphs. Given a hypergraph $H$ and a linear ordering of the vertices
$v_{1},\ldots,v_{n}$, define the _backward degree_ of vertex $v_{i}$ as the
number of _different_ hyperedges of the form $\\{v_{j}\\}\cup A$, with
$A\subseteq\\{v_{1},\ldots,v_{i-1}\\}$ (we allow $A$ to be empty). The
_coloring number_ $\mathop{\mathrm{c}ol}(H)$ of hypergraph $H$ is the minimum
$k$ such that in some linear ordering of the vertices all backward degrees are
at most $k-1$. This definition differs slightly from the one given in [8], but
it is appropriate for our purposes.
###### Lemma 2
Let $H$ be a hypergraph with $\mathop{\mathrm{c}ol}(H)=k$. Then there is a
function $f:V(H)\rightarrow\mathbb{Z}_{k}$ such that every hyperedge $B$
satisfies
$\mathop{\displaystyle\sum}\limits_{v\in B}f(v)\neq
0(\mathop{\mathrm{m}od}k).$
Proof. Start with a linear ordering of the vertices realizing
$\mathop{\mathrm{c}ol}(H)$ and proceed greedily in that order. At each step
there are at most $k-1$ partial sums we have to account, and each of them is
reset by exactly one value. Hence, there is always a good choice for the next
value of $f$.
Now we give an upper bound for the coloring number of hypergraphs arising from
bipartite planar graphs.
###### Lemma 3
Let $G$ be a bipartite planar graph with bipartition classes $X$ and $Y$. Let
$H$ be a hypergraph on the set of vertices $X$ whose incidence graph is $G$.
Then $\mathop{\mathrm{c}ol}(H)\leqslant 12$. In particular, there exists a
coloring $f:X\rightarrow$ $\mathbb{Z}_{12}$ satisfying condition:
$\mathop{\displaystyle\sum}\limits_{x\in N(y)}f(x)\neq
0(\mathop{\mathrm{m}od}12)$
for every non-isolated vertex $y\in Y$.
Proof. We may assume that no two vertices in $Y$ are _twins_ (have exactly the
same nonempty neighborhood), as multiple hyperedges do not count in backward
degree. We shall prove that hypergraph $H$ always contains a vertex of the
usual degree at most $11$. This is sufficient since a hypergraph $H-x$ still
does not contain multiple hyperedges, (therefore the incidence graph of $H-x$
does not contain twins) and we may order the vertices of $H$ by sequential
deletion of such vertices.
Fix an embedding of $G$ in the plane. Transform this embedding into a new
plane graph $P$ in the following way. For every vertex $y\in Y$, draw a simple
closed curve $C(y)$ through the neighbors of $y$ within $\varepsilon$-distance
from the connecting edges, so that a simply connected region $F(y)$ arises
with the following properties:
1. 1.
All neighbors of $y$ belong to $C(y)$.
2. 2.
All other points of the edges connecting $y$ to its neighbors (and $y$ itself)
are in the interior of $F(y)$.
3. 3.
No other points of the embedding of $G$ are in $F(y)$.
Forget now about $y$’s and their edges inside regions $F(y)$. In this way we
get a plane (pseudo)graph $P$ on the set of vertices $X$ whose faces can be
properly $2$-colored: color the faces $F(y)$ by black and all other faces by
white. Notice that hypergedes of $H$ turned into black faces in $P$. Hence,
$\deg_{H}(v)$ is just the number of black faces incident to $v$.
We claim that there is always a vertex in $P$ incident to at most $11$ black
faces. First, shrink all loops and all $2$-sided faces of $P$ to get a new
pseudograph $Q$ whose faces have at least three vertices. Let $v$, $e$, and
$f$ denote the number of vertices, edges, and faces in $Q$, respectively. So,
we have $3f\leqslant 2e$, and by Euler’s formula we get $e\leqslant 3v-6$.
Hence, there must be a vertex $x$ of degree at most $5$ in $Q$. Now, by the
lack of twins in $G$, each edge incident to $x$ in $Q$ has multiplicity at
most $4$ in $P$. Also, there can be at most one loop at each vertex in $P$, by
the same reason. Therefore degree of $x$ in $P$ is at most $22$, and there are
at most $11$ black faces incident to $x$. The proof of the lemma is complete.
It is worth noticing that the above lemma is tight. To see this take the
icosahedron on the vertex set $X$ and modify it in the following way: (1)
subdivide each edge and each face of the icosahedron with one new vertex, (2)
append a hanging edge to each vertex from $X$. The resulting graph is a twin-
free planar bipartite graph in which every vertex in $X$ has degree $11$.
## 3 Combinatorial Nullstellensatz
For the proof of our main result we will need a simple consequence of the
celebrated Combinatorial Nullstellensatz of Alon. For the sake of completeness
we provide also an elegant, simple proof due to Michałek [9].
###### Theorem 4
_(Combinatorial Nullstellensatz)_ Let $\mathbb{F}$ be an arbitrary field, and
let $P(x_{1},\ldots,x_{n})$ be a polynomial in the ring of polynomials
$\mathbb{F}[x_{1},\ldots,x_{n}]$. Suppose that there is a nonvanishing
monomial $x_{1}^{k_{1}}\ldots x_{n}^{k_{n}}$ in $P$ such that
$k_{1}+\ldots+k_{n}=\deg(P)$. Then for every subsets $A_{1},\ldots,A_{n}$ of
the field $\mathbb{F}$, with $\left|A_{i}\right|\geqslant k_{i}+1$, there are
elements $a_{i}\in A_{i}$ such that $P(a_{1},\ldots,a_{n})\neq 0$.
Proof. We will proceed by induction on the degree of polynomial $P$. If
$\deg(P)=0$, then $P$ is a nonzero constant polynomial and the assertion holds
trivially. Let $\deg(P)\geqslant 1$ and suppose the theorem is true for all
polynomials of strictly smaller degree. Hence, for at least one
$i\in\\{1,\ldots,n\\}$ we must have $k_{i}\geqslant 1$. Assume, for
simplicity, that $k_{1}\geqslant 1$, and let $a\in A_{1}$ be a fixed element.
Using the algorithm of long division of polynomials, we may write
$P=(x_{1}-a)Q+R.$ (*)
Indeed, we may treat $P$ as a polynomial in one variable $x_{1}$ with
coefficients in the ring $\mathbb{F}[x_{2},\ldots,x_{n}]$ and perform long
division by the polynomial $(x_{1}-a)$ to determine uniquely quotient $Q$ and
remainder $R$. Since $\deg(x_{1}-a)=1$, the remainder $R$ must be a constant
in $\mathbb{F}[x_{2},\ldots,x_{n}]$, which means that it does not contain
variable $x_{1}$. Hence, by the assumption on the nonvanishing monomial in
$P$, the quotient $Q$ must have a nonvanishing monomial
$x^{k_{1}-1}x_{2}^{k_{2}}\ldots x_{n}^{k_{n}}$ and
$\deg(Q)=(k_{1}-1)+k_{2}+\ldots+k_{n}$.
Suppose on the contrary that $P(x)$ vanishes on the set
$A_{1}\times\ldots\times A_{n}$. Take any element $x\in\\{a\\}\times
A_{2}\times\ldots\times A_{n}$ and substitute to equation $(\ast)$. Since
$P(x)=0$, we get that $R(x)=0$. But $R$ does not contain variable $x_{1}$, so
it follows that $R$ also vanishes on the whole set $A_{1}\times\ldots\times
A_{n}$. Take now any $x\in(A_{1}\setminus\\{a\\})\times
A_{2}\times\cdots\times A_{n}$ and substitute to equation $(\ast)$. Since
$P(x)=0$, $R(x)=0$, and $(x_{1}-a)\neq 0$, it follows that $Q(x)=0$. This
means that $Q$ vanishes on the whole set $(A_{1}\setminus\\{a\\})\times
A_{2}\times\cdots\times A_{n}$, which contradicts the inductive assumption.
The above theorem has many surprising applications in geometry, combinatorics,
and number theory [1]. We used it in [4] to prove that every planar bipartite
graph has an additive coloring from arbitrary lists of size at least three.
Below we give a slight extension of this result, which will be useful later.
###### Theorem 5
Let $G$ be a bipartite graph whose edges can be oriented so that each vertex
has indegree at most $k$. Suppose that each vertex $v$ is assigned with a list
$L(v)$ of $k+1$ real numbers. Then for every function
$q:V(G)\rightarrow\mathbb{R}$ there is a coloring $f$ of the vertices such
that
$q(u)+\mathop{\displaystyle\sum}\limits_{x\in N(u)}f(x)\neq
q(v)+\mathop{\displaystyle\sum}\limits_{x\in N(v)}f(x)$
for every pair of adjacent vertices $u$ and $v$.
Proof. Let $U=\\{u_{1},\ldots,u_{m}\\}$ and $V=\\{v_{1},\ldots,v_{n}\\}$ be
the bipartition classes of a graph $G$. Let $\\{x_{1},\ldots,x_{m}\\}$ and
$\\{y_{1},\ldots,y_{n}\\}$ be the variables assigned to the vertices of these
classes, respectively. Denote by $S(u)$ the sum of variables assigned to the
neighbors of $u$. Consider a polynomial $P$ over the field of reals defined by
$P(x_{1},\ldots,x_{m},y_{1},\ldots,y_{n})=\mathop{\displaystyle\prod}\limits_{u_{i}v_{j}\in
E(G)}(q(u_{i})+S(u_{i})-q(v_{j})-S(v_{j})).$
We claim that $P$ contains a nonvanishing monomial with exponents bounded by
$k$. Let $\overrightarrow{G}$ be an orientation of $G$ with indegrees bounded
by $k$. In every factor of $P$ corresponding to edge $u_{i}v_{j}$ choose one
of the variables $x_{i}$ or $y_{j}$–the one that corresponds to the vertex on
which the arrow points. In this way we obtain monomial $M=x_{1}^{k_{1}}\ldots
x_{m}^{k_{m}}y_{1}^{l_{1}}\ldots y_{n}^{l_{n}}$ satisfying $0\leqslant
k_{i},l_{j}\leqslant k$. Why is this monomial nonvanishing in $P$? It is
because each variable occurs in factors of $P$ with uniform sign ($x_{i}$ with
minus sign, $y_{j}$ with plus sign). Hence, the sign of monomial $M$ in $P$ is
uniquely determined by the sequence of exponents, and therefore its copies
cannot cancel. Finally, to apply Combinatorial Nullstellensatz, notice that
$\deg(P)$ equals the number of edges in $G$, which is the same as
$k_{1}+\ldots+k_{m}+l_{1}+\ldots+l_{n}$ since $q(u_{i})-q(v_{j})$ are
constants.
###### Corollary 6
Every tree has an additive coloring from arbitrary lists of size two. Every
bipartite planar graph has an additive coloring from arbitrary lists of size
three.
Proof. Every tree has an orientation with at most one incoming edge to every
vertex. Every bipartite planar graph has an orientation with indegrees bounded
by two.
## 4 Main results
Let us start with a simpler case of planar $3$-colorable graphs.
###### Theorem 7
Every planar graph $G$ with $\chi(G)\leqslant 3$ satisfies $\eta(G)\leqslant
36$.
Proof. Let $V(G)=A\cup B\cup C$ be a partition of the vertex set of $G$ into
three independent sets. Let $H$ be a subgraph of $G$ on the set of vertices
$V(H)=A\cup B\cup C$ with the edge set
$E(H)=\\{uv\in E(G):u\in A\cup B\text{ and }v\in C\\}.$
Clearly $H$ is a bipartite graph. Hence, by Theorem 3, there is a function
$h:C\rightarrow\\{1,2,\ldots,12\\}$ such that the sum
$S_{h}(u)=\mathop{\displaystyle\sum}\limits_{x\in N_{H}(u)}h(x)$
satisfies $S_{h}(u)\neq 0(\mathop{\mathrm{m}od}12)$ for every vertex $u\in
A\cup B$ having at least one neighbor in $C$. For other vertices the above sum
is empty and we adopt $S_{h}(u)=0$ by convention.
Consider now a bipartite subgraph $F$ of $G$ induced by the vertices $A\cup
B$. Assign to each vertex $u$ in $F$ the list $L(u)=\\{12,24,36\\}$, and apply
Theorem 5 with function $q(u)=S_{h}(u)$. Let $f$ be a coloring satisfying the
assertion of Theorem 5. That is, $f$ satisfies condition
$S_{f}(u)+S_{h}(u)\neq S_{f}(v)+S_{h}(v)$ for every edge $uv\in E(F)$, where
$S_{f}(u)=\mathop{\displaystyle\sum}\limits_{x\in N_{F}(u)}f(x).$
Finally, let $g$ be a function defined on the whole set of vertices $V(G)$ by
joining $f$ and $h$:
$g(x)=\left\\{\begin{array}[]{l}h(x)\text{ if }x\in C\\\ f(x)\text{ if }x\in
A\cup B\end{array}\right..$
We claim that $g$ is an additive coloring of $G$ over the set
$\\{1,2,\ldots,36\\}$. Let $S(u)$ be the sum of $g$-labels over the whole
neighborhood $N(u)$, that is, $S(u)=S_{h}(u)+S_{f}(u)$. Let $uv$ be any edge
in $G$. If $u\in A\cup B$ and $v\in C$, then $S_{h}(u)\neq
0(\mathop{\mathrm{m}od}12)$ and $S_{f}(u)=0(\mathop{\mathrm{m}od}12)$, thus
$S(u)\neq 0(\mathop{\mathrm{m}od}12)$. On the other hand,
$S_{h}(v)=S_{f}(v)=0(\mathop{\mathrm{m}od}12)$, so
$S(v)=0(\mathop{\mathrm{m}od}12)$. In the other case, if $u\in A$ and $v\in
B$, condition $S(u)\neq S(v)$ is guaranteed by construction of $f$. This
completes the proof.
The proof for $4$-colorable planar graphs is similar in spirit, though a bit
more technical.
###### Theorem 8
Every planar graph satisfies $\eta(G)\leqslant 468$.
Proof. Let $V(G)=A\cup B\cup C\cup D$ be a partition of the vertex set of $G$
into four independent sets. Let $H_{1}$ be a subgraph of $G$ on the set of
vertices $(A\cup B)\cup C$ with the edge set
$E(H_{1})=\\{uv\in E(G):u\in A\cup B\text{ and }v\in C\\}.$
Clearly $H_{1}$ is a bipartite graph. Hence, by Theorem 3, there is a function
$h_{1}:C\rightarrow\mathbb{Z}_{12}$ such that the sum
$S_{h_{1}}(u)=\mathop{\displaystyle\sum}\limits_{x\in N_{H_{1}}(u)}h_{1}(x)$
satisfies $S_{h_{1}}(u)\neq 0(\mathop{\mathrm{m}od}12)$ for every vertex $u\in
A\cup B$ with at least one neighbor in $C$. Now, Let $H_{2}$ be a subgraph of
$G$ on the set of vertices $(A\cup B\cup C)\cup D$ with the edge set
$E(H_{2})=\\{uv\in E(G):u\in A\cup B\cup C\text{ and }v\in D.$
Clearly $H_{2}$ is a bipartite graph. Hence, by Theorem 3, there is a function
$h_{2}:D\rightarrow$ $\mathbb{Z}_{13}$ such that the sum
$S_{h_{2}}(u)=\mathop{\displaystyle\sum}\limits_{x\in N_{H_{2}}(u)}h_{2}(x)$
satisfies $S_{h_{2}}(u)\neq 0(\mathop{\mathrm{m}od}13)$ for every vertex
$u\in(A\cup B\cup C)$ having a neighbor in $D$.
Now, using functions $h_{1}$ and $h_{2}$, we define a new function $h:C\cup
D\rightarrow\\{1,2,\ldots,156\\}$ as follows. First we extend $h_{1}$ and
$h_{2}$ to the whole set $C\cup D$ by putting $h_{1}(x)=0$ for $x\in D$ and
$h_{2}(x)=0$ for $x\in C$. Let $\sigma$ be a group isomorphism from
$\mathbb{Z}_{12}\times\mathbb{Z}_{13}$ to $\mathbb{Z}_{156}$. For each $x\in
C\cup D$ define $h(x)$ as the unique number in the range
$\\{1,2,\ldots,156\\}$ satisfying congruence
$h(x)\equiv\sigma((h_{1}(x),h_{2}(x))(\mathop{\mathrm{m}od}156).$
Let
$S_{h}(u)=\mathop{\displaystyle\sum}\limits_{x\in N(u)\cap(C\cup D)}h(x)$
for every $u\in A\cup B$, where, as before, $S_{h}(u)=0$ if $N(u)\cap(C\cup
D)=\emptyset$. First we claim that $S_{h}(u)\neq 0(\mathop{\mathrm{m}od}156)$
for every vertex $u\in A\cup B$ which has at least one neighbor in $C\cup D$.
Indeed, since $\sigma$ is a group isomorphism we may write
$\displaystyle S_{h}(u)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\sum}\limits_{x\in N(u)\cap(C\cup
D)}h(x)=\mathop{\displaystyle\sum}\limits_{x\in N(u)\cap(C\cup
D)}\sigma((h_{1}(x),h_{2}(x))$ $\displaystyle=$
$\displaystyle\sigma\left(\left(\mathop{\displaystyle\sum}\limits_{x\in
N(u)\cap C}h_{1}(x),\mathop{\displaystyle\sum}\limits_{x\in N(u)\cap
D}h_{2}(x)\right)\right)=\sigma((S_{h_{1}}(u),S_{h_{2}}(u))).$
Hence, $S_{h}(u)$ cannot be zero in $\mathbb{Z}_{156}$, since at least one of
the sums $S_{h_{1}}(u)$ or $S_{h_{2}}(u)$ is non-zero in its respective group.
Notice also that $S_{h}(u)\neq 0(\mathop{\mathrm{m}od}156)$ for every vertex
$u\in C$ and having a neighbor in $D$, as in this case we have
$S_{h}(u)=\sigma((0,S_{h_{2}}(u)))$ and $S_{h_{2}}(u)\neq 0$ in
$\mathbb{Z}_{13}$.
Consider now a bipartite subgraph $F$ of $G$ induced by the vertices $A\cup
B$. Assign to each vertex $u$ in $F$ the list $L(u)=\\{156,312,468\\}$, and
apply Theorem 5 with function $q(u)=S_{h}(u)$. Let $f$ be a coloring
satisfying the assertion of Theorem 5. That is, $f$ satisfies condition
$S_{f}(u)+S_{h}(u)\neq S_{f}(v)+S_{h}(v)$ for every edge $uv\in E(F)$, where
$S_{f}(u)=\mathop{\displaystyle\sum}\limits_{x\in N_{F}(u)}f(x).$
Putting things together we define a function $g$ on the whole set of vertices
$V(G)$ by joining $f$ and $h$:
$g(x)=\left\\{\begin{array}[]{l}h(x)\text{ if }x\in C\cup D\\\ f(x)\text{ if
}x\in A\cup B\end{array}\right..$
We claim that $g$ is an additive coloring of $G$ over the set
$\\{1,2,\ldots,468\\}$. Let $S(u)$ be the sum of $g$-labels over the whole
neighborhood $N(u)$, that is, $S(u)=S_{h}(u)+S_{f}(u)$. Let $uv$ be any edge
in $G$. If $u\in A\cup B$ and $v\in C\cup D$, then $S_{h}(u)\neq
0(\mathop{\mathrm{m}od}156)$ while $S_{f}(u)=0(\mathop{\mathrm{m}od}156)$,
thus $S(u)\neq 0(\mathop{\mathrm{m}od}156)$. The other end of the edge
satisfies $S_{h}(v)=S_{f}(v)=0(\mathop{\mathrm{m}od}156)$, so
$S(v)=0(\mathop{\mathrm{m}od}156)$. If $u\in A$ and $v\in B$, condition
$S(u)\neq S(v)$ is guaranteed by construction of $f$. We are left with the
last case $u\in C$ and $v\in D$. Suppose on the contrary that $S(u)=S(v)$.
Since $S_{f}(u)=S_{f}(v)=0(\mathop{\mathrm{m}od}156)$, we get
$S_{h}(u)=S_{h}(v)$ in $\mathbb{Z}_{156}$. But
$S_{h}(u)=\sigma((0,S_{h_{2}}(u)))$ and $S_{h_{2}}(u)\neq 0$ in
$\mathbb{Z}_{13}$, while $S_{h}(v)=\sigma((S_{h_{1}}(v),0))$. This
contradiction completes the proof.
A set of vertices $I$ in a graph $G$ is called _two-independent_ if the
distance between any two vertices of $I$ is at least three. In [3] it was
proved that every planar graph of girth at least $13$ has a vertex
decomposition into two sets $I$ and $F$ such that $I$ is two-independent and
$F$ induces a forest. Our last theorem follows easily from this result.
###### Theorem 9
Every planar graph of girth at least $13$ satisfies $\eta(G)\leqslant 4$.
Proof. Let $V(G)=I\cup F$, where $I$ is $2$-independent and $F$ induces a
forest. By Corollary 6 there is an additive coloring $f$ of the forest $F$
using labels $\\{2,4\\}$. Extend this coloring to the whole graph $G$ by
putting $f(i)=1$ for each vertex $i\in I$. It is easy to see that $f$ is an
additive coloring of $G$.
## 5 Finite abelian groups
The problem of additive coloring can be considered in a more general setting
of Abelian (additive) groups. We may use elements of any such group $\Gamma$
as the labels of vertices and define the additive coloring the same way as
before. Accordingly to our main conjecture, as well as to the methods we
develop so far, one could expect that perhaps every graph has an additive
coloring over some group whose order is equal to the chromatic number of the
graph. We prove below that this is not true.
###### Theorem 10
For every $r\geqslant 2$ there is a graph $G_{r}$ such that $\chi(G_{r})=r$,
and there is no additive coloring of $G_{r}$ over any finite Abelian group of
order $r$. But there is an additive coloring of $G_{r}$ in $\mathbb{Z}_{r+1}$.
Proof. Let $P$ denote a path on five vertices $a,x,b,y,c$ (in that order).
Consider a graph $H=H(r)$ obtained by blowing up each of the two vertices $x$
and $y$ to the clique $K_{r-1}$. Now, take $r$ copies of $H$, chose one vertex
$v_{i}$ in any of the two cliques $K_{r-1}$ in each copy of $H$, and join all
these vertices mutually to form a new clique $K_{r}$. We claim that in this
way we constructed a graph $G_{r}$ satisfying the assertion of the theorem. It
is not hard to see that $\chi(G_{r})=r$. To prove the first part of the
theorem, suppose that $\Gamma$ is any Abelian group of order $r$, and there is
a coloring $f:V(G_{r})\rightarrow\Gamma$ such that the sums $S(v)$ form a
proper coloring of $G_{r}$. Notice that in any proper coloring of $H$ with $r$
colors, the vertices $a$, $b$, and $c$ must have the same color. Thus
$s(a)=s(b)=s(c)$. Notice also that, by the definition of additive coloring we
have $S(b)=S(a)+S(c)$, which implies that $S(a)=S(b)=S(c)=0$ in every copy of
$H$ in $G_{r}$. This implies in turn that $S(v)\neq 0$ for all other vertices
of $G_{r}$. In particular, we get a proper coloring of the clique $K_{r}$ by
non-zero elements of $\mathbb{Z}_{r}$, which is not possible.
For the second assertion we define explicitly an additive coloring
$f:V(G_{r})\rightarrow\mathbb{Z}_{r+1}$ as follows. Denote by $H_{i}$ the
$i$th copy of the graph $H$ in $G_{r}$. Let $X_{i}$ and $Y_{i}$ denote the two
cliques $K_{r-1}$ in $H_{i}$ obtained by blowing up the vertices $x$ and $y$,
respectively. Also, let $a_{i}$, $b_{i}$, and $c_{i}$ be the respective copies
of the end vertices and the middle vertex of the path $P$ in $H_{i}$. Finally,
let $v_{i}$ denote the unique vertex of $H_{i}$ belonging to the clique
$K_{r}$. We may assume that $v_{i}\in V(X_{i})$. We have to distinguish two
cases.
1. 1.
_(The number_ $r+1$_is odd.)_ Put $f(v_{i})=f(b_{i})=0$ and
$f(a_{i})=f(c_{i})=i$ for all $i=1,2,\ldots,r$. Then extend injectively the
coloring using all labels from the set $\\{1,2,\ldots,r\\}\setminus\\{i,-i\\}$
on each of the two cliques $X_{i}$ and $Y_{i}$. So, the total sum of labels in
each of the cliques $X_{i}$ and $Y_{i}$ is zero. Hence, we get $S(v_{i})=i$
and $S(a_{i})=S(b_{i})=S(c_{i})=0$. For any other vertex $u$ we get $S(u)\neq
0$. Also, we cannot have conflicts inside cliques $X_{i}$ and $Y_{i}$ by
injectivity.
2. 2.
_(The number_ $r+1$_is even.)_ Let $r+1=2k$. First we construct our coloring
on all copies $H_{i}$ for $i\neq k$. Put $f(v_{i})=f(b_{i})=f(c_{i})=0$ and
$f(a_{i})=i$. Extend injectively the coloring on the clique $X_{i}$ using all
labels from the set $\\{1,2,\ldots,r\\}\setminus\\{i,-i\\}$. So, the total sum
of labels on $X_{i}$ is equal to $k$. Next, extend the coloring injectively to
cliques $Y_{i}$ using all labels from the set
$\\{1,2,\ldots,r\\}\setminus\\{k\\}$. Hence, the total sum of labels over
$Y_{i}$ is zero. Thus we get $S(v_{i})=k+i$, $S(a_{i})=S(b_{i})=k$, and
$S(c_{i})=0$ for all $i\neq k$. For $u\in X_{i}$ we have $S(u)=k+i-f(u)\neq
k$, since $f(u)\neq i$. For $u\in Y_{i}$ we have $S(u)=-f(u)\neq 0,k$. Also
there are no conflicts inside cliques $X_{i}$ and $Y_{i}$ by injectivity. It
remains to extend the coloring to the copy $H_{k}$. Put $f(v_{k})=0$,
$f(a_{k})=1$, $f(b_{k})=k$, and $f(c_{k})=k-1$. Next put injectively all
labels from the set $\\{1,2,\ldots,r\\}\setminus\\{k,k+1\\}$ to the vertices
of $X_{k}$, and similarly for $Y_{k}$ using the set
$\\{0,1,\ldots,r\\}\setminus\\{k,r\\}$. So, the total sum over $X_{k}$ is
$k-1$ and the total sum over $Y_{k}$ is $1$. Hence, we get $S(a_{k})=k-1$,
$S(c_{k})=1$, $S(b_{k})=k$, and $S(v_{k})=0$. Since each vertex $u\in
X_{k}\cup Y_{k}$ satisfies $S(u)=-f(u)$, no other conflicts could appear.
The proof is complete.
Notice that graph $G_{4}$ from the above proof is planar, so we cannot get our
main conjecture for planar graphs using finite groups. Notice also, that
$G_{2}$ is a tree, and $G_{3}$ is an outer planar graph, so the same
difficulty is true for planar graphs with smaller chromatic number. Perhaps
every $r$-colorable graph has an additive coloring modulo $r+1$.
We conclude this section with the following simple result.
###### Theorem 11
Let $A$ be a fixed Abelian group. The problem of deciding whether a given
graph $G$ has an additive coloring over $A$ is NP-complete if
$\left|A\right|\geqslant 3$, and polynomial for $A=\mathbb{Z}_{2}$.
Proof. Let $\left|A\right|=k\geqslant 3$. For a given graph $G$, whose vertex
set is $V(G)=\\{v_{1},\ldots,v_{n}\\}$, consider a new graph $G^{\prime}$
obtained by adding $n$ new vertices
$\\{v_{1}^{\prime},\ldots,v_{n}^{\prime}\\}$ and $n$ new edges
$v_{i}v_{i}^{\prime}$ for $i=1,\ldots,n$. We prove that $G$ is $k$-colorable
(in the usual sense) if and only if $G^{\prime}$ is additively colorable over
$A$. This will prove the first assertion of the theorem.
Obviously, if $G^{\prime}$ has an additive coloring over $A$, then $G$ is
$k$-colorable in the usual sense. For the other implication, assume that $G$
is $k$-colorable, and fix a proper coloring $c$ of $G$ using $A$ as the set of
colors. Now fix a nonzero element $a\in A$ and define a new coloring $f$ of
$G^{\prime}$ in the following way:
1. 1.
If $c(v_{i})=0$, then $f(v_{i})=a$.
2. 2.
If $c(v_{i})\neq 0$, then $f(v_{i})=0$.
3. 3.
$f(v_{i}^{\prime})=c(v_{i})-\mathop{\displaystyle\sum}\limits_{x\in
N_{G}(v_{i})}f(v_{i})$.
We claim that $f$ is a desired additive coloring of $G^{\prime}$ over $A$.
Indeed, the sum of colors around each vertex $v_{i}$ satisfies
$S(v_{i})=\mathop{\displaystyle\sum}\limits_{x\in
N_{G}(v_{i})}f(v_{i})+f(v_{i}^{\prime})=c(v_{i}),$
so there are no conflicts in $G$. Also by definition of $f$ we have
$S(v_{i}^{\prime})=f(v_{i})\neq c(v_{i})=S\left(v_{i}\right)$
for each vertex $v_{i}^{\prime}$. This prove the claim.
For the second assertion just notice that the problem reduces to recognizing
if a given graph $G$ is bipartite, and then checking solvability of a system
of linear equations of the form $Mx=y$ over $\mathbb{Z}_{2}$, where $M$ is the
adjacency matrix of $G$, and $y$ is binary vector encoding a proper coloring
of $G$. There are actually two possible such vectors for a connected bipartite
graph $G$. This completes the proof.
## 6 Open problems
We conclude the paper with a short list of open questions concerning additive
coloring of graphs.
###### Conjecture 12
Every graph $G$ satisfies $\eta(G)\leqslant\chi(G)$.
It is not known whether this is true for bipartite graphs. It is not even
known if $\eta(G)$ is bounded for bipartite graphs. A heuristic argument is
that the statement of the conjecture holds trivially if we extend the set of
labels to real numbers. Indeed, any proper coloring of a $k$-colorable graph
$G$ with a set of $k$ real numbers which is independent over rationals, gives
an additive coloring of $G$. Another direction is to consider additive
colorings in finite Abelian groups.
###### Conjecture 13
Every graph $G$ has an additive coloring modulo $\chi(G)+1$.
If true this is best possible, as we proved in section 5.
Our last problem arose as a vertex analog of the famous _antimagic labeling
conjecture_ of Ringel [5].
###### Conjecture 14
Let $G$ be a simple graph on $n$ vertices in which no two vertices have the
same neighborhood. Then there is a bijection
$f:V(G)\rightarrow\\{1,2,\ldots,n\\}$ such that
$\mathop{\displaystyle\sum}\limits_{x\in
N(u)}f(x)\neq\mathop{\displaystyle\sum}\limits_{x\in N(v)}f(x)$
for any two distinct vertices $u$ and $v$.
###### Acknowledgement 15
Sebastian Czerwiński and Jarosław Grytczuk acknowledge a partial support from
Polish Ministry of Science and Higher Education Grants (MNiSW) (N N201 271335)
and (MNiSW) (N N206 257035).
## References
* [1] N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput., 8 (1999), 7–29.
* [2] T. Bartnicki, J. Grytczuk, S. Niwczyk, Weight choosability of graphs, J. Graph Theory, 60 (2009), 242–256.
* [3] Y. Bu, D. W. Cranston, M. Montassier, A. Raspaud, W. Wang, Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201–219.
* [4] S. Czerwiński, J. Grytczuk, W. Żelazny, Lucky labelings of graphs, Inform. Process. Letters 109 (2009), 1078–1081.
* [5] N. Hartsfield, G. Ringel, Pearls of graph theory, Academic Press (1990).
* [6] M. Kalkowski, M. Karoński, F. Pfender, Vertex-coloring edge-weightings: towards 1-2-3-conjecture, J. Comb. Theory, Ser. B 100 (2010), 347–349.
* [7] M. Karoński, T. Łuczak, A. Thomason, Edge veights and vertex colours, J. Combin. Theory Ser. B 91 (2004), 151–157.
* [8] H. A. Kierstead, G. Konjevod, Coloring number and on-line Ramsey theory for graphs and hypergraphs, Combinatorica 29 (2009) 49–64.
* [9] M. Michałek, A short proof of Combinatorial Nullstellensatz, American Math. Monthly, 117 (2010), 821–823.
|
arxiv-papers
| 2012-02-03T11:26:52 |
2024-09-04T02:49:27.001636
|
{
"license": "Public Domain",
"authors": "Tomasz Bartnicki, Bart{\\l}omiej Bosek, Sebastian Czerwi\\'nski,\n Jaros{\\l}aw Grytczuk, Grzegorz Matecki, Wiktor \\.Zelazny",
"submitter": "Sebastian Czerwi\\'nski",
"url": "https://arxiv.org/abs/1202.0667"
}
|
1202.0681
|
$Id:espcrc1.tex,v1.22004/02/2411:22:11speppingExp$ On maximum matchings in
almost regular graphsPetros A. Petrosyan
# On maximum matchings in almost regular graphs
Petros A. Petrosyan email: pet_petros@{ipia.sci.am, ysu.am, yahoo.com}
Institute for Informatics and Automation Problems,
National Academy of Sciences, 0014, Armenia Department of Informatics and
Applied Mathematics,
Yerevan State University, 0025, Armenia
###### Abstract
In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with
$2\leq\delta(G)\leq\Delta(G)\leq 3$ contains a maximum matching whose
unsaturated vertices do not have a common neighbor, where $\Delta(G)$ and
$\delta(G)$ denote the maximum and minimum degrees of vertices in $G$,
respectively. In the same paper they suggested the following conjecture: every
graph $G$ with $\Delta(G)-\delta(G)\leq 1$ contains a maximum matching whose
unsaturated vertices do not have a common neighbor. Recently, Picouleau
disproved this conjecture by constructing a bipartite counterexample $G$ with
$\Delta(G)=5$ and $\delta(G)=4$. In this note we show that the conjecture is
false for graphs $G$ with $\Delta(G)-\delta(G)=1$ and $\Delta(G)\geq 4$, and
for $r$-regular graphs when $r\geq 7$.
Keywords: bipartite graph, regular graph, maximum matching
## 1 Introduction
Throughout this note all graphs are finite, undirected, and have no loops, but
may contain multiple edges. Let $V(G)$ and $E(G)$ denote the sets of vertices
and edges of $G$, respectively. For a graph $G$, let $\Delta(G)$ and
$\delta(G)$ denote the maximum and minimum degrees of vertices in $G$,
respectively. For two distinct vertices $u$ and $v$ of a graph $G$, let
$E(uv)$ denote the set of all edges of $G$ joining $u$ with $v$. An
$(a,b)$-biregular bipartite graph $G$ is a bipartite graph $G$ with the
vertices in one part all having degree $a$ and the vertices in the other part
all having degree $b$. The terms and concepts that we do not define can be
found in [3].
In [1], Mkrtchyan, Petrosyan and Vardanyan proved the following result.
###### Theorem 1
Every graph $G$ with $2\leq\delta(G)\leq\Delta(G)\leq 3$ contains a maximum
matching whose unsaturated vertices do not have a common neighbor.
###### Corollary 2
Every cubic graph $G$ contains a maximum matching whose unsaturated vertices
do not have a common neighbor.
In the same paper they posed the following
###### Conjecture 3
Every graph $G$ with $\Delta(G)-\delta(G)\leq 1$ contains a maximum matching
whose unsaturated vertices do not have a common neighbor.
Also, they noted that they do not even know, whether this conjecture holds for
$r$-regular graphs with $r\geq 4$. In [2], Picouleau showed that Conjecture 3
is false when $G$ is a $(5,4)$-biregular bipartite graph. However, the
question is still open when $\Delta(G)=4$ and $\delta(G)=3$, and when
$\Delta(G)-\delta(G)=1$ with $\Delta(G)\geq 6$. Also, the question remains
open for $r$-regular graphs when $r\geq 4$.
In this note we prove that for any $r\geq 2$, there exists a
$(2r,2r-1)$-biregular bipartite graph $G$ such that for any maximum matching
of $G$ and any pair of unsaturated vertices with respect to this maximum
matching, these vertices have a common neighbor. Next we show that for any
$r\geq 3$, there exists a graph $G$ with $\Delta(G)=2r+1$ and $\delta(G)=2r$
such that for any maximum matching of $G$, there is a pair of unsaturated
vertices with a common neighbor. Finally, we prove that for any $r\geq 3$,
there exists a $(2r+1)$-regular graph $G$ such that for any maximum matching
of $G$, there is a pair of unsaturated vertices with a common neighbor. We
also construct the $8$-regular graph with the same property and prove that for
any $r\geq 5$, there exists a $2r$-regular graph $G$ such that for any maximum
matching of $G$ and any pair of unsaturated vertices with respect to this
maximum matching, these vertices have a common neighbor.
## 2 Results
First we consider graphs $G$ with $\Delta(G)-\delta(G)=1$.
###### Theorem 4
For any $r\geq 2$, there exists a $(2r,2r-1)$-biregular bipartite graph $G$
such that for any maximum matching of $G$ and any pair of unsaturated vertices
with respect to this maximum matching, these vertices have a common neighbor.
* Proof.
For the proof, we construct a graph $B_{2r,2r-1}$ for $r\geq 2$ that satisfies
the specified conditions. We define a graph $B_{2r,2r-1}$ as follows:
$V\left(B_{2r,2r-1}\right)=U\cup V$, where
$U=\left\\{u_{(i,j)}\colon\,1\leq i<j\leq 2r\right\\}$,
$V=\left\\{v_{1}^{(i)},\ldots,v_{r}^{(i)}\colon\,1\leq i\leq 2r\right\\}$, and
$E\left(B_{2r,2r-1}\right)=\left\\{v_{k}^{(i)}u_{(i,j)},v_{k}^{(j)}u_{(i,j)}\colon\,1\leq
i<j\leq 2r,1\leq k\leq r\right\\}$.
Clearly, $B_{2r,2r-1}$ is a $(2r,2r-1)$-biregular bipartite graph with a
bipartition $(U,V)$. Moreover, let us note that $|U|=\binom{2r}{2}=2r^{2}-r$
and $|V|=2r^{2}$. Thus, $B_{2r,2r-1}$ has no perfect matching. On the other
hand, by Hall’s theorem, it is not hard to see that each maximum matching $M$
saturates $U$. This implies that for any maximum matching $M$ of
$B_{2r,2r-1}$, we have $r$ unsaturated vertices from $V$. Now let
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ be any pair of unsaturated vertices
from $V$ with respect to some maximum matching $M$. We consider two cases.
Case 1: $i=j$.
In this case, by the construction of $B_{2r,2r-1}$, $v_{k_{0}}^{(i)}$ and
$v_{k_{1}}^{(i)}$ have a common neighbor $u_{(i,l)}$ with $i<l$ or $u_{(l,i)}$
with $l<i$.
Case 2: $i\neq j$.
In this case, by the construction of $B_{2r,2r-1}$, $v_{k_{0}}^{(i)}$ and
$v_{k_{1}}^{(j)}$ have a common neighbor $u_{(i,j)}$ if $i<j$ or $u_{(j,i)}$
if $j<i$. $\square$
###### Theorem 5
For any $r\geq 3$,
(1)
there exists a $(2r+1)$-regular graph $G$ such that for any maximum matching
of $G$, there is a pair of unsaturated vertices with a common neighbor,
(2)
there exists a graph $H$ with $\Delta(H)=2r+1$ and $\delta(H)=2r$ such that
for any maximum matching of $H$, there is a pair of unsaturated vertices with
a common neighbor.
* Proof.
(1) For the proof, we are going to construct a graph $G_{2r+1}$ for $r\geq 3$
that satisfies the specified conditions. We define a graph $G_{2r+1}$ as
follows:
$V\left(G_{2r+1}\right)=\left\\{x,y,z\right\\}\cup\left\\{v_{1}^{(i)},v_{2}^{(i)},v_{3}^{(i)}\colon\,1\leq
i\leq 2r+1\right\\}$ and
$E\left(G_{2r+1}\right)=\left\\{xv_{1}^{(i)},yv_{2}^{(i)},zv_{3}^{(i)}\colon\,1\leq
i\leq
2r+1\right\\}\cup\left\\{v_{1}^{(i)}v_{2}^{(i)},v_{2}^{(i)}v_{3}^{(i)},v_{3}^{(i)}v_{1}^{(i)}\colon\,\left|E\left(v_{1}^{(i)}v_{2}^{(i)}\right)\right|=\left|E\left(v_{2}^{(i)}v_{3}^{(i)}\right)\right|=\left|E\left(v_{3}^{(i)}v_{1}^{(i)}\right)\right|=r,1\leq
i\leq 2r+1\right\\}$.
Clearly, $G_{2r+1}$ is a $(2r+1)$-regular graph with
$\left|V\left(G_{2r+1}\right)\right|=6r+6$. By Tutte’s theorem, $G_{2r+1}$ has
no perfect matching. On the other hand, it is not hard to see that each
maximum matching $M$ saturates $x,y$ and $z$. This implies that for any
maximum matching $M$ of $G_{2r+1}$, we have $2r-2$ unsaturated vertices from
$V\left(G_{2r+1}\right)\setminus\\{x,y,z\\}$. Since $r\geq 3$, we have that
for any maximum matching $M$ of $G_{2r+1}$, the graph $G_{2r+1}$ has at least
four unsaturated vertices from $V\left(G_{2r+1}\right)\setminus\\{x,y,z\\}$.
However, by the construction of $G_{2r+1}$, the vertices from
$V\left(G_{2r+1}\right)\setminus\\{x,y,z\\}$ have only three possible values
of the subindex; thus there are two vertices with the same subindex. Let
$v_{k}^{(i)}$ and $v_{k}^{(j)}$ be these unsaturated vertices from
$V\left(G_{2r+1}\right)\setminus\\{x,y,z\\}$ with respect to some maximum
matching $M$. If $k=1$, then, by the construction of $G_{2r+1}$, $v_{1}^{(i)}$
and $v_{1}^{(j)}$ have a common neighbor $x$, if $k=2$, then, by the
construction of $G_{2r+1}$, $v_{2}^{(i)}$ and $v_{2}^{(j)}$ have a common
neighbor $y$, and if $k=3$, then, by the construction of $G_{2r+1}$,
$v_{3}^{(i)}$ and $v_{3}^{(j)}$ have a common neighbor $z$.
(2) For the proof, it suffices to define a graph $H_{2r+1,2r}$ for $r\geq 3$
as follows: $V\left(H_{2r+1,2r}\right)=V\left(G_{2r+1}\right)$ and
$E\left(H_{2r+1,2r}\right)=E\left(G_{2r+1}\right)\setminus\left\\{v_{3}^{(i)}v_{1}^{(i)}\colon\,1\leq
i\leq 2r+1\right\\}$. Clearly, $H_{2r+1,2r}$ is a graph with
$\Delta\left(H_{2r+1,2r}\right)=2r+1$ and $\delta\left(H_{2r+1,2r}\right)=2r$.
Similarly as in the proof of (1) it can be shown that for any maximum matching
of $H_{2r+1,2r}$, there are two unsaturated vertices with a common neighbor.
$\square$
These results combined with the result of Picouleau show that Conjecture 3 is
false for graphs $G$ with $\Delta(G)-\delta(G)=1$ and $\Delta(G)\geq 4$. Also,
Conjecture 3 is false for $(2r+1)$-regular graphs when $r\geq 3$. Next we
consider even regular graphs. First we consider the $8$-regular graph $G$
shown in Fig. 1. Similarly as in the proof of Theorem 5 it can be shown that
for any maximum matching of $G$, there are two unsaturated vertices with a
common neighbor.
Figure 1: The $8$-regular graph $G$.
###### Theorem 6
For any $r\geq 5$, there exists a $2r$-regular graph $G$ such that for any
maximum matching of $G$ and any pair of unsaturated vertices with respect to
this maximum matching, these vertices have a common neighbor.
* Proof.
For the proof, we construct a graph $F_{2r}$ for $r\geq 5$ that satisfies the
specified conditions. We define a graph $F_{2r}$ as follows:
$V\left(F_{2r}\right)=\left\\{x,y,z\right\\}\cup\left\\{v_{1}^{(i)},v_{2}^{(i)},v_{3}^{(i)}\colon\,1\leq
i\leq r\right\\}$ and
$E\left(F_{2r}\right)=\left\\{xv_{1}^{(i)},xv_{2}^{(i)},yv_{1}^{(i)},yv_{3}^{(i)},zv_{2}^{(i)},zv_{3}^{(i)}\colon\,1\leq
i\leq
r\right\\}\cup\left\\{v_{1}^{(i)}v_{2}^{(i)},v_{2}^{(i)}v_{3}^{(i)},v_{3}^{(i)}v_{1}^{(i)}\colon\,\left|E\left(v_{1}^{(i)}v_{2}^{(i)}\right)\right|=\left|E\left(v_{2}^{(i)}v_{3}^{(i)}\right)\right|=\left|E\left(v_{3}^{(i)}v_{1}^{(i)}\right)\right|=r-1,1\leq
i\leq r\right\\}$.
Clearly, $F_{2r}$ is a $2r$-regular graph with
$\left|V\left(F_{2r}\right)\right|=3r+3$. By Tutte’s theorem, $F_{2r}$ has no
perfect matching. On the other hand, it is not hard to see that each maximum
matching $M$ saturates $x,y$ and $z$. This implies that for any maximum
matching $M$ of $F_{2r}$, we have $r-3$ unsaturated vertices from
$V\left(F_{2r}\right)\setminus\\{x,y,z\\}$. Since $r\geq 5$, we have that for
any maximum matching $M$ of $F_{2r}$, the graph $F_{2r}$ has at least two
unsaturated vertices from $V\left(F_{2r}\right)\setminus\\{x,y,z\\}$. Now let
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ be any pair of unsaturated vertices
from $V\left(F_{2r}\right)\setminus\\{x,y,z\\}$ with respect to some maximum
matching $M$. We consider two cases.
Case 1: $k_{0}=k_{1}$.
If $k_{0}=k_{1}=1$ or $k_{0}=k_{1}=2$, then, by the construction of $F_{2r}$,
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ have a common neighbor $x$. If
$k_{0}=k_{1}=3$, then, by the construction of $F_{2r}$, $v_{k_{0}}^{(i)}$ and
$v_{k_{1}}^{(j)}$ have a common neighbor $y$.
Case 2: $k_{0}\neq k_{1}$.
If $(k_{0},k_{1})=(1,2)$, then, by the construction of $F_{2r}$,
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ have a common neighbor $x$, if
$(k_{0},k_{1})=(1,3)$, then, by the construction of $F_{2r}$,
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ have a common neighbor $y$, and if
$(k_{0},k_{1})=(2,3)$, then, by the construction of $F_{2r}$,
$v_{k_{0}}^{(i)}$ and $v_{k_{1}}^{(j)}$ have a common neighbor $z$. $\square$
Our results show that Conjecture 3 is also false for $r$-regular graphs when
$r\geq 7$. Thus, the question remains open only for $4,5$ and $6$-regular
graphs.
* Acknowledgement
We would like to thank the anonymous referees for useful suggestions.
## References
* [1] V.V. Mkrtchyan, S.S. Petrosyan, G.N. Vardanyan, On disjoint matchings in cubic graphs, Discrete Math. 310 (2010) 1588-1613.
* [2] C. Picouleau, A note on a conjecture on maximum matching in almost regular graphs, Discrete Math. 310 (2010) 3646-3647.
* [3] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 2001.
|
arxiv-papers
| 2012-02-03T12:39:14 |
2024-09-04T02:49:27.009773
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Petros A. Petrosyan",
"submitter": "Petros Petrosyan",
"url": "https://arxiv.org/abs/1202.0681"
}
|
1202.0757
|
# Optimal frames and Newton’s method
Matthew Fickus Air Force Institute of Technology, Department of Mathematics,
Wright-Patterson AFB, OH 45433, Matthew.Fickus@afit.edu Dustin G. Mixon
Princeton University, Program in Applied and Computational Mathematics,
Princeton, NJ 08544
###### Abstract
Given a parametrized family of finite frames, we consider the optimization
problem of finding the member of this family whose coefficient space most
closely contains a given data vector. This nonlinear least squares problem
arises naturally in the context of a certain type of radar system. We derive
analytic expressions for the first and second partial derivatives of the
objective function in question, permitting this optimization problem to be
efficiently solved using Newton’s method. We also consider how sensitive the
location of this minimizer is to noise in the data vector. We further provide
conditions under which one should expect the minimizer of this objective
function to be unique. We conclude by discussing a related variational-
calculus-based approach for solving this frame optimization problem over an
interval of time.
## 1 Introduction
In frame theory, the synthesis operator of a finite sequence of vectors
$\\{{f_{n}}\\}_{n=1}^{N}$ in $\mathbb{R}^{M}$ is the operator
$F:\mathbb{R}^{N}\rightarrow\mathbb{R}^{M}$, $Fw:=\sum_{n=1}^{N}w(n)f_{n}$.
That is, $F$ is an $M\times N$ matrix whose $n$th column is $f_{n}$. Taking
the transpose of the synthesis operator yields the analysis operator
$F^{*}:\mathbb{R}^{M}\rightarrow\mathbb{R}^{N}$ given by
$(F^{*}v)(n)=\langle{v},{f_{n}}\rangle$. The sequence
$\\{{f_{n}}\\}_{n=1}^{N}$ is a frame for $\mathbb{R}^{M}$ if there exist frame
bounds $0<A\leq B<\infty$ such that $A\|{v}\|^{2}\leq\|{F^{*}v}\|^{2}\leq
B\|{v}\|^{2}$ for all $v\in\mathbb{R}^{M}$. In this finite-dimensional
setting, we have that $\\{{f_{n}}\\}_{n=1}^{N}$ is a frame for
$\mathbb{R}^{M}$ if and only if it spans $\mathbb{R}^{M}$, which is equivalent
to having its frame operator $FF^{*}$ be invertible.
Frame theory was born of the theory of linear least squares: given an
overdetermined system $F^{*}v=w$, the goal is to reconstruct $v$ from $F$ and
$w$ by minimizing $\|{F^{*}v-w}\|^{2}$. The minimizers $v$ are characterized
as the solutions to the normal equations $FF^{*}v=Fw$, which have a unique
solution of $v=(FF^{*})^{-1}Fw$ if and only if $\\{{f_{n}}\\}_{n=1}^{N}$ is a
frame for $\mathbb{R}^{M}$. The minimum value of $\|{F^{*}v-w}\|^{2}$ is thus:
$\min_{v\in\mathbb{R}^{M}}\|{F^{*}v-w}\|^{2}=\|{F^{*}(FF^{*})^{-1}Fw-w}\|^{2}=\bigl{\|}{\bigl{[}{\mathrm{I}-F^{*}(FF^{*})^{-1}F}\bigr{]}w}\bigr{\|}^{2}.$
(1)
In this paper, we focus on a generalization of these ideas that, as detailed
below, arises naturally in a certain radar problem. In this generalization, we
are not given a single frame $\\{{f_{n}}\\}_{n=1}^{N}$ for $\mathbb{R}^{M}$
but rather a parametrized family of frames $\\{{f_{n}(x)}\\}_{n=1}^{N}$ for
$\mathbb{R}^{M}$, where the parameter vector $x$ lies in some subset $\Omega$
of $\mathbb{R}^{P}$. That is, we have an $M\times N$ synthesis matrix $F(x)$,
each of whose entries depend on $P$ real parameters. In particular, given
$F(x)$ and $w$, our goal is to solve:
$\underset{x\in\Omega}{\mathrm{argmin}}\min_{v\in\mathbb{R}^{M}}\|{F^{*}(x)v-w}\|^{2}.$
(2)
That is, we want to find the parameters $x$ for which the range of the
corresponding analysis operator $F^{*}(x)$ is as close as possible to $w$. In
a radar application detailed below, the optimal $x$ and $v$ correspond to the
unknown position and velocity of a target of interest, $F(x)$ has an explicit
formula in terms of the locations of a set of radar transmitters and
receivers, and $w$ is a collection of Doppler measurements; solving (2)
corresponds to determining the target’s location based on Doppler information
alone. More generally, solving (2) corresponds to finding the particular frame
from the family of frames $\\{{f_{n}(x)}\\}_{n=1}^{N}$ which is most
consistent with the measured data $w$.
The first step in solving the nonlinear least squares problem (2) is to
recognize that for any fixed $x\in\Omega$, the minimization over $v$ is but a
simple linear least squares problem. That is, in light of (1), solving (2)
reduces to minimizing the error function $E:\Omega\rightarrow[0,\infty)$,
$E(x):=\bigl{\|}{\\{{\mathrm{I}-F^{*}(x)[F(x)F^{*}(x)]^{-1}F(x)}\\}w}\bigr{\|}^{2}.$
(3)
The bulk of this paper is devoted to the study of the minimization of (3). Our
first priority is to develop a practical method for finding the optimal $x$.
However, since $F$ can depend nonlinearly on $x$, it is unrealistic to expect
a nice closed-form solution for this optimal $x$ in terms of $F$ and $w$. We
thus settle for a good numerical algorithm to iteratively compute $x$, namely
Newton’s method. This method relies on explicit expressions for the gradient
and Hessian of (3), which are presented in the following section. In Section
3, we provide some results on the uniqueness of the optimal $x$ as well as the
sensitivity of (3) to noise in $w$. In the final section, we discuss an
alternative variational-calculus-based approach for solving a time-varying
version of (2) in which $x$ and $w$ are functions of time and $v$ is the time-
derivative of $x$. In the remainder of the introduction, we detail our
motivating radar application and highlight the relevant literature.
### 1.1 A radar application
In the radar community, the process of determining a target’s location is
known as localization. To be precise, let
$x:\mathbb{R}\rightarrow\mathbb{R}^{M}$ denote the trajectory of the target.
That is, $x(t)$ denotes the location of the target at any given time $t$, and
the dimension $M$ is typically either $2$ or $3$. The goal of a given radar
system is to determine $x(t)$ by bouncing electromagnetic signals off of the
target.
Here, we focus on static radar: the target is mobile, but the hardware that
transmits and receives our radar signals is not. More specifically, we
consider multistatic radar, which makes use of multiple fixed pairs of
transmitters and receivers. Let $N$ denote the number of these pairs, and let
$\\{{a_{n}}\\}_{n=1}^{N}$ and $\\{{b_{n}}\\}_{n=1}^{N}$ in $\mathbb{R}^{M}$
denote the locations of the transmitters and receivers, respectively. A signal
broadcast from $a_{n}$, bounced off of the target located at $x(t)$, and then
received at $b_{n}$ travels a total distance of $\varphi_{n}(x(t))$, where
$\varphi_{n}:\mathbb{R}^{M}\rightarrow\mathbb{R}$ is the $n$th bistatic
distance function:
$\varphi_{n}(x):=\|{x-a_{n}}\|+\|{x-b_{n}}\|.$ (4)
In the most simple radar systems, each transmitter broadcasts a sequence of
pulses. In the radar literature, the time lag between the transmitted and
received pulses is known as the time difference of arrival (TDOA). By
multiplying the TDOA by the speed of light, one obtains a set of $N$ real-
valued measurements $\\{{y_{n}(t)}\\}_{n=1}^{N}$ which serve as estimates of
$\\{{\varphi_{n}(x(t))}\\}_{n=1}^{N}$. The target’s true position $x(t)$ is
then known to lie on the intersection of prolate spheroids of the form
$\\{{x\in\mathbb{R}^{M}:\varphi_{n}(x)=y_{n}(t)}\\}$.
We focus on a different class of radar systems in which each transmitter
broadcasts a continuous wave of constant frequency. The periodic nature of
such signals makes it nearly impossible to accurately measure TDOA. Rather,
one instead measures the frequency difference of arrival (FDOA), namely the
change in frequency between the transmitted and received wave. This FDOA is
proportional to the time-derivative of the bistatic distance:
$\frac{\mathrm{d}}{\mathrm{d}t}\varphi_{n}(x(t))=\langle{\dot{x}(t)},{\nabla\varphi_{n}(x(t))}\rangle,$
a fact known as the Doppler effect.
In short, we focus on FDOA multistatic radar, in which the goal is to
determine $x(t)$ from a number of FDOA measurements
$\\{{w_{n}(t)}\\}_{n=1}^{N}$, where each $w_{n}(t)$ should equal
$\nabla\varphi_{n}(x(t))\cdot\dot{x}(t)$, up to measurement error and noise.
Here, one’s first instinct is to try to explicitly solve the following system
of $N$ first-order ordinary differential equations:
$\langle{\dot{x}(t)},{\nabla\varphi_{n}(x(t))}\rangle=w_{n}(t),\quad\forall
n=1,\dotsc,N.$ (5)
The problem with this approach is that when $M<N$, the system (5) is
overdetermined and so it is not likely to be solvable. This may be remedied
with a standard least-squares approach, namely minimizing:
$\sum_{n=1}^{N}|{\langle{\dot{x}(t)},{\nabla\varphi_{n}(x(t))}\rangle-
w_{n}(t)}|^{2}.$ (6)
Indeed, in the final section of this paper, we apply a variational approach to
determine the Euler-Lagrange equation that any minimizer $x$ of the time-
integral of (6) must satisfy. However, the bulk of this paper is devoted to a
more simple approach, namely using frame theory to tackle the minimization of
(6). Here, the key idea is to minimize (6) individually at each time $t$.
To be precise, let us now assume that FDOA measurements
$\\{{w_{n}(t)}\\}_{n=1}^{N}$ are only available at a single time $t_{0}$. In
this setting, the target’s velocity at that instant is essentially an unknown
variable which is independent from its position. To simplify notation, we let
$x:=x(t_{0})$ and $v:=\dot{x}(t_{0})$ in $\mathbb{R}^{M}$ denote this unknown
position and velocity, and let $w\in\mathbb{R}^{N}$ denote the vector of FDOA
measurements $\\{{w_{n}(t_{0})}\\}_{n=1}^{N}$. Denoting the vector
$\nabla\varphi_{n}(x)$ as $f_{n}(x)$, the quantity (6) at $t=t_{0}$ can then
be rewritten as:
$\sum_{n=1}^{N}|{\langle{v},{f_{n}(x)}\rangle-
w_{n}}|^{2}=\|{F^{*}(x)v-w}\|^{2}.$
Thus, our problem of minimizing (6) for $t=t_{0}$ reduces to our frame
optimization problem (2) in the special case where $P=M$ and the $n$th frame
element $f_{n}(x)$ is the gradient of the $n$th bistatic distance function
(4); one may quickly show that this gradient is the sum of two unit vectors,
namely the vectors pointing to the target from the $n$th transmitter and
receiver, respectively:
$f_{n}(x)=\nabla\varphi_{n}(x)=\frac{x-a_{n}}{\|{x-a_{n}}\|}+\frac{x-b_{n}}{\|{x-b_{n}}\|}.$
(7)
With perfect FDOA measurements, the target’s true location corresponds to the
global minimizer of the error function (3). In the third section, we consider
the localization error that results from imperfect measurements. We also
consider the uniqueness of this minimizer: the results suggest that in order
to uniquely localize a target, one wants the number of FDOA measurements $N$
to be at least $2M$. This makes intuitive sense since these measurements
depend on $2M$ real-variable unknowns: $M$ unknowns for the target’s position,
and an additional $M$ unknowns for the target’s velocity.
### 1.2 Relevant literature
Optimization is a significant tool in frame theory, see [1, 7] for example.
The first derivatives of certain frame-theoretic quantities are presented in
[2, 3]; we build on these results here, computing both first and second
derivatives of the more complicated quantity (3).
The motivating FDOA radar localization problem has long been a subject of
interest in the radar community. Early work focused on the radar-based
tracking of transmitters moving in ballistic trajectories [12], and a formal
analysis of the underlying theory [11]. A least-squares formulation of this
localization problem is given in [4, 5, 6]. There, an approximate solution to
(2) was found by explicitly evaluating the error function (3) over a large
grid. It has since been suggested that the location of aircraft may be
determined solely by measuring the Doppler effect that their motion induces in
the carrier waves of television broadcasts [10].
## 2 Newton’s method
Newton’s method is a popular algorithm for solving nonlinear least squares
problems. We use it to minimize our error function $E(x)$ defined in (3).
Newton’s method is an iterative algorithm and will converge to the minimizer
at a quadratic rate provided $E(x)$ is sufficiently well-behaved and we make a
good initial guess [9]. A formal analysis of when $E(x)$ meets the necessary
criteria for convergence is given in [8]. In practice, we make our initial
guess $x_{0}$ by evaluating $E(x)$ over a grid, and choosing the grid point
which yields the smallest value.
Given a current guess $x_{k}$, Newton’s method approximates $E(x)$ as a
paraboloid (second-order Taylor multinomial) on a neighborhood of $x_{k}$ and
then moves in the direction of that paraboloid’s vertex. Explicitly, we let:
$x_{k+1}:=x_{k}-\gamma[(\nabla^{2}E)(x_{k})]^{-1}(\nabla E)(x_{k}),$ (8)
where $0<\gamma<1$ is some experimentally chosen step-size parameter and
$(\nabla E)(x)$ and $(\nabla^{2}E)(x)$ are the gradient and Hessian of (3),
respectively. We compute this gradient and Hessian using a type of
differential calculus for matrix-valued functions.
To be precise, for an open subset $\Omega$ of $\mathbb{R}^{P}$, let
$\mathrm{C}^{1}(\Omega,\mathbb{R}^{N_{1}\times N_{2}})$ denote the set of
matrix-valued functions $A(x)$ from $\Omega$ into the set of all $N_{1}\times
N_{2}$ matrices with the property that each of the $P$ partial derivatives of
each of the $N_{1}N_{2}$ entries of $A(x)$ exist and is continuous on
$\Omega$. Here, the partial derivative $\frac{\partial A}{\partial x_{p}}(x)$
of $A(x)$ with respect to the $p$th variable $x_{p}$ is obtained by computing
the $p$-partial derivative of each entry of $A(x)$ independently.
Equivalently, letting $\\{{\delta_{p}}\\}_{p=1}^{P}$ be the identity basis for
$\mathbb{R}^{P}$, we have:
$\frac{\partial A}{\partial x_{p}}(x):=\lim_{t\rightarrow
0}\frac{1}{t}[A(x+t\delta_{p})-A(x)].$
One can easily show that this derivative is linear and has a matrix-product
rule:
$\frac{\partial}{\partial x_{p}}[A_{1}(x)A_{2}(x)]=\frac{\partial
A_{1}}{\partial x_{p}}(x)A_{2}(x)+A_{1}(x)\frac{\partial A_{2}}{\partial
x_{p}}(x),$
for all $A_{1}\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{N_{1}\times N_{2}})$,
$A_{2}\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{N_{2}\times N_{3}})$. Writing inner
products on $\mathbb{R}^{N}$ in terms of matrix products, this further yields
the inner-product rule:
$\frac{\partial}{\partial
x_{p}}\langle{w_{1}(x)},{w_{2}(x)}\rangle=\biggl{\langle}{\frac{\partial
w_{1}}{\partial
x_{p}}(x)},{w_{2}(x)}\biggr{\rangle}+\biggl{\langle}{w_{1}(x)},{\frac{\partial
w_{2}}{\partial x_{p}}(x)}\biggr{\rangle},$
for all $w_{1},w_{2}\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{N})$. We shall also
make use of a type of quotient rule. To be precise, if
$A\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{N\times N})$, then its entries, and
therefore determinant, are continuous in $x$. In particular, if $A(x_{0})$ is
invertible for some $x_{0}\in\Omega$, then $A(x)$ is also invertible on a
neighborhood of $x_{0}$. Moreover, the cofactor form of $A^{-1}(x)$ implies
that it is itself continuous in $x$. As such,
$\displaystyle-A^{-1}(x)\frac{\partial A}{\partial x_{p}}(x)A^{-1}(x)$
$\displaystyle=-\lim_{t\rightarrow 0}A^{-1}(x+t\delta_{p})\lim_{t\rightarrow
0}\frac{1}{t}[A(x+t\delta_{p})-A(x)]A^{-1}(x)$
$\displaystyle=\lim_{t\rightarrow
0}\frac{1}{t}[A^{-1}(x+t\delta_{p})-A^{-1}(x)].$
In particular, if $A(x)$ is invertible for all $x\in\Omega$, then
$A^{-1}\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{N\times N})$ with:
$\frac{\partial A^{-1}}{\partial x_{p}}(x)=-A^{-1}(x)\frac{\partial
A}{\partial x_{p}}(x)A^{-1}(x).$ (9)
These facts in hand, we are ready to compute the partial derivatives of (3).
Let:
$\Pi(x):=\mathrm{I}-F^{*}(x)(F(x)F^{*}(x))^{-1}F(x).$
It is well known that $\Pi(x)$ is the orthogonal projection operator onto the
null space of $F(x)$. As such, our error function (3) can be simplified as:
$E(x)=\|{\Pi(x)w}\|^{2}=\langle{\Pi(x)w},{\Pi(x)w}\rangle=\langle{w},{\Pi^{*}(x)\Pi(x)w}\rangle=\langle{w},{\Pi(x)w}\rangle.$
By the product rule, the $p$th derivative of $E(x)$ is thus $\frac{\partial
E}{\partial x_{p}}(x)=\langle{w},{\frac{\partial\Pi(x)}{\partial
x_{p}}w}\rangle$. To compute this derivative of $\Pi$, we first use (9) and
the product rule to find the corresponding partial derivative of
$[F(x)F^{*}(x)]^{-1}$; here, for the sake of succinctness and readability, we
shorten “$F(x)$” to simply “$F$”:
$\frac{\partial}{\partial
x_{p}}(FF^{*})^{-1}=-(FF^{*})^{-1}\Bigl{(}{\frac{\partial F}{\partial
x_{p}}F^{*}+F\frac{\partial F^{*}}{\partial x_{p}}}\Bigr{)}(FF^{*})^{-1}.$
(10)
The product rule then gives:
$\displaystyle\frac{\partial\Pi}{\partial x_{p}}$
$\displaystyle=\frac{\partial}{\partial
x_{p}}\bigl{[}{\mathrm{I}-F^{*}(FF^{*})^{-1}F}\bigr{]}$
$\displaystyle=-\frac{\partial F^{*}}{\partial
x_{p}}(FF^{*})^{-1}F-F^{*}(FF^{*})^{-1}\frac{\partial F}{\partial x_{p}}$
$\displaystyle\qquad+F^{*}(FF^{*})^{-1}\Bigl{(}{\frac{\partial F}{\partial
x_{p}}F^{*}+F\frac{\partial F^{*}}{\partial x_{p}}}\Bigr{)}(FF^{*})^{-1}F.$
Combining the first and fourth terms above, as well as the second and third,
gives:
$\displaystyle\frac{\partial\Pi}{\partial x_{p}}$
$\displaystyle=-\bigl{[}{\mathrm{I}-F^{*}(FF^{*})^{-1}F}\bigr{]}\frac{\partial
F^{*}}{\partial x_{p}}(FF^{*})^{-1}F-F^{*}(FF^{*})^{-1}\frac{\partial
F}{\partial x_{p}}\bigl{[}{\mathrm{I}-F^{*}(FF^{*})^{-1}F}\bigr{]}$
$\displaystyle=-\Pi\,\Pi_{p}^{*}-\Pi_{p}\Pi,$ (11)
where we adopt the notation $\Pi_{p}:=F^{*}(FF^{*})^{-1}\frac{\partial
F}{\partial x_{p}}$. Since $\Pi^{*}=\Pi$ and we are working with real-valued
inner products, the $p$th partial derivative of $E$ is thus:
$\frac{\partial E}{\partial
x_{p}}=\biggl{\langle}{w},{\frac{\partial\Pi}{\partial
x_{p}}w}\biggr{\rangle}=\langle{w},{-\Pi\,\Pi_{p}^{*}-\Pi_{p}\Pi
w}\rangle=-2\langle{w},{\Pi_{p}\Pi w}\rangle.$ (12)
In light of (12), computing second derivatives of $E$ involves computing the
first derivatives of $\Pi_{p}$. This computation parallels that of (11),
making renewed use of (10). To be precise, for any $p,q=1,\dotsc,P$,
$\displaystyle\frac{\partial\Pi_{p}}{\partial x_{q}}$
$\displaystyle=\frac{\partial}{\partial x_{q}}F^{*}(FF^{*})^{-1}\frac{\partial
F}{\partial x_{p}}$ $\displaystyle=\frac{\partial F^{*}}{\partial
x_{q}}(FF^{*})^{-1}\frac{\partial F}{\partial
x_{p}}+F^{*}(FF^{*})^{-1}\frac{\partial^{2}F}{\partial x_{q}\partial x_{p}}$
$\displaystyle\qquad-F^{*}(FF^{*})^{-1}\Bigl{(}{\frac{\partial F}{\partial
x_{q}}F^{*}+F\frac{\partial F^{*}}{\partial
x_{q}}}\Bigr{)}(FF^{*})^{-1}\frac{\partial F}{\partial x_{p}}.$
Combining the first and fourth terms above yields:
$\frac{\partial\Pi_{p}}{\partial x_{q}}=\Pi\frac{\partial F^{*}}{\partial
x_{q}}(FF^{*})^{-1}\frac{\partial F}{\partial
x_{p}}+\Pi_{q,p}-\Pi_{q}\Pi_{p},$
where $\Pi_{q,p}:=F^{*}(FF^{*})^{-1}\frac{\partial^{2}F}{\partial
x_{q}\partial x_{p}}$. Putting $(FF^{*})^{-1}(FF^{*})$ in the first term
gives:
$\displaystyle\frac{\partial\Pi_{p}}{\partial x_{q}}$
$\displaystyle=\Pi\frac{\partial F^{*}}{\partial
x_{q}}(FF^{*})^{-1}FF^{*}(FF^{*})^{-1}\frac{\partial F}{\partial
x_{p}}+\Pi_{q,p}-\Pi_{q}\Pi_{p}$
$\displaystyle=\Pi\,\Pi_{q}^{*}\Pi_{p}+\Pi_{q,p}-\Pi_{q}\Pi_{p}.$ (13)
Having (11) and (13), we take the $q$th partial derivative of (12):
$\displaystyle-\frac{1}{2}\frac{\partial^{2}E}{\partial x_{q}\partial x_{p}}$
$\displaystyle=\biggl{\langle}{w},{\frac{\partial\Pi_{p}}{\partial x_{q}}\Pi
w}\biggr{\rangle}+\biggl{\langle}{w},{\Pi_{p}\frac{\partial\Pi}{\partial
x_{q}}w}\biggr{\rangle}$
$\displaystyle=\bigl{\langle}{w},{(\Pi\,\Pi_{q}^{*}\Pi_{p}+\Pi_{q,p}-\Pi_{q}\Pi_{p})\Pi
w}\bigr{\rangle}+\bigl{\langle}{w},{\Pi_{p}(-\Pi\,\Pi_{q}^{*}-\Pi_{q}\Pi)w}\bigr{\rangle}$
$\displaystyle=\bigl{\langle}{w},{(\Pi_{q,p}\Pi+\Pi\,\Pi_{q}^{*}\Pi_{p}\Pi-\Pi_{q}\Pi_{p}\Pi-\Pi_{p}\Pi_{q}\Pi-\Pi_{p}\Pi\,\Pi_{q}^{*})w}\bigr{\rangle}.$
We now rearrange this statement to better indicate the symmetry between $p$
and $q$ in this expression, which is consistent with the symmetry of mixed
partial derivatives:
$\displaystyle\frac{\partial^{2}E}{\partial x_{q}\partial x_{p}}$
$\displaystyle=2\bigl{\langle}{w},{(\Pi_{p}\Pi_{q}+\Pi_{q}\Pi_{p})\Pi
w}\bigr{\rangle}+2\bigl{\langle}{\Pi\,\Pi_{p}^{*}w},{\Pi\,\Pi_{q}^{*}w}\bigr{\rangle}$
$\displaystyle\qquad-2\bigl{\langle}{\Pi_{q}\Pi w},{\Pi_{p}\Pi
w}\bigr{\rangle}-2\bigl{\langle}{w},{\Pi_{q,p}\Pi w}\bigr{\rangle}.$ (14)
We summarize (12) and (14) as the following result:
###### Theorem 1.
Let $\Omega$ be an open subset of $\mathbb{R}^{P}$ and let
$F\in\mathrm{C}^{2}(\Omega,\mathbb{R}^{M\times N})$ where the columns
$\\{{f_{n}(x)}\\}_{n=1}^{N}$ of $F(x)$ always form a frame for
$\mathbb{R}^{M}$. For any $w\in\mathbb{R}^{N}$, the first and second partial
derivatives of (3) are:
$\displaystyle\frac{\partial E}{\partial x_{p}}$
$\displaystyle=-2\langle{w},{\Pi_{p}\Pi w}\rangle,$
$\displaystyle\frac{\partial^{2}E}{\partial x_{q}\partial x_{p}}$
$\displaystyle=2\bigl{\langle}{w},{(\Pi_{p}\Pi_{q}+\Pi_{q}\Pi_{p})\Pi
w}\bigr{\rangle}+2\bigl{\langle}{\Pi\,\Pi_{p}^{*}w},{\Pi\,\Pi_{q}^{*}w}\bigr{\rangle}$
$\displaystyle\qquad-2\bigl{\langle}{\Pi_{q}\Pi w},{\Pi_{p}\Pi
w}\bigr{\rangle}-2\bigl{\langle}{w},{\Pi_{q,p}\Pi w}\bigr{\rangle},$
for all $p,q=1,\ldots,P$, where:
$\Pi:=\mathrm{I}-F^{*}(FF^{*})^{-1}F,\quad\Pi_{p}:=F^{*}(FF^{*})^{-1}\frac{\partial
F}{\partial
x_{p}},\quad\Pi_{q,p}:=F^{*}(FF^{*})^{-1}\frac{\partial^{2}F}{\partial
x_{q}\partial x_{p}}.$
Though the expressions for the gradient and Hessian of $E(x)$ in Theorem 1 are
complicated, they are nevertheless straightforward to implement in the
Newton’s method iteration (8). To be precise, given a current guess $x_{k}$,
our first task is to compute $\frac{\partial F}{\partial x_{p}}(x_{k})$ and
$\frac{\partial^{2}F}{\partial x_{q}\partial x_{p}}(x_{k})$ for all
$p,q=1,\dotsc,P$. Calculating these derivatives columnwise, this is equivalent
to finding $\frac{\partial f_{n}}{\partial x_{p}}(x_{k})$ and
$\frac{\partial^{2}f_{n}}{\partial x_{q}\partial x_{p}}(x_{k})$ for all
$n=1,\dotsc,N$ and $p,q=1,\dotsc,P$. For the particular frame (7) that arises
in FDOA multistatic radar, the first derivatives can be found by substituting
$x_{k}-a_{n}$ and $x_{k}-b_{n}$ into the following easily-derived formula:
$\frac{\partial}{\partial
x_{p}}\frac{x}{\|{x}\|}=\frac{1}{\|{x}\|}\pi(x)\delta_{p},$
and summing the results; here, $\pi(x):=\mathrm{I}-\frac{xx^{*}}{\|{x}\|^{2}}$
is the projection operator onto the orthogonal complement of the line passing
through $x$, where $x^{*}$ denotes the transpose of the column vector $x$. The
second derivatives of $f_{n}(x)$ at $x_{k}$ can similarly be found using the
relation:
$\frac{\partial^{2}}{\partial x_{q}\partial
x_{p}}\frac{x}{\|{x}\|}=-\frac{1}{\|{x}\|^{3}}\bigl{[}{\pi(x)(\delta_{p}\delta_{q}^{*}+\delta_{q}\delta_{p}^{*})+(\delta_{p}^{*}\pi(x)\delta_{q})\mathrm{I}}\bigr{]}x.$
With $\frac{\partial F}{\partial x_{p}}(x_{k})$ and
$\frac{\partial^{2}F}{\partial x_{q}\partial x_{p}}(x_{k})$ in hand for all
$p,q=1,\dotsc,P$, our second task in any given iteration (8) of Newton’s
method is to compute $(\nabla E)(x_{k})$ and $(\nabla^{2}E)(x_{k})$ using
Theorem 1; we now briefly outline an efficient means for doing so. We begin by
computing the synthesis operator
$\tilde{F}(x_{k}):=[F(x_{k})F^{*}(x_{k})]^{-1}F(x_{k})$ of the canonical dual
frame $\\{{\tilde{f}_{n}(x_{k})}\\}_{n=1}^{N}$. Here, we emphasize that it is
not necessary to explicitly compute the inverse of the frame operator
$F(x_{k})F^{*}(x_{k})$. Rather, the best algorithms for numerically computing
$\tilde{F}(x_{k})$, such as Matlab’s “pinv” command, rely on methods of
numerical linear algebra, such as QR factorization. We then write the relevant
operators in terms of $\tilde{F}^{*}(x_{k})$:
$\displaystyle\Pi(x_{k})$
$\displaystyle=\mathrm{I}-\tilde{F}^{*}(x_{k})F(x_{k}),$
$\displaystyle\Pi_{p}(x_{k})$
$\displaystyle=\tilde{F}^{*}(x_{k})\tfrac{\partial F}{\partial x_{p}}(x_{k}),$
$\displaystyle\Pi_{q,p}(x_{k})$
$\displaystyle=\tilde{F}^{*}(x_{k})\tfrac{\partial^{2}F}{\partial
x_{q}\partial x_{p}}(x_{k}),$
and compute, in the following order, the quantities:
$\displaystyle\Pi(x_{k})w,\quad\\{{\Pi_{p}(x_{k})\Pi(x_{k})w}\\}_{p=1}^{P},\quad\\{{\Pi_{p}^{*}(x_{k})w}\\}_{p=1}^{P},$
$\displaystyle\\{{\Pi(x_{k})\Pi_{p}^{*}(x_{k})w}\\}_{p=1}^{P},\qquad\\{{\Pi_{q,p}(x_{k})\Pi(x_{k})w}\\}_{p,q=1}^{P}.$
(15)
Here, to be efficient, we make use of previous computations and exploit
associativity to avoid costly matrix-matrix multiplications. For example, to
compute $\Pi_{p}(x_{k})\Pi(x_{k})w$, we take the previously computed $N\times
1$ vector $\Pi(x_{k})w$, multiply it by the $M\times N$ matrix
$\tfrac{\partial F}{\partial x_{p}}(x_{k})$ and then multiply the resulting
$M\times 1$ vector by the previously computed $N\times M$ matrix
$\tilde{F}^{*}(x_{k})$; this avoids the $\mathrm{O}(M^{2}N)$ cost of computing
$\tilde{F}^{*}(x_{k})\tfrac{\partial F}{\partial x_{p}}(x_{k})$ directly.
By rearranging some of the operators in Theorem 1, we see that every entry of
$(\nabla E)(x_{k})$ and $(\nabla^{2}E)(x_{k})$ can be found by computing inner
products of the quantities (15). Once this gradient and Hessian are found, we
then compute $x_{k+1}$ according to (8). By iterating this process, we produce
a sequence $\\{{x_{k}}\\}_{k=0}^{\infty}$ which hopefully converges to the
minimizer of $E(x)$. For the FDOA multistatic radar problem in particular,
this approach seems to work well, often successfully localizing the target;
see [8] for extensive experimentation on simulated data.
## 3 The sensitivity and uniqueness of minimizers
For any given parameters $x\in\Omega\subseteq\mathbb{R}^{P}$, the quantity
$E(x)$, as defined in (3), is the squared-distance of a given
$w\in\mathbb{R}^{N}$ from the range of the analysis operator of the frame
$\\{{f_{n}(x)}\\}_{n=1}^{N}$. In the previous section, we discussed a
numerical method for minimizing $E(x)$, that is, for finding the particular
frame(s) which are most likely to have generated a given $w$. In this section,
we consider the uniqueness of such a minimizer $x$, as well as how sensitive
it is to changes in $w$.
These two issues—sensitivity and uniqueness—are very important in real-world
applications of this minimization problem. For example, let us recall FDOA
multistatic radar where $P=M$, $f_{n}(x)$ is given by (7), and $w$ is a list
of Doppler-effect measurements, one for each of $N$ distinct pairs of
transmitters and receivers. Let $x_{0}$ and $v_{0}$ in $\mathbb{R}^{M}$ denote
a target’s position and velocity at a given instant, respectively. By (5), the
$n$th component of $F^{*}(x_{0})v_{0}\in\mathbb{R}^{N}$ is the instantaneous
rate of change of the bistatic distance (4). In a perfect radar system, the
FDOA measurements $w$ would equal $F^{*}(x_{0})v_{0}$. However, due to various
real-world issues, such as noise, quantization and an oversimplified physics
model, our actual FDOA measurements are $w=F^{*}(x_{0})v_{0}+\varepsilon$,
where $\varepsilon\in\mathbb{R}^{N}$ is some hopefully small error vector. The
value of $E(x)$ at the target’s true location $x_{0}$ is thus:
$\displaystyle E(x_{0})$
$\displaystyle=\bigl{\|}{\\{{\mathrm{I}-F^{*}(x_{0})[F(x_{0})F^{*}(x_{0})]^{-1}F(x_{0})}\\}(F^{*}(x_{0})v_{0}+\varepsilon)}\bigr{\|}^{2}$
$\displaystyle=\bigl{\|}{\\{{\mathrm{I}-F^{*}(x_{0})[F(x_{0})F^{*}(x_{0})]^{-1}F(x_{0})}\\}\varepsilon}\bigr{\|}^{2}.$
(16)
In particular, if we make the unrealistic assumption that $\varepsilon=0$,
then the target’s true location $x_{0}$ is a global minimizer of $E(x)$,
having value zero. This begs the question: for $\varepsilon\neq 0$, how far
away is the minimizer of:
$E(x):=\bigl{\|}{\\{{\mathrm{I}-F^{*}(x)[F(x)F^{*}(x)]^{-1}F(x)}\\}(F^{*}(x_{0})v_{0}+\varepsilon)}\bigr{\|}^{2}$
(17)
from $x_{0}$? Though a complete answer to this question eludes us, we are
nevertheless able to make two meaningful points.
First, since $\mathrm{I}-F^{*}(x_{0})[F(x_{0})F^{*}(x_{0})]^{-1}F(x_{0})$ is
the projection operator onto the null space of $F(x_{0})$ we have
$E(x_{0})\leq\|{\varepsilon}\|^{2}$, with equality precisely when
$\varepsilon$ lies in this null space. Thus, the target must lie somewhere in
the level set $\\{{x\in\mathbb{R}^{P}:E(x)\leq\|{\varepsilon}\|^{2}}\\}$, the
size of which is based both on the size of our measurement error $\varepsilon$
and the geometry of the surface $E(x)$. Indeed, if $E(x)$ has small curvature
at its minimizer, then even a slight error in $w$ may result in a large error
in $x_{0}$. The expressions for the gradient and Hessian of this surface,
given in Theorem 1, are our first steps towards a better understanding of this
geometry.
Second, we note that the minimizer of (17) need not be unique even when
$\varepsilon=0$. For example if $N=M$, then for any $x\in\Omega$ the frame
$\\{{f_{n}(x)}\\}_{n=1}^{N}$ is actually a basis for $\mathbb{R}^{M}$; this
implies $F(x)$ is invertible and so
$F^{*}(x)[F(x)F^{*}(x)]^{-1}F(x)=\mathrm{I}$. In this case, we therefore have
that (17) is identically zero, meaning every $x$ is a minimizer. This is not
good: for FDOA multistatic radar, this means that for any $x$, there exists a
vector $v$ such that a target with that position and velocity would yield the
measured Doppler vector $w=F^{*}(x_{0})v_{0}+\varepsilon$; it is therefore
impossible to localize the target.
In practice, we address this uniqueness problem by adding more measurements.
Indeed, even when $\varepsilon=0$, the $N$-dimensional measurement vector
$w=F^{*}(x_{0})v_{0}$ depends on $M+P$ unknowns—the $P$-dimensional vector
$x_{0}$ and the $M$-dimensional vector $v_{0}$—and so it’s reasonable to
believe that we need at least $N\geq M+P$ in order to guarantee that (17) has
$x_{0}$ as its unique minimizer. To be precise, note that in this
$\varepsilon=0$ case, the set of all minimizers of (17) is equal to its set of
zeros, namely:
$\Bigl{\\{}{x\in\Omega:\bigl{[}{\mathrm{I}-F^{*}(x)[F(x)F^{*}(x)]^{-1}F(x)}\bigr{]}w=0}\Bigr{\\}},$
(18)
which contains $x_{0}$. And though (18) equals $\Omega$ for $N=M$, our
numerical experiments [8] indicate that making $Q$ additional measurements,
that is, increasing $N$ to $M+Q$ for some $Q=1,\dotsc,P$, shrinks (18) down to
a $(P-Q)$-dimensional submanifold of $\Omega$. In particular, for $N\geq M+P$,
the set of minimizers (18) always seems to be discrete. Though we are unable
to formally prove that such behavior always holds, we are able to give the
following partial result, which shows that when a certain set of $N$ vectors
spans $\mathbb{R}^{M+P}$, then it is impossible for $E(x)$ to have a smooth
continuum of minimizers.
###### Theorem 2.
Let $\Omega$ be an open subset of $\mathbb{R}^{P}$ and let
$F\in\mathrm{C}^{1}(\Omega,\mathbb{R}^{M\times N})$, where the columns
$\\{{f_{n}(x)}\\}_{n=1}^{N}$ of $F(x)$ always form a frame for
$\mathbb{R}^{M}$. Given some $w\in\mathbb{R}^{N}$, if the vectors:
$\\{{f_{n}(x)\oplus Df_{n}(x)^{*}[F(x)F^{*}(x)]^{-1}F(x)w}\\}_{n=1}^{N}$ (19)
form a frame for $\mathbb{R}^{M}\oplus\mathbb{R}^{P}$ for every $x\in\Omega$,
then the zero set (18) of the error function (3) cannot contain a nonconstant
smooth curve. Here, $Df_{n}(x)$ denotes the $M\times P$ Jacobian matrix of
$f_{n}$ at $x$.
###### Proof.
We prove by contrapositive, assuming the set of zeros of $E(x)$ contains a
nonconstant smooth curve and proving (19) is not always a frame for
$\mathbb{R}^{M}\oplus\mathbb{R}^{P}$. To be precise, let
$x\in\mathrm{C}^{1}((-\delta,\delta),\mathbb{R}^{P})$ be a smooth curve of
zeros of $E(x)$ and let $\hat{x}:=x(0)$, $\hat{u}:=\dot{x}(0)\neq 0$.
Adopting the shorthand
$\Pi(x):=\mathrm{I}-F^{*}(x)[F(x)F^{*}(x)]^{-1}F^{*}(x)$ of the previous
section, we have $0=E(x(t))=\|{\Pi(x(t))w}\|^{2}$ for all
$t\in(-\delta,\delta)$ and so $\Pi(x(t))w=0$ for all such $t$. For any
$n=1,\dotsc,N$, letting $\psi_{n}:\Omega\rightarrow\mathbb{R}$,
$\psi_{n}(x)=\langle{\Pi(x)w},{\delta_{n}}\rangle$, we therefore have that
$\psi_{n}(x(t))=0$ for all $t\in(-\delta,\delta)$. By the chain rule, the
derivative of this equation at $t=0$ is thus:
$0=\frac{\mathrm{d}}{\mathrm{d}t}\psi_{n}(x(t))\Bigr{|}_{t=0}=\sum_{p=1}^{P}\frac{\partial\psi_{n}}{\partial
x_{p}}(x(0))\frac{\mathrm{d}x_{p}}{\mathrm{d}t}(0)=\sum_{p=1}^{P}\frac{\partial\psi_{n}}{\partial
x_{p}}(\hat{x})\hat{u}(p),$ (20)
where $\hat{u}(p)$ denotes the $p$th entry of $\hat{u}\in\mathbb{R}^{P}$. Now,
the product rule gives:
$\frac{\partial\psi_{n}}{\partial x_{p}}(\hat{x})=\frac{\partial}{\partial
x_{p}}\langle{\Pi(x)w},{\delta_{n}}\rangle\Bigr{|}_{x=\hat{x}}=\biggl{\langle}{\frac{\partial\Pi}{\partial
x_{p}}(x)w},{\delta_{n}}\biggr{\rangle}\Bigr{|}_{x=\hat{x}}.$
At this point, a computation (11) from the previous section gives:
$\frac{\partial\psi_{n}}{\partial
x_{p}}(\hat{x})=-\bigl{\langle}{[\Pi(\hat{x})\Pi_{p}^{*}(\hat{x})+\Pi_{p}(\hat{x})\Pi(\hat{x})]w},{\delta_{n}}\bigr{\rangle}.$
Moreover, since $\hat{x}$ is a zero of $E(x)$ we have $\Pi(\hat{x})w=0$, and
so this simplifies to:
$\frac{\partial\psi_{n}}{\partial
x_{p}}(\hat{x})=-\langle{\Pi(\hat{x})\Pi_{p}^{*}(\hat{x})w},{\delta_{n}}\rangle.$
(21)
Substituting (21) into (20) gives:
$0=-\sum_{p=1}^{P}\langle{\Pi(\hat{x})\Pi_{p}^{*}(\hat{x})w},{\delta_{n}}\rangle\hat{u}(p)=-\biggl{\langle}{\Pi(\hat{x})\sum_{p=1}^{P}\hat{u}(p)\Pi_{p}^{*}(\hat{x})w},{\delta_{n}}\biggr{\rangle}.$
Since $n$ is arbitrary, we have that
$\Pi(\hat{x})\sum_{p=1}^{P}\hat{u}(p)\Pi_{p}^{*}(\hat{x})=0$. This implies
that $\sum_{p=1}^{P}\hat{u}(p)\Pi_{p}^{*}(\hat{x})$ lies in the null space of
$\Pi(\hat{x})=\mathrm{I}-F^{*}(\hat{x})[F(\hat{x})F(\hat{x})^{*}]^{-1}F(\hat{x})$,
which is known to equal the range (column space) of the analysis operator
$F^{*}(\hat{x})$. In particular, there necessarily exists
$\hat{v}\in\mathbb{R}^{M}$ such that:
$F^{*}(\hat{x})\hat{v}=\sum_{p=1}^{P}\hat{u}(p)\Pi_{p}^{*}(\hat{x})w.$
As such, for any $n=1,\dots,N$,
$0=\biggl{\langle}{F^{*}(\hat{x})\hat{v}-\sum_{p=1}^{P}\hat{u}(p)\Pi_{p}^{*}(\hat{x})w},{\delta_{n}}\biggr{\rangle}=\langle{\hat{v}},{F(\hat{x})\delta_{n}}\rangle-\sum_{p=1}^{P}\hat{u}(p)\langle{w},{\Pi_{p}(\hat{x})\delta_{n}}\rangle.$
(22)
To simplify this statement, note that $F(\hat{x})\delta_{n}=f_{n}(\hat{x})$
and so:
$\displaystyle\langle{w},{\Pi_{p}(\hat{x})\delta_{n}}\rangle$
$\displaystyle=\biggl{\langle}{w},{F^{*}(\hat{x})[F(\hat{x})F^{*}(\hat{x})]^{-1}\frac{\partial
F}{\partial x_{p}}(\hat{x})\delta_{n}}\biggr{\rangle}$
$\displaystyle=\biggl{\langle}{[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w},{\frac{\partial
f_{n}}{\partial x_{p}}(\hat{x})}\biggr{\rangle}$
$\displaystyle=\biggl{\langle}{[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w},{Df_{n}(\hat{x})\delta_{p}}\biggr{\rangle}$
$\displaystyle=\bigl{(}{Df_{n}(\hat{x})^{*}[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w}\bigr{)}(p).$
As such, (22) becomes:
$\displaystyle 0$
$\displaystyle=\langle{\hat{v}},{f_{n}(\hat{x})}\rangle_{\mathbb{R}^{M}}-\langle{\hat{u}},{Df_{n}(\hat{x})^{*}[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w}\rangle_{\mathbb{R}^{P}}$
$\displaystyle=\bigl{\langle}{\hat{v}\oplus(-\hat{u})},{f_{n}(\hat{x})\oplus
Df_{n}(\hat{x})^{*}[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w}\bigr{\rangle}_{\mathbb{R}^{M}\oplus\mathbb{R}^{P}}.$
In particular, since $\hat{u}\neq 0$ then $\hat{v}\oplus(-\hat{u})$ is a
nonzero vector which is orthogonal to:
$f_{n}(\hat{x})\oplus
Df_{n}(\hat{x})^{*}[F(\hat{x})F^{*}(\hat{x})]^{-1}F(\hat{x})w$
for all $n=1,\dotsc,N$, meaning this set of vectors is not a frame for
$\mathbb{R}^{M}\oplus\mathbb{R}^{P}$. ∎
## 4 A variational approach
In the previous sections, we studied the problem of minimizing (3). As noted
in the introduction, such minimization can be used to “solve” an
overdetermined system of differential equations (5) in the least-squares sense
of minimizing (6), even when the needed data $w(t)$ is only available at a
single instant $t_{0}$. There, we treated the unknown quantities $x(t_{0})$
and $\dot{x}(t_{0})$ as independent unknowns, which reduces the problem of
minimizing (6) to (2) in the special case where $P=M$. However, this
simplification comes at a cost: in light of Theorem 2, we expect that at least
$M+P=2M$ measurements must be made in order to uniquely determine $x(t_{0})$.
When only $N<2M$ measurements are available, we are therefore led to consider
alternative approaches to the minimization of (6) in which $\dot{x}(t)$ is
properly treated as a quantity that depends on $x(t)$.
In particular, in this section, we assume that the data $w(t)$ is given over
an interval of time $[t_{0},t_{1}]$, and we seek a parameterized curve $x(t)$
that minimizes the integral of (6) over time, namely:
$\hat{E}(x):=\int_{t_{0}}^{t_{1}}\sum_{n=1}^{N}|{\langle{\dot{x}(t)},{\nabla\varphi_{n}(x(t))}\rangle-
w_{n}(t)}|^{2}\,\mathrm{d}t.$
More formally, letting $F^{*}(x)$ denote the analysis operator of the vectors
$\\{{f_{n}(x)}\\}_{n=1}^{N}$, $f_{n}(x):=\nabla\varphi_{n}(x)$ and letting
$\Omega\subseteq\mathbb{R}^{M}$ denote the set of points $x$ at which these
vectors form a frame for $\mathbb{R}^{M}$, we seek the minimizer of the
functional:
$\hat{E}:\mathrm{C}^{1}([t_{0},t_{1}],\Omega)\rightarrow\mathbb{R},\qquad\hat{E}(x):=\int_{t_{0}}^{t_{1}}\|{F^{*}(x(t))\dot{x}(t)-w(t)}\|^{2}\,\mathrm{d}t.$
(23)
In the following result, we use techniques of variational calculus to find the
Euler-Lagrange equation that any minimizer of (23) necessarily satisfies.
###### Theorem 3.
Let $\varphi_{n}\in\mathrm{C}^{2}(\Omega)$ for each $n=1,\dotsc,N$, and
$w\in\mathrm{C}^{1}([t_{0},t_{1}],\mathbb{R})$. Then any minimizer $x$ of (23)
necessarily satisfies the Euler-Lagrange equation:
$F(x(t))\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}{F^{*}(x(t))\dot{x}(t)-w(t)}\bigr{]}=0,$
(24)
for all $t\in[t_{0},t_{1}]$.
###### Proof.
We write the functional (23) as:
$\hat{E}(x)=\int_{t_{0}}^{t_{1}}\hat{e}(t,x(t),\dot{x}(t))\,\mathrm{d}t,$
where the integrand function is:
$\hat{e}:[t_{0},t_{1}]\times\Omega\times\mathbb{R}^{M}\rightarrow\mathbb{R},\qquad\hat{e}(t,x,v)=\|{F^{*}(x)v-w(t)}\|^{2}.$
Note that since each entry of $F^{*}(x)$ is a first partial derivative of some
$\varphi_{n}\in\mathrm{C}^{2}(\Omega)$, we necessarily have that $F^{*}(x)$ is
continuously differentiable over $\Omega$. This, combined with the assumption
that $w\in\mathrm{C}^{1}([t_{0},t_{1}],\mathbb{R})$ and the fact that
$\hat{e}$ is quadratic in the entries of $v$, implies that $\hat{e}$ is
continuously differentiable over
$[t_{0},t_{1}]\times\Omega\times\mathbb{R}^{M}$. Classical results from the
calculus of variations then tell us that the functional $\hat{E}$ is Fréchet-
differentiable over $\mathrm{C}^{1}([t_{0},t_{1}],\Omega)$, and that any
minimizer of $\hat{E}$ necessarily satisfies the following system of $M$
Euler-Lagrange equations:
$0=\frac{\partial\hat{e}}{\partial
x_{m}}(t,x(t),\dot{x}(t))-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\hat{e}}{\partial
v_{m}}(t,x(t),\dot{x}(t)),\quad\forall m=1,\dotsc,M.$ (25)
For any $(t,x,v)\in[t_{0},t_{1}]\times\Omega\times\mathbb{R}^{M}$, the product
rule gives the partial derivative of $\hat{e}$ with respect to the $m$th
spatial variable to be:
$\displaystyle\frac{\partial\hat{e}}{\partial x_{m}}(t,x,v)$
$\displaystyle=\frac{\partial}{\partial x_{m}}\|{F^{*}(x)v-w(t)}\|^{2}$
$\displaystyle=\frac{\partial}{\partial
x_{m}}\bigl{\langle}{F^{*}(x)v-w(t)},{F^{*}(x)v-w(t)}\bigr{\rangle}$
$\displaystyle=2\biggl{\langle}{F^{*}(x)v-w(t)},{\frac{\partial}{\partial
x_{m}}\bigl{[}{F^{*}(x)v-w(t)}\bigr{]}}\biggr{\rangle}$
$\displaystyle=2\biggl{\langle}{F^{*}(x)v-w(t)},{\frac{\partial
F^{*}}{\partial x_{m}}(x)v}\biggr{\rangle}.$ (26)
Similarly, the partial derivative with respect to the $m$th momentum variable
is:
$\displaystyle\frac{\partial\hat{e}}{\partial v_{m}}(t,x,v)$
$\displaystyle=2\biggl{\langle}{F^{*}(x)v-w(t)},{\frac{\partial}{\partial
v_{m}}\bigl{[}{F^{*}(x)v-w(t)}\bigr{]}}\biggr{\rangle}$
$\displaystyle=2\bigl{\langle}{F^{*}(x)v-w(t)},{F^{*}(x)\delta_{m}}\bigr{\rangle}.$
(27)
For a parameterized curve $x\in\mathrm{C}^{1}([t_{0},t_{1}],\Omega)$ that
minimizes $\hat{E}$, evaluating (26) and (27) at triples of the form
$(t,x,v)=(t,x(t),\dot{x}(t))$ and then substituting the results into (25)
gives:
$\displaystyle 0$
$\displaystyle=\biggl{\langle}{F^{*}(x(t))\dot{x}(t)-w(t)},{\frac{\partial
F^{*}}{\partial x_{m}}(x(t))\dot{x}(t)}\biggr{\rangle}$
$\displaystyle\qquad-\frac{\mathrm{d}}{\mathrm{d}t}\bigl{\langle}{F^{*}(x(t))\dot{x}(t)-w(t)},{F^{*}(x(t))\delta_{m}}\bigr{\rangle},\quad\forall
m=1,\dotsc,M.$ (28)
To simplify, note that the product rule gives:
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bigl{\langle}{F^{*}(x(t))\dot{x}(t)-w(t)},{F^{*}(x(t))\delta_{m}}\bigr{\rangle}$
$\displaystyle=\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}{F^{*}(x(t))\dot{x}(t)-w(t)}\bigr{]}},{F^{*}(x(t))\delta_{m}}\biggr{\rangle}$
$\displaystyle\quad+\biggl{\langle}{F^{*}(x(t))\dot{x}(t)-w(t)},{\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}}\biggr{\rangle}.$
Substituting this expression into (28) and collecting common terms gives:
$\displaystyle 0$
$\displaystyle=\biggl{\langle}{F^{*}(x(t))\dot{x}(t)-w(t)},{\frac{\partial
F^{*}}{\partial
x_{m}}(x(t))\dot{x}(t)-\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}}\biggr{\rangle}$
$\displaystyle\qquad-\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}{F^{*}(x(t))\dot{x}(t)-w(t)}\bigr{]}},{F^{*}(x(t))\delta_{m}}\biggr{\rangle},\quad\forall
m=1,\dotsc,M.$ (29)
At this point, it suffices for us to show that:
$\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}=\frac{\partial
F^{*}}{\partial x_{m}}(x(t))\dot{x}(t).$ (30)
Indeed, substituting (30) into (29) yields:
$\displaystyle 0$
$\displaystyle=\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}{F^{*}(x(t))\dot{x}(t)-w(t)}\bigr{]}},{F^{*}(x(t))\delta_{m}}\biggr{\rangle}$
$\displaystyle=\biggl{\langle}{F(x(t))\frac{\mathrm{d}}{\mathrm{d}t}\bigl{[}{F^{*}(x(t))\dot{x}(t)-w(t)}\bigr{]}},{\delta_{m}}\biggr{\rangle},\quad\forall
m=1,\dotsc,M,$
which is equivalent to our claim (24). To show that the vector equation (30)
holds, note that the $n$th coordinate of the left-hand side is:
$\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}},{\delta_{n}}\biggr{\rangle}=\frac{\mathrm{d}}{\mathrm{d}t}\bigl{\langle}{F^{*}(x(t))\delta_{m}},{\delta_{n}}\bigr{\rangle}=\frac{\mathrm{d}}{\mathrm{d}t}\bigl{\langle}{\delta_{m}},{f_{n}(x(t))}\bigr{\rangle}.$
Now recall that the $n$th frame vector $f_{n}(x)$ is defined as the gradient
of the $n$th scalar-valued function $\varphi_{n}(x)$. As such, its $m$th entry
is $\frac{\partial\varphi_{n}}{\partial x_{m}}(x)$, implying:
$\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}},{\delta_{n}}\biggr{\rangle}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\varphi_{n}}{\partial
x_{m}}(x(t))=\biggl{\langle}{\dot{x}(t)},{\Bigl{(}{\nabla\frac{\partial\varphi_{n}}{\partial
x_{m}}}\Bigr{)}(x(t))}\biggr{\rangle}.$ (31)
Now, since $\varphi_{n}\in\mathrm{C}^{2}(\Omega)$ by assumption, we know its
second partial derivatives are symmetric, implying:
$\Bigl{(}{\nabla\frac{\partial\varphi_{n}}{\partial
x_{m}}}\Bigr{)}(x)=\Bigl{(}{\frac{\partial}{\partial
x_{m}}\nabla\varphi_{n}}\Bigr{)}(x)=\frac{\partial f_{n}}{\partial
x_{m}}(x),\quad\forall x\in\Omega.$
Using this fact in (31) yields:
$\biggl{\langle}{\frac{\mathrm{d}}{\mathrm{d}t}F^{*}(x(t))\delta_{m}},{\delta_{n}}\biggr{\rangle}=\biggl{\langle}{\dot{x}(t)},{\frac{\partial
f_{n}}{\partial x_{m}}(x(t))}\biggr{\rangle}=\biggl{\langle}{\frac{\partial
F^{*}}{\partial x_{m}}(x(t))\dot{x}(t)},{\delta_{n}}\biggr{\rangle},$
which, since it holds for all $n=1,\dotsc,N$, implies our claim (30). ∎
We conclude by noting that Theorem 3 does not give an explicit algorithm for
finding the minimizer of (23). Rather, it provides a first-order, nonlinear
differential equation that any minimizer must satisfy. In practice, one way to
make use of this result is to guess a large number of initial position and
velocity combinations $(x(t_{0}),\dot{x}(t_{0}))$; for each combination, we
can then use a numerical differential equation solver to extrapolate the
target’s trajectory $x:[t_{0},t_{1}]\rightarrow\mathbb{R}^{M}$ according to
(24); we then choose the particular curve $x$ whose corresponding value
$\hat{E}(x)$ is minimal. We leave further developments of these ideas for
future research.
## Acknowledgments
We thank Laura Suzuki, William Sturgis and the anonymous reviewer for their
enlightening comments and suggestions. This work was supported by NSF DMS
1042701, NSF CCF 1017278, AFOSR F1ATA01103J001, AFOSR F1ATA00183G003 and the
A.B. Krongard Fellowship. The views expressed in this article are those of the
authors and do not reflect the official policy or position of the United
States Air Force, Department of Defense or the U.S. Government.
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|
arxiv-papers
| 2012-02-03T16:35:34 |
2024-09-04T02:49:27.016968
|
{
"license": "Public Domain",
"authors": "Matthew Fickus and Dustin G. Mixon",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1202.0757"
}
|
1202.0890
|
# Non-equilibrium spatial distribution of Rashba spin torque in ferromagnetic
metal layer
N. L. Chung g0600187@nus.edu.sg Computational Nanoelectronics and Nano-device
Laboratory, Electrical and Computer Engineering Department National University
of Singapore, 4 Engineering Drive 3, Singapore 117576. M. B. A. Jalil
Computational Nanoelectronics and Nano-device Laboratory, Electrical and
Computer Engineering Department National University of Singapore, 4
Engineering Drive 3, Singapore 117576. Information Storage Materials
Laboratory, Electrical and Computer Engineering Department National University
of Singapore, 4 Engineering Drive 3, Singapore 117576. S. G. Tan
Computational Nanoelectronics and Nano-device Laboratory, Electrical and
Computer Engineering Department National University of Singapore, 4
Engineering Drive 3, Singapore 117576. Data Storage Institute, A*STAR (Agency
for Science, Technology and Research), DSI Building, 5 Engineering Drive 1,
Singapore 117608.
###### Abstract
We study the spatial distribution of spin torque induced by a strong Rashba
spin-orbit coupling (RSOC) in a ferromagnetic (FM) metal layer, using the
Keldysh non-equilibrium Green’s function method. In the presence of the
$s$-$d$ interaction between the non-equilibrium conduction electrons and the
local magnetic moments, the RSOC effect induces a torque on the moments, which
we term the Rashba spin torque.
A correlation between the Rashba spin torque and the spatial spin current is
presented in this work, clearly mapping the spatial distribution of Rashba
spin torque in a nano-sized ferromagnetic device. When local magnetism is
turned on, the out-of-plane ($S_{z}$) Spin Hall effect (SHE) is disrupted, but
rather unexpectedly an in-plane ($S_{y}$) SHE is detected. We also study the
effect of Rashba strength ($\alpha_{R}$) and splitting exchange ($\Delta$) on
the non-equilibrium Rashba spin torque averaged over the device. Rashba spin
torque allows an efficient transfer of spin momentum such that a typical
switching field of 20 mT can be attained with a low current density of less
than $10^{7}$A/$\mathrm{cm}^{2}$.
###### pacs:
72.25.-b,72.25.Ba,74.78.Na,75.75.-c
## I Introduction
Ever since the theoretical prediction of the spin transfer torque (STT)
slonczewski:jmmm159 ; berger:9353 , there has been much research effort in
utilizing the STT phenomenon to induce magnetization switching and precession
in ferromagnetic (FM) nanostructures without the need for an externally
applied magnetic field. Devices which rely on the STT effect for magnetization
switching offer the advantages of lower power consumption and reduced device
dimension, which are crucial factors for nanoscale and high-density spintronic
applications. The STT effect has been studied in conventional magnetic
nanostructures such as spin valves katine:prl3149 and magnetic tunneling
junctions huai:apl84 . For STT to occur in these magnetic multilayers, one
requires a pair of FM layers, i.e., a reference spin layer to generate a spin-
polarized current for injection into the second free (switchable) layer. The
two layers are magnetized in a noncollinear configuration so as to induce the
transfer of the transverse spin momentum from the reference to the free layer,
which is mediated by conduction electrons flowing between the two layers. In
the above process, the role of the spin-orbit coupling (SOC) effect is
neglected. However, it is well-established that SOC can generate a
nonequilibrium spin accumulation under the passage of current. Thus, it is
conceivable that, in the presence of strong SOC effect, one can induce a STT
without the need for an additional reference FM layer. This is corroborated by
previous theoretical work which showed that the presence of Rashba spin-orbit
coupling (RSOC) whose strength is denoted by $\alpha_{R}$, and exchange
interaction $\Delta$ between conduction electrons and local spins, can give
rise to domain wall motion via spin momentum transfer obata:prb77 . The same
spin transfer mechanism can also occur in a FM layer with a large $\alpha_{R}$
and $\Delta$ values tan:annal326 ; manchon:prb78 . The predicted RSOC-induced
spin momentum transfer was experimentally demonstrated in a nanowire array
mihai:natm9 ; pi:apl97 . The above findings suggest that by utilizing Rashba-
induced STT, one can achieve magnetization switching within a single FM layer,
without an additional non-collinear FM layer. Such _single layer switching_
holds several potential advantages over conventional STT devices, such as a
more symmetric current switching profile and the reduced influence of spin
depolarization at the interfaces.
A key element which determines the feasibility of Rashba STT is the presence
of a strong Rashba SOC in the FM metal layer. Initial studies on the Rashba
effect were focused on semiconductor (SC) materials miller:prl90 ; sato:jap89
; giglberger:prb75 ; larionov:prb78 ; akabori:prb77 , especially in two-
dimensional electron gas (2DEG) heterojunction structures, which consist of
two SC layers with different energy bandgaps. The conduction electrons in the
2DEG experience a strong RSOC effect due to the large potential gradient, as a
result of the band-bending at the heterojunction interface. However, utilizing
the Rashba-induced STT in SC materials is not an attractive proposition as SCs
are intrinsically non-magnetic. Even if ferromagnetic behavior can be induced
in them via doping (e.g. in dilute magnetic semiconductors or DMS), the
resulting Curie temperature lies well below room temperature. Recent studies
have shown, however, that a strong RSOC effect can also be induced in metallic
nanostructures, both of the FM and non-FM types lashell:prl77 ; ast:prl98 ;
cercellier:prb73 ; krupin:prb71 . It is known that the Rashba SOC requires a
structural inversion asymmetry (SIA), which gives rise to an internal electric
field. In a metallic FM layer, the SIA can be enhanced by adjacent layers of
heavy metals and oxides, which create the requisite band structure mismatch
and large potential gradient at the interfaces premper:prb76 ; christian:prb77
; abdelouahed:prb82 ; dil.prl101 . By engineering the interfaces of the
metallic FM layer, one can control the strength of the RSOC effect within the
layer. The ability to enhance the RSOC coupling via interfacial effects has
led to the experimental demonstration of the effect of Rashba-induced STT, as
mentioned previously mihai:natm9 ; pi:apl97 . However, to effectively harness
this effect in future magnetic memory applications, it is essential to have an
understanding of the microscopic spin transport in the presence of the RSOC
effect, and the resulting non-equilibrium spatial distribution of the Rashba-
induced STT.
Thus, in this paper, we apply the Keldysh nonequilibrium Green’s function
(NEGF) technique to study the spin torque generated by the Rashba SOC on the
local magnetization in a metallic FM layer. The NEGF method is suitable for
the study of the Rashba STT, which is essentially driven by nonequilibrium
spin accumulation generated by the passage of current in the presence of RSOC.
In addition, the NEGF method can systematically incorporate the effects of the
leads, and interactions (RSOC and exchange coupling) as self-energy terms. In
Section II of the paper, we introduce the system Hamiltonian, consisting of
the Rashba term $H_{SO}=\alpha_{R}(\hat{\bm{z}}\times\hat{\bm{p}})$, where
$\hat{\bm{z}}$ is a unit vector parallel to the internal electric field
${\bm{E}}$, which acts perpendicular to the FM layer, $\hat{\bm{p}}$ is the
electron wavevector and $\alpha_{R}$ is the RSOC strength. The Hamiltonian
also includes the s-d interaction characterized by the exchange energy
$\Delta$, which couples the nonequilibrium spin density due to RSOC effect to
the local moments. Based on the second-quantized form of the Hamiltonian, we
apply the tight-binding NEGF formalism, and calculate various microscopic
transport quantities in the system, such as the local spin current and spin
density, and the overall spin torque generated. In Section III, we numerically
investigate (i) the spin torque efficiency as a function of the strengths of
the RSOC effect $(\alpha_{R})$, and the $s$-$d$ exchange interaction
($\Delta$), (ii) the relationship between the spin torque distribution and the
local spin currents, and (iii) the in-plane spin Hall effect arising from
RSOC. Finally, the summary of results and conclusion are presented in Section
IV.
## II Theory and Model
Figure 1: Schematic diagram of a ferromagnetic (FM) layer sandwiched between
two dissimilar materials (oxides or heavy elements) to increase the vertical
electric field $E_{z}$ and thus enhance the Rashba SOC effect. Current
$\bm{j}_{e}$ flows in the in-plane $x$-direction. The magnetization of the FM
layer $\bm{M}$ is oriented in the vertical $z$-direction.
The structure under consideration is depicted in Fig. 1. It consists of a
metallic FM layer, sandwiched between two dissimilar materials (oxides or
heavy elements) to enhance the RSOC interaction at the interfaces and within
the FM layer. The local magnetization $\bm{M}$ is oriented along the vertical
$z$-direction. A charge current $\bm{\hat{j}}_{e}$ is injected in the
$x$-direction, which generates a field $H_{\mathrm{eff}}$ along the
$\bm{\hat{y}}=(\bm{\hat{z}}\times\bm{\hat{j}}_{e}$) direction. The Hamiltonian
for the system can be expressed as:
$\displaystyle\hat{H}$ $\displaystyle=\hat{H}_{0}+\hat{H}_{so},$ (1)
$\displaystyle\hat{H}_{0}$
$\displaystyle=\frac{\hat{\bm{p}}^{2}}{2m}-\Delta({\bm{M}}\cdot\bm{\hat{S}}),$
(2) $\displaystyle\hat{H}_{so}$
$\displaystyle=\frac{\alpha_{R}}{\hbar}(\hat{\bm{p}}\times\hat{\bm{z}})\cdot\hat{\bm{S}},$
(3)
where $m$ is the free electron mass, and $\hbar$ is the reduced Planck’s
constant. Here, $H_{0}$ denotes the kinetic energy of the conduction electrons
in the FM layer, ${\bm{M}}$ is the magnetization direction, $\Delta$ is the
exchange coupling between the free electron spin and the local moments, and
$\hat{\bm{S}}=\\{\hat{S}_{j}\\}$ (where $j=\\{x,y,z\\}$) is the vector of
Pauli spin matrices. $\hat{H}_{so}$ denotes the Rashba interaction which
couples the electron spin with its momentum, with $\hat{\bm{p}}$ being the
electron momentum, and the potential gradient inducing the RSOC effect being
assumed to be in the direction $\bm{z}$ normal to the FM layer. The potential
gradient may arise from a variety of sources such as impurities, host atoms,
and structural confinement sih.nat1 ; kim.apl012504 ; szunyogh.prl96 ;
castro.prl103 ; matos.prb81 . In order to apply the many-body NEGF formalism,
the above Hamiltonian has to be recast into the second quantized form:
$\displaystyle\hat{H}_{0}$
$\displaystyle=-t_{0}\sum_{\bm{r},\sigma}\sum_{\pm}c^{{\dagger}}_{\bm{r}\sigma}(c_{\bm{r}\pm\bm{a}\sigma}+c_{\bm{r}\pm\bm{b}\sigma})+\sum_{\bm{r},\sigma}\varepsilon_{\bm{r}\sigma}c^{{\dagger}}_{\bm{r}\sigma}c_{\bm{r}\sigma}$
(4) $\displaystyle\hat{H}_{so}$
$\displaystyle=-it_{SO}\sum_{\bm{r},\sigma,\sigma^{\prime}}\sum_{\pm}c^{{\dagger}}_{\bm{r}\sigma}(\pm(\hat{S}_{x})_{\sigma\sigma^{\prime}}c_{\bm{r}\pm\bm{b}\sigma^{\prime}}\mp(\hat{S}_{y})_{\sigma\sigma^{\prime}}c_{\bm{r}\pm\bm{a}\sigma^{\prime}}).$
(5)
where $c_{\bm{r}\sigma}$($c_{\bm{r}\sigma}^{{\dagger}}$) is the fermionic
annihilation(creation) operator of an electron with spin
$\sigma=\uparrow,\downarrow$ at position $\bm{r}$. Here, $a$ is the lattice
spacing representation on a square lattice in the tight-binding NEGF
formulation , $\bm{a}=a\bm{e}_{x}$ and $\bm{b}=a\bm{e}_{y}$ are the unit
lattice vectors. $t_{0}$ represents the hopping energy between lattice points,
and is obtained by $t_{0}=\hbar/2ma^{2}$. The terms
$\varepsilon_{\bm{r}\uparrow}=4t_{0}+\Delta/2$ and
$\varepsilon_{\bm{r}\downarrow}=4t_{0}-\Delta/2$ represent the on-site energy
at the lattice site, and $t_{SO}=\alpha_{R}/2a$ is the SO coupling energy due
to the Rashba interaction.
In order to perform numerical analysis through the NEGF, the retarded
($G^{r}$) and lesser ($G^{<}$) Green’s functions are required. These are
defined as
$\displaystyle
G^{r}_{\bm{r}\sigma,\bm{r}^{\prime}\sigma^{\prime}}(t,t^{\prime})$
$\displaystyle=i\langle\\{c_{\bm{r}\sigma}(t),c^{{\dagger}}_{\bm{r^{\prime}}\sigma^{\prime}}(t^{\prime})\\}\rangle\theta(t-t^{\prime}),$
(6) $\displaystyle
G^{<}_{\bm{r}\sigma,\bm{r}^{\prime}\sigma^{\prime}}(t,t^{\prime})$
$\displaystyle=i\langle
c^{{\dagger}}_{\bm{r^{\prime}}\sigma^{\prime}}(t^{\prime})c_{\bm{r}\sigma}(t)\rangle$
(7)
After Fourier transformation, the expression for $G^{r}$ in energy space is
given by
$\displaystyle G^{r}(\epsilon)$
$\displaystyle=[\epsilon-\hat{H}-\Sigma^{r}(\epsilon)]^{-1}.$ (8)
In the above, $\Sigma^{r}=\sum_{\alpha}\Sigma_{\alpha}^{r}$ is the retarded
self-energy incurred by the lead $\alpha$, where $\alpha=L(R)$ represents the
left (right) lead. $\Sigma_{\alpha}^{r}$ can be determined by
$\Sigma_{\alpha}^{r}=V_{\alpha}g_{\alpha}^{r}V^{\dagger}_{\alpha}$, where
$V_{\alpha}$ is the coupling matrix between the lead $\alpha$ and the FM
layer, and $g_{\alpha}^{r}$ is the retarded Green’s function of the lead
$\alpha$ and can be calculated numerically by the renormalization method
nikolic:prb73 . $G^{<}$ can be calculated from the relation
$\displaystyle G^{<}(\epsilon)$
$\displaystyle=G^{r}(\epsilon)\Sigma^{<}(\epsilon)G^{a}(\epsilon).$ (9)
where $G^{a}=(G^{r})^{{\dagger}}$.
$\Sigma^{<}=\sum_{\alpha}\Sigma_{\alpha}^{<}$, where
$\Sigma_{\alpha}^{<}=if_{\alpha}\Gamma_{\alpha}$ is the lesser self-energy due
to lead $\alpha$, $f_{\alpha}$ is the Fermi function in lead $\alpha$, and
$\Gamma_{\alpha}=-2$Im$\Sigma^{r}_{\alpha}$ is the linewidth function
representing the coupling between the lead $\alpha$ and the central FM region.
Various transport properties can be evaluated once the different Green’s
functions ($G^{r}$, $G^{a}$, and $G^{<}$) have been solved via Eqs. (8) and
(9). The charge current through the system can be expressed in terms of the
different Green’s functions, as follows
$\displaystyle I_{\alpha}=$
$\displaystyle\frac{e}{h}\int^{+\infty}_{-\infty}d\epsilon\medspace\mathrm{Tr}\left\\{\left[\Sigma^{<}_{\alpha}(\epsilon)A(\epsilon)\right]-\left[\Gamma(\epsilon)_{\alpha}G^{<}(\epsilon)\right]\right\\},$
(10)
where $A(\epsilon)=i[G^{r}(\epsilon)-G^{a}(\epsilon)]$ is spectral function.
Likewise, the current-driven local spin density in the central region is
related to $G^{<}$ as follows
$\displaystyle\langle s_{i}\rangle_{m}$
$\displaystyle=\frac{\hbar}{2}\sum_{\sigma\sigma^{\prime}}(\hat{S}_{i})_{\sigma\sigma^{\prime}}\langle\hat{c}^{\dagger}_{m\sigma}\hat{c}_{m\sigma^{\prime}}\rangle$
$\displaystyle=\frac{\hbar}{4\pi
i}\int^{\infty}_{-\infty}d\epsilon\sum_{\sigma\sigma^{\prime}}(\hat{S}_{i})_{\sigma\sigma^{\prime}}G^{<}_{mm,\sigma\sigma^{\prime}}(\epsilon),$
$\displaystyle=\frac{\hbar}{4\pi
i}\int^{\infty}_{-\infty}d\epsilon\medspace\mathrm{Tr}[\hat{S}_{i}G^{<}_{mm}(\epsilon)],$
(11)
where the subscript $m$ refers to the site index.
The spin torque exerted on the local magnetization can be defined as the
difference between spin current going into and coiming out of the lattice
point. We express the spin torque as the divergence of spin
currentsalahuddin:arxiv :
$\displaystyle{\bm{\tau}}$ $\displaystyle=\mu_{B}\int
dV\nabla\cdot\bm{j}^{S_{i}},$ (12)
where $V$ is the volume, $\mu_{B}$ is the Bohr magneton, and $\bm{j}^{S_{i}}$
is the spin current density between lattice points. The spin torque
${\bm{\tau}}$ can be also be defined as
$\bm{\tau}=-\gamma\bm{M}\times{\bm{H}}$, where $\gamma$ is the gyromagnetic
ratio. We focus on the effective field $H_{\mathrm{eff}}$ induced by Rashba
SOC which acts on the local moments along the
$\bm{\hat{y}}=(\bm{\hat{z}}\times\bm{\hat{j}}_{e}$) direction. Thus, the
effective field due to RSOC is
$\displaystyle H_{\mathrm{eff}}$ $\displaystyle=H_{y}=\frac{\tau_{x}}{\gamma
M_{s}V},$ (13)
where $M_{s}$ is the saturation magnetization, and $\tau_{x}$ is obtained from
Eq. (12). The torque efficiency is then given by ratio of
$H_{\mathrm{eff}}/{j}_{e}$.
Under steady-state condition and in the absence of dissipative processes, the
spin torque ${\bm{\tau}}$, as defined according to Eq. (12), is related to the
divergence of the spin current. By considering the Heisenberg equation of
motion, the local spin bond current between sites $m$ and $m^{\prime}$ can be
expressed in terms of $G^{<}$ nikolic:prb73 ; hattori:JPSJ78 , i.e.
$\displaystyle\langle\hat{j}^{s_{i}}_{mm^{\prime}}\rangle$
$\displaystyle=\langle\hat{j}^{s_{i}(kin)}_{mm^{\prime}}\rangle+\langle\hat{j}^{s_{i}(SO)}_{mm^{\prime}}\rangle,$
$\displaystyle\langle\hat{j}^{s_{i(kin)}}_{mm^{\prime}}\rangle$
$\displaystyle=\frac{et_{0}}{2\hbar}\int^{\infty}_{-\infty}\frac{d\epsilon}{2\pi}\medspace\mathrm{Tr}[\hat{S}_{i}(G^{<}_{m^{\prime}m}(\epsilon)-G^{<}_{mm^{\prime}}(\epsilon))]$
$\displaystyle\langle\hat{j}^{s_{i(SO)}}_{mm^{\prime}}\rangle$
$\displaystyle=[\bm{e}_{i}\times(\bm{m^{\prime}}-\bm{m})]_{z}\frac{et_{SO}}{2\hbar}\int^{\infty}_{-\infty}\frac{d\epsilon}{2\pi
i}\medspace\mathrm{Tr}[(G^{<}_{m^{\prime}m}(\epsilon)+G^{<}_{mm^{\prime}}(\epsilon))],$
(14)
where $(\bm{m^{\prime}}-\bm{m})$ represents the unit vector between
neighbouring sites on the $x$-$y$ plane and $\bm{e}_{i}$ represents the unit
vector of spin $\langle S_{i}\rangle$. The above expression for the bond spin
current comprises of two terms, i.e., the kinetic and SO coupling terms,
arising from the corresponding terms in the Hamiltonian of Eqs. (2) and (3).
By considering Eqs. (12) and (14) together, the spin torque is then given by
the divergence of the spin bond current $(\nabla\cdot\bm{j}^{s})$, which in
the discretized tight-binding model is approximated as:nikolic:prb73 ;
hattori:JPSJ78
$\displaystyle\tau_{x(\bm{m})}$
$\displaystyle=-\mu_{B}\left(\langle\hat{j}^{s_{z}}_{\bm{m},\bm{m}+\bm{e}_{x}}\rangle+\langle\hat{j}^{s_{z}}_{\bm{m}-\bm{e}_{x},\bm{m}}\rangle\right)/L_{SO},$
(15) $\displaystyle\tau_{y(\bm{m})}$
$\displaystyle=-\mu_{B}\left(\langle\hat{j}^{s_{z}}_{\bm{m},\bm{m}+\bm{e}_{y}}\rangle+\langle\hat{j}^{s_{z}}_{\bm{m}-\bm{e}_{y},\bm{m}}\rangle\right)/L_{SO},$
(16) $\displaystyle\tau_{z(\bm{m})}$
$\displaystyle=\mu_{B}\left(\langle\hat{j}^{s_{x}}_{\bm{m},\bm{m}+\bm{e}_{x}}\rangle+\langle\hat{j}^{s_{x}}_{\bm{m}-\bm{e}_{x},\bm{m}}\rangle+\langle\hat{j}^{s_{y}}_{\bm{m},\bm{m}+\bm{e}_{y}}\rangle+\langle\hat{j}^{s_{y}}_{\bm{m}-\bm{e}_{y},\bm{m}}\rangle\right)/L_{SO},$
(17)
where $L_{SO}$ is the spin precession length (over which spin precesses by 1
radian), and can be expressed as $L_{SO}=\frac{\pi at_{0}}{2t_{SO}}$. The
above constitutes to the spin torque expression of Eq. (12).
## III Results and Discussion
Based on the tight-binding NEGF formulation presented in the above section, we
performed numerical calculations of transport parameters such as the local
spin density, bond spin current, and the effective field $H_{\mathrm{eff}}$ in
order to analyze the effect of RSOC induced non-equilibrium spatial spin
torque on the FM layer structure. In our calculations, the following parameter
values are assumed, unless otherwise stated:
$\alpha_{R}=10^{-11}~{}\mathrm{eVm}$, $m=9.1\times 10^{-31}$ kg,
$a=0.05~{}\mathrm{nm}$, $E_{F}=7.83~{}\mathrm{eV}$, $M_{s}=1.09\times
10^{6}~{}\mathrm{Am^{-1}}$, $\Delta=1.6~{}\mathrm{eV}$ Miranda.surf117 , and
room temperature $T=300$ K. The Fermi energy $E_{F}$ and saturation
magnetization $M_{s}$ assume exemplary values corresponding to that of Co.
Figure 2: The dependence of the effective current induced field
($H_{\mathrm{eff}}$) due to the Rashba spin torque is plotted as a function of
charge current density (${j}_{e}$) for (a) varying Rashba strength
$\alpha_{R}$ with a fixed exchange coupling $\Delta=1.6$ eV, and (b) varying
exchange coupling $\Delta$ with a fixed $\alpha_{R}=10^{-10}$ eVm. In (c), the
spin torque efficiency ($H_{\mathrm{eff}}/j_{e}$) is plotted as a function of
both $\Delta$ and $\alpha_{R}$. In the calculations, we assume the dimension
of the sample to be $50a\times 50a$, where $a=0.05$ nm.
We first analyze the role of two key parameters $\alpha_{R}$ and $\Delta$ in
determining the strength of the effective field $H_{\mathrm{eff}}$ and the
torque efficiency of the system. Figs. 2 and 2 show that, with a fixed
$\alpha_{R}$ and $\Delta$ respectively, $H_{\mathrm{eff}}$ increases linearly
with $j_{e}$. This trend is consistent with the prediction that
$\displaystyle
H_{\mathrm{eff}}=\frac{\alpha_{R}P}{\mu_{0}\mu_{B}}(\hat{\bm{z}}\times\bm{j}_{e}),$
(18)
derived from either gauge formulation tan:annal326 or from semiclassical
(Boltzmann) transport equation manchon:prb78 in the strong coupling limit.
Eq. (18) is a global expression of spin torque under linear response. In the
gauge formulation, the factor $P$ assumes a value of $\frac{1}{2}$ in the
adiabatic limit, while in the Boltzmann model, it refers to the spin
polarization of current. We now consider the torque efficiency, which is given
by the gradient of $H_{\mathrm{eff}}$ with respect to $j_{e}$. As can be seen
from Figs. 2 and 2, the torque efficiency is generally enhanced with increase
in either $\alpha_{R}$ and $\Delta$. However, in our non-equilibrium spatial
treatment, it is clear from the plot in Fig. 2 that the torque efficiency does
not vary linearly with $\alpha_{R}$, unlike the prediction of Eq. (18). The
difference can be accounted for by noting that the global expression of Eq.
(18) is derived in the limit of large coupling $\Delta$, i.e., up to only the
linear order in $\frac{\alpha_{R}}{\Delta}$. In our model, as can be seen from
Fig. 2, the torque efficiency shows a slight oscillatory dependence
superimposed upon a general increase with respect to $\alpha_{R}$, especially
at the region of $\alpha_{R}<10^{-10}$ eVm. However, at the region where
$\alpha_{R}\geq 10^{-10}$ eVm, its behavior is similar to the prediction
derived from the Boltzmann semiclassical model for arbitrary coupling strength
manchon:prb78 . From the effective field $H_{\mathrm{eff}}$, one can estimate
the critical current density required for magnetization switching. In Figs. 2
and 2, we consider RSOC strengths ranging from $10^{-11}$ to $10^{-10}$ eVm,
which roughly corresponds to the practical values observed at the interfaces
with heavy metal or oxide layers. Assuming an exemplary spin polarization of
$P=\frac{\Delta}{E_{F}}\approx 0.5$, RSOC strength of $\alpha_{R}=10^{-10}$
eVm, and a switching field of $H_{s}\approx 0.02$ T applicable for Co nanowire
structures mihai:natm9 , we find that the critical current density for
switching is approximately $10^{6}$ A/$\mathrm{cm}^{2}$ [see Fig. 2]. This is
significantly lower than the critical current density of the order of $10^{7}$
A/$\mathrm{cm}^{2}$ for the case of the conventional Slonczewski spin torque
in spin valve structures jiang:prl92 ; sukegawa:apl96 .
Figure 3: The spatial distribution of the (a) Rashba effect spin torque
$\tau_{x}$ and its correlation with the local spin current $\langle
j^{s_{z}}_{mm^{\prime}}\rangle$ by setting $\alpha_{R}$ to $0.5\times
10^{-10}$ eVm, (b) $\tau_{x}$ and its correlation with $\langle
j^{s_{z}}_{mm^{\prime}}\rangle$ by setting $\alpha_{R}$ to $1.5\times
10^{-10}$ eVm. The spin torque density is expressed in units of
$\mu_{B}/L_{SO}$. The sample has a lateral size of $50a\times 50a$.
Next, we examine the relationship between the Rashba-induced torque $\tau$ and
the spatial distribution of the spin currents. In Fig. 3, we plot the spin
torque component $\tau_{x}$ based on the torque definition of Eq. (15), which
relates it to the divergence of the local spin bond current
$j^{s_{z}}_{mm^{\prime}}$. For comparison, we plot the spatial distribution of
the spin bond current $j^{s_{z}}_{mm^{\prime}}$ in Fig. 3. We observe a close
correlation between the spatial distribution of $\tau_{x}$ and the flow of the
$z$-polarized spin current $j^{s_{z}}_{mm^{\prime}}$. The presence of RSOC
causes a vortex-like flow of the bond spin current $j^{s_{z}}_{mm^{\prime}}$
as shown in Fig. 3. Regions where $j^{s_{z}}_{mm^{\prime}}$ is flowing in the
$+x$ ($-x$) direction corresponds to a large positve (negative) $\tau_{x}$.
Conversely, in regions where the positive and negative spin current fluxes
meet and cancel each other, the spin torque $\tau_{x}$ becomes small. When the
Rashba coupling strength $\alpha_{R}$ is increased, the magnitude of
$\tau_{x}$ is generally larger since it scales with $\alpha_{R}$, as shown in
Fig. 2. In addition, the vortices associated with the spin current become
spatially smaller. This may be attributed to the increase in the rate of spin
precession of the conduction electrons with $\alpha_{R}$. The increased
density of the vortices result in some cancelation of the bond spin currents
near the center of the FM layer, so that more of the bond spin current flows
at the boundaries, as shown in Fig. 3.
(a) $\Delta=0$, $\alpha_{R}=10^{-10}$ eVm, $\langle s_{z}\rangle_{m}$. SHE
recovered.
(b) $\Delta=0$, $\alpha_{R}=10^{-10}$ eVm, $\langle s_{y}\rangle_{m}$. Absence
of SHE.
(c) $\Delta=1.6$ eV, $\alpha_{R}=5\times 10^{-11}$ eVm, $\langle
s_{z}\rangle_{m}$. SHE disrupted
(d) $\Delta=1.6$ eV, $\alpha_{R}=5\times 10^{-11}$ eVm, $\langle
s_{y}\rangle_{m}$. SHE detecteded.
(e) $\Delta=1.6$ eV, $\alpha_{R}=1.5\times 10^{-10}$ eVm, $\langle
s_{y}\rangle_{m}$. SHE disrupted.
Figure 4: The spatial distribution of the spin density (a) $\langle
s_{z}\rangle_{m}$, (b) $\langle s_{y}\rangle_{m}$, both with $\Delta=0$ eV,
$\alpha_{R}=1\times 10^{-10}$ eVm, (c) $\langle s_{z}\rangle_{m}$, (d)
$\langle s_{y}\rangle_{m}$, both with $\Delta=1.6$ eV, $\alpha_{R}=1.5\times
10^{-10}$ eVm. In (e) $\langle s_{y}\rangle_{m}$ is plotted with a larger
$\alpha_{R}=1.5\times 10^{-10}$ eVm, and $\Delta=1.6$ eV. The sample has a
lateral size of $50a\times 50a$.
Finally, we analyze the spin density distribution and its dependence on the
exchange strength $\Delta$. Figs. 4(a) and 4(b) plot the spin density of
$\langle s_{z}\rangle_{m}$ in the absence and presence of $\Delta$,
respectively. In the absence of exchange coupling ($\Delta=0$), the
distribution profile of $\langle s_{z}\rangle_{m}$ clearly indicates a
transverse separation of the $z$-spins, i.e. an out-of-plane spin Hall effect.
This agrees with previous calculations based on the multimode scattering
matrix method which predict a spin-Hall like separation of the out-of-plane
spin component in the presence of Rashba effect brusheim:prb74 . However, the
clear out-of-plane spin Hall separation disappears when a sizable exchange
$\Delta$ is present, as shown in Fig. 4(c). It is found that the magnitude of
$\langle s_{z}\rangle_{m}$ assumes a much larger value throughout the FM
layer. This increase may be attributed to the alignment of the electron spin
to the local moments oriented along the $z$-direction. We also analyze the in-
plane spin density $\langle s_{y}\rangle_{m}$ distribution, as shown in Figs.
4(b), 4(d), and 4(e). There is no transverse separation of the in-plane spin
density in the absence of $\Delta$ [Fig. 4(b)]. This is in line with
theoretical prediction where the spin Hall effect induced by RSOC applies only
to out-of-plane spins. However, in the presence of strong exchange coupling
$\Delta$, an “in-plane” spin Hall effect is present [Fig. 4(d)]. This in-plane
spin Hall effect is destroyed in the presence of a strong Rashba strength,
i.e. when $\alpha_{R}$ is increased to $1.5\times 10^{-10}$ eVm [Fig. 4(e)].
This may be explained by noting that a large RSOC strength increases the rate
of spin precession. Thus, the in-plane spin density $\langle s_{y}\rangle_{m}$
oscillates and changes signs along the direction of electron propagation
($x$-direction), as can be seen in Fig. 4(d).
## IV Conclusion
In summary, we have studied the non-equilibrium spatial intrinsic spin torque
induced by Rashba spin orbit coupling in a ferromagnetic metal layer. Unlike
the conventional Slonczewski spin torque, the Rashba induced torque is
generated within a single layer, i.e. it does not require spin injection from
an another ferromagnetic reference layer. We analyze the effect of two crucial
parameters determining the strength of the Rashba spin torque: (i) the
strength $\alpha_{R}$ of the RSOC effect which is responsible for polarizing
the injected charge current, and (ii) the exchange splitting $\Delta$ which
couples the conduction electron to the local FM moments, thus allowing the
transfer of spin momentum to the latter. The spin transport through the system
is modeled via the tight-binding non-equilibrum Green’s function (NEGF)
formalism. The NEGF theory systematically incorporates many-body effects
including interactions with the leads as self-energy terms, and enables
current and spin density to be evaluated spatially under nonequilibrium (bias-
driven) conditions. Based on the NEGF theory, we numerically evaluate various
transport parameters of the system, such as the effective field
$H_{\mathrm{eff}}$ due to the spin torque, and the spatial distribution of the
non-equilibrium spin current and spin accumulation. We found that
$H_{\mathrm{eff}}$ generally increases with both the RSOC strength
$\alpha_{R}$ and the exchange coupling $\Delta$. However, the dependence of
$H_{\mathrm{eff}}$ on both parameters is not totally linear, unlike previous
predictions based on gauge formulation or semiclassical Boltzmann which are
global and only partially non-equilibrium (linear response), and in the strong
coupling limits. For practical values of $\Delta$ and $\alpha_{R}$, the
calculated critical current density corresponding to a typical switching field
of 200 mT is calculated to be lower than $10^{7}$ A/$\mathrm{cm}^{2}$,
comparable to that obtained via the conventional Slonczewski spin torque. For
the structure under consideration where net current is in the $x$-direction
and the local moments are aligned in the vertical $z$-direction, the net
effective field (spin torque) is in the $y$ ($x$)-direction. We plot the
spatial profile of the $x$-component of the spin torque $\tau_{x}$, which
bears a close correlation to that of the $z$-polarized bond spin current. It
is also observed that the Rashba torque $\tau_{x}$ is concentrated near the
boundaries of the FM layer. We also found that the combined presence of RSOC
effect and exchange coupling $\Delta$ induces a Hall separation of in-plane
spins, whereas the spin Hall effect for out-of-plane spins disappear with the
introduction of $\Delta$. Our calculations predict an effective field
$H_{\mathrm{eff}}$ of the order of 1 Tesla for a current density of $10^{7}$
$\mathrm{A/cm}^{2}$, thus indicating the feasibility of utilizing the Rashba
induced spin torque to achieve magnetization switching in spintronic
applications.
## References
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|
arxiv-papers
| 2012-02-04T10:42:51 |
2024-09-04T02:49:27.028291
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. L. Chung, M. B. A. Jalil and S. G. Tan",
"submitter": "Nyukleong Chung",
"url": "https://arxiv.org/abs/1202.0890"
}
|
1202.0895
|
# Causal Rate Distortion Function on Abstract Alphabets: Optimal
Reconstruction and Properties
Photios A. Stavrou, Charalambos D. Charalambous and Christos K. Kourtellaris
ECE Department, University of Cyprus, Green Park, Aglantzias 91,
P.O. Box 20537, 1687, Nicosia, Cyprus
e-mail: stavrou.fotios@ucy.ac.cy, chadcha@ucy.ac.cy,
kourtellaris.christos@ucy.ac.cy
###### Abstract
A causal rate distortion function with a general fidelity criterion is
formulated on abstract alphabets and a coding theorem is derived. Existence of
the minimizing kernel is shown using the topology of weak convergence of
probability measures. The optimal reconstruction kernel is derived, which is
causal, and certain properties of the causal rate distortion function are
presented.
## I INTRODUCTION
Given a distortion or fidelity constraint between source sequences
$X^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{i}\\}_{i=0}^{\infty}\in{\cal
X}^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{\infty}{\cal
X}_{i}$ and reproduction sequences
$Y^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{i}\\}_{i=0}^{\infty}\in{\cal
Y}^{\infty}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{\infty}{\cal
Y}_{i}$, non-causal codes achieve the rate distortion function (RDF) of the
source, which is the optimal performance theoretically attainable. The RDF is
described in [1] for memoryless sources, in [2] for stationary ergodic
sources, in [3] for information and distortion stable processes, and in [4]
using the information spectrum method. The RDF for general sources on Polish
spaces (complete separable metric spaces) and its properties are discussed
extensively in [5].
Causal codes as defined in [6] are a sub-class of non-causal codes, with the
addition constraint on the reproduction coder (cascade of encoder-decoder)
such that $Y_{i}$ depends on the past and present source symbols
$\\{X_{0},X_{1},\ldots,X_{i}\\}$ but not on the future symbols
$\\{X_{i+1},X_{i+2},\ldots\\}$, thus, $Y_{i}=f_{i}(X_{0},X_{1},\ldots,X_{i})$
$\forall{i}$, where $\\{f_{i}\\}_{i=0}^{\infty}$ are measurable functions
called reproduction coders.
Causal codes are extensively analyzed in [6] using entropy type criteria
(entropy of reproduction coder), further investigated in [7] where side
information is present, while [8] consider stationary sources at high
resolution. The rate loss due to causality for Gaussian stationary sources
with memory and mean square distortion is analyzed in [9], and recently in
[10].
Zero-delay codes are a sub-class of causal codes, with the additional
constraint on the reproduction coder such that the reproduction $Y_{i}$ is
done at the same time the corresponding source symbol $X_{i}$ is encoded, that
is, both encoding and decoding are done causally. Sequential codes as defined
in [11] and applied in [12] are causal zero-delay codes such that the
reproduction of each source symbol is done sequentially following the time
ordering $X_{0},Y_{0},X_{1},Y_{1},\ldots$.
The objective of this paper is to impose a causal constraint on the
reproduction coder, and formulate the causal source coding problem with
fidelity criterion via rate distortion theory, on general alphabets using the
topology of weak convergence of probability measures. The results include the
following.
* 1)
Information theoretic definition of causal rate distortion function as an
optimization problem in which the reproduction conditional distribution
satisfies a causality constraint.
* 2)
Source coding theorem for directed information and distortion stable
processes.
* 3)
Expression of the optimal causal reconstruction distribution and properties of
the causal RDF.
Causal Rate Distortion Function (CRDF).
The precise definition of causal codes is stated below and it is found in [6].
###### Definition I.1
(Causal Reproduction Coder) A reproduction coder is called causal if for all
$i\leq{n}$
$\displaystyle
f_{i}(x^{n})=f_{i}(\tilde{x}^{n})~{}\mbox{whenever}~{}x^{i}=\tilde{x}^{i}$
A source code is called causal if its induced reproduction coder is causal.
From Definition I.1, it follows that the reproduction coder is causal if and
only if the following Markov chain holds
$(X_{i+1},X_{i+2},\ldots)\Leftrightarrow(X^{i},Y^{i-1})\Leftrightarrow{Y_{i}}$,
$i=0,1,\ldots$.
Assume an average distortion constraint
$\displaystyle
E\big{\\{}d_{0,n}(X^{n},Y^{n})\big{\\}}\leq{D},\>d_{0,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\frac{1}{n+1}\sum^{n}_{i=0}\rho_{i}(x^{i},y^{i})$
where $D\geq 0$, $d_{0,n}(\cdot,\cdot)$ a non-negative distortion function.
Consider causal reproduction coders defined in Definition I.1. Define the
causal convolution of conditional distributions by
$\displaystyle{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes^{n}_{i=0}P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})$
Since a reproduction coder is causal if and only if the above Markov chain
holds, then the reproduction conditional distribution of a causal coder
satisfies
$\displaystyle{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n}),~{}P-a.s$
(1)
Substituting (1) into mutual information $I(X^{n};Y^{n})$ it follows that for
causal coders the information theoretic RDF for which an operational meaning
will be saught, is given by
$\displaystyle R^{c}(D)$
$\displaystyle=\lim_{n\rightarrow\infty}\inf_{{\overrightarrow{P}}_{Y^{n}|X^{n}}:E\big{\\{}d_{0,n}(X^{n},Y^{n})\big{\\}}\leq{D}}\frac{1}{n+1}\int_{{\cal
X}_{0,n}\times{\cal Y}_{0,n}}$
$\displaystyle\log\Big{(}\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{{P}_{Y^{n}}(dy^{n})}\Big{)}{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})P_{X^{n}}(dx^{n})$
$\displaystyle=\lim_{n\rightarrow\infty}\inf\frac{1}{n+1}{\mathbb{I}}_{X^{n}{\rightarrow}Y^{n}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$
(2)
where the joint distribution $P_{X^{n},Y^{n}}(dx^{n},dy^{n})$ for causal codes
is uniquely defined by
$P_{X^{n},Y^{n}}(dx^{n},dy^{n})={\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\otimes{P}_{X^{n}}(dx^{n})$.
Note that (2) is precisely the expression consider in [11] to derive coding
theorem for sequential codes. It is easy to verify that
${\mathbb{I}}_{X^{n}{\rightarrow}Y^{n}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})$
is the directed information from $X^{n}$ to $Y^{n}$,
$I(X^{n}\rightarrow{Y^{n}})\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{n}{I}(X^{i};Y_{i}|Y^{i-1})$,
subject to the requirement that the source is not affected by past
reconstruction symbols, that is,
$\otimes_{i=0}^{n}{P}_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1})=\otimes_{i=0}^{n}{P}_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1})=P_{X^{n}}(dx^{n})$.
However, if the causality constraint (1) is not imposed, the conditional
distribution ${\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$ in (2) should
be replaced by ${P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$, and the resulting expression
is the classical RDF. Since, by the chain rule
${P}_{Y^{n}|X^{n}}(dy^{n}|X^{n}=x^{n})=\otimes^{n}_{i=0}P_{Y_{i}|Y^{i-1}=y^{i-1},X^{n}=x^{n}}(dy_{i}|Y^{i-1}=y^{i-1},X^{n}=x^{n})$,
in general the classical RDF solution yields reconstructions of $Y_{i}=y_{i}$
which depends on future values of the source symbols
$(X_{i+1}=x_{i+1},\ldots,X_{n}=x_{n})$, in addition to its past reconstruction
symbols $Y^{i-1}=y^{i-1}$, and past and present symbols $X^{i}=x^{i}$. On the
other hand (1) implies causality.
## II PROBLEM FORMULATION AND CODING THEOREMS
Let $\mathbb{N}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,\ldots,n\\}$,
$n\in\mathbb{N}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,2,\ldots\\}$. The
source and reconstruction alphabets are sequences of Polish spaces [13]
$\\{{\cal X}_{t}:t\in\mathbb{N}\\}$ and $\\{{\cal Y}_{t}:t\in\mathbb{N}\\}$,
respectively, (e.g., ${\cal Y}_{t},{\cal X}_{t}$ are complete separable metric
spaces), associated with their corresponding measurable spaces $({\cal
X}_{t},{\cal B}({\cal X}_{t}))$ and $({\cal Y}_{t},{\cal B}({\cal Y}_{t}))$.
Sequences of alphabets are identified with the product spaces $({\cal
X}_{0,n},{\cal B}({\cal
X}_{0,n}))\stackrel{{\scriptstyle\triangle}}{{=}}\times_{k=0}^{n}({\cal
X}_{k},{\cal B}({\cal X}_{k}))$, and $({\cal Y}_{0,n},{\cal B}({\cal
Y}_{0,n}))\stackrel{{\scriptstyle\triangle}}{{=}}\times_{k=0}^{n}({\cal
Y}_{k},{\cal B}({\cal Y}_{k}))$. The source and reconstruction are processes
denoted by
$X^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{n}:n\in\mathbb{N}^{n}\\}$,
$X_{n}\in{\cal X}_{n}$, and by
$Y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{n}:n\in\mathbb{N}^{n}\\}$,
$Y_{n}\in{\cal Y}_{n}$, respectively. Probability measures on any measurable
space $({\cal Z},{\cal B}({\cal Z}))$ are denoted by ${\cal M}_{1}({\cal Z})$.
It is assumed that the $\sigma$-algebras
$\sigma\\{X^{-1}\\}=\sigma\\{Y^{-1}\\}=\\{\emptyset,\Omega\\}$.
###### Definition II.1
Let $({\cal X},{\cal B}({\cal X})),({\cal Y},{\cal B}({\cal Y}))$ be
measurable spaces in which $\cal Y$ is a Polish Space.
A stochastic Kernel on $\cal Y$ given $\cal X$ is a mapping $q:{\cal B}({\cal
Y})\times{\cal X}\rightarrow[0,1]$ satisfying the following two properties:
1) For every $x\in{\cal X}$, the set function $q(\cdot;x)$ is a probability
measure (possibly finitely additive) on ${\cal B}({\cal Y}).$
2) For every $F\in{\cal B}({\cal Y})$, the function $q(F;\cdot)$ is ${\cal
B}({\cal X})$-measurable.
The set of all such stochastic Kernels is denoted by ${\cal Q}({\cal Y};{\cal
X})$.
Stochastic kernels are classified into non-causal and causal as follows.
###### Definition II.2
Given measurable spaces $({\cal X}_{0,n},{\cal B}({\cal X}_{0,n}))$, $({\cal
Y}_{0,n},{\cal B}({\cal Y}_{0,n}))$, and their product spaces, data
compression channels are defined as follows.
1. 1.
A Non-Causal Data Compression Channel is a stochastic kernel
$q_{0,n}(dy^{n};x^{n})\in{\cal Q}({\cal Y}_{0,n};{\cal
X}_{0,n}),n\in\mathbb{N}$.
2. 2.
A Causal Product Data Compression Channel is a convolution of a sequence of
causal stochastic kernels defined by
$\displaystyle{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{i=0}^{n}q_{i}(dy_{i};y^{i-1},x^{i})$
where $q_{i}\in{\cal Q}({\cal Y}_{i};{\cal Y}_{0,i-1}\times{\cal
X}_{0,i}),i=0,\ldots,n,~{}n\in\mathbb{N}$. The set of such convolution of
causal kernels is denoted by $\overrightarrow{\cal Q}({\cal Y}_{0,n};{\cal
X}_{0,n})$.
### II-A Information Theoretic Causal Rate Distortion Function
This section gives the abstract formulation of $R^{c}(D)$. Given a source
probability measure ${\cal\mu}_{0,n}\in{\cal M}_{1}({\cal X}_{0,n})$ and a
reconstruction kernel ${\overrightarrow{q}}_{0,n}\in\overrightarrow{\cal
Q}({\cal Y}_{0,n};{\cal X}_{0,n})$ consistent with causal reproduction coder,
define the following probability measures.
P1: The joint measure $P_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n}\times{\cal
X}_{0,n})$:
$\displaystyle P_{0,n}(G_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}(\mu_{0,n}\otimes{\overrightarrow{q}}_{0,n})(G_{0,n}),\>G_{0,n}\in{\cal
B}({\cal X}_{0,n})\times{\cal B}({\cal Y}_{0,n})$ $\displaystyle=\int_{{\cal
X}_{0,n}}{\overrightarrow{q}}_{0,n}(G_{0,n,x^{n}};x^{n})\mu_{0,n}(d{x^{n}})$
where $G_{0,n,x^{n}}$ is the $x^{n}-$section of $G_{0,n}$ at point ${x^{n}}$
defined by
$G_{0,n,x^{n}}\stackrel{{\scriptstyle\triangle}}{{=}}\\{y^{n}\in{\cal
Y}_{0,n}:(x^{n},y^{n})\in G_{0,n}\\}$ and $\otimes$ denotes the convolution.
P2: The marginal measure $\nu_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n})$:
$\displaystyle\nu_{0,n}(F_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}P_{0,n}({\cal
X}_{0,n}\times F_{0,n}),~{}F_{0,n}\in{\cal B}({\cal Y}_{0,n})$
$\displaystyle=\int_{{\cal
X}_{0,n}}{\overrightarrow{q}}_{0,n}(F_{0,n};x^{n})\mu_{0,n}(dx^{n})$
P3: The product measure $\pi_{0,n}:{\cal B}({\cal X}_{0,n})\times{\cal
B}({\cal Y}_{0,n})\mapsto[0,1]$ of $\mu_{0,n}\in{\cal M}_{1}({\cal X}_{0,n})$
and $\nu_{0,n}\in{\cal M}_{1}({\cal Y}_{0,n})$:
$\displaystyle\pi_{0,n}(G_{0,n})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}(\mu_{0,n}\times\nu_{0,n})(G_{0,n}),~{}G_{0,n}\in{\cal
B}({\cal X}_{0,n})\times{\cal B}({\cal Y}_{0,n})$ $\displaystyle=\int_{{\cal
X}_{0,n}}\nu_{0,n}(G_{0,n,x^{n}})\mu_{0,n}(dx^{n})$
The precise information measure used to define CRDF is the mutual information
between two sequences of random processes $X^{n}$ and $Y^{n}$ whose
distributions are consistent with the definition of the causal reproduction
coder, e.g., generated via ${\bf P1}$-${\bf P3}$. Hence, by the construction
of probability measures ${\bf P1}$-${\bf P3}$, and the chain rule of relative
entropy [13]:
$\displaystyle
I(X^{n};Y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{D}(P_{0,n}||\pi_{0,n})$
(3) $\displaystyle=\int_{{\cal X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{d(\mu_{0,n}\otimes{\overrightarrow{q}}_{0,n})}{d(\mu_{0,n}\times\nu_{0,n})}\Big{)}d(\mu_{0,n}\otimes{\overrightarrow{q}}_{0,n})$
$\displaystyle=\int\log\Big{(}\frac{{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})}{\nu_{0,n}(dy^{n})}\Big{)}{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})\mu_{0,n}(dx^{n})$
(4)
$\displaystyle\equiv\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
(5)
Note that $(\ref{re3})$ states that mutual information is expressed as a
functional of $\\{\mu_{0,n},{\overrightarrow{q}}_{0,n}\\}$ denoted by
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$.
Also, if the causality assumption on the reproduction coder is not imposed,
then $I(X^{n};Y^{n})=\mathbb{I}(\mu_{0,n},{q}_{0,n})$, which is how classical
RDF is defined.
The next lemma gives equivalent statements which are consistent with causal
reproduction coders in terms of causal convolution of reconstruction kernels,
mutual information, directed information, and conditional independence.
###### Lemma II.3
The following are equivalent for each $n\in\mathbb{N}$.
1. 1.
$q_{0,n}(dy^{n};x^{n})={\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$ a.s., where
${\overrightarrow{q}}_{0,n}$ is given in Definition II.2-2).
2. 2.
For each $i=0,1,\ldots,n-1$,
$Y_{i}\Leftrightarrow(X^{i},Y^{i-1})\Leftrightarrow(X_{i+1},X_{i+2},\ldots,X_{n})$,
forms a Markov chain.
3. 3.
$I(X^{n};Y^{n})=I(X^{n}\rightarrow Y^{n})$.
4. 4.
For each $i=0,1,\ldots,n-1$, $Y^{i}\Leftrightarrow X^{i}\Leftrightarrow
X_{i+1}$ forms a Markov chain.
###### Proof:
The prove is omitted due to space limitation.∎
$I(X^{n};Y^{n})=I(X^{n}\rightarrow{Y}^{n})\equiv{\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},\overrightarrow{q}_{0,n})$
is a functional of $\\{\mu_{0,n},{\overrightarrow{q}}_{0,n}\\}$. Hence, the
information definition of a causal rate distortion is defined by optimizing
${\mathbb{I}}(\mu_{0,n},\overrightarrow{q}_{0,n})$ over
${\overrightarrow{q}}_{0,n}$ which satisfies a distortion constraint.
###### Definition II.4
(Causal Information Rate Distortion Function) Suppose
$d_{0,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\frac{1}{n+1}\sum^{n}_{i=0}\rho_{i}(x^{i},y^{i})$,
where $\rho_{i}:{\cal X}_{i}\times{\cal Y}_{i}\rightarrow[0,\infty)$, is a
sequence of ${\cal B}({\cal X}_{i})\times{\cal B}({\cal Y}_{i})$-measurable
distortion functions, and let $\overrightarrow{Q}_{0,n}(D)$ (assuming is non-
empty) denotes the average distortion or fidelity constraint defined by
$\displaystyle\overrightarrow{Q}_{0,n}$
$\displaystyle(D)\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}\overrightarrow{q}_{0,n}\in\overrightarrow{\cal
Q}({\cal Y}_{0,n};{\cal
X}_{0,n}):\ell({\overrightarrow{q}}_{0,n})\stackrel{{\scriptstyle\triangle}}{{=}}$
$\displaystyle\frac{1}{n+1}\int_{{\cal X}_{0,n}\times{{\cal
Y}}_{0,n}}d_{0,n}({x^{n}},{y^{n}})(\overrightarrow{q}_{0,n}\otimes\mu_{0,n})(d{x}^{n},d{y}^{n})$
$\displaystyle\leq D\Big{\\}},~{}D\geq 0$ (6)
Define
$\displaystyle{R}_{0,n}^{c}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{{\overrightarrow{q}_{0,n}\in\overrightarrow{Q}_{0,n}(D)}}\frac{1}{n+1}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
(7)
The operational meaning of CRDF is established via
${R}^{c}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\lim_{n\rightarrow{\infty}}{R}_{0,n}^{c}(D)$,
provided the limit exists.
Clearly, ${R}_{0,n}^{c}(D)$ is characterized by minimizing
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
over the causal convolution measure
${\overrightarrow{q}}_{0,n}\in{\overrightarrow{Q}}_{0,n}(D)$.
### II-B Coding Theorems for Causal and Sequential Codes
This section gives an operational meaning to ${R}^{c}_{0,n}(D)$ via coding
theorems. There are two cases, sequential codes and causal codes.
Sequential Codes. Coding theorems for sequential codes are established in [11]
for the finite alphabet case, and two-dimensional source
$X^{T,N}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{t,n}:t=0,\ldots,T,n=0,\ldots,N\\}$,
where $t$ represents time index and $n$ represents spatial index, under the
assumption that $P(X^{T,N})=\otimes_{n=0}^{N}{P}(X_{n}^{T})$, and
$\\{X_{n}^{T}:n=0,\ldots,N\\}$ are identically distributed, and the distortion
constraint is
$E_{X^{T,N}}\big{\\{}\frac{1}{N+1}\sum_{n=0}^{N}\rho(x_{t,n},y_{t,n})\leq{D}_{t},~{}t=0,1,\ldots,T\big{\\}}$.
With a slight modification of the per-letter distortion function above, it can
be shown that the coding theorem in [12] is still valid, and that the
corresponding sequential RDF is given by $R^{SRD}(D)\equiv{R}^{c}(D)$. The
coding theorem is derived using strong typicality.
Causal Codes. Here we describe a coding theorem for causal codes.
###### Definition II.5
(Causal Code) A $(n,2^{nR},D)$ causal source code of block length $n$, and
rate R consists of an encoding mapping $e(\cdot)$, $e:{\cal
X}_{0,n}\longrightarrow{\cal
W}\stackrel{{\scriptstyle\triangle}}{{=}}\\{1,2,\ldots,2^{nR}\\}$ and a
sequence of decoder mapping $\\{g_{i}\\}_{i=0}^{n}(\cdot)$,
$g_{i}:\\{1,2,\ldots,2^{nR}\\}\longrightarrow{\cal Y}_{i},~{}i=0,1,\ldots,n$
such that the sequence of reproduction coders
$\\{f_{i}=g_{i}\circ{e}\\}_{i=0}^{n}$ are causal.
###### Definition II.6
(Achievable Rate) A rate distortion pair $(R,D)$ is called achievable if
$\forall\epsilon>0$ and sufficiently large $n$ there exists a $(n,2^{nR},D)$
causal code such that
$\displaystyle\frac{1}{n+1}E\big{\\{}d_{0,n}(X^{n},Y^{n})\big{\\}}\leq{D}+\epsilon$
###### Definition II.7
(Causal Rate Distortion Function) The CRDF $R(D)$ is the infimum of rates $R$
such that $(R,D)$ is achievable.
The definition of the coding theorem can be done for i) stationary ergodic
processes $\big{\\{}(X_{i},Y_{i}):i=0,1,\ldots\big{\\}}$ by invoking versions
of Shannon-McMillan-Breimann Theorem, ii) for information and distortion
stable processes by invoking versions of Dobrushin’s conditions, and iii) for
processes with information spectrum via variants of the methods in [4].
Here we discuss ii) since the distortion function $d_{0,n}(x^{n},y^{n})$ is
general and does not fall under the special case discussed in [[2], Section
9.8] for ergodic sources.
Define the information density consistent with the causal reproduction coder
by
${\Lambda}_{0,n}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\log\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{P_{Y^{n}}(dy^{n})}$
where it is assumed absolute continuity
${\overrightarrow{P}}_{Y^{n}|X^{n}}(\cdot|x^{n})\ll{P}_{Y^{n}}(\cdot)$,
$\mu_{0,n}-a.s.$ for almost all $x^{n}\in{\cal X}_{0,n}$. Then
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})=E\big{\\{}\Lambda_{0,n}(x^{n},y^{n})\big{\\}}$,
where the joint distribution is
$P_{X^{n},Y^{n}}=P_{X^{n}}\otimes{\overrightarrow{P}}_{Y^{n}|X^{n}}$.
###### Definition II.8
(Information and Distortion Stable) For each $\epsilon>0$ define the
$\epsilon$-typical set of directed information density
$\displaystyle{\cal T}_{\epsilon}^{(n)}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\bigg{\\{}(x^{n},y^{n})\in{\cal
X}_{0,n}\times{\cal
Y}_{0,n}:\bigg{|}\frac{1}{n+1}\log\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{P_{Y^{n}}(dy^{n})}$
$\displaystyle-\frac{1}{n+1}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(P_{X^{n}},{\overrightarrow{P}}_{Y^{n}|X^{n}})\bigg{|}<\epsilon\bigg{\\}}$
and the $\epsilon$-typical set of the distortion by
$\displaystyle{\cal D}_{\epsilon}^{(n)}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\bigg{\\{}(x^{n},y^{n})\in{\cal
X}_{0,n}\times{\cal Y}_{0,n}:\bigg{|}\frac{1}{n+1}d_{0,n}(x^{n},y^{n})$
$\displaystyle-\frac{1}{n+1}E\big{\\{}d_{0,n}(x^{n},y^{n})\big{\\}}\bigg{|}<\epsilon\bigg{\\}}$
The process $\big{\\{}(X_{n},Y_{n}):n\in\mathbb{N}\big{\\}}$ is called
directed information and distortion stable if
$\lim_{n\rightarrow\infty}Prob({\cal T}_{\epsilon}^{(n)})=1,$ and
$\lim_{n\rightarrow\infty}Prob({\cal D}_{\epsilon}^{(n)})=1$, respectively,
for every $\epsilon>0$.
Note that for stationary ergodic process
$\big{\\{}(X_{n},Y_{n}):n\in\mathbb{N}\big{\\}}$ and certain distortion
functions (see [2], Section 9.8) information and distortion stability follows.
Before the statements leading to coding theorem are introduced, the notion of
stability of the source is required.
###### Definition II.9
The source $\\{X_{n}:n\in\mathbb{N}\\}$ is called stable if for any given
$D>0$ and $\epsilon>0$ there exists $\\{Y_{n}:n\in\mathbb{N}\\}$ such that
$\big{\\{}(X_{n},Y_{n}):n\in\mathbb{N}\big{\\}}$ is directed information and
distortion stable, and
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n+1}E\big{\\{}d_{0,n}(x^{n},y^{n})\big{\\}}\leq{D}$
(8)
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n+1}E\Big{\\{}\log\frac{{\overrightarrow{P}}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{P_{Y^{n}}(dy^{n})}\Big{\\}}\leq{R^{c}(D)}+\epsilon$
(9)
where $E\\{\cdot\\}$ is with respect to
$P_{X^{n},Y^{n}}=\overrightarrow{P}_{Y^{n}|X^{n}}\otimes{P}_{X^{n}}$.
Note that by specializing $d_{0,n}(x^{n},y^{n})$ to distortion functions that
satisfy sub-additivity property the limit in (8) exists. Utilizing Definition
II.9, it can be shown that the following statements hold, which are vital in
establishing the coding theorem.
###### Lemma II.10
Assume $\\{X_{n}:n\in\mathbb{N}\\}$ is stable and the joint distribution
$P_{X^{n},Y^{n}}$ is defined by
$P_{X^{n},Y^{n}}(dx^{n},dy^{n})=\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\otimes{P}_{X^{n}}(dx^{n})$.
Then
* 1)
$\lim_{n\rightarrow\infty}P_{X^{n},Y^{n}}({\cal
T}_{\epsilon}^{(n)})=\lim_{n\rightarrow\infty}P_{X^{n},Y^{n}}({\cal
D}_{\epsilon}^{(n)})=1$
* 2)
For sufficiently large $n$, there exists $\epsilon>0$ such that
$\displaystyle\frac{\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})}{{P}_{X^{n}}(dx^{n})}\leq{2}^{n\big{(}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}({P}_{X^{n}},\overrightarrow{P}_{Y^{n}|X^{n}})+3\epsilon\big{)}}$
Using Lemma II.10, the source coding theorem stated below can be established.
###### Theorem II.11
(Source Coding Theorem) Assume $\\{X_{n}:n\in\mathbb{N}\\}$ is stable and
$\sup_{(x^{i},y^{i})\in{\cal X}_{0,i}\times{\cal
Y}_{0,i}}\rho_{i}(x^{i},y^{i})<k$, $k<\infty$ for all $i$. If $R>{R}^{c}(D)$
then for any $\delta>0$ and sufficiently large $n$, there exists an
$(n,2^{nR},D)$ causal code which satisfies the average distortion
$\frac{1}{n+1}E_{P_{X^{n},Y^{n}}}\big{\\{}\frac{1}{n+1}d_{0,n}(x^{n},y^{n})\big{\\}}\leq{D}+\delta$.
###### Proof:
The derivation utilizes Lemma II.10, and random codebook generation. Fix
$\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})$, which achieves the equality
in $R^{c}(D)$ $\big{(}$e.g., (7)$\big{)}$. Calculate
${\overrightarrow{P}}_{Y^{n}}(dy^{n})=\int_{{\cal
X}_{0,n}}\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n}){P}_{X^{n}}(dx^{n})$.
Randomly generate rate distortion codebook ${\cal C}$ of $2^{nR}$ sequences
$Y^{n}$ according to ${\overrightarrow{P}}_{Y^{n}}(dy^{n})$ and reveal the
codebook to encoder and decoder. Utilizing Lemma II.10 and Definition II.9,
the result is obtained following [3].∎
## III EXISTENCE OF OPTIMAL CAUSAL RECONSTRUCTION
In this section, the existence of the minimizing causal product kernel in
$(\ref{ex12})$ is shown by using the topology of weak convergence of
probability measures on Polish spaces. The only assumptions required are 1)
${\cal Y}_{0,n}$ is a compact Polish space, 2) ${\cal X}_{0,n}$ is a Polish
space, and 3) $d_{0,n}(x^{n},\cdot)$ is continuous on ${\cal Y}_{0,n}$.
### III-A Weak Compactness and Existence of Optimal Reconstruction Kernel
Define the family of measures
$\displaystyle{\overrightarrow{\cal Q}}({\cal Y}_{0,n};{\cal X}_{0,n})$
$\displaystyle=\big{\\{}{\overrightarrow{q}}_{0,n}(dy^{n};x^{n}):{\overrightarrow{q}}_{0,n}(dy^{n};x^{n})$
$\displaystyle=\otimes_{i=0}^{n}{q}_{i}(dy_{i};y^{i-1},x^{i})\big{\\}}$
###### Lemma III.1
Let ${\cal Y}_{0,n}$ be a compact Polish space and ${\cal X}_{0,n}$ a Polish
space.
Then
* 1)
The family of measures
${\overrightarrow{q}}_{0,n}(dy^{n};x^{n})\in{\overrightarrow{\cal Q}}({\cal
Y}_{0,n};{\cal X}_{0,n})$ is compact.
* 2)
Under the assumption that $d_{0,n}(x^{n},\cdot)$ is continuous in ${\cal
Y}_{0,n}$ the set ${\overrightarrow{Q}}_{0,n}(D)$ is a closed subset of
${\overrightarrow{\cal Q}}({\cal Y}_{0,n};{\cal X}_{0,n})$.
###### Proof:
1) This follows from the fact that any
${\overrightarrow{q}}_{0,n}(dy^{n};x^{n})\in{\overrightarrow{\cal Q}}({\cal
Y}_{0,n};{\cal X}_{0,n})$ is factorized as
${\overrightarrow{q}}_{0,n}(dy^{n};x^{n})=\otimes_{i=0}^{n}{q}_{i}(dy_{i};y^{i-1},x^{i})$,
where $q_{i}(dy_{i};y^{i-1},x^{i})\in{\cal Q}({\cal Y}_{i};{\cal
Y}_{0,i-1}\times{\cal X}_{0,i})$, $1\leq{i}\leq{n}$, and ${\cal Y}_{0,n}$
compact Polish space implies that
$\\{q_{i}(\cdot;y^{i-1},x^{i}):y^{i-1}\in{\cal Y}_{0,i-1},x^{i}\in{\cal
X}_{0,i}\\}$ is compact, $\forall{i}$. Utilizing this, by induction it can be
shown that the family of convolution measures ${\overrightarrow{\cal Q}}({\cal
Y}_{0,n};{\cal X}_{0,n})$ is compact.
2) Utilizing compactness of ${\overrightarrow{\cal Q}}({\cal Y}_{0,n};{\cal
X}_{0,n})$ and the assumption on $d_{0,n}(x^{n},\cdot)$ it can be shown that
${\overrightarrow{Q}}_{0,n}(D)$ is a closed subset of ${\overrightarrow{\cal
Q}}({\cal Y}_{0,n};{\cal X}_{0,n})$. ∎
The next theorem establishes existence of the minimizing reconstruction kernel
for (7).
###### Theorem III.2
Suppose ${\cal Y}_{0,n}$ is compact Polish space and $d_{0,n}(x^{n},\cdot)$ is
continuous in ${\cal Y}_{0,n}$. Then ${R}^{c}_{0,n}(D)$ has a minimum.
###### Proof:
The assumptions are sufficient to show lower semicontinuity of the functional
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
with respect to ${\overrightarrow{q}}_{0,n}$ for a fixed ${\mu}_{0,n}$.
Moreover, by Lemma III.1, 2) since $\overrightarrow{Q}_{0,n}(D)$ is a closed
subset of a compact set $\overrightarrow{\cal Q}({\cal Y}_{0,n};{\cal
X}_{0,n})$, then $\overrightarrow{Q}_{0,n}(D)$ is also compact. By Weiestrass
theorem existence follows. ∎
## IV OPTIMAL CAUSAL RECONSTRUCTION
In this section the form of the optimal causal reconstruction kernel is
derived and the properties of $R_{0,n}^{c}(D)$ are discussed under a
stationarity assumption.
### IV-A Optimal Reconstruction
###### Assumption IV.1
The family of measures that admits the factorization
${\overrightarrow{q}}(dy^{n}|x^{n})=\otimes^{n}_{i=0}q_{i}(dy_{i}|y^{i-1},x^{i})$
is the convolution of stationary conditional distributions.
Assumption IV.1 holds for stationary ergodic process
$\\{(X_{i},Y_{i}):i\in\mathbb{N}\\}$ and $\rho_{i}(x^{i},y^{i})$, which is
stationary and time-invariant $\forall{i}$. The method is based on calculus of
variations on the space of measures [14]. Utilizing Assumption IV.1, which
holds for stationary ergodic processes $\\{(X_{i},Y_{i}):i=0,1,\ldots,n\\}$
and single letter distortion function or distortion function discussed in
[[2], Section 9.8], the Gateaux differential of
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
is done in only one direction $\big{(}$since $q_{i}(dy_{i};y^{i-1},x^{i})$ are
stationary$\big{)}$. This simplifies the calculations of Gateaux derivative of
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$.
###### Theorem IV.2
Suppose
${\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
is well defined for every
${\overrightarrow{q}}_{0,n}\in\overrightarrow{Q}_{0,n}(D)$ possibly taking
values from the set $[0,\infty).$ Then
${\overrightarrow{q}}_{0,n}\rightarrow{\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n})$
is Gateaux differentiable at every point in $\overrightarrow{Q}_{0,n}(D)$, and
the Gateaux derivative at the point ${\overrightarrow{q}}_{0,n}^{0}$ in the
direction ${\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0}$ is given
by
$\displaystyle\delta{\mathbb{I}}_{\mu_{0,n}}({\overrightarrow{q}}_{0,n}^{0};{\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0})=\int_{{\cal
X}_{0,n}\times{\cal
Y}_{0,n}}\log\Big{(}\frac{{\overrightarrow{q}}_{0,n}^{0}(dy^{n};x^{n})}{\nu_{0,n}^{0}(dy^{n})}\Big{)}$
$\displaystyle({\overrightarrow{q}}_{0,n}-{\overrightarrow{q}}_{0,n}^{0})(dy^{n};x^{n})\mu_{0,n}(dx^{n})$
(10)
where $\nu_{0,n}^{0}\in{\cal M}_{1}({\cal Y}_{0,n})$ is the marginal measure
corresponding to ${\overrightarrow{q}}_{0,n}^{0}\otimes\mu_{0,n}\in{\cal
M}_{1}({\cal Y}_{0,n}\times{\cal X}_{0,n})$.
###### Proof:
The proof utilizes Assumption IV.1.∎
The constrained problem defined by (7) can be reformulated using Lagrange
multipliers as follows (equivalence of constrained and unconstrained problems
follows from [14]).
$\displaystyle{R}_{0,n}^{c}(D)$ $\displaystyle=\sup_{s\leq
0}\inf_{{\overrightarrow{q}}_{0,n}\in\overrightarrow{\cal Q}({\cal
Y}_{0,n};{\cal
X}_{0,n})}\Big{\\{}\frac{1}{n+1}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\mu_{0,n},{\overrightarrow{q}}_{0,n})$
$\displaystyle-s\big{(}\ell({\overrightarrow{q}}_{0,n})-D\big{)}\Big{\\}}$
(11)
and $s\in(-\infty,0]$ is the Lagrange multiplier.
Note that $\overrightarrow{\cal Q}({\cal Y}_{0,n};{\cal X}_{0,n})$ represents
the causality constraint set. Therefore, one should introduce another set of
Lagrange multipliers to obtain an optimization without constraints. This
process is involved hence we state the main results.
###### Theorem IV.3
Suppose Assumption IV.1 and
$d_{0,n}(x^{n},y^{n})=\sum_{i=0}^{n}\rho_{i}(x^{i},y^{i})$ hold. The infimum
in $(\ref{ex13})$ is attained at
$\overrightarrow{q}^{*}_{0,n}\in\overrightarrow{Q}_{0,n}(D)$ given by
$\displaystyle\overrightarrow{q}^{*}_{0,n}(dy^{n};x^{n})=\otimes_{i=0}^{n}\frac{e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy_{i};y^{i-1})}{\int_{{\cal
Y}_{i}}e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy_{i};y^{i-1})}$ (12)
and $\nu^{*}_{i}(dy_{i};y^{i-1})\in{\cal Q}({\cal Y}_{i};{\cal Y}_{0,{i-1}})$.
The causal rate distortion function is given by
$\displaystyle{R}_{0,n}^{c}(D)=sD-\frac{1}{n+1}\sum_{i=0}^{n}\int_{{{\cal
X}_{0,i}}\times{{\cal Y}_{0,i-1}}}\log\Big{(}$ $\displaystyle\int_{{\cal
Y}_{i}}e^{s\rho_{i}(x^{i},y^{i})}\nu^{*}_{i}(dy_{i};y^{i-1})\Big{)}{{\overrightarrow{q}}^{*}_{0,i-1}}(dy^{i-1};x^{i-1})\otimes\mu_{0,i}(dx^{i})$
(13)
If ${R}_{0,n}^{c}(D)>0$ then $s<0$ and
$\displaystyle\frac{1}{n+1}\sum_{i=0}^{n}\int_{{\cal X}_{0,i}}\int_{{\cal
Y}_{0,i}}\rho_{i}(x^{i},y^{i}){\overrightarrow{q}}^{*}_{0,i}(dy^{i};x^{i})\mu_{0,i}(dx^{i})=D$
###### Proof:
The fully unconstraint problem of (11) is obtained by introducing another set
of Lagrange multipliers. Using this and Theorem IV.2 we obtain (12) and (13).∎
Note that according to Assumption IV.1, the terms appear in the right side of
(12) are identical.
### IV-B PROPERTIES OF $R_{0,n}^{c}(D)$
In this section, we present some important properties of the CRDF as it is
defined in (7).
###### Theorem IV.4
1. 1.
${R}_{0,n}^{c}(D)$ is a convex, non-increasing function of $D$
2. 2.
If $\rho_{i}\in L^{1}(\pi_{i})$ then
a) ${R}_{0,n}^{c}(\frac{1}{n+1}\sum_{i=0}^{n}E_{\pi_{i}}(\rho_{i}))=0$;
b) ${R}_{0,n}^{c}(D)$ is non-increasing for $D\in[0,D_{max}]$ where
$D_{max}=\frac{1}{n+1}\sum_{i=0}^{n}E_{\pi_{i}}(\rho_{i})$ and
${R}_{0,n}^{c}(D)=0$ for any $D\geq D_{max}$
3. 3.
${R}_{0,n}^{c}(D)>0$ for all $D<D_{max}$ and ${R}_{0,n}^{c}(D)=0$ for all
$D\geq D_{max}$, where
$\displaystyle D_{max}=\min_{\\{y^{n}\\}\in{\cal
Y}_{0,n}}\frac{1}{n+1}\sum_{i=0}^{n}\int_{{\cal
X}_{0,i}}\rho_{i}(x^{i},y^{i})\mu_{0,i}(dx^{i})$
if such a minimum exists.
###### Proof:
Omitted due to space limitation. ∎
## V CONCLUSION
The solution of the CRDF subject to a reconstruction kernel which is a
convolution of causal kernels is presented, on abstract alphabets. Some of its
properties are also presented. It is believed that the optimal reconstruction
kernel as a convolution of causal kernels has several implications in
applications where causality of the decoder as a function of the source is of
concern. Specific example by invoking (11) will be part of the final paper.
## References
* [1] T. M. Cover and J. A. Thomas, _Elements of Information Theory_ , 2nd ed. John Wiley & Sons, Inc., Hoboken, New Jersey, 2006.
* [2] R. T. Gallager, _Information theory and Reliable Communication_. John Wiley & Sons, Inc., New York, 1968.
* [3] S. Ihara, _Information theory - for continuous systems_. World Scientific, 1993.
* [4] T. S. Han and S. Verdu, “Approximation theory of output statistics,” _IEEE Transactions on Information Theory_ , vol. 39, no. 3, pp. 752 – 772, may 1993.
* [5] I. Csiszár, “On an extremum problem of information theory,” _Studia Scientiarum Mathematicarum Hungarica_ , vol. 9, pp. 57 – 71, 1974.
* [6] D. Neuhoff and R. Gilbert, “Causal source codes,” _IEEE Transactions on Information Theory_ , vol. 28, no. 5, pp. 701 – 713, sep 1982.
* [7] T. Weissman and N. Merhav, “On competitive prediction and its relation to rate-distortion theory,” _IEEE Transactions on Information Theory_ , vol. 49, no. 12, pp. 3185–3194, dec. 2003.
* [8] T. Linder and R. Zamir, “Causal coding of stationary sources and individual sequences with high resolution,” _IEEE Transactions on Information Theory_ , vol. 52, no. 2, pp. 662–680, feb. 2006.
* [9] A. K. Gorbunov and M. S. Pinsker, “Asymptotic behavior of nonanticipative epsilon-entropy for Gaussian processes,” _Problems of Information Transmission_ , vol. 27, no. 4, pp. 361 – 365, 1991.
* [10] M. S. Derpich and J. Østergaard, “Improved upper bounds to the causal quadratic rate-distortion function for gaussian stationary sources,” _Accepted in IEEE Transactions on Information Theory_ , [Online:] Available at http://arxiv.org/abs/1001.4181v2, 2011.
* [11] S. C. Tatikonda, “Control over communication constraints,” Ph.D. dissertation, Mass. Inst. of Tech. (M.I.T.), Cambridge, MA, 2000.
* [12] S. Tatikonda, A. Sahai, and S. Mitter, “Stochastic linear control over a communication channel,” _IEEE Transactions on Automatic Control_ , vol. 49, no. 9, pp. 1549 – 1561, sept. 2004.
* [13] P. Dupuis and R. S. Ellis, _A weak Convergence Approach to the Theory of Large Deviations_. John Wiley & Sons, Inc., New York, 1997.
* [14] D. G. Luenberger, _Optimization by Vector Space Methods_. John Wiley & Sons, Inc., New York, 1969.
|
arxiv-papers
| 2012-02-04T13:23:32 |
2024-09-04T02:49:27.036231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Photios A. Stavrou, Charalambos D. Charalambous and Christos K.\n Kourtellaris",
"submitter": "Photios Stavrou",
"url": "https://arxiv.org/abs/1202.0895"
}
|
1202.0958
|
# Directed Information on Abstract Spaces: Properties and Extremum Problems
Charalambos D. Charalambous and Photios A. Stavrou ECE Department, University
of Cyprus, Nicosia, Cyprus
e-mail:{ chadcha, stavrou.fotios}@ucy.ac.cy
###### Abstract
This paper describes a framework in which directed information is defined on
abstract spaces. The framework is employed to derive properties of directed
information such as convexity, concavity, lower semicontinuity, by using the
topology of weak convergence of probability measures on Polish spaces. Two
extremum problems of directed information related to capacity of channels with
memory and feedback, and non-anticipative and sequential rate distortion are
analyzed showing existence of maximizing and minimizing distributions,
respectively.
## I Introduction
Directed information from a sequence of Random Variables (RV’s)
$X^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{X_{0},X_{1},\ldots,X_{n}\\}\in{\cal
X}_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{n}{\cal X}_{i}$,
to another sequence
$Y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{Y_{0},Y_{1},\ldots,Y_{n}\\}\in{\cal
Y}_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{n}{\cal Y}_{i}$
is often defined via [1, 2]111Unless otherwise, integrals with respect to
measures are over the spaces on which these are defined.
$\displaystyle
I(X^{n}\rightarrow{Y}^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\sum_{i=0}^{n}I(X^{i};Y_{i}|Y^{i-1})$
(1)
$\displaystyle=\sum_{i=0}^{n}\int\log\bigg{(}\frac{P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})}{P_{Y_{i}|Y^{i-1}}(dy_{i}|y^{i-1})}\bigg{)}{P}_{X^{i},Y^{i}}(dx^{i},dy^{i})$
(2)
$\displaystyle\equiv\mathbb{I}_{X^{n}\rightarrow{Y}^{n}}(P_{X_{i}|X^{i-1},Y^{i-1}},P_{Y_{i}|Y^{i-1},X^{i}}:i=0,\ldots,n)$
(3)
Since the joint distribution in (2) is decomposed via
$P_{X^{i},Y^{i}}(dx^{i},dy^{i})=\otimes_{j=0}^{i}{P}_{X_{j}|X^{j-1},Y^{j-1}}(dx_{j}|x^{j-1},y^{j-1})\otimes{P}_{Y_{j}|Y^{j-1},X^{j}}(dy_{j}|y^{j-1},x^{j})$,
the notation $\mathbb{I}_{X^{n}\rightarrow{Y}^{n}}(\cdot,\cdot)$ denotes the
functional dependence on two collections of non-anticipative or causal
conditional distributions
$\\{P_{X_{i}|X^{i-1},Y^{i-1}}(\cdot|\cdot,\cdot),~{}P_{Y_{i}|Y^{i-1},X^{i}}(\cdot|\cdot,\cdot)~{}:~{}i=0,1,\ldots,n\\}$.
In information theory, directed information (1)-(3) or its variants are used
to characterize capacity of channels with memory and feedback [3, 4, 5], lossy
data compression with feedforward information at the decoder [6], lossy data
compression of sequential codes [4], lossy data compression of non-
anticipative codes [7], and capacity of networks such as the two-way channel,
multiple access channel [8, 9], etc. The previous references derive coding
theorems based on $a)$ stationary ergodic processes
$\\{(X_{i},Y_{i})\\}_{i=0}^{\infty}$, $b)$ Dobrushin’s stability of the
information density
$\log\otimes_{i=0}^{n}\frac{P_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i})}{P_{Y_{i}|Y^{i-1}}(dy_{i}|y^{i-1})}$,
and $c)$ via information spectrum methods [10].
Capacity of Channels with Feedback. Based on $a)$ or $b)$ the operational
definition of channels with memory and feedback is given by
$\displaystyle
C^{f}(P)\stackrel{{\scriptstyle\triangle}}{{=}}\lim_{n\rightarrow\infty}\sup_{\overleftarrow{P}_{X^{n}|Y^{n-1}}(\cdot|\cdot)\in\overleftarrow{\cal{P}}(P)}\frac{1}{n+1}I(X^{n}\rightarrow{Y^{n}})$
(4)
where $\overleftarrow{\cal{P}}(P)$ denotes the power constraint set, and
$\displaystyle\overleftarrow{P}_{X^{n}|Y^{n-1}}(dx^{n}|dy^{n-1})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{i=0}^{n}{P}_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1})$
Sequential and Non-Anticipative Rate Distortion Function. Based on $a)$ or
$b)$ the operational definition of sequential and non-anticipative rate
distortion function is given by expression
$\displaystyle
R^{c}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\lim_{n\rightarrow\infty}\inf_{\overrightarrow{P}_{Y^{n}|X^{n}}(\cdot|\cdot)\in\overrightarrow{Q}(D)}\frac{1}{n+1}I({X^{n}}\rightarrow{Y^{n}})$
(5)
where $\overrightarrow{Q}(D)$ is the distortion fidelity constraint and
$\displaystyle\overrightarrow{P}_{Y^{n}|X^{n}}(dy^{n}|x^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\otimes_{i=0}^{n}{P}_{Y_{i}|Y^{i-1},X^{i}}(dy_{i}|y^{i-1},x^{i}),$
$\displaystyle{P}_{X_{i}|X^{i-1},Y^{i-1}}(dx_{i}|x^{i-1},y^{i-1})=P_{X_{i}|X^{i-1}}(dx_{i}|x^{i-1})-a.s.$
The complete investigation of existence, characterization, and properties of
the above extremum problems requires extensive analysis of the functional
$\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}(\cdot,\cdot)$ as defined in (3). This is
analogous to capacity of channels without feedback which involves maximization
of mutual information $I({X^{n}};{Y^{n}})$ over the power constraint set, and
to classical rate distortion function which involves minimization of mutual
information $I({X^{n}};{Y^{n}})$ over the fidelity constraint. However, mutual
information
$I(X^{n};Y^{n})\equiv\mathbb{I}_{X^{n};Y^{n}}(P_{X^{n}},P_{Y^{n}|X^{n}})$,
inherits from its information divergence definition
$I(X^{n};Y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{D}(P_{X^{n},Y^{n}}||P_{X^{n}}\times{P}_{Y^{n}})$,
several important functional properties such as convexity, concavity, lower
semicontinuity, etc. These properties are vital both for finite alphabet
spaces, as well as abstract alphabet spaces [11, 12]. The difficulty
associated with directed information $I(X^{n}\rightarrow{Y^{n}})$, rises from
the fact that this information measure (1)-(3) is a functional
$\mathbb{I}_{X^{n}\rightarrow{Y}^{n}}(\cdot,\cdot)$ of the collection of
conditional distributions
$\\{P_{X_{i}|X^{i-1},Y^{i-1}}(\cdot|\cdot,\cdot):i=0,1,\ldots,n\\}$ and
$\\{P_{Y_{i}|Y^{i-1},X^{i}}(\cdot|\cdot,\cdot):i=0,1,\ldots,n\\}$.
The objective of this paper is to address the following questions, when ${\cal
X}_{0,n}$ and ${\cal Y}_{0,n}$ are complete separable metric spaces (Polish
spaces).
* 1.
Is there an equivalent directed information definition expressed via
information divergence $\mathbb{D}(\cdot||\cdot)$ as a functional of two
appropriate conditional distributions ${\bf P}(\cdot|{\bf y})$ on ${\cal
X}^{\mathbb{N}}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{\infty}{\cal
X}_{i}$ for ${\bf y}=(y_{0},y_{1},\ldots)\in{\cal
Y}^{\mathbb{N}}\stackrel{{\scriptstyle\triangle}}{{=}}\times_{i=0}^{\infty}{\cal
Y}_{i}$ and ${\bf Q}(\cdot|{\bf x})$ on ${\cal Y}^{\mathbb{N}}$ for ${\bf
x}\in{\cal X}^{\mathbb{N}}$ which uniquely define
$\\{P_{X_{i}|X^{i-1},Y^{i-1}}:i=0,1,\ldots\\}$ and
$\\{P_{Y_{i}|Y^{i-1},X^{i}}:i=0,1,\ldots\\}$, respectively, and vice-versa?
* 2.
Is directed information convex and concave functional with respect to the
conditional distributions ${\bf P}(\cdot|{\bf y})$ and ${\bf Q}(\cdot|{\bf
x})$?
* 3.
Is directed information a lower semicontinuous functional of the conditional
distributions ${\bf P}(\cdot|{\bf y})$ and ${\bf Q}(\cdot|{\bf x})$?
* 4.
What are appropriate conditions for existence of the maximizing encoder
admissible distributions and minimizing distortion admissible distributions?
This paper answers the above questions by invoking the topology of weak
convergence of probability measures on Polish spaces and Prohorov’s theorems.
The paper is organized as follows. Section II provides the construction of two
equivalent definitions of causal channels on abstract spaces while in Section
III the main properties of directed information are given. Finally, in Section
IV the extremum problems (4), (5) are discussed. The derivations of theorems
are outlined since they are quite lengthy.
## II Causal Channels on Abstract Spaces
In this section, the aim is to establish two equivalent definitions of
conditional distributions or basic processes, which define any probabilistic
channel with causal feedback, that relates causally the input-output behavior
of any channel. This formulation is necessary to investigate questions 1.–4.
Let $\mathbb{N}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,2,\ldots\\},$ and
$\mathbb{N}^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{0,1,2,\ldots,n\\}.$
Introduce two sequence of spaces $\\{({\cal X}_{n},{\cal B}({\cal
X}_{n})):n\in\mathbb{N}\\}$ and $\\{({\cal Y}_{n},{\cal B}({\cal
Y}_{n})):n\in\mathbb{N}\\},$ where ${\cal X}_{n},{\cal Y}_{n},n\in\mathbb{N}$
are topological spaces, and ${\cal B}({\cal X}_{n})$ and ${\cal B}({\cal
Y}_{n})$ are Borel $\sigma-$algebras of subsets of ${\cal X}_{n}$ and ${\cal
Y}_{n},$ respectively. Points in ${\cal
X}^{\mathbb{N}}\stackrel{{\scriptstyle\triangle}}{{=}}{{\times}_{n\in\mathbb{N}}}{\cal
X}_{n},$ ${\cal
Y}^{\mathbb{N}}\stackrel{{\scriptstyle\triangle}}{{=}}{\times_{n\in\mathbb{N}}}{\cal
Y}_{n}$ are denoted by ${\bf
x}\stackrel{{\scriptstyle\triangle}}{{=}}\\{x_{0},x_{1},\ldots\\}\in{\cal
X}^{\mathbb{N}},$ ${\bf
y}\stackrel{{\scriptstyle\triangle}}{{=}}\\{y_{0},y_{1},\ldots\\}\in{\cal
Y}^{\mathbb{N}},$ respectively, while their restrictions to finite coordinates
by
$x^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{x_{0},x_{1},\ldots,x_{n}\\}\in{\cal
X}_{0,n},$
$y^{n}\stackrel{{\scriptstyle\triangle}}{{=}}\\{y_{0},y_{1},\ldots,y_{n}\\}\in{\cal
Y}_{0,n},$ for $n\in\mathbb{N}$.
Let ${\cal B}({\cal
X}^{\mathbb{N}})\stackrel{{\scriptstyle\triangle}}{{=}}\odot_{i\in\mathbb{N}}{\cal
B}({\cal X}_{i})$ denote the $\sigma-$algebra on ${\cal X}^{\mathbb{N}}$
generated by cylinder sets $\\{{\bf x}=(x_{0},x_{1},\ldots)\in{\cal
X}^{\mathbb{N}}:x_{0}\in{A}_{0},x_{1}\in{A}_{1},\ldots,x_{n}\in{A}_{n}\\},A_{i}\in{\cal
B}({\cal X}_{i}),0\leq{i}\leq{n},n\geq 1$, and similarly for ${\cal B}({\cal
Y}^{\mathbb{N}})\stackrel{{\scriptstyle\triangle}}{{=}}\odot_{i\in\mathbb{N}}{\cal
B}({\cal Y}_{i})$. Hence, ${\cal B}({\cal X}_{0,n})$ and ${\cal B}({\cal
Y}_{0,n})$ denote the $\sigma-$algebras of cylinder sets in ${\cal
X}^{\mathbb{N}}$ and ${\cal Y}^{\mathbb{N}},$ respectively, with bases over
$A_{i}\in{\cal B}({\cal X}_{i})$, and $B_{i}\in{\cal B}({\cal
Y}_{i}),~{}0\leq{i}\leq{n}$, respectively.
Backward or Feedback Channel. Suppose for each $n\in\mathbb{N},$ the
distributions $\\{p_{n}(dx_{n}|x^{n-1},y^{n-1}):n\in\mathbb{N}\\}$ with
$p_{0}(dx_{0}|x^{-1},y^{-1})\stackrel{{\scriptstyle\triangle}}{{=}}{p}_{0}(x_{0})$
satisfy the following conditions.
i) For $n\in\mathbb{N},$ $p_{n}(\cdot|x^{n-1},y^{n-1})$ is a probability
measure on ${\cal B}({\cal X}_{n});$
ii) For $n\in\mathbb{N},$ $A_{n}\in{\cal B}({\cal X}_{n})$,
$p_{n}(A_{n}|x^{n-1},y^{n-1})$ is $\odot^{n-1}_{i=0}{\cal B}({\cal
X}_{i})\odot{\cal B}({\cal Y}_{i})-$measurable in $x^{n-1}\in{\cal
X}_{0,n-1},$ $y^{n-1}\in{\cal Y}_{0,n-1}.$
Given the collection $\\{p_{n}(dx_{n}|x^{n-1},y^{n-1}):n\in\mathbb{N}\\}$
satisfying conditions i), ii), one can construct a family of distributions on
$({\cal X}^{\mathbb{N}},{\cal B}({\cal
X}^{\mathbb{N}}))\stackrel{{\scriptstyle\triangle}}{{=}}\big{(}\times_{i\in\mathbb{N}}{\cal
X}_{i},\odot_{i\in\mathbb{N}}{\cal B}({\cal X}_{i})\big{)}$ as follows.
Let $C\in{\cal B}({\cal X}_{0,n})$ be a cylinder set of the form
$C\stackrel{{\scriptstyle\triangle}}{{=}}\big{\\{}{\bf x}\in{\cal
X}^{\mathbb{N}}:x_{0}\in{C_{0}},x_{1}\in{C_{1}},\ldots,x_{n}\in{C_{n}}\big{\\}},~{}C_{i}\in{\cal
B}({\cal X}_{i}),~{}0\leq i\leq n$. Define a family of measures ${\bf
P}(\cdot|{\bf y})$ on ${\cal B}({\cal X}^{\mathbb{N}})$ by
$\displaystyle{\bf P}(C|{\bf y})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\int_{C_{0}}p_{0}(dx_{0})\ldots\int_{C_{n}}p_{n}(dx_{n}|x^{n-1},y^{n-1})$
(6)
$\displaystyle\equiv{\overleftarrow{P}}_{0,n}(C_{0,n}|y^{n-1}),~{}C_{0,n}=\times_{i=0}^{n}{C_{i}}$
(7)
The notation ${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})$ is used to denote the
restriction of the measure ${\bf P}(\cdot|{\bf y})$ on cylinder sets
$C\in{\cal B}({\cal X}_{0,n})$, for $n\in\mathbb{N}$.
Thus, if conditions i) and ii) hold then for each ${\bf y}\in{\cal
Y}^{\mathbb{N}},$ the right hand side of $(\ref{equation2})$ defines a
consistent family of finite-dimensional distribution on $({\cal
X}^{\mathbb{N}},{\cal B}({\cal X}^{\mathbb{N}}))$, and hence there exists a
unique measure on $({\cal X}^{\mathbb{N}},{\cal B}({\cal X}^{\mathbb{N}})),$
from which $p_{n}(dx_{n}|x^{n-1},y^{n-1})$ is obtained. This leads to the
first, usual definition of a feedback channel, as a family of functions
$p_{n}(dx_{n}|x^{n-1},y^{n-1})$ satisfying conditions ${\bf i)}$ and ${\bf
ii)}.$
An alternative, equivalent definition of a feedback channel is established as
follows. Introduce the assumption
${\bf iii)}$ $\\{{\cal X}_{n}:n\in\mathbb{N}\\}$ are complete separable metric
spaces (Polish Spaces) and $\\{{\cal B}({\cal X}_{n}):n\in\mathbb{N}\\}$ are
the $\sigma-$algebras of Borel sets.
Consider a family of measures ${\bf P}(\cdot|{\bf y})$ on $({\cal
X}^{\mathbb{N}},{\cal B}({\cal X}^{\mathbb{N}}))$ satisfying the following
consistency condition.
C1: If $E\in{\cal B}({\cal X}_{0,n})$, then ${\bf P}(E|{\bf y})$ is ${\cal
B}({\cal Y}_{0,n-1})-$measurable function of ${\bf y}\in{\cal
Y}^{\mathbb{N}}$.
Then, by assumption ${\bf iii)}$, for any family of measures ${\bf
P}(\cdot|{\bf y})$ satisfying C1 one can construct a collection of versions of
conditional distributions $\\{p_{n}(dx_{n}|x^{n-1},y^{n-1}):n\in\mathbb{N}\\}$
satisfying conditions ${\bf i)}$ and ${\bf ii)}$ which are connected with
${\bf P}(\cdot|{\bf y})$ via relation $(\ref{equation2}).$
Therefore, for Polish Spaces $\\{{\cal X}_{n}:n\in\mathbb{N}\\}$ the second
equivalent definition is given by a family of measures ${\bf P}(\cdot|{\bf
y})$ on $({\cal X}^{\mathbb{N}},{\cal B}({\cal X}^{\mathbb{N}}))$ depending
parametrically on ${\bf y}\in{\cal Y}^{\mathbb{N}}$ and satisfying the
consistency condition C1.
The point to be made here is that the second equivalent definition of a
feedback channel, together with similar definition for the forward channel is
convenient to define directed information via relative entropy, similar to the
mutual information definition, and extend well-known functional properties of
mutual information to directed information.
Forward Channel. The previous methodology is repeated for the collection of
functions $\\{q_{n}(dy_{n}|y^{n-1},x^{n}):n\in\mathbb{N}\\}$ which satisfy the
following conditions.
iv) For $n\in\mathbb{N},$ $q_{n}(\cdot|y^{n-1},x^{n})$ is a probability
measure on ${\cal B}({\cal Y}_{n});$
v) For $n\in\mathbb{N}$, $B_{n}\in{\cal B}({\cal Y}_{n})$,
$q_{n}(B_{n}|y^{n-1},x^{n})$ is $\odot^{n-1}_{i=0}{\cal B}({\cal
Y}_{i})\odot_{i=0}^{n}{\cal B}({\cal X}_{i})-$measurable function of
$x^{n}\in{\cal X}_{0,n},$ $y^{n-1}\in{\cal Y}_{0,n-1}$.
Similarly as before, given a cylinder set
$D\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}{\bf y}\in{\cal
Y}^{\mathbb{N}}:y_{0}{\in}D_{0},y_{1}{\in}D_{1},\ldots,y_{n}{\in}D_{n}\Big{\\}},~{}D_{i}\in{\cal
B}({\cal Y}_{i}),~{}0\leq i\leq n$, define a family of measures on ${\cal
B}({\cal Y}^{\mathbb{N}})$ by
$\displaystyle{\bf Q}(D|{\bf x})$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\int_{D_{0}}q_{0}(dy_{0}|x_{0})\ldots\int_{D_{n}}q_{n}(dy_{n}|y^{n-1},x^{n})$
(8)
$\displaystyle\equiv{\overrightarrow{Q}}_{0,n}(D_{0,n}|x^{n}),~{}D_{0,n}=\times_{i=0}^{n}{D_{i}}$
(9)
Similarly as before, there exists a unique measure on $({\cal
Y}^{\mathbb{N}},{\cal B}({\cal Y}^{\mathbb{N}}))$ for which the family of
distributions $\\{q_{n}(dy_{n}|y^{n-1},x^{n}):n\in\mathbb{N}\\}$ is obtained.
Introduced the assumption
${\bf vi)}$ $\\{{\cal Y}_{n}:n\in\mathbb{N}\\}$ are Polish Spaces and
$\\{{\cal B}({\cal Y}_{n}):n\in\mathbb{N}\\}$ are the $\sigma-$algebras of
Borel sets.
Consider a family of measures ${\bf Q}(D|{\bf x})$ satisfying the following
consistency condition.
C2: If $F\in{\cal B}({\cal Y}_{0,n}),$ then ${\bf Q}(F|{\bf x})$ is ${\cal
B}({\cal X}_{0,n})-$measurable function of ${\bf x}\in{\cal X}^{\mathbb{N}}.$
Then, by assumption ${\bf vi)}$, for any family of measures ${\bf
Q}(\cdot|{\bf x})$ on $({\cal Y}^{\mathbb{N}},{\cal B}({\cal
Y}^{\mathbb{N}}))$ satisfying consistency condition C2 one can construct a
collection of functions $\\{q_{n}(dy_{n}|y^{n-1},x^{n}):n\in\mathbb{N}\\}$
satisfying conditions ${\bf iv)}$ and ${\bf v)}$ which are connected with
${\bf Q}(\cdot|{\bf x})$ via relation $(\ref{equation4})$. Note that
Kolmogorov’s extension theorem guarantees the construction of countable
additive probability measures for both ${\bf P}(\cdot|{\bf y})$ and ${\bf
Q}(\cdot|{\bf x})$.
Given the basic measures ${\bf P}(\cdot|{\bf y})$ on ${\cal X}^{\mathbb{N}}$
and ${\bf Q}(\cdot|{\bf x})$ on ${\cal Y}^{\mathbb{N}}$ satisfying consistency
condition ${\bf C1}$ and ${\bf C2}$, respectively, construct the collections
of conditional distributions as follows.
Let $A^{(n)}=\\{{\bf x}:x_{n}{\in}A\\},$ $A\in{\cal B}({\cal X}_{n})$ and
$B^{(n)}=\\{{\bf y}:y_{n}{\in}B\\},$ $B\in{\cal B}({\cal Y}_{n}).$ In
addition, let ${\bf P}(A^{(n)}|{\bf y}|{{\cal B}({\cal X}_{0,n-1})})$ denote
the conditional probability of $A^{(n)}$ with respect to ${\cal B}({\cal
X}_{0,n-1})$ calculated on the probability space $\big{(}{\cal
X}^{\mathbb{N}},{\cal B}({\cal X}^{\mathbb{N}}),{\bf P}(\cdot|{\bf
y})\big{)},$ and similarly for ${\bf Q}(B^{(n)}|{\bf x}|{{\cal B}({\cal
Y}_{0,n-1})})$. Then
$\displaystyle\mathbb{P}\big{\\{}X_{n}{\in}A|X^{n-1}=x^{n-1},Y^{n-1}=y^{n-1}\big{\\}}$
$\displaystyle={\bf P}\big{(}\\{{\bf x}:x_{n}{\in}A\\}|{\bf y}|{{\cal B}({\cal
X}_{0,n-1})}\big{)}=p_{n}(A_{n};x^{n-1},y^{n-1})-a.s.$
$\displaystyle\mathbb{P}\big{\\{}Y_{n}{\in}B|Y^{n-1}=y^{n-1},X^{n}=x^{n}\big{\\}}$
$\displaystyle={\bf Q}\big{(}\\{{\bf y}:y_{n}{\in}B\\}|{\bf x}|{{\cal B}({\cal
Y}_{0,n-1})}\big{)}=q_{n}(B_{n};y^{n-1},x^{n})-a.s.$
Note that $p_{n}(\cdot;\cdot,\cdot)\in{\cal Q}({\cal X}_{n};{\cal
X}_{0,n-1}\times{\cal Y}_{0,n-1})$ and $q_{n}(\cdot;\cdot,\cdot)\in{\cal
Q}({\cal Y}_{n};{\cal Y}_{0,n-1}\times{\cal X}_{0,n})$ are stochastic kernels
[13], determined from ${\bf P}(\cdot|\cdot)$ and ${\bf Q}(\cdot|\cdot),$
respectively, (e.g., related via $(\ref{equation2})$, $(\ref{equation4})$).
The distribution of RV’s $\\{(X_{i},Y_{i}):i\in\mathbb{N}\\}$ is defined by
$\displaystyle{P}$
$\displaystyle\big{\\{}X_{0}{\in}A_{0},Y_{0}\in{B}_{0},\ldots,X_{n}{\in}A_{n},Y_{n}{\in}B_{n}\big{\\}}$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\int_{A_{0}}p_{0}(dx_{0})\int_{B_{0}}q_{0}(dy_{0};x_{0})\ldots\int_{B_{n}}q_{n}(dy_{n};y^{n-1},x^{n})$
Hence, for any ${\bf P}(\cdot|\cdot)$ and ${\bf Q}(\cdot|\cdot)$ satisfying
consistency conditions there exist a probability space and a sequence of RV’s
$\\{(X_{i},Y_{i}):i\in\mathbb{N}\\}$ defined on it, whose joint probability
distribution is defined uniquely via ${\bf P}(\cdot|\cdot)$ and ${\bf
Q}(\cdot|\cdot)$.
## III Directed Information Properties and Compactness
In this section, directed information $I(X^{n}\rightarrow{Y^{n}})$ will be
defined via relative entropy, using the basic measures ${\bf P}(\cdot|{\bf
y})$ and ${\bf Q}(\cdot|{\bf x})$, and identify its properties. Define
$\displaystyle{\cal Q}^{\bf C1}({\cal X}^{\mathbb{N}};{\cal
Y}^{\mathbb{N}})\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}{\bf
P}(\cdot|{\bf y})\in{\cal M}_{1}({\cal X}^{\mathbb{N}}):{\bf P}(\cdot|{\bf
y})~{}\mbox{are regular}$ $\displaystyle\mbox{ probability measures and
satisfy consistency condition {\bf C1}}\Big{\\}}.$ $\displaystyle{\cal Q}^{\bf
C2}({\cal Y}^{\mathbb{N}};{\cal
X}^{\mathbb{N}})\stackrel{{\scriptstyle\triangle}}{{=}}\Big{\\{}{\bf
Q}(\cdot|{\bf x})\in{\cal M}_{1}({\cal Y}^{\mathbb{N}}):{\bf Q}(\cdot|{\bf
x})~{}\mbox{ are regular}$ $\displaystyle\mbox{probability measures and
satisfy consistency condition {\bf C2}}\Big{\\}}.$
Given conditional distributions ${\bf P}(\cdot|\cdot)\in{\cal Q}^{\bf
C1}({\cal X}^{\mathbb{N}};{\cal Y}^{\mathbb{N}})$ and ${\bf
Q}(\cdot|\cdot)\in{\cal Q}^{\bf C2}({\cal Y}^{\mathbb{N}};{\cal
X}^{\mathbb{N}})$ define the following measures.
P1: The joint distribution on ${\cal X}^{\mathbb{N}}\times{\cal
Y}^{\mathbb{N}}$ defined uniquely by
$\displaystyle({\overleftarrow{P}}_{0,n}$
$\displaystyle\otimes{\overrightarrow{Q}}_{0,n})(\times^{n}_{i=0}A_{i}{\times}B_{i}),A_{i}\in{\cal
B}({\cal X}_{i}),~{}B_{i}\in{\cal B}({\cal Y}_{i}),$
$\displaystyle{\stackrel{{\scriptstyle\triangle}}{{=}}}\mathbb{P}\big{\\{}X_{0}{\in}A_{0},Y_{0}\in{B}_{0},\ldots,X_{n}{\in}A_{n},Y_{n}{\in}B_{n}\big{\\}}$
(10)
P2: The marginal distributions on ${\cal X}^{\mathbb{N}}$ defined uniquely by
$\displaystyle\mu_{0,n}(\times^{n}_{i=0}A_{i}),~{}A_{i}\in{\cal B}({\cal
X}_{i}),~{}1\leq i\leq n$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}\mathbb{P}\\{X_{0}\in{A}_{0},Y_{0}\in{\cal
Y}_{0},\ldots,X_{n}\in{A}_{n},Y_{n}\in{\cal Y}_{n}\\},$
$\displaystyle=({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n})(\times^{n}_{i=0}(A_{i}\times{\cal
Y}_{i}))$
P3: The marginal distributions on ${\cal Y}^{\mathbb{N}}$ defined uniquely for
$B_{i}\in{\cal B}({\cal Y}_{i}),~{}1\leq i\leq n$ by
$\displaystyle\nu_{0,n}(\times^{n}_{i=0}B_{i})=({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n})(\times^{n}_{i=0}({\cal
X}_{i}\times{B}_{i}))$
P4: The measure ${\overrightarrow{\Pi}}_{0,n}:{\cal B}({\cal
X}_{0,n})\odot{\cal B}({\cal Y}_{0,n})\mapsto[0,1]$ defined uniquely for
$A_{i}\in{\cal B}({\cal X}_{i})$, $B_{i}\in{\cal B}({\cal Y}_{i})$, $1\leq
i\leq n$ by
$\displaystyle{\overrightarrow{\Pi}}_{0,n}(\times^{n}_{i=0}(A_{i}{\times}B_{i}))$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}({\overleftarrow{P}}_{0,n}\otimes\nu_{0,n})(\times^{n}_{i=0}(A_{i}{\times}B_{i}))$
P5: The measure ${\overleftarrow{\Pi}}_{0,n}:{\cal B}({\cal
Y}_{0,n})\odot{\cal B}({\cal X}_{0,n})\mapsto[0,1]$ defined uniquely for
$A_{i}\in{\cal B}({\cal X}_{i})$, $B_{i}\in{\cal B}({\cal Y}_{i})$, $1\leq
i\leq n$ by
$\displaystyle{\overleftarrow{\Pi}}_{0,n}(\times^{n}_{i=0}(A_{i}{\times}B_{i}))$
$\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}(\mu_{0,n}\otimes{\overrightarrow{Q}}_{0,n})(\times^{n}_{i=0}(A_{i}{\times}B_{i}))$
### III-A Directed Information
Let ${\bf P}(\cdot|\cdot)\in{\cal Q}^{\bf C1}({\cal X}^{\mathbb{N}};{\cal
Y}^{\mathbb{N}})$ and ${\bf Q}(\cdot|\cdot)\in{\cal Q}^{\bf C2}({\cal
X}^{\mathbb{N}};{\cal Y}^{\mathbb{N}}).$ By invoking the definition of
directed information (1) or (2), it can be shown that
$\displaystyle
I(X^{n}\rightarrow{Y}^{n})=\mathbb{D}({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}||{\overrightarrow{\Pi}}_{0,n})$
(11)
$\displaystyle=\int\log\Big{(}\frac{{\overrightarrow{Q}}_{0,n}(dy^{n}|x^{n})}{\nu_{0,n}(dy^{n})}\Big{)}({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n})(dx^{n},dy^{n})$
$\displaystyle\equiv{\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
(12)
The right hand side of (11) follows from repeated application of chain rule of
relative entropy [13], while (12) follows from the fact that
${\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}<<{\overleftarrow{P}}_{0,n}\otimes\nu_{0,n}$
if and only if ${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})<<\nu_{0,n}(\cdot)$ for
$\mu_{0,n}-$almost all $x^{n}\in{\cal X}_{0,n}$. Further, if
${\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}<<{\overleftarrow{P}}_{0,n}\otimes\nu_{0,n}$
then the Radon-Nikodym derivative
$\frac{({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n})}{({\overleftarrow{P}}_{0,n}\otimes\nu_{0,n})}(x^{n},y^{n})$
represents a version of
$\frac{{\overrightarrow{Q}}_{0,n}(\cdot|x^{n})}{\nu_{0,n}(\cdot)}(y^{n})$,
$\mu_{0,n}-a.s$ for all $x^{n}\in{\cal X}_{0,n}$.
The notation ${\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}(\cdot,\cdot)$ indicates
the functional dependence of $I(X^{n}\rightarrow{Y^{n}})$ on
$\\{{\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n}\\}$.
### III-B Convexity and Concavity of Directed Information
Let ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$, ${\cal Q}^{\bf
C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ be the restrictions of ${\cal Q}^{\bf
C1}({\cal X}^{\mathbb{N}};{\cal Y}^{\mathbb{N}})$ and ${\cal Q}^{\bf C2}({\cal
Y}^{\mathbb{N}};{\cal X}^{\mathbb{N}})$, respectively, to cylinder sets with
bases over $A_{i}\in{\cal B}({\cal X}_{i})$, and $B_{i}\in{\cal B}({\cal
Y}_{i})$, $i=0,1,\ldots,n$. These are regular conditional distributions.
###### Theorem 1.
Let $\\{({\cal X}_{n},{\cal B}({\cal X}_{n})):n\in{\mathbb{N}}\\},$ $\\{({\cal
Y}_{n},{\cal B}({\cal Y}_{n})):n\in{\mathbb{N}}\\}$ be Polish spaces. Then
* 1)
${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$, ${\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n})$ are convex sets.
* 2)
${\mathbb{I}}_{X^{n}\rightarrow{Y}^{n}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
is a convex functional of ${\overrightarrow{Q}}_{0,n}\in{\cal Q}^{\bf
C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ for a fixed
${\overleftarrow{P}}_{0,n}\in{\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal
Y}_{0,n-1})$.
* 3)
${\mathbb{I}}_{X^{n}\rightarrow{Y}^{n}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
is a concave functional of ${\overleftarrow{P}}_{0,n}\in{\cal Q}^{\bf
C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$ for a fixed
${\overrightarrow{Q}}_{0,n}\in{\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal
X}_{0,n}).$
###### Proof:
1) Utilize the convexity of regular conditional distributions, and then the
consistency condition ${\bf C1}$, ${\bf C2}$. 2), 3), follow from log-sum
formulae. ∎
### III-C Lower semicontinuity-Continuity of Directed Information
This part discusses the lower-semicontinuity and continuity of directed
information as a functional of
${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal Q}^{\bf C1}({\cal
X}_{0,n};{\cal Y}_{0,n-1})$ and $\overrightarrow{Q}_{0,n}(\cdot|x^{n})\in{\cal
Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n})$. Before establishing the main
results, sufficient conditions for weak compactness of the set of measures
${{\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})}$, ${{\cal Q}}^{\bf
C2}({\cal Y}_{0,n};{\cal X}_{0,n}),$ and joint and marginal measures are
given.
###### Theorem 2.
Part A. Let ${\cal Y}_{0,n}$ be a compact Polish space and ${\cal X}_{0,n}$ a
Polish space. Assume ${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal Q}^{\bf
C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$ satisfy the following condition.
CA: For all $g(\cdot){\in}BC({\cal X}_{0,n})$, where $BC({\cal X}_{0,n})$
denotes the set of bounded continuous real-valued functions on ${\cal
X}_{0,n}$,
$\displaystyle(x^{n-1},y^{n-1})\longmapsto\int_{{\cal
X}_{n}}g(x)p_{n}(dx;x^{n-1},y^{n-1})\in\mathbb{R}$ (13)
is jointly continuous in $(x^{n-1},y^{n-1})\in{\cal X}_{0,n-1}\times{\cal
Y}_{0,n-1}$.
Then the following weak convergence results hold.
* A1)
Let ${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal Q}^{\bf C1}({\cal
X}_{0,n};{\cal Y}_{0,n-1})$ and
$\big{\\{}{\overrightarrow{Q}}_{0,n}^{\alpha}(\cdot|{x^{n}})\big{\\}}_{\alpha\geq
1}\in{\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n}).$ Then the joint measure
$({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}^{\alpha})(dx^{n},dy^{n})\buildrel
w\over{\Longrightarrow}({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}^{0})(dx^{n},dy^{n}),$
where ${\overrightarrow{Q}}_{0,n}^{0}(\cdot|x^{n})\in{\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n}).$
* A2)
Let ${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal Q}^{\bf C1}({\cal
X}_{0,n};{\cal Y}_{0,n-1})$ and
$\big{\\{}{\overrightarrow{Q}}_{0,n}^{\alpha}(\cdot|{x^{n}})\big{\\}}_{\alpha\geq
1}\in{\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ and define the family
of joint measures
$\big{\\{}({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}^{\alpha})(dx^{n},dy^{n})\big{\\}}_{\alpha\geq
1}$ having marginals $\\{\nu_{0,n}^{\alpha}\\}_{\alpha\geq 1}$ on ${\cal
Y}_{0,n}$ and $\\{\mu_{0,n}^{\alpha}\\}_{\alpha\geq 1}$ on ${\cal X}_{0,n}$.
Then $\nu_{0,n}^{\alpha}(dy^{n})\buildrel
w\over{\Longrightarrow}\nu_{0,n}^{0}(dy^{n})$ and
$\mu_{0,n}^{\alpha}(dx^{n})\buildrel
w\over{\Longrightarrow}\mu_{0,n}^{0}(dx^{n})$ where $\nu_{0,n}^{0}\in{\cal
M}_{1}({\cal Y}_{0,n})$ and $\mu_{0,n}^{0}\in{\cal M}_{1}({\cal X}_{0,n})$ are
the marginals of
$({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}^{0})(dx^{n},dy^{n}).$
* A3)
The sets of measures ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$, and
${\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ are weakly compact.
* A4)
Let ${\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal Q}^{\bf C1}({\cal
X}_{0,n};{\cal Y}_{0,n-1}),$
$\big{\\{}{\overrightarrow{Q}}_{0,n}^{\alpha}(\cdot|{x^{n}})\big{\\}}_{\alpha\geq
1}\in{\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n}),$ and
$\\{\nu_{0,n}^{\alpha}\\}_{\alpha\geq 1}$ the marginals of
$\big{\\{}({\overleftarrow{P}}_{0,n}\otimes{\overrightarrow{Q}}_{0,n}^{\alpha})(dx^{n},dy^{n})\big{\\}}_{\alpha\geq
1}.$ Then
${\overrightarrow{\Pi}}_{0,n}^{\alpha}(dx^{n},dy^{n})\equiv{\overleftarrow{P}}_{0,n}(dx^{n}|dy^{n-1})\otimes\nu_{0,n}^{\alpha}(dy^{n})\buildrel
w\over{\Longrightarrow}{\overleftarrow{P}}_{0,n}(dx^{n}|dy^{n-1})\otimes\nu_{0,n}^{0}(dy^{n})\equiv{\overrightarrow{\Pi}}_{0,n}^{0}(dx^{n},dy^{n}),$
where $\nu_{0,n}^{0}\in{\cal M}_{1}({\cal Y}_{0,n})$ is the weak limit of
$\nu_{0,n}^{\alpha}\in{\cal M}_{1}({\cal Y}_{0,n})$.
Part B. Let ${\cal X}_{0,n}$ be a compact Polish space and ${\cal Y}_{0,n}$ a
Polish space. Assume ${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in{\cal Q}^{\bf
C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ satisfy the following condition.
CB: For all $h(\cdot){\in}BC({\cal Y}_{0,n})$, the function
$\displaystyle(x^{n},y^{n-1})\longmapsto\int_{{\cal
Y}_{n}}h(y)q_{n}(dy;y^{n-1},x^{n})\in\mathbb{R}$ (14)
is jointly continuous in $(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}$.
The statements of Part A hold by interchanging ${\overleftarrow{Q}}_{0,n}$
with ${\overleftarrow{P}}_{0,n}$, $\nu_{0,n}$ with $\mu_{0,n}$,
${\overrightarrow{\Pi}}_{0,n}$ with ${\overleftarrow{\Pi}}_{0,n}$.
###### Proof:
The proof is quite lengthy and it is based on Prohorov’s theorem relating
tightness and weak compactness of a family of probability measures [13].∎ The
results of Theorem 2 are sufficient to establish lower semicontinuity of
directed information
$I(X^{n}\rightarrow{Y}^{n})\equiv{\mathbb{I}}_{X^{n}\rightarrow{Y}^{n}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$.
###### Theorem 3.
1) Suppose the conditions in Theorem 2, Part A hold. Then
${\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
is lower semicontinuous on ${\overrightarrow{Q}}_{0,n}\in{\cal Q}^{\bf
C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ for fixed
${\overleftarrow{P}}_{0,n}\in{\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal
Y}_{0,n-1})$.
2) Suppose the conditions in Theorem 2, Part B hold. Then
${\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
is lower semicontinuous on ${\overleftarrow{P}}_{0,n}\in{\cal Q}^{\bf
C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$ for fixed
${\overrightarrow{Q}}_{0,n}\in{\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal
X}_{0,n})$.
###### Proof:
Utilizes (11), Theorem 2, and lower semicontinuity of relative entropy.∎ For
capacity problems, it is desirable to identify conditions so that
${\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
as a function of $\overleftarrow{P}_{0,n}$ for fixed
$\overrightarrow{Q}_{0,n}$ is either upper semicontinuous or continuous.
###### Theorem 4.
Consider a forward channel ${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in{\cal
Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n}),$ and a closed family of feedback
channels ${\cal Q}^{c,\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})\subseteq{\cal
Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1}).$ Suppose there exists a family
of measures $\bar{\nu}_{0,n}(dy^{n})$ on $({\cal Y}_{0,n},{\cal B}({\cal
Y}_{0,n}))$ such that
${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\ll{\bar{\nu}}_{0,n}(dy^{n})$ with
Radon-Nikodym derivatives
$\xi_{\bar{\nu}_{0,n}}(x^{n},y^{n})\stackrel{{\scriptstyle\triangle}}{{=}}\frac{{\overrightarrow{Q}}_{0,n}(\cdot|x^{n})}{\bar{\nu}_{0,n}(\cdot)}(y^{n}),$
and
1) the family of Radon-Nikodym derivatives
$\xi_{\bar{\nu}_{0,n}}(x^{n},y^{n})$ is continuous on ${\cal
X}_{0,n}\times{\cal Y}_{0,n},$ and
$\xi_{\bar{\nu}_{0,n}}(x^{n},y^{n})\log\xi_{\bar{\nu}_{0,n}}(x^{n},y^{n})$ is
uniformly integrable over
$\\{\bar{\nu}_{0,n}\otimes{\overleftarrow{P}}_{0,n}:{\overleftarrow{P}}_{0,n}\in{\cal
Q}^{c,\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})\\}$;
2) for a fixed $y^{n}\in{\cal Y}_{0,n},$ the Radon-Nikodym derivative
$\xi_{\bar{\nu}_{0,n}}(x^{n},y^{n})$ is uniformly integrable over ${\cal
Q}^{c,\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1}).$
Then, the directed information
${\mathbb{I}}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
as a functional of
$\\{{\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n}\\}\in{\cal Q}^{c,\bf
C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})\times{\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n})$ is bounded and weakly continuous over ${\cal
Q}^{c,\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$.
###### Proof:
Invoke (11) and generalize related results in [14]. ∎
## IV Extremum Problems of Directed Information
### IV-A Existence of Capacity Achieving Distribution
Consider a communication channel with memory and feedback
${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in{\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n})$ and power constraint
$\displaystyle\overleftarrow{\cal{P}}_{0,n}(P)\stackrel{{\scriptstyle\triangle}}{{=}}\big{\\{}{\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in{\cal
Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1}):$
$\displaystyle\int{g}_{0,n}(x^{n},y^{n-1})({\overleftarrow{Q}}_{0,n}\otimes{\overrightarrow{P}}_{0,n})(dx^{n},dy^{n})\leq{P}\big{\\}}$
where for any $n\in\mathbb{N}$, $g_{0,n}:{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}\longmapsto[0,\infty]$ is Borel measurable, and
$\overleftarrow{\cal{P}}_{0,n}(P)$ non-empty. In the absence of any power
constraints the set of input conditional distributions is ${\cal Q}^{\bf
C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$.
The finite horizon maximization of directed information over
$\overleftarrow{\cal{P}}_{0,n}(P)$ or ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal
Y}_{0,n-1})$ (e.g., with or without power constraints) is defined by
$\displaystyle
C^{f}_{0,n}\stackrel{{\scriptstyle\triangle}}{{=}}\sup_{\begin{subarray}{c}{\overleftarrow{P}}_{0,n}(\cdot|y^{n-1})\in\overleftarrow{\cal{P}}_{0,n}(P)\\\
~{}{or}~{}{\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal
Y}_{0,n-1})\end{subarray}}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}({\overleftarrow{P}}_{0,n},{\overrightarrow{Q}}_{0,n})$
(15)
The next theorem establishes existence of the maximizer.
###### Theorem 5.
Suppose the assumptions of Theorem 2, Part A are satisfied.
* 1)
The set ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$ is compact.
* 2)
Suppose $g_{0,n}:{\cal X}_{0,n}\times{\cal Y}_{0,n-1}\longmapsto[0,\infty]$ is
measurable and continuous in $(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}$. Then the set $\overleftarrow{\cal{P}}_{0,n}(P)$ is a closed
subset of ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$.
* 3)
If in addition the assumptions of Theorem 4 are satisfied (here the assumption
on ${\cal Q}^{\bf C1}({\cal X}_{0,n};{\cal Y}_{0,n-1})$ is satisfied by 1) and
2) ) then $C_{0,n}^{f}$ has a maximum in ${\cal Q}^{\bf C1}({\cal
X}_{0,n};{\cal Y}_{0,n-1})$ (without constraints) or in
$\overleftarrow{\cal{P}}_{0,n}(P)$ (with power constraints).
###### Proof:
1) Utilize the fact that probability measures on compact Polish spaces are
compact. 2) Utilize the fact that closed subset of weakly compact set is
compact. 3) Follows from Weierstrass theorem.∎
### IV-B Existence of Non-Anticipative Rate Distortion Achieving Distribution
Consider a reconstruction channel
${\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in{\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n})$, a fixed source $\mu_{0,n}(dx^{n})\in{\cal
M}_{1}({\cal X}_{0,n})$, and define the fidelity constraint by
$\displaystyle\overrightarrow{Q}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\big{\\{}{\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in{\cal
Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n}):$
$\displaystyle\int{d}_{0,n}(x^{n},y^{n})({\mu}_{0,n}\otimes{\overrightarrow{Q}}_{0,n})(dx^{n},dy^{n})\leq{D}\big{\\}}$
(16)
where $D\geq{0}$, and for each $n\in\mathbb{N}$, $d_{0,n}:{\cal
X}_{0,n}\times{\cal Y}_{0,n}\longmapsto[0,\infty]$ is Borel measurable, and
$\overrightarrow{Q}_{0,n}(D)$ is non-empty.
The finite horizon minimization of directed information over
$\overrightarrow{Q}_{0,n}(D)$ is defined by
$\displaystyle
R^{c}_{0,n}(D)\stackrel{{\scriptstyle\triangle}}{{=}}\inf_{{\overrightarrow{Q}}_{0,n}(\cdot|x^{n})\in\overrightarrow{Q}_{0,n}(D)}\mathbb{I}_{X^{n}\rightarrow{Y^{n}}}({\mu}_{0,n},{\overrightarrow{Q}}_{0,n})$
(17)
The next theorem establishes existence of the minimizer.
###### Theorem 6.
Let ${\cal X}_{0,n}$ be a Polish space and ${\cal Y}_{0,n}$ a compact Polish
space. Assume $\forall~{}h(\cdot){\in}BC({\cal Y}_{0,n})$, the function
$\displaystyle(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}\longmapsto\int_{{\cal
Y}_{n}}h(y)q_{n}(dy;y^{n-1},x^{n})\in\mathbb{R}$
is continuous jointly in $(x^{n},y^{n-1})\in{\cal X}_{0,n}\times{\cal
Y}_{0,n-1}$.
Then
* 1)
The set ${\cal Q}^{\bf C2}({\cal Y}_{0,n};{\cal X}_{0,n})$ is compact.
* 2)
Assume $d_{0,n}:{\cal X}_{0,n}\times{\cal Y}_{0,n}\longmapsto[0,\infty]$ is
measurable and continuous on $y^{n}\in{\cal Y}_{0,n}$. Then
$\overrightarrow{Q}_{0,n}(D)$ is a closed subset of ${\cal Q}^{\bf C2}({\cal
Y}_{0,n};{\cal X}_{0,n})$.
* 3)
$R_{0,n}^{c}(D)$ has a minimum in $\overrightarrow{Q}_{0,n}(D)$.
###### Proof:
Utilize Theorem 2, Part A, and generalize the derivation in [12] to
$(n+1)-$fold convolution measures. ∎
## V Conclusion
In this paper we have provided a general framework through which the
properties of mutual information are extended to directed information on
Polish spaces. The existence of extremums to capacity problems with memory and
feedback, and to lossy non-anticipative data compression problems are
discussed.
## References
* [1] H. Marko, “The bidirectional communication theory–A generalization of information theory,” _IEEE Transactions on Communications_ , vol. 21, no. 12, pp. 1345–1351, Dec. 1973.
* [2] J. L. Massey, “Causality, feedback and directed information,” in _International Symposium on Information Theory and its Applications (ISITA ’90)_ , Nov. 27-30 1990, pp. 303–305.
* [3] S. Tatikonda and S. Mitter, “The capacity of channels with feedback,” _IEEE Transactions on Information Theory_ , vol. 55, no. 1, pp. 323–349, Jan. 2009.
* [4] S. C. Tatikonda, “Control over communication constraints,” Ph.D. dissertation, Mass. Inst. of Tech. (M.I.T.), Cambridge, MA, 2000.
* [5] J. Chen and T. Berger, “The capacity of finite-state Markov channels with feedback,” _IEEE Transactions on Information Theory_ , vol. 51, no. 3, pp. 780–798, March 2005.
* [6] R. Venkataramanan, “Information-theoretic results on communication problems with feed-forward and feedback,” Ph.D. dissertation, University of Michigan-Ann Arbor, December 2008.
* [7] P. A. Stavrou, C. D. Charalambous, and C. K. Kourtellaris, “Causal rate distortion function on abstract alphabets: Optimal reconstruction and properties,” _CoRR_ , vol. abs/1202.0895, 2012. [Online]. Available: http://arxiv.org/abs/1202.0895v1
* [8] G. Kramer, “Directed information for channels with feedback,” Ph.D. dissertation, Swiss Federal Institute of Technology (ETH), December 1998.
* [9] ——, “Capacity results for the discrete memoryless network,” _IEEE Transactions on Information Theory_ , vol. 49, no. 1, pp. 4–21, Jan. 2003.
* [10] T. S. Han and S. Verdu, “Approximation theory of output statistics,” _IEEE Transactions on Information Theory_ , vol. 39, no. 3, pp. 752–772, May 1993.
* [11] I. Csiszár, “Arbitrarily varying channels with general alphabets and states,” _IEEE Transactions on Information Theory_ , vol. 38, no. 6, pp. 1725–1742, Nov. 1992.
* [12] ——, “On an extremum problem of information theory,” _Studia Scientiarum Mathematicarum Hungarica_ , vol. 9, pp. 57–71, 1974.
* [13] P. Dupuis and R. S. Ellis, _A Weak Convergence Approach to the Theory of Large Deviations_. John Wiley & Sons, Inc., New York, 1997.
* [14] M. Fozunbal, S. McLaughlin, and R. Schafer, “Capacity analysis for continuous-alphabet channels with side information, part I: A general framework,” _IEEE Transactions on Information Theory_ , vol. 51, no. 9, pp. 3075–3085, Sept. 2005.
|
arxiv-papers
| 2012-02-05T12:45:48 |
2024-09-04T02:49:27.046781
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Charalambos D. Charalambous and Photios A. Stavrou",
"submitter": "Photios Stavrou",
"url": "https://arxiv.org/abs/1202.0958"
}
|
1202.1054
|
# Considering a resource-light approach to learning verb valencies
Alex Rudnick
School of Informatics and Computing, Indiana University
Bloomington, Indiana, USA
alexr@cs.indiana.edu
###### Abstract
Here we describe work on learning the subcategories of verbs in a
morphologically rich language using only minimal linguistic resources. Our
goal is to learn verb subcategorizations for Quechua, an under-resourced
morphologically rich language, from an unannotated corpus. We compare results
from applying this approach to an unannotated Arabic corpus with those
achieved by processing the same text in treebank form. The original plan was
to use only a morphological analyzer and an unannotated corpus, but
experiments suggest that this approach by itself will not be effective for
learning the combinatorial potential of Arabic verbs in general. The lower
bound on resources for acquiring this information is somewhat higher,
apparently requiring a a part-of-speech tagger and chunker for most languages,
and a morphological disambiguater for Arabic.
## 1 Introduction
When constructing NLP systems for a new language, we often want to know the
valence of its verbs, which is to say how many and which types of arguments
each verb may combine with. Some dictionaries may provide this information,
but even assuming a broad-coverage machine-readable dictionary exists a given
language, that dictionary may not say whether arguments are optional for a
given verb, or how likely they are to occur.
Knowing the selectional preferences and requirements of verbs is useful for
systems that have explicit lexicalized grammars of the languages they cover,
whether for parsing, generation, or both [1997], and of linguistic interest on
its own [2004]. The aim of this work was to build resources for use in L3
[2011], a rule-based machine translation system based on dependency grammars,
which records the combinatorial possibilities for every word in its lexicon,
and during parsing and generation constructs a graph describing the structure
of the input and output sentences. We are particularly interested in
linguistic resources for Quechua, which is spoken by roughly 10 million people
in the Andean region of South America, and is thus the largest indigenous
language of the Americas. However, evaluating the approach for Quechua is
difficult, due to a lack of existing lexica and treebanks, so initial
experiments have been carried out with Arabic.
An empirical approach based on a corpus or treebank allows us to learn the
relative frequency with which a given verb takes specific types of arguments.
As a simple example from English, we would like to be able to learn that while
“eat” usually has a direct object, “put” nearly always has one. Verbs may also
occur with clausal dependents in various ways. For some examples in English,
see Figure 1.
In order to automatically learn this information for resource-scarce,
morphologically rich languages, we set out to implement a system that requires
only an unannotated corpus and a morphological analyzer; other recent
approaches have made use of more linguistic knowledge, in the form of
treebanks, parsers, or chunkers. In practice, our we will also require more
resources to be fruitful; this may be addressed in the future.
* I put. | I believe that he is tall.
---|---
* I put the potato. | I consider him tall.
I put the potato on the table. | * I consider that he is tall.
Figure 1: “What did you do yesterday?”, and _believe_ vs. _consider_
## 2 Related work
Many other researchers have addressed the problem of documenting the
properties of the different verbs in a given language, using evidence from
corpora and manual lexicography. Automatic approaches have the potential to
involve less manual work avoid human biases, giving a more objective measure
of the behavior of a given verb.
We see in the literature a few different terms that describe the combinatorial
potential of a verb, including _subcategorization_ , _subcategorization
frames_ , _valence_ or _valency_. In any case, these terms describe which
arguments and adjuncts may appear with a given verb, how often, and which ones
are obligatory. While describing similar notions, these terms do not seem to
be interchangeable; while this work is concerned with with “surface level”
syntax, looking for arguments that are present in practice (such that a parser
could find them), in Functional Generative Description (FGD), the term valency
refers to a tectogrammatical notion; arguments might be known to the speaker
but not expressed. This deeper notion cannot be readily observed from text
alone, as pointed out by Bojar [2003].
### 2.1 Valency lexica for English
Gahl et al. [2004] describe a study in which they had a team of linguists
annotate thousands of English sentences – 200 sentences for each of 281
English verbs of interest – and build a table of distributions of
subcategorization frames that they observed for each of the verbs. They
describe the difficulties that may be faced in trying to learn
subcategorizations from a corpus: in a given body of text (even one as big as
the Brown corpus), it may be that not all possible subcategorizations will be
observed. Additionally, different genres of text may exhibit different verb
usage. This paper also gives a good overview of the uses of valency
information and a view on verb subcategorization from psycholinguistics,
including elicitation experiments that psycholinguists have used to learn the
relative frequencies of different uses of verbs.
Gahl’s group has made their results available in machine-readable form,
providing a potentially useful resource for those interested in English verbs.
However, their approach was very labor-intensive and required a large corpus.
Ushioda et al. [1993] describe earlier work on acquiring verb
subcategorizations for English. Their method requires a tagged corpus,
although an untagged corpus and an accurate tagger would work as well. On the
basis of the tags, they perform partial parsing to identify noun phrases
(chunking), and then use some simple rules specified in terms of regular
grammars to identify common patterns of constituents in the sentences, which
are marked with corresponding subcategorization frames. This approach does not
require the use of a deep parser, but the rules had to be crafted specifically
for English.
Ushioda explores the WSJ corpus with this extractor, and reports results on 33
randomly selected common verbs: the extraction rules achieve 86% accuracy over
sentences from a test set taken from WSJ, where the correct subcategorization
frames for the test set had been determined manually.
Brent [1993] addresses the seeming impasse that in order to get accurate
parses automatically, one needs to know about the syntactic frames of
different verbs, but in order to get the frame information from a corpus, the
sentences must be parsed accurately. He handles this problem by crafting
language-specific rules that initially only refer to closed-class words and do
not require complete parses. This approach is somewhere between having no
syntactic knowledge at all, and requiring a large grammar of the target
language: to start out, it must first figure out which words in the corpus are
verbs. He then uses statistics to infer previously unknown facts about the
language, for example, which English verbs can occur with each of six
different kinds of arguments.
While this appears to be an effective approach, one wonders how hard it would
be to apply to an unfamiliar language. Producing the initial rules may require
a lot of linguistic insight; for example, Brent relies on the fact that in
English, verbs typically do not appear immediately after determiners or
prepositions. In a language with more free word order, or one without
determiners or prepositions, what sorts of rules might one use?
Briscoe and Carroll [1997] describe a system that finds subcategorization
frames for verbs in English, including relative frequencies for each class for
a given verb. They adopt a very detailed scheme for verb subcategories, in
which each usage falls into one of 160 different classes, where each class
includes specific information about particles and control of the arguments of
the verb. Their system requires the use of a POS tagger, a lemmatizer, and a
pre-trained probabilistic parser; after identifying and classifying the
different subcategories of verb usages, they incorporate this information into
another parser and demonstrate that it improves parsing accuracy.
### 2.2 Valency lexica for Slavic languages
The VALLEX project [2007] has produced a large hand-curated database of
valency frames for verbs in Czech, covering roughly the 2500 most common verbs
in Czech and cataloging their various senses. VALLEX makes use of Functional
Generative Description as the background linguistic theory for its account of
verbs, and so records, at least, whether a verb sense takes an _actor_ ,
_addressee_ , _patient_ , _effect_ , and _origin_ , and whether these must be
specified, as well as a large number of other “quasi-valency complementations”
and “free modifications”. VALLEX provides a very detailed account of the
potential uses of each verb in its lexicon, much more detailed than what can
currently be produced with automatic methods.
More recently, Przepiòrkowski has done work focusing on Polish, comparing
valence dictionaries built with the use of shallow parsing to those built with
deep parsing [2009]. Because his shallow parser may not handle all of the
sentences in the corpus, his approach ends up ignoring more than half of the
training data, but from the remaining 41% of the IPI PAN Corpus, he collects
counts of the different frames in which each verb was observed, and uses a
small number of Polish-specific rules to post-process the observations, then
does statistical filtering to try to reduce noisy observations.
Przepiòrkowski evaluates the extracted lexical information in two different
ways, making use of both pre-existing valence dictionaries and sentences hand-
annotated by linguists, finding that his shallow-parsing technique actually
produces results that agree more closely with frames that were observed in the
texts by linguists than the existing valence dictionaries.
Debowski [2009] presents a procedure for extracting valence information and
frame weights for Polish that makes use of a non-probabilistic deep parser and
a novel use of EM, which he says is simpler than the more traditional repeated
inside-outside approach to optimizing weights for a probabilistic grammar.
Additionally, in his EM formulation, the weight-optimization problem is
convex, so he can start with uniform prior probabilities and be guaranteed to
get a globally optimum solution. Debowski also includes an approach for
filtering incorrect frames that were found in the parsed text.
When analyzing his results, Debowski notes that some of his observed “false
positives” described valid uses of the verbs in question, but were not
included in the compiled valence dictionaries that he used in evaluating his
approach.
### 2.3 Valency Lexica for Arabic
Informed by the Prague Arabic Dependency Treebank and the Functional
Generative Description (FGD) theory of syntax, Bielický and Smrž [2008]
describe desiderata for a valency lexicon for Arabic. They do not describe the
production of such a lexicon in practice, but lay out a framework for
discussing one, proposing a structure for lexical entries in the valence
dictionary. Their structure is based on VALLEX, which seems to have a broadly
applicable formalism for describing verbs. They also describe some tools
useful for the task, including an FST-based morphological analyzer for Arabic,
and explain FGD’s account of verbal arguments/adjuncts.
### 2.4 Resources for Quechua
Rios et al. [2009] address the more general problem of acquiring enough
linguistic knowledge to build effective NLP systems for under-resourced
languages such as Quechua, with a more labor-intensive approach. They describe
their construction of a phrase-aligned treebank for Quechua and Spanish, which
covers about 200 sentences, with text from the _Declaration of Human Rights_
(available in many languages, including Spanish and Quechua) and the website
of _La Defensoría del Pueblo_ , a Peruvian government organization that
advocates for citizens rights. Aside from the morphological analysis of
Quechua, the treebanking and alignment process currently require human
attention, though this may be partially automated in the future.
The treebank so far is small, but it may be increasingly useful for machine
translation as their treebanking process becomes more automated. Rios _et al._
note a surprising number of available bitexts for Spanish/Quechua, including
political texts, news, translated novels and poetry.
## 3 Proposed Approach
Our approach starts by processing each sentence in the corpus with the
morphological analyzer, thus finding all of the verbs. For sentences with only
one verb, we then count the occurrences of nouns that seem to be, because of
inflection, the arguments of the verb. Here plausible verb arguments will need
to be identified with a small number of language-specific heuristics. For
example, a noun inflected with the accusative case in a sentence with a verb
and a clear subject will likely be the object of that verb. This approach
throws away the information from sentences with multiple verbs and embedded
clauses, but it does not require syntactic analysis. We had hoped that the
frequencies learned with this approach will approximate the frequencies that
would be learned using deeper syntactic analysis, but this does not bear out
empirically.
Noisy observations could be filtered out using an approach similar to the one
described in [2009]. In the long run, for consistency, we would like to build
a lexicon in the VALLEX style, discovering whether each given verb usage
contains an explicit Actor, Addressee, Patient, Effect, and Origin, when these
roles can be identified by the morphological cues.
### 3.1 Evaluating Valency Learning Techniques
When building a system that builds valency lexica for the verbs of a given
language, we would like both good recall, meaning that the system identifies a
many of the verb usages that are actually present in the training text, and
high precision, meaning that the answers the system returns are actually
correct. To measure both of these, we can take some preexisting lexicon to be
the gold standard, but good valency lexica are not available for most
languages.
What we can do instead is take the verb usages in a treebank, and consider the
subcategorization lexicon constructed in that way to be the gold standard. We
have an Arabic treebank (Arabic Treebank Part 1, v3.0) available from the LDC
[2005], so for this work we make use both of that treebank and the associated
flat text. We chose Arabic for its rich morphology, and for the somewhat
convenient, though not freely redistributable, treebank. If the results were
good for Arabic, then that would be evidence that it might be helpful for
constructing valency lexica for other languages as well.
## 4 Experiments with Arabic
We carried out experiments with the text of the Arabic treebank, using both
the transliterated text with syntactic annotations and the unannotated Unicode
text in Arabic script. Given the treebank annotations, we can find the verbs
in each sentence, as well as the other components of the verb phrases, quite
easily by traversing the parse trees. For initial experiments due to the
sparsity of the data, we pass over the problem of deciding whether a
constituent is an argument or adjunct of the verb.
To find all of the verb subcategory frames in the treebank, we traverse the
tree of each sentence and record the immediate children of the verb phrase
that are not the verb itself. These are considered a set, and recorded with
the stem of the verb in question. The process is described in more detail in
Figure 2.
* •
For each sentence in the treebank…
* –
For each verb phrase in that sentence…
* *
Look for a word in the VP with a tag that contains one of IV, PV, IV_PASS or
PV_PASS (one may not be present; if so, skip this VP)
* *
Find the stem for the verb, if present
* *
Record the verb stem, along with the tags of the sibling constituents.
Figure 2: Process for finding verbs and arguments in the treebank
Considering the entire treebank, which consists of 734 news documents, there
are 5845 sentences, containing 14115 verb phrases. The majority (92%) of these
verb phrases have a verb that can be found with the rules described. The most
common verb stems, presented in Buckwalter transliteration, were: “kAn”,
“qAl”, “>aEolan”, “>aDAf”, “>ak ad”, “kuwn”, “>awoDaH”, “*akar”, “mokin”,
“>afAd”. Each of these occurred at least 100 times in the corpus. Not all of
these can be translated sensibly to English without context by Google
Translate, but using it as a glossing tool, we get: _was_ , _declared_ ,
_added_ , _confirmed_ , _fact_ , _clear_ , _enabled_ , and _reported_. There
were 1747 different verb stems observed altogether.
Adapting the approach of Przepiòrkowski [2009], we then focus on the sentences
from the corpus that contain only one verb. This allows us to avoid making
attachment decisions, since deep parsers may not be available or reliable.
Bojar [2003] does something similar, with the addition of a chunker that can
find subordinate clause boundaries. This approach also seems sensible
particularly for Quechua and Arabic, since case is typically marked on nouns
for both languages, although this still leaves the problem of dropped
arguments.
On its own, filtering out a large number of sentences is not a problem; that
we include a nontrivial fraction of the sentences at all is promising. To
improve coverage, we could simply feed the system more unannotated text. For
Arabic, we could use the very large supply of Arabic news available on the
web. This approach would be less plausible for Quechua, although of course
unannotated Quechua text is more plentiful than Quechua treebanks.
| count | fraction
---|---|---
sentences in PATB part 1 | 5845 | 1.0
sentences with only one VP | 926 | 0.16
unique verb stems observed | 1747 | 1.0
unique verb stems in sentences with only one VP | 376 | 0.22
Figure 3: Filtering results on sentences from the Penn Arabic Treebank
### 4.1 Sentence Selection in Practice
We might wonder, however, whether our sampling of sentences leads to biases in
the observed verbs and their usages. Considering English verbs such as
“think”, “believe”, or “request”, which usually occur with some clausal
argument that includes some other verb, we imagine that the analogous verbs in
the language we are investigating would be under-represented or simply not
learned at all.
Experiments showed that both of these worries are well-founded: sentences that
had only one verb had a substantially different distribution of verb stems.
The verbs that were most common in the one-verb sentences were “saj al”,
“qAl”, “>aEolan”, “$Arik”, “kAn”, “daEA”, “fAz”, “balag”, “>aHoraz”, and
“lotaqiy”. These are glossed by Google Translate as: _record_ , _said_ ,
_announced_ , _was_ , _called_ , _beat_ , _was_ , _made_ , and _assess_ ,
definitely a different sort of verb than the ones we see commonly in the in
the text generally.
Even among the verbs that happen commonly in both the text in general and the
one-verb sentences, we observe different usages. The most common verb in
general, “kAn” (_was_), occurs with another verb phrase as an argument about
400 times in the treebank. The second-most common verb in general and in the
one-verb sentences, “qAl” (“declared/said”), most often occurs with an SBAR
(indicating a nested clause) in general, but of course these usages do not
occur in the one-verb sentences.
### 4.2 Morphological ambiguity
In experiments with the nearly-unannotated111very lightly marked-up with SGML
text distributed with the treebank, we made use of AraMorph [2004], a Free
Software version of the Buckwalter morphological analyzer that handles Unicode
text. The goal with the unannotated text was to see which subcategory frames
we could observe in sentences with a single verb – the rich Arabic morphology
usually marks case on nouns, which should allow us to find many arguments to
verbs. We could then compare the valencies learned from the unannotated corpus
with those that are more easily observable from the treebank. If the valencies
that we discover with the unannotated approach are close to those learned from
the treebank, and we get a broad coverage over the verbs observed in the
corpus, then this would provide an argument that the technique works fairly
well for Arabic, and we could continue using it as we acquire more textual
data for more under-resourced languages.
However, Buckwalter-style morphological analyzers do not account for
morphological ambiguity, which would present difficulties in the long run.
This problem is particularly dire because the Arabic script is an _abjad_ ,
which is to say that it only records the consonants for each word. There is an
optional system for annotating the vowel sounds as well, but it is often not
used in practice. This presents a problem for Arabic-language NLP systems,
though, since a word without context rarely has a unique morphological
analysis. In fact, within the corpus, we observed a mean of about 7.5 possible
Buckwalter analyses per word, with $\sigma=8.4$, and a maximum of 86. 222The
word with 86 Buckwalter analyses can mean, at least, “one”, “and scrutinize”,
“and sharpen”, or “and be furious”.
We also briefly experimented with a Quechua morphological analyzer, Michael
Gasser’s AntiMorfo system [2010]; it can analyze Quechua verbs, nouns, and
adjectives. We also see morphological ambiguity in Quechua words, although it
is not nearly so striking, largely due to the orthography with vowels. We saw
a mean of 1.7 analyses per word, with $\sigma=1.3$, and a maximum of 10. For
Quechua, we used a small corpus produced by the AVENUE project, described in
[2006], which includes bitext elicited from native speakers, and monolingual
text, both from UN documents and local stories. Interestingly, with the FST-
based morphological analysis, we cannot tell whether a word is definitely a
verb. For example, _waqaychu_ may be a verb, noun, or infinitive, as “waqa” is
in AntiMorfo’s lexicon as a verb root, and “waqay” as a noun root. So to get
good results for Quechua, we would need a POS tagger or other means to choose
between analyses. As far as treebanks of Quechua, for evaluating an
automatically-extracted Quechua verb lexicon, to our knowledge there is no
large one available, although Rios et al. are developing a small one [2009].
## 5 Conclusions and future work
While the problem of discovering grammatical subcategories of verbs remains
interesting, and solutions to that problem would be of practical use, it is
almost definitely not enough to use only a morphological analyzer and a
medium-sized unannotated corpus for this purpose; disregarding sentences with
more than one verb leads to misleading view of the language as a whole, and
selection preferences learned from these sentences would not be suitable for
parsing most sentences. To make better use of the existing data, it would be
helpful to have a chunker to find the boundaries of clauses and noun phrases,
as in [2009]. For Czech verbs, Bojar similarly used finite-state rules to find
coordinated and subordinate clauses [2003].
While ambiguity in morphological analysis can be a hurdle in any language, the
vowel-free nature of typical Arabic text presents a particularly serious one;
we would like to be able to use any available unannotated text, but without
morphological disambiguation, we are left with many possible interpretations
for most tokens. This could be mitigated with software like MADA+TOKAN [2009],
which chooses the most likely morphological analysis for a given context; for
other languages, such as Quechua, similar tools will need to be developed.
Even with proper POS tagging, morphological disambiguation, and chunking, we
are still faced with the problem of negative evidence; without a very large
corpus, we cannot say with confidence that a given verb cannot appear with
certain arguments, simply because it has not yet been observed with those
arguments. This difficulty suggests an active learning approach, perhaps
coupled with crowdsourcing. We could imagine a system that generates sentences
to test the hypothesis that a given verb may be used with a given
subcategorization frame, then presents those sentences to human users for
grammaticality judgments.
## References
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|
arxiv-papers
| 2012-02-06T06:33:03 |
2024-09-04T02:49:27.056856
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alex Rudnick",
"submitter": "Alex Rudnick",
"url": "https://arxiv.org/abs/1202.1054"
}
|
1202.1080
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2011-227 LHCb-PAPER-2011-019
Measurement of the cross-section ratio
${\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})}$ for prompt ${\chi_{c}}$ production
at ${\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-2.07413ptV}}$}}$
The LHCb Collaboration 111Authors are listed on the following pages.
The prompt production of the charmonium $\chi_{c1}$ and $\chi_{c2}$ mesons has
been studied in proton-proton collisions at the Large Hadron Collider at a
centre-of-mass energy of $\sqrt{s}=7$ TeV. The $\chi_{c}$ mesons are
identified through their decays $\chi_{c}\,\rightarrow\,J/\psi\,\gamma$ with
$J/\psi\,\rightarrow\,\mu^{+}\,\mu^{-}$ using 36 $\mathrm{pb^{-1}}$ of data
collected by the LHCb detector in 2010. The ratio of the prompt production
cross-sections for the two $\chi_{c}$ spin states,
$\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$, has been determined as a function
of the $J/\psi$ transverse momentum, $p_{\mathrm{T}}^{J/\psi}$, in the range
from 2 to 15 GeV/$c$. The results are in agreement with the next-to-leading
order non-relativistic QCD model at high $p_{\mathrm{T}}^{J/\psi}$ and lie
consistently above the pure leading-order colour singlet prediction.
Submitted to Phys. Lett. B
LHCb Collaboration
R. Aaij23, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi42, C. Adrover6, A.
Affolder48, Z. Ajaltouni5, J. Albrecht37, F. Alessio37, M. Alexander47, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr22, S. Amato2, Y. Amhis38, J.
Anderson39, R.B. Appleby50, O. Aquines Gutierrez10, F. Archilli18,37, L.
Arrabito53, A. Artamonov 34, M. Artuso52,37, E. Aslanides6, G. Auriemma22,m,
S. Bachmann11, J.J. Back44, D.S. Bailey50, V. Balagura30,37, W. Baldini16,
R.J. Barlow50, C. Barschel37, S. Barsuk7, W. Barter43, A. Bates47, C. Bauer10,
Th. Bauer23, A. Bay38, I. Bediaga1, S. Belogurov30, K. Belous34, I.
Belyaev30,37, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S. Benson46, J.
Benton42, R. Bernet39, M.-O. Bettler17, M. van Beuzekom23, A. Bien11, S.
Bifani12, T. Bird50, A. Bizzeti17,h, P.M. Bjørnstad50, T. Blake37, F. Blanc38,
C. Blanks49, J. Blouw11, S. Blusk52, A. Bobrov33, V. Bocci22, A. Bondar33, N.
Bondar29, W. Bonivento15, S. Borghi47,50, A. Borgia52, T.J.V. Bowcock48, C.
Bozzi16, T. Brambach9, J. van den Brand24, J. Bressieux38, D. Brett50, M.
Britsch10, T. Britton52, N.H. Brook42, H. Brown48, A. Büchler-Germann39, I.
Burducea28, A. Bursche39, J. Buytaert37, S. Cadeddu15, O. Callot7, M.
Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, A. Carbone14, G.
Carboni21,k, R. Cardinale19,i,37, A. Cardini15, L. Carson49, K. Carvalho
Akiba2, G. Casse48, M. Cattaneo37, Ch. Cauet9, M. Charles51, Ph.
Charpentier37, N. Chiapolini39, K. Ciba37, X. Cid Vidal36, G. Ciezarek49,
P.E.L. Clarke46,37, M. Clemencic37, H.V. Cliff43, J. Closier37, C. Coca28, V.
Coco23, J. Cogan6, P. Collins37, A. Comerma-Montells35, F. Constantin28, G.
Conti38, A. Contu51, A. Cook42, M. Coombes42, G. Corti37, G.A. Cowan38, R.
Currie46, B. D’Almagne7, C. D’Ambrosio37, P. David8, P.N.Y. David23, I. De
Bonis4, S. De Capua21,k, M. De Cian39, F. De Lorenzi12, J.M. De Miranda1, L.
De Paula2, P. De Simone18, D. Decamp4, M. Deckenhoff9, H. Degaudenzi38,37, M.
Deissenroth11, L. Del Buono8, C. Deplano15, D. Derkach14,37, O. Deschamps5, F.
Dettori24, J. Dickens43, H. Dijkstra37, P. Diniz Batista1, F. Domingo
Bonal35,n, S. Donleavy48, F. Dordei11, A. Dosil Suárez36, D. Dossett44, A.
Dovbnya40, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, S. Easo45, U.
Egede49, V. Egorychev30, S. Eidelman33, D. van Eijk23, F. Eisele11, S.
Eisenhardt46, R. Ekelhof9, L. Eklund47, Ch. Elsasser39, D. Elsby55, D.
Esperante Pereira36, L. Estève43, A. Falabella16,14,e, E. Fanchini20,j, C.
Färber11, G. Fardell46, C. Farinelli23, S. Farry12, V. Fave38, V. Fernandez
Albor36, M. Ferro-Luzzi37, S. Filippov32, C. Fitzpatrick46, M. Fontana10, F.
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Furcas20, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini51, Y.
Gao3, J-C. Garnier37, J. Garofoli52, J. Garra Tico43, L. Garrido35, D.
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Ghez4, V. Gibson43, V.V. Gligorov37, C. Göbel54, D. Golubkov30, A.
Golutvin49,30,37, A. Gomes2, H. Gordon51, M. Grabalosa Gándara35, R. Graciani
Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E.
Greening51, S. Gregson43, B. Gui52, E. Gushchin32, Yu. Guz34, T. Gys37, G.
Haefeli38, C. Haen37, S.C. Haines43, T. Hampson42, S. Hansmann-Menzemer11, R.
Harji49, N. Harnew51, J. Harrison50, P.F. Harrison44, J. He7, V. Heijne23, K.
Hennessy48, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, E.
Hicks48, K. Holubyev11, P. Hopchev4, W. Hulsbergen23, P. Hunt51, T. Huse48,
R.S. Huston12, D. Hutchcroft48, D. Hynds47, V. Iakovenko41, P. Ilten12, J.
Imong42, R. Jacobsson37, A. Jaeger11, M. Jahjah Hussein5, E. Jans23, F.
Jansen23, P. Jaton38, B. Jean-Marie7, F. Jing3, M. John51, D. Johnson51, C.R.
Jones43, B. Jost37, M. Kaballo9, S. Kandybei40, M. Karacson37, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon55, U. Kerzel37, T. Ketel24, A. Keune38, B. Khanji6,
Y.M. Kim46, M. Knecht38, P. Koppenburg23, A. Kozlinskiy23, L. Kravchuk32, K.
Kreplin11, M. Kreps44, G. Krocker11, P. Krokovny11, F. Kruse9, K.
Kruzelecki37, M. Kucharczyk20,25,37,j, T. Kvaratskheliya30,37, V.N. La Thi38,
D. Lacarrere37, G. Lafferty50, A. Lai15, D. Lambert46, R.W. Lambert24, E.
Lanciotti37, G. Lanfranchi18, C. Langenbruch11, T. Latham44, C. Lazzeroni55,
R. Le Gac6, J. van Leerdam23, J.-P. Lees4, R. Lefèvre5, A. Leflat31,37, J.
Lefrançois7, O. Leroy6, T. Lesiak25, L. Li3, L. Li Gioi5, M. Lieng9, M.
Liles48, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu38,3, J. Luisier38, A. Mac Raighne47, F.
Machefert7, I.V. Machikhiliyan4,30, F. Maciuc10, O. Maev29,37, J. Magnin1, S.
Malde51, R.M.D. Mamunur37, G. Manca15,d, G. Mancinelli6, N. Mangiafave43, U.
Marconi14, R. Märki38, J. Marks11, G. Martellotti22, A. Martens8, L. Martin51,
A. Martín Sánchez7, D. Martinez Santos37, A. Massafferri1, Z. Mathe12, C.
Matteuzzi20, M. Matveev29, E. Maurice6, B. Maynard52, A. Mazurov16,32,37, G.
McGregor50, R. McNulty12, C. Mclean14, M. Meissner11, M. Merk23, J. Merkel9,
R. Messi21,k, S. Miglioranzi37, D.A. Milanes13,37, M.-N. Minard4, J. Molina
Rodriguez54, S. Monteil5, D. Moran12, P. Morawski25, R. Mountain52, I. Mous23,
F. Muheim46, K. Müller39, R. Muresan28,38, B. Muryn26, B. Muster38, M. Musy35,
J. Mylroie-Smith48, P. Naik42, T. Nakada38, R. Nandakumar45, I. Nasteva1, M.
Nedos9, M. Needham46, N. Neufeld37, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5,
N. Nikitin31, A. Nomerotski51, A. Novoselov34, A. Oblakowska-Mucha26, V.
Obraztsov34, S. Oggero23, S. Ogilvy47, O. Okhrimenko41, R. Oldeman15,d, M.
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Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A.
Sciubba18,l, M. Seco36, A. Semennikov30, K. Senderowska26, I. Sepp49, N.
Serra39, J. Serrano6, P. Seyfert11, B. Shao3, M. Shapkin34, I. Shapoval40,37,
P. Shatalov30, Y. Shcheglov29, T. Shears48, L. Shekhtman33, O. Shevchenko40,
V. Shevchenko30, A. Shires49, R. Silva Coutinho44, T. Skwarnicki52, A.C.
Smith37, N.A. Smith48, E. Smith51,45, K. Sobczak5, F.J.P. Soler47, A.
Solomin42, F. Soomro18, B. Souza De Paula2, B. Spaan9, A. Sparkes46, P.
Spradlin47, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S.
Stone52,37, B. Storaci23, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S.
Swientek9, M. Szczekowski27, P. Szczypka38, T. Szumlak26, S. T’Jampens4, E.
Teodorescu28, F. Teubert37, C. Thomas51, E. Thomas37, J. van Tilburg11, V.
Tisserand4, M. Tobin39, S. Topp-Joergensen51, N. Torr51, E. Tournefier4,49,
M.T. Tran38, A. Tsaregorodtsev6, N. Tuning23, M. Ubeda Garcia37, A. Ukleja27,
P. Urquijo52, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez35, P.
Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis42, M. Veltri17,g, B. Viaud7, I.
Videau7, X. Vilasis-Cardona35,n, J. Visniakov36, A. Vollhardt39, D.
Volyanskyy10, D. Voong42, A. Vorobyev29, H. Voss10, S. Wandernoth11, J.
Wang52, D.R. Ward43, N.K. Watson55, A.D. Webber50, D. Websdale49, M.
Whitehead44, D. Wiedner11, L. Wiggers23, G. Wilkinson51, M.P. Williams44,45,
M. Williams49, F.F. Wilson45, J. Wishahi9, M. Witek25, W. Witzeling37, S.A.
Wotton43, K. Wyllie37, Y. Xie46, F. Xing51, Z. Xing52, Z. Yang3, R. Young46,
O. Yushchenko34, M. Zavertyaev10,a, F. Zhang3, L. Zhang52, W.C. Zhang12, Y.
Zhang3, A. Zhelezov11, L. Zhong3, E. Zverev31, A. Zvyagin37.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
24Nikhef National Institute for Subatomic Physics and Vrije Universiteit,
Amsterdam, The Netherlands
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraców, Poland
26AGH University of Science and Technology, Kraców, Poland
27Soltan Institute for Nuclear Studies, Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
43Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
44Department of Physics, University of Warwick, Coventry, United Kingdom
45STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
46School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
47School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
48Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
49Imperial College London, London, United Kingdom
50School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
51Department of Physics, University of Oxford, Oxford, United Kingdom
52Syracuse University, Syracuse, NY, United States
53CC-IN2P3, CNRS/IN2P3, Lyon-Villeurbanne, France, associated member
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55University of Birmingham, Birmingham, United Kingdom
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
Explaining heavy quarkonium production remains a challenging problem for
Quantum Chromodynamics (QCD). At the energies of the proton-proton ($pp$)
collisions at the Large Hadron Collider, $c\overline{}c$ pairs are expected to
be produced predominantly via Leading Order (LO) gluon-gluon interactions,
followed by the formation of the bound charmonium states. While the former can
be calculated using perturbative QCD, the latter is described by non-
perturbative models. Other, more recent, approaches make use of non-
relativistic QCD factorization (NRQCD) which assumes a combination of the
colour-singlet (CS) and colour-octet (CO) $c\overline{}c$ and soft gluon
exchange for the production of the final bound state [1]. To describe previous
experimental data, it was found to be necessary to include Next-to-Leading
Order (NLO) QCD corrections for the description of charmonium production [2,
3].
The study of the production of $P$-wave charmonia $\chi_{cJ}(1P)$, with
$J\,{=}\,0,1,2$, is important, since these resonances give substantial feed-
down contributions to the prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ production through their radiative decays
$\chi_{c}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$
and can have significant impact on the measurement of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ polarisation. Furthermore, the
ratio of the production rate of $\chi_{c2}$ to that of $\chi_{c1}$ is
interesting because it is sensitive to the CS and CO production mechanisms.
Measurements of $\chi_{c}$ production and the relative amounts of the
$\chi_{c1}$ and $\chi_{c2}$ spin states, have previously been made using
different particle beams and energies [4, 5, 6]. In this Letter, we report a
measurement from the LHCb experiment of the ratio of the prompt cross-sections
for the two $\chi_{c}$ spin states, $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$,
as a function of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ transverse
momentum in the range
$2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$ and in
the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,4.5$. The $\chi_{c}$ candidates are reconstructed through their
radiative decay $\chi_{c}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma$, with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\rightarrow\,\mu^{+}\,\mu^{-}$, using a data sample with an integrated
luminosity of ${36}\>{\mbox{\,pb}^{-1}}$ collected during 2010. In this
Letter, prompt production of $\chi_{c}$ refers to $\chi_{c}$ mesons that are
produced at the interaction point and do not arise from the decay of a
$b$-hadron. The sample therefore includes $\chi_{c}$ from the decay of short-
lived resonances, such as $\psi{(2S)}$, which are also produced at the
interaction point. All three $\chi_{cJ}$ states are considered in the
analysis. Since the
$\chi_{c0}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$
branching fraction is $\sim{30}$ (17) times smaller than that of the
$\chi_{c1}$ ($\chi_{c2}$), the yield of $\chi_{c0}$ is not significant. The
measurements extend the $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ coverage with respect to previous experiments.
## 2 LHCb detector and selection requirements
The LHCb detector [7] is a single-arm forward spectrometer with an angular
coverage from approximately ${10}\>{\rm\,mrad}$ to ${300}\>{\rm\,mrad}$
(${250}\>{\rm\,mrad}$) in the bending (non-bending) plane. The detector
consists of a vertex detector (VELO), a dipole magnet, a tracking system, two
ring-imaging Cherenkov (RICH) detectors, a calorimeter system and a muon
system.
Of particular importance in this measurement are the calorimeter and muon
systems. The calorimeter consists of a scintillating pad detector (SPD) and a
pre-shower, followed by electromagnetic (ECAL) and hadronic calorimeters. The
SPD and pre-shower are designed to distinguish between signals from photons
and electrons. The ECAL is constructed from scintillating tiles interleaved
with lead tiles. Muons are identified using hits in detectors interleaved with
iron filters.
The signal simulation sample used for this analysis was generated using the
Pythia $6.4$ generator [8] configured with the parameters detailed in Ref.
[9]. The EvtGen [10], Photos [11] and Geant4 [12] packages were used to decay
unstable particles, generate QED radiative corrections and simulate
interactions in the detector, respectively. The sample consists of events in
which at least one ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\rightarrow\,\mu^{+}\,\mu^{-}$ decay takes place with no constraint on
the production mechanism.
The trigger consists of a hardware stage followed by a software stage which
applies a full event reconstruction. For this analysis the trigger selects a
pair of oppositely charged muon candidates, where either one of the muons has
a transverse momentum
$p_{\mathrm{T}}\,{>}\,\mbox{${1.8}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
or one of the pair has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.56}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$
and the other has
$p_{\mathrm{T}}\,{>}\,\mbox{${0.48}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$.
The invariant mass of the candidates is required to be greater than
${2.9}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. The photons are not
involved in the trigger decision for this analysis.
Photons are identified and reconstructed using the calorimeter and tracking
systems. The identification algorithm provides an estimator for the hypothesis
that a calorimeter cluster originates from a photon. This is a likelihood-
based estimator constructed from variables that rely on calorimeter and
tracking information. For example, in order to reduce the electron background,
candidate photon clusters are required not to be matched to a track
extrapolated into the calorimeter. For each photon candidate a likelihood
($\mathrm{CL}_{\gamma}$) is calculated based on simulated signal and
background samples. The photons identified by the calorimeter and used in this
analysis can be classified as two types: those that have converted in the
material after the dipole magnet and those that have not. Converted photons
are identified as clusters in the ECAL with correlated activity in the SPD. In
order to account for the different energy resolutions of the two types of
photons, the analysis is performed separately for converted and non-converted
photons and the results combined as described in Sect. 3. Photons that convert
before the magnet require a different analysis strategy and are not considered
here. The photons used to reconstruct the $\chi_{c}$ candidates are required
to have a transverse momentum
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$,
a momentum
$p^{\gamma}\,{>}\,\mbox{${5}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$ and a
likelihood $\mathrm{CL}_{\gamma}\,{>}\,0.5$.
The muon and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ identification
criteria are identical to those used in Ref. [13]: each track must be
identified as a muon with
$p_{\mathrm{T}}\,{>}\,700{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and a quality
of the track fit $\chi^{2}/\mathrm{ndf}\,{<}\,4$, where $\mathrm{ndf}$ is the
number of degrees of freedom. The two muons must originate from a common
vertex with a probability of the vertex fit ${>}\,0.5\%$. In addition, in this
analysis the $\mu^{+}\,\mu^{-}$ invariant mass is required to be in the range
${3062}\,{-}\,{\mbox{${3120}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$}}$.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ pseudo-decay time, $t_{z}$,
is used to reduce the contribution from non-prompt decays, by requiring
$t_{z}\,{=}\,(z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{-}\,z_{\mathrm{PV}})M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{/}\,p_{z}\,{<}\,\mbox{${0.1}\>{{\rm\,ps}}$}$, where
$M_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is the reconstructed dimuon
invariant mass, $z_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{-}\,z_{\mathrm{PV}}$ is the $z$ separation of the reconstructed
production (primary) and decay vertices of the dimuon, and $p_{z}$ is the
$z$-component of the dimuon momentum with the $z$-axis parallel to the beam
line. Simulation studies show that, with this requirement applied, the
remaining fraction of $\chi_{c}$ from $b$-hadron decays is about $0.1\%$. This
introduces an uncertainty much smaller than any of the other systematic or
statistical uncertainties evaluated in this analysis and is not considered
further.
In the data, the average $\chi_{c}$ candidate multiplicity per selected event
is $1.3$ and the percentage of events with more than one genuine $\chi_{c}$
candidate (composed of a unique ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and photon) is estimated to be $0.23\%$ from the simulation. All $\chi_{c}$
candidates are considered for further analysis. The mass difference, $\Delta
M\,{=}\,M\left(\mu^{+}\,\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\,\mu^{-}\right)$,
of the selected candidates is shown in Fig. 1 for the converted and non-
converted samples; the overlaid fits are described in Sect. 3.
Figure 1: Distribution of $\Delta
M\,{=}\,M\left(\mu^{+}\,\mu^{-}\,\gamma\right)\,{-}\,M\left(\mu^{+}\,\mu^{-}\right)$
for selected candidates with
$3\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}}$}$ for (a)
converted and (b) non-converted photons. The lower solid curves correspond to
the $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ peaks from left to right,
respectively (the $\chi_{c0}$ peak is barely visible). The background
distribution is shown as a dashed curve. The upper solid curve corresponds to
the overall fit function.
## 3 Experimental method
The production cross-section ratio of the $\chi_{c2}$ and $\chi_{c1}$ states
is measured as
$\displaystyle\frac{\sigma(\chi_{c2})}{\sigma(\chi_{c1})}\,{=}\,\frac{N_{\chi_{c2}}}{N_{\chi_{c1}}}\,{\cdot}\,\frac{\epsilon^{\chi_{c1}}}{\epsilon^{\chi_{c2}}}\,{\cdot}\,\frac{{\cal
B}(\chi_{c1}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)}{{\cal
B}(\chi_{c2}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)},$ (1)
where ${\cal B}(\chi_{c1}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)$ and ${\cal
B}(\chi_{c2}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)$ are the $\chi_{c1}$ and $\chi_{c2}$ branching fractions to
the final state ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$, and
$\displaystyle\frac{\epsilon^{\chi_{c1}}}{\epsilon^{\chi_{c2}}}\,{=}\,\frac{\epsilon^{\chi_{c1}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,\epsilon^{\chi_{c1}}_{\gamma}\,\epsilon^{\chi_{c1}}_{\mathrm{sel}}}{\epsilon^{\chi_{c2}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,\epsilon^{\chi_{c2}}_{\gamma}\,\epsilon^{\chi_{c2}}_{\mathrm{sel}}},$
(2)
where $\epsilon^{\chi_{cJ}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is
the efficiency to trigger, reconstruct and select a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ from a $\chi_{cJ}$ decay,
$\epsilon^{\chi_{cJ}}_{\gamma}$ is the efficiency to reconstruct and select a
photon from a $\chi_{cJ}$ decay and $\epsilon^{\chi_{cJ}}_{\mathrm{sel}}$ is
the efficiency to subsequently select the $\chi_{cJ}$ candidate.
Since the mass difference between the $\chi_{c1}$ and $\chi_{c2}$ states is
${{45.54}\,{\pm}\,{0.11}}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$, the
signal peaks cannot be separately isolated using the calorimeter information.
An unbinned maximum likelihood fit to the $\Delta M$ mass difference
distribution is performed to obtain the three $N_{\chi_{cJ}}$ yields
simultaneously. The determination of the efficiency terms in Eq. 2 is
described in Sect. 3.1.
The signal mass distribution is parametrised using three Gaussian functions
($\mathcal{F}_{\mathrm{sig}}^{J}$ for $J\,{=}\,0,1,2$). The combinatorial
background is described by
$\displaystyle\mathcal{F}_{\mathrm{bgd}}\,{=}\,x^{a}\\!\left(1-e^{\frac{m_{0}}{c}\left(1-x\right)}\right)+b\left(x-1\right),$
(3)
where $x\,{=}\,\Delta M\,{/}\,m_{0}$ and $m_{0}$, $a$, $b$ and $c$ are free
parameters.
A possible source of background from partially reconstructed decays is due to
$\psi{(2S)}\,{\rightarrow}\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{0}\,\pi^{0}$ decays where the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and a photon from one of the
neutral pions are reconstructed and selected as a $\chi_{c}$ candidate.
Simulation studies show that the expected yield is $\sim{0.1\%}$ of the signal
yield and this background is therefore neglected for this analysis.
The overall fit function is
$\displaystyle\mathcal{F}\,{=}\,\sum_{J=0}^{2}f_{\chi_{cJ}}\,\mathcal{F}_{\mathrm{sig}}^{J}\,{+}\,\left[1\,{-}\,\sum_{J=0}^{2}f_{\chi_{cJ}}\right]\mathcal{F}_{\mathrm{bgd}},$
(4)
where $f_{\chi_{cJ}}$ are the signal fractions. The mass differences between
the $\chi_{c1}$ and $\chi_{c2}$ states and the $\chi_{c1}$ and $\chi_{c0}$
states are fixed to the values from Ref. [14]. The mass resolutions for the
$\chi_{c}$ states, $\sigma_{\mathrm{res}}^{\chi_{cJ}}$, are given by the
widths of the Gaussian functions for each state. The ratios of the mass
resolutions,
$\sigma_{\mathrm{res}}^{\chi_{c2}}\,{/}\,\sigma_{\mathrm{res}}^{\chi_{c1}}$
and
$\sigma_{\mathrm{res}}^{\chi_{c0}}\,{/}\,\sigma_{\mathrm{res}}^{\chi_{c1}}$,
are taken from simulation. The value of
$\sigma_{\mathrm{res}}^{\chi_{c2}}\,{/}\,\sigma_{\mathrm{res}}^{\chi_{c1}}$ is
consistent with the value measured from data, fitting in a reduced $\Delta M$
range and with a simplified background parametrisation.
With the mass differences and the ratio of the mass resolutions fixed, a fit
is performed to the data in the range
$3\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$, in order
to determine the $\chi_{c1}$ mass resolution
$\sigma_{\mathrm{res}}^{\chi_{c1}}$. This range is chosen because the
background has a different shape in the
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin
${{2}\,{-}\,{3}}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ and is not well
described by $\mathcal{F}_{\mathrm{bgd}}$ when combined with the rest of the
sample. Simulation studies show that the signal parameters for the $\chi_{cJ}$
states in the $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
bin ${{2}\,{-}\,{3}}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ are consistent
with the parameters in the rest of the sample. The distributions of $\Delta M$
for the fits to the converted and non-converted candidates are shown in Fig.
1. The mass resolution, $\sigma_{\mathrm{res}}^{\chi_{c1}}$, is measured to be
${{21.8}\,{\pm}\,{0.8}}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${{18.3}\,{\pm}\,{0.4}}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ for
converted and non-converted candidates respectively. The corresponding values
in the simulation are
${{19.0}\,{\pm}\,{0.2}}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${{17.5}\,{\pm}\,{0.1}}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and show
a weak dependence of $\sigma_{\mathrm{res}}^{\chi_{c1}}$ on
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ which is
accounted for in the systematic uncertainties.
In order to measure the $\chi_{c}$ yields, the fit is then performed in bins
of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the
range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$. For each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin, the mass
differences, the ratio of the mass resolutions and
$\sigma_{\mathrm{res}}^{\chi_{c1}}$ are fixed as described above. In total,
there are eight free parameters for each fit in each bin in
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and the
results are summarized in Table 1; the fit $\chi^{2}/\mathrm{ndf}$ for the
converted and non-converted samples is good in all bins. The total observed
yields of $\chi_{c0}$, $\chi_{c1}$ and $\chi_{c2}$ are ${820}\,{\pm}\,{650}$,
${38\,630}\,{\pm}\,{550}$ and ${26\,110}\,{\pm}\,{620}$, respectively,
calculated from the signal fractions $f_{\chi_{cJ}}$ and the number of
candidates in the sample. The raw $\chi_{c}$ yields for converted and non-
converted candidates are combined, corrected for efficiency (as described in
Sect. 3.1) and the cross-section ratio is determined using Eq. 1.
Table 1: Signal $\chi_{c}$ yields and fit quality from the fit to the converted and non-converted candidates in each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | Converted photons | Non-converted photons
---|---|---
$\chi_{c1}$ yield | $\chi_{c2}$ yield | $\chi^{2}\,{/}\,\mathrm{ndf}$ | $\chi_{c1}$ yield | $\chi_{c2}$ yield | $\chi^{2}\,{/}\,\mathrm{ndf}$
${2}\,{-}\,{3}$ | ${3120}\,{\pm}\,{248}$ | ${2482}\,{\pm}\,{301}$ | 0.91 | ${4080}\,{\pm}\,{246}$ | ${3927}\,{\pm}\,{280}$ | 1.02
${3}\,{-}\,{4}$ | ${3462}\,{\pm}\,{224}$ | ${3082}\,{\pm}\,{249}$ | 0.81 | ${4919}\,{\pm}\,{183}$ | ${3443}\,{\pm}\,{207}$ | 1.02
${4}\,{-}\,{5}$ | ${3235}\,{\pm}\,{146}$ | ${1769}\,{\pm}\,{174}$ | 1.03 | ${4497}\,{\pm}\,{134}$ | ${2718}\,{\pm}\,{143}$ | 1.08
${5}\,{-}\,{6}$ | ${2476}\,{\pm}\,{110}$ | ${1443}\,{\pm}\,{121}$ | 0.84 | ${3203}\,{\pm}\,{105}$ | ${1999}\,{\pm}\,{107}$ | 1.45
${6}\,{-}\,{7}$ | ${1497}\,{\pm}\,{80}$ | ${736}\,{\pm}\,{89}$ | 1.05 | ${1946}\,{\pm}\,{78}$ | ${1338}\,{\pm}\,{83}$ | 0.78
${7}\,{-}\,{8}$ | ${933}\,{\pm}\,{77}$ | ${658}\,{\pm}\,{86}$ | 0.77 | ${1342}\,{\pm}\,{59}$ | ${747}\,{\pm}\,{60}$ | 1.15
${8}\,{-}\,{9}$ | ${660}\,{\pm}\,{47}$ | ${302}\,{\pm}\,{51}$ | 0.90 | ${817}\,{\pm}\,{43}$ | ${395}\,{\pm}\,{42}$ | 0.78
${9}\,{-}\,{10}$ | ${451}\,{\pm}\,{34}$ | ${142}\,{\pm}\,{35}$ | 0.82 | ${501}\,{\pm}\,{32}$ | ${256}\,{\pm}\,{31}$ | 1.09
${10}\,{-}\,{11}$ | ${255}\,{\pm}\,{25}$ | ${86}\,{\pm}\,{26}$ | 1.13 | ${317}\,{\pm}\,{26}$ | ${188}\,{\pm}\,{25}$ | 0.85
${11}\,{-}\,{12}$ | ${129}\,{\pm}\,{28}$ | ${99}\,{\pm}\,{30}$ | 0.87 | ${222}\,{\pm}\,{19}$ | ${103}\,{\pm}\,{18}$ | 0.93
${12}\,{-}\,{13}$ | ${129}\,{\pm}\,{16}$ | ${46}\,{\pm}\,{15}$ | 1.09 | ${154}\,{\pm}\,{15}$ | ${50}\,{\pm}\,{13}$ | 0.98
${13}\,{-}\,{15}$ | ${127}\,{\pm}\,{18}$ | ${42}\,{\pm}\,{20}$ | 0.91 | ${158}\,{\pm}\,{18}$ | ${63}\,{\pm}\,{17}$ | 1.05
### 3.1 Efficiencies
The efficiency ratios to reconstruct and select $\chi_{c}$ candidates are
obtained from simulation. Since the photon interaction with material is not
part of the event generation procedure, the individual efficiencies for
converted and non-converted candidates are not separated. Therefore, the
combined efficiencies are calculated. The ratios of the overall efficiency for
the detection of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons
originating from the decay of a $\chi_{c1}$ compared to a $\chi_{c2}$,
$\epsilon^{\chi_{c2}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{/}\,\epsilon^{\chi_{c1}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$, are consistent with unity for all
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins, as shown
in Fig. 2. The ratios of the efficiencies for reconstructing and selecting
photons from $\chi_{c}$ decays and then selecting the $\chi_{c}$,
$\epsilon^{\chi_{c2}}_{\gamma}\epsilon^{\chi_{c2}}_{\mathrm{sel}}\,{/}\,\epsilon^{\chi_{c1}}_{\gamma}\epsilon^{\chi_{c1}}_{\mathrm{sel}}$,
are also shown in Fig. 2. In general these efficiency ratios are consistent
with unity, except in the
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins
${{2}\,{-}\,{3}}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ and
${{3}\,{-}\,{4}}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ where the
reconstruction and detection efficiencies for $\chi_{c1}$ are smaller than for
$\chi_{c2}$. The increase in the efficiency ratio in these bins arises because
the photon $p_{\mathrm{T}}$ spectra are different for $\chi_{c1}$ and
$\chi_{c2}$. The photon
$p_{\mathrm{T}}^{\gamma}\,{>}\,\mbox{${650}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c}}$}$
requirement cuts harder in the case of the $\chi_{c1}$ and therefore lowers
this efficiency. The increase in the efficiency ratio is a kinematic effect,
rather than a reconstruction effect, and is well modelled by the simulation.
Figure 2: Reconstruction and selection efficiency ratios in bins of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The ratio of
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ efficiency
($\epsilon^{\chi_{c2}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{/}\,\epsilon^{\chi_{c1}}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$) is shown with red circles. The ratio of the photon reconstruction
and selection efficiency times the $\chi_{c}$ selection efficiency
($\epsilon^{\chi_{c2}}_{\gamma}\epsilon^{\chi_{c2}}_{\mathrm{sel}}\,{/}\,\epsilon^{\chi_{c1}}_{\gamma}\epsilon^{\chi_{c1}}_{\mathrm{sel}}$)
is shown with blue triangles.
### 3.2 Polarisation
The production of polarised $\chi_{c}$ states would modify the efficiencies
calculated from the simulation, which assumes unpolarised $\chi_{c}$. A
measurement of the $\chi_{c}$ polarisation would require an angular analysis,
which is not feasible with the present amount of data. Various polarisation
scenarios are considered in Table 2. Assuming no azimuthal dependence in the
production process, the
$\chi_{c}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$
system is described by three angles:
$\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$, $\theta_{\chi_{c}}$
and $\phi$, where $\theta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is
the angle between the directions of the positive muon in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame,
$\theta_{\chi_{c}}$ is the angle between the directions of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ in the $\chi_{c}$ rest frame
and the $\chi_{c}$ in the laboratory frame, and $\phi$ is the angle between
the plane formed from the $\chi_{c}$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ momentum vectors in the
laboratory frame and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay
plane in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame. The
angular distributions are independent of the choice of polarisation axis (the
direction of the $\chi_{c}$ in the laboratory frame) and are detailed in Ref.
[5]. For each simulated event in the unpolarised sample, a weight is
calculated from the distribution of these angles in the various polarisation
hypotheses compared to the unpolarised distribution. The weights in Table 2
are then the average of these per-event weights in the simulated sample. For a
given ($|m_{\chi_{c1}}|$, $|m_{\chi_{c2}}|$) polarisation combination, the
central value of the determined cross-section ratio in each
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin should be
multiplied by the number in the table. The maximum effect from the possible
polarisation of the $\chi_{c1}$ and $\chi_{c2}$ mesons is given separately
from the systematic uncertainties in Table 4 and Fig. 3.
Table 2: Polarisation weights in $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins for different combinations of $\chi_{c1}$ and $\chi_{c2}$ polarisation states $|J,m_{\chi_{cJ}}\rangle$ with $|m_{\chi_{cJ}}|=0,\cdots J$. The polarisation axis is defined as the direction of the $\chi_{c}$ in the laboratory frame. Unpol. means the $\chi_{c}$ is unpolarised. ($|m_{\chi_{c1}}|,|m_{\chi_{c2}}|$) | $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ (${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$)
---|---
${2}\\!-\\!{3}$ | ${3}\\!-\\!{4}$ | ${4}\\!-\\!{5}$ | ${5}\\!-\\!{6}$ | ${6}\\!-\\!{7}$ | ${7}\\!-\\!{8}$ | ${8}\\!-\\!{9}$ | ${9}\\!-\\!{10}$ | ${10}\\!-\\!{11}$ | ${11}\\!-\\!{12}$ | ${12}\\!-\\!{13}$ | ${13}\\!-\\!{15}$
(Unpol,0) | 0.99 | 0.97 | 0.94 | 0.91 | 0.88 | 0.87 | 0.86 | 0.86 | 0.86 | 0.85 | 0.85 | 0.88
(Unpol,1) | 0.97 | 0.98 | 0.97 | 0.95 | 0.94 | 0.94 | 0.93 | 0.93 | 0.93 | 0.93 | 0.93 | 0.93
(Unpol,2) | 1.03 | 1.04 | 1.07 | 1.11 | 1.14 | 1.17 | 1.18 | 1.18 | 1.19 | 1.18 | 1.19 | 1.16
(0,Unpol) | 1.01 | 0.99 | 0.97 | 0.93 | 0.90 | 0.89 | 0.87 | 0.86 | 0.85 | 0.87 | 0.86 | 0.84
(1,Unpol) | 0.99 | 1.00 | 1.02 | 1.04 | 1.05 | 1.06 | 1.06 | 1.07 | 1.08 | 1.07 | 1.07 | 1.08
(0,0) | 1.00 | 0.97 | 0.91 | 0.84 | 0.80 | 0.77 | 0.75 | 0.74 | 0.72 | 0.74 | 0.74 | 0.74
(0,1) | 0.98 | 0.97 | 0.93 | 0.88 | 0.85 | 0.83 | 0.81 | 0.81 | 0.79 | 0.81 | 0.81 | 0.78
(0,2) | 1.04 | 1.03 | 1.03 | 1.03 | 1.03 | 1.03 | 1.03 | 1.02 | 1.00 | 1.03 | 1.03 | 0.98
(1,0) | 0.99 | 0.97 | 0.96 | 0.94 | 0.93 | 0.92 | 0.92 | 0.92 | 0.92 | 0.91 | 0.91 | 0.95
(1,1) | 0.97 | 0.98 | 0.98 | 0.99 | 0.99 | 0.99 | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | 1.01
(1,2) | 1.03 | 1.04 | 1.09 | 1.15 | 1.20 | 1.23 | 1.26 | 1.26 | 1.28 | 1.26 | 1.27 | 1.25
## 4 Systematic uncertainties
The branching fractions used in the analysis are ${\cal
B}(\chi_{c1}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)\,{=}\,{0.344}\,{\pm}\,{0.015}$ and ${\cal
B}(\chi_{c2}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\gamma)\,{=}\,{0.195}\,{\pm}\,{0.008}$, taken from Ref. [14]. The
relative systematic uncertainty on the cross-section ratio resulting from the
$\chi_{c}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$
branching fractions is 6%; the absolute uncertainty is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 3.
The simulation sample used to calculate the efficiencies has approximately the
same number of $\chi_{c}$ candidates as are observed in the data. The
statistical errors from the finite number of simulated events are included as
a systematic uncertainty in the final results. The uncertainty associated to
this is determined by sampling the efficiencies used in Eq. 1 according to
their errors. The relative systematic uncertainty due to the limited size of
the simulation sample is found to be in the range (${0.6}\,{-}\,{7.2}$)% and
is given for each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ bin in Table 3.
The measured $\chi_{c}$ yields depend on the values of the fixed parameters
and the fit range used. The associated systematic uncertainty has been
evaluated by repeating the fit many times, changing the values of the fixed
parameters and the fit range. Since the uncertainties arising from the fixed
parameters are expected to be correlated, a single procedure is used
simultaneously varying all these parameters. The $\chi_{c}$ mass difference
parameters are sampled from two Gaussian distributions with widths taken from
the errors on the masses given in Ref. [14]. The mass resolution ratios,
$\sigma_{\mathrm{res}}^{\chi_{c2}}\,{/}\,\sigma_{\mathrm{res}}^{\chi_{c1}}$
and
$\sigma_{\mathrm{res}}^{\chi_{c0}}\,{/}\,\sigma_{\mathrm{res}}^{\chi_{c1}}$,
are varied according to the error matrix of the fit to the simulated sample in
the range $3\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$.
The mass resolution $\sigma_{\mathrm{res}}^{\chi_{c1}}$ is also determined
using a simplified background model and fitting in a reduced range. Simulation
studies show that the value of $\sigma_{\mathrm{res}}^{\chi_{c1}}$ also has a
weak dependence on $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$. The mass resolution $\sigma_{\mathrm{res}}^{\chi_{c1}}$ is randomly
sampled from the values obtained from the default fit (described in Sect. 3)
according to its error, the simplified fit, again according to its error, and
by modifying it in each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ bin according to the variation observed in the simulation.
The systematic uncertainty associated with the shape of the fitted background
function is incorporated by including or excluding the $\chi_{c0}$ signal
shape, which peaks in the region where the background shape is most sensitive.
The background shape is also sensitive to the rise in the $\Delta M$
distribution. The systematic uncertainty from this is included by varying the
lower edge of the fit range in the interval
${\pm}\,{\mbox{${10}\>{{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$}}$ around
its nominal value for each bin in
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$.
The overall systematic uncertainty from the fit is then determined from the
distribution of the $\chi_{c2}\,/\,\chi_{c1}$ cross-section ratios by
repeating the sampling procedure described above many times. The relative
uncertainty is found to be in the range (${2.2}\,{-}\,{14.6}$)% and is given
for each bin of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ in Table 3.
A systematic uncertainty related to the calibration of the simulation is
evaluated by performing the analysis on simulated events and comparing the
efficiency-corrected ratio of yields,
$(N_{\chi_{c2}}\,{/}\,N_{\chi_{c1}})\,{\cdot}\,(\epsilon^{\chi_{c1}}\,{/}\,{\epsilon^{\chi_{c2}}})$,
to the true ratio generated in the sample. A deviation of $-9.6\%$ is
observed, caused by non-Gaussian signal shapes in the simulation from the
calorimeter calibration. These are not seen in the data, which is well
described by Gaussian signal shapes. The deviation is included as a systematic
error, by sampling from the negative half of a Gaussian with zero mean and a
width of $9.6\%$. The relative uncertainty on the cross-section ratio is found
to be less than $6.0\%$ and is given for each bin of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in Table 3. A
second check of the procedure was performed using simulated events generated
according to the distributions observed in the data, i.e. three overlapping
Gaussians and a background shape similar to that in Fig. 1. In this case no
evidence for a deviation was observed. Other systematic uncertainties due to
the modelling of the detector in the simulation are negligible.
In summary, the overall systematic uncertainty, excluding that due to the
branching fractions, is evaluated by simultaneously sampling the deviation of
the cross-section ratio from the central value, using the distributions of the
cross-section ratios described above. The separate systematic uncertainties
are shown in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ in Table 3 and the combined uncertainties are shown in Table 4.
Table 3: Summary of the systematic uncertainties (absolute values) on $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$ in each $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | ${2}\,{-}\,{3}$ | ${3}\,{-}\,{4}$ | ${4}\,{-}\,{5}$ | ${5}\,{-}\,{6}$ | ${6}\,{-}\,{7}$ | ${7}\,{-}\,{8}$
---|---|---|---|---|---|---
Branching fractions | ${}^{+0.08}_{-0.08}$ | ${}^{+0.08}_{-0.08}$ | ${}^{+0.06}_{-0.06}$ | ${}^{+0.07}_{-0.07}$ | ${}^{+0.07}_{-0.07}$ | ${}^{+0.06}_{-0.06}$
Size of simulation sample | ${}^{+0.01}_{-0.01}$ | ${}^{+0.01}_{-0.01}$ | ${}^{+0.01}_{-0.01}$ | ${}^{+0.01}_{-0.01}$ | ${}^{+0.02}_{-0.01}$ | ${}^{+0.02}_{-0.02}$
Fit model | ${}^{+0.04}_{-0.05}$ | ${}^{+0.05}_{-0.04}$ | ${}^{+0.03}_{-0.03}$ | ${}^{+0.03}_{-0.03}$ | ${}^{+0.03}_{-0.04}$ | ${}^{+0.05}_{-0.04}$
Simulation calibration | ${}^{+0.00}_{-0.08}$ | ${}^{+0.00}_{-0.07}$ | ${}^{+0.00}_{-0.05}$ | ${}^{+0.00}_{-0.05}$ | ${}^{+0.00}_{-0.06}$ | ${}^{+0.00}_{-0.06}$
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$(${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$) | ${8}\,{-}\,{9}$ | ${9}\,{-}\,{10}$ | ${10}\,{-}\,{11}$ | ${11}\,{-}\,{12}$ | ${12}\,{-}\,{13}$ | ${13}\,{-}\,{15}$
Branching fractions | ${}^{+0.05}_{-0.05}$ | ${}^{+0.05}_{-0.05}$ | ${}^{+0.05}_{-0.05}$ | ${}^{+0.06}_{-0.06}$ | ${}^{+0.04}_{-0.04}$ | ${}^{+0.04}_{-0.04}$
Size of simulation sample | ${}^{+0.02}_{-0.02}$ | ${}^{+0.02}_{-0.02}$ | ${}^{+0.04}_{-0.04}$ | ${}^{+0.06}_{-0.06}$ | ${}^{+0.05}_{-0.05}$ | ${}^{+0.05}_{-0.05}$
Fit model | ${}^{+0.03}_{-0.04}$ | ${}^{+0.03}_{-0.03}$ | ${}^{+0.03}_{-0.03}$ | ${}^{+0.02}_{-0.13}$ | ${}^{+0.02}_{-0.02}$ | ${}^{+0.08}_{-0.03}$
Simulation calibration | ${}^{+0.00}_{-0.04}$ | ${}^{+0.00}_{-0.04}$ | ${}^{+0.00}_{-0.05}$ | ${}^{+0.00}_{-0.06}$ | ${}^{+0.00}_{-0.04}$ | ${}^{+0.00}_{-0.03}$
## 5 Results and conclusions
The cross-section ratio, $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$, measured
in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ is
given in Table 4 and shown in Fig. 3. Previous measurements from WA11 in
$\pi^{-}$Be collisions at ${185}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ gave
$\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})\,{=}\,{1.4}\,{\pm}\,{0.6}$ [4], and
from HERA-B in $p\mathrm{A}$ collisions at
$\sqrt{s}\,{=}\,\mbox{${41.6}\>{\mathrm{GeV}}$}$ with
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ below roughly
${5}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$ gave
$\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})\,{=}\,{1.75}\,{\pm}\,{0.7}$ [5]. The
data points from CDF [6] at
$\sqrt{s}\,{=}\,\mbox{${1.96}\>{\mathrm{\,Te\kern-1.00006ptV}}$}$ in
$p\bar{p}$ collisions are also shown in Fig. 3a).
Table 4: Ratio $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$ in bins of $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ in the range $2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}}$}$ and in the rapidity range $2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$. The first error is the statistical error, the second is the systematic uncertainty (apart from the branching fraction and polarisation) and the third is due to the $\chi_{c}\,\rightarrow\,{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\gamma$ branching fractions. Also given is the maximum effect of the unknown $\chi_{c}$ polarisations on the result as described in Sect. 3.2. $p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ (${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$ | Polarisation effects
---|---|---
${2}\,{-}\,{3}$ | ${1.39}^{+0.12\;+0.06\;+0.08}_{-0.13\;-0.09\;-0.08}$ | ${}^{+0.06}_{-0.05}$
${3}\,{-}\,{4}$ | ${1.32}^{+0.10\;+0.03\;+0.08}_{-0.09\;-0.09\;-0.08}$ | ${}^{+0.06}_{-0.05}$
${4}\,{-}\,{5}$ | ${1.02}^{+0.07\;+0.04\;+0.06}_{-0.06\;-0.06\;-0.06}$ | ${}^{+0.09}_{-0.09}$
${5}\,{-}\,{6}$ | ${1.08}^{+0.07\;+0.04\;+0.07}_{-0.06\;-0.06\;-0.07}$ | ${}^{+0.16}_{-0.17}$
${6}\,{-}\,{7}$ | ${1.09}^{+0.08\;+0.03\;+0.07}_{-0.09\;-0.07\;-0.07}$ | ${}^{+0.22}_{-0.22}$
${7}\,{-}\,{8}$ | ${1.08}^{+0.13\;+0.05\;+0.06}_{-0.10\;-0.07\;-0.06}$ | ${}^{+0.25}_{-0.25}$
${8}\,{-}\,{9}$ | ${0.86}^{+0.10\;+0.04\;+0.05}_{-0.10\;-0.06\;-0.05}$ | ${}^{+0.22}_{-0.21}$
${9}\,{-}\,{10}$ | ${0.75}^{+0.11\;+0.04\;+0.05}_{-0.11\;-0.06\;-0.05}$ | ${}^{+0.20}_{-0.19}$
${10}\,{-}\,{11}$ | ${0.91}^{+0.16\;+0.05\;+0.05}_{-0.15\;-0.07\;-0.05}$ | ${}^{+0.25}_{-0.25}$
${11}\,{-}\,{12}$ | ${0.91}^{+0.19\;+0.09\;+0.06}_{-0.17\;-0.10\;-0.06}$ | ${}^{+0.24}_{-0.24}$
${12}\,{-}\,{13}$ | ${0.68}^{+0.18\;+0.05\;+0.04}_{-0.16\;-0.07\;-0.04}$ | ${}^{+0.19}_{-0.18}$
${13}\,{-}\,{15}$ | ${0.69}^{+0.20\;+0.07\;+0.04}_{-0.18\;-0.07\;-0.04}$ | ${}^{+0.18}_{-0.18}$
Figure 3: Ratio $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$ in bins of
$2\,{<}\,p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{<}\,\mbox{${15}\>{{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}}$}$. The LHCb
results, in the rapidity range
$2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$ and
assuming the production of unpolarised $\chi_{c}$ mesons, are shown with solid
black circles and the internal error bars correspond to the statistical error;
the external error bars include the contribution from the systematic
uncertainties (apart from the polarisation). The lines surrounding the data
points show the maximum effect of the unknown $\chi_{c}$ polarisations on the
result. The upper and lower limits correspond to the spin states as described
in the text. The CDF data points, at
$\sqrt{s}\,{=}\,\mbox{${1.96}\>{\mathrm{\,Te\kern-0.90005ptV}}$}$ in
$p\bar{p}$ collisions and in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ pseudo-rapidity range $|\eta^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}|<1.0$, are shown in (a) with open blue circles [6]. The two hatched
bands in (b) correspond to the ChiGen Monte Carlo generator [15] and NLO NRQCD
[3] predictions.
Theoretical predictions, calculated in the LHCb rapidity range
$2.0\,{<}\,y^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\,{<}\,4.5$, from
the ChiGen Monte Carlo generator [15], which is an implementation of the
leading-order colour-singlet model described in Ref. [16], and from the NLO
NRQCD calculations [3] are shown in Fig. 3b). The hatched bands represent the
uncertainties in the theoretical predictions.
Figure 3 also shows the maximum effect of the unknown $\chi_{c}$ polarisations
on the result, shown as the lines surrounding the data points. In the first
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bin, the upper
limit corresponds to the spin state combination
$(|m_{\chi_{c1}}|,|m_{\chi_{c2}}|)\,{=}\,(0,2)$ and the lower limit
corresponds to the spin state combination $(1,1)$. In all subsequent
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ bins, the
upper limit corresponds to spin state combination $(1,2)$ and the lower limit
corresponds to $(0,0)$.
In summary, the ratio of the $\sigma(\chi_{c2})\,/\,\sigma(\chi_{c1})$ prompt
production cross-sections has been measured as a function of
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ using
${36}\>{\mbox{\,pb}^{-1}}$ of data collected by LHCb during 2010 at a centre-
of-mass energy $\sqrt{s}\,{=}\,\mbox{${7}\>{\mathrm{\,Te\kern-1.00006ptV}}$}$.
The ChiGen generator describes the shape of the distribution reasonably well,
although the data lie consistently above the model prediction. This could be
explained by important higher order perturbative corrections and/or sizeable
colour octet terms not included in the calculation. The results are in
agreement with the NLO NRQCD model for
$p_{\mathrm{T}}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\,{>}\,\mbox{${8}\>{{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}}$}$.
## Acknowledgments
We would like to thank L. A. Harland-Lang, W. J. Stirling and K. Chao for
supplying the theory predictions for comparison to our data and for many
helpful discussions.
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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|
arxiv-papers
| 2012-02-06T09:37:33 |
2024-09-04T02:49:27.065574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr., S. Amato, Y. Amhis, J.\n Anderson, R. B. Appleby, O. Aquines Gutierrez, F. Archilli, L. Arrabito, A.\n Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J. J. Back, D.\n S. Bailey, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, I. Bediaga, S. Belogurov, K.\n Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S. Benson, J.\n Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S. Bifani, T.\n Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F. Blanc, C. Blanks, J.\n Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T. Brambach, J. van den Brand,\n J. Bressieux, D. Brett, M. Britsch, T. Britton, N. H. Brook, H. Brown, A.\n B\\\"uchler-Germann, I. Burducea, A. Bursche, J. Buytaert, S. Cadeddu, O.\n Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, A. Carbone, G.\n Carboni, R. Cardinale, A. Cardini, L. Carson, K. Carvalho Akiba, G. Casse, M.\n Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, N. Chiapolini, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J.\n Closier, C. Coca, V. Coco, J. Cogan, P. Collins, A. Comerma-Montells, F.\n Constantin, G. Conti, A. Contu, A. Cook, M. Coombes, G. Corti, G. A. Cowan,\n R. Currie, B. D'Almagne, C. D'Ambrosio, P. David, P. N. Y. David, I. De\n Bonis, S. De Capua, M. De Cian, F. De Lorenzi, J. M. De Miranda, L. De Paula,\n P. De Simone, D. Decamp, M. Deckenhoff, H. Degaudenzi, M. Deissenroth, L. Del\n Buono, C. Deplano, D. Derkach, O. Deschamps, F. Dettori, J. Dickens, H.\n Dijkstra, P. Diniz Batista, F. Domingo Bonal, S. Donleavy, F. Dordei, A.\n Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A.\n Dziurda, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, F.\n Eisele, S. Eisenhardt, R. Ekelhof, L. Eklund, Ch. Elsasser, D. Elsby, D.\n Esperante Pereira, L. Est\\'eve, A. Falabella, E. Fanchini, C. F\\\"arber, G.\n Fardell, C. Farinelli, S. Farry, V. Fave, V. Fernandez Albor, M. Ferro-Luzzi,\n S. Filippov, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, N. Gauvin, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson,\n V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado Cardoso, E. Graug\\'es,\n G. Graziani, A. Grecu, E. Greening, S. Gregson, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S. Hansmann-Menzemer,\n R. Harji, N. Harnew, J. Harrison, P. F. Harrison, J. He, V. Heijne, K.\n Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E. Hicks, K.\n Holubyev, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S. Huston, D.\n Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A.\n Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B. Jean-Marie, F.\n Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo, S. Kandybei, M.\n Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A.\n Keune, B. Khanji, Y. M. Kim, M. Knecht, P. Koppenburg, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, K.\n Kruzelecki, M. Kucharczyk, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\'evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, L. Li\n Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. H. Lopes,\n E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, D. Martinez Santos, A. Massafferri, Z. Mathe, C. Matteuzzi, M.\n Matveev, E. Maurice, B. Maynard, A. Mazurov, G. McGregor, R. McNulty, C.\n Mclean, M. Meissner, M. Merk, J. Merkel, R. Messi, S. Miglioranzi, D. A.\n Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P.\n Morawski, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn,\n B. Muster, M. Musy, J. Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I.\n Nasteva, M. Nedos, M. Needham, N. Neufeld, C. Nguyen-Mau, M. Nicol, V. Niess,\n N. Nikitin, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, S. K. Paterson, G. N. Patrick, C. Patrignani, C.\n Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M. Pepe\n Altarelli, S. Perazzini, D. L. Perego, E. Perez Trigo, A. P\\'erez-Calero\n Yzquierdo, P. Perret, M. Perrin-Terrin, G. Pessina, A. Petrella, A.\n Petrolini, A. Phan, E. Picatoste Olloqui, B. Pie Valls, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M. Plo Casasus, G. Polok, A.\n Poluektov, E. Polycarpo, D. Popov, B. Popovici, C. Potterat, A. Powell, T. du\n Pree, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian, J. H.\n Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, G. Raven, S.\n Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi, K. Rinnert, D. A. Roa\n Romero, P. Robbe, E. Rodrigues, F. Rodrigues, P. Rodriguez Perez, G. J.\n Rogers, S. Roiser, V. Romanovsky, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz,\n G. Sabatino, J. J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C.\n Salzmann, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, S. Schleich, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, B. Shao, M. Shapkin, I. Shapoval, P. Shatalov, Y.\n Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires,\n R. Silva Coutinho, T. Skwarnicki, A. C. Smith, N. A. Smith, E. Smith, K.\n Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, B. Souza De Paula, B. Spaan,\n A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S.\n Stone, B. Storaci, M. Straticiuc, U. Straumann, V. K. Subbiah, S. Swientek,\n M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Topp-Joergensen, N. Torr, E. Tournefier, M. T. Tran, A. Tsaregorodtsev, N.\n Tuning, M. Ubeda Garcia, A. Ukleja, P. Urquijo, U. Uwer, V. Vagnoni, G.\n Valenti, R. Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M.\n Veltri, B. Viaud, I. Videau, X. Vilasis-Cardona, J. Visniakov, A. Vollhardt,\n D. Volyanskyy, D. Voong, A. Vorobyev, H. Voss, S. Wandernoth, J. Wang, D. R.\n Ward, N. K. Watson, A. D. Webber, D. Websdale, M. Whitehead, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, K. Wyllie, Y. Xie, F. Xing, Z. Xing, Z.\n Yang, R. Young, O. Yushchenko, M. Zavertyaev, F. Zhang, L. Zhang, W. C.\n Zhang, Y. Zhang, A. Zhelezov, L. Zhong, E. Zverev, A. Zvyagin",
"submitter": "Valerie Gibson",
"url": "https://arxiv.org/abs/1202.1080"
}
|
1202.1223
|
A new balance index for phylogenetic trees
uib1]Arnau Mir
uib1]Francesc Rossellócor1
uib2]Lucía Rotger
[cor1]Corresponding author
[uib1]Research Institute of Health Science (IUNICS) and Department of Mathematics and Computer Science,
University of the Balearic Islands,
E-07122 Palma de
Mallorca, Spain
[uib2]Department of Mathematics and Computer Science,
University of the Balearic Islands,
E-07122 Palma de
Mallorca, Spain
Several indices that measure the degree of balance of a rooted phylogenetic tree have been proposed so far in the literature. In this work we define and study a new index of this kind, which we call the total cophenetic index: the sum, over all pairs of different leaves, of the depth of their least common ancestor. This index makes sense for arbitrary trees, can be computed in linear time and it has a larger range of values and a greater resolution power than other indices like Colless' or Sackin's. We compute its maximum and minimum values for arbitrary and binary trees, as well as exact formulas for its expected value for binary trees under the Yule and the uniform models of evolution. As a byproduct of this study, we obtain an exact formula for the expected value of the Sackin index under the uniform model, a result that seems to be new in the literature.
Phylogenetic treeImbalance indexCophenetic valueSackin index
§ INTRODUCTION
A phylogenetic tree is a representation of the shared evolutionary history of a set of extant species. From the mathematical point a view, it is a leaf-labeled rooted tree, with its leaves representing the extant species under study, its internal nodes representing common ancestors of some of them, the root representing the most recent common ancestor of all of them, and the arcs representing direct descendants through mutations.
One of the most thoroughly studied shape properties of phylogenetic trees is their balance, that is, the degree to which the children of internal nodes tend to have the same number of descendant taxa. This global degree of balance of a tree is usually quantified by means of a single number generically called an balance index. The two most popular balance indices are Sackin's [21] and Colless' [6] (see 2.2), but there are many more
<cit.>, and Shao and Sokal <cit.> explicitly advise to use more than one such index to quantify tree balance.
Such balance indices only depend on the topology of the trees, not on the branch lengths or the actual taxa labeling their leaves. Since it is believed that the raw topology of a phylogenetic tree already reflects, at least to some extent, the evolutionary processes that have produced it <cit.>, these indices have also been widely used as tools to test stochastic models of evolution [14, 22].
Two of the most popular stochastic models of evolutionary tree growth are the Yule and the uniform models. The Yule, or Equal-Rate Markov model [8, 28], starts with a single node and, at every step, a leaf is chosen randomly and uniformly, and it is replaced by a cherry, i.e., a phylogenetic tree consisting only of a root and two leaves. Finally, once the desired number of leaves is reached, the labels are assigned randomly and uniformly to the leaves. This corresponds to a model of evolution where, at each step, each currently extant species can give rise with the same probability to two new species. Under this model different trees with the same number of leaves may have different probabilities. In contrast, the main feature of the uniform, or Proportional to Distinguishable Arrangements model [20] is that all phylogenetic trees with the same number of leaves have the same probability. From the point of view of tree growth [5, 24], this corresponds to a process where, starting with a node labeled 1, at the $k$-th step a new pendant arc, ending in the leaf labeled $k+1$, is added either to a new root or to some edge (being all possible locations of this new pendant arc equiprobable). Notice that this is not an explicit model of evolution, only of tree growth. Several properties of the distributions of Sackin's and Colless' indices have been studied in the literature under these models [2, 3, 9, 10, 16, 17, 18, 19, 25].
In this paper we propose a new balance index, the total cophenetic index. It is defined as the sum of the cophenetic values [23] of all pairs of different leaves. The main features of our index are that, unlike Colless' index, it makes sense for arbitrary (i.e., not necessarily fully resolved) trees; as Colless' and Sackin's indices, it can be easily computed in linear time; its range of values is larger than Colless' and Sackin's (up to $O(n^3)$, instead of $O(n^2)$), and it has a greater resolution power than those indices.
We compute the maximum and minimum values of our index, both in the arbitrary and the binary cases, and explicit formulas for its average value under the Yule and the uniform models for binary trees. We actually deduce its average value under the uniform model from an explicit formula for the average value of the Sackin index. This average value was known until now only for its limit distribution [3], and our formula seems thus to be new in the literature.
The rest of this paper is organized as follows. In a first section we introduce the basic notations and facts on phylogenetic trees that will be used henceforth, and we recall some basic facts on the Sackin and the Colless indices. Then, in Section 3, we define our total cophenetic index $\Phi$ and we establish its basic properties. In Section 4 we compute its maximum and minimum values, and then, in subsequent sections, we compute its expected value under the Yule and the uniform models. We finally devote a last section to conclusions and the discussion of two preliminary numerical experiments involving $\Phi$.
§ PRELIMINARIES
§.§ Phylogenetic trees
In this paper, by a phylogenetic tree on a set $S$ of taxa we mean a rooted tree with its leaves bijectively labeled in the set $S$. To simplify the language, we shall always identify a leaf of a phylogenetic tree with its label. We shall use the term phylogenetic tree with $n$ leaves to refer to a phylogenetic tree on the set $\{1,\ldots,n\}$. We shall denote by $L(T)$ the set of leaves of a phylogenetic tree $T$ and by $V_{int}(T)$ its set of internal nodes.
A phylogenetic tree is binary, or fully resolved, when all its internal nodes are bifurcating, that is, when every internal node has exactly two children.
Whenever there exists a path from $u$ to $v$ in a phylogenetic tree $T$, we shall say that $v$ is a descendant of $u$ and also that $u$ is an ancestor of $v$. The cluster of a node $v$ in $T$ is the set $C_T(v)$ of its descendant leaves, an we shall denote by $\kappa_T(v)$ the cardinal $|C_T(v)|$, that is, the number of descendant leaves of $v$.
Given a node $v$ of a phylogenetic tree $T$, the subtree of $T$ rooted at $v$ is the subgraph of $T$ induced on the set of descendants of $v$. It is a phylogenetic tree on $C_T(v)$ with root this node $v$.
The lowest common ancestor (LCA) of a pair of nodes $u,v$ of a phylogenetic tree $T$, in symbols $LCA_T(u,v)$, is the unique common ancestor of them that is a descendant of every other common ancestor of them.
The depth $\delta_T(v)$ of a node $v$ in a phylogenetic tree $T$ is the length (in number of arcs) of the unique path from the root $r$ to $v$.
A rooted caterpillar is a binary phylogenetic tree all whose internal nodes have a leaf child: see Fig. <ref>.(a). A rooted star is a phylogenetic tree such that all its leaves have depth 1: see Fig. <ref>.(b).
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(6,0) node $\ldots$;
(8,0) node [tre] (n) ; n
(1,2) node[tre] (a) ;
(3,3) node[tre] (b) ;
(7,5) node[tre] (r) ;
(5,4) node .;
(5.3,4.15) node .;
(5.6,4.3) node .;
(b)– (4.5,3.75);
(r)– (6,4.5);
(r)– (n);
(4,-2) node (a);
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(6,0) node $\ldots$;
(8,0) node [tre] (n) ; n
(4,3) node[tre] (r) ;
(4,-2) node (b);
(a) A rooted caterpillar with $n$ leaves. (b) The rooted star with $n$ leaves.
Let $T$ be a binary phylogenetic tree. For every $v\in V_{int}(T)$, say with children $v_1,v_2$, the balance value of $v$ is $bal_T(v)=|\kappa_T(v_1)-\kappa_T(v_2)|$. An internal node $v$ of $T$ is balanced when $bal_T(v)\leq 1$. So, a node $v$ with children $v_1$ and $v_2$ is balanced if, and only if, $\{\kappa_T(v_1),\kappa_T(v_2)\}=\{\lfloor \kappa_T(v)/2\rfloor,\lceil \kappa_T(v)/2\rceil\}$.
We shall say that a binary phylogenetic tree $T$ is maximally balanced when all its internal nodes are balanced. Recurrently, a binary phylogenetic tree is maximally balanced when its root is balanced and both subtrees rooted at the children of the root are maximally balanced. Notice that, for any number $n$ of nodes, the topology of a maximally balanced tree with $n$ leaves is fixed, and therefore two maximally balanced trees with the same number of leaves differ only in their labeling. Fig. <ref> depicts the maximally balanced trees with $n=2,\ldots,6$ leaves, up to relabelings.
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(1,2) node[tre] (a) ;
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(1,2) node[tre] (a) ;
(2,4) node[tre] (b) ;
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(6,0) node [tre] (4) ; 4
(1,2) node[tre] (a) ;
(5,2) node[tre] (c) ;
(3,4) node[tre] (b) ;
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(6,0) node [tre] (4) ; 4
(8,0) node [tre] (5) ; 5
(1,2) node[tre] (a) ;
(7,2) node[tre] (c) ;
(2,4) node[tre] (b) ;
(4,6) node[tre] (d) ;
(0,0) node [tre] (1) ; 1
(2,0) node [tre] (2) ; 2
(4,0) node [tre] (3) ; 3
(6,0) node [tre] (4) ; 4
(8,0) node [tre] (5) ; 5
(10,0) node [tre] (6) ; 6
(1,2) node[tre] (a) ;
(9,2) node[tre] (c) ;
(2,4) node[tre] (b) ;
(8,4) node[tre] (d) ;
(5,6) node[tre] (r) ;
Maximally balanced trees.
Let $\TT_n$ (resp., $\TB_n$) be the set of isomorphism classes of phylogenetic trees (resp, binary phylogenetic trees) with $n$ leaves. It is well known <cit.> that $|\TB_1|=1$ and, for every $n\geq 2$,
|\TB_n|=(2n-3)!!=(2n-3)(2n-5)\cdots 3 \cdot 1.
No closed formula is known for the cardinal $|\TT_n|$, only recurrences or generating functions (see again <cit.> and the references therein).
An ordered $m$-forest on a set $S$ is an ordered sequence of $m$ phylogenetic trees $(T_1,T_2,\ldots,T_m)$, each $T_i$ on a set $S_i$ of taxa, such that these sets $S_i$ are pairwise disjoint and their union is $S$. An ordered forest is binary when it consists of binary trees.
Let $\FF_{m,n}$ (resp., $\FB_{m,n}$) be the set of isomorphism classes of ordered $m$-forests
(resp., binary ordered $m$-forests) on a set $S$ with $|S|=n$. It is known (see, for instance, <cit.>) that for every $n\geq m\geq 1$,
|\FB_{m,n}|= \frac{(2n-m-1)!m}{(n-m)!2^{n-m}}.
Again, no closed formula is known for $|\FF_{m,n}|$.
§.§ Balance indices
Several balance indices have been proposed so far in the literature <cit.>. The two most popular ones are the Sackin index [21] and the Colless index [6]. The Sackin index of a phylogenetic tree $T\in \TT_n$ is defined as the sum of the depths of its leaves:
Alternatively [2],
S(T)=\sum_{v\in V_{int}(T)} \kappa_T(v).
On the other hand, the Colless index of a binary phylogenetic tree $T$ is defined as
C(T)=\sum_{v\in V_{int}(T)} bal_T(v).
This Colless index has been extended to non-binary trees by defining $bal_T(v)=0$ for every non-bifurcating internal node [22].
It is straightforward to notice that these two indices depend only on the topology of the tree, and they are invariant under isomorphisms and relabelings of leaves. This is desirable in a balance index, because the degree of symmetry of a tree depends only on its shape.
Both Sackin's and Colless's indices reach their maximum value exactly at caterpillars, which are clearly the more imbalanced trees, and they reach their minimum on $\TB_n$ at the maximally balanced trees [9, 22]. In both cases, the maximum value is in $O(n^2)$.
But they may also reach their minimum on $\TB_n$ at other trees. For instance, for $n=6$, both indices take their minimum value at the two trees $T,T'$ depicted in Fig. <ref>.
$T'$ is maximally balanced, but $T$ is not so. Actually, it is easy to check that Sackin's index is invariant under interchanges of cousins, which may produce trees with different degrees of symmetry but the same Sackin index.
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(10,0) node[tre] (6) ; 6
(1,2) node[tre] (a) ;
(5,2) node[tre] (b) ;
(9,2) node[tre] (c) ;
(3,4) node[tre] (d) ;
(5,6) node[tre] (r) ;
(5,-1.5) node $T$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(10,0) node[tre] (6) ; 6
(1,2) node[tre] (a) ;
(2,4) node[tre] (b) ;
(9,2) node[tre] (c) ;
(8,4) node[tre] (d) ;
(5,6) node[tre] (r) ;
(5,-1.5) node $T'$;
Two trees having minimum Sackin's and Colless' indices on $\TB_6$: $S(T)=S(T')=16$ and $C(T)=C(T')=2$.
The main drawback with Colless' index is its difficult meaningful generalization to non-binary trees. Moreover, as Fig. <ref> shows, although not every interchange of cousins yields trees with the same Colless index, there are still interchanges of cousins that modify the symmetry of the trees but preserve this index.
The expected values of these indices on $\BT_n$ have been studied under the Yule and the uniform models. Recall that, under the Yule model, different trees in $\BT_n$ may have different probabilities: namely, a tree $T$ with $n$ leaves has probability [4, 25]
P_Y(T)=\frac{2^{n-1}}{n!}\prod_{v\in V_{int}(T)}\frac{1}{\kappa_T(v)-1}.
Under the uniform model, all trees in $\BT_n$ are equiprobable, and thus they have probability
Let $C_n$ and $S_n$ be the random variables defined by choosing a tree $T\in \BT_n$ and computing $C(T)$ or $S(T)$, respectively. The following facts are known about the expected values of these random variables:
* Under the Yule model,
* $E_Y(C_n)=(n\hspace*{-1ex} \mod 2)+n\sum\limits _{j=2}^{\lfloor n/2\rfloor} 1/j$ [9].
* $E_Y(S_n)=2n\sum\limits_{j=2}^{n}1/j$ [10].
* Under the uniform model,
E_U(C_n), E_U(S_n)\sim \sqrt{\pi}n^{3/2}\quad \mbox{[3]}.
We shall actually prove in this paper (see Theorem <ref>) that
E_U(S_n)=\frac{n }{2n-3}\, {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array};2\bigg),
${}_3 F_2$ is a hypergeometric function [1].
§ THE TOTAL COPHENETIC INDEX
For every pair of leaves $i,j$ in a phylogenetic tree $T$, their cophenetic value [23] is the depth of their least common ancestor:
\varphi_T(i,j)=\delta_T(LCA_T(i,j)).
The total cophenetic index of a phylogenetic tree $T\in\TT_n$ is the sum of the cophenetic values of its pairs of different leaves:
\Phi(T)=\sum_{1\leq i<j\leq n} \varphi_T(i,j).
This index can be seen as an extension of Sackin's: instead of adding up the depths of the leaves (that is, the depths of the LCA of every leaf and itself), $\Phi(T)$ adds up the depths of the LCA of every pair of leaves in $T$. Notice also that, as Sackin's and Colless' indices, $\Phi(T)$ only depends on the topology of $T$, and in particular it is invariant under permutations of its labels.
Fig. <ref> shows all possible topologies of phylogenetic trees with 5 leaves, and their total cophenetic indices. Although we shall return on it later for trees with an arbitrary number $n$ of leaves, notice that the rooted star has the smallest total cophenetic value, 0; the binary tree with the smallest total cophenetic value is the maximally balanced; and the tree with the largest total cophenetic value is the caterpillar.
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(4,5) node[tre] (r) ;
(3,-2) node $\Phi(T)=0$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(1,3) node[tre] (z1) ;
(3,5) node[tre] (r) ;
(3,-2) node $\Phi(T)=1$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(1,2) node[tre] (z1) ;
(5,2) node[tre] (z2) ;
(4.5,5) node[tre] (r) ;
(3,-2) node $\Phi(T)=2$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(2,2) node[tre] (z1) ;
(4.5,5) node[tre] (r) ;
(3,-2) node $\Phi(T)=3$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(1,2) node[tre] (z1) ;
(2.5,4) node[tre] (z2) ;
(5,6) node[tre] (r) ;
(3,-2) node $\Phi(T)=4$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(2,3) node[tre] (z1) ;
(4.5,6) node[tre] (r) ;
(7,3) node[tre] (z2) ;
(3,-2) node $\Phi(T)=4$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(1,2) node[tre] (z1) ;
(2.5,4) node[tre] (z2) ;
(5,6) node[tre] (r) ;
(7,3) node[tre] (z3) ;
(3,-2) node $\Phi(T)=5$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(8,0) node[tre] (5) ; 5
(4,6) node[tre] (r) ;
(3,3) node[tre] (z) ;
(3,-2) node $\Phi(T)=6$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(1,2) node[tre] (z) ;
(2.5,4) node[tre] (r) ;
(8,0) node[tre] (5) ; 5
(4,6) node[tre] (r0) ;
(3,-2) node $\Phi(T)=7$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(1,2) node[tre] (z) ;
(5,2) node[tre] (x) ;
(2.5,4) node[tre] (r) ;
(8,0) node[tre] (5) ; 5
(4,6) node[tre] (r0) ;
(3,-2) node $\Phi(T)=8$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(2,2) node[tre] (z) ;
(3,4) node[tre] (r) ;
(8,0) node[tre] (5) ; 5
(4,6) node[tre] (r0) ;
(3,-2) node $\Phi(T)=9$;
(0,0) node[tre] (1) ; 1
(2,0) node[tre] (2) ; 2
(4,0) node[tre] (3) ; 3
(6,0) node[tre] (4) ; 4
(1,2) node[tre] (z) ;
(2.5,3.33) node[tre] (x) ;
(4,4.66) node[tre] (r) ;
(8,0) node[tre] (5) ; 5
(5.5,6) node[tre] (r0) ;
(3,-2) node $\Phi(T)=10$;
All phylogenetic trees with 5 leaves, up to relabelings, and their total cophenetic index.
The following alternative expression for $\Phi(T)$ will be useful in many proofs.
Let $T\in \TT_n$ be a phylogenetic tree with root $r$. Then,
\Phi(T)=\sum_{v\in V_{int}(T)-\{r\}} \binom{\kappa_T(v)}{2}.
For every $v\in V_{int}(T)-\{r\}$ and for every $i,j\in L(T)$, let
\gamma_{v}(i,j)=\left\{\begin{array}{ll}
1 & \mbox{ if $i,j\in C_T(v)$}\\
0 & \mbox{ otherwise}
\end{array}\right.
Then, $\phi_T(i,j)=\sum\limits_{v\in V_{int}(T)-\{r\}} \gamma_{v}(i,j)$ and thus
\begin{array}{rl}
\Phi(T)& \displaystyle =\sum_{1\leq i<j\leq n}\sum_{V_{int}(T)-\{r\}} \gamma_{v}(i,j)
=\sum_{v\in V_{int}(T)-\{r\}}\sum_{1\leq i<j\leq n} \gamma_{v}(i,j)\\[1ex] & \displaystyle
=\sum_{v\in V_{int}(T)-\{r\}}\binom{|C_T(v)|}{2}.
\end{array}
For every $T\in \TT_n$, $\Phi(T)$ can be computed in time $O(n)$.
The vector $(\kappa_T(v))_{v\in V_{int}(T)-\{r\}}$ can be computed in linear time by traversing in post order the tree $T$ <cit.>, and then, by the last lemma, $\Phi(T)$ is computed in linear time from this vector.
Let $T\in \TT_n$ be a phylogenetic tree with root $r$, and let $T_1,\ldots,T_k$, $k\geq 2$, be the subtrees rooted at the children of $r$; cf. Fig <ref>. Then,
\Phi(T)=\sum_{i=1}^k\Phi(T_i)+\sum_{i=1}^k\binom{|L(T_i)|}{2}.
(0,0) node[tre] (z1) ;
(0,-2) node $T_1$;
(5,0) node[tre] (z2) ;
(5,-2) node $T_2$;
(8,-2.8) node .;
(8.4,-2.8) node .;
(8.8,-2.8) node .;
(11.5,0) node[tre] (z3) ;
(11.5,-2) node $T_k$;
(7.25,4) node[tre] (z) ;
Let $z_i$ be the root of $T_i$, $i=1,\ldots,k$, and $r$ the root of $T$. Then, by Lemma <ref>,
\begin{array}{rl}
\Phi(T) & \displaystyle =\sum_{v\in V_{int}(T)-\{r\}} \binom{\kappa_T(v)}{2}
=\sum_{i=1}^{k}\sum_{v\in V_{int}(T_i)} \binom{\kappa_{T_i}(v)}{2}\\ & \displaystyle
=\sum_{i=1}^{k}\Big(\binom{\kappa_{T_i}(z_i)}{2}+\sum_{v\in V_{int}(T_i)-\{z_i\}} \binom{\kappa_{T_i}(v)}{2}\Big) \\ & \displaystyle
=\sum_{i=1}^{k}\Big( \binom{|L(T_i)|}{2}+ \Phi(T_i) \Big).
\end{array}
This shows that the total cophenetic index is a recursive tree shape statistic in the sense of [11].
Next lemma shows that the total cophenetic index is local, in the sense that if two trees differ only on a rooted subtree, then the difference between their total cophenetic values is equal to that of these subtrees. Sackin's and Colless' indices also satisfy this property.
Let $T_0$ and $T_0'$ be two phylogenetic trees with $L(T_0)=L(T_0')\subseteq \{1,\ldots,n\}$, let $T\in \TT_n$ be such that its subtree rooted at some node $z$ is $T_0$, and let $T'\in\TT_n$ be the tree obtained from $T$ by replacing $T_0$ by $T_0'$ as its subtree rooted at $z$. Then
\Phi(T)-\Phi(T')=\Phi(T_0)-\Phi(T_0').
Without any loss of generality, assume that $L(T_0)=L(T_0')=\{1,\ldots,m\}$, with $m\leq n$. Let $k=\delta_T(z)=\delta_{T'}(z)$. Then, for every $i,j\leq m$,
\varphi_T(i,j)=k+\varphi_{T_0}(i,j),\
\varphi_{T'}(i,j)=k+\varphi_{T'_0}(i,j).
On the other hand, $\varphi_T(i,j)=\varphi_{T'}(i,j)$ if $i>m$ or $j>m$. Therefore
\begin{array}{rl}
\Phi(T)-\Phi(T') & \displaystyle =
\sum_{1\leq i<j\leq m}(\varphi_T(i,j)-\varphi_{T'}(i,j))\\ & \displaystyle =
\sum_{1\leq i<j\leq m}(\varphi_{T_0}(i,j)-\varphi_{T_0'}(i,j))=
\Phi(T_0)-\Phi(T_0').
\end{array}
The nodal distance $d_T(i,j)$ between a pair of leaves $i,j$ is the length of the unique undirected path connecting them; equivalently, it is the sum of the lengths of the paths from $LCA(i,j)$ to $i$ and $j$. The total area [12] of a tree $T\in \TT_n$ is defined as
D(T)=\sum_{1\leq i<j\leq n} d_T(i,j).
There is an easy relation between $\Phi(T)$, $S(T)$ and $D(T)$, which will be used several times in this paper.
For every $T\in \TT_n$,
It is straightforward to check that, for every $i,j\in L(T)$,
\delta_T(i)+\delta_T(j)=d_T(i,j)+2\varphi_T(i,j).
\begin{array}{rl}
2\phi(T)+D(T) & \displaystyle =\sum_{1\leq i<j\leq n}(2\varphi_T(i,j)+d_T(i,j))
=\sum_{1\leq i<j\leq n}(\delta_T(i)+\delta_T(j))\\ & \displaystyle =(n-1)\sum_{i=1}^{n}\delta_T(i)=(n-1)S(T).
\end{array}
§ TREES WITH MAXIMUM AND MINIMUM $\PHI$
In this section we determine which trees in $\TT_n$ and $\TB_n$ have the largest and smallest total cophenetic indices. We begin by establishing two lemmas that will allow us to find the trees with the maximum $\Phi$ on $\TT_n$.
Let $T_1, \ldots,T_k$, with $k\geq 3$, be an ordered forest on $\{1,\ldots,m\}$. Consider the trees $T_0,T_0'\in \TT_n$ described in Fig. <ref>. Then, $\Phi(T_0')-\Phi(T_0)> 0$.
(0,0) node[tre] (z1) ;
(0,-2) node $T_1$;
(5,0) node[tre] (z2) ;
(5,-2) node $T_2$;
(8,-2.8) node .;
(8.4,-2.8) node .;
(8.8,-2.8) node .;
(11.5,0) node[tre] (z3) ;
(12,-2) node $T_{k-1}$;
(17.5,0) node[tre] (z4) ;
(17.5,-2) node $T_{k}$;
(8.75,4) node[tre] (r) ;
(8,-4) node $T_0$;
(0,0) node[tre] (z1) ;
(0,-2) node $T_1$;
(5,0) node[tre] (z2) ;
(5,-2) node $T_2$;
(8,-2.8) node .;
(8.4,-2.8) node .;
(8.8,-2.8) node .;
(11.5,0) node[tre] (z3) ;
(12,-2) node $T_{k-1}$;
(17.5,0) node[tre] (z4) ;
(17.5,-2) node $T_{k}$;
(6,2) node[tre] (x) ; x
(12,4) node[tre] (r) ;
(8,-3.9) node $T_0'$;
The trees $T_0$ and $T_0'$ in the statement of Lemma <ref>.
With the notations of Fig. <ref>, notice that
\begin{array}{rl}
\Phi(T_0')-\Phi(T_0) & \displaystyle = \sum_{v\in V_{int}(T'_0)-\{r\}} \binom{\kappa_{T_0'}(v)}{2}-
\sum_{v\in V_{int}(T_0)-\{r\}} \binom{\kappa_{T_0}(v)}{2}\\ & \displaystyle =\binom{\kappa_{T'_0}(x)}{2}> 0.
\end{array}
For every non-binary phylogenetic tree $T\in \TT_n$, there always exists a binary phylogenetic tree $T'$ such that $\Phi(T')>\Phi(T)$.
Let $T\in \TT_n$ be a non-binary phylogenetic tree. Then it contains an internal node $z$ whose rooted subtree looks like the tree $T_0$ in the previous lemma, for some $k\geq 3$. By Lemma <ref> and the last lemma, if $T'\in\TT_n$ is the tree obtained from $T$ by replacing $T_0$ by $T_0'$ as its subtree rooted at $z$, then $\Phi(T')-\Phi(T)> 0$.
Therefore, the maximum total cophenetic index is reached at a binary tree.
Let $m\geq 4$, let $2\leq k\leq m-2$, let $T_1$ be any binary tree on $\{k+1,\ldots,m\}$, and let $T_0$ and $T_0'$ be the phylogenetic trees in $\TB_m$ depicted in Fig. <ref>. Then,
(0,0) node[tre] (z) ;
(0,-2) node $T_1$;
(2,6) node[tre] (x) ;
(4,4.6) node[tre] (a) ;
(6,3.2) node[tre] (b) ;
(4,-2.5) node [tre] (1) ; 1
(6,-2.5) node [tre] (2) ; 2
(8.3,1.8-0.7*0.3) node .;
(8.6,1.8-0.7*0.6) node .;
(8.9,1.8-0.7*0.9) node .;
(11,-0.3) node[tre] (c) ;
(9,-2.5) node [tre] (k-1) ;
(k-1) node $k\!\!\!-\!\!\!1$;
(13,-2.5) node [tre] (k) ; k
(6,-4) node $T_0$;
(0,0) node[tre] (z) ;
(0,-2) node $T_1$;
(2,1) node[tre] (a) ;
(3,-2.5) node [tre] (1) ;
(1) node $k$;
(4,2) node[tre] (b) ;
(5,-2.5) node [tre] (2) ;
(2) node $k\!\!\!-\!\!\!1$;
(5.3,2.65) node .;
(5.6,2.8) node .;
(5.9,2.95) node .;
(7.5,3.75) node[tre] (c) ;
(8.5,-2.5) node [tre] (k) ;
(k) node $2$;
(9.5,4.75) node[tre] (r) ;
(10.5,-2.5) node [tre] (k0) ;
(k0) node $1$;
(6,-4) node $T_0'$;
The trees $T_0$ and $T_0'$ in the statement of Lemma <ref>.
By Lemma <ref>, and recalling that $|L(T_1)|=m-k$, we have that
\begin{array}{l}
\displaystyle \Phi(T_0)=\phi(T_1)+\binom{m-k}{2}+\binom{k}{2}+\binom{k-1}{2}+\cdots+\binom{3}{2}+\binom{2}{2}\\
\displaystyle \Phi(T'_0)=\binom{m-1}{2}+\binom{m-2}{2}+\cdots+\binom{m-k+1}{2}+\binom{m-k}{2}+\phi(T_1)
\end{array}
and hence, since $m-k\geq 2$,
\Phi(T_0')-\Phi(T_0)=\sum_{j=1}^{k-1}\binom{m-k+j}{2}-\binom{j+1}{2}>0.
The trees in $\TT_n$ with maximum total cophenetic index are exactly the rooted caterpillars $K_n$, and this maximum is
By Corollary <ref>, any tree in $\TT_n$ with maximum total cophenetic index will be binary. Let now $T\in\TB_n$ and assume that it is not a caterpillar. Therefore, it has an internal node $z$ of largest depth without any leaf child; in particular, all internal descendant nodes of $z$ have some leaf child. Thus, and up to a relabeling of its leaves, the subtree of $T$ rooted at $z$ has the form of the tree $T_0$ in Fig. <ref>, for some $k\geq 2$ and some $l\geq k+2$. But then, by Lemma <ref> (taking as $T_1$ the caterpillar subtree rooted at the parent of the leaf $k$), the tree $T_0'$ also depicted in Fig. <ref> has a strictly larger total cophenetic index.
Then, by Lemma <ref>, if we replace in $T$ the subtree rooted at $z$ by this tree $T_0'$, we obtain a new tree $T'$ with $\Phi(T')> \Phi(T)$.
This implies that no tree other than a caterpillar can have the largest
total cophenetic index.
(0,0) node[tre] (1) ; 1
(2,2) node[tre] (a) ;
(3,0) node [tre] (2) ; 2
(4,4) node[tre] (b) ;
(5,2) node [tre] (3) ; 3
(5.3,5.3) node .;
(5.6,5.6) node .;
(5.9,5.9) node .;
(7.5,7.5) node[tre] (c) ;
(8.5,4.5) node [tre] (k) ; k
(9.5,9.5) node[tre] (d) ;
(d) node $z$;
(19,0) node[tre] (l) ; l
(17,2) node[tre] (a1) ;
(16,0) node [tre] (l-1) ;
(l-1) node $l\!\!\!-\!\!\!1$;
(15,4) node[tre] (b1) ;
(14,2) node [tre] (l-2) ;
(l-2) node $l\!\!\!-\!\!\!2$;
(13.7,5.3) node .;
(13.4,5.6) node .;
(13.1,5.9) node .;
(11.5,7.5) node[tre] (c1) ;
(10.5,4.5) node [tre] (k+1) ;
(k+1) node $k\!\!\!+\!\!\!1$;
(9.5,-2) node $T_0$;
(0,0) node[tre] (1) ; 1
(2,2) node[tre] (a) ;
(4,0) node[tre] (2) ; 2
(4,3) node[tre] (b) ;
(6,0) node[tre] (3) ; 3
(5.3,3.65) node .;
(5.6,3.8) node .;
(5.9,3.95) node .;
(7.5,4.75) node[tre] (c) ;
(10,0) node[tre] (k) ; k
(9.5,5.75) node[tre] (d) ;
(12,0) node[tre] (k+1) ;
(k+1) node $l$;
(10.3,6.15) node .;
(10.6,6.3) node .;
(10.9,6.45) node .;
(12.5,7.25) node[tre] (e) ;
(e) node $z$;
(16,0) node[tre] (l) ;
(l) node $k\!\!\!+\!\!\!1$;
(9.5,-2) node $T_0'$;
The trees $T_0$ and $T_0'$ in the proof of Proposition <ref>.
As far as the total cophenetic index of the rooted caterpillar $K_n$ with $n$ leaves depicted in Fig. <ref>.(a) goes, since the parent of the leaf labelled $j$, for $j=2,\ldots,n$, has $j$ descendant leaves, by Lemma <ref> we have that
It is obvious that minimum total cophenetic index is 0, and it is attained only at the rooted star trees, depicted in Fig. <ref>.(b). Therefore, the range of $\Phi$ on $\TT_n$ goes from 0 to $\binom{n}{3}$. This is one order of magnitude larger than the range of Sackin's and Colless' indices, whose maximum value, reached also at the rooted caterpillars, has order $O(n^2)$ [9, 19, 22].
Let us characterize now those binary phylogenetic trees with smallest total cophenetic index.
Let $T_1,T_2,T_3,T_4$ be an ordered binary forest on $\{1,\ldots,m\}$, let
$x_i=|L(T_i)|$, for $i=1,2,3,4$, and assume that $x_1\geq x_2$, $x_3\geq x_4$ and $x_1> x_3$. Let $T_0$ the phylogenetic tree depicted in Fig <ref>.(a), and let $T\in \TB_n$ ($n\geq m$) be a binary phylogenetic tree having $T_0$ as a subtree rooted at some node. If $\Phi(T)$ is minimum in $\TB_n$, then $x_4\geq x_2$.
(0,2) node[tre] (v1) ;
(6,2) node[tre] (v2) ;
(12,2) node[tre] (v3) ;
(18,2) node[tre] (v4) ;
(3,4) node[tre] (a) ; a
(15,4) node[tre] (b) ; b
(9,6) node[tre] (z) ; z
(0,0) node $T_1$;
(6,0) node $T_2$;
(12,0) node $T_3$;
(18,0) node $T_4$;
(9,-2.5) node (a) $T_0$;
(0,2) node[tre] (v1) ;
(6,2) node[tre] (v2) ;
(12,2) node[tre] (v3) ;
(18,2) node[tre] (v4) ;
(3,4) node[tre] (a) ; a
(15,4) node[tre] (b) ; b
(9,6) node[tre] (z) ; z
(0,0) node $T_1$;
(6,0) node $T_4$;
(12,0) node $T_3$;
(18,0) node $T_2$;
(9,-2.5) node (b) $T'_0$;
(a) The tree $T_0$ in the statement of Lemma <ref>.
(b) The tree $T_0'$ in the proof of Lemma <ref>.
Assume that $x_2>x_4$. We shall show that, in this case, a suitable interchange of cousins in $T_0$ produces a tree with smaller total cophenetic index, which in particular will imply that $\Phi(T)$ cannot be the minimum in $\TB_n$.
Assume that the tree $T$ in the statement has the subtree $T_0$ rooted at a node $z$. Consider the tree $T_0'$ obtained by interchanging in $T_0$ the subtrees $T_2$ and $T_4$ (see Fig. <ref>.(b)) and let $T'$ be the tree obtained from $T$ by replacing $T_0$ by $T_0'$ as its subtree rooted at $z$. Then, by Lemma <ref>,
\begin{array}{rl}
\Phi(T')-\Phi(T) & =\Phi(T_0')-\Phi(T_0)\\ & \displaystyle =\binom{\kappa_{T_0'}(a)}{2}+\binom{\kappa_{T_0'}(b)}{2}-\binom{\kappa_{T_0}(a)}{2}-\binom{\kappa_{T_0}(b)}{2}\\
& \displaystyle =\binom{x_1+x_4}{2}+\binom{x_2+x_3}{2}-\binom{x_1+x_2}{2}-\binom{x_3+x_4}{2}\\ \displaystyle
& =x_1x_4+x_2x_3-x_1x_2-x_3x_4=(x_1-x_3)(x_4-x_2)<0
\end{array}
which shows that $\Phi(T')<\Phi(T)$.
From the proof of the last lemma we deduce that if, in the tree $T_0$ in Fig. <ref>.(a), $|L(T_1)|\neq |L(T_3)|$ and $|L(T_2)|\neq |L(T_4)|$, and if we interchange $T_2$ and $T_4$, then the resulting tree has always a different total cophenetic index.
Let $T_1,T_2,$ be an ordered binary forest on $\{1,\ldots,m-1\}$, let
$x_i=|L(T_i)|$, for $i=1,2$, and assume that $x_1\geq x_2$. Let $T_0$ the phylogenetic tree depicted in Fig <ref>.(a), and let $T\in \TB_n$ be a binary phylogenetic tree having $T_0$ as a subtree rooted at some node. If $\Phi(T)$ is minimum in $\TB_n$, then $x_1=x_2=1$.
(0,2) node[tre] (v1) ;
(6,2) node[tre] (v2) ;
(3,4) node[tre] (a) ; a
(15,-0.5) node[tre] (m) ; m
(9,6) node[tre] (z) ; z
(0,0) node $T_1$;
(6,0) node $T_2$;
(7.5,-2.5) node (a) $T_0$;
(0,2) node[tre] (v1) ;
(0,0) node $T_1$;
(8,6) node[tre] (z) ; z
(6,2) node[tre] (v2) ;
(6,0) node $T_2$;
(11,4) node[tre] (b) ; b
(16,-0.5) node[tre] (m) ; m
(7.5,-2.5) node (b) $T'_0$;
(a) The tree $T_0$ in the statement of Lemma <ref>. (b) The tree $T_0'$ in the proof of Lemma <ref>.
Assume that $x_1>1$. We shall show that, again in this case, a suitable interchange of cousins in $T_0$ produces a tree with smaller total cophenetic index.
Assume that the tree $T$ in the statement has the subtree $T_0$ rooted at a node $z$. Let $T'$ by the tree obtained from $T$ by replacing $T_0$ by the subtree $T_0'$ described in Fig. <ref>.(b). Then:
\begin{array}{rl}
\Phi(T')-\Phi(T) & =\Phi(T_0')-\Phi(T_0)\\ & \displaystyle =\binom{\kappa_{T_0'}(b)}{2}-\binom{\kappa_{T_0}(a)}{2}=
\binom{x_2+1}{2}-\binom{x_1+x_2}{2}<0
\end{array}
which shows that $\Phi(T')<\Phi(T)$.
The last two lemmas show that, unlike what happens with Sackin's and Colless' indices, any interchange of cousins that changes the balance of their grandparent always changes the total cophenetic index of a tree.
For every $T\in \TB_n$, $\Phi(T)$ is minimum on $\TB_n$ if, and only if, $T$ is maximally balanced.
Assume that $T\in \TB_n$ is not maximally balanced, and let $z$ be a non-balanced internal node in $T$ with largest depth. Assume that $a$ and $b$ are its children, with $\kappa_T(a)\geq \kappa_T(b)+2$.
If $b$ is a leaf, then, by Lemma <ref>, $\kappa_T(a)=2$ and therefore $\kappa_T(a)\not\geq 3=\kappa_T(b)+2$. Therefore, $a$ and $b$ are internal, and hence balanced. Let $T_0$ be the subtree of $T$ rooted at $z$, represented in Fig. <ref>.(a), and let $x_i=|L(T_i)|$, for $i=1,2,3,4$; without any loss of generality, we shall assume that $x_1\geq x_2$ and $x_3\geq x_4$ and thus, since $a$ and $b$ are balanced, $x_2=x_1$ or $x_1-1$ and $x_4=x_3$ or $x_3-1$. Then, $x_1+x_2=\kappa_T(a)\geq \kappa_T(b)+2=x_3+x_4+2$ implies that
$2x_1\geq 2x_3+1$, and hence that $x_1>x_3$.
Therefore, by Lemma <ref>, if $\Phi(T)$ is minimum in $\TB_n$, it must happen that $x_1>x_3\geq x_4\geq x_2$. Since it forbids the equality $x_1=x_2$, it implies that $x_1=x_2+1$ and therefore $x_2=x_3=x_4$. But then $x_1+x_2=2x_2+1\not\geq x_3+x_4+2=2x_2+2$, against the assumption that $z$ is not balanced.
So, the only binary trees with minimum $\Phi$ are the maximally balanced.
Let us compute now this minimum value of $\Phi$ on $\BT_n$.
For every $n$, let $f(n)$ be the minimum of $\Phi$ on $\TB_n$. Then, $f(1)=f(2)=0$ and
f(n)=f(\lceil n/2\rceil)+f(\lfloor n/2\rfloor)+\binom{\lceil n/2\rceil}{2}+\binom{\lfloor n/2\rfloor}{2},\quad \mbox{ for }n\geq 3.
This recurrence for $f(n)$ is a direct consequence of Lemma <ref> and the fact that the root of a maximally balanced tree in $\TB_n$ is balanced and the subtrees rooted at their children are maximally balanced.
For every $n\geq 0$, let $a(n)$ is the highest power of 2 that divides $n!$. Then, for every $n\geq 1$,
f(n)= \sum_{k=0}^{n-1} a(n).
The sequence $(a(n))_n$ is sequence A011371 in Sloane's On-Line Encyclopedia of Integer Sequences [26], where we learn that it satisfies the recurrence
a(n)=\lfloor n/2\rfloor+ a(\lfloor n/2\rfloor).
Let now $(x(n))_n$ denote the sequence of partial sums of $(a(n))_n$, which is
sequence A174605 in Sloane's Encyclopedia. Then,
the sequence $(x(n))_n$ starts with $x(0)=x(1)=0$ and it satisfies the recurrence
x(n)-x(n-1)=a(n)=\lfloor n/2\rfloor+ a(\lfloor n/2\rfloor)=\lfloor n/2\rfloor+ x(\lfloor n/2\rfloor)-x(\lfloor n/2\rfloor-1).
We want to prove that $f(n+1)=x(n)$, for every $n\geq 0$.
Since $f(1)=f(2)=0$, it remains to check the equality
f(n+1)-f(n)=\lfloor n/2\rfloor+ f(\lfloor n/2\rfloor+1)-f(\lfloor n/2\rfloor),\quad \mbox{for $n\geq 2$}.
We prove this equality with the help of Lemma <ref> and by distinguishing four cases, depending on the residue of $n$ mod 4.
* If $n=4m$, then
\begin{array}{rl}
f(n+1)-f(n) & = f(2m+1)+f(2m)+\binom{2m+1}{2}+\binom{2m}{2}\\ & \qquad -(f(2m)+f(2m)+\binom{2m}{2}+\binom{2m}{2})\\
& = f(2m+1)-f(2m)+\binom{2m+1}{2}-\binom{2m}{2}\\
& = f(2m+1)-f(2m)+2m\\
&= f(\lfloor n/2\rfloor+1)-f(\lfloor n/2\rfloor)+\lfloor n/2\rfloor
\end{array}
* If $n=4m+1$, then
\begin{array}{rl}
f(n+1)-f(n) & = f(2m+1)+f(2m+1)+\binom{2m+1}{2}+\binom{2m+1}{2}\\ & \qquad -(f(2m+1)+f(2m)+\binom{2m+1}{2}+\binom{2m}{2})\\
& = f(2m+1)-f(2m)+\binom{2m+1}{2}-\binom{2m}{2}\\
& = f(2m+1)-f(2m)+2m\\
&= f(\lfloor n/2\rfloor+1)-f(\lfloor n/2\rfloor)+\lfloor n/2\rfloor
\end{array}
* If $n=4m+2$, then
\begin{array}{rl}
f(n+1)-f(n) & = f(2m+2)+f(2m+1)+\binom{2m+2}{2}+\binom{2m+1}{2}\\ & \qquad -(f(2m+1)+f(2m+1)+\binom{2m+1}{2}+\binom{2m+1}{2})\\
& = f(2m+2)-f(2m+1)+\binom{2m+2}{2}-\binom{2m+1}{2}\\
& = f(2m+2)-f(2m+1)+2m+1\\
&= f(\lfloor n/2\rfloor+1)-f(\lfloor n/2\rfloor)+\lfloor n/2\rfloor
\end{array}
* If $n=4m+3$, then
\begin{array}{rl}
f(n+1)-f(n) & = f(2m+2)+f(2m+2)+\binom{2m+2}{2}+\binom{2m+2}{2}\\ & \qquad -(f(2m+2)+f(2m+1)+\binom{2m+2}{2}+\binom{2m+1}{2})\\
& = f(2m+2)-f(2m+1)+\binom{2m+2}{2}-\binom{2m+1}{2}\\
& = f(2m+2)-f(2m+1)+2m+1\\
&= f(\lfloor n/2\rfloor+1)-f(\lfloor n/2\rfloor)+\lfloor n/2\rfloor
\end{array}
This completes the proof.
In particular, this yields a new meaning and a new recurrence for sequence A174605 in Sloane's Encyclopedia.
§ EXPECTED VALUE OF $\PHI$ UNDER THE YULE MODEL
Let $\Phi_n$ be the random variable that chooses a tree $T\in \TB_n$ and computes its total cophenetic index $\Phi(T)$. In this section we determine the expected value of $\Phi_n$ under the Yule model.
To do this, we shall make use of the following lemma, which can be useful to study the expected value under the Yule model of other binary recursive tree shape statistics in the sense of [11].
Let $I$ be a mapping that associates to each phylogenetic tree a real number $\mathbb{R}$ satisfying the following two conditions:
(a) It is invariant under tree isomorphisms and relabelings of leaves.
(b) There exists a mapping $f:\NN\times \NN\to \RR$ such that, for every phylogenetic trees $T,T'$ on disjoint sets of taxa $S,S'$, respectively,
I(T\,\widehat{\ }\, T')=I(T)+I(T')+f(|S|,|S'|).
For every $n\geq 1$, let $I_n$ be the random variable that chooses a tree $T\in \TB_n$ and computes $I(T)$, and let $E_Y(I_n)$ be its expected value under the Yule model. Then,
E_Y(I_n)=\frac{1}{n-1}\Big(2\sum_{k=1}^{n-1} E_Y(I_k) + \sum_{k=1}^{n-1}f(k,n-k)\Big).
First of all, notice that if $T_k\in \TT(S_k)$, with $S_k\subsetneq \{1,\ldots,n\}$ with $|S_k|=k$, and $T'_{n-k}\in\TT( \{1,\ldots,n\}\setminus S_k)$, then
P_Y(T_k\widehat{\ }\, {}T'_{n-k})=\dfrac{2}{(n-1)\binom{n}{k}} P_Y(T_{k})P_Y(T'_{n-k})
where $P_Y$ denotes the probability of a phylogenetic tree under the Yule model. This assertion is a direct consequence of the explicit probabilities of $T_k$, $T'_{n-k}$ and $T_k\widehat{\ }\, {}T'_{n-k}$ under the Yule model given in 2.2, and the fact that $V_{int}(T_k\widehat{\ }\, {}T'_{n-k})=V_{int}(T_{k})\cup V_{int}(T'_{n-k})\cup\{r\}$ (where $r$ denotes the root of $T_k\widehat{\ }\, {}T'_{n-k}$), these unions being disjoint.
Let us compute now $E_Y(I_n)$ using its very definition:
\begin{array}{l}
E_Y(I_n) \displaystyle =\sum_{T\in \TB_n} I(T)\cdot p_Y(T)
\\
\quad \displaystyle =\sum_{k=1}^{n-1}\sum_{S_k\subsetneq\{1,\ldots,n\}\atop |S_k|=k}
\sum_{T_k\in \TB(S_k)}\sum_{T'_{n-k}\in \TB(S_k^c)}I(T_k\widehat{\ }\, {}T'_{n-k})\cdot p_Y(T_k\widehat{\ }\, {}T'_{n-k})
\\
\quad \displaystyle =\frac{1}{2} \sum_{k=1}^{n-1}\binom{n}{k}
\sum_{T_k\in \TB_k}\sum_{T'_{n-k} \in \TB_{n-k}}(I(T_k)+I(T'_{n-k})\\
\quad \displaystyle \qquad\qquad\qquad\qquad\qquad +f(k,n-k))\cdot \frac{2}{(n-1)\binom{n}{k}} P_Y(T_{k})P_Y(T'_{n-k})\\
\quad \displaystyle =\frac{1}{n-1}\sum_{k=1}^{n-1}
\sum_{T_k}\sum_{T'_{n-k}}(I(T_k)+I(T'_{n-k})+f(k,n-k))P_Y(T_{k})P_Y(T'_{n-k})
\\
\quad \displaystyle =\frac{1}{n-1}\sum_{k=1}^{n-1}
\Big( \sum_{T_k}\sum_{T'_{n-k} }I(T_k)P_Y(T_{k})P_Y(T'_{n-k}) \\
\quad \displaystyle\qquad\qquad\qquad\qquad + \sum_{T_k}\sum_{T'_{n-k} } I(T'_{n-k}) P_Y(T_{k})P_Y(T'_{n-k}) \\
\quad \displaystyle\qquad\qquad\qquad\qquad
+ \sum_{T_k}\sum_{T'_{n-k}} f(k,n-k)P_Y(T_{k})P_Y(T'_{n-k}) \Big)
\\
\quad \displaystyle =\frac{1}{n-1}\sum_{k=1}^{n-1}
\Big( \sum_{T_k} I(T_k)P_Y(T_{k}) + \sum_{T'_{n-k}} I(T'_{n-k}) P_Y(T'_{n-k}) + f(k,n-k) \Big)
\\
\quad \displaystyle =\frac{1}{n-1}\sum_{k=1}^{n-1}
(E_Y(I_k) + E_Y(I_{n-k}) + f(k,n-k))
\\
\quad \displaystyle =\frac{1}{n-1}\Big(2\sum_{k=1}^{n-1}
E_Y(I_k) + \sum_{k=1}^{n-1}f(k,n-k)\Big)
\end{array}
Under the Yule model, the expected value of $\Phi_n$ is
Lemma <ref> implies that $\Phi$ satisfies the hypothesis of Lemma <ref> with $f(k,n-k)=\binom{k}{2}+\binom{n-k}{2}$.
\sum_{k=1}^{n-1}f(k,n-k)=\sum_{k=1}^{n-1}\Big(\binom{k}{2}+\binom{n-k}{2}\Big)=
2\sum_{k=1}^{n-1} \binom{k}{2},
and hence
E_Y(\Phi_k) + \sum_{k=1}^{n-1}\binom{k}{2}\Big).
\begin{array}{rl}
E_Y(\Phi_n) & \displaystyle =
\frac{2}{n-1} \Big(\sum_{k=1}^{n-1} E_Y(\Phi_k) + \sum_{k=1}^{n-1}\binom{k}{2}\Big) \\
& \displaystyle =
\frac{2}{n-1} E_Y(\Phi_{n-1})+\frac{n-2}{n-1}\cdot \frac{2}{n-2} \sum_{k=1}^{n-2} E_Y(\Phi_k)\\
& \displaystyle\qquad\qquad\qquad
+\frac{2}{n-1}\binom{n-1}{2}+\frac{n-2}{n-1}\cdot \frac{2}{n-2} \sum_{k=1}^{n-2} \binom{k}{2}
\\
& \displaystyle =
\frac{2}{n-1} E_Y(\Phi_{n-1})+\frac{n-2}{n-1}E_Y(\Phi_{n-1})+\frac{2}{n-1}\binom{n-1}{2}\\
& \displaystyle
= \frac{n}{n-1} E_Y(\Phi_{n-1})+n-2
\end{array}
To solve this equation, rewrite it as
\frac{1}{n}E_Y(\Phi_n)=\frac{1}{n-1} E_Y(\Phi_{n-1})+\frac{n-2}{n}
Setting $x_n=E_Y(\Phi_n)/n$, the sequence $(x_n)_n$ satisfies
x_n=x_{n-1}+\frac{n-2}{n}\mbox{, starting with $x_2=0$}.
x_n=\sum_{i=3}^n \frac{i-2}{i}=(n-2)-2\sum_{i=3}^n\frac{1}{i}=(n-1)-2\sum_{i=2}^n\frac{1}{i}
and thus, finally,
Let $S_n$ stand for the random variable that chooses a tree $T_n\in \TB_n$ and computes its Sackin index $S(T_n)$; cf. 2.2.
Notice that, since $E_Y(S_n)=2n\sum\limits_{j=2}^{n}1/j$ [10], we have that
We have not been able to find a direct reason for this equality.
$E_Y(\Phi_n)=n^2+(1-2 \gamma)n -2 n \ln(n)+o(n)$.
So, the order $O(n^2)$ of the expected value under the Yule model of the total cophenetic index on $\BT_n$ is larger than the order $O(n\log(n))$ of the expected values of Sackin's and Colless' indices [3].
From the expected values of the Sackin and the total cophenetic indices, we can deduce the expected value of the total area $D$ on $\TB_n$ under the Yule model.
Let $D_n$ be the random variable that chooses a tree $T\in \TB_n$ and computes its total area $D(T)$. Under the Yule model, its expected value is
From Lemma <ref> we deduce that
and therefore
E_Y(D_n =(n-1)E_Y(S_n)-2E_Y(\Phi_n) =2n(n+1)\sum_{i=2}^n\frac{1}{i}-2n(n-1).
In <cit.>, it is claimed that
which cannot be correct: since all three trees $T\in \BT_3$ have $D(T)=8$, it must happen that $E_Y(D_3)=8$, while the expression given in loc. cit. yields
$E_Y(D_3)=5$. And incidentally, our formula does yield the correct value in this case.
§ EXPECTED VALUE OF $\PHI$ UNDER THE UNIFORM MODEL
In this section we determine the expected value of $\Phi_n$ under the uniform model.
This expected value of $\Phi_n$ will be easily deduced, through Lemma <ref>, from the expected value of the total area, which was obtained in [12], and the expected value of the Sackin index, which we obtain in Theorem <ref> below. This last formula is, to our knowledge, new.
Since, under the uniform model, all trees in $\TT_n$ have the same probability, $1/(2n-3)!!$, the expected value of $S_n$ under the uniform model is
E_U(S_n)=\frac{\sum_{T\in \TB_n} S(T)}{(2n-3)!!}.
So, we need to compute the numerator in this fraction.
For every $n\geq 3$, $\displaystyle \sum_{T\in \TB_n} S(T)=n\sum_{k=1}^{n-1} \dfrac{(2n-k-3)!k^2}{(n-k-1)!2^{n-k-1}}$.
For every $k=1,\ldots,n-1$, let
c_{k,n}=|\{ T\in \TB_n\mid \delta_T(1)=k\}|=|\{ T\in \TT_n\mid \delta_T(i)=k\}|\mbox{ for every $1\leq i\leq n$}.
\begin{array}{rl}
\displaystyle \sum_{T\in \TB_n} S(T) & \displaystyle =\sum_{T\in \TB_n}\sum_{i=1}^{n} \delta_T(i) =\sum_{i=1}^{n}\sum_{T\in \TB_n} \delta_T(i)\\
& \displaystyle =\sum_{i=1}^{n}\sum_{k=1}^{n-1} k\cdot |\{ T\in \TT_n\mid \delta_T(i)=k\}|\\
& \displaystyle =\sum_{i=1}^{n}\sum_{k=1}^{n-1} k\cdot |\{ T\in \TT_n\mid \delta_T(1)=k\}| =n\sum_{k=1}^{n-1} k\cdot c_{k,n}.
\end{array}
It remains to compute $c_{k,n}$ for $k\geq 1$. To do so, notice that every tree $T\in \TB_n$ such that $\delta(1)=k$ will have the form described in Fig. <ref>. Therefore, it is determined by the ordered $k$-forest $T_1,T_2,\ldots,T_{k}$ on $\{2,\ldots,n\}$, and thus
c_{k,n}=|\FF_{k,n-1}|= \frac{(2n-k-3)!k}{(n-k-1)!2^{n-k-1}},
from which the expression in the statement follows.
(-1,3) node[tre] (1) ; 1
(2,4) node[tre] (x1) ;
(3.3,5.3) node .;
(3.6,5.6) node .;
(3.9,5.9) node .;
(5,7) node[tre] (x2) ;
(7,9) node[tre] (r) ;
(5,3) node[tre] (tk) ;
(5,1) node $T_{k}$;
(8,6) node[tre] (t2) ;
(8,4) node $T_{2}$;
(10,8) node[tre] (t1) ;
(10,6) node $T_{1}$;
The structure of a tree $T$ with $\delta_T(1)=k$.
Now, recall that the (generalized) hypergeometric function ${}_pF_q$ is defined [1] as
{}_pF_q\bigg(\begin{array}{rrr} a_1,&\ldots, &a_p \\[-0.5ex] b_1,& \ldots, &b_q\end{array};z\bigg)=\sum_{k\geq 0} \frac{(a_1)_k\cdots (a_p)_k}{(b_1)_k\cdots (b_q)_k}\cdot \frac{z^k}{k!},
where $(a)_k := a\cdot (a+1)\cdots (a+k-1)$. Many popular software systems, like Mathematica or R, have implementations of these functions.
The expected value of the random variable $S_n$ under the uniform model is
E_U(S_n)=\frac{n}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array};2\bigg)
By the last lemma, we have that
E_U(S_n)=\frac{\sum_{T\in \TB_n} S(T)}{(2n-3)!!}=\frac{n}{(2n-3)!!} \sum_{k=1}^{n-1} \frac{(2n-k-3)!k^2}{(n-k-1)!2^{n-k-1}}.
\begin{array}{rl}
\displaystyle \frac{nk^2(2n-k-3)!}{(2n-3)!!(n-k-1)!2^{n-k-1}} & \displaystyle =
\frac{nk^2(2n-k-3)!2^{n-2}(n-2)!}{(2n-3)!(n-k-1)!2^{n-k-1}}\\[2ex] & \displaystyle=
\frac{nk^2(2n-k-3)!2^{n-2}(n-2)!k!}{(2n-3)!(n-k-1)!2^{n-k-1}k!} \\[2ex] & \displaystyle=
\frac{nk^22^{k-1}\binom{n-1}{k}}{(n-1)\binom{2n-3}{k}}
\end{array}
and thus
\begin{array}{l}
E_U(S_n) = \displaystyle
\frac{n}{n-1}\sum_{k=1}^{n-1} k^22^{k-1}\cdot \frac{\binom{n-1}{k}}{\binom{2n-3}{k}}\\
\qquad = \displaystyle\frac{n}{n-1} \sum_{k=1}^{n-1} \frac{k^22^{k-1}(n-1)(n-2)(n-3) \cdots (n-k)}{(2n-3)(2n-4)(2n-5)\cdots (2n-k-2)}\\
\qquad = \displaystyle\frac{n}{2n-3} \sum_{k=1}^{n-1} \frac{k^22^{k-1}(n-2)(n-3) \cdots (n-k)}{(2n-4)(2n-5)\cdots (2n-k-2)}\\
\qquad = \displaystyle\frac{n}{2n-3} \sum_{k=1}^{n-1} \frac{k^22^{k-1}(2-n)(2-n+1)\cdots (-n+k)}{(4-2n)(4-2n+1)\cdots (2-2n+k)}\\
\qquad = \displaystyle\frac{n}{2n-3} \sum_{k=0}^{n-2} \frac{(k+1)^22^k(2-n)(2-n+1)\cdots (1-n+k)}{(4-2n)(4-2n+1)\cdots (3-2n+k)}\\
\qquad = \displaystyle\frac{n}{2n-3} \sum_{k\geq 0} \frac{((k+1)!)^2(2-n)(2-n+1)\cdots(1-n+k)\cdot 2^k}{(k!)^2(4-2n)(4-2n+1)\cdots (3-2n+k)}\\
\qquad = \displaystyle\frac{n}{2n-3} \sum_{k\geq 0} \frac{(2)_k(2)_k(2-n)_k}{(1)_k(4-2n)_k}\cdot \frac{2^k}{k!} =\dfrac{n}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[ -0.5ex] 1,\ 4-2n\end{array};2\bigg)
\end{array}
as we claimed.
We have now the following result.
Under the uniform model, the expected value of $\Phi_n$ is
E_U(\Phi_n)=\binom{n}{2}\Bigg(\frac{1}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array}; 2\bigg)-\frac{1}{2}\cdot \frac{(2n-2)!!}{(2n-3)!!}\Bigg)\sim \frac{\sqrt{\pi}}{4}n^{5/2}
The expected values under the uniform model of $S_n$ and $D_n$ are:
\begin{array}{ll}
\displaystyle E_U(S_n)=\frac{n}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array};2\bigg)& \quad \mbox{ by Theorem \ref{th:sackin}}
\\
\displaystyle E_U(D_n)=\binom{n}{2} \cdot\frac{(2n-2)!!}{(2n-3)!!}& \quad \mbox{\cite {MirR10}}
\end{array}
Then, by Lemma <ref>,
\begin{array}{rl}
E_U(\Phi) & \displaystyle =\frac{n-1}{2}E_U(S_n)-\frac{1}{2}E_U(D_n)\\ & \displaystyle =
\frac{n-1}{2}\cdot \frac{n}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array};2\bigg)
-\frac{1}{2}\binom{n}{2} \cdot\frac{(2n-2)!!}{(2n-3)!!}\\
& \displaystyle =
\binom{n}{2}\Bigg(\frac{1}{2n-3} {}_3 F_2
\bigg(\begin{array}{l} 2,\ 2,\ 2-n \\[-0.5ex] 1,\ 4-2n\end{array}; 2\bigg)-\frac{1}{2}\cdot \frac{(2n-2)!!}{(2n-3)!!}\Bigg)
\end{array}
The assertion $E_U(\Phi_n) \sim \frac{\sqrt{\pi}}{4}n^{5/2}$ comes easily from Lemma <ref>, and the facts that
$E(S_n) \sim \sqrt{\pi}n^{3/2}$ [3], and that, using Stirling's approximation for large factorials,
$E_U(D_n)\sim \frac{\sqrt{\pi}}{2}n^{5/2}$ <cit.>.
§ DISCUSSION AND CONCLUSIONS
In this paper we have introduced a new balance index for phylogenetic trees, the total cophenetic index $\Phi$. This index makes sense for arbitrary phylogenetic trees, it can be computed in linear time, and it has a larger range of values than Sackin's or Colless' indices. We have computed its maximum and minimum values for binary and arbitrary phylogenetic trees, and its expected value under the Yule and the uniform models. In a future work we plan to study other statistical properties of $\Phi$, like its variance, its limiting distribution or its correlation to other balance indices.
From the point of view of the measurement of the degree of symmetry of a tree, our index outperforms the resolution power of Sackin's and Colless' indices. We already saw some hints of this property in the previous sections: for instance, in Theorem <ref>, where we proved that the only trees $T\in \TB_n$ that have minimum $\Phi(T)$ are the maximally balanced, something that is not true in general for Sackin's and Colless' indices (recall Fig. <ref>); or in the lemmas previous to the proof of this theorem, where we saw that any interchange of cousins that modifies the balance of their grandparent also modifies the value of $\Phi$. As a further evidence of this greater resolution power, we have estimated the probability that a pair of trees $T_1,T_2\in \BT_n$ have $I(T_1)=I(T_2)$, for $I=C,S,\Phi$. To do so, for every $n=2,\ldots,10^4$ we have chosen randomly a number $N$ of pairs of trees in $\TB_n$ (for the first few values of $n$, $N$ was taken to be $|\TB_n|$, but starting at $n=8$, we took $N=3000$), and computed, for $I=C,S,\Phi$,
\hat{p}_n(I)=\frac{\mbox{number of pairs $(T_1,T_2)$ with $n$ leaves such that $I(T_1)=I(T_2)$}}{N}.
Fig. <ref> summarizes the results. It plots $\log(\hat{p}_n(I))$ for the three balance indices as a function of $\log(n)$. We can see that that the total cophenetic index has the lowest such estimated probability of a tie. We plan to perform a deeper study of the probability of ties for the different balance indices in a future paper.
$\leftarrow$ Sackin
$\leftarrow$ Colless
$\leftarrow$ $\Phi$
Log-log plot of the estimated probability of a tie for three balance indices.
This greater resolution power of $\Phi$ makes it a better candidate to be used to test evolutionary hypotheses.
We have performed a preliminary such test on the TreeBASE database [15]. We have considered the numbers $n$ of leaves for which the TreeBASE contains at least 20 binary phylogenetic trees with $n$ leaves, and for each such $n$ we have computed the mean of the total cophenetic indices of the corresponding binary trees. Fig. <ref> plots the log of these means as a function of $\log(n)$. We have added the curves of the log of the expected values of $\Phi_n$ under the Yule distribution (lower curve) and under the uniform distribution (upper curve), again as a function of $\log(n)$. This figure shows that the total cophenetic indices of the binary phylogenetic trees in TreeBASE are better explained by the uniform model than by the Yule model. We also plan to report in a future paper on more extensive tests on stochastic models of evolutionary processes using the total cophenetic index.
Log-log plots of the mean of the total cophenetic index of the binary trees in TreeBASE with a fixed number $n$ of leaves, of $E_Y(\Phi_n)$ (lower curve) and $E_U(\Phi_n)$ (upper curve).
§ ACKNOWLEDGEMENTS
The research reported in this paper has been partially supported by the Spanish government and the UE FEDER program, through projects MTM2009-07165 and TIN2008-04487-E/TIN. We thank G. Cardona, E. Hernández-García, and J. Miró for several comments on previous versions of this work.
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[20] D. E. Rosen, Vicariant Patterns and Historical Explanation in Biogeography.
Syst. Biol. 27 (1978), 159–188.
[21] M. J. Sackin, “Good” and “bad” phenograms. Sys. Zool, 21 (1972), 225–226.
[22] K.T. Shao, R. Sokal, Tree balance. Sys. Zool, 39 (1990), 226–276.
[23] R. Sokal, F. Rohlf,
The Comparison of Dendrograms by Objective Methods.
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[24] M. Steel, A. McKenzie, Distributions of cherries for two models of trees. Math. Biosc. 164 (2000), 81–92.
[25] M. Steel, A. McKenzie, Properties of phylogenetic trees generated by Yule-type speciation models. Math. Biosc. 170 (2001), 91–112.
[26]
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|
arxiv-papers
| 2012-02-06T17:47:01 |
2024-09-04T02:49:27.081698
|
{
"license": "Public Domain",
"authors": "Arnau Mir, Francesc Rossello, Lucia Rotger",
"submitter": "Francesc Rossell\\'o",
"url": "https://arxiv.org/abs/1202.1223"
}
|
1202.1234
|
# The road to deterministic matrices with the restricted isometry property
Afonso S. Bandeira Program in Applied and Computational Mathematics,
Princeton University, Princeton, New Jersey 08544, USA; E-mail:
ajsb@math.princeton.edu , Matthew Fickus Department of Mathematics and
Statistics, Air Force Institute of Technology, Wright-Patterson Air Force
Base, OH 45433, USA; matthew.fickus@afit.edu , Dustin G. Mixon Program in
Applied and Computational Mathematics, Princeton University, Princeton, New
Jersey 08544, USA; E-mail: dmixon@princeton.edu and Percy Wong Program in
Applied and Computational Mathematics, Princeton University, Princeton, New
Jersey 08544, USA; E-mail: pakwong@math.princeton.edu
###### Abstract.
The restricted isometry property (RIP) is a well-known matrix condition that
provides state-of-the-art reconstruction guarantees for compressed sensing.
While random matrices are known to satisfy this property with high
probability, deterministic constructions have found less success. In this
paper, we consider various techniques for demonstrating RIP deterministically,
some popular and some novel, and we evaluate their performance. In evaluating
some techniques, we apply random matrix theory and inadvertently find a simple
alternative proof that certain random matrices are RIP. Later, we propose a
particular class of matrices as candidates for being RIP, namely, equiangular
tight frames (ETFs). Using the known correspondence between real ETFs and
strongly regular graphs, we investigate certain combinatorial implications of
a real ETF being RIP. Specifically, we give probabilistic intuition for a new
bound on the clique number of Paley graphs of prime order, and we conjecture
that the corresponding ETFs are RIP in a manner similar to random matrices.
###### Key words and phrases:
restricted isometry property, compressed sensing, equiangular tight frames
###### 2000 Mathematics Subject Classification:
Primary: 15A42. Secondary: 05E30, 15B52, 60F10, 94A12
The authors thank Prof. Peter Sarnak and Joel Moreira for insightful
discussions and helpful suggestions. ASB was supported by NSF Grant No.
DMS-0914892, MF was supported by NSF Grant No. DMS-1042701 and AFOSR Grant
Nos. F1ATA01103J001 and F1ATA00183G003, and DGM was supported by the A.B.
Krongard Fellowship. The views expressed in this article are those of the
authors and do not reflect the official policy or position of the United
States Air Force, Department of Defense, or the U.S. Government.
## 1\. Introduction
Let $x$ be an unknown $N$-dimensional vector with the property that at most
$K$ of its entries are nonzero, that is, $x$ is $K$_-sparse_. The goal of
compressed sensing is to construct relatively few non-adaptive linear
measurements along with a stable and efficient reconstruction algorithm that
exploits this sparsity structure. Expressing each measurement as a row of an
$M\times N$ matrix $\Phi$, we have the following noisy system:
$y=\Phi x+z.$ (1)
In the spirit of _compressed_ sensing, we only want a few measurements: $M\ll
N$. Also, in order for there to exist an inversion process for (1), $\Phi$
must map $K$-sparse vectors injectively, or equivalently, every subcollection
of $2K$ columns of $\Phi$ must be linearly independent. Unfortunately, the
natural reconstruction method in this general case, i.e., finding the sparsest
approximation of $y$ from the dictionary of columns of $\Phi$, is known to be
NP-hard [21]. Moreover, the independence requirement does not impose any sort
of dissimilarity between columns of $\Phi$, meaning distinct identity basis
elements could lead to similar measurements, thereby bringing instability in
reconstruction.
To get around the NP-hardness of sparse approximation, we need more structure
in the matrix $\Phi$. Instead of considering linear independence of all
subcollections of $2K$ columns, it has become common to impose a much stronger
requirement: that every submatrix of $2K$ columns of $\Phi$ be well-
conditioned. To be explicit, we have the following definition:
###### Definition 1.
The matrix $\Phi$ has the _$(K,\delta)$ -restricted isometry property (RIP)_
if
$(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}\leq(1+\delta)\|x\|^{2}$
for every $K$-sparse vector $x$. The smallest $\delta$ for which $\Phi$ is
$(K,\delta)$-RIP is the _restricted isometry constant (RIC)_ $\delta_{K}$.
In words, matrices which satisfy RIP act as a near-isometry on sufficiently
sparse vectors. Note that a $(2K,\delta)$-RIP matrix with $\delta<1$
necessarily has that all subcollections of $2K$ columns are linearly
independent. Also, the well-conditioning requirement of RIP forces
dissimilarity in the columns of $\Phi$ to provide stability in reconstruction.
Most importantly, the additional structure of RIP allows for the possibility
of getting around the NP-hardness of sparse approximation. Indeed, a
significant result in compressed sensing is that RIP sensing matrices enable
efficient reconstruction:
###### Theorem 2 (Theorem 1.3 in [8]).
Suppose an $M\times N$ matrix $\Phi$ has the $(2K,\delta)$-restricted isometry
property for some $\delta<\sqrt{2}-1$. Assuming $\|z\|\leq\varepsilon$, then
for every $K$-sparse vector $x\in\mathbb{R}^{N}$, the following reconstruction
from (1):
$\tilde{x}=\arg\min\|\hat{x}\|_{1}\qquad\mbox{s.t.
}\|y-\Phi\hat{x}\|\leq\varepsilon$
satisfies $\|\tilde{x}-x\|\leq C\varepsilon$, where $C$ only depends on
$\delta$.
The fact that RIP sensing matrices convert an NP-hard reconstruction problem
into an $\ell_{1}$-minimization problem has prompted many in the community to
construct RIP matrices. Among these constructions, the most successful have
been random matrices, such as matrices with independent Gaussian or Bernoulli
entries [4], or matrices whose rows were randomly selected from the discrete
Fourier transform matrix [25]. With high probability, these random
constructions support sparsity levels $K$ on the order of
$\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$. Intuitively, this
level of sparsity is near-optimal because $K$ cannot exceed
$\smash{\frac{M}{2}}$ by the linear independence condition. Unfortunately, it
is difficult to check whether a particular instance of a random matrix is
$(K,\delta)$-RIP, as this involves the calculation of singular values for all
$\smash{\binom{N}{K}}$ submatrices of $K$ columns of the matrix. For this
reason, and for the sake of reliable sensing standards, many have become
interested in finding deterministic RIP matrix constructions.
In the next section, we review the well-understood techniques that are
commonly used to analyze the restricted isometry of deterministic
constructions: the Gershgorin circle theorem, and the _spark_ of a matrix.
Unfortunately, neither technique demonstrates RIP for sparsity levels as large
as what random constructions are known to support; rather, with these
techniques, a deterministic $M\times N$ matrix $\Phi$ can only be shown to
have RIP for sparity levels on the order of $\sqrt{M}$. This limitation has
become known as the “square-root bottleneck,” and it poses an important
problem in matrix design [30].
To date, the only deterministic construction that manages to go beyond this
bottleneck is given by Bourgain et al. [7]; in Section 3, we discuss what they
call _flat RIP_ , which is the technique they use to demonstrate RIP. It is
important to stress the significance of their contribution: Before [7], it was
unclear how deterministic analysis might break the bottleneck, and as such,
their result is a major theoretical achievement. On the other hand, their
improvement over the square-root bottleneck is notably slight compared to what
random matrices provide. However, by our Theorem 14, their technique can
actually be used to demonstrate RIP for sparsity levels much larger than
$\sqrt{M}$, meaning one could very well demonstrate random-like performance
given the proper construction. Our result applies their technique to random
matrices, and it inadvertently serves as a simple alternative proof that
certain random matrices are RIP. In Section 4, we introduce an alternate
technique, which by our Theorem 17, can also demonstrate RIP for large
sparsity levels.
After considering the efficacy of these techniques to demonstrate RIP, it
remains to find a deterministic construction that is amenable to analysis. To
this end, we discuss various properties of a particularly nice matrix which
comes from _frame theory_ , called an _equiangular tight frame (ETF)_.
Specifically, real ETFs can be characterized in terms of their Gram matrices
using strongly regular graphs [32]. By applying the techniques of Sections 3
and 4 to real ETFs, we derive equivalent combinatorial statements in graph
theory. By focussing on the ETFs which correspond to Paley graphs of prime
order, we are able to make important statements about their clique numbers and
provide some intuition for an open problem in number theory. We conclude by
conjecturing that the Paley ETFs are RIP in a manner similar to random
matrices.
## 2\. Well-understood techniques
### 2.1. Applying Gershgorin’s circle thoerem
Take an $M\times N$ matrix $\Phi$. For a given $K$, we wish to find some
$\delta$ for which $\Phi$ is $(K,\delta)$-RIP. To this end, it is useful to
consider the following expression for the restricted isometry constant:
$\delta_{K}=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K}\|_{2}.$
(2)
Here, $\Phi_{\mathcal{K}}$ denotes the submatrix consisting of columns of
$\Phi$ indexed by $\mathcal{K}$. Note that we are not tasked with actually
computing $\delta_{K}$; rather, we recognize that $\Phi$ is $(K,\delta)$-RIP
for every $\delta\geq\delta_{K}$, and so we seek an upper bound on
$\delta_{K}$. The following classical result offers a particularly easy-to-
calculate bound on eigenvalues:
###### Theorem 3 (Gershgorin circle theorem [17]).
For each eigenvalue $\lambda$ of a $K\times K$ matrix $A$, there is an index
$i\in\\{1,\ldots,K\\}$ such that
$\Big{|}\lambda-A[i,i]\Big{|}\leq\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{K}\Big{|}A[i,j]\Big{|}.$
To use this theorem, take some $\Phi$ with unit-norm columns. Note that
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is the Gram matrix of the columns
indexed by $\mathcal{K}$, and as such, the diagonal entries are $1$, and the
off-diagonal entries are inner products between distinct columns of $\Phi$.
Let $\mu$ denote the worst-case coherence of
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$:
$\mu:=\max_{\begin{subarray}{c}i,j\in\\{1,\ldots,N\\}\\\ i\neq
j\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|.$
Then the size of each off-diagonal entry of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is $\leq\mu$, regardless of our
choice for $\mathcal{K}$. Therefore, for every eigenvalue $\lambda$ of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K}$, the Gershgorin
circle theorem gives
$|\lambda|=|\lambda-0|\leq\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{K}|\langle\varphi_{i},\varphi_{j}\rangle|\leq(K-1)\mu.$ (3)
Since (3) holds for every eigenvalue $\lambda$ of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K}$ and every choice of
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$, we conclude from (2) that
$\delta_{K}\leq(K-1)\mu$, i.e., $\Phi$ is $(K,(K-1)\mu)$-RIP. This process of
using the Gershgorin circle theorem to demonstrate RIP for deterministic
constructions has become standard in the community [3, 14, 16].
Recall that random RIP constructions support sparsity levels $K$ on the order
of $\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$. To see how well
the Gershgorin circle theorem demonstrates RIP, we need to express $\mu$ in
terms of $M$ and $N$. To this end, we consider the following result:
###### Theorem 4 (Welch bound [33]).
Every $M\times N$ matrix with unit-norm columns has worst-case coherence
$\mu\geq\sqrt{\frac{N-M}{M(N-1)}}.$
To use this result, we consider matrices whose worst-case coherence achieves
equality in the Welch bound. These are known as equiangular tight frames [29],
which can be defined as follows:
###### Definition 5.
A matrix is said to be an _equiangular tight frame (ETF)_ if
* (i)
the columns have unit norm,
* (ii)
the rows are orthogonal with equal norm, and
* (iii)
the inner products between distinct columns are equal in modulus.
To date, there are three general constructions that build several families of
ETFs [16, 32, 34]. Since ETFs achieve equality in the Welch bound, we can
further analyze what it means for an $M\times N$ ETF $\Phi$ to be
$(K,(K-1)\mu)$-RIP. In particular, since Theorem 2 requires that $\Phi$ be
$(2K,\delta)$-RIP for $\delta<\sqrt{2}-1$, it suffices to have
$\smash{\frac{2K}{\sqrt{M}}<\sqrt{2}-1}$, since this implies
$\delta=(2K-1)\mu=(2K-1)\sqrt{\frac{N-M}{M(N-1)}}\leq\frac{2K}{\sqrt{M}}<\sqrt{2}-1.$
(4)
That is, ETFs form sensing matrices that support sparsity levels $K$ on the
order of $\sqrt{M}$. Most other deterministic constructions have identical
bounds on sparsity levels [3, 14, 16]. In fact, since ETFs minimize coherence,
they are necessarily optimal constructions in terms of the Gershgorin
demonstration of RIP, but the question remains whether they are actually RIP
for larger sparsity levels; the Gershgorin demonstration fails to account for
cancellations in the sub-Gram matrices
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$, and so this technique is too weak
to indicate either possibility.
### 2.2. Spark considerations
Recall that, in order for an inversion process for (1) to exist, $\Phi$ must
map $K$-sparse vectors injectively, or equivalently, every subcollection of
$2K$ columns of $\Phi$ must be linearly independent. This linear independence
condition can be nicely expressed in more general terms, as the following
definition provides:
###### Definition 6.
The _spark_ of a matrix $\Phi$ is the size of the smallest linearly dependent
subset of columns, i.e.,
$\mathrm{Spark}(\Phi)=\min\Big{\\{}\|x\|_{0}:Fx=0,~{}x\neq 0\Big{\\}}.$
This definition was introduced by Dohono and Elad [15] to help build a theory
of sparse representation that later gave birth to modern compressed sensing.
The concept of spark is also found in matroid theory, where it goes by the
name _girth_ [1]. The condition that every subcollection of $2K$ columns of
$\Phi$ is linearly independent is equivalent to $\mathrm{Spark}(\Phi)>2K$.
Relating spark to RIP, suppose $\Phi$ is $(K,\delta)$-RIP with
$\mathrm{Spark}(\Phi)\leq K$. Then there exists a nonzero $K$-sparse vector
$x$ such that
$(1-\delta)\|x\|^{2}\leq\|\Phi x\|^{2}=0,$
and so $\delta\geq 1$. The reason behind this stems from our necessary linear
independence condition: RIP implies linear independence, and so small spark
implies linear dependence, which in turn implies not RIP.
As an example of using spark to analyze RIP, we now consider a construction
that dates back to Seidel [27], and was recently developed further in [16].
Here, a special type of block design is used to build an ETF. Let’s start with
a definition:
###### Definition 7.
A $(t,k,v)$-_Steiner system_ is a $v$-element set $V$ with a collection of
$k$-element subsets of $V$, called _blocks_ , with the property that any
$t$-element subset of $V$ is contained in exactly one block. The
$\\{0,1\\}$-_incidence matrix_ $A$ of a Steiner system has entries $A[i,j]$,
where $A[i,j]=1$ if the $i$th block contains the $j$th element, and otherwise
$A[i,j]=0$.
One example of a Steiner system is a set with all possible two-element blocks.
This forms a $(2,2,v)$-Steiner system because every pair of elements is
contained in exactly one block. The following theorem details how to construct
ETFs using Steiner systems.
###### Theorem 8 (Theorem 1 in [16]).
Every $(2,k,v)$-Steiner system can be used to build a
$\smash{\frac{v(v-1)}{k(k-1)}\times v(1+\frac{v-1}{k-1})}$ equiangular tight
frame $\Phi$ according the following procedure:
* (i)
Let $A$ be the $\frac{v(v-1)}{k(k-1)}\times v$ incidence matrix of a
$(2,k,v)$-Steiner system.
* (ii)
Let $H$ be a $(1+\frac{v-1}{k-1})\times(1+\frac{v-1}{k-1})$ (possibly complex)
Hadamard matrix.
* (iii)
For each $j=1,\ldots,v$, let $\Phi_{j}$ be a
$\frac{v(v-1)}{k(k-1)}\times(1+\frac{v-1}{k-1})$ matrix obtained from the
$j$th column of $A$ by replacing each of the one-valued entries with a
distinct row of $H$, and every zero-valued entry with a row of zeros.
* (iv)
Concatenate and rescale the $\Phi_{j}$’s to form
$\Phi=(\frac{k-1}{v-1})^{\frac{1}{2}}[\Phi_{1}\cdots\Phi_{v}]$.
As an example, we build an ETF from a (2,2,4)-Steiner system. In this case, we
make use of the corresponding incidence matrix $A$ along with a $4\times 4$
Hadamard matrix $H$:
$A=\left[\begin{array}[]{cccc}+&+&&\\\ +&&+&\\\ +&&&+\\\ &+&+&\\\ &+&&+\\\
&&+&+\end{array}\right],\qquad H=\left[\begin{array}[]{cccc}+&+&+&+\\\
+&-&+&-\\\ +&+&-&-\\\ +&-&-&+\end{array}\right].$
In both of these matrices, pluses represent $1$’s, minuses represent $-1$’s,
and blank spaces represent $0$’s. For the matrix $A$, each row represents a
block. Since each block contains two elements, each row of the matrix has two
ones. Also, any two elements determines a unique common row, and so any two
columns have a single one in common. To form the corresponding $6\times 16$
ETF $\Phi$, we replace the three ones in each column of $A$ with the second,
third, and fourth rows of $H$. Normalizing the columns gives the following
$6\times 16$ ETF:
$\Phi=\frac{1}{\sqrt{3}}\left[\begin{array}[]{cccccccccccccccc}+&-&+&-&+&-&+&-&&&&&&&&\\\
+&+&-&-&&&&&+&-&+&-&&&&\\\ +&-&-&+&&&&&&&&&+&-&+&-\\\
&&&&+&+&-&-&+&+&-&-&&&&\\\ &&&&+&-&-&+&&&&&+&+&-&-\\\
&&&&&&&&+&-&-&+&+&-&-&+\end{array}\right].$ (5)
It is easy to verify that $\Phi$ satisfies Definition 5. Several infinite
families of $(2,k,v)$-Steiner systems are already known, and Theorem 8 says
that each one can be used to build a different ETF. Recall from the previous
subsection that Steiner ETFs, being ETFs, are optimal constructions in terms
of the Gershgorin demonstration of RIP. We now use the notion of spark to
further analyze Steiner ETFs. Specifically, note that the first four columns
in (5) are linearly dependent. As such, $\mathrm{Spark}(\Phi)\leq 4$. In
general, the spark of a Steiner ETF is
$\smash{\leq\frac{v-1}{k-1}\leq\sqrt{2M}}$ (see Theorem 3 of [16] and
discussion thereafter), and so having $K$ on the order of $\sqrt{M}$ is
_necessary_ for a Steiner ETF to be $(K,\delta)$-RIP for some $\delta<1$. This
answers the closing question of the previous subsection: in general, ETFs are
not RIP for sparsity levels larger than the order of $\sqrt{M}$. This
contrasts with random constructions, which support sparsity levels as large as
the order of $\smash{\frac{M}{\log^{\alpha}N}}$ for some $\alpha\geq 1$. That
said, are there techniques to demonstrate that certain deterministic matrices
are RIP for sparsity levels larger than the order of $\sqrt{M}$?
## 3\. Flat restricted orthogonality
In [7], Bourgain et al. provided a deterministic construction of $M\times N$
RIP matrices that support sparsity levels $K$ on the order of
$M^{1/2+\varepsilon}$ for some small value of $\varepsilon$. To date, this is
the only known deterministic RIP construction that breaks the so-called
“square-root bottleneck.” In this section, we analyze their technique for
demonstrating RIP, but first, we provide some historical context. We begin
with a definition:
###### Definition 9.
The matrix $\Phi$ has _$(K,\theta)$ -restricted orthogonality (RO)_ if
$|\langle\Phi x,\Phi y\rangle|\leq\theta\|x\|\|y\|$
for every pair of $K$-sparse vectors $x,y$ with disjoint support. The smallest
$\theta$ for which $\Phi$ has $(K,\theta)$-RO is the _restricted orthogonality
constant (ROC)_ $\theta_{K}$.
In the past, restricted orthogonality was studied to produce reconstruction
performance guarantees for both $\ell_{1}$-minimization and the Dantzig
selector [9, 10]. Intuitively, restricted orthogonality is important to
compressed sensing because any stable inversion process for (1) would require
$\Phi$ to map vectors of disjoint support to particularly dissimilar
measurements. For the present paper, we are interested in upper bounds on
RICs; in this spirit, the following result illustrates some sort of
equivalence between RICs and ROCs:
###### Lemma 10 (Lemma 1.2 in [9]).
$\theta_{K}\leq\delta_{2K}\leq\theta_{K}+\delta_{K}$.
To be fair, the above upper bound on $\delta_{2K}$ does not immediately help
in estimating $\delta_{2K}$, as it requires one to estimate $\delta_{K}$.
Certainly, we may iteratively apply this bound to get
$\delta_{2K}\leq\theta_{K}+\theta_{\lceil K/2\rceil}+\theta_{\lceil
K/4\rceil}+\cdots+\theta_{1}+\delta_{1}\leq(1+\lceil\log_{2}K\rceil)\theta_{K}+\delta_{1}.$
(6)
Note that $\delta_{1}$ is particularly easy to calculate:
$\delta_{1}=\max_{n\in\\{1,\ldots,N\\}}\Big{|}\|\varphi_{n}\|^{2}-1\Big{|},$
which is zero when the columns of $\Phi$ have unit norm. In pursuit of a
better upper bound on $\delta_{2K}$, we use techniques from [7] to remove the
log factor from (6):
###### Lemma 11.
$\delta_{2K}\leq 2\theta_{K}+\delta_{1}$.
###### Proof.
Given a matrix $\Phi=[\varphi_{1}\cdots\varphi_{N}]$, we want to upper-bound
the smallest $\delta$ for which $(1-\delta)\|x\|^{2}\leq\|\Phi
x\|^{2}\leq(1+\delta)\|x\|^{2}$, or equivalently:
$\delta\geq\Big{|}\|\Phi\tfrac{x}{\|x\|}\|^{2}-1\Big{|}$ (7)
for every nonzero $2K$-sparse vector $x$. We observe from (7) that we may take
$x$ to have unit norm without loss of generality. Letting $\mathcal{K}$ denote
a size-$2K$ set that contains the support of $x$, and letting
$\\{x_{k}\\}_{k\in\mathcal{K}}$ denote the corresponding entries of $x$, the
triangle inequality gives
$\displaystyle\Big{|}\|\Phi x\|^{2}-1\Big{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{K}}x_{i}\varphi_{i},\sum_{j\in\mathcal{K}}x_{j}\varphi_{j}\bigg{\rangle}-1\bigg{|}$
$\displaystyle=\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle+\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}$
$\displaystyle\leq\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}+\bigg{|}\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}.$
(8)
Since $\sum_{i\in\mathcal{K}}|x_{i}|^{2}=1$, the second term of (8) satisfies
$\bigg{|}\sum_{i\in\mathcal{K}}\|x_{i}\varphi_{i}\|^{2}-1\bigg{|}\leq\sum_{i\in\mathcal{K}}|x_{i}|^{2}\Big{|}\|\varphi_{i}\|^{2}-1\Big{|}\leq\sum_{i\in\mathcal{K}}|x_{i}|^{2}\delta_{1}=\delta_{1},$
(9)
and so it remains to bound the first term of (8). To this end, we note that
for each $i,j\in\mathcal{K}$ with $j\neq i$, the term $\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle$ appears in
$\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle$
as many times as there are size-$K$ subsets of $\mathcal{K}$ which contain $i$
but not $j$, i.e., $\binom{2K-2}{K-1}$ times. Thus, we use the triangle
inequality and the definition of restricted orthogonality to get
$\displaystyle\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}$
$\displaystyle=\bigg{|}\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}$
$\displaystyle\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{K}\setminus\mathcal{I}}x_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\theta_{K}\bigg{(}\sum_{i\in\mathcal{I}}|x_{i}|^{2}\bigg{)}^{1/2}\bigg{(}\sum_{j\in\mathcal{K}\setminus\mathcal{I}}|x_{j}|^{2}\bigg{)}^{1/2}.$
At this point, $x$ having unit norm implies
$(\sum_{i\in\mathcal{I}}|x_{i}|^{2})^{1/2}(\sum_{j\in\mathcal{K}\setminus\mathcal{I}}|x_{j}|^{2})^{1/2}\leq\frac{1}{2}$,
and so
$\bigg{|}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}\langle
x_{i}\varphi_{i},x_{j}\varphi_{j}\rangle\bigg{|}\leq\frac{1}{\binom{2K-2}{K-1}}\sum_{\begin{subarray}{c}\mathcal{I}\subseteq\mathcal{K}\\\
|\mathcal{I}|=K\end{subarray}}\frac{\theta_{K}}{2}=\frac{\binom{2K}{K}}{\binom{2K-2}{K-1}}\frac{\theta_{K}}{2}=\bigg{(}4-\frac{2}{K}\bigg{)}\frac{\theta_{K}}{2}.$
Applying both this and (9) to (8) gives the result. ∎
Having discussed the relationship between restricted isometry and restricted
orthogonality, we are now ready to introduce the property used in [7] to
demonstrate RIP:
###### Definition 12.
The matrix $\Phi=[\varphi_{1}\cdots\varphi_{N}]$ has _$(K,\hat{\theta})$ -flat
restricted orthogonality_ if
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every disjoint pair of subsets
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$.
Note that $\Phi$ has $(K,\theta_{K})$-flat restricted orthogonality (FRO) by
taking $x$ and $y$ in Definition 9 to be the characteristic functions
$\chi_{\mathcal{I}}$ and $\chi_{\mathcal{J}}$, respectively. Also to be clear,
_flat restricted orthogonality_ is called _flat RIP_ in [7]; we feel the name
change is appropriate considering the preceeding literature. Moreover, the
definition of flat RIP in [7] required $\Phi$ to have unit-norm columns,
whereas we strengthen the corresponding results so as to make no such
requirement. Interestingly, FRO bears some resemblence to the cut-norm of the
Gram matrix $\Phi^{*}\Phi$, defined as the maximum value of
$|\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\langle\varphi_{i},\varphi_{j}\rangle|$
over _all_ subsets $\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$; the
cut-norm has received some attention recently for the hardness of its
approximation [2]. The following theorem illustrates the utility of flat
restricted orthogonality as an estimate of the RIC:
###### Theorem 13.
A matrix with $(K,\hat{\theta})$-flat restricted orthogonality has a
restricted orthogonality constant $\theta_{K}$ which is $\leq
C\hat{\theta}\log K$, and we may take $C=75$.
Indeed, when combined with Lemma 11, this result gives an upper bound on the
RIC: $\delta_{2K}\leq 2C\hat{\theta}\log K+\delta_{1}$. The noteworthy benefit
of this upper bound is that the problem of estimating singular values of
submatrices is reduced to a combinatorial problem of bounding the coherence of
disjoint sums of columns. Furthermore, this reduction comes at the price of a
mere log factor in the estimate. In [7], Bourgain et al. managed to satisfy
this combinatorial coherence property using techniques from additive
combinatorics. While we will not discuss their construction, we find the proof
of Theorem 13 to be instructive; our proof is valid for all values of $K$ (as
opposed to sufficiently large $K$ in the original [7]), and it has near-
optimal constants where appropriate. The proof can be found in the Appendix.
To reiterate, Bourgain et al. [7] used flat restricted orthogonality to build
the only known deterministic construction of $M\times N$ RIP matrices that
support sparsity levels $K$ on the order of $M^{1/2+\varepsilon}$ for some
small value of $\varepsilon$. We are particularly interested in the efficacy
of FRO as a technique to demonstrate RIP in general. Certainly, [7] shows that
FRO can produce at least an $\varepsilon$ improvement over the Gershgorin
technique discussed in the previous section, but it remains to be seen whether
FRO can do better.
In the remainder of this section, we will show that flat restricted
orthogonality is actually capable of demonstrating RIP with much higher
sparsity levels than indicated by [7]. Hopefully, this realization will spur
further research in deterministic constructions which satisfy FRO. To evaluate
FRO, we investigate how well it performs with random matrices; in doing so, we
give an alternative proof that certain random matrices satisfy RIP with high
probability:
###### Theorem 14.
Construct an $M\times N$ matrix $\Phi$ by drawing each of its entries
independently from a Gaussian distribution with mean zero and variance
$\frac{1}{M}$, take $C$ to be the constant from Theorem 13, and set
$\alpha=0.01$. Then $\Phi$ has $(K,\frac{(1-\alpha)\delta}{2C\log K})$-flat
restricted orthogonality and $\delta_{1}\leq\alpha\delta$, and therefore the
$(2K,\delta)$-restricted isometry property, with high probability provided
$M\geq\frac{33C^{2}}{\delta^{2}}K\log^{2}K\log N$.
In proving this result, we will make use of the following Bernstein
inequality:
###### Theorem 15 (see [5, 35]).
Let $\\{Z_{m}\\}_{m=1}^{M}$ be independent random variables of mean zero with
bounded moments, and suppose there exists $L>0$ such that
$\mathbb{E}|Z_{m}|^{k}\leq\frac{\mathbb{E}|Z_{m}|^{2}}{2}L^{k-2}k!$ (10)
for every $k\geq 2$. Then
$\mathrm{Pr}\bigg{[}\sum_{m=1}^{M}Z_{m}\geq
2t\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2}\bigg{]}\leq
e^{-t^{2}}$ (11)
provided
$\displaystyle{t\leq\frac{1}{2L}\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2}}$.
###### Proof of Theorem 14.
Considering Lemma 11, it suffices to show that $\Phi$ has restricted
orthogonality and that $\delta_{1}$ is sufficiently small. First, to
demonstrate restricted orthogonality, it suffices to demonstrate FRO by
Theorem 13, and so we will ensure that the following quantity is small:
$\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}=\sum_{m=1}^{M}\bigg{(}\sum_{i\in\mathcal{I}}\varphi_{i}[m]\bigg{)}\bigg{(}\sum_{j\in\mathcal{J}}\varphi_{j}[m]\bigg{)}.$
(12)
Notice that $X_{m}:=\sum_{i\in\mathcal{I}}\varphi_{i}[m]$ and
$Y_{m}:=\sum_{j\in\mathcal{J}}\varphi_{j}[m]$ are mutually independent over
all $m=1,\ldots,M$ since $\mathcal{I}$ and $\mathcal{J}$ are disjoint. Also,
$X_{m}$ is Gaussian with mean zero and variance $\frac{|\mathcal{I}|}{M}$,
while $Y_{m}$ similarly has mean zero and variance $\frac{|\mathcal{J}|}{M}$.
Viewed this way, (12) being small corresponds to the sum of independent random
variables $Z_{m}:=X_{m}Y_{m}$ having its probability measure concentrated at
zero. To this end, Theorem 15 is naturally applicable, as the absolute central
moments of a Gaussian random variable $X$ with mean zero and variance
$\sigma^{2}$ are well known:
$\mathbb{E}|X|^{k}=\left\\{\begin{array}[]{rl}\sqrt{\frac{2}{\pi}}\sigma^{k}(k-1)!!&\mbox{
if $k$ odd},\\\ \sigma^{k}(k-1)!!&\mbox{ if $k$ even}.\end{array}\right.$
Since $Z_{m}=X_{m}Y_{m}$ is a product of independent Gaussian random
variables, this gives
$\mathbb{E}|Z_{m}|^{k}=\mathbb{E}|X_{m}|^{k}~{}\mathbb{E}|Y_{m}|^{k}\leq\Big{(}\frac{|\mathcal{I}|}{M}\Big{)}^{k/2}\Big{(}\frac{|\mathcal{J}|}{M}\Big{)}^{k/2}\Big{(}(k-1)!!\Big{)}^{2}\leq\bigg{(}\frac{(|\mathcal{I}||\mathcal{J}|)^{1/2}}{M}\bigg{)}^{k}k!.$
Further since
$\mathbb{E}|Z_{m}|^{2}=\frac{|\mathcal{I}||\mathcal{J}|}{M^{2}}$, we may
define $L:=2\frac{(|\mathcal{I}||\mathcal{J}|)^{1/2}}{M}$ to get (10). Later,
we will take $\hat{\theta}<\delta<\sqrt{2}-1<\frac{1}{2}$. Considering
$t:=\frac{\hat{\theta}\sqrt{M}}{2}<\frac{\sqrt{M}}{4}=\frac{1}{2L}\Big{(}M\frac{|\mathcal{I}||\mathcal{J}|}{M^{2}}\Big{)}^{1/2}=\frac{1}{2L}\bigg{(}\sum_{m=1}^{M}\mathbb{E}|Z_{m}|^{2}\bigg{)}^{1/2},$
we therefore have (11), which in this case has the form
$\mathrm{Pr}\Bigg{[}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}\geq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}\Bigg{]}\leq
2e^{-M\hat{\theta}^{2}/4},$
where the probability is doubled due to the symmetric distribution of
$\sum_{m=1}^{M}Z_{m}$. Since we need to account for all possible choices of
$\mathcal{I}$ and $\mathcal{J}$, we will perform a union bound. The total
number of choices is given by
$\sum_{|\mathcal{I}|=1}^{K}\sum_{|\mathcal{J}|=1}^{K}\binom{N}{|\mathcal{I}|}\binom{N-|\mathcal{I}|}{|\mathcal{J}|}\leq
K^{2}\binom{N}{K}^{2}\leq N^{2K},$
and so the union bound gives
$\mathrm{Pr}\Big{[}\mbox{$\Phi$ does not have
$(K,\hat{\theta})$-FRO}\Big{]}\leq
2e^{-M\hat{\theta}^{2}/4}~{}N^{2K}=2\exp\Big{(}-\frac{M\hat{\theta}^{2}}{4}+2K\log
N\Big{)}.$ (13)
Thus, Gaussian matrices tend to have FRO, and hence restricted orthogonality
by Theorem 13; this is made more precise below.
Again by Lemma 11, it remains to show that $\delta_{1}$ is sufficiently small.
To this end, we note that $M\|\varphi_{n}\|^{2}$ has chi-squared distribution
with $M$ degrees of freedom, and so we can use another (simpler)
concentration-of-measure result; see Lemma 1 of [20]:
$\mathrm{Pr}\bigg{[}\Big{|}\|\varphi_{n}\|^{2}-1\Big{|}\geq
2\Big{(}\sqrt{\frac{t}{M}}+\frac{t}{M}\Big{)}\bigg{]}\leq 2e^{-t}$
for any $t>0$. Specifically, we pick
$\delta^{\prime}:=2\Big{(}\sqrt{\frac{t}{M}}+\frac{t}{M}\Big{)}\leq\frac{4t}{M},$
and we perform a union bound over the $N$ choices for $\varphi_{n}$:
$\mathrm{Pr}\Big{[}\delta_{1}>\delta^{\prime}\Big{]}\leq
2\exp\Big{(}-\frac{M\delta^{\prime}}{4}+\log N\Big{)}.$ (14)
To summarize, Lemma 11, the union bound, Theorem 13, and (13) and (14) give
$\displaystyle\mathrm{Pr}\Big{[}\delta_{2K}>\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\theta_{K}>\frac{(1-\alpha)\delta}{2}\mbox{
or }\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\theta_{K}>\frac{(1-\alpha)\delta}{2}\Big{]}+\mathrm{Pr}\Big{[}\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq\mathrm{Pr}\Big{[}\mbox{$\Phi$ does not have
$\displaystyle{\Big{(}K,\frac{(1-\alpha)\delta}{2C\log
K}\Big{)}}$-FRO}\Big{]}+\mathrm{Pr}\Big{[}\delta_{1}>\alpha\delta\Big{]}$
$\displaystyle\leq
2\exp\Big{(}-\frac{M}{4}\Big{(}\frac{(1-\alpha)\delta}{2C\log
K}\Big{)}^{2}+2K\log N\Big{)}+2\exp\Big{(}-\frac{M\alpha\delta}{4}+\log
N\Big{)},$
and so $M\geq\frac{33C^{2}}{\delta^{2}}K\log^{2}K\log N$ gives that $\Phi$ has
$(2K,\delta)$-RIP with high probability. ∎
We note that a version of Theorem 14 also holds for matrices whose entries are
independent Bernoulli random variables taking values $\pm\frac{1}{\sqrt{M}}$
with equal probability. In this case, one can again apply Theorem 15 by
comparing moments with those of the Gaussian distribution; also, a union bound
with $\delta_{1}$ will not be necessary since the columns have unit norm,
meaning $\delta_{1}=0$.
## 4\. Restricted isometry by the power method
In the previous section, we established the efficacy of flat restricted
orthogonality as a technique to demonstrate RIP. While flat restricted
orthogonality has proven useful in the past [7], future deterministic RIP
constructions might not use this technique. Indeed, it would be helpful to
have other techniques available that demonstrate RIP beyond the square-root
bottleneck. In pursuit of such techniques, we recall that the smallest
$\delta$ for which $\Phi$ is $(K,\delta)$-RIP is given in terms of operator
norms in (2). In addition, we notice that for any self-adjoint matrix $A$,
$\|A\|_{2}=\|\lambda(A)\|_{\infty}\leq\|\lambda(A)\|_{p},$
where $\lambda(A)$ denotes the spectrum of $A$ with multiplicities. Let
$A=UDU^{*}$ be the eigenvalue decomposition of $A$. When $p$ is even, we can
express $\|\lambda(A)\|_{p}$ in terms of an easy-to-calculate trace:
$\|\lambda(A)\|_{p}^{p}=\mathrm{Tr}[D^{p}]=\mathrm{Tr}[(UDU^{*})^{p}]=\mathrm{Tr}[A^{p}].$
Combining these ideas with the fact that
$\|\cdot\|_{p}\rightarrow\|\cdot\|_{\infty}$ pointwise leads to the following:
###### Theorem 16.
Given an $M\times N$ matrix $\Phi$, define
$\delta_{K;q}:=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]^{\frac{1}{2q}}.$
Then $\Phi$ has the $(K,\delta_{K;q})$-restricted isometry property for every
$q\geq 1$. Moreover, the restricted isometry constant of $\Phi$ is approached
by these estimates: $\lim_{q\rightarrow\infty}\delta_{K;q}=\delta_{K}$.
Similar to flat restricted orthogonality, this _power method_ has a
combinatorial aspect that prompts one to check every sub-Gram matrix of size
$K$; one could argue that the power method is slightly _less_ combinatorial,
as flat restricted orthogonality is a statement about all pairs of disjoint
subsets of size $\leq K$. Regardless, the work of Bourgain et al. [7]
illustrates that combinatorial properties can be useful, and there may exist
constructions to which the power method would be naturally applied. Moreover,
we note that since $\delta_{K;q}$ approaches $\delta_{K}$, a sufficiently
large choice of $q$ should deliver better-than-$\varepsilon$ improvement over
the Gershgorin analysis. How large should $q$ be? If we assume $\Phi$ has
unit-norm columns, taking $q=1$ gives
$\delta_{K;1}^{2}=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2}]=\max_{\begin{subarray}{c}\mathcal{K}\subseteq\\{1,\ldots,N\\}\\\
|\mathcal{K}|=K\end{subarray}}\sum_{i\in\mathcal{K}}\sum_{\begin{subarray}{c}j\in\mathcal{K}\\\
j\neq i\end{subarray}}|\langle\varphi_{i},\varphi_{j}\rangle|^{2}\leq
K(K-1)\mu^{2},$ (15)
where $\mu$ is the worst-case coherence of $\Phi$. Equality is achieved above
whenever $\Phi$ is an ETF, in which case (15) along with reasoning similar to
(4) demonstrates that $\Phi$ is RIP with sparsity levels on the order of
$\sqrt{M}$, as the Gershgorin analysis established. It remains to be shown how
$\delta_{K;2}$ compares. To make this comparison, we apply the power method to
random matrices:
###### Theorem 17.
Construct an $M\times N$ matrix $\Phi$ by drawing each of its entries
independently from a Gaussian distribution with mean zero and variance
$\frac{1}{M}$, and take $\delta_{K;q}$ to be as defined in Theorem 16. Then
$\delta_{K;q}\leq\delta$, and therefore $\Phi$ has the $(K,\delta)$-restricted
isometry property, with high probability provided
$M\geq\frac{81}{\delta^{2}}K^{1+1/q}\log\frac{eN}{K}$.
While flat restricted orthogonality comes with a negligible penalty of
$\log^{2}K$ in the number of measurements, the power method has a penalty of
$K^{1/q}$. As such, the case $q=1$ uses the order of $K^{2}$ measurements,
which matches our calculation in (15). Moreover, the power method with $q=2$
can demonstrate RIP with $K^{3/2}$ measurements, i.e., $K\sim M^{1/2+1/6}$,
which is considerably better than an $\varepsilon$ improvement over the
Gershgorin technique.
###### Proof of Theorem 17.
Take $t:=\frac{\delta}{3K^{1/2q}}-(\frac{K}{M})^{1/2}$ and pick
$\mathcal{K}\subseteq\\{1,\ldots,N\\}$. Then Theorem II.13 of [13] states
$\mathrm{Pr}\bigg{[}1-\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\leq\sigma_{\min}(\Phi_{\mathcal{K}})\leq\sigma_{\max}(\Phi_{\mathcal{K}})\leq
1+\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]}\geq 1-2e^{-Mt^{2}/2}.$
Continuing, we use the fact that
$\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})=\sigma(\Phi_{\mathcal{K}})^{2}$
to get
$\displaystyle 1-2e^{-Mt^{2}/2}$
$\displaystyle\leq\mathrm{Pr}\bigg{[}\bigg{(}1-\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{)}^{2}\leq\lambda_{\min}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\lambda_{\max}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\bigg{(}1+\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{)}^{2}\bigg{]}$
$\displaystyle\leq\mathrm{Pr}\bigg{[}1-3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\leq\lambda_{\min}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq\lambda_{\max}(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})\leq
1+3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]},$ (16)
where the last inequality follows from the fact that
$(\frac{K}{M})^{1/2}+t<1$. Since $\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$
and $\mathrm{I}_{K}$ are simultaneously diagonalizable, the spectrum of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K}$ is given by
$\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})=\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}})-1$.
Combining this with (16) then gives
$\mathrm{Pr}\bigg{[}\Big{\|}\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})\Big{\|}_{\infty}\leq
3\bigg{(}\sqrt{\frac{K}{M}}+t\bigg{)}\bigg{]}\geq 1-2e^{-Mt^{2}/2}.$
Considering $\mathrm{Tr}[A^{2q}]^{\frac{1}{2q}}=\|\lambda(A)\|_{2q}\leq
K^{\frac{1}{2q}}\|\lambda(A)\|_{\infty}$, we continue:
$\mathrm{Pr}\bigg{[}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]^{\frac{1}{2q}}\leq\delta\bigg{]}\geq\mathrm{Pr}\bigg{[}K^{\frac{1}{2q}}\Big{\|}\lambda(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})\Big{\|}_{\infty}\leq\delta\bigg{]}\geq
1-2e^{-Mt^{2}/2}.$
From here, we perform a union bound over all possible choices of
$\mathcal{K}$:
$\displaystyle\mathrm{Pr}\bigg{[}\exists\mathcal{K}\mbox{ s.t.
}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]^{\frac{1}{2q}}>\delta\bigg{]}$
$\displaystyle\leq\binom{N}{K}\mathrm{Pr}\bigg{[}\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]^{\frac{1}{2q}}>\delta\bigg{]}$
$\displaystyle\leq 2\exp\Big{(}-\frac{Mt^{2}}{2}+K\log\frac{eN}{K}\Big{)}.$
(17)
Rearranging $M\geq\frac{81}{\delta^{2}}K^{1+1/q}\log\frac{eN}{K}$ gives
$K^{1/2}\leq\frac{\delta M^{1/2}}{9K^{1/2q}\log^{1/2}(eN/K)}\leq\frac{\delta
M^{1/2}}{9K^{1/2q}}$, and so
$\frac{Mt^{2}}{2}=\frac{1}{2}\bigg{(}\frac{\delta
M^{1/2}}{3K^{1/2q}}-K^{1/2}\bigg{)}^{2}\geq\frac{1}{2}\bigg{(}\frac{2\delta
M^{1/2}}{9K^{1/2q}}\bigg{)}^{2}\geq 2K\log\frac{eN}{K}.$ (18)
Combining (17) and (18) gives the result. ∎
## 5\. Equiangular tight frames as RIP candidates
In Section 2, we observed that equiangular tight frames (ETFs) are optimal RIP
matrices under the Gershgorin analysis. In the present section, we reexamine
ETFs as prospective RIP matrices. Specifically, we consider the possibility
that certain classes of $M\times N$ ETFs support sparsity levels $K$ larger
than the order of $\sqrt{M}$. Before analyzing RIP, let’s first observe some
important features of ETFs. Recall that Definition 5 characterized ETFs in
terms of their rows and columns. Interestingly, _real_ ETFs have a natural
alternative characterization.
Let $\Phi$ be a real $M\times N$ ETF, and consider the corresponding Gram
matrix $\Phi^{*}\Phi$. Observing Definition 5, we have from (i) that the
diagonal entries of $\Phi^{*}\Phi$ are 1’s. Also, (iii) indicates that the
off-diagonal entries are equal in absolute value (to the Welch bound); since
$\Phi$ has real entries, the phase of each off-diagonal entry of
$\Phi^{*}\Phi$ is either positive or negative. Letting $\mu$ denote the
absolute value of the off-diagonal entries, we can decompose the Gram matrix
as $\Phi^{*}\Phi=\mathrm{I}_{N}+\mu S$, where $S$ is a matrix of zeros on the
diagonal and $\pm 1$’s on the off-diagonal. Here, $S$ is referred to as a
_Seidel adjacency matrix_ , as $S$ encodes the adjacency rule of a simple
graph with $i\leftrightarrow j$ whenever $S[i,j]=-1$; this correspondence
originated in [31].
There is an important equivalence class amongst ETFs: given an ETF $\Phi$, one
can negate any of the columns to form another ETF $\Phi^{\prime}$. Indeed, the
ETF properties in Definition 5 are easily verified to hold for this new
matrix. For obvious reasons, $\Phi$ and $\Phi^{\prime}$ are called _flipping
equivalent_. This equivalence plays a key role in the following result, which
characterizes real ETFs in terms of a particular class of strongly regular
graphs:
###### Definition 18.
We say a simple graph $G$ is _strongly regular_ of the form
$\mathrm{srg}(v,k,\lambda,\mu)$ if
* (i)
$G$ has $v$ vertices,
* (ii)
every vertex has $k$ neighbors (i.e., $G$ is $k$_-regular_),
* (iii)
every two adjacent vertices have $\lambda$ common neighbors, and
* (iv)
every two non-adjacent vertices have $\mu$ common neighbors.
###### Theorem 19 (Corollary 5.6 in [32]).
Every real $M\times N$ equiangular tight frame with $N>M+1$ is flipping
equivalent to a frame whose Seidel adjacency matrix corresponds to the join of
a vertex with a strongly regular graph of the form
$\mathrm{srg}\bigg{(}N-1,L,\frac{3L-N}{2},\frac{L}{2}\bigg{)},\qquad
L:=\frac{N}{2}-1+\bigg{(}1-\frac{N}{2M}\bigg{)}\sqrt{\frac{M(N-1)}{N-M}}.$
Conversely, every such graph corresponds to flipping equivalence classes of
equiangular tight frames in the same manner.
The previous two sections illustrated the main issue with the Gershgorin
analysis: it ignores important cancellations in the sub-Gram matrices. We
suspect that such cancellations would be more easily observed in a real ETF,
since Theorem 19 neatly represents the Gram matrix’s off-diagonal oscillations
in terms of adjacencies in a strongly regular graph. The following result
gives a taste of how useful this graph representation can be:
###### Theorem 20.
Take a real equiangular tight frame $\Phi$ with worst-case coherence $\mu$,
and let $G$ denote the corresponding strongly regular graph in Theorem 19.
Then the restricted isometry constant of $\Phi$ is given by
$\delta_{K}=(K-1)\mu$ for every $K\leq\omega(G)+1$, where $\omega(G)$ denotes
the size of the largest clique in $G$.
###### Proof.
The Gershgorin analysis (3) gives the bound $\delta_{K}\leq(K-1)\mu$, and so
it suffices to prove $\delta_{K}\geq(K-1)\mu$. Since $K\leq\omega(G)+1$, there
exists a clique of size $K$ in the join of $G$ with a vertex. Let
$\mathcal{K}$ denote the vertices of this clique, and take $S_{\mathcal{K}}$
to be the corresponding Seidel adjacency submatrix. In this case,
$S_{\mathcal{K}}=\mathrm{I}_{K}-\mathrm{J}_{K}$, where $\mathrm{J}_{K}$ is the
$K\times K$ matrix of all 1’s. Observing the decomposition
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}=\mathrm{I}_{K}+\mu S_{\mathcal{K}}$,
it follows from (2) that
$\delta_{K}\geq\|\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K}\|_{2}=\|\mu
S_{\mathcal{K}}\|_{2}=\mu\|\mathrm{I}_{K}-\mathrm{J}_{K}\|_{2}=(K-1)\mu,$
which concludes the proof. ∎
This result indicates that the Gershgoin analysis is tight for all real ETFs,
at least for sufficiently small values of $K$. In particular, in order for a
real ETF to be RIP beyond the square-root bottleneck, its graph must have a
small clique number. As an example, note that the first four columns of the
Steiner ETF in (5) have negative inner products with each other, and thus the
corresponding subgraph is a clique. In general, each block of an $M\times N$
Steiner ETF, whose size is guaranteed to be $\mathrm{O}(\sqrt{M})$, is a
lower-dimensional simplex and therefore has this property; this is an
alternative proof that the Gershgorin analysis of Steiner ETFs is tight for
$K=\mathrm{O}(\sqrt{M})$.
### 5.1. Equiangular tight frames with flat restricted orthogonality
To find ETFs that are RIP beyond the square-root bottleneck, we must apply
better techniques than Gershgorin. We first consider what it means for an ETF
to have $(K,\hat{\theta})$-flat restricted orthogonality. Take a real ETF
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$ with worst-case coherence $\mu$, and
note that the corresponding Seidel adjacency matrix $S$ can be expressed in
terms of the usual $\\{0,1\\}$-adjacency matrix $A$ of the same graph:
$S[i,j]=1-2A[i,j]$ whenever $i\neq j$. Therefore, for every disjoint
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$, we want
$\displaystyle\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
$\displaystyle\geq\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}=\bigg{|}\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\mu
S[i,j]\bigg{|}$
$\displaystyle\qquad=\mu\bigg{|}|\mathcal{I}||\mathcal{J}|-2\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}A[i,j]\bigg{|}=2\mu\bigg{|}E(\mathcal{I},\mathcal{J})-\frac{1}{2}|\mathcal{I}||\mathcal{J}|\bigg{|},$
(19)
where $E(\mathcal{I},\mathcal{J})$ denotes the number of edges between
$\mathcal{I}$ and $\mathcal{J}$ in the graph. This condition bears a striking
resemblence to the following well-known result in graph theory:
###### Lemma 21 (Expander mixing lemma [19]).
Given a $d$-regular graph of $n$ vertices, the second largest eigenvalue
$\lambda$ of its adjacency matrix satisfies
$\bigg{|}E(\mathcal{I},\mathcal{J})-\frac{d}{n}|\mathcal{I}||\mathcal{J}|\bigg{|}\leq\lambda(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every pair of vertex subsets $\mathcal{I},\mathcal{J}$.
In words, the expander mixing lemma says that the number of edges between
vertex subsets of a regular graph is roughly what you would expect in a
_random_ regular graph. For this lemma to be applicable to (19), we need the
strongly regular graph of Theorem 19 to satisfy
$\frac{L}{N-1}=\frac{d}{n}\approx\frac{1}{2}$. Using the formula for $L$, it
is not difficult to show that
$|\frac{L}{N-1}-\frac{1}{2}|=\mathrm{O}(M^{-1/2})$ provided $N=\mathrm{O}(M)$
and $N\geq 2M$. Furthermore, the second largest eigenvalue of the strongly
regular graph will be $\lambda\approx\frac{1}{2}N^{1/2}$, and so the expander
mixing lemma says the optimal $\hat{\theta}$ is $\leq
2\mu\lambda\approx(\frac{N-M}{M})^{1/2}$ since
$\mu=(\frac{N-M}{M(N-1)})^{1/2}$. This is a rather weak estimate for
$\hat{\theta}$ because the expander mixing lemma does not account for the
sizes of $\mathcal{I}$ and $\mathcal{J}$ being $\leq K$. Put in this light, a
real ETF that has flat restricted orthogonality corresponds to a strongly
regular graph that satisfies a particularly strong version of the expander
mixing lemma.
### 5.2. Equiangular tight frames and the power method
Next, we try applying the power method to ETFs. Given a real ETF
$\Phi=[\varphi_{1}\cdots\varphi_{N}]$, let $H:=\Phi^{*}\Phi-\mathrm{I}_{N}$
denote the “hollow” Gram matrix. Also, take $E_{\mathcal{K}}$ to be the
$N\times K$ matrix built from the columns of $\mathrm{I}_{N}$ that are indexed
by $\mathcal{K}$. Then
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]=\mathrm{Tr}[(E_{\mathcal{K}}^{*}\Phi^{*}\Phi
E_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]=\mathrm{Tr}[(E_{\mathcal{K}}^{*}HE_{\mathcal{K}})^{2q}]=\mathrm{Tr}[(HE_{\mathcal{K}}E_{\mathcal{K}}^{*})^{2q}].$
Since
$E_{\mathcal{K}}E_{\mathcal{K}}^{*}=\sum_{k\in\mathcal{K}}\delta_{k}\delta_{k}^{*}$,
where $\delta_{k}$ is the $k$th identity basis element, we continue:
$\displaystyle\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]$
$\displaystyle=\mathrm{Tr}\bigg{[}\bigg{(}H\sum_{k\in\mathcal{K}}\delta_{k}\delta_{k}^{*}\bigg{)}^{2q}\bigg{]}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\cdots\sum_{k_{2q-1}\in\mathcal{K}}\mathrm{Tr}[H\delta_{k_{0}}\delta_{k_{0}}^{*}\cdots
H\delta_{k_{2q-1}}\delta_{k_{2q-1}}^{*}]$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\cdots\sum_{k_{2q-1}\in\mathcal{K}}\delta_{k_{0}}^{*}H\delta_{k_{1}}\cdots\delta_{k_{2q-1}}^{*}H\delta_{k_{0}},$
(20)
where the last step used the cyclic property of the trace. From here, note
that $H$ has a zero diagonal, meaning several of the terms in (20) are zero,
namely, those for which $k_{\ell+1}=k_{\ell}$ for some
$\ell\in\mathbb{Z}_{2q}$. To simplify (20), take $\mathcal{K}^{(2q)}$ to be
the set of $2q$-tuples satisfying $k_{\ell+1}\neq k_{\ell}$ for every
$\ell\in\mathbb{Z}_{2q}$:
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{2q}]=\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(2q)}}\prod_{\ell\in\mathbb{Z}_{2q}}\langle\varphi_{k_{\ell}},\varphi_{k_{\ell+1}}\rangle=\mu^{2q}\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(2q)}}\prod_{\ell\in\mathbb{Z}_{2q}}S[k_{\ell},k_{\ell+1}],$
(21)
where $\mu$ is the wost-case coherence of $\Phi$, and $S$ is the corresponding
Seidel adjacency matrix. Note that the left-hand side is necessarily
nonnegative, while it is not immediate why the right-hand side should be. This
indicates that more simplification can be done, but for the sake of clarity,
we will perform this simplification in the special case where $q=2$; the
general case is very similar. When $q=2$, we are concerned with 4-tuples
$\\{k_{0},k_{1},k_{2},k_{3}\\}\in\mathcal{K}^{(4)}$. Let’s partition these
4-tuples according to the value taken by $k_{0}$ and $k_{q}=k_{2}$. Note, for
a fixed $k_{0}$ and $k_{2}$, that $k_{1}$ can be any value other than $k_{0}$
or $k_{2}$, as can $k_{3}$. This leads to the following simplification:
$\displaystyle\sum_{\\{k_{\ell}\\}\in\mathcal{K}^{(4)}}\prod_{\ell\in\mathbb{Z}_{4}}S[k_{\ell},k_{\ell+1}]$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\sum_{k_{2}\in\mathcal{K}}\bigg{(}\sum_{\begin{subarray}{c}k_{1}\in\mathcal{K}\\\
k_{0}\neq k_{1}\neq
k_{2}\end{subarray}}S[k_{0},k_{1}]S[k_{1},k_{2}]\bigg{)}\bigg{(}\sum_{\begin{subarray}{c}k_{3}\in\mathcal{K}\\\
k_{2}\neq k_{3}\neq k_{0}\end{subarray}}S[k_{2},k_{3}]S[k_{3},k_{0}]\bigg{)}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\sum_{k_{2}\in\mathcal{K}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}$
$\displaystyle=\sum_{k_{0}\in\mathcal{K}}\bigg{|}\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k\neq
k_{0}\end{subarray}}S[k_{0},k]S[k,k_{0}]\bigg{|}^{2}+\sum_{k_{0}\in\mathcal{K}}\sum_{\begin{subarray}{c}k_{2}\in\mathcal{K}\\\
k_{2}\neq
k_{0}\end{subarray}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}.$
The first term above is $K(K-1)^{2}$, while the other term is not as easy to
analyze, as we expect a certain degree of cancellation. Substituting this
simplification into (21) gives
$\mathrm{Tr}[(\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}-\mathrm{I}_{K})^{4}]=\mu^{4}\bigg{(}K(K-1)^{2}+\sum_{k_{0}\in\mathcal{K}}\sum_{\begin{subarray}{c}k_{2}\in\mathcal{K}\\\
k_{2}\neq
k_{1}\end{subarray}}~{}~{}~{}~{}\bigg{|}\\!\\!\\!\\!\sum_{\begin{subarray}{c}k\in\mathcal{K}\\\
k_{0}\neq k\neq k_{2}\end{subarray}}S[k_{0},k]S[k,k_{2}]\bigg{|}^{2}\bigg{)}.$
If there were no cancellations in the second term, then it would equal
$K(K-1)(K-2)^{2}$, thereby dominating the expression. However, if oscillations
occured as a $\pm 1$ Bernoulli random variable, we could expect this term to
be on the order of $K^{3}$, matching the order of the first term. In this
hypothetical case, since $\mu\leq M^{-1/2}$, the parameter $\delta_{K;2}^{4}$
defined in Theorem 16 scales as $\frac{K^{3}}{M^{2}}$, and so $M\sim K^{3/2}$;
this corresponds to the behavior exhibited in Theorem 17. To summarize, much
like flat restricted orthogonality, applying the power method to ETFs leads to
interesting combinatorial questions regarding subgraphs, even when $q=2$.
### 5.3. The Paley equiangular tight frame as an RIP candidate
Pick some prime $p\equiv 1\bmod 4$, and build an $M\times p$ matrix $H$ by
selecting the $M:=\frac{p+1}{2}$ rows of the $p\times p$ discrete Fourier
transform matrix which are indexed by $Q$, the quadratic residues modulo $p$
(including zero). To be clear, the entries of $H$ are scaled to have unit
modulus. Next, take $D$ to be an $M\times M$ diagonal matrix whose zeroth
diagonal entry is $p^{-1/2}$, and whose remaining $M-1$ entries are
$(\frac{2}{p})^{1/2}$. Now build the matrix $\Phi$ by concatenating $DH$ with
the zeroth identity basis element; for example, when $p=5$, we have a $3\times
6$ matrix:
$\Phi=\left[\begin{array}[]{llllll}\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&\sqrt{\frac{1}{5}}&1\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&0\\\
\sqrt{\frac{2}{5}}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}4/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}3/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}2/5}&\sqrt{\frac{2}{5}}e^{-2\pi\mathrm{i}/5}&0\\\
\end{array}\right].$
We claim that in general, this process produces an $M\times 2M$ equiangular
tight frame, which we call the _Paley ETF_ [24]. Presuming for the moment that
this claim is true, we have the following result which lends hope for the
Paley ETF as an RIP matrix:
###### Lemma 22.
An $M\times 2M$ Paley equiangular tight frame has restricted isometry constant
$\delta_{K}<1$ for all $K\leq M$.
###### Proof.
First, we note that Theorem 6 of [1] used Chebotarëv’s theorem [28] to prove
that the spark of the $M\times 2M$ Paley ETF $\Phi$ is $M+1$, that is, every
size-$M$ subcollection of columns of $\Phi$ forms a spanning set. Thus, for
every $\mathcal{K}\subseteq\\{1,\ldots,2M\\}$ of size $\leq M$, the smallest
singular value of $\Phi_{\mathcal{K}}$ is positive. It remains to show that
the square of the largest singular value is strictly less than 2. Let $x$ be a
unit vector for which
$\|\Phi_{\mathcal{K}}^{*}x\|=\|\Phi_{\mathcal{K}}^{*}\|_{2}$. Then since the
spark of $\Phi$ is $M+1$, the columns of $\Phi_{\mathcal{K}^{\mathrm{c}}}$
span, and so
$\|\Phi_{\mathcal{K}}\|_{2}^{2}=\|\Phi_{\mathcal{K}}^{*}\|_{2}^{2}=\|\Phi_{\mathcal{K}}^{*}x\|^{2}<\|\Phi_{\mathcal{K}}^{*}x\|^{2}+\|\Phi_{\mathcal{K}^{\mathrm{c}}}^{*}x\|^{2}=\|\Phi^{*}x\|^{2}\leq\|\Phi^{*}\|_{2}^{2}=\|\Phi\Phi^{*}\|_{2}=2,$
where the final step follows from Definition 5(i)-(ii), which imply
$\Phi\Phi^{*}=2\mathrm{I}_{M}$. ∎
Now that we have an interest in the Paley ETF $\Phi$, we wish to verify that
it is, in fact, an ETF. It suffices to show that the columns of $\Phi$ have
unit norm, and that the inner products between distinct columns equal the
Welch bound in absolute value. Certainly, the zeroth identity basis element is
unit-norm, while the squared norm of each of the other columns is given by
$\frac{1}{p}+(M-1)\frac{2}{p}=\frac{2M-1}{p}=1$. Also, the inner product
between the zeroth identity basis element and any other column equals the
zeroth entry of that column: $p^{-1/2}=(\frac{N-M}{M(N-1)})^{1/2}$. It remains
to calculate the inner product between distinct columns which are not identity
basis elements. To this end, note that since $a^{2}=b^{2}$ if and only if
$a=\pm b$, the sequence $\\{k^{2}\\}_{k=1}^{p-1}\subseteq\mathbb{Z}_{p}$
doubly covers $Q\setminus\\{0\\}$, and so
$\langle\varphi_{n},\varphi_{n^{\prime}}\rangle=\frac{1}{p}+\sum_{m\in
Q\setminus\\{0\\}}\bigg{(}\sqrt{\frac{2}{p}}e^{-2\pi\mathrm{i}mn/p}\bigg{)}\bigg{(}\sqrt{\frac{2}{p}}e^{2\pi\mathrm{i}mn^{\prime}/p}\bigg{)}=\frac{1}{p}\sum_{k=0}^{p-1}e^{2\pi\mathrm{i}(n^{\prime}-n)k^{2}/p}.$
This well-known expression is called a quadratic Gauss sum, and since $p\equiv
1\bmod 4$, its value is determined by the Legendre symbol in the following
way:
$\langle\varphi_{n},\varphi_{n^{\prime}}\rangle=\frac{1}{\sqrt{p}}(\frac{n^{\prime}-n}{p})$
for every $n,n^{\prime}\in\mathbb{Z}_{p}$ with $n\neq n^{\prime}$, where
$\bigg{(}\frac{k}{p}\bigg{)}:=\left\\{\begin{array}[]{rl}+1&\mbox{ if $k$ is a
nonzero quadratic residue modulo $p$,}\\\ 0&\mbox{ if $k=0$,}\\\ -1&\mbox{
otherwise.}\end{array}\right.$
Having established that $\Phi$ is an ETF, we notice that the inner products
between distinct columns of $\Phi$ are real. This implies that the columns of
$\Phi$ can be unitarily rotated to form a real ETF $\Psi$; indeed, one may
take $\Psi$ to be the $M\times 2M$ matrix formed by taking the nonzero rows of
$L^{\mathrm{T}}$ in the Cholesky factorization $\Phi^{*}\Phi=LL^{\mathrm{T}}$.
As such, we consider the Paley ETF to be real. From here, Theorem 19 prompts
us to find the corresponding strongly regular graph. First, we can flip the
identity basis element so that its inner products with the other columns of
$\Phi$ are all negative. As such, the corresponding vertex in the graph will
be adjacent to each of the other vertices; naturally, this will be the vertex
to which the strongly regular graph is joined. For the remaining vertices,
$n\leftrightarrow n^{\prime}$ precisely when $(\frac{n^{\prime}-n}{p})=-1$,
that is, when $n^{\prime}-n$ is not a quadratic residue. The corresponding
subgraph is therefore the complement of the Paley graph, namely, the Paley
graph [26]. In general, Paley graphs of order $p$ necessarily have $p\equiv
1\bmod 4$, and so this correspondence is particularly natural.
One interesting thing about the Paley ETF’s restricted isometry is that it
lends insight into important properties of the Paley graph. The following is
the best known upper bound for the clique number of the Paley graph of prime
order (see Theorem 13.14 of [6] and discussion thereafter), and we give a new
proof of this bound using restricted isometry:
###### Theorem 23.
Let $G$ denote the Paley graph of prime order $p$. Then the size of the
largest clique is $\omega(G)<\sqrt{p}$.
###### Proof.
We start by showing $\omega(G)+1\leq M$. Suppose otherwise: that there exists
a clique $\mathcal{K}$ of size $M+1$ in the join of a vertex with $G$. Then
the corresponding sub-Gram matrix of the Paley ETF has the form
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}=(1+\mu)\mathrm{I}_{M+1}-\mu\mathrm{J}_{M+1}$,
where $\mu=p^{-1/2}$ is the worst-case coherence and $\mathrm{J}_{M+1}$ is the
$(M+1)\times(M+1)$ matrix of 1’s. Since the largest eigenvalue of
$\mathrm{J}_{M+1}$ is $M+1$, the smallest eigenvalue of
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is
$1+p^{-1/2}-(M+1)p^{-1/2}=1-\frac{1}{2}(p+1)p^{-1/2}$, which is negative when
$p\geq 5$, contradicting the fact that
$\Phi_{\mathcal{K}}^{*}\Phi_{\mathcal{K}}$ is positive semidefinite.
Since $\omega(G)+1\leq M$, we can apply Lemma 22 and Theorem 20 to get
$1>\delta_{\omega(G)+1}=\Big{(}\omega(G)+1-1\Big{)}\mu=\frac{\omega(G)}{\sqrt{p}},$
(22)
and rearranging gives the result. ∎
It is common to apply probabilistic and heuristic reasoning to gain intuition
in number theory. For example, consecutive entries of the Legendre symbol are
known to mimic certain properties of a $\pm 1$ Bernoulli random variable [22].
Moreover, Paley graphs enjoy a certain quasi-random property that was studied
in [11]. On the other hand, Graham and Ringrose [18] showed that, while random
graphs of size $p$ have an expected clique number of $(1+o(1))2\log p/\log 2$,
Paley graphs of prime order deviate from this random behavior, having a clique
number $\geq c\log p\log\log\log p$ infinitely often. The best known universal
lower bound, $(1/2+o(1))\log p/\log 2$, is given in [12], which indicates that
the random graph analysis is at least tight in some sense. Regardless, this
has a significant difference from the upper bound $\sqrt{p}$ in Theorem 23,
and it would be nice if probabilistic arguments could be leveraged to improve
this bound, or at least provide some intuition.
Note that our proof (22) hinged on the fact that $\delta_{\omega(G)+1}<1$,
courtesy of Lemma 22. Hence, any improvement to our estimate for
$\delta_{\omega(G)+1}$ would directly lead to the best known upper bound on
the Paley graph’s clique number. To approach such an improvement, note that
for large $p$, the Fourier portion of the Paley ETF $DH$ is not significatly
different from the normalized partial Fourier matrix $(\frac{2}{p+1})^{1/2}H$;
indeed,
$\|H_{\mathcal{K}}^{*}D^{2}H_{\mathcal{K}}-\frac{2}{p+1}H_{\mathcal{K}}^{*}H_{\mathcal{K}}\|_{2}\leq\frac{2}{p}$
for every $\mathcal{K}\subseteq\mathbb{Z}_{p}$ of size $\leq\frac{p+1}{2}$,
and so the difference vanishes. If we view the quadratic residues modulo $p$
(the row indices of $H$) as random, then a random partial Fourier matrix
serves as a proxy for the Fourier portion of the Paley ETF. This in mind, we
appeal to the following:
###### Theorem 24 (Theorem 3.2 in [23]).
Draw rows from the $N\times N$ discrete Fourier transform matrix uniformly at
random with replacement to construct an $M\times N$ matrix, and then normalize
the columns to form $\Phi$. Then $\Phi$ has restricted isometry constant
$\delta_{K}\leq\delta$ with probability $1-\varepsilon$ provided
$\frac{M}{\log M}\geq\frac{C}{\delta^{2}}K\log^{2}K\log
N\log\varepsilon^{-1}$, where $C$ is a universal constant.
In our case, both $M$ and $N$ scale as $p$, and so picking $\delta$ to achieve
equality above gives
$\delta^{2}=\frac{C^{\prime}}{p}K\log^{2}K\log^{2}p\log\varepsilon^{-1}.$
Continuing as in (22), denote $\omega=\omega(G)$ and take $K=\omega$ to get
$\frac{C^{\prime}}{p}\omega\log^{2}\omega\log^{2}p\log\varepsilon^{-1}\geq\delta_{\omega}^{2}=\frac{(\omega-1)^{2}}{p}\geq\frac{\omega^{2}}{2p},$
and then rearranging gives $\omega/\log^{2}\omega\leq
C^{\prime\prime}\log^{2}p\log\varepsilon^{-1}$ with probability
$1-\varepsilon$. Interestingly, having
$\omega/\log^{2}\omega=\mathrm{O}(\log^{3}p)$ with high probability (again,
under the model that quadratic residues are random) agrees with the results of
Graham and Ringrose [18]. This gives some intuition for what we can expect the
size of the Paley graph’s clique number to be, while at the same time
demonstrating the power of Paley ETFs as RIP candidates. We conclude with the
following, which can be reformulated in terms of both flat restricted
orthogonality and the power method:
###### Conjecture 25.
The Paley equiangular tight frame has the $(K,\delta)$-restricted isometry
property with some $\delta<\sqrt{2}-1$ whenever
$K\leq\frac{Cp}{\log^{\alpha}p}$, for some universal constants $C$ and
$\alpha$.
## 6\. Appendix
In this section, we prove Theorem 13, which states that a matrix with
$(K,\hat{\theta})$-flat restricted orthogonality has $\theta_{K}\leq
C\hat{\theta}\log K$, that is, it has restricted orthogonality. The proof
below is adapted from the proof of Lemma 3 in [7]. Our proof has the benefit
of being valid for all values of $K$ (as opposed to sufficiently large $K$ in
the original [7]), and it has near-optimal constants where appropriate.
Moreover in this version, the columns of the matrix are not required to have
unit norm.
###### Proof of Theorem 13.
Given arbitrary disjoint subsets
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$, we will bound the following quantity
three times, each time with different constraints on
$\\{x_{i}\\}_{i\in\mathcal{I}}$ and $\\{y_{j}\\}_{j\in\mathcal{J}}$:
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(23)
To be clear, our third bound will have no constraints on
$\\{x_{i}\\}_{i\in\mathcal{I}}$ and $\\{y_{j}\\}_{j\in\mathcal{J}}$, thereby
demonstrating restricted orthogonality. Note that by assumption, (23) is
$\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$ whenever the $x_{i}$’s
and $y_{j}$’s are in $\\{0,1\\}$. We first show that this bound is preserved
when we relax the $x_{i}$’s and $y_{j}$’s to lie in the interval $[0,1]$.
Pick a disjoint pair of subsets
$\mathcal{I}^{\prime},\mathcal{J}^{\prime}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}^{\prime}|,|\mathcal{J}^{\prime}|\leq K$. Starting with some
$k\in\mathcal{I}^{\prime}$, note that flat restricted orthogonality gives that
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2},$
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\hat{\theta}(|\mathcal{I}\setminus\\{k\\}||\mathcal{J}|)^{1/2}\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
for every disjoint $\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$ and $k\in\mathcal{I}$. Thus, we may take
any $x_{k}\in[0,1]$ to form a convex combination of these two expressions, and
then the triangle inequality gives
$\displaystyle\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
$\displaystyle\geq
x_{k}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}+(1-x_{k})\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\geq\bigg{|}x_{k}\bigg{\langle}\sum_{i\in\mathcal{I}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}+(1-x_{k})\bigg{\langle}\sum_{i\in\mathcal{I}\setminus\\{k\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\bigg{\\{}\begin{array}[]{cc}x_{k},&i=k\\\
1,&i\neq
k\end{array}\bigg{\\}}\varphi_{i},\sum_{j\in\mathcal{J}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(26)
Since (26) holds for every disjoint
$\mathcal{I},\mathcal{J}\subseteq\\{1,\ldots,N\\}$ with
$|\mathcal{I}|,|\mathcal{J}|\leq K$ and $k\in\mathcal{I}$, we can do the same
thing with an additional index $i\in\mathcal{I}^{\prime}$ or
$j\in\mathcal{J}^{\prime}$, and replace the corresponding unit coefficient
with some $x_{i}$ or $y_{j}$ in $[0,1]$. Continuing in this way proves the
claim that (23) is $\leq\hat{\theta}(|\mathcal{I}||\mathcal{J}|)^{1/2}$
whenever the $x_{i}$’s and $y_{j}$’s lie in the interval $[0,1]$.
For the second bound, we assume the $x_{i}$’s and $y_{j}$’s are nonnegative
with unit norm:
$\sum_{i\in\mathcal{I}}x_{i}^{2}=\sum_{j\in\mathcal{J}}y_{j}^{2}=1$. To bound
(23) in this case, we partition $\mathcal{I}$ and $\mathcal{J}$ according to
the size of the corresponding coefficients:
$\mathcal{I}_{k}:=\\{i\in\mathcal{I}:2^{-(k+1)}<x_{i}\leq
2^{-k}\\},\qquad\mathcal{J}_{k}:=\\{j\in\mathcal{J}:2^{-(k+1)}<y_{j}\leq
2^{-k}\\}.$
Note the unit-norm constraints ensure that
$\mathcal{I}=\bigcup_{k=0}^{\infty}\mathcal{I}_{k}$ and
$\mathcal{J}=\bigcup_{k=0}^{\infty}\mathcal{J}_{k}$. The triangle inequality
thus gives
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{k_{1}=0}^{\infty}\sum_{i\in\mathcal{I}_{k_{1}}}x_{i}\varphi_{i},\sum_{k_{2}=0}^{\infty}\sum_{j\in\mathcal{J}_{k_{2}}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}2^{-(k_{1}+k_{2})}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}_{k_{1}}}\frac{x_{i}}{2^{-k_{1}}}\varphi_{i},\sum_{j\in\mathcal{J}_{k_{2}}}\frac{y_{j}}{2^{-k_{2}}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
(27)
By the definitions of $\mathcal{I}_{k_{1}}$ and $\mathcal{J}_{k_{2}}$, the
coefficients of $\varphi_{i}$ and $\varphi_{j}$ in (27) all lie in $[0,1]$. As
such, we continue by applying our first bound:
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{\infty}2^{-(k_{1}+k_{2})}\hat{\theta}(|\mathcal{I}_{k_{1}}||\mathcal{J}_{k_{2}}|)^{1/2}$
$\displaystyle=\hat{\theta}\bigg{(}\sum_{k=0}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}\bigg{)}\bigg{(}\sum_{k=0}^{\infty}2^{-k}|\mathcal{J}_{k}|^{1/2}\bigg{)}.$
(28)
We now observe from the definition of $\mathcal{I}_{k}$ that
$1=\sum_{i\in\mathcal{I}}x_{i}^{2}=\sum_{k=0}^{\infty}\sum_{i\in\mathcal{I}_{k}}x_{i}^{2}>\sum_{k=0}^{\infty}4^{-(k+1)}|\mathcal{I}_{k}|.$
Thus for any positive integer $t$, the Cauchy-Schwarz inequality gives
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}$
$\displaystyle=\sum_{k=0}^{t-1}2^{-k}|\mathcal{I}_{k}|^{1/2}+\sum_{k=t}^{\infty}2^{-k}|\mathcal{I}_{k}|^{1/2}$
$\displaystyle\leq
t^{1/2}\bigg{(}\sum_{k=0}^{t-1}4^{-k}|\mathcal{I}_{k}|\bigg{)}^{1/2}+\sum_{k=t}^{\infty}2^{-k}K^{1/2}$
$\displaystyle<2(t^{1/2}+K^{1/2}2^{-t}),$ (29)
and similarly for the $\mathcal{J}_{k}$’s. For a fixed $K$, we note that (29)
is minimized when $K^{1/2}2^{-t}=\frac{t^{-1/2}}{2\log 2}$, and so we pick $t$
to be the smallest positive integer such that
$K^{1/2}2^{-t}\leq\frac{t^{-1/2}}{2\log 2}$. With this, we continue (28):
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle<\hat{\theta}\Big{(}2(t^{1/2}+K^{1/2}2^{-t})\Big{)}^{2}$
$\displaystyle\leq 4\hat{\theta}\bigg{(}t^{1/2}+\frac{t^{-1/2}}{2\log
2}\bigg{)}^{2}=4\hat{\theta}\bigg{(}t+\frac{1}{\log 2}+\frac{1}{(2\log
2)^{2}t}\bigg{)}.$ (30)
From here, we claim that $t\leq\lceil\frac{\log K}{\log 2}\rceil$. Considering
the definition of $t$, this is easily verified for $K=2,3,\ldots,7$ by showing
$K^{1/2}2^{-s}\leq\frac{s^{-1/2}}{2\log 2}$ for $s=\lceil\frac{\log K}{\log
2}\rceil$. For $K\geq 8$, one can use calculus to verify the second inequality
of the following:
$K^{1/2}2^{-\lceil\frac{\log K}{\log 2}\rceil}\leq K^{1/2}2^{-\frac{\log
K}{\log 2}}\leq\frac{1}{2\log 2}\bigg{(}\frac{\log K}{\log
2}+1\bigg{)}^{-1/2}\leq\frac{1}{2\log 2}\bigg{\lceil}\frac{\log K}{\log
2}\bigg{\rceil}^{-1/2},$
meaning $t\leq\lceil\frac{\log K}{\log 2}\rceil$. Substituting
$t\leq\frac{\log K}{\log 2}+1$ and $t\geq 1$ into (30) then gives
$\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}<4\hat{\theta}\bigg{(}\frac{\log
K}{\log 2}+1+\frac{1}{\log 2}+\frac{1}{(2\log 2)^{2}}\bigg{)}\\\
=\hat{\theta}(C_{0}\log K+C_{1}),$
with $C_{0}\approx 5.77$, $C_{1}\approx 11.85$. As such, (23) is $\leq
C^{\prime}\hat{\theta}\log K$ with $C^{\prime}=C_{0}+\frac{C_{1}}{\log 2}$ in
this case.
We are now ready for the final bound on (23) in which we apply no constraints
on the $x_{i}$’s and $y_{j}$’s. To do this, we consider the positive and
negative real and imaginary parts of these coefficients:
$x_{i}=\sum_{k=0}^{3}x_{i,k}\mathrm{i}^{k}\quad\mbox{s.t.}\quad x_{i,k}\geq
0\quad\forall k,$
and similarly for the $y_{j}$’s. With this decomposition, we apply the
triangle inequality to get
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}\sum_{k_{1}=0}^{3}x_{i,k_{1}}\mathrm{i}^{k_{1}}\varphi_{i},\sum_{j\in\mathcal{J}}\sum_{k_{2}=0}^{3}y_{j,k_{2}}\mathrm{i}^{k_{2}}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{3}\sum_{k_{2}=0}^{3}\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i,k_{1}}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j,k_{2}}\varphi_{j}\bigg{\rangle}\bigg{|}.$
Finally, we normalize the coefficients by
$(\sum_{i\in\mathcal{I}}x_{i,k_{1}}^{2})^{1/2}$ and
$(\sum_{j\in\mathcal{J}}y_{j,k_{2}}^{2})^{1/2}$ so we can apply our second
bound:
$\displaystyle\bigg{|}\bigg{\langle}\sum_{i\in\mathcal{I}}x_{i}\varphi_{i},\sum_{j\in\mathcal{J}}y_{j}\varphi_{j}\bigg{\rangle}\bigg{|}$
$\displaystyle\leq\sum_{k_{1}=0}^{3}\sum_{k_{2}=0}^{3}\bigg{(}\sum_{i\in\mathcal{I}}x_{i,k_{1}}^{2}\bigg{)}^{1/2}\bigg{(}\sum_{j\in\mathcal{J}}y_{j,k_{2}}^{2}\bigg{)}^{1/2}C^{\prime}\hat{\theta}\log
K$ $\displaystyle\leq(C\hat{\theta}\log K)\|x\|\|y\|,$
where $C=4C^{\prime}\approx 74.17$ by the Cauchy-Schwarz inequality, and so we
are done. ∎
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|
arxiv-papers
| 2012-02-06T18:32:56 |
2024-09-04T02:49:27.092088
|
{
"license": "Public Domain",
"authors": "Afonso S. Bandeira, Matthew Fickus, Dustin G. Mixon and Percy Wong",
"submitter": "Dustin Mixon",
"url": "https://arxiv.org/abs/1202.1234"
}
|
1202.1309
|
A Decidable Fragment of Strategy Logic
A Decidable Fragment of Strategy Logic
Fabio Mogavero$^{1}$, Aniello Murano$^{1}$, Giuseppe Perelli$^{1}$, and
Moshe Y. Vardi$^{2}$
F. Mogavero, A. Murano, G. Perelli, and M.Y. Vardi
{mogavero, murano}@na.infn.it perelli.gi@gmail.com
$^{1}$Universitá degli Studi di Napoli "Federico II", Napoli, Italy.
$^{2}$Rice University, Houston, Texas, USA.
Strategy Logic (, for short) has been recently introduced by
Mogavero, Murano, and Vardi as a useful formalism for reasoning explicitly
about strategies, as first-order objects, in multi-agent concurrent games.
This logic turns to be very powerful, subsuming all major previously studied
modal logics for strategic reasoning, including , , and the like.
Unfortunately, due to its expressiveness, has a non-elementarily
decidable model-checking problem and a highly undecidable satisfiability
problem, specifically, 11.
In order to obtain a decidable sublogic, we introduce and study here
One-Goal Strategy Logic (, for short).
This logic is a syntactic fragment of , strictly subsuming , which
encompasses formulas in prenex normal form having a single temporal goal at a
time, for every strategy quantification of agents.
is known to have an elementarily decidable model-checking problem.
Here we prove that, unlike , it has the bounded tree-model property and its
satisfiability problem is decidable in 2, thus not harder than the one
for .
In open-system verification <cit.>, an important area of research
is the study of modal logics for strategic reasoning in the setting of
multi-agent games <cit.>.
An important contribution in this field has been the development of
Alternating-Time Temporal Logic (, for short), introduced by
Alur, Henzinger, and Kupferman <cit.>.
allows reasoning about strategic behavior of agents with temporal
Formally, it is obtained as a generalization of the branching-time temporal
logic <cit.>, where the path quantifiers there exists
“$\E$” and for all “$\A$” are replaced with strategic modalities
of the form “$\EExs{\ASet}$” and “$\AAll{\ASet}$”, for a set $\ASet$ of
Such strategic modalities are used to express cooperation and competition
among agents in order to achieve certain temporal goals.
In particular, these modalities express selective quantifications over those
paths that are the results of infinite games between a coalition and its
formulas are interpreted over concurrent game structures (,
for short) <cit.>, which model interacting processes.
Given a $\GName$ and a set $\ASet$ of agents, the formula
$\EExs{\ASet} \psi$ holds at a state $\sElm$ of $\GName$ if there is a set of
strategies for the agents in $\ASet$ such that, no matter which strategy is
executed by the agents not in $\ASet$, the resulting outcome of the
interaction in $\GName$ satisfies $\psi$ at $\sElm$.
Several decision problems have been investigated about ; both its
model-checking and satisfiability problems are decidable in
2 <cit.>.
The complexity of the latter is just like the one for <cit.>.
Despite its powerful expressiveness, suffers from the strong limitation
that strategies are treated only implicitly through modalities that refer to
games between competing coalitions.
To overcome this problem, Chatterjee, Henzinger, and Piterman introduced
Strategy Logic (, for short) <cit.>, a logic that treats
strategies in two-player turn-based games as first-order
The explicit treatment of strategies in this logic allows the expression of
many properties not expressible in .
Although the model-checking problem of is known to be decidable, with
a non-elementary upper bound, it is not known if the satisfiability problem is
decidable <cit.>.
While the basic idea exploited in <cit.> of explicitly quantify over
strategies is powerful and useful <cit.>, still suffers from
various limitations.
In particular, it is limited to two-player turn-based games.
Furthermore, does not allow different players to share the same
strategy, suggesting that strategies have yet to become truly first-class
objects in this logic.
For example, it is impossible to describe the classic strategy-stealing
argument of combinatorial games such as Chess, Go, Hex, and the
like <cit.>.
These considerations led us to introduce and investigate a new Strategy
Logic, denoted , as a more general framework than , for explicit
reasoning about strategies in multi-agent concurrent games <cit.>.
Syntactically, extends the linear-time temporal-logic <cit.>
by means of strategy quantifiers, the existential $\EExs{\xElm}$ and
the universal $\AAll{\xElm}$, as well as agent binding $(\aElm,
\xElm)$, where $\aElm$ is an agent and $\xElm$ a variable.
Intuitively, these elements can be read as “there exists a strategy
$\xElm$”, “for all strategies $\xElm$”, and “bind agent
$\aElm$ to the strategy associated with $\xElm$”, respectively.
For example, in a $\GName$ with agents $\alpha$, $\beta$, and $\gamma$,
consider the property “$\alpha$ and $\beta$ have a common strategy to avoid a
This property can be expressed by the formula $\EExs{\xSym} \AAll{\ySym}
(\alpha, \xSym) (\beta, \xSym) (\gamma, \ySym) (\G \neg \mathit{fail})$.
The variable $\xSym$ is used to select a strategy for the agents $\alpha$ and
$\beta$, while $\ySym$ is used to select another one for agent $\gamma$ such
that their composition, after the binding, results in a play where
$\mathit{fail}$ is never met.
Further examples, motivations, and results can be found in a technical
report <cit.>.
The price that one has to pay for the expressiveness of w.r.t. is
the lack of important model-theoretic properties and an increased complexity
of related decision problems.
In particular, in <cit.>, it was shown that does not have the
bounded-tree model property and the related satisfiability problem is
The contrast between the undecidability of the satisfiability problem for and the elementary decidability of the same problem for , provides
motivation for an investigation of decidable fragments of that subsume
In particular, we would like to understand why is computationally more
difficult than .
We introduce here the syntactic fragment One-Goal Strategy Logic
(, for short), which encompasses formulas in a special prenex normal form
having a single temporal goal at a time.
This means that every temporal formula $\psi$ is prefixed with a
quantification-binding prefix that quantifies over a tuple of strategies and
bind strategies to all agents.
With one can express, for example, visibility constraints on
strategies among agents, i.e., only some agents from a coalition have
knowledge of the strategies taken by those in the opponent coalition.
Also, one can describe the fact that, in the Hex game, the
strategy-stealing argument does not let the player who adopts it to win.
Observe that both the above properties cannot be expressed neither in nor in .
In a technical report <cit.>, we showed that is strictly more
expressive that , yet its model-checking problem is 2, just like
the one for , while the same problem for is non-elementarily
Our main result here is that the satisfiability problem for is also
Thus, in spite of its expressiveness, has the same computational
properties of , which suggests that the one-goal restriction is the key
to the elementary complexity of the latter logic too.
To achieve our main result, we use a fundamental property of the semantics of
called elementariness, which allows us to simplify reasoning
about strategies by reducing it to a set of reasonings about actions.
This intrinsic characteristic of means that, to choose an existential
strategy, we do not need to know the entire structure of
universally-quantified strategies, as it is the case for , but only their
values on the histories of interest.
Technically, to formally describe this property, we make use of the machinery
of dependence maps, which is introduced to define a Skolemization
procedure for , inspired by the one in first-order logic.
Using elementariness, we show that satisfies the bounded
tree-model property.
This allows us to efficiently make use of a tree automata-theoretic
approach <cit.> to solve the satisfiability problem.
Given a formula $\varphi$, we build an alternating co-Büchi tree
automaton <cit.>, whose size is only exponential in the size of
$\varphi$, accepting all bounded-branching tree models of the formula.
Then, together with the complexity of automata-nonemptiness checking, we get
that the satisfiability procedure for is 2.
We believe that our proof techniques are of independent interest and
applicable to other logics as well.
Related works.
Several works have focused on extensions of to incorporate more
powerful strategic constructs.
Among them, we recall the Alternating-Time (,
for short) <cit.>, Game Logic (, for short) <cit.>,
Quantified Decision Modality (qD$\mu$, for
short) <cit.>, Coordination Logic (, for short) <cit.>,
and some other extensions considered in <cit.>, <cit.>,
and <cit.>.
and qD$\mu$ are intrinsically different from (as well as from and ) as they are obtained by extending the
propositional $\mu$-calculus <cit.> with strategic modalities.
is similar to qD$\mu$, but with temporal operators
instead of explicit fixpoint constructors.
and are orthogonal to .
Indeed, they both use more than a temporal goal, has quantifier
alternation fixed to one, and only works for two agents.
The paper is almost self contained; all proofs are reported in the appendixes.
In Appendix <ref>, we recall standard mathematical notation and
some basic definitions that are used in the paper.
Additional details on can be found in the technical
report <cit.>.
A concurrent game structure (, for short) <cit.> is a
tuple $\GName \defeq \CGSStruct$, where $\APSet$ and $\AgSet$ are finite
non-empty sets of atomic propositions and agents, $\AcSet$ and
$\StSet$ are enumerable non-empty sets of actions and states,
$\sElm[0] \in \StSet$ is a designated initial state, and $\labFun :
\StSet \to \pow{\APSet}$ is a labeling function that maps each state to
the set of atomic propositions true in that state.
Let $\DecSet \defeq \AcSet^{\AgSet}$ be the set of decisions, i.e.,
functions from $\AgSet$ to $\AcSet$ representing the choices of an action for
each agent.
Then, $\trnFun : \StSet \times \DecSet \to \StSet$ is a transition
function mapping a pair of a state and a decision to a state.
If the set of actions is finite, i.e., $b = \card{\AcSet} < \omega$, we say
that $\GName$ is $b$-bounded, or simply bounded.
If both the sets of
A track (resp., path) in a $\GName$ is a finite (resp., an
infinite) sequence of states $\trkElm \in \StSet^{*}$ (resp., $\pthElm \in
\StSet^{\omega}$) such that, for all $i \in \numco{0}{\card{\trkElm} - 1}$
(resp., $i \in \SetN$), there exists a decision $\decFun \in \DecSet$ such
that $(\trkElm)_{i + 1} = \trnFun((\trkElm)_{i}, \decFun)$ (resp.,
$(\pthElm)_{i + 1} = \trnFun((\pthElm)_{i}, \decFun)$).
A track $\trkElm$ is non-trivial if $\card{\trkElm} > 0$, i.e.,
$\trkElm \neq \epsilon$.
$\TrkSet \subseteq \StSet^{+}$ (resp., $\PthSet \subseteq \StSet^{\omega}$)
denotes the set of all non-trivial tracks (resp., paths).
Moreover, $\TrkSet(\sElm) \defeq \set{ \trkElm \in \TrkSet }{ \fst{\trkElm} =
\sElm }$ (resp., $\PthSet(\sElm) \defeq \set{ \pthElm \in \PthSet }{
\fst{\pthElm} = \sElm }$) indicates the subsets of tracks (resp., paths)
starting at a state $\sElm \in \StSet$.
A strategy is a partial function $\strFun : \TrkSet \pto \AcSet$ that
maps each non-trivial track in its domain to an action.
For a state $\sElm \in \StSet$, a strategy $\strFun$ is said
$\sElm$-total if it is defined on all tracks starting in $\sElm$, i.e.,
$\dom{\strFun} = \TrkSet(\sElm)$.
$\StrSet \defeq \TrkSet \pto \AcSet$ (resp., $\StrSet(\sElm) \defeq
\TrkSet(\sElm) \to \AcSet$) denotes the set of all (resp., $\sElm$-total)
For all tracks $\rho \in \TrkSet$, by $(\strFun)_{\trkElm} \in \StrSet$ we
denote the translation of $\strFun$ along $\trkElm$, i.e., the strategy
with $\dom{(\strFun)_{\trkElm}} \defeq \set{ \lst{\trkElm} \cdot \trkElm' }{
\trkElm \cdot \trkElm' \in \dom{\strFun} }$ such that
$(\strFun)_{\trkElm}(\lst{\trkElm} \cdot \trkElm') \defeq \strFun(\trkElm
\cdot \trkElm')$, for all $\trkElm \cdot \trkElm' \in \dom{\strFun}$.
Let $\VarSet$ be a fixed set of variables.
An assignment is a partial function $\asgFun : \VarSet \cup \AgSet \pto
\StrSet$ mapping variables and agents in its domain to a strategy.
An assignment $\asgFun$ is complete if it is defined on all agents,
i.e., $\AgSet \subseteq \dom{\asgFun}$.
For a state $\sElm \in \StSet$, it is said that $\asgFun$ is
$\sElm$-total if all strategies $\asgFun(\lElm)$ are $\sElm$-total,
for $\lElm \in \dom{\asgFun}$.
$\AsgSet \defeq \VarSet \cup \AgSet \pto \StrSet$ (resp., $\AsgSet(\sElm)
\defeq \VarSet \cup \AgSet \pto \StrSet(\sElm)$) denotes the set of all
(resp., $\sElm$-total) assignments.
Moreover, $\AsgSet(\XSet) \defeq \XSet \to \StrSet$ (resp.,
$\AsgSet(\XSet, \sElm) \defeq \XSet \to \StrSet(\sElm)$) indicates the
subset of $\XSet$-defined (resp., $\sElm$-total) assignments, i.e.,
(resp., $\sElm$-total) assignments defined on the set $\XSet \subseteq
\VarSet \cup \AgSet$.
For all tracks $\rho \in \TrkSet$, by $(\asgFun)_{\trkElm} \in
\AsgSet(\lst{\trkElm})$ we denote the translation of $\asgFun$ along
$\trkElm$, i.e., the $\lst{\trkElm}$-total assignment with
$\dom{(\asgFun)_{\trkElm}} \defeq \dom{\asgFun}$, such that
$(\asgFun)_{\trkElm}(\lElm) \defeq (\asgFun(\lElm))_{\trkElm}$, for all $\lElm
\in \dom{\asgFun}$.
For all elements $\lElm \in \VarSet \cup \AgSet$, by $\asgFun[][\lElm \mapsto
\strFun] \in \AsgSet$ we denote the new assignment defined on
$\dom{\asgFun[][\lElm \mapsto \strFun]} \defeq \dom{\asgFun} \cup \{ \lElm \}$
that returns $\strFun$ on $\lElm$ and $\asgFun$ otherwise, i.e.,
$\asgFun[][\lElm \!\mapsto\! \strFun](\lElm) \!\defeq\! \strFun$ and
$\asgFun[][\lElm \!\mapsto\! \strFun](\lElm') \!\defeq\! \asgFun(\lElm')$, for
all $\lElm' \!\in\! \dom{\asgFun} \!\setminus\! \{ \lElm \}$.
A path $\playElm \in \PthSet(\sElm)$ starting at a state $\sElm \in \StSet$ is
a play w.r.t. a complete $\sElm$-total assignment $\asgFun \in
\AsgSet(\sElm)$ ($(\asgFun, \sElm)$-play, for short) if, for all $i \in
\SetN$, it holds that $(\playElm)_{i + 1} = \trnFun((\playElm)_{i}, \decFun)$,
where $\decFun(\aElm) \defeq \asgFun(\aElm)((\playElm)_{\leq i})$, for each
$\aElm \in \AgSet$.
The partial function $\playFun : \AsgSet \times \StSet \pto \PthSet$, with
$\dom{\playFun} \defeq \set{ (\asgFun, \sElm) }{ \AgSet \subseteq
\dom{\asgFun} \land \asgFun \in \AsgSet(\sElm) \land \sElm \in \StSet }$,
returns the $(\asgFun, \sElm)$-play $\playFun(\asgFun, \sElm) \in
\PthSet(\sElm)$, for all $(\asgFun, \sElm)$ in its domain.
For a state $\sElm \in \StSet$ and a complete $\sElm$-total assignment
$\asgFun \in \AsgSet(\sElm)$, the $i$-th global translation of
$(\asgFun, \sElm)$, with $i \in \SetN$, is the pair of a complete assignment
and a state $(\asgFun, \sElm)^{i} \defeq ((\asgFun)_{(\playElm)_{\leq i}},
(\playElm)_{i})$, where $\playElm = \playFun(\asgFun, \sElm)$.
From now on, we use the name of a as a subscript to extract the
components from its tuple-structure.
Accordingly, if $\GName = \CGSStruct$, we have $\AcSet[\GName] = \AcSet$,
$\labFun[\GName] = \labFun$, $\sElm[0\GName] = \sSym[0]$, and so on.
Also, we use the same notational concept to make explicit to which the
sets $\DecSet$, $\TrkSet$, $\PthSet$, etc. are related to.
Note that, we omit the subscripts if the structure can be unambiguously
individuated from the context.
One-Goal Strategy Logic
In this section, we introduce syntax and semantics of One-Goal Strategy Logic
(, for short), as a syntactic fragment of , which we also report here
for technical reasons.
For more about , see <cit.>.
syntactically extends by means of two strategy
quantifiers, existential $\EExs{\xElm}$ and universal $\AAll{\xElm}$, and
agent binding $(\aElm, \xElm)$, where $\aElm$ is an agent and $\xElm$
is a variable.
Intuitively, these elements can be read, respectively, as “there
exists a strategy $\xElm$”, “for all strategies $\xElm$”, and
“bind agent $\aElm$ to the strategy associated with the variable
formulas are built inductively from the sets of atomic
propositions $\APSet$, variables $\VarSet$, and agents $\AgSet$, by using
the following grammar, where $\pElm \in \APSet$, $\xElm \in \VarSet$, and
$\aElm \in \AgSet$:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \EExs{\xElm} \varphi \mid \AAll{\xElm} \varphi
\mid (\aElm, \xElm) \varphi$.
By $\mthfun{sub}(\varphi)$ we denote the set of all subformulas of
the formula $\varphi$.
For instance, with $\varphi = \EExs{\xSym} (\alpha, \xSym) (\F \pSym)$, we
have that $\sub{\varphi} = \{ \varphi, (\alpha, \xSym) (\F \pSym), (\F
\pSym), \pSym, \Tt \}$.
By $\free{\varphi}$ we indicate the set of free agents/variables
of $\varphi$ defined as the subset of $\AgSet \cup \VarSet$ containing
(i) all the agents for which there is no variable application before
the occurrence of a temporal operator and (ii) all the variables for
which there is an application but no quantification.
For example, let $\varphi = \EExs{\xElm} (\alpha, \xElm)(\beta, \yElm)(\F
\pElm)$ be the formula on agents $\AgSet = \{ \alpha, \beta, \gamma \}$.
Then, we have $\free{\varphi} = \{ \gamma, \yElm \}$, since $\gamma$ is an
agent without any application before $\F \pElm$ and $\yElm$ has no
quantification at all.
A formula $\varphi$ without free agents (resp., variables), i.e., with
$\free{\varphi} \cap \AgSet = \emptyset$ (resp., $\free{\varphi} \cap
\VarSet = \emptyset)$, is named agent-closed (resp.,
If $\varphi$ is both agent- and variable-closed, it is named
By $\mthfun{snt}(\varphi)$ we denote the set of all sentences that are
subformulas of $\varphi$.
As for , we define the semantics of w.r.t. concurrent game
For a $\GName$, a state $\sElm$, and an $\sElm$-total assignment
$\asgFun$ with $\free{\varphi} \subseteq \dom{\asgFun}$, we write $\GName,
\asgFun, \sElm \models \varphi$ to indicate that the formula $\varphi$ holds
at $\sElm$ under the assignment $\asgFun$.
The semantics of formulas involving $\pElm$, $\neg$, $\wedge$, and
$\vee$ is defined as usual in and we omit it here (see <cit.>,
for the full definition).
The semantics of the remaining part, which involves quantifications,
bindings, and temporal operators follows.
Given a $\GName$, for all formulas $\varphi$, states $\sElm \in
\StSet$, and $\sElm$-total assignments $\asgFun \in \AsgSet(\sElm)$ with
$\free{\varphi} \subseteq \dom{\asgFun}$, the relation $\GName, \asgFun,
\sElm \models \varphi$ is inductively defined as follows.
$\GName, \asgFun, \sElm \models \EExs{\xElm} \varphi$ iff there
exists an $\sElm$-total strategy $\strFun \in \StrSet(\sElm)$ such
that $\GName, \asgFun[][\xElm \mapsto \strFun], \sElm \models
\varphi$;
$\GName, \asgFun, \sElm \models \AAll{\xElm} \varphi$ iff for all
$\sElm$-total strategies $\strFun \in \StrSet(\sElm)$ it holds
that $\GName, \asgFun[][\xElm \mapsto \strFun], \sElm \models
\varphi$.
Moreover, if $\free{\varphi} \cup \{\xElm \} \subseteq \dom{\asgFun}
\cup \{\aElm \}$ for an agent $\aElm \in \AgSet$, it holds that:
$\GName, \asgFun, \sElm \models (\aElm, \xElm) \varphi$ iff $\GName,
\asgFun[][\aElm \mapsto \asgFun(\xElm)], \sElm \models \varphi$.
Finally, if $\asgFun$ is also complete, it holds that:
$\GName, \asgFun, \sElm \models \X \varphi$ if $\GName, (\asgFun,
\sElm)^{1} \models \varphi$;
$\GName, \asgFun, \sElm \models \varphi_{1} \U \varphi_{2}$ if there
is an index $i \in \SetN$ with $k \!\leq\! i$ such that $\GName,
(\asgFun, \sElm)^{i} \!\models\! \varphi_{2}$ and, for all indexes
$j \!\in\! \SetN$ with $k \!\leq j \!<\! i$, it holds that $\GName,
(\asgFun, \sElm)^{j} \!\models\! \varphi_{1}$;
$\GName, \asgFun, \sElm \models \varphi_{1} \R \varphi_{2}$ if, for
all indexes $i \in \SetN$ with $k \!\leq\! i$, it holds that $\GName,
(\asgFun, \sElm)^{i} \!\models\! \varphi_{2}$ or there is an index
$j \!\in\! \SetN$ with $k \!\leq\! j \!<\! i$ such that
Intuitively, at Items <ref>
and <ref>, respectively, we evaluate the existential
$\EExs{\xElm}$ and universal $\AAll{\xElm}$ quantifiers over strategies, by
associating them to the variable $\xElm$.
Moreover, at Item <ref>, by means of an agent binding
$(\aElm, \xElm)$, we commit the agent $\aElm$ to a strategy associated with
the variable $\xElm$.
It is evident that the semantics is simply embedded into the one.
A $\GName$ is a model of an sentence $\varphi$, denoted by
$\GName \models \varphi$, iff $\GName , \emptyset, \sElm_{0} \models
\varphi$, where $\emptyset$ is the empty assignment.
Moreover, $\varphi$ is satisfiable iff there is a model for it.
Given two s $\GName_{1}$, $\GName_{2}$ and a sentence $\varphi$, we
say that $\varphi$ is invariant under $\GName_{1}$ and $\GName_{2}$
iff it holds that: $\GName_{1} \models \varphi$ iff $\GName_{2} \models
\varphi$.
Finally, given two formulas $\varphi_{1}$ and $\varphi_{2}$ with
$\free{\varphi_{1}} = \free{\varphi_{2}}$, we say that $\varphi_{1}$
implies $\varphi_{2}$, in symbols $\varphi_{1} \implies \varphi_{2}$,
if, for all s $\GName$, states $\sElm \in \StSet$, and
$\free{\varphi_{1}}$-defined $\sElm$-total assignments $\asgFun \in
\AsgSet(\free{\varphi_{1}}, \sElm)$, it holds that if $\GName, \asgFun,
\sElm \models \varphi_{1}$ then $\GName, \asgFun, \sElm \models
\varphi_{2}$.
Accordingly, we say that $\varphi_{1}$ is equivalent to
$\varphi_{2}$, in symbols $\varphi_{1} \equiv \varphi_{2}$, if $\varphi_{1}
\implies \varphi_{2}$ and $\varphi_{2} \implies \varphi_{1}$.
As an example, consider the sentence $\varphi \!=\! \EExs{\xSym} \!
\AAll{\ySym} \allowbreak \EExs{\zSym} ((\alpha, \xSym) (\beta, \ySym) (\X
\pSym) \!\wedge\! (\alpha, \ySym) (\beta, \zSym) (\X \qSym))$.
Note that both agents $\alpha$ and $\beta$ use the strategy associated with
$\ySym$ to achieve simultaneously the goals $\X \pSym$ and $\X \qSym$,
A model for $\varphi$ is the $\GName \defeq \CGSTuple {\{ \pSym, \qSym
\}} {\{ \alpha, \beta\}} {\{ 0, 1 \}} {\{ \sSym[0], \sSym[1], \sSym[2],
\sSym[3] \}} {\labFun} {\trnFun} {\sSym[0]}$, where $\labFun(\sSym[0])
\defeq \emptyset$, $\labFun(\sSym[1]) \defeq \{ \pSym \}$,
$\labFun(\sSym[2]) \defeq \{ \pSym, \qSym \}$, $\labFun(\sSym[3]) \defeq \{
\qSym \}$, $\trnFun(\sSym[0]$, $ (0, 0)) \defeq \sSym[1]$,
$\trnFun(\sSym[0], (0, 1)) \defeq \sSym[2]$, $\trnFun(\sSym[0], (1, 0))
\defeq \sSym[3]$, and all the remaining transitions go to $\sSym[0]$.
See the representation of $\GName$ depicted in Figure <ref>, in
which vertexes are states of the game and labels on edges represent
decisions of agents or sets of them, where the symbol $*$ is used in place
of every possible action.
Clearly, $\GName \models \varphi$ by letting, on $\sSym[0]$, the variables
$\xSym$ to chose action $0$ (the formula $(\alpha, \xSym) (\beta, \ySym) (\X
\pSym)$ is satisfied for any choice of $\ySym$, since we can move from
$\sSym[0]$ to either $\sSym[1]$ or $\sSym[2]$, both labeled with $\pSym$)
and $\zSym$ to choose action $1$ when $\ySym$ has action $0$ and, vice
versa, $0$ when $\ySym$ has $1$ (in both cases, the formula $(\alpha, \ySym)
(\beta, \zSym) (\X \qSym)$ is satisfied, since one can move from $\sSym[0]$
to either $\sSym[2]$ or $\sSym[3]$, both labeled with $\qSym$).
To formalize the syntactic fragment of , we need first to define
the concepts of quantification and binding prefixes.
A quantification prefix over a set $\VSet \!\subseteq\! \VarSet$ of
variables is a finite word $\qpElm \!\in\! \set{ \EExs{\xElm},
\AAll{\xElm} }{ \xElm \in \VSet }^{\card{\VSet}}$ of length $\card{\VSet}$
such that each variable $\xElm \in \VSet$ occurs just once in
A binding prefix over a set $\VSet \subseteq \VarSet$ of variables
is a finite word $\bpElm \in \set{ (\aElm, \xElm) }{ \aElm \in \AgSet
\land \xElm \in \VSet }^{\card{\AgSet}}$ of length $\card{\AgSet}$ such
that each agent $\aElm \in \AgSet$ occurs just once in $\bpElm$.
Finally, $\QPSet(\VSet) \subseteq \set{ \EExs{\xElm}, \AAll{\xElm} }{
\xElm \in \VSet }^{\card{\VSet}}$ and $\BPSet(\VSet) \subseteq \set{
(\aElm, \xElm) }{ \aElm \in \AgSet \land \xElm \in \VSet
}^{\card{\AgSet}}$ denote, respectively, the sets of all quantification
and binding prefixes over variables in $\VSet$.
We can now define the syntactic fragment we want to analyze.
The idea is to force each group of agent bindings, represented by a binding
prefix, to be coupled with a quantification prefix.
formulas are built inductively from the sets of atomic propositions
$\APSet$, quantification prefixes $\QPSet(\VSet)$, for $\VSet \subseteq
\VarSet$, and binding prefixes $\BPSet(\VarSet)$, by using the following
grammar, with $\pElm \in \APSet$, $\qpElm \in \cup_{\VSet \subseteq
\VarSet} \QPSet(\VSet)$, and $\bpElm \in \BPSet(\VarSet)$:
$\varphi ::= \pElm \mid \neg \varphi \mid \varphi \wedge \varphi \mid
\varphi \vee \varphi \mid \X \varphi \mid \varphi \:\U \varphi \mid
\varphi \:\R \varphi \mid \qpElm \bpElm \varphi$,
with $\qpElm \in \QPSet(\free{\bpElm \varphi})$, in the formation rule
$\qpElm \bpElm \varphi$.
In the following, for a goal we mean an agent-closed formula of
the kind $\bpElm \psi$, where $\psi$ is variable-closed and $\bpElm \in
\BPSet(\free{\psi})$.
Note that, since $\bpElm \varphi$ is a goal, it is agent-closed, so,
$\free{\bpElm \varphi} \subseteq \VarSet$.
Moreover, an sentence $\varphi$ is principal if it is of the
form $\varphi = \qpElm \bpElm \psi$, where $\bpElm \psi$ is a goal and
$\qpElm \in \QPSet(\free{\bpElm \psi})$.
By $\psnt{\varphi} \subseteq \snt{\varphi}$ we denote the set of
principal subsentences of the formula $\varphi$.
As an example, let $\varphi_{1} = \qpSym \bpSym[1] \psi_{1}$ and
$\varphi_{2} = \qpSym (\bpSym[1] \psi_{1} \wedge \bpSym[2] \psi_{2})$, where
$\qpSym = \AAll{\xSym} \EExs{\ySym} \AAll{\zSym}$, $\bpSym[1] = (\alpha,
\xSym) (\beta, \ySym) (\gamma, \zSym)$, $\bpSym[2] = (\alpha,
\ySym) (\beta, \zSym) (\gamma, \ySym)$, $\psi_{1} = \X \pSym$, and $\psi_{2}
= \X \qSym$.
Then, it is evident that $\varphi_{1} \in \OGSL$ but $\varphi_{2} \not\in
\OGSL$, since the quantification prefix $\qpSym$ of the latter does not have
in its scope a unique goal.
It is fundamental to observe that the formula $\varphi_{1}$ of the above
example cannot be expressed in , as proved in <cit.> and
reported in the following theorem, since its $2$-quantifier alternation
cannot be encompassed in the $1$-alternation modalities.
On the contrary, each formula of the type $\EExs{\ASet} \psi$, where
$\ASet = \{\alpha_{1}, \ldots, \alpha_{n}\} \subseteq \AgSet = \{\alpha_{1},
\ldots, \alpha_{n}, \beta_{1}, \ldots, \beta_{m}\}$ can be expressed in
as follows: $\EExs{\xSym[1]} \cdots \EExs{\xElm[n]} \AAll{\ySym[1]}
\allowbreak \cdots \allowbreak \AAll{\ySym[m]} (\alpha_{1}, \xSym[1]) \cdots
(\alpha_{n}, \xSym[n]) (\beta_{1}, \ySym[1]) \cdots (\beta_{m}, \ySym[m])
\psi$.
is strictly more expressive than .
We now give two examples in which we show the importance of the ability to
write specifications with alternation of quantifiers greater than $1$ along
with strategy sharing.
[Escape from Alcatraz[We thank Luigi Sauro for
having pointed out this example.]]
Consider the situation in which an Alcatraz prisoner tries to escape from
jail with the help of an external accomplice of him, by helicopter.
Due to his panoramic point of view, assume that the accomplice has the
full visibility on the behaviors of guards, while the prisoner does
not have the same ability.
Therefore, the latter has to put in practice an escape strategy that,
independently from guards moves, can be supported by his accomplice to
We can formalize such an intricate situation by means of an sentence as follows.
First, let $\GName[A]$ be a modeling the possible situations in
which the agents “$\pSym$” prisoner, “$\gSym$” guards, and “$\aSym$”
accomplice can reside, together with all related possible moves.
Then, we can verify the existence of an escape strategy by checking the
assertion $\GName[A] \models \EExs{\xSym} \AAll{\ySym} \EExs{\zSym}
(\pSym, \xSym) (\gSym, \ySym) (\aSym, \zSym) (\F \mthfun[\PSym]{free})$.
[Stealing-Strategy in Hex]
Hex is a two-player game, red vs blue, in which each player in
turn places a stone of his color on a single empty hexagonal cell of the
rhomboidal playing board having opposite sides equally colored, either red
or blue.
The goal of each player is to be the first to form a path connecting the
opposing sides of the board marked by his color.
It is easy to prove that the stealing-strategy argument does not lead to a
winning strategy in Hex, i.e., if the player that moves second copies the
moves of the opponent, he surely loses the play.
It is possible to formalize this fact in as follows.
First model Hex with a $\GName[H]$ whose states represent a possible
possible configurations reached during a play between “$\rSym$” red and
“$\bSym$” blue.
Then, express the negation of the stealing-strategy argument by asserting
that $\GName[H] \models \EExs{\xSym} (\rSym, \xSym) (\bSym, \xSym) (\F
\mthfun[\rSym]{cnc})$.
Intuitively, this sentence says that agent $\rSym$ has a strategy that,
once it is copied (binded) by $\bSym$ it allows the former to win, i.e.,
to be the first to connect the related red edges ($\F
\mthfun[\rSym]{cnc}$).
Strategy Quantifications
We now define the concept of dependence map.
The key idea is that every quantification prefix occurring in an formula
can be represented by a suitable choice of a dependence map over strategies.
Such a result is at the base of the definition of the elementariness
property and allows us to prove that is elementarily satisfiable, i.e.,
we can simplify a reasoning about strategies by reducing it to a set of local
reasonings about actions <cit.>.
Dependence map
First, we introduce some notation regarding quantification prefixes.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\QPVSet(\qpElm) \defeq \VSet \subseteq \VarSet$ of variables.
By $\QPEVSet{\qpElm} \defeq \set{ \xElm \in \VSet }{ \exists i \in
\numco{0}{\card{\qpElm}} .\: (\qpElm)_{i} = \EExs{\xElm} }$ and
$\QPAVSet{\qpElm} \defeq \VSet \setminus \QPEVSet{\qpElm}$ we denote,
respectively, the sets of existential and universal variables
quantified in $\qpElm$.
For two variables $\xElm, \yElm \in \VSet$, we say that $\xElm$
precedes $\yElm$ in $\qpElm$, in symbols $\xElm \qpordRel[\qpElm]
\yElm$, if $\xElm$ occurs before $\yElm$ in $\qpElm$.
Moreover, by $\QPDepSet(\qpElm) \defeq \set{ (\xElm, \yElm) \in \VSet \times
\VSet }{ \xElm \in \QPAVSet{\qpElm}, \yElm \in \QPEVSet{\qpElm} \land \xElm
\qpordRel[\qpElm] \yElm}$ we denote the set of dependence pairs,
i.e., a dependence relation, on which we derive the parameterized version
$\QPDepSet(\qpElm, \yElm) \defeq \set{ \xElm \in \VSet }{ (\xElm, \yElm) \in
\QPDepSet(\qpElm)}$ containing all variables from which $\yElm$ depends.
Also, we use $\dual{\qpElm} \in \QPSet(\VSet)$ to indicate the
quantification derived from $\qpElm$ by dualizing each quantifier
contained in it, i.e., for all $i \in \numco{0}{\card{\qpElm}}\!$, it holds
that $(\dual{\qpElm})_{i} = \EExs{\xElm}$ iff $(\qpElm)_{i} = \AAll{\xElm}$,
with $\xElm \in \VSet$.
Clearly, $\QPEVSet{\dual{\qpElm}} = \QPAVSet{\qpElm}$ and
$\QPAVSet{\dual{\qpElm}} = \QPEVSet{\qpElm}$.
Finally, we define the notion of valuation of variables over a
generic set $\DSet$ as a partial function $\valFun : \VarSet \pto \DSet$
mapping every variable in its domain to an element in $\DSet$.
By $\ValSet[\DSet](\VSet) \defeq \VSet \to \DSet$ we denote the set of all
valuation functions over $\DSet$ defined on $\VSet \subseteq \VarSet$.
We now give the semantics for quantification prefixes via the following
definition of dependence map.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set of
variables $\VSet \subseteq \VarSet$, and $\DSet$ a set.
Then, a dependence map for $\qpElm$ over $\DSet$ is a function
$\spcFun : \ValSet[\DSet](\QPAVSet{\qpElm}) \to \ValSet[\DSet](\VSet)$
satisfying the following properties:
$\spcFun(\valFun)_{\rst \QPAVSet{\qpElm}} \!=\! \valFun$, for all
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$;
$\spcFun(\valFun[1])(\xElm) \!=\! \spcFun(\valFun[2])(\xElm)$, for all
$\valFun[1], \valFun[2] \in \ValSet[\DSet](\QPAVSet{\qpElm})$ and
$\xElm \in \QPEVSet{\qpElm}$ such that $\valFun[1]_{\rst
\QPDepSet(\qpElm, \xElm)} \!=\! \valFun[2]_{\rst \QPDepSet(\qpElm,
\xElm)}$.
$\SpcSet[\DSet](\qpElm)$ denotes the set of all dependence maps for
$\qpElm$ over $\DSet$.
Intuitively, Item <ref> asserts that $\spcFun$ takes the
same values of its argument w.r.t. the universal variables in $\qpElm$ and
Item <ref> ensures that the value of $\spcFun$ w.r.t. an
existential variable $\xElm$ in $\qpElm$ does not depend on variables not in
$\QPDepSet(\qpElm, \xElm)$.
To get better insight into this definition, a dependence map $\spcFun$ for
$\qpElm$ can be considered as a set of Skolem functions that, given a
value for each variable in $\VSet$ that is universally quantified in
$\qpElm$, returns a possible value for all the existential variables in
$\qpElm$, in a way that is consistent w.r.t. the order of quantifications.
We now state a fundamental theorem that describes how to eliminate
strategy quantifications of an formula via a choice of a dependence map
over strategies.
This procedure, easily proved to be correct by induction on the structure of
the formula in <cit.>, can be seen as the equivalent of the
Skolemization in first order logic <cit.>.
Let $\GName$ be a and $\varphi = \qpElm \psi$ an sentence,
where $\psi$ is agent-closed and $\qpElm \in \QPSet(\free{\psi})$.
Then, $\GName \models \varphi$ iff there exists a dependence map
$\spcFun \in \SpcSet[ {\StrSet(\sElm[0])} ](\qpElm)$ such that $\GName,
\spcFun(\asgFun), \sElm[0] \models \psi$, for all $\asgFun \in
\AsgSet(\QPAVSet{\qpElm}, \sElm[0])$.
The above theorem substantially characterizes the semantics by means of
the concept of dependence map.
In particular, it shows that if a formula is satisfiable then it is always
possible to find a suitable dependence map returning the existential
strategies in response to the universal ones.
Such a characterization lends itself to define alternative
semantics of , based
on the choice of a subset of dependence maps that meet a certain given
We do this on the aim of identifying semantic fragments of having
better model properties and easier decision problems.
With more details, given a $\GName$, one of its states $\sElm$, and a
property $\PNot$, we say that a sentence $\qpElm \psi$ is
$\PNot$-satisfiable, in symbols $\GName \models_{\PNot} \qpElm \psi$, if
there exists a dependence map $\spcFun$ meeting $\PNot$ such that, for all
assignment $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$, it holds that
$\GName, \spcFun(\asgFun), \sElm \models \psi$.
Alternative semantics identified by a property $\PNot$ are even more
interesting if there exists a syntactic fragment corresponding to it, i.e.,
each satisfiable sentence of such a fragment is $\PNot$-satisfiable and vice
In the following, we put in practice this idea in order to show that has the same complexity of w.r.t. the satisfiability problem.
Elementary quantifications
According to the above description, we now introduce a suitable property
of dependence maps, called elementariness, together with the related
alternative semantics.
Then, in Theorem <ref>, we state that has the
elementariness property, i.e., each sentence is satisfiable iff it is
elementary satisfiable.
Intuitively, a dependence map $\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$
over functions from a set $\TSet$ to a set $\DSet$ is elementary if it can
be split into a set of dependence maps over $\DSet$, one for each element of
$\TSet$, represented by a function $\adj{\spcFun}: \TSet \to
\SpcSet[\DSet](\qpElm)$.
This idea allows us to enormously simplify the reasoning about strategy
quantifications, since we can reduce them to a set of quantifications over
actions, one for each track in their domains.
Note that sets $\DSet$ and $\TSet$, as well as $\USet$ and $\VSet$ used in
the following, are generic and in our framework they may refer to actions
and strategies ($\DSet$), tracks ($\TSet$), and variables ($\USet$ and
In particular, observe that functions from $\TSet$ to $\DSet$ represent
We prefer to use abstract name, as the properties we describe hold
To formally develop the above idea, we have first to introduce the generic
concept of adjoint function.
From now on, we denote by $\flip{\gFun}: \YSet \to (\XSet \to \ZSet)$ the
operation of flipping of a generic function $\gFun : \XSet \to (\YSet
\to \ZSet)$, i.e., the transformation of $\gFun$ by swapping the order of
its arguments.
Such a flipping is well-grounded due to the following chain of isomorphisms:
$\XSet \to (\YSet \to \ZSet) \cong (\XSet \times \YSet) \to \ZSet \cong
(\YSet \times \XSet) \to \ZSet \cong \YSet \to (\XSet \to \ZSet)$.
Let $\DSet$, $\TSet$, $\USet$, and $\VSet$ be four sets, and
$\mFun : (\TSet \to \DSet)^{\USet} \to (\TSet \to \DSet)^{\VSet}$ and
$\adj{\mFun} : \TSet \to (\DSet^{\USet} \to \DSet^{\VSet})$ two functions.
Then, $\adj{\mFun}$ is the adjoint of $\mFun$ if
$\adj{\mFun}(\tElm)(\flip{\gFun}(\tElm))(\xElm) =
\mFun(\gFun)(\xElm)(\tElm)$, for all $\gFun \in (\TSet \to
\DSet)^{\USet}$, $\xElm \in \VSet$, and $\tElm \in \TSet$.
Intuitively, a function $\mFun$ transforming a map of kind $(\TSet \to
\DSet)^{\USet}$ into a new map of kind $(\TSet \to \DSet)^{\VSet}$ has an
adjoint $\adj{\mFun}$ if such a transformation can be done pointwisely
w.r.t. the set $\TSet$, i.e., we can put out as a common domain the set
$\TSet$ and then transform a map of kind $\DSet^{\USet}$ in a map of kind
Observe that, if a function has an adjoint, this is unique.
Similarly, from an adjoint function it is possible to determine the original
function unambiguously.
Thus, it is established a one-to-one correspondence between functions
admitting an adjoint and the adjoint itself.
The formal meaning of the elementariness of a dependence map over
generic functions follows.
Let $\qpElm \in \QPSet(\VSet)$ be a quantification prefix over a set
$\VSet \subseteq \VarSet$ of variables, $\DSet$ and $\TSet$ two sets, and
$\spcFun \in \SpcSet[\TSet \to \DSet](\qpElm)$ a dependence map
for $\qpElm$ over $\TSet \to \DSet$.
Then, $\spcFun$ is elementary if it admits an adjoint function.
$\ESpcSet[\TSet \to \DSet](\qpElm)$ denotes the set of all
elementary dependence maps for $\qpElm$ over $\TSet \to \DSet$.
As mentioned above, we now introduce the important variant of semantics based on the property of elementariness of dependence maps over
We refer to the related satisfiability concept as elementary
satisfiability, in symbols $\emodels$.
The new semantics of formulas involving atomic propositions, Boolean
connectives, temporal operators, and agent bindings is defined as for the
classic one, where the modeling relation $\models$ is substituted with
$\emodels$, and we omit to report it here.
In the following definition, we only describe the part concerning the
quantification prefixes.
Observe that by $\bpFun[\bpElm] : \AgSet \to \VarSet$, for $\bpElm \in
\BndSet(\VarSet)$, we denote the function associating to each agent the
variable of its binding in $\bpElm$.
Let $\GName$ be a , $\sElm \in \StSet$ one of its states, and $\qpElm
\bpElm \psi$ an principal sentence.
Then $\GName, \emptyfun, \sElm \emodels \qpElm \bpElm \psi$ iff there is
an elementary dependence map $\spcFun \in \ESpcSet[
{\StrSet(\sElm)} ](\qpElm)$ for $\qpElm$ over $\StrSet(\sElm)$ such that
$\GName, \spcFun(\asgFun) \cmp \bpFun[\bpElm], \sElm \emodels \psi$, for
all $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$.
It is immediate to see a strong similarity between the statement of
Theorem <ref> of strategy quantification and the
previous definition.
The only crucial difference resides in the choice of the kind of dependence
Moreover, observe that, differently from the classic semantics, the
quantifications in a prefix are not treated individually but as an atomic
This is due to the necessity of having a strict correlation between the
point-wise structure of the quantified strategies.
Finally, we state the following fundamental theorem which is a key step in
the proof of the bounded model property and decidability of the
satisfiability for , whose correctness has been proved
in <cit.>.
The idea behind the proof of the elementariness property resides in the
strong similarity between the statement of Theorem <ref> of
strategy quantification and the definition of the winning condition in
a classic turn-based two-player game.
Indeed, on one hand, we say that a sentence is satisfiable iff “there
exists a dependence map such that, for all all assignments, it
holds that ...”.
On the other hand, we say that the first player wins a game iff “there
exists a strategy for him such that, for all strategies of the other player,
it holds that ...”.
The gap between these two formulations is solved in by using the
concept of elementary quantification.
So, we build a two-player turn-based game in which the two players are
viewed one as a dependence map and the other as a valuation over universal
quantified variables, both over actions, such that the formula is satisfied
iff the first player wins the game.
This construction is a deep technical evolution of the proof method used for
the dualization of alternating automata on infinite objects <cit.>.
Precisely, it uses Martin's Determinacy Theorem <cit.> on the
auxiliary turn-based game to prove that, if there is no dependence map of a
given prefix that satisfies the given property, there is a dependence map of
the dual prefix satisfying its negation.
Let $\GName$ be a and $\varphi$ an sentence.
Then, $\GName \!\models\! \varphi$ iff $\GName \!\emodels\! \varphi$.
In order to understand what elementariness means from a syntactic point of
view, note that in it holds that $\qpElm \bpElm \X \psi \equiv \qpElm
\bpElm \X \qpElm \bpElm\psi$, i.e., we can requantify the strategies to
satisfy the inner subformula $\psi$.
This equivalence is a generalization of what is well know to hold for :
$\E \X \psi \equiv \E \X \E \psi$.
Moreover, note that, as reported in <cit.>, elementariness does not
hold for more expressive fragments of , such as .
Bounded Dependence Maps
Here we prove a boundedness property for dependence maps crucial to get, in
Section <ref>, the bounded tree-model property for ,
which is a preliminary step towards our decidability proof for the logic.
As already mentioned, on reasoning about the satisfiability of an sentence, one can simplify the process, via elementariness, by splitting a
dependence map over strategies in a set of dependence maps over actions.
Thus, to gain the bounded model property, it is worth understanding how to
build dependence maps over a predetermined finite set of actions, while
preserving the satisfiability of the sentence of interest.
The main difficulty here is that, the verification process of a sentence
$\varphi$ over an (unbounded) $\TName$ may require some of its
subsentences, perhaps in contradiction among them, to be checked on disjoint
subtrees of $\TName$.
For example, consider the formula $\varphi = \phi_{1} \wedge \phi_{2}$,
where $\phi_{1} = \qpElm[1] \bpElm \X \pSym$ and $\phi_{2} = \qpElm[2]
\bpElm \X \neg \pSym$ with $\bpElm = (\alpha, \xSym) (\beta, \ySym) (\gamma,
\zSym)$.
It is evident that, if $\TName \models \varphi$, the two strategy
quantifications made via the prefixes $\qpElm[1]$ and $\qpElm[2]$ have to
select two disjoint subtrees of $\TName$ on which verify the temporal
properties $\X \pSym$ and $\X \neg \pSym$, respectively.
So, a correct pruning of $\TName$ in a bounded tree-model has to keep the
satisfiability of the subsentences $\phi_{1}$ and $\phi_{2}$ separated, by
avoiding the collapse of the relative subtrees, which can be ensured via the
use of an appropriate number of actions.
By means of characterizing properties named overlapping (see
Definitions <ref> and <ref>) on quantification-binding
prefixes and sets of dependence maps, called signatures (see
Definition <ref>) and signature dependences (see
Definition <ref>), respectively, we ensure that the set of
required actions is finite.
Practically, we prove that sentences with overlapping signatures necessarily
share a common subtree, independently from the number of actions in the
model (see Corollary <ref>).
Conversely, sentences with non-overlapping signatures may need different
So, a model must have a sufficient big set of actions, which we prove to
be finite anyway (see Theorem <ref>).
Note that, in the previous example, $\varphi$ to be satisfiable needs to
have non-overlapping signatures, since otherwise there is at least a shared
outcome on which verify the incompatible temporal properties $\X \pSym$ and
$\X \neg \pSym$.
We now give few more details on the idea behind the properties described
Suppose to have a set of quantification prefixes $\QSet \subseteq
\QPSet(\VSet)$ over a set of variables $\VSet$.
We ask whether there is a relation among the elements of $\QSet$ that forces
a set of related dependence maps to intersect their ranges in at least one
valuation of variables.
For instance, consider in the previous example the prefixes to be
set as follows: $\qpElm[1] \defeq \AAll{\xSym} \EExs{\ySym} \EExs{\zSym}$
and $\qpElm[2] \defeq \AAll{\zSym} \EExs{\ySym} \AAll{\xSym}$.
Then, we want to know whether an arbitrary pair of dependence maps
$\spcFun[1] \in \SpcSet[\DSet](\qpElm[1])$ and $\spcFun[2] \in
\SpcSet[\DSet](\qpElm[2])$ has intersecting ranges, for a set $\DSet$.
In this case, since $\ySym$ is existentially quantified in both prefixes, we
can build $\spcFun[1]$ and $\spcFun[2]$ in such a way they choose different
elements of $\DSet$ on $\ySym$, when they do the same choices on the
other variables, supposed that $\card{\DSet} > 1$.
Thus, if the prefixes share at least an existential variable, it is
possible to find related dependence maps that are non-overlapping.
Indeed, in this case, the formula $\varphi$ is satisfied on the $\GName[SA]$ of Figure <ref>, since we can allow $\ySym$ on
$\sElm[0]$ to chose $0$ for $\qpElm[1]$ and $1$ for $\qpElm[2]$.
Now, let consider the following prefixes: $\qpElm[1] \defeq \AAll{\xSym}
\EExs{\zSym} \AAll{\ySym}$ and $\qpElm[2] \defeq \AAll{\zSym} \AAll{\ySym}
\EExs{\xSym}$.
Although, in this case, each variable is existentially quantified at most
once, we have that $\xSym$ and $\zSym$ mutually depend in the different
So, there is a cyclic dependence that can make two related non-overlapping
dependence maps.
Indeed, suppose to have $\DSet = \{ 0, 1\}$.
Then, we can choose $\spcFun[1] \in \SpcSet[\DSet](\qpElm[1])$ and
$\spcFun[2] \in \SpcSet[\DSet](\qpElm[2])$ in the way that, for all
valuations $\valFun[1] \in \dom{\spcFun[1]}$ and $\valFun[2] \in
\dom{\spcFun[2]}$, it holds that $\spcFun[1](\valFun[1])(\zSym) \defeq
\valFun[1](\xSym)$ and
$\spcFun[2](\valFun[2])(\xSym) \defeq 1 - \valFun[2](\zSym)$.
Thus, $\spcFun[1]$ and $\spcFun[2]$ do not intersect their ranges.
Indeed, with the considered prefixes, the formula $\varphi$ is satisfied on
the $\GName[CD]$ of
Finally, consider a set of prefixes in which there is neither a shared
existential quantified variable nor a cyclic dependence, such as the
following: $\qpElm[1] \!\defeq\! \AAll{\xSym} \AAll{\ySym} \EExs{\zSym}$,
$\qpElm[2] \!\defeq\! \EExs{\ySym} \AAll{\xSym} \AAll{\zSym}$, and
$\qpElm[3] \!\defeq\! \AAll{\ySym} \EExs{\xSym} \AAll{\zSym}$.
We now show that an arbitrary choice of dependence maps $\spcFun[1] \!\in\!
\SpcSet[\DSet](\qpElm[1])$, $\spcFun[2] \!\in\! \SpcSet[\DSet](\qpElm[2])$,
and $\spcFun[3] \!\in\! \SpcSet[\DSet](\qpElm[3])$ must have intersecting
ranges, for every set $\DSet$.
Indeed, since $\ySym$ in $\qpElm[2]$ does not depend from any other
variable, there is a value $\dElm[\ySym] \!\in\! \DSet$ such that, for all
$\valFun[2] \!\in\! \dom{\spcFun[2]}$, it holds that
$\spcFun[2](\valFun[2])(\ySym) \!=\! \dElm[\ySym]$.
Now, since $\xSym$ in $\qpElm[3]$ depends only on $\ySym$, there is a
value $\dElm[\xSym] \!\in\! \DSet$ such that, for all $\valFun[3] \!\in\!
\dom{\spcFun[3]}$ with $\valFun[3](\ySym) \!=\! \dElm[\ySym]$, it holds that
$\spcFun[3](\valFun[3])(\xSym) \!=\! \dElm[\xSym]$.
Finally, we can determine the value $\dElm[\zSym] \!\in\! \DSet$ of $\zSym$
in $\qpElm[1]$ since $\xSym$ and $\ySym$ are fixed.
So, for all $\valFun[1] \!\in\! \dom{\spcFun[1]}$ with $\valFun[1](\xSym)
\!=\! \dElm[\xSym]$ and $\valFun[1](\ySym) \!=\! \dElm[\ySym]$, it holds
that $\spcFun[1](\valFun[1])(\zSym) \!=\! \dElm[\zSym]$.
Thus, the valuation $\valFun \!\in\! \ValSet[\DSet](\VSet)$, with
$\valFun(\xSym) \!=\! \dElm[\xSym]$, $\valFun(\ySym) \!=\! \dElm[\ySym]$,
and $\valFun(\zSym) \!=\! \dElm[\zSym]$, is such that $\valFun \in
\rng{\spcFun[1]} \cap \rng{\spcFun[2]} \cap \rng{\spcFun[3]}$.
Note that we run this procedure since we can find at each step an
existential variable that depends only on universal variables
previously determined.
In order to formally define the above procedure, we need to introduce some
preliminary definitions.
As first thing, we generalize the described construction by taking into
account not only quantification prefixes but binding prefixes too.
This is due to the fact that different principal subsentences of the
specification can share the same quantification prefix by having different
binding prefixes.
Moreover, we need to introduce a tool that gives us a way to
differentiate the check of the satisfiability of a given sentence in
different parts of the model, since it can use different actions when starts
the check from different states.
For this reason, we introduce the concepts of signature and
labeled signature.
The first is used to arrange opportunely prefixes with bindings, represented
in a more general form through the use of a generic support set $\ESet$,
while the second allows us to label signatures, by means of a set $\LSet$,
to maintain an information on different instances of the same sentence.
A signature on a set $\ESet$ is a pair $\sigElm \defeq (\qpElm,
\bFun) \in \QPSet(\VSet) \times \VSet^{\ESet}$ of a quantification prefix
$\qpElm$ over $\VSet$ and a surjective function $\bFun$ from $\ESet$ to
$\VSet$, for a given set of variables $\VSet \subseteq \VarSet$.
A labeled signature on $\ESet$ w.r.t. a set $\LSet$ is a pair
$(\sigElm, \lElm) \in (\QPSet(\VSet) \times \VSet^{\ESet}) \times \LSet$
of a signature $\sigElm$ on $\ESet$ and a labeling $\lElm$ in $\LSet$.
The sets $\SigSet(\ESet) \defeq \bigcup_{\VSet \subseteq \VarSet}
\QPSet(\VSet) \!\times\! \VSet^{\ESet}$ and $\LSigSet(\ESet, \LSet) \defeq
\SigSet(\ESet) \times \LSet$ contain, respectively, all signatures
on $\ESet$ and labeled signatures on $\ESet$ w.r.t. $\LSet$.
We now extend the concepts of existential quantification and functional
dependence from prefixes to signatures.
By $\QPEVSet{\sigElm} \defeq \set{ \eElm \in \ESet }{ \bFun(\eElm) \in
\QPEVSet{\qpElm} }$, $\QPDepSet(\sigElm) \defeq \set{ (\eElm', \eElm'') \in
\ESet \times \ESet }{ (\bFun(\eElm'), \bFun(\eElm'')) \in \QPDepSet(\qpElm)
}$, and $\BPColSet(\sigElm) \defeq \set{ (\eElm', \eElm'') \in \ESet \times
\ESet }{ \bFun(\eElm') = \bFun(\eElm'') \in \QPAVSet{\qpElm} }$, with
$\sigElm = (\qpElm, \bFun) \in \SigSet(\ESet)$, we denote the set of
existential elements, and the relation sets of functional
dependent and collapsing elements, respectively.
Moreover, for a set $\SSet \subseteq \SigSet(\ESet)$ of signatures, we
define $\BPColSet(\SSet) \defeq (\bigcup_{\sigElm \in \SSet}
\BPColSet(\sigElm))^{+}$ as the transitive relation set of collapsing
elements and $\QPEVSet{\SSet} \defeq \bigcup_{\sigElm \in \SSet}
\QPEVSet{\SSet, \sigElm}$, with $\QPEVSet{\SSet, \sigElm} \defeq \set{
\eElm \in \QPEVSet{\sigElm} }{ \exists \sigElm' \in \SSet, \eElm' =
(\qpElm', \bFun') \in \QPEVSet{\sigElm'} \:.\: (\sigElm \neq \sigElm' \lor
\bFun(\eElm) \neq \bFun'(\eElm')) \land (\eElm, \eElm') \in
\BPColSet(\SSet)}$, as the set of elements that are existential in two
signatures, either directly or via a collapsing chain.
Finally, by $\QPDepSet'(\sigElm) \defeq \set{(\eElm', \eElm'') \in \ESet
\times \ESet}{ \exists \eElm''' \in \ESet \:.\: (\eElm', \eElm''') \in
\BPColSet(\SSet) \land (\eElm''', \eElm'') \in \QPDepSet(\sigElm)}$ we
indicate the relation set of functional dependent elements connected via a
collapsing chain.
As described above, if a set of prefixes has a cyclic dependence
between variables, we are sure to find a set of dependence maps,
bijectively related to such prefixes, that do not share any total assignment
in their codomains.
Here, we formalize this concept of dependence by considering bindings too.
In particular, the check of dependences is not done directly on variables,
but by means of the associated elements of the support set $\ESet$.
Note that, in the case of labeled signatures, we do not take into account
the labeling component, since two instances of the same signature with
different labeling cannot have a mutual dependent variable.
To give the formal definition of cyclic dependence, we first provide the
definition of $\SSet$-chain.
An $\SSet$-chain for a set of signatures $\SSet \subseteq
\SigSet(\ESet)$ on $\ESet$ is a pair $(\vec{\eElm}, \vec{\sigElm}) \in
\ESet^{k} \times \SSet^{k}$, with $k \in \numco{1}{\omega}$, for which the
following hold:
$\lst{\vec{\eElm}} \in \QPAVSet{\lst{\vec{\sigElm}}}$;
$((\vec{\eElm})_{i}, (\vec{\eElm})_{i+1}) \in
\QPDepSet'((\vec{\sigElm})_{i})$, for all $i \in \numco{0}{k - 1}$;
$(\vec{\sigElm})_{i} \neq (\vec{\sigElm})_{j}$, for all $i, j \in
\numco{0}{k}$ with $i < j$.
It is important to observe that, due to Item <ref>, each
$\SSet$-chain cannot have length greater than $\card{\SSet}$.
Now we can give the definition of cyclic dependence.
A cyclic dependence for a set of signatures $\SSet \subseteq
\SigSet(\ESet)$ on $\ESet$ is an $\SSet$-chain $(\vec{\eElm},
\vec{\sigElm})$ such that $(\lst{\vec{\eElm}}, \fst{\vec{\eElm}}) \in
\QPDepSet'(\lst{\vec{\sigElm}})$.
Moreover, it is a cyclic dependence for a set of labeled signatures
$\PSet \subseteq \LSigSet(\ESet, \LSet)$ on $\ESet$ w.r.t. $\LSet$ if it
is a cyclic dependence for the set of signatures $\set{ \sigElm \in
\SigSet(\ESet) }{ \exists \lElm \in \LSet \:.\: (\sigElm, \lElm) \in \PSet
The sets $\CSet(\SSet), \CSet(\PSet) \subseteq \ESet^{+} \times \SSet^{+}$
contain, respectively, all cyclic dependences for signatures in $\SSet$
and labeled signatures in $\PSet$.
Observe that $\card{\CSet(\SSet)} \!\!\leq\!\! \card{\ESet}^{\card{\SSet}}
\!\cdot\! \card{\SSet}!$, so, $\card{\CSet(\PSet)} \!\!\leq\!\!
\card{\ESet}^{\card{\PSet}} \!\cdot\! \card{\PSet}!$.
At this point, we can formally define the property of overlapping for
According to the above description, this implies that dependence maps
related to prefixes share at least one total variable valuation in their
Thus, we say that a set of signatures is overlapping if they do not have
common existential variables and there is no cyclic dependence.
Observe that, if there are two different instances of the same signature
having an existential variable, we can still construct a set of
dependence maps that do not share any valuation, so we have to avoid
this possibility too.
A set $\SSet \subseteq \SigSet(\ESet)$ of signatures on $\ESet$ is
overlapping if $\QPEVSet{\SSet} = \emptyset$ and $\CSet(\SSet) =
\emptyset$.
A set $\PSet \!\subseteq\! \LSigSet(\ESet, \LSet)$ of labeled signatures
on $\ESet$ w.r.t. $\LSet$ is overlapping if the derived set of
signatures $\set{ \sigElm \in \SigSet(\ESet) }{ \exists \lElm \in \LSet
\:.\: (\sigElm, \lElm) \in \PSet }$ is overlapping and, for all
$(\sigElm, \lElm'), (\sigElm, \lElm'') \in \PSet$, if $\QPEVSet{\sigElm}
\neq \emptyset$ then $\lElm' = \lElm''$.
Finally, to manage the one-to-one connection between signatures and related
dependence maps, it is useful to introduce the simple concept of signature
dependence, which associates to every signature a related dependence map.
We also define, as expected, the concept of overlapping for these
functions, which intuitively states that the contained dependence maps have
identical valuations of variables in their codomains, once they are composed
with the related functions on the support set.
A signature dependence for a set of signatures $\SSet \subseteq
\SigSet(\ESet)$ on $\ESet$ over $\DSet$ is a function
$\spcmapFun : \SSet \to \cup_{(\qpElm, \bFun) \in \SSet}
\SpcSet[\DSet](\qpElm)$ such that, for all $(\qpElm, \bFun) \in \SSet$, it
holds that $\spcmapFun((\qpElm, \bFun)) \in \SpcSet[\DSet](\qpElm)$.
A signature dependence for a set of labeled signatures $\PSet
\subseteq \LSigSet(\ESet, \LSet)$ on $\ESet$ w.r.t. $\LSet$ over $\DSet$
is a function $\spcmapFun : \PSet \to \cup_{((\qpElm, \bFun), \lElm) \in
\PSet} \SpcSet[\DSet](\qpElm)$ such that, for all $((\qpElm, \bFun),
\lElm) \in \PSet$, it holds that $\spcmapFun(((\qpElm, \bFun), \lElm)) \in
\SpcSet[\DSet](\qpElm)$.
The sets $\SpcMapSet[\DSet](\SSet)$ and $\LSpcMapSet[\DSet](\PSet)$
contain, respectively, all signature dependences for $\SSet$ and labeled
signature dependences for $\PSet$ over $\DSet$.
A signature dependence $\spcmapFun \in \SpcMapSet[\DSet](\SSet)$ is
overlapping if $\cap_{(\qpElm, \bFun) \in \SSet} \set{ \valFun
\circ \bFun }{ \valFun \in \rng{\spcmapFun(\qpElm, \bFun)}} \neq
\emptyset$.
A labeled signature dependence $\spcmapFun \in \LSpcMapSet[\DSet](\PSet)$
is overlapping if $\cap_{((\qpElm, \bFun), \lElm) \in \PSet} \set{
\valFun \circ \bFun }{ \valFun \in \rng{\spcmapFun((\qpElm, \bFun),
\lElm)}} \neq \emptyset$.
As explained above, signatures and signature dependences have a strict
correlation w.r.t. the concept of overlapping.
Indeed, the following result holds.
The idea here is to find, at each step of the construction of the common
valuation, a variable, called pivot, that does not depend on other
variables whose value is not already set.
This is possible if there are no cyclic dependences and each variable is
existential in at most one signature.
Let $\SSet \subseteq \SigSet(\ESet)$ be a finite set of overlapping
signatures on $\ESet$.
Then, for all signature dependences $\spcmapFun \in
\SpcMapSet[\DSet](\SSet)$ for $\SSet$ over a set $\DSet$, it holds that
$\spcmapFun$ is overlapping.
This theorem can be easily lifted to labeled signatures, as stated in the
following corollary.
Let $\PSet \subseteq \LSigSet(\ESet, \allowbreak \LSet)$ be a finite set
of overlapping labeled signatures on $\ESet$ w.r.t. $\LSet$.
Then, for all labeled signature dependences $\spcmapFun \in
\LSpcMapSet[\DSet](\PSet)$ for $\PSet$ over a set $\DSet$, it
holds that $\spcmapFun$ is overlapping.
Finally, if the set $\DSet$ is sufficiently large, in the case of
non-overlapping labeled signatures, we can find a signature dependence that
is non-overlapping too, as reported in following theorem.
The high-level combinatorial idea behind the proof is to assign to each
existential variable, related to a given element of the support set in a
signature, a value containing a univocal flag in $\PSet \times
\QPVSet(\PSet)$, where $\QPVSet(\PSet) \defeq \bigcup_{((\qpElm, \bFun),
\lElm) \in \PSet} \QPVSet(\qpElm)$, representing the signature itself.
Thus, signatures sharing an existential element surely have related
dependence maps that cannot share a common valuation.
Moreover, for each cyclic dependence, we choose a particular element whose
value is the inversion of that assigned to the element from which it
depends, while all other elements preserve the related values.
In this way, in a set of signature having cyclic dependences, there is one
of them whose associated dependence maps have valuations that differ from
those in the dependence maps of the other signatures, since it is the unique
that has an inversion of the values.
Let $\PSet \subseteq \LSigSet(\ESet, \LSet)$ be a set of labeled
signatures on $\ESet$ w.r.t. $\LSet$.
Then, there exists a labeled signature dependence $\spcmapFun \in
\LSpcMapSet[\DSet](\PSet)$ for $\PSet$ over $\DSet \!\defeq\! \PSet
\!\times\! \QPVSet(\PSet) \!\times\! \{ 0, 1 \}^{\CSet(\PSet)}$ such
that, for all $\PSet' \subseteq \PSet$, it holds that $\spcmapFun_{\rst
\PSet'} \in \LSpcMapSet[\DSet](\PSet')$ is non-overlapping, if $\PSet'$
is non-overlapping.
Model Properties
We now investigate basic model properties of that turn out to be
important on their own and useful to prove the decidability of the
satisfiability problem.
First, recall that the satisfiability problem for branching-time logics can be
solved via tree automata, once a kind of bounded tree-model property holds.
Indeed, by using it, one can build an automaton accepting all models of
formulas, or their encoding.
So, we first introduce the concepts of concurrent game tree,
decision tree, and decision-unwinding and then show that is invariant under decision-unwinding, which directly implies that it
satisfies a unbounded tree-model property.
Finally, by using the techniques previously introduced, we further prove that
the above property is actually a bounded tree-model property.
Tree-model property
We now introduce two particular kinds of whose structure is a directed
As already explained, we do this since the decidability procedure we give in
the last section of the paper is based on alternating tree automata.
A concurrent game tree (, for short) is a $\TName \defeq
\CGSStruct[\epsilon]$, where (i) $\StSet \subseteq \DirSet^{*}$ is
a $\DirSet$-tree for a given set $\DirSet$ of directions and (ii)
if $\tElm \cdot \eElm \in \StSet$ then there is a decision $\decFun \in
\DecSet$ such that $\trnFun(\tElm, \decFun) = \tElm \cdot \eElm$, for all
$\tElm \in \StSet$ and $\eElm \in \DirSet$.
Furthermore, $\TName$ is a decision tree (, for short) if
(i) $\StSet = \DecSet^{*}$ and (ii) if $\tElm \cdot \decFun
\in \StSet$ then $\trnFun(\tElm, \decFun) = \tElm \cdot \decFun$, for all
$\tElm \in \StSet$ and $\decFun \in \DecSet$.
Intuitively, s are s with a tree-shaped transition relation and
s have, in addition, states uniquely determining the history of
computation leading to them.
At this point, we can define a generalization for s of the classic
concept of unwinding of labeled transition systems, namely
Note that, in general and differently from , is not invariant
under decision-unwinding, as we show later.
On the contrary, satisfies such an invariance property.
This fact allows us to show that this logic has the unbounded tree-model
Let $\GName$ be a .
Then, the decision-unwinding of $\GName$ is the $\GName[DU]
\defeq \CGSTuple {\APSet} {\AgSet} {\AcSet[\GName]} {\DecSet[\GName]^{*}}
{\labFun} {\trnFun} {\epsilon}$ for which there is a surjective function
$\unwFun : \DecSet[\GName]^{*} \to \StSet[\GName]$ such that (i)
$\unwFun(\epsilon) = \sElm[0\GName]$, (ii) $\unwFun(\trnFun(\tElm,
\decFun)) = \trnFun[\GName](\unwFun(\tElm), \decFun)$, and (iii)
$\labFun(\tElm) = \labFun[\GName](\unwFun(\tElm))$, for all $\tElm \in
\DecSet[\GName]^{*}$ and $\decFun \in \DecSet[\GName]$.
Note that each $\GName$ has a unique associated decision-unwinding
We say that a sentence $\varphi$ has the decision-tree
model property if, for each $\GName$, it holds that $\GName \models
\varphi$ iff $\GName[DU] \models \varphi$.
By using a standard proof by induction on the structure of formulas,
we can show that this logic is invariant under decision-unwinding, i.e.,
each sentence has decision-tree model property, and, consequently,
that it satisfies the unbounded tree-model property.
For the case of the combined quantification and binding
prefixes $\qpElm \bpElm \psi$, we can use a technique that allows to build,
given an elementary dependence map $\spcFun$ satisfying the formula
on a $\GName$, an elementary dependence map $\spcFun'$
satisfying the same formula over the $\GName[DU]$, and vice versa.
This construction is based on a step-by-step transformation of the adjoint
of a dependence maps into another, which is done for each track of the
original model.
This means that we do not actually transform the strategy quantifications
but the equivalent infinite set of action quantifications.
is invariant under decision-unwinding;
has the decision-tree model property.
Although this result is a generalization of that proved to hold for ,
it actually represents an important demarcation line between and .
Indeed, as we show in the following theorem, does not satisfy neither
the tree-model property nor, consequently, the invariance under
does not have the decision-tree model property;
is not invariant under decision-unwinding.
Bounded tree-model property
We now have all tools we need to prove the bounded tree-model property for
, which we recall does not satisfy <cit.>.
Actually, we prove here a stronger property, which we name bounded
disjoint satisfiability.
To this aim, we first introduce the new concept regarding the satisfiability
of different instances of the same subsentence of the original
specification, which intuitively states that these instances can be checked
on disjoint subtrees of the tree model.
With more detail, this property asserts that, if two instances use part of
the same subtree, they are forced to use the same dependence map as well.
This intrinsic characteristic of is fundamental to build a unique
automaton that checks the truth of all subsentences, by simply merging their
respective automata, without using a projection operation that eliminates
their proper alphabets, which otherwise can be in conflict.
In this way, we are able to avoid an exponential blow-up.
A clearer discussion on this point is reported later in the paper.
Let $\TName$ be a , $\varphi \defeq \qpElm \bpElm \psi$ an principal sentence, and $\SSet \defeq \set{ \sElm \in \StSet }{ \TName,
\emptyset, \sElm \models \varphi }$.
Then, $\TName$ satisfies $\varphi$ disjointly over $\SSet$ if there
exist two functions $\headFun : \SSet \to \SpcSet[\AcSet](\qpElm)$ and
$\bodyFun : \TrkSet(\epsilon) \to \SpcSet[\AcSet](\qpElm)$ such that, for
all $\sElm \in \SSet$ and $\asgFun \in \AsgSet(\QPAVSet{\qpElm}, \sElm)$,
it holds that $\TName, \spcFun(\asgFun), \sElm \models \bpElm \psi$, where
the elementary dependence maps $\spcFun \in
\ESpcSet[\StrSet(\sElm)](\qpElm)$ is defined as follows: (i)
$\adj{\spcFun}(\sElm) \defeq \headFun(\sElm)$; (ii)
$\adj{\spcFun}(\trkElm) \defeq \bodyFun(\trkElm' \cdot \trkElm)$, for all
$\trkElm \in \TrkSet(\sElm)$ with $\card{\trkElm} > 1$, where $\trkElm'
\in \TrkSet(\epsilon)$ is the unique track such that $\trkElm' \cdot
\trkElm \in \TrkSet(\epsilon)$.
In the following theorem, we finally describe the crucial step behind our
automata-theoretic decidability procedure for .
At an high-level, the proof proceeds as follows.
We start from the satisfiability of the specification $\varphi$ over a $\TName$, whose existence is ensured by
Item <ref> of
Theorem <ref> of positive model properties.
Then, we construct an intermediate $\TName[\sharp]$, called
flagged model, which is used to check the satisfiability of all
subsentences of $\varphi$ in a disjoint way.
Intuitively, the flagged model adds a controller agent, named sharp
that decides on which subtree a given subsentence has to be verified.
Now, by means of Theorem <ref> on the elementariness,
we construct the adjoint functions of the dependence maps used to
verify the satisfiability of the sentences on $\TName[\sharp]$.
Then, by applying Corollary <ref> and Theorem <ref>
of overlapping and non-overlapping dependence maps, respectively, we
transform the dependence maps over actions, contained in the ranges of the
adjoint functions, in a bounded version, which preserves the satisfiability
of the sentences on a bounded pruning $\TName[\sharp]'$ of $\TName[\sharp]$.
Finally, we remove the additional agent $\sharp$ obtaining the required
bounded $\TName'$.
Observe that, due to the particular construction of the bounded dependence
maps, the disjoint
Let $\varphi$ be an satisfiable sentence and $\PSet \defeq \set{
((\qpElm, \bpElm), (\psi, i)) \in \LSigSet(\AgSet, \SL \times \{ 0, 1 \})
}{ \qpElm \bpElm \psi \in \psnt{\varphi} \land i \in \{ 0, 1 \} }$ the set
of all labeled signatures on $\AgSet$ w.r.t. $\SL \times \{ 0, 1 \}$ for
Then, there exists a $b$-bounded $\TName$, with $b = \card{\PSet}
\cdot \card{\QPVSet(\PSet)} \cdot 2^{\card{\CSet(\PSet)}}$, such that
$\TName \models \varphi$.
Moreover, for all $\phi \in \psnt{\varphi}$, it holds that $\TName$
satisfies $\phi$ disjointly over the set $\set{ \sElm \in \StSet }{
\TName, \emptyset, \sElm \models \phi }$.
Satisfiability Procedure
We finally solve the satisfiability problem for and show that it is
2, as for .
The algorithmic procedures is based on an automata-theoretic approach, which
reduces the decision problem for the logic to the emptiness problem of a
suitable universal Co-Büchi tree automaton (, for short) <cit.>.
From an high-level point of view, the automaton construction seems similar to
what was proposed in literature for <cit.> and <cit.>.
However, our technique is completely new, since it is based on the novel
notions of elementariness and disjoint satisfiability.
Principal sentences
To proceed with the satisfiability procedure, we have to introduce a concept
of encoding for an assignment and the labeling of a .
Let $\TName$ be a , $\tElm \in \StSet[\TName]$ one of its states, and
$\asgFun \in \AsgSet[\TName](\VSet, \tElm)$ an assignment defined on the
set $\VSet \subseteq \VarSet$.
A $(\ValSet[ {\AcSet[\TName]} ](\VSet) \times \pow{\APSet})$-labeled
$\DecSet[\TName]$-tree $\TName' \defeq
\LTTuple{}{}{\StSet[\TName]}{\uFun}$ is an assignment-labeling
encoding for $\asgFun$ on $\TName$ if $\uFun(\lst{(\trkElm)_{\geq 1}})
\!=\! (\flip{\asgFun}(\trkElm), \labFun[\TName](\lst{\trkElm}))$, for all
$\trkElm \in \TrkSet[\TName](\tElm)$.
Observe that there is a unique assignment-labeling encoding for each
assignment over a given .
Now, we prove the existence of a $\UName[\bpElm \psi | ^{\AcSet}]$ for
each goal $\bpElm \psi$ having no principal subsentences.
$\UName[\bpElm \psi | ^{\AcSet}]$ recognizes all the assignment-labeling
encodings $\TName'$ of an a priori given assignment $\asgFun$ over a generic
$\TName$, once the goal is satisfied on $\TName$ under $\asgFun$.
Intuitively, we start with a , recognizing all infinite words on the
alphabet $\pow{\APSet}$ that satisfy the formula $\psi$, obtained by a
simple variation of the Vardi-Wolper construction <cit.>.
Then, we run it on the encoding tree $\TName'$ by following the directions
imposed by the assignment in its labeling.
Let $\bpElm \psi$ an goal without principal subsentences and
$\AcSet$ a finite set of actions.
Then, there exists an $\UName[\bpElm \psi | ^{\AcSet}] \defeq
\TATuple {\ValSet[\AcSet](\free{\bpElm \psi}) \times \pow{\APSet}}
{\DecSet} {\QSet[\bpElm \psi]} {\atFun[\bpElm \psi]} {\qElm[0\bpElm\psi]}
{\aleph_{\bpElm \psi}}$ such that, for all s $\TName$ with
$\AcSet[\TName] = \AcSet$, states $\tElm \in \StSet[\TName]$, and
assignments $\asgFun \in \AsgSet[\TName](\free{\bpElm \psi}, \tElm)$, it
holds that $\TName, \asgFun, \tElm \models \bpElm \psi$ iff $\TName' \in
\LangSet(\UName[\bpElm \psi | ^{\AcSet}])$, where $\TName'$ is the
assignment-labeling encoding for $\asgFun$ on $\TName$.
We now introduce a new concept of encoding regarding the elementary
dependence maps over strategies.
Let $\TName$ be a , $\tElm \in \StSet[\TName]$ one of its states, and
$\spcFun \in \ESpcSet[ {\StrSet[\TName](\tElm)} ](\qpElm)$ an elementary
dependence map over strategies for a quantification prefix
$\qpElm \in \QPSet(\VSet)$ over the set $\VSet \subseteq \VarSet$.
A $(\SpcSet[ {\AcSet[\TName]} ](\qpElm) \times \pow{\APSet})$-labeled
$\DirSet$-tree $\TName' \defeq \LTTuple{}{}{\StSet[\TName]}{\uFun}$ is an
elementary dependence-labeling encoding for $\spcFun$ on $\TName$
if $\uFun(\lst{(\trkElm)_{\geq 1}}) \!=\! (\adj{\spcFun}(\trkElm),
\labFun[\TName](\lst{\trkElm}))$, for all $\trkElm \!\in\!
\TrkSet[\TName](\tElm)$.
Observe that also in this case there exists a unique elementary
dependence-model encoding for each elementary dependence map over
Finally, in the next lemma, we show how to handle locally the strategy
quantifications on each state of the model, by simply using a quantification
over actions modeled by the choice of an action dependence map.
Intuitively, we guess in the labeling what is the right part of the
dependence map over strategies for each node of the tree and then verify
that, for all assignments of universal variables, the corresponding complete
assignment satisfies the inner formula.
Let $\qpElm \bpElm \psi$ be an principal sentence without principal
subsentences and $\AcSet$ a finite set of actions.
Then, there exists an $\UName[\qpElm \bpElm \psi | ^{\AcSet}]
\defeq \TATuple {\SpcSet[\AcSet](\qpElm) \times \pow{\APSet}} {\DecSet}
{\QSet[\qpElm \bpElm \psi]} {\atFun[\qpElm \bpElm \psi]}
{\qElm[0\qpElm\bpElm\psi]} {\aleph_{\qpElm \bpElm \psi}}$ such that, for
all s $\TName$ with $\AcSet[\TName] = \AcSet$, states $\tElm \in
\StSet[\TName]$, and elementary dependence maps over strategies
$\spcFun \in \ESpcSet[ {\StrSet[\TName](\tElm)} ](\qpElm)$, it holds that
$\TName, \spcFun(\asgFun), \tElm \emodels \bpElm \psi$, for all $\asgFun
\in \AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$, iff $\TName' \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\AcSet}])$, where $\TName'$ is
the elementary dependence-labeling encoding for $\spcFun$ on $\TName$.
Full sentences
By summing up all previous results, we are now able to solve the
satisfiability problem for the full fragment.
To construct the automaton for a given sentence $\varphi$, we first
consider all $\UName[\phi | ^{\AcSet}]$, for an assigned bounded set
$\AcSet$, previously described for the principal sentences $\phi \in
\psnt{\varphi}$, in which the inner subsentences are considered as atomic
Then, thanks to the disjoint satisfiability property of
Definition <ref>, we can merge them into a unique $\UName[\varphi]$ that supplies the dependence map labeling of internal
components $\UName[\phi | ^{\AcSet}]$, by using the two functions $\headFun$
and $\bodyFun$ contained into its labeling.
Moreover, observe that the final automaton runs on a $b$-bounded
decision tree, where $b$ is obtained from
Theorem <ref> on the bounded-tree model property.
Let $\varphi$ be an sentence.
Then, there exists an $\UName[\varphi]$ such that $\varphi$ is
satisfiable iff $\LangSet(\UName[\varphi]) \neq \emptyset$.
Finally, by a simple calculation of the size of $\UName[\varphi]$ and the
complexity of the related emptiness problem, we state in the next theorem
the precise computational complexity of the satisfiability problem for
The satisfiability problem for is 2.
Mathematical Notation
In this short reference appendix, we report the classical mathematical
notation and some common definitions that are used along the whole work.
*Classic objects
We consider $\SetN$ as the set of natural numbers and $\numcc{m}{n}
\defeq \set{ k \in \SetN }{ m \leq k \leq n }$, $\numco{m}{n} \defeq \set{ k
\in \SetN }{ m \leq k < n }$, $\numoc{m}{n} \defeq \set{ k \in \SetN }{ m <
k \leq n }$, and $\numoo{m}{n} \defeq \set{ k \in \SetN }{ m < k < n }$ as
its interval subsets, with $m \in \SetN$ and $n \in \SetNI \defeq
\SetN \cup \{ \omega \}$, where $\omega$ is the numerable infinity,
i.e., the least infinite ordinal.
Given a set $\XSet$ of objects, we denote by $\card{\XSet} \in
\SetNI \cup \{ \infty \}$ the cardinality of $\XSet$, i.e., the
number of its elements, where $\infty$ represents a more than
countable cardinality, and by $\pow{\XSet} \defeq \set{ \YSet }{ \YSet
\subseteq \XSet }$ the powerset of $\XSet$, i.e., the set of all its
By $\RRel \subseteq \XSet \times \YSet$ we denote a relation between
the domain $\dom{\RRel} \defeq \XSet$ and codomain
$\cod{\RRel} \defeq \YSet$, whose range is indicated by $\rng{\RRel}
\defeq \set{ \yElm \in \YSet }{ \exists \xElm \in \XSet .\: (\xElm, \yElm)
\in \RRel }$.
We use $\RRel^{-1} \defeq \set{ (\yElm, \xElm) \in \YSet \times \XSet }{
(\xElm, \yElm) \in \RRel }$ to represent the inverse of $\RRel$
Moreover, by $\SRel \cmp \RRel$, with $\RRel \subseteq \XSet \times \YSet$
and $\SRel \subseteq \YSet \times \ZSet$, we denote the composition
of $\RRel$ with $\SRel$, i.e., the relation $\SRel \cmp \RRel \defeq \set{
(\xElm, \zElm) \in \XSet \times \ZSet }{ \exists \yElm \in \YSet .\: (\xElm,
\yElm) \in \RRel \land (\yElm, \zElm) \in \SRel }$.
We also use $\RRel^{n} \defeq \RRel^{n - 1} \cmp \RRel$, with $n \in
\numco{1}{\omega}\!$, to indicate the $n$-iteration of $\RRel
\subseteq \XSet \times \YSet$, where $\YSet \subseteq \XSet$ and $\RRel^{0}
\defeq \set{ (\yElm, \yElm) }{ \yElm \in \YSet }$ is the identity on
With $\RRel^{+} \defeq \bigcup_{n = 1}^{< \omega} \RRel^{n}$ and $\RRel^{*}
\defeq \RRel^{+} \cup \RRel^{0}$ we denote, respectively, the
transitive and reflexive-transitive closure of $\RRel$.
Finally, for an equivalence relation $\RRel \subseteq \XSet \times
\XSet$ on $\XSet$, we represent with $\class{ \XSet }{ \:\RRel } \defeq
\set{ [\xElm]_{\RRel} }{ \xElm \in \XSet }$, where $[\xElm]_{\RRel} \defeq
\set{ \xElm' \in \XSet }{ (\xElm, \xElm') \in \RRel }$, the quotient
set of $\XSet$ w.r.t. $\RRel$, i.e., the set of all related equivalence
classes $[\cdot]_{\RRel}$.
We use the symbol $\YSet^{\XSet} \subseteq \pow{\XSet \times \YSet}$ to
denote the set of total functions $\fFun$ from $\XSet$ to $\YSet$,
i.e., the relations $\fFun \subseteq \XSet \times \YSet$ such that for all
$\xElm \in \dom{\fFun}$ there is exactly one element $\yElm \in \cod{\fFun}$
such that $(\xElm, \yElm) \in \fFun$.
Often, we write $\fFun : \XSet \to \YSet$ and $\fFun : \XSet \pto \YSet$ to
indicate, respectively, $\fFun \in \YSet^{\XSet}$ and $\fFun \in
\bigcup_{\XSet' \subseteq \XSet} \YSet^{\XSet'}$.
Regarding the latter, note that we consider $\fFun$ as a partial
function from $\XSet$ to $\YSet$, where $\dom{\fFun} \subseteq \XSet$
contains all and only the elements for which $\fFun$ is defined.
Given a set $\ZSet$, by $\fFun[\rst \ZSet] \defeq \fFun \cap (\ZSet \times
\YSet)$ we denote the restriction of $\fFun$ to the set $\XSet \cap
\ZSet$, i.e., the function $\fFun[\rst \ZSet] : \XSet \cap \ZSet \pto \YSet$
such that, for all $\xElm \in \dom{\fFun} \cap \ZSet$, it holds that
$\fFun[\rst \ZSet](\xElm) = \fFun(\xElm)$.
Moreover, with $\emptyfun$ we indicate a generic empty function,
i.e., a function with empty domain.
Note that $\XSet \cap \ZSet = \emptyset$ implies $\fFun[\rst \ZSet] =
\emptyfun$.
Finally, for two partial functions $\fFun, \gFun : \XSet \pto \YSet$, we use
$\fFun \Cup \gFun$ and $\fFun \Cap \gFun$ to represent, respectively, the
union and intersection of these functions defined as follows:
$\dom{\fFun \Cup \gFun} \defeq \dom{\fFun} \cup \dom{\gFun} \setminus \set{
\xElm \in \dom{\fFun} \cap \dom{\gFun} }{ \fFun(\xElm) \neq \gFun(\xElm) }$,
$\dom{\fFun \Cap \gFun} \defeq \set{ \xElm \in \dom{\fFun} \cap \dom{\gFun}
}{ \fFun(\xElm) = \gFun(\xElm) }$, $(\fFun \Cup \gFun)(\xElm) =
\fFun(\xElm)$ for $\xElm \in \dom{\fFun \Cup \gFun} \cap \dom{\fFun}$,
$(\fFun \Cup \gFun)(\xElm) = \gFun(\xElm)$ for $\xElm \in \dom{\fFun \Cup
\gFun} \cap \dom{\gFun}$, and $(\fFun \Cap \gFun)(\xElm) = \fFun(\xElm)$ for
$\xElm \in \dom{\fFun \Cap \gFun}$.
By $\XSet^{n}$, with $n \in \SetN$, we denote the set of all
$n$-tuples of elements from $\XSet$, by $\XSet^{*} \defeq \bigcup_{n
= 0}^{< \omega} \XSet^{n}$ the set of finite words on the
alphabet $\XSet$, by $\XSet^{+} \defeq \XSet^{*} \setminus \{
\epsilon \}$ the set of non-empty words, and by $\XSet^{\omega}$ the
set of infinite words, where, as usual, $\epsilon \in \XSet^{*}$ is
the empty word.
The length of a word $\wElm \in \XSet^{\infty} \defeq \XSet^{*} \cup
\XSet^{\omega}$ is represented with $\card{\wElm} \in \SetNI$.
By $(\wElm)_{i}$ we indicate the $i$-th letter of the finite word
$\wElm \in \XSet^{*}$, with $i \in \numco{0}{\card{\wElm}}$.
Furthermore, by $\fst{\wElm} \defeq (\wElm)_{0}$ (resp., $\lst{\wElm} \defeq
(\wElm)_{\card{\wElm} - 1}$), we denote the first (resp.,
last) letter of $\wElm$.
In addition, by $(\wElm)_{\leq i}$ (resp., $(\wElm)_{> i}$), we indicate the
prefix up to (resp., suffix after) the letter of index $i$ of
$\wElm$, i.e., the finite word built by the first $i + 1$ (resp., last
$\card{\wElm} - i - 1$) letters $(\wElm)_{0}, \ldots, (\wElm)_{i}$ (resp.,
$(\wElm)_{i + 1}, \ldots, (\wElm)_{\card{\wElm} - 1}$).
We also set, $(\wElm)_{< 0} \defeq \epsilon$, $(\wElm)_{< i} \defeq
(\wElm)_{\leq i - 1}$, $(\wElm)_{\geq 0} \defeq \wElm$, and $(\wElm)_{\geq
i} \defeq (\wElm)_{> i - 1}$, for $i \in \numco{1}{\card{\wElm}}$.
Mutatis mutandis, the notations of $i$-th letter, first, prefix, and suffix
apply to infinite words too.
Finally, by $\pfx{\wElm[1], \wElm[2]} \in \XSet^{\infty}$ we denote the
maximal common prefix of two different words $\wElm[1], \wElm[2] \in
\XSet^{\infty}$, i.e., the finite word $\wElm \in \XSet^{*}$ for which
there are two words $\wElm[1|'], \wElm[2|'] \in \XSet^{\infty}$ such that
$\wElm[1] = \wElm \cdot \wElm[1|']$, $\wElm[2] = \wElm \cdot \wElm[2|']$,
and $\fst{\wElm[1|']} \neq \fst{\wElm[2|']}$.
By convention, we set $\pfx{\wElm, \wElm} \defeq \wElm$.
For a set $\DirSet$ of objects, called directions, a
$\DirSet$-tree is a set $\TSet \subseteq \DirSet^{*}$ closed under
prefix, i.e., if $\tElm \cdot \dElm \in \TSet$, with $\dElm \in \DirSet$,
then also $\tElm \in \TSet$.
We say that it is complete if it holds that $\tElm \cdot \dElm' \in
\TSet$ whenever $\tElm \cdot \dElm \in \TSet$, for all $\dElm' < \dElm$,
where $< \: \subseteq \DirSet \times \DirSet$ is an a priori fixed strict
total order on the set of directions that is clear from the context.
Moreover, it is full if $\TSet = \DirSet^{*}$.
The elements of $\TSet$ are called nodes and the empty word
$\epsilon$ is the root of $\TSet$.
For every $\tElm \in \TSet$ and $\dElm \in \DirSet$, the node $\tElm \cdot
\dElm \in \TSet$ is a successor of $\tElm$ in $\TSet$.
The tree is $b$-bounded if the maximal number $b$ of its successor
nodes is finite, i.e., $b = \max_{\tElm \in \TSet} \card{\set{ \tElm \cdot
\dElm \in \TSet }{ \dElm \in \DirSet }} < \omega$.
A branch of the tree is an infinite word $\wElm \in \DirSet^{\omega}$
such that $(\wElm)_{\leq i} \in \TSet$, for all $i \in \SetN$.
For a finite set $\LabSet$ of objects, called symbols, a
$\LabSet$-labeled $\DirSet$-tree is a quadruple
$\LTDef{\LabSet}{\DirSet}$, where $\TSet$ is a $\DirSet$-tree and $\vFun :
\TSet \to \LabSet$ is a labeling function.
When $\DirSet$ and $\LabSet$ are clear from the context, we call $\LTStruct$
simply a (labeled) tree.
Proofs of Section <ref>
In this appendix, we give the proofs of Theorem <ref> and
Corollary <ref> of overlapping dependence maps and
Theorem <ref> of non-overlapping dependence maps.
In particular, to prove the first two results, we need to introduce the
concept of pivot for a given set of signatures and then show some useful
related properties.
Moreover, for the latter result, we define an apposite ad-hoc signature
dependence, based on a sharp combinatorial construction, in order to maintain
separated the dependence maps associated to the components of a
non-overlapping set of signatures.
To proceed with the definitions, we have first to introduce some additional
Let $\ESet$ be a set and $\sigElm \in \SigSet(\ESet)$ a signature.
Then, $\QPAVSet{\sigElm} \defeq \ESet \setminus \QPEVSet{\sigElm}$
indicates the set of elements in $\ESet$ associated to universal quantified
Moreover, for an element $\eElm \in \ESet$, we denote by $\QPDepSet(\sigElm,
\eElm) \defeq \set{\eElm' \in \ESet}{(\eElm', \eElm) \in
\QPDepSet(\sigElm)}$ the set of elements from which $\eElm$ is functional
Given another element $\eElm' \in \ESet$, we say that $\eElm$
precedes $\eElm'$ in $\sigElm$, in symbols $\eElm \qpordRel[\sigElm]
\eElm'$, if $\bFun(\eElm) \qpordRel[\qpElm] \bFun(\eElm')$, where $\sigElm =
(\qpElm, \bFun)$.
Observe that this kind of order is, in general, not total, due to the fact
that $\bFun$ is not necessarily injective.
Consequently, by $\min_{\qpordRel[\sigElm]} \FSet$, with $\FSet \subseteq
\ESet$, we denote the set of minimal elements of $\FSet$ w.r.t. $\qpordRel[\sigElm]$.
Finally, for a given set of signatures $\SSet \subseteq \SigSet(\ESet)$, we
indicate by $\QPAVSet{\SSet} \defeq \bigcap_{\sigElm \in \SSet}
\QPAVSet{\sigElm}$ the set of elements that are universal in all signatures
of $\SSet$, by $\BPColSet(\SSet, \eElm) \defeq \set {\eElm' \in \ESet
\setminus \QPAVSet{\SSet} }{ (\eElm', \eElm) \in \BPColSet(\SSet)}$ the set
of existential elements that form a collapsing chain with $\eElm$, and by
$\BPColSet(\SSet, \sigElm) \defeq \set{ \eElm \in \ESet }{ \exists \eElm'
\in \QPEVSet{\sigElm} \:.\: (\eElm', \eElm) \in \BPColSet(\SSet) }$ the set
of elements that form a collapsing chain with at least one existential
element in $\sigElm$.
Intuitively, a pivot is an element on which we can extend a partial
assignment that is shared by a set of dependence maps related to signatures
via a signature dependence, in order to find a total assignment by an
iterative procedure.
Let $\FSet$ the domain of a partial function $\dFun : \ESet \to \DSet$ and
$\eElm$ an element not yet defined, i.e., $\eElm \in \ESet \setminus \FSet$.
If, on one hand, $\eElm$ is existential quantified over a signature $\sigElm
= (\qpElm, \bFun)$ and all the elements from which $\eElm$ depends on that
signature are in the domain $\FSet$, then the value of $\eElm$ is uniquely
determined by the related dependence map.
So, $\eElm$ is a pivot.
If, on the other hand, $\eElm$ is universal quantified over all signatures
$\sigElm \in \SSet$ and all elements that form a collapsing chain with
$\eElm$ are in the domain $\FSet$, then, also in this case we can define
the value of $\eElm$ being sure to leave the possibility to build a total
So, also in this case $\eElm$ is a pivot.
For this reason, pivot plays a fundamental role in the construction of such
shared assignments.
The existence of a pivot for a given finite set of signatures $\SSet
\subseteq \SigSet(\ESet)$ w.r.t. a fixed domain $\FSet$ of a partial
assignment is ensured under the hypothesis that there are no cyclic
dependences in $\SSet$.
The existence proof passes through the development of three lemmas
describing a simple seeking procedure.
With the previous description and the examples of Section <ref>
in mind, we now formally describe the properties that an element of the
support set has to satisfy in order to be a pivot for a set of
signatures w.r.t. an a priori given subset of elements.
Let $\SSet \subseteq \SigSet(\ESet)$ be a set of signatures on $\ESet$ and
$\FSet \subset \ESet$ a subset of elements.
Then, an element $\eElm \in \ESet$ is a pivot for $\SSet$ w.r.t. $\FSet$ if $\eElm \not\in \FSet$ and either one of the following items
$\eElm \in \QPAVSet{\SSet}$ and $\BPColSet(\SSet, \eElm) \subseteq
\FSet$;
there is a signature $\sigElm \in \SSet$ such that $\eElm \in
\QPEVSet{\sigElm}$ and $\QPDepSet(\sigElm, \eElm) \subseteq \FSet$.
Intuitively, Item <ref> asserts that the pivot is universal
quantified over all signatures and all existential elements that form a
collapsing chain starting in the pivot itself are already defined.
On the contrary, Item <ref> asserts that the pivot is
existential quantified and, on the relative signature, it depends only on
already defined elements.
Before continuing, we provide the auxiliary definition of minimal
Let $\SSet \subseteq \SigSet(\ESet)$ be a set of signatures on $\ESet$ and
$\FSet \subset \ESet$ a subset of elements.
A pair $(\vec{\eElm}, \vec{\sigElm}) \in \ESet^{k} \times \SSet^{k}$,
with $k \in \numco{1}{\omega}$, is a minimal $\SSet$-chain w.r.t. $\FSet$ if it is an $\SSet$-chain such that $(\vec{\eElm})_{i} \in
\min_{(\vec{\sigElm})_{i}} (\ESet \setminus \FSet)$, for all $i \in
\numco{0}{k}$.
In addition to the definition of pivot, we also give the formal concept of
pivot seeker that is used, in an iterative procedure, to find a pivot
if this exists.
Let $\SSet \subseteq \SigSet(\ESet)$ be a set of signatures on $\ESet$ and
$\FSet \subset \ESet$ a subset of elements.
Then, a pair $(\eElm \cdot \vec{\eElm}, \sigElm \cdot \vec{\sigElm}) \in
\EElm^{k} \times \SSet^{k}$ of sequences of elements and signatures of
length $k \in \numco{1}{\omega}$ is a pivot seeker for $\SSet$
w.r.t. $\FSet$ if the following hold:
$\eElm \in \min_{\sigElm} (\ESet \setminus \FSet)$;
$\fst{\vec{\eElm}} \in (\QPEVSet{\sigElm} \cup \BPColSet(\SSet,
\sigElm)) \setminus \FSet$, if $k > 1$;
$(\vec{\eElm}, \vec{\sigElm})$ is a minimal $\SSet$-chain, if $k > 1$.
Intuitively, a pivot seeker is a snapshot of the seeking procedure at a
certain step.
Item <ref> ensures that the element $\eElm$ we are going to
consider as a candidate for pivot depends only on the elements defined
in $\FSet$.
Item <ref> builds a link between the signature $\sigElm$ of
the present candidate and the head element $\fst{\vec{\eElm}}$ of the
previous step, in order to maintain information about the dependences that
are not yet satisfied.
Finally, Item <ref> is used to ensure that the procedure
avoids loops by checking pivots on signature already considered.
As shown through the above mentioned examples, in the case of overlapping
signatures, we can always find a pivot w.r.t. a given set of elements
already defined, by means of a pivot seeker.
The following lemma ensures that we can always start the iterative procedure
over pivot seekers to find a pivot.
Let $\SSet \subseteq \SigSet(\ESet)$ be a set of signatures on $\ESet$ and
$\FSet \subset \ESet$ a subset of elements.
Then, there exists a pivot seeker for $\SSet$ w.r.t. $\FSet$ of length
Let $\sigElm \in \SSet$ be a generic signature and $\eElm \in \ESet$ an
element such that $\eElm \in \min_{\sigElm} (\ESet \setminus \FSet)$.
Then, it is immediate to see that the pair $(\eElm, \sigElm) \in \ESet^{1}
\times \SSet^{1}$ is a pivot seeker for $\SSet$ w.r.t. $\FSet$ of length
$1$, since Item <ref> of Definition <ref> of
pivot seekers is verified by construction and
Items <ref> and <ref> are vacuously
Now, suppose to have a pivot seeker of length not greater than the size of
the support set $\ESet$ and that no pivot is already found.
Then, in the case of signatures without cyclic dependences, we can always
continue the iterative procedure, by extending the previous pivot seeker of
just one further element.
Let $\SSet \subseteq \SigSet(\ESet)$ be a set of signatures on $\ESet$
with $\CSet(\SSet) = \emptyset$ and $\FSet \subset \ESet$ a subset of
Moreover, let $(\eElm \cdot \vec{\eElm}, \sigElm \cdot \vec{\sigElm}) \in
\EElm^{k} \times \SSet^{k}$ be a pivot seeker for $\SSet$ w.r.t. $\FSet$
of length $k \in \numco{1}{\omega}$.
Then, if $\eElm$ is not a pivot for $\SSet$ w.r.t. $\FSet$, there exists
a pivot seeker for $\SSet$ w.r.t. $\FSet$ of length $k + 1$.
By Item <ref> of Definition <ref> of pivot
seekers, we deduce that $\eElm \notin \FSet$ and $\QPDepSet(\sigElm,
\eElm) \subseteq \FSet$.
Thus, if $\eElm$ is not a pivot for $\SSet$ w.r.t. $\FSet$, by
Definition <ref> of pivot, we have that $\eElm \not\in
\QPAVSet{\SSet}$ or $\BPColSet(\SSet, \eElm) \not\subseteq \FSet$ and, in
both cases, $\eElm \in \QPAVSet{\sigElm}$.
We now distinguish the two cases.
* $\eElm \notin \QPAVSet{\SSet}$.
There exists a signature $\sigElm' \in \SSet$ such that $\eElm \in
\QPEVSet{\sigElm'}$.
So, consider an element $\eElm' \in \min_{\sigElm'} (\ESet \setminus
\FSet)$.
We now show that the pair of sequences $(\eElm' \cdot \eElm \cdot
\vec{\eElm}, \sigElm' \cdot \sigElm \cdot \vec{\sigElm}) \in \ESet^{k
+ 1} \times \SSet^{k + 1}$ of length $k + 1$ satisfies
Items <ref> and <ref> of
Definition <ref>.
The first item is trivially verified by construction.
Moreover, $\fst{\eElm \cdot \vec{\eElm}} = \eElm \in
\QPEVSet{\sigElm'} \setminus \FSet$.
Hence, the second item holds as well.
* $\eElm \in \QPAVSet{\SSet}$.
We necessarily have that $\BPColSet(\SSet, \eElm) \not\subseteq
\FSet$.
Thus, there is an element $\eElm' \in \ESet \setminus (\QPAVSet{\SSet}
\cup \FSet)$ such that $(\eElm', \eElm) \in \BPColSet(\SSet)$.
Consequently, there exists also a signature $\sigElm' \in \SSet$ such
that $\eElm' \in \QPEVSet{\sigElm'} \setminus \FSet$.
So, consider an element $\eElm'' \in \min_{\sigElm'} (\ESet \setminus
\FSet)$.
We now show that the pair of sequences $(\eElm'' \cdot \eElm \cdot
\vec{\eElm}, \sigElm' \cdot \sigElm \cdot \vec{\sigElm}) \in \ESet^{k
+ 1} \times \SSet^{k + 1}$ of length $k + 1$ satisfies
Items <ref> and <ref> of
Definition <ref>.
The first item is trivially verified by construction.
Moreover, since $(\eElm', \eElm) \in \BPColSet(\SSet)$, by the
definition of $\BPColSet(\SSet, \sigElm')$, we have that $\fst{\eElm
\cdot \vec{\eElm}} = \eElm \in \BPColSet(\SSet, \sigElm') \setminus
\FSet$.
Hence, the second item holds as well.
At this point, it only remains to show that Item <ref>
of Definition <ref> holds, i.e., that $(\eElm \cdot
\vec{\eElm}, \sigElm \cdot \vec{\sigElm})$ is a minimal $\SSet$-chain
w.r.t. $\FSet$.
For $k = 1$, we have that Items <ref>
and <ref> of Definition <ref> of $\SSet$-chain
are vacuously verified.
Moreover, since $\eElm \in \QPAVSet{\sigElm}$, also
Item <ref> of the previous definition holds.
Finally, the $\SSet$-chain is minimal w.r.t. $\FSet$, due to the fact
that $\eElm \in \min_{\sigElm} (\ESet \setminus \FSet)$.
Now, suppose that $k > 1$.
Since $(\vec{\eElm}, \vec{\sigElm})$ is already an $\SSet$-chain, to prove
Items <ref> and <ref> of
Definition <ref> of $\SSet$-chain, we have only to show that
$(\eElm, \fst{\vec{\eElm}}) \in \QPDepSet'(\sigElm)$ and $\sigElm \neq
(\vec{\sigElm})_{i}$, for all $i \in \numco{0}{k - 1}$, respectively.
By Items <ref> and <ref> of
Definition <ref>, we have that $\eElm \in \min_{\sigElm}(\ESet
\setminus \FSet)$ and $\fst{\vec{\eElm}} \in (\QPEVSet{\sigElm} \cup
\BPColSet(\SSet, \sigElm)) \setminus \FSet$.
So, two cases arise.
* $\fst{\vec{\eElm}} \in \QPEVSet{\sigElm} \setminus \FSet$.
Since $\eElm \in \QPAVSet{\sigElm} \cap \min_{\sigElm}(\ESet \setminus
\FSet)$, we can deduce that $(\eElm, \fst{\vec{\eElm}}) \in
\QPDepSet(\sigElm) \subseteq \QPDepSet'(\sigElm)$.
* $\fst{\vec{\eElm}} \in \BPColSet(\SSet, \sigElm) \setminus \FSet$.
By the definition of $\BPColSet(\SSet, \sigElm)$, there exists $\eElm'
\in \QPEVSet{\sigElm} \setminus \FSet$ such that $(\eElm',
\fst{\vec{\eElm}}) \in \BPColSet(\SSet)$.
Now, since $\eElm \in \QPAVSet{\sigElm} \cap \min_{\sigElm}(\ESet
\setminus \FSet)$, we can deduce that $(\eElm, \eElm') \in
\QPDepSet(\sigElm)$.
Thus, by definition of $\QPDepSet'(\sigElm)$, it holds that $(\eElm,
\fst{\vec{\eElm}}) \in \QPDepSet'(\sigElm)$.
Finally, suppose by contradiction that there exists $i \in \numco{0}{k -
1}$ such that $\sigElm = (\vec{\sigElm})_{i}$.
Two cases can arise.
* $i = k - 2$.
Then, by Item <ref> of Definition <ref>, we
have that $(\vec{\eElm})_{i} = \lst{\vec{\eElm}} \in
\QPAVSet{\lst{\vec{\sigElm}}} = \QPAVSet{(\vec{\sigElm})_{i}}$;
* $i < k - 2$.
Then, by Item <ref> of Definition <ref>, we
have that $((\vec{\eElm})_{i}, (\vec{\eElm})_{i + 1}) \in
\QPDepSet'((\vec{\sigElm})_{i})$.
Consequently, $(\vec{\eElm})_{i} \in \QPAVSet{(\vec{\sigElm})_{i}}$.
By Definition <ref> of minimal $\SSet$-chain, since
$(\vec{\eElm}, \vec{\sigElm})$ is minimal w.r.t. $\FSet$, it holds that
$(\vec{\eElm})_{i} \in \min_{(\vec{\sigElm})_{i}} (\ESet \setminus
\FSet)$.
So, $(\vec{\eElm})_{i} \in \QPAVSet{(\vec{\sigElm})_{i}} \cap
\min_{(\vec{\sigElm})_{i}} (\ESet \setminus \FSet)$.
Moreover, by Item <ref> of Definition <ref>,
we have that $(\vec{\eElm})_{0} \in (\QPEVSet{\sigElm} \cup
\BPColSet(\SSet, \sigElm)) \setminus \FSet =
(\QPEVSet{(\vec{\sigElm})_{i}} \cup \BPColSet(\SSet, (\vec{\sigElm})_{i}))
\setminus \FSet$.
Thus, by applying a reasoning similar to the one used above to prove that
$(\eElm, \fst{\vec{\eElm}}) \in \QPDepSet'(\sigElm)$, we obtain that
$((\vec{\eElm})_{i}, (\vec{\eElm})_{0}) \in
\QPDepSet'((\vec{\sigElm})_{i})$
Hence, $((\vec{\eElm})_{\leq i}, (\vec{\sigElm})_{\leq i})$ satisfies
Definition <ref> of cyclic dependences.
So, $((\vec{\eElm})_{\leq i}, (\vec{\sigElm})_{\leq i}) \in \CSet(\SSet)
\neq \emptyset$, which is a contradiction.
Finally, if we have run the procedure until all elements in $\ESet$ are
visited, the first one of the last pivot seeker is necessarily a pivot.
Let $\SSet \subseteq \SigSet(\ESet)$ be a finite set of signatures on
$\ESet$ with $\CSet(\SSet) = \emptyset$ and $\FSet \subset \ESet$ a subset
of elements.
Moreover, let $(\eElm \cdot \vec{\eElm}, \sigElm \cdot \vec{\sigElm}) \in
\EElm^{k} \times \SSet^{k}$ be a pivot seeker for $\SSet$ w.r.t. $\FSet$
of length $k \defeq \card{\SSet} + 1$.
Then, $\eElm$ is a pivot for $\SSet$ w.r.t. $\FSet$.
Suppose by contradiction that $\eElm$ is not a pivot for $\SSet$ w.r.t. $\FSet$.
Then, by Lemma <ref> of pivot seeker extension, there exists
a pivot seeker for $\SSet$ w.r.t. $\FSet$ of length $k + 1$, which is
impossible due to Item <ref> of
Definition <ref> of pivot seekers, since an $\SSet$-chain of
length $k$ does not exist.
By appropriately combining the above lemmas, we can prove the existence of a
pivot for a given set of signatures having no cyclic dependences.
Let $\SSet \subseteq \SigSet(\ESet)$ be a finite set of signatures on
$\ESet$ with $\CSet(\SSet) = \emptyset$ and $\FSet \subset \ESet$ a subset
of elements.
Then, there exists a pivot for $\SSet$ w.r.t. $\FSet$.
By Lemma <ref> of pivot seeker existence, there is a pivot
seeker of length $1$ for $\SSet$ w.r.t. $\FSet$, which can be extended,
by using Lemma <ref> of pivot seeker extension, at most
$\card{\SSet} < \omega$ times, due to Lemma <ref> of seeking
termination, before the reach of a pivot $\eElm$ for $\SSet$ w.r.t. $\FSet$.
Big signature dependences
In order to prove Theorem <ref>, we first introduce big
signature map $\spcmapFun$.
Let $\PSet \subseteq \LSigSet(\ESet)$ be a set of labeled signatures over
a set $\ESet$, and $\DSet = \PSet \times \QPVSet(\PSet) \times \{0, 1
\}^{C(\PSet)}$.
Then, the big signature dependence $\spcmapFun \in
\SpcMapSet[\DSet](\PSet)$ for $\PSet$ over $\DSet$ is defined as follow.
For all $(\sigElm, \lElm) = ((\qpElm, \bFun), \lElm) \in \PSet$, and
$\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$, we have that:
$\spcmapFun((\sigElm, \lElm))(\valFun)(\xElm) \defeq \valFun(\xElm)$,
for all $\xElm \in \QPAVSet{\qpElm}$;
$\spcmapFun((\sigElm, \lElm))(\valFun)(\xElm) \defeq ((\sigElm,
\lElm), \xElm, \hFun)$, for all $\xElm \in \QPEVSet{\qpElm}$,
where $\hFun \in \{0, 1 \}^{C(\PSet)}$ is such that, for all
$(\vec{\eElm}, \vec{\sigElm})$, the following hold:
if $\sigElm = \fst{\vec{\sigElm}}$ and $\xElm =
\bFun(\fst{\vec{\eElm}})$ then $\hFun((\vec{\eElm},
\vec{\sigElm})) \defeq 1 - \hFun'((\vec{\eElm}, \vec{\sigElm}))$,
where $\hFun' \in \{0, 1 \}^{C(\PSet)}$ is such that
$\valFun(\bFun(\lst{\vec{\eElm}})) = ((\sigElm', \lElm'),
\xElm', \hFun')$, for some $(\sigElm', \lElm') \in \PSet$ and
$\xElm' \in \QPVSet(\PSet)$;
if there exists $i \in \numco{1}{\card{\vec{\sigElm}}}$ such
that $\sigElm = (\vec{\sigElm})_{i}$ and $\xElm =
\bFun((\vec{\eElm})_{i})$, then $\hFun((\vec{\eElm},
\vec{\sigElm})) \defeq \hFun'((\vec{\eElm}, \vec{\sigElm}))$,
where $\hFun' \in \{0, 1 \}^{C(\PSet)}$ is such that
$\valFun(\bFun((\vec{\eElm})_{i})) = ((\sigElm', \lElm'),
\xElm', \hFun')$, for some $(\sigElm', \lElm') \in \PSet$ and
$\xElm' \in \QPVSet(\PSet)$;
if none of the above cases apply, set $\hFun((\vec{\eElm},
\vec{\sigElm})) \defeq 0$.
Note that Items <ref>
and <ref> are mutually exclusive since, by
definition of cyclic dependence, each signature $(\vec{\sigElm})_{i}$ occurs
only once in $\vec{\sigElm}$.
It is easy to see that the previous definition is well formed, i.e., that
$\spcmapFun$ is actually a labeled signature dependence.
Indeed the following lemma holds.
Let $\PSet \subseteq \LSigSet(\ESet)$ be a set of labeled signatures over
a set $\ESet$ and $\DSet = \PSet \times \QPVSet(\PSet) \times \{0, 1
\}^{C(\PSet)}$.
Then the big signature dependence $\spcmapFun$ for $\PSet$ over $\DSet$ is
a labeled signature dependence for $\PSet$ over $\DSet$.
We have to show that $\wFun(((\qpElm, \bFun), \lElm))$ is a dependence map
for $\qpElm$ over $\DSet$, for all $(\sigElm, \lElm) \in \PSet$.
* By Item <ref> of
Definition <ref> it holds that $\wFun((\sigElm,
\lElm))(\valFun)(\xElm) = \valFun(\xElm)$, for all $\xElm \in
\QPAVSet{\qpElm}$ and $\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$,
which means $\wFun((\sigElm, \lElm))(\valFun)_{\rst
\QPAVSet{\qpElm}} = \valFun$, that means that
Item <ref> of Definition <ref> holds.
* For the Item <ref> of
Definition <ref>, let $\valFun[1], \valFun[2] \in
\ValSet[\DSet](\QPAVSet{\qpElm})$ and $\xElm \in \QPEVSet{\qpElm}$
such that $(\valFun_{1})_{\rst \QPDepSet(\qpElm,\xElm)} =
(\valFun_{2})_{\rst \QPDepSet(\qpElm, \xElm)}$.
We have to prove that $\spcmapFun((\sigElm,
\lElm))(\valFun[1])(\xElm) = \spcmapFun((\sigElm,
\lElm))(\valFun[2])(\xElm)$.
By definition, we have that $\spcmapFun((\sigElm,
\lElm))(\valFun[1])(\xElm) = ((\sigElm, \lElm), \xElm, \hFun[1])$
and $\spcmapFun((\sigElm, \lElm))(\valFun[2])(\xElm) = ((\sigElm,
\lElm), \xElm, \hFun[2])$.
So, we have only to show that $\hFun[1] = \hFun[2]$.
To do this, consider a cyclic dependence $(\vec{\eElm},
\vec{\sigElm}) \in C(\PSet)$ for which there exists $i \in
\numco{0}{\card{\vec{\sigElm}}}$ such that $\sigElm =
(\vec{\sigElm})_{i}$ and $\xElm = \bFun((\vec{\eElm})_{i})$.
Then, we have that $\valFun[1](\ySym) = \valFun[2](\ySym) =
((\sigElm', \lElm'), \ySym', \hFun')$ for $\ySym =
\bFun((\vec{\eElm})_{(i-1) \mod \card{\vec{\sigElm}}})$.
Then, we have the following:
* by Item <ref> of
Definition <ref>, if $i = 1$ then
$\hFun[1]((\vec{\eElm}, \vec{\sigElm})) = 1 -
\hFun[1]'((\vec{\eElm}, \vec{\sigElm})) = \hFun[2]((\vec{\eElm},
\vec{\sigElm}))$;
* by Item <ref> of
Definition <ref>, if $i \in
\numoo{1}{\card{\vec{\sigElm}}}$ then $\hFun[1]((\vec{\eElm},
\vec{\sigElm})) = \hFun[1]'((\vec{\eElm}, \vec{\sigElm})) =
\hFun[2]((\vec{\eElm}, \vec{\sigElm}))$.
On the other side, consider a cyclic dependence $(\vec{\eElm},
\vec{\sigElm}) \in C(\PSet)$ such that $\sigElm \neq
(\vec{\sigElm})_{i}$ or $\xElm \neq \bFun((\vec{\eElm})_{i})$, for
all $i \in \numco{0}{\card{\vec{\sigElm}}}$.
In this case, by Item <ref> of
Definition <ref>, we have that $\hFun[1]((\vec{\eElm},
\vec{\sigElm})) = 0 = \hFun[2]((\vec{\eElm}, \vec{\sigElm}))$.
Proofs of theorems
We are finally able to show the proofs of the above mentioned results.
Let $\SSet \subseteq \SigSet(\ESet)$ be a finite set of overlapping
signatures on $\ESet$.
Then, for all signature dependences $\spcmapFun \in
\SpcMapSet[\DSet](\SSet)$ for $\SSet$ over a set $\DSet$, it holds that
$\spcmapFun$ is overlapping.
By Definition <ref> of signature dependence, to prove the
statement, i.e., that $\cap_{(\qpElm, \bFun) \in \SSet} \set{ \valFun
\circ \bFun }{ \valFun \in \rng{\spcmapFun(\qpElm, \bFun)} } \neq
\emptyset$, we show the existence of a function $\dFun \in \DSet^{\ESet}$
such that, for all signatures $\sigElm = (\qpElm, \bFun) \in \SSet$, there
is a valuation $\valFun[\sigElm] \!\in\! \rng{\spcmapFun(\sigElm)}$ for
which $\dFun = \valFun[\sigElm] \cmp \bFun$.
We build $\dFun$ iteratively by means of a succession of partial functions
$\dFun_{j} \!:\! \ESet \pto \DSet$, with $j \in \numcc{0}{\card{\ESet}}$,
satisfying the following invariants:
$\dFun[j](\eElm') = \dFun[j](\eElm'')$, for all $(\eElm', \eElm'') \in
\BPColSet(\SSet) \cap (\dom{\dFun[j]} \times \dom{\dFun[j]})$;
for all $\eElm \in \dom{\dFun[j]}$, there is $i \in \numco{0}{j}$ such
that $\eElm$ is a pivot for $\SSet$ w.r.t. $\dom{\dFun[i]}$;
$\dom{\dFun[j]} \subset \dom{\dFun[j + 1]}$, where $j < \card{\ESet}$;
$\dFun[j] = \dFun[j + 1]_{\rst \dom{\dFun[j]}}$, where $j <
\card{\ESet}$.
Before continuing, observe that, since $\QPEVSet{\SSet} = \emptyset$, for
each element $\eElm \in \ESet \setminus \QPAVSet{\SSet}$, there exists
exactly one signature $\sigElm[\eElm] = (\qpElm[\eElm], \bFun[\eElm]) \in
\SSet$ such that $\eElm \in \QPEVSet{\sigElm[\eElm]}$.
As base case, we simply set $\dFun[0] \defeq \emptyfun$.
It is immediate to see that the invariants are vacuously satisfied.
Now, consider the iterative case $j \in \numco{0}{\card{\ESet}}$.
By Lemma <ref> of pivot existence, there is a pivot $\eElm[j]
\in \ESet$ for $\SSet$ w.r.t. $\dom{\dFun[j]}$.
Remind that $\eElm[j] \not\in \dom{\dFun[j]}$.
At this point, two cases can arise.
* $\eElm[j] \in \QPAVSet{\SSet}$.
If there is an element $\eElm \in \dom{\dFun[j]}$ such that $(\eElm,
\eElm[j]) \in \BPColSet(\SSet)$ then set $\dFun[j + 1] \defeq
\dFun[j][ {\eElm[j]} \mapsto \dFun[j](\eElm) ]$.
By Invariant <ref> at step $j$, the choice of such
an element is irrelevant.
Otherwise, choose a value $\cElm \in \DSet$, and set $\dFun[j + 1]
\defeq \dFun[j][ {\eElm[j]} \mapsto \cElm ]$.
In both cases, all invariants at step $j + 1$ are trivially satisfied
by construction.
* $\eElm[j] \not\in \QPAVSet{\SSet}$.
Consider a valuation $\valFun[j] \in \ValSet[\DSet](\QPAVSet{\qpElm[
{\eElm[j]} ])}$ such that $\valFun[j](\bFun[ {\eElm[j]} ](\eElm)) =
\dFun[j](\eElm)$, for all $\eElm \in \dom{\dFun[j]} \cap
\QPAVSet{\sigElm[ {\eElm[j]} ]}$.
The existence of such a valuation is ensured by
Invariant <ref> at step $j$, since
$\dFun[j](\eElm') = \dFun[j](\eElm'')$, for all $\eElm', \eElm'' \in
\dom{\dFun[j]}$ with $\bFun[ {\eElm[j]} ](\eElm') = \bFun[ {\eElm[j]}
Now, set $\dFun[j + 1] \defeq \dFun[j][ {\eElm[j]} \mapsto
\spcmapFun(\sigElm[ {\eElm[j]} ])(\valFun[j])(\bFun[ {\eElm[j]}
](\eElm[j])) ]$.
It remains to verify the invariants at step $j + 1$.
Invariants <ref>, <ref>,
and <ref> are trivially satisfied by construction.
For Invariant <ref>, instead, suppose that there
exists $(\eElm[j], \eElm) \in \BPColSet(\SSet) \cap (\dom{\dFun[j +
1]} \times \dom{\dFun[j + 1]})$ with $\eElm[j] \neq \eElm$.
By Invariant <ref> at step $j$, there is $i \in
\numco{0}{j}$ such that $\eElm$ is a pivot for $\SSet$ w.r.t. $\dom{\dFun[i]}$, i.e., $\eElm = \eElm[i]$.
At this point, two subcases can arise, the first of which results to
be impossible.
* $\eElm[i] \in \QPAVSet{\SSet}$.
By Item <ref> of Definition <ref> of pivot,
it holds that $\BPColSet(\SSet, \eElm[i]) \subseteq
\dom{\dFun[i]}$.
Moreover, since $\eElm[j] \not\in \QPAVSet{\SSet}$ and $(\eElm[j],
\eElm[i]) \in \BPColSet(\SSet)$, it holds that $\eElm[j] \in
\BPColSet(\SSet, \eElm[i])$.
Thus, by a repeated application of
Invariant <ref> from step $i$ to step $j$, we
have that $\eElm[j] \in \dom{\dFun[i]} \subset \dom{\dFun[j]}
\not\ni \eElm[j]$, which is a contradiction.
* $\eElm[i] \not\in \QPAVSet{\SSet}$.
Since $\eElm[j], \eElm[i] \not\in \QPAVSet{\SSet}$ and $(\eElm[j],
\eElm[i]) \in \BPColSet(\SSet)$, it is easy to see that $\sigElm[
{\eElm[j]} ] = \sigElm[ {\eElm[i]} ]$ and $\bFun[ {\eElm[j]}
](\eElm[j]) = \bFun[ {\eElm[i]} ](\eElm[i])$.
Otherwise, we have that $\eElm[j] \in \QPEVSet{\SSet} =
\emptyset$, which is impossible.
Hence, it follows that $\dFun[j + 1](\eElm[j]) =
\spcmapFun(\sigElm[ {\eElm[j]} ])(\valFun[j])(\bFun[ {\eElm[j]}
](\eElm[j])) = \spcmapFun(\sigElm[ {\eElm[i]}
])(\valFun[j])(\bFun[ {\eElm[i]} ](\eElm[i]))$.
Moreover, $\dFun[i + 1](\eElm[i]) = \spcmapFun(\sigElm[ {\eElm[i]}
])(\valFun[i])(\bFun[ {\eElm[i]} ](\eElm[i]))$.
Now, it is easy to observe that $\QPDepSet(\qpElm[j], \bFun[
{\eElm[j]} ](\eElm[j])) \!=\! \QPDepSet(\qpElm[i], \bFun[
{\eElm[i]} ](\eElm[i]))$, from which we derive that
$\valFun[j]_{\rst \QPDepSet(\qpElm[j], \bFun[
{\eElm[j]} ](\eElm[j]))} \!=\! \valFun[i]_{\rst
\QPDepSet(\qpElm[i], \bFun[ {\eElm[i]} ](\eElm[i]))}$.
At this point, by Item <ref> of
Definition <ref> of dependence maps, it holds
that $\spcmapFun(\sigElm[ {\eElm[i]} ])(\valFun[j])(\bFun[
{\eElm[i]} ](\eElm[i])) \!=\! \spcmapFun(\sigElm[ {\eElm[i]}
])(\valFun[i]) \allowbreak (\bFun[ {\eElm[i]} ](\eElm[i]))$, so,
$\dFun[j + 1](\eElm[j]) \!=\! \dFun[i + 1](\eElm[i])$.
Finally, by a repeated application of
Invariant <ref> from step $i + 1$ to step $j$,
we obtain that $\dFun[i + 1](\eElm[i]) = \dFun[j + 1](\eElm[i])$.
Hence, $\dFun[j + 1](\eElm[j]) = \dFun[j + 1](\eElm[i])$.
By a repeated application of Invariant <ref> from step
$0$ to step $\card{\ESet} - 1$, we have that $\dFun[\card{\ESet}]$ is a
total function.
So, we can now prove that $\dFun \defeq \dFun[\card{\ESet}]$ satisfies the
statement, i.e., $\dFun \in \cap_{(\qpElm, \bFun) \in \SSet} \set{ \valFun
\circ \bFun }{ \valFun \in \rng{\spcmapFun(\qpElm, \bFun)} }$.
For each signature $\sigElm = (\qpElm, \bFun) \in \SSet$, consider the
universal valuation $\valFun[\sigElm | '] \in
\ValSet[\DSet](\QPAVSet{\qpElm})$ such that $\valFun[\sigElm |
'](\bFun(\eElm)) = \dFun(\eElm)$, for all $\eElm \in \QPAVSet{\sigElm}$.
The existence of such a valuation is ensured by
Invariant <ref> at step $\card{\ESet}$.
Moreover, let $\valFun[\sigElm] \defeq \spcmapFun(\sigElm)(\valFun[\sigElm
| '])$.
It remains to prove that $\dFun = \valFun[\sigElm] \cmp \bFun$, by showing
separately that $\dFun_{\rst \QPAVSet{\sigElm}} = (\valFun[\sigElm] \cmp
\bFun)_{\rst \QPAVSet{\sigElm}}$ and $\dFun_{\rst \QPEVSet{\sigElm}} =
(\valFun[\sigElm] \cmp \bFun)_{\rst \QPEVSet{\sigElm}}$ hold.
On one hand, by Item <ref> of
Definition <ref>, for each $\xElm \in \QPAVSet{\qpElm}$, it
holds that $\valFun[\sigElm | '](\xElm) =
\spcmapFun(\sigElm)(\valFun[\sigElm | '])(\xElm)$.
Thus, for each $\eElm \in \QPAVSet{\sigElm}$, we have that
$\valFun[\sigElm | '](\bFun(\eElm)) = \spcmapFun(\sigElm)(\valFun[\sigElm
| '])(\bFun(\eElm))$, which implies $\dFun(\eElm) = \valFun[\sigElm |
'](\bFun(\eElm)) = \spcmapFun(\sigElm)(\valFun[\sigElm | '])(\bFun(\eElm))
= \valFun[\sigElm](\bFun(\eElm)) = (\valFun[\sigElm] \cmp \bFun)(\eElm)$.
So, $\dFun_{\rst \QPAVSet{\sigElm}} = (\valFun[\sigElm] \cmp \bFun)_{\rst
\QPAVSet{\sigElm}}$.
On the other hand, consider an element $\eElm \in \QPEVSet{\sigElm}$.
By Invariant <ref> at step $\card{\ESet}$, there is $i
\in \numco{0}{\card{\ESet}}$ such that $\eElm$ is a pivot for $\SSet$
w.r.t. $\dom{\dFun[i]}$.
This means that $\eElm[i] = \eElm$ and so $\sigElm[ {\eElm[i]} ] =
\sigElm$.
So, by construction, we have that $\dFun[i + 1](\eElm) =
\spcmapFun(\sigElm)(\valFun[i])(\bFun(\eElm))$.
Moreover, $\spcmapFun(\sigElm)(\valFun[\sigElm | '])(\bFun(\eElm)) =
\valFun[\sigElm](\bFun(\eElm)) = (\valFun[\sigElm] \cmp \bFun)(\eElm)$.
Thus, to prove the required statement, we have only to show that
$\dFun(\eElm) = \dFun[i + 1](\eElm)$ and
$\spcmapFun(\sigElm)(\valFun[i])(\bFun(\eElm)) =
\spcmapFun(\sigElm)(\valFun[\sigElm | '])(\bFun(\eElm))$.
By a repeated application of Invariants <ref>
and <ref> from step $i$ to step $\card{\ESet} - 1$, we
obtain that $\dom{\dFun[i]} \subset \dom{\dFun}$, $\dFun[i] = \dFun_{\rst
\dom{\dFun[i]}}$, and $\dFun[i + 1](\eElm) = \dFun(\eElm)$.
Thus, by definition of $\valFun[i]$ and $\valFun[\sigElm | ']$, it follows
that $\valFun[i](\bFun(\eElm')) = \dFun[i](\eElm') = \dFun(\eElm') =
\valFun[\sigElm | '](\bFun(\eElm'))$, for all $\eElm' \in \dom{\dFun[i]}$.
At this point, by Item <ref> of Definition <ref>, it
holds that $\QPDepSet(\sigElm, \eElm) \subseteq \dom{\dFun[i]}$, which
implies that $\valFun[i]_{\rst \QPDepSet(\qpElm, \bFun(\eElm))} =
\valFun[\sigElm | ']_{\rst \QPDepSet(\qpElm, \bFun(\eElm))}$.
Hence, by Item <ref> of Definition <ref>, we
have that $\spcmapFun(\sigElm)(\valFun[i])(\bFun(\eElm)) =
\spcmapFun(\sigElm)(\valFun[\sigElm | '])(\bFun(\eElm))$.
So, $\dFun_{\rst \QPEVSet{\sigElm}} = (\valFun[\sigElm] \cmp \bFun)_{\rst
\QPEVSet{\sigElm}}$.
Let $\PSet \subseteq \LSigSet(\ESet, \allowbreak \LSet)$ be a finite set
of overlapping labeled signatures on $\ESet$ w.r.t. $\LSet$.
Then, for all labeled signature dependences $\spcmapFun \in
\LSpcMapSet[\DSet](\PSet)$ for $\PSet$ over a set $\DSet$, it holds that
$\spcmapFun$ is overlapping.
Consider the set $\PSet' \defeq \set{ (\sigElm, \lElm) \in \PSet }{
\QPEVSet{\sigElm} \neq \emptyset }$ of all labeled signatures in $\PSet$
having at least one existential element.
Since $\PSet$ is overlapping, by Definition <ref> of
overlapping signatures, we have that, for all $(\sigElm, \lElm[1]),
(\sigElm, \lElm[2]) \in \PSet'$, it holds that $\lElm[1] = \lElm[2]$.
So, let $\SSet \defeq \set{ \sigElm \in \SigSet(\ESet) }{ \exists \lElm
\in \LSet \:.\: (\sigElm, \lElm) \in \PSet' }$ be the set of first
components of labeled signatures in $\PSet'$ and $\hFun : \SSet \to
\PSet'$ the bijective function such that $\hFun(\sigElm) \defeq (\sigElm,
\lElm)$, for all $\sigElm \in \SSet$, where $\lElm \in \LSet$ is the
unique label for which $(\sigElm, \lElm) \in \PSet'$ holds.
Now, since $\SSet$ is overlapping, by Theorem <ref> of
overlapping dependence maps, we have that the signature dependence
$\spcmapFun \cmp \hFun \in \SpcMapSet[\DSet](\SSet)$ is overlapping as
Thus, it is immediate to see that $\spcmapFun_{\rst \PSet'}$ is also
overlapping, i.e., by Definition <ref> of signature
dependences, there exists $\dFun \in \DSet^{\ESet}$ such that $\dFun \in
\cap_{((\qpElm, \bFun), \lElm) \in \PSet'} \set{ \valFun \circ \bFun }{
\valFun \in \rng{\spcmapFun((\qpElm, \bFun), \lElm)}} \neq \emptyset$.
At this point, consider the labeled signatures $(\sigElm, \lElm) =
((\qpElm, \bFun), \lElm) \in \PSet \setminus \PSet'$.
Since $\QPEVSet{\sigElm} = \emptyset$, i.e., $\QPEVSet{\qpElm} =
\emptyset$, we derive that $\spcmapFun((\sigElm, \lElm)) \in
\SpcSet[\DSet](\qpElm)$ is the identity dependence map, i.e., it is the
identity function on $\ValSet[\DSet](\QPVSet(\qpElm))$.
Thus, $\rng{\spcmapFun((\sigElm, \lElm))} =
\ValSet[\DSet](\QPVSet(\qpElm))$.
So, we have that $\dFun \in \cap_{((\qpElm, \bFun), \lElm) \in \PSet}
\set{ \valFun \circ \bFun }{ \valFun \in \rng{\spcmapFun((\qpElm, \bFun),
\lElm)}} \neq \emptyset$.
Hence, again by Definition <ref>, it holds that $\spcmapFun$ is
Let $\PSet \subseteq \LSigSet(\ESet, \LSet)$ be a set of labeled
signatures on $\ESet$ w.r.t. $\LSet$.
Then, there exists a labeled signature dependence $\spcmapFun \in
\LSpcMapSet[\DSet](\PSet)$ for $\PSet$ over $\DSet \!\defeq\! \PSet \times
\{ 0, 1 \}^{\CSet(\PSet)}$ such that, for all $\PSet' \!\subseteq\!
\PSet$, it holds that $\spcmapFun_{\rst \PSet'} \in
\LSpcMapSet[\DSet](\PSet')$ is non-overlapping, if $\PSet'$ is
Let $\SSet' \defeq \set{ \sigElm \in \SigSet(\ESet) }{ \exists \lElm \in
\LSet \:.\: (\sigElm, \lElm) \in \PSet' }$ be the set of signatures that
occur in some labeled signature in $\PSet'$.
If $\PSet'$ is non-overlapping, we distinguish the following three cases.
* There exist $(\sigElm, \lElm[1]), (\sigElm, \lElm[2]) \in \PSet'$,
with $\sigElm = (\qpElm, \bFun)$, such that $\QPEVSet{\sigElm} \neq
\emptyset$ and $\lElm[1] \neq \lElm[2]$.
Then, for all valuations $\valFun \in
\ValSet[\DSet](\QPAVSet{\qpElm})$ and variables $\xElm \in
\QPEVSet{\qpElm}$, we have that $\spcmapFun((\sigElm,
\lElm[1]))(\valFun)(\xElm) = ((\sigElm, \lElm[1]), \xElm, \hFun[1])
\neq ((\sigElm, \lElm[2]), \xElm, \allowbreak \hFun[2]) =
\spcmapFun((\sigElm, \lElm[1]))(\valFun)(\xElm)$.
Thus, $\spcmapFun((\sigElm, \lElm[1])) (\valFun)(\xElm) \cmp \bFun
\neq \spcmapFun((\sigElm, \lElm[2]))(\valFun)(\xElm) \cmp \bFun$, for
all $\valFun \in \ValSet[\DSet](\QPAVSet{\qpElm})$.
Hence, $\spcmapFun$ is non-overlapping.
* $\QPEVSet{\SSet'} \neq \emptyset$.
Then, there exist $\sigElm' = (\qpElm', \bFun')$, $\sigElm'' =
(\qpElm'', \bFun'') \in \SSet'$, $\eElm' \in \QPEVSet{\sigElm'}$, and
$\eElm'' \in \QPEVSet{\sigElm''}$ such that $\sigElm' \neq \sigElm''$
or $\bFun'(\eElm') \neq \bFun''(\eElm'')$ and, in both cases,
$(\eElm', \eElm'') \in \BPColSet(\SSet')$.
By contradiction, let $\dFun \in \cap_{((\qpElm, \bFun), \lElm) \in
\PSet'}\set{\valFun \cmp \bFun}{\valFun \in \rng{\spcmapFun(((\qpElm,
\bFun), \lElm))}}$.
Observe that $\dFun(\eElm') = \dFun(\eElm'')$, for all $(\eElm',
\eElm'') \in \BPColSet(\SSet')$.
So, there exist $\valFun' \in \ValSet[\DSet](\QPAVSet{\qpElm'})$ and
$\valFun'' \in \ValSet[\DSet](\QPAVSet{\qpElm''})$ such that
$\valFun'(\bFun'(\eElm)) = \dFun(\eElm)$, for all $\eElm \in
\QPAVSet{\sigElm'}$, and $\valFun''(\bFun''(\eElm)) = \dFun(\eElm)$,
for all $\eElm \in \QPAVSet{\sigElm''}$.
Observe that there are $\lElm', \lElm'' \in \LSet$ such that
$(\sigElm', \lElm'), (\sigElm'', \lElm'') \in \PSet'$.
So, by the hypothesis of the existence of $\dFun$, we have that
$\spcmapFun((\sigElm', \lElm'))(\valFun')(\bFun'(\eElm')) =
\dFun(\eElm') = \dFun(\eElm'') = \spcmapFun((\sigElm'',
\lElm''))(\valFun'')(\bFun''(\eElm''))$.
Now, the following cases arise.
* $\sigElm' \neq \sigElm''$.
By Definition <ref> of big signature dependence, it
holds that $\spcmapFun((\sigElm',
\lElm'))(\valFun')(\bFun'(\eElm')) = ((\sigElm', \lElm'),
\bFun'(\eElm'), \hFun') \neq ((\sigElm'', \lElm''),
\bFun''(\eElm''), \hFun'') \allowbreak = \spcmapFun((\sigElm'',
\lElm'')) (\valFun'')(\bFun''(\eElm''))$, which is a
* $\sigElm' = \sigElm''$.
Then, we have that $\bFun'(\eElm') \neq \bFun''(\eElm'')$.
By Definition <ref>, it holds that
$\spcmapFun((\sigElm', \lElm')) \allowbreak
(\valFun')(\bFun'(\eElm')) = ((\sigElm', \lElm'), \bFun'(\eElm'),
\hFun') \neq ((\sigElm'', \allowbreak \lElm''), \bFun''(\eElm''),
\hFun'') = \spcmapFun((\sigElm'', \lElm''))(\valFun'') \allowbreak
(\bFun''(\eElm''))$, which is a contradiction.
* $\CSet(\SSet') \neq \emptyset$.
Then, there exists $(\vec{\eElm}, \vec{\sigElm}) \in C(\SSet')$.
Let $n \defeq \card{\vec{\sigElm}} - 1$.
Assume, by contradiction, that there exists $\dFun \in \cap_{((\qpElm,
\bFun), \lElm) \in \PSet'}\set{\valFun \cmp \bFun}{\valFun \in
\rng{\spcmapFun(((\qpElm, \bFun), \lElm))}}$.
Observe again that $\dFun(\eElm') = \dFun(\eElm'')$, for all $(\eElm',
\eElm'') \in \BPColSet(\SSet')$.
Now, for all $(\vec{\sigElm})_{i} = (\qpElm[i], \bFun[i]) \in \SSet'$
there exists $\lElm[i] \in \LSet$ such that $((\vec{\sigElm})_{i},
\lElm[i]) \in \PSet'$.
Moreover, let $\valFun[i] \in \ValSet[\DSet](\QPAVSet{\qpElm[i]})$
such that $\valFun[i](\bFun[i](\eElm)) = \dFun(\eElm)$, for all $\eElm
\in \QPAVSet{\sigElm[i]}$.
Then, there exist $n + 1$ functions $\hFun[0], ..., \hFun[n] \in
\{0,1 \}^{C(\PSet)}$ such that, for all $i \in \numcc{0}{n}$, we have
that $\dFun((\vec{\eElm})_{i}) = \spcmapFun(((\vec{\sigElm})_{i},
\lElm[i]))(\valFun[i])(\bFun[i]((\vec{\eElm})_{i})) =
(((\vec{\sigElm})_{i}, \lElm[i]), \bFun[i]((\vec{\eElm})_{i}),
\hFun[i])$.
Observe that, by Item <ref> of
Definition <ref>, for all $i \in \numco{0}{n}$, it holds
that $\hFun[i + 1]((\vec{\eElm}, \vec{\sigElm})) =
\hFun[i]((\vec{\eElm}, \vec{\sigElm}))$ and, in particular,
$\hFun[0]((\vec{\eElm}, \vec{\sigElm})) = \hFun[n]((\vec{\eElm},
\vec{\sigElm}))$.
However, by Item <ref> of
Definition <ref>, it holds that $\hFun[0]((\vec{\eElm},
\vec{\sigElm})) = 1 - \hFun[n]((\vec{\eElm}, \vec{\sigElm}))$.
So, $\hFun[0]((\vec{\eElm}, \vec{\sigElm})) \neq
\hFun[n]((\vec{\eElm}, \vec{\sigElm}))$, which is a contradiction.
Proofs of Section <ref>
In this appendix, we prove Theorem <ref> on the negative
properties for .
Successively, we introduce
the concept of flagged model and flagged formulas.
Finally, we prove Theorem <ref>.
For , it holds that:
* it is not invariant under decision-unwinding;
* it does not have the decision-tree model property.
[Item (1)].
Assume by contradiction that is invariant under
decision-unwinding and consider the two s $\GName[1] \defeq
\CGSTuple{\APSet} {\AgSet} {\AcSet} {\StSet} {\labFun}
{\trnFun[\GName_{1}]} {\sElm[0]}$ and
$\GName[2] \defeq \CGSTuple{\APSet} {\AgSet} {\AcSet} {\StSet}
{\labFun} {\trnFun[\GName_{2}]} {\sElm[0]}$, with $\APSet = \{
\pSym \}$, $\AgSet = \{ \alpha, \beta \}$, $\AcSet = \{ 0, 1 \}$,
$\StSet = \{ \sSym[0], \sSym[1 | '], \sSym[1 | ''], \sSym[2 |
'], \sSym[2 | ''], \sSym[3 | '], \sSym[3 | ''] \}$,
$\labFun(\sElm[2]') = \labFun(\sElm[2]'') = \{\pSym \}$ and
$\labFun(\sElm) = \emptyset$, for all $\sElm \in \StSet \setminus
\{\sElm[2]', \sElm[2]'' \}$, and $\trnFun[\GName_{1}]$ and
$\trnFun[\GName_{2}]$ given as follow.
If by $\aSym \bSym$ we indicate the decision in which agent $\alpha$
takes the action $\aSym$ and agent $\beta$ the action $\bSym$,
then we set $\trnFun[\GName_{1}]$ and $\trnFun[\GName_{2}]$ as follow:
$\trnFun[\GName_{1}](\sElm[0], 0*) = \trnFun[\GName_{2}](\sElm[0],
*0)= \sElm[1]'$,
$\trnFun[\GName_{1}](\sElm[0], 1*)= \trnFun[\GName_{2}](\sElm[0], *1)
= \sElm[1]''$,
$\trnFun[\GName_{1}](\sElm[1]', 0*) = \trnFun[\GName_{2}](\sElm[1]',
0*) = \sElm[2]'$,
$\trnFun[\GName_{1}](\sElm[1], 1*) = \trnFun[\GName_{2}](\sElm[1], 1*)
= \sElm[3]'$,
$\trnFun[\GName_{1}](\sElm[1]'', 0*) = \trnFun[\GName_{2}](\sElm[1]'',
0*)= \sElm[2]''$,
$\trnFun[\GName_{1}](\sElm[1]'', 1*) = \trnFun[\GName_{2}](\sElm[1]'',
1*)= \sElm[3]''$, and
$\trnFun[\GName_{1}](\sElm, **) = \trnFun[\GName_{2}](\sElm, **) =
\sElm$, for all $\sElm \in \{\sElm[2]', \sElm[2]'', \sElm[3]',
\sElm[3]'' \}$.
Observe that $\GName[1DU] = \GName[2DU]$.
Then, it is evident that $\GName[1] \models \varphi$ iff $\GName[1DU]
\models \varphi$ iff $\GName[2DU] \models \varphi$ iff $\GName[2]
\models \varphi$.
In particular, the property does have to hold for the sentence
$\varphi = \EExs{\xSym} \EExs{\ySym[\pSym]} \EExs{\ySym[\neg \pSym]}
((\alpha, \xSym) (\beta, \ySym[\pSym]) \allowbreak (\X \X \pSym))
\wedge ((\alpha, \xSym) (\beta, \ySym[\neg \pSym]) (\X \X \neg
\pSym))$.
It is easy to see that $\GName[1] \not\models \varphi$, while
$\GName[2] \models \varphi$.
Thus, cannot be invariant under decision-unwinding.
Indeed, each strategy $\strFun[\xElm]$ of the agent $\alpha$ in
$\GName[1]$ forces to reach only one state at a time among $\sSym[2 |
']$, $\sSym[2 | '']$, $\sSym[3 | ']$, and $\sSym[3 | '']$.
Formally, for each strategy $\strFun[\xElm] \in \StrSet[ {\GName[1]}
](\sSym[0])$, there is a state $\sElm \in \{ \sSym[2 | '], \sSym[2 |
''], \sSym[3 | '], \sSym[3 | ''] \}$ such that, for all strategies
$\strFun[\yElm] \in \StrSet[ {\GName[1]} ](\sSym[0])$, it holds that
$(\playElm)_{2} = \sElm$, where $\playElm \defeq
\playFun(\emptyfun[\alpha \mapsto \strFun[\xElm]][ {\beta \mapsto
\strFun[\yElm]} ], \sSym[0])$.
Thus, it is impossible to satisfy both the goals $\X \X \pSym$ and $\X
\X \neg \pSym$ with the same strategy of $\alpha$.
On the contrary, since $\sSym[0]$ in $\GName[2]$ is owned by the agent
$\beta$, we may reach both $\sSym[1 | ']$ and $\sSym[1 | '']$ with the
same strategy $\strFun[\xElm]$ of $\alpha$.
Thus, if $\strFun[\xElm](\sSym[0] \cdot \sSym[1 | ']) \neq
\strFun[\xElm](\sSym[0] \cdot \sSym[1 | ''])$, we reach, at the same
time, either the pair of states $\sSym[2 | ']$ and $\sSym[3 | '']$ or
$\sSym[2 | ']$ and $\sSym[3 | ']$.
Formally, there are a strategy $\strFun[\xElm] \in \StrSet[
{\GName[2]} ](\sSym[0])$, with $\strFun[\xElm](\sSym[0] \cdot \sSym[1
| ']) \neq \strFun[\xElm](\sSym[0] \cdot \sSym[1 | ''])$, a pair of
states $(\sElm[\pSym], \sElm[\neg \pSym]) \in \{ (\sSym[2 | '],
\sSym[3 | '']), (\sSym[2 | ''], \sSym[3 | ']) \}$, and two strategies
$\strFun[ {\yElm[\pSym]} ], \strFun[ {\yElm[\neg \pSym]} ] \in
\StrSet[ {\GName[2]} ](\sSym[0])$ such that $(\playElm[\pSym])_{2} =
\sElm[\pSym]$ and $(\playElm[\neg \pSym])_{2} = \sElm[\neg \pSym]$,
where $\playElm[\pSym] \defeq \playFun(\emptyfun[\alpha \mapsto
\strFun[\xElm]][ {\beta \mapsto \strFun[ {\yElm[\pSym]} ]} ],
\sSym[0])$ and $\playElm[\neg \pSym] \defeq \playFun(\emptyfun[\alpha
\mapsto \strFun[\xElm]][ {\beta \mapsto \strFun[ {\yElm[\neg \pSym]}
]} ], \sSym[0])$.
Hence, we can satisfy both the goals $\X \X \pSym$ and $\X \X \neg
\pSym$ with the same strategy of $\alpha$.
[Item (2)].
To prove the statement we have to show that there exists a satisfiable
sentence that does not have a model.
Consider the sentence $\varphi \defeq \varphi_{1} \wedge
\varphi_{2}$, where $\varphi_{1}$ is the negation of the sentence
$\varphi$ used in Item (1) and $\varphi_{2} \defeq \AAll{\xSym}
\AAll{\ySym} (\alpha, \xSym) (\beta, \ySym) \X ((\EExs{\xSym}
\EExs{\ySym} (\alpha, \xSym) (\beta, \ySym) \X \pSym) \wedge
(\EExs{\xSym} \EExs{\ySym} (\alpha, \xSym) (\beta, \ySym) \X \neg
\pSym))$.
Moreover, note that the sentence $\varphi_{2}$ is equivalent to the
formula $\A \X ((\E \X \pSym) \wedge (\E \X \neg \pSym))$.
Then, consider the $\GName \defeq \CGSStruct$
with $\APSet = \{\pSym\}$, $\AgSet = \{\alpha, \beta\}$, $\AcSet =
\{0,1 \}$, $\StSet = \{\sElm[0], \sElm[1], \sElm[2], \sElm[3]\}$,
$\labFun(\sElm[0]) \labFun(\sElm[1]) = \labFun(\sElm[3]) = \emptyset$
and $\labFun(\sElm[2]) = \{ \pSym \}$, and $\trnFun(\sElm[0],
**) = \sElm[1]$, $\trnFun(\sElm[1], 0*) = \sElm[1]$,
$\trnFun(\sElm[1], 1*) = \sElm[3]$, and $\trnFun(\sElm, **) =
\sElm$, for all $\sElm \in \{\sElm[2], \sElm[3]\}$.
It is easy to see that $\GName$ satisfies $\varphi$.
At this point, let $\TName$ be a model of $\varphi_{2}$.
Then, such a tree has necessarily at least two actions and,
consequently, two different successors $\tElm[1], \tElm[2] \in
\DecSet^{*}$ of the root $\epsilon$, where $\tElm[1], \tElm[2] \in
\DecSet$ and $\tElm[1](\alpha) = \tElm[2](\alpha)$.
Moreover, there are two decisions $\decFun[1], \decFun[2] \in \DecSet$
such that $\pSym \in \labFun(\tElm[1] \cdot \decFun[1])$ and $\pSym
\not\in \labFun(\tElm[2] \cdot \decFun[2])$.
Now, let $\strFun[\xSym], \strFun[ {\ySym[\pSym]} ], \strFun[
{\ySym[\neg \pSym]} ] \in \StrSet(\epsilon)$ be three strategies for
which the following holds: $\strFun[\xSym](\epsilon) =
\tElm[1](\alpha)$, $\strFun[ {\ySym[\pSym]} ](\epsilon) =
\tElm[1](\beta)$, $\strFun[ {\ySym[\neg \pSym]} ](\epsilon) =
\tElm[2](\beta)$, $\strFun[\xSym](\tElm[1]) = \decFun[1](\alpha)$,
$\strFun[ {\ySym[\pSym]} ](\tElm[1]) = \decFun[1](\beta)$,
$\strFun[\xSym](\tElm[2]) = \decFun[2](\alpha)$, and $\strFun[
{\ySym[\neg \pSym]} ](\tElm[2]) = \decFun[2](\beta)$.
Then, it is immediate to see that $\TName, \emptyfun[\xSym \mapsto
\strFun[\xSym]][\ySym[\pSym] \mapsto \strFun[ {\ySym[\pSym]}
]][\ySym[\neg \pSym] \mapsto \strFun[ {\ySym[\neg \pSym]} ]], \epsilon
\models ((\alpha, \xSym) (\beta, \ySym[\pSym]) \allowbreak (\X \X
\pSym)) \wedge ((\alpha, \xSym) (\beta, \ySym[\neg \pSym]) (\X \X \neg
\pSym))$.
Thus, we obtain that $\TName \not\models \varphi_{1}$.
Hence, $\varphi$ does not have a model.
Flagged features
A flagged model of a given $\GName$ is obtained adding a
so-called $\sharp$-agent to the set of agents and flagging every state with
two flags.
Intuitively, the $\sharp$-agent takes control of the flag to use in order to
establish which part of a given formula is checked in the .
We start giving first the definition of plan and then the concepts of
flagged model and flagged formulas.
A track (resp., path) plan in a $\GName$ is a finite (resp.,
an infinite) sequence of decisions $\plnElm \in \DecSet^{*}$ (resp.,
$\plnElm \in \DecSet^{\omega}$).
$\TPlnSet \defeq \DecSet^{*}$ (resp., $\PPlnSet \defeq\DecSet^{\omega}$)
denotes the set of all track (resp., path) plans.
Moreover, with each non-trivial track $\trkElm \in \TrkSet$ (resp., path
$\pthElm \in \PthSet$) it is associated the set $\TPlnSet(\trkElm) \defeq
\set{ \plnElm \in \DecSet^{\card{\trkElm} - 1} }{ \forall i \in
\numco{0}{\card{\plnElm}} \,.\, (\trkElm)_{i + 1} = \trnFun((\trkElm)_{i},
(\plnElm)_{i}) } \subseteq \TPlnSet$ (resp., $\PPlnSet(\pthElm) \defeq
\set{ \plnElm \in \DecSet^{\omega} }{ \forall i \in \SetN \,.\,
(\pthElm)_{i + 1} = \trnFun((\pthElm)_{i}, (\plnElm)_{i}) } \subseteq
\PPlnSet$) of track (resp., path) plans that are consistent with
$\trkElm$ (resp., $\pthElm$).
Let $\GName = \CGSStruct$ be a with $\card{\AcSet} \geq 2$.
Let $\sharp \notin \AgSet$ and $\cElm[\sharp] \in \AcSet$.
Then, the flagged is defined as follows:
$\GName[\sharp] = \CGSTuple {\APSet} {\AgSet \cup \{ \sharp \}} {\AcSet}
{\StSet \times \{ 0, 1 \}} {\labFun[\sharp]} {\trnFun[\sharp]}
{(\sElm[0], 0)}$
where $\labFun[\sharp](\sElm, \iota) \defeq \labFun(\sElm)$, for all
$\sElm \in \StSet$ and $\iota \in \{ 0, 1 \}$, and
$\trnFun[\sharp]((\sElm, \iota), \decFun) \defeq (\trnFun(\sElm,
\decFun_{\rst \AgSet}), \iota')$ with $\iota' = 0$ iff $\decFun(\sharp)
= \cElm[\sharp]$.
Since $\GName$ and $\GName[\sharp]$ have a different set of agents, an
agent-closed formula $\varphi$ w.r.t. $\AgSet[\GName]$ is clearly not
agent-closed w.r.t. $\AgSet[\GName_{\sharp}]$.
For this reason, we introduce the concept of flagged formulas, that
represent, in some sense, the agent-closure of formulas.
Let $\varphi \in \OGSL$.
The universal flagged formula of $\varphi$, in symbol
$\varphi_{A\sharp}$, is obtained by replacing every principal subsentence
$\phi \in \psnt{\varphi}$ with the formula $\phi_{A \sharp} \defeq
\AAll{\xElm[\sharp]} (\sharp, \xElm_{\sharp}) \phi$.
The existential flagged formula of $\varphi$, in symbol
$\varphi_{E\sharp}$, is obtained by replacing every principal subsentence
$\phi \in \psnt{\varphi}$ with the formula $\phi_{E \sharp} \defeq
\EExs{\xElm[\sharp]} (\sharp, \xElm_{\sharp}) \phi$.
Substantially, these definitions help us to check satisfiability of
principal subsentences in a separate way.
The special agent $\sharp$ takes control, over the flagged model, of which
branch to walk on the satisfiability of some $\phi \in \psnt{\varphi}$.
Obviously, there is a strict connection between satisfiability of flagged
formulas over $\GName[\sharp]$ and $\varphi$ over $\GName$.
Indeed, the following lemma holds.
Let $\varphi \in \OGSL$ and let $\varphi_{A\sharp}$ and
$\varphi_{E\sharp}$ the flagged formulas.
Moreover, let $\GName$ be a $\CGS$ and $\GName[\sharp]$ his relative
flagged $\CGS$. Then, for all $\sElm \in \StSet$, it holds that:
if $\GName, \emptyset, \sElm \models \varphi$ then $ \GName[\sharp],
\emptyset (s, \iota) \models \varphi_{A \sharp}$, for all $\iota \in
\{0,1\}$;
if, for all $\iota \in \{0, 1\}$ it holds that $\GName[\sharp],
\emptyset, (\sElm, \iota) \models \varphi_{E \sharp}$, then $\GName,
\emptyset, \sElm \models \varphi$.
On the first case, let $\spcFun \in
\SpcSet_{\StrSet_{\GName}}(\qpElm)$, we consider $\spcFun_{A \sharp}
\in \SpcSet_{\StrSet_{\GName[\sharp]}}(\AAll{\xElm[\sharp]} \qpElm)$ such
that if $\xElm \neq \xElm[\sharp]$ then $\spcFun_{A
\sharp}(\asgFun)(\xElm) = \spcFun(\asgFun)(\xElm)$, otherwise
$\spcFun_{A \sharp}(\asgFun)(\xElm) = \asgFun(\xElm[\sharp])$.
On the second case, let $\spcFun_{E \sharp} \in
\SpcSet_{\StrSet_{\GName[\sharp]}}(\EExs{\xElm[\sharp]} \qpElm)$, we
consider $\spcFun \in \SpcSet_{\StrSet_{\GName}}(\qpElm)$ such that
$\spcFun(\asgFun)(\xElm)) = \spcFun_{E \sharp}(\asgFun)(\xElm))$ (note
that $\dom{\spcFun(\asgFun)}$ is strictly included in
$\dom{\spcFun_{E \sharp}(\asgFun)}$).
Now, given a binding $\bpElm$ and its relative function $\bpFun_{\bpElm}$,
consider $\bpElm_{\sharp} \defeq (\sharp, \xElm[\sharp]) \bpElm$ and its
relative function $\bpFun_{\bpElm, \sharp}$.
We show that in both cases considered above there is some useful relation
between $\pthElm_{\bpElm} \defeq \playFun(\spcFun(\asgFun) \circ
\bpFun_{\bpElm}, \sElm)$ and $\pthElm_{\bpElm,\sharp} \defeq
\playFun(\spcFun_{\sharp}(\asgFun) \circ \bpFun_{\bpElm,\sharp}, (\sElm,
\iota))$.
Indeed, let $\plnElm_{\bpElm}$ the plan such that, for all $i \in \SetN$,
we have that $(\pthElm_{\bpElm})_{i+1} = \trnFun((\pthElm_{\bpElm})_{i},
(\plnElm_{\bpElm})_{i})$ and let $\plnElm_{\bpElm,\sharp}$ the plan such
that, for all $i \in \SetN$, we have that $(\pthElm_{\bpElm,\sharp})_{i+1}
= \trnFun((\pthElm_{\bpElm,\sharp})_{i},(\plnElm_{\bpElm, \sharp})_{i})$.
By the definition of play, for each $i \in \SetN$ and $\aElm \in \AgSet$,
we have that $(\plnElm_{\bpElm})_{i}(\aElm) = (\spcFun(\asgFun) \circ
\bpFun_{\bpElm})(\aElm)((\pthElm_{\bpElm})_{i})$ and $(\plnElm_{\bpElm,
\sharp})_{i}(\aElm) = (\spcFun_{\sharp}(\asgFun) \circ \bpFun_{\bpElm,
\sharp})(\aElm)((\pthElm_{\bpElm, \sharp})_{i})$.
Clearly, for all $i \in \SetN$, we have that $(\plnElm_{\bpElm})_{i} =
((\plnElm_{\bpElm,\sharp})_{i})_{\rst \AgSet}$.
Due to these facts, we can prove by induction that for each $i \in \SetN$
there exists $\iota \in \{0, 1\}$ such that $(\pthElm_{\bpElm,
\sharp})_{i} = ((\pthElm_{\bpElm})_{i}, \iota)$.
The base case is trivial and we omit it here.
As inductive case, suppose that $(\pthElm_{\bpElm, \sharp})_{i} =
((\pthElm_{\bpElm})_{i}, \iota)$, for some $i$.
Then, by definition we have that $(\pthElm_{\bpElm, \sharp})_{i+1} =
\trnFun[\sharp]((\pthElm_{\bpElm,
\sharp})_{i},(\plnElm_{\bpElm,\sharp})_{i})$.
Moreover, by definition of $\trnFun[\sharp]$, we have that
$(\pthElm_{\bpElm, \sharp})_{i+1} =
(\trnFun((\pthElm_{\bpElm})_{i},((\plnElm_{\bpElm, \sharp})_{i})_{\rst
\AgSet}),\iota')$, for some $\iota' \in \{0,1 \}$.
Since $(\plnElm_{\bpElm})_{i} = ((\plnElm_{\bpElm, \sharp})_{i})_{\rst
\AgSet}$, we have that $(\pthElm_{\bpElm, \sharp})_{i+1} =
(\trnFun((\pthElm_{\bpElm})_{i}, (\plnElm_{\bpElm})_{i}), \iota') =
((\pthElm_{\bpElm})_{i+1}, \iota')$, which is the assert.
It follows, by definition of $\labFun[\sharp]$, that
$\labFun((\pthElm_{\bpElm})_{i}) =
\labFun[\sharp]((\pthElm_{\bpElm,\sharp})_{i})$, for each $i \in \SetN$.
So, every sentence satisfied on $\pthElm_{\bpElm}$ is satisfied also on
Now we proceed to prove Items <ref>
and <ref>, separately.
Item <ref>.
First, consider the case that $\phi$ is of the form
$\qpElm \psi$, where $\qpElm$ is a quantification prefix and $\psi$ is
a boolean composition of goals.
Since $\GName,\emptyset, \sElm \models \phi$, there exists $\spcFun
\in \SpcSet_{\StrSet_{\GName}}(\qpElm)$ such that we have
$\GName,\spcFun(\asgFun),\sElm \models \psi$, for all assignment
$\asgFun \in \AsgSet_{\GName}(\sElm)$.
Now, consider $\phi_{A, \sharp} \defeq \AAll{\xElm[\sharp]}(\sharp,
\xElm[\sharp]) \phi$, which is equivalent to $\AAll{\xElm[\sharp]}
\qpElm(\sharp, \xElm[\sharp]) \psi$.
Then, consider $\spcFun_{A \sharp} \in
\SpcSet_{\StrSet_{\GName[\sharp]}}(\AAll{\xElm[\sharp]} \qpElm)$
such that $\spcFun_{A \sharp}(\asgFun)(\xElm) =
\spcFun(\asgFun)(\xElm)$, if $\xElm \neq \xElm[\sharp]$, and
$\spcFun_{\sharp}(\asgFun)(\xElm) = \asgFun(\xElm[\sharp])$,
Clearly, $\spcFun_{A \sharp}$ is build starting from $\spcFun$ as
described above.
Then, from the fact that $\GName, \emptyset, \sElm \models \phi$,
it follows that $\GName[\sharp], \emptyset, (\sElm, \iota) \models
\phi_{A \sharp}$.
Now, if we have a formula $\varphi$ embedding some proper principal
subsentence, then by the induction hypothesis every $\phi \in
\psnt{\varphi}$ is satisfied by $\GName$ if and only if $\phi_{A,
\sharp}$ is satisfied by $\GName[\sharp]$.
By working on the structure of the formula it follows that
the result holds for $\varphi$ and $\varphi_{A \sharp}$ too, so the
proof for this Item is done.
Item <ref>.
First, consider the case of $\phi$ is of the form $\qpElm \psi$,
where $\qpElm$ is a quantification prefix and $\psi$ is a
boolean composition of goals.
Let $\GName[\sharp], \emptyset,(\sElm, \iota) \models \phi_{E,
\sharp}$.
Note that $\phi_{E, \sharp} \defeq \EExs{\xElm[\sharp]}(\sharp,
\xElm[\sharp]) \qpElm \psi$ is equivalent to $\EExs{\xElm[\sharp]}
\qpElm(\sharp, \xElm[\sharp]) \psi$, so there exists
$\spcFun_{E \sharp} \in
\SpcSet_{\StrSet_{\GName[\sharp]}}(\EExs{\xElm[\sharp]}\qpElm)$ such
that, for all assignment $\asgFun \in
\AsgSet_{\GName[\sharp]}(\EExs{\xElm[\sharp]}\qpElm)$, we have that
$\GName[\sharp], \spcFun_{E \sharp}(\asgFun), (\sElm, \iota)
\models(\sharp, \xElm[\sharp]) \psi$.
Then, consider $\spcFun \in \SpcSet_{\StrSet_{\GName}}$ given by
$\spcFun(\asgFun)(\xElm)) = \spcFun_{\sharp}(\asgFun)(\xElm))$.
Clearly, $\spcFun$ is build starting from $\spcFun_{E \sharp}$ as
described above.
Then, from $\GName[\sharp], \spcFun_{E \sharp}(\asgFun), (\sElm,
\iota) \models (\sharp, \xElm[\sharp]) \psi$ it follows that $\GName,
\emptyset, \sElm \models \phi$.
Now, if we have a formula $\varphi$ embedding some proper principal
subsentence, then by the induction hypothesis every $\phi \in
\psnt{\varphi}$ is satisfied by $\GName$ if and only if $\phi_{A,
\sharp}$ is satisfied by $\GName[\sharp]$.
By working on the structure of the formula it follows that
the result holds for $\varphi$ and $\varphi_{E \sharp}$ too, so the
proof for this Item is done.
Proof of Theorem <ref>
From now on, by using Item <ref> of
Theorem <ref>, we can assume to work exclusively on
Let $\SSet_{\phi} \defeq \set{\sElm \in \StSet_{\TName}}{\TName, \emptyset,
\sElm \models \phi}$ and $\TSet[\phi] \defeq \SSet[\phi] \times \{0, 1\}$.
By Item <ref> of Lemma <ref>, we have that
$\TName_{\sharp},\emptyset, \tElm \models \phi_{A \sharp}$, for all $\tElm
\in \TSet[\phi]$.
Moreover, for all $\tElm \in \TSet[\phi]$, consider a strategy
$\strFun[\sharp]^{\tElm} \in \StrSet[\TName_{\sharp}](\tElm)$ given by
$\strFun[\sharp]^{\tElm}(\rho) = \cElm[\sharp]$ iff $\rho = \tElm$.
Moreover, for all $\phi \in \psnt{\varphi}$, consider the
function $\AFun_{\phi}: \TrkSet_{\TName_{\sharp}}(\varepsilon) \rightarrow
2^{(\StSet_{\TName_{\sharp}} \times \TrkSet_{\TName_{\sharp}})}$ given by
$\AFun_{\phi}(\rho) \defeq \set{(\rho_{i}, \rho')}{i \in
\numco{0}{\card{\rho}} \land \rho'\in
\TrkSet_{\TName_{\sharp}}(\emptyset[\sharp \rightarrow
\strFun^{\rho_{i}}_{\sharp}], \rho_{i}) \land \lst{\rho}=\lst{\rho'}}$.
Note that $(\lst{\rho}, \lst{\rho}) \in \AFun_{\phi}(\rho)$.
Indeed: (i) $\lst{\rho} = \rho_{\card{\rho}}$; (ii) $\lst{\rho)}
\in \TrkSet_{\TName_{\sharp}}(\emptyset[\sharp \rightarrow
\strFun^{\lst{\rho}}_{\sharp}], \lst{\rho})$; and (iii) $\lst{\rho}
= \lst{\lst{\rho}}$.
Observe that if $(\rho_{i}, \rho') \in \AFun_{\phi}(\rho)$ then $\rho' =
\rho_{\geq i}$.
Hence, except for $(\lst{\rho},\lst{\rho})$, there exists at most one pair
in $\AFun_{\phi}(\rho)$.
Indeed, by contradiction let $(\rho_{i},\rho_{\geq i})$ and $(\rho_{j},
\rho_{\geq j})$ both in $\AFun_{\phi}(\rho)$ with $i \lneqq j$ and $j \neq
\card{\rho}$.
Then, by the definition of compatible tracks
$\TrkSet_{\TName_{\sharp}}(\emptyset[\sharp \rightarrow
\strFun^{\rho}_{\sharp}], \rho_{i})$, there exists a plan $\plnElm \in
\PlnSet(\rho_{\geq i})$ such that for all $h \in \numco{0} {\card{\rho}-i}$
we have $\plnElm_{h}(\sharp)=\strFun^{\rho}_{\sharp}((\rho_{\geq i})_{\leq
Then, by the definition of $\strFun^{\rho}_{\sharp}$, $\plnElm_{h}(\sharp)
\neq \cElm_{\sharp}$.
So, by the definition of plan and $\trnFun_{\sharp}$, we have
that $\rho_{j+1} = (\sElm, 1)$.
On the other hand, since $(\rho_j, \rho_{\geq j}) \in \AFun_{\phi}(\rho)$,
then there exists a plan $\plnElm' \in \PlnSet(\rho_{\geq j})$ such that
$(\plnElm')_{0}(\sharp) = \strFun^{\rho_{\geq j}}_{\sharp}(\rho_{\geq j}) =
\cElm_{\sharp}$.
Which implies, by the definition of plan and $\trnFun_{\sharp}$, we have
that $\rho_{j+1} = (\sElm',0)$, which is in contradiction with the fact that
the second coordinate of $\rho_{j+1}$ is 1, as shown above.
This reasoning allows us to build the functions $head_{\phi}$ and
$body_{\phi}$ for the disjoint satisfiability of $\phi$ over
$\TName[\sharp]$ on the set $\TSet[\phi]$.
Indeed, the unique element $(\trkElm[i], \trkElm') \in
\AFun_{\phi}(\trkElm) \setminus \{ (\lst{\trkElm}, \lst{\trkElm}) \}$
can be used to define opportunely the elementary dependence map used for
such disjoint satisfiability.
Let $\varphi$ be an satisfiable sentence and $\PSet \defeq \set{
((\qpElm, \bpElm), (\psi, i)) \in \LSigSet(\AgSet, \SL \times \{ 0, 1 \})
}{ \qpElm \bpElm \psi \in \psnt{\varphi} \land i \in \{ 0, 1 \} }$ the set
of all labeled signatures on $\AgSet$ w.r.t. $\SL \times \{ 0, 1 \}$ for
Then, there exists a $b$-bounded $\TName$, with $b = \card{\PSet}
\cdot \card{\QPVSet(\PSet)} \cdot 2^{\card{\CSet(\PSet)}}$, such that
$\TName \models \varphi$.
Moreover, for all $\phi \in \psnt{\varphi}$, it holds that $\TName$
satisfies $\phi$ disjointly over the set $\set{ \sElm \in \StSet }{
\TName, \emptyset, \sElm \models \phi }$.
Since $\varphi$ is satisfiable, then, by
Item <ref> of
Theorem <ref>, we have that there exists a $\TName$, such that $\TName \models \varphi$.
We now prove that there exists a bounded $\TName' \defeq
\CGSTuple{\APSet}{\AgSet}{\AcSet[
\TName']}{\StSet[\TName']]}{\labFun[\TName']}{ \trnFun [ \TName
']}{\epsilon}$ with $\AcSet[\TName'] \defeq \numco{0}{n}$ and $n =
\card{\PSet} \cdot \card{\QPVSet(\PSet)} \cdot 2^{\card{\CSet(\PSet)}}$.
Since $\TName'$ is a , we have to define only the labeling function
To do this, we need two auxiliary functions $\hFun: \StSet[\TName]
\times \DecSet[\TName'] \to \DecSet[\TName]$ and $\gFun: \StSet[\TName']
\to \StSet[\TName]$ that lift correctly the labeling function
$\labFun[\TName]$ to $\labFun[\TName']$.
Function $\gFun$ is defined recursively as follows: (i)
$\gFun(\epsilon) \defeq \epsilon$, (ii) $\gFun(\tElm' \cdot
\decFun') \defeq \gFun(\tElm') \cdot \hFun(\gFun(\tElm'), \decFun')$.
Then, for all $\tElm' \in \StSet[\TName']$, we define
$\labFun[\TName'](\tElm') \defeq \labFun[\TName](\gFun(\tElm'))$.
It remains to define the function $\hFun$.
By Item <ref> of Lemma <ref>, we have
that $\TName[\sharp] \models \varphi_{A \sharp}$ and consequently that
$\TName[\sharp] \models \varphi_{E \sharp}$.
Moreover, applying the reasoning explained above, $\TName[\sharp]$
satisfies disjointly $\phi$ over $\SSet[\phi]$, for all $\phi \in
\psnt{\varphi}$.
Then, for all $\phi \in \psnt{\varphi}$, we have that there exist a
function $head_{\phi}: \SSet[\phi] \to \SpcSet[\AcSet_{\TName}](\qpElm)$
and a function $body_{\phi}: \TrkSet[\TName](\epsilon) \to
\SpcSet[\AcSet_{\TName}](\qpElm)$ that allow $\TName$ to satisfy $\phi$ in
a disjoint way over $\SSet[\phi]$.
Now, by Theorem <ref>, there exists a signature dependence
$\spcmapFun \in \LSpcMapSet[\AcSet_{\TName'}](\PSet)$ such that, for all
$\PSet' \subseteq \PSet$, we have that $\spcmapFun_{\rst \PSet'} \in
\LSpcMapSet[\AcSet_{\TName'}](\PSet)$ is non-overlapping, if $\PSet$ is
Moreover, by Corollary <ref>, for all $\PSet' \subseteq
\PSet$, we have that $\spcmapFun_{\rst \PSet'} \in
\LSpcMapSet[\AcSet_{\TName'}](\PSet)$ is overlapping, if $\PSet$ is
At this point, consider the function $\DFun: \DecSet[\TName'] \to
2^{\PSet}$ that, for all $\decFun' \in \DecSet[\TName']$, is given by
$\DFun(\decFun') \defeq \set{((\qpElm, \bpElm), (\psi, i)) = \sigElm \in
\PSet}{\exists \eElm' \in \AcSet[\TName']^{\QPAVSet(\qpElm)}. \decFun' =
\spcmapFun(\sigElm)(\eElm') \cmp \bpFun}$.
Note that, for all $\decFun' \in \DecSet[\TName']$, we have that
$\DFun(\decFun) \subseteq \PSet$ is overlapping.
Now, consider the functions $\WFun: \StSet[\TName_{\sharp}] \to
\LSpcMapSet[\AcSet_{\TName}](\PSet)$ such that, for all $\tElm \in
\StSet[\TName_{\sharp}]$ and $\sigElm = ((\qpElm, \bpElm), (\psi, i)) \in
\PSet$, is such that
$\WFun(\tElm)(\sigElm) = \left\{
\begin{array}{ll}
head_{\phi}(\tElm) &, \tElm \in
\TSet_{\phi} \\
body_{\phi}(\trkElm') &,
\mbox{otherwise}
\end{array}
\right.
where $\phi = \qpElm \bpElm \psi$ and $\trkElm' \in
\TrkSet[\TName_{\sharp}](\epsilon)$ is the unique track such that
$\lst{\trkElm'} = \tElm$.
Moreover, consider the function $\TFun: \StSet[\TName_{\sharp}] \times
\DecSet[\TName] \to 2^{\PSet}$ such that, for all $\tElm \in
\StSet[\TName_{\sharp}]$ and $\decFun \in \DecSet[\TName]$, it is given by
$\TFun(\tElm, \decFun) \defeq \set{\sigElm = ((\qpElm, \bpElm), (\psi, i))
\in \PSet}{\exists \eElm \in \AcSet[\TName]^{\QPAVSet{\qpElm}}. \decFun =
\WFun(\tElm)(\eElm) \cmp \bpFun}$.
It is easy to see that, for all $\decFun' \in \DecSet[\TName']$ and
$\tElm \in \StSet[\TName_{\sharp}]$, there exists $\decFun \in
\DecSet[\TName]$ such that $\DFun(\decFun') \subseteq \TFun(\tElm,
\decFun)$.
By Corollary <ref>, for all $\tElm \in
\StSet[\TName_{\sharp}]$, we have that $\WFun(\tElm)_{\rst
\DFun(\decFun')}$ is overlapping.
So, by Definition <ref>, for all $\tElm \in
\StSet[\TName_{\sharp}]$ and $\decFun' \in \DecSet[\TName']$, there
exists $\decFun \in \AcSet[\TName_{\sharp}]^{\AgSet}$ such that $\decFun
\in \cap_{\sigElm = (\qpElm, \bpFun), (\psi, i) \in
\DFun(\decFun')}\set{\zFun \circ \bFun}{\zFun \in
\rng{\WFun(\tElm)(\sigElm)}}$, which implies $\TFun(\tElm, \decFun)
\supseteq \DFun(\decFun')$.
Finally, by applying the previous reasoning we obtain the function
$\hFun$ such that, for all $(\tElm, \decFun') \in \StSet[\TName]
\times \DecSet[\TName']$, it associates a decision $\hFun(\tElm, \decFun')
\defeq \decFun \in \DecSet[\TName]$.
The proof that $\TName' \models \varphi$ proceeds naturally by
induction and it is omitted here.
Proofs of Section <ref>
In this appendix, we give the proofs of Lemmas <ref>
and <ref> of goal and sentence automaton and
Theorems <ref> and <ref> of automaton and
Alternating tree automata
Nondeterministic tree automata are a generalization to infinite trees
of the classical nondeterministic word automata on infinite words.
Alternating tree automata are a further generalization of
nondeterministic tree automata <cit.>.
Intuitively, on visiting a node of the input tree, while the latter sends
exactly one copy of itself to each of the successors of the node, the former
can send several own copies to the same successor.
Here we use, in particular, alternating parity tree automata, which
are alternating tree automata along with a parity acceptance
condition (see <cit.>, for a survey).
We now give the formal definition of alternating tree automata.
An alternating tree automaton (, for short) is a tuple
$\AName \defeq \ATAStruct$, where $\LabSet$, $\DirSet$, and $\QSet$ are,
respectively, non-empty finite sets of input symbols,
directions, and states, $\qElm[0] \in \QSet$ is an
initial state, $\aleph$ is an acceptance condition to be
defined later, and $\delta : \QSet \times \LabSet \to \PBoolSet(\DirSet
\times \QSet)$ is an alternating transition function that maps each
pair of states and input symbols to a positive Boolean combination on the
set of propositions of the form $(\dElm, \qElm) \in \DirSet \times \QSet$,
a.k.a. moves.
On one side, a nondeterministic tree automaton (, for
short) is a special case of in which each conjunction in the
transition function $\delta$ has exactly one move $(\dElm, \qElm)$
associated with each direction $\dElm$.
This means that, for all states $\qElm \in \QSet$ and symbols $\sigma \in
\LabSet$, we have that $\atFun(\qElm, \sigma)$ is equivalent to a Boolean
formula of the form $\bigvee_{i} \bigwedge_{\dElm \in \DirSet} (\dElm,
\qElm[i, \dElm])$.
On the other side, a universal tree automaton (, for
short) is a special case of in which all the Boolean combinations that
appear in $\delta$ are conjunctions of moves.
Thus, we have that $\atFun(\qElm, \sigma) = \bigwedge_{i} (\dElm[i],
\qElm[i])$, for all states $\qElm \in \QSet$ and symbols $\sigma \in
\LabSet$.
The semantics of the s is given through the following concept of run.
A run of an $\AName = \ATAStruct$ on a $\LabSet$-labeled
$\DirSet$-tree $\TName = \LTStruct$ is a $(\DirSet \times \QSet)$-tree
$\RSet$ such that, for all nodes $\xElm \in \RSet$, where $\xElm =
\prod_{i = 1}^{n} (\dElm[i], \qElm[i])$ and $\yElm \defeq \prod_{i =
1}^{n} \dElm[i]$ with $n \in \numco{0}{\omega}$, it holds that (i)
$\yElm \in \TSet$ and (ii), there is a set of moves $\SSet
\subseteq \DirSet \times \QSet$ with $\SSet \models \delta(\qElm[n],
\vFun(\yElm))$ such that $\xElm \cdot (\dElm, \qElm) \in \RSet$, for all
$(\dElm, \qElm) \in \SSet$.
In the following, we consider s along with the parity acceptance
condition (, for short) $\aleph \defeq (\FSet_{1}, \ldots,
\FSet_{k}) \in (\pow{\QSet})^{+}$ with $\FSet_{1} \subseteq \ldots \subseteq
\FSet_{k} = \QSet$ (see <cit.>, for more).
The number $k$ of sets in the tuple $\aleph$ is called the index of
the automaton.
We also consider s with the co-Büchi acceptance condition
(, for short) that is the special parity condition with index
Let $\RSet$ be a run of an $\AName$ on a tree $\TName$ and $\wElm$ one
of its branches.
Then, by $\infFun(\wElm) \defeq \set{ \qElm \in \QSet }{ \card{\set{ i \in
\SetN }{ \exists \dElm \in \DirSet . (w)_{i} = (\dElm, \qElm) }} = \omega }$
we denote the set of states that occur infinitely often as the second
component of the letters along the branch $w$.
Moreover, we say that $w$ satisfies the parity acceptance condition $\aleph
= (\FSet_{1}, \ldots, \FSet_{k})$ if the least index $i \in \numcc{1}{k}$
for which $\infFun(w) \cap \FSet_{i} \neq \emptyset$ is even.
Finally, we can define the concept of language accepted by an .
An $\AName = \ATAStruct$ accepts a $\LabSet$-labeled
$\DirSet$-tree $\TName$ iff is there exists a run $\RSet$ of $\AName$ on
$\TName$ such that all its infinite branches satisfy the acceptance
condition $\aleph$.
By $\LangSet(\AName)$ we denote the language accepted by the $\AName$,
i.e., the set of trees $\TName$ accepted by $\AName$.
Moreover, $\AName$ is said to be empty if $\LangSet(\AName) =
\emptyset$.
The emptiness problem for $\AName$ is to decide whether
$\LangSet(\AName) = \emptyset$.
Proofs of theorems
We are finally able to show the proofs of the above mentioned results.
Let $\bpElm \psi$ an goal without principal subsentences and
$\AcSet$ a finite set of actions.
Then, there exists an $\UName[\bpElm \psi | ^{\AcSet}] \defeq
\TATuple {\ValSet[\AcSet](\free{\bpElm \psi}) \times \pow{\APSet}}
{\DecSet} {\QSet[\bpElm \psi]} {\atFun[\bpElm \psi]} {\qElm[0\bpElm\psi]}
{\aleph_{\bpElm \psi}}$ such that, for all s $\TName$ with
$\AcSet[\TName] = \AcSet$, states $\tElm \in \StSet[\TName]$, and
assignments $\asgFun \in \AsgSet[\TName](\free{\bpElm \psi}, \tElm)$, it
holds that $\TName, \asgFun, \tElm \models \bpElm \psi$ iff $\TName' \in
\LangSet(\UName[\bpElm \psi | ^{\AcSet}])$, where $\TName'$ is the
assignment-labeling encoding for $\asgFun$ on $\TName$.
A first step in the construction of the $\UName[\bpElm \psi |
^{\AcSet}]$, is to consider the $\UName[\psi] \defeq \WATuple
{\pow{\APSet}} {\QSet[\psi]} {\atFun[\psi]} {\QSet[0\psi]}
{\aleph_{\psi}}$ obtained by dualizing the resulting from the
application of the classic Vardi-Wolper construction to the formula
$\neg \psi$ <cit.>.
Observe that $\LangSet(\UName[\psi]) = \LangSet(\psi)$, i.e., this
automaton recognizes all infinite words on the alphabet $\pow{\APSet}$
that satisfy the formula $\psi$.
Then, define the components of $\UName[\bpElm \psi | ^{\AcSet}] \defeq
\TATuple {\ValSet[\AcSet](\free{\bpElm \psi}) \times \pow{\APSet}}
{\DecSet} {\QSet[\bpElm \psi]} {\atFun[\bpElm \psi]} {\qElm[0\bpElm\psi]}
{\aleph_{\bpElm \psi}}$, as follows:
* $\QSet[\bpElm \psi] \defeq \{ \qElm[0\bpElm\psi] \} \cup \QSet[\psi]$,
with $\qElm[0\bpElm\psi] \not\in \QSet[\psi]$;
* $\atFun[\bpElm \psi](\qElm[0\bpElm\psi], (\valFun, \sigma)) \defeq
\bigwedge_{\qElm \in \QSet[0\psi]} \atFun[\bpElm \psi](\qElm,
(\valFun, \sigma))$, for all $(\valFun, \sigma) \in
\ValSet[\AcSet](\free{\bpElm \psi}) \times \pow{\APSet}$;
* $\atFun[\bpElm \psi](\qElm, (\valFun, \sigma)) \!\defeq\!
\bigwedge_{\qElm' \!\in\! \atFun[\psi](\qElm, \sigma)}
(\valFun \cmp \bndFun[\bpElm], \qElm')$, for all $\qElm \!\in\!
\QSet[\psi]$ and $(\valFun, \sigma) \in \ValSet[\AcSet](\free{\bpElm
\psi}) \times \pow{\APSet}$;
* $\aleph_{\bpElm \psi} \defeq \aleph_{\psi}$.
Intuitively, the $\UName[\bpElm \psi | ^{\AcSet}]$ simply runs the
$\UName[\psi]$ on the branch of the encoding individuated by the
assignment in input.
Thus, it is easy to see that, for all states $\tElm \in \StSet[\TName]$
and assignments $\asgFun \in \AsgSet[\TName](\free{\bpElm \psi}, \tElm)$,
it holds that $\TName, \asgFun, \tElm \models \bpElm \psi$ iff $\TName'
\in \LangSet(\UName[\bpElm \psi | ^{\AcSet}])$, where $\TName'$ is the
assignment-labeling encoding for $\asgFun$ on $\TName$.
Let $\qpElm \bpElm \psi$ be an principal sentence without principal
subsentences and $\AcSet$ a finite set of actions.
Then, there exists an $\UName[\qpElm \bpElm \psi | ^{\AcSet}]
\defeq \TATuple {\SpcSet[\AcSet](\qpElm) \times \pow{\APSet}} {\DecSet}
{\QSet[\qpElm \bpElm \psi]} {\atFun[\qpElm \bpElm \psi]}
{\qElm[0\qpElm\bpElm\psi]} {\aleph_{\qpElm \bpElm \psi}}$ such that, for
all s $\TName$ with $\AcSet[\TName] = \AcSet$, states $\tElm \in
\StSet[\TName]$, and elementary dependence maps over strategies
$\spcFun \in \ESpcSet[ {\StrSet[\TName](\tElm)} ](\qpElm)$, it holds that
$\TName, \spcFun(\asgFun), \tElm \emodels \bpElm \psi$, for all $\asgFun
\in \AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$, iff $\TName' \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\AcSet}])$, where $\TName'$ is
the elementary dependence-labeling encoding for $\spcFun$ on $\TName$.
By Lemma <ref> of goal automaton, there is an
$\UName[\bpElm \psi | ^{\AcSet}] \defeq
\TATuple {\ValSet[\AcSet](\free{\bpElm \psi}) \times \pow{\APSet}}
{\DecSet} {\QSet[\bpElm \psi]} {\atFun[\bpElm \psi]} {\qElm[0\bpElm\psi]}
{\aleph_{\bpElm \psi}}$ such that, for all s $\TName$ with
$\AcSet[\TName] = \AcSet$, states $\tElm \in \StSet[\TName]$, and
assignments $\asgFun \in \AsgSet[\TName](\free{\bpElm \psi}, \tElm)$, it
holds that $\TName, \asgFun, \tElm \models \bpElm \psi$ iff $\TName' \in
\LangSet(\UName[\bpElm \psi | ^{\AcSet}])$, where $\TName'$ is the
assignment-labeling encoding for $\asgFun$ on $\TName$.
Now, transform $\UName[\bpElm \psi | ^{\AcSet}]$ into the new $\UName[\qpElm \bpElm \psi | ^{\AcSet}] \defeq \TATuple
{\SpcSet[\AcSet](\qpElm) \times \pow{\APSet}} {\DecSet} {\QSet[\qpElm
\bpElm \psi]} {\atFun[\qpElm \bpElm \psi]} {\qElm[0\qpElm\bpElm\psi]}
{\aleph_{\qpElm \bpElm \psi}}$, with $\QSet[\qpElm \bpElm \psi] \defeq
\QSet[\bpElm \psi]$, $\qElm[0\qpElm\bpElm\psi] \defeq \qElm[0\bpElm\psi]$,
and $\aleph_{\qpElm \bpElm \psi} \defeq \aleph_{\bpElm \psi}$, which is
used to handle the quantification prefix $\qpElm$ atomically, where the
transition function is defined as follows: $\atFun[\qpElm \bpElm
\psi](\qElm, (\spcFun, \sigma)) \defeq \bigwedge_{\valFun \in
\ValSet[\AcSet](\QPAVSet{\qpElm})} \atFun[\bpElm \psi](\qElm,
(\spcFun(\valFun), \sigma))$, for all $\qElm \in \QSet[\qpElm \bpElm
\psi]$ and $(\spcFun, \sigma) \in \SpcSet[\AcSet](\qpElm) \times
\pow{\APSet}$.
Intuitively, $\UName[\qpElm \bpElm \psi | ^{\AcSet}]$ reads an action
dependence map $\spcFun$ on each node of the input tree $\TName'$ labeled
with a set of atomic propositions $\sigma$ and simulates the execution of
the transition function $\atFun[\bpElm \psi](\qElm, (\valFun, \sigma))$ of
$\UName[\bpElm \psi | ^{\AcSet}]$, for each possible valuation $\valFun =
\spcFun(\valFun')$ on $\free{\bpElm \psi}$ obtained from $\spcFun$ by a
universal valuation $\valFun' \in \ValSet[\AcSet](\QPAVSet{\qpElm})$.
It is worth observing that we cannot move the component set
$\SpcSet[\AcSet](\qpElm)$ from the input alphabet to the states of
$\UName[\qpElm \bpElm \psi | ^{\AcSet}]$ by making a related guessing of
the dependence map $\spcFun$ in the transition function, since we have to
ensure that all states in a given node of the tree $\TName'$, i.e., in
each track of the original model $\TName$,
Finally, it remains to prove that, for all states $\tElm \in
\StSet[\TName]$ and elementary dependence maps over strategies $\spcFun \in
\ESpcSet[ {\StrSet[\TName](\tElm)} ](\qpElm)$, it holds that $\TName,
\spcFun(\asgFun), \tElm \emodels \bpElm \psi$, for all $\asgFun \in
\AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$, iff $\TName' \in
\LangSet(\UName[\qpElm \bpElm \psi | ^{\AcSet}])$, where $\TName'$ is the
elementary dependence-labeling encoding for $\spcFun$ on $\TName$.
[Only if].
Suppose that $\TName, \spcFun(\asgFun), \tElm \emodels \bpElm \psi$, for
all $\asgFun \in \AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$.
Since $\psi$ does not contain principal subsentences, we have that
$\TName, \spcFun(\asgFun), \tElm \models \bpElm \psi$.
So, due to the property of $\UName[\bpElm \psi | ^{\AcSet}]$,
it follows that there exists an assignment-labeling encoding
$\TName[\asgFun | '] \in \LangSet(\UName[\bpElm \psi | ^{\AcSet}])$, which
implies the existence of a $(\DecSet \times \QSet[\bpElm \psi])$-tree
$\RSet[\asgFun]$ that is an accepting run for $\UName[\bpElm \psi |
^{\AcSet}]$ on $\TName[\asgFun | ']$.
At this point, let $\RSet \defeq \bigcup_{\asgFun \in
\AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)} \RSet[\asgFun]$ be the union of
all runs.
Then, due to the particular definition of the transition function of
$\UName[\qpElm \bpElm \psi | ^{\AcSet}]$, it is not hard to see that
$\RSet$ is an accepting run for $\UName[\qpElm \bpElm \psi | ^{\AcSet}]$
on $\TName'$.
Hence, $\TName' \in \LangSet(\UName[\qpElm \bpElm \psi | ^{\AcSet}])$.
Suppose that $\TName' \in \LangSet(\UName[\qpElm \bpElm
\psi | ^{\AcSet}])$.
Then, there exists a $(\DecSet \times \QSet[\qpElm \bpElm \psi])$-tree
$\RSet$ that is an accepting run for $\UName[\qpElm \bpElm \psi |
^{\AcSet}]$ on $\TName'$.
Now, for each $\asgFun \in \AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$, let
$\RSet[\asgFun]$ be the run for $\UName[\bpElm \psi | ^{\AcSet}]$ on the
assignment-state encoding $\TName[\asgFun | ']$ for $\spcFun(\asgFun)$ on
Due to the particular definition of the transition function of
$\UName[\qpElm \bpElm \psi | ^{\AcSet}]$, it is not hard to see that
$\RSet[\asgFun] \subseteq \RSet$.
Thus, since $\RSet$ is accepting, we have that $\RSet[\asgFun]$ is
accepting as well.
So, $\TName[\asgFun | '] \in \LangSet(\UName[\bpElm \psi | ^{\AcSet}])$.
At this point, due to the property of $\UName[\bpElm \psi | ^{\AcSet}]$,
it follows that $\TName, \spcFun(\asgFun), \tElm \models \bpElm \psi$.
Since $\psi$ does not contain principal subsentences, we have that
$\TName, \spcFun(\asgFun), \tElm \!\emodels\! \bpElm \psi$, for all
$\asgFun \!\in\! \AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$.
Let $\varphi$ be an sentence.
Then, there exists an $\UName[\varphi]$ such that $\varphi$ is
satisfiable iff $\LangSet(\UName[\varphi]) \neq \emptyset$.
By Theorem <ref> of bounded tree-model
property, if an sentence $\varphi$ is satisfiable, it is
satisfiable in a disjoint way on a $b$-bounded with $b \defeq
\card{\PSet} \cdot \card{\QPVSet(\PSet)} \cdot 2^{\card{\CSet(\PSet)}}$,
where $\PSet \defeq \set{ ((\qpElm, \bpElm), (\psi, i)) \in
\LSigSet(\AgSet, \SL \times \{ 0, 1 \}) }{ \qpElm \bpElm \psi \in
\psnt{\varphi} \land i \in \{ 0, 1 \} }$ is the set of all labeled
signatures on $\AgSet$ w.r.t. $\SL \times \{ 0, 1 \}$.
Thus, we can build an automaton that accepts only $b$-bounded tree
To do this, in the following, we assume $\AcSet \defeq \numco{0}{b}$.
Consider each principal subsentence $\phi \in \psnt{\varphi}$ of $\varphi$
as a sentence with atomic propositions in $\APSet \cup \psnt{\varphi}$
having no inner principal subsentence.
This means that these subsentences are considered as fresh atomic propositions.
Now, let $\UName[\phi | ^{\AcSet}] \defeq \TATuple
{\SpcSet[\AcSet](\qpElm) \times \pow{\APSet \cup \psnt{\varphi}}}
{\DecSet} {\QSet[\phi]} {\atFun[\phi]} {\qElm[0\phi]} {\aleph_{\phi}}$ be
the s built in Lemma <ref>.
Moreover, set $\MSet \defeq \set{ \mFun \in \psnt{\varphi} \to
\bigcup_{\qpElm \in \QPSet(\VSet), \VSet \subseteq \VarSet}
\SpcSet[\AcSet](\qpElm) }{ \forall \phi = \qpElm \bpElm \psi \in
\psnt{\varphi} \:.\: \mFun(\phi) \in \SpcSet[\AcSet](\qpElm) }$.
Then, we define the components of the $\UName[\varphi] \defeq
\TATuple {\MSet \times \MSet \times \pow{\APSet \cup \psnt{\varphi}}}
{\DecSet} {\QSet} {\atFun} {\qElm[0]} {\aleph}$, as follows:
* $\QSet \defeq \{ \qElm[0], \qElm[c] \} \cup \bigcup_{\phi \in
\psnt{\varphi}} \{ \phi \} \times \QSet[\phi]$;
* $\atFun(\qElm[0], (\mFun[h], \mFun[b], \sigma)) \defeq
\atFun(\qElm[c], (\mFun[h], \mFun[b], \sigma))$, if $\sigma
\models \varphi$, and $\atFun(\qElm[0], (\mFun[h], \mFun[b],
\sigma)) \defeq \Ff$, otherwise, where $\varphi$ is considered here as
a Boolean formula on $\APSet \cup \psnt{\varphi}$;
* $\atFun(\qElm[c], (\mFun[h], \mFun[b], \sigma)) \defeq
\bigwedge_{\decFun \in \DecSet} (\decFun, \qElm[c]) \wedge
\bigwedge_{\phi \in \sigma \cap \psnt{\varphi}} \allowbreak
\atFun[\phi](\qElm[0\phi], (\mFun[h](\phi), \sigma)) [(\decFun, \qElm)
/ (\decFun, (\phi, \qElm))]$;
* $\atFun((\phi, \qElm), (\mFun[h], \mFun[b], \sigma)) \defeq
\atFun[\phi](\qElm, (\mFun[b](\phi), \sigma)) [(\decFun, \qElm') /
\allowbreak (\decFun, (\phi, \qElm'))]$;
* $\aleph \defeq \bigcup_{\phi \in \psnt{\varphi}} \{ \phi \} \times
\aleph_{\phi}$.
Intuitively, $\UName[\varphi]$ checks whether there are principal
subsentences $\phi$ of $\varphi$ contained into the labeling, for all
nodes of the input tree, by means of the checking state $\qElm[c]$.
In the affirmative case, it runs the related automata $\UName[\phi |
^{\AcSet}]$ by supplying them, as dependence maps on actions, the heading
part $\mFun[h]$, when it starts, and the body part $\mFun[b]$,
In this way, it checks that the disjoint satisfiability is verified.
We now prove that the above construction is correct.
[Only if].
Suppose that $\varphi$ is satisfiable.
Then, by Theorem <ref> there exists a $b$-bounded
$\TName$ such that $\TName \models \varphi$.
In particular, w.l.o.g., assume that $\AcSet[\TName] = \AcSet$.
Moreover, for all $\phi = \qpElm \bpElm \psi \in \psnt{\varphi}$, it holds
that $\TName$ satisfies $\phi$ disjointly over the set $\SSet[\phi] \defeq
\set{ \tElm \in \StSet[\TName] }{ \TName, \emptyset, \tElm \models \phi
This means that, by Definition <ref> of disjoint
satisfiability, there exist two functions $\headFun[\phi] : \SSet[\phi]
\to \SpcSet[\AcSet](\qpElm)$ and $\bodyFun[\phi] :
\TrkSet[\TName](\epsilon) \to \SpcSet[\AcSet](\qpElm)$ such that, for all
$\tElm \in \SSet[\phi]$ and $\asgFun \in \AsgSet[\TName](\QPAVSet{\qpElm},
\tElm)$, it holds that $\TName, \spcFun[\phi, \tElm](\asgFun), \tElm
\models \bpElm \psi$, where the elementary dependence map $\spcFun[\phi,
\tElm] \in \ESpcSet[ {\StrSet[\TName](\tElm)} ](\qpElm)$ is defined as
follows: (i) $\adj{\spcFun[\phi, \tElm]}(\tElm) \defeq
\headFun[\phi](\tElm)$; (ii) $\adj{\spcFun[\phi, \tElm]}(\trkElm)
\defeq \bodyFun[\phi](\trkElm' \cdot \trkElm)$, for all $\trkElm \in
\TrkSet[\TName](\tElm)$ with $\card{\trkElm} > 1$, where $\trkElm' \in
\TrkSet[\TName](\epsilon)$ is the unique track such that $\trkElm' \cdot
\trkElm \in \TrkSet[\TName](\epsilon)$.
Now, let $\TName[\varphi]$ be the over $\APSet \cup
\psnt{\varphi}$ with $\AcSet[ {\TName[\varphi]} ] = \AcSet$ such that
(i) $\labFun[ {\TName[\varphi]} ](\tElm) \cap \APSet =
\labFun[\TName](\tElm)$ and (ii) $\phi \in \labFun[
{\TName[\varphi]} ](\tElm)$ iff $\tElm \in \SSet[\phi]$, for all $\tElm
\in \StSet[ {\TName[\varphi]} ] = \StSet[\TName]$ and $\phi \in
\psnt{\varphi}$.
By Lemma <ref>, we have that $\TName[\phi, \tElm | '] \in
\LangSet(\UName[\phi | ^{\AcSet}])$, where $\TName[\phi, \tElm | ']$ is
the elementary dependence-labeling encoding for $\spcFun[\phi, \tElm]$ on
Thus, there is a $(\DecSet \times \QSet[\phi])$-tree $\RSet[\phi, \tElm]$
that is an accepting run for $\UName[\phi | ^{\AcSet}]$ on $\TName[\phi,
\tElm | ']$.
So, let $\RSet[\phi, \tElm | ']$ be the $(\DecSet \times \QSet)$-tree
defined as follows: $\RSet[\phi, \tElm | '] \defeq \set{ (\tElm \cdot
\tElm', (\phi, \qElm)) }{ (\tElm', \qElm) \in \RSet[\phi, \tElm] }$.
At this point, let $\RSet \defeq \RSet[c] \cup \bigcup_{\phi \in
\psnt{\varphi}, \tElm \in \SSet[\phi]} \RSet[\phi, \tElm | ']$ be the
$(\DecSet \times \QSet)$-tree, where $\RSet[c] \defeq \{ \epsilon \} \cup
\set{ (\tElm, \qElm[c]) }{ \tElm \in \StSet[\TName] \land \tElm \neq
\epsilon }$, and $\TName' \defeq \LTTuple{}{}{\StSet[\TName]}{\uFun}$ one
of the $(\MSet \times \MSet \times \pow{\APSet \cup
\psnt{\varphi}})$-labeled $\DecSet$-tree satisfying the following
property: for all $\tElm \in \StSet[\TName]$ and $\phi \in
\psnt{\varphi}$, it holds that $\uFun(\tElm) = (\mFun[h], \mFun[b],
\sigma)$, where (i) $\sigma \cap \APSet = \labFun[\TName](\tElm)$,
(ii) $\phi \in \sigma$ iff $\tElm \in \SSet[\phi]$, (iii)
$\mFun[h](\phi) = \headFun[\phi](\tElm)$, if $\tElm \in \SSet[\phi]$, and
(iv) $\mFun[b](\phi) = \bodyFun[\phi](\trkElm[\tElm])$ with
$\trkElm[\tElm] \in \TrkSet[\TName](\epsilon)$ the unique track such that
$\lst{\trkElm[\tElm]} = \tElm$.
Moreover, since $\TName \models \varphi$, we have that $\labFun[
{\TName[\varphi]} ](\epsilon) \models \varphi$, where, in the last
expression, $\varphi$ is considered as a Boolean formula on $\APSet \cup
\psnt{\varphi}$.
Then, it is easy to prove that $\RSet$ is an accepting run for
$\UName[\varphi]$ on $\TName'$, i.e., $\TName' \in
\LangSet(\UName[\varphi])$.
Hence, $\LangSet(\UName[\varphi]) \neq \emptyset$.
Suppose that there is an $(\MSet \times \MSet \times
\pow{\APSet \cup \psnt{\varphi}})$-labeled $\DecSet$-tree $\TName' \defeq
\LTTuple{}{}{\DecSet^{*}}{\uFun}$ such that $\TName' \in
\LangSet(\UName[\varphi])$ and let the $(\DecSet \times \QSet)$-tree
$\RSet$ be the accepting run for $\UName[\varphi]$ on $\TName'$.
Moreover, let $\TName$ be the over $\APSet \cup \psnt{\varphi}$ with
$\AcSet[\TName] = \AcSet$ such that, for all $\tElm \in \StSet[\TName]$,
it holds that $\uFun(\tElm) = (\mFun[h], \mFun[b],
\labFun[\TName](\tElm))$, for some $\mFun[h], \mFun[b] \in \MSet$.
Now, for all $\phi = \qpElm \bpElm \psi \in \psnt{\varphi}$, we make the
following further assumptions:
* $\SSet[\phi] \defeq \set{ \tElm \in \StSet[\TName] }{ \exists
\mFun[h], \mFun[b] \in \MSet, \sigma \in \pow{\APSet \cup
\psnt{\varphi}} \:.\: \allowbreak \uFun(\tElm) = (\mFun[h], \mFun[b],
\sigma) \land \phi \in \sigma }$;
* let $\RSet[\phi, \tElm]$ be the $(\DecSet \times \QSet[\phi])$-tree
such that $\RSet[\phi, \tElm] \defeq \{ \epsilon \} \cup \set{
(\tElm', \qElm) }{ (\tElm \cdot \tElm', (\phi, \qElm)) \in \RSet }$,
for all $\tElm \in \SSet[\phi]$;
* let $\TName[\phi, \tElm | ']$ be the elementary dependence-labeling
encoding for $\spcFun[\phi, \tElm] \in \ESpcSet[
{\StrSet[\TName](\tElm)} ](\qpElm)$ on $\TName$, for all $\tElm \in
\SSet[\phi]$, where $\adj{\spcFun[\phi, \tElm]}(\tElm) \defeq
\mFun[h](\phi)$, with $\uFun(\tElm) = (\mFun[h], \mFun[b], \sigma)$
for some $\mFun[b] \in \MSet$ and $\sigma \in \pow{\APSet \cup
\psnt{\varphi}}$, and $\adj{\spcFun[\phi, \tElm]}(\trkElm) \defeq
\mFun[b](\phi)$, with $\uFun(\lst{\trkElm}) = (\mFun[h], \mFun[b],
\sigma)$ for some $\mFun[h] \in \MSet$ and $\sigma \in \pow{\APSet
\cup \psnt{\varphi}}$, for all $\trkElm \in \TrkSet[\TName](\tElm)$
with $\card{\trkElm} > 1$.
Since $\RSet$ is an accepting run, it is easy to prove that
$\RSet[\phi, \tElm]$ is an accepting run for $\UName[\phi | ^{\AcSet}]$ on
$\TName[\phi, \tElm | ']$.
Thus, $\TName[\phi, \tElm | '] \in \LangSet(\UName[\phi | ^{\AcSet}])$.
So, by Lemma <ref>, it holds that $\TName, \spcFun[\phi,
\tElm](\asgFun), \tElm \models \bpElm \psi$, for all $\tElm \in
\SSet[\phi]$ and $\asgFun \!\in \!\AsgSet[\TName](\QPAVSet{\qpElm}, \tElm)$,
which means that $\SSet[\phi] \!=\! \set{ \tElm \in \StSet[\TName] }{ \TName,
\emptyset, \tElm \models \phi }$.
Finally, since $\labFun[ {\TName[\varphi]} ](\epsilon) \models \varphi$,
we have that $\TName \models \varphi$, where, in the first expression,
$\varphi$ is considered as a Boolean formula on $\APSet \cup
\psnt{\varphi}$.
The satisfiability problem for is 2.
By Theorem <ref> of automaton, to verify whether an
sentence $\varphi$ is satisfiable we can calculate the emptiness of
the $\UName[\varphi]$.
This automaton is obtained by merging all s $\UName[\phi |
^{\AcSet}]$, with $\phi = \qpElm \bpElm \psi \in \psnt{\varphi}$, which
in turn are based on the s $\UName[\bpElm \psi | ^{\AcSet}]$ that
embed the s $\UName[\psi]$.
By a simple calculation, it is easy to see that $\UName[\varphi]$ has
$2^{\AOmicron{\card{\varphi}}}$ states.
Now, by using a well-known nondeterminization procedure for s <cit.>, we obtain an equivalent $\NName[\varphi]$ with
$2^{2^{\AOmicron{\card{\varphi}}}}$ states and index
The emptiness problem for such a kind of automaton with $n$ states and
index $h$ is solvable in time $\AOmicron{n^{h}}$.
Thus, we get that the time complexity of checking whether $\varphi$ is
satisfiable is $2^{2^{\AOmicron{\card{\varphi}}}}$.
Hence, the membership of the satisfiability problem for in
2 directly follows.
Finally the thesis is proved, by getting the relative lower bound from
the same problem for
|
arxiv-papers
| 2012-02-06T22:43:55 |
2024-09-04T02:49:27.112282
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabio Mogavero, Aniello Murano, Giuseppe Perelli, and Moshe Y. Vardi",
"submitter": "Fabio Mogavero PhD",
"url": "https://arxiv.org/abs/1202.1309"
}
|
1202.1331
|
# Integer Subsets with High Volume and Low Perimeter
Patrick Devlin – prd41@math.rutgers.edu
## 1 Introduction
One of the most widely-known classical geometry problems is the so-called
isoperimetric problem, one equivalent variation of which is:
> If a figure in the plane has area $A$, what is the smallest possible value
> for its perimeter?
In the Euclidean plane, the optimal configuration is a circle, implying that
any figure with area $A$ has perimeter at least $2\sqrt{A\pi}$, and this lower
bound may be obtained if and only if the figure is a circle.
####
In 2011, Miller et al. [2] extended the isoperimetric problem in a new
direction, in which integer subsets took the role of geometric figures. For
any integer subset $A$, they defined its volume as the sum over all its
elements, and they defined its perimeter as the sum of all elements $x\in A$
such that $\\{x-1,x+1\\}\not\subset A$. Thus, the volume can be thought of as
the sum of all the elements of $A$, and the perimeter can be thought of as the
sum of all the elements on the “boundary” of $A$ (that is to say, the elements
of $A$ whose successor and predecessor are not both in $A$).
####
The main focus of [2] was to examine the relationships between a set’s
perimeter and its volume. More specifically, the authors wanted to answer the
corresponding “isoperimetric question”111They focused on this question in
particular because it turns out that all of the related extremal questions are
trivial.:
> If a subset of $\\{0,1,\ldots\\}$ has volume $n$, what is the smallest
> possible value for its perimeter?
Adopting their notation, we will let $P(n)$ denote this value through the
duration of this paper222This is sequence A186053 in OEIS..
####
Because their work is so recent, Miller et al. are the only ones who have
published on this variation of the isoperimetric problem or on the function
$P(n)$. Their work was to provide bounds for $P(n)$, by which they were able
to determine its asymptotic behavior. Specifically, their main result was
###### Theorem 1.
_(Miller et al., 2011)_ Let $P(n)$ be as defined. Then
$P(n)\thicksim\sqrt{2}n^{1/2}$. Moreover, for all $n\geq 1$,
$\sqrt{2}n^{1/2}-1/2<P(n)<\sqrt{2}n^{1/2}+(2n^{1/4}+8)\log_{2}\log_{2}n+58.$
(1)
####
Their proof of the lower bound will be reproduced in following sections.
However, their proof of the upper bound was found by a construction argument,
which we will not reproduce here since we will analytically derive a tighter
bound in Theorem 12.
####
Beyond the inequalities in (1) provided by Miller et al., nothing else has
been published on $P(n)$ except for some values for small $n$. It should be
noted that [2] provides very good bounds on a related function, in which the
sets of interest are allowed to have both negative as well as positive
elements. However, this result was also obtained by a construction argument,
and it is not relevant to this paper.
### Outline of Results
In this paper, we focus on improving the few results known on $P(n)$,
including deriving multiple exact formulas and developing an understanding of
its interesting long-term behavior. Many of these results are stated in terms
of an intimately related function, $Q(n)$, which may be briefly defined as
$Q(n):=\min_{A\subseteq\\{0,1,\ldots\\}}\Big{\\{}per(A^{c}):vol(A)=n\Big{\\}}.$
Since it proves to be so closely related to $P(n)$, we also provide results on
$Q(n)$ throughout the paper.
####
We begin in Section 3 with several prelimary lemmas including those used in
[2]. Then in Section 4, we define auxilary functions, with which we
combinatorially derive several recursive formulas for $P(n)$. We then
introduce the function $Q(n)$ and derive similar recursive formulas for it as
well.
####
In Section 5, we relate the functions $P(n)$ and $Q(n)$ by providing yet more
recurrence relations for both of them, from which we see that each function
completely determines the other. With this in place, we move on to Section 6,
in which we use these recurrences to determine several analytic results for
$P(n)$ and $Q(n)$, including upper and lower bounds and derivations of their
asymptotic behavior.
####
Our work then culminates in Section 7, in which we state and prove the
strongest results of the paper. By appealing to our analytic bounds on $P(n)$
and $Q(n)$, we show that for all sufficiently large values of $n$, the
recurrences of Section 5 admit certain drastic simplifications. By then
combining this result with rigorous computer calculations, we arrive at the
main theorem of the paper333More adequate introductions of the functions $f$
and $g$ are given in Section 6.:
###### Theorem 18.
Let $P(n)$ and $Q(n)$ be as given. Then if $n\geq 0$ is not one of the $177$
known counterexamples tabulated in Table 1 of the appendix (in particular, for
all $n>149,894$), we have
$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle
f(n)+Q(g(n))\qquad\qquad\text{and}$ $\displaystyle Q(n)$ $\displaystyle=$
$\displaystyle 1+f(n)+P(g(n)),$
where the functions $f(n)$ and $g(n)$, given by
$f(n):=\left\lfloor\sqrt{2n}+1/2\right\rfloor=\left[\sqrt{2n}\right],\qquad\qquad\text{and}\qquad\qquad
g(n):=\dfrac{f(n)[f(n)+1]}{2}-n,$
are the smallest nonnegative integers satisfying $[1+2+3+\cdots+f(n)]-g(n)=n$.
With this, we derive several other satisfying and revealing reccurence
relations and quasi-explicit representations for $P(n)$ and $Q(n)$. We also
breifly demonstate and discuss the intricate fractal-like symmetry of the
graphs of these functions. We then conclude in Section 8 by noting
applications in the design of algorithms related to this problem and with some
open questions for future research.
####
For an earlier version of this paper with somewhat more detailed proofs and
expositions, see [1].
## 2 Definitions and Notation
For the reader’s possible convenience, a brief list of definitions used
throughout the paper is given here. In each definition, $A$ is assumed to be a
subset of $\\{0,1,2,\ldots\\}$, and $n$ and $k$ are assumed to be nonnegative
integers.
* •
The boundary of $A$, $\partial A$, is $\partial A:=\\{z\in
A:\\{z-1,z+1\\}\not\subseteq A\\}.$ In words, it is the set of elements of $A$
whose successor or predecessor is not in $A$.
* •
The volume and perimeter of $A$ are defined as
$vol(A):=\sum_{z\in A}z,\qquad\text{and}\qquad per(A):=\sum_{z\in\partial
A}z,$
respectively. (For convention, the volume and perimeter of the empty set is
0.)
* •
$P(n):=\min_{A\subseteq\\{0,1,\ldots\\}}\Big{\\{}per(A):vol(A)=n\Big{\\}}.$
* •
The complement of $A$ is $A^{c}:=\\{0,1,\ldots\\}\setminus
A=\\{z\in\\{0,1,\ldots\\}:z\notin A\\}.$
* •
$Q(n):=\min_{A\subseteq\\{0,1,\ldots\\}}\Big{\\{}per(A^{c}):vol(A)=n\Big{\\}}.$
* •
The helper functions $p(n;k)$ and $q(n;k)$ are defined as
$p(n;k):=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A):vol(A)=n\Big{\\}},\quad\qquad
q(n;k):=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A^{c}):vol(A)=n\Big{\\}}.$
* •
The special helper function $\sigma(n;k)$ is
$\sigma(n;k):=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A^{c}):vol(A)=n,\quad\text{and}\quad
k\in A\Big{\\}}.$
* •
The functions $f(n)$ and $g(n)$ are given by
$f(n)=\left[\sqrt{2n}\right],\qquad\text{and}\qquad
g(n)=\dfrac{f(n)[f(n)+1]}{2}-n=\dfrac{\left[\sqrt{2n}\
\right]^{2}+\left[\sqrt{2n}\ \right]}{2}-n,$
where $[x]$ denotes the nearest integer function. In Proposition 8, we show
that $f(n)$ and $g(n)$ are also the smallest nonnegative integers satisfying
$[1+2+\cdots+f(n)]-g(n)=n$.
* •
For all $N$ (and particularly, for $N=149,894$), we define
$\phi(n;N)=\phi(n):=\min_{i\geq 0}\\{g^{i}(n)\leq N\\}$.
## 3 Preliminary Results
The following lemma is used throughout [2] and is essential in proving their
lower bound on $P(n)$.
###### Lemma 2.
_(Miller et al., 2011)_ Assume $A$ is a finite nonempty subset of
$\\{0,1,\ldots\\}$, and let $m$ denote its maximum element. Then
$m\leq per(A)\leq vol(A)\leq\dfrac{m(m+1)}{2}.$
Using this lemma, the following lower bound is immediately attained.
###### Proposition 3.
Assume $A\subseteq\\{0,1,\ldots\\}$ is finite. Then we have
$\sqrt{2vol(A)}-1/2\leq\dfrac{-1+\sqrt{1+8vol(A)}}{2}\leq per(A).$
Moreover, for any positive integer $n$, this implies
$\sqrt{2}n^{1/2}-1/2\leq P(n).$
As stated before, except for the previously mentioned constructive upper bound
on $P(n)$, these two results are all that has been published about $P(n)$. The
remainder of the paper is devoted to new results.
### Miscellaneous Lemmas
###### Lemma 4.
Let $A\neq\emptyset$ be a finite subset of $\\{0,1,\ldots\\}$, and let $m$
denote its maximum element. Then
$m+1\leq per(A^{c})$
with equality if and only if $\\{1,\ldots,m\\}\subseteq A$.
###### Proof.
Let $A$ be as given. Then $m\in A$, but we know $m+1\notin A$. Therefore,
$m+1\in\partial A^{c}$ implying that $m+1\leq per(A^{c})$. Now since
$m+1\in\partial A^{c}$, we know that $m+1=per(A^{c})$ if and only if $\partial
A^{c}$ is equal to either $\\{m+1\\}$ or $\\{0,m+1\\}$. But this happens if
and only if $\\{1,2,\ldots,m\\}\subseteq A$, as desired. ∎
###### Proposition 5.
Assume $A\subseteq\\{0,1,\ldots\\}$ is finite. Then we have
$\sqrt{2vol(A)}+1/2\leq\dfrac{-1+\sqrt{1+8vol(A)}}{2}+1\leq per(A^{c}).$
Moreover, for any positive integer $n$, this implies
$\sqrt{2}n^{1/2}+1/2\leq Q(n).$
###### Proof.
This follows from the previous lemma in the same way as Proposition 3. ∎
## 4 Recurrence Relations using Auxilary Functions
We now derive our first set of recurrence relations for $P(n)$ and $Q(n)$.
Although the relations derived in Section 5 are actually more revealing, the
relations presented here follow naturally, and they motivate the introduction
of important auxilary functions. Moreover, because of their convenient
structure, these relations are used extensively in the design of algorithms
for computing values, as we breifly discuss in Section 8.
### First Recurrence for $P(n)$
As is often the case in analyzing discrete functions, we may obtain an exact
recurrence relation for $P(n)$ in terms of a related auxillary function. In
our case, recall that $P(n)$ is the minimum perimeter among all subsets of
$\\{0,1,\ldots\\}$ having volume $n$. This suggests defining an auxilary
function, $p(n;k)$, as
$p(n;k)=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A):vol(A)=n\Big{\\}}.$
Then for all $n\geq 0$, we have
$P(n)=\min_{k\in\\{0,1,\ldots\\}}\Bigg{\\{}\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A):vol(A)=n\Big{\\}}\Bigg{\\}}=\min_{k\in\\{0,1,\ldots\\}}\Bigg{\\{}p(n;k)\Bigg{\\}}.$
####
From its definition, it is clear that for all fixed $n$, the function $p(n;k)$
is monotonically decreasing with $k$. Moreover, for all $K\geq n$, we have
$p(n;K)=p(n;n)$ since any subset of $\\{0,1,\ldots\\}$ having volume $n$ must
necessarily be a subset of $\\{0,1,2,\ldots,n\\}$. Therefore the above
equation simplifies to
$P(n)=\min_{k\in\\{0,1,\ldots\\}}\Bigg{\\{}p(n;k)\Bigg{\\}}=\lim_{k\to\infty}p(n;k)=p(n;n).$
(2)
Thus, we now seek a recurrence for $p(n;k)$, which will provide us with $P(n)$
by calculating $p(n;n)$.
####
For notational convenience, let $S(n;k)$ denote the set of all subsets of
$\\{0,1,\ldots,k\\}$ having volume $n$. Then to obtain our desired recurrence
for $p(n;k)$, we will consider the following paritition of $S(n;k)$
$S(n;k)=\bigcup_{l=0}^{k+1}\Big{\\{}A\in S(n;k):\\{l,\ldots,k\\}\subseteq
A\quad\text{and}\quad l-1\notin A\Big{\\}}.$
From this partition, it follows that
$p(n;k)=\min_{l\in\\{0,1,\ldots,k+1\\}}\Bigg{\\{}\min_{A\in
S(n;k)}\Big{\\{}per(A):\\{l,\ldots,k\\}\subseteq A\quad\text{and}\quad
l-1\notin A\Big{\\}}\Bigg{\\}}.$ (3)
####
Now let $0\leq l\leq k+1$ be fixed. Then we have
$\displaystyle\min_{A\in S(n;k)}\Big{\\{}per(A):\\{l,\ldots,k\\}\subseteq
A\quad\text{and}\quad l-1\notin A\Big{\\}}$
$\displaystyle\qquad=\min_{B\subseteq\\{0,1,\ldots,l-2\\}}\Big{\\{}per(B\cup\\{l,l+1,\ldots,k\\}):vol(B\cup\\{l,l+1,\ldots,k\\})=n\Big{\\}}$
$\displaystyle\qquad=\begin{cases}\displaystyle\min_{B\in\\{0,1,\ldots,k-1\\}}\Big{\\{}per(B):vol(B)=n\Big{\\}}\quad&\text{if
$l=k+1$},\\\
\displaystyle\min_{B\in\\{0,1,\ldots,k-2\\}}\Big{\\{}k+per(B):vol(B)=n-k\Big{\\}}\quad&\text{if
$l=k$},\\\
\displaystyle\min_{B\in\\{0,1,\ldots,l-2\\}}\Big{\\{}k+l+per(B):vol(B)=n-\left[k(k+1)/2-l(l-1)/2\right]\Big{\\}}\quad&\text{if
$0\leq l<k$},\end{cases}$
$\displaystyle\qquad=\begin{cases}p(n;k-1)\quad&\text{if $l=k+1$},\\\
k+p(n-k;k-2)\quad&\text{if $l=k$},\\\
k+l+p\big{(}n-[k(k+1)-l(l-1)]/2;l-2\big{)}\quad&\text{if $0\leq
l<k$}.\end{cases}$
Therefore, by substituting into (3), we are able to obtain the recurrence
$\displaystyle p(n;k)$ $\displaystyle=$
$\displaystyle\min\Bigg{\\{}p(n;k-1),k+p(n-k;k-2),$ (4)
$\displaystyle\qquad\qquad
k+\min_{l\in\\{0,\ldots,k-1\\}}\Big{\\{}l+p\big{(}n-[k(k+1)-l(l-1)]/2;l-2\big{)}\Big{\\}}\Bigg{\\}},$
which is valid for all $n\geq 1$ and for all $k\geq 1$. Moreover, as boundary
conditions, which are clear from its definition, we have that $p(n;k)$
satisfies
$p(n;k)=\begin{cases}0\qquad&\text{if $n=0$,}\\\ \infty\qquad&\text{if $n<0$
or $k\leq 0<n$.}\end{cases}$
Thus, this recurrence for $p(n;k)$ gives the following compact recursive
representation for $P(n)$ for all $n\geq 0$:
$P(n)=\min\Big{\\{}p(n;n-1),n\Big{\\}}.$ (5)
### Introduction of $Q(n)$ and Derivation of First Recurrences
Because of its intimate connections with the function $P(n)$ that will be
explored in subsequent sections, we now introduce the function $Q(n)$, which
is defined as
$Q(n)=\min_{A\subseteq\\{0,1,\ldots\\}}\Big{\\{}per(A^{c}):vol(A)=n\Big{\\}}.$
The difference between this function and the function $P(n)$ is subtle, and
based on how similarly the two functions are defined, one would expect their
behavior to be very close. As we will see, this is indeed the case, and the
connections between $P(n)$ and $Q(n)$ are actually of fundamental importance.
However, it is important for the reader to keep in mind the difference in how
these functions are defined.
####
As with the function $P(n)$, we define the auxilary function $q(n;k)$ as
$q(n;k)=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A^{c}):vol(A)=n\Big{\\}},$
and just as before, for all $n\geq 0$, we have that
$Q(n)=q(n;n).$ (6)
####
Because of the difference between how the functions $P(n)$ and $Q(n)$ are
defined, we now need to define a special auxilary function, $\sigma(n;k)$, in
order to obtain a compact recurrence for $q(n)$. This function is defined by
$\sigma(n;k)=\min_{A\subseteq\\{0,1,\ldots,k\\}}\Big{\\{}per(A^{c}):vol(A)=n\quad\text{and}\quad
k\in A\Big{\\}}.$
Note the similarities between $\sigma(n;k)$ and $q(n;k)$. In fact, it is clear
that for all $n\geq 1$ and $k\geq 0$, we have
$q(n;k)=\min_{l\in\\{1,2,\ldots,k\\}}\Big{\\{}\sigma(n;l)\Big{\\}}.$ (7)
Using this equation and (6), we obtain that for all $n\geq 1$
$Q(n)=\min_{l\in\\{1,2,\ldots,n\\}}\Big{\\{}\sigma(n;l)\Big{\\}},$ (8)
with $Q(0)=0$.
####
Just as was the case for $P(n)$, in order to obtain a useful recurrence
relation for $Q(n)$, it now only remains to find a recurrence for
$\sigma(n;k)$. As before, we accomplish this by a simple partition yielding
$\sigma(n;k)=k+1+\min\Big{\\{}\sigma(n-k;k-1)-k,\sigma(n-k;k-2),k-1+q(n-k;k-3)\Big{\\}},$
which we obtain by partitioning the subsets of interest into the three groups
(I) sets containing $k-1$, (II) sets containing $k-2$ but not $k-1$, and (III)
sets containing neither $k-2$ nor $k-1$.
####
At this point, we need to note that some care must be given to the
interpretation of the above equation, which depends on how we define
$\sigma(0;0)$. However, if we note and state as a boundary condition that
$\sigma(n,n)=2n$ for all $n\geq 1$, then these concerns are effectively
removed.
####
We then have a recurrence relation for $\sigma$. As boundary conditions for
$\sigma(n;k)$, we have
$\sigma(n;k)=\begin{cases}0\qquad&\text{if $n=k=0$,}\\\ 2n\qquad&\text{if
$n=k\geq 1$,}\\\ \infty\qquad&\text{if $n<0$ or if $k\in\\{0,1\\}$ and
$n>k$,}\\\ \infty\qquad&\text{if $0\leq k>n\geq 0$.}\end{cases}$
Then for all $n\geq 2$, and $2\leq k<n$, we have
$\sigma(n;k)=k+1+\min\Big{\\{}k-1+q(n-k;t-3),\sigma(n-k;k-2),\sigma(n-k;k-1)-k\Big{\\}}.$
Thus, by using (8) we have a recurrence for $Q(n)$ as well.
## 5 More Direct Recurrence Relations
Now by making use of different partitions of the sets of interest, we derive
the following recurrence relations, from which we see the first connections
between the functions $P(n)$ and $Q(n)$.
### Recurrence for $P(n)$ involving $q(n;k)$ and $\sigma(n;k)$
We may calculate $P(n)$ by a “more direct” recurrence relation, which is found
by partitioning all sets of volume $n$ first according to their maximum
element, $m$, and then according to the largest integer smaller than $m$ not
contained in each set.
####
Let $A$ be a set of volume $n$, let $m$ be its maximum element, and let $l$ be
the largest element of $\\{-1,0,\ldots,m\\}$ not contained in $A$. Then $A$
may be written uniquely as $A=\\{0,1,2,\ldots,m\\}\setminus B$ for some set
$B\subseteq\\{0,1,\ldots,l\\}$, where the volume of $B$ is equal to
$(1+2+\cdots+m)-n$ and $l\in B$. If $l=m-1$, then $per(A)=per(B^{c})$. Else,
we have $per(A)=m+per(B^{c})$.
####
From this observation, we obtain that for all $n\geq 2$
$P(n)=\min_{m\geq
1}\Big{\\{}m+q([1+2+\cdots+m]-n;m-2),\sigma([1+2+\cdots+m]-n;m-1)\Big{\\}},$
(9)
where $q(n;k)$ and $\sigma(n;k)$ are defined as earlier.
### Recurrence for $Q(n)$ involving $p(n;k)$
As before, we also have a simple recurrence that can be used to calculate
$Q(n)$ “more directly”. Let $A$ be a set of volume $n$ and maximum element
$m$. Then the set $A$ may be written uniquely in the form
$A=\\{0,1,2,\ldots,m\\}\setminus B$ for some set
$B\subseteq\\{0,1,\ldots,m-1\\}$, where the volume of $B$ is equal to
$(1+2+\cdots+m)-n$. Now we know that for all such sets $A$ and $B$, we have
$per(A^{c})=per(B)+(m+1)$.
####
This observation leads to the simple and beautiful recurrence that for all
$n\geq 2$,
$Q(n)=1+\min_{m\geq 1}\Big{\\{}m+p([1+2+\cdots+m]-n;m-1)\Big{\\}},$ (10)
where $p(n;k)$ is as defined earlier.
## 6 Analysis of Recurrences
Although equations (9) and (10) appear somewhat intractible (and they offer
little or no computational advantage over the first recurrences of Section 4),
they turn out to be crucial in understanding the behavior of $P(n)$ (and of
$Q(n)$ as well). In Section 7, we are able to greatly simplify these
recurrence, but in order to do so, we must first derive some analytic bounds
on $P(n)$ and $Q(n)$.
### Relevant Lemmas and Notions
###### Lemma 6.
Let $n$ be a positive integer. Then there exist unique nonnegative integers
$f(n)$ and $g(n)$ satisfying
$n=[0+1+\cdots+f(n)]-g(n),$
where $0\leq g(n)<f(n)$. Moreover, $f(n)$ and $g(n)$ are given by444We will
use these explicit functional representations for $f(n)$ and $g(n)$ so that
$f(0)=g(0)=0$ is well-defined.
$f(n)=\left\lceil\dfrac{-1+\sqrt{1+8n}}{2}\right\rceil,\qquad\text{and}\qquad
g(n)=\dfrac{f(n)[f(n)+1]}{2}-n.$
####
Having defined these functions, we may now restate previous lemmas involving
$P(n)$ and $Q(n)$ in these terms. The most important result we will use
combines Propositions 3 and 5 as follows:
###### Corollary 7.
Restating earlier results in new notation, for all $n\geq 1$, we have that
$P(n)\geq f(n),\qquad\text{and}\qquad Q(n)\geq f(n)+1.$
####
Finally, before moving on, we must present two more results on the functions
$f(x)$ and $g(x)$.
###### Proposition 8.
Let $f(n)=\left\lceil\dfrac{-1+\sqrt{1+8n}}{2}\right\rceil$ as before. Then
for all integers $n\geq 0$, we have
$f(n)=\left\lceil\dfrac{-1+\sqrt{1+8n}}{2}\right\rceil=\left\lceil\sqrt{2n}-1/2\right\rceil=\left[\sqrt{2n}\right],$
where $[x]$ is the nearest integer function.
###### Proof.
It suffices to show the first part of the stated equation holds, and the fact
that $\sqrt{2n}$ is never a half-integer will complete the proof. Now by way
of contradiction, suppose that the first two representations are not equal.
Then this would imply that there exist integers $p\in\mathbb{Z}$ and
$n\in\\{0,1,\ldots\\}$ such that
$\sqrt{2n}-1/2\leq p<\dfrac{\sqrt{1+8n}-1}{2},$
which implies $8n\leq(2p+1)^{2}<8n+1.$ But since $n$ and $p$ are integers,
this forces $8n=(2p+1)^{2}$, which taken modulo 2 yields a contradiction. ∎
###### Proposition 9.
Let $f(n)$ and $g(n)$ be defined as before. Then for all integers $L\geq 0$
and $n\geq 0$, we have
$g^{L}(n)\leq 2\cdot(n/2)^{1/2^{L}}.$
###### Proof.
The proof is by induction on $L$. If $L=0$, then the claim is trivially true,
which establishes the base case. Now suppose the claim holds for $L=m$. Then
for all $n\geq 0$, we have
$g(n)\leq f(n)-1<\sqrt{2n}-1/2<\sqrt{2n},$
which implies $g^{m+1}(n)=g(g^{m}(n))<\sqrt{2\cdot g^{m}(n)}.$ Then using the
induction hypothesis and that the square root function is increasing completes
the proof. ∎
### Upper Bounds and Asymptotics for $P(n)$ and $Q(n)$
Using the recurrences of Section 5, we now obtain simple upper bounds on
$P(n)$ and $Q(n)$, which taken with the last few lemmas, yield good absolute
upper bounds in terms of $n$.
###### Theorem 10.
Let $f(n)$ and $g(n)$ be defined as before. Then for all $n\geq 0$, we have
the bounds
$\displaystyle P(n)$ $\displaystyle\leq$ $\displaystyle
f(n)+Q(g(n)),\qquad\text{and}$ $\displaystyle Q(n)$ $\displaystyle\leq$
$\displaystyle 1+f(n)+P(g(n)).$
###### Proof.
For $n=0$ and $n=1$, the two inequalities hold. Then for all $n\geq 2$, we may
appeal to (9) to obtain
$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle\min_{m\geq
1}\Big{\\{}m+q([1+2+\cdots+m]-n;m-2),\sigma([1+2+\cdots+m]-n;m-1)\Big{\\}}$
$\displaystyle\leq$ $\displaystyle
f(n)+\min\Big{\\{}q(g(n);f(n)-2),\sigma(g(n);f(n)-1)\Big{\\}}=f(n)+q(g(n);f(n)-1)=f(n)+Q(g(n)),$
and the corresponding inequality for $Q(n)$ is proven analogously. ∎
###### Corollary 11.
For all nonnegative integers $n$ and $L$, we have that
$\displaystyle P(n)$ $\displaystyle\leq$ $\displaystyle
L+P(g^{2L}(n))+\sum_{i=0}^{2L-1}f(g^{i}(n)),\qquad\text{and}$ $\displaystyle
Q(n)$ $\displaystyle\leq$ $\displaystyle
L+Q(g^{2L}(n))+\sum_{i=0}^{2L-1}f(g^{i}(n)),$
where $g^{i}(n)$ is the $i$-fold composition of $g$ evaluated at $n$, and by
convention we take $g^{0}(n)=n$.
###### Theorem 12.
Let $P(n)$ and $Q(n)$ be as given. Then $P(n)\sim Q(n)\sim\sqrt{2}n^{1/2}$.
Moreover, for all $n>2$,
$\displaystyle\sqrt{2}n^{1/2}-1/2<$ $\displaystyle P(n)$
$\displaystyle\leq\sqrt{2}n^{1/2}+(2^{3/4}\cdot
n^{1/4}+1)[\log_{2}(\log_{2}(n/2))-1]+7,\qquad\text{and}$
$\displaystyle\sqrt{2}n^{1/2}+1/2<$ $\displaystyle Q(n)$
$\displaystyle\leq\sqrt{2}n^{1/2}+(2^{3/4}\cdot
n^{1/4}+1)[\log_{2}(\log_{2}(n/2))-1]+7.$
###### Proof.
The lower bounds in the asserted inequalities have already been proven. To
prove the upper bounds, we merely combine the results in the last corollary
with the past few bounds on $f(n)$ and $g(n)$. More specifically, assuming
$n>2$, we know from Proposition 9 that if
$L\geq(\log_{2}(\log_{2}(n/2))-1)/2$, then
$g^{2L}(n)\leq 2\cdot(n/2)^{1/2^{(\log_{2}(\log_{2}(n/2))-1)}}=\cdots=8.$
By considering values of $P(n)$ and $Q(n)$ for $n\leq 8$, we see that
$g^{2L}(n)\leq 8$ implies $P(g^{2L}(n))\leq 7$ and $Q(g^{2L}(n))\leq 7$. Now
by the last corollary and the past few lemmas, we have
$\displaystyle P(n)$ $\displaystyle\leq$ $\displaystyle
L+P(g^{2L}(n))+\sum_{i=0}^{2L-1}f(g^{i}(n))\leq
L+P(g^{2L}(n))+\sum_{i=0}^{2L-1}\sqrt{2g^{i}(n)}+1/2$ $\displaystyle\leq$
$\displaystyle
2L+P(g^{2L}(n))+\sum_{i=0}^{2L-1}\sqrt{4\cdot(n/2)^{1/2^{i}}}\leq
2L+P(g^{2L}(n))+\sqrt{2n}+2\sum_{i=1}^{2L-1}\sqrt{(n/2)^{1/2^{i}}}$
$\displaystyle\leq$ $\displaystyle 2L+P(g^{2L}(n))+\sqrt{2n}+4L(n/2)^{1/4}.$
Then taking $L=(\log_{2}(\log_{2}(n/2))-1)/2$ proves the bound. The inequality
for $Q(n)$ is proven analogously. ∎
Note that these bounds on $P(n)$ are slightly better than those of [2] stated
in Theorem 1. Also note that the upper bound on the summation is very crude.
However, these bounds are sufficient for our purposes.
## 7 Obtaining Good Recurrences for $P(n)$ and $Q(n)$
Although the bounds in Theorem 12 are rather good, they reveal nothing about
the actual fluctuations of $P(n)$ and $Q(n)$. And although we have already
obtained multiple recurrence relations for finding exact values, these
relations all involve auxilary helper functions, multiple variables, and
unweildy minimum functions. In this section, we combine our analytic bounds
and combinatorial results to obtain surprisingly simple and satisfying
recurrence relations for $P(n)$ and $Q(n)$ and even quasi-explicit formulae.
### New Lower Bounds on $P(n)$ and $Q(n)$
###### Lemma 13.
Let $n$ and $k$ be positive integers with $k<f(n)$. Then $p(n;k),$ $q(n;k)$,
and $\sigma(n;k)$ are all infinite.
###### Proof.
This follows from the fact that if $k<f(n)$, there are no subsets of
$\\{0,1,\ldots,k\\}$ with volume $n$. ∎
###### Lemma 14.
Let $n$ and $m$ be positive integers with $m>f(n)$. Then we have
$\displaystyle m+p([1+2+\cdots+m]-n;m-1)$ $\displaystyle\geq$ $\displaystyle
f(n)+\sqrt{2(g(n)+f(n)+1)}+1/2\qquad\qquad\text{and}$ $\displaystyle
m+q([1+2+\cdots+m]-n;m-2)$ $\displaystyle\geq$ $\displaystyle
f(n)+\sqrt{2(g(n)+f(n)+1)}+3/2.$
###### Proof.
Consider the following chain of inequalities, which uses the simple lower
bound in Theorem 12
$\displaystyle p([1+2+\cdots+m]-n;m-1)$ $\displaystyle\geq$ $\displaystyle
P([1+2+\cdots+m]-n)\geq\sqrt{2([1+2+\cdots+m]-n)}-1/2$ $\displaystyle\geq$
$\displaystyle\sqrt{2(g(n)+[f(n)+1]+[f(n)+2]+\cdots+m)}-1/2$
$\displaystyle\geq$ $\displaystyle\sqrt{2(g(n)+f(n)+1)}-1/2.$
Adding $m\geq f(n)+1$ to both sides proves the first inequality, and the
second is proven in the same way. ∎
###### Lemma 15.
Let $n$ and $m$ be positive integers with $m\geq f(n)$. Then we have
$\sigma([1+2+\cdots+m]-n;m-1)\geq 2f(n)-2.$
###### Proof.
We may assume $f(n)\geq 2$, or the claim is trivially true. Let
$A\subseteq\\{0,1,\ldots,m-1\\}$ be such that $vol(A)=[1+2+\cdots+m]-n$ and
$m-1\in A$. By way of contradiction, suppose that $per(A^{c})<2f(n)-2$.
####
If $m\geq 2f(n)-2$, then since $m-1\in\partial A$, this would imply that
$per(A^{c})\geq m\geq 2f(n)-2$. Therefore, we may assume that $m\leq 2f(n)-3$.
Now since $m\geq f(n)$, the volume of $A$ may be written as
$vol(A)[1+2+\cdots+m]-n=g(n)+[(f(n)+1)+(f(n)+2)+\cdots+m]<f(n)+[f(n)+1]+\cdots+m,$
and because $m\leq 2f(n)-3=[f(n)-2]+[f(n)-1]$, we also have
$vol(A)<[f(n)]+[f(n)+1]+\cdots+[m-1]+[f(n)-2]+[f(n)-1]=\sum_{i=f(n)-2}^{m-1}i.$
####
From this, we know that there is at least one element of
$\\{f(n)-2,f(n)-1,\ldots,m-2\\}$ that is not contained in $A$, because
otherwise the volume of $A$ would be too large. Let $l\in A^{c}$ be the
largest integer satisfying $f(n)-2\leq l\leq m-2$. Then since $m-1\in A$, we
know that $l\in\partial A^{c}$, which implies
$per(A^{c})\geq l+m\geq f(n)-2+m\geq f(n)-2+f(n)=2f(n)-2.$
But this contradicts the assumption that $per(A^{c})<2f(n)-2$, thus completing
the proof. ∎
####
With these lemmas, we are now able to prove the following lower bounds.
###### Theorem 16.
Let $P(n)$ and $Q(n)$ be as given. Then for all $n\geq 2$, we have
$\displaystyle P(n)$ $\displaystyle\geq$ $\displaystyle
f(n)+\min\Big{\\{}Q(g(n)),\sqrt{2(g(n)+f(n)+1)}+3/2,f(n)-2\Big{\\}}\qquad\qquad\text{and}$
$\displaystyle Q(n)$ $\displaystyle\geq$ $\displaystyle
1+f(n)+\min\Big{\\{}P(g(n)),\sqrt{2(g(n)+f(n)+1)}+1/2\Big{\\}}.$
###### Proof.
Starting with (9) and applying Lemmas 13, 14, and 15, we obtain
$\displaystyle P(n)$ $\displaystyle=$
$\displaystyle\min_{m>f(n)}\Big{\\{}f(n)+q(g(n);f(n)-2),m+q([1+2+\cdots+m]-n;m-2),$
$\displaystyle\qquad\qquad\sigma(g(n);f(n)-1),\sigma([1+2+\cdots+m]-n;m-1)\Big{\\}}$
$\displaystyle\geq$ $\displaystyle
f(n)+\min\Big{\\{}Q(g(n)),\sqrt{2(g(n)+f(n)+1)}+3/2,f(n)-2\Big{\\}}.$
The second inequality is proven analogously by starting with (10). ∎
### Squeezing an Equation from Inequalities (Eventually)
At this point, we have simple upper bounds on $P(n)$ and $Q(n)$ provided by
Theorem 10 and nearly simple lower bounds from Theorem 16, which are
complicated by the “min” operators. Suppose we could show that eventually
$P(g(n))$ and $Q(g(n))$ happen to be the smallest terms in each minimum. Then
our lower bounds would simplify drastically and our lower and upper bounds
would squeeze together, yielding a simple pair of mutually recursive equations
that would hold for all sufficiently large $n$.
####
As it turns out, we can in fact prove this claim, which is the content of the
following proposition:
###### Proposition 17.
Let $P(n)$ and $Q(n)$ be as given. Then there exists an $N\in\mathbb{Z}$ such
that for all $n\geq N$
$\displaystyle P(g(n))$ $\displaystyle=$
$\displaystyle\min\Big{\\{}P(g(n)),\sqrt{2(g(n)+f(n)+1)}+1/2\Big{\\}}\qquad\qquad\text{and}$
$\displaystyle Q(g(n))$ $\displaystyle=$
$\displaystyle\min\Big{\\{}Q(g(n)),\sqrt{2(g(n)+f(n)+1)}+3/2,f(n)-2\Big{\\}}.$
Moreover, these claims hold if we take $N$ to be $2,500,000$.
###### Proof.
We will first prove there is such an $N\in\mathbb{Z}$. Then we will discuss
why we may take $N$ to be $2,500,000$.
####
We need to show that eventually $P(g(n))\leq\sqrt{2(g(n)+f(n)+1)}+1/2$. From
Theorem 12, we know
$P(r)\leq\sqrt{2r}+o(\sqrt{r}).$
Therefore, there exists a constant $G$ such that for all $r\geq G$, we have
$P(r)\leq\sqrt{2r}+o(\sqrt{r})\leq\sqrt{4r}.$
From this, it follows that for all $n$, if $g(n)\geq G$, then we have
$P(g(n))\leq\sqrt{4g(n)}\leq\sqrt{2(g(n)+f(n)+1)}+1/2.$
####
Let $M$ be the maximum value taken by $P(k)$ for $0\leq k\leq G$, and let
$n\geq M^{2}(M^{2}+1)/2$ be arbitrary. Now if $g(n)\geq G$, then we know the
claim holds. Therefore, we can assume $g(n)<G$. But if this is the case, then
we know $P(g(n))\leq M$, which implies
$P(g(n))\leq M\leq\sqrt{f(n)}\leq\sqrt{2(g(n)+f(n)+1)}+1/2.$
####
Therefore, for all $n\geq M^{2}(M^{2}+1)/2=:N_{P}$, the first equation holds.
In the same way, we may find a constant $N_{Q}$ after which the second
inequality holds. Thus, taking $N:=\max\\{N_{P},N_{Q}\\}$ proves the existence
of such an integer $N$.
####
Now proving that we may in fact take $N$ to be $2,500,000$, follows from
somewhat lengthy but routine refinements of the previous argument. In the
above notation, the main idea is to first obtain any analytic upper bound on
$G$. This upper bound on $G$ is then refined by using computer calculated data
to compare $P(r)$ with $\sqrt{4r}$ to make $G$ as small as possible. Using
this technique for both $N_{P}$ and $N_{Q}$ then proves the claim. ∎ With this
proposition, we are able to prove our main result.
###### Theorem 18.
Let $P(n)$ and $Q(n)$ be as given. Then if $n\geq 0$ is not one of the $177$
known counterexamples tabulated in Table 1 of the appendix (in particular, for
all $n>149,894$), we have
$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle
f(n)+Q(g(n))\qquad\qquad\text{and}$ $\displaystyle Q(n)$ $\displaystyle=$
$\displaystyle 1+f(n)+P(g(n)),$
where as before, the functions $f(n)$ and $g(n)$, given by
$f(n):=\left\lfloor\sqrt{2n}+1/2\right\rfloor=\left[\sqrt{2n}\right],\qquad\qquad\text{and}\qquad\qquad
g(n):=\dfrac{f(n)[f(n)+1]}{2}-n,$
are also the smallest nonnegative integers satisfying
$[1+2+3+\cdots+f(n)]-g(n)=n$.
###### Proof.
If $n\geq 2,500,000$, then the result follows by using the previous
proposition to simplify the lower bounds of Theorem 16 and comparing these to
the upper bounds in Theorem 10.
####
On the other hand, if $0\leq n<2,500,000$, then the result holds by performing
an exhaustive computer seach for counterexamples555A brief discussion of the
algorithms used for this search is provided in Section 8. Code is available on
request.. There are only $177$ counterexamples in this range, as tabulated in
Table 1 of the appendix. In particular, if $n>149,894$, then the claim holds
since $149,894$ is the largest counterexample. ∎
### Corollaries and Remarks
There are many interesting implications of Theorem 18; from this result, many
things can be discovered about the behavior of $P(n)$ and $Q(n)$, and the
intimate connection between these two functions is made evident. Although
these results can be formulated simply as algebraic statements about the
recurrence relations, the corresponding geometric statements about the graphs
of these functions is perhaps more enlightening.
Figure 1: Graph of $P(n)$ (_red_) and $P(n)-f(n)=P(n)-[\sqrt{2n}]$ (_brown_)
####
Examining Figures 1 and 2 suggests several apparent patterns of the graphs of
these functions. For example, we see that the graphs $P(n)$ and $Q(n)$ are
each “drifting” upwards by a translation of $f(n)$. After compensating for
this drift, the patterns in the graphs become more apparent.
Figure 2: Graph of $Q(n)$ (blue) and $Q(n)-f(n)-1=Q(n)-[\sqrt{2n}]-1$ (green)
####
Now the curves $P(n)-f(n)$ and $Q(n)-f(n)-1$ (shown in brown and green
respectively) appear to be almost “periodic” in a sense, with zeroes at
$0,1,3,6,10,\ldots$. This apparent behavior is even more pronounced when the
values of these functions are laid out in the following triangular array
$\begin{tabular}[]{cccccc}\lx@intercol\hfil\text{$\\{a_{n}\\}_{n=0}^{\infty}$}\hfil\lx@intercol\\\
&&&&$a_{0}$\\\ &&&&$a_{1}$\\\ &&&$a_{2}$&$a_{3}$\\\
&&$a_{4}$&$a_{5}$&$a_{6}$\\\ &$a_{7}$&$a_{8}$&$a_{9}$&$a_{10}$\\\
$a_{11}$&$a_{12}$&$a_{13}$&$a_{14}$&$a_{15}$\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \end{tabular},\qquad\text{which yields
for
example}\qquad\begin{tabular}[]{cccccc}\lx@intercol\hfil\text{$\\{(f(n),g(n))\\}_{n=0}^{\infty}$}\hfil\lx@intercol\\\
&&&&$(0,0)$\\\ &&&&$(1,0)$\\\ &&&$(2,1)$&$(2,0)$\\\
&&$(3,2)$&$(3,1)$&$(3,0)$\\\ &$(4,3)$&$(4,2)$&$(4,1)$&$(4,0)$\\\
$(5,4)$&$(5,3)$&$(5,2)$&$(5,1)$&$(5,0)$\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \end{tabular}.$
####
Then arranging values in this triangular manner, we have
$\begin{tabular}[]{cccccccccc}\lx@intercol\hfil\text{$\\{P(n)-f(n)\\}_{n=0}^{\infty}$}\hfil\lx@intercol\\\
&&&&&&&&&0\\\ &&&&&&&&&0\\\ &&&&&&&&0&0\\\ &&&&&&&1&2&0\\\ &&&&&&2&3&2&0\\\
&&&&&3&3&4&2&0\\\ &&&&4&5&3&4&2&0\\\ &&&4&5&6&3&4&2&0\\\ &&6&4&5&6&3&4&2&0\\\
&7&7&4&5&6&3&4&2&0\\\
6&7&7&4&5&6&3&4&2&0\end{tabular}\qquad\qquad\qquad\begin{tabular}[]{cccccccccc}\lx@intercol\hfil\text{$\\{Q(n)-f(n)-1\\}_{n=0}^{\infty}$}\hfil\lx@intercol\\\
&&&&&&&&&-1\\\ &&&&&&&&&0\\\ &&&&&&&&1&0\\\ &&&&&&&2&1&0\\\ &&&&&&2&2&1&0\\\
&&&&&4&2&2&1&0\\\ &&&&5&4&2&2&1&0\\\ &&&3&5&4&2&2&1&0\\\ &&6&3&5&4&2&2&1&0\\\
&7&6&3&5&4&2&2&1&0\\\ 6&7&6&3&5&4&2&2&1&0\end{tabular}.$
Then it appears that the rows (read from right to left) of $\\{P(n)-f(n)\\}$
‘approach’ $0,2,4,3,6,5,4,7,7,6,\ldots$, and the rows of $\\{Q(n)-f(n)-1\\}$
‘approach’ $0,1,2,2,4,5,3,6,7,6,\ldots$. Moreover, these two sequences seem to
be just $\\{Q(n)\\}$ and $\\{P(n)\\}$, respectively. In fact, this follows as
our first corollary of Theorem 18:
###### Corollary 19.
Let $\\{P(n)-f(n)\\}_{n=0}^{\infty}$ and $\\{Q(n)-f(n)-1\\}_{n=0}^{\infty}$ be
arranged in the triangular manner previously discussed. Then unless $n$ is one
of the 177 counterexamples in Table 1 of the appendix, reading the rows of
$\\{P(n)-f(n)\\}$ from to right to left exactly agrees with $Q(t)$, and
reading the rows of $\\{Q(n)-f(n)-1\\}$ exactly agrees with $P(t)$.
###### Proof.
This follows immediately from Theorem 18 by how the triangular array was
constructed. ∎
####
Formulating this as a geometric statement is to say that except for 177
particular points, each “lump” in the graphs of $P(n)-f(n)$ and $Q(n)-f(n)-1$
is simply a reflection of a partial copy of $Q(n)$ or $P(n)$, respectively.
Thus, the graph of $P(n)$ eventually consists solely of “shifted” and
reflected partial copies of $Q(n)$, and similarly the graph of $Q(n)$
eventually consists solely of “shifted” and reflected partial copies of
$P(n)$. This mutual similarity of the two functions also induces self-
similarity as shown in the following results.
###### Corollary 20.
If $g(n)<f(n)-1$, and if $n$ and $n-f(n)$ are not one of the 177 values in
Table 1,
$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle
1+P(n-f(n))\qquad\qquad\text{and}$ $\displaystyle Q(n)$ $\displaystyle=$
$\displaystyle 1+Q(n-f(n)).$
###### Proof.
This follows from Theorem 18 and the fact that if $g(n)\neq f(n)-1$, then
$g(n)=g(n-f(n))$. ∎
This corollary is the statement that with a finite number of exceptions,
unless $n$ is one of the values at the far left of a row, then the value for
$n$ in the triangle for $\\{P(n)\\}_{n=0}^{\infty}$ (or in
$\\{Q(n)\\}_{n=0}^{\infty}$) is simply one more than the value directly above
that entry in the triangle.
###### Corollary 21.
If $n$ and $g(n)$ are not one of the 177 values listed in Table 1 of the
appendix (and in particular, if $g(n)>149,894$), then we have
$\displaystyle P(n)$ $\displaystyle=$ $\displaystyle
1+f(n)+f(g(n))+P(g^{2}(n))\qquad\qquad\text{and}$ $\displaystyle Q(n)$
$\displaystyle=$ $\displaystyle 1+f(n)+f(g(n))+Q(g^{2}(n)).$
###### Proof.
This follows immediately by applying Theorem 18 twice. ∎
####
This last recurrence is readily ‘solved’ yielding the following quasi-explicit
equations.
###### Proposition 22.
For all $n\geq 0$, let $\phi(n;149,894)=\phi(n)$ denote the smallest
nonnegative integer satisfying $g^{\phi(n)}(n)\leq 149,894$. Then for all
$n\geq 0$, we have
$\displaystyle P(n)$ $\displaystyle=$
$\displaystyle\begin{cases}P(g^{\phi(n)}(n))+\sum_{i=1}^{\phi(n)}f(g^{i-1}(n))+\phi(n)/2\qquad&\text{if
$\phi(n)$ is even}\\\
Q(g^{\phi(n)}(n))+\sum_{i=1}^{\phi(n)}f(g^{i-1}(n))+[\phi(n)-1]/2\qquad&\text{if
$\phi(n)$ is odd,}\end{cases}\qquad\qquad\text{and}$ $\displaystyle Q(n)$
$\displaystyle=$
$\displaystyle\begin{cases}Q(g^{\phi(n)}(n))+\sum_{i=1}^{\phi(n)}f(g^{i-1}(n))+\phi(n)/2\qquad&\text{if
$\phi(n)$ is even}\\\
P(g^{\phi(n)}(n))+\sum_{i=1}^{\phi(n)}f(g^{i-1}(n))+[\phi(n)+1]/2\qquad&\text{if
$\phi(n)$ is odd.}\end{cases}$
###### Proof.
This follows easily from the previous corollary. Although the function
$\phi(n)$ is much too elusive for most honest mathematicians to call these
equations truly “explicit”, they ought not be considered recursive. This is
because even though $P$ and $Q$ are referenced on the right-hand side, their
arguments are bounded; therefore, by appealing to Table 1, those terms are
effectively known. ∎
This gives rise to the following, perhaps surprising fact:
###### Corollary 23.
Let $P(n)$ and $Q(n)$ be as given. Then for all $n\geq 0$, we have
$-1\leq Q(n)-P(n)\leq 2.$
###### Proof.
For all $n\geq 0$, we can appeal to Proposition 22 to obtain that
$Q(n)-P(n)=\begin{cases}Q(g^{\phi(n)}(n))-P(g^{\phi(n)}(n)),\qquad&\text{if
$\phi(n)$ is even},\\\ P(g^{\phi(n)}(n))-Q(g^{\phi(n)}(n))+1,\qquad&\text{if
$\phi(n)$ is odd.}\end{cases}$
Moreover, for our purposes, we can assume that $\phi(n)$ is one of the 177
counterexamples tabulated in Table 1 or else we could continue to appeal to
Theorem 18 until this is the case. But looking at a table of these 177 values,
we see that if $k$ is one of those exceptions, then $0\leq Q(k)-P(k)\leq 2$,
which completes the proof. ∎
## 8 Conclusion
We conclude by discussing applications for computing $P(n)$ and $Q(n)$ and by
listing some open questions.
### “Sufficiently Large” and Computer Algorithms
In Proposition 17, we state results that hold for all sufficiently large
values of $n$ (in particular, for all $n\geq 2,500,000$). We then use this
result to prove Theorem 18, and we use a computer aided search to completely
classify all counterexamples, which brings up a brief discussion of
algorithms.
####
The most naïve approach to compute $P(n)$ would be simply to list all sets of
volume $n$ and find which has the smallest perimeter. This would require
roughly $\mathcal{O}\left(2^{n}\right)$ time and $\mathcal{O}\left(n\right)$
memory, which is much too slow for large $n$, and a different approach is
needed.
####
Using the recurrence relations in Section 4, dynamic programming enables us to
design algorithms for computing $P(n)$ and $Q(n)$ taking
$\mathcal{O}\left(n^{2}f(n)\right)=\mathcal{O}\left(n^{2.5}\right)$ time and
using $\mathcal{O}\left(n^{2}\right)$ memory. We can reduce this memory
requirement to roughly $\mathcal{O}\left(n\right)$ by employing a custom data
structure, which benefits from the fact that for fixed $n$, functions such as
$p(n;k)$ seem to take very few distinct values. Using these algorithms, the
author was able to check all values of $P(n)$ and $Q(n)$ for $n\leq
3,500,000$, which is more than enough to obtain the results of Theorem 18.
####
Now that we have proven the recurrences in Theorem 18 and Proposition 22, we
may use these to compute $P(n)$ or $Q(n)$ in
$\mathcal{O}\left(\Phi(n)\right)\leq\mathcal{O}\left(\log_{2}\log_{2}(n/2)\right)$
time using no additional memory. Moreover, we can compute a list of
$P(0),P(1),\ldots,P(n)$ [or $Q(0),Q(1),\ldots,Q(n)$] in
$\mathcal{O}\left(n\right)$ time using the required
$\mathcal{O}\left(n\right)$ memory.
####
Thus, one can now simply use Theorem 18 and the 177 values in Table 1 to
compute $P(n)$ and $Q(n)$ extremely quickly, and $P(n)$ and $Q(n)$ can be
tabulated essentially as far out as desired. The author is more than willing
to provide anyone interested with code and calculated results.
### Open Questions
There are several possible areas of future research. Because the function
$P(n)$ was first introduced so recently, this paper serves as a comprehensive
overview of all that is known.
* –
Little is known about the behavior of the functions $p(n;k)$, $q(n;k)$, and
$\sigma(n;k)$.
* –
It appears that for any fixed $n\leq 100,000$ the function $p(n;k)$ takes at
most two finite values as $k$ varies. This may be interesting and might be
proveable by focusing on Proposition 17.
* –
Very little or nothing whatsoever is known about $\phi(n;N)$ from Proposition
22.
* –
Characterizing sets for which $P(n)$ is obtained may be interesting. It seems
likely that the partitions used and the code developed in this paper would
help with that. Moreover, the result of Theorem 18 seems likely to help with
this.
* –
Providing more direct (i.e., less analytic) proofs for these results would
likely be quite enlightening.
* –
There seems to be no pattern or unifying properties for the 177
counterexamples tabulated in Table 1. Alternate proofs of the main results may
shed light on these seemingly sporadic values.
## References
* [1] Patrick Devlin. Sets with high volume and low perimeter. arXiv:1107.2954v1 [math.CO].
* [2] Steven J. Miller, Frank Morgan, Edward Newkirk, Lori Pedersen, and Deividas Seferis. Isoperimetric sets of integers. Mathematics Magazine, 2011.
## 9 Appendix
The 177 counterexamples to Theorem 18 are tabulated below. Entries of the form
(_123_) are not actually counterexamples to the theorem, and they are included
here only for completeness.
n | P(n) | Q(n)
---|---|---
0 | 0 | 0
2 | 2 | (_4_)
4 | 4 | (_6_)
7 | 6 | (_7_)
8 | 7 | (_7_)
11 | 8 | (_10_)
16 | 10 | (_12_)
17 | 11 | (_11_)
29 | 14 | (_15_)
92 | (_22_) | 23
125 | 25 | (_25_)
154 | 28 | 28
155 | 29 | (_29_)
174 | 29 | 29
361 | (_38_) | 38
390 | 39 | (_39_)
441 | (_42_) | 42
473 | 43 | 43
529 | (_46_) | 46
564 | 47 | 47
601 | 49 | (_50_)
637 | 49 | 49
704 | 54 | (_55_)
742 | 53 | 53
743 | 54 | 55
783 | 54 | 54
837 | (_53_) | 54
1003 | (_58_) | 59
1147 | 62 | 62
1184 | (_63_) | 64
1340 | 67 | 67
1341 | 68 | (_69_)
1380 | (_68_) | 69
1394 | 68 | 68
1548 | 72 | 72
1549 | 73 | (_74_)
1606 | 73 | 73
1665 | 74 | 74
1771 | 77 | 77
1772 | 78 | (_79_)
1833 | 78 | 78
1896 | 79 | 79
2173 | (_82_) | 82
2241 | 83 | 83
2279 | 86 | 86
n | P(n) | Q(n)
---|---|---
2508 | (_88_) | 88
2581 | 89 | 89
2867 | (_94_) | 94
2945 | 95 | 95
3250 | (_100_) | 100
3333 | 101 | 101
3336 | (_103_) | 104
3503 | 104 | 105
3588 | 104 | 104
3657 | (_106_) | 106
3745 | 107 | 107
3748 | (_109_) | 110
3925 | 110 | 111
4015 | 110 | 110
4016 | 111 | (_112_)
4107 | 111 | 111
4466 | 116 | 116
4467 | 117 | (_118_)
4563 | 117 | 117
4564 | 118 | (_119_)
4661 | 118 | 118
5186 | (_123_) | 124
5289 | 123 | 123
5806 | (_130_) | 131
5915 | 130 | 130
6026 | 131 | 131
6461 | (_137_) | 138
6576 | 137 | 137
6693 | 138 | 138
6811 | 139 | 139
7151 | (_144_) | 145
7272 | 144 | 144
7395 | 145 | 145
7396 | 146 | (_146_)
7436 | (_143_) | 143
7519 | 146 | 146
8003 | 151 | 151
8132 | 152 | 152
8133 | 153 | (_153_)
8262 | 153 | 153
8305 | (_151_) | 151
9222 | (_159_) | 159
9454 | 163 | 163
10086 | (_163_) | 164
10187 | (_167_) | 167
n | P(n) | Q(n)
---|---|---
10478 | 169 | (_169_)
11200 | (_175_) | 175
11245 | (_172_) | 173
11505 | 177 | (_177_)
12261 | (_183_) | 183
12467 | (_181_) | 182
12580 | 185 | (_185_)
12583 | (_187_) | 188
12904 | 188 | 189
13066 | 188 | 188
13370 | (_191_) | 191
13703 | 193 | (_193_)
13752 | (_190_) | 191
14041 | 196 | 197
14210 | 196 | 196
14381 | 197 | 197
15052 | 204 | (_205_)
15227 | 205 | (_206_)
15402 | 204 | 204
15403 | 205 | 206
15580 | 205 | 205
15759 | 206 | 206
16511 | (_208_) | 209
17254 | (_214_) | 215
17441 | 214 | 214
17985 | (_217_) | 218
18955 | 223 | 223
19152 | (_224_) | 224
19522 | (_226_) | 227
20532 | (_232_) | 232
20533 | 233 | (_234_)
20737 | 233 | 233
21122 | (_235_) | 236
21961 | (_241_) | 242
22172 | 241 | 241
22173 | 242 | (_243_)
22385 | 242 | 242
22654 | (_241_) | 241
22814 | 244 | (_244_)
23656 | (_250_) | 251
23875 | 250 | 250
23876 | 251 | (_252_)
24096 | 251 | 251
24541 | 253 | (_253_)
24598 | (_251_) | 251
n | P(n) | Q(n)
---|---|---
26855 | 262 | 262
28726 | (_271_) | 271
28783 | (_268_) | 269
28968 | 272 | 272
30910 | (_281_) | 281
31161 | 282 | 282
33174 | (_291_) | 291
33434 | 292 | 292
35518 | (_301_) | 301
35787 | 302 | 302
36391 | (_301_) | 302
37147 | 307 | 307
39125 | (_312_) | 313
39625 | 317 | 317
39626 | 318 | 319
39909 | 318 | 318
41958 | (_323_) | 324
44890 | (_334_) | 335
47921 | (_345_) | 346
50126 | 353 | 353
51051 | (_356_) | 357
53326 | 364 | 364
53327 | 365 | (_365_)
53655 | 365 | 365
56625 | 375 | 375
56626 | 376 | (_376_)
56964 | 376 | 376
61851 | (_389_) | 389
65764 | (_401_) | 401
66129 | 402 | (_402_)
69797 | (_413_) | 413
70173 | 414 | (_414_)
73950 | (_425_) | 425
74337 | 426 | (_426_)
78223 | (_437_) | 437
78621 | 438 | (_438_)
108375 | 510 | 510
114014 | 523 | 523
129359 | (_554_) | 554
136036 | (_568_) | 568
142881 | (_582_) | 582
149894 | (_596_) | 596
Table 1: Comprehensive list of exceptions to Theorem 18.
|
arxiv-papers
| 2012-02-07T02:02:16 |
2024-09-04T02:49:27.138462
|
{
"license": "Public Domain",
"authors": "Patrick Devlin",
"submitter": "Patrick Devlin",
"url": "https://arxiv.org/abs/1202.1331"
}
|
1202.1367
|
# Visualizing Communication on Social Media
Making Big Data Accessible
Karissa McKelvey, Alex Rudnick, Michael D. Conover, Filippo Menczer
Center for Complex Networks and Systems Research
Indiana University School of Informatics and Computing
{krmckelv, alexr, midconov, fil}@indiana.edu
###### Abstract
The broad adoption of the web as a communication medium has made it possible
to study social behavior at a new scale. With social media networks such as
Twitter, we can collect large data sets of online discourse. Social science
researchers and journalists, however, may not have tools available to make
sense of large amounts of data or of the structure of large social networks.
In this paper, we describe our recent extensions to Truthy, a system for
collecting and analyzing political discourse on Twitter. We introduce several
new analytical perspectives on online discourse with the goal of facilitating
collaboration between individuals in the computational and social sciences.
The design decisions described in this article are motivated by real-world use
cases developed in collaboration with colleagues at the Indiana University
School of Journalism.
###### keywords:
Visualization; Social Media; Communication Networks; HCID; Online Discourse;
Computational Social Science.
###### category:
H.5.3 Group and Organization Interfaces Computer-Supported Cooperative Work
††terms: Design; Human Factors
## 1 Introduction
Online social networking platforms provide a rich and detailed picture of
complex sociological phenomena at multiple scales. Recent studies have
demonstrated that digital trace data can be combined with sophisticated
statistical tools to produce insights into the behavior and interactivity
patterns of hundreds of thousands of individual actors [3]. Despite these
advances, computational techniques such as machine learning and large-scale
network analysis frequently remain beyond the reach of scholars trained in the
fields of communication and social science. Consequently, much of the research
on sociotechnical systems suffers a lack of theory-driven insight, and is
often limited to descriptive accounts of the phenomena under study. Clearly
there exists a need for technologies which bridge the gap between quantitative
and qualitative means of understanding of social systems.
We argue that interactive visualization tools are a natural solution to this
problem, as they surface large volumes of information about a system, allowing
users to make use of human vision in the development of theory-driven insight.
Information visualization pioneer Ben Schneiderman identifies several key
features of an information visualization tool. Such a tool should allow users
to gain an overview of the data under study, provide zoom & filtering
capabilities, item-level details-on-demand, allow users to see relationships
among items in a collection, and extract target data about specific subsets
within the collection [6].
To this end, we introduce an interactive dashboard for the study of
communication processes on the Twitter microblogging platform. Based on the
computational architecture of Truthy111http://truthy.indiana.edu, a system
designed to facilitate the tracking and detection of coordinated political
deception campaigns, we have created a platform to address the information
needs of researchers in the social sciences by providing real-time,
interactive visualizations of information diffusion processes on Twitter. Key
features of this tool include the ability to produce high-level statistical
and visual overviews of large-scale communication networks; filter for
discussions about specific topics; visualize relationships among users in core
communication networks; leverage statistical inferences about users’
influence, demographics and affect; and examine and extract details about
individuals and collections of users based on a variety of filtering criteria.
## 2 The Truthy Platform
The _Truthy_ system was originally designed to analyze and detect the
emergence of coordinated misinformation campaigns on Twitter [5, 2]. Now
tasked with the study of information diffusion in general, Truthy monitors a
real-time, high-throughput feed of 140-character messages known as tweets.
Truthy clusters tweets into groups of related messaged called ‘memes.’ Memes
typically correspond to discussion topics, communication channels, or
information resources shared among Twitter users. Here we outline the criteria
for the grouping of content into memes:
Hashtags
Hashtags are tokens used to identify the topic or intended audience of a
tweet. For example, #taxes or #occupy.
Mentions
A Twitter user can include another user’s screen name in a post, prefixed with
the @ symbol. These “mentions” are used to denote that a particular Twitter
user is being discussed or to address a post to that user.
Hyperlinks
We extract URLs from tweets by matching strings of valid URL characters that
begin with http://.
Phrases
Finally, we consider the entire text of the tweet itself to be a meme once all
Twitter metadata, punctuation, and URLs have been removed. Substrings of
tweets may also be matched.
As such, we define each meme as the set of all tweets containing a common
hashtag, mentioned username, hyperlink, or substring. Using these signifiers
as the atomic units of information transfer, we are able to create large-
scale, high-resolution models of information propagation dynamics. Though we
refer the reader to previous works for a detailed description of the Truthy
system, here we give a brief overview of some of the key analytical elements
afforded by the original framework.
For ease of navigation, collections of related memes are algorithmically
grouped into top-level categories (‘themes’) representing the most coarse-
grained level of analysis available on the Truthy platform. Within a given
theme, users can search for memes containing specific keywords or sort content
based on a variety of statistical features. Navigating a theme, users are
presented with a concise visual representation of each meme, characterized in
terms of a multiplex information diffusion network (Figure 1) and sparklines
representing activity levels over time.
Figure 1: Diffusion network associated with the #usa hashtag. Nodes represent
individual users and edges represent two types of tweets, either mentions
(orange) or retweets (blue).
At the level of an individual meme the user is presented with a high-
resolution image of the meme’s information diffusion network and a variety of
statistics about the activity and connectivity of users who have produced
content associated with that meme. These features include the number of users
and tweets, diffusion network statistics such as mean degree and large
connected component size, and user-specific statistics such as the most
retweeted user and the number of unique injection points. Additionally, users
can interact with a zoomable historical timeseries of activity volume and
produce animations of relevant meme-meme co-occurrence patterns.
Together these interface elements provide multiple, complementary perspectives
on the activity associated with clusters of related content. Despite the
benefits associated with the approach outlined above, this design is largely
geared towards providing high-level characterizations of each meme’s structure
and dynamics. Often, these broad accounts of the system’s behavior are
inappropriate for users trained in the qualitative analysis of social
dynamics. To address this shortcoming, we propose a series of design
affordances targeting users who seek to examine the data in an interactive
fashion. In doing so, we hope to provide a useful case study for researchers
interested in developing technologies that bridge the methodological divide
between qualitative and quantitative epistemologies.
## 3 Design Goals
The target users of the system we describe include social scientists,
political scientists, journalists and researchers engaged in the study of
communication. In order to clarify the requirements of this target audience,
we worked with colleagues at the Indiana University School of Journalism to
identify use cases that might be typical of an individual using this system
for research purposes. Below we describe a series of prototypical research
questions that motivate the design of these visualization interfaces.
1. 1.
Are there well-defined communication behaviors that characterize the
activities of influential actors?
2. 2.
What is the role of bridging users in facilitating information transfer
between ideologically opposed communities?
3. 3.
Which Twitter accounts act as opinion leaders, and how do they engage in
frame-making and agenda-setting?
These examples evoke a number of common themes that help to clarify the
requirements of our prototypical user. At the meme-wide level, users may tend
to be interested in the structural positions occupied by individual actors in
a given communication network. At the individual level, our target audience
requires fine-grained information, including access to measures of influence,
affect and ideology, information on users’ communication choices, and access
to detailed historical data on content production.
To address these requirements, we created interfaces containing several key
analytical components. These elements include an interactive layout of the
communication network shared among a meme’s most retweeted users and detailed
user-level metrics on activity volume, sentiment, inferred ideology, language,
communication channel choices, and a real-time feed of each individuals recent
activity. Additionally, we provide a filterable, searchable, and sortable
table-based interface that allows researchers to make rapid comparisons
between the statistical attributes associated with arbitrary sets of actors in
a given meme. Finally, we provide a novel mechanism to facilitate the
acquisition of user- and meme-level tweet content in a way that falls within
the activities permitted of the Twitter Terms of Service.
## 4 Interface Elements
### 4.1 Overview of Calculated Metrics
Here we provide an overview of the data users can expect to encounter while
using this system to explore a specific meme. For a finite subset of high-
profile users we compute a number of descriptive statistics including their
total tweets, retweets and mentions, probable language, affective sentiment
with respect to the meme, date of most recent activity, account creation date,
and for users engaged in discussions about U.S. Politics, their inferred
partisan affiliation.
Sentiment is calculated using OpinionFinder, a system that performs coarse-
grained subjectivity analysis by searching for substrings in a text to
identify phrases that express positive or negative sentiments [7]. We also
attempt to identify the dominant language of each Twitter user with the
Compact Language Detector library222http://code.google.com/p/chromium-compact-
language-detector/, which was developed by Google for use in the Chrome web
browser. This library makes use of character-level n-grams to estimate the
language of a given text. This may surface useful information; as Twitter is
used worldwide, we have anecdotally observed communities clustering along
language boundaries.
To infer the partisan leanings of individual users we apply machine-learning
techniques that leverage network and text features to make predictions of
political ideology. Originally developed by Conover et al., this technique
relies on the highly-clustered structure of a network of political retweets
and the hashtag usage of individuals in these clusters. Using a data set of
1,000 manually-annotated users, the authors found that membership in a
specific network cluster accurately predicted users’ political affiliation
with 87.3% accuracy. Additionally, the authors found that an individual’s
hashtag choices could be used to correctly predict political affiliation with
83.5% accuracy [1].
Here, we combine these two approaches to build a classification model that
allows for the prediction of the political affiliation of hundreds of
thousands of individuals. To this end we relied on a corpus containing all
tweets produced by the 18,470 individuals in either of the two retweet network
communities identified in [1]. After extracting hashtags from these tweets, we
trained a support vector machine to classify users as belonging to one of the
two politically homogeneous network clusters. Using training and testing sets
of 925 and 935 distinct users, respectively, randomly selected from the
network of political retweets we report that this SVM correctly predicts the
cluster association of each user with 85.0% accuracy. Consequently, assuming
that network cluster membership is predictive of political affiliation in
87.3% of cases (as demonstrated in [1]), we report an estimated overall
accuracy of 85.0*87.3=74.5%. We report the “partisanship” of each user in
terms of the confidence of the SVM-based prediction, as measured by the
distance of the user vector from the classification hyperplane. For users
below some tunable confidence threshold $\epsilon$, we refrain from reporting
a prediction. While a complete analysis of this methodology is beyond the
scope of this article, we propose that this inference may simply act as a
supplementary guide for users of the system, and does not replace prudent
case-by-case judgements.
### 4.2 Interactive Meme Diffusion Network
Visible in the right-most portion of Figure 2, the meme diffusion network
visualization addresses a key shortcoming of previous work by allowing users
to identify the accounts associated with specific nodes in the communication
network. As many of the motivating research questions dealt with the role of
influential users in these communication networks, we filter the original
layout (Figure 1) to include only the top twenty most retweeted users and
their neighbors. This promotes visual clarity and allows for a detailed
examination of the relationships between the individuals our algorithm
estimates to be influential in terms of broadcast reach.
Edges between nodes in this network represent at least one retweet event, and
the width of the line increases with the number of retweets between each pair
of users. Nodes in this network have an area that scales logarithmically with
out-degree, and in the context of themes about U.S. Political Discourse, take
a fill color according to their inferred partisan affiliation. As per the
requirements outlined above, a user can hover over any node in the network to
see the associated account name; clicking a node brings up an interface
detailing a number of features about that user.
The interactive meme diffusion network is created using _d3.js_ , a JavaScript
library that binds arbitrary data to a Document Object Model (DOM), and then
applies transformations using Scalable Vector Graphics
333http://mbostock.github.com/d3.
### 4.3 User Data Exploration Interface
In addition to the data exploration framework described above, we provide a
filterable, searchable, and sortable table based on the Google Chart Tools API
444http://code.google.com/apis/chart/interactive/docs/gallery/table.html. In
this interface (Figure 3), a data table is presented containing fields
corresponding to several of the properties described in the calculated metrics
section. This interface allows researchers to investigate the behavioral and
demographic characteristics of substantially larger collections of individuals
compared to the Interactive Meme Diffusion Network. For a given set of search
criteria, the entire statistical and demographic information contained in this
data table can be exported as comma-separated values for further study with
other analysis platforms.
### 4.4 Exporting Data and Statistics to Truthy Users
The Twitter Terms of Service prohibit services that provide direct access to
historical tweet content, instead requiring that requests for data be made
through the official Twitter API. One of the key demands of our users is the
ability to access and export historical tweet data for individuals and
collections of users. Consequently, we developed a novel approach to this
problem that allows users to download Twitter content by means of client-side
API calls. To this end, we generate locally-executable JavaScript that
automatically downloads tweets corresponding to specific memes or sets of
users from the Twitter API. These data are available in a variety of popular
file formats, such as .csv or Gephi network files.
## 5 Related Work
In The Guardian’s recent investigation of the UK riot rumors
555http://www.guardian.co.uk/uk/interactive/2011/dec/07/london-riots-twitter,
each meme was identified by hand, expanded by a parametrized Levenshtein
distance algorithm, and then independently coded by three sociology PhD
students. These visualizations are helpful in the study of the spread of
misinformation; however, this approach requires significant human labor and
thus is more difficult to scale.
Twitinfo 666http://www.twitinfo.csail.mit.edu is a website presenting research
on network analysis and visualizations of Twitter data. Its content is
collected in automatically identified “bursts” of tweets [4]. Twitinfo also
calculates the top tweeted URLs in each burst, and plots each tweet on a map,
colored according to sentiment. Twitinfo focuses on specific memes, identified
by the researchers, and is thus somewhat limited for users who might wish to
investigate arbitrary topics.
Ripples is a feature of Google’s social network, useful for visualizing the
spread of posts among users; Google+ has a reposting mechanism similar to
Twitter’s retweeting. When a user reposts content, Ripples tracks the
intermediate users along the diffusion path, information which is not
available via the Twitter API. Users are represented as colorful bubbles,
which recursively contain the intermediate users who have also shared the
post, and are scaled according to estimated influence.
## 6 Conclusion
We have outlined the structure of an information visualization platform for
the exploration of communication networks on the Twitter microblogging
platform. We motivate our design decisions in terms of traditional benchmarks
for effective interface design as well as in terms of insights gleaned from
close collaboration with journalism scholars actively engaged in research on
the role of social media and the public sphere. The product of this
collaboration is a visual analytics tool that allows non-technical users to
explore the results of large-scale computational and statistical analyses in
an intuitive and informative fashion.
As increasing amounts of data on online discourse and deliberation are
collected, we see a rising demand for technologies that make these data
intelligible to the broader research community. We envision that systems which
leverage in equal measure the strengths of the computational and social
sciences will act as one of the primary drivers of research innovation for
years to come. Work continues on the Truthy system, and in the near future, we
will extend the visualizations with temporal and geographic information. The
visualizations described in this paper are all currently available on the
Truthy website, and we encourage readers to explore them.
#### Acknowledgements.
We are grateful to Emily Metzgar and Hans Ibold for their invaluable insights
and support. This work is supported in part by NSF REU award CCF-1101743.
Figure 2: Interactive meme diffusion network visualization and user data
exploration interface for content associated with user @ronpaul. Figure 3:
Filterable, sortable, searchable data table for users associated with the
#tcot meme, a conservative communication channel.
## References
* [1] Michael D. Conover, Bruno Gonçalves, Jacob Ratkiewicz, Alessandro Flammini, and Filippo Menczer. Predicting the political alignment of twitter users. In Proceedings of the IEEE Third International Conference on Social Computing (SocialCom), pages 192–199. IEEE, October 2011.
* [2] Michael D. Conover, Jacob Ratkiewicz, Bruno Gonçalves, Alessandro Flammini, and Filippo Menczer. Political polarization on twitter. In International Conference on Weblogs and Social Media 2011, 2011\.
* [3] David Lazer, Alex Pentland, Lada Adamic, Sinan Aral, Albert-László Barabási, Devon Brewer, Nicholas Christakis, Noshir Contractor, James Fowler, Myron Gutmann, Tony Jebara, Gary King, Michael Macy, Deb Roy, and Marshall Van Alstyne. Computational social science. Science, 323(5915):721–723, 2009.
* [4] Adam Marcus, Michael S. Bernstein, Osama Badar, David R. Karger, Samuel Madden, and Robert C. Miller. Twitinfo: Aggregating and visualizing microblogs for event exploration. In ACM CHI Conference on Human Factors in Computing Systems, 2011\.
* [5] J. Ratkiewicz, M. D. Conover, M. Meiss, B. Goncalves, A. Flammini, and F. Menczer. Detecting and tracking political abuse in social media. In Fifth International AAAI Conference on Weblogs and Social Media, 2011.
* [6] B. Shneiderman. The eyes have it: A task by data type taxonomy for information visualizations. In Visual Languages, 1996. Proceedings., IEEE Symposium on, pages 336–343. IEEE, 1996.
* [7] Theresa Wilson, Paul Hoffmann, Swapna Somasundaran, Jason Kessler, Janyce Wiebe, Yejin Choi, Claire Cardie, Ellen Riloff, and Siddharth Patwardhan. Opinionfinder: a system for subjectivity analysis. In Proceedings of HLT/EMNLP on Interactive Demonstrations, HLT-Demo ’05, pages 34–35, Stroudsburg, PA, USA, 2005. Association for Computational Linguistics.
|
arxiv-papers
| 2012-02-07T08:05:05 |
2024-09-04T02:49:27.148738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Karissa McKelvey and Alex Rudnick and Michael D. Conover and Filippo\n Menczer",
"submitter": "Alex Rudnick",
"url": "https://arxiv.org/abs/1202.1367"
}
|
1202.1382
|
# Global well-posedness of 2D compressible Navier-Stokes equations with large
data and vacuum
Quansen Jiu,1,3 Yi Wang2,3 and Zhouping Xin3 The research is partially
supported by National Natural Sciences Foundation of China (No. 11171229) and
Project of Beijing Education Committee. E-mail: jiuqs@mail.cnu.edu.cnThe
research is partially supported by Hong Kong RGC Earmarked Research Grant
CUHK4042/08P and CUHK4041/11P, and National Natural Sciences Foundation of
China (No. 11171326). E-mail: wangyi@amss.ac.cn.The research is partially
supported by Zheng Ge Ru Funds, Hong Kong RGC Earmarked Research Grant
CUHK4042/08P and CUHK4041/11P, and a Focus Area Grant at The Chinese
University of Hong Kong. Email: zpxin@ims.cuhk.edu.hk
1 School of Mathematical Sciences, Capital Normal University, Beijing 100048,
P. R. China
2 Institute of Applied Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of
Mathematics, CAS, Beijing 100190, P. R. China
3The Institute of Mathematical Sciences, Chinese University of HongKong,
HongKong
Abstract: In this paper, we study the global well-posedness of the 2D
compressible Navier-Stokes equations with large initial data and vacuum. It is
proved that if the shear viscosity $\mu$ is a positive constant and the bulk
viscosity $\lambda$ is the power function of the density, that is,
$\lambda(\rho)=\rho^{\beta}$ with $\beta>3$, then the 2D compressible Navier-
Stokes equations with the periodic boundary conditions on the torus
$\mathbb{T}^{2}$ admit a unique global classical solution $(\rho,u)$ which may
contain vacuums in an open set of $\mathbb{T}^{2}$. Note that the initial data
can be arbitrarily large to contain vacuum states.
Key Words: compressible Navier-Stokes equations, density-dependent viscosity,
global well-posedness, vacuum,
## 1 Introduction
In this paper, we consider the following compressible and isentropic Navier-
Stokes equations with density-dependent viscosities
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{t}\rho+{\rm div}(\rho
u)=0,\\\ \partial_{t}(\rho u)+{\rm div}(\rho u\otimes u)+\nabla
P(\rho)=\mu\Delta u+\nabla((\mu+\lambda(\rho)){\rm div}u),&\quad
x\in\mathbb{T}^{2},t>0,\end{array}\right.$ (1.3)
where $\rho(t,x)\geq 0$, $u(t,x)=(u_{1},u_{2})(t,x)$ represent the density and
the velocity of the fluid, respectively. And $\mathbb{T}^{2}$ is the
2-dimensional torus $[0,1]\times[0,1]$ and $t\in[0,T]$ for any fixed $T>0$. We
denote the right hand side of $\eqref{CNS}_{2}$ by
$\mathcal{L}_{\rho}u=\mu\Delta u+\nabla((\mu+\lambda(\rho)){\rm div}u).$
Here, it is assumed that
$\mu={\rm const.}>0,\qquad\lambda(\rho)=\rho^{\beta},\quad\beta>3,$ (1.4)
such that the operator $\mathcal{L}_{\rho}$ is strictly elliptic.
Let the pressure function be given by
$\displaystyle P(\rho)=A\rho^{\gamma},$ (1.5)
where $\gamma>1$ denotes the adiabatic exponent and $A>0$ is the constant.
Without loss of generality, $A$ is normalized to be $1$. The initial values
are given by
$(\rho,u)(t=0,x)=(\rho_{0},u_{0})(x).$ (1.6)
Here the periodic boundary conditions on the unit torus $\mathbb{T}^{2}$ on
$(\rho,u)(t,x)$ are imposed to the system (1.3). This model problem,
(1.3)-(1.6), was first proposed by Vaigant-Kazhikhov in [51] where they showed
the well-posedness of the classical solution to this problem provided the
initial density is uniformly away from vacuum. In this paper, we study the
global well-posedness of the classical solution to this problem (1.3)-(1.6)
with general nonnegative initial densities.
There are extensive studies on global well-posedness of the compressible
Navier-Stokes equations in the case that both the shear and the bulk viscosity
are positive constants satisfying the physical restrictions. In particular,
the one-dimensional theory is rather satisfactory, see [20, 38, 33, 34] and
the references therein. In multi-dimensional case, the local well-posedness
theory of classical solutions to both initial-value and initial-boundary-value
problems was established by Nash [44], Itaya [26] and Tani [50] in the absence
of vacuum. The short time well-posedness of either strong or classical
solutions containing vacuum was studied recently by Cho-Kim [8] and Luo[40] in
3D and 2D case, respectively. In particular, Cho-Kim [8] obtained the short
existence and uniqueness of the classical solution to the Cauchy problem for
the isentropic CNS with general nonnegative initial density under the
assumption that the initial data satisfies a natural compatibility condition
[8]. One of the fundamental questions is whether these local (in time)
solutions can be extended globally in time. The first pioneering work along
this line is the well-known theory of Matsumura-Nishida [41], where they
obtained a unique global classical solution to the CNS in
$H^{s}(\mathbb{R}^{3})$ $(s\geq 3)$ for initial data close to its far field
state which is a non-vacuum equilibrium state, and furthermore, the solution
behaves diffusively toward the far field state. The proof in [41] consists of
elaborate energy estimates based on the dissipative structure of the CNS and
spectrum analysis for the linearized of CNS at the non-vacuum far field state.
This theory has been generalized to data with discontinuities by Hoff [18] and
data in Besov spaces by Danchin in [9]. It should be noted that this theory
[41, 18, 9] requires that the solution has small oscillations from the uniform
non-vacuum far field state so that the density is strictly away from the
vacuum uniformly in time. A natural and important long standing open problem
is whether a similar theory holds for the initial data containing vacuums. In
this direction, the major breakthrough is due to P. L. Lions [37], where he
obtained the existence of a renormalized weak solution with finite energy and
large initial data which can contain vacuums for the isentropic CNS when the
exponent $\gamma$ is suitably large, see also the refinements and
generalizations in [15, 29]. However, little is known on the structure,
regularity, and uniqueness of such a weak solution except the partial
regularity estimates for 2-dimensional periodic problems in Desjardins [10]
where a stronger estimate is obtained under the assumption of uniform
boundedness of the density. Recently, under some additional assumptions on the
viscosity coefficients, and the far fields state is a non-vacuum state, Hoff
[18, 19] obtained a new type of global weak solution with small total energy
for the isentropic CNS, which have extra structure and regularity information
(such as Lagrangian structure in the non-vacuum region) compared with the
renormalized weak solutions in [37, 15, 29]. However, the uniqueness and
regularity of those weak solutions whose existence has been proved in [37, 15,
29] remain completely open in general. By the weak-strong uniqueness of P. L.
Lions [37], this is equivalent to the problem of global (in time) well-
posedness of classical solution in the presence of vacuum. It should be
pointed out that this important question is a very difficult and subtle issue
since, in general, one would not expect a positive answer to this question due
to the finite time blow-up results of Xin in [52], where it is shown that in
the case that the initial density has compact support, any smooth solution to
the Cauchy problem of the CNS without heat conduction blows up in finite time
for any space dimension, see also the recent generalizations to the case for
non-compact but rapidly decreasing (at far fields) initial density [46]. The
mechanism for such a blow-up has also been investigated recently and various
blow-up criterion have been derived in [13, 14, 22, 23, 25, 48, 49]. More
recently, Huang-Li-Xin[24] proved the global well-posedness of classical
solutions with small energy but large oscillations which can contain vacuums
to 3D isentropic compressible Navier-Stokes equations. See also the recent
generalization to 3D full compressible Navier-Stokes equations [21], the
isentropic Navier-Stokes equations with potential forces [35], and 1D or
spherically symmetric isentropic Navier-Stokes equations with large initial
data [11, 12].
The case that the viscosity coefficients depend on the density and vanish at
the vacuum has received a lot attention recently, see [2, 3, 4, 5, 9, 17, 27,
28, 29, 30, 31, 32, 36, 39, 42, 43, 47, 53, 54, 55] and the references
therein. Liu, Xin and Yang first proposed in [39] some models of the
compressible Navier-Stokes equations with density-dependent viscosities to
investigate the dynamics of the vacuum. On the other hand, when deriving by
Chapman-Enskog expansions from the Boltzmann equation, the viscosity of the
compressible Navier-Stokes equations depends on the temperature and thus on
the density for isentropic flows. Also, the viscous Saint-Venant system for
the shallow water, derived from the incompressible Navier-Stokes equations
with a moving free surface, is expressed exactly as in (1.3) $N=2$,
$\mu=\rho$, $\lambda=0$, and $\gamma=2$ (see [16]). For the special case,
(1.4), the global well-posedness result of Vaigant-Kazhikhov [51] is the first
important surprising result for general large initial data with the only
constraint that it is initially away from vacuum. However, in the presence of
vacuum, there appear new mathematical challenges in dealing with such systems.
In particular, these systems become highly degenerate. The velocity cannot
even be defined in the presence of vacuum and hence it is difficult to get
uniform estimates for the velocity near vacuum. Substantial achievements have
been made for the one-dimensional case, such as both short time and long time
existence and uniqueness for the problem of a compact of viscous fluid expands
into vacuum with either stress free condition or continuity condition have
been established with $\mu=\rho^{\alpha}$ for suitable $\alpha$, see [39, 32,
53, 54] etc. Li-Li-Xin [36] recently proved the global existence of weak
solutions to the initial-boundary value problem for such a system on a finite
internal with general initial data which may contain vacuum and discovered the
phenomena that all the vacuum states must vanish in finite time and any smooth
solution blows up near the time of vacuum vanishing which are in sharp
contrast to the case of constant viscosity coefficients, which have been
extended to the Cauchy problem on $\mathbb{R}^{1}$ for arbitrary initial data
with a uniform non-vacuum state at far fields by Jiu-Xin [32]. In the case
that a basic nonlinear wave pattern is the rarefaction wave, whose nonlinear
asymptotic stability has been proved in [30, 31] for the one-dimensional
isentropic CNS system with density-dependent viscosity in the framework of
weak solutions even the rarefaction wave is connecting to the vacuum[38]. Note
also that in the case that the initial data is strictly away from vacuum,
Mellet and Vasseur has obtained the existence and uniqueness of global strong
solution to the one-dimensional Cauchy problem [43]. However, the progress is
very limited for multi-dimensional problems. Even the short time well-
posedness of classical solutions has not been established for such a system in
the presence of vacuum. The global existence of general weak solutions to the
compressible Navier-Stokes equations with density-dependent viscosities or the
viscous Saint-Venant system for the shallow water model in the multi-
dimensional case remains open, and one can refer to [4], [5], [17], [42] for
recent developments along this line. Note also that Zhang-Fang [55] proved the
existence of global weak solution with small energy to 2D Vaigant-Kazhikhov
model [51] in the framework of [19] and presented the vanishing vacuum
behavior. However, the uniqueness of this weak solution is open.
In this paper, we investigate the global existence of the classical solution
to 2-dimensional Vaigant-Kazhikhov model [51], that is, CNS system (1.3)-(1.6)
with periodic boundary condition and general nonnegative initial density. It
should be noted that for the 2-dimensional problem, the basic reformulation of
Vaigant-Kazhikhov [51] and the formulation in terms of the material derivative
used in [18, 24] are equivalent. Following some of the key ideas developed by
Vaigant-Kazhikhov [51], we are able to derive the uniform upper bound of the
density under the assumptions that the initial density is nonnegative. Then we
can derive the higher order estimates to the solution to guarantee the
existence of the global classical solution.
The main results of the present paper can be stated in the following.
###### Theorem 1.1
If the initial values $(\rho_{0},u_{0})(x)$ satisfy that
$0\leq(\rho_{0}(x),P(\rho_{0})(x))\in W^{2,q}(\mathbb{T}^{2})\times
W^{2,q}(\mathbb{T}^{2}),\quad u_{0}(x)\in
H^{2}(\mathbb{T}^{2}),\quad\int_{\mathbb{T}^{2}}\rho_{0}(x)dx>0$ (1.7)
for some $q>2$ and the compatibility condition
$\mathcal{L}_{\rho_{0}}u_{0}-\nabla P(\rho_{0})=\sqrt{\rho}_{0}g(x)$ (1.8)
with some $g\in L^{2}(\mathbb{T}^{2})$, then there exists a unique global
classical solution $(\rho,u)(t,x)$ to the compressible Navier-Stokes equations
(1.3)-(1.6) with
$\begin{array}[]{ll}\displaystyle 0\leq\rho(t,x)\leq
C,\quad\forall(t,x)\in[0,T]\times\mathbb{T}^{2},~{}~{}~{}~{}(\rho,P(\rho))(t,x)\in
C([0,T];W^{2,q}(\mathbb{T}^{2})),\\\ u\in C([0,T];H^{2}(\mathbb{T}^{2}))\cap
L^{2}(0,T;H^{3}(\mathbb{T}^{2})),~{}~{}\sqrt{t}u\in
L^{\infty}(0,T;H^{3}(\mathbb{T}^{2})),\\\ tu\in
L^{\infty}(0,T;W^{3,q}(\mathbb{T}^{2})),~{}~{}u_{t}\in
L^{2}(0,T;H^{1}(\mathbb{T}^{2}))\\\ \sqrt{t}u_{t}\in
L^{2}(0,T;H^{2}(\mathbb{T}^{2}))\cap
L^{\infty}(0,T;H^{1}(\mathbb{T}^{2})),~{}~{}tu_{t}\in
L^{\infty}(0,T;H^{2}(\mathbb{T}^{2})),\\\ \sqrt{t}\sqrt{\rho}u_{tt}\in
L^{2}(0,T;L^{2}(\mathbb{T}^{2})),~{}~{}t\sqrt{\rho}u_{tt}\in
L^{\infty}(0,T;L^{2}(\mathbb{T}^{2})),~{}~{}t\nabla u_{tt}\in
L^{2}(0,T;L^{2}(\mathbb{T}^{2})).\end{array}$ (1.9)
###### Remark 1.1
From the regularity of the solution $(\rho,u)(t,x)$, it can be shown that
$(\rho,u)$ is a classical solution of the system (1.3) in
$[0,T]\times\mathbb{T}^{2}$ (see the details in Section 5).
###### Remark 1.2
If the initial data contains vacuum, then it is natural to impose the
compatibility (1.8) as the case of constant viscosity coefficients in [8].
###### Remark 1.3
In Theorem 1.1, it is not clear whether or not $u_{tt}\in
L^{2}(0,T;L^{2}(\mathbb{T}^{2}))$ even though one has the regularity $t\nabla
u_{tt}\in L^{2}(0,T;L^{2}(\mathbb{T}^{2}))$.
###### Remark 1.4
It is open to get the similar theory to the Cauchy problem or the Dirichlet
problem to the 2D compressible Navier-Stokes equations (1.3).
If the initial values are much more regular, based on Theorem 1.1, we can
prove
###### Theorem 1.2
If the initial values $(\rho_{0},u_{0})(x)$ satisfy that
$0\leq(\rho_{0}(x),P(\rho_{0})(x))\in H^{3}(\mathbb{T}^{2})\times
H^{3}(\mathbb{T}^{2}),\quad u_{0}(x)\in
H^{3}(\mathbb{T}^{2}),\quad\int_{\mathbb{T}^{2}}\rho_{0}(x)dx>0$ (1.10)
and the compatibility condition (1.8), then there exists a unique global
classical solution $(\rho,u)(t,x)$ to the compressible Navier-Stokes equations
(1.3)-(1.6) satisfying all the properties listed in (1.9) in Theorem 1.1 with
any $2<q<\infty$. Furthermore, it holds that
$\begin{array}[]{ll}\displaystyle u\in
L^{2}(0,T;H^{4}(\mathbb{T}^{2})),~{}~{}(\rho,P(\rho))\in
C([0,T];H^{3}(\mathbb{T}^{2})),\\\ \displaystyle\rho u\in
C([0,T];H^{3}(\mathbb{T}^{2})),~{}~{}\sqrt{\rho}\nabla^{3}u\in
C([0,T];L^{2}(\mathbb{T}^{2})).\end{array}$ (1.11)
###### Remark 1.5
In fact, the conditions on the initial velocity $u_{0}$ can be weakened to
$u_{0}\in H^{2}(\mathbb{T}^{2})$ and $\sqrt{\rho_{0}}\nabla^{3}u_{0}\in
L^{2}(\mathbb{T}^{2})$ to get (1.11).
###### Remark 1.6
In Theorem 1.2, it is not clear whether or not $u\in
C([0,T];H^{3}(\mathbb{T}^{2}))$ even though one has $\rho u\in
C([0,T];H^{3}(\mathbb{T}^{2}))$.
###### Remark 1.7
It is noted that in Theorem 1.2, the compatibility condition (1.8) is exactly
same as in Theorem 1.1.
_Notations._ Throughout this paper, positive generic constants are denoted by
$c$ and $C$, which are independent of $\delta$, $m$ and $t\in[0,T]$, without
confusion, and $C(\cdot)$ stands for some generic constant(s) depending only
on the quantity listed in the parenthesis. For function spaces,
$L^{p}(\mathbb{T}^{2}),1\leq p\leq\infty$, denote the usual Lebesgue spaces on
$\mathbb{T}^{2}$ and $\|\cdot\|_{p}$ denotes its $L^{p}$ norm.
$W^{k,p}(\mathbb{T}^{2})$ denotes the $k^{th}$ order Sobolev space and
$H^{k}(\mathbb{T}^{2}):=W^{k,2}(\mathbb{T}^{2})$.
## 2 Preliminaries
As in [51], we introduce the following variables. First denote the effective
viscous flux by
$F=(2\mu+\lambda(\rho)){\rm div}u-P(\rho),$
and the vorticity by
$\omega=\partial_{x_{1}}u_{2}-\partial_{x_{2}}u_{1}.$
Also, we define that
$H=\frac{1}{\rho}(\mu\omega_{x_{1}}+F_{x_{2}}),\qquad\qquad
L=\frac{1}{\rho}(-\mu\omega_{x_{2}}+F_{x_{1}}).$
Then the momentum equation $\eqref{CNS}_{2}$ can be rewritten as
$\left\\{\begin{array}[]{ll}u_{1t}+u\cdot\nabla
u_{1}=\frac{1}{\rho}(-\mu\omega_{x_{2}}+F_{x_{1}})=L,\\\ u_{2t}+u\cdot\nabla
u_{2}=\frac{1}{\rho}(\mu\omega_{x_{1}}+F_{x_{2}})=H.\end{array}\right.$ (2.1)
Then the effective viscous flux $F$ and the vorticity $\omega$ solve the
following system:
$\left\\{\begin{array}[]{ll}\omega_{t}+u\cdot\nabla\omega+\omega{\rm
div}u=H_{x_{1}}-L_{x_{2}},\\\
(\frac{F+P(\rho)}{2\mu+\lambda(\rho)})_{t}+u\cdot\nabla(\frac{F+P(\rho)}{2\mu+\lambda(\rho)})+(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}=H_{x_{2}}+L_{x_{1}}.\end{array}\right.$
(2.2)
Due to the continuity equation $\eqref{CNS}_{1}$, it holds that
$\left\\{\begin{array}[]{ll}\omega_{t}+u\cdot\nabla\omega+\omega{\rm
div}u=H_{x_{1}}-L_{x_{2}},\\\ F_{t}+u\cdot\nabla
F-\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\\\
\qquad+(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]=(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}}).\end{array}\right.$
(2.3)
Furthermore, the system for $(H,L)$ can be derived as
$\left\\{\begin{array}[]{ll}\rho H_{t}+\rho u\cdot\nabla H-\rho H{\rm
div}u+u_{x_{2}}\cdot\nabla F+\mu u_{x_{1}}\cdot\nabla\omega+\mu(\omega{\rm
div}u)_{x_{1}}\\\
\qquad-\big{\\{}\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\big{\\}}_{x_{2}}\\\
\qquad+\big{\\{}(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]\big{\\}}_{x_{2}}\\\
\qquad=[(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})]_{x_{2}}+\mu(H_{x_{1}}-L_{x_{2}})_{x_{1}},\\\
\rho L_{t}+\rho u\cdot\nabla L-\rho L{\rm div}u+u_{x_{1}}\cdot\nabla F-\mu
u_{x_{2}}\cdot\nabla\omega-\mu(\omega{\rm div}u)_{x_{2}}\\\
\qquad-\big{\\{}\rho(2\mu+\lambda(\rho))[F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}]{\rm
div}u\big{\\}}_{x_{1}}\\\
\qquad+\big{\\{}(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]\big{\\}}_{x_{1}}\\\
\qquad=[(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})]_{x_{1}}-\mu(H_{x_{1}}-L_{x_{2}})_{x_{2}}.\end{array}\right.$
(2.4)
In the following, we will utilize the above systems in different steps. Note
that these systems are equivalent to each other for the smooth solution to the
original system (1.3).
Several elementary Lemmas are needed later. The first one is the Gagliardo-
Nirenberg inequality which can be found in [45].
###### Lemma 2.1
$\forall h\in W_{0}^{1,m}(\mathbb{T}^{2})$ or $h\in W^{1,m}(\mathbb{T}^{2})$
with $\displaystyle\int_{\mathbb{T}^{2}}hdx=0$, it holds that
$\|h\|_{q}\leq C\|\nabla h\|_{m}^{\alpha}\|h\|_{r}^{1-\alpha},$ (2.5)
where
$\alpha=(\frac{1}{r}-\frac{1}{q})(\frac{1}{r}-\frac{1}{m}+\frac{1}{2})^{-1}$,
and if $m<2,$ then $q$ is between $r$ and $\frac{2m}{2-m}$, that is,
$q\in[r,\frac{2m}{2-m}]$ if $r<\frac{2m}{2-m}$, $q\in[\frac{2m}{2-m},r]$ if
$r\geq\frac{2m}{2-m},$ if $m=2,$ then $q\in[r,+\infty)$, if $m>2$, then
$q\in[r,+\infty].$ Consequently, $\forall h\in W^{1,m}(\mathbb{T}^{2})$, one
has
$\|h\|_{q}\leq C(\|h\|_{1}+\|\nabla h\|_{m}^{\alpha}\|h\|_{r}^{1-\alpha}),$
(2.6)
where $C$ is a constant which may depend on $q$.
The following Lemma is the Poicare inequality.
###### Lemma 2.2
$\forall h\in W_{0}^{1,m}(\mathbb{T}^{2})$ or $h\in W^{1,m}(\mathbb{T}^{2})$
with $\displaystyle\int_{\mathbb{T}^{2}}hdx=0$, if $1\leq m<2,$ then
$\|h\|_{\frac{2m}{2-m}}\leq C(2-m)^{-\frac{1}{2}}\|\nabla h\|_{m},$ (2.7)
where the positive constant $C$ is independent of $m.$
The following Lemma follows from Lemma 2.2, of which proof can be found in
[51].
###### Lemma 2.3
$\forall h\in W^{1,\frac{2m}{m+\eta}}(\mathbb{T}^{2})$ with $m\geq 2$ and
$0<\eta\leq 1$, we have
$\|h\|_{2m}\leq
C(\|h\|_{1}+m^{\frac{1}{2}}\|h\|_{2(1-\varepsilon)}^{s}\|\nabla
h\|_{\frac{2m}{m+\eta}}^{1-s}),$ (2.8)
where
$\varepsilon\in[0,\frac{1}{2}],s=\frac{(1-\varepsilon)(1-\eta)}{m-\eta(1-\varepsilon)}$
and the positive constant $C$ is independent of $m.$
## 3 Approximate solutions
In this section, we construct a sequence of approximate solutions by making
use of the theory of Vaigant-Kazhikhov [51] and derive some uniform a-priori
estimates which are necessary to prove Theorem 1.1. To this end, we need a
careful approximation of the initial data.
Step 1. Approximation of initial data: To apply the theory of Vaigant-
Kazhikhov [51], we approximate of the initial data in (1.10) as follows.
First. the initial density and pressure can be approximated as
$\rho_{0}^{\delta}=\rho_{0}+\delta,\qquad P_{0}^{\delta}=P(\rho_{0})+\delta,$
(3.1)
for any small positive constant $\delta>0$. To approximate the initial
velocity, we define $u_{0}^{\delta}$ to be the unique solution to the
following elliptic problem
$\mathcal{L}_{\rho_{0}^{\delta}}u_{0}^{\delta}=\nabla
P_{0}^{\delta}+\sqrt{\rho_{0}}g$ (3.2)
with the periodic boundary conditions on $\mathbb{T}^{2}$ and
$\displaystyle\int_{\mathbb{T}^{2}}u_{0}^{\delta}dx=\int_{\mathbb{T}^{2}}u_{0}dx:=\bar{u}_{0}.$
It should be noted that $u_{0}^{\delta}$ is uniquely determined due to the
compatibility condition (1.8).
It follows from (3.2) that
$\mathcal{L}_{\rho_{0}}u_{0}^{\delta}=-\nabla\big{[}(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})){\rm
div}u_{0}^{\delta}\big{]}+\nabla P_{0}^{\delta}+\sqrt{\rho_{0}}g.$ (3.3)
By the elliptic regularity, it holds that
$\begin{array}[]{ll}\displaystyle\|u_{0}^{\delta}-\bar{u}_{0}\|_{H^{2}(\mathbb{T}^{2})}\\\
\displaystyle\leq
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{\infty}\|\nabla({\rm
div}u_{0}^{\delta})\|_{2}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{\infty}\|{\rm
div}u_{0}^{\delta}\|_{2}+\|\nabla
P_{0}^{\delta}\|_{2}+\|\sqrt{\rho_{0}}g\|_{2}\Big{]}\\\ \displaystyle\leq
C\Big{[}\delta\|u_{0}^{\delta}\|_{H^{2}(\mathbb{T}^{2})}+\|P_{0}\|_{H^{1}(\mathbb{T}^{2})}+\|\sqrt{\rho_{0}}\|_{L^{\infty}(\mathbb{T}^{2})}\|g\|_{2}\Big{]}\\\
\leq
C\Big{[}\delta\|u_{0}^{\delta}\|_{H^{2}(\mathbb{T}^{2})}+1\Big{]}.\end{array}$
(3.4)
where the generic positive constant $C$ is independent of $\delta>0.$
Therefore, if $\delta\ll 1$, then (3.4) yields that
$\|u_{0}^{\delta}\|_{H^{2}(\mathbb{T}^{2})}\displaystyle\leq C$ (3.5)
where the positive constant $C$ is independent of $0<\delta\ll 1.$
Due to the compatibility condition (1.8) and (3.2), it holds that
$\begin{array}[]{ll}\displaystyle\mathcal{L}_{\rho_{0}}(u_{0}^{\delta}-u_{0})&\displaystyle=-\nabla\big{[}(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})){\rm
div}u_{0}^{\delta}\big{]}:=\Theta^{\delta}.\end{array}$ (3.6)
Therefore, by the elliptic regularity, (3.1) and (3.5), one can get that
$\begin{array}[]{ll}\|u_{0}^{\delta}-u_{0}\|_{H^{2}(\mathbb{T}^{2})}\leq
C\|\Theta^{\delta}\|_{2}\\\ \qquad\displaystyle\leq\displaystyle
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{L^{\infty}(\mathbb{T}^{2})}\|\nabla^{2}u_{0}^{\delta}\|_{2}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{L^{\infty}(\mathbb{T}^{2})}\|{\rm
div}u_{0}^{\delta}\|_{2}\Big{]}\\\ \qquad\displaystyle\leq
C\Big{[}\|\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0})\|_{L^{\infty}(\mathbb{T}^{2})}+\|\nabla(\lambda(\rho_{0}^{\delta})-\lambda(\rho_{0}))\|_{L^{\infty}(\mathbb{T}^{2})}\Big{]}\\\
\qquad\displaystyle\leq C\delta~{}\rightarrow 0,\qquad{\rm
as}~{}~{}\delta\rightarrow 0.\end{array}$ (3.7)
For the initial data $(\rho_{0}^{\delta},P_{0}^{\delta},u_{0}^{\delta})$
constructed above for each fixed $\delta>0$, it is proved in [51] that the
compressible Navier-Stokes equations (1.3) with $\beta>3$ has a unique global
strong solution $(\rho^{\delta},u^{\delta})$ such that
$c_{\delta}\leq\rho^{\delta}\leq C_{\delta}$ for some positive constants
$c_{\delta},C_{\delta}$ depending on $\delta$. In the following, we will
derive the uniform bound to $(\rho^{\delta},u^{\delta})$ with respect to
$\delta$ and then pass the limit $\delta\rightarrow 0$ to get the classical
solution which may contain vacuum states in an open set of $\mathbb{T}^{2}$.
It should be noted that in comparison with estimates presented in [51], we
will obtain uniform estimates with respective to the lower bound of the
density such that vacuum is permitted in these estimates. To this end, the
compatibility condition (1.8) will be crucial.
For simplicity of notations, we will omit the superscript δ of
$(\rho^{\delta},u^{\delta})$ in the following in the case of no confusions.
Step 2. Elementary energy estimates:
###### Lemma 3.1
There exists a positive constant $C$ depending on $(\rho_{0},u_{0})$, such
that
$\sup_{t\in[0,T]}\big{(}\|\sqrt{\rho}u\|^{2}_{2}+\|\rho\|^{\gamma}_{\gamma}\big{)}+\int_{0}^{T}\big{(}\|\nabla
u\|_{2}^{2}+\|\omega\|_{2}^{2}+\|(2\mu+\lambda(\rho))^{\frac{1}{2}}{\rm
div}u\|_{2}^{2}\big{)}dt\leq C.$ (3.8)
Proof: Multiplying the equation $\eqref{ns1}_{i}$ by $\rho u_{i},(i=1,2)$,
summing the resulting equations and then integrating over $\mathbb{T}^{2}$ and
using the continuity equation $\eqref{CNS}_{1}$, it holds that
$\frac{d}{dt}\int\rho|u|^{2}dx+\int(\mu\omega^{2}+(2\mu+\lambda(\rho))({\rm
div}u)^{2})dx+\int u\cdot\nabla Pdx=0.$
Multiplying the continuity equation $\eqref{CNS}_{1}$ by
$\frac{1}{\gamma-1}\rho^{\gamma-1}$ and then integrating over $\mathbb{T}^{2}$
yields that
$\frac{d}{dt}\int\frac{\rho^{\gamma}}{\gamma-1}dx+\int P{\rm div}udx=0.$
Therefore, combining the above two estimates and then integrating over $[0,t]$
with respect to $t$, we obtain
$\begin{array}[]{ll}\displaystyle\int(\frac{1}{2}\rho|u|^{2}+\frac{1}{\gamma-1}\rho^{\gamma})dx+\int_{0}^{T}\int(\mu\omega^{2}+(2\mu+\lambda(\rho))({\rm
div}u)^{2})dxdt\\\
\displaystyle=\int(\frac{1}{2}\rho_{0}^{\delta}|u_{0}^{\delta}|^{2}+\frac{1}{\gamma-1}(\rho_{0}^{\delta})^{\gamma})dx\\\
\displaystyle\leq
C\Big{[}\|\rho_{0}^{\delta}\|_{H^{3}(\mathbb{T}^{2})}\|u_{0}^{\delta}\|^{2}_{H^{2}(\mathbb{T}^{2})}+\|\rho_{0}^{\delta}\|_{H^{3}(\mathbb{T}^{2})}^{\gamma}\Big{]}\leq
C.\end{array}$ (3.9)
Denote
$\phi(t)=\int(\mu\omega^{2}+(2\mu+\lambda(\rho))({\rm div}u)^{2})dx,\qquad
t\in[0,T].$ (3.10)
Then
$\|\nabla u\|_{2}^{2}(t)\leq C\big{[}\|\omega\|_{2}^{2}(t)+\|{\rm
div}u\|_{2}^{2}(t)\big{]}\leq C\phi(t)\in L^{1}(0,T).$
Thus the proof of Lemma 3.1 is completed. $\hfill\Box$
Step 3. Density estimates: Applying the operator $div$ to the momentum
equation $\eqref{CNS}_{2}$, we have
$[{\rm div}(\rho u)]_{t}+{\rm div}[{\rm div}(\rho u\otimes u)]=\Delta F.$
(3.11)
Consider the following two elliptic problems:
$\Delta\xi={\rm div}(\rho u),\qquad\int\xi dx=0,$ (3.12) $\Delta\eta={\rm
div}[{\rm div}(\rho u\otimes u)],\qquad\int\eta dx=0,$ (3.13)
both with the periodic boundary condition on the torus $\mathbb{T}^{2}.$
By the elliptic estimates and H${\rm\ddot{o}}$lder inequality, it holds that
###### Lemma 3.2
* (1)
$\|\nabla\xi\|_{2m}\leq Cm\|\rho\|_{\frac{2mk}{k-1}}\|u\|_{2mk},$ for any
$k>1,m\geq 1;$
* (2)
$\|\nabla\xi\|_{2-r}\leq
C\|\sqrt{\rho}u\|_{2}\|\rho\|^{\frac{1}{2}}_{\frac{2-r}{r}},$ for any $0<r<1;$
* (3)
$\|\eta\|_{2m}\leq Cm\|\rho\|_{\frac{2mk}{k-1}}\|u\|^{2}_{4mk},$ for any
$k>1,m\geq 1;$
where $C$ are positive constants independent of $m,k$ and $r$.
Proof: (1) By the elliptic estimates to the equation (3.12) and then using the
H${\rm\ddot{o}}$lder inequality, we have for any $k>1,m\geq 1$,
$\|\nabla\xi\|_{2m}\leq Cm\|\rho u\|_{2m}\leq
Cm\|\rho\|_{\frac{2mk}{k-1}}\|u\|_{2mk}.$
Similarly, the statements (2) and (3) can be proved. $\hfill\Box$
Based on Lemmas 2.1-2.3 and Lemma 3.2, it holds that
###### Lemma 3.3
* (1)
$\|\xi\|_{2m}\leq Cm^{\frac{1}{2}}\|\nabla\xi\|_{\frac{2m}{m+1}}\leq
Cm^{\frac{1}{2}}\|\rho\|_{m}^{\frac{1}{2}},$ for any $m\geq 2;$
* (2)
$\|u\|_{2m}\leq C\left[m^{\frac{1}{2}}\|\nabla u\|_{2}+1\right],$ for any
$m\geq 2;$
* (3)
$\|\nabla\xi\|_{2m}\leq
C\left[m^{\frac{3}{2}}k^{\frac{1}{2}}\|\rho\|_{\frac{2mk}{k-1}}\phi(t)^{\frac{1}{2}}+m\|\rho\|_{\frac{2mk}{k-1}}\right],$
for any $k>1,m\geq 1;$
* (4)
$\|\eta\|_{2m}\leq
C\left[m^{2}k\|\rho\|_{\frac{2mk}{k-1}}\phi(t)+m\|\rho\|_{\frac{2mk}{k-1}}\right],$
for any $k>1,m\geq 1;$
where $C$ are positive constants independent of $m,k$.
Proof: (1) By Lemma 2.2 and Lemma 3.2 (2), it holds that
$\begin{array}[]{ll}\displaystyle\|\xi\|_{2m}\leq
Cm^{\frac{1}{2}}\|\nabla\xi\|_{\frac{2m}{m+1}}&\displaystyle\leq
Cm^{\frac{1}{2}}\|\sqrt{\rho}u\|_{2}\|\rho\|_{m}^{\frac{1}{2}}\\\
&\displaystyle\leq Cm^{\frac{1}{2}}\|\rho\|_{m}^{\frac{1}{2}},\end{array}$
where in the last inequality one has used the elementary energy estimates
(3.9).
(2). From the conservative form of the compressible Navier-Stokes equations
(1.3) and the periodic boundary conditions, we have
$\frac{d}{dt}\int\rho(t,x)dx=\frac{d}{dt}\int\rho u(t,x)dx=0,$
that is,
$\int\rho(t,x)dx=\int\rho_{0}(x)dx,\qquad\int\rho
u(t,x)dx=\int\rho_{0}u_{0}(x)dx,\qquad\forall t\in[0,T].$
By Lemma 2.2, it follows that
$\|u\|_{2m}\leq\|u-\bar{u}\|_{2m}+\|\bar{u}\|_{2m}\leq
Cm^{\frac{1}{2}}\|\nabla u\|_{\frac{2m}{m+1}}+|\bar{u}|,$ (3.14)
where $m>2$ and $\bar{u}=\bar{u}(t)=\displaystyle\int u(t,x)dx.$
On the other hand, we have
$|\int\rho(u-\bar{u})dx|\leq\|\rho\|_{\gamma}\|u-\bar{u}\|_{\frac{\gamma}{\gamma-1}}\leq
C\|\nabla u\|_{2},$ (3.15)
where in the last inequality we have used the elementary energy estimates
(3.9) and the Poincare inequality.
Note that
$|\int\rho(u-\bar{u})dx|=|\int\rho_{0}u_{0}dx-\bar{u}\int\rho_{0}(x)dx|\geq|\bar{u}|\int\rho_{0}dx-|\int\rho_{0}u_{0}dx|.$
(3.16)
Combining (3.15) with (3.16) implies that
$|\bar{u}|\leq\frac{|\int\rho_{0}u_{0}dx|}{\int\rho_{0}dx}+\frac{C\|\nabla
u\|_{2}}{\int\rho_{0}dx}.$ (3.17)
Substituting (3.17) into (3.14) completes the proof of Lemma 3.3 (2).
The assertions (3) and (4) in Lemma 3.3 are direct consequences of Lemma 3.3
(2) and Lemma 3.2 (1), (3), respectively. Thus the proof of Lemma 3.3 is
completed. $\hfill\Box$
Substituting (3.12) and (3.13) into (3.11) yields that
$\Delta\Big{(}\xi_{t}+\eta-F+\int F(t,x)dx\Big{)}=0.$ (3.18)
Thus, it holds that
$\xi_{t}+\eta-F+\int F(t,x)dx=0.$ (3.19)
It follows from the definition of the effective viscous flux $F$ that
$\xi_{t}-(2\mu+\lambda(\rho)){\rm div}u+P(\rho)+\eta+\int F(t,x)dx=0.$ (3.20)
Then the continuity equation $\eqref{CNS}_{1}$ yields that
$\xi_{t}+\frac{2\mu+\lambda(\rho)}{\rho}(\rho_{t}+u\cdot\nabla\rho)+P(\rho)+\eta+\int
F(t,x)dx=0.$ (3.21)
Define
$\theta(\rho)=\int_{1}^{\rho}\frac{2\mu+\lambda(s)}{s}ds=2\mu\ln\rho+\frac{1}{\beta}(\rho^{\beta}-1).$
(3.22)
Then we obtain the following transport equation
$(\xi+\theta(\rho))_{t}+u\cdot\nabla(\xi+\theta(\rho))+P(\rho)+\eta-u\cdot\nabla\xi+\int
F(t,x)dx=0.$ (3.23)
###### Lemma 3.4
For any $k\geq 1,$ it holds that
$\sup_{t\in[0,T]}\|\rho(t,\cdot)\|_{k}\leq Ck^{\frac{2}{\beta-1}}.$ (3.24)
Proof: Multiplying the equation (3.23) by
$\rho[(\xi+\theta(\rho))_{+}]^{2m-1}$ with $m\geq 4$ being integer, here and
in what follows, the notation $(\cdots)_{+}$ denotes the positive part of
$(\cdots)$, one can get that
$\begin{array}[]{ll}\displaystyle\frac{1}{2m}\frac{d}{dt}\int\rho[(\xi+\theta(\rho))_{+}]^{2m}dx+\int\rho
P(\rho)[(\xi+\theta(\rho))_{+}]^{2m-1}dx=-\int\rho\eta[(\xi+\theta(\rho))_{+}]^{2m-1}dx\\\
\displaystyle+\int\rho u\cdot\nabla\xi[(\xi+\theta(\rho))_{+}]^{2m-1}dx-\int
F(t,x)dx\int\rho[(\xi+\theta(\rho))_{+}]^{2m-1}dx.\end{array}$ (3.25)
Denote
$f(t)=\big{\\{}\int\rho[(\xi+\theta(\rho))_{+}]^{2m}dx\big{\\}}^{\frac{1}{2m}},\qquad
t\in[0,T].$ (3.26)
Now we estimate the terms on the right hand side of (3.25). First,
$\begin{array}[]{ll}\displaystyle|-\int\rho\eta[(\xi+\theta(\rho))_{+}]^{2m-1}dx|\leq\int\rho^{\frac{1}{2m}}|\eta|\big{[}\rho(\xi+\theta(\rho))^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\qquad\displaystyle\leq\|\rho\|_{2m\beta+1}^{\frac{1}{2m}}\|\eta\|_{2m+\frac{1}{\beta}}\|\rho(\xi+\theta(\rho))^{2m}_{+}\|_{1}^{\frac{2m-1}{2m}}\\\
\qquad\displaystyle\leq
C\|\rho\|_{2m\beta+1}^{\frac{1}{2m}}\Big{[}(m+\frac{1}{2\beta})^{2}k\|\rho\|_{\frac{2(m+\frac{1}{2\beta})k}{k-1}}\phi(t)+(m+\frac{1}{2\beta})\|\rho\|_{\frac{2(m+\frac{1}{2\beta})k}{k-1}}\Big{]}f(t)^{2m-1}\\\
\qquad\displaystyle\leq
C\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}f(t)^{2m-1}\big{[}m^{2}\phi(t)+m\big{]},\end{array}$
(3.27)
where $\phi(t)$ is defined as in (3.10) and in the last inequality we have
taken $k=\frac{\beta}{\beta-1}.$
Next, for $\frac{1}{2m\beta+1}+\frac{1}{p}+\frac{1}{q}=1$ with $p,q\geq 1$,
one has
$\begin{array}[]{ll}\displaystyle|\int\rho
u\cdot\nabla\xi[(\xi+\theta(\rho))_{+}]^{2m-1}dx|\leq\int\rho^{\frac{1}{2m}}|u||\nabla\xi|\big{[}\rho(\xi+\theta(\rho))^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\qquad\displaystyle\leq\|\rho\|_{2m\beta+1}^{\frac{1}{2m}}\|u\|_{2mp}\|\nabla\xi\|_{2mq}\|\rho(\xi+\theta(\rho))^{2m}_{+}\|_{1}^{\frac{2m-1}{2m}}\\\
\qquad\displaystyle\leq
C\|\rho\|_{2m\beta+1}^{\frac{1}{2m}}\Big{[}(mp)^{\frac{1}{2}}\|\nabla
u\|_{2}+1\Big{]}\Big{[}(mq)^{\frac{3}{2}}k^{\frac{1}{2}}\|\rho\|_{\frac{2mqk}{k-1}}\phi(t)^{\frac{1}{2}}+m\|\rho\|_{\frac{2mqk}{k-1}}\Big{]}f(t)^{2m-1}\\\
\qquad\displaystyle\leq
C\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}f(t)^{2m-1}\big{[}m^{\frac{1}{2}}\phi(t)^{\frac{1}{2}}+1\big{]}\big{[}m^{\frac{3}{2}}\phi(t)^{\frac{1}{2}}+m\big{]}\\\
\qquad\displaystyle\leq
C\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}f(t)^{2m-1}\big{[}m^{2}\phi(t)+m\big{]},\end{array}$
(3.28)
where in the third inequality one has chosen $p=q=\frac{2m\beta+1}{m\beta}$
and $k=\frac{\beta}{\beta-1}.$
Then it follows that
$\begin{array}[]{ll}\displaystyle|-\int
F(t,x)dx\int\rho[(\xi+\theta(\rho))_{+}]^{2m-1}dx|\\\
\displaystyle\leq\int|(2\mu+\lambda(\rho)){\rm
div}u-P(\rho)|dx\int\rho^{\frac{1}{2m}}\big{[}\rho(\xi+\theta(\rho))^{2m}_{+}\big{]}^{\frac{2m-1}{2m}}dx\\\
\displaystyle\leq\Big{[}(\int(2\mu+\lambda(\rho))({\rm
div}u)^{2}dx)^{\frac{1}{2}}(\int(2\mu+\lambda(\rho))dx)^{\frac{1}{2}}+\int
P(\rho)dx\Big{]}\|\rho\|_{1}^{\frac{1}{2m}}\|\rho(\xi+\theta(\rho))^{2m}_{+}\|_{1}^{\frac{2m-1}{2m}}\\\
\displaystyle\leq
C\Big{[}\phi(t)^{\frac{1}{2}}+\phi(t)^{\frac{1}{2}}(\int\rho^{\beta}dx)^{\frac{1}{2}}+1\Big{]}f(t)^{2m-1}\\\
\displaystyle\leq
C\Big{[}\phi(t)^{\frac{1}{2}}+\phi(t)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}+1\Big{]}f(t)^{2m-1}.\end{array}$
(3.29)
Substituting (3.27), (3.28) and (3.29) into (3.25) yields that
$\begin{array}[]{ll}\displaystyle\frac{1}{2m}\frac{d}{dt}(f^{2m}(t))+\int\rho
P(\rho)[(\xi+\theta(\rho))_{+}]^{2m-1}dx\\\ \displaystyle\leq
C\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}f(t)^{2m-1}\big{[}m^{2}\phi(t)+m\big{]}+C\Big{[}\phi(t)^{\frac{1}{2}}+\phi(t)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}+1\Big{]}f(t)^{2m-1}.\end{array}$
(3.30)
Then it holds that
$\frac{d}{dt}f(t)\leq
C\Big{[}1+\phi(t)^{\frac{1}{2}}+\phi(t)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}+\big{(}m^{2}\phi(t)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}\Big{]}.$
(3.31)
Integrating the above inequality over $[0,t]$ gives that
$f(t)\leq
f(0)+C\Big{[}1+\int_{0}^{t}\phi(\tau)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}(\tau)d\tau+\int_{0}^{t}\big{(}m^{2}\phi(\tau)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}(\tau)d\tau\Big{]}.$
(3.32)
Now we calculate the quantity
$f(0)=\Big{(}\int\rho^{\delta}_{0}[(\xi^{\delta}_{0}+\theta(\rho^{\delta}_{0}))_{+}]^{2m}dx\Big{)}^{\frac{1}{2m}}.$
By Lemma 3.2 (1) with $t=0$, we can easily get
$\|\xi^{\delta}_{0}\|_{L^{\infty}}\leq C.$
Furthermore, by the definition of
$\theta(\rho^{\delta}_{0})=2\mu\ln\rho^{\delta}_{0}+\frac{1}{\beta}((\rho^{\delta}_{0})^{\beta}-1)$,
we have
$\xi^{\delta}_{0}+\theta(\rho^{\delta}_{0})\rightarrow-\infty,\quad{\rm
as}\quad\rho^{\delta}_{0}\rightarrow 0+.$
Thus there exists a positive constant $\sigma$, such that if
$0\leq\rho^{\delta}_{0}\leq\sigma$, then
$(\xi^{\delta}_{0}+\theta(\rho^{\delta}_{0}))_{+}\equiv 0.$
Now one has
$\begin{array}[]{ll}f(0)&\displaystyle=\Big{[}\Big{(}\int_{[0\leq\rho_{0}\leq\sigma]}+\int_{[\sigma\leq\rho^{\delta}_{0}\leq
M]}\Big{)}\rho^{\delta}_{0}(\xi^{\delta}_{0}+\theta(\rho^{\delta}_{0}))_{+}^{2m}dx\Big{]}^{\frac{1}{2m}}\\\
&\displaystyle=\Big{[}\int_{[\sigma\leq\rho^{\delta}_{0}\leq
M]}\rho^{\delta}_{0}(\xi^{\delta}_{0}+\theta(\rho^{\delta}_{0}))_{+}^{2m}dx\Big{]}^{\frac{1}{2m}}\leq
C(\sigma,M),\end{array}$ (3.33)
where the positive constant $C(\sigma,M)$ is independent of $\delta$ and $m$.
It follows from (3.32) and (3.33) that
$f(t)\leq
C\Big{[}1+\int_{0}^{t}\phi(\tau)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}(\tau)d\tau+\int_{0}^{t}\big{(}m^{2}\phi(\tau)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}(\tau)d\tau\Big{]}.$
(3.34)
Set $\Omega_{1}(t)=\\{x\in\mathbb{T}^{2}|\rho(t,x)>2\\}$ and
$\Omega_{2}(t)=\\{x\in\Omega_{1}(t)|\xi(t,x)+\theta(\rho)(t,x)>0\\}$. Then one
has
$\begin{array}[]{ll}&\displaystyle\|\rho\|_{2m\beta+1}^{\beta}(t)=\Big{(}\int\rho^{2m\beta+1}dx\Big{)}^{\frac{\beta}{2m\beta+1}}=\Big{(}\int_{\Omega_{1}(t)}\rho^{2m\beta+1}dx+\int_{\mathbb{T}^{2}\setminus\Omega_{1}(t)}\rho^{2m\beta+1}dx\Big{)}^{\frac{\beta}{2m\beta+1}}\\\
&\displaystyle~{}~{}\leq\Big{(}\int_{\Omega_{1}(t)}\rho^{2m\beta+1}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\leq
C\Big{(}\int_{\Omega_{1}(t)}\rho|\theta(\rho)|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
&\displaystyle~{}~{}=C\Big{(}\int_{\Omega_{2}(t)}\rho|\theta(\rho)+\xi-\xi|^{2m}dx+\int_{\Omega_{1}(t)\setminus\Omega_{2}(t)}\rho|\theta(\rho)|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
&\displaystyle~{}~{}\leq
C\Big{(}\int_{\Omega_{2}(t)}\rho(\theta(\rho)+\xi)^{2m}dx+\int_{\Omega_{2}(t)}\rho|\xi|^{2m}dx+\int_{\Omega_{1}(t)\setminus\Omega_{2}(t)}\rho|\xi|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\\\
&\displaystyle~{}~{}\leq
C\Big{(}f(t)^{2m}+\int_{\mathbb{T}^{2}}\rho|\xi|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+C\leq
C\Big{[}f(t)+\Big{(}\int_{\mathbb{T}^{2}}\rho|\xi|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}+1\Big{]}.\end{array}$
(3.35)
Note that
$\begin{array}[]{ll}\displaystyle\Big{(}\int_{\mathbb{T}^{2}}\rho|\xi|^{2m}dx\Big{)}^{\frac{\beta}{2m\beta+1}}&\displaystyle\leq\|\rho\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\||\xi|^{2m}\|^{\frac{\beta}{2m\beta+1}}_{\frac{2m\beta+1}{2m\beta}}=\|\rho\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\|\xi\|^{\frac{2m\beta}{2m\beta+1}}_{2m+\frac{1}{\beta}}\\\
&\displaystyle\leq\|\rho\|_{2m\beta+1}^{\frac{\beta}{2m\beta+1}}\Big{[}C(m+\frac{1}{2\beta})^{\frac{1}{2}}\|\rho\|_{m+\frac{1}{2\beta}}^{\frac{1}{2}}\Big{]}^{\frac{2m\beta}{2m\beta+1}}\leq
Cm^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta(m+1)}{2m\beta+1}},\end{array}$
(3.36)
Then one can get
$\begin{array}[]{ll}\|\rho\|_{2m\beta+1}^{\beta}(t)&\displaystyle\leq
C\Big{[}1+f(t)+m^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta(m+1)}{2m\beta+1}}(t)\Big{]}\\\
&\displaystyle\leq\frac{1}{2}\|\rho\|_{2m\beta+1}^{\beta}(t)+C\Big{(}1+f(t)+m^{\frac{m\beta+\frac{1}{2}}{m(2\beta-1)}}\Big{)}.\end{array}$
(3.37)
Thus it holds that
$\begin{array}[]{ll}\|\rho\|_{2m\beta+1}^{\beta}(t)&\displaystyle\leq
C\Big{[}f(t)+m^{\frac{\beta}{2\beta-1}}\Big{]}\\\ &\displaystyle\leq
C\Big{[}m^{\frac{\beta}{2\beta-1}}+\int_{0}^{t}\phi(\tau)^{\frac{1}{2}}\|\rho\|_{2m\beta+1}^{\frac{\beta}{2}}(\tau)d\tau+\int_{0}^{t}\big{(}m^{2}\phi(\tau)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}(\tau)d\tau\Big{]}\\\
&\displaystyle\leq
C\Big{[}m^{\frac{\beta}{2\beta-1}}+\int_{0}^{t}\|\rho\|_{2m\beta+1}^{\beta}(\tau)d\tau+\int_{0}^{t}\big{(}m^{2}\phi(\tau)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}(\tau)d\tau\Big{]}.\end{array}$
(3.38)
Applying Gronwall’s inequality yields that
$\|\rho\|_{2m\beta+1}^{\beta}(t)\leq
C\Big{[}m^{\frac{\beta}{2\beta-1}}+\int_{0}^{t}\big{(}m^{2}\phi(\tau)+m\big{)}\|\rho\|_{2m\beta+1}^{1+\frac{1}{2m}}(\tau)d\tau\Big{]}.$
(3.39)
Denote
$y(t)=m^{-\frac{2}{\beta-1}}\|\rho\|_{2m\beta+1}(t).$
Then it holds that
$\begin{array}[]{ll}\displaystyle y^{\beta}(t)&\displaystyle\leq
C\Big{[}m^{\frac{\beta(1-3\beta)}{(2\beta-1)(\beta-1)}}+m^{\frac{1}{m(\beta-1)}}\int_{0}^{t}\phi(\tau)y(\tau)^{1+\frac{1}{2m}}d\tau+m^{\frac{1}{m(\beta-1)}-1}\int_{0}^{t}y(\tau)^{1+\frac{1}{2m}}d\tau\Big{]}\\\
&\displaystyle\leq
C\Big{[}1+\int_{0}^{t}(\phi(\tau)+1)y^{\beta}(\tau)d\tau\Big{]}.\end{array}$
So applying the Gronwall’s inequality again yields that
$y(t)\leq C,\quad\forall t\in[0,T],$
that is,
$\|\rho\|_{2m\beta+1}(t)\leq Cm^{\frac{2}{\beta-1}},\quad\forall t\in[0,T].$
Equivalently, (3.24) holds. Thus Lemma 3.4 is proved. $\hfill\Box$
Step 4: First-order derivative estimates of the velocity.
###### Lemma 3.5
There exists a positive constant $C$, such that
$\sup_{t\in[0,T]}\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx+\int_{0}^{T}\int\rho(H^{2}+L^{2})dxdt\leq
C.$ (3.40)
Proof: Multiplying the equation $\eqref{F-omega}_{1}$ by $\mu\omega$, the
equation $\eqref{F-omega}_{2}$ by $\frac{F}{2\mu+\lambda(\rho)}$,
respectively, and then summing the resulted equations together, one has
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx+\frac{\mu}{2}\int\omega^{2}{\rm
div}udx-\frac{1}{2}\int\rho F^{2}(\frac{1}{2\mu+\lambda(\rho)})^{\prime}{\rm
div}udx\\\ \displaystyle\quad-\frac{1}{2}\int F^{2}\frac{{\rm
div}u}{2\mu+\lambda(\rho)}dx-\int\rho F({\rm
div}u)(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}dx+\int
F[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]dx\\\
\displaystyle\quad=-\int\rho(H^{2}+L^{2})dx.\end{array}$ (3.41)
Notice that
$\begin{array}[]{ll}\displaystyle(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}&\displaystyle=(u_{1x_{1}}+u_{2x_{2}})^{2}+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})\\\
&\displaystyle=({\rm
div}u)^{2}+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})\\\
&\displaystyle=({\rm
div}u)\left(\frac{F}{2\mu+\lambda(\rho)}+\frac{P(\rho)}{2\mu+\lambda(\rho)}\right)+2(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}}),\end{array}$
then one has
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx+\int\rho(H^{2}+L^{2})dx=-\frac{\mu}{2}\int\omega^{2}{\rm
div}udx\\\ \displaystyle\quad+\frac{1}{2}\int F^{2}({\rm
div}u)\Big{[}\rho(\frac{1}{2\mu+\lambda(\rho)})^{\prime}-\frac{1}{2\mu+\lambda(\rho)}\Big{]}dx+\int
F({\rm
div}u)\Big{[}\rho(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}-\frac{P(\rho)}{2\mu+\lambda(\rho)}\Big{]}dx\\\
\displaystyle\quad-\int
2F(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})dx.\end{array}$ (3.42)
Set
$Z^{2}(t)=\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx,$
and
$\varphi^{2}(t)=\int\rho(H^{2}+L^{2})dx=\int\frac{1}{\rho}\big{[}(\mu\omega_{x_{1}}+F_{x_{2}})^{2}+(-\mu\omega_{x_{2}}+F_{x_{1}})^{2}\big{]}dx.$
Then it follows that for $0<r\leq\frac{1}{2},$
$\displaystyle\|\nabla(F,\omega)\|_{2(1-r)}\displaystyle\leq
C\varphi(t)\|\rho\|_{\frac{1-r}{r}}^{\frac{1}{2}}\leq
C\varphi(t)(\frac{1-r}{r})^{\frac{1}{\beta-1}}\leq
C\varphi(t)r^{\frac{1}{1-\beta}},$ (3.43)
and
$\begin{array}[]{ll}\displaystyle\|\nabla u\|_{2}+\|\omega\|_{2}+\|{\rm
div}u\|_{2}+\|(2\mu+\lambda(\rho))^{\frac{1}{2}}{\rm div}u\|_{2}\\\
\displaystyle\leq
C\Big{[}Z(t)+\Big{(}\int\frac{P(\rho)^{2}}{2\mu+\lambda(\rho)}dx\Big{)}^{\frac{1}{2}}\Big{]}\leq
C(Z(t)+1).\end{array}$ (3.44)
Now we estimate the four terms on the right hand side of (3.42). First, by the
interpolation inequality and Lemma 2.2, (3.43) and (3.44), for
$0<\varepsilon\leq\frac{1}{4}$, it holds that
$\begin{array}[]{ll}\displaystyle|-\frac{\mu}{2}\int\omega^{2}{\rm
div}udx|\leq C\|{\rm div}u\|_{2}\|\omega\|_{4}^{2}\leq
C(Z(t)+1)\|\omega\|_{2}^{\frac{1-3\varepsilon}{1-2\varepsilon}}\|\nabla\omega\|_{2(1-\varepsilon)}^{\frac{1-\varepsilon}{1-2\varepsilon}}\\\
\displaystyle\qquad\qquad\leq
C(Z(t)+1)Z(t)^{\frac{1-3\varepsilon}{1-2\varepsilon}}\varphi(t)^{\frac{1-\varepsilon}{1-2\varepsilon}}\varepsilon^{\frac{1-\varepsilon}{(1-\beta)(1-2\varepsilon)}}\\\
\displaystyle\qquad\qquad\leq\alpha\varphi^{2}(t)+C_{\alpha}Z(t)^{2}(Z(t)+1)^{\frac{2(1-2\varepsilon)}{1-3\varepsilon}}\varepsilon^{\frac{2}{1-\beta}\frac{1-\varepsilon}{1-3\varepsilon}}\\\
\displaystyle\qquad\qquad\leq\alpha\varphi^{2}(t)+C_{\alpha}(Z(t)^{2}+1)^{2+\frac{\varepsilon}{1-3\varepsilon}}\varepsilon^{\frac{2}{1-\beta}\frac{1-\varepsilon}{1-3\varepsilon}},\end{array}$
(3.45)
where and in the sequel $\alpha>0$ is a small positive constant to be
determined and $C_{\alpha}$ is a positive constant depending on $\alpha$.
Next, one has
$\begin{array}[]{ll}\displaystyle|\frac{1}{2}\int F^{2}{\rm
div}u\Big{[}\rho(\frac{1}{2\mu+\lambda(\rho)})^{\prime}-\frac{1}{2\mu+\lambda(\rho)}\Big{]}dx|\\\
\displaystyle=|\frac{1}{2}\int
F^{2}\left(\frac{F}{2\mu+\lambda(\rho)}+\frac{P(\rho)}{2\mu+\lambda(\rho)}\right)\frac{2\mu+\lambda(\rho)+\rho\lambda^{\prime}(\rho)}{(2\mu+\lambda(\rho))^{2}}dx|\\\
\displaystyle\leq
C\int|F|^{2}\left(\frac{|F|}{2\mu+\lambda(\rho)}+\frac{P(\rho)}{2\mu+\lambda(\rho)}\right)dx\leq
C\left(1+\int\frac{|F|^{3}}{2\mu+\lambda(\rho)}dx\right),\end{array}$ (3.46)
and
$\begin{array}[]{ll}\displaystyle|\frac{1}{2}\int F{\rm
div}u\Big{[}\rho(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}-\frac{P(\rho)}{2\mu+\lambda(\rho)}\Big{]}dx|\\\
\displaystyle=|\frac{1}{2}\int
F\left(\frac{F}{2\mu+\lambda(\rho)}+\frac{P(\rho)}{2\mu+\lambda(\rho)}\right)\frac{P(\rho)(2\mu+\lambda(\rho))+\rho\lambda^{\prime}(\rho)P(\rho)-\rho
P^{\prime}(\rho)(2\mu+\lambda(\rho))}{(2\mu+\lambda(\rho))^{2}}dx|\\\
\displaystyle\leq
C\int|F|\left(\frac{|F|}{2\mu+\lambda(\rho)}+\frac{P(\rho)}{2\mu+\lambda(\rho)}\right)P(\rho)dx\displaystyle\leq
C\left(1+\int\frac{|F|^{3}}{2\mu+\lambda(\rho)}dx\right).\end{array}$ (3.47)
On the other hand, it holds that
$|-\int 2F(u_{1x_{2}}u_{2x_{1}}-u_{1x_{1}}u_{2x_{2}})dx|\leq C\int|F||\nabla
u|^{2}dx.$ (3.48)
Substituting (3.45)-(3.48) into (3.42) yields that
$\frac{1}{2}\frac{d}{dt}Z^{2}(t)+\varphi(t)^{2}\leq\alpha\varphi(t)^{2}+C_{\alpha}(Z(t)^{2}+1)^{2+\frac{\varepsilon}{1-3\varepsilon}}\varepsilon^{\frac{2}{1-\beta}}+C\left[1+\int\frac{|F|^{3}}{2\mu+\lambda(\rho)}dx+\int|F||\nabla
u|^{2}dx\right].$ (3.49)
Now it remains to estimate the terms
$\displaystyle\int\frac{|F|^{3}}{2\mu+\lambda(\rho)}dx$ and
$\displaystyle\int|F||\nabla u|^{2}dx$ on the right hand side of (3.49). By
Lemma 2.3, for $\varepsilon\in[0,\frac{1}{2}]$ and $\eta=\varepsilon$, it
holds that
$\|F\|_{2m}\leq C\Big{[}\|F\|_{1}+m^{\frac{1}{2}}\|\nabla
F\|_{\frac{2m}{m+\varepsilon}}^{1-s}\|F\|^{s}_{2(1-\varepsilon)}\Big{]},$
(3.50)
where $\displaystyle
s=\frac{(1-\varepsilon)^{2}}{m-\varepsilon(1-\varepsilon)}$ and the positive
constant $C$ is independent of $m$ and $\varepsilon.$
Choose the positive constant $\varepsilon=2^{-m}$ with $m>2$ being integer in
the inequalities (3.49) and (3.50). By the density estimate (3.24) in Lemma
3.4, one can get
$\begin{array}[]{ll}\|F\|_{1}&\displaystyle=\int(2\mu+\lambda(\rho))^{-\frac{1}{2}}|F|(2\mu+\lambda(\rho))^{\frac{1}{2}}dx\\\
&\displaystyle\leq\left(\frac{|F|^{2}}{2\mu+\lambda(\rho)}\right)^{\frac{1}{2}}\left(\int(2\mu+\lambda(\rho))dx\right)^{\frac{1}{2}}\leq
CZ(t),\end{array}$ (3.51)
and
$\begin{array}[]{ll}\|F\|_{2(1-\varepsilon)}^{s}&\displaystyle=\left(\int(2\mu+\lambda(\rho))^{-(1-\varepsilon)}|F|^{2(1-\varepsilon)}(2\mu+\lambda(\rho))^{1-\varepsilon}dx\right)^{\frac{s}{2(1-\varepsilon)}}\\\
&\displaystyle\leq\left(\frac{|F|^{2}}{2\mu+\lambda(\rho)}\right)^{\frac{s}{2}}\left(\int(2\mu+\lambda(\rho))^{\frac{1-\varepsilon}{\varepsilon}}dx\right)^{\frac{s\varepsilon}{2(1-\varepsilon)}}\\\
&\displaystyle\leq
CZ(t)^{s}\left(\|\rho\|^{\frac{s\beta}{2}}_{\frac{\beta(1-\varepsilon)}{\varepsilon}}+1\right)\leq
CZ(t)^{s}\left[\left(\frac{\beta(1-\varepsilon)}{\varepsilon}\right)^{\frac{s\beta}{\beta-1}}+1\right]\\\
&\displaystyle\leq
CZ(t)^{s}\left(\varepsilon^{-\frac{s\beta}{\beta-1}}+1\right)=CZ(t)^{s}\left(2^{\frac{ms\beta}{\beta-1}}+1\right)\leq
CZ(t)^{s},\end{array}$ (3.52)
where in the last inequality one has used the fact that
$ms=\frac{m(1-\varepsilon)^{2}}{m-\varepsilon(1-\varepsilon)}\rightarrow 1$ as
$m\rightarrow+\infty.$
Substituting (3.43) with $r=\frac{\varepsilon}{m+\varepsilon}$, (3.51) and
(3.52) into (3.50) yields that
$\begin{array}[]{ll}\displaystyle\|F\|_{2m}&\displaystyle\leq
C\Big{[}Z(t)+m^{\frac{1}{2}}\|\nabla
F\|_{\frac{2m}{m+\varepsilon}}^{1-s}Z(t)^{s}\Big{]}\leq
C\Big{[}Z(t)+m^{\frac{1}{2}}(\frac{m+\varepsilon}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\Big{]}\\\
&\displaystyle\leq
C\Big{[}Z(t)+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\Big{]}.\end{array}$
(3.53)
Thus it follows that
$\begin{array}[]{ll}\displaystyle\int\frac{|F|^{3}}{2\mu+\lambda(\rho)}dx\displaystyle=\int\frac{|F|^{2-\frac{1}{m-1}}}{(2\mu+\lambda(\rho))^{1-\frac{1}{2(m-1)}}}(\frac{1}{2\mu+\lambda(\rho)})^{\frac{1}{2(m-1)}}|F|^{1+\frac{1}{m-1}}dx\\\
\qquad\quad\displaystyle\leq\int\left(\frac{|F|^{2}}{2\mu+\lambda(\rho)}\right)^{1-\frac{1}{2(m-1)}}|F|^{\frac{m}{m-1}}dx\\\
\qquad\quad\displaystyle\leq\left(\int\frac{|F|^{2}}{2\mu+\lambda(\rho)}dx\right)^{\frac{2m-3}{2(m-1)}}\left(\int|F|^{2m}dx\right)^{\frac{1}{2(m-1)}}\\\
\qquad\quad\displaystyle\leq
Z(t)^{\frac{2m-3}{m-1}}\|F\|_{2m}^{\frac{m}{m-1}}\leq
CZ(t)^{\frac{2m-3}{m-1}}\Big{[}Z(t)+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\Big{]}^{\frac{m}{m-1}}\\\
\qquad\quad\displaystyle\leq
C\Big{[}Z(t)^{3}+m^{\frac{m}{2(m-1)}}(\frac{m}{\varepsilon})^{\frac{(1-s)m}{(\beta-1)(m-1)}}\varphi(t)^{\frac{(1-s)m}{m-1}}Z(t)^{\frac{(2+s)m-3}{m-1}}\Big{]}\\\
\qquad\quad\displaystyle\leq\alpha\varphi(t)^{2}+C_{\alpha}\Big{[}Z(t)^{3}+m^{\frac{m}{m(1+s)-2}}(\frac{m}{\varepsilon})^{\frac{2(1-s)m}{(\beta-1)(m(1+s)-2)}}Z(t)^{\frac{2((2+s)m-3)}{m(1+s)-2}}\Big{]}\\\
\qquad\quad\displaystyle\leq\alpha\varphi(t)^{2}+C_{\alpha}\Big{[}(1+Z(t)^{2})^{2}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+\frac{1-ms}{m(1+s)-2}}\Big{]}\end{array}$
(3.54)
where in the last inequality one has used the fact that
$ms=\frac{m(1-\varepsilon)^{2}}{m-\varepsilon(1-\varepsilon)}\rightarrow 1$
with $\varepsilon=2^{-m}$ as $m\rightarrow+\infty$.
Furthermore, it holds that
$\begin{array}[]{ll}\displaystyle\int|F||\nabla
u|^{2}dx&\displaystyle\leq\|F\|_{2m}\|\nabla u\|_{\frac{4m}{2m-1}}^{2}\leq
C\|F\|_{2m}\Big{(}\|{\rm
div}u\|_{\frac{4m}{2m-1}}^{2}+\|\omega\|_{\frac{4m}{2m-1}}^{2}\Big{)}\\\
&\displaystyle\leq
C\|F\|_{2m}\Big{(}\|\frac{F}{2\mu+\lambda(\rho)}\|_{\frac{4m}{2m-1}}^{2}+\|\omega\|_{\frac{4m}{2m-1}}^{2}+1\Big{)}.\end{array}$
(3.55)
Note that
$\begin{array}[]{ll}\displaystyle\|\frac{F}{2\mu+\lambda(\rho)}\|_{\frac{4m}{2m-1}}^{2}&\displaystyle=\left(\int\frac{|F|^{\frac{4m}{2m-1}}}{(2\mu+\lambda(\rho))^{\frac{4m}{2m-1}}}dx\right)^{\frac{2m-1}{2m}}\\\
&\displaystyle=\left(\int\frac{|F|^{\frac{2m(2m-3)}{(2m-1)(m-1)}}}{(2\mu+\lambda(\rho))^{\frac{4m}{2m-1}}}|F|^{\frac{2m}{(2m-1)(m-1)}}dx\right)^{\frac{2m-1}{2m}}\\\
&\displaystyle\leq\|F\|_{2m}^{\frac{1}{m-1}}\left(\int\frac{|F|^{2}}{(2\mu+\lambda(\rho))^{\frac{4(m-1)}{2m-3}}}dx\right)^{\frac{2m-3}{2(m-1)}}\\\
&\displaystyle\leq
C\|F\|_{2m}^{\frac{1}{m-1}}\left(\int\frac{|F|^{2}}{2\mu+\lambda(\rho)}dx\right)^{\frac{2m-3}{2(m-1)}}\leq
C\|F\|_{2m}^{\frac{1}{m-1}}Z(t)^{\frac{2m-3}{m-1}},\end{array}$ (3.56)
and from $\displaystyle\int\omega dx=0$, Lemma 2.2 and (3.43), one has
$\begin{array}[]{ll}\displaystyle\|\omega\|_{\frac{4m}{2m-1}}^{2}&\displaystyle\leq
C\|\omega\|_{2}^{2-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\|\nabla\omega\|_{2(1-\varepsilon)}^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}\leq
CZ(t)^{2-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\Big{[}\varepsilon^{\frac{1}{1-\beta}}\varphi(t)\Big{]}^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}\\\
&\displaystyle\leq
C2^{\frac{m(1-\varepsilon)}{(\beta-1)m(1-2\varepsilon)}}Z(t)^{2-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\varphi(t)^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}\leq
CZ(t)^{2-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\varphi(t)^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}.\end{array}$
(3.57)
Now substituting (3.53), (3.56) and (3.57) into (3.55) gives that
$\begin{array}[]{ll}&\displaystyle\int|F||\nabla u|^{2}dx\displaystyle\leq
C\Big{[}Z(t)+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\Big{]}\Big{[}1+Z(t)^{2-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\varphi(t)^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}\Big{]}\\\
&\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+C\Big{[}Z(t)+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\Big{]}^{1+\frac{1}{m-1}}Z(t)^{\frac{2m-3}{m-1}}\\\
&\displaystyle\leq
C\Big{[}Z(t)+Z(t)^{3}+Z(t)^{3-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\varphi(t)^{\frac{1-\varepsilon}{m(1-2\varepsilon)}}+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s}Z(t)^{s}\\\
&\displaystyle~{}+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}\varphi(t)^{1-s+\frac{1-\varepsilon}{m(1-2\varepsilon)}}Z(t)^{2+s-\frac{1-\varepsilon}{m(1-2\varepsilon)}}+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{(1-s)m}{(\beta-1)(m-1)}}\varphi(t)^{\frac{(1-s)m}{m-1}}Z(t)^{\frac{ms+2m-3}{m-1}}\Big{]}\\\
&\displaystyle\leq\alpha\varphi(t)^{2}+C_{\alpha}\Big{[}(1+Z^{2}(t))^{2}+\Big{(}m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}Z(t)^{s}\Big{)}^{\frac{2}{1+s}}\\\
&\displaystyle~{}+\Big{(}m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{1-s}{\beta-1}}Z(t)^{2+s-\frac{1-\varepsilon}{m(1-2\varepsilon)}}\Big{)}^{\frac{2}{1+s-\frac{1-\varepsilon}{m(1-2\varepsilon)}}}+\Big{(}m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{(1-s)m}{(\beta-1)(m-1)}}Z(t)^{2+\frac{ms-1}{m-1}}\Big{)}^{\frac{2(m-1)}{m(s+1)-2}}\Big{]}\\\
&\displaystyle\leq\alpha\varphi(t)^{2}+C_{\alpha}\Big{[}(1+Z^{2}(t))^{2}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})\\\
&\displaystyle~{}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+\frac{1-ms+(2ms-1)\varepsilon}{(1+s)m(1-2\varepsilon)-1+\varepsilon}}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+\frac{1-ms}{m(s+1)-2}}\Big{]}.\end{array}$
(3.58)
Substituting (3.54) and (3.58) into (3.49) and choosing $\alpha$ sufficiently
small yield that
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}(Z^{2}(t))+\frac{1}{2}\varphi(t)^{2}\leq
C(Z(t)^{2}+1)^{2+\frac{\varepsilon}{1-3\varepsilon}}\varepsilon^{\frac{2}{1-\beta}}+C\Big{[}(1+Z^{2}(t))^{2}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})\\\
\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+m(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+\frac{1-ms+(2ms-1)\varepsilon}{(1+s)m(1-2\varepsilon)-1+\varepsilon}}+m^{\frac{1}{2}}(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+\frac{1-ms}{m(s+1)-2}}\Big{]}.\end{array}$
(3.59)
Note that $\lim_{m\rightarrow+\infty}[2^{m}(1-ms)]=2$, and so $1-ms\sim
2\varepsilon$ as $m\rightarrow+\infty$. Thus for $m$ sufficiently large, one
has
$\frac{1-ms}{m(1+s)-2}\sim\frac{2\varepsilon}{1-2\varepsilon+m-2}=\frac{2\varepsilon}{m-1-2\varepsilon}\leq
4\varepsilon,$
and
$\frac{1-ms+(2ms-1)\varepsilon}{(1+s)m(1-2\varepsilon)-1+\varepsilon}=\frac{(1-ms)(1-2\varepsilon)+\varepsilon}{(1+s)m(1-2\varepsilon)-1+\varepsilon}\sim\frac{3\varepsilon-4\varepsilon^{2}}{(m+1-2\varepsilon)(1-2\varepsilon)-1+\varepsilon}\leq
4\varepsilon.$
Then (3.59) yields the following inequality for suitably large $m$,
$\frac{1}{2}\frac{d}{dt}(Z^{2}(t))+\frac{1}{2}\varphi(t)^{2}\leq
Cm(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}(1+Z(t)^{2})^{2+4\varepsilon}.$
(3.60)
Note that
$\begin{array}[]{ll}Z^{2}(t)&\displaystyle=\int(\mu\omega^{2}+\frac{F^{2}}{2\mu+\lambda(\rho)})dx\\\
&\displaystyle\leq C\int[\mu\omega^{2}+(2\mu+\lambda(\rho))({\rm
div}u)^{2}+\frac{P^{2}(\rho)}{2\mu+\lambda(\rho)})]dx\\\ &\displaystyle\leq
C\big{(}\phi(t)+\int P^{2}(\rho)dx\big{)}\in L^{1}(0,T).\end{array}$ (3.61)
Applying the Gronwall’s inequality to (3.60) and using (3.61) show that
$\frac{1}{(1+Z^{2}(t))^{4\varepsilon}}-\frac{1}{(1+Z^{2}(0))^{4\varepsilon}}+Cm\varepsilon(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}\geq
0.$ (3.62)
Then we have the inequality
$\frac{1}{(1+Z^{2}(t))^{4\varepsilon}}\geq\frac{1}{2(1+Z^{2}(0))^{4\varepsilon}},$
(3.63)
provided that
$Cm\varepsilon(\frac{m}{\varepsilon})^{\frac{2}{\beta-1}}\leq\frac{1}{2(1+Z^{2}(0))^{4\varepsilon}}.$
(3.64)
This condition, (3.64), is satisfied if
$Cm^{1+\frac{2}{\beta-1}}2^{-m(1-\frac{2}{\beta-1})}\leq\frac{1}{2},$ (3.65)
since
$\begin{array}[]{ll}\displaystyle
Z^{2}(0)&\displaystyle=\int\Big{[}\mu(\omega^{\delta}_{0})^{2}+\frac{(F^{\delta}_{0})^{2}}{2\mu+\lambda(\rho^{\delta}_{0})}\Big{]}dx\\\
&\displaystyle\leq
C\Big{[}\|u_{0}^{\delta}\|^{2}_{H^{2}(\mathbb{T}^{2})}+\|\rho_{0}^{\delta}\|_{H^{3}(\mathbb{T}^{2})}^{\beta}\|u_{0}^{\delta}\|^{2}_{H^{2}(\mathbb{T}^{2})}+\|\rho_{0}^{\delta}\|_{H^{3}(\mathbb{T}^{2})}^{2\gamma}\Big{]}\leq
C.\end{array}$
Now if $\beta>3,$ that is, $1-\frac{2}{\beta-1}>0$, then we can choose
sufficiently large $m>2$ to guarantee the condition (3.65). Consequently, the
inequality (3.63) is satisfied with $\beta>3$ and sufficiently large $m>2$.
Then
$Z^{2}(t)\leq 2^{2^{m-1}}(1+Z^{2}(0))-1\leq C,$ (3.66)
and
$\int_{0}^{T}\varphi(t)dt\leq C.$ (3.67)
Thus the proof of Lemma 3.5 is completed. $\hfill\Box$
Step 5: Second order derivative estimates for the velocity:
###### Lemma 3.6
There exists a positive constant $C$ independent of $\delta$, such that
$\sup_{t\in[0,T]}\int\rho(H^{2}+L^{2})dx+\int_{0}^{T}\int\mu(H_{x_{1}}-L_{x_{2}})^{2}+(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})^{2}dxdt\leq
C.$ (3.68)
Proof: Multiplying the equations, $\eqref{H-L}_{1}$ and $\eqref{H-L}_{2}$, by
$H$ and $L$, respectively, summing the resulted equations together, and
integrating with respect to $x$ over $\mathbb{T}^{2}$ lead to
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int\rho(H^{2}+L^{2})dx+\int\mu(H_{x_{1}}-L_{x_{2}})^{2}+(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})^{2}dxdx\\\
\displaystyle=\int\rho(H^{2}+L^{2}){\rm div}udx-\int\mu\omega{\rm
div}u(L_{x_{2}}-H_{x_{1}})dx\\\
\displaystyle~{}~{}-\int\rho(2\mu+\lambda(\rho))\big{[}F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}\big{]}{\rm
div}u(H_{x_{2}}+L_{x_{1}})dx\\\
\displaystyle~{}~{}-\int\big{[}H(u_{x_{2}}\cdot\nabla F+\mu
u_{x_{1}}\cdot\nabla\omega)+L(u_{x_{1}}\cdot\nabla F-\mu
u_{x_{2}}\cdot\nabla\omega)\big{]}dx\\\
\displaystyle~{}~{}+\int(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}](H_{x_{2}}+L_{x_{1}})dx.\end{array}$
(3.69)
Set
$Y(t)=\left(\int\rho(H^{2}+L^{2})dx\right)^{\frac{1}{2}},$ (3.70)
and
$\psi(t)=\left(\int\mu(H_{x_{1}}-L_{x_{2}})^{2}+(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})^{2}dx\right)^{\frac{1}{2}}.$
(3.71)
Note that
$\begin{array}[]{ll}\displaystyle\int(|\nabla H|^{2}+|\nabla
L|^{2})dx&\displaystyle=\int(H_{x_{1}}^{2}+H_{x_{2}}^{2}+L_{x_{1}}^{2}+L_{x_{2}}^{2})dx\\\
&\displaystyle=\int\big{[}(H_{x_{1}}-L_{x_{2}})^{2}+(H_{x_{2}}+L_{x_{1}})^{2}\big{]}dx\leq\frac{1}{\mu}\psi^{2}(t).\end{array}$
Thus it holds that
$\|\nabla(H,L)\|_{2}(t)\leq C\psi(t),\qquad\forall t\in[0,T].$ (3.72)
Then it follows from the elliptic system
$\mu\omega_{x_{1}}+F_{x_{2}}=\rho
H,\qquad\qquad-\mu\omega_{x_{2}}+F_{x_{1}}=\rho L,$
that
$\|\nabla(F,\omega)\|_{p}\leq C\|\rho(H,L)\|_{p},\qquad\forall 1<p<+\infty.$
(3.73)
Furthermore, since $\displaystyle\int(\mu\omega_{x_{1}}+F_{x_{2}})dx=0,$ by
the mean value theorem, there exists a point $x_{*}\in\mathbb{T}^{2}$, such
that $(\mu\omega_{x_{1}}+F_{x_{2}})(x_{*},t)=0,$ and so $H(x_{*},t)=0.$
Similarly, there exists a point $x_{*}^{\prime}$, such that
$L(x_{*}^{\prime},t)=0.$ Therefore, by the Poincare inequality, it holds that
$\|(H,L)\|_{p}\leq C\|\nabla(H,L)\|_{2},\qquad\forall 1\leq p<+\infty,$ (3.74)
where $C$ may depend on $p$.
Now we estimate the right hand side of (3.69) term by term. First, by the
H${\rm\ddot{o}}$lder inequality, (3.74) and the density estimate (3.24), it
holds that
$\begin{array}[]{ll}\displaystyle|\int\rho(H^{2}+L^{2}){\rm
div}udx|=|\int\rho(H^{2}+L^{2})\frac{F+P(\rho)}{2\mu+\lambda(\rho)}dx|\\\
\displaystyle\qquad\quad\leq\|\sqrt{\rho}(H,L)\|_{2}\|(H,L)\|_{4}\|\frac{\sqrt{\rho}(F+P(\rho))}{2\mu+\lambda(\rho)}\|_{4}\leq
CY(t)\psi(t)(1+\|F\|_{4}).\end{array}$ (3.75)
Note that
$\begin{array}[]{ll}\|(F,\omega)\|_{4}\leq
C(\|\nabla(F,\omega)\|_{\frac{3}{2}}+\|(F,\omega)\|_{1})\\\
\displaystyle~{}\leq
C\Big{[}\|\nabla(F,\omega)\|_{\frac{3}{2}}+\Big{(}\int\frac{F^{2}}{2\mu+\lambda(\rho)}dx\Big{)}^{\frac{1}{2}}\Big{(}\int(2\mu+\lambda(\rho))dx\Big{)}^{\frac{1}{2}}+\|\omega\|_{2}\Big{]}\leq
C\Big{[}Y(t)+1\Big{]},\end{array}$ (3.76)
where in the last inequality one has used the estimate (3.43) with
$r=\frac{1}{4}$ and the estimate (3.66).
Substituting (3.76) into (3.75) yields that
$\begin{array}[]{ll}\displaystyle|\int\rho(H^{2}+L^{2}){\rm
div}udx|&\displaystyle\leq
CY(t)\psi(t)(Y(t)+1)\leq\alpha\psi^{2}(t)+C_{\alpha}(Y(t)+1)^{4}.\end{array}$
(3.77)
Second, direct estimates give
$\begin{array}[]{ll}\displaystyle|-\int\mu\omega{\rm
div}u(L_{x_{2}}-H_{x_{1}})dx|\leq\mu\left(\int(L_{x_{2}}-H_{x_{1}})^{2}dx\right)^{\frac{1}{2}}\left(\int\omega^{2}({\rm
div}u)^{2}dx\right)^{\frac{1}{2}}\\\\[5.69054pt]
\displaystyle\leq\alpha\psi^{2}(t)+C_{\alpha}\int\omega^{2}({\rm
div}u)^{2}dx\leq\alpha\psi^{2}(t)+C_{\alpha}\|\omega\|_{4}^{2}\|\frac{F+P(\rho)}{2\mu+\lambda(\rho)}\|_{4}^{2}\\\
\displaystyle\leq\alpha\psi^{2}(t)+C_{\alpha}\|\omega\|_{4}^{2}(1+\|F\|_{4}^{2})\leq\alpha\psi^{2}(t)+C_{\alpha}(Y(t)+1)^{4}.\end{array}$
(3.78)
Similarly, one has
$\begin{array}[]{ll}\displaystyle|-\int\rho(2\mu+\lambda(\rho))\big{[}F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}\big{]}{\rm
div}u(H_{x_{2}}+L_{x_{1}})dx|\\\
\displaystyle\leq\alpha\int(2\mu+\lambda(\rho))(H_{x_{2}}+L_{x_{1}})^{2}dx\\\
\displaystyle\qquad\qquad+C_{\alpha}\int\rho^{2}(2\mu+\lambda(\rho))\big{[}F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}\big{]}^{2}({\rm
div}u)^{2}dx\\\
\displaystyle\leq\alpha\psi^{2}(t)+C_{\alpha}\int\rho^{2}\big{[}F(\frac{1}{2\mu+\lambda(\rho)})^{\prime}+(\frac{P(\rho)}{2\mu+\lambda(\rho)})^{\prime}\big{]}^{2}\frac{|F|^{2}+P^{2}(\rho)}{2\mu+\lambda(\rho)}dx\\\
\displaystyle\leq\alpha\psi^{2}(t)+C_{\alpha}(1+\|F\|_{4}^{4})\leq\alpha\psi^{2}(t)+C_{\alpha}(Y(t)+1)^{4}.\end{array}$
(3.79)
Next,
$\begin{array}[]{ll}\displaystyle|-\int\big{[}H(u_{x_{2}}\cdot\nabla F+\mu
u_{x_{1}}\cdot\nabla\omega)+L(u_{x_{1}}\cdot\nabla F-\mu
u_{x_{2}}\cdot\nabla\omega)\big{]}dx|\\\ \displaystyle\leq C\int|(H,L)||\nabla
u||\nabla(F,\omega)|dx\\\ \displaystyle\leq C\|(H,L)\|_{8}\|\nabla
u\|_{2}\|\nabla(F,\omega)\|_{\frac{8}{3}}\leq
C\|\nabla(H,L)\|_{2}\|\rho(H,L)\|_{\frac{8}{3}},\end{array}$ (3.80)
where one has used the fact that
$\|\nabla u\|_{2}\leq C(\|{\rm div}u\|_{2}+\|\omega\|_{2})\leq
C(\|\frac{F+P(\rho)}{2\mu+\lambda(\rho)}\|_{2}+\|\omega\|_{2})\leq C.$
Note that
$\begin{array}[]{ll}\displaystyle\|\rho(H,L)\|_{\frac{8}{3}}&\displaystyle=\left(\int\rho^{\frac{8}{3}}|(H,L)|^{\frac{8}{3}}dx\right)^{\frac{3}{8}}=\left(\int\sqrt{\rho}|(H,L)||(H,L)|^{\frac{5}{3}}\rho^{\frac{13}{6}}dx\right)^{\frac{3}{8}}\\\
&\displaystyle\leq\|\sqrt{\rho}(H,L)\|_{2}^{\frac{3}{8}}\|(H,L)\|_{4}^{\frac{5}{8}}\|\rho\|_{26}^{\frac{13}{16}}\leq
CY(t)^{\frac{3}{8}}\|\nabla(H,L)\|_{2}^{\frac{5}{8}}.\end{array}$ (3.81)
It follows from (3.80) and (3.81) that
$\begin{array}[]{ll}\displaystyle|-\int\big{[}H(u_{x_{2}}\cdot\nabla F+\mu
u_{x_{1}}\cdot\nabla\omega)+L(u_{x_{1}}\cdot\nabla F-\mu
u_{x_{2}}\cdot\nabla\omega)\big{]}dx|\\\ \displaystyle\leq
CY(t)^{\frac{3}{8}}\|\nabla(H,L)\|_{2}^{\frac{13}{8}}\leq
CY(t)^{\frac{3}{8}}\psi(t)^{\frac{13}{8}}\leq\alpha\psi(t)^{2}+C_{\alpha}Y(t)^{2}.\end{array}$
(3.82)
Moreover,
$\begin{array}[]{ll}\displaystyle|\int(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}](H_{x_{2}}+L_{x_{1}})dx|\\\
\displaystyle\leq\alpha\psi(t)^{2}+C_{\alpha}\int(2\mu+\lambda(\rho))[(u_{1x_{1}})^{2}+2u_{1x_{2}}u_{2x_{1}}+(u_{2x_{2}})^{2}]^{2}dx\\\
\displaystyle\leq\alpha\psi(t)^{2}+C_{\alpha}\|2\mu+\lambda(\rho)\|_{2}\|\nabla
u\|_{8}^{4}\\\ \displaystyle\leq\alpha\psi(t)^{2}+C_{\alpha}(\|{\rm
div}u\|_{8}^{4}+\|\omega\|_{8}^{4})\leq\alpha\psi(t)^{2}+C_{\alpha}(\|(F,\omega)\|_{8}^{4}+1)\\\
\displaystyle\leq\alpha\psi(t)^{2}+C_{\alpha}(\|\nabla(F,\omega)\|_{\frac{8}{5}}^{4}+1)\leq\alpha\psi(t)^{2}+C_{\alpha}(1+Y(t))^{4}.\end{array}$
(3.83)
Substituting the estimates (3.77)-(3.79), (3.82) and (3.83) into (3.69), one
can arrive at
$\frac{1}{2}\frac{d}{dt}(Y^{2}(t))+\psi^{2}(t)\leq
5\alpha\psi^{2}(t)+C_{\alpha}(1+Y^{2}(t))^{2}.$ (3.84)
Choosing $5\alpha=\frac{1}{2}$, noting that $Y^{2}(t)=\varphi^{2}(t)\in
L^{1}(0,T)$, and then using Gronwall’s inequality yield that
$Y^{2}(t)+\int_{0}^{T}\psi^{2}(t)dt\leq Y^{2}(0)+C.$ (3.85)
Now we calculate the initial values $Y^{2}(0).$ By the approximate
compatibility condition (3.2), one has
$\mathcal{L}_{\rho^{\delta}_{0}}u^{\delta}_{0}-\nabla
P^{\delta}_{0}=\sqrt{\rho}_{0}g,~{}~{}{\rm with}~{}~{}g\in
L^{2}(\mathbb{T}^{2}).$
On the other hand, it holds that
$\begin{array}[]{ll}\mathcal{L}_{\rho^{\delta}_{0}}u^{\delta}_{0}&\displaystyle=\mu\Delta
u^{\delta}_{0}+\nabla((\mu+\lambda(\rho^{\delta}_{0})){\rm
div}u^{\delta}_{0})=\mu\Delta u^{\delta}_{0}+\nabla(F^{\delta}_{0}-\mu{\rm
div}u^{\delta}_{0}+P^{\delta}_{0})\\\ &\displaystyle=[\mu\nabla({\rm
div}u^{\delta}_{0})-\mu\nabla\times(\nabla\times
u^{\delta}_{0})]+\nabla(F^{\delta}_{0}-\mu{\rm
div}u^{\delta}_{0}+P^{\delta}_{0})\end{array}$ (3.86)
where $F_{0}^{\delta}=(2\mu+\lambda(\rho_{0}^{\delta})){\rm
div}u_{0}^{\delta}-P_{0}^{\delta}$ and similarly one can define
$\omega_{0}^{\delta},L_{0}^{\delta},H_{0}^{\delta}$, $\nabla\times$ denotes
the 3-dimensional curl operator, and
$\nabla\times(\nabla\times
u^{\delta}_{0})=(\partial_{x_{2}}\omega^{\delta}_{0},-\partial_{x_{1}}\omega^{\delta}_{0},0)$
is regarded as the 2-dimensional vector
$(\partial_{x_{2}}\omega^{\delta}_{0},-\partial_{x_{1}}\omega^{\delta}_{0})^{t}$.
Thus
$\begin{array}[]{ll}\mathcal{L}_{\rho^{\delta}_{0}}u^{\delta}_{0}-\nabla
P^{\delta}_{0}&\displaystyle=\nabla
F^{\delta}_{0}-\mu(\partial_{x_{2}}\omega^{\delta}_{0},-\partial_{x_{1}}\omega^{\delta}_{0})^{t}\\\
&\displaystyle=(F^{\delta}_{0x_{1}}-\mu\partial_{x_{2}}\omega^{\delta}_{0},F^{\delta}_{0x_{2}}+\mu\partial_{x_{1}}\omega^{\delta}_{0})^{t}=\rho^{\delta}_{0}(L^{\delta}_{0},H^{\delta}_{0})^{t}.\end{array}$
(3.87)
Therefore
$\sqrt{\rho_{0}}g=\rho_{0}^{\delta}(L^{\delta}_{0},H^{\delta}_{0})^{t}.$
(3.88)
Consequently, it holds that
$Y^{2}(0)=\|\sqrt{\rho^{\delta}_{0}}(L^{\delta}_{0},H^{\delta}_{0})\|_{2}^{2}=\|\frac{\sqrt{\rho_{0}}}{\sqrt{\rho_{0}^{\delta}}}g\|_{2}^{2}\leq
C.$ (3.89)
This, together with (3.85), shows that
$Y^{2}(t)+\int_{0}^{T}\psi^{2}(t)dt\leq C.$ (3.90)
This completes the proof of Lemma 3.6. $\hfill\Box$
###### Remark 3.1
Similar to the derivation of (3.87), one can get that for any $t\in[0,T]$,
$\mathcal{L}_{\rho}u-\nabla P(\rho)=\rho(L,H)^{t}.$
Then it follows from the momentum equation $\eqref{CNS}_{2}$ that
$u_{t}=(L,H)^{t}-u\cdot\nabla u.$ (3.91)
The above identity can also be obtained directly from (2.1).
Step 6. Upper bound of the density: We are now ready to derive the upper bound
for the density in the super-norm independent of $\delta$, which is crucial
for the proof of Theorem 1.1 as in [25, 22, 23]. First, we have
###### Lemma 3.7
It holds that
$\int_{0}^{T}\|(F,\omega)\|_{\infty}^{3}dt\leq C.$ (3.92)
Proof: By (3.73) with $p=3$, one has
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|\nabla(F,\omega)\|_{3}^{3}dt&\displaystyle\leq
C\int_{0}^{T}\|\rho(H,L)\|_{3}^{3}dt=C\int\int\rho^{3}|(H,L)|^{3}dxdt\\\
&\displaystyle=C\int\int\sqrt{\rho}|(H,L)||(H,L)|^{2}\rho^{\frac{5}{2}}dxdt\\\
&\displaystyle\leq
C\int\|\sqrt{\rho}(H,L)\|_{2}\|(H,L)\|_{8}^{2}\|\rho\|^{\frac{5}{2}}_{10}dt\\\
&\displaystyle\leq C\int_{0}^{T}\|\nabla(H,L)\|_{2}^{2}dt\leq
C\int_{0}^{T}\psi^{2}(t)\leq C,\end{array}$ (3.93)
which, combined with the estimates in Lemma 2.3, yields that
$\int_{0}^{T}\|(F,\omega)\|_{\infty}^{3}dt\leq\int_{0}^{T}\|(F,\omega)\|_{W^{1,3}(\mathbb{T}^{2})}^{3}dt\leq
C.$ (3.94)
The proof of Lemma 3.7 is finished. $\hfill\Box$
With Lemma 3.7 in hand, we can obtain the uniform upper bound for the density.
###### Lemma 3.8
It holds that
$\rho(t,x)\leq C,\qquad\forall(t,x)\in[0,T]\times\mathbb{T}^{2}.$ (3.95)
Proof: From the continuity equation $\eqref{CNS}_{1}$, we have
$\theta(\rho)_{t}+u\cdot\nabla\theta(\rho)+P(\rho)+F=0,$ (3.96)
where $\theta(\rho)$ is defined in (3.22).
Along the particle path $\vec{X}(\tau;t,x)$ through the point
$(t,x)\in[0,T]\times\mathbb{T}^{2}$ defined by
$\left\\{\begin{array}[]{ll}\displaystyle\frac{d\vec{X}(\tau;t,x)}{d\tau}=u(\tau,\vec{X}(\tau;t,x)),\\\
\displaystyle\vec{X}(\tau;t,x)|_{\tau=t}=x,\end{array}\right.$ (3.97)
there holds the following ODE
$\frac{d}{d\tau}\theta(\rho)(\tau,\vec{X}(\tau;t,x))=-P(\rho)(\tau,\vec{X}(\tau;t,x))-F(\tau,\vec{X}(\tau;t,x)),$
(3.98)
which is integrated over $[0,t]$ to yield that
$\theta(\rho)(t,x)-\theta(\rho_{0})(\vec{X}_{0})=-\int_{0}^{t}(P(\rho)+F)(\tau,\vec{X}(\tau;t,x))d\tau,$
(3.99)
with $\vec{X}_{0}=\vec{X}(\tau;t,x)|_{\tau=0}$.
It follows from (3.99) that
$2\mu\ln\frac{\rho(t,x)}{\rho_{0}(\vec{X}_{0})}+\frac{1}{\beta}\rho^{\beta}(t,x)+\int_{0}^{t}P(\rho)(\tau,\vec{X}(\tau;t,x))d\tau=\frac{1}{\beta}\rho_{0}(\vec{X}_{0})^{\beta}-\int_{0}^{t}F(\tau,\vec{X}(\tau;t,x))d\tau.$
(3.100)
So
$2\mu\ln\frac{\rho(t,x)}{\rho_{0}(\vec{X}_{0})}\leq\frac{1}{\beta}\|\rho_{0}\|_{\infty}^{\beta}+\int_{0}^{t}\|F(\tau,\cdot)\|_{\infty}d\tau\leq
C,$ (3.101)
which implies that
$\frac{\rho(t,x)}{\rho_{0}(\vec{X}_{0})}\leq C.$
Therefore, we have
$\rho(t,x)\leq C,\qquad\forall(t,x)\in[0,T]\times\mathbb{T}^{2}.$ (3.102)
Hence the Lemma is proved. $\hfill\Box$
As an immediate consequence of the upper bound of the density, one has
###### Lemma 3.9
It holds that for any $1<p<\infty$,
$\int_{0}^{T}\big{(}\|{\rm
div}u\|_{\infty}^{3}+\|\nabla(F,\omega)\|_{p}^{2}\big{)}dt\leq C.$ (3.103)
Proof: First, note that
$\int_{0}^{T}\|{\rm div}u\|_{\infty}^{3}dt\leq
C\int_{0}^{T}(\|F\|_{\infty}^{3}+\|P(\rho)\|_{\infty}^{3})dt\leq C.$ (3.104)
Then for any $1<p<\infty$,
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|\nabla(F,\omega)\|_{p}^{2}dt&\displaystyle\leq
C\int_{0}^{T}\|\rho(H,L)\|_{p}^{2}dt\\\ &\displaystyle\leq
C\int_{0}^{T}\|(H,L)\|_{p}^{2}dt\leq
C\int_{0}^{T}\|\nabla(H,L)\|_{2}^{2}dt\leq C.\end{array}$ (3.105)
Thus Lemma 3.9 is proved. $\hfill\Box$
## 4 Higher order estimates
With the approximate solutions and basic estimates at hand, we can derive some
uniform estimates on their higher order derivatives easily as in [25, 22, 23].
We start with estimates on first order derivatives.
###### Lemma 4.1
It holds that for any $1\leq p<+\infty$,
$\sup_{t\in[0,T]}\|(\nabla\rho,\nabla
P(\rho))(t,\cdot)\|_{p}+\int_{0}^{T}\|\nabla u\|_{\infty}^{2}dt\leq C.$ (4.1)
Proof: Applying the operator $\nabla$ to the continuity equation
$\eqref{CNS}_{1}$, one has
$(\nabla\rho)_{t}+\nabla
u\cdot\nabla\rho+u\cdot\nabla(\nabla\rho)+\nabla\rho{\rm div}u+\rho\nabla({\rm
div}u)=0.$ (4.2)
Multiplying the equation (4.2) by $p|\nabla\rho|^{p-2}\nabla\rho$ with $p\geq
2$ implies that
$(|\nabla\rho|^{p})_{t}+{\rm div}(u|\nabla\rho|^{p})+(p-1)|\nabla\rho|^{p}{\rm
div}u+p|\nabla\rho|^{p-2}\nabla\rho\cdot(\nabla
u\cdot\nabla\rho)+p\rho|\nabla\rho|^{p-2}\nabla\rho\cdot\nabla({\rm div}u)=0.$
(4.3)
Integrating over $\mathbb{T}^{2}$ gives
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla\rho\|_{p}^{p}\\\
\displaystyle=-(p-1)\int|\nabla u|^{p}{\rm
div}udx-p\int|\nabla\rho|^{p-2}\nabla\rho\cdot(\nabla
u\cdot\nabla\rho)dx-p\int\rho|\nabla\rho|^{p-2}\nabla\rho\cdot\nabla({\rm
div}u)dx\\\ \displaystyle\leq(p-1)\|{\rm
div}u\|_{\infty}\|\nabla\rho\|_{p}^{p}+p\|\nabla
u\|_{\infty}\|\nabla\rho\|_{p}^{p}+p\|\rho\|_{\infty}\|\nabla\rho\|_{p}^{p-1}\|\nabla{\rm
div}u\|_{p}.\end{array}$ (4.4)
This implies that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla\rho\|_{p}&\displaystyle\leq
C\Big{[}\|\nabla u\|_{\infty}\|\nabla\rho\|_{p}+\|\nabla{\rm
div}u\|_{p}\Big{]}\\\ &\displaystyle\leq C\Big{[}\|\nabla
u\|_{\infty}\|\nabla\rho\|_{p}+\|\nabla\big{(}\frac{F+P(\rho)}{2\mu+\lambda(\rho)}\big{)}\|_{p}\Big{]}\\\
&\displaystyle\leq C\Big{[}\Big{(}\|\nabla
u\|_{\infty}+\|F\|_{\infty}+1\Big{)}\|\nabla\rho\|_{p}+\|\nabla
F\|_{p}\Big{]}.\end{array}$ (4.5)
By Remark 3.1, one has
$\mathcal{L}_{\rho}u=\nabla P(\rho)+\rho(L,H)^{t}.$ (4.6)
Thus the elliptic estimates and (3.74) yields that for any $1<p<\infty,$
$\begin{array}[]{ll}\|\nabla^{2}u\|_{p}&\displaystyle\leq C\big{[}\|\nabla
P(\rho)\|_{p}+\|\rho(L,H)\|_{p}\big{]}\\\ &\displaystyle\leq
C\big{[}\|\nabla\rho\|_{p}+\|(L,H)\|_{p}\big{]}\leq
C\big{[}\|\nabla\rho\|_{p}+\|\nabla(L,H)\|_{2}\big{]}.\end{array}$ (4.7)
By Beal-Kato-Majda type inequality (see [23]-[25] or [51]), it holds that
$\begin{array}[]{ll}\displaystyle\|\nabla u\|_{\infty}&\displaystyle\leq
C\big{(}\|{\rm
div}u\|_{\infty}+\|\omega\|_{\infty}\big{)}\ln(e+\|\nabla^{2}u\|_{3})\\\
&\displaystyle\leq C\big{(}\|{\rm
div}u\|_{\infty}+\|\omega\|_{\infty}\big{)}\ln(e+\|\nabla\rho\|_{3})+C\big{(}\|{\rm
div}u\|_{\infty}+\|\omega\|_{\infty}\big{)}\ln(e+\|\nabla(H,L)\|_{2}).\end{array}$
(4.8)
The combination of (4.5) with $p=3$ and (4.8) yields that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla\rho\|_{3}&\displaystyle\leq
C\Big{[}\big{(}\|{\rm
div}u\|_{\infty}+\|\omega\|_{\infty}\big{)}\ln(e+\|\nabla(H,L)\|_{2})+\|F\|_{\infty}+1\Big{]}\|\nabla\rho\|_{3}\\\
&\displaystyle~{}~{}~{}+C\big{(}\|{\rm
div}u\|_{\infty}+\|\omega\|_{\infty}\big{)}\|\nabla\rho\|_{3}\ln(e+\|\nabla\rho\|_{3})+C\|\nabla
F\|_{3}.\end{array}$ (4.9)
By the estimates (3.94), (3.104), (3.105) and the Gronwall’s inequality, it
holds that
$\sup_{t\in[0,T]}\|\nabla\rho\|_{3}\leq C,$ (4.10)
which, together with (3.94), (3.104), (4.7) and (4.8), yields that
$\int_{0}^{T}\|\nabla u\|_{\infty}^{2}dt\leq C.$ (4.11)
Therefore, by (4.11), Lemma 3.7, Lemma 3.9 and Gronwall inequality, one can
derive from (4.5) that
$\sup_{t\in[0,T]}\|\nabla\rho\|_{p}\leq
C(\|\nabla\rho_{0}\|_{p}+1),\qquad\forall p\in[1,+\infty).$ (4.12)
Thus the proof of Lemma 4.1 is completed. $\hfill\Box$
###### Lemma 4.2
It holds that for any $1\leq p<+\infty$,
$\sup_{t\in[0,T]}\Big{[}\|u(t,\cdot)\|_{\infty}+\|\nabla
u\|_{p}+\|(\rho_{t},P_{t})\|_{p}+\|(\rho_{t},P(\rho)_{t})\|_{H^{1}}+\|(\rho,u)\|_{H^{2}}\Big{]}+\int_{0}^{T}\|u\|_{H^{3}}^{2}dt\leq
C.$ (4.13)
Proof: By $L^{2}-$estimates to the elliptic system (4.6), one has
$\begin{array}[]{ll}\displaystyle\sup_{t\in[0,T]}\|u\|_{H^{2}}&\displaystyle\leq
C\sup_{t\in[0,T]}\big{(}\|\nabla P(\rho)\|_{2}+\|\rho(H,L)\|_{2}\big{)}\\\
&\displaystyle\leq C\sup_{t\in[0,T]}\big{(}\|\nabla
P(\rho)\|_{2}+\|\sqrt{\rho}(H,L)\|_{2}\big{)}\leq C.\end{array}$ (4.14)
It follows from the Sobolev embedding theorem that
$\sup_{[0,T]\times\mathbb{T}^{2}}|u(t,x)|\leq C,\qquad\sup_{t\in[0,T]}\|\nabla
u\|_{p}\leq C,~{}~{}\forall 1\leq p<+\infty.$ (4.15)
Due to $\eqref{CNS}_{1}$, one can get $\rho_{t}=-u\cdot\nabla\rho-\rho~{}{\rm
div}u$ and $P_{t}=-u\cdot\nabla P-\rho P^{\prime}(\rho)~{}{\rm div}u$, which,
together with the uniform upper bound of the density and the estimates in
Lemma 4.1 and (4.15), yields that
$\sup_{t\in[0,T]}\|(\rho_{t},P_{t})\|_{p}\leq C,\qquad\forall
p\in[1,+\infty).$ (4.16)
Applying $\nabla^{2}$ to the continuity equation $\eqref{CNS}_{1}$, then
multiplying the resulted equation by $\nabla^{2}\rho$, and then integrating
over the torus $\mathbb{T}^{2}$, one can get that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla^{2}\rho\|_{2}^{2}&\displaystyle\leq
C\Big{[}\|\nabla
u\|_{\infty}\|\nabla^{2}\rho\|^{2}_{2}+\|\nabla\rho\|_{4}\|\nabla^{2}\rho\|_{2}\|\nabla^{2}u\|_{4}+\|\rho\|_{\infty}\|\nabla^{2}\rho\|_{2}\|\nabla^{3}u\|_{2}\Big{]}\\\
&\displaystyle\leq C\Big{[}\big{(}\|\nabla
u\|_{\infty}+1)\|\nabla^{2}\rho\|^{2}_{2}+\|\nabla^{3}u\|^{2}_{2}+1\Big{]}.\end{array}$
(4.17)
Similarly,
$\frac{d}{dt}\|\nabla^{2}P(\rho)\|_{2}^{2}\leq C\Big{[}\big{(}\|\nabla
u\|_{\infty}+1)\|\nabla^{2}P(\rho)\|^{2}_{2}+\|\nabla^{3}u\|^{2}_{2}+1\Big{]}.$
(4.18)
Note that (4.6) implies that
$\mathcal{L}_{\rho}(\nabla
u)=\nabla^{2}P(\rho)+\nabla[\rho(H,L)]+\nabla(\nabla\lambda(\rho){\rm
div}u):=\Phi.$
Then the standard elliptic estimates give that
$\begin{array}[]{ll}\displaystyle\|u\|_{H^{3}}&\displaystyle\leq
C\Big{[}\|u\|_{H^{1}}+\|\Phi\|_{2}\Big{]}\\\ &\displaystyle\leq
C\Big{[}\|u\|_{H^{1}}+\|\nabla^{2}P(\rho)\|_{2}+\|\rho\|_{\infty}\|\nabla(H,L)\|_{2}+\|\nabla\rho\|_{4}\|(H,L)\|_{4}\\\
&\displaystyle~{}~{}~{}~{}~{}+\|\nabla^{2}\rho\|_{2}\|{\rm
div}u\|_{\infty}+\|\nabla\rho\|_{4}\|\nabla^{2}u\|_{4}\Big{]},\end{array}$
(4.19)
and
$\|\nabla^{2}u\|_{4}\leq C\|\nabla P(\rho)\|_{4}+\|\rho(L,H)\|_{4}\leq
C(1+\|\nabla(H,L)\|_{2}).$
Consequently,
$\displaystyle\|u\|_{H^{3}}\displaystyle\leq
C\Big{[}1+\|\nabla^{2}P(\rho)\|_{2}+\|\nabla(H,L)\|_{2}+\|\nabla^{2}\rho\|_{2}\|{\rm
div}u\|_{\infty}\Big{]}.$ (4.20)
Substituting (4.19) into (4.17) and (4.18) yields that
$\frac{d}{dt}\|(\nabla^{2}\rho,\nabla^{2}P(\rho))\|_{2}^{2}\leq
C\Big{[}\big{(}\|\nabla
u\|^{2}_{\infty}+1\big{)}\|(\nabla^{2}\rho,\nabla^{2}P(\rho))\|^{2}_{2}+\|\nabla(H,L)\|^{2}_{2}+1\Big{]}.$
(4.21)
Then the Gronwall’s inequality yields that
$\begin{array}[]{ll}\displaystyle\|(\nabla^{2}\rho,\nabla^{2}P(\rho))\|_{2}^{2}(t)&\displaystyle\leq\Big{(}\|(\nabla^{2}\rho_{0},\nabla^{2}P_{0})\|_{2}^{2}+C\int_{0}^{T}(\|\nabla(H,L)\|_{2}^{2}+1)dt\Big{)}e^{\displaystyle
C\int_{0}^{T}\big{(}\|\nabla u\|^{2}_{\infty}+1\big{)}dt}\\\
&\displaystyle\leq C,\end{array}$ (4.22)
which also implies that
$\sup_{t\in[0,T]}\big{(}\|(\rho,P(\rho))\|_{H^{2}}+\|(\rho_{t},P(\rho)_{t})\|_{H^{1}}\big{)}+\int_{0}^{T}\|u\|_{H^{3}}^{2}dt\leq
C.$ (4.23)
The proof of Lemma 4.2 is completed. $\hfill\Box$
###### Lemma 4.3
It holds that
$\sup_{t\in[0,T]}\|\sqrt{\rho}u_{t}\|_{2}^{2}(t)+\int_{0}^{T}\|u_{t}\|_{H^{1}}^{2}dt\leq
C.$ (4.24)
Proof: The momentum equation $\eqref{CNS}_{2}$ can be written as
$\rho u_{t}+\rho u\cdot\nabla u+\nabla P(\rho)=\mathcal{L}_{\rho}u:=\mu\Delta
u+\nabla((\mu+\lambda(\rho)){\rm div}u).$ (4.25)
Applying $\partial_{t}$ to the above equation gives that
$\rho u_{tt}+\rho u\cdot\nabla u_{t}+\nabla P(\rho)_{t}=\mu\Delta
u_{t}+\nabla((\mu+\lambda(\rho)){\rm
div}u_{t})-\rho_{t}u_{t}-\rho_{t}u\cdot\nabla u-\rho u_{t}\cdot\nabla
u+\nabla(\lambda(\rho)_{t}{\rm div}u).$ (4.26)
Multiplying the equation (4.26) by $u_{t}$ and integrating the resulting
equation with respect to $x$ over $\mathbb{T}^{2}$ imply that
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int\rho|u_{t}|^{2}dx+\int\Big{(}\mu|\nabla
u_{t}|^{2}+(\mu+\lambda(\rho))|{\rm div}u_{t}|^{2}\Big{)}dx\\\
\displaystyle=-\int\nabla P(\rho)_{t}\cdot
u_{t}dx-\int\rho_{t}|u_{t}|^{2}dx-\int\rho_{t}(u\cdot\nabla u)\cdot u_{t}dx\\\
\displaystyle~{}~{}-\int\rho(u_{t}\cdot\nabla u)\cdot
u_{t}dx+\int\nabla(\lambda(\rho)_{t}{\rm div}u)\cdot u_{t}dx.\end{array}$
(4.27)
Notice that
$\begin{array}[]{ll}\displaystyle-\int\nabla P(\rho)_{t}\cdot
u_{t}dx&\displaystyle=\int P(\rho)_{t}{\rm div}u_{t}dx\\\
&\displaystyle\leq\frac{\mu}{4}\int|{\rm
div}u_{t}|^{2}dx+C\int|P_{t}|^{2}dx\leq\frac{\mu}{4}\int|{\rm
div}u_{t}|^{2}dx+C,\end{array}$ (4.28)
$\begin{array}[]{ll}\displaystyle-\int\rho_{t}|u_{t}|^{2}dx=\int{\rm div}(\rho
u)|u_{t}|^{2}dx=-2\int\rho(u\cdot\nabla u_{t})\cdot u_{t}dx\\\
\qquad\qquad\displaystyle\leq\frac{\mu}{8}\int|\nabla
u_{t}|^{2}dx+C\|u\|_{\infty}^{2}\|\sqrt{\rho}\|_{\infty}^{2}\|\sqrt{\rho}u_{t}\|_{2}^{2}\leq\frac{\mu}{4}\int|\nabla
u_{t}|^{2}dx+C\|\sqrt{\rho}u_{t}\|_{2}^{2},\end{array}$ (4.29)
$\begin{array}[]{ll}\displaystyle-\int\rho_{t}(u\cdot\nabla u)\cdot
u_{t}dx=\int{\rm div}(\rho u)[(u\cdot\nabla u)\cdot u_{t}]dx=-\int\rho
u\cdot\nabla[(u\cdot\nabla u)\cdot u_{t}]dx\\\
\displaystyle\qquad\quad\leq\|\rho\|_{\infty}\|u\|_{\infty}^{2}\|\nabla
u_{t}\|_{2}\|\nabla
u\|_{2}+\|u\|_{\infty}\|\sqrt{\rho}\|_{\infty}\|\sqrt{\rho}u_{t}\|_{2}\big{(}\|\nabla
u\|_{4}^{2}+\|u\|_{\infty}\|\nabla^{2}u\|_{2}\big{)}\\\
\displaystyle\qquad\quad\leq\frac{\mu}{4}\int|\nabla
u_{t}|^{2}dx+C\big{(}\|\sqrt{\rho}u_{t}\|_{2}^{2}+\|(\nabla
u,\nabla^{2}u)\|_{2}^{2}+\|\nabla u\|_{4}^{4}\big{)}\\\
\displaystyle\qquad\quad\leq\frac{\mu}{4}\int|\nabla
u_{t}|^{2}dx+C\big{(}\|\sqrt{\rho}u_{t}\|_{2}^{2}+1\big{)},\end{array}$ (4.30)
$|-\int\rho(u_{t}\cdot\nabla u)\cdot u_{t}dx|\leq\|\nabla
u\|_{\infty}\|\sqrt{\rho}u_{t}\|_{2}^{2},$ (4.31)
and
$\begin{array}[]{ll}\displaystyle\int\nabla(\lambda(\rho)_{t}{\rm div}u)\cdot
u_{t}dx&\displaystyle=-\int\lambda(\rho)_{t}{\rm div}u{\rm div}u_{t}dx\\\
&\displaystyle\leq\frac{\mu}{4}\int|{\rm
div}u_{t}|^{2}dx+C\|\lambda(\rho)_{t}\|_{4}^{2}\|{\rm
div}u\|_{4}^{2}\displaystyle\leq\frac{\mu}{4}\int|{\rm
div}u_{t}|^{2}dx+C.\end{array}$ (4.32)
Substituting the above estimates into (4.27) and then integrating with respect
to $t$ over $[0,t]$ yield that
$\|\sqrt{\rho}u_{t}\|_{2}^{2}(t)+\int_{0}^{t}\|\nabla
u_{t}\|_{2}^{2}dt\leq\|\sqrt{\rho^{\delta}_{0}}u^{\delta}_{t}(0)\|_{2}^{2}+C\int_{0}^{t}(\|\nabla
u\|_{\infty}+1)\|\sqrt{\rho}u_{t}\|^{2}_{2}dt+C.$ (4.33)
By the compatibility condition (3.2), it holds that
$\sqrt{\rho^{\delta}_{0}}u^{\delta}_{t}(0)=\frac{\sqrt{\rho}_{0}}{\sqrt{\rho^{\delta}_{0}}}g-\sqrt{\rho^{\delta}_{0}}u^{\delta}_{0}\cdot\nabla
u^{\delta}_{0},$
thus we have
$\|\sqrt{\rho^{\delta}_{0}}u^{\delta}_{t}(0)\|_{2}^{2}\leq\|\frac{\sqrt{\rho}_{0}}{\sqrt{\rho^{\delta}_{0}}}g\|_{2}^{2}+\|\rho^{\delta}_{0}\|_{\infty}\|u^{\delta}_{0}\|^{2}_{\infty}\|\nabla
u^{\delta}_{0}\|_{2}^{2}\leq C,$
which, together with (4.33) and the Gronwall’s inequality, yields that
$\sup_{t\in[0,T]}\|\sqrt{\rho}u_{t}\|_{2}^{2}(t)+\int_{0}^{T}\|\nabla
u_{t}\|_{2}^{2}dt\leq C.$ (4.34)
By (3.91), for any $1\leq p<+\infty,$
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|u_{t}\|_{p}^{2}dt&\displaystyle\leq\int_{0}^{T}(\|(H,L)\|_{p}^{2}+\|u\|_{\infty}^{2}\|\nabla
u\|_{p}^{2})dt\\\
&\displaystyle\leq\int_{0}^{T}(\|\nabla(H,L)\|_{2}^{2}+\|u\|_{\infty}^{2}\|\nabla
u\|_{p}^{2})dt\leq C.\end{array}$
Therefore, one can arrive at
$\int_{0}^{T}\|u_{t}\|_{H^{1}}^{2}dt\leq C.$
Thus the proof of Lemma 4.3 is completed. $\hfill\Box$
###### Lemma 4.4
It holds that
$\sup_{t\in[0,T]}\|(\rho_{t},P(\rho)_{t},\lambda(\rho)_{t})\|_{H^{1}}(t)+\int_{0}^{T}\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}dt\leq
C.$ (4.35)
Proof: From the continuity equation, it holds that
$\rho_{t}=-u\cdot\nabla\rho-\rho{\rm div}u$ and
$\rho_{tt}=-u_{t}\cdot\nabla\rho-u\cdot\nabla\rho_{t}-\rho_{t}{\rm
div}u-\rho{\rm div}u_{t},$ and thus
$\sup_{t\in[0,T]}\|\nabla\rho_{t}\|_{2}(t)\leq\sup_{t\in[0,T]}\Big{[}\|\nabla\rho\|_{4}\|\nabla
u\|_{4}+\|u\|_{\infty}\|\nabla^{2}\rho\|_{2}+\|\rho\|_{\infty}\|\nabla^{2}u\|_{2}\Big{]}\leq
C.$ (4.36)
and
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|\rho_{tt}\|_{2}^{2}dt&\displaystyle\leq\int_{0}^{T}\Big{[}\|u_{t}\|_{4}^{2}\|\nabla\rho\|_{4}^{2}+\|u\|_{\infty}^{2}\|\nabla\rho_{t}\|_{2}^{2}+\|\rho_{t}\|_{4}^{2}\|\nabla
u\|_{4}^{2}+\|\rho\|_{\infty}^{2}\|\nabla u_{t}\|_{2}^{2}\Big{]}dt\\\
&\displaystyle\leq C\int_{0}^{T}(\|u_{t}\|_{H^{1}}^{2}+1)dt\leq C.\end{array}$
(4.37)
Similarly, we have
$\sup_{t\in[0,T]}\|\nabla(P(\rho)_{t},\lambda(\rho)_{t})\|_{2}(t)+\int_{0}^{T}\|(P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}dt\leq
C.$ (4.38)
Thus the proof of Lemma 4.4 is completed. $\hfill\Box$
###### Lemma 4.5
It holds that
$\begin{array}[]{ll}\displaystyle\sup_{t\in[0,T]}\Big{[}t\|u_{t}\|^{2}_{H^{1}}+t\|u\|^{2}_{H^{3}}+t\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|^{2}_{2}+\|(\rho,P(\rho))\|_{W^{2,q}}\Big{]}\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad+\int_{0}^{T}t\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+\|u_{t}\|_{H^{2}}^{2}(t)+\|u\|_{H^{4}}^{2}\Big{]}dt\leq
C.\end{array}$ (4.39)
Proof: Now multiplying the equation (4.26) by $u_{tt}$ and then integrating
with respect to $x$ over $\mathbb{T}^{2}$ yield that
$\begin{array}[]{ll}\displaystyle\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+\frac{1}{2}\frac{d}{dt}\int\Big{(}\mu|\nabla
u_{t}|^{2}+(\mu+\lambda(\rho))|{\rm
div}u_{t}|^{2}\Big{)}dx=\frac{1}{2}\int\lambda(\rho)_{t}|{\rm
div}u_{t}|^{2}dx\\\ \displaystyle-\int\big{(}\nabla
P_{t}+\rho_{t}u_{t}+\rho_{t}u\cdot\nabla u+\rho u\cdot\nabla u_{t}+\rho
u_{t}\cdot\nabla u\big{)}\cdot u_{tt}dx+\int\nabla(\lambda(\rho)_{t}{\rm
div}u)\cdot u_{tt}dx.\end{array}$ (4.40)
Note that
$\begin{array}[]{ll}\displaystyle\int\nabla(\lambda(\rho)_{t}{\rm div}u)\cdot
u_{tt}dx\displaystyle=-\int\lambda(\rho)_{t}{\rm div}u{\rm div}u_{tt}dx\\\
~{}~{}~{}\displaystyle=-\frac{d}{dt}\int\lambda(\rho)_{t}{\rm div}u{\rm
div}u_{t}dx+\int\big{(}\lambda(\rho)_{t}|{\rm
div}u|^{2}+\lambda(\rho)_{tt}{\rm div}u{\rm div}u_{t}\big{)}dx.\end{array}$
Substituting the above identity into (4.40) yields that
$\begin{array}[]{ll}\displaystyle\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+\frac{1}{2}\frac{d}{dt}\int\Big{(}\mu|\nabla
u_{t}|^{2}+(\mu+\lambda(\rho))|{\rm div}u_{t}|^{2}+\lambda(\rho)_{t}{\rm
div}u{\rm div}u_{t}\Big{)}dx=\frac{3}{2}\int\lambda(\rho)_{t}|{\rm
div}u_{t}|^{2}dx\\\ \displaystyle~{}~{}~{}-\int\big{(}\nabla
P_{t}+\rho_{t}u_{t}+\rho_{t}u\cdot\nabla u+\rho u\cdot\nabla u_{t}+\rho
u_{t}\cdot\nabla u\big{)}\cdot u_{tt}dx+\int\lambda(\rho)_{tt}{\rm div}u{\rm
div}u_{t}dx.\end{array}$ (4.41)
Note that $\lambda(\rho)$ satisfies the transport equation
$\lambda(\rho)_{t}=-u\cdot\nabla\lambda(\rho)-\rho\lambda^{\prime}(\rho){\rm
div}u,$ and then it holds that
$\begin{array}[]{ll}\displaystyle|\frac{3}{2}\int\lambda(\rho)_{t}|{\rm
div}u_{t}|^{2}dx|&\displaystyle=|-\frac{3}{2}\int
u\cdot\nabla\lambda(\rho)|{\rm
div}u_{t}|^{2}dx-\frac{3}{2}\int\rho\lambda^{\prime}(\rho){\rm div}u|{\rm
div}u_{t}|^{2}dx|\\\ &\displaystyle=|3\int\lambda(\rho){\rm
div}u_{t}u\cdot\nabla({\rm
div}u_{t})dx+\frac{3}{2}\int(\lambda(\rho)-\rho\lambda^{\prime}(\rho)){\rm
div}u|{\rm div}u_{t}|^{2}dx|\\\ &\displaystyle\leq
C\|\lambda(\rho)u\|_{\infty}\|{\rm div}u_{t}\|_{2}\|\nabla({\rm
div}u_{t})\|_{2}+C\|\lambda(\rho)-\rho\lambda^{\prime}(\rho)\|_{\infty}\|\nabla
u\|_{\infty}\|{\rm div}u_{t}\|_{2}^{2}\\\ &\displaystyle\leq C\|{\rm
div}u_{t}\|_{2}\|\nabla({\rm div}u_{t})\|_{2}+C\|\nabla u\|_{\infty}\|{\rm
div}u_{t}\|_{2}^{2}.\end{array}$ (4.42)
It follows from (4.26) that
$\mathcal{L}_{\rho}u_{t}=\rho u_{tt}+\rho_{t}u_{t}+(\rho u\cdot\nabla
u)_{t}+\nabla P(\rho)_{t}+\nabla(\lambda(\rho)_{t}{\rm div}u).$
Then the standard elliptic estimates show that
$\begin{array}[]{ll}\displaystyle\|\nabla^{2}u_{t}\|_{2}&\displaystyle\leq
C\Big{[}\|\sqrt{\rho}\|_{\infty}\|\sqrt{\rho}u_{tt}\|_{2}+\|\rho_{t}\|_{4}\|u_{t}\|_{4}+\|\rho_{t}\|_{4}\|u\|_{\infty}\|\nabla
u\|_{4}+\|\rho\|_{\infty}\|u_{t}\|_{4}\|\nabla u\|_{4}\\\
&\displaystyle~{}~{}~{}+\|\rho u\|_{\infty}\|\nabla u_{t}\|_{2}+\|\nabla
P(\rho)_{t}\|_{2}+\|\nabla\lambda(\rho)_{t}\|_{2}\|{\rm
div}u\|_{\infty}+\|\lambda(\rho)_{t}\|_{4}\|\nabla^{2}u\|_{4}\Big{]}\\\
&\displaystyle\leq C\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}+\|u_{t}\|_{4}+1+\|\nabla
u_{t}\|_{2}+\|{\rm div}u\|_{\infty}+\|\nabla^{2}u\|_{4}\Big{]}\\\
&\displaystyle\leq C\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}+\|u_{t}\|_{4}+1+\|\nabla
u_{t}\|_{2}+\|{\rm div}u\|_{\infty}+\|\nabla^{3}u\|_{2}\Big{]}.\end{array}$
(4.43)
Substituting (4.43) into (4.42) yields that
$|\frac{3}{2}\int\lambda(\rho)_{t}|{\rm
div}u_{t}|^{2}dx|\leq\frac{1}{8}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+C(\|\nabla
u\|_{\infty}+1)\|\nabla
u_{t}\|_{2}^{2}+C\big{(}\|u_{t}\|_{4}^{2}+\|\nabla^{3}u\|_{2}^{2}+\|\nabla
u\|_{\infty}^{2}).$ (4.44)
At the same time, it holds that
$\begin{array}[]{ll}\displaystyle-\int\nabla P(\rho)_{t}\cdot
u_{tt}dx&\displaystyle=\int P(\rho)_{t}{\rm div}u_{tt}dx=\frac{d}{dt}\int
P(\rho)_{t}{\rm div}u_{t}dx-\int P(\rho)_{tt}{\rm div}u_{t}dx\\\
&\displaystyle\leq\frac{d}{dt}\int P(\rho)_{t}{\rm
div}u_{t}dx+\|P(\rho)_{tt}\|_{2}^{2}+\|{\rm div}u_{t}\|_{2}^{2},\end{array}$
(4.45) $\begin{array}[]{ll}\displaystyle-\int\rho_{t}u_{t}\cdot
u_{tt}dx&\displaystyle=\int\rho_{t}\big{(}\frac{|u_{t}|^{2}}{2}\big{)}_{t}dx=\frac{d}{dt}\int\rho_{t}\frac{|u_{t}|^{2}}{2}dx-\int\rho_{tt}\frac{|u_{t}|^{2}}{2}dx,\end{array}$
(4.46)
while
$\begin{array}[]{ll}\displaystyle-\int\rho_{tt}\frac{|u_{t}|^{2}}{2}dx&\displaystyle=\int{\rm
div}(\rho u)_{t}\frac{|u_{t}|^{2}}{2}dx=\int(\rho u)_{t}\cdot\nabla u_{t}\cdot
u_{t}dx\\\
&\displaystyle\leq\|\sqrt{\rho}\|_{\infty}\|\sqrt{\rho}u_{t}\|_{2}\|u_{t}\|_{4}\|\nabla
u_{t}\|_{4}+\|u\|_{\infty}\|\rho_{t}\|_{4}\|u_{t}\|_{4}\|\nabla u_{t}\|_{2}\\\
&\displaystyle\leq C\Big{[}\|u_{t}\|_{4}\|\nabla
u_{t}\|_{4}+\|u_{t}\|_{4}\|\nabla u_{t}\|_{2}\Big{]}\leq
C\Big{[}\|u_{t}\|_{4}\|\nabla^{2}u_{t}\|_{2}+\|u_{t}\|_{4}\|\nabla
u_{t}\|_{2}\Big{]}\\\ &\displaystyle\leq
C\|u_{t}\|_{4}\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}+\|u_{t}\|_{4}+1+\|\nabla
u_{t}\|_{2}+\|{\rm div}u\|_{\infty}+\|\nabla^{3}u\|_{2}\Big{]}\\\
&\displaystyle\leq\frac{1}{8}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+C\Big{[}\|u_{t}\|^{2}_{4}+1+\|\nabla
u_{t}\|^{2}_{2}+\|{\rm
div}u\|^{2}_{\infty}+\|\nabla^{3}u\|^{2}_{2}\Big{]}.\end{array}$ (4.47)
Moreover, it follows that
$\begin{array}[]{ll}\displaystyle-\int\rho_{t}u\cdot\nabla u\cdot u_{tt}dx\\\
\displaystyle=-\frac{d}{dt}\int\rho_{t}u\cdot\nabla u\cdot
u_{t}dx+\int\rho_{tt}u\cdot\nabla u\cdot u_{t}dx\int\rho_{t}u_{t}\cdot\nabla
u\cdot u_{t}dx+\int\rho_{t}u\cdot\nabla u_{t}\cdot u_{t}dx\\\
\displaystyle=-\frac{d}{dt}\int\rho_{t}u\cdot\nabla u\cdot
u_{t}dx+\|\rho_{tt}\|_{2}\|u\|_{\infty}\|\nabla u\|_{4}\|u_{t}\|_{4}\\\
\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\|\rho_{t}\|_{4}\|u_{t}\|_{4}^{2}\|\nabla
u\|_{4}+\|\rho_{t}\|_{4}\|u\|_{\infty}\|\nabla u_{t}\|_{2}\|u_{t}\|_{4}\\\
\displaystyle\leq-\frac{d}{dt}\int\rho_{t}u\cdot\nabla u\cdot
u_{t}dx+C\Big{[}\|\rho_{tt}\|_{2}\|u_{t}\|_{4}+\|u_{t}\|_{4}^{2}+\|\nabla
u_{t}\|_{2}\|u_{t}\|_{4}\Big{]}\\\
\displaystyle\leq-\frac{d}{dt}\int\rho_{t}u\cdot\nabla u\cdot
u_{t}dx+C\Big{[}\|\rho_{tt}\|_{2}^{2}+\|u_{t}\|_{4}^{2}+\|\nabla
u_{t}\|_{2}^{2}\Big{]},\end{array}$ (4.48)
$\begin{array}[]{ll}\displaystyle-\int\rho u\cdot\nabla u_{t}\cdot
u_{tt}dx&\displaystyle\leq\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}u\|_{\infty}\|\nabla
u_{t}\|_{2}\leq\frac{1}{8}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+C\|\nabla
u_{t}\|_{2}^{2},\end{array}$ (4.49) $\begin{array}[]{ll}\displaystyle-\int\rho
u_{t}\cdot\nabla u\cdot
u_{tt}dx&\displaystyle\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}\|_{\infty}\|\nabla
u\|_{4}\|u_{t}\|_{4}\leq\frac{1}{8}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+C\|u_{t}\|_{4}^{2},\end{array}$
(4.50)
and
$\begin{array}[]{ll}\displaystyle-\int\lambda(\rho)_{tt}{\rm div}u{\rm
div}u_{t}dx&\displaystyle\leq\|\lambda(\rho)_{tt}\|_{2}\|\nabla
u\|_{\infty}\|{\rm
div}u_{t}\|_{2}\leq\frac{1}{2}\Big{[}\|\lambda(\rho)_{tt}\|_{2}^{2}+\|\nabla
u\|^{2}_{\infty}\|{\rm div}u_{t}\|_{2}^{2}\Big{]}.\end{array}$ (4.51)
Collecting all the above estimates and substituting them into (4.41) yield
that
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+\frac{d}{dt}G(t)\\\
\displaystyle\leq
C\Big{[}\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}+\|u_{t}\|_{4}^{2}+\|\nabla^{3}u\|_{2}^{2}+(\|\nabla
u\|_{\infty}^{2}+1)(\|\nabla u_{t}\|_{2}^{2}+1)\Big{]}\end{array}$ (4.52)
where
$G(t)=\int\Big{(}\mu|\nabla u_{t}|^{2}+(\mu+\lambda(\rho))|{\rm
div}u_{t}|^{2}+\lambda(\rho)_{t}{\rm div}u{\rm div}u_{t}-P(\rho)_{t}{\rm
div}u_{t}+\rho_{t}\frac{|u_{t}|^{2}}{2}+\rho_{t}u\cdot\nabla u\cdot
u_{t}\Big{)}dx.$ (4.53)
Note that
$\begin{array}[]{ll}\displaystyle|\int\lambda(\rho)_{t}{\rm div}u{\rm
div}u_{t}dx|&\displaystyle\leq\|\lambda(\rho)_{t}\|_{4}\|{\rm
div}u\|_{4}\|{\rm div}u_{t}\|_{2}\\\ &\displaystyle\leq\frac{\mu}{8}\|\nabla
u_{t}\|_{2}^{2}+C\|\lambda(\rho)_{t}\|^{2}_{4}\|{\rm
div}u\|_{4}^{2}\leq\frac{\mu}{8}\|\nabla u_{t}\|_{2}^{2}+C,\end{array}$
$|-\int P(\rho)_{t}{\rm div}u_{t}dx|\leq\frac{\mu}{8}\|\nabla
u_{t}\|_{2}^{2}+C\|P(\rho)_{t}\|_{2}^{2}\leq\frac{\mu}{8}\|\nabla
u_{t}\|_{2}^{2}+C,$
$\begin{array}[]{ll}\displaystyle|\int\rho_{t}\frac{|u_{t}|^{2}}{2}dx|&\displaystyle=|\int{\rm
div}(\rho u)\frac{|u_{t}|^{2}}{2}dx|=|\int\rho u\cdot\nabla u_{t}\cdot
u_{t}dx|\\\
&\displaystyle\leq\|\sqrt{\rho}u_{t}\|_{2}\|\sqrt{\rho}u\|_{\infty}\|\nabla
u_{t}\|_{2}\leq\frac{\mu}{8}\|\nabla u_{t}\|_{2}^{2}+C,\end{array}$
and
$\begin{array}[]{ll}\displaystyle|\int\rho_{t}u\cdot\nabla u\cdot
u_{t}dx|&\displaystyle=|\int{\rm div}(\rho u)(u\cdot\nabla u\cdot
u_{t})dx|=|\int\rho u\cdot\nabla(u\cdot\nabla u\cdot u_{t})dx|\\\
&\displaystyle\leq\|\sqrt{\rho}u_{t}\|_{2}\|\sqrt{\rho}u\|_{\infty}\big{(}\|\nabla
u\|_{4}^{2}+\|u\|_{\infty}\|\nabla^{2}u\|_{2}\big{)}+\|\rho|u|^{2}\|_{\infty}\|\nabla
u_{t}\|_{2}\|\nabla u\|_{2}\\\ &\displaystyle\leq\frac{\mu}{8}\|\nabla
u_{t}\|_{2}^{2}+C.\end{array}$
Therefore, it holds that
$C_{1}(\|\nabla u_{t}\|_{2}^{2}-1)\leq G(t)\leq C(\|\nabla
u_{t}\|_{2}^{2}+1),$ (4.54)
for some positive constants $C,C_{1}$.
Now from (6.9), we can arrive at
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+\frac{d}{dt}G(t)\\\
\displaystyle\leq
C\Big{[}\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}+\|u_{t}\|_{4}^{2}+\|\nabla^{3}u\|_{2}^{2}+(\|\nabla
u\|_{\infty}^{2}+1)(G(t)+1)\Big{]}.\end{array}$ (4.55)
Multiplying the above inequality by $t$ and then integrating the resulting
inequality with respect to $t$ over the interval $[\tau,t_{1}]$ with
$\tau,t_{1}\in[0,T]$ give that
$\begin{array}[]{ll}\displaystyle\int_{\tau}^{t_{1}}t\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)dt+t_{1}G(t_{1})\leq
C\tau G(\tau)+C\int_{\tau}^{t_{1}}\Big{[}(\|\nabla
u\|_{\infty}^{2}+1)(tG(t)+1)\Big{]}dt\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad+C\int_{\tau}^{t_{1}}\Big{[}\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}+\|u_{t}\|_{4}^{2}+\|\nabla^{3}u\|_{2}^{2}+G(t)\Big{]}dt.\end{array}$
(4.56)
It follows from Lemma 4.3 and (4.54) that $G(t)\in L^{1}(0,T)$. Thus, due to
[6], there exists a subsequence $\tau_{k}$ such that
$\tau_{k}\rightarrow 0,\qquad\tau_{k}G(\tau_{k})\rightarrow 0,\qquad{\rm
as}~{}~{}k\rightarrow+\infty.$ (4.57)
Taking $\tau=\tau_{k}$ in (4.56), then $k\rightarrow+\infty$ and using the
Gronwall’s inequality, one gets that
$\sup_{t\in[0,T]}\big{[}t\|\nabla
u_{t}\|_{2}^{2}(t)\big{]}+\int_{0}^{T}t\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)dt\leq
C.$ (4.58)
Note that (4.43) implies that
$\sup_{t\in[0,T]}\big{[}t\|(\rho_{tt},P(\rho)_{tt},\lambda(\rho)_{tt})\|_{2}^{2}(t)\big{]}+\int_{0}^{T}t\|\nabla^{2}u_{t}\|_{2}^{2}(t)dt\leq
C.$ (4.59)
It follows from (3.91) that
$\nabla(L,H)^{t}=\nabla u_{t}-\nabla(u\cdot\nabla u).$
Consequently, it holds that
$\sup_{t\in[0,T]}\big{[}t\|\nabla(L,H)^{t}\|^{2}_{2}(t)\big{]}\leq C,$ (4.60)
which, together with (3.74), implies that
$\sup_{t\in[0,T]}\big{[}t\|(L,H)^{t}\|^{2}_{2}(t)\big{]}\leq
C\sup_{t\in[0,T]}\big{[}t\|\nabla(L,H)^{t}\|^{2}_{2}(t)\big{]}\leq C.$ (4.61)
Therefore, it follows that
$\sup_{t\in[0,T]}\big{[}t\|u_{t}\|^{2}_{2}(t)\big{]}\leq
C\sup_{t\in[0,T]}\big{[}t\|(L,H)^{t}\|^{2}_{2}(t)+t\|u\cdot\nabla
u\|_{2}^{2}(t)\big{]}\leq C.$ (4.62)
So one can infer further that
$\sup_{t\in[0,T]}\big{[}t\|u_{t}\|^{2}_{H^{1}}(t)\big{]}+\int_{0}^{T}t\|u_{t}\|_{H^{2}}^{2}(t)dt\leq
C.$ (4.63)
Applying $\partial_{x_{j}x_{k}}$, $j,k=1,2$, to $\eqref{CNS}_{1}$ gives
$\begin{array}[]{ll}\displaystyle(\rho_{x_{j}x_{k}})_{t}+u\cdot\nabla(\rho_{x_{j}x_{k}})+u_{x_{j}x_{k}}\cdot\nabla\rho+u_{x_{j}}\cdot\nabla\rho_{x_{k}}+u_{x_{k}}\cdot\nabla\rho_{x_{j}}\\\
\displaystyle\qquad~{}~{}~{}+\rho_{x_{j}x_{k}}{\rm div}u+\rho_{x_{j}}({\rm
div}u)_{x_{k}}+\rho_{x_{k}}({\rm div}u)_{x_{j}}+\rho({\rm
div}u)_{x_{j}x_{k}}=0.\end{array}$
Multiplying the above equation by $q|\nabla^{2}\rho|^{q-2}\rho_{x_{j}x_{k}}$
with $q>2$ given in Theorem 1.1 and summing over $j,k=1,2$ give that
$\begin{array}[]{ll}\displaystyle(|\nabla^{2}\rho|^{q})_{t}+{\rm
div}(u|\nabla^{2}\rho|^{q})+(q-1)|\nabla^{2}\rho|^{q}{\rm
div}u+q|\nabla^{2}\rho|^{q-2}\rho_{x_{j}x_{k}}\Big{[}u_{x_{j}x_{k}}\cdot\nabla\rho\\\
\displaystyle+u_{x_{j}}\cdot\nabla\rho_{x_{k}}+u_{x_{k}}\cdot\nabla\rho_{x_{j}}+\rho_{x_{j}}({\rm
div}u)_{x_{k}}+\rho_{x_{k}}({\rm div}u)_{x_{j}}+\rho({\rm
div}u)_{x_{j}x_{k}}\Big{]}=0.\end{array}$
Integrating the above equality with respect to $x$ over $\mathbb{T}^{2}$ leads
to that
$\frac{d}{dt}\|\nabla^{2}\rho\|_{q}^{q}\leq(q-1)\|{\rm
div}u\|_{\infty}\|\nabla^{2}\rho\|_{q}^{q}+Cq\|\nabla^{2}\rho\|_{q}^{q-1}\Big{[}\|\nabla\rho\|_{2q}\|\nabla^{2}u\|_{2q}+\|\nabla
u\|_{\infty}\|\nabla^{2}\rho\|_{q}+\|\rho\|_{\infty}\|\nabla^{3}u\|_{q}\Big{]}.$
Thus one can get
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla^{2}\rho\|_{q}&\displaystyle\leq
C\Big{[}\|\nabla
u\|_{\infty}\|\nabla^{2}\rho\|_{q}+\|\nabla\rho\|_{2q}\|\nabla^{2}u\|_{2q}+\|\rho\|_{\infty}\|\nabla^{3}u\|_{q}\Big{]}\\\
&\displaystyle\leq C\Big{[}\|\nabla
u\|_{\infty}\|\nabla^{2}\rho\|_{q}+\|\nabla^{2}u\|_{W^{1,q}}\Big{]},\end{array}$
(4.64)
where $q>2$. Similarly, one can obtain
$\displaystyle\frac{d}{dt}\|\nabla^{2}P\|_{q}\leq C\Big{[}\|\nabla
u\|_{\infty}\|\nabla^{2}P\|_{q}+\|\nabla^{2}u\|_{W^{1,q}}\Big{]}.$ (4.65)
Apply $\partial_{x_{i}}$ with $i=1,2$ to the elliptic system
$\mathcal{L}_{\rho}u=\rho u_{t}+\rho u\cdot\nabla u+\nabla P(\rho)$ to get
$\mathcal{L}_{\rho}u_{x_{i}}=-\nabla(\lambda(\rho)_{x_{i}}{\rm
div}u)+\rho_{x_{i}}u_{t}+\rho u_{x_{i}t}+\rho_{x_{i}}u\cdot\nabla u+\rho
u_{x_{i}}\cdot\nabla u+\rho u\cdot\nabla u_{x_{i}}+\nabla
P(\rho)_{x_{i}}:=\Psi.$
Then the standard elliptic regularity estimates imply that
$\begin{array}[]{ll}\displaystyle\|\nabla u\|_{W^{2,q}}&\displaystyle\leq
C\Big{[}\|\nabla u\|_{q}+\|\Psi\|_{q}\Big{]}\\\ &\displaystyle\leq
C\Big{[}1+(\|\nabla
u\|_{\infty}+1)\|(\nabla^{2}\rho,\nabla^{2}P)\|_{q}+\|\nabla
u\|_{W^{1,q}}+\|u_{t}\|_{W^{1,q}}\Big{]}\\\ &\displaystyle\leq
C\Big{[}1+(\|\nabla
u\|_{\infty}+1)\|(\nabla^{2}\rho,\nabla^{2}P)\|_{q}+\|u\|_{H^{3}}+\|u_{t}\|_{H^{1}}+\|\nabla
u_{t}\|_{q}\Big{]}.\end{array}$ (4.66)
Thus it follows from (4.64), (4.65) and (4.66) that
$\frac{d}{dt}\|(\nabla^{2}\rho,\nabla^{2}P)\|_{q}\leq C\Big{[}1+(\|\nabla
u\|_{\infty}+1)\|(\nabla^{2}\rho,\nabla^{2}P)\|_{q}+\|u\|_{H^{3}}+\|u_{t}\|_{H^{1}}+\|\nabla
u_{t}\|_{q}\Big{]}.$ (4.67)
Note that Lemma 2.2 implies that
$\begin{array}[]{ll}\displaystyle\int_{0}^{T}\|\nabla
u_{t}\|_{q}(t)dt&\displaystyle\leq
C\int_{0}^{T}\|\nabla^{2}u_{t}\|_{2}(t)dt\leq
C\sup_{t\in[0,T]}\big{[}\sqrt{t}\|\nabla^{2}u_{t}\|_{2}(t)\big{]}\int_{0}^{T}t^{-\frac{1}{2}}dt\leq
C.\end{array}$
Therefore, it follows from (4.67) and the Gronwall’s inequality that
$\begin{array}[]{ll}\displaystyle\|(\nabla^{2}\rho,\nabla^{2}P(\rho))\|_{q}(t)\leq\Big{(}\|(\nabla^{2}\rho_{0},\nabla^{2}P(\rho_{0}))\|_{q}\\\
\displaystyle\qquad\qquad+C\int_{0}^{t}(1+\|u\|_{H^{3}}+\|u_{t}\|_{H^{1}}+\|\nabla
u_{t}\|_{q})ds\Big{)}e^{\displaystyle C\int_{0}^{t}(\|\nabla
u\|_{\infty}(s)+1)ds}\leq C,\end{array}$ (4.68)
which then gives
$\sup_{t\in[0,T]}\|(\rho,P(\rho))\|_{W^{2,q}(\mathbb{T}^{2})}\leq C.$ (4.69)
So the proof of Lemma 4.5 is completed. $\hfill\Box$
###### Lemma 4.6
It holds that for any $0<\tau\leq T$,
$\sup_{t\in[0,T]}\big{[}t^{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+t^{2}\|u_{t}\|_{H^{2}}^{2}+t^{2}\|u\|_{W^{3,q}}^{2}\big{]}+\int_{0}^{T}t^{2}\|\nabla
u_{tt}\|_{2}^{2}(t)dt\leq C.$ (4.70)
Proof: Applying $\partial_{t}$ to the equation (4.26) gives that
$\begin{array}[]{ll}&\displaystyle\rho u_{ttt}+\rho u\cdot\nabla
u_{tt}-\mathcal{L}_{\rho}u_{tt}=-\nabla p_{tt}-\rho_{tt}(u_{t}+u\cdot\nabla
u)-2\rho_{t}(u_{tt}+u_{t}\cdot\nabla u+u\cdot\nabla u_{t})\\\
&\displaystyle-2\rho u_{t}\cdot\nabla u_{t}-\rho u_{tt}\cdot\nabla
u+2\nabla((\lambda(\rho))_{t}{\rm div}u_{t})+\nabla(\lambda(\rho)_{tt}{\rm
div}u).\end{array}$ (4.71)
Multiplying the equation (4.71) by $u_{tt}$ and integrating the resulting
equation with respect to $x$ over $\mathbb{T}^{2}$ yield that
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\int\rho|u_{tt}|^{2}dx+\int\mu|\nabla
u_{tt}|^{2}+(\mu+\lambda(\rho))({\rm div}u_{tt})^{2}dx=\int p_{tt}{\rm
div}u_{tt}dx\\\ \displaystyle\qquad-\int\rho_{tt}(u_{t}+u\cdot\nabla u)\cdot
u_{tt}dx-2\int\rho_{t}(u_{tt}+u_{t}\cdot\nabla u+u\cdot\nabla u_{t})\cdot
u_{tt}dx-2\int\rho u_{t}\cdot\nabla u_{t}\cdot u_{tt}dx\\\
\displaystyle\qquad-\int\rho u_{tt}\cdot\nabla u\cdot
u_{tt}dx-2\int\lambda(\rho)_{t}{\rm div}u_{t}{\rm
div}u_{tt}dx-\int\lambda(\rho)_{tt}{\rm div}u{\rm div}u_{tt}dx.\end{array}$
Multiply the above equality by $t^{2}$ to get that
$\begin{array}[]{ll}\displaystyle\frac{1}{2}\frac{d}{dt}\Big{(}t^{2}\int\rho|u_{tt}|^{2}dx\Big{)}-t\int\rho|u_{tt}|^{2}dx+t^{2}\int\mu|\nabla
u_{tt}|^{2}+(\mu+\lambda(\rho))({\rm div}u_{tt})^{2}dx=t^{2}\int P_{tt}{\rm
div}u_{tt}dx\\\ \displaystyle-t^{2}\int\rho_{tt}(u_{t}+u\cdot\nabla u)\cdot
u_{tt}dx-2t^{2}\int\rho_{t}(u_{tt}+u_{t}\cdot\nabla u+u\cdot\nabla u_{t})\cdot
u_{tt}dx-2t^{2}\int\rho u_{t}\cdot\nabla u_{t}\cdot u_{tt}dx\\\
\displaystyle-t^{2}\int\rho u_{tt}\cdot\nabla u\cdot
u_{tt}dx-2t^{2}\int\lambda(\rho)_{t}{\rm div}u_{t}{\rm
div}u_{tt}dx-t^{2}\int\lambda(\rho)_{tt}{\rm div}u{\rm
div}u_{tt}dx:=\sum_{i=1}^{7}I_{i}.\end{array}$ (4.72)
Clearly,
$|I_{1}|\leq\alpha t^{2}\|{\rm
div}u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\|P_{tt}\|_{2}^{2}.$
Now we estimate $I_{2}$, which is a little more delicate due to the absence of
estimates for $u_{tt}$. First, rewrite $I_{2}$ as
$\begin{array}[]{ll}\displaystyle I_{2}&\displaystyle=t^{2}\int{\rm div}(\rho
u)_{t}(L,H)^{t}\cdot u_{tt}dx=-t^{2}\int(\rho
u)_{t}\cdot\nabla\big{[}(L,H)^{t}\cdot u_{tt}\big{]}dx\\\
&\displaystyle=-t^{2}\int\rho u_{t}\cdot\nabla\big{[}(L,H)^{t}\cdot
u_{tt}\big{]}dx-t^{2}\int\rho_{t}u\cdot\nabla
u_{tt}\cdot(L,H)^{t}dx-t^{2}\int\rho_{t}u\cdot\nabla(L,H)^{t}\cdot u_{tt}dx\\\
&\displaystyle=-t^{2}\int\rho u_{t}\cdot\nabla\big{[}(L,H)^{t}\cdot
u_{tt}\big{]}dx-t^{2}\int\rho_{t}u\cdot\nabla u_{tt}\cdot(L,H)^{t}dx\\\
&\displaystyle\qquad\qquad\qquad\qquad\qquad-t^{2}\int\rho
u\cdot\nabla\big{[}u\cdot\nabla(L,H)^{t}\cdot
u_{tt}\big{]}dx:=I_{21}+I_{22}+I_{23}\end{array}$
where the superscript t means the transpose of the vector $(L,H)$.
Now, direct estimates yields that
$\begin{array}[]{ll}|I_{21}|&\displaystyle\leq
t^{2}\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}\|_{\infty}\|u_{t}\|_{\infty}\|\nabla(L,H)^{t}\|_{2}+t^{2}\|\rho\|_{\infty}\|\nabla
u_{tt}\|_{2}\|u_{t}\|_{4}\|(L,H)^{t}\|_{4}\\\ &\displaystyle\leq
Ct^{2}\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}\|u_{t}\|_{H^{2}}\|\nabla(L,H)^{t}\|_{2}+\|\nabla
u_{tt}\|_{2}\|u_{t}\|_{H^{1}}\|\nabla(L,H)^{t}\|_{2}\Big{]}\\\
&\displaystyle\leq\alpha t^{2}\|\nabla
u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}^{2}\|\nabla(L,H)^{t}\|^{2}_{2}+\|u_{t}\|_{H^{2}}^{2}+\|u_{t}\|_{H^{1}}^{2}\|\nabla(L,H)^{t}\|^{2}_{2}\Big{]}\\\
&\displaystyle\leq\alpha t^{2}\|\nabla
u_{tt}\|_{2}^{2}+C_{\alpha}\Big{[}t^{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}\|\nabla(L,H)^{t}\|^{2}_{2}+t^{2}\|u_{t}\|_{H^{2}}^{2}+\|\nabla(L,H)^{t}\|^{2}_{2}\Big{]}\end{array}$
(4.73)
where in the last inequality one has used Lemma 4.5.
Similarly, one can obtain
$\begin{array}[]{ll}|I_{22}|&\displaystyle\leq t^{2}\|\nabla
u_{tt}\|_{2}\|u\|_{\infty}\|\rho_{t}\|_{4}\|(L,H)^{t}\|_{4}\\\
&\displaystyle\leq\alpha t^{2}\|\nabla
u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\|\rho_{t}\|_{H^{1}}^{2}\|\nabla(L,H)^{t}\|_{2}^{2}\leq\alpha
t^{2}\|\nabla
u_{tt}\|_{2}^{2}+C_{\alpha}\|\nabla(L,H)^{t}\|_{2}^{2}\end{array}$ (4.74)
and
$\begin{array}[]{ll}|I_{23}|&\displaystyle\leq
t^{2}\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}u\|_{\infty}\|\nabla
u\|_{\infty}\|\nabla(L,H)^{t}\|_{2}+\|\nabla u_{tt}\|_{2}\|\rho
u^{2}\|_{\infty}\|\nabla(L,H)^{t}\|_{2}\\\
&\displaystyle\qquad\qquad\qquad+\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}u^{2}\|_{\infty}(\|\nabla^{2}u_{t}\|_{2}+\|u\|_{\infty}\|\nabla^{3}u\|_{2}+\|\nabla
u\|_{\infty}\|\nabla^{2}u\|_{2})\Big{]}\\\ &\displaystyle\leq\alpha
t^{2}\|\nabla u_{tt}\|_{2}^{2}+C_{\alpha}\|\nabla(L,H)^{t}\|_{2}^{2}\\\
&\displaystyle\qquad+C\Big{[}t^{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t\|\nabla
u\|_{\infty}^{2}+1)+t^{2}\|\nabla^{2}u_{t}\|_{2}^{2}+t^{2}\|u\|_{H^{3}}^{2}+\|\nabla(L,H)^{t}\|_{2}^{2}\Big{]}.\end{array}$
(4.75)
Continuing, using the lemmas obtained so far, one can get that
$\begin{array}[]{ll}|I_{3}|&\displaystyle\leq
t^{2}\Big{[}\|\sqrt{\rho}u\|_{\infty}\|\nabla
u_{tt}\|_{2}\|\sqrt{\rho}u_{tt}\|_{2}+\|\rho u\|_{\infty}\|\nabla
u_{tt}\|_{2}(\|u_{t}\cdot\nabla u\|_{2}+\|u\cdot\nabla u_{t}\|_{2})\\\
&\displaystyle\qquad\quad+\|\sqrt{\rho}u\|_{\infty}\|\sqrt{\rho}u_{tt}\|_{2}(\|\nabla(u_{t}\cdot\nabla
u)\|_{2}+\|\nabla(u\cdot\nabla u_{t})\|_{2})\Big{]}\\\ &\displaystyle\leq
t^{2}\Big{[}\|\sqrt{\rho}u\|_{\infty}\|\nabla
u_{tt}\|_{2}\|\sqrt{\rho}u_{tt}\|_{2}+\|\rho u\|_{\infty}\|\nabla
u_{tt}\|_{2}(\|u_{t}\|_{4}\|\nabla u\|_{4}+\|u\|_{\infty}\|\nabla
u_{t}\|_{2})\\\
&\displaystyle\qquad\quad+\|\sqrt{\rho}u\|_{\infty}\|\sqrt{\rho}u_{tt}\|_{2}(\|u_{t}\|_{4}\|\nabla^{2}u\|_{4}+\|\nabla
u_{t}\|_{4}\|\nabla u\|_{4}+\|u\|_{\infty}\|\nabla^{2}u_{t}\|_{2})\Big{]}\\\
&\displaystyle\leq\alpha t^{2}\|\nabla
u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\Big{[}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+\|u_{t}\|_{H^{1}}^{2}(\|\nabla^{3}u\|^{2}_{2}+1)+\|u_{t}\|_{H^{2}}^{2}\Big{]},\end{array}$
(4.76) $\begin{array}[]{ll}\displaystyle|I_{4}|&\displaystyle\leq
t^{2}\|\sqrt{\rho}u_{tt}\|_{2}\|\sqrt{\rho}\|_{\infty}\|u_{t}\|_{4}\|\nabla
u_{t}\|_{4}\\\ &\displaystyle\leq
Ct^{2}\|\sqrt{\rho}u_{tt}\|_{2}\|u_{t}\|_{H^{1}}\|\nabla^{2}u_{t}\|_{2}\leq
Ct^{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}+Ct^{2}\|u_{t}\|^{2}_{H^{1}}\|\nabla^{2}u_{t}\|^{2}_{2},\end{array}$
(4.77) $|I_{5}|\leq Ct^{2}\|\sqrt{\rho}u_{tt}\|^{2}_{2}\|\nabla u\|_{\infty},$
(4.78)
and
$\begin{array}[]{ll}\displaystyle|I_{6}|&\displaystyle\leq t^{2}\|{\rm
div}u_{tt}\|_{2}\|\lambda(\rho)_{t}\|_{4}\|\nabla u_{t}\|_{4}\leq\alpha
t^{2}\|{\rm
div}u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\|\nabla^{2}u_{t}\|_{2}^{2},\end{array}$
(4.79) $\begin{array}[]{ll}\displaystyle|I_{7}|&\displaystyle\leq t^{2}\|{\rm
div}u_{tt}\|_{2}\|\lambda(\rho)_{tt}\|_{2}\|\nabla u\|_{\infty}\leq\alpha
t^{2}\|{\rm
div}u_{tt}\|_{2}^{2}+C_{\alpha}t^{2}\|\lambda(\rho)_{tt}\|_{2}^{2}\|u\|_{H^{3}}^{2}.\end{array}$
(4.80)
Substituting the above estimates on $I_{i}~{}(i=1,2,\cdots,7)$ into (4.72) and
then integrating the resulting inequality with respect $t$ over $[\tau,t_{1}]$
with $\tau,t_{1}\in[0,T]$ give that
$t_{1}^{2}\|\sqrt{\rho}u_{tt}(t_{1})\|_{2}^{2}+\int_{\tau}^{t_{1}}t^{2}\|\nabla
u_{tt}\|_{2}^{2}dt\leq C+C\tau^{2}\|\sqrt{\rho}u_{tt}(\tau)\|_{2}^{2}.$ (4.81)
Since $t\sqrt{\rho}u_{tt}\in L^{2}([0,T]\times\mathbb{T}^{2})$ due to Lemma
4.5, there exists a subsequence $\tau_{k}$ such that
$\tau_{k}\rightarrow
0,\qquad\tau_{k}^{2}\|\sqrt{\rho}u_{tt}(\tau_{k})\|_{2}^{2}\rightarrow
0,\qquad{\rm as}~{}~{}k\rightarrow+\infty.$ (4.82)
Letting $\tau=\tau_{k}$ in (4.81) and $k\rightarrow+\infty$, one gets that
$t^{2}\|\sqrt{\rho}u_{tt}(t)\|_{2}^{2}+\int_{0}^{t}s^{2}\|\nabla
u_{tt}(s)\|_{2}^{2}dt\leq C.$ (4.83)
By, (4.43), it holds that
$\sup_{t\in[0,T]}\big{[}t^{2}\|\nabla^{2}u_{t}\|_{2}^{2}(t)\big{]}\leq
C\sup_{t\in[0,T]}\Big{[}t^{2}\|\sqrt{\rho}u_{tt}\|_{2}^{2}(t)+t^{2}\|u_{t}\|_{H^{1}}^{2}+t^{2}\|u\|_{H^{3}}^{2}+1\Big{]}\leq
C.$ (4.84)
Finally, by (4.66), we can obtain
$\sup_{t\in[0,T]}\big{[}t^{2}\|\nabla u\|_{W^{2,q}}^{2}(t)\big{]}\leq
C\sup_{t\in[0,T]}\Big{[}t^{2}\|u\|_{H^{3}}^{2}+t^{2}\|u_{t}\|_{H^{2}}^{2}+1\Big{]}\leq
C.$ (4.85)
So the proof of Lemma 4.6 is completed. $\hfill\Box$
## 5 The proof of Theorem 1.1
With the uniform-in-$\delta$ bounds of the solution
$(\rho^{\delta},u^{\delta})$ in Lemmas 3.1-3.7 and Lemma 4.1-4.6, one can
prove the convergence of the sequence $(\rho^{\delta},u^{\delta})$ to a limit
$(\rho,u)$ satisfying the same bounds as $(\rho^{\delta},u^{\delta})$ as
$\delta$ tends to zero and the limit $(\rho,u)$ is a unique solution to the
original problem (1.3)-(1.6). The details are omitted for brevity and one can
refer to Cho-Kim [7] for the routine proofs. In the following, we will show
that $(\rho,u)$ satisfy the bounds in Theorem 1.1 and $(\rho,u)$ is a
classical solution to (1.3). Since $u\in L^{2}(0,T;H^{3}(\mathbb{T}^{2}))$ and
$u_{t}\in L^{2}(0,T;H^{1}(\mathbb{T}^{2}))$, so the Sobolev’s embedding
theorem implies that
$u\in C([0,T];H^{2}(\mathbb{T}^{2}))\hookrightarrow
C([0,T]\times\mathbb{T}^{2}).$
Then it follows from $(\rho,P(\rho))\in
L^{\infty}(0,T;W^{2,q}(\mathbb{T}^{2}))$ and $(\rho,P(\rho))_{t}\in
L^{\infty}(0,T;H^{1}(\mathbb{T}^{2}))$ that $(\rho,P(\rho))\in
C([0,T];W^{1,q}(\mathbb{T}^{2}))\cap C([0,T];W^{2,q}(\mathbb{T}^{2})-weak)$.
This and (4.68) then imply that
$(\rho,P(\rho))\in C([0,T];W^{2,q}(\mathbb{T}^{2})).$
Since for any $\tau\in(0,T)$,
$(\nabla u,\nabla^{2}u)\in
L^{\infty}(\tau,T;W^{1,q}(\mathbb{T}^{2})),\qquad(\nabla
u_{t},\nabla^{2}u_{t})\in L^{\infty}(\tau,T;L^{2}(\mathbb{T}^{2})).$
Therefore,
$(\nabla u,\nabla^{2}u)\in C([\tau,T]\times\mathbb{T}^{2}),$
Due to the fact that
$\nabla(\rho,P(\rho))\in C([0,T];W^{1,q}(\mathbb{T}^{2}))\hookrightarrow
C([0,T]\times\mathbb{T}^{2})$
and the continuity equation $\eqref{CNS}_{1}$, it holds that
$\rho_{t}=u\cdot\nabla\rho+\rho{\rm div}u\in C([\tau,T]\times\mathbb{T}^{2}).$
It follows from the momentum equation $\eqref{CNS}_{2}$ that
$\begin{array}[]{ll}(\rho u)_{t}&\displaystyle=\mathcal{L}_{\rho}u-{\rm
div}(\rho u\otimes u)-\nabla P(\rho)\\\ &\displaystyle=\mu\Delta
u+(\mu+\lambda(\rho))\nabla({\rm div}u)+({\rm div}u)\nabla\lambda(\rho)+\rho
u\cdot\nabla u+\rho u{\rm div}u+(u\cdot\nabla\rho)u-\nabla P(\rho)\\\
&\displaystyle\in C([\tau,T]\times\mathbb{T}^{2}).\end{array}$
Thus we completed the proof of Theorem 1.1.
## 6 The proof of Theorem 1.2
Based on Theorem 1.1, one can prove Theorem 1.2 easily as follows. Since
$\rho_{0}\in H^{3}(\mathbb{T}^{2})\hookrightarrow W^{2,q}(\mathbb{T}^{2})$
for any $2<q<+\infty$, it follows that under the conditions of Theorem 1.2,
Theorem 1.1 holds for any $2<q<+\infty$. Thus, we need only to prove the
higher order regularity presented in Theorem 1.2.
###### Lemma 6.1
It holds that
$\begin{array}[]{ll}\displaystyle\sup_{t\in[0,T]}\Big{[}\|\sqrt{\rho}\nabla^{3}u\|_{2}(t)+\|(\rho,P(\rho),\lambda(\rho))\|_{H^{3}}(t)\Big{]}+\int_{0}^{T}\|u\|_{H^{4}}^{2}dt\leq
C.\end{array}$
Proof: Applying $\partial_{x_{j}x_{k}}$, $j,k=1,2$, to $\eqref{moment-1}$
yields that
$\begin{array}[]{ll}\displaystyle\rho u_{x_{j}x_{k}t}+\rho u\cdot\nabla
u_{x_{j}x_{k}}+\rho_{x_{j}x_{k}}u_{t}+\rho_{x_{j}}u_{x_{k}t}+\rho_{x_{k}}u_{x_{j}t}+\rho_{x_{j}x_{k}}u\cdot\nabla
u+\rho u_{x_{j}x_{k}}\cdot\nabla u\\\
\displaystyle+\rho_{x_{j}}u_{x_{k}}\cdot\nabla u+\rho_{x_{j}}u\cdot\nabla
u_{x_{k}}+\rho_{x_{k}}u_{x_{j}}\cdot\nabla u+\rho_{x_{k}}u\cdot\nabla
u_{x_{j}}+\rho u_{x_{j}}\cdot\nabla u_{x_{k}}+\rho u_{x_{k}}\cdot\nabla
u_{x_{j}}\\\ \displaystyle+\nabla P(\rho)_{x_{j}x_{k}}=\mu\Delta
u_{x_{j}x_{k}}+\nabla((\mu+\lambda(\rho)){\rm div}u)_{x_{j}x_{k}}.\end{array}$
(6.1)
Then multiplying $\eqref{Jan-19-2}$ by $\Delta u_{x_{j}x_{k}}$ and integrating
with respect to $x$ over $\mathbb{T}^{2}$ imply that
$\begin{array}[]{ll}\displaystyle\int\big{[}\mu|\Delta
u_{x_{j}x_{k}}|^{2}+\nabla((\mu+\lambda(\rho)){\rm
div}u)_{x_{j}x_{k}}\cdot\Delta u_{x_{j}x_{k}}\big{]}dx=\int\big{(}\rho
u_{x_{j}x_{k}t}+\rho u\cdot\nabla u_{x_{j}x_{k}}\big{)}\cdot\Delta
u_{x_{j}x_{k}}dx\\\
\displaystyle+\int\big{[}\rho_{x_{j}x_{k}}u_{t}+\rho_{x_{j}}u_{x_{k}t}+\rho_{x_{k}}u_{x_{j}t}+\rho_{x_{j}x_{k}}u\cdot\nabla
u+\rho u_{x_{j}x_{k}}\cdot\nabla u+\rho_{x_{j}}u_{x_{k}}\cdot\nabla
u+\rho_{x_{j}}u\cdot\nabla u_{x_{k}}\\\
\displaystyle\qquad\quad+\rho_{x_{k}}u_{x_{j}}\cdot\nabla
u+\rho_{x_{k}}u\cdot\nabla u_{x_{j}}+\rho u_{x_{j}}\cdot\nabla u_{x_{k}}+\rho
u_{x_{k}}\cdot\nabla u_{x_{j}}+\nabla P(\rho)_{x_{j}x_{k}}\big{]}\cdot\Delta
u_{x_{j}x_{k}}dx.\end{array}$ (6.2)
Integrations by part several times yield
$\begin{array}[]{ll}\displaystyle\int\big{(}\rho u_{x_{j}x_{k}t}+\rho
u\cdot\nabla u_{x_{j}x_{k}}\big{)}\cdot\Delta u_{x_{j}x_{k}}dx\\\
\displaystyle=-\int\Big{[}\rho\Big{(}\frac{|\nabla
u_{x_{j}x_{k}}|^{2}}{2}\Big{)}_{t}+\rho u\cdot\nabla\Big{(}\frac{|\nabla
u_{x_{j}x_{k}}|^{2}}{2}\Big{)}+\sum_{i=1}^{2}\rho_{x_{i}}u_{x_{j}x_{k}t}\cdot
u_{x_{i}x_{j}x_{k}}\\\
\displaystyle\qquad\qquad\qquad+\sum_{i=1}^{2}\rho_{x_{i}}u\cdot\nabla
u_{x_{j}x_{k}}\cdot u_{x_{i}x_{j}x_{k}}+\sum_{i=1}^{2}\rho
u_{x_{i}}\cdot\nabla u_{x_{j}x_{k}}\cdot u_{x_{i}x_{j}x_{k}}\Big{]}dx\\\
\displaystyle=-\frac{d}{dt}\int\rho\frac{|\nabla
u_{x_{j}x_{k}}|^{2}}{2}dx+\int\Big{[}\sum_{i=1}^{2}\rho_{x_{i}x_{j}}u_{x_{k}t}\cdot
u_{x_{i}x_{j}x_{k}}+\sum_{i=1}^{2}\rho_{x_{i}}u_{x_{k}t}\cdot
u_{x_{i}x_{j}x_{j}x_{k}}\\\
\displaystyle\qquad\qquad\qquad-\sum_{i=1}^{2}\rho_{x_{i}}u\cdot\nabla
u_{x_{j}x_{k}}\cdot u_{x_{i}x_{j}x_{k}}-\sum_{i=1}^{2}\rho
u_{x_{i}}\cdot\nabla u_{x_{j}x_{k}}\cdot u_{x_{i}x_{j}x_{k}}\Big{]}dx\\\
\end{array}$ (6.3)
and
$\begin{array}[]{ll}\displaystyle\int\nabla((\mu+\lambda(\rho)){\rm
div}u)_{x_{j}x_{k}}\cdot\Delta u_{x_{j}x_{k}}dx\\\
\displaystyle=\int\Big{[}(\mu+\lambda(\rho))|\nabla({\rm
div}u)_{x_{j}x_{k}}|^{2}+\Big{(}\lambda(\rho)_{x_{j}x_{k}}\nabla({\rm
div}u)+\lambda(\rho)_{x_{j}}\nabla({\rm
div}u)_{x_{k}}+\lambda(\rho)_{x_{k}}\nabla({\rm div}u)_{x_{j}}\\\
\displaystyle~{}+\nabla\lambda(\rho)_{x_{j}x_{k}}{\rm
div}u+\nabla\lambda(\rho)({\rm
div}u)_{x_{j}x_{k}}+\nabla\lambda(\rho)_{x_{j}}({\rm
div}u)_{x_{k}}+\nabla\lambda(\rho)_{x_{k}}({\rm
div}u)_{x_{j}}\Big{)}\cdot\nabla({\rm
div}u)_{x_{j}x_{k}}\Big{]}dx.\end{array}$ (6.4)
Then substituting (6.3) and (6.4) into (6.2), summing over $j,k=1,2$ and using
the Cauchy and Young inequalities and the estimates in Sections 3-4, one has
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\sqrt{\rho}\nabla^{3}u\|_{2}^{2}+2\mu\|\nabla^{2}\Delta
u\|_{2}^{2}(t)\leq
C\Big{[}(\|u\|^{2}_{H^{3}}+1)\|(\nabla^{3}P(\rho),\nabla^{3}\lambda(\rho))\|^{2}_{2}+1\Big{]}.\end{array}$
(6.5)
Next, applying $\partial_{x_{i}x_{j}x_{k}}$, $i,j,k=1,2$, to $\eqref{CNS}_{1}$
gives
$\rho_{x_{i}x_{j}x_{k}t}+\partial_{x_{i}x_{j}x_{k}}({\rm div}(\rho u))=0.$
(6.6)
Multiplying (6.6) by $\rho_{x_{i}x_{j}x_{k}}$ and summing over $i,j,k=1,2$ and
then integrating with respect to $x$ over $\mathbb{T}^{2}$, one gets that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|\nabla^{3}\rho\|_{2}^{2}&\displaystyle\leq
C\|\nabla^{3}\rho\|_{2}\Big{[}\|\nabla\rho\|_{\infty}\|\nabla^{3}u\|_{2}+\|\nabla^{2}u\|_{4}\|\nabla^{2}\rho\|_{4}+\|\nabla
u\|_{\infty}\|\nabla^{3}\rho\|_{2}+\|\rho\|_{\infty}\|\nabla^{4}u\|_{2}\Big{]}\\\
&\displaystyle\leq C\|\nabla^{3}\rho\|_{2}\Big{[}\|\nabla^{3}u\|_{2}+\|\nabla
u\|_{\infty}\|\nabla^{3}\rho\|_{2}+\|\rho\|_{\infty}\|\nabla^{2}\Delta
u\|_{2}\Big{]}\\\ &\displaystyle\leq\alpha\|\nabla^{2}\Delta
u\|^{2}_{2}+C_{\alpha}(\|u\|_{H^{3}}+1)\|\nabla^{3}\rho\|^{2}_{2},\end{array}$
(6.7)
where $\alpha>0$ is a constant to be determined.
Similarly, one can obtain
$\displaystyle\frac{d}{dt}\|(\nabla^{3}P(\rho),\nabla^{3}\lambda(\rho))\|^{2}_{2}\leq\alpha\|\nabla^{2}\Delta
u\|^{2}_{2}+C_{\alpha}(\|u\|_{H^{3}}+1)\|(\nabla^{3}P(\rho),\nabla^{3}\lambda(\rho))\|^{2}_{2}.$
(6.8)
Let $\alpha=\frac{\mu}{3}$. It follows from inequalities (6.5), (6.7) and
(6.8), that
$\begin{array}[]{ll}\displaystyle\frac{d}{dt}\|(\sqrt{\rho}\nabla^{3}u,\nabla^{3}\rho,\nabla^{3}P(\rho),\nabla^{3}\lambda(\rho))\|_{2}^{2}(t)+\mu\|\nabla^{2}\Delta
u\|_{2}^{2}(t)\\\ \displaystyle\qquad\qquad\qquad\leq
C\Big{[}(\|u\|^{2}_{H^{3}}+1)\|(\nabla^{3}\rho,\nabla^{3}P(\rho),\nabla^{3}\lambda(\rho))\|^{2}_{2}+1\Big{]}.\end{array}$
(6.9)
Then integrating (6.9) over $[0,t]$ and using the Gronwall’s inequality lead
to that
$\begin{array}[]{ll}\displaystyle\sup_{t\in[0,T]}\Big{[}\|\sqrt{\rho}\nabla^{3}u\|_{2}(t)+\|\nabla^{3}(\rho,P(\rho),\lambda(\rho))\|_{2}\Big{]}+\int_{0}^{T}\|\nabla^{2}\Delta
u\|_{2}^{2}dt\leq C.\end{array}$
So the proof of Lemma 6.1 is completed. $\hfill\Box$
Now we prove other higher regularities presented in (1.11) of Theorem 1.2. It
follows easily from (6.9) and (1.9) that for any $t_{1},t_{2}\in[0,T]$,
$\|\sqrt{\rho}\nabla^{3}u\|_{2}^{2}(t_{1})-\|\sqrt{\rho}\nabla^{3}u\|_{2}^{2}(t_{2})\to
0,$ (6.10)
as $t_{1}\to t_{2}$.
Thanks to Theorem 1.1, one has
$\rho\in C([0,T];H^{2}(\mathbb{T}^{2}))\hookrightarrow
C([0,T]\times\mathbb{T}^{2}).$ (6.11)
It holds that
$\begin{array}[]{ll}\displaystyle|\|\rho\nabla^{3}u\|^{2}_{2}(t_{1})-\|\rho\nabla^{3}u\|^{2}_{2}(t_{2})|=|\int\rho^{2}|\nabla^{3}u|^{2}(t_{1},x)dx-\int\rho^{2}|\nabla^{3}u|^{2}(t_{2},x)dx|\\\
\displaystyle\leq|\int\rho(t_{1},x)\big{[}\rho|\nabla^{3}u|^{2}(t_{1},x)-\rho|\nabla^{3}u|^{2}(t_{2},x)\big{]}dx|+|\int\rho|\nabla^{3}u|^{2}(t_{2},x)\big{[}\rho(t_{1},x)-\rho(t_{2},x)\big{]}dx|\\\
\displaystyle\leq\sup_{[0,T]\times\mathbb{T}^{2}}\rho(t,x)~{}|\int\big{[}\rho|\nabla^{3}u|^{2}(t_{1},x)-\rho|\nabla^{3}u|^{2}(t_{2},x)\big{]}dx|\\\
\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sup_{t\in[0,T]}\int\rho|\nabla^{3}u|^{2}(t,x)dx\sup_{x\in\mathbb{T}^{2}}|\rho(t_{1},x)-\rho(t_{2},x)|\\\
\displaystyle\leq
C\Big{[}\Big{|}\int\rho|\nabla^{3}u|^{2}(t_{1},x)dx-\int\rho|\nabla^{3}u|^{2}(t_{2},x)dx\Big{|}+\sup_{x\in\mathbb{T}^{2}}|\rho(t_{1},x)-\rho(t_{2},x)|\Big{]}\\\
\displaystyle\rightarrow 0,~{}~{}{\rm as}~{}~{}t_{1}\rightarrow
t_{2},\end{array}$ (6.12)
where one has used (6.10) and (6.11).
Moreover, due to the facts that $\rho\nabla^{3}u\in
L^{\infty}([0,T];L^{2}(\mathbb{T}^{2})),\rho\in
C([0,T];H^{2}(\mathbb{T}^{2}))$ and $u\in C([0,T];H^{2}(\mathbb{T}^{2}))$, it
follows that $\rho\nabla^{3}u\in C([0,T];H^{3}-w)$ which means that
$\rho\nabla^{3}u$ is weakly continuous with values in
$H^{3}((\mathbb{T}^{2}))$. This, together with (6.12), leads to
$\rho\nabla^{3}u\in C([0,T];L^{2}(\mathbb{T}^{2})).$ (6.13)
In a similar way, one can prove that
$(\rho,P(\rho))\in C([0,T];H^{3}(\mathbb{T}^{2}))\hookrightarrow
C([0,T];C^{1}(\mathbb{T}^{2})).$ (6.14)
Moreover, since $u\in C([0,T];H^{2}(\mathbb{T}^{2}))$ by Theorem 1.1 and
$\rho\in C([0,T];H^{3}(\mathbb{T}^{2}))$ by (6.14), one can prove that for any
$t_{1},t_{2}\in[0,T]$,
$\displaystyle\|\nabla^{3}\rho u(t_{1},\cdot)-\nabla^{3}\rho
u(t_{2},\cdot)\|^{2}_{2}\to 0,$ (6.15)
$\displaystyle\|\nabla\rho\nabla^{2}u(t_{1},\cdot)-\nabla\rho\nabla^{2}u(t_{2},\cdot)\|^{2}_{2}\to
0,$ (6.16) $\displaystyle\|\nabla^{2}\rho\nabla
u(t_{1},\cdot)-\nabla^{2}\rho\nabla u(t_{2},\cdot)\|^{2}_{2}\to 0$ (6.17)
respectively as $t_{1}\to t_{2}$. In fact, to prove (6.17), one has
$\displaystyle\int|\nabla^{2}\rho\nabla u(t_{1},x)-\nabla^{2}\rho\nabla
u(t_{2},x)|^{2}dx$
$\displaystyle\leq\int|\nabla^{2}\rho(t_{1},x)-\nabla^{2}\rho(t_{2},x)|^{2}|\nabla
u(t_{1},x)|^{2}dx+\int|\nabla^{2}\rho(t_{2},x)|^{2}|\nabla u(t_{1},x)-\nabla
u(t_{2},x)|^{2}dx$ $\displaystyle\leq
C\|\nabla^{2}\rho(t_{1},\cdot)-\nabla^{2}\rho(t_{2},\cdot)\|_{2}^{2}+\|\nabla^{2}\rho\|^{2}_{4}\|\nabla
u(t_{1},\cdot)-\nabla u(t_{2},\cdot)\|^{2}_{4}$ $\displaystyle\leq
C(\|\nabla^{2}\rho(t_{1},\cdot)-\nabla^{2}\rho(t_{2},\cdot)\|_{2}^{2}+\|\nabla^{2}u(t_{1},\cdot)-\nabla^{2}u(t_{2},\cdot)\|^{2}_{2})\to
0,$
as $t_{1}\to t_{2}$. Similarly, (6.15) and (6.16) can be proved. In view of
(6.13) and (6.15)-(6.17), we have proved that
$\rho u\in C([0,T];H^{3}(\mathbb{T}^{2})).$ (6.18)
The proof of Theorem 1.2 is completed.
## Acknowledgments
Parts of this work were done when Y. Wang was a postdoctoral fellow at the IMS
of Chinese University of Hong Kong during the academic year 2010-2011; Q.S.
Jiu was visiting the IMS of The Chinese University of Hong Kong, and when Q.S.
Jiu and Z.P. Xin were visiting the IMS of National University of Singapore.
The authors would like to thank these institutions for their supports and
hospitality.
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|
arxiv-papers
| 2012-02-07T09:29:26 |
2024-09-04T02:49:27.157385
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quansen Jiu, Yi Wang, Zhouping Xin",
"submitter": "Yi Wang",
"url": "https://arxiv.org/abs/1202.1382"
}
|
1202.1383
|
2012-012 J/$\psi$ suppression at forward rapidity in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV
The ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members The ALICE Collaboration
The ALICE experiment has measured the inclusive J/$\psi$ production in Pb-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV down to zero transverse
momentum in the rapidity range $2.5<y<4$. A suppression of the inclusive
J/$\psi$ yield in Pb-Pb is observed with respect to the one measured in pp
collisions scaled by the number of binary nucleon-nucleon collisions. The
nuclear modification factor, integrated over the 0%–80% most central
collisions, is $0.545\pm 0.032\rm{(stat.)}\pm 0.083\rm{(syst.)}$ and does not
exhibit a significant dependence on the collision centrality. These features
appear significantly different from measurements at lower collision energies.
Models including J/$\psi$ production from charm quarks in a deconfined
partonic phase can describe our data.
Ultra-relativistic collisions of heavy nuclei aim at producing nuclear matter
at high temperature and pressure. Under such conditions Quantum Chromodynamics
predicts the existence of a deconfined state of partonic matter, the quark-
gluon plasma (QGP). Among the possible probes of the QGP, heavy quarks are of
particular interest since they are expected to be produced in the primary
partonic scatterings and to coexist with the surrounding medium. Therefore,
the measurement of quarkonium states and hadrons with open heavy flavor is
expected to provide essential information on the properties of the strongly-
interacting system formed in the early stages of heavy-ion collisions [1]. In
particular, according to the color-screening model [2], measuring the in-
medium dissociation probability of the different quarkonium states is expected
to provide an estimate of the initial temperature of the system. In the past
two decades, J/$\psi$ production in heavy-ion collisions was intensively
studied at the Super Proton Synchrotron (SPS) and at the Relativistic Heavy
Ion Collider (RHIC), from approximately $20$ to $200$ GeV center of mass
energy per nucleon pair ($\sqrt{s_{\mathrm{NN}}}$). At the SPS, a strong
J/$\psi$ suppression was found in the most central Pb-Pb collisions [3]. The
observed suppression is larger than the one expected from Cold Nuclear Matter
(CNM) effects, which include nuclear absorption and (anti-) shadowing. The
dissociation of excited $\rm{c\overline{c}}$ states like $\chi_{\rm{c}}$ and
$\psi\rm{(2S)}$, which in pp collisions constitute about 40% of the inclusive
J/$\psi$ yield [1], is one possible interpretation of the observed
suppression. A J/$\psi$ suppression was also observed at RHIC, in central Au-
Au collisions [4, 5], at a level similar to the one observed at the SPS when
measured at mid-rapidity although it is larger at forward rapidity. Several
models [6, 7, 8, 9] attempt to reproduce the RHIC data by adding to the direct
J/$\psi$ production a regeneration component from deconfined charm quarks in
the medium, which counteracts the J/$\psi$ dissociation in a QGP. A
quantitative description of these final-state effects is however difficult at
the present time because of the lack of precision in the CNM effects and in
the open charm cross section determination. The measurement of charmonium
production is especially promising at the Large Hadron Collider (LHC) where
the high-energy density of the medium and the large number of c$\bar{\rm{c}}$
pairs produced in central Pb-Pb collisions should help to disentangle between
the different suppression and regeneration scenarios. At the LHC, a
suppression of inclusive J/$\psi$ with high transverse momentum was observed
in central Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV with respect
to peripheral collisions or pp collisions at the same energy by ATLAS [10] and
CMS [11], respectively.
In this Letter, we report ALICE results on inclusive J/$\psi$ production in
Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at forward rapidity,
measured via the $\mu^{+}\mu^{-}$ decay channel. Our measurement encloses the
low transverse momentum region that is not accessible to other LHC experiments
and thus complements their observations. The J/$\psi$ corrected yield in Pb-Pb
collisions is combined with the one measured in pp collisions at the same
center-of-mass energy [12] to form the J/$\psi$ nuclear modification factor
$R_{\rm{AA}}$. The results are presented as a function of collision centrality
and rapidity ($y$), and in intervals of transverse momentum ($p_{\rm{t}}$).
The ALICE detector is described in [13]. At forward rapidity ($2.5<y<4$) the
production of quarkonium states is measured in the muon spectrometer 111In the
ALICE reference frame, the muon spectrometer covers a negative $\eta$ range
and consequently a negative $y$ range. We have chosen to present our results
with a positive $y$ notation. down to $p_{\rm{t}}=0$. The spectrometer
consists of a ten interaction length thick absorber filtering the muons in
front of five tracking stations comprising two planes of cathode pad chambers
each, with the third station inside a dipole magnet with a 3 Tm field
integral. The tracking apparatus is completed by a triggering system made of
four planes of resistive plate chambers downstream of a 1.2 m thick iron wall,
which absorbs secondary hadrons escaping from the front absorber and low
momentum muons coming mainly from $\pi$ and K decays. In addition, the silicon
pixel detector (SPD) and scintillator arrays (VZERO) were used in this
analysis. The VZERO counters, two arrays of 32 scintillator tiles each, cover
$2.8\leq\eta\leq 5.1$ (VZERO-A) and $-3.7\leq\eta\leq-1.7$ (VZERO-C). The SPD
consists of two cylindrical layers covering $|\eta|\leq 2.0$ and $|\eta|\leq
1.4$ for the inner and outer layers, respectively. All these detectors have
full azimuthal coverage. The minimum bias (MB) trigger requirement used for
this analysis consists of a logical AND of the three following conditions: (i)
a signal in two readout chips in the outer layer of the SPD, (ii) a signal in
VZERO-A, (iii) a signal in VZERO-C, providing a high triggering efficiency for
hadronic interactions. The beam induced background was further reduced by
timing cuts on the signals from the VZERO and from the zero degree
calorimeters (ZDC). The contribution from electromagnetic processes was
removed with a cut on the energy deposited in the neutron ZDCs. The centrality
determination is based on a fit of the VZERO amplitude distribution as
described in [14]. A cut corresponding to the most central 80% of the nuclear
cross section was applied; for these events the MB trigger is fully efficient.
A data sample of 17.7 $\times$ 106 Pb-Pb collisions collected in 2010
satisfying all the above conditions is used in the following analysis. It
corresponds to an integrated luminosity $\mathcal{L}_{\rm int}\approx 2.9$
$\mu$b-1. This data sample was further divided into five centrality classes
from 0%–10% (central collisions) to 50%–80% (peripheral collisions).
J/$\psi$ candidates are formed by combining pairs of opposite-sign (OS) tracks
reconstructed in the geometrical acceptance of the muon spectrometer.
Figure 1: (Color online) Invariant mass spectrum of $\mu^{+}\mu^{-}$ pairs
(solid black circles) with $p_{\rm{t}}\geq 0$ and $2.5<y<4$ in the 0%–80% most
central Pb-Pb collisions.
To reduce the combinatorial background, the reconstructed tracks in the muon
tracking chambers are required to match a track segment in the muon trigger
system. The resulting invariant mass distribution of OS muon pairs for the
0%–80% most central Pb-Pb collisions is shown in Fig. 1, where a J/$\psi$
signal above the combinatorial background is clearly visible. The J/$\psi$ raw
yield was extracted by using two different methods. The OS dimuon invariant
mass distribution was fitted with a Crystal Ball (CB) function to reproduce
the J/$\psi$ line shape, and a sum of two exponentials to describe the
underlying continuum. The CB function connects a Gaussian core with a power-
law tail [15] at low mass to account for energy loss fluctuations and
radiative decays. At high transverse momenta ($p_{\rm{t}}\geq 3\ {\rm
GeV}/c$), the sum of two exponentials does not describe correctly the
underlying continuum; it was replaced by a third order polynomial.
Alternatively, the combinatorial background was subtracted using an event-
mixing technique. The resulting mass distribution was fitted with a CB
function and an exponential or a first order polynomial to describe the
remaining background. The event mixing was preferred to the like-sign
subtraction technique since it is less sensitive to correlated signal pairs
present in the like-sign spectra and gives better statistical precision. The
$\psi\rm{(2S)}$ was not included in the signal line shape since its
contribution is negligible. The width of the J/$\psi$ mass peak depends on the
resolution of the spectrometer which can be affected by the detector occupancy
that increases with centrality. This effect, evaluated by embedding simulated
J/$\psi\rightarrow\mu^{+}\mu^{-}$ decays into real events, was found to be
less than 2%. This conclusion was confirmed by a direct measurement of the
tracking chamber resolution versus centrality using reconstructed tracks.
Therefore, the same CB line shape can be used for all centrality classes. The
parameters of the CB tails were fixed to the values obtained either in
simulations or in pp collisions where the signal to background ratio is much
higher. For each of these choices, the mean and width of the CB Gaussian part
were fixed to the value obtained by fitting the mass distribution in the
centrality range 0%–80%. In addition, a variation of the width of $\pm$ 1
standard deviation was applied to account for uncertainties (varying the mean
has turned out to have a negligible effect in comparison). The raw J/$\psi$
yield in each centrality class was determined as the average of the results
obtained with the two fitting approaches and the various CB parametrizations,
while the corresponding systematic uncertainties were defined as the RMS of
these results. It was also checked that every individual result differs from
the mean value by less than three RMS. The raw J/$\psi$ yield in our Pb-Pb
sample is $2350\pm 139\rm{(stat.)}\pm 189\rm{(syst.)}$. The invariant mass
resolution is around 78 MeV/$c^{2}$, in very good agreement with the embedded
J/$\psi$ simulations. The signal to background ratio integrated over $\pm\ 3\
\sigma$ of the mass resolution varies from 0.1 for central collisions to 1.5
for peripheral collisions.
The measured number of J/$\psi$ ($N_{\rm{J/}\psi}^{i}$) was normalized to the
number of events in the corresponding centrality class ($N_{\rm{events}}^{i}$)
and further corrected for the branching ratio (BR) of the dimuon decay
channel, the acceptance $A$ and the efficiency $\epsilon^{i}$ of the detector.
The inclusive J/$\psi$ yield in each centrality class $i$ for our measured
$p_{\rm{t}}$ and $y$ ranges ($\Delta p_{\rm{t}}$, $\Delta y$) is then given
by:
$\displaystyle Y_{\rm{J/}\psi}^{i}(\Delta p_{\rm{t}},\Delta
y)=\frac{N_{\rm{J/}\psi}^{i}}{{\rm
BR}_{\rm{J/}\psi\rightarrow\mu^{+}\mu^{-}}N_{\rm{events}}^{i}A\epsilon^{i}}.$
(1)
The product $A\epsilon$ was determined from Monte Carlo simulations. The
generated J/$\psi$ $p_{\rm{t}}$ and $y$ distributions were extrapolated from
existing measurements [16], including shadowing effects from EKS98
calculations [17]. As the measured J/$\psi$ polarization in pp collisions at
$\sqrt{s}=7$ TeV is compatible with zero [18], and J/$\psi$ mesons produced
from charm quarks in the medium are expected to be unpolarized, we presume
J/$\psi$ production is unpolarized. For the tracking chambers, the time-
dependent status of each electronic channel during the data taking period was
taken into account as well as the residual misalignment of the detection
elements. The efficiencies of the muon trigger chambers were determined from
data and were then applied in the simulations [19]. Finally, the dependence of
the efficiency on the detector occupancy was included using the embedding
technique. For J/$\psi$ mesons emitted at $2.5<y<4$ and $p_{\rm{t}}\geq 0$, a
run-averaged value of $A\epsilon=0.176$ with a 8% relative systematic
uncertainty was obtained. A $8\%\pm 2\%({\rm syst.})$ relative decrease of the
efficiency was observed when going from peripheral to central collisions.
The J/$\psi$ yield measured in Pb-Pb collisions in centrality class $i$ is
combined with the inclusive J/$\psi$ cross section measured in pp collisions
at the same energy to form the nuclear modification factor $R_{\rm{AA}}$
defined as:
$\displaystyle R_{\rm{AA}}^{i}=\frac{Y_{\rm{J/}\psi}^{i}(\Delta
p_{\rm{t}},\Delta y)}{\langle
T_{\rm{AA}}^{i}\rangle\times\sigma_{\rm{J/}\psi}^{\rm{pp}}(\Delta
p_{\rm{t}},\Delta y)}.$ (2)
The inclusive J/$\psi$ cross section in pp collisions
$\sigma_{\rm{J/}\psi}^{\rm{pp}}(\Delta p_{\rm{t}},\Delta y)$ was measured
using the same apparatus and analysis technique within the corresponding
$p_{\rm{t}}$ and $y$ range [12]. The reference value
$\sigma_{\rm{J/}\psi}^{\rm{pp}}$ used for the calculation of $R_{\rm{AA}}$
integrated over $p_{\rm{t}}$ and $y$ is $3.34\pm 0.13\rm{(stat.)}\pm
0.24\rm{(syst.)}\pm 0.12\rm{(lumi.)}^{+0.53}_{-1.07}\rm{(pol.)}\;\mu b$. The
centrality intervals used in this analysis, the average number of
participating nucleons $\langle N_{\rm{part}}\rangle$ and average value of the
nuclear overlap function $\langle T_{\rm{AA}}\rangle$ derived from a Glauber
model calculation [14] are summarized in Table 1.
Table 1: The average number of participating nucleons $\langle N_{\rm{part}}\rangle$ without and with $\langle N_{\rm{coll}}\rangle$ weighting, the mid-rapidity charged-particle density d$N_{\rm{ch}}^{\rm{w}}/$d$\eta|_{\eta=0}$ with $\langle N_{\rm{coll}}\rangle$ weighting and the average value of the nuclear overlap function $\langle T_{\rm{AA}}\rangle$ for the centrality classes expressed in percentages of the nuclear cross section [14]. Centrality | $\langle N_{\rm{part}}\rangle$ | $\langle N_{\rm{part}}^{\rm{w}}\rangle$ | $\frac{{\rm d}N_{\rm{ch}}^{\rm{w}}}{\rm{d}\eta|_{\eta=0}}$ | $\langle T_{\rm{AA}}\rangle$ (mb-1)
---|---|---|---|---
0%–10% | 356$\pm$4 | 361$\pm$4 | 1463$\pm$60 | 23.5$\pm$1.0
10%–20% | 260$\pm$4 | 264$\pm$4 | 979$\pm$37 | 14.4$\pm$0.6
20%–30% | 186$\pm$4 | 189$\pm$4 | 658$\pm$23 | 8.74$\pm$0.37
30%–50% | 107$\pm$3 | 117$\pm$3 | 369$\pm$13 | 3.87$\pm$0.18
50%–80% | 32$\pm$2 | 47$\pm$2 | 110$\pm$5 | 0.72$\pm$0.05
0%–80% | 139$\pm$3 | 264$\pm$4 | – | 7.03$\pm$0.27
Since our most peripheral bin is rather large, the variables $\langle
N_{\rm{part}}^{\rm{w}}\rangle$ and the charged-particle density measured at
mid-rapidity ${\rm d}N_{\rm{ch}}^{\rm{w}}/{\rm d}\eta|_{\eta=0}$ were weighted
by the number of binary collisions $\langle N_{\rm{coll}}\rangle$. Indeed in
absence of nuclear matter effects, the J/$\psi$ production cross section in
nucleus-nucleus is expected to scale with $\langle N_{\rm{coll}}\rangle$. The
weighted values are given in Table 1 and are used for the ALICE data points in
the following figures. All systematic uncertainties entering the $R_{\rm{AA}}$
calculation are listed in Table 2. In the figures below, the point to point
uncorrelated systematic uncertainties are represented as boxes at the position
of the data points while the statistical ones are indicated by vertical bars.
Correlated systematic uncertainties are quoted directly on the figures.
Table 2: Summary of the systematic uncertainties entering the $R_{\rm{AA}}$ calculation. The type I (II) stands for correlated (uncorrelated) uncertainties. The centrality dependence for the type II is given as a range. source | value | type
---|---|---
signal extraction | 5%–12% | II
input MC parametrization | 5% | I
tracking efficiency | 5% and 0%–1% | I and II
trigger efficiency | 4% and 0%–2% | I and II
matching efficiency | 2% | I
$T_{\rm{AA}}$ | 4%–8% | II
$\sigma_{\rm{J/}\psi}^{\rm{pp}}$ at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV | 9% | I
The inclusive J/$\psi$ $R_{\rm{AA}}$ measured by ALICE at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV in the range $2.5<y<4$ and $p_{\rm{t}}\geq
0$ is shown in Fig. 2 as a function of ${\rm d}N_{\rm{ch}}/{\rm
d}\eta|_{\eta=0}$ (left) and $N_{\rm{part}}$ (right). The charged-particle
density closely relates to the energy density of the created medium whereas
the number of participants reflects the collision geometry. The centrality
integrated J/$\psi$ $R_{\rm{AA}}$ is $R^{0\%-80\%}_{\rm{AA}}=0.545\pm
0.032\rm{(stat.)}\pm 0.083\rm{(syst.)}$, indicating a clear J/$\psi$
suppression.
Figure 2: (Color online) Inclusive J/$\psi$ $R_{\rm{AA}}$ as a function of the
mid-rapidity charged-particle density (top) and the number of participating
nucleons (bottom) measured in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV compared to PHENIX results in Au-Au
collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV at mid-rapidity and forward
rapidity [4, 5, 20]. The ALICE data points are placed at the ${\rm
d}N_{\rm{ch}}^{\rm{w}}/{\rm d}\eta|_{\eta=0}$ and $\langle
N_{\rm{part}}^{\rm{w}}\rangle$ values defined in Table 1.
The contribution from beauty hadron feed-down to the inclusive J/$\psi$ yield
in our $y$ and $p_{\rm{t}}$ domain was measured by the LHCb collaboration to
be about 10% in pp collisions at $\sqrt{s}=7$ TeV [21]. Therefore, the
difference between the prompt J/$\psi$ $R_{\rm{AA}}$ and our inclusive
measurement is expected not to exceed 11% if $N_{\rm{coll}}$ scaling of beauty
production is assumed and shadowing effects are neglected. All $R_{\rm{AA}}$
results are presented assuming unpolarized J/$\psi$ production in pp and Pb-Pb
collisions. The comparison with the PHENIX measurements 222 The PHENIX mid-
rapidity J/$\psi$ $R_{\rm{AA}}$ was measured in centrality classes wider than
the ones in which the mid-rapidity charged-particle density is given [20].
Therefore a linear interpolation was done to extract the mid-rapidity charged-
particle density in the three most peripheral classes. at
$\sqrt{s_{\mathrm{NN}}}=200$ GeV at forward rapidity $1.2<|y|<2.2$ [5, 20]
shows that our inclusive J/$\psi$ $R_{\rm{AA}}$ is almost a factor of three
larger for d$N_{\rm{ch}}/$d$\eta|_{\eta=0}\gtrsim 600$ ($N_{\rm{part}}\gtrsim
180$). In addition, our results do not exhibit a significant centrality
dependence.
The rapidity dependence of the J/$\psi$ $R_{\rm{AA}}$ is presented in Fig. 3
for two $p_{\rm{t}}$ domains, $p_{\rm{t}}\geq$ 0 and $p_{\rm{t}}\geq 3\ {\rm
GeV}/c$. The J/$\psi$ reference cross sections in pp collisions 333We report
here $\sigma_{\rm{J/}\psi}^{\rm{pp}}(p_{\rm{t}}\geq 3\rm{GeV}/c,\;2.5<y\leq
3.25)=0.34\pm 0.03\rm{(stat.)}\pm 0.03\rm{(syst.)}\pm 0.02\rm{(lumi.)}\;\mu b$
and $\sigma_{\rm{J/}\psi}^{\rm{pp}}(p_{\rm{t}}\geq
3\rm{GeV}/c,\;3.25<y<4)=0.50\pm 0.04\rm{(stat.)}\pm 0.04\rm{(syst.)}\pm
0.02\rm{(lumi.)}\;\mu b$ that can not directly be extracted from [12]. and the
$R_{\rm{AA}}$ total systematic uncertainties, indicated as open boxes in the
figure, were evaluated in the same kinematic range.
Figure 3: (Color online) Centrality integrated inclusive J/$\psi$
$R_{\rm{AA}}$ measured in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$
TeV as a function of rapidity for two $p_{\rm{t}}$ ranges. The open boxes
contain the total systematic uncertainties except the ones on the integrated
luminosity in the pp reference and on the $T_{\rm{AA}}$, i.e. 5.2% (8.3%) for
the ALICE (CMS [11]) data. The two models [22, 23] predict the $R_{\rm{AA}}$
due only to shadowing effects for nDSg (shaded areas) and EPS09 (lines) nPDF
respectively.
Our results are shown together with a measurement from CMS [11] of the
inclusive J/$\psi$ $R_{\rm{AA}}$ in the rapidity range $1.6<|y|<2.4$ with
$p_{\rm{t}}\geq 3\ {\rm GeV}/c$. No significant rapidity dependence can be
seen in the J/$\psi$ $R_{\rm{AA}}$ for $p_{\rm{t}}\geq 0$. For $p_{\rm{t}}\geq
3\ {\rm GeV}/c$, a decrease of $R_{\rm{AA}}$ is observed with increasing
rapidity reaching a value of $0.289\pm 0.061\rm{(stat.)}\pm 0.078\rm{(syst.)}$
for $3.25<y<4$. At LHC energies, J/$\psi$ nuclear absorption is likely to be
negligible and the modification of the gluon distribution function is
dominated by shadowing effects [24]. An estimate of shadowing effects is shown
in Fig. 3 within the Color Singlet Model at Leading Order [22] and the Color
Evaporation Model at Next to Leading Order [23]. The shadowing is respectively
calculated with the nDSg and the EPS09 parametrizations [23] of the nuclear
Parton Distribution Function (nPDF). For nDSg (EPS09) the upper and lower
limits correspond to the uncertainty in the factorization scale (uncertainty
of the nPDF). The effect of shadowing shows no dependence with rapidity and
its overall amount is reduced by the addition of a transverse momentum cut. At
most, shadowing effects are expected to lower the $R_{\rm{AA}}$ from 1 to 0.7.
Recent Color Glass Condensate (CGC) calculations for LHC energies may indicate
a larger initial state suppression ($R_{\rm{AA}}\approx 0.5$) [25]. However,
any J/$\psi$ suppression due to initial state effects, CGC or shadowing, will
be stronger at lower $p_{\rm{t}}$ contrary to the data behavior.
Figure 4: (Color online) Inclusive J/$\psi$ $R_{\rm{AA}}$ measured in Pb-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV compared to the predictions by
Statistical Hadronization Model [26], Transport Model I [27] and II [28], see
text for details. The ALICE data points are placed at the $\langle
N_{\rm{part}}^{\rm{w}}\rangle$ values defined in Table 1.
In Fig. 4, our measurement is compared with theoretical models that include a
J/$\psi$ regeneration component from deconfined charm quarks in the medium.
The Statistical Hadronization Model [6, 26] assumes deconfinement and a
thermal equilibration of the bulk of the c$\bar{\rm{c}}$ pairs. Then
charmonium production occurs only at phase boundary by statistical
hadronization of charm quarks. The prediction is given for two values of ${\rm
d}\sigma_{\rm{c}\bar{\rm{c}}}/{\rm d}y$ in absence of a measurement for Pb-Pb
collisions. The two transport model results [27, 28] presented in the same
figure differ mostly in the rate equation controlling the J/$\psi$
dissociation and regeneration. Both are shown as a band which connects the
results obtained with (lower limit) and without (higher limit) shadowing. The
width of the band can be interpreted as the uncertainty of the prediction. In
both transport models, the amount of regenerated J/$\psi$ in the most central
collisions contributes to about 50% of the production yield, the rest being
from initial production.
In summary, we have presented the first measurement of inclusive J/$\psi$
nuclear modification factor down to $p_{\rm{t}}=0$ at forward rapidity in Pb-
Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV. The J/$\psi$ $R_{\rm{AA}}$
is larger than the one measured at the SPS and at RHIC for most central
collisions and does not exhibit a significant centrality dependence.
Statistical hadronization and transport models which respectively feature a
full and a partial J/$\psi$ production from charm quarks in the QGP phase can
describe the data. Towards a definitive conclusion about the role of J/$\psi$
production from deconfined charm quarks in a partonic phase, the amount of
shadowing needs to be measured precisely in pPb collisions. In this context,
the measurement of open charm and J/$\psi$ elliptic flow will also help to
determine the degree of thermalization for charm quarks.
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## 1 Acknowledgements
The ALICE collaboration would like to thank all its engineers and technicians
for their invaluable contributions to the construction of the experiment and
the CERN accelerator teams for the outstanding performance of the LHC complex.
The ALICE collaboration acknowledges the following funding agencies for their
support in building and running the ALICE detector:
Calouste Gulbenkian Foundation from Lisbon and Swiss Fonds Kidagan, Armenia;
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq),
Financiadora de Estudos e Projetos (FINEP), Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP);
National Natural Science Foundation of China (NSFC), the Chinese Ministry of
Education (CMOE) and the Ministry of Science and Technology of China (MSTC);
Ministry of Education and Youth of the Czech Republic;
Danish Natural Science Research Council, the Carlsberg Foundation and the
Danish National Research Foundation;
The European Research Council under the European Community’s Seventh Framework
Programme;
Helsinki Institute of Physics and the Academy of Finland;
French CNRS-IN2P3, the ‘Region Pays de Loire’, ‘Region Alsace’, ‘Region
Auvergne’ and CEA, France;
German BMBF and the Helmholtz Association;
General Secretariat for Research and Technology, Ministry of Development,
Greece;
Hungarian OTKA and National Office for Research and Technology (NKTH);
Department of Atomic Energy and Department of Science and Technology of the
Government of India;
Istituto Nazionale di Fisica Nucleare (INFN) of Italy;
MEXT Grant-in-Aid for Specially Promoted Research, Japan;
Joint Institute for Nuclear Research, Dubna;
National Research Foundation of Korea (NRF);
CONACYT, DGAPA, México, ALFA-EC and the HELEN Program (High-Energy physics
Latin-American–European Network);
Stichting voor Fundamenteel Onderzoek der Materie (FOM) and the Nederlandse
Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands;
Research Council of Norway (NFR);
Polish Ministry of Science and Higher Education;
National Authority for Scientific Research - NASR (Autoritatea Naţională
pentru Cercetare Ştiinţifică - ANCS);
Federal Agency of Science of the Ministry of Education and Science of Russian
Federation, International Science and Technology Center, Russian Academy of
Sciences, Russian Federal Agency of Atomic Energy, Russian Federal Agency for
Science and Innovations and CERN-INTAS;
Ministry of Education of Slovakia;
Department of Science and Technology, South Africa;
CIEMAT, EELA, Ministerio de Educación y Ciencia of Spain, Xunta de Galicia
(Consellería de Educación), CEADEN, Cubaenergía, Cuba, and IAEA (International
Atomic Energy Agency);
Swedish Research Council (VR) and Knut $\&$ Alice Wallenberg Foundation (KAW);
Ukraine Ministry of Education and Science;
United Kingdom Science and Technology Facilities Council (STFC);
The United States Department of Energy, the United States National Science
Foundation, the State of Texas, and the State of Ohio.
## Appendix A The ALICE Collaboration
B. Abelevorg1234&J. Adamorg1274&D. Adamováorg1283&A.M. Adareorg1260&M.M.
Aggarwalorg1157&G. Aglieri Rinellaorg1192&A.G. Agocsorg1143&A.
Agostinelliorg1132&S. Aguilar Salazarorg1247&Z. Ahammedorg1225&A. Ahmad
Masoodiorg1106&N. Ahmadorg1106&S.U. Ahnorg1160,org1215&A. Akindinovorg1250&D.
Aleksandrovorg1252&B. Alessandroorg1313&R. Alfaro Molinaorg1247&A.
Aliciorg1133,org1335&A. Alkinorg1220&E. Almaráz Aviñaorg1247&J. Almeorg1122&T.
Altorg1184&V. Altiniorg1114&S. Altinpinarorg1121&I. Altsybeevorg1306&C.
Andreiorg1140&A. Andronicorg1176&V. Anguelovorg1200&J. Anielskiorg1256&C.
Ansonorg1162&T. Antičićorg1334&F. Antinoriorg1271&P. Antonioliorg1133&L.
Aphecetcheorg1258&H. Appelshäuserorg1185&N. Arbororg1194&S. Arcelliorg1132&A.
Arendorg1185&N. Armestoorg1294&R. Arnaldiorg1313&T. Aronssonorg1260&I.C.
Arseneorg1176&M. Arslandokorg1185&A. Asryanorg1306&A. Augustinusorg1192&R.
Averbeckorg1176&T.C. Awesorg1264&J. Äystöorg1212&M.D. Azmiorg1106&M.
Bachorg1184&A. Badalàorg1155&Y.W. Baekorg1160,org1215&R. Bailhacheorg1185&R.
Balaorg1313&R. Baldini Ferroliorg1335&A. Baldisseriorg1288&A. Balditorg1160&F.
Baltasar Dos Santos Pedrosaorg1192&J. Bánorg1230&R.C. Baralorg1127&R.
Barberaorg1154&F. Barileorg1114&G.G. Barnaföldiorg1143&L.S. Barnbyorg1130&V.
Barretorg1160&J. Bartkeorg1168&M. Basileorg1132&N. Bastidorg1160&B.
Bathenorg1256&G. Batigneorg1258&B. Batyunyaorg1182&C. Baumannorg1185&I.G.
Beardenorg1165&H. Beckorg1185&I. Belikovorg1308&F. Belliniorg1132&R.
Bellwiedorg1205&E. Belmont-Morenoorg1247&G. Bencediorg1143&S. Beoleorg1312&I.
Berceanuorg1140&A. Bercuciorg1140&Y. Berdnikovorg1189&D. Berenyiorg1143&C.
Bergmannorg1256&D. Berzanoorg1313&L. Betevorg1192&A. Bhasinorg1209&A.K.
Bhatiorg1157&N. Bianchiorg1187&L. Bianchiorg1312&C. Bianchinorg1270&J.
Bielčíkorg1274&J. Bielčíkováorg1283&A. Bilandzicorg1109&S.
Bjelogrlicorg1320&F. Blancoorg1205&F. Blancoorg1242&D. Blauorg1252&C.
Blumeorg1185&M. Boccioliorg1192&N. Bockorg1162&A. Bogdanovorg1251&H.
Bøggildorg1165&M. Bogolyubskyorg1277&L. Boldizsárorg1143&M. Bombaraorg1229&J.
Bookorg1185&H. Borelorg1288&A. Borissovorg1179&S. Boseorg1224&F.
Bossúorg1312&M. Botjeorg1109&S. Böttgerorg27399&B. Boyerorg1266&P. Braun-
Munzingerorg1176&M. Bregantorg1258&T. Breitnerorg27399&T.A. Browningorg1325&M.
Brozorg1136&R. Brunorg1192&E. Brunaorg1312,org1313&G.E. Brunoorg1114&D.
Budnikovorg1298&H. Bueschingorg1185&S. Bufalinoorg1312,org1313&K.
Bugaievorg1220&O. Buschorg1200&Z. Butheleziorg1152&D. Caballero
Ordunaorg1260&D. Caffarriorg1270&X. Caiorg1329&H. Cainesorg1260&E. Calvo
Villarorg1338&P. Cameriniorg1315&V. Canoa Romanorg1244,org1279&G. Cara
Romeoorg1133&F. Carenaorg1192&W. Carenaorg1192&N. Carlin Filhoorg1296&F.
Carminatiorg1192&C.A. Carrillo Montoyaorg1192&A. Casanova Díazorg1187&J.
Castillo Castellanosorg1288&J.F. Castillo Hernandezorg1176&E.A.R.
Casulaorg1145&V. Catanescuorg1140&C. Cavicchioliorg1192&J. Cepilaorg1274&P.
Cerelloorg1313&B. Changorg1212,org1301&S. Chapelandorg1192&J.L.
Charvetorg1288&S. Chattopadhyayorg1224&S. Chattopadhyayorg1225&M.
Cherneyorg1170&C. Cheshkovorg1192,org1239&B. Cheynisorg1239&E.
Chiavassaorg1313&V. Chibante Barrosoorg1192&D.D. Chinellatoorg1149&P.
Chochulaorg1192&M. Chojnackiorg1320&P. Christakoglouorg1109,org1320&C.H.
Christensenorg1165&P. Christiansenorg1237&T. Chujoorg1318&S.U. Chungorg1281&C.
Cicaloorg1146&L. Cifarelliorg1132,org1192&F. Cindoloorg1133&J.
Cleymansorg1152&F. Coccettiorg1335&F. Colamariaorg1114&D. Colellaorg1114&G.
Conesa Balbastreorg1194&Z. Conesa del Valleorg1192&P. Constantinorg1200&G.
Continorg1315&J.G. Contrerasorg1244&T.M. Cormierorg1179&Y. Corrales
Moralesorg1312&P. Corteseorg1103&I. Cortés Maldonadoorg1279&M.R.
Cosentinoorg1125,org1149&F. Costaorg1192&M.E. Cotalloorg1242&E.
Crescioorg1244&P. Crochetorg1160&E. Cruz Alanizorg1247&E. Cuautleorg1246&L.
Cunqueiroorg1187&A. Daineseorg1270,org1271&H.H. Dalsgaardorg1165&A.
Danuorg1139&I. Dasorg1224,org1266&K. Dasorg1224&D. Dasorg1224&S.
Dashorg1254,org1313&A. Dashorg1149&S. Deorg1225&A. De Azevedo
Moregulaorg1187&G.O.V. de Barrosorg1296&A. De Caroorg1290,org1335&G. de
Cataldoorg1115&J. de Cuvelandorg1184&A. De Falcoorg1145&D. De
Gruttolaorg1290&H. Delagrangeorg1258&E. Del Castillo Sanchezorg1192&A.
Delofforg1322&V. Demanovorg1298&N. De Marcoorg1313&E. Dénesorg1143&S. De
Pasqualeorg1290&A. Deppmanorg1296&G. D Erasmoorg1114&R. de Rooijorg1320&D. Di
Bariorg1114&T. Dietelorg1256&C. Di Giglioorg1114&S. Di Libertoorg1286&A. Di
Mauroorg1192&P. Di Nezzaorg1187&R. Diviàorg1192&Ø. Djuvslandorg1121&A.
Dobrinorg1179,org1237&T. Dobrowolskiorg1322&I. Domínguezorg1246&B.
Dönigusorg1176&O. Dordicorg1268&O. Drigaorg1258&A.K. Dubeyorg1225&L.
Ducrouxorg1239&P. Dupieuxorg1160&A.K. Dutta Majumdarorg1224&M.R. Dutta
Majumdarorg1225&D. Eliaorg1115&D. Emschermannorg1256&H. Engelorg27399&H.A.
Erdalorg1122&B. Espagnonorg1266&M. Estienneorg1258&S. Esumiorg1318&D.
Evansorg1130&G. Eyyubovaorg1268&D. Fabrisorg1270,org1271&J. Faivreorg1194&D.
Falchieriorg1132&A. Fantoniorg1187&M. Faselorg1176&R. Fearickorg1152&A.
Fedunovorg1182&D. Fehlkerorg1121&L. Feldkamporg1256&D. Feleaorg1139&G.
Feofilovorg1306&A. Fernández Téllezorg1279&R. Ferrettiorg1103&A.
Ferrettiorg1312&J. Figielorg1168&M.A.S. Figueredoorg1296&S.
Filchaginorg1298&D. Finogeevorg1249&F.M. Fiondaorg1114&E.M. Fioreorg1114&M.
Florisorg1192&S. Foertschorg1152&P. Fokaorg1176&S. Fokinorg1252&E.
Fragiacomoorg1316&M. Fragkiadakisorg1112&U. Frankenfeldorg1176&U.
Fuchsorg1192&C. Furgetorg1194&M. Fusco Girardorg1290&J.J. Gaardhøjeorg1165&M.
Gagliardiorg1312&A. Gagoorg1338&M. Gallioorg1312&D.R. Gangadharanorg1162&P.
Ganotiorg1264&C. Garabatosorg1176&E. Garcia-Solisorg17347&I.
Garishviliorg1234&J. Gerhardorg1184&M. Germainorg1258&C. Geunaorg1288&A.
Gheataorg1192&M. Gheataorg1192&B. Ghidiniorg1114&P. Ghoshorg1225&P.
Gianottiorg1187&M.R. Girardorg1323&P. Giubellinoorg1192&E. Gladysz-
Dziadusorg1168&P. Glässelorg1200&R. Gomezorg1173&E.G. Ferreiroorg1294&L.H.
González-Truebaorg1247&P. González-Zamoraorg1242&S. Gorbunovorg1184&A.
Goswamiorg1207&S. Gotovacorg1304&V. Grabskiorg1247&L.K. Graczykowskiorg1323&R.
Grajcarekorg1200&A. Grelliorg1320&A. Grigorasorg1192&C. Grigorasorg1192&V.
Grigorievorg1251&A. Grigoryanorg1332&S. Grigoryanorg1182&B. Grinyovorg1220&N.
Grionorg1316&P. Grosorg1237&J.F. Grosse-Oetringhausorg1192&J.-Y.
Grossiordorg1239&R. Grossoorg1192&F. Guberorg1249&R. Guernaneorg1194&C. Guerra
Gutierrezorg1338&B. Guerzoniorg1132&M. Guilbaudorg1239&K.
Gulbrandsenorg1165&T. Gunjiorg1310&A. Guptaorg1209&R. Guptaorg1209&H.
Gutbrodorg1176&Ø. Haalandorg1121&C. Hadjidakisorg1266&M. Haiducorg1139&H.
Hamagakiorg1310&G. Hamarorg1143&B.H. Hanorg1300&L.D. Hanrattyorg1130&A.
Hansenorg1165&Z. Harmanovaorg1229&J.W. Harrisorg1260&M. Hartigorg1185&D.
Haseganorg1139&D. Hatzifotiadouorg1133&A. Hayrapetyanorg1192,org1332&S.T.
Heckelorg1185&M. Heideorg1256&H. Helstruporg1122&A. Herghelegiuorg1140&G.
Herrera Corralorg1244&N. Herrmannorg1200&K.F. Hetlandorg1122&B.
Hicksorg1260&P.T. Hilleorg1260&B. Hippolyteorg1308&T. Horaguchiorg1318&Y.
Horiorg1310&P. Hristovorg1192&I. Hřivnáčováorg1266&M. Huangorg1121&S.
Huberorg1176&T.J. Humanicorg1162&D.S. Hwangorg1300&R. Ichouorg1160&R.
Ilkaevorg1298&I. Ilkivorg1322&M. Inabaorg1318&E. Incaniorg1145&G.M.
Innocentiorg1312&P.G. Innocentiorg1192&M. Ippolitovorg1252&M. Irfanorg1106&C.
Ivanorg1176&M. Ivanovorg1176&A. Ivanovorg1306&V. Ivanovorg1189&O.
Ivanytskyiorg1220&A. Jachołkowskiorg1192&P. M. Jacobsorg1125&L.
Jancurováorg1182&H.J. Jangorg20954&S. Jangalorg1308&M.A. Janikorg1323&R.
Janikorg1136&P.H.S.Y. Jayarathnaorg1205&S. Jenaorg1254&R.T. Jimenez
Bustamanteorg1246&L. Jirdenorg1192&P.G. Jonesorg1130&H. Jungorg1215&W.
Jungorg1215&A. Juskoorg1130&A.B. Kaidalovorg1250&V. Kakoyanorg1332&S.
Kalcherorg1184&P. Kaliňákorg1230&M. Kaliskyorg1256&T. Kalliokoskiorg1212&A.
Kalweitorg1177&K. Kanakiorg1121&J.H. Kangorg1301&V. Kaplinorg1251&A. Karasu
Uysalorg1192,org15649&O. Karavichevorg1249&T. Karavichevaorg1249&E.
Karpechevorg1249&A. Kazantsevorg1252&U. Kebschullorg27399&R.
Keidelorg1327&S.A. Khanorg1225&M.M. Khanorg1106&P. Khanorg1224&A.
Khanzadeevorg1189&Y. Kharlovorg1277&B. Kilengorg1122&M. Kimorg1301&T.
Kimorg1301&S. Kimorg1300&D.J. Kimorg1212&J.H. Kimorg1300&J.S. Kimorg1215&S.H.
Kimorg1215&D.W. Kimorg1215&B. Kimorg1301&S. Kirschorg1184,org1192&I.
Kiselorg1184&S. Kiselevorg1250&A. Kisielorg1192,org1323&J.L. Klayorg1292&J.
Kleinorg1200&C. Klein-Bösingorg1256&M. Kliemantorg1185&A. Klugeorg1192&M.L.
Knichelorg1176&A.G. Knospeorg17361&K. Kochorg1200&M.K. Köhlerorg1176&A.
Kolojvariorg1306&V. Kondratievorg1306&N. Kondratyevaorg1251&A.
Konevskikhorg1249&A. Korneevorg1298&C. Kottachchi Kankanamge Donorg1179&R.
Kourorg1130&M. Kowalskiorg1168&S. Koxorg1194&G. Koyithatta
Meethaleveeduorg1254&J. Kralorg1212&I. Králikorg1230&F. Kramerorg1185&I.
Krausorg1176&T. Krawutschkeorg1200,org1227&M. Krelinaorg1274&M.
Kretzorg1184&M. Krivdaorg1130,org1230&F. Krizekorg1212&M. Krusorg1274&E.
Kryshenorg1189&M. Krzewickiorg1109,org1176&Y. Kucheriaevorg1252&C.
Kuhnorg1308&P.G. Kuijerorg1109&P. Kurashviliorg1322&A.B. Kurepinorg1249&A.
Kurepinorg1249&A. Kuryakinorg1298&V. Kushpilorg1283&S. Kushpilorg1283&H.
Kvaernoorg1268&M.J. Kweonorg1200&Y. Kwonorg1301&P. Ladrón de Guevaraorg1246&I.
Lakomovorg1266,org1306&R. Langoyorg1121&C. Laraorg27399&A. Lardeuxorg1258&P.
La Roccaorg1154&C. Lazzeroniorg1130&R. Leaorg1315&Y. Le Bornecorg1266&K.S.
Leeorg1215&S.C. Leeorg1215&F. Lefèvreorg1258&J. Lehnertorg1185&L.
Leistamorg1192&M. Lenhardtorg1258&V. Lentiorg1115&H. Leónorg1247&I. León
Monzónorg1173&H. León Vargasorg1185&P. Lévaiorg1143&J. Lienorg1121&R.
Lietavaorg1130&S. Lindalorg1268&V. Lindenstruthorg1184&C.
Lippmannorg1176,org1192&M.A. Lisaorg1162&L. Liuorg1121&P.I. Loenneorg1121&V.R.
Logginsorg1179&V. Loginovorg1251&S. Lohnorg1192&D. Lohnerorg1200&C.
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Luoorg1329&G. Luparelloorg1320&L. Luquinorg1258&C. Luzziorg1192&K.
Maorg1329&R. Maorg1260&D.M. Madagodahettige-Donorg1205&A. Maevskayaorg1249&M.
Magerorg1177,org1192&D.P. Mahapatraorg1127&A. Maireorg1308&M. Malaevorg1189&I.
Maldonado Cervantesorg1246&L. Malininaorg1182,M.V.Lomonosov Moscow State
University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia&D.
Mal’Kevichorg1250&P. Malzacherorg1176&A. Mamonovorg1298&L. Manceauorg1313&L.
Mangotraorg1209&V. Mankoorg1252&F. Mansoorg1160&V. Manzariorg1115&Y.
Maoorg1194,org1329&M. Marchisoneorg1160,org1312&J. Marešorg1275&G.V.
Margagliottiorg1315,org1316&A. Margottiorg1133&A. Marínorg1176&C.A. Marin
Tobonorg1192&C. Markertorg17361&I. Martashviliorg1222&P.
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Garcíaorg1258&Y. Martynovorg1220&A. Masorg1258&S. Masciocchiorg1176&M.
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Mazzoniorg1286&F. Meddiorg1285&A. Menchaca-Rochaorg1247&J. Mercado
Pérezorg1200&M. Meresorg1136&Y. Miakeorg1318&L. Milanoorg1312&J.
Milosevicorg1268,Institute of Nuclear Sciences, Belgrade, Serbia&A.
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Mussoorg1313&B.K. Nandiorg1254&R. Naniaorg1133&E. Nappiorg1115&C.
Nattrassorg1222&N.P. Naumovorg1298&S. Navinorg1130&T.K. Nayakorg1225&S.
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Velasquezorg1237,org1246&G. Ortonaorg1312&A. Oskarssonorg1237&P.
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Pérez Laraorg1109&E. Perez Lezamaorg1246&D. Periniorg1192&D. Perrinoorg1114&W.
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Teixidoorg27399&A. Pulvirentiorg1154,org1192&V. Puninorg1298&M.
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Stassinakiorg1112&B.K. Srivastavaorg1325&J. Stachelorg1200&I. Stanorg1139&I.
Stanorg1139&G. Stefanekorg1322&G. Stefaniniorg1192&T. Steinbeckorg1184&M.
Steinpreisorg1162&E. Stenlundorg1237&G. Steynorg1152&D. Stoccoorg1258&M.
Stolpovskiyorg1277&K. Strabykinorg1298&P. Strmenorg1136&A.A.P.
Suaideorg1296&M.A. Subieta Vásquezorg1312&T. Sugitateorg1203&C.
Suireorg1266&M. Sukhorukovorg1298&R. Sultanovorg1250&M. Šumberaorg1283&T.
Susaorg1334&A. Szanto de Toledoorg1296&I. Szarkaorg1136&A. Szostakorg1121&C.
Tagridisorg1112&J. Takahashiorg1149&J.D. Tapia Takakiorg1266&A.
Tauroorg1192&G. Tejeda Muñozorg1279&A. Telescaorg1192&C. Terrevoliorg1114&J.
Thäderorg1176&D. Thomasorg1320&R. Tieulentorg1239&A.R. Timminsorg1205&D.
Tlustyorg1274&A. Toiaorg1184,org1192&H. Toriiorg1203,org1310&L.
Toscanoorg1313&F. Toselloorg1313&D. Truesdaleorg1162&W.H. Trzaskaorg1212&T.
Tsujiorg1310&A. Tumkinorg1298&R. Turrisiorg1271&T.S. Tveterorg1268&J.
Uleryorg1185&K. Ullalandorg1121&J. Ulrichorg1199,org27399&A. Urasorg1239&J.
Urbánorg1229&G.M. Urciuoliorg1286&G.L. Usaiorg1145&M. Vajzerorg1274,org1283&M.
Valaorg1182,org1230&L. Valencia Palomoorg1266&S. Valleroorg1200&N. van der
Kolkorg1109&P. Vande Vyvreorg1192&M. van Leeuwenorg1320&L. Vannucciorg1232&A.
Vargasorg1279&R. Varmaorg1254&M. Vasileiouorg1112&A. Vasilievorg1252&V.
Vecherninorg1306&M. Veldhoenorg1320&M. Venaruzzoorg1315&E. Vercellinorg1312&S.
Vergaraorg1279&D.C. Vernekohlorg1256&R. Vernetorg14939&M. Verweijorg1320&L.
Vickovicorg1304&G. Viestiorg1270&O. Vikhlyantsevorg1298&Z. Vilakaziorg1152&O.
Villalobos Baillieorg1130&A. Vinogradovorg1252&L. Vinogradovorg1306&Y.
Vinogradovorg1298&T. Virgiliorg1290&Y.P. Viyogiorg1225&A. Vodopyanovorg1182&K.
Voloshinorg1250&S. Voloshinorg1179&G. Volpeorg1114,org1192&B. von
Hallerorg1192&D. Vranicorg1176&G. Øvrebekkorg1121&J. Vrlákováorg1229&B.
Vulpescuorg1160&A. Vyushinorg1298&B. Wagnerorg1121&V. Wagnerorg1274&R.
Wanorg1308,org1329&D. Wangorg1329&M. Wangorg1329&Y. Wangorg1200&Y.
Wangorg1329&K. Watanabeorg1318&J.P. Wesselsorg1192,org1256&U.
Westerhofforg1256&J. Wiechulaorg21360&J. Wikneorg1268&M. Wildeorg1256&G.
Wilkorg1322&A. Wilkorg1256&M.C.S. Williamsorg1133&B. Windelbandorg1200&L.
Xaplanteris Karampatsosorg17361&H. Yangorg1288&S. Yangorg1121&S.
Yasnopolskiyorg1252&J. Yiorg1281&Z. Yinorg1329&H. Yokoyamaorg1318&I.-K.
Yooorg1281&J. Yoonorg1301&W. Yuorg1185&X. Yuanorg1329&I. Yushmanovorg1252&C.
Zachorg1274&C. Zampolliorg1133,org1192&S. Zaporozhetsorg1182&A.
Zarochentsevorg1306&P. Závadaorg1275&N. Zaviyalovorg1298&H.
Zbroszczykorg1323&P. Zelnicekorg27399&I.S. Zguraorg1139&M. Zhalovorg1189&X.
Zhangorg1160,org1329&Y. Zhouorg1320&D. Zhouorg1329&F. Zhouorg1329&X.
Zhuorg1329&A. Zichichiorg1132,org1335&A. Zimmermannorg1200&G.
Zinovjevorg1220&Y. Zoccaratoorg1239&M. Zynovyevorg1220
## Affiliation notes
M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear
Physics, Moscow, RussiaAlso at: M.V.Lomonosov Moscow State University,
D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia
Institute of Nuclear Sciences, Belgrade, SerbiaAlso at: ”Vinča” Institute of
Nuclear Sciences, Belgrade, Serbia
## Collaboration Institutes
org1279Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
org1220Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
org1262Budker Institute for Nuclear Physics, Novosibirsk, Russia
org1292California Polytechnic State University, San Luis Obispo, California,
United States
org14939Centre de Calcul de l’IN2P3, Villeurbanne, France
org1197Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN),
Havana, Cuba
org1242Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
org1244Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico
City and Mérida, Mexico
org1335Centro Fermi – Centro Studi e Ricerche e Museo Storico della Fisica
“Enrico Fermi”, Rome, Italy
org17347Chicago State University, Chicago, United States
org1288Commissariat à l’Energie Atomique, IRFU, Saclay, France
org1294Departamento de Física de Partículas and IGFAE, Universidad de Santiago
de Compostela, Santiago de Compostela, Spain
org1106Department of Physics Aligarh Muslim University, Aligarh, India
org1121Department of Physics and Technology, University of Bergen, Bergen,
Norway
org1162Department of Physics, Ohio State University, Columbus, Ohio, United
States
org1300Department of Physics, Sejong University, Seoul, South Korea
org1268Department of Physics, University of Oslo, Oslo, Norway
org1145Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari,
Italy
org1270Dipartimento di Fisica dell’Università and Sezione INFN, Padova, Italy
org1315Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
org1132Dipartimento di Fisica dell’Università and Sezione INFN, Bologna, Italy
org1285Dipartimento di Fisica dell’Università ‘La Sapienza’ and Sezione INFN,
Rome, Italy
org1154Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
org1290Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
org1312Dipartimento di Fisica Sperimentale dell’Università and Sezione INFN,
Turin, Italy
org1103Dipartimento di Scienze e Tecnologie Avanzate dell’Università del
Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy
org1114Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
org1237Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden
org1192European Organization for Nuclear Research (CERN), Geneva, Switzerland
org1227Fachhochschule Köln, Köln, Germany
org1122Faculty of Engineering, Bergen University College, Bergen, Norway
org1136Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia
org1274Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
org1229Faculty of Science, P.J. Šafárik University, Košice, Slovakia
org1184Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
org1215Gangneung-Wonju National University, Gangneung, South Korea
org1212Helsinki Institute of Physics (HIP) and University of Jyväskylä,
Jyväskylä, Finland
org1203Hiroshima University, Hiroshima, Japan
org1329Hua-Zhong Normal University, Wuhan, China
org1254Indian Institute of Technology, Mumbai, India
org36378Indian Institute of Technology Indore (IIT), Indore, India
org1266Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud,
CNRS-IN2P3, Orsay, France
org1277Institute for High Energy Physics, Protvino, Russia
org1249Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
org1320Nikhef, National Institute for Subatomic Physics and Institute for
Subatomic Physics of Utrecht University, Utrecht, Netherlands
org1250Institute for Theoretical and Experimental Physics, Moscow, Russia
org1230Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia
org1127Institute of Physics, Bhubaneswar, India
org1275Institute of Physics, Academy of Sciences of the Czech Republic,
Prague, Czech Republic
org1139Institute of Space Sciences (ISS), Bucharest, Romania
org27399Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
org1185Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
org1177Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt,
Germany
org1256Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
Münster, Germany
org1246Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de
México, Mexico City, Mexico
org1247Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico
org23333Institut of Theoretical Physics, University of Wroclaw
org1308Institut Pluridisciplinaire Hubert Curien (IPHC), Université de
Strasbourg, CNRS-IN2P3, Strasbourg, France
org1182Joint Institute for Nuclear Research (JINR), Dubna, Russia
org1143KFKI Research Institute for Particle and Nuclear Physics, Hungarian
Academy of Sciences, Budapest, Hungary
org1199Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany
org20954Korea Institute of Science and Technology Information
org1160Laboratoire de Physique Corpusculaire (LPC), Clermont Université,
Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France
org1194Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Université
Joseph Fourier, CNRS-IN2P3, Institut Polytechnique de Grenoble, Grenoble,
France
org1187Laboratori Nazionali di Frascati, INFN, Frascati, Italy
org1232Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
org1125Lawrence Berkeley National Laboratory, Berkeley, California, United
States
org1234Lawrence Livermore National Laboratory, Livermore, California, United
States
org1251Moscow Engineering Physics Institute, Moscow, Russia
org1140National Institute for Physics and Nuclear Engineering, Bucharest,
Romania
org1165Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
org1109Nikhef, National Institute for Subatomic Physics, Amsterdam,
Netherlands
org1283Nuclear Physics Institute, Academy of Sciences of the Czech Republic,
Řež u Prahy, Czech Republic
org1264Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
org1189Petersburg Nuclear Physics Institute, Gatchina, Russia
org1170Physics Department, Creighton University, Omaha, Nebraska, United
States
org1157Physics Department, Panjab University, Chandigarh, India
org1112Physics Department, University of Athens, Athens, Greece
org1152Physics Department, University of Cape Town, iThemba LABS, Cape Town,
South Africa
org1209Physics Department, University of Jammu, Jammu, India
org1207Physics Department, University of Rajasthan, Jaipur, India
org1200Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany
org1325Purdue University, West Lafayette, Indiana, United States
org1281Pusan National University, Pusan, South Korea
org1176Research Division and ExtreMe Matter Institute EMMI, GSI
Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany
org1334Rudjer Bošković Institute, Zagreb, Croatia
org1298Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
org1252Russian Research Centre Kurchatov Institute, Moscow, Russia
org1224Saha Institute of Nuclear Physics, Kolkata, India
org1130School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
org1338Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru
org1316Sezione INFN, Trieste, Italy
org1271Sezione INFN, Padova, Italy
org1313Sezione INFN, Turin, Italy
org1286Sezione INFN, Rome, Italy
org1146Sezione INFN, Cagliari, Italy
org1133Sezione INFN, Bologna, Italy
org1115Sezione INFN, Bari, Italy
org1155Sezione INFN, Catania, Italy
org1322Soltan Institute for Nuclear Studies, Warsaw, Poland
org1258SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France
org1304Technical University of Split FESB, Split, Croatia
org1168The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland
org17361The University of Texas at Austin, Physics Department, Austin, TX,
United States
org1173Universidad Autónoma de Sinaloa, Culiacán, Mexico
org1296Universidade de São Paulo (USP), São Paulo, Brazil
org1149Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
org1239Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, France
org1205University of Houston, Houston, Texas, United States
org1222University of Tennessee, Knoxville, Tennessee, United States
org1310University of Tokyo, Tokyo, Japan
org1318University of Tsukuba, Tsukuba, Japan
org21360Eberhard Karls Universität Tübingen, Tübingen, Germany
org1225Variable Energy Cyclotron Centre, Kolkata, India
org1306V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia
org1323Warsaw University of Technology, Warsaw, Poland
org1179Wayne State University, Detroit, Michigan, United States
org1260Yale University, New Haven, Connecticut, United States
org1332Yerevan Physics Institute, Yerevan, Armenia
org15649Yildiz Technical University, Istanbul, Turkey
org1301Yonsei University, Seoul, South Korea
org1327Zentrum für Technologietransfer und Telekommunikation (ZTT),
Fachhochschule Worms, Worms, Germany
|
arxiv-papers
| 2012-02-07T09:49:10 |
2024-09-04T02:49:27.176562
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "ALICE Collaboration",
"submitter": "Alice Publications",
"url": "https://arxiv.org/abs/1202.1383"
}
|
1202.1391
|
arxiv-papers
| 2012-02-07T10:29:07 |
2024-09-04T02:49:27.187671
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Pomarol (Barcelona, Autonoma U.)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1202.1391"
}
|
|
1202.1629
|
¡html¿¡head¿ ¡meta http-equiv=”content-type” content=”text/html;
charset=ISO-8859-1”¿
¡title¿CERN-2012-001¡/title¿
¡/head¿
¡body¿
¡h1¿¡a
href=”http://physicschool.web.cern.ch/PhysicSchool/ESHEP/ESHEP2010/default.html”¿2010
European School of High-energy Physics¡/a¿¡/h1¿
¡h2¿Raseborg, Finland , 20 Jun - 3 Jul 2010¡/h2¿
¡h2¿Proceedings - CERN Yellow Report
¡a href=”https://cdsweb.cern.ch/record/1226997”¿CERN-2012-001¡/a¿¡/h2¿
¡h3¿editors: C. Grojean and M. Spiropulu¡/h3¿
The European School of High-Energy Physics is intended to give young
physicists an introduction to the theoretical aspects of recent advances in
elementary particle physics. These proceedings contain lecture notes on the
Standard Model of electroweak interactions, quantum chromodynamics, heavy ion
physics, physics beyond the Standard Model, neutrino physics, and cosmology.
¡h2¿Lectures¡/h2¿
¡p¿ LIST:arXiv:1201.0537¡br¿
LIST:arXiv:1104.2863¡br¿
LIST:arXiv:1012.0186¡br¿
LIST:arXiv:1202.1391¡br¿
LIST:arXiv:1201.6158¡br¿
LIST:arXiv:1012.5204¡br¿
LIST:arXiv:1201.6164 ¡br¿ ¡/p¿ ¡/body¿¡/html¿
|
arxiv-papers
| 2012-02-08T08:59:50 |
2024-09-04T02:49:27.199514
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. Grojean and M. Spiropulu (eds.) (CERN)",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1202.1629"
}
|
1202.1656
|
# Open Data:
Reverse Engineering and Maintenance Perspective
Holger M. Kienle hkienle@acm.org
http://holgerkienle.wikispaces.com/
###### Abstract
Open data is an emerging paradigm to share large and diverse
datasets—primarily from governmental agencies, but also from other
organizations—with the goal to enable the exploitation of the data for
societal, academic, and commercial gains. There are now already many datasets
available with diverse characteristics in terms of size, encoding and
structure. These datasets are often created and maintained in an ad-hoc
manner. Thus, open data poses many challenges and there is a need for
effective tools and techniques to manage and maintain it.
In this paper we argue that software maintenance and reverse engineering have
an opportunity to contribute to open data and to shape its future development.
From the perspective of reverse engineering research, open data is a new
artifact that serves as input for reverse engineering techniques and
processes. Specific challenges of open data are document scraping, image
processing, and structure/schema recognition. From the perspective of
maintenance research, maintenance has to accommodate changes of open data
sources by third-party providers, traceability of data transformation
pipelines, and quality assurance of data and transformations. We believe that
the increasing importance of open data and the research challenges that it
brings with it may possibly lead to the emergence of new research streams for
reverse engineering as well as for maintenance.
Figure 1: Example of publication of Romanian ERDF spending data (PDF file)
## I Introduction and Background
Open data is an approach to data management based on the tenet that “certain
data should be freely available to everyone to use and republish” (Wikipedia).
Open data has increasingly gained traction over the years and by now is
supported by parts of academia, government and business.
Open data can be characterized as information processing with the goal to
create knowledge and to manipulate that knowledge effectively (e.g., via
collaborative tagging and interactive mash-up visualizations). Arguably, the
idea of open data was first brought to the attention of a broader audience
with an article from UK’s The Guardian in 2006, which had the opening line:
“Our taxes fund the collection of public data – yet we pay again to access it.
Make the data freely available to stimulate innovation” [1]. However, it
should be noted that efforts started earlier than that—the Australian
government has been moving towards more open data management since at least
2001 [10].
Besides open data, there are related concepts such as open content
(opencontent.org/definition/), open access
(www.earlham.edu/~peters/fos/overview.htm) and open knowledge
(opendefinition.org/okd/), but their boundaries are blurry; in the following
we use only the term open data with the understanding that it should be
interpreted in a broad sense.
In academia there is the recognition that scientific data should be freely
available to speed up scientific advances and to enable new forms of
collaborative research (e.g., science 2.0 and open notebook science [3]). In
government, data is made available to increase transparency how government
operates and to encourage participation of citizens. Open data in the
governmental domain is encouraged by laws such as the European Directive on
the Re-Use of Public Sector Information (PSI Directive) and the Freedom of
Information Act (FOIA) in the US. The Obama administration pursues the Open
Government Initiative to “ensure the public trust and establish a system of
transparency, public participation, and collaboration”
(www.whitehouse.gov/open) while the Digital Agenda for Europe calls for action
to “open up public data sources for re-use”
(ec.europa.eu/information_society/digital-agenda).
One can argue for open data from many angles, including societal and economic
benefits; conversely, there are also concerns such as potential privacy risks
and the fear that raw data can be misinterpreted [14] [8] [27] [18].
Regardless of its perceived potential and risks, it is a fact that
increasingly data is made available in an open manner. This trend is also
apparent by emerging events such as the Open Government Data Camp
(ogdcamp.org/) and the Open Knowledge Conference (OKCon) (okcon.org).
Open data is already reality. The UK government has made available so far more
than 7,000 datasets at data.gov.uk. Other examples of dataset providers are
the Open Knowledge Foundation’s publicdata.eu, the US government
(www.data.gov, over 3,700 datasets), and The World Bank
(data.worldbank.org/data-catalog, over 7,000 datasets). Furthermore, states,
regions, and cities have also started open data initiatives; to give one of
many examples, the city of Munich has started to publish information as open
data and has held the Munich Open Government Day
(www.muenchen.de/Rathaus/dir/limux/mogdy/Programmierwettbewerb/).
The Open Government Data (OGD) Stakeholder Survey (survey.lod2.eu) conducted
in 2010 has collected 329 responses from citizens, politicians, public
administrators, industry, media and science that are producers, publishers
and/or consumers of open data [16]. The survey revealed that national datasets
are most desirable before regional and worldwide ones and that important
(quality) criteria for open data are provenance/source of data, format,
completeness of metadata, and official certificates. Users of open data are
most interested in geospacial, economic and financial data and want to do
research/analysis, visualization, and simply consuming of the data.
It is expected that open data will continue to be implemented by a growing
number of governments and organizations. Thus, the handling of open data will
increase and with it the need to have effective tools and techniques to manage
and maintain it. In this paper we argue that software maintenance and reverse
engineering has an opportunity to contribute to open data and to shape its
future development. The baseline for this observation is that from the
viewpoint of reverse engineering open data is just another new artifact as
input to the reverse engineering process. Reverse engineering has continuously
broadened its artifacts going beyond source code and databases [17] to, for
instance, images (CAPCHAs) [12] and (business) processes [25] [13]. Of course,
all these artifacts can be treated as data (including source code).
Similarly, open data and its infrastructure has several maintenance challenges
that need to be studied so that domain-specific techniques and tools can be
developed to meet key requirements such as verifiability and traceability. The
mining of software artifacts and their interdependencies [11] can be extended
and adapted towards open data with the goal to, for instance, improving on
detecting and correction of “buggy” data items and data
extractors/transformers, and studying of open data maintenance processes and
collaboration patterns among different groups of contributors (e.g., people
concerned with data scraping, manipulating/abstracting and visualizing).
The reminder of the paper is organized as follows. Section II provides a real-
world example (ERDF data) to illustrate the current state of open data and its
challenges. ERDF data is distributed over various locations using different
formats and inconsistent meta-data. Other examples of open data exhibit
similar challenges. Drawing from this example, Sections III and IV discuss the
reverse engineering and maintenance perspectives of open data, respectively.
For each perspective we identify challenges and research opportunities.
Section V concludes the paper.
## II Illustrative Example
The European Regional Development Fund (ERDF) distributes money to regions in
Europe with the objective “to help reinforce economic and social cohesion by
redressing regional imbalances”
(europa.eu/legislation_summaries/agriculture/general_framework/g24234_en.htm).
Its current funding round runs from 2007–2013, has a budget of EUR 201
billion, and is governed by various regulations. The implementing regulation,
Commission Regulation (EC) No 1828/2006, states in Article 7 that “the
managing authority shall be responsible for … the publication, electronically
or otherwise, of the list of beneficiaries, the names of the operations and
the amount of public funding allocated to the operations.” This requirement
has been newly introduced in a push towards increasing transparency. As a
consequence, the managing governmental authorities of ERDF funds typically
make this information available on public Web sites.
The European Commission maintains a collection of links that point to the
individual data sources
(ec.europa.eu/regional_policy/country/commu/beneficiaries/index_en.htm).
Depending on the country, there can be a single, centralized access point or
multiple access points of a country’s (groups of) regions, provinces, states,
etc. For example, Romania has a central site, each German state maintains its
own Web site, and The Netherlands has four Web sites, each encompassing
several provinces.
Figure 2: Publication of ERDF spending for Saxony (single HTML table) Figure
3: ERDF spending for The Netherlands (Web app)
The Romanian site (www.fonduri-ue.ro/proiecte-contractate-236) publishes data
monthly in a RAR archive that contains a set of seven PDF files. Figure 1
gives an example how the PDF is organized. The site of the German state of
Saxony (www.statistik.sachsen.de/foerderportal/) provides a single HTML table
of all spending data ordered by the name of the beneficiary (cf. Figure 2).
Besides having four separate sites for different groups of regions, The
Netherlands has a dedicated site (www.europaomdehoek.nl) that provides a Web
application for interactive exploration (cf. Figure 3).
Open data activists have the goal to collect, abstract and visualize all ERDF
data in a consistent manner. The Financial Times and the Bureau of
Investigative Journalism did work on a consolidated spending database in 2010
because “there has been little transparency about how the funds are used”
[20]. They found a number of misuses and abuses of funds that let them to
conclude “the concepts of what EU representatives think of as transparency and
what actually allows citizens to easily understand how the 27-member bloc
spends the Structural Funds are worlds apart.”
CREATE TABLE erdf ( ’Country’ text, ’Region’ text, ’Operational program’ text,
’Op name’ text, ’Co financing rate’ text, ’Program’ text, ’Sub region or
county’ text, ’District’ text, ’Beneficiary’ text, ’Normalized beneficiary’
text, ’Subcontractor’ text, ’Project title’ text, ’Classification category’
text, ’Sector code’ text, ’Parent company or owner’ text, ’Trade description’
text, ’Ft category’ text, ’Description’ text, ’Operational program name’ text,
’Amount estimated eu funding in euro’ text, ’Amount paid in euro’ text,
’Amount allocated eu funds in euro’ text, ’Amount allocated eu funds and
public funds combined in euro’ text, ’Amount allocated public funds in euro’
text, ’Amount allocated private funds in euro’ text, ’Amount allocated
voluntary funds in euro’ text, ’Amount allocated other public funds in euro’
text, ’Amount total project cost in euro’ text, ’Amount unknown source in
euro’ text, ’Amount eligible in euro’ text, ’Currency’ text, ’Amount estimated
eu funding’ text, ’Amount paid’ text, ’Amount allocated eu funds’ text,
’Amount allocated eu funds and public funds combined’ text, ’Amount allocated
public funds’ text, ’Amount allocated private funds’ text, ’Amount allocated
voluntary funds’ text, ’Amount allocated other public funds’ text, ’Amount
total project cost’ text, ’Amount unknown source’ text, ’Amount eligible’
text, ’Intermediate body’ text, ’Date’ text, ’Year’ text, ’Start year’ text,
’Final payment year’ text, ’Sub program name’ text, ’Sub sub program name’
text, ’Objective’ text, ’Category’ text, ’Legal entity’ text, ’Match funded’
text, ’Eu fund percentage’ text, ’Sub program information’ text, ’Min percent
funded by eu funds’ text, ’Max percent funded by eu funds’ text, ’Next update’
text, ’Parsed data file’ text, ’Original file name’ text, ’Direct link’ text,
’Uri to landing page’ text );
Figure 4: Schema of ERDF database
A query interface to the database is available at eufunds.ftdata.co.uk/. As a
single zipped file in SQLite format the database is about 600MB.111Personal
email communication with Friedrich Lindenberg (pudo.org). The database has a
flat schema (i.e., a single table; cf. Figure 4) so that it be can easily
mapped to spreadsheet/CSV formats. All fields in the schema are text and there
are many entries that are not directly available from the data sources.
Constructing the consolidated database was a major effort because “the data
were published on more than 100 websites, in nearly 600 documents and in 21
languages. So, while the information was, in principle, freely available, it
was not presented in a way that could be meaningfully analyzed” [20]. Also,
data was not always available (Greece published blank tables in PDF files),
incomplete (Belgium), outdated (some German states), wrong (UK), or password
protected [20] [2].
For the next funding round (2014–2020) the European Commission is working on
regulations to encourage open data such as a centralized database that
contains more project details (e.g., EU co-financing rate and total spending)
and adheres to the _8 Principles of Open Government Data_
(www.opengovdata.org/home/8principles).
## III Reverse Engineering Perspective
A prerequisite for open data is to obtain (raw) data in a form such that it
can be effectively used for information processing and visualization,
knowledge generation and decision making. Unfortunately, even if data is
accessible it still needs to be transformed to enable its (effective) use.
This step is essentially very similar to traditional software reverse
engineering—using Chikofsky and Cross’s classical definition [5] as a
baseline, we can define reverse engineering for open data as the creation of
data representations that (1) transform the original data to another form
and/or (2) transform it into a higher level of abstraction. Note that the
original data is not changed; in fact, it often resides at an authoritative
source that provides read-only access.
Transforming of the original data into another form is typically required
because the data is not efficiently machine-readable, queryable or storable.
Open data is made available in many different formats such as XML, HTML, Word
documents, Excel spreadsheets, comma-separated values format (CSV), and PDF
files. According to the OGD Stakeholder Survey the most popular current
formats are HMTL (52%), PDF (50%), CSV/XLS (37%), DOC/RTF (32%), XML (27%),
APIs (22%) and RDF (18%) [16]. Thus, formats that are easily machine
processable are currently loosing out to other formats. Indeed, according to
the survey the most requested (future) formats are APIs, XML and RDF.
Figure 5: Example of raw API access for ERDF data of The Netherlands
For open data publishing the following quality levels can be distinguished
(based on Shadbolt [23]):
available:
data is accessible on the Web (in any form or format)
structured:
data is structured (e.g., CSV and Microsoft Office binary formats)
standardized:
data uses open, standardized formats (e.g., XML, RDF and JSON)
addressable:
individual data-points are denoted by a unique URL
linked:
data links to data from external sources (other data providers)
In order to process data at the lower levels (available, structured, and
standardized) some kind of reverse engineering is typically required. At this
point in time, open data is accessible mostly at the two lowest levels
(available and structured). ERDF data (cf. Section II) is almost exclusively
published as PDF or HTML, and “no data is currently published in XML or JSON
or RDF” [21].
Even if data is structured, the encoding may vary (e.g., for CSV different
conventions are used to denote field/record separators, string entities, date
and time, and so on). If data is structured, the data’s schema (or metadata)
may not be available. In this case, it may be desirable to
(semi-)automatically recover the data’s structure (_schema recovery_).
In practice, processing of PDF files are a major concern. They are very
inconvenient to process while being surprisingly common in practice. For
example, all departments of the UK government publish their annual reports as
PDFs and “those PDFs are full of tables, however not one department publishes
these as a spreadsheet or any accessible format” [22]. PDF is a complex
format222The ISO standard 32000-1:2008 that covers version 1.7 of the PDF
format has almost 800 pages
(www.adobe.com/devnet/acrobat/pdfs/PDF32000_2008.pdf). that can include
PostScript and JavaScript code, forms, and vector/raster images; so far there
are nine different official versions of PDF with differing capabilites. Also,
a PDF may contain features that are only supported by Adobe software and that
cannot be processed by other PDF viewers. As a result each PDF file may pose
different challenges when extracting its content. This causes many practical
problems, for instance in a PDF containing the spending of a UK department
“the core tables were impossible to export” and for another department “the
tables were so badly formatted in the original PDFs that we had to copy the
data out by hand” [22].
In the following subsections we briefly outline reverse engineering challenges
and techniques that are needed for open data.
### III-A Scraping
Open data is typically made available on the Web. Often there is a permanent
URL that points to a self-enclosed document, but it is also common that data
is embedded within static HTML or a dynamic Web application. ScraperWiki
(www.scraperwiki.com) is an example of a portal that allows to develop and run
scripts in Python, Ruby and PHP. Scripts scrape Web sites that contain open
data and make the results available for simple interactive exploration or for
download (CSV, JSON, or SQLite). ScraperWiki scripts can use libraries that
simplify processing (e.g., lxml.html for HTML parsing).
An example of a static HTML page is the ERDF data for Saxony. There is a
ScraperWiki
script333scraperwiki.com/scrapers/eu_regional_development_fund_recipients_-
_saxony_g/ (37 lines of Python code) that processes this data. As typical for
such a scrapers, the script would break if layout and/or names change in the
HTML encoding. Another similar example is the WHO’s Global Alert and Response
information that can be obtained for the years 1996–2011 with differently
URLs: www.who.int/csr/don/archive/year/yyyy/en/. The ScraperWiki script
(scraperwiki.com/scrapers/who_outbreaks/) that processes this data is 63 lines
of Python code.
The ERDF Web application for The Netherlands has neither static HMTL nor an
official API to obtain the data. Via reverse engineering of the Web
application (e.g., with the help of a JavaScript debugger) a query-URL can be
obtained, which takes a project ID and in return provides the raw data for the
corresponding project in JSON. Figure 5 shows an example of the query-URL with
project ID 7096 (which provides the raw data for the visualization in Figure
3).444A ScraperWiki script can be found at
scraperwiki.com/scrapers/dutch_european-funded_regional_development_project/.
For Web sites that provide a query interface only, it can be difficult or
impossible to determine the size of the underlying database and to assure
exhaustive extraction of the available data. In such cases a semi-automatic
approach for filling out search queries is desirable. Interestingly, this
problem is also encountered by search engines that have to cope with the so-
called _hidden Web_ [4].
### III-B Image Processing
Reverse engineering of open data can require the transformation of bitmaps
towards characters and vector data. This typically entails optical character
recognition (OCR). But layout and lines may also need to be processed, for
instance for tables or multi-column text, which can be handled by _document
image analysis_ [19]. An example that requires this approach is the ERDF data
of Bulgaria, which is provided as bitmaps embedded in PDFs.
In the reverse engineering literature there are examples of image processing
techniques and OCR in the areas of CAPCHAs [12], UML diagrams [15] and GUI
testing [6] that might be applicable for the processing of open data as well.
For instance, GUI testing of a Web site for different browsers can be
accomplished by “graphical diffing” of the rendered pages. Similarly, table
structures of different PDF documents could be graphically differenced.
There are generic tools available that provide functionality for converting
PDFs to text such as Adobe Reader and pdftotext (part of Xpdf). However,
depending on the complexity and PDF-representation of information its
structure can get lost. For example, when Acrobat Reader 9 extracts content
for the PDF in Figure 1 text is in the wrong order and column boundaries are
lost. The pdftotext tool provides a much more usable extraction for this PDF
file, but the table header is not correctly recognized.
A general problem is that converters are not customizable. It would be
desirable, for instance, to specific table layouts also with the help of
graphical regions. Imagine the typical scenario of a PDF file that contains a
single table spread over many pages. If the columns of the table are
consistently located at the same horizontal offsets a geometric specification
could be easily used as guidance for data extraction.
### III-C Structure and Schema Recognition
Open data may be made available as a single table without much structure. As
the ERDF database (cf. Figure 4)) illustrates, there can be a large number of
fields/columns with many rows of data.
From a database perspective, this data is only in first normal form (1NF).
While this format permits SQL-style queries, it has little structure. Field
information needs to be repeated on each row. For example, in the ERDF
database if a certain beneficiary has multiple projects then all of the
beneficiary’s information is repeated, possibly with variations (e.g., with or
without diacritical marks or different capitalization). Such variations can
introduce mistakes when data is transformed and consolidated.
Open data is typically published without a description of the schema, formally
or informally. The meaning of “schema” should be interpreted broadly in the
sense that Word-style and HTML documents can have structures as well (e.g., a
certain combination of font attributes could have a certain meaning). There
are _hypertext-based data models_ “in which page authors use combinations of
HTML elements (such as a list of hyperlinks), perform certain data-model tasks
(such as indicate that all entities pointed to by the hyperlinks belong to the
same set)” [4].
For such data as well as for collaboratively constructed and mashed-up open
data one cannot assume “centralized data design” as it is found at traditional
databases. Reverse engineering techniques could be used to recover and
complete schema information and to infer constraints/structures of the data.
There is promising research in that direction for Web-embedded structured data
[4] [26]. Another example is the OpenII open source tool set
(openii.sourceforge.net/) for data integration tasks such as clustering and
visualization of schemas as well as matching of source/target schemas.
Figure 6: Model of open data processing
## IV Maintenance Perspective
While the reverse engineering perspective is mostly concerned with the
transformation of information in individual documents/databases, the
maintenance perspective addresses the management and flow of information as
well as quality attributes of the whole process.
We propose the following model of open data processing. Figure 6 gives an
overview of the model with the most important elements and the data flow among
them. The output of processing is to present information in a novel form that
allows to gain unique insights that would not have been possible with the raw
data sources. Typical examples are (interactive) visualizations that provide
abstractions and expose dependencies of data. The inputs to the process are
multiple, diverse data sources. These data sources are typically from
independent third parties. Thus, data needs to be pulled from sources and it
is expected that the data provider may modify data in the future in a manner
that is more or less unpredictable for the consumer. Data is kept in a
(central) repository, which needs to support versioning, data provenance and
differencing (discussed below). This model shares similarities with the ones
that are found in reverse engineering (extract, abstract and present) and data
warehousing (extract, transformation and load (ETL)).
The data processing is organized as a transformation pipeline.555Instead of a
linear pipeline one can image a more complex model based on _flow graphs_
where nodes in the graph represent operators, which can be flow manipulations
or transformations. The SPADE programming language for stream processing is
based on this principle [7]. A transformation step may, for instance, (1)
change the format, data representation, and/or schema, (2) augment data (e.g.,
aggregation of data items or annotation/cross-referencing of data items), and
(3) perform sanity checks and validate (schema) constraints.
Note that each transformation may be manual, semi-automatic or fully-
automatic. Since data sources are often only semi-structured, human
verification and corrections are not uncommon. For instance, the OCR
recognition of a number may we wrong for a certain data item. Once this is
recognized (via manual inspection or violation of a sanity constraint) a hand-
written transformation could be added to the transformation pipeline to fix
this data item.
To analyze the (transformed) data in an effective manner by its users
visualizations are needed. Visualizations can be text-based, graphical, or
both and are typically made available as a web interface. Visualizations need
a query interface to the repository (which may differ from the way that
transformations access the database). Visualizations may support user-
generated content (UGC) that enhances the “baseline” repository with
additional knowledge (e.g., URIs to external data sources).
Since the outputs of the process are expected to be used by research,
businesses and governments to advance their understanding, trust in the data
and transparency in the processing is essential. In this context, key
requirements to support are versioning and traceability for quality control. A
closely related research field is _data provenance_ , which can be defined as
“information that helps determine the derivation history of a data product,
starting from its original sources;” and furthermore “the two important
features of the provenance of a data product are the ancestral data product(s)
from which this data product evolved, and the process of transformation of
these ancestral data product(s), possibly through workflows, that helped
derive this data product” [24].
For each data item at the output it should be possible to trace its
dependencies through the transformations back to the source data. This is
important for debugging and assurance. All data sources, transformations, etc.
need to be versioned so that output can be faithfully reproduced later on if
needed. If a data provider makes a modification (e.g., change of an existing
PDF file) or addition (e.g., a new PDF becomes available) it needs to be
properly versioned.
If a data source has been modified there needs to be effective tools support
to analyze the differences (“deltas”). It may be that the underlying
information has changed, that the representations or encoding has changed, or
both. Think of a PDF file that looks identical to the eye, but whose encoding
has changed such that pdftotext produces now different output that breaks
assumptions in the transformation pipeline. Generally, it is desirable to be
able to analyze deltas for two (or more) configurations of runs.
Since open data infrastructure is just starting to emerge there are no
dominant technologies and infrastructures yet. Once they are emerging one can
expect that projects will have to be migrated to more established platforms
(i.e., both data and software migration). The W3C’s SPARQL (www.w3.org/TR/rdf-
sparql-query/) is currently discussed as a possible query end point for open
data [9].666A collection of SPARQL endpoints for open data is available at
labs.mondeca.com/sparqlEndpointsStatus/. To accomplish this, Web applications
such as the ERDF app from The Netherlands (cf. Section II) will have to be
migrated towards a SPARQL API.
## V Conclusions
In this paper we have outlined the push for open data and described its
current state with the help of an example—open data for the beneficiaries of
European Regional Development Fund money. We then described research
challenges for open data in the areas of reverse engineering and maintenance.
Open data presents not only worthwhile research opportunities, but promises to
benefit society as well. It is our hope that this paper will inspire other
researchers within the reverse engineering and maintenance communities to take
up the open data challenge.
## References
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|
arxiv-papers
| 2012-02-08T11:08:37 |
2024-09-04T02:49:27.204869
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Holger M. Kienle",
"submitter": "Holger Kienle",
"url": "https://arxiv.org/abs/1202.1656"
}
|
1202.1665
|
# Persistent electrical doping of Bi2Sr2CaCu2O8+x mesa structures
Holger Motzkau Thorsten Jacobs Sven-Olof Katterwe Andreas Rydh Vladimir M.
Krasnov Vladimir.Krasnov@fysik.su.se Department of Physics, Stockholm
University, AlbaNova University Center, SE – 106 91 Stockholm, Sweden
###### Abstract
Application of a significantly large bias voltage to small Bi2Sr2CaCu2O8+x
mesa structures leads to persistent doping of the mesas. Here we employ this
effect for analysis of the doping dependence of the electronic spectra of
Bi-2212 single crystals by means of intrinsic tunneling spectroscopy. We are
able to controllably and reversibly change the doping state of the same single
crystal from underdoped to overdoped state, without changing its chemical
composition. It is observed that such physical doping is affecting
superconductivity in Bi-2212 similar to chemical doping by oxygen impurities:
with overdoping the critical temperature and the superconducting gap decrease,
with underdoping the $c$-axis critical current rapidly decreases due to
progressively more incoherent interlayer tunneling and the pseudogap rapidly
increases, indicative for the presence of the critical doping point. We
distinguish two main mechanisms of persistent electric doping: (i) even in
voltage contribution, attributed to a charge transfer effect, and (ii) odd in
voltage contribution, attributed to reordering of oxygen impurities.
###### pacs:
74.72.Gh, 74.62.Dh, 74.55.+v, 74.72.Kf
## I Introduction
High temperature superconductivity (HTSC) in cuprates occurs as a result of
doping of a parent antiferromagnetic Mott insulator and properties of cuprates
change significantly with doping. Superconductivity in overdoped cuprates is
fairly well described by the conventional BCS-type second-order phase
transition TallonPhC ; SecondOrder ; MR . But properties of underdoped
cuprates are abnormal due to the persistence of the normal state pseudogap,
strong superconducting fluctuations, or possibly preformed pairing, Kaminski
and magnetism Kerr_Shen at $T>T_{\text{c}}$. There are indications that the
transition from the normal to the abnormal behavior occurs abruptly at a
critical doping point TallonPhC ; Doping ; Balakirev ; Kartsovnik ;
LeBoeuf2011 . This may be a consequence of the quantum phase transition - a
phase transition, which occurs at $T=0$, in frustrated systems as a result of
a competition of coexisting order parameters. The coexistence of
superconductivity at $T<T_{\text{c}}$ with the pseudogap KrasnovPRL2000 ;
Lee2007 ; Bernhard , charge and spin density order Kartsovnik ; LeBoeuf2011 ;
Nernst2010 was indeed reported by several techniques. Clearly, detailed
doping-dependent studies are needed both for understanding the puzzling nature
of HTSC in cuprates and for the development of novel HTSC materials.
Usually, the mobile carrier concentration is controlled by chemical doping via
chemical substitution, or in case of cuprates also by variation of the oxygen
content via appropriate annealing and subsequent quenching to room
temperature. This allows an accurate control of the chemical composition, but
less so of the local arrangement of impurities and disorder, which is equally
important for cuprates DisorderBi2201 . For example, it is well established
that properties of the $\mathrm{YBa_{2}Cu_{3}O_{6+x}}$ compound strongly
depend not only on the average oxygen concentration but also crucially on the
order/disorder of oxygen atoms in Cu-O-Cu chains Chains ; Raman . Therefore,
analysis of the doping phase diagram of cuprates requires accurate control of
both, the concentration and the microscopic structure of impurities.
The carrier concentration can be also varied via two physical doping
processes, well established for semiconductors: photo-doping Kudinov ;
Schuller ; PhotoScience2011 and through the electric-field effect Ahn ;
Moeckly ; Sorkin ; Tulina ; KovalCurrentInj ; KovalMemres . In case of
cuprates, physical doping may be persistent at low temperatures in a sense
that it is relaxing very slowly after removing the light Kudinov ; Schuller
or field Moeckly ; Tulina ; KovalCurrentInj ; KovalMemres . Recently, a
persistent electric doping via strong current injection was discovered
KovalCurrentInj ; KovalMemres . It is resembling a resistive switching
phenomenon in memristor devices Memristor1 ; Memristor2 and is related to
previous similar observations in point contact experiments on Bi2Sr2CaCu2O8+x
(Bi-2212) Tulina . Such an electric doping is reversible, reproducible and
easily controllable. It opens a possibility to analyze the doping dependence
of HTSC on one sample without changing its chemical composition
PhotoScience2011 . Despite that, there were very few direct spectroscopic
studies of cuprates employing physical doping techniques.
Intrinsic Tunneling Spectroscopy (ITS) provides an unique opportunity to probe
bulk electronic properties of HTSC SecondOrder . This technique utilizes weak
interlayer ($c$-axis) coupling in quasi two dimensional HTSC compounds, in
which mobile charge carriers are confined in CuO2 planes separated by some
blocking layer (e.g. SrO-2BiO-SrO in case of Bi-2212). This leads to a
formation of atomic scale intrinsic tunnel junctions, and to an appearance of
the intrinsic Josephson effect at $T<T_{\mathrm{c}}$ Kleiner94 ;
KrasnovPRL2000 ; Suzuki ; Katterwe2009 ; Superluminal .
In this work we employ the persistent electric doping for analysis of the
doping dependence of electronic spectra of Bi-2212 single crystals by means of
intrinsic tunneling spectroscopy SecondOrder ; Doping ; MR ; KrasnovPRL2000 ;
KatterwePRL2008 . Controllable and reversible persistent physical doping is
achieved by applying a $c$-axis voltage of a few volts to small Bi-2212 mesa
structures. Thus we are able to change the doping state of Bi-2212 single
crystals without changing its chemical composition. A wide doping range from a
moderately underdoped to strongly overdoped state could be reached. It is
observed that the physical doping is affecting the intrinsic tunneling
characteristics of Bi-2212 similar to chemical doping Doping . With overdoping
the critical temperature and the superconducting gap decrease. With
underdoping the pseudogap rapidly increases, indicative for the presence of
the quantum critical doping point in the phase diagram, and the $c$-axis
critical current density rapidly decreases, indicating a progressively more
incoherent interlayer tunneling. We distinguish two main mechanisms of
persistent electric doping: (i) an even in voltage contribution, attributed to
a charge transfer effect, and (ii) an odd in voltage contribution, attributed
to reordering of oxygen impurities.
The paper is organized as follows. In Sec. II we make a brief overview of
physical doping mechanisms of cuprates. Sec. III provides experimental
details. In Sec. IV we present the main experimental results and in Sec. V we
discuss possible mechanisms of persistent electric doping, followed by
conclusions.
## II Physical doping of cuprates
### II.1 Photo-doping
Photo-doping allows a wide-range variation of doping in the same sample
Kudinov ; Schuller ; PhotoScience2011 . The ordinary non-equilibrium photo-
doping is quickly relaxing because of a very short life time ($\sim$ps) of
photoinduced charge carriers PhotoScience2011 . However, in underdoped
$\mathrm{YBa_{2}Cu_{3}O_{6+x}}$ and some other cuprates a different type of
persistent photo-doping takes place Kudinov ; Schuller . It involves
significant energies $\sim$$1\,\mathrm{eV}$, which makes it metastable at low
temperatures. Several mechanisms are contributing to the persistent photo-
doping PhDopTheory , such as charge transfer, which changes the redox state of
the impurity atom Kudinov , and ordering of oxygen impurities in the lattice
Schuller . Photo-doping always leads to an increase of the doping level with
respect to the initial state.
### II.2 Electric field effects
Electric fields may both increase or decrease the number of mobile charge
carriers, depending on the direction of the applied field Ahn . The ordinary
electric field effect is not persistent and exists only during the time an
electric field is applied. Since the electric field penetrates only to the
Thomas-Fermi charge screening length, $\lambda_{\mathrm{TF}}\lesssim
1\,\mathrm{nm}$, just a thin surface layer can be modified Shapiro .
A persistent electrostatic field-effect due to net electric polarization or
trapped charges can be realized at the interface between a superconductor and
a ferroelectric SuperFerro or polar insulator ElStatic . This is also a
surface phenomenon, but in case of layered cuprates, which represent stacks of
metallic CuO planes sandwiched between polar-insulating layers Polariton ,
electrostatic charging of insulating layers may in principle lead to the bulk
persistent electrostatic field-effect.
Another type of a persistent and bulk electric-field effect has been observed
at large current densities Moeckly . Similar to photo-doping, it was
attributed to a charge transfer Salluzzo2008 and reordering of oxygen
impurities Sorkin . Significant oxygen mobility in intense electric fields
also leads to a resistive switching phenomenon Tulina .
### II.3 Resistive switching in complex oxides
The resistive switching phenomenon occurs in many complex oxides and is the
basis for the development of resistive memory devices. Several mechanisms may
be involved in the resistive switching phenomenon Memristor1 ; Memristor2 ,
including a change of the redox state of some of the elements, oxygen
migration, and filament formation. Resistive switching has been observed on
depleted surfaces of Bi-2212 cuprates Tulina and attributed to oxygen
migration. Recently, it was demonstrated that the resistive switching-like
phenomenon can be used for controllable and reversible doping of small Bi-2212
micro-structures over a wide doping range KovalCurrentInj ; KovalMemres .
## III Experimental
Mesas were fabricated on top of freshly cleaved Bi-2212 single crystals by
means of optical lithography, Ar ion milling and focused ion beam trimming.
Four batches of crystals were used: pure near optimally doped (OP) Bi-2212,
pure strongly underdoped (UD) Bi-2212, lead-substituted
$\mathrm{Bi_{1.75}Pb_{0.25}Sr_{2}CaCu_{2}O}_{8+\delta}$ [Bi(Pb)-2212], and
yttrium-doped $\mathrm{Bi_{2}Sr_{2}Ca_{1-x}Y_{x}Cu_{2}O}_{8+\delta}$
[Bi(Y)-2212]. Mesas of different sizes from $5\times 5$ to $1\times
0.5~{}\muup\mathrm{m}^{2}$ and with a different number of junctions $N=8-56$
were studied. Details of the sample fabrication can be found in Ref. Submicron
. All studied mesas exhibited a persistent doping effect upon application of a
sufficiently large bias voltage.
The samples were placed in a flowing gas cryostat and measured in a three-
probe configuration with a common top gold contact. The ground contacts for
current and voltage were provided through other mesas on the same crystal. A
Keithley K6221 current source and a FPGA-based arbitrary waveform generator
and lock-in amplifier were used to bias and measure the samples. Biasing was
done at pseudo-constant voltage, with an optional small superimposed ac
voltage to simultaneously measure the high-bias differential resistance in
addition to the dc resistance. Positive bias is defined as current going into
the mesa through the common top contact, as sketched in Fig. 1 (a).
Figure 1: (Color online). Dynamics of electric doping (a-c) for a Bi(Pb)-2212
mesa at $T=135\,\mathrm{K}$ and (d-f) for an optimally doped Bi-2212 mesa at
$T=2\,\mathrm{K}$. Panel (a) shows time evolution of the dc resistance at a
bias of $\approx 1.9\,\mathrm{V}$. The dashed line represents a stretched
exponential decay. The inset schematically shows a sketch of a mesa structure
and the direction of electric field at positive bias. (b) Demonstration of
odd-in-voltage doping: the mesa resistance increases/decreases at
positive/negative bias. Panel (c) shows the predominantly odd voltage
dependence of the logarithmic rate of the dc resistance change $\mathrm{d}\ln
R_{\mathrm{dc}}/\mathrm{d}t$. Different symbols represent different runs. The
inset in (c) demonstrates the sign change of the doping direction at higher
bias. Panel (d) demonstrates a gradual change from a negative to positive
doping rate with increasing bias voltage and time. It also demonstrates a
history dependence of the doping rate, i.e., a different sign of resistance
change at the same bias voltage, depending on the former doping treatment. The
history dependence upon several sweeps of the bias voltage are shown in (e).
Panel (f) demonstrates the predominantly even in voltage doping for OP
Bi-2212. However an asymmetry indicates the presence of a subdominant odd in
voltage doping. Note that despite a significant difference between the two
mesas, the threshold doping voltage is similar $\sim\pm 1.7\,\mathrm{V}$ (c).
## IV Results
From chemical (oxygen) doping studies it is known that doping/undoping of
Bi-2212 is accompanied by a proportional decrease/increase of the $c$-axis
resistivity Doping . Therefore, we can control the doping state by tracing the
mesa resistance.
### IV.1 Dynamics of electric doping
The basic features of the dynamics of the persistent electric doping are shown
in Fig. 1. Panel (a) shows the time-evolution of the Bi(Pb)-2212 mesa
resistance at a bias of $\approx$$1.9\,\mathrm{V}$ at $T=135\,\mathrm{K}$. It
is seen that the mesa resistance is decreasing with time, indicating a gradual
doping of the mesa. The doping rate decreases with time, following a stretched
exponential decay, $R=R_{0}+R_{\mathrm{d}}\exp[-(t/\tau)^{\beta}]$, shown by
the dashed line in in Fig. 1 (a), which is also typical for persistent photo-
doping Kudinov ; Schuller . The doping can be equally well performed at any
temperature from 4 to 300 K, but the rate is increasing with $T$. In most
cases we perform doping at low $T$ in order to be able to immediately probe
the superconducting characteristics. Upon reduction of the bias below the
threshold voltage the state of the mesa remains stable even at room
temperature on the time scale of several days.
The resistive change is reversed upon voltage reversal, as illustrated in
Figs. 1 (b) and (c) for the same Bi(Pb)-2212 mesa. The resistance decreases
for positive bias (electric field into the crystal) and increases for negative
bias (electric field towards the top contact), which indicates that we can
controllably and reversibly dope and undope the mesa. The doping rate and
direction depend both on the sign and the absolute value of bias voltage.
Figure 2: (Color online) Intrinsic tunneling characteristics of the same OP
Bi-2212 (a-d) and Bi(Pb)-2212 (e-g) mesas as in Fig. 1 at different doping
states. (a) $T$ dependence of the ac resistance $R_{\text{ac}}(T)$ in the
initial high-resistive state (1) and subsequent low resistance doping states
(4-6). The inset illustrates the significant enhancement of $T_{\text{c}}$ of
the depleted surface junction. Panels (b-d) represent different parts of $I-V$
characteristics in the doping states 1, 2, 3 and 6 at $T=2$ K. A variation of
the tunnel resistance, sum-gap kink and the critical current is clearly seen.
Panels (e) and (f) show $R_{\text{ac}}(T)$ for Bi(Pb)-2212 in the initial
high-resistive state (1) and subsequent low resistance doping states (4) and
(6). This mesa with a large number of junctions $N=56$ exhibits a significant
spread in the $T_{\text{c}}$ of individual junctions, seen as small resistance
drops in (f). Panel (g) shows $I-V$s of the Bi(Pb)-2212 mesa in the doping
states 2, 4 and 5. A progressive increase of the critical current and decrease
of the sum-gap kink voltage is seen.
Figure 1 (c) summarizes the bias dependence of the doping rate for the
Bi(Pb)-2212 mesa. Below the threshold voltages,
$\left|V_{\mathrm{dc}}\right|\lesssim 1.7\,\mathrm{V}$, the mesa resistance is
stable. Upon increasing the bias voltage, the resistance of the mesa starts to
gradually change at a rate that increases drastically up to $|V|\sim 2.2\,V$
as shown in the inset of Fig. 1 (c). A further voltage increase reduces the
rate and then reverses the resistance alteration rate (see the inset at
$V=-2.32\,\mathrm{V}$). The sign change of the alteration rate at high bias is
in agreement with the observations by Koval _et al._ KovalCurrentInj . The
behavior in this regime is, however, history dependent, as may be seen from
Fig. 1 (c), and the final state depends on how long time the mesas was biased
at every bias voltage. The doping process for the Bi(Pb)-2212 mesa, Fig. 1
(c), is predominantly odd in bias voltage, i.e., the direction of doping is
changed when the sign of the bias voltage is changed.
Figures 1 (d) and (e) show a detailed view of the time and voltage dependence
of doping for a near optimally doped pure Bi-2212 mesa. The top panel of (d)
shows the time evolution of the resistance of an OP Bi-2212 mesa for different
bias voltages, shown in the bottom panel. It is seen that the resistance is
constant at $V=1.7\,\mathrm{V}$, and starts to increase slowly at 1.9 V.
However, at 2 V the resistance initially increases but then starts to decrease
after a few minutes. This clearly shows that there are two counteracting
processes: a positive rate mechanism that saturates quickly and a mechanism
with negative rates that is dominating at longer times and at higher voltages.
The second process also saturates with time, which is clear from Fig. 1 (a)
and history dependent rates of Fig. 1 (e). At larger voltage, the resistance
steadily decreases at a rate which is strongly bias dependent, as shown in
Fig. 1 (f).
Figures 1 (e,f) show the bias dependence of doping and a history dependence
upon sequential voltage sweeps (e). It is seen that for the OP Bi-2212 mesa
the electric doping is predominantly even in voltage, i.e., the direction of
doping does not depend on the sign of the bias voltage. However, certain
asymmetry of the doping rate versus bias voltage characteristics in Figs. 2
(c) and (f) indicates the presence of sub-dominant even- and odd-in-voltage
contributions for Bi(Pb)-2212 and OP Bi-2212 mesas, respectively.
Figure 3: (Color online). Doping of a strongly underdoped Bi-2212 mesa. (a)
$I$-$V$ characteristics in the initial state 1 (black) and two successive
doping states 2 (red) and 3 (green). The increase of the critical current by a
factor five is seen. (b) The corresponding $\mathrm{d}I$-$\mathrm{d}V$
characteristics. The superconducting sum-gap peak moves to slightly lower
voltages and becomes sharper with doping. The $c$-axis pseudogap hump voltage
rapidly decreases with doping. (c) Zero bias ac resistivity
$R_{\text{ac}}(T)$. The $T_{\text{c}}$ has increased by about 15 K in the
state 3.
### IV.2 Doping dependence of ITS characteristics
Using the described method, the superconducting properties of the mesas have
been altered to different intermediate doping states denoted by a successive
number. The electric doping changes all mesa characteristics: the $c$-axis
resistivity, the critical temperature $T_{\mathrm{c}}$, the $c$-axis critical
current density $J_{\mathrm{c}}$, the superconducting energy gap, $\Delta$,
the $c$-axis pseudogap and the $c$-axis resistivity in a manner very similar
to chemical (oxygen) doping Doping .
Figure 2 shows temperature dependencies of low bias ac-resistances for the
initial, high resistance state (HRS), and doped, lower resistance states
(LRS), for (a) an OP Bi-2212 and (e,f) a Bi(Pb)-2212 mesa. The $I$-$V$
characteristics of those mesas at different doping states are presented in
panels (b-d) and (g), respectively.
From Fig. 2 (a) it is seen that the initial state was characterized by the
strong thermal-activation-type increase of resistance with decreasing $T$,
typical for underdoped Bi-2212 Katterwe2009 . The general shape of the
resistive transition was described in Ref. SecondOrder . At
$T_{\mathrm{c}}\sim 82\,\mathrm{K}$ the resistance dropped to the top-contact
resistance, which originates from the first deteriorated junction between the
top CuO plane, shortly exposed to atmosphere after cleavage, and the second,
un-deteriorated CuO plane SecondOrder . Initially this junction had a very low
$T_{\mathrm{c}}^{\prime}$ and a very small critical current, $I_{\mathrm{c}}$,
as can be seen from the corresponding $I$-$V$ in panel (d). After electric
doping, the resistance in the normal state dropped almost three times and
became less semiconducting. The main $T_{\mathrm{c}}$ of the mesa changed only
slightly, indicating that the doping was changing around the optimal doping
level with the flat $T_{\mathrm{c}}$ vs. doping dependence. However, the
properties of the top junction changed drastically: the
$T_{\mathrm{c}}^{\prime}$ increased to $\sim 50\,\mathrm{K}$, and the critical
current increased ten-fold as shown in panel (d), even though it still remains
$\sim 20$ times smaller than for the rest of the junctions in the mesa, as can
be seen from panel (c). This indicates that the surface CuO plane was
initially strongly underdoped and, therefore, responded much stronger to
variation of doping, due to the steep $T_{\mathrm{c}}$ vs. doping dependence
at the underdoped side of the doping phase diagram of cuprates Doping . A
similar trend was also observed in photo-doping Kudinov ; Schuller .
Figure 4: (Color online). Intrinsic tunneling characteristics of a small
Bi(Y)-2212 mesas at low $T$ before (slightly underdoped, HRS) and after
(optimally doped, LRS) doping by a short voltage pulse. Panels (a) and (b)
show $I$-$V$ curves at a large scale and at the quasiparticle branches,
respectively. A reduction of the high bias tunnel resistance and a
simultaneous increase of the critical current in the doped state is seen.
Panel (c) shows $\mathrm{d}I/\mathrm{d}V$ ITS characteristics. A change in the
shape of the curves is seen: the peak becomes sharper and the dip-hump less
pronounced in the optimally doped state, compared to the initial underdoped
state.
Another possible reason for the stronger response of the underdoped top
junction is the larger $c$-axis resistivity of underdoped intrinsic Josephson
junctions Doping . Because of that the electric field is not uniformly
distributed along the mesa but is larger in the high-resistive top junction.
This together with the strong voltage dependence of the electric doping leads
to a faster doping of the top junction and the doping may even go in the
opposite direction with respect to the rest of the mesa.
The effect of non-uniform doping along the height of the mesa becomes more
pronounced in higher mesas with a larger number of intrinsic Josephson
junctions. This is seen from $R(T)$ for the Bi(Pb)-2212 mesa from Fig. 2 (e),
which contained a fairly large number of junctions $N\approx 56$. It is seen
that after doping some junctions retained the initial $T_{\mathrm{c}}\sim
90\,\mathrm{K}$, but some were very strongly overdoped to $T_{\mathrm{c}}\sim
30\,\mathrm{K}$. Fig. 2 (f) shows $R(T)$ at the intermediate doping state 4.
Small drops represent critical temperatures of individual junctions in the
mesa. Apparently there is a gradual distribution of $T_{\mathrm{c}}$ along the
height of the mesa.
Doping of Bi-2212 leads to a rapid increase of the $c$-axis critical current
density Doping . This is clearly seen from $I$-$V$ curves for the Bi-2212 (OP)
mesa shown in Figs. 2 (b-d). The increase of $I_{\mathrm{c}}$ of the surface
junction is shown in panel (d). One-by-one switching of the rest of the
junctions into the resistive state at $I>I_{\mathrm{c}}$ leads to the
appearance of multiple-quasiparticle (QP) branch structures in the $I$-$V$
curves. The corresponding critical current at the first QP branch is shown in
panel (c). It also strongly increases with doping. The same effect doubles
$I_{\mathrm{c}}$ in the Bi(Pb)-2212 mesa, in Fig. 2 (g).
From Fig. 2 (b) it is seen that the $I$-$V$ curves exhibit a kink at large
bias, followed by an ohmic tunnel resistance. The kink represents the sum-gap
singularity in superconducting tunnel junctions at $V=2\Delta/e$ per junction
SecondOrder ; MR . This is the basis of the ITS technique, which allows
analysis of the superconducting energy gap $\Delta$ in the bulk of the Bi-2212
single crystal.
Accurate analysis of the electronic spectra with the ITS technique requires
mesas with a small area and a small number of identical junctions. This is
needed for avoiding possible artifacts, associated with self-heating, in-plane
non-equipotentiality, and spread in junction parameters Heating_PRL2005 ;
SecondOrder . This is particularly important for the analysis of the
genuineshape of tunneling characteristics, which remains a controversial issue
Comment2 . Even though the $I$-$V$ characteristics in Figs. 2 (b) and (g) are
distorted by self-heating, as evident from a back-bending at large bias, the
general trend for a variation of the sum-gap kink with doping is clearly seen:
The superconducting gap decreases and the sum-gap kink becomes sharper with
(over-)doping. This qualitative conclusion is not affected by self-heating
because the dissipation power at the kink decreases with decreasing resistance
and becomes smaller with subsequent doping. Thus, with over-doping the
superconducting sum-gap singularity becomes sharper and moves to lower
voltages despite the progressive reduction of self-heating. This clearly
reveals the doping-variation of the genuine $c$-axis tunneling characteristics
Comment2 . A similar tendency was observed by other techniques, including the
angular resolved photoemission spectroscopy ARPES_doping , scanning tunneling
spectroscopy STS_inhomo , and tunneling spectroscopy on point contacts
Zasad_doping , as well as in previous ITS studies involving chemical (oxygen)
doping Doping ; KatterwePRL2008 ; SecondOrder .
Figure 3 shows the electric doping of an initially strongly underdoped Bi-2212
mesa. A five-fold increase in critical current in the $I$-$V$ characteristics
(a), and an increase of $T_{\mathrm{c}}$ of about 15 K (c) can be seen. The
tunneling conductance $\mathrm{d}I/\mathrm{d}V(V)$ curves are shown in panel
(b). It is seen that the sum-gap peak shifts to slightly lower voltages and
becomes sharper with doping. The hump voltage, attributed to the $c$-axis
pseudogap SecondOrder , rapidly decreases with increasing conductance,
pointing towards an abrupt opening of the pseudogap at the critical doping
point, consistent with chemical doping studies Doping ; Balakirev ; Bernhard .
Note that in all studied cases the electric doping has lead to a significant
increase of the critical current, while the superconducting gap was
decreasing. The anticorrelation between $I_{\text{c}}R_{\text{n}}$ and
$\Delta$ in underdoped Bi-2212 has been reported before Doping and was
attributed to progressively more incoherent $c$-axis transport in combination
with the $d$-wave symmetry of the superconducting order parameter.
### IV.3 Short-pulse doping
Figure 5: (Color online) Short pulse resistive switching in a strongly
underdoped Bi-2212 mesa. (a) $I$-$V$ characteristics in the initial high-
resistive state (HRS) and the doped low-resistive state (LRS) at $T=2$ K.
Panels (b) and (c) demonstrate the resistive switching sequence
$R_{\text{ac}}$ versus time at $T=100$ K. The switching was made by a train of
short positive and negative pulses, shown in the bottom of panel (b). It is
seen that the HRS is stable, but the LRS is initially relaxing and then
saturates at a resistance below the HRS. Several hundred reproducible
resistive switching events can be achieved without visible degradation (c).
So far we were discussing a gradual electric doping of Bi-2212 mesas at the
time scale of an hour, as shown in Fig. 1. Such a long time doping allows a
very strong variation of the doping state, but often leads to inhomogeneous
doping within the mesa height, as shown in Fig. 2. Koval et.al. KovalMemres
demonstrated that a short-pulse doping strategy leads to highly reversible and
reproducible doping, similar to resistive switching in point contacts Tulina .
This is probably related to the lack of significant electromigration during
the short pulse, which may eventually lead to an irreversible destruction of
the crystal structure Moeckly .
Figure 4 represents the ITS characteristics at $T\sim 30\,\mathrm{K}$ for a
small $\sim 2\times 2\,\mathrm{\muup m^{2}}$ Bi(Y)-2212 mesa with a small
amount of junctions and small self-heating Heating_PRL2005 . In the initial
high-resistive state the mesa is slightly underdoped with $T_{\mathrm{c}}\sim
91\,\mathrm{K}$. The mesa was switched to a low-resistive state by a short
voltage pulse $V\gtrsim 2\,\mathrm{V}$ of about a milliseconds width. The
periodicity of QP branches in Fig. 4 (b) demonstrates that after switching
into the LRS the mesa remains highly uniform. From comparison of $I$-$V$
curves in panels (a) and (b) it is seen that the decrease of resistance by
$\sim 1/3$ is accompanied by an almost four-fold increase of $I_{\mathrm{c}}$.
The $T_{\mathrm{c}}$ increased to $\sim 93\,\mathrm{K}$ being an indication
that the mesa became near optimally doped.
Fig. 4 (c) represents the tunneling conductance $\mathrm{d}I/\mathrm{d}V$ in
the high-resistive (underdoped) and the low-resistive (optimally doped)
states. The following main changes in ITS spectra are seen: the
superconducting sum-gap peak voltage decreased in the doped low resistive
state and the shapes of spectra are changed. The relative sum-gap peak height
$\mathrm{d}I/\mathrm{d}V(V_{\mathrm{p}})R_{\mathrm{n}}$ is increased by about
50% in the low-resistive state. The high-resistive state exhibits a peak-dip-
hump structure, which is less obvious in the low-resistive state. All this is
similar to the slow-doping case, Figs. 2 and 3, and consistent with the change
of doping from the slightly underdoped to near optimally-doped state Doping ,
in accordance with other spectroscopic studies involving chemical (oxygen)
doping ARPES_doping ; STS_inhomo ; Zasad_doping .
As already discussed above, the increase of sharpness of the sum-gap kink in
the $I$-$V$ curve and the peak in the $\mathrm{d}I/\mathrm{d}V$ curve in the
low-resistive state, reported in Figs. 2 and 4, and the change of the shape of
the peak-dip-hump feature in Fig. 4 (c) can not be attributed to self-heating,
because the actual power dissipation at the peak is decreasing in the low-
resistive state. For example, the corresponding powers at the peaks in Fig. 4
(c) are $P=0.21$ and 0.17 mW, respectively. This observation supports the
conclusion of Ref. Comment2 that the appearance and the shape of the peak-
dip-hump structure in the ITS characteristics of small mesas is determined
primarily by the doping level.
A detailed analysis of short pulsed resistive switching has been performed on
strongly underdoped Bi-2212 mesas. For negative pulses with
$50\,\mathrm{\muup{}s}$ length and compliance voltages up to
$-2.5\,\mathrm{V}$ there was no resistive switching. At $-3.0\,\mathrm{V}$, a
reduction of the quasiparticle resistance to 99.3% is observed (LRS). Nine
subsequent pulses reduce the resistance further to 98.4%. A single positive
pulse switches the resistance back to the initial HRS, while more positive
pulses do not result in additional changes. A single negative pulse with a
higher compliance voltage of $-3.5\,\mathrm{V}$ instead reduces the resistance
of the LRS to about 93.4% of the HRS, and the corresponding positive pulses
switches it back. Pulses with lower compliance voltage lead to a partial
switching but additional pulses with the same compliance do not lead to a
significant change. Fig. 5 (a) shows $I$-$V$ curves of another UD Bi-2212 mesa
at $T=2\,\mathrm{K}$ in the HRS and the LRS obtained with a $3.5\,\mathrm{V}$
pulse of $50\,\mathrm{\muup{}s}$ width. The general difference between
$I$-$V$s after the pulsed doping in Figs. 4 (a) and 5 (a) is the same as for
the slow doping in Figs. 2 (a) and 3 (a).
Fig. 5 (b) demonstrates a reproducible resistive switching between HRS and LRS
at elevated $T=100\,\mathrm{K}$ for another UD Bi-2212 mesa. The switching was
performed using a similar positive and negative pulse sequence with $\pm
3.5\,\mathrm{V}$, a pulse width of $100\,\mathrm{\muup{}s}$ and an interval
between pulses of 3 s, shown in the bottom panel of Fig. 5 (b). Panels (b) and
(c) show the corresponding time sequence of the measured zero-bias resistance
of the mesa. It is seen that negative voltage pulses lead to switching into
the LRS while a subsequent positive pulse switches the mesa back into the HRS.
The corresponding resistance change rates are of the order of $\mathrm{d}\ln
R/\mathrm{d}t\approx\mp 1000\,\mathrm{s^{-1}}$, which are somewhat higher than
for the slow doping shown in Fig. 1 (d), but not inconsistent with that data,
taking into account that the compliance voltage is also significantly higher.
It is seen that the HRS is stable and shows no visible relaxation at
$T=100\,\mathrm{K}$, while the LRS is initially relaxing with the
characteristic time $\tau_{\mathrm{LRS}}=(0.77\pm 0.30)\,\mathrm{s}$ and then
saturates before reaching the HRS, as shown in panel (b). At
$T=280\,\mathrm{K}$, the behavior is similar with a time constant of
$\tau_{\mathrm{LRS}}=(0.75\pm 0.34)\,\mathrm{s}$.
## V Discussion: mechanisms of persistent electric doping
In Sec. II we briefly reviewed known mechanisms of persistent physical doping
of cuprates. It is likely that some of them are playing a role in persistent
electric doping, studied here. Indeed, the phenomenon is clearly related to
the persistent electric-field effect Moeckly , observed in
$\mathrm{YBa_{2}Cu_{3}O_{6+x}}$, which in turn is clearly related to
persistent photo-doping observed for various cuprates Kudinov ; Schuller .
To identify possible mechanisms, we first summarize characteristic features of
the persistent electric doping:
i) Observed different voltage dependencies (odd and even) indicate that
several distinct mechanisms are involved.
ii) The doping rate shows a threshold-like behavior as a function of bias
voltage (see Fig. 1). The threshold voltage depends on $T$ in a thermal-
activation manner, i.e. decreases with increasing $T$. Remarkably, the
threshold voltage is weakly dependent on the number of junctions in the mesa,
consistent with previous reports KovalCurrentInj ; KovalMemres , and is
comparable to that for a single point contact Tulina . This suggests that the
phenomenon is connected to some characteristic energy $\sim 1\,\mathrm{eV}$,
rather than directly to the electric field. Indeed, for a given voltage, the
latter should scale inversely proportional to the number of junctions in the
mesa, i.e., would not be universal for different mesas (that is why we
hesitate to refer to the phenomenon as a persistent electric field effect, and
rather call it persistent electric doping).
iii) However, the role of the electric field should not be underestimated.
Indeed, the displacement field is given by $D=V\varepsilon_{\mathrm{r}}/Nt$,
where $\varepsilon_{\mathrm{r}}$ is the dielectric constant and $t$ is the
thickness of the insulating barrier between CuO bi-layers. The ratio
$t/\varepsilon_{\mathrm{r}}\simeq 0.1\,\mathrm{nm}$ was estimated from an
analysis of Fiske (geometrical resonance) step voltages Superluminal .
Therefore, the displacement field in intrinsic junctions is
$D\simeq\frac{V}{N}\times 10^{8}\,\mathrm{cm^{-1}}.$ (1)
For $V\simeq 2\,\mathrm{V}$ and $N\simeq 10$, which corresponds to the case of
Fig. 4, one would get a very large value $D\simeq 2\times
10^{7}\,\mathrm{V/cm}$, which is certainly capable of seriously polarizing and
displacing ions in complex oxides Moeckly ; Tulina ; Memristor1 .
iv) The phenomenon is not associated with a net change of the oxygen content,
which may only decrease within the cryostat. To the contrary, mesas can be
repeatedly and reversibly doped and undoped, as shown in Figs. 1 (b) and 5
(b).
### V.1 Charge transfer and electrostatic charge trapping
If an injected electron has a high enough energy, it may join one of the ions,
leading to a change of the redox state and the effective doping. It was
suggested that such a “charge transfer” mechanism is involved both in photo-
doping of cuprates Kudinov and in the resistive switching phenomenon in other
complex oxides Memristor1 .
Alternatively, the electron may be trapped (localized) in dielectric parts,
leading to electrostatic charging of the sample. The Bi-2212 compound has a
layered structure with metallic CuO planes sandwiched between polar insulator
BiO layers Polariton . In this case the electrostatic charging will take place
in BiO layers, which may affect the doping state of the neighboring CuO planes
via the electrostatic field-effect. The electrostatic charging of means by
current injection takes place uniformly within the whole structure. Therefore,
unlike the conventional electric field effect Ahn and the electrostatic field
effect at the interface between a superconductor and a ferroelectric material
SuperFerro ; ElStatic , the current injection may lead to a persistent bulk
electrostatic field-effect doping of Bi-2212. Koval et. al. KovalCurrentInj
emphasized the similarity of the phenomenon with the floating-gate effect
utilized in Flash memory devices.
Both types of charging effects have common similarities:
i) The charge transfer requires a certain energy ($\sim$eV for
$\mathrm{YBa_{2}Cu_{3}O_{6+x}}$), rather than electric field.
ii) The sign of the current and the direction of the electric field does not
matter. Therefore, such doping should be even with respect to the voltage
sign, consistent with the observations in Ref. KovalCurrentInj .
Therefore, we attribute even in voltage persistent electric doping to charge
transfer and/or charge trapping mechanisms.
### V.2 Oxygen reorientation and reordering
It is well established that the doping state of cuprates depends not only on
the amount of off-stoichiometric oxygen, but also on the relative orientation
of the oxygen bonds Chains . Therefore, the doping state can be changed by
oxygen reordering. Since the required energy is large $\sim$$\mathrm{eV}$,
compared to thermal energies, oxygen reordering is a slow process and does not
take place spontaneously at low enough $T$. Oxygen reordering is considered as
one of the main mechanisms of the persistent photo Schuller and electric
field Moeckly doping.
In the case of persistent electric-field doping, the oxygen reordering is
steered by the polarization. Therefore, the direction of doping should depend
on the direction of the electric field, i.e., should be odd with respect to
the bias voltage. We clearly see such a contribution in our experiment, see
Fig. 1 (b). Note that the odd-in-voltage contribution was reported in the
point-contact case Tulina , but not reported in previous related works made on
zig-zag type Bi-2212 microstructures KovalCurrentInj ; KovalMemres . The
geometry of the latter samples is symmetric with respect to the electric field
direction (changing the sign of the electric field is equivalent to flipping
their sample up-side down). This is not the case in point contacts and mesa
structures, studied here, for which the fields down (into the crystal) and up
(into the top electrode) are not equivalent. Therefore, the difference may
partly be due to the difference in sample geometry, or to the observed sample-
dependence of the relative strength of odd and even in voltage doping
contributions, as shown in Figs. 1 (d) and (e).
Thus, we attribute the odd in voltage persistent electric doping mechanism to
field-induced oxygen reorientation/reordering.
### V.3 Irreversible processes: electromigration, filament and arc formation
An increase of the bias voltage above $\sim 2.5-3.0\,\mathrm{V}$ leads to a
gradual increase of the current and irreversible change of the mesa
properties. A similar phenomenon was observed in
$\mathrm{YBa_{2}Cu_{3}O_{6+x}}$ thin films and attributed to electromigration
and field-induced diffusion of oxygen, which is even in bias voltage. The
increase of conductance is probably due to a dielectric breakthrough in the
insulating BiO layers, which leads to a pin-hole and filament formation. Thus,
we attribute the slow and irreversible drift of the mesa characteristics at
large bias voltages to electromigration in the mesas. This destructive process
is, however, distinctly different from the reversible and reproducible
electric doping effect, reported above.
After deterioration by electromigration, the mesa characteristics become
similar to resistive switching characteristics for point contacts on top of
oxygen-depleted, Bi-2212 surfaces Tulina . At even higher bias the resistance
becomes very high (infinite). But an inspection in a microscope shows that the
mesa itself remains intact. There is no physical evaporation of material or a
crater at the place of the mesa, as in the case of a violent electric
discharge. Instead there are clear indications of an arc formation at one of
the sharp corners of the mesa, which probably leads to delamination of the
structure and mechanical disattachment of the mesa from the base crystal.
### V.4 Mechanisms of energy accumulation
The most puzzling property of the reported persistent electric doping is that
the required bias voltage is weakly dependent on the number of junctions in
the mesa KovalCurrentInj ; KovalMemres . This is clearly seen from the
presented data, for which the threshold voltage is always $\sim 2\,\mathrm{V}$
for $N=9$ in Fig. 4, $N=11$ in Fig. 1 (f) and $N\sim 56$ in Fig. 1 (d), which
is also similar to that for a single point-contact $\sim 1.5$ V Tulina . It
is, therefore, clear that electric doping requires a certain electron energy,
rather than electric field. However, for stacked tunnel junctions, the energy
acquired by the injected electron in every tunneling event is proportional to
the voltage drop across the junction, $\delta E\simeq eV/N$, and is
significantly smaller than the threshold energy. The main question is,
therefore, how the electrons accumulate a sufficiently large energy, required
for doping.
In Ref. KovalMemres it was suggested that an electron can accumulate energy
upon sequential tunneling through several junctions without relaxation.
However, in this case only electrons in the last junction will have enough
energy. This would result in a strongly inhomogeneous doping in different
junctions. Moreover, the probability of sequential tunneling without
relaxation is small, because of the small ratio of relaxation time
($\tau\sim\mathrm{ps}$) Cascade to the tunnel time, $t_{\mathrm{tun}}$,
$t_{\mathrm{tun}}/\tau\gg 1$. The probability of sequential tunneling through
$N$ junctions is decreasing rapidly $\propto(\tau/t_{\mathrm{tun}})^{N}$ with
increasing $N$. This should lead to a dramatic increase of the doping time
with increasing $N$. Indeed, suppose that it takes $t_{N}\sim
10\,\mathrm{min}$ for $N$ junctions at $I=1\,\mathrm{mA}$ to dope the mesa.
This will involve $N_{e}=It_{N}/e$ tunneling events in each junction ($e$ is
the electron charge). Since the probability of sequential tunneling through
$2N$ junctions is decreasing quadratically, it would require $N_{e}^{2}$
tunneling events per junction, which will take $t_{2N}=N_{e}t_{N}=3.75\times
10^{19}\,\mathrm{min}$. However, such a dramatic increase of the doping time
with increasing mesa height is inconsistent with experiments.
For the sequential tunneling scenario to be relevant, the ratio
$t_{\mathrm{tun}}/\tau$ should rapidly drop with increasing electron energy
and become of the order of unity at $E\sim 1\,\mathrm{eV}$. This may be caused
by resonant tunneling, which increases the tunneling rate of quasiparticles
with a certain energy, and/or by a drastic slowing down of the high energy
quasiparticle relaxation, which may be caused by a rapid decrease of the
Eliashberg’s electron-boson spectral function and a gap in the corresponding
bosonic density of states at high energies Cascade . Such a scenario is
interesting to investigate because it may give an information about the
bosonic spectrum, involved in Cooper pairing, and thus provide a clue about
the electron-boson coupling mechanism, responsible for high-$T_{\mathrm{c}}$
superconductivity in cuprates Cascade .
We also want to propose an alternative mechanism for the energy accumulation
of electrons: the formation of electric-field domains in the natural atomic
superlattice formed by the mesa. Electric field domains are well studied in
semiconducting superlattices Superlattice . They appear in weakly coupled
superlattices close to the resonant tunneling condition. The corresponding
non-linearity leads to an instability and a multiple-valued current-voltage
characteristics. As a result, the electric field distribution in the
superlattice becomes nonuniform and is concentrated in one or several
junctions.
A possibility of a formation of electric field domains in Bi-2212 mesas is not
just a hypothesis. In fact, the multiple-branch $I$-$V$ of Bi-2212 mesas due
to one-by-one switching of intrinsic junctions from the superconducting to the
resistive state, shown in Fig. 4 (b), is due to a formation of electric field
domains in individual tunnel junctions. The formation of electric field
domains in Bi-2212 mesas at high bias would explain many of the features of
the studied persistent electric doping. In this case electrons in the domain
may gain an energy close to eV without sequential tunneling through the whole
mesa. Furthermore, since domains are typically dynamic and propagate through
the whole superlattice Superlattice , this would also explain the uniformity
of doping in the whole mesa, and not just in the outermost junction.
## VI Conclusions
We have studied the effect of persistent electric doping on intrinsic
tunneling characteristics of small Bi-2212 mesa structures. It was shown that
the application of a sufficiently large voltage to the mesas leads to a
controllable and reversible physical doping of the mesas, without a
modification of their chemical composition. This allows the analysis of bulk
electronic spectra in Bi-2212 in a wide doping range on one and the same mesa.
This physical doping has the same effect as chemical (oxygen) doping on the
intrinsic tunneling characteristics of Bi-2212: the $c$-axis resistivity
decreases, the critical current increases and the energy gap is decreasing
together with $T_{\text{c}}$ with over-doping. The anticorrelation between
$I_{\text{c}}R_{\text{n}}$ and $\Delta$ indicates that the $c$-axis transport
becomes progressively more incoherent at the underdoped side of the phase
diagram. An analysis of the doping variation of the intrinsic tunneling
characteristics of the same mesa provides a clue about its genuine shape: with
subsequent doping, the sum-gap peak in the tunneling conductance becomes
sharper and the pseudogap hump rapidly decreases with doping, suggesting the
presence of a critical doping point, in agreement with previous chemical
doping studies Doping .
By analyzing the bias- and time dependence we could identify different
mechanisms involved in the persistent electric doping: i) The even-in-voltage
process via charge transfer and/or charge trapping. ii) The odd-in-voltage
process via oxygen reordering. Those are distinct from the irreversible
electromigration and oxygen electrodiffusion, observed at higher bias.
We confirm the previous report KovalMemres that the threshold voltage for the
electric doping is weakly dependent of the number of junctions in the mesas
and is similar to that for a single surface point contact Tulina . This
indicates that it is the energy of injected electrons, rather than electric
field, that determines the phenomenon. We suggest that the required energy
accumulation by tunnel electrons may be due to a formation of electric field
domains in the natural atomic superlattice formed in the Bi-2212 single
crystal.
Acknowledgments We are grateful to the Swedish Research Council and the SU-
Core Facility in Nanotechnology for financial and technical support.
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|
arxiv-papers
| 2012-02-08T11:48:32 |
2024-09-04T02:49:27.215997
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Holger Motzkau, Thorsten Jacobs, Sven-Olof Katterwe, Andreas Rydh, and\n Vladimir M. Krasnov",
"submitter": "Holger Motzkau",
"url": "https://arxiv.org/abs/1202.1665"
}
|
1202.1725
|
-titleHadron Collider Physics Symposium 2011 11institutetext: Nikhef, Amsterdam, The Netherlands
# QCD studies in the forward region @ LHCb
Victor Coco victor.coco@cern.ch on behalf of LHCb collaboration
###### Abstract
The LHCb experiment at the LHC is fully instrumented over a unique
pseudorapidity range in the forward region. Although it has been designed for
$b$-physics, LHCb is able to provide valuable informations on particle
production in this region of phase space. Therefore QCD studies have been
performed with the LHCb detector on $pp$ collisions at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$. The measurement
of charged particles multiplicity at $\sqrt{s}=7~{}\mathrm{TeV}$,
$\bar{\Lambda}/\Lambda$, $\bar{\Lambda}/K_{s}^{0}$ production ratios at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$, as well as
light hadrons ($p$,$K$,$pi$) production ratios at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$ are reported.
## 1 Introduction
The LHCb experiment is dedicated to CP violation and rare decay measurements
involving b and c hadrons at the LHC lhcb . Since most of the correlated
$b\bar{b}$ pair are produced at small angle with respect to the beam line, the
LHCb detector has been designed as a single arm spectrometer instrumented over
the $2<\eta<5$ pseudo-rapidity range. This unique coverage of the forward
region, together with excellent tracking and particle identification
performances allows the probing particle production at the LHC in an uncharted
region of phase space.
The forward hadron production measurements gives important inputs to tune the
hadronisation models in the LHCb acceptance, since these models have been
extrapolated from measurements performed in a different energy and rapidity
regime. Production cross-section of several particles have been measured at
LHCb. $K_{s}^{0}$ cross section measurement at $\sqrt{s}=900~{}\mathrm{GeV}$
ks and $\phi$ cross section measurement at $\sqrt{s}=7~{}\mathrm{TeV}$ phi
probe strangeness production. $J/\psi$ jpsi , $\psi(2S)$ psi , $\Upsilon$
upsilon , prompt charm c and $b$ b cross section measurements at
$\sqrt{s}=7~{}\mathrm{TeV}$ probe heavy flavour production. The measurements
of charged particles multiplicity at $\sqrt{s}=7~{}\mathrm{TeV}$ chmult , as
well as $V_{0}$ production ratios:
$\bar{\Lambda}/\Lambda$,$\bar{\Lambda}/K_{s}^{0}$ v0 and light hadrons
($\pi$,$K$,$p$) production ratio p both at $\sqrt{s}=900~{}\mathrm{GeV}$ and
$\sqrt{s}=7~{}\mathrm{TeV}$ are presented in the following.
## 2 Charged particles multiplicity at $\sqrt{s}=7~{}\mathrm{TeV}$
The measurement of charged particles multiplicity at the LHCb experimentchmult
, provides input for modelling the underlying event structure in high energy
$pp$ collisions in the forward region.
The charged particles are reconstructed in the vertex locator (VELO) detector
situated close to the interaction region. It provides a high reconstruction
efficiency and purity for charged particles in the $\eta$ range
$-2.5<\eta<-2.0$ and $2.0<\eta<4.5$. Since the magnetic field in the VELO is
negligible, no momentum information is to VELO-reconstructed charged
particles. In order to probe more energetic collision, the charged particles
multiplicity is also measured in events where at least one track in the
pseudo-rapidity range of $2.5<\eta<4.5$ went through the whole tracking
system, which allow the measurement of its momentum. This track is required to
have a transverse momentum greater than $1~{}GeV/c$.
Figure 1: The multiplicity distribution in $\eta$ bins (shown as points with
statistical error bars) with predictions of different event generators. The
inner error bar represents the statistical uncertainty and the outer error bar
represents the systematic and statistical uncertainty on the measurements.
Predictions are from Pythia 6, Phojet and Pythia 8, comparison with other
predictions can be found in chmult . Left is for all events, right is for
events with at least one track with $p_{T}>1~{}GeV/c$ in $2.5<\eta<4.5$.
The analysis is performed on $1.5$ million events for each magnet polarity,
which were triggered by requiring at least one reconstructed VELO track. This
dataset has low pile up, with only $3.7\pm 0.4\%$ of events having more than
one interaction. Monte Carlo simulation is used to correct for acceptance,
resolution effects and to estimate the secondary charged particles
contamination. The main source of systematic uncertainty is due to the
tracking efficiency determination and is estimated to be $4\%$ overall. Other
source of systematics uncertainty are small compared to it. The event
multiplicity is obtained by unfolding of the migration due to reconstruction
inefficiencies.
Figure 1 left, shows the unfolded multiplicity distribution in $\eta$ bins for
all events, while figure 1 right shows it for harder events with at least one
track with $p_{T}>1~{}GeV/c$ in $2.5<\eta<4.5$. Several event generator have
been compared with the data. None are fully able to describe the multiplicity
distributions, even though the agreement is better for hard QCD events.
## 3 $\bar{\Lambda}/\Lambda$ and $\bar{\Lambda}/K_{s}^{0}$ ratios
The production ratios of baryon/anti-baryon, such as $\bar{\Lambda}/\Lambda$,
allow to study the baryon-number transport from the beam particles to the
final state. $\bar{\Lambda}/K_{s}^{0}$ ratio on the other hand is a measure of
the baryon to meson suppression, which is a good test for different
fragmentation models.
High purity samples of prompt $K_{s}^{0}$ decaying into $\pi^{+}\pi^{-}$ and
$\Lambda$, $\bar{\Lambda}$ decaying into $p\pi$ are selected based on a Fisher
discriminant combining the impact parameter of the $V^{0}$ particles and their
daughters. Only events with one primary vertex are selected to avoid
diffractive event contribution. No particle identification is used here v0 .
Figure 2: The production cross-section ratio $\bar{\Lambda}/\Lambda$ at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$ as a function of
rapidity compared with the predictions of the LHCb MC tune lhcbmc , Perugia 0
and Perugia NOCR perugia . Vertical lines show the combined statistical and
systematic uncertainties and the short horizontal bars (where visible) show
the statistical component.
Figure 3: The production cross-section ratio $\bar{\Lambda}/K_{s}^{0}$ at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$ as a function of
rapidity compared with the predictions of the LHCb MC tune lhcbmc and Perugia
0 perugia . Vertical lines show the combined statistical and systematic
uncertainties and the short horizontal bars (where visible) show the
statistical component.
Figure 4: The ratios $\bar{\bar{\Lambda}}/\Lambda$ (top) and
$\bar{\Lambda}/K_{s}^{0}$ (bottom) from LHCb are compared at both at
$\sqrt{s}=900~{}\mathrm{GeV}$ (red) and $\sqrt{s}=7~{}\mathrm{TeV}$ (blue)
with the published results from STAR star (black) as a function of rapidity
loss. Vertical lines show the combined statistical and systematic
uncertainties and the short horizontal bars (where visible) show the
statistical component.
The systematic uncertainty is reduced since the major uncertainty on the
production cross-sections, that comes from absolute luminosity measurement,
cancels through the ratio. The remaining sources of systematic uncertainty
come from kinematic correction of the Monte Carlo simulation used to evaluate
selection efficiency, the uncertainty in the interaction with material and the
diffractive events pollution. Depending on the bins in $\eta$ and $p_{T}$,
they add up to 0.02-0.06 for $\bar{\Lambda}/\Lambda$ and 0.02-0.03 for
$\bar{\Lambda}/K_{s}^{0}$.
The measurement is performed with $0.3~{}\mathrm{nb^{-1}}$ at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $1.38~{}\mathrm{nb^{-1}}$ at
$\sqrt{s}=7~{}\mathrm{TeV}$. Figure 2 shows that $\bar{\Lambda}/\Lambda$ is in
rather good agreement with Perugia 0 tune for Pythia at low rapidity while at
high rapidity, extreme models of baryon transport seams to be favoured perugia
. This behaviour is observable both at $\sqrt{s}=900~{}\mathrm{GeV}$ and
$\sqrt{s}=7~{}\mathrm{TeV}$. In Figure 3, $\bar{\Lambda}/K_{s}^{0}$ shows an
excess over the whole range of rapidity, at both center of mass energy.
Another way to present the baryon number transport results is to show the
production ratio of anti-baryon to baryon as a function of the rapidity loss,
$\Delta y=y_{beam}-y$ where $y_{beam}$ is the rapidity of the incoming beam,
Figure 4. It allows to compare the LHCb results with the measurements of the
previous experiments. It also shows that there is no significant energy scale
violation between the results at $\sqrt{s}=900~{}\mathrm{GeV}$ and
$\sqrt{s}=7~{}\mathrm{TeV}$.
## 4 Light hadrons production ratios
Figure 5: Distribution of the $\bar{p}/p$ production ratio against rapidity,
$0.8<p_{T}<1.2~{}\mathrm{GeV/c}$. Top for $\sqrt{s}=900~{}\mathrm{GeV}$,
bottom for $\sqrt{s}=7\mathrm{TeV}$. Data are compared with the predictions of
the LHCb MC tune lhcbmc and Perugia 0 and Preugia NOCR tunes perugia
.
The measurement of light hadron production ratios at LHC is another input that
can be used to validate hadronisation models. In addition $\bar{p}/p$
production ratio probes the baryon number transport. The production ratios
$\bar{p}/p$, $K^{-}/K^{+}$, $\pi^{-}/\pi^{+}$,
$(\bar{p}+p)/(K^{-}+K^{+})$,$(\bar{p}+p)/(\pi^{-}+\pi^{+})$ and
$(K^{-}+K^{+})/(\pi^{-}+\pi^{+})$ have been measured in the forward region,
both at $\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$ center
of mass energy. The preliminary results shown here are an update of p . The
measurement is performed in three bins of transverse momentum,
$p_{T}<0.8~{}\mathrm{GeV/c}$,$0.8<p_{T}<1.2~{}\mathrm{GeV/c}$, and
$p_{T}>1.2~{}\mathrm{GeV/c}$ and five bins of rapidity, in the range $2<y<5$,
with $0.3~{}\mathrm{nb^{-1}}$ at $\sqrt{s}=900~{}\mathrm{GeV}$ and
$1.8~{}\mathrm{nb^{-1}}$ at $\sqrt{s}=7~{}\mathrm{TeV}$.
Prompt light hadrons with $p>5~{}\mathrm{GeV/c}$ are selected in events with
at least one reconstructed primary vertex. Exploiting the powerful hadronic
separation capabilities of the LHCb RICH system, the cross contamination
between species have been evaluated. The performance of the particle
identification is calibrated on kinematically isolated samples of
$\phi\rightarrow K^{+}K^{-}$, $K_{s}\rightarrow\pi^{+}\pi^{-}$ and
$\Lambda\rightarrow\pi p$. The main systematic comes from cross contamination
between $p$, $K$ and $\pi$. It depends on which ratio is considered and varies
from the percent level up to tens percent in the extreme region of rapidity.
Results have been compared with different MC models. If the general behaviour
is reproduced by the models, there is still room for improvements. Figure 5
and figure 6 show as illustration the results of the analysis for $\bar{p}/p$
and $(K^{-}+K^{+})/(\pi^{-}+\pi^{+})$, for the $[0.8;12~{}\mathrm{GeV/c}]$
$p_{T}$ bin.
Figure 6: Distribution of the $(K^{-}+K^{+})/(\pi^{-}+\pi^{+})$ production
ratio against rapidity, $0.8<p_{T}<1.2~{}\mathrm{GeV/c}$. Top for
$\sqrt{s}=900~{}\mathrm{GeV}$, bottom for $\sqrt{s}=7\mathrm{TeV}$. Data are
compared with the predictions of the LHCb MC tune lhcbmc and Perugia 0 and
Preugia NOCR tunes perugia
.
In the following, highlight is put on the $\bar{p}/p$ production ratio since
it allows further tests of the baryon number transport. Results are consistent
with Monte Carlo models at $\sqrt{s}=7~{}\mathrm{TeV}$ but differ
significantly at $\sqrt{s}=900~{}\mathrm{GeV}$, especially at low $p_{T}$, as
shown in Figure 5. Like for the $\bar{\Lambda}/\Lambda$ production ratio
measurement, $\bar{p}/p$ production ratio is shown as function of the rapidity
loss $\Delta y$, Figure 7. The LHCb measurements are compatible with those of
the previous experiments povp , but significantly more precise. The ALICE
experiment measurement cover the high rapidity loss region with good
precision, showing the complementarity at LHC between experiments covering the
central region and LHCb for production measurements.
Figure 7: $\Delta y$ from LHCb p and other experiments povp , weighted
average in $p_{T}$. Uncertainties are statistical + systematics, except in the
case of data collected at the ISR where only statistical uncertainties are
shown.
## 5 Conclusions
The charged particle multiplicity measured at LHCb shows that the number of
charged particles produced forward is underestimated by most of the Monte
Carlo models in the forward region of $pp$ collisions at
$\sqrt{s}=7~{}\mathrm{TeV}$. The measurements of $V_{0}$ production ratios
suggest lower strange baryon suppression and higher baryon number transport
than in the Monte Carlo models that have been investigated. The LHCb data are
consistent with data from lower energy experiments. Light hadron production
ratios of $\bar{p}/p$, $K^{-}/K^{+}$, $\pi^{-}/\pi^{+}$,
$(\bar{p}+p)/(K^{-}+K^{+})$,$(\bar{p}+p)/(\pi^{-}+\pi^{+})$ and
$(K^{-}+K^{+})/(\pi^{-}+\pi^{+})$ have been measured both at
$\sqrt{s}=900~{}\mathrm{GeV}$ and $\sqrt{s}=7~{}\mathrm{TeV}$, and compared to
some expectation from Monte Carlo models. It suggests that improvements are
needed to fully reproduce their behaviour as function of rapidity and
transverse momentum.
## References
* (1) Alves Jr., A. A. and others (LHCb Collaboration), JINST (2008) 3 S08005.
* (2) R. Aaij et al. (LHCb Collaboration), Phys. Lett. B (2010) 693 69-80, arXiv:1008.3105.
* (3) R. Aaij et al. (LHCb Collaboration), Phys. Lett. B (2011) 703 267-273, arXiv:1107.3935
* (4) R. Aaij et al. (LHCb Collaboration), Eur. Phys. J. C (2011) 71 1645\.
* (5) LHCb Collaboration, LHCb-CONF-2011-026.
* (6) LHCb Collaboration, LHCb-CONF-2011-016
* (7) LHCb Collaboration, LHCb-CONF-2010-013.
* (8) R. Aaij et al. (LHCb Collaboration), Phys. Lett. B (2010) 694 209-216, arXiv:1009.2731.
* (9) R. Aaij et al. (LHCb Collaboration), arXiv:1112.4592, submitted to EPJ
* (10) R. Aaij et al. (LHCb Collaboration), J. High Energy Phys. (2011) 08 034, arXiv:1107.0882
* (11) LHCb Collaboration, LHCb-CONF-2010-009.
* (12) Clemencic, M and others, Journal of Physics: Conference Series (2011) 331 3
* (13) P. Z. Skands, Phys. Rev. D 82 (2010) 074018.
* (14) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 75 (2007) 064901 .
* (15) A. K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. 105 (2010) 072002, arXiv:1006.5432, 072002, I. G. Bearden et al. (BRAHMS Collaboration), Phys. Lett. B (2005) 607 42, B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C (2009) 79 034909, A. M. Rossi, G. Vannini, A. Bussiere, E. Albini, D. DAlessandro and G. Giacomelli, Nucl. Phys. B (1975) 84 269, M. Banner et al., Phys. Lett. B (1972) 41 547, B. Alper et al., Phys. Lett. B (1973) 46 265, T. Anticic et al. (NA49 Collaboration), Eur. Phys. J. C (2010) 65 9\.
|
arxiv-papers
| 2012-02-08T15:03:13 |
2024-09-04T02:49:27.230236
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Coco Victor (for the LHCb Collaboration)",
"submitter": "Victor Coco",
"url": "https://arxiv.org/abs/1202.1725"
}
|
1202.1726
|
# Magneto-transport in a quantum network: Evidence of a mesoscopic switch
Srilekha Saha Theoretical Condensed Matter Physics Division, Saha Institute
of Nuclear Physics, Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
Santanu K. Maiti santanu@post.tau.ac.il School of Chemistry, Tel Aviv
University, Ramat-Aviv, Tel Aviv-69978, Israel S. N. Karmakar Theoretical
Condensed Matter Physics Division, Saha Institute of Nuclear Physics,
Sector-I, Block-AF, Bidhannagar, Kolkata-700 064, India
###### Abstract
We investigate magneto-transport properties of a $\theta$ shaped three-arm
mesoscopic ring where the upper and lower sub-rings are threaded by Aharonov-
Bohm fluxes $\phi_{1}$ and $\phi_{2}$, respectively, within a non-interacting
electron picture. A discrete lattice model is used to describe the quantum
network in which two outer arms are subjected to binary alloy lattices while
the middle arm contains identical atomic sites. It is observed that the
presence of the middle arm provides localized states within the band of
extended regions and lead to the possibility of switching action from a high
conducting state to a low conducting one and vice versa. This behavior is
justified by studying persistent current in the network. Both the total
current and individual currents in three separate branches are computed by
using second-quantized formalism and our idea can be utilized to study
magnetic response in any complicated quantum network. The nature of localized
eigenstates are also investigated from probability amplitudes at different
sites of the quantum device.
###### pacs:
73.23.-b, 73.23.Ra.
## I Introduction
Theoretical and experimental investigations in low-dimensional systems lead to
the opportunity of visualizing various novel quantum mechanical effects Jaya1
; Jaya2 in a tunable way. Persistent current being one such exotic quantum
mechanical phenomenon observed in normal metal mesoscopic rings and nanotubes
pierced by Aharonov-Bohm (AB) flux
Figure 1: (Color online). Schematic view of a quantum network where the upper
and lower sub-rings threaded by AB fluxes $\phi_{1}$ and $\phi_{2}$,
respectively. Both the upper and lower arms are subjected to binary alloy
lattices, while the middle arm contains identical lattice sites. The filled
colored circles correspond to the positions of the atomic sites.
$\phi$. Prior to its experimental evidence, the possibility of a non-decaying
current in normal metal rings was first predicted by Büttiker, Imry and
Landauer Butti in a pioneering work, and, in the sub-sequent years
theoretical attempts were made gefen ; altshu ; schmid ; maiti1 ; san5 ; bell
; chen ; wu ; li1 ; li2 ; san1 ; san3 ; san4 ; san6 to understand the actual
mechanism behind it. The experimental realization of this phenomenon of non-
decaying current in metallic rings/cylinders has been established quite in
late. It has been first examined by Levy et al. levy and later many other
experiments chand ; mailly ; blu have confirmed the existence of non-
dissipative currents in such quantum systems.
Although the studies involving simple mesoscopic rings have already generated
a wealth of literature there is still need to look deeper into the problem to
address several important issues those have not yet been explored, as for
example the understanding of the behavior of persistent current in multiply
connected quantum network, specially in presence of disorder. It is well known
that in presence of random site-diagonal disorder in an one-dimensional ($1$D)
mesoscopic ring all the energy levels are localized ander , and accordingly,
the persistent current gets reduced enormously in presence of disorder
compared to that of an ordered ring. But there are some $1$D disordered
systems which support extended eigenstates along with the localized energy
levels, and these materials may provide several interesting issues, mainly to
provide a localization to delocalization transition and vice versa. For
example, in a pioneering work Dunlap et al. Phillip1 have shown that even in
1D disordered systems extended eigenstates are possible for certain kind of
topological correlations among the atoms. They have proposed that any physical
system which can be described by the random dimer model should exhibit the
transmission resonances and a huge enhancement in the transmission takes place
when the Fermi level coincides with the unscattered states. In a consecutive
year Wu et al. Phillip2 have argued that the random dimer model can also be
used to explain the insulator to metal transition in polyaniline as a result
of the movement of the Fermi level to extended region. Later, several other
works liu ; hu1 ; hu2 ; arun1 ; arun2 have also been carried out in such type
of materials to exhibit many important physical results.
The existence of localized energy eigenstates together with the extended
states in a simple ring geometry has been explored in some recent works by
Jiang et al. hu1 ; hu2 . They have analyzed the nature of these states by
evaluating persistent current and the wave amplitudes at different sites of
the ring. In these systems localized states appear by virtue of disorder. But,
in our present work we make an attempt to establish localized eigenstates,
even in the absence of disorder, along with extended states simply by
considering the effect of topology of the system. To the best of our
knowledge, this behavior has not been addressed earlier in the literature.
Here we consider a three-arm mesoscopic ring in which two outer arms are
subjected to binary alloy lattices and the middle one contains identical
lattice sites, and, we show that due to the presence of the middle arm quasi-
localized energy eigenstates are observed within the band of extend regions.
It leads to the possibility of getting switching action from a high conducting
state to a low one and vice versa as a result of the movement of the Fermi
level. We illustrate this behavior by studying persistent current in the
quantum network and explore the nature of energy eigenstates in terms of the
probability amplitude in different lattice sites of the geometry. Our present
analysis can be utilized to study magnetic response in any complicated quantum
network and we believe that this work offers an excellent opportunity to study
the simultaneous effects of topology and the magnetic fields threaded by two
sub-rings in our three-arm ring system.
With an introduction in Section I we organize the paper as follows. In Section
II, first we present the model, then describe the theoretical formalism which
include the Hamiltonian and the formulation of persistent currents in
individual branches of the network. In Section III we analyze the results and
finally in Section IV we draw our conclusions.
## II Model and Theoretical Formulation
### II.1 The model and the Hamiltonian
Let us refer to Fig. 1. A three-arm mesoscopic ring where the upper and lower
sub-rings are threaded by AB fluxes $\phi_{1}$ and $\phi_{2}$, respectively.
The outer arms are subjected to binary alloy lattices (consisting of A and B
types of atoms) and the middle arm contains identical lattice sites (atomic
sites labeled by C) except those on the boundaries. The filled colored circles
correspond to the positions of the atomic sites. Within a tight-binding
framework the Hamiltonian for such a network reads as,
$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j}\epsilon_{j}c_{j}^{\dagger}c_{j}+t\sum_{j}\left(c_{j}^{{\dagger}}c_{j+1}e^{-i\theta_{1}}+h.c.\right)$
(1) $\displaystyle+$
$\displaystyle\sum_{l}\epsilon_{l}c_{l}^{\dagger}c_{l}+v\sum_{l}\left(c_{l}^{{\dagger}}c_{l+1}e^{-i\theta_{2}}+h.c.\right)$
where, $\epsilon_{j}$ represents the site energy for the outer arms, while for
the middle arm it is assigned by $\epsilon_{l}$. In the outer ring
$\epsilon_{j}=\epsilon_{A}$ or $\epsilon_{B}$ alternately so that it forms a
binary alloy. On the other hand, $\epsilon_{l}=\epsilon_{C}$ for the atomic
sites those are referred by C atoms. $t$ and $v$ are the nearest-neighbor
hopping integrals in the outer and middle arms, respectively. Due to the
presence of magnetic fluxes $\phi_{1}$ and $\phi_{2}$ in two sub-rings, phase
factors $\theta_{1}$ and $\theta_{2}$ appears into the Hamiltonian. They are
expressed as follows: $\theta_{1}=2\pi(\phi_{1}+\phi_{2})/(N_{U}+N_{L})$ and
$\theta_{2}=2\pi(\phi_{1}-\phi_{2})/2N_{M}$. Here the fluxes are measured in
units of the elementary flux-quantum $\phi_{0}$ ($=ch/e$), and, $N_{U}$,
$N_{M}$ and $N_{L}$ represent the total number of single bonds (each single
bond is formed by connecting two neighboring lattice sites) in the upper,
middle and lower arms, respectively. It reveals that $N_{U}+N_{M}+N_{L}-1$
number of atomic sites in the quantum network. $c_{j}^{{\dagger}}$ ($c_{j}$)
corresponds to the creation (annihilation) operator of an electron at the
$j~{}\mbox{th}$ site, and, a similar definition also goes for the atomic sites
$l$.
### II.2 Calculation of persistent current
In the second quantized notation the general expression of charge current
operator is in the form san2 ,
$I=\frac{2\pi
ie\alpha}{L}\sum_{n}\left(c_{n}^{{\dagger}}c_{n+1}-c_{n+1}^{{\dagger}}c_{n}\right).$
(2)
Here, $L$ is the length of the arm in which we are interested to calculate the
current and $\alpha$ represents the nearest-neighbor hopping strength. The
nearest-neighbor hopping strength ($\alpha$) is equal to $t$ for the outer
arms, while for the middle arm it becomes identical to $v$. Therefore, for a
particular eigenstate $|\psi_{k}\rangle$ the persistent current becomes,
$I^{k}=\langle\psi_{k}|I|\psi_{k}\rangle$, where
$|\psi_{k}\rangle=\sum_{p}a_{p}^{k}|p\rangle$. Here $|p\rangle$’s are the
Wannier states and $a_{p}^{k}$’s are the corresponding coefficients.
Following the above relations, now we can write down the expressions for
persistent currents in the individual branches for a given eigenstate
$|\psi_{k}\rangle$. They are as follows.
For the upper arm:
$I_{U}^{k}=\frac{2\pi
iet}{N_{U}+N_{L}}\sum_{j}\left(a_{j}^{k*}a_{j+1}^{k}e^{-i\theta_{1}}-h.c.\right)$
(3)
where, summation over $j$ spans from $1$ to $N_{U}$. In the case of middle-
arm:
$I_{M}^{k}=\frac{\pi
iev}{N_{M}}\sum_{l}\left(a_{l}^{k*}a_{l+1}^{k}e^{-i\theta_{2}}-h.c.\right)$
(4)
here, the net contribution comes from $N_{M}$ bonds. Finally, for the case of
lower arm:
$I_{L}^{k}=\frac{2\pi
iet}{N_{U}+N_{L}}\sum_{j}\left(a_{j}^{k*}a_{j+1}^{k}e^{-i\theta_{1}}-h.c.\right)$
(5)
In this case the net contribution comes from the lower bonds. The lattice
constant $a$ is set equal to $1$.
At absolute zero temperature ($T=0\,$K), the net persistent current in any
branch of the quantum network for a particular electron filling can be
obtained by taking sum of the individual contributions from the energy levels
with energies less than or equal to Fermi energy $E_{F}$. Therefore, for
$N_{e}$ electron system total persistent in any branch becomes
$I_{\beta}=\sum_{k}^{N_{e}}I^{k}_{\beta}$, where $\beta=U$, $M$ and $L$, for
the upper, middle and lower arms, respectively. Once $I_{U}$, $I_{M}$ and
$I_{L}$ are known, the net persistent current for the full network can be
easily obtained simply adding the contributions of the individual arms, and
hence the total current is given by $I_{T}=I_{U}+I_{M}+I_{L}$.
The net persistent current ($I_{T}$) can also be determined in some other ways
as available in the literature. Most probably the easiest way of calculating
persistent current is to take first order derivative of ground state energy
with respect to magnetic flux maiti1 ; san5 . However in this method it is not
possible to find the distribution of persistent current in individual arms of
the network with a high degree of accuracy. On the other hand in our present
scheme, the so-called second-quantized approach, there are certain advantages
compared to other available procedures. Firstly, we can easily calculate
persistent current in any branch of a network. Secondly, the determination of
individual responses in separate arms provides much deeper insight to the
actual mechanism of electron transport in a transparent way.
In the present work we investigate all the essential feature of magneto-
transport at absolute zero temperature and choose the units where $c=e=h=1$.
Throughout the numerical work we set $t=v=-1$ and measure the energy scale in
unit of $t$.
## III Numerical results and discussion
### III.1 Quantum network with $\epsilon_{A}=\epsilon_{B}=0$
We first start with a perfect quantum system where $\epsilon_{A}$ and
$\epsilon_{B}$ and $\epsilon_{C}$ are all identical to each other and we set
$\epsilon_{A}=\epsilon_{B}\epsilon_{C}=0$. To have a
Figure 2: (Color online). Energy levels as a function of flux $\phi_{1}$ for a
three-arm ring with $\epsilon_{A}=\epsilon_{B}=0$ considering $N_{U}+N_{L}=10$
and $N_{M}=3$, where (a) and (b) correspond to $\phi_{2}=0$ and $\phi_{0}/4$,
respectively.
clear idea about the magnetic response of the model quantum system, first we
illustrate the behavior of energy spectra as a function of flux $\phi_{1}$ for
different values of flux $\phi_{2}$ threaded by the lower sub-ring. The
results are presented in Fig. 2, where (a) and (b) correspond to $\phi_{2}=0$
and $\phi_{0}/4$, respectively. In the absence of flux $\phi_{2}$, energy
levels near the edges of the spectrum become more dispersive than those lying
in the central region (see Fig. 2(a)) and near the center of the spectrum the
energy levels are almost non-dispersive with respect to flux $\phi_{1}$. This
feature implies that the persistent current amplitude becomes highly sensitive
to the electron feeling i.e., the Fermi energy $E_{F}$ of the system, since
the current is directly proportional to the slope of the energy levels maiti1
. The situation becomes much more interesting when we add a magnetic flux in
the lower sub-ring. Here, the energy levels near the central region of the
spectrum becomes more dispersive in nature
Figure 3: (Color online). Persistent current in different arms as a function
of $\phi_{1}$ for a three-arm ring with $\epsilon_{A}=\epsilon_{B}=0$ in the
half-filled band case considering $N_{U}+N_{L}=60$ and $N_{M}=25$.
than the energy levels near the edges (Fig. 2(b)), and it increases gradually
with flux $\phi_{2}$, which gives a possibility of getting higher current
amplitude with increasing the total number of electrons $N_{e}$ in the system.
A similar kind of energy spectrum is also observed if we plot the energy
levels as a function of flux $\phi_{2}$ instead of $\phi_{1}$, keeping
$\phi_{1}$ as a constant. All these energy levels exhibit $\phi_{0}$ ($=1$, in
our choice of units $c=e=h=1$) flux-quantum periodicity. Thus, for such a
simple quantum network persistent current amplitude might be regulated for a
particular filling simply by tuning the magnetic flux threaded by anyone of
two such sub-rings, and, its detailed descriptions are available in the sub-
sequent parts.
In Fig. 3 we present the variation of persistent current in individual arms of
the three-arm quantum network as a function of flux $\phi_{1}$ for some fixed
values of $\phi_{2}$. The panels from the top correspond to the results for
the upper, middle and lower arms, respectively,
Figure 4: (Color online). Persistent current in different arms as a function
of $\phi_{1}$ for a three-arm ring with $\epsilon_{A}=\epsilon_{B}=0$
considering $N_{U}+N_{L}=40$ and $N_{M}=17$. The red, green and blue curves
correspond to $N_{e}=10$, $15$ and $20$, respectively. For all these spectra
$\phi_{2}$ is set at $\phi_{0}/4$.
and in all these cases the current is determined for the half-filled band case
i.e., $N_{e}=42$. The left column represents the current for $\phi_{2}=0$, and
the right column gives the current when $\phi_{2}$ is fixed at $\phi_{0}/4$.
From the spectra we notice that in some cases current shows continuous like
behavior while in some other cases it exhibits saw-tooth like nature as a
function of flux $\phi_{1}$ threaded by the upper sub-ring. This saw-tooth or
continuous like feature solely depends on the behavior of the ground state
energy for a particular filling ($N_{e}$). It is to be noted that in a
conventional ordered AB ring we always get saw-tooth like behavior of
persistent current irrespective of the filling of the electrons maiti1 . In
the saw-tooth variation a sudden change in direction of persistent current
takes place across a particular value of magnetic flux which corresponds to a
phase reversal from the diamagnetic nature to the paramagnetic one or vice
versa. In our three-arm geometry we also observe that though the current in
the upper arm or in the lower arm is not so sensitive to the flux $\phi_{2}$,
but the current amplitude in the middle arm changes remarkably, even an order
of magnitude, in presence of flux $\phi_{2}$, which leads to a net larger
current since the total current is obtained by adding the contributions from
the individual arms.
To explore the filling dependent behavior of persistent current, in Fig. 4 we
display persistent currents for three different arms as a function of flux
$\phi_{1}$ for a typical value of $\phi_{2}$. Here, $\phi_{2}$ is set at
$\phi_{0}/4$. The red, green and blue lines represent the currents for
$N_{e}=10$, $15$ and $20$, respectively. The current in different arms shows
quite a complex structure which strongly depends on the electron filling as
well as magnetic flux $\phi_{2}$. In all these cases persistent current
provides $\phi_{0}$ flux-quantum periodicity, like a traditional single-
channel mesoscopic ring or a multi-channel cylinder.
### III.2 Quantum network with $\epsilon_{A}\neq\epsilon_{B}$
Now we focus our attention to the geometry where site energies in the outer
arms are no longer identical to each other i.e.,
$\epsilon_{A}\neq\epsilon_{B}$. In this case energy spectrum gets modified
significantly compared to the
Figure 5: (Color online). Energy levels as a function of flux $\phi_{1}$ for a
three-arm ring with $\epsilon_{A}=-\epsilon_{B}=1$ considering
$N_{U}+N_{L}=10$ and $N_{M}=3$, where (a) and (b) correspond to $\phi_{2}=0$
and $\phi_{0}/4$, respectively.
previous one where site energies are uniform ($\epsilon_{A}=\epsilon_{B}$). To
illustrate it in Fig. 5 we plot the energy-flux characteristics for a three-
arm quantum network considering $\epsilon_{A}=-\epsilon_{B}=1$, where (a) and
(b) correspond to the identical meaning as given in Fig. 2. Since the upper
and lower arms of the network are subjected to the binary alloy lattices we
get two sets of discrete energy levels spaced by a finite gap around $E=0$
(see Fig. 5). Quite interestingly we see that the energy levels near the two
extreme edges of the spectrum are more dispersive in nature than those
situated along the inner region. With increasing the difference in site
energies ($|\epsilon_{A}-\epsilon_{B}|$), we get more less dispersive energy
levels in the inner region and for large enough value of
$|\epsilon_{A}-\epsilon_{B}|$ these levels become almost non-dispersive and
they practically contribute nothing to the current. Thus, for such a system a
mixture of quasi-extended and quasi-localized energy levels are found out and
it can provide a very large or almost zero current depending on the electron
filling. For a very
Figure 6: (Color online). Current-flux characteristics of a three-arm ring
with $\epsilon_{A}=-\epsilon_{B}=1$ considering $N_{U}+N_{L}=60$, $N_{M}=25$
and $\phi_{2}=\phi_{0}/4$ in the quarter-filled ($N_{e}=21$) band case, where
(a)-(d) correspond to the currents in the upper, middle and lower arms and in
the full system, respectively.
Figure 7: (Color online). Current-flux characteristics of a three-arm ring
with $\epsilon_{A}=-\epsilon_{B}=1$ in the half-filled ($N_{e}=42$) band case
for the same parameter values used in Fig. 6.
large system size, the energy separation between two successive levels in each
set of discrete energy levels gets reduced and we get two quasi-band of
energies separated by a finite gap, where the gap is controlled by the
parameter values. It is important to note that, unlike the previous one (Fig.
2), for this case the energy spectrum is not so sensitive to flux $\phi_{2}$
(Fig. 5). The presence of C-type of atoms in the middle arm which divides the
binary alloy ring into two sub-rings is responsible for the existence of
quasi-localized energy levels near the inside edges of two quasi-band of
energies. Thus we get more non-dispersive energy levels with increasing the
length of the middle arm.
The existence of nearly extended and localized states becomes much more
clearly visible from our current-flux spectra. As illustrative example, in
Fig. 6 we display the variation of persistent current in
Figure 8: (Color online). Charge stiffness constant ($D$) as a function of
electron filling ($N_{e}$) for a three-arm ring with
$\epsilon_{A}=-\epsilon_{B}=1$ considering $N_{U}+N_{L}=60$, $N_{M}=25$ and
$\phi_{2}=\phi_{0}/4$.
Figure 9: (Color online). Probability amplitude (P.A.) as a function of site
index ($n$) for a three-arm ring with $\epsilon_{A}=-\epsilon_{B}=1$
considering $N_{U}+N_{L}=60$ and $N_{M}=25$ when $\phi_{1}$ and $\phi_{2}$ are
set at $\phi_{0}/4$. (a) and (b) correspond to the results of $21$-st and
$42$-nd eigenstates, respectively.
individual arms including the total current of a three-arm ring considering
$\epsilon_{A}=-\epsilon_{B}=1$ for the quarter-filled ($N_{e}=21$) band case.
The flux $\phi_{2}$ is set equal to $\phi_{0}/4$. From the spectra it is
clearly observed that the current in each arm provides a non-zero value (Figs.
6(a)-(c)), and accordingly, the system supports a finite current as shown in
Fig. 6(d). The situation becomes completely opposite when the filling factor
is changed. Quite remarkably we notice that persistent current almost vanishes
in three separate branches which provides almost vanishing net current in the
half-filled band case. The results are illustrated in Fig. 7, where (a)-(d)
correspond to the identical meaning as in Fig. 6. The vanishing nature at
half-filling and the non-vanishing behavior of current when the system is
quarterly filled can be easily understood from the following argument. The
total current in any branch or in the complete system mainly depends on the
contributions coming from the higher occupied energy levels, while the
contributions from the other occupied energy levels cancel with each other.
Therefore, for the quarter-filled band case, the net contribution comes from
the energy levels which are quasi-extended in nature and a non-zero current
appears. On the other hand, for the half-filled band case, the net
contribution arises from the levels those are almost localized, and hence,
nearly vanishing current is obtained. Thus, we can emphasize that the three-
arm ring leads to a possibility of getting high-amplitude to low-amplitude
(almost zero) persistent current simply by tuning the filling factor $N_{e}$
i.e., the Fermi energy $E_{F}$, and, hence the network can be used as a
mesoscopic switch.
The high-conducting to low-conducting switching action with the change of
electron filling $N_{e}$ in our topology can also be very well explained from
the spectrum given in Fig. 8, where we measure the conducting nature by
calculating charge stiffness constant, the so-called Drude weight ($D$), in
accordance with the idea originally put forward by Kohn kohn . The Drude
weight for the system can be easily determined by taking the second order
derivative of the ground state energy for a particular filling with respect to
flux $\phi_{1}$ ($\phi_{1}\rightarrow 0$) threaded by the ring kohn ; san3 ;
san33 . Kohn has shown that $D$ decays exponentially to zero for an insulating
system, while it becomes finite for a conducting system. A nice feature of the
result shown in Fig. 8 is that, the charge stiffness constant almost drops to
zero around the half-filled region which reveals the insulating phase, while
away from this region it ($D$) has a finite non-zero value that indicates a
conducting nature. This feature corroborates the findings presented in Figs. 6
and 7.
To ensure the extended or localized nature of energy eigenstates, finally we
demonstrate the variation of probability amplitude (P.A.) of the eigenstates
as a function site index $n$. The probability amplitude of getting an electron
at any site $n$ for a particular eigenstate $|\psi_{k}\rangle$ is obtained
from the factor $|a_{n}^{k}|^{2}$. Here we analyze the localization behavior
for two different energy eigenstates, viz, $21$-st and $42$-nd states. For the
first one the energy is located well inside a quasi-band, while for the other
the energy is placed at the edge of this band. The results are given in Fig. 9
for a three-arm ring with $84$ atomic sites. The red dashed lines are used to
separate the three distinct regions of the network. In Fig. 9(a) we present
the probability amplitudes of the 21st eigenstates and see that the
probability amplitude becomes finite for any site $n$ which indicates that the
energy eigenstate is quasi-extended. While, for the other state ($42$-nd) the
probability amplitude almost vanishes at every site of the upper and lower
arms of the network. Only at the atomic sites of the middle arm we have finite
probability amplitudes. This state does not contribute anything to the current
and we can refer the state as a localized one.
## IV Conclusion
To summarize, in the present work we have explored the magneto-transport
properties of a $\theta$ shaped three-arm quantum ring in the non-interacting
electron framework. The upper and lower sub-rings of the network are threaded
by magnetic fluxes $\phi_{1}$ and $\phi_{2}$, respectively. We have used a
single-band tight-binding Hamiltonian to illustrate the model quantum system,
where the outer arms are subjected to the binary alloy lattices and the middle
arm has identical lattice sites. In the absence of the middle arm, all the
energy eigenstates are extended, but the inclusion of the middle arm produces
some quasi localized states within the band of extended states and provides a
possibility of getting a high conducting state to the low conducting one upon
the movement of the Fermi energy. Thus, the system can be used a mesoscopic
switch. We have verified the switching action from high- to low-conducting
state and vice versa by investigating the persistent current and charge
stiffness constant in the network for different band fillings. We have
numerically computed both the total current and the individual currents in
separate branches by using second-quantized approach. We hope our present
analysis may be helpful for studying magneto-transport properties in any
complicated quantum network. Finally, we have also examined the nature of the
energy eigenstates in terms of the probability amplitude in different sites of
the geometry.
## References
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|
arxiv-papers
| 2012-02-08T15:08:06 |
2024-09-04T02:49:27.235906
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Srilekha Saha, Santanu K. Maiti and S. N. Karmakar",
"submitter": "Santanu Maiti K.",
"url": "https://arxiv.org/abs/1202.1726"
}
|
1202.1811
|
10.1080/10652460YYxxxxxxx 1476-8291 1065-2469 00 00 2009 January
# Fourier expansions for a logarithmic fundamental solution of the
polyharmonic equation
Howard S. Cohla,b∗
aApplied and Computational Mathematics Division, Information Technology
Laboratory, National Institute of Standards and Technology, Gaithersburg,
Maryland, U.S.A. ;
bDepartment of Mathematics, University of Auckland, Auckland, New Zealand
∗Corresponding author. Email: howard.cohl@nist.gov
( v3.5 released October 2008)
###### Abstract
In even-dimensional Euclidean space for integer powers of the Laplacian
greater than or equal to the dimension divided by two, a fundamental solution
for the polyharmonic equation has logarithmic behavior. We give two approaches
for developing a Fourier expansion of this logarithmic fundamental solution.
The first approach is algebraic and relies upon the construction of two-
parameter polynomials. We describe some of the properties of these
polynomials, and use them to derive the Fourier expansion for a logarithmic
fundamental solution of the polyharmonic equation. The second approach depends
on the computation of parameter derivatives of Fourier series for a power-law
fundamental solution of the polyharmonic equation. The resulting Fourier
series is given in terms of sums over associated Legendre functions of the
first kind. We conclude by comparing the two approaches and giving the
azimuthal Fourier series for a logarithmic fundamental solution of the
polyharmonic equation in rotationally-invariant coordinate systems.
35A08; 31B30; 31C12; 33C05; 42A16
###### keywords:
fundamental solutions; polyharmonic equation; Fourier series; polynomials;
associated Legendre functions
## 1 Introduction
Solutions of the polyharmonic equation (powers of the Laplacian operator) are
ubiquitous in many areas of computational, pure, applied mathematics, physics
and engineering. We concern ourselves, in this paper, with a fundamental
solution of the polyharmonic equation (Laplace, biharmonic, etc.), which by
convolution yields a solution to the inhomogeneous polyharmonic equation.
Solutions to inhomogeneous polyharmonic equations are useful in many physical
applications including those areas related to Poisson’s equation such as
Newtonian gravity, electrostatics, magnetostatics, quantum direct and exchange
interactions (cf. §1 in Cohl & Dominici (2010) [4]), etc. Furthermore,
applications of higher-powers of the Laplacian include such varied areas as
minimal surfaces [13], Continuum Mechanics [9], Mesh deformation [7],
Elasticity [10], Stokes Flow [8], Geometric Design [20], Cubature formulae
[17], mean value theorems (cf. Pizzetti’s formula) [14], and Hartree-Fock
calculations of nuclei [21].
It is a well-known fact (see for instance Schwartz (1950) ([16], p. 45),
Gel′fand & Shilov (1964) ([6], p. 202) that a fundamental solution of the
polyharmonic equation on $d$-dimensional Euclidean space is given by
combinations of power-law and logarithmic functions of the global distance
between two points. In a recent paper (Cohl & Dominici (2010) [4]), we derived
a complex identity which determined the Fourier coefficients of a power-law
fundamental solution of the polyharmonic equation. The Fourier coefficients
were seen to be given in terms of associated Legendre functions. The present
work is concerned with computing the Fourier coefficients of a logarithmic
fundamental solution of the polyharmonic equation. One obtains a logarithmic
fundamental solution for the polyharmonic equation only in even-dimensional
Euclidean space and only when the power of the Laplacian is greater than or
equal to the dimension divided by two. The most familiar example of a
logarithmic fundamental solution of the polyharmonic equation occurs in two-
dimensions, for a single-power of the Laplacian, i.e., Laplace’s equation.
We present two different approaches for obtaining Fourier series of a
logarithmic fundamental solution for the polyharmonic equation. The first
approach is algebraic and involves the generation of a certain set of
naturally arising two-index polynomials which we refer to as logarithmic
polynomials. The second approach starts with the main result from Cohl &
Dominici (2010) [4] and determines the Fourier series expansion for a
logarithmic fundamental solution of the polyharmonic equation through
parameter differentiation. Series expansions for fundamental solutions of
linear partial differential equations such as the polyharmonic equation are
extremely useful in determining Dirichlet boundary values for solutions on
interior domains (see for example Cohl & Tohline (1999) [5]).
This paper is organized as follows. In §2 we introduce the problem. In §3 we
describe our algebraic approach to computing a Fourier series of a logarithmic
fundamental solution of the polyharmonic equation. In §4 we give our limit
derivative approach for computing the Fourier series of a logarithmic
fundamental solution of the polyharmonic equation. In §5 we give some
comparisons between the two approaches. In §6 we use the results presented in
the previous sections to obtain azimuthal Fourier expansions for a logarithmic
fundamental solution of the polyharmonic equation in rotationally-invariant
coordinate systems which parametrize points in $d$-dimensional Euclidean
space. In Appendix 7 we present some necessary formulae relating to
differentiation of associated Legendre functions of the first kind with
respect to the degree. In Appendix 8 we present some of the properties of the
logarithmic polynomials.
Throughout this paper we rely on the following definitions. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, the set of
integers is given by ${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\},$ and the set
${\mathbf{Q}}$ represents the rational numbers. For
$a_{1},a_{2},\ldots\in{\mathbf{C}}$, if $i,j\in{\mathbf{Z}}$ and $j<i$ then
$\sum_{n=i}^{j}a_{n}=0$ and $\prod_{n=i}^{j}a_{n}=1$, where ${\mathbf{C}}$
represents the complex numbers. The set ${\mathbf{R}}$ represents the real
numbers.
## 2 Fundamental solution of the polyharmonic equation and the non-
logarithmic Fourier series
If $\Phi$ satisfies the polyharmonic equation given by
$(-\Delta)^{k}\Phi({\bf x})=0,$ (1)
where ${\bf x}\in{\mathbf{R}}^{d},$ $\Delta:C^{p}({\mathbf{R}}^{d})\to
C^{p-2}({\mathbf{R}}^{d})$, for $p\geq 2$ is the Laplacian operator defined by
$\Delta:=\frac{\partial^{2}}{\partial
x_{1}^{2}}+\ldots+\frac{\partial^{2}}{\partial x_{d}^{2}},$ $k\in{\mathbf{N}}$
and $\Phi\in C^{2k}({\mathbf{R}}^{d}),$ then $\Phi$ is called polyharmonic. If
the power $k$ of the Laplacian equals two, then (1) is called the biharmonic
equation and $\Phi$ is called biharmonic. The inhomogeneous polyharmonic
equation is given by
$(-\Delta)^{k}\Phi({\bf x})=\rho({\bf x}),$ (2)
where we take $\rho$ to be an integrable function so that a solution to (2)
exists. A fundamental solution for the polyharmonic equation on
${\mathbf{R}}^{d}$ is a function
${{\mathfrak{g}}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ which satisfies the
equation
$(-\Delta)^{k}{{\mathfrak{g}}}_{k}^{d}({\bf x},{\bf x}^{\prime})=c\delta({\bf
x}-{\bf x}^{\prime}),$ (3)
for some $c\in{\mathbf{R}},c\neq 0$, where $\delta$ is the Dirac delta
function and ${{\bf x}^{\prime}}\in{\mathbf{R}}^{d}$. When $c=1$, we call a
fundamental solution of the polyharmonic equation normalized, and denote it by
${\mathcal{G}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$. The Euclidean inner
product $(\cdot,\cdot):{\mathbf{R}}^{d}\times{\mathbf{R}}^{d}\to{\mathbf{R}}$
defined by $({\bf x},{{\bf
x}^{\prime}}):=x_{1}x_{1}^{\prime}+\ldots+x_{d}x_{d}^{\prime},$ induces a norm
(the Euclidean norm) $\|\cdot\|:{\mathbf{R}}^{d}\to[0,\infty)$, on the finite-
dimensional vector space ${\mathbf{R}}^{d}$, given by $\|{\bf
x}\|:=\sqrt{({\bf x},{\bf x})}.$ In the rest of this paper, we will use the
gamma function $\Gamma:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$,
which is a natural generalization of the factorial function (see for instance
Chapter 5 in Olver et al. (2010) [15]). A fundamental solution of the
polyharmonic equation is given by the following theorem.
###### Theorem 2.1.
Let $d,k\in{\mathbf{N}}$. Define
${\mathcal{G}}_{k}^{d}({\bf x},{\bf
x}^{\prime})=\left\\{\begin{array}[]{ll}{\displaystyle\frac{(-1)^{k+d/2+1}\
\|{\bf x}-{\bf x}^{\prime}\|^{2k-d}}{(k-1)!\ \left(k-d/2\right)!\
2^{2k-1}\pi^{d/2}}\left(\log\|{\bf x}-{\bf
x}^{\prime}\|-\beta_{k-d/2,d}\right)}\\\\[2.0pt] \hskip
210.55022pt\mathrm{if}\ d\,\,\mathrm{even},\ k\geq d/2,\\\\[10.0pt]
{\displaystyle\frac{\Gamma(d/2-k)\|{\bf x}-{\bf x}^{\prime}\|^{2k-d}}{(k-1)!\
2^{2k}\pi^{d/2}}}\hskip 95.88564pt\mathrm{otherwise},\end{array}\right.$
where $\beta_{p,d}\in{\mathbf{Q}}$ is defined as
$\beta_{p,d}:=\frac{1}{2}\left[H_{p}+H_{d/2+p-1}-H_{d/2-1}\right],$ with
$H_{j}$ being the $j$th harmonic number
$H_{j}:=\sum_{i=1}^{j}\frac{1}{i},$
then ${\mathcal{G}}_{k}^{d}$ is a normalized fundamental solution for
$(-\Delta)^{k}$ on Euclidean space ${\mathbf{R}}^{d}$.
Proof. See Cohl (2010) [3] and Boyling (1996) [2].
A separable rotationally-invariant coordinate system for the polyharmonic
equation (1) on ${\mathbf{R}}^{d}$ is given by
$\left.\begin{array}[]{rcl}x_{1}&=&R(\xi_{1},\ldots,\xi_{d-1})\cos\phi\\\\[2.84544pt]
x_{2}&=&R(\xi_{1},\ldots,\xi_{d-1})\sin\phi\\\\[2.84544pt]
x_{3}&=&x_{3}(\xi_{1},\ldots,\xi_{d-1})\\\\[2.84544pt] &\vdots&\\\\[2.84544pt]
x_{d}&=&x_{d}(\xi_{1},\ldots,\xi_{d-1})\end{array}\quad\right\\},$ (4)
which is described by an angle $\phi\in{\mathbf{R}}$ and $(d-1)$-curvilinear
coordinates $(\xi_{1},\ldots,\xi_{d-1})$. A separable rotationally-invariant
coordinate system transforms the polyharmonic equation into a set of
$d$-uncoupled ordinary differential equations with separation constants
$m\in{\mathbf{Z}}$ and $k_{j}$ for $1\leq j\leq d-2$. For a separable
rotationally-invariant coordinate system, this uncoupling is accomplished, in
general, by assuming a solution to (1) of the form
$\Phi(x)=e^{im\phi}\,{\mathcal{R}}(\xi_{1},\ldots,\xi_{d-1})\prod_{i=1}^{d-1}A_{i}(\xi_{i},m,k_{1},\ldots,k_{d-2}),$
where the domains of the functions ${\mathcal{R}}$ and $A_{i}$, for $1\leq
i\leq d-1$, and the constants $k_{j}$ for $1\leq j\leq d-1$, depend on the
specific rotationally-invariant coordinate system. A rotationally-invariant
coordinate system parametrizes points on the $(d-1)$-dimensional half-
hyperplane given by $\phi=const.$ and $R\geq 0$ using curvilinear coordinates
$(\xi_{1},\ldots,\xi_{d-1})$. (For a general description of the theory of
separation of variables see Miller (1977) [12].) The Euclidean distance
between two points ${\bf x},{{\bf x}^{\prime}}\in{\mathbf{R}}^{d}$, expressed
in a rotationally-invariant coordinate system, is given by
$\displaystyle\|{\bf x}-{{\bf
x}^{\prime}}\|=\sqrt{2RR^{\prime}}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{1/2},$
where the toroidal parameter $\chi>1$, is given by
$\chi:=\frac{R+{R^{\prime}}^{2}+{\displaystyle\sum_{i=3}^{d}(x_{i}-x_{i}^{\prime})^{2}}}{\displaystyle
2RR^{\prime}},$ (5)
where $R,R^{\prime}\in[0,\infty)$ are defined in (4) for ${\bf x},{{\bf
x}^{\prime}}\in{\mathbf{R}}^{d}$. The hypersurfaces given by $\chi>1$ equals
constant are independent of coordinate system and represent hyper-tori of
revolution.
From Theorem 2.1 we see that, apart from multiplicative constants, the
algebraic expression
${\mathfrak{l}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ of an unnormalized
fundamental solution for the polyharmonic equation in Euclidean space
${\mathbf{R}}^{d}$ for $d$ even, $k\geq d/2,$ is given by
${\mathfrak{l}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}\left(\log\|{\bf x}-{{\bf
x}^{\prime}}\|-\beta_{k-d/2,d}\right).$ (6)
By expressing ${\mathfrak{l}}_{k}^{d}$ in a rotationally-invariant coordinate
system (4) we obtain
$\displaystyle{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf x}^{\prime}})$
$\displaystyle=$
$\displaystyle\left(2RR^{\prime}\right)^{p}\left[\frac{1}{2}\log\left(2RR^{\prime}\right)-\beta_{p,d}\right]\left[\chi-\cos(\phi-\phi^{\prime})\right]^{p}$
(7)
$\displaystyle+\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{p}\log\left[\chi-\cos(\phi-\phi^{\prime})\right],$
where $p=k-d/2\in{\mathbf{N}}_{0}$. For the polyharmonic equation in even-
dimensional Euclidean space ${\mathbf{R}}^{d}$ with $1\leq k\leq d/2-1,$ apart
from multiplicative constants, the algebraic expression for an unnormalized
fundamental solution of the polyharmonic equation
${\mathfrak{h}}_{k}^{d}:({\mathbf{R}}^{d}\times{\mathbf{R}}^{d})\setminus\\{({\bf
x},{\bf x}):{\bf x}\in{\mathbf{R}}^{d}\\}\to{\mathbf{R}}$ is given by
${\mathfrak{h}}_{k}^{d}({\bf x},{{\bf x}^{\prime}}):=\|{\bf x}-{{\bf
x}^{\prime}}\|^{2k-d}.$
By expressing ${\mathfrak{h}}_{k}^{d}$ in a rotationally-invariant coordinate
system we obtain
${\mathfrak{h}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})=\left(2RR^{\prime}\right)^{-q}\left[\chi-\cos(\phi-\phi^{\prime})\right]^{-q},$
(8)
where $q=2k-d$.
By examining (7) and (8), we see that for computation of Fourier expansions
about the azimuthal separation angle $(\phi-\phi^{\prime})$ of
${\mathfrak{l}}_{k}^{d}$ and ${\mathfrak{h}}_{k}^{d}$, all that is required is
to compute the Fourier cosine series for the following three functions
$f_{\chi},h_{\chi}:{\mathbf{R}}\to(0,\infty)$ and
$g_{\chi}:{\mathbf{R}}\to{\mathbf{R}}$ defined as
$\displaystyle f_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{p},$ $\displaystyle
g_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{p}\log\left(\chi-\cos\psi\right),\qquad\mbox{and}$
$\displaystyle h_{\chi}(\psi):=\left(\chi-\cos\psi\right)^{-q},$
where $p\in{\mathbf{N}}_{0}$, $q\in{\mathbf{N}}$ and $\chi>1$ is a fixed
parameter.
The Fourier series of $f_{\chi}$ is given in Cohl & Dominici (2010) [4] (cf.
(4.4) therein), namely
$(z-\cos\psi)^{p}=(z^{2}-1)^{p/2}\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(9)
where the Neumann factor $\epsilon_{n}=2-\delta_{n,0}$ commonly occurs in
Fourier series, $\delta_{n,0}$ is the Kronecker delta, and
$(z)_{n}:=\prod_{i=1}^{n}(z+i-1),$
for $z\in{\mathbf{C}}$ and $n\in{\mathbf{N}}_{0}$, is the Pochhammer symbol
(rising factorial). We have used Whipple’s formula in (9) (see for instance,
(8.2.7) in Abramowitz & Stegun (1972) [1]) to convert the associated Legendre
function of the second kind $Q_{\nu}^{\mu}:(1,\infty)\to{\mathbf{C}}$
appearing in [4] to the associated Legendre function of the first kind
$P_{\nu}^{\mu}:(1,\infty)\to{\mathbf{R}}$. The associated Legendre function of
the first kind can be defined using the Gauss hypergeometric function, namely
(Magnus, Oberhettinger & Soni (1966) [11], p. 153)
$P_{\nu}^{\mu}(z):=\frac{1}{\Gamma(1-\mu)}\left(\frac{z+1}{z-1}\right)^{\mu/2}{}_{2}F_{1}\left(-\nu,\nu+1;1-\mu;\frac{1-z}{2}\right).$
The Gauss hypergeometric function
${}_{2}F_{1}:{\mathbf{C}}\times{\mathbf{C}}\times({\mathbf{C}}\setminus-{\mathbf{N}}_{0})\times\\{z\in{\mathbf{C}}:|z|<1\\}\to{\mathbf{C}}$
can be defined in terms of the following infinite series
${}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{n!(c)_{n}}z^{n}$
(see for instance Chapter 15 in Olver et al. (2010) [15]).
The Fourier series of $h_{\chi}$ is given in Cohl & Dominici (2010) [4]
(Whipple formula (8.2.7) in Abramowitz & Stegun (1972) [1] and cf. (4.5)
therein), namely
$\frac{1}{(z-\cos\psi)^{q}}=\frac{(z^{2}-1)^{-q/2}}{(q-1)!}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)(n+q-1)!P_{q-1}^{-n}\left(\frac{z}{\sqrt{z^{2}-1}}\right),$
(10)
where $q\in{\mathbf{N}}$. Since the Fourier series of $h_{\chi}$ is computed
in Cohl & Dominici (2010) [4], we understand how to compute Fourier expansions
of ${\mathfrak{h}}_{k}^{d}$ (8) in separable rotationally-invariant coordinate
systems. In order to compute Fourier expansion of ${\mathfrak{l}}_{k}^{d}$ (7)
in separable rotationally-invariant coordinate systems, all that remains is to
determine the Fourier series of $g_{\chi}$. This is the goal of the next two
sections.
## 3 Algebraic approach to the logarithmic Fourier series
Since $\chi>1$, one may make the substitution $\chi=\cosh\eta$ to evaluate the
Fourier series of $g_{\chi}$. For instance, it is given in the form of
$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi),$ where
$p\in{\mathbf{N}}_{0}$. For $p=0$ the result is well-known (see for instance
Magnus, Oberhettinger & Soni (1966) [11], p. 259)
$\log(\cosh\eta-\cos\psi)=\eta-\log{2}-2\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi),$
(11)
which as we will see, should be compared with (10) for $q=1$, namely
$\displaystyle\frac{1}{\cosh\eta-\cos\psi}=\frac{1}{\sinh\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}.$
(12)
Note that for $\eta>0$ we may write $e^{\eta}$ and therefore $\eta$ as a
function of $\cosh\eta$ since $\sinh\eta=\sqrt{\cosh^{2}\eta-1},$
$e^{\eta}=\cosh\eta+\sqrt{\cosh^{2}\eta-1},$ and therefore
$\eta=\log\left(\cosh\eta+\sqrt{\cosh^{2}\eta-1}\right).$ Now examine the
$p=1$ case for $g_{\chi}$. If we multiply both sides of (11) by
$(\cosh\eta-\cos\psi)$ and take advantage of the formula
$\cos(n\psi)\cos\psi=\frac{1}{2}\Bigl{\\{}\cos[(n+1)\psi]+\cos[(n-1)\psi]\Bigr{\\}},$
(13)
then we have
$\displaystyle(\cosh\eta-\cos\psi)\log(\cosh\eta-\cos\psi)$ $\displaystyle=$
$\displaystyle(\eta-\log 2)\cosh\eta$ (14) $\displaystyle{}-(\eta-\log
2)\cos\psi-2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi)$
$\displaystyle{}+\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+1)\psi]+\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-1)\psi].$
Collecting the contributions to the Fourier cosine series, we obtain
$\displaystyle(\cosh\eta-\cos\psi)\log(\cosh\eta-\cos\psi)$ $\displaystyle=$
$\displaystyle(1+\eta-\log 2)\cosh\eta$ (15)
$\displaystyle{}-\sinh\eta+\cos\psi\left(\log
2-1-\eta-\frac{1}{2}e^{-2\eta}\right)$
$\displaystyle{}+2\sum_{n=2}^{\infty}\frac{e^{-n\eta}\cos
n\psi}{n(n^{2}-1)}(\cosh\eta+n\sinh\eta).$
If we compare (15) with (10) for $q=1$, namely
$\displaystyle\frac{1}{(\cosh\eta-\cos\psi)^{2}}=\frac{1}{\sinh^{3}\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}(\cosh\eta+n\sinh\eta),$
(16)
we notice that the factor $(\cosh\eta+n\sinh\eta)$ appears in both series.
For $p=2$ in $g_{\chi}$, we use (13) and similarly have
$\displaystyle(\cosh\eta-\cos\psi)^{2}\log(\cosh\eta-\cos\psi)$
$\displaystyle{}=(\eta-\log 2)\cosh^{2}\eta-2(\eta-\log 2)\cosh\eta\cos\psi$
$\displaystyle{}+(\eta-\log
2)\cos^{2}\psi-(2\cosh^{2}\eta+1)\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos(n\psi)$
$\displaystyle{}+2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+1)\psi]+2\cosh\eta\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-1)\psi]{}$
$\displaystyle{}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+2)\psi]-\frac{1}{2}\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n-2)\psi].$
If we collect the contributions of the Fourier cosine series, we obtain
$\displaystyle(\cosh\eta-\cos\psi)^{2}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log
2)\left(\cosh^{2}\eta+\frac{1}{2}\right)$ (17)
$\displaystyle{}+2\cosh\eta\,e^{-\eta}-\frac{1}{4}e^{-2\eta}+\Biggl{[}-2(\eta-\log
2)\cosh\eta-\left(2\cosh^{2}\eta+\frac{3}{2}\right)e^{-\eta}$
$\displaystyle{}+\cosh\eta\,e^{-2\eta}-\frac{1}{6}e^{-3\eta}\Biggr{]}\cos\psi+\Biggl{[}\frac{1}{2}(\eta-\log
2)+2\cosh\eta\,e^{-\eta}$
$\displaystyle{}-\frac{1}{2}(2\cosh^{2}\eta+1)e^{-2\eta}+\frac{2}{3}\cosh\eta\,e^{-3\eta}-\frac{1}{8}e^{-4\eta}\Biggr{]}\cos
2\psi$
$\displaystyle{}-4\sum_{n=3}^{\infty}\frac{e^{-n\eta}\cos(n\psi)}{n(n^{2}-1)(n^{2}-4)}\left[(n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right].$
By comparing (17) with (10) for $q=2$, namely
$\displaystyle\displaystyle\frac{1}{(\cosh\eta-\cos\psi)^{3}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\sinh^{5}\eta}$ (18)
$\displaystyle{}\times\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}\left[(n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right],$
then we notice that the factor
$\left((n^{2}-1)\sinh^{2}\eta+3n\sinh\eta\cosh\eta+3\cosh^{2}\eta\right)$
appears in both series. We will demonstrate in §5, why the identification
mentioned in (16) and (18) occurs.
This algebraic approach for determining the Fourier series of $g_{\chi}$ will
now be generalized. By starting with (11) and repeatedly multiplying by
factors of $(\cosh\eta-\cos\psi)$, we see that the general Fourier series of
$g_{\chi}$ can be given in terms of a sequence of polynomials
$R_{p}^{k}:(1,\infty)\to{\mathbf{R}}$, with $p\in{\mathbf{N}}_{0}$ and
$k\in{\mathbf{Z}}$, as
$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (19)
$\displaystyle{}+2\sum_{k=-p}^{p}(-1)^{k+1}R_{p}^{k}(\cosh\eta)\sum_{n=1}^{\infty}\frac{e^{-n\eta}}{n}\cos[(n+k)\psi].$
We will refer to $R_{p}^{k}$ as logarithmic polynomials with argument
$x=\cosh\eta$ (in our notation $p$ and $k$ are both indices) (See Appendix 8
for a description of some of the properties of the logarithmic polynomials).
The double sum in (19) is simplified by making the replacement $n+k\mapsto n$.
It then follows that the resulting double sum naturally breaks into two
disjoint regions, one triangular
${\mathcal{A}}:=\\{(k,n):-p\leq k\leq p-1,\ k+1\leq n\leq p\\},$
with $p(2p+1)$ terms and the other infinite rectangular
${\mathcal{B}}:=\\{(k,n):-p\leq k\leq p,\ p+1<n<\infty\\}.$
By rearranging the order of the $k$ and $n$ summations in (19), we derive
$\displaystyle\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (20)
$\displaystyle\displaystyle+\sum_{n=0}^{p}\cos(n\psi)e^{-n\eta}\,\mathfrak{r}_{n,p}^{-p,n-1}(\cosh\eta)+\sum_{n=1}^{p-1}\cos(n\psi)e^{n\eta}\,\mathfrak{r}_{-n,p}^{-p,-n-1}(\cosh\eta)$
$\displaystyle\displaystyle+\sum_{n=p+1}^{\infty}\frac{\cos(n\psi)e^{-n\eta}}{n(n^{2}-1)\cdots(n^{2}-p^{2})}\,\Re_{n,p}(\cosh\eta),$
where $\mathfrak{r}_{n,p}^{k_{1},k_{2}},\Re_{n,p}:(1,\infty)\to{\mathbf{R}}$
are defined as
$\mathfrak{r}_{n,p}^{k_{1},k_{2}}(\cosh\eta):=2\sum_{k=k_{1}}^{k_{2}}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k},$
and
$\Re_{n,p}(\cosh\eta):=\frac{(n+p)!}{(n-p-1)!}\mathfrak{r}_{n,p}^{-p,p}(\cosh\eta),\quad(n\geq
p+1),$
respectively. We can also write the Fourier series directly in terms of the
logarithmic polynomials $R_{p}^{k}$ as follows
$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (21)
$\displaystyle+2\sum_{n=0}^{p}\cos(n\psi)e^{-n\eta}\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}$
$\displaystyle-2\sum_{n=1}^{p-1}\cos(n\psi)e^{n\eta}\sum_{k=-p}^{-n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n+k}$
$\displaystyle+2\sum_{n=p+1}^{\infty}\cos(n\psi)e^{-n\eta}\sum_{k=-p}^{p}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}.$
Note that by using (9), then we can express $(\cosh\eta-\cos\psi)^{p}$ as a
Fourier series, namely
$(\cosh\eta-\cos\psi)^{p}=\sinh^{p}\eta\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}(\coth\eta),$
(22)
where $p\in{\mathbf{N}}_{0}$.
If we define the function $\mathfrak{P}_{n,p}:(1,\infty)\to{\mathbf{R}}$ such
that
$\displaystyle\mathfrak{P}_{n,p}(\cosh\eta):=\left\\{\begin{array}[]{ll}\displaystyle\mathfrak{r}_{0,p}^{-p,-1}(\cosh\eta)\hskip
130.25679pt\mathrm{if}\ n=0,\\\\[5.69046pt] \displaystyle
D_{n,p}(\cosh\eta)+E_{n,p}(\cosh\eta)\hskip 62.3116pt\mathrm{if}\ 1\leq n\leq
p-1,\\\\[5.69046pt] \displaystyle
e^{-p\eta}\mathfrak{r}_{p,p}^{-p,p-1}(\cosh\eta)\hskip 104.70593pt\mathrm{if}\
1\leq n=p,\\\\[5.69046pt]
\displaystyle\frac{e^{-n\eta}}{(n^{2}-p^{2})\cdots(n^{2}-1)n}\,\Re_{n,p}(\cosh\eta)\hskip
21.33955pt\mathrm{if}\ n\geq p+1,\end{array}\right.$ (24)
where $D_{n,p},E_{n,p}:(1,\infty)\to{\mathbf{R}}$ are defined as
$D_{n,p}(\cosh\eta)=\left\\{\begin{array}[]{ll}\displaystyle
e^{n\eta}\mathfrak{r}_{-n,p}^{-p,-n-1}(\cosh\eta)&\qquad\mathrm{if}\ p\geq
2,\\\\[5.69046pt] \displaystyle 0&\qquad\mathrm{if}\
p\in\\{0,1\\},\end{array}\right.$
and
$E_{n,p}(\cosh\eta)=\left\\{\begin{array}[]{ll}\displaystyle
e^{-n\eta}\mathfrak{r}_{n,p}^{-p,n-1}(\cosh\eta)&\qquad\mathrm{if}\ p\geq
1,\\\\[5.69046pt] \displaystyle 0&\qquad\mathrm{if}\ p=0,\end{array}\right.$
respectively, then we can write (20) as
$\displaystyle\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$
$\displaystyle+\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{P}_{n,p}(\cosh\eta).$
In fact, if we use (22), then we can express $g_{\chi}$ as
$\displaystyle\hskip
5.69046pt\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{P}_{n,p}(\cosh\eta)$
$\displaystyle{}+(\eta-\log
2)\sinh^{p}\eta\sum_{n=0}^{p}\epsilon_{n}\cos(n\psi)\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}(\coth\eta).$
Furthermore, if we define $\mathfrak{Q}_{n,p}:(1,\infty)\to{\mathbf{R}}$ as
$\mathfrak{Q}_{n,p}(\cosh\eta):=\mathfrak{P}_{n,p}(\cosh\eta)+\frac{\epsilon_{n}(-p)_{n}(p-n)!}{(p+n)!}(\eta-\log
2)\sinh^{p}\eta P_{p}^{n}(\coth\eta),$
then we can express $g_{\chi}$ as
$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)=\sum_{n=0}^{\infty}\cos(n\psi)\mathfrak{Q}_{n,p}(\cosh\eta).$
(25)
## 4 Limit derivative approach to the logarithmic Fourier series
We now use a second approach to compute the Fourier series for a logarithmic
fundamental solution of the polyharmonic equation (6). We would like to match
our results to the computations in §3, which clearly demonstrate different
behaviors for the two regimes, $0\leq n\leq p$ and $n\geq p+1$. By applying
the identity
$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)=\lim_{\nu\rightarrow
0}\frac{\partial}{\partial\nu}(\cosh\eta-\cos\psi)^{\nu+p},$ (26)
where $p\in{\mathbf{N}}_{0}$, to
$\displaystyle(\cosh\eta-\cos\psi)^{\nu}$ $\displaystyle=$
$\displaystyle\sinh^{\nu}\eta\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(\nu+1)_{n}}P_{\nu}^{n}(\coth\eta),$
(27)
where $\nu\in{\mathbf{C}}\setminus-{\mathbf{N}}$ (cf. (3.11b) in Cohl &
Dominici (2010) [4]), one can compute the Fourier cosine series of
$(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$, provided availability of
the necessary parameter derivatives.
Applying (26) to (27), we obtain
$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\sinh^{\nu+p}\eta\right]\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+1)_{n}}P_{p}^{n}(\coth\eta)$
$\displaystyle+\sinh^{p}\eta\sum_{n=0}^{\infty}(-1)^{n}\epsilon_{n}\cos(n\psi)\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{1}{(\nu+p+1)_{n}}\right]P_{p}^{n}(\coth\eta)$
$\displaystyle+\sinh^{p}\eta\sum_{n=0}^{\infty}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+1)_{n}}\left[\lim_{\nu\to
0}\frac{\partial}{\partial\nu}P_{\nu+p}^{n}(\coth\eta)\right].$
Note that for $p\in{\mathbf{Z}}$ and $n\in{\mathbf{N}}_{0}$, the associated
Legendre function of the first kind $P_{p}^{n}$ vanishes if $n\geq p+1$. This
is easily seen using the Rodrigues-type formula (cf. (14.7.11) in Olver et al.
(2010) [15])
$P_{p}^{n}(z)=(z^{2}-1)^{n/2}\frac{d^{n}P_{p}(z)}{dz^{n}},$
and the fact that $P_{p}(z)$ is a polynomial in $z$ of degree $p$. The
derivatives are given as follows:
$\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\sinh^{\nu+p}\eta=\sinh^{p}\eta\log\sinh\eta,$
(28) $\lim_{\nu\to
0}\frac{\partial}{\partial\nu}\frac{1}{(\nu+p+1)_{n}}=\frac{p!}{(p+n)!}\left[\psi(p+1)-\psi(p+1+n)\right],$
(29)
and
$\lim_{\nu\to
0}\frac{\partial}{\partial\nu}P_{\nu+p}^{n}(\coth\eta)=\left[\frac{\partial}{\partial\nu}P_{\nu}^{n}(\coth\eta)\right]_{\nu=p},$
(30)
where $\psi:{\mathbf{C}}\setminus-{\mathbf{N}}_{0}\to{\mathbf{C}}$ is the
digamma function defined in terms of the derivative of the gamma function
$\frac{d}{dz}\Gamma(z)=:\psi(z)\Gamma(z)$
(see for instance (5.2.2) in Olver et al. (2010) [15]). The degree-derivative
of the associated Legendre function of the first kind in (30) is determined
using (34) and (35). By collecting terms and using (28), (29), and (30), we
obtain
$\displaystyle(\cosh\eta-\cos\psi)^{p}\log(\cosh\eta-\cos\psi)$
$\displaystyle=$ $\displaystyle(\eta-\log 2)(\cosh\eta-\cos\psi)^{p}$ (31)
$\displaystyle+p!\sinh^{p}\eta\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}\cos(n\psi)}{(p+n)!}$
$\displaystyle\times\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}(\coth\eta)$
$\displaystyle+(-1)^{p}p!\sinh^{p}\eta\sum_{n=0}^{p-1}\frac{\epsilon_{n}\cos(n\psi)}{(p+n)!}$
$\displaystyle\times\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}(\coth\eta)$
$\displaystyle+2(-1)^{p}p!\sinh^{p}\eta\sum_{n=1}^{p}\frac{(-1)^{n}\cos(n\psi)}{(p-n)!}$
$\displaystyle\times\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}(\coth\eta)$
$\displaystyle+2(-1)^{p+1}p!\sinh^{p}\eta\sum_{n=p+1}^{\infty}\cos(n\psi)(n-p-1)!P_{p}^{-n}(\coth\eta).$
## 5 Comparison of the two approaches
The limit derivative approach presented in §4 might be considered, of the two
methods, preferred for computing the azimuthal Fourier series for a
logarithmic fundamental solution of the polyharmonic equation. This is because
it produces azimuthal Fourier coefficients in terms of the well-known special
functions, associated Legendre functions. On the other hand, the algebraic
approach presented in §3 produces results in terms of the two-parameter
logarithmic polynomials $R_{p}^{k}(x)$. As far as the author is aware, these
polynomials are previously unencountered in the literature. By comparison of
the two approaches we see how the logarithmic polynomials $R_{p}^{k}(x)$
(potentially a new type of special function) are intimately related to the
associated Legendre functions. In this section we make this comparison
concrete. We should also mention that the following comparison equations
resolve to become quite complicated as $p$ increases, and they have been
checked for $0\leq p\leq 10$ using Mathematica with the assistance of an
algorithm generated using (37) and (38) from Appendix 8.
By equating the Fourier coefficients using the two approaches we can obtain
summation formulae which are satisfied by the logarithmic polynomials. For
$n=0$ and $p\geq 1$ we have
$\displaystyle\sum_{k=1}^{p}\frac{(-1)^{k+1}e^{-k\eta}R_{p}^{k}(\cosh\eta)}{k}$
$\displaystyle{}=\sinh^{p}\eta\left[\psi(2p+1)-\psi(p+1)\right]P_{p}(\coth\eta)$
$\displaystyle{}+(-1)^{p}\sinh^{p}\eta\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}(\coth\eta),$
for $1\leq n\leq p-1$, $p\geq 2$ we derive
$\displaystyle\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}+e^{2n\eta}\sum_{k=-p}^{-n-1}\frac{(-1)^{k}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n+k}$
$\displaystyle=$ $\displaystyle p!e^{n\eta}\sinh^{p}\eta$
$\displaystyle{}\times\Biggl{\\{}\frac{(-1)^{n}}{(p+n)!}\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}(\coth\eta)$
$\displaystyle{}+\frac{(-1)^{p}}{(p+n)!}\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}(\coth\eta)$
$\displaystyle{}+\frac{(-1)^{p+n}}{(p-n)!}\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}(\coth\eta)\Biggr{\\}},$
for $n=p$, $p\geq 1$ we obtain
$\displaystyle\sum_{k=-p}^{p-1}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{p-k}$
$\displaystyle=$ $\displaystyle p!e^{p\eta}\sinh^{p}\eta$
$\displaystyle{}\times\Biggl{\\{}\frac{(-1)^{p}}{(2p)!}\left[\psi(2p+1)-\psi(1)\right]P_{p}^{p}(\coth\eta)$
$\displaystyle{}+\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-p}(\coth\eta)\Biggr{\\}},$
and for $n\geq p+1$, $p\geq 0,$ we see that
$\displaystyle\sum_{k=-p}^{p}\frac{(-1)^{k+1}e^{k\eta}R_{p}^{k}(\cosh\eta)}{n-k}$
(32) $\displaystyle=(-1)^{p+1}p!(n-p-1)!e^{n\eta}\sinh^{p}\eta
P_{p}^{-n}(\coth\eta).$
We now have closed-form expressions for the finite terms given by (20), (25)
and (20) in terms of associated Legendre functions of the first kind. We also
have a proof of the correspondence for the “ending” function $\Re_{n,q-1}$
mentioned in §3. Through (32), the function $\Re_{n,p}$ (cf. (15), (16), (17),
(18), and (20)) is directly related to the associated Legendre function of the
first kind, namely
$\Re_{n,p}(\cosh\eta)=2(-1)^{p+1}p!(p+n)!e^{n\eta}\sinh^{p}\eta
P_{p}^{-n}(\coth\eta).$
Therefore through (10) we have
$\frac{1}{(\cosh\eta-\cos\psi)^{q}}=\frac{(-1)^{q}}{2[(q-1)!]^{2}\sinh^{2q-1}\eta}\sum_{n=0}^{\infty}\epsilon_{n}\cos(n\psi)e^{-n\eta}\Re_{n,q-1}(\cosh\eta),$
which demonstrates the correspondences which was mentioned near (12), (16),
and (18) for (20) and (31).
## 6 Fourier expansion for a logarithmic fundamental solution of the
polyharmonic equation
Now that we have computed the Fourier series for $g_{\chi}$, namely (20) (cf.
(25)) and (31), we are in a position to compute the azimuthal Fourier series
for a logarithmic fundamental solution of the polyharmonic equation (6).
For instance, using (7), (22) and (31), we have
$\displaystyle\hskip 2.84544pt{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\log\left(RR^{\prime}\right)+\eta-\beta_{p,d}\right](\chi^{2}-1)^{p/2}$
$\displaystyle\times\sum_{n=0}^{p}\epsilon_{n}\cos[n(\phi-\phi^{\prime})]\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+\frac{1}{2}(2RR^{\prime})^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=0}^{p}\frac{(-1)^{n}\epsilon_{n}\cos[n(\phi-\phi^{\prime})]}{(p+n)!}$
$\displaystyle\times\left[2\psi(2p+1)-\psi(p+1+n)-\psi(p+1-n)\right]P_{p}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+\frac{1}{2}(2RR^{\prime})^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=0}^{p-1}\frac{\epsilon_{n}\cos[n(\phi-\phi^{\prime})]}{(p+n)!}$
$\displaystyle\times\sum_{k=0}^{p-n-1}\frac{(-1)^{k}(2n+2k+1)\left[1+\frac{k!(p+n)!}{(2n+k)!(p-n)!}\right]}{(p-n-k)(p+n+k+1)}P_{n+k}^{n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+(2RR^{\prime})^{p}(-1)^{p}p!(\chi^{2}-1)^{p/2}\sum_{n=1}^{p}\frac{(-1)^{n}\cos[n(\phi-\phi^{\prime})]}{(p-n)!}$
$\displaystyle\times\sum_{k=0}^{n-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}^{-n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle+(2RR^{\prime})^{p}(-1)^{p+1}p!(\chi^{2}-1)^{p/2}$
$\displaystyle\times\sum_{n=p+1}^{\infty}\cos[n(\phi-\phi^{\prime})](n-p-1)!P_{p}^{-n}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}.$
Alternatively, using (7), (9), and (21) we obtain
$\displaystyle\hskip 14.22636pt{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(2RR^{\prime}\right)^{p}\left[\log\left(RR^{\prime}\right)+\eta-\beta_{p,d}\right](\chi^{2}-1)^{p/2}$
$\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{p}\epsilon_{n}\cos[n(\phi-\phi^{\prime}]\frac{(-p)_{n}(p-n)!}{(p+n)!}P_{p}^{n}\left(\frac{\chi}{\sqrt{\chi^{2}-1}}\right),$
$\displaystyle+(2RR^{\prime})^{p}\sum_{n=0}^{p}\cos[n(\phi-\phi^{\prime})]\left(\chi-\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{n-1}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n-k}$
$\displaystyle-(2RR^{\prime})^{p}\sum_{n=1}^{p-1}\cos[n(\phi-\phi^{\prime})]\left(\chi+\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{-n-1}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n+k}$
$\displaystyle+(2RR^{\prime})^{p}\sum_{n=p+1}^{\infty}\cos[n(\phi-\phi^{\prime})]\left(\chi-\sqrt{\chi^{2}-1}\right)^{n}$
$\displaystyle\hskip
56.9055pt\times\sum_{k=-p}^{p}\frac{(-1)^{k+1}\left(\chi+\sqrt{\chi^{2}-1}\right)^{k}R_{p}^{k}(\chi)}{n-k}.$
Using these formulae, we can, for instance, obtain the axisymmetric component
of a logarithmic fundamental solution of the polyharmonic equation, namely
$\displaystyle\left.{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})\right|_{n=0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}$
$\displaystyle{}\times\left[\log(RR^{\prime})+\eta-\beta_{p,d}+2\psi(2p+1)-2\psi(p+1)\right]P_{p}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle{}+(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}\sum_{k=0}^{p-1}\frac{(-1)^{k}(2k+1)}{(p-k)(p+k+1)}P_{k}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)},$
or
$\displaystyle\left.{\mathfrak{l}}_{k}^{d}({\bf x},{{\bf
x}^{\prime}})\right|_{n=0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(2RR^{\prime})^{p}(\chi^{2}-1)^{p/2}\left[\log(RR^{\prime})+\eta-\beta_{p,d}\right]P_{p}\Biggl{(}\frac{\chi}{\sqrt{\chi^{2}-1}}\Biggr{)}$
$\displaystyle{}\hskip
51.21504pt+(2RR^{\prime})^{p}\sum_{k=1}^{p}\frac{(-1)^{k+1}(\chi-\sqrt{\chi^{2}-1})^{k}R_{p}^{k}(\chi)}{k}.$
The above expressions for the axisymmetric component of a fundamental solution
of the polyharmonic equation is one type of expression sought after in Tsai,
Chen & Hsu (2009) [19].
## 7 Derivatives with respect to the degree of certain integer-order
associated Legendre functions of the first kind
The derivative with respect to its degree for the associated Legendre function
of the first kind evaluated at the zero degree is given in §4.4.3 of Magnus,
Oberhettinger & Soni (1966) [11] as
$\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=0}=\frac{z-1}{2}{}_{2}F_{1}\left(1,1;2;\frac{1-z}{2}\right).$
(33)
An important generalization of this formula has recently been derived in
Szmytkowski (2011) [18]. The degree-derivative of the associated Legendre
function of the first kind for $p,m\in{\mathbf{N}}_{0}$ and $0\leq m\leq p$
(cf. (5.12) in Szmytkowski (2011) [18]) is given by
$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle P_{p}^{m}(z)\log\frac{z+1}{2}$ (34)
$\displaystyle{}+\left[2\psi(2p+1)-\psi(p+1)-\psi(p-m+1)\right]P_{p}^{m}(z)$
$\displaystyle{}+(-1)^{p+m}\sum_{k=0}^{p-m-1}(-1)^{k}\frac{(2k+2m+1)\left[1+\frac{k!(p+m)!}{(k+2m)!(p-m)!}\right]}{(p-m-k)(p+m+k+1)}P_{k+m}^{m}(z)$
$\displaystyle{}+(-1)^{p}\frac{(p+m)!}{(p-m)!}\sum_{k=0}^{m-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}^{-m}(z),$
and for $m\geq p+1$ (cf. (5.16) in Szmytkowski (2011) [18]) there is
$\left[\frac{\partial}{\partial\nu}P_{\nu}^{m}(z)\right]_{\nu=p}=(-1)^{p+n+1}(p+n)!(n-p-1)!P_{p}^{-n}(z).$
(35)
Some special cases of (34) include for $m=0$
$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle
P_{p}(z)\log\frac{z+1}{2}+2\left[\psi(2p+1)-\psi(p+1)\right]P_{p}(z)$
$\displaystyle{}+2(-1)^{p}\sum_{k=0}^{p-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}(z),$
for $m=p$
$\displaystyle\left[\frac{\partial}{\partial\nu}P_{\nu}^{p}(z)\right]_{\nu=p}$
$\displaystyle=$ $\displaystyle
P_{p}^{p}(z)\log\frac{z+1}{2}+\left[2\psi(2p+1)-\psi(p+1)+\gamma\right]P_{p}^{p}(z)$
$\displaystyle{}+(-1)^{p}(2p)!\sum_{k=0}^{p-1}(-1)^{k}\frac{2k+1}{(p-k)(p+k+1)}P_{k}^{-p}(z),$
where $\gamma=-\psi(1)$ is Euler’s constant $\gamma\approx 0.57721566490$. Of
course we also have for $m=p=0$
$\left[\frac{\partial}{\partial\nu}P_{\nu}(z)\right]_{\nu=0}=\log\frac{z+1}{2},$
which exactly matches (33).
## 8 The logarithmic polynomials
The logarithmic polynomials $R_{p}^{k}$ are nonvanishing only for $-p\leq
k\leq p$ and by construction, they are satisfied by the following recurrence
relation
$R_{p}^{k}(x)=\frac{1}{2}R_{p-1}^{k-1}(x)+xR_{p-1}^{k}(x)+\frac{1}{2}R_{p-1}^{k+1}(x).$
(36)
From (11) we have that $R_{0}^{0}(x)=1$. This gives us the starting point for
the recursion. We conjecture that the derivative of the logarithmic
polynomials $R_{p}^{k}$ is given by
$\frac{d}{dx}R_{p}^{\pm k}(x)=pR_{p-1}^{\pm k}(x).$
It is evident by construction that these polynomials are even in the index
$k$, i.e.,
$R_{p}^{k}(x)=R_{p}^{-k}(x).$
Some of the first few logarithmic polynomials are given by
$\displaystyle R_{0}^{0}(x)=1,$ $\displaystyle R_{1}^{0}(x)=x,\ R_{1}^{\pm
1}(x)=\frac{1}{2}$ $\displaystyle R_{2}^{0}(x)=\frac{1}{2}+x^{2},\ R_{2}^{\pm
1}(x)=x,\ R_{2}^{\pm 2}(x)=\frac{1}{4},$ $\displaystyle
R_{3}^{0}(x)=\frac{3}{2}x+x^{3},\ R_{3}^{\pm
1}(x)=\frac{3}{8}+\frac{3}{2}x^{2},\ R_{3}^{\pm 2}(x)=\frac{3}{4}x,\
R_{3}^{\pm 3}(x)=\frac{1}{8}.$
We can find the generating function for the logarithmic polynomials as
follows. Let
$F(x,y,z)=\sum_{p=0}^{\infty}\sum_{k=-\infty}^{\infty}R_{p}^{k}(x)y^{k}z^{p}$
be the generating function for the logarithmic polynomials $R_{p}^{k}$. If we
define the function
$S_{p}(x,y)=\sum_{k=-\infty}^{\infty}R_{p}^{k}(x)y^{k},$
then using the recurrence relation for $R_{p}^{k}$ (36) we can show
$S_{p}(x,y)={\displaystyle\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)}S_{p-1}(x,y).$
Combining this result along with the fact that $R_{0}^{0}(x)=1$, we have
$S_{p}(x,y)={\displaystyle\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)^{p}},$
so therefore the generating function for the logarithmic polynomials
$R_{p}^{k}$ is given by
$F(x,y,z)=\frac{1}{\displaystyle
1-z\left(x+\frac{1}{2}\left(y+\frac{1}{y}\right)\right)}.$
An algorithm for generating the logarithmic polynomials can be obtained by
solving the following set of difference equations
$\left.\begin{array}[]{rcl}a_{0}(p)&=&\frac{1}{2}a_{0}(p-1)\\\\[8.5359pt]
a_{1}(p)&=&\frac{1}{2}a_{1}(p-1)+xa_{0}(p-1)\\\\[8.5359pt]
a_{2}(p)&=&\frac{1}{2}a_{2}(p-1)+xa_{1}(p-1)+\frac{1}{2}a_{0}(p-1)\\\\[8.5359pt]
&\vdots&\\\\[8.5359pt]
a_{n}(p)&=&\frac{1}{2}a_{n}(p-1)+xa_{n-1}(p-1)+\frac{1}{2}a_{n-2}(p-1)\end{array}\quad\right\\},$
(37)
subject to the boundary conditions
$\left.\begin{array}[]{rcl}a_{0}(0)&=&1\\\\[5.69046pt]
a_{1}(1)&=&xa_{0}(0)\\\\[5.69046pt]
a_{2}(2)&=&xa_{1}(1)+a_{0}(1)\\\\[5.69046pt] &\vdots&\\\\[5.69046pt]
a_{n}(n)&=&xa_{n-1}(n-1)+a_{n-2}(n-1)\end{array}\quad\right\\},$ (38)
where $R_{p}^{k}=a_{p-|k|}(p)$ is given along diagonals for a fixed $p-|k|$.
For instance, one can obtain $R_{p}^{\pm p}(x)=\frac{1}{2^{p}},$ for $p\geq
0$, $R_{p}^{\pm(p-1)}(x)=\frac{p}{2^{p-1}}x,$ for $p\geq 1$ and
$R_{p}^{\pm(p-2)}(x)=\frac{p}{2^{p}}+\frac{p(p-1)}{2^{p-1}}x^{2},$ for $p\geq
2,$ etc.
## Acknowledgements
I would like to thank Tom ter Elst, Matthew Auger, and Radosław Szmytkowski
for valuable discussions. I would also like to thank Shenghui Yang at Wolfram
Research for valuable assistance in generating an algorithm to symbolically
compute the logarithmic polynomials. I acknowledge funding for time to write
this paper from the Dean of the Faculty of Science at the University of
Auckland in the form of a three month stipend to enhance University of
Auckland 2012 PBRF Performance. Part of this work was conducted while H. S.
Cohl was a National Research Council Research Postdoctoral Associate in the
Information Technology Laboratory at the National Institute of Standards and
Technology, Gaithersburg, Maryland, U.S.A.
## References
* [1] M. Abramowitz and I. A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C., 1972.
* [2] J. B. Boyling. Green’s functions for polynomials in the Laplacian. Zeitschrift für Angewandte Mathematik und Physik, 47(3):485–492, 1996.
* [3] H. S. Cohl. Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers in ${\mathbf{R}}^{d}$ and ${\mathbf{H}}^{d}$. PhD thesis, The University of Auckland, 2010. xiv+190 pages.
* [4] H. S. Cohl and D. E. Dominici. Generalized Heine’s identity for complex Fourier series of binomials. Proceedings of the Royal Society A, 467:333–345, 2010.
* [5] H. S. Cohl and J. E. Tohline. A Compact Cylindrical Green’s Function Expansion for the Solution of Potential Problems. The Astrophysical Journal, 527:86–101, December 1999.
* [6] I. M. Gel’fand and G. E. Shilov. Generalized functions. Vol. 1. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Properties and operations, Translated from the Russian by Eugene Saletan.
* [7] BT Helenbrook. Mesh deformation using the biharmonic operator. International Journal for Numerical Methods in Engineering, 56(7):1007–1021, 2003.
* [8] B. J. Kirby. Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press, Cambridge, 2010.
* [9] W.M. Lai, D. Rubin, and E. Krempl. Introduction to continuum mechanics, revised edition, SI/metric units. Pergamon, Oxford, UK, 1978.
* [10] S. A. Lurie and V. V. Vasiliev. The biharmonic problem in the theory of elasticity. Gordon and Breach Publishers, Luxembourg, 1995.
* [11] W. Magnus, F. Oberhettinger, and R. P. Soni. Formulas and theorems for the special functions of mathematical physics. Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52. Springer-Verlag New York, Inc., New York, 1966.
* [12] W. Miller, Jr. Symmetry and separation of variables. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. With a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, Vol. 4.
* [13] J. Monterde and H. Ugail. On harmonic and biharmonic Bézier surfaces. Computer Aided Geometric Design, 21(7):697–715, 2004.
* [14] M. Nicolescu. Les Fonctions Polyharmoniques. Hermann, Paris, 1936.
* [15] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. Cambridge University Press, Cambridge, 2010.
* [16] L. Schwartz. Théorie des distributions. Tome I. Actualités Sci. Ind., no. 1091 = Publ. Inst. Math. Univ. Strasbourg 9. Hermann, Paris, 1950.
* [17] S. L. Sobolev. Cubature formulas and modern analysis: an introduction. Oxonian, New Delhi, 1992.
* [18] R. Szmytkowski. On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). Journal of Mathematical Chemistry, 49(7):1436–1477, June 2011.
* [19] C. S. Tsai, C.-C. and Chen and T.-W. Hsu. The method of particular solutions for solving axisymmetric polyharmonic and poly-Helmholtz equations. Engineering Analysis with Boundary Elements, 33(12):1396–1402, 2009\.
* [20] H. Ugail. Partial Differential Equations for Geometric Design. Springer-Verlag, London, 2011.
* [21] D. Vautherin. Hartree-Fock Calculations with Skyrme’s Interaction. II. Axially Deformed Nuclei. Physical Review C, 7:296–316, 1973.
|
arxiv-papers
| 2012-02-08T20:48:22 |
2024-09-04T02:49:27.245239
|
{
"license": "Public Domain",
"authors": "Howard S. Cohl",
"submitter": "Howard Cohl",
"url": "https://arxiv.org/abs/1202.1811"
}
|
1202.1849
|
# Quantum transport anomalies in DNA containing mispairs
Xue-Feng Wang xf˙wang1969@yahoo.com Department of physics, Soochow
University, 1 Shizi Street, Suzhou, 215006 China Tapash Chakraborty and J.
Berashevich Department of Physics and Astronomy, The University of Manitoba,
Winnipeg, Canada, R3T 2N2
###### Abstract
The effect of mispair on charge transport in a DNA of sequence (GC)(TA)N(GC)3
connected to platinum electrodes is studied using the tight-binding model.
With parameters derived from ab initio density functional result, we calculate
the current versus bias voltage for DNA with and without mispair and for
different numbers of (TA) basepairs $N$ between the single and triple (GC)
basepairs. The current decays exponentially with $N$ under low bias but
reaches a minimum under high bias when a multichannel transport mechanism is
established. A (GA) mispair substituting a (TA) basepair near the middle of
the (TA)N sequence usually enhances the current by one order due to its low
ionization energy but may decrease the current significantly when an
established multichannel mechanism is broken.
## I introduction
Longitudinal charge transport along DNA has been the subject of extensive
study in the last decade. schu ; chak ; gene Charge transport occurs in the
oxidative and reductive DNA damage or repair processes and can happen in the
long distance range. brow ; loft ; schu Study of transport properties may
lead to a better understanding of the fundamental driving processes in
biological evolution. Furthermore, the charge transport process might have
been used naturally for basepair mismatch detection during the DNA repairing
process. It is already known that, due to chemical reaction and radiative
ionization, mispairs or gene mutations happen quite often in the cell.
Fortunately, almost all of the mispairs can be detected and repaired during
the replication process to keep the material genetically stable. However, some
of the mutations may escape from the detecting and repairing processes and
result in various genetic diseases including cancer. A recent study indicates
a negative correlation between the cancer risk and sensitivity of charge
transport property of the gene to a mutation. cancer Understanding how
mispairs modify the electric properties of DNA then becomes very important
scha ; okad ; edir and, together with the usage of other properties, apal ;
bera may improve mutation detecting techniques. fixe ; marr ; tian ; zhan In
addition, thanks to its perfect self-assembling and self-recognition
properties found in nature, DNA is also expected to be a potentially
functional material for molecular devices. In this case mispairs may be used
to obtain unique functions of the devices.
The charge transport through a DNA sequence can be measured by chemical or
physical methods. schu ; chak ; gene In one of the typical chemical
experiments, Giese et al. used a DNA of sequence (GC)(TA)N(GC)3. gies They
measured the charge transfer rate from the (GC) basepair to the (GC)3 triple
basepair for different number $N$ of (TA) basepairs, and found a crossover
from a rapid decay of the charge transfer rate vs $N$ to an almost zero decay
around $N=3$. As an alternative to other explanations, berl ; bixo ; jort ;
reng ; cram ; bask we have proposed this as a crossover from one dominant
channel transport to a multichannel transport. wang2 An example of physical
experiments is the one performed by Porath et al.pora1 where a DNA sequence
(GC)m is located between two platinum electrodes and the current versus
voltage is directly measured. This result has also been simulated by simple
tight-binding models. li ; cuni ; wang1 It is known that G.A mispair in
various conformations bera is the most stable mispair and often present in
the DNA. brow The magnetic properties of DNA was studied earlier and found to
be significantly influenced by the presence of the G.A mispair. apal In this
paper, we will study the effect of mispairs, such as G(anti)$\cdot$A(anti)
indicated in the following as (GaAa), and G(anti)$\cdot$A(syn) indicated as
(GaAs), leon on charge transport when a Watson-Crick (TA) basepair is
replaced by a mispair in the DNA sequence (GC)(TA)N(GC)3 connected to platinum
electrodes.
## II method
We consider a $p$-type semiconductor DNA duplex chain of basepairs connected
to a circuit via two platinum electrodes suitable for experimental realization
pora1 . Each platinum electrode is modeled as a semi-infinite one-dimensional
(1D) electrode wang1 connected to the G base at one end of the first strand
as illustrated in 1(a). The tight-binding Hamiltonian of the system reads
Figure 1: (a) Schematic illustration of the equilibrium energy band across the
system with a DNA of sequence (GC)(TA)N(GC)3 connect to two platinum
electrodes. (b) The energy dependence of $\epsilon_{m}$ (solid curve) and
$t_{m}$ (dashed) for electrons in the platinum electrode $X$, with $X=L$ or
$R$ for the left or right electrode.
$H=2\sum_{n=-\infty}^{\infty}[\varepsilon_{n}c_{n}^{\dagger}c_{n}-t_{n,n+1}(c_{n}^{\dagger}c_{n+1}+c_{n+1}^{\dagger}c_{n})]$
$+2\sum_{n=1}^{N}u_{n}d_{n}^{\dagger}d_{n}-2\sum_{n=1}^{N-1}h_{n,n+1}(d_{n}^{\dagger}d_{n+1}+d_{n+1}^{\dagger}d_{n})$
$-2\sum_{n=1}^{N}\lambda_{n}(c_{n}^{\dagger}d_{n}+d_{n}^{\dagger}c_{n}).$ (1)
Here $c_{n}^{\dagger}$ ($d_{n}^{\dagger}$) is the creation operator of holes
in the first (second) strand on site $n$ of the DNA chain (for $1\leq n\leq
N$), the left electrodes ($n\leq 0$), and the right electrodes ($n\geq N+1$).
The on-site energy of site $n$ in the first (second) strand is denoted by
$\varepsilon_{n}$ ($u_{n}$), which is equal to the highest occupied molecular
orbit (HOMO) energy of the base on this site in the DNA chain and the center
of conduction band in the electrodes. The coupling parameter of the first
(second) strand $t_{n,n+1}$ ($h_{n,n+1}$) is equal to the intra-strand
coupling parameter between neighboring sites $n$ and $n+1$ of the DNA for
$1\leq n\leq N-1$, one-fourth of the conduction band-width in the electrodes
$t_{m}$ for $n\leq-1$ and $n\geq N+1$, and the coupling strength between the
electrodes and the DNA strands for $n=0$ and $n=N$. The inter-strand coupling
between sites in the same basepair is described by $\lambda_{n}$. The factor 2
multiplied to each sum in Eq. (1) arises from the spin degeneracy.
In transport experiments,pora1 a high bias voltage can be applied to drive
the system far from equilibrium and holes in wide energy range may contribute
to the current. Since the carriers usually come from various energy bands and
the profile of band distribution is energy dependent, the effective parameters
$\varepsilon_{m}$ and $t_{m}$, which are averages over the profiles, are then
energy dependent. We assume that the parameters for the 1D tight-binding model
have a similar dependence on energy as in bulk platinum roge and the
dependence is extracted from its 3D band structure. Near the Fermi energy,
there are six bands located approximately at $-5.8$, $-4.7$, $-3.7$, $-2.2$,
$-0.2$, and $2.0$eV above the Fermi energy with band width $1.9$, $1.3$,
$1.5$, $3.1$, $1.4$, and $6.0$ eV respectively. Using Lorentzian broadening,
we can mimic the bulk DOS and extract the parameters $\varepsilon_{m}$ and
$t_{m}$ as shown in 1(b). The parameters are then scaled to match the known
values at the Fermi energy as was done in Ref.wang1 . For electrons at the
Fermi energy, the on-site energy is $\epsilon^{0}_{m}=-0.33$ eV with a
coupling parameter $t^{0}_{m}=0.55$ eV. As estimated from the experimental
data li ; wang1 the equilibrium Fermi energy is 1.73 eV higher than the HOMO
on-site energy of the G base when the (G$\cdot$C) basepair makes contact with
the platinum electrodes. Here we assume that the first DNA strand is coupled
to the electrodes with a contact parameter of $t_{dm}=0.1$ eV while the second
strand does not contact the electrodes directly. Note that our main result is
not sensitive to the choice of the electrodes and the contact parameters.
Figure 2: Illustration of a three-basepair DNA used in the ADF program to
obtain the tight-binding parameters. In the first (second) strand, the on-site
energy of a HOMO orbital is $\varepsilon_{n}$ ($u_{n}$) and the intrastrand
coupling parameter between neighboring sites is $t_{n,n+1}$ ($h_{n,n+1}$) with
$n$ the base index. The interstrand coupling parameter is denoted by
$\lambda_{n}$.
The tight-binding parameters of DNA are estimated based on the HOMO energies
of isolated nucleobases and the charge transfer integral between the HOMO
orbitals calculated by the ab initio density functional method integrated in
the ADF (Amsterdam Density Functional) program. adf ; apal ; bera The on-site
energies for bases G, C, T, and A are $-9.40$, $-10.27$, $-10.46$, and
$-9.79$, respectively. The hopping coupling parameters are listed in Table 1.
Table 1: The values of the intra- and interstrand tight-binding parameters
(in eV) for different DNA sequences, where the middle pair (X1$\cdot$X2) can
be G$\cdot$C and T$\cdot$A basepairs or G(anti)$\cdot$A(anti) and
G(anti)$\cdot$A(syn) mispairs. apal 5’-G$-$X1$-$G-3’
---
3’-C$-$X2$-$C-5’
(X1$\cdot$X2) | $t^{\vphantom{\dagger}}_{\rm 12}$ | $t^{\vphantom{\dagger}}_{\rm 23}$ | $\lambda^{\vphantom{\dagger}}_{\rm 2}$ | $h^{\vphantom{\dagger}}_{\rm 12}$ | $h^{\vphantom{\dagger}}_{\rm 23}$
G$\cdot$C | 0.133 | 0.133 | 0.028 | 0.14 | 0.14
T$\cdot$A | 0.164 | 0.400 | 0.070 | 0.154 | 0.099
5’-T$-$X1$-$T-3’
3’-A$-$X2$-$A-5’
(X1$\cdot$X2) | $t^{\vphantom{\dagger}}_{\rm 12}$ | $t^{\vphantom{\dagger}}_{\rm 23}$ | $\lambda^{\vphantom{\dagger}}_{\rm 2}$ | $h^{\vphantom{\dagger}}_{\rm 12}$ | $h^{\vphantom{\dagger}}_{\rm 23}$
T$\cdot$A | 0.330 | 0.330 | 0.070 | 0.011 | 0.011
Ga$\cdot$Aa | 0.400 | 0.164 | 0.029 | 0.055 | 0.002
Ga$\cdot$As | 0.250 | 0.167 | 0.057 | 0.207 | 0.027
The current $I$ when a voltage bias $V$ is applied over the two platinum
electrodes is then evaluated by the transfer matrix method wang2 ; chak ; yan
; marc . For an open system, the secular equation is expressed as a group of
equations of the form
$\displaystyle
t_{n-1,n}\Psi^{+}_{n-1}+(\varepsilon_{n}-E)\Psi^{+}_{n}+\lambda_{n}\Psi^{-}_{n}+t_{n,n+1}\Psi^{+}_{n+1}=0$
$\displaystyle
h_{n-1,n}\Psi^{-}_{n-1}+(u_{n}-E)\Psi^{-}_{n}+\lambda_{n}\Psi^{+}_{n}+h_{n,n+1}\Psi^{-}_{n+1}=0$
with $\Psi^{+}_{n}$ ($\Psi^{-}_{n}$) the wave function of the first (second)
strand on site $n$. The wave functions of the sites $n+1$ and $n$ are related
to those of the sites $n$ and $n-1$ by a transfer matrix $\hat{M}$,
$\left(\begin{array}[]{c}\Psi^{+}_{n+1}\\\ \Psi^{-}_{n+1}\\\ \Psi^{+}_{n}\\\
\Psi^{-}_{n}\end{array}\right)=\hat{M}\left(\begin{array}[]{c}\Psi^{+}_{n}\\\
\Psi^{-}_{n}\\\ \Psi^{+}_{n-1}\\\ \Psi^{-}_{n-1}\end{array}\right),\,\,\,$ (2)
with
$\hat{M}=\left[\begin{array}[]{cccc}\frac{(E-\varepsilon_{n})}{t_{n,n+1}}&\frac{-\lambda_{n}}{t_{n,n+1}}&-\frac{t_{n-1,n}}{t_{n,n+1}}&0\\\
\frac{-\lambda_{n}}{h_{n,n+1}}&\frac{(E-\varepsilon_{n})}{h_{n,n+1}}&0&-\frac{h_{n-1,n}}{h_{n,n+1}}\\\
1&0&0&0\\\ 0&1&0&0\end{array}\right].$
The transmission is then calculated by assuming the plane waves propagating in
the electrodes for the holes $\Psi_{n}=Ae^{ik_{L}na}+Be^{-ik_{L}na}$ for
$n\leq 0$ and $\Psi_{n}=Ce^{ik_{L}na}$ for $n\geq N+1$ in the left and right
electrodes, respectively. Expressing the output wave amplitude $C$ in terms of
the input wave amplitude $A$ and the transmission,
$T(E)=\frac{|C|^{2}\sin(k_{R}a)}{|A|^{2}\sin(k_{L}a)}.$
The distance between two neighboring bases along any DNA strand is $a=3.4$ Å.
The net current primarily comes from the hole transmission between the
electrodes’ Fermi energies and is calculated as datt
$I=\frac{e^{2}}{h}\sum_{\sigma}\int_{-\infty}^{\infty}dE\,T^{\sigma}(E)[f^{L}(E)-f^{R}(E)].$
Here the Fermi function is $f^{X}(E)=1/\exp[(E-E^{X}_{F})/k_{B}T]$ with $X=L$
or $R$ and the room temperature $T=300$ K. When a bias voltage $V$ is applied
between the two electrodes, the left (right) Fermi energy is assumed
$E^{L}_{F}=V/2$ ($E^{R}_{F}=-V/2$).
## III results and discussions
Figure 3: (a) Current $I$ versus bias voltage $V$ of the DNA sequence
(GC)(TA)N(GC)3. $N=1-11$ for curves from the top. (b) Current I versus (TA)
basepair number $N$ at fixed bias voltage $V$. The value of $V$ is indicated
beside each curve.
The results for the current ($I$) versus voltage ($V$) for DNA of sequence
(GC)(TA)N(GC)3 with $N=1,2,...,11$ are shown in 3(a). Each curve except for
$N=1$ has steps at voltages 1.45, 1.64, 1.85, and 2.02 eV indicating that the
transport channels are mainly formed at four threshold voltages related to the
four kinds of bases. In the $N=1$ curve, the second step is split into two at
$V=1.62$ and 1.7 V and there is no step at 2.02 V. In a log scale, the two
lower-energy steps have almost the same height for all the curves while the
height of the two higher-energy steps increases with the number of (TA) base-
pairs. At a bias voltage lower than the first threshold, the DNA works as an
energy barrier for electron transport since the Fermi energy of both
electrodes is located between the HOMO and LUMO energy. At a bias near the
first and the second threshold ($1.45<1.8$ eV), HOMO channels of the (GC)
basepairs become available for charge transport but the (TA) basepairs behave
as energy barriers for transport. At a bias higher than the third threshold
($V>1.85$ eV), transport channels of (TA) basepair also participate in the
charge transport across the DNA molecule. Along the curve $N=1$, we can hardly
see the third and the fourth steps. The addition of (TA) basepairs can
establish a network of bases due to the interstrand coupling and introduces
additional transport channels, as was reported in Ref.wang2 . Consequently, we
observe the height increase of the third and fourth steps in the log scale.
The current enhancement due to additional channels may compete with the
exponential current decay with the length of the molecule. This results in a
current minimum for $N>5$, as clearly shown in 3(b) where the current is
plotted versus $N$ at various $V$. At a bias less than 1.8V, the current
decays exponentially with $N$ with almost the same exponent. For a bias higher
than 1.8V, the exponent decreases with $N$ until it is almost zero for $N>5$
under the bias $V=2.1$V or higher. This crossover from a rapid to almost zero
decay of the charge transfer versus the (TA) basepair numbers has been
observed in Ref. gies with the chemical method and can also be observed in
physical experiments as described in Ref. pora1 . Different from our previous
simplified model wang2 where uniform parameters and virtual electrodes are
assumed, here we employ a more realistic model with the tight-binding
parameters of DNA and electrode extracted from ab initio calculations. In
addition, a variable bias voltage is applied between the two electrodes to
obtain the I-V curve. Note that the role of diagonal interstrand hopping is
relevant to the electron transport in DNA and its inclusion in the calculation
might shift the I-V curve but not the conclusion. well ; wang3
Figure 4: (a) The $I-V$ curves of DNA sequence
(GC)(TA)N/2(GaAa)(TA)(N-1)/2(GC)3 for $N=3,4,...,11$. Here $N/2$ and $(N-1)/2$
take only the interger part of the value. (b) Current I versus the number $N$
at several selected bias voltage $V$ indicated by the values beside each
curve. The $I-V$ curves and $I-N$ curves for DNA sequence
(GC)(TA)(N-1)/2(GaAa)(TA)N/2(GC)3 are plotted in (c) and (d).
To estimate the effect of mispairs (GaAa) and (GaAs) on the charge transport,
we replace one of the (TA) basepairs near the middle of the (TA)N sequence by
a mispair and calculate the corresponding $I-V$ curve. In 4(a) and (b), the
result for the [$(N+2)/2$]th (TA) basepair replaced by a (GaAa) mispair is
shown and in (c) and (d) the [$(N+1)/2$]th basepair is replaced by a mispair.
Here [$R$] means extracting the integer part of a real number $R$. Note that
the curves of odd $N$ are the same in 4(a) and (c), corresponding to equal
number of (TA) basepairs to the left and right of the (GaAa) mispair. For even
$N$, there are one more (TA) to the left of the mispair in (a) and one less in
(b). With the replacement of the mispair, the current is greatly enhanced
after the first threshold voltage, indicating that the G base in the mispair
works as a tunneling bridge. However, for $N>5$, the current decreases with
the mispair at a bias higher than the fourth threshold voltage (2 eV) as shown
in 4(a) and (c). This happens because the mispair destroys the resonant
transport network formed by the periodic (TA) basepairs series, i.e. the
mispair works as an impurity. A weak $N$ dependent current appears only for
higher $N$ and at a lower current value, as shown in 4(b) and (d) when both
the (TA) sequences besides the mispair form resonant transport network.
In the presentce of a mispair, some steps shift and extra steps appear along
the $I-V$ curves. For $N=3$ or a DNA of sequence (GC)(TA)(GaAa)(TA)(GC)3, the
simplest case with a mispair, three steps at $V=1.41$, 1.55, and 1.68 V appear
in the $I-V$ curve and the current is enhanced by more than one order at high
voltage with the substitution by a mispair. For $N\geq 5$, however, the first
three steps shift to $V=1.45$, $1.53$, and $1.64$ V with the last one decaying
over $N$. Compared to the case without a mispair, one extra step appears at
$V=1.53$ V. In the bias range 1.45 V $<V<$ 1.53 V, the curves for $N>4$ are
almost equally separated in the vertical log scale, indicating an
exponentially decaying current with $N$ as also shown in 4(b) and (d).
Furthermore, the mispair location also affects significantly the current. The
$N=4$ curve has steps at the same position as for $N=3$ in 4(a) but as the
$N\geq 5$ curves in 4(b). In the range 1.58 V $<V<$ 2.2V, the even $N$ curves
are near the curve of $N+1$ in (a) but near the curve $N-1$ in (b). The change
is also represented by shifted steps along the $V=1.6$ V curves when comparing
4(b) to (d). This observation suggests that the current in this bias range is
mainly determined by the length of the (TA)n sequence between the left single
(GC) basepair and the (GaAa) mispair. Under stronger bias, the current at
first decays exponentially with $N$ and then fluctuates near a value slightly
below 1 nA. This long-DNA current limit is about five times smaller than that
observed in the system without a mispair.
When the (GaAa) mispair is replaced by a (GaAs) mispair, the $I-V$ curves show
fewer steps as illustrated in Fig. 5. For $N=3$ there are three steps at
$V=1.45$, $1.58$, and $1.68$V while for $N\geq 5$ there are only two steps at
$V=1.45$ and $1.61$V below bias 2V. Similar to the case of (GaAa) mispair
shown in 4, the $N=4$ curve also has steps at the position of the $N=3$ curve
in (a) but at the positions of the $N\geq 5$ curves in (b). In addition, the
current is mainly limited by the number of (TA) basepairs between the left
(GC) basepair and the (GaAs) mispair in the range 1.6V$<V<2.2$V. For $V>2.2$V,
no step in curves of $N\leq 7$ suggests again that one transport channel
dominates in the short (TA)n DNA sequence. For large $N$, a series of steps
appear in the $I-V$ curves and the current do not decrease monotonically with
$N$, indicating the enhancement of current due to the increase of transport
channel number with $N$ can compensate the current decay with the length of
each channel. The current decays exponentially at $V<1.6$ V but decays with
steps in the range $1.6<V<2.2$ V. At $V>2.2$ V the current decays in short DNA
and then fluctuates when multichannel tunneling mechanism dominates in the
long (TA)n sequence. Overall a (GaAs) mispair substitution changes less the
current than that of a (GaAa) mispair especially in high bias since the
intrastrand coupling parameter of a (GaAs) mispair is closer to that of a (TA)
basepair.
Figure 5: The same as in 4 is plotted for basepair mismatch (GaAs).
## IV summary
In summary, we have studied charge transport through DNA connected to two
platinum electrodes and contains mispairs within a realistic tight-binding
scheme. The energy dependent tight-binding parameters for the electrodes are
obtained by fitting the density of states near the Fermi energy of the
material. The parameters of DNA are derived from the ab initio density
functional calculation of the coupling between HOMO states in neighbor bases.
When a (TA) basepair in the (GC)(TA)N(GC)3 sequence is replaced by (GaAa) or
(GaAs) mispairs, the current is usually enhanced due to the lower ionization
energy of the mispairs. In DNA with a long (TA)N sequence, multichannel
tunneling mechanism set a minimal current at a high bias, similar to a
previous experimental observation. The substitution of the mispair in this
case, however, will break the multichannel tunneling mechanism and decrease
the current significantly.
## V acknowledgements
X.F.W. acknowledges support from the startup fund in Soochow University and
NSFC in China. The work was supported in part by the Canada Research Chairs
Program (T.C.).
## References
* (1) Schuster, G. B. (Ed.), Long-range charge transfer in DNA, Springer-Verlag: Berlin Heidlberg, 2004.
* (2) Chakraborty, T. (Ed.) 2007 Charge migration in DNA (Perspectives from Physics, Chemistry, and Biology) (New York: Springer).
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* (35) V. Apalkov, X. F. Wang, and T. Chakraborty 2007 Charge Migration in DNA: perspectives from Physics, Chemistry, and Biology ed T. Chakraborty (New York:Springer) Chapter 5 (Physics Aspects of Charge Migration Through DNA).
|
arxiv-papers
| 2012-02-08T22:27:38 |
2024-09-04T02:49:27.257250
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-Feng Wang, Tapash Chakraborty and J. Berashevich",
"submitter": "Xue Feng Wang",
"url": "https://arxiv.org/abs/1202.1849"
}
|
1202.1869
|
# A note on a generalized circular summation formula of theta functions
Jun-Ming Zhu Department of Mathematics, Luoyang Normal University, Luoyang
City, Henan Province 471022, China Department of Mathematics, East China
Normal University, Shanghai City 200241, China junming_zhu@163.com
###### Abstract.
In this note, we make a correction of the imaginary transformation formula of
Chan and Liu’s circular formula of theta functions. We also get the imaginary
transformation formulaes for a type of generalized cubic theta functions.
Keywords: Theta function, Circular summation, Jacobi imaginary transformation,
Cubic theta function.
MSC (2010): 11B65, 11F27, 05A19.
The author is supported by the Natural Science Foundation of China (Grant No.
11171107 ) and the Foundation of Fundamental and Advanced Research of Henan
Province (Grant No. 112300410024 ).
## 1\. Introduction
Throughout we put $q=e^{2\pi i\tau}$, where $\mbox{Im}\ \tau>0$. As usual, the
Jacobi theta functions $\theta_{k}(z|\tau)$ for $k\in\\{1,2,3,4\\}$ are
defined as:
$\displaystyle\theta_{1}(z|\tau)$ $\displaystyle=$ $\displaystyle-iq^{1\over
8}\sum\limits_{n=-\infty}^{\infty}(-1)^{n}q^{n(n+1)\over 2}e^{(2n+1)iz},$
$\displaystyle\theta_{2}(z|\tau)$ $\displaystyle=$ $\displaystyle q^{1\over
8}\sum\limits_{n=-\infty}^{\infty}q^{n(n+1)\over 2}e^{(2n+1)iz},$
$\displaystyle\theta_{3}(z|\tau)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=-\infty}^{\infty}q^{n^{2}\over 2}e^{2niz},$
$\displaystyle\theta_{4}(z|\tau)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=-\infty}^{\infty}(-1)^{n}q^{n^{2}\over
2}e^{2niz}.$
Circular summation of theta functions has been an interesting topic. In their
paper [4], S.H. Chan and Z.-G. Liu proved the following remarkable circular
summation formula of theta functions [4, Theorem 4].
###### Theorem 1.1.
Suppose $y_{1},y_{2},\cdots,y_{n}$ are $n$ complex numbers such that
$y_{1}+y_{2}+\cdots+y_{n}=0$. Then we have
(1.1) $\sum_{k=0}^{mn-1}\prod_{j=1}^{n}\theta_{3}\left(z+y_{j}+{k\pi\over
mn}\big{|}\tau\right)=G_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)\theta_{3}(mnz|m^{2}n\tau),$
where
(1.2)
$G_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)=mn\sum_{\begin{subarray}{c}r_{1},\cdots,r_{n}=-\infty\\\
r_{1}+\cdots+r_{n}=0\end{subarray}}^{\infty}q^{{1\over
2}(r_{1}^{2}+\cdots+r_{n}^{2})}e^{2i(r_{1}y_{1}+\cdots+r_{n}y_{n})}.$
This theorem generalizes the fundamental result in Zeng’s [8]. Zeng [8] is
motivated by Ramanujan’s circular summation formula [6, page 54] (see also
[3]) and Boon et al. [2, Eq. (7)]. For detailed account of the topic of the
circular summation of theta functions, we refer the readers to [3] and [4].
Applying the Jacobi imaginary transformation to (1.1), S.H. Chan and Z.-G. Liu
got the dual form of Theorem 1.1, i.e. [4, Theorem 5].
###### Theorem 1.2.
, Suppose $y_{1},y_{2},\cdots,y_{n}$ are $n$ complex numbers such that
$y_{1}+y_{2}+\cdots+y_{n}=0$. Then we have
(1.3) $\sum_{k=0}^{mn-1}q^{k^{2}\over
2}e^{2kiz}\prod_{j=1}^{n}\theta_{3}\left(mz+(y_{j}+km)\pi\tau\big{|}m^{2}n\tau\right)=F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)\theta_{3}(z|\tau),$
where
$F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)=\frac{(-i\tau)^{1-n\over
2}}{(m^{2}n)^{n\over
2}}q^{-\frac{y_{1}^{2}+y_{2}^{2}+\cdots+y_{n}^{2}}{2m^{2}n}}G_{m,n}\left({y_{1}\pi\over
m^{2}n},\cdots,{y_{n}\pi\over m^{2}n}\Big{|}-{1\over m^{2}n\tau}\right).$
There is also another formula ([4, Eq. (1.10)]) for
$F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)$ in [4, Theorem 5] without proof.
Substituting [4, Eq. (1.10)] back into (1.3) (i.e. [4, Eq. (1.8)] ), and then,
taking $m=2$ and $n=1$, we obtain
(1.4) $\theta_{3}(2z|4\tau)+\theta_{2}(2z|4\tau)=(1+q)\theta_{3}(z|\tau).$
But
$\displaystyle\theta_{3}(z|\tau)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=-\infty}^{\infty}q^{n^{2}\over 2}e^{2niz}$
$\displaystyle=$ $\displaystyle\sum_{n=-\infty}^{\infty}q^{(2n)^{2}\over
2}e^{4niz}+\sum_{n=-\infty}^{\infty}q^{(2n+1)^{2}\over 2}e^{2(2n+1)iz}$
$\displaystyle=$ $\displaystyle\theta_{3}(2z|4\tau)+\theta_{2}(2z|4\tau).$
Thus (1.4) does not hold unless $q=0$, so we can conclude that the formula [4,
Eq. (1.10)] is incorrect.
In Section 2, we will make a correction of the formula [4, Eq. (1.10)] with
proof.
In Section 3, we will use Theorem 2.1 to deduce the imaginary transformation
formulaes of a type of generalized cubic theta functions.
## 2\. Correction of the formula [4, Eq. (1.10)]
We rewrite Theorem 1.2 plus the formula for
$F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)$ [4, Theorem 5] in the following
version, where (2.3) is the correction of [4, Eq. (1.10)].
###### Theorem 2.1.
, Suppose $y_{1},y_{2},\cdots,y_{n}$ are $n$ complex numbers such that
$y_{1}+y_{2}+\cdots+y_{n}=0$. Then we have
(2.1) $\sum_{k=0}^{mn-1}q^{k^{2}\over
2}e^{2kiz}\prod_{j=1}^{n}\theta_{3}\left(mz+y_{j}+km\pi\tau\big{|}m^{2}n\tau\right)=F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)\theta_{3}(z|\tau),$
where
(2.2) $\displaystyle F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)$ $\displaystyle=$
$\displaystyle\frac{(-i\tau)^{1-n\over 2}}{(m^{2}n)^{n\over
2}}e^{\frac{y_{1}^{2}+y_{2}^{2}+\cdots+y_{n}^{2}}{m^{2}n\pi\tau
i}}G_{m,n}\left({y_{1}\over m^{2}n\tau},{y_{2}\over
m^{2}n\tau},\cdots,{y_{n}\over m^{2}n\tau}\Big{|}-{1\over m^{2}n\tau}\right)$
(2.3) $\displaystyle=$ $\displaystyle\sum_{k=0}^{n-1}q^{-{m^{2}k^{2}\over
2}}\sum_{\begin{subarray}{c}r_{1},r_{2},\cdots,r_{n}=-\infty\\\
r_{1}+r_{2}+\cdots+r_{n}=k\end{subarray}}^{\infty}q^{{1\over
2}m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})}e^{-2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}.$
###### Proof.
For the proof of (2.1) and (2.2), see [4]. Obviously, (2.1) can also be proved
directly by the theory of Elliptic functions like the proof of [4, Theorem 4].
We only prove (2.3) in detail. Applying the series expansion of
$\theta_{3}(z|\tau)$ to (2.1), the left hand side of (2.1) equals
$\displaystyle\sum_{k=0}^{mn-1}q^{k^{2}\over
2}e^{2kiz}\prod_{j=1}^{n}\sum_{r_{j}=-\infty}^{\infty}q^{{m^{2}nr_{j}^{2}\over
2}}e^{2ir_{j}(mz+y_{j}+km\pi\tau)}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{mn-1}\sum_{r_{1},\cdots,r_{n}=-\infty}^{\infty}q^{{1\over
2}[k^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]+km(r_{1}+\cdots+r_{n})}$
$\displaystyle\cdot e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}e^{2imz\
(r_{1}+r_{2}+\cdots+r_{n})+2kiz}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{m-1}\sum_{k=0}^{n-1}\sum_{r_{1},\cdots,r_{n}=-\infty}^{\infty}q^{{1\over
2}[(km+l)^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]+(km+l)m(r_{1}+\cdots+r_{n})}$
$\displaystyle\cdot e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}e^{2imz\
(r_{1}+r_{2}+\cdots+r_{n}+k)+2ilz}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\sum_{r_{1},\cdots,r_{n}=-\infty}^{\infty}q^{{1\over
2}[k^{2}m^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]+km^{2}(r_{1}+\cdots+r_{n})}$
$\displaystyle\cdot e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}e^{2imz\
(r_{1}+r_{2}+\cdots+r_{n}+k)}$
$\displaystyle+\sum_{k=0}^{n-1}\sum_{r_{1},\cdots,r_{n}=-\infty}^{\infty}q^{{1\over
2}[(km+1)^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]+(km+1)m(r_{1}+\cdots+r_{n})}$
$\displaystyle\cdot e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}e^{2imz\
(r_{1}+r_{2}+\cdots+r_{n}+k)+2iz}$
$\displaystyle+\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots$
$\displaystyle+\sum_{k=0}^{n-1}\sum_{r_{1},\cdots,r_{n}=-\infty}^{\infty}q^{{1\over
2}[(km+m-1)^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]+(km+m-1)m(r_{1}+\cdots+r_{n})}$
$\displaystyle\cdot e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}e^{2imz\
(r_{1}+r_{2}+\cdots+r_{n}+k)+2iz(m-1)}.$
Note that, in the last identity above, the term independent of $z$ is produced
only from the first sum. The right hand side of (2.1) equals
$F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)\sum_{n=-\infty}^{\infty}q^{n^{2}\over
2}e^{2niz}.$
Then equating the terms that are independent of $z$ on both sides of (2.1), we
find that
$\displaystyle F_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\sum_{\begin{subarray}{c}r_{1},r_{2},\cdots,r_{n}=-\infty\\\
r_{1}+r_{2}+\cdots+r_{n}=-k\end{subarray}}^{\infty}q^{{1\over
2}[k^{2}m^{2}+m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})]-k^{2}m^{2}}e^{2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n-1}q^{-{m^{2}k^{2}\over
2}}\sum_{\begin{subarray}{c}r_{1},r_{2},\cdots,r_{n}=-\infty\\\
r_{1}+r_{2}+\cdots+r_{n}=k\end{subarray}}^{\infty}q^{{1\over
2}m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})}e^{-2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}.$
This completes the proof.
## 3\. The imaginary transformation formulaes for a type of generalized cubic
theta functions
We rewrite (2.2) and (2.3) as the following corollary.
###### Corollary 3.1.
For $n$ complex numbers $y_{1},y_{2},\cdots,y_{n}$ such that
$y_{1}+y_{2}+\cdots+y_{n}=0$ and the function
$G_{m,n}(y_{1},y_{2},\cdots,y_{n}|\tau)$ defined by (1.2), we have
$\displaystyle G_{m,n}\left({y_{1}\over m^{2}n\tau},\cdots,{y_{n}\over
m^{2}n\tau}\Big{|}-{1\over m^{2}n\tau}\right)$ $\displaystyle=$
$\displaystyle\frac{(m^{2}n)^{n\over 2}}{(-i\tau)^{1-n\over
2}}e^{\frac{(y_{1}^{2}+\cdots+y_{n}^{2})i}{m^{2}n\pi\tau}}\sum_{k=0}^{n-1}q^{-{m^{2}k^{2}\over
2}}\sum_{\begin{subarray}{c}r_{1},r_{2},\cdots,r_{n}=-\infty\\\
r_{1}+r_{2}+\cdots+r_{n}=k\end{subarray}}^{\infty}q^{{1\over
2}m^{2}n(r_{1}^{2}+r_{2}^{2}+\cdots+r_{n}^{2})}e^{-2i(r_{1}y_{1}+r_{2}y_{2}+\cdots+r_{n}y_{n})}.$
Setting $m=1$ and $n=3$ in Corollary 3.1, and then, replacing $\tau$ by
$\tau\over 3$ , we obtain
$\displaystyle
G_{1,3}\left({y_{1}\over\tau},{y_{2}\over\tau},{-y_{1}-y_{2}\over\tau}\Big{|}-{1\over\tau}\right)$
$\displaystyle=$ $\displaystyle-\sqrt{3}i\tau\
e^{\frac{2i(y_{1}^{2}+y_{2}^{2}+y_{1}y_{2})}{\pi\tau}}\sum_{k=0}^{2}q^{-{k^{2}\over
6}}\sum_{\begin{subarray}{c}r_{1},r_{2},r_{3}=-\infty\\\
r_{1}+r_{2}+r_{3}=k\end{subarray}}^{\infty}q^{{1\over
2}(r_{1}^{2}+r_{2}^{2}+r_{3}^{2})}e^{-2i[r_{1}y_{1}+r_{2}y_{2}-r_{3}(y_{1}+y_{2})]}$
$\displaystyle=$ $\displaystyle-\sqrt{3}i\tau\
e^{\frac{2i(y_{1}^{2}+y_{2}^{2}+y_{1}y_{2})}{\pi\tau}}\left[\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}e^{-2ir_{1}(2y_{1}+y_{2})-2ir_{2}(y_{1}+2y_{2})}\right.$
$\displaystyle+\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}-r_{1}-r_{2}+{1\over
3}}e^{-2ir_{1}(2y_{1}+y_{2})-2ir_{2}(y_{1}+2y_{2})+2i(y_{1}+y_{2})}$
$\displaystyle+\left.\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}-2r_{1}-2r_{2}+{4\over
3}}e^{-2ir_{1}(2y_{1}+y_{2})-2ir_{2}(y_{1}+2y_{2})+4i(y_{1}+y_{2})}\right].$
In the square brackets of the last identity above, replace $r_{1}$ and $r_{2}$
in the first and second sums by $-r_{1}$ and $-r_{2}$, respectively, and in
the last sum, replace $r_{1}$ and $r_{2}$ by $r_{1}+1$ and $r_{2}+1$,
respectively. Then we obtain
$\displaystyle
G_{1,3}\left({y_{1}\over\tau},{y_{2}\over\tau},{-y_{1}-y_{2}\over\tau}\Big{|}-{1\over\tau}\right)$
$\displaystyle=$ $\displaystyle-\sqrt{3}i\tau\
e^{\frac{2i(y_{1}^{2}+y_{2}^{2}+y_{1}y_{2})}{\pi\tau}}\left[\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}}e^{2ir_{1}(2y_{1}+y_{2})+2ir_{2}(y_{1}+2y_{2})}\right.$
$\displaystyle+e^{2i(y_{1}+y_{2})}\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}+r_{1}+r_{2}+{1\over
3}}e^{2ir_{1}(2y_{1}+y_{2})+2ir_{2}(y_{1}+2y_{2})}$
$\displaystyle+\left.e^{-2i(y_{1}+y_{2})}\sum_{r_{1},r_{2}=-\infty}^{\infty}q^{r_{1}^{2}+r_{2}^{2}+r_{1}r_{2}+r_{1}+r_{2}+{1\over
3}}e^{-2ir_{1}(2y_{1}+y_{2})-2ir_{2}(y_{1}+2y_{2})}\right].$
###### Definition 3.2.
The cubic theta functions are defined as
(3.3) $\displaystyle a(x,y|\tau)$ $\displaystyle:=$
$\displaystyle\sum_{m,n=-\infty}^{\infty}q^{m^{2}+n^{2}+mn}e^{2im(2x+y)+2in(x+2y)},$
(3.4) $\displaystyle b(x,y|\tau)$ $\displaystyle:=$
$\displaystyle\sum_{m,n=-\infty}^{\infty}\omega^{m-n}q^{m^{2}+n^{2}+mn}e^{2im(2x+y)+2in(x+2y)},$
(3.5) $\displaystyle c(x,y|\tau)$ $\displaystyle:=$
$\displaystyle\sum_{m,n=-\infty}^{\infty}q^{m^{2}+n^{2}+mn+m+n+{1\over
3}}e^{2im(2x+y)+2in(x+2y)},$
where $\omega:=e^{2i\pi\over 3}$.
We have followed Chan and Liu [4, Definition 1] in the definition of
$a(x,y|\tau)$ above. For more discussions on cubic theta functions of three
variables, see [1, 5, 7]
Direct computation can verify that
$b(x,y|\tau)=a\left(x,y+{\pi\over 3}\big{|}\tau\right)\quad\mbox{and}\quad
c(x,y|\tau)=q^{1\over 3}a\left(x+{\pi\tau\over 3},y+{\pi\tau\over
3}\big{|}\tau\right).$
By (1.2) and (3.3), we obtain
$G_{1,3}(y_{1},y_{2},-y_{1}-y_{2}|\tau)=a(y_{1},y_{2}|\tau).$
Then from (3) and the discussion above, we get the following identities.
###### Proposition 3.3.
For any complex numbers $x$ and $y$, we have
$\displaystyle a\left({x\over\tau},{y\over\tau}\big{|}-{1\over\tau}\right)$
$\displaystyle=$ $\displaystyle-\sqrt{3}i\tau\
e^{\frac{2i(x^{2}+y^{2}+xy)}{\pi\tau}}\left[a(x,y|\tau)+e^{2i(x+y)}c(x,y|\tau)+e^{-2i(x+y)}c(-x,-y|\tau)\right],$
$\displaystyle c\left({x\over\tau},{y\over\tau}\big{|}-{1\over\tau}\right)$
$\displaystyle=$ $\displaystyle\mbox{\small$-\sqrt{3}i\tau\
e^{\frac{2i(x^{2}+y^{2}+xy)}{\pi\tau}-{2i(x+y)\over\tau}}\left[a(x,y|\tau)+\omega
e^{2i(x+y)}c(x,y|\tau)+\omega^{2}e^{-2i(x+y)}c(-x,-y|\tau)\right]$}.$
## References
* [1] S. Bhargava, Unification of the cubic analogues of the Jacobian theta function, J. Math. Anal. Appl. 193 (1995) 543–558.
* [2] M. Boon, M.L. Glasser, J. Zak, J. Zucker, Additive decompositions of $\theta$-functions of multiple arguments, J. Phys. A 15 (1982) 3439–3440.
* [3] H.H. Chan, Z.-G. Liu, S.T. Ng, Circular summation of theta functions in Ramanujan’s lost notebook, J. Math. Anal. Appl. 316 (2006) 628–641.
* [4] S.H. Chan, Z.-G. Liu, On a new circular summation of theta functions, J. Number Theory 130 (2010) 1190–1196.
* [5] R. Chapman, Cubic identities for theta series in three variables, Ramanujan J. 8 (2005) 459–465
* [6] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988.
* [7] X.-M. Yang, The products of three theta functions and the general cubic theta functions, Acta Math. Sin. (English Series) 26 (2010) no. 6 1115–1124.
* [8] X.-F. Zeng, A generalized circular summation of theta function and its application, J. Math. Anal. Appl. 356 (2009) 698–703.
|
arxiv-papers
| 2012-02-09T01:55:21 |
2024-09-04T02:49:27.264617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-Ming Zhu",
"submitter": "Jun-Ming Zhu",
"url": "https://arxiv.org/abs/1202.1869"
}
|
1202.1900
|
# Slowing light with a coupled optomechanical crystal array
Zhenglu Duan and Bixuan Fan Center for quantum science and engineering,
Jiangxi Normal University, Nanchang, 330022, China Center for Engineered
Quantum System, School of Mathematics and Physics, University of Queensland,
St Lucia, 4072, Qld, Australia
###### Abstract
We study the propagation of light in a resonator optical waveguide consisting
of evanescently coupled optomechanical crystal array. In the strong driving
limit, the Hamiltonian of system can be linearized and diagonalized. In this
case we obtain the polaritons, which is formed by the interaction of photons
and the collective excitation of mechanical resonators. By analyzing the
dispersion relations of polaritons, we find that the band structure can be
controlled by changing the related parameters. It has been suggested an
engineerable band structure can be used to slow and stop light pulses.
###### pacs:
42.50.Wk, 42.70.Qs, 73.20.Mf,03.67.-a
## I Introduction
Owing to its important application in many fields, such as, low-threshold
lasing lasing , pulse delaydelay and optical memories memory01 ; memory02 ;
memory03 , slow light attracts a great deal of practical interest. A number of
schemes to delay and store light have been suggested, such as
electromagnetically induced transparency (EIT) in the atomic ensembles EIT01 ;
EIT02 , photonic crystal waveguide band edgesPC01 ; PC02 , solid-state
multilayer semiconductor structuresolid , coupled resonator optical waveguide
(CROW)CROW ; Fan01 ; Fan02 ; FPcavity , more complicated hybrid structure,
e.g., coupled resonator optical waveguide doped with atomsSun01 ; Sun02 ;
atom01 ; atom02 .
For a static photonic structure, for example a bare CROW, due to the
limitation of delay-bandwidth product constraint, it is not suitable to stop
light. To dynamically stop and release the light, a dynamically tunable system
is required. Fan suggested that, if there are extra resonators side coupling
to the optical cavity cells of the CROW, Fano interference can lead to a large
change of bandwidth of the system when a small refractive index modulation is
employedFan01 . The velocity of light can therefore be dynamically slowed down
and even stopped. Unlike the case of EIT the light is coherently stored in a
static way in the resonance cavity array. Based on this idea researchers have
replaced optical resonators with atoms to couple to the resonators in the CROW
and found that the light can be converted to collective excitations of atoms
and then reversely converted and releasedSun01 ; atom02 .
Optomechanics opens a door to directly control the mechanical motion with
light Review . Many applications of optomechanics have been proposed, for
example, using cooled nanomechanical oscillators to test quantum mechanics QM
, ultra-sensitive detection of force and dispalcementforce ; position ,
quantum optics and quantum information processing squeezing ; entanglement ;
statepreparation .
Meanwhile, as a new quantum system, optomechanics is also used to stop light.
EIT effect in cavity optomechanical system with a Bose-Einstein condensate
(BEC) is suggested to slow the lightzhu . Research groups led respectively by
PainterEITQM02 and KippenbergEITQM01 proposed slowing down light based on
EIT in optomechanics. The photons are mapped onto the phonon modes instead of
internal atomic degrees of freedom in the case of EIT in atomic ensembles. In
fact, like the case of a CROW, an optical waveguide coupled to an
optomechanical crystal array has been suggested to slow and stop light
pulsePainter1 .
Motivated by the work mentioned above, this paper investigates the photon
transmission in a homogeneous side coupled optomechanical crystal array, in
which each optical resonator in the bare CROW couples to an extra mechanical
resonator. The interaction between the mechanical mode and optical modes
allows the photonic band structure of CROW to be modulated, allowing stopping
and releasing light possible in our model. By adjusting the refractive index
of the photonic crystal, the photons can be mapped onto the collective
excitation of mechanical modes and then be stopped. Our scheme offers a
patternable, compact and on-chip platform to slow and stop light.
The paper is organized as follows: In Sec. II we present our model of spatial
periodic optomechanical crystal arrays, which consists of optomechanical
crystal cells side-coupled each other. After linearizing the Hamiltonian, in
Sec. III, we transform the system into momentum space and then use Bogoliubov
transformation to diagonalise the Hamiltonian. Based on the dispersion
relations of upper and lower branch polaritons obtained in previous section,
take lower polariton for example, we investigate the band structure and
demonstrate how slow down the velocity of polariton by compressing the
bandwidth in Sec. IV. The final section concludes the paper.
## II Derivation of model
Figure 1: The schematic of 1-D periodic array of optomechanical crystals. Each
optomechanical crystal cell is made up of photonic crystal and phononic
crystal defect cavities coplanarPainter . The optomechanical crystal cells
interact with each other by evanescent light field between them. The light
pulse drops in and out by side-coupling wave guides.
We consider an 1-D periodic array of optomechanical crystals, which consists
of $N$ evanescently coupled optomechanical crystals, shown in Figure 1. The
single optomechanical crystal cell, co-localizing photonic and phononic
resonances in a quasi one-dimensional optomechanical crystal, is proposed by
the research group led by PainterPainter . The mechanical modes of the
optomechanical crystal cell can be divided into common and differential modes
of in-plane and out-plane motion of these nanobeams. For simplicity, we just
consider the case that the gaps between the nanobeams are time-independent,
i.e. the common mode case. Therefore the coupling between neighboring optical
cavities is constant. To excite the system, a probe optical signal is dropped
in the optomechanical array in a side-coupled configuration, and the output
signal is dropped out in a similar manner. With this consideration, the
Hamiltonian of the system in the reference frame rotating with probe laser
frequency $\omega_{p}$ can be written as
$\displaystyle H$ $\displaystyle=H_{0}+H_{int}$ (1a) $\displaystyle H_{0}$
$\displaystyle=\sum_{i}\hbar\delta\hat{a}_{j}^{\dagger}\hat{a}_{j}+\sum_{j}\hbar\omega_{m}\hat{b}_{j}^{\dagger}\hat{b}_{j}$
(1b) $\displaystyle H_{int}$ $\displaystyle=\sum_{j}\hbar
g\left(\hat{b}_{j}^{\dagger}+\hat{b}_{j}\right)\hat{a}_{j}^{\dagger}\hat{a}_{j}$
$\displaystyle-\sum_{j}\hbar
G\left(\hat{a}_{j}^{\dagger}\hat{a}_{j+1}+\hat{a}_{j+1}^{\dagger}\hat{a}_{j}\right).$
(1c) here $\hat{a}_{j}^{\dagger}\left(\hat{a}_{j}\right)$ and
$\hat{b}_{j}^{\dagger}\left(\hat{b}_{j}\right)$ are creation (annihilation)
operators of optical cavity mode and mechanical mode in the $j$-th
optomechanical cell, respectively. $\delta=\omega_{c}-\omega_{p}$ is the
detuning between cavity field and probe laser, $\omega_{m}$ is the mechanical
resonator angular frequency, the constant $g$ is the coupling strength between
cavity and mechanical resonator and $G$ denotes the nearest neighboring
evanescently coupling of intercavity.
When the intracavity fields have a large amplitude, i.e. in the strong-driving
limit, we can linearize the Hamiltonian by setting $\hat{f}=\left\langle
f\right\rangle+\delta\hat{f}$
($f=\hat{a}_{j},\hat{a}_{j}^{\dagger},\hat{b}_{j},\hat{b}_{j}^{\dagger}$),
where $\left\langle f\right\rangle$ is the steady mean value and
$\delta\hat{f}$ is the corresponding fluctuation around its steady value. With
this ansatz, we then obtain the linearized Hamiltonian
$\displaystyle H_{0}$ $\displaystyle=\sum_{j}\hbar\omega_{m}\delta
b_{j}^{\dagger}\delta b_{j}+\sum_{i}\hbar\tilde{\delta}\delta
a_{j}^{\dagger}\delta a_{j}\allowbreak$ (2) $\displaystyle H_{int}$
$\displaystyle=\sum_{j}\hbar\tilde{g}\left(\allowbreak\delta
a_{j}^{\dagger}\allowbreak\delta b_{j}+\delta a_{j}\delta
b_{j}^{\dagger}\allowbreak\right)$ $\displaystyle-\sum_{j}\hbar
G\left(\delta\hat{a}_{j}^{\dagger}\delta\hat{a}_{i+1}+\delta\hat{a}_{j+1}^{\dagger}\delta\hat{a}_{j}\right)$
(3)
where the effective detuning is $\tilde{\delta}=\delta+g\left(\left\langle
b\right\rangle+\left\langle b^{\ast}\right\rangle\right)/2$ and the effective
coupling between light and mechanical vibration is $\tilde{g}=g\left\langle
a\right\rangle$. In Eq. (3) we have omitted the counter-rotational wave term
in the interaction between the cavity field and mechanical vibration.
## III Dispersion of polariton
Let us study the Hamiltonian in $k$-representation. Taking into account the
periodic properties of the system, we can make Fourier transformations
$\displaystyle A_{k}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j}\delta\hat{a}_{j}e^{ikjL}$ (4a)
$\displaystyle A_{k}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j}\delta\hat{a}_{j}^{\dagger}e^{-ikjL}$
(4b) $\displaystyle B_{k}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j}\delta\hat{b}_{j}e^{ikjL}$ (4c)
$\displaystyle B_{k}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{j}\delta\hat{b}_{j}^{\dagger}e^{-ikjL}$
(4d) where $A_{k}\left(A_{k}^{\dagger}\right)$ are the normal mode operators
of the coupled optical cavity, $B_{k}\left(B_{k}^{\dagger}\right)$ are the
boson operators to describe the collective excitation (phonon) of mechanical
resonators, $k=2\pi n/LN$ with $n=0,1...N-1$, and $L$ is the distance of
inter-cavity. Inserting above transformation relation into Eqs. (1a), we
arrive at the new Hamiltonian $\displaystyle H$
$\displaystyle=\sum_{k}\hbar\omega_{m}B_{k}^{\dagger}B_{k}+\sum_{k}\hbar\omega_{ph}\left(k\right)A_{k}^{\dagger}A_{k}$
$\displaystyle+\sum_{k}\hbar\tilde{g}\allowbreak\left(\allowbreak
A_{k}^{\dagger}\allowbreak B_{k}+B_{k}^{\dagger}\allowbreak
A_{k}\allowbreak\right)$ (5)
here $\omega_{ph}\left(k\right)=\tilde{\delta}-2G\cos\left(kL\right)$ is the
original dispersion property of photon dependent on quasimomentum $k$ in the
side-coupled photonic crystal cavity array. Hamiltonian (5) describes the
interaction of the photonic and phononic modes.
To decouple the Hamiltonian (5), we introduce the Bogoliubov transformation
$\displaystyle A_{k}$ $\displaystyle=uC_{k}+vD_{k}$ (6a) $\displaystyle
A_{k}^{\dagger}$ $\displaystyle=uC_{k}^{\dagger}+vD_{k}^{\dagger}$ (6b)
$\displaystyle B_{k}$ $\displaystyle=vC_{k}-uD_{k}$ (6c) $\displaystyle
B_{k}^{\dagger}$ $\displaystyle=vC_{k}^{\dagger}-uD_{k}^{\dagger}.$ (6d) and
the inverse transformation is $\displaystyle C_{k}$
$\displaystyle=uA_{k}+vB_{k}$ (7a) $\displaystyle C_{k}^{\dagger}$
$\displaystyle=uA_{k}^{\dagger}+vB_{k}^{\dagger}$ (7b) $\displaystyle D_{k}$
$\displaystyle=vA_{k}-uB_{k}$ (7c) $\displaystyle D_{k}^{\dagger}$
$\displaystyle=vA_{k}^{\dagger}-uB_{k}^{\dagger}$ (7d) Since operators $C_{k}$
and $D_{k}$ must satisfy Bosonic commutation relations
$\displaystyle\left[C_{k},C_{k}^{\dagger}\right]$ $\displaystyle=1$ (8)
$\displaystyle\left[D_{k},D_{k}^{\dagger}\right]$ $\displaystyle=1,$ (9)
transformation coefficients $u$ and $v$ have the relation $u^{2}+v^{2}=1$.
Substituting Eqs. (6) into Eq. (5), the Hamiltonian can be rewritten as
$\displaystyle H$
$\displaystyle=\left(\omega_{ph}\left(k\right)v^{2}+\omega_{m}u^{2}-2\tilde{g}uv\right)D_{k}D_{k}^{\dagger}\allowbreak$
$\displaystyle+\left(\omega_{ph}\left(k\right)u^{2}+\omega_{m}v^{2}+2\tilde{g}uv\right)C_{k}C_{k}^{\dagger}$
$\displaystyle+\left(\left(\omega_{ph}\left(k\right)-\omega_{m}\right)uv+\tilde{g}\left(v^{2}-u^{2}\right)\right)D_{k}C_{k}^{\dagger}$
$\displaystyle+\left(\left(\omega_{ph}\left(k\right)-\omega_{m}\right)uv+\tilde{g}\left(v^{2}-u^{2}\right)\right)D_{k}^{\dagger}C_{k}.$
(10)
Obviously, if
$\left(\omega_{ph}\left(k\right)\allowbreak-\omega_{m}\right)uv+\tilde{g}\left(v^{2}\allowbreak-u^{2}\right)=0$
(11)
the Hamiltonian will have the diagonalized form
$H=\sum_{k}\hbar\omega_{D}\left(k\right)D_{k}^{\dagger}D_{k}+\sum_{k}\hbar\omega_{C}\left(k\right)C_{k}^{\dagger}C_{k}.$
(12)
From Eqs. (7) and (12) one can note that $C_{k}$ and $D_{k}$ operators
represent two types of elementary excitations (phonon-photon polaritons) in
coupled optomechanical array system, which is the result of the coherently
mixing of photons and phonons through coupling in each optomechanics cell. The
dispersion relations of lower and upper branches polaritons are determined by
$\displaystyle\omega_{C}\left(k\right)$
$\displaystyle=\frac{1}{2}\left(\omega_{m}+\omega_{ph}\left(k\right)-\sqrt{4\tilde{g}^{2}+\left(\omega_{ph}\left(k\right)-\omega_{m}\right)^{2}}\right)$
(13) $\displaystyle\omega_{D}\left(k\right)$
$\displaystyle=\frac{1}{2}\left(\omega_{m}+\omega_{ph}\left(k\right)+\sqrt{4\tilde{g}^{2}+\left(\omega_{ph}\left(k\right)-\omega_{m}\right)^{2}}\right).$
(14)
It is found that the original single optical band structure is split into two
bands owing to the interaction between photons and phonons.
## IV Slowing light with tunable band structure
The bandwidths of lower and upper branches, respectively, are
$\displaystyle\Delta_{WC}$
$\displaystyle=\omega_{C}\left(\pi\right)-\omega_{C}\left(0\right)$
$\displaystyle=2G+\Delta$ (15) $\displaystyle\Delta_{WD}$
$\displaystyle=\omega_{D}\left(\pi\right)-\omega_{D}\left(0\right)$
$\displaystyle=2G-\Delta$ (16)
where
$\Delta=\sqrt{\tilde{g}^{2}+\left(\Delta_{OM}/2-G\right)^{2}}-\sqrt{\tilde{g}^{2}+\left(\Delta_{OM}/2+G\right)^{2}}$
(17)
with the detuning $\Delta_{OM}=\tilde{\delta}-\omega_{m}$. Compared to the
original optical band, the bandwidth of the lower branch polariton is enlarged
and the upper one is compressed. Moreover, the most important thing is that
both of these bandwidths of lower and upper bands can be modulated by changing
parameters, such as $\tilde{g}$, $G$ and $\Delta_{OM}$. Figure 2 shows a
typical picture of the bandwidth of lower branch polariton dependent on
$\Delta_{OM}$, from which one can note that the bandwidth decreases with
increasing $\Delta_{OM}$. In more detail, when $\Delta_{OM}/2\ll-\tilde{g}$,
the bandwidth of lower band $\Delta_{WC}\simeq 4G$, corresponding to the
maximum bandwidth; when $\Delta_{OM}/2\gg\tilde{g}$, the bandwidth is
approximately equal to zero.
Figure 2: The bandwidth of lower branch polariton as a function of
$\Delta_{0}$. The insets are the corresponding band structures of polariton
when detuning (a)$\Delta_{0}=-100$; (b) $\Delta_{0}=0$ and (c)
$\Delta_{0}=100$. $G=1$, $g=5$, $\omega_{m}=100$. Here $\Delta_{OM}$ is in
units of $G$.
On the other hand, it is known that the group velocity of polaritons in a
lattice is related with dispersion
$\displaystyle v_{C,D}$ $\displaystyle=$
$\displaystyle\frac{\partial\omega_{C,D}\left(k\right)}{\partial k}$ (18)
$\displaystyle=$ $\displaystyle-
GL\sin\left(kL\right)\left(1\pm\frac{\Delta_{OM}-2G\cos\left(kL\right)}{\sqrt{4\tilde{g}^{2}+\left(\Delta_{OM}-2G\cos\left(kL\right)\right)^{2}}}\right).$
which is also dependent on parameters $\tilde{g}$, $G$ and $\Delta_{OM}$ and
therefore can be tuned. Such a tunable band structure leads to a tunable group
velocity. Figure 3 shows the lower branch polariton with a momentum
$kL=\pi/2$. We observe its group velocity decreases rapidly from its maximum
value $G$ and vanishes as $\Delta_{OM}$ increases. In fact, such tunable band
structure can play an important role in optical communication and quantum
memory, for example, Fan suggested using a tunable CROW to slow and stop light
pulseslowinglight .
Figure 3: Velocity of lower branch polariton. Other parameters are the same as
in Fig. 2.
Here we briefly demonstrate the process, taking the lower branch polariton as
an example, to slow the light pulse in optomechanical crystal array. To begin
with, the optical cavity is adjusted to be resonant with laser frequency, so
the detuning is $\Delta_{OM}/2=-\omega_{m}/2\ll-\tilde{g}$. At this point, the
bandwidth of lower branch is largest and can accommodate the entire light
pulse, and the lower branch polariton is made up of photons, shown in figure
4. After the pulse enters completely into the optomechanical array, we then
compress the bandwidth of lower branch polariton adiabatically by tuning the
resonance frequency of the optical cavity until
$\Delta_{OM}/2=\omega_{m}/2\gg\tilde{g}$. Further compressing the bandwidth,
more and more photons are converted to mechanical modes in the lower branch,
meanwhile, the velocity of polaritons slows down and approaches to zero.
From the point of conversion between photons and mechanical collective
excitations, one can also understand the mechanism of stoping light. Because
the total excitations number $N_{k}=B_{k}^{\dagger}B_{k}+A_{k}^{\dagger}A_{k}$
commutes with Hamiltonian $H$, when adjusting some parameters, such as
$\Delta_{OM}$, the total excitations number is conserved, while the numbers of
photons and mechanical collective excitations $A_{k}^{\dagger}A_{k}$ and
$B_{k}^{\dagger}B_{k}$ are not conserved due to not commuting with the
Hamiltonian. Hence the photons and mechanical collective excitations are
mutually convertible, which results in mapping the light onto the mechanical
vibration and vice versa. Figure 4 illustrates that the conversion between the
photons and mechanical collective excitations. The number of mechanical
collective excitation increases, from zero to unity, while the number of
photons decrease from unity to zero, with increasing the detuning
$\Delta_{OM}$.
Figure 4: Transformation coefficients in the lower branch polariton versus
detuning. The amplitude of $u$ and $v$ represent the ratio of photons and
phonons in the lower branch polariton. Solid and dashed lines represents $v$
and $u$ respectively. Other parameters are the same as in Fig. 2.
We keep in mind that the rate of tuning cavity frequency should be less than
the band gap between upper and lower branches, which is given by
$\Delta_{WCD}\left(k\right)=\omega_{D}\left(k\right)-\omega_{C}\left(k\right)$
(19)
This limitation avoids the polaritons in lower branch jumping up to the upper
branch. Figure 5 shows that the band gap first linearly decreases and then
linearly rises with increasing the detuning $\Delta_{OM}$. The minimum value
of band gap is about $2G$ when the detuning of optical cavity is resonant with
mechanical resonator.
Figure 5: Band gap between upper and lower branches versus detuning. There is
a minimum value when $\Delta_{OM}=0$. Other parameters are the same as in Fig.
2.
To practically stop light in the coupled optomechanics array, we tune the
detuning between optical cavity field and probe laser by adjusting the
refractive index of material,e.g. Si, which makes up of the optomechanical
crystal. In our scheme, the amplitude of detuning modulation is the same order
of magnitude of the mechanical resonance frequency, so the refractive index
shift should be
$\frac{\delta n}{n}\approx\frac{\delta\omega}{\omega_{0}}\sim 10^{-5}$ (20)
which is feasible in practical optoelectronic devicesmodulation . Here we have
taken the typical parameters $\omega_{c}/2\pi=200$ THz, $\omega_{m}/2\pi=10$
GHzPainter .
## V Conclusion
We have studied the model of light transmission in a spatial periodic
optomechanical crystal array. The optical cavities of the array evanescently
couple to each other one by one to form a CROW. In the strong driving limit,
we linearized the system and obtained the dispersion relations of lower and
upper branch polaritons with Bogoliubov transformation in the momentum space.
Our results show that the modulation of detuning between optical cavity and
laser light can vary the bandwidths of polaritons, which has been demonstrated
to be able to stop and release a light pulse.
## VI Acknowledgments
We thank Andrew Bolt for polishing English.
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|
arxiv-papers
| 2012-02-09T07:29:47 |
2024-09-04T02:49:27.273433
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhenglu Duan and Bixuan Fan",
"submitter": "Zhenglu Duan",
"url": "https://arxiv.org/abs/1202.1900"
}
|
1202.1969
|
# A note on the weighted $q$-Hardy-littlewood-type maximal operator with
respect to $q$-Volkenborn integral in the $p$-adic integer ring
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com and Mehmet
Acikgoz University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr
###### Abstract.
The essential aim of this paper is to define weighted $q$-Hardy-littlewood-
type maximal operator by means of $p$-adic $q$-invariant distribution on
$\mathbb{Z}_{p}$. Moreover, we give some interesting properties concerning
this type maximal operator.
###### Key words and phrases:
$q$-Volkenborn integral, Hardy-littlewood theorem, $p$-adic analysis,
$q$-analysis
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. Introduction, Definitions and Notations
Recently, $q$-analysis has served as a structure between mathematics and
physics. Therefore, there is a significant increase of activity in the area of
the $q$-analysis due to applications of the $q$-analysis in mathematics,
statistics and physics.
$p$-adic numbers also play a vital and important role in mathematics. $p$-adic
numbers were invented by the German mathematician Kurt Hensel [10], around the
end of the nineteenth century. In spite of their being already one hundred
years old, these numbers are still today enveloped in an aura of mystery
within the scientific community.
The $p$-adic $q$-integral (or $q$-Volkenborn integral) are originally
constructed by Kim [4]. The $q$-Volkenborn integral is used in Mathematical
Physics for example the functional equation of the $q$-Zeta function, the
$q$-Stirling numbers and $q$-Mahler theory of integration with respect to the
ring $\mathbb{Z}_{p}$ together with Iwasawa’s $p$-adic $q$-$L$ function.
T. Kim, by using $q$-Volkenborn integral, introduced a novel Lebesgue-Radon-
Nikodym type theorem. He is given some interesting properties concerning
Lebesgue-Radon-Nikodym theorem. After, Jang [9] defined $q$-extension of
Hardy-Littlewood-type maximal operator by means of $q$-Volkenborn integral on
$\mathbb{Z}_{p}.$ Next, Kim, Choi and Kim [7] defined weighted Lebesgue-Radon-
Nikodym theorem. They also gave some interesting properties of this type
theorem.
By the same motivation of the above studies, in this paper, we construct
weighted $q$-Hardy-littlewood-type maximal operator by the means of $p$-adic
$q$-integral on $\mathbb{Z}_{p}$. We also give some interesting properties of
this type operator.
Imagine that $p$ be a fixed prime number. Let $\mathbb{Q}_{p}$ be the field of
$p$-adic rational numbers and let $\mathbb{C}_{p}$ be the completion of
algebraic closure of $\mathbb{Q}_{p}$.
Thus,
$\mathbb{Q}_{p}=\left\\{x=\sum_{n=-k}^{\infty}a_{n}p^{n}:0\leq a_{n}\leq
p-1\right\\}.$
Then $\mathbb{Z}_{p}$ is an integral domain, which is defined by
$\mathbb{Z}_{p}=\left\\{x=\sum_{n=0}^{\infty}a_{n}p^{n}:0\leq a_{n}\leq
p-1\right\\},$
or
$\mathbb{Z}_{p}=\left\\{x\in\mathbb{Q}_{p}:\left|x\right|_{p}\leq 1\right\\}.$
In this paper, we assume that $q\in\mathbb{C}_{p}$ with
$\left|1-q\right|_{p}<1$ as an indeterminate.
The $p$-adic absolute value $\left|.\right|_{p}$, is normally defined by
$\left|x\right|_{p}=\frac{1}{p^{r}}\text{,}$
where $x=p^{r}\frac{s}{t}$ with
$\left(p,s\right)=\left(p,t\right)=\left(s,t\right)=1$ and $r\in\mathbb{Q}$.
A $p$-adic Banach space $B$ is a $\mathbb{Q}_{p}$-vector space with a lattice
$B^{0}$ ($\mathbb{Z}_{p}$-module) separated and complete for $p$-adic
topology, ie.,
$B^{0}\simeq\lim_{\overleftarrow{n\in\mathbb{N}}}B^{0}/p^{n}B^{0}\text{.}$
For all $x\in B$, there exists $n\in\mathbb{Z}$, such that $x\in p^{n}B^{0}$.
Define
$v_{B}\left(x\right)=\sup_{n\in\mathbb{N}\cup\left\\{+\infty\right\\}}\left\\{n:x\in
p^{n}B^{0}\right\\}\text{.}$
It satisfies the following properties:
$\displaystyle v_{B}\left(x+y\right)$ $\displaystyle\geq$
$\displaystyle\min\left(v_{B}\left(x\right),v_{B}\left(y\right)\right)\text{,}$
$\displaystyle v_{B}\left(\beta x\right)$ $\displaystyle=$ $\displaystyle
v_{p}\left(\beta\right)+v_{B}\left(x\right)\text{, if
}\beta\in\mathbb{Q}_{p}\text{.}$
Then, $\left\|x\right\|_{B}=p^{-v_{B}\left(x\right)}$ defines a norm on $B,$
such that $B$ is complete for $\left\|.\right\|_{B}$ and $B^{0}$ is the unit
ball.
A measure on $\mathbb{Z}_{p}$ with values in a $p$-adic Banach space $B$ is a
continuous linear map
$f\mapsto\int
f\left(x\right)\mu=\int_{\mathbb{Z}_{p}}f\left(x\right)\mu\left(x\right)$
from $C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$, (continuous function
on $\mathbb{Z}_{p}$) to $B$. We know that the set of locally constant
functions from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{p}$ is dense in
$C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ so.
Explicitly, for all $f\in C^{0}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$,
the locally constant functions
$f_{n}=\sum_{i=0}^{p^{n}-1}f\left(i\right)1_{i+p^{n}\mathbb{Z}_{p}}\rightarrow\text{
}f\text{ in }C^{0}\text{.}$
Now if $\mu\in\boldsymbol{D}_{0}\left(\mathbb{Z}_{p},\mathbb{Q}_{p}\right)$,
set
$\mu\left(i+p^{n}\mathbb{Z}_{p}\right)=\int_{\mathbb{Z}_{p}}1_{i+p^{n}\mathbb{Z}_{p}}\mu$.
Then $\int_{\mathbb{Z}_{p}}f\mu$ is given by the following “Riemann sums”
$\int_{\mathbb{Z}_{p}}f\mu=\lim_{n\rightarrow\infty}\sum_{i=0}^{p^{n}-1}f\left(i\right)\mu\left(i+p^{n}\mathbb{Z}_{p}\right)\text{.}$
T. Kim defined $\mu_{q}$ as follows:
$\mu_{q}\left(\xi+dp^{n}\mathbb{Z}_{p}\right)=\frac{q^{\xi}}{\left[dp^{n}\right]_{q}}$
and this can be extended to a distribution on $\mathbb{Z}_{p}$. This
distribution yields an integral in the case $d=1$.
So, $q$-Volkenborn integral was defined by T. Kim as follows:
$\displaystyle
I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q}}\sum_{\xi=0}^{p^{n}-1}f\left(\xi\right)q^{\xi}\text{,
(for details, see \cite[cite]{[\@@bibref{}{Kim 4}{}{}]},
\cite[cite]{[\@@bibref{}{Kim 5}{}{}]}). }$
Where $\left[x\right]_{q}$ is a $q$-extension of $x$ which is defined by
$\left[x\right]_{q}=\frac{1-q^{x}}{1-q}\text{,}$
note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x$ cf. [2], [3], [4], [5],
[9].
Let $d$ be a fixed positive integer with $\left(p,d\right)=1$. We now set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{n}}\mathbb{Z}/dp^{n}\mathbb{Z},$ $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle\mathbb{Z}_{p},$ $\displaystyle
X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p},$
$\displaystyle a+dp^{n}\mathbb{Z}_{p}$ $\displaystyle=$
$\displaystyle\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}p^{n}\right)\right\\},$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{n}$. For $f\in
UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$,
$\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{q}\left(x\right)=\int_{X}f\left(x\right)d\mu_{q}\left(x\right),$
(for details, see [8]).
By the meaning of $q$-Volkenborn integral, we consider below strongly $p$-adic
$q$-invariant distribution $\mu_{q}$ on $\mathbb{Z}_{p}$ in the form
$\left|\left[p^{n}\right]_{q}\mu_{q}\left(a+p^{n}\mathbb{Z}_{p}\right)-\left[p^{n+1}\right]_{q}\mu_{q}\left(a+p^{n+1}\mathbb{Z}_{p}\right)\right|<\delta_{n},$
where $\delta_{n}\rightarrow 0$ as $n\rightarrow\infty$ and $\delta_{n}$ is
independent of $a$. Let $f\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$,
for any $a\in\mathbb{Z}_{p}$, we assume that the weight function
$\omega\left(x\right)$ is defined by $\omega\left(x\right)=\omega^{x}$ where
$\omega\in\mathbb{C}_{p}$ with $\left|1-\omega\right|_{p}<1$. We define the
weighted measure on $\mathbb{Z}_{p}$ as follows:
(1.2)
$\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)=\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\xi}f\left(\xi\right)d\mu_{q}\left(\xi\right)$
where the integral is the $q$-Volkenborn integral. By (1.2), we easily note
that $\mu_{f,q}^{\left(\omega\right)}$ is a strongly weighted measure on
$\mathbb{Z}_{p}$. That is,
$\displaystyle\left|\left[p^{n}\right]_{q}\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)-\left[p^{n+1}\right]_{q}\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n+1}\mathbb{Z}_{p}\right)\right|_{p}$
$\displaystyle=$
$\displaystyle\left|\sum_{x=0}^{p^{n}-1}\omega^{x}f\left(x\right)q^{x}-\sum_{x=0}^{p^{n}}\omega^{x}f\left(x\right)q^{x}\right|_{p}$
$\displaystyle\leq$
$\displaystyle\left|\frac{f\left(p^{n}\right)\omega^{p^{n}}q^{p^{n}}}{p^{n}}\right|_{p}\left|p^{n}\right|_{p}$
$\displaystyle\leq$ $\displaystyle Cp^{-n}$
Thus, we get the following proposition.
###### Proposition 1.
For $f,g\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$, we have
$\mu_{\alpha f+\beta
g,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)=\alpha\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)+\beta\mu_{g,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)\text{.}$
where $\alpha,\beta$ are positive constants. Moreover,
$\left|\left[p^{n}\right]_{q}\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)-\left[p^{n+1}\right]_{q}\mu_{f,q}^{\left(\omega\right)}\left(a+p^{n+1}\mathbb{Z}_{p}\right)\right|\leq
Cp^{-n}$
where $C$ is positive constant.
Let $UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ be the space of uniformly
differentiable functions on $\mathbb{Z}_{p}$ with supnorm
$\left\|f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|f\left(x\right)\right|_{p}.$
The difference quotient $\Delta_{1}f$ of $f$ is the function of two variables
given by
$\Delta_{1}f\left(m,x\right)=\frac{f\left(x+m\right)-f\left(x\right)}{m},\text{
for all }x\text{, }m\in\mathbb{Z}_{p}\text{, }m\neq 0\text{.}$
A function $f:\mathbb{Z}_{p}\rightarrow\mathbb{C}_{p}$ is said to be a
Lipschitz function if there exists a constant $M>0$ $\left(\text{the Lipschitz
constant of }f\right)$ such that
$\left|\Delta_{1}f\left(m,x\right)\right|\leq M\text{ for all
}m\in\mathbb{Z}_{p}\backslash\left\\{0\right\\}\text{ and
}x\in\mathbb{Z}_{p}.$
The $\mathbb{C}_{p}$ linear space consisting of all Lipschitz function is
denoted by $Lip\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$. This space is a
Banach space with the respect to the norm
$\left\|f\right\|_{1}=\left\|f\right\|_{\infty}\mathop{\textstyle\bigvee}\left\|\Delta_{1}f\right\|_{\infty}$
(for more informations, see [1], [2], [3], [4], [5], [6], [9]). The main aim
of this paper is to define weighted $q$-extension of Hardy Littlewood type
maximal operator. Moreover, we show the boundedness of the weighted $q$-Hardy-
littlewood-type maximal operator in the $p$-adic integer ring.
## 2\. The weighted $q$-Hardy-littlewood-type maximal operator
In view of (1.2) and the definition of $p$-adic $q$-integral on
$\mathbb{Z}_{p}$, we now start with the following theorem.
###### Theorem 1.
Let $\mu_{q}^{\left(w\right)}$ be a strongly $p$-adic $q$-invariant in the
$p$-adic integer ring and $f\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$.
Then for any $n\in\mathbb{Z}$ and any $\xi\in\mathbb{Z}_{p}$, we have
$(1)$
$\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{\xi}{p^{n}}}f\left(\xi\right)q^{-\frac{\xi}{p^{n}}}d\mu_{q^{p^{-n}}}\left(\xi\right)=\frac{\omega^{\frac{a}{p^{n}}}}{\left[p^{n}\right]_{q^{p^{-n}}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right),$
$(2)$
$\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{\xi}{p^{n}}}d\mu_{q^{p^{-n}}}\left(\xi\right)=\frac{\left(1-q\right)\omega^{\frac{a}{p^{n}}}q^{\frac{a}{p^{n}}}}{\left(1-\omega
q\right)\left[p^{n}\right]_{q^{p^{-n}}}}\left(1+\frac{\log\omega}{\log
q}\right).$
###### Proof.
(1) We see use of (1) and (1.2)
$\displaystyle\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{\xi}{p^{n}}}f\left(\xi\right)q^{-\frac{\xi}{p^{n}}}d\mu_{q^{p^{-n}}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m+n}\right]_{q^{p^{-n}}}}\sum_{\xi=0}^{p^{m}-1}\omega^{\frac{a+p^{n}\xi}{p^{n}}}f\left(a+p^{n}\xi\right)q^{-\frac{a+p^{n}\xi}{p^{n}}}q^{\frac{a+p^{n}\xi}{p^{n}}}$
$\displaystyle=$
$\displaystyle\omega^{\frac{a}{p^{n}}}\lim_{m\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q^{p^{-n}}}\left[p^{m}\right]_{q}}\sum_{\xi=0}^{p^{m}-1}\omega^{\xi}q^{-\xi}f\left(a+p^{n}\xi\right)q^{\xi}$
$\displaystyle=$
$\displaystyle\frac{\omega^{\frac{a}{p^{n}}}}{\left[p^{n}\right]_{q^{p^{-n}}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right).$
(2) By the same method of (1), we easily see the following
$\displaystyle\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{\xi}{p^{n}}}d\mu_{q^{p^{-n}}}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m+n}\right]_{q^{p^{-n}}}}\sum_{\xi=0}^{p^{m}-1}\omega^{\frac{a+\xi
p^{n}}{p^{n}}}q^{\frac{a+\xi p^{n}}{p^{n}}}$ $\displaystyle=$
$\displaystyle\frac{\omega^{\frac{a}{p^{n}}}q^{\frac{a}{p^{n}}}}{\left[p^{n}\right]_{q^{p^{-n}}}}\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{q}}\sum_{\xi=0}^{p^{m}-1}\omega^{\xi}q^{\xi}$
$\displaystyle=$
$\displaystyle\frac{\omega^{\frac{a}{p^{n}}}q^{\frac{a}{p^{n}}}\left(1-q\right)}{\left[p^{n}\right]_{q^{p^{-n}}}\left(1-\omega
q\right)}\lim_{m\rightarrow\infty}\frac{1-\left(\omega
q\right)^{p^{m}}}{1-q^{p^{m}}}$ $\displaystyle=$
$\displaystyle\frac{\omega^{\frac{a}{p^{n}}}q^{\frac{a}{p^{n}}}\left(1-q\right)}{\left[p^{n}\right]_{q^{p^{-n}}}\left(1-\omega
q\right)}\left(1+\frac{\log\omega}{\log q}\right)$
Since $\underset{m\rightarrow\infty}{\lim}q^{p^{m}}=1$ for
$\left|1-q\right|_{p}<1,$ our assertion follows.
Now, we are ready to give definition of weighted $q$-extension of Hardy-
littlewood-type maximal operator related to $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ with a strong $p$-adic $q$-invariant distribution $\mu_{q}$
in the $p$-adic integer ring.
###### Definition 1.
Let $\mu_{q}^{\left(\omega\right)}$ be a strongly $p$-adic $q$-invariant
distribution in the $p$-adic integer ring and $f\in
UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$. Then weighted $q$-extension of
the Hardy-littlewood-type maximal operator with respect to $p$-adic
$q$-integral on $a+p^{n}\mathbb{Z}_{p}$ is defined by
$M_{p,q}^{\left(\omega\right)}f\left(a\right)=\underset{n\in\mathbb{Z}}{\sup}\frac{1}{\mu_{1,q^{p^{-n}}}^{\left(w^{p^{-n}}\right)}\left(x+p^{n}\mathbb{Z}_{p}\right)}\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{x}{p^{n}}}q^{-\frac{x}{p^{n}}}f\left(x\right)d\mu_{q^{\frac{1}{p^{n}}}}\left(x\right)$
for all $a\in\mathbb{Z}_{p}$.
We recall that famous Hardy-littlewood maximal operator $M_{\mu}$ is defined
by
(2.1) $M_{\mu}f\left(a\right)=\underset{a\in
Q}{\sup}\frac{1}{\mu\left(Q\right)}\int_{Q}\left|f\left(x\right)\right|d\mu\left(x\right),$
where $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a locally bounded
Lebesgue measurable function, $\mu$ is a Lebesgue measure on
$\left(-\infty,\infty\right)$ and the supremum is taken over all cubes $Q$
which are parallel to the coordinate axes. Note that the boundedness of the
Hardy-Littlewood maximal operator serves as one of the most important tools
used in the investigation of the properties of variable exponent spaces (see
[9]). The essential aim of Theorem 1 is to deal with the weighted
$q$-extension of the classical Hardy-Littlewood maximal operator in the space
of $p$-adic Lipschitz functions on $\mathbb{Z}_{p}$ and to find the
boundedness of them. By the meaning of Definition 1, we get the following
theorem.
###### Theorem 2.
Let $f\in UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ and
$x\in\mathbb{Z}_{p}$, we get
(1) $M_{p,q}^{\left(\omega\right)}f\left(a\right)=\frac{1-\omega
q}{1-q}\frac{\log q}{\log\left(\omega
q\right)}\underset{n\in\mathbb{Z}}{\sup}q^{-\frac{x}{p^{n}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(x+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right)$,
(2)
$\left|M_{p,q}^{\left(\omega\right)}f\left(a\right)\right|_{p}\leq\underset{n\in\mathbb{Z}}{\sup}\left|\frac{1-wq}{q^{\frac{x}{p^{n}}}}\frac{\log
q}{\log\left(\omega
q\right)}\right|_{p}\left\|f\right\|_{1}\left\|\left(\frac{q}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}$,
where
$\left\|\left(\frac{q}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}=\int_{\mathbb{Z}_{p}}\left(\frac{q}{\omega}\right)^{-\xi}d\mu_{q}\left(\xi\right)$.
###### Proof.
(1) Because of Theorem 1 and Definition 1, we see
$\displaystyle M_{p,q}^{\left(\omega\right)}f\left(a\right)$ $\displaystyle=$
$\displaystyle\underset{n\in\mathbb{Z}}{\sup}\frac{1}{\mu_{1,q^{p^{-n}}}^{\left(w^{p^{-n}}\right)}\left(x+p^{n}\mathbb{Z}_{p}\right)}\int_{a+p^{n}\mathbb{Z}_{p}}\omega^{\frac{x}{p^{n}}}q^{-\frac{x}{p^{n}}}f\left(x\right)d\mu_{q^{\frac{1}{p^{n}}}}\left(x\right)$
$\displaystyle=$
$\displaystyle\underset{n\in\mathbb{Z}}{\sup}\frac{\left(1-\omega
q\right)\left[p^{n}\right]_{q^{p^{-n}}}\log
q}{\left(1-q\right)\omega^{\frac{x}{p^{n}}}q^{\frac{x}{p^{n}}}\log\left(\omega
q\right)}\frac{\omega^{\frac{x}{p^{n}}}}{\left[p^{n}\right]_{q^{p^{-n}}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right)$
$\displaystyle=$ $\displaystyle\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\underset{r\in\mathbb{Z}}{\sup}\frac{1}{q^{\frac{x}{p^{n}}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right)$
(2) On account of (1), we can derive the following
$\displaystyle\left|M_{p,q}^{\left(\omega\right)}f\left(a\right)\right|_{p}$
$\displaystyle=$ $\displaystyle\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\underset{n\in\mathbb{Z}}{\sup}\frac{1}{q^{\frac{x}{p^{n}}}}\int_{\mathbb{Z}_{p}}\omega^{\xi}f\left(a+p^{n}\xi\right)q^{-\xi}d\mu_{q}\left(\xi\right)\right|_{p}$
$\displaystyle\leq$ $\displaystyle\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|q^{-\frac{x}{p^{n}}}\int_{\mathbb{Z}_{p}}f\left(a+p^{n}\xi\right)\left(\frac{q}{\omega}\right)^{-\xi}d\mu_{q}\left(\xi\right)\right|_{p}$
$\displaystyle\leq$ $\displaystyle\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|q^{-\frac{x}{p^{n}}}\right|_{p}\int_{\mathbb{Z}_{p}}\left|f\left(a+p^{n}\xi\right)\right|_{p}\left|\left(\frac{q}{\omega}\right)^{-\xi}\right|_{p}d\mu_{q}\left(\xi\right)$
$\displaystyle\leq$ $\displaystyle\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|q^{-\frac{x}{p^{n}}}\right|_{p}\left\|f\right\|_{1}\int_{\mathbb{Z}_{p}}\left|\left(\frac{q}{\omega}\right)^{-\xi}\right|_{p}d\mu_{q}\left(\xi\right)$
$\displaystyle=$ $\displaystyle\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|q^{-\frac{x}{p^{n}}}\right|_{p}\left\|f\right\|_{1}\left\|\left(\frac{q}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}.$
Thus, we complete the proof of theorem.
We note that Theorem 2 (2) shows the supnorm-inequality for the weighted
$q$-Hardy-Littlewood-type maximal operator in the $p$-adic integer ring, in a
word, Theorem 2 (2) shows the following inequality
(2.2)
$\left\|M_{p,q}^{\left(\omega\right)}f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|M_{p,q}^{\left(\omega\right)}f\left(x\right)\right|_{p}\leq
K\left\|f\right\|_{1}\left\|\left(\frac{q}{\omega}\right)^{-\left(.\right)}\right\|_{L^{1}}$
where $K=\left|\frac{\left(1-\omega q\right)\log
q}{\left(1-q\right)\log\left(\omega
q\right)}\right|_{p}\underset{n\in\mathbb{Z}}{\sup}\left|q^{-\frac{x}{p^{n}}}\right|_{p}$.
By the equation (2.2), we get the following Corollary, which is the
boundedness for weighted $q$-Hardy-Littlewood-type maximal operator in the
$p$-adic integer ring.
###### Corollary 1.
$M_{p,q}^{\left(\omega\right)}$ is a bounded operator from
$UD\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ into
$L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$, where
$L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$ is the space of all
$p$-adic supnorm-bounded functions with the
$\left\|f\right\|_{\infty}=\underset{x\in\mathbb{Z}_{p}}{\sup}\left|f\left(x\right)\right|_{p}\text{,}$
for all $f\in L^{\infty}\left(\mathbb{Z}_{p},\mathbb{C}_{p}\right)$.
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* [8] T. Kim, Non-archimedean q-integrals associated with multiple Changhee q-Bernoulli Polynomials, Russ. J. Math Phys. 10 (2003) 91-98.
* [9] L-C. Jang, On the $q$-extension of the Hardy-littlewood-type maximal operator related to $q$-Volkenborn integral in the $p$-adic integer ring, Journal of Chungcheon Mathematical Society, Vol. 23, No. 2, June 2010.
* [10] K. Hensel, Theorie der Algebraischen Zahlen I. Teubner, Leipzig, 1908.
* [11] N. Koblitz, $p$-adic Numbers, $p$-adic Analysis and Zeta Functions, Springer-Verlag, New York Inc, 1977.
|
arxiv-papers
| 2012-02-09T13:06:35 |
2024-09-04T02:49:27.284652
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci and Mehmet Acikgoz",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1202.1969"
}
|
1202.1974
|
###### Abstract
Let $K_{m[n]}$ be the complete multipartite graph with $m$ parts, while each
part contains $n$ vertices. The orientably-regular embeddings of complete
graphs $K_{m[1]}$ have been determined by Biggs (1971) [1], James and Jones
(1985) [14]. During the past twenty years, several papers such as Du et
al.(2007, 2010) [6, 7], Jones et al. (2007, 2008) [16, 17], Kwak and Kwon
(2005, 2008) [18, 19] and Nedela et al. (1997, 2002)[21, 22] contributed to
the orientably-regular embeddings of complete bipartite graphs $K_{2[n]}$ and
the final classification was given by Jones [15] in 2010. Based on our former
paper [25], this paper gives a complete classification of orientably-regular
embeddings of graphs $K_{m[n]}$ for the general cases $m\geq 3$ and $n\geq 2$.
A Classification of Orientably-Regular Embeddings
of Complete Multipartite Graphs
Shaofei Dua and Jun-Yang Zhanga,b,
aSchool of Mathematical Sciences,
Capital Normal University, Beijing 100048, P.R.China
b Department of Mathematics and Information Science,
Zhangzhou Normal University, Zhangzhou, Fujian 363000, P.R.China
††footnotetext: Supported by NNSF(10971144).
## 1 Introduction
A (topological) _map_ is a cellular decomposition of a closed surface. A
common way to describe such a map is to view it as a 2-cell embedding of a
connected graph or multigraph $\Gamma$ into the surface $S$. The components of
the complement $S\setminus\Gamma$ are simply-connected regions called the
_faces_ of the map (or the embedding). An _automorphism_ of a map ${\cal M}$
is an automorphism of the underlying (multi)graph $\Gamma$ which extends to a
self-homeomorphism of the supporting surface $S$. It is well known that the
automorphism group $\hbox{\rm Aut}({\cal M})$ of a map ${\cal M}$ acts semi-
regularly on the set of all incident vertex-edge-face triples (or _flags_ of
$\Gamma$). In particular, if $\hbox{\rm Aut}({\cal M})$ acts regularly on the
flags, we call it a regular map. In the orientable case, if the group of all
orientation-preserving automorphisms of ${\cal M}$ acts regularly on the set
of all incident vertex-edge pairs (or _arcs_) of ${\cal M}$, then we call
${\cal M}$ an _orientably regular_ map. Such maps fall into two classes: those
that admit also orientation-reversing automorphisms, which are called
_reflexible_ , and those that do not, which are _chiral_. Therefore, a
reflexible map is a regular map but a chiral map is not.
One of the central problems in topological graph theory is to classify all the
regular embeddings in orientable or nonorientble surfaces of a given graph. In
a general setting, the classification problem was treated by Gardiner, Nedela,
Širáň and Škoviera in [10]. However, for particular classes of graphs, it has
been solved only in a few cases. Let $K_{m[n]}$ be the complete multipartite
graph with $m$ parts, while each part contains $n$ vertices. All the regular
embeddings of complete graphs $K_{m[1]}$ have been determined by Biggs, James
and Jones [1, 14] for orientably case and by Wilson [24] for nonorentably
case. As for the complete bipartite graphs $K_{2[n]}$, the nonoritenably
regular embeddings of these graphs have recently been classified by Kwak and
Kwon [20]; during the past twenty years, several papers [6, 7, 16, 17, 18, 19,
22] contributed to the orientably case, and the final classification was given
by Jones [15] in 2010. Since then, the classification for general case $m\geq
3$ and $n\geq 2$ has become an attractive topic in this research field. The
only known result is the determination of such embeddings for $n=p$ a prime,
given by Du, Kwak and Nedela in [9].
In this paper, we shall classify the orientably-regular embeddings of complete
multipartite graphs. A start point is the main result in our former paper
[25], namely the following reduction theorem.
###### Proposition 1.1 (reduction theorem).
Let $\mathcal{M}$ be an orientably-regular embedding of $K_{m[n]}$ where
$m\geq 3$ and $n\geq 2$, with the group $\hbox{\rm Aut}^{+}(\mathcal{M})$ of
all orientation-preserving automorphisms. Let $\hbox{\rm
Aut}^{+}_{0}(\mathcal{M})$ be the normal subgroup of $\hbox{\rm
Aut}^{+}(\mathcal{M})$ consisting of automorphisms preserving each part
setwise. Then $\hbox{\rm Aut}^{+}_{0}(\mathcal{M})$ is an isobicyclic group.
Moreover, we have
1. (1)
if $m\geq 4$, then $m=p$ and $n=p^{e}$ for some prime $p$; or
2. (2)
if $m=3$, then $\hbox{\rm Aut}^{+}_{0}(\mathcal{M})=Q\times K$, where $Q$ is a
3-subgroup (may be trivial) and $K$ is an abelian $3^{\prime}$-subgroup.
In Proposition 1.1, an isobicyclic group means a group $H=\langle
x\rangle\langle y\rangle$, where $|x|=|y|=n$, $\langle x\rangle\cap\langle
y\rangle=1$ and there exists an involution $\alpha\in\hbox{\rm Aut}(H)$ such
that $x^{\alpha}=y$. Throughout the paper, we call $(H,x,y)$ a $n$-isobicyclic
triple and it plays an important role in the classification of orientably-
regular embeddings of $K_{m[n]}$.
As usual, the orientably-regular map will be presented by a triple $(G;a,b)$
for a group $G$ generated by an element $a$ and an involution $b$ (see Section
2 for the details). The following is the main theorem of this paper.
###### Theorem 1.2 (classification theorem).
For $m\geq 3$ and $n\geq 2$, let $K_{m[n]}$ be the complete multipartite graph
with $m$ parts, while each part contains $n$ vertices. Suppose that ${\cal M}$
is an orientably-regular embedding of $K_{m[n]}$ with the group $G$ of all
orientation-preserving automorphisms. Then $G$ and ${\cal M}$ are given by
1. (1)
$m=p\geq 5$, $n=p^{e}$ for a prime $p$:
$G_{1}(p,e)=\langle a,c|a^{p^{e}(p-1)}=c^{p^{e+1}}=1,c^{a}=c^{r}\rangle$,
where $r$ is a given generator of $\mathbb{Z}_{p^{e+1}}^{*}$;
$\mathcal{M}_{1}(p,e,j)={\cal
M}\big{(}G_{1}(p,e);a^{j},a^{\frac{p^{e}(p-1)}{2}}c\big{)}$, where
$j\in\mathbb{Z}_{p^{e}(p-1)}^{*}$.
2. (2)
$m=n=p\geq 5$ for a prime $p$:
$\begin{array}[]{rl}G_{2}(p)=&\langle w,z\rangle\rtimes\langle
c,g\rangle=\langle w,z,c,g\mid
w^{p}=z^{p}=c^{p}=g^{p-1}=1,[w,z]=1,c^{g}=c^{t},\\\ &\hskip
136.57323ptw^{c}=wz,z^{c}=z,w^{g}=w,z^{g}=z^{t}\rangle,\end{array}$
where $t$ is a given generator of $\mathbb{Z}_{p}^{*}$;
$\mathcal{M}_{2}(p,j)=\mathcal{M}\big{(}G_{2}(p);wg^{j},cg^{\frac{p-1}{2}}\big{)}$,
where $j\in\mathbb{Z}_{p-1}^{*}$.
3. (3)
$m=p=3$, $n=k3^{e}$ for $3\nmid k$ and $e\neq 0$:
$\begin{array}[]{rl}G_{3}(k,e)=&\langle a,b\mid a^{2\cdot
3^{e}k}=b^{2}=1,c=a^{3^{e}}b,a^{2\cdot
3^{e}}=x_{1},x_{1}^{b}=y_{1},[x_{1},y_{1}]=1,c^{3^{e+1}}=1,\\\ &\hskip
28.45274pty_{1}^{a}=x_{1}^{-1}y_{1}^{-1},c^{a}=c^{2}x_{1}^{u}y_{1}^{\frac{1-3^{e}}{2}u}\rangle,\end{array}$
where, if $k=1$, then $c^{a}=c^{2}$; if $k\geq 2$, then $u3^{e}\equiv
1\pmod{k}$;
$\mathcal{M}_{3}(k,e,j)=\mathcal{M}(G_{3}(k,e);a^{j},b)$, where
$j\in\mathbb{Z}_{2\cdot 3^{e}k}^{*}$ and
$\mathcal{M}_{3}(k,e,j_{1})\cong\mathcal{M}_{3}(k,e,j_{2})$ if and only if
$j_{1}\equiv j_{2}\pmod{2\cdot 3^{e}}$.
4. (4)
$m=p=3$, $n=3^{e}k$ for $3\nmid k$:
$\begin{array}[]{rl}G_{4}(k,e,i,l)=&\langle a,b\mid
a^{2n}=b^{2}=1,a^{2}=x,x^{b}=y,[x,y]=x^{\frac{in}{3}}y^{-\frac{in}{3}},y^{a}=x^{-1}y^{-1},\\\
&\hskip
25.60747pt(ab)^{3}=x^{\frac{ln}{3}}y^{-\frac{ln}{3}}\rangle,\end{array}$
where, if $e=0$ then $(i,l)=0$; if $e=1$ then $(i,l)=(0,0)$, $(0,1)$; if
$e\geq 2$ then $(i,l)=(0,0),(0,1),(1,0),(1,1)$ or $(1,-1)$;
$\mathcal{M}_{4}(k,e,i,l,j)=\mathcal{M}(G_{4}(k,e,i,l);a^{j},b)$, where $j=1$
for $(i,l)=(0,0)$ and $j=\pm 1$ for other cases;
The above maps are unique determined by the given parameters. The following
two tables give the enumerations for these maps.
Table 1: Enumerations of the resulting maps
---
Maps | Number | Reflexible | Type
| | or Chiral |
$\mathcal{M}_{1}(p,e,j)$ | $p^{e-1}(p-1)\phi(p-1)$ | C | $\\{p^{e}(p-1),p^{e}(p-1)\\}$,
| | | if $p\equiv 1~{}\pmod{4}$;
| | | $\\{\frac{p^{e}(p-1)}{2},p^{e}(p-1)\\}$,
| | | if $p\equiv 3~{}\pmod{4}$.
$\mathcal{M}_{2}(p,j)$, $p\geq 5$ | $\phi(p-1)$ | C | $\\{p(p-1),p(p-1)\\}$,
| | | if $p\equiv 1~{}\pmod{4}$;
| | | $\\{\frac{p(p-1)}{2},p(p-1)\\}$,
| | | if $p\equiv 3~{}\pmod{4}$.
$\mathcal{M}_{3}(k,e,j)$ | $2\cdot 3^{e-1}$ | C | $\\{3^{e+1},2n\\}$
$\mathcal{M}_{4}(k,e,0,0,1)$ | $1$ | R | $\\{3,2n\\}$
$\mathcal{M}_{4}(k,e,0,1,\pm 1)$ | $2$ | C | $\\{9,2n\\}$
$\mathcal{M}_{4}(k,e,1,0,\pm 1)$ | $2$ | C | $\\{3,2n\\}$
$\mathcal{M}_{4}(k,e,1,\pm 1,\pm 1)$ | $4$ | C | $\\{9,2n\\}$
Table 2: Total numbers of regular embeddings of $K_{m[n]}$
---
m | n | Reflexible | Chiral | Total
$3$ | $k$ | $1$ | $0$ | $1$
| $3k$ | $1$ | $2$ | $3$
| $3^{e}k(e\geq 2)$ | $1$ | $2\cdot 3^{e-1}+8$ | $2\cdot 3^{e-1}+9$
$p\geq 5$ | $p$ | $0$ | $p\phi(p-1)$ | $p\phi(p-1)$
| $p^{e}(e\geq 2)$ | $0$ | $p^{e-1}(p-1)\phi(p-1)$ | $p^{e-1}(p-1)\phi(p-1)$
###### Remark 1.3.
From the Theorem 1.2 and its proof in Sections 3 and 4, we may obtain some
remarks as follows.
1. (1)
$\mathcal{M}_{1}(p,e,j)$ and $\mathcal{M}_{3}(1,e,j)$ are two families of
Cayley maps.
2. (2)
Let $P$ be a Sylow $m$-subgroup of $G=\hbox{\rm Aut}^{+}({\cal M})$ ($m=p\geq
5$ or $m=3$) and let $\hbox{\rm Exp}(P)$ be the exponent of $P$. Then
$\hbox{\rm Exp}(P)=m^{e+1}$ for the groups $G_{1}(p,e)$ or $G_{3}(k,e)$; and
$\hbox{\rm Exp}(P)=m^{e}$ for the groups $G_{2}(p)$ or $G_{4}(k,e,i,l)$.
3. (3)
For $m=p\geq 5$, $H=\hbox{\rm Aut}^{+}_{0}(\mathcal{M})$ is a
$p^{e}$-isobicyclic group and hence $H^{\prime}$ must be a cyclic group of
order at most $p^{e-1}$. It is interesting that either
$H^{\prime}\cong\mathbb{Z}_{p^{e-1}}$ (with the biggest possible order) or
$H^{\prime}=1$ (with the smallest possible order).
Similarly, for $m=3$, $\hbox{\rm Aut}^{+}_{0}(\mathcal{M})=Q\times K$ where
$K$ is abelian, and we have that either $Q^{\prime}\cong\mathbb{Z}_{3^{e-1}}$
(with the biggest possible order), or $Q^{\prime}\lesssim\mathbb{Z}_{3}$ (with
the smallest or the second smallest possible order).
4. (4)
The orientably-regular embeddings of $K_{m[p]}$ for prime $p$ have been
classified in [8], and they are precisely the maps $\mathcal{M}_{1}(p,1,j)$,
$\mathcal{M}_{2}(p,j)$, ${\cal M}_{3}(1,1,j)$, $\mathcal{M}_{4}(1,1,0,0,j)$
and $\mathcal{M}_{4}(p,0,0,0,j)$.
The orientably-regular embeddings of $K_{3[n]}$ when $H=\hbox{\rm
Aut}^{+}_{0}(\mathcal{M})$ is abelian have been classified in [25], and they
are precisely the maps $\mathcal{M}_{4}(k,e,0,l,j)$.
By examining the Conder’s lists of orientably-regular maps of type from 2 to
101 (see [4]), one can see that there are 15 orientably-regular embeddings of
$K_{3[9]}$, which exactly coincides with our results here.
The paper is organized as follows. After this induction section, we describe
the orientably-regular maps in more details and give some preliminary results
for later use in Section 2. To classify the orientably-regular embeddings of
$K_{m[n]}$ for the general cases $m\geq 3$ and $n\geq 2$, by Proposition 1.1
we only need to consider the graphs $K_{p[p^{e}]}$ for $p\geq 5$ a prime and
the graphs $K_{3[n]}$ separately. These two cases will be dealt with in
Section 3 and Section 4 respectively. Finally, the proof of Theorem 1.2 is
summarized in Section 5.
## 2 Preliminaries
Throughout this paper, all graphs are finite, simple and undirected. For a
graph $\Gamma$, by $\mathrm{V}(\Gamma)$, $\mathrm{E}(\Gamma)$ and
$\mathrm{D}(\Gamma)$ we denote the vertex set, edge set and arc set of
$\Gamma$, respectively. For any positive integer $n$, set
$[n]=\\{1,\cdots,n\\}.$ For a prime divisor $p$ of $n$, by $p^{d}\|n$ we
denote that $p^{d}$ but $p^{d+1}\nmid n.$ For a ring $R$, we use $R^{*}$ to
denote the multiplicative group of $R$. The cyclic group of order $n$ as well
as the integer residue ring modulo $n$ will be denoted by $\mathbb{Z}_{n}$ and
the dihedral group of order $n$ will be denoted by $\mathbb{D}_{n}$. By
$\mathrm{Z}(G)$, $N_{G}(K)$ and $C_{G}(K)$ we denote the center of the group
$G$, the normalizer and the centralizer of a subgroup $K$ in $G$,
respectively. By $N\rtimes K$, we denote a semidirect product of $N$ by $K$
where $N$ is normal. For the notions not defined here, please refer [2, 12].
It is well known that the automorphism group $G=\hbox{\rm Aut}(\mathcal{M})$
of a regular map $\mathcal{M}$ is generated by a generator $a$ of the
stabilizer of a vertex $\gamma$ (which is necessarily cyclic) and an
involution $b$ inverting an edge incident with $\gamma$, see [10]. Moreover,
the embedding is determined by the group $G$ and the choice of generators $a$
and $b$ [21, 8].
If the underlying graph is simple, then we may describe it by so called coset
graphs. Let $G=\langle a,b\rangle$ be a group where $a^{t}=b^{2}=1$ and
$\langle a\rangle$ is core-free. Let $\Gamma=\hbox{\rm Cos}(G,\langle
a\rangle,b)$ be the coset graph with vertex set $\hbox{\rm
V}(\Gamma)=\\{\langle a\rangle g\mid g\in G\\}$ and arc set
$\mathrm{D}(\Gamma)=\\{(\langle a\rangle g,\langle a\rangle bg)\mid g\in
G\\}$. Then $G$ acts regularly on $\mathrm{D}(\Gamma)$ by right
multiplication, the stabilizer of the vertex $\langle a\rangle 1$ is the
subgroup $\langle a\rangle$ of $G$ and $b$ is an involution inverting the arc
$(\langle a\rangle 1,\langle a\rangle b)$. By defining the local rotation $R$
by $(\langle a\rangle g,\langle a\rangle bg)^{R}=(\langle a\rangle g,\langle
a\rangle bag)$, we get an orientably-regular map, called algebraic map,
denoted by $\mathcal{M}(G;a,b)$.
It is easy to show that two algebraic maps $\mathcal{M}(G;a,b)$ and
$\mathcal{M}(G;a^{\prime},b^{\prime})$ are isomorphic if and only if there is
a group automorphism in $\hbox{\rm Aut}(G)$ taking $a\mapsto a^{\prime}$ and
$b\mapsto b^{\prime}$. If the order of $ab$ and $a$ are $s$ and $t$
respectively, then $\mathcal{M}(G;a,b)$ has type $\\{s,t\\}$ in the notation
of Coxeter and Moser [3], meaning that the faces are all $s$-gons and the
vertices all have valency $t$. If $\mathcal{M}(G;a,b)$ and
$\mathcal{M}(G;a^{-1},b)$ are isomorphic maps, then $\mathcal{M}(G;a,b)$ is
reflexible, otherwise $\mathcal{M}(G;a,b)$ is chiral.
From the above arguments, one can transfer the classification problem of
regular embeddings of a given graph into a purely group theoretical problem.
More precisely, one may classify all the regular maps with a given underlying
graph $\Gamma$ of valency $t$ in the following two steps:
1. (1)
Find the representatives $G$ (as abstract groups) of the isomorphism classes
of arc-regular subgroups of $\hbox{\rm Aut}(\Gamma)$ with cyclic vertex-
stabilizers.
2. (2)
For each group $G$ given in (1), determine all the algebraic regular maps
$\mathcal{M}(G;a,b)$ with underlying graphs isomorphic to $\Gamma$, or
equivalently, determine the representatives of the orbits of $\hbox{\rm
Aut}(\Gamma)$ on the set of generating pairs $(a,b)$ of $G$ such that
$|\langle a\rangle|=t$, $|\langle b\rangle|=2$ and $\hbox{\rm Cos}(G,\langle
a\rangle,b)\cong\Gamma$.
Now we give two lemmas for later use.
###### Lemma 2.1.
Suppose that $m$ is an odd prime and $n>2$ is an integer. Let $G=\langle
a,b\rangle$ and $H=\langle x,y\rangle$ where $a^{m-1}=x$ and $x^{b}=y$. If
$(H,x,y)$ is a $n$-isobicyclic triple, $H\unlhd G$, $G/H\cong\hbox{\rm
AGL}(1,m)$ and $C_{G}(H)=\mathrm{Z}(H)$, then $\mathcal{M}(G;a,b)$ is a
regular embedding of $K_{m[n]}$.
###### Proof.
It suffices to show that the coset graph $\Gamma=\hbox{\rm Cos}(G,\langle
a\rangle,b)$ is a complete $m$-partite graph. Since $(H,x,y)$ is a
$n$-isobicyclic triple, we have $H=\langle x\rangle\langle y\rangle$, $\langle
x\rangle\bigcap\langle y\rangle=1$ and $|H|=n^{2}$. Noting that
$|G|=|G/H||H|=|\hbox{\rm AGL}(1,m)||H|=m(m-1)n^{2}$, we have $|G:\langle
a\rangle|=mn$. Since $a^{m-1}=x$, $(a^{b})^{m-1}=(a^{m-1})^{b}=y$, $H=\langle
x,y\rangle$ and $C_{G}(H)=\mathrm{Z}(H)$, we have $\langle
a\rangle\bigcap\langle a^{b}\rangle\leq C_{G}(x)\cap
C_{G}(y)=C_{G}(H)=\mathrm{Z(}H)\leq H$. It follows that $\langle
a\rangle\bigcap\langle a^{b}\rangle\leq H\cap\langle a\rangle\bigcap\langle
a^{b}\rangle=\langle x\rangle\bigcap\langle y\rangle=1$, that is $\langle
a\rangle\bigcap\langle a^{b}\rangle=1$. Therefore $\Gamma$ is a simple graph
of order $mn$ and valency $(m-1)n$.
Set $\Delta=\\{\langle a\rangle h\mid h\in H\\}$ and $\Sigma=\\{\Delta g\mid
g\in G\\}.$ Then $\Sigma$ is a block system for $G$ acting on $\hbox{\rm
V}(\Gamma)$. Since $|\langle a\rangle H:\langle a\rangle|=|\langle
a\rangle\langle x\rangle\langle y\rangle:\langle a\rangle|=|\langle
a\rangle\langle y\rangle:\langle a\rangle|=n$, we have $|\Delta|=n$ and then
$|\Sigma|=m$. Clearly, $y^{i-j}\notin b\langle a\rangle$ for any two integers
$i$ and $j$ in $[n]$. Noting that $\Delta=\\{\langle a\rangle h\mid h\in
H\\}=\\{\langle a\rangle y^{i}\mid i\in[n]\\}$, $\Delta$ contains no pair of
adjacent vertices. Therefore $\Gamma$ is a complete $m$-partite graph.∎
###### Lemma 2.2.
Let $q=1+p^{f}$ where $p$ is a prime and $f\geq 1$ . If $p^{d}|k$ where $d\geq
1$, then
1. (1)
$(q^{k}-1)/(q-1)\equiv k\pmod{p^{d+f}}$;
2. (2)
$(q^{k+1}-1)/(q-1)\equiv k+1\pmod{p^{d+f}}$.
###### Proof.
(1) Since $(q^{k}-1)/(q-1)=k+\sum\limits_{i=2}^{k}\binom{k}{i}p^{(i-1)f}$, it
suffices to prove that $p^{d+f}|\binom{k}{i}p^{(i-1)f}$ for any $2\leq i\leq
k$.
The conclusion is clear for $i-1\geq\frac{d+f}{f}$ and so we assume that
$i-1<\frac{d+f}{f}$. Then $2\leq i<2+\frac{d}{f}\leq p^{d}$. Set
$k=k^{\prime}p^{d}$ and $i=i^{\prime}p^{d_{i}}$ where $(p,i^{\prime})=1$. Then
$0\leq d_{i}<d$ and
$\binom{k}{i}=\binom{k-1}{i-1}\frac{k}{i}=\binom{k-1}{i-1}\frac{k^{\prime}}{i^{\prime}}p^{d-d_{i}}.$
Since $\binom{k}{i}$ is an integer and $(i^{\prime},p)=1$, we have
$i^{\prime}\mid\binom{k-1}{i-1}k^{\prime}$ and hence
$\binom{k-1}{i-1}\frac{k^{\prime}}{i^{\prime}}$ is an integer as well. Noting
that
$\binom{k}{i}p^{(i-1)f}=\binom{k-1}{i-1}\frac{k^{\prime}}{i^{\prime}}p^{(i-1)f+d-d_{i}},$
we have $p^{(i-1)f+d-d_{i}}\mid\binom{k}{i}p^{(i-1)f}$. Noting
$i^{\prime}p^{d_{i}}=i\geq 2$, one may check that
$i^{\prime}p^{d_{i}}-d_{i}-1\geq 1$. Then
$(i-1)f+d-d_{i}=(i^{\prime}p^{d_{i}}-1)f+d-d_{i}\geq(i^{\prime}p^{d_{i}}-d_{i}-1)f+d\geq
f+d,$
which implies $p^{d+f}|\binom{k}{i}p^{(i-1)f}$.
(2) By the Binomial Theorem, we have
$q^{k}-1=(1+p^{f})^{k}-1=\binom{k}{1}p^{f}+\binom{k}{2}p^{2f}+\cdots+\binom{k}{k}p^{kf}.$
Then we get $p^{d+f}\mid q^{k}-1$ since $p^{d}\mid k$. It follows that
$q^{k}\equiv 1\pmod{p^{d+f}}$. Since it has been proved that
$(q^{k}-1)/(q-1)\equiv k\pmod{p^{d+f}},$
we get
$(q^{k+1}-1)/(q-1)=(q^{k}-1)/(q-1)+q^{k}\equiv k+1\pmod{p^{d+f}}.$
∎
## 3 Regular embeddings of $K_{p[p^{e}]}$
In this section, we manly consider the case $m=p$ and $n=p^{e}$ where $p\geq
5$ is a prime. However, to obtain some results used in Section 4, we allow
$p=3$ in this section if no state explicitly.
Set $\Gamma=K_{p[p^{e}]}$, with the vertex set
$V(\Gamma)=\bigcup_{i=1}^{p}\Delta_{i},~{}\mbox{where}~{}\Delta_{i}=\\{\gamma_{i1},\gamma_{i2},\cdots,\gamma_{i{p^{e}}}\\}$
and the edges are all pairs $\\{\gamma_{ij},\gamma_{kl}\\}$ of vertices with
$i\neq k$. Then $\hbox{\rm Aut}(\Gamma)=S_{p^{e}}\wr S_{p}$, which has blocks
$\Delta_{i}$ where $1\leq i\leq p$.
Let ${\cal M}$ be an orientably-regular map with the underlying graph $\Gamma$
and set $G=\hbox{\rm Aut}^{+}({\cal M})=\langle a,b\rangle$, where $\langle
a\rangle=G_{\gamma_{11}}$ and $b$ reverses the arc
$(\gamma_{11},\gamma_{21})$. We use $H=\hbox{\rm Aut}_{0}^{+}({\cal M})$ to
denote the normal subgroup of $G$ consisting of automorphisms preserving each
part setwise. By Proposition 1.1, $H$ is a $p^{e}$-isobicyclic group.
By a result of Hupert [11], $H$ is metacyclic. One can see from [16] that
$H=\langle x,z|x^{p^{e}}=z^{p^{e}}=1,z^{x}=z^{q}\rangle,$ (1)
where $q=1+p^{f}$ for $f\in[e]$ and different $f$ give nonisomorphic groups.
Particularly, $H\cong\mathbb{Z}_{p^{e}}\times\mathbb{Z}_{p^{e}}$ if $f=e$ and
$H$ is nonabelian if $f\in[e-1]$. Each element of $H$ can be written uniquely
in the form $x^{i}z^{j}$ where $i,j\in\mathbb{Z}_{p^{e}}$, with the rules
$(x^{i}z^{j})(x^{k}z^{l})=x^{i+k}z^{jq^{k}+l}\quad\mbox{and}\quad(x^{i}z^{j})^{k}=x^{ik}z^{j(q^{ik}-1)/(q^{i}-1)}.$
(2)
The center, derived subgroup and Frattini subgroup of $H$ are
$\mathrm{Z}(H)=\langle x^{p^{e-f}},z^{p^{e-f}}\rangle$, $H^{\prime}=\langle
z^{p^{f}}\rangle$ and $\Phi(H)=\langle h^{p}\mid h\in H\rangle$, respectively.
The exponent $\hbox{\rm Exp}(H)$ of $H$ is $p^{e}$. Moreover, $H$ is a regular
$p$-group, that is, all elements $h_{1},h_{2}\in H$ satisfy
$(h_{1}h_{2})^{p}=h_{1}^{p}h_{2}^{p}c_{1}^{p}\cdots c_{k}^{p}$ where
$c_{1},\cdots,c_{k}\in\langle h_{1},h_{2}\rangle^{\prime}$.
Since $H_{\gamma_{11}}=\langle a^{m-1}\rangle$, one can set
$a^{m-1}=x^{i}z^{j}$ where $p\nmid i$. By Eq.(2), we have
$a^{i^{-1}(m-1)}=xz^{j^{\prime}}$ for some $j^{\prime}\in\mathbb{Z}_{p^{e}}$
and then $z^{a^{i^{-1}(m-1)}}=z^{x}=z^{1+q}$. Replacing $a$ and $x$ by
$a^{i^{-1}}$ and $xz^{j^{\prime}}$ respectively, we may assume that
$a^{m-1}=x$. Then $(H,x,y)$ is a $p^{e}$-isobicyclic triple by setting
$y=x^{b}$.
Let $P$ be a Sylow $p$-subgroup of $G$. Then $P$ is an extension of $H$ by
$\mathbb{Z}_{p}$. Since $\hbox{\rm Exp}(H)=p^{e}$, we have $\hbox{\rm
Exp}(P)=p^{e+1}$ or $p^{e}$ and then we shall discuss these two cases in the
following two subsections, separately.
### 3.1 $\hbox{\rm Exp}(P)=p^{e+1}$
###### Theorem 3.1.
Suppose that $\hbox{\rm Exp}(P)=p^{e+1}$ where $p\geq 5$. Then ${\cal M}$ is
isomorphic to one of the maps $\mathcal{M}_{1}(p,e,j)$ where
$j\in\mathbb{Z}_{p^{e}(p-1)}^{*}$. Moreover, all of the maps
$\mathcal{M}_{1}(p,e,j)$ are orientably-regular embeddings of $K_{p[p^{e}]}$
and such maps are uniquely determined by the parameter $j$.
###### Proof.
The proof is divided into two steps.
(1) Determination of the group $G$.
Recalling that for $p\geq 5$,
$G_{1}(p,e)=\langle a,c|a^{p^{e}(p-1)}=c^{p^{e+1}}=1,c^{a}=c^{r}\rangle,$
where $r$ is a given generator of $\mathbb{Z}_{p^{e+1}}^{*}$. If we allow
$p=3$ for the groups $G_{1}(p,e)$ and maps $\mathcal{M}_{1}(p,e,j)$, then
$G_{1}(3,e)$ and $\mathcal{M}_{1}(3,e,j)$ are exactly $G_{3}(1,e)$ and
$\mathcal{M}_{3}(1,e,j)$ by choosing $r=2$ respectively. Therefore, we allow
$p=3$ in the following arguments.
Since $\hbox{\rm Exp}(P)=p^{e+1}$, there exists an element $g$ of order
$p^{e+1}$ in $G\setminus H$. Clearly $\langle g\rangle$ permutates the $p$
parts of $\Gamma$ and hence is regular on $V(\Gamma)$. Since
$G_{\gamma_{11}}=\langle a\rangle$, we get $\langle a\rangle\bigcap\langle
g\rangle=1$ and $G=\langle a\rangle\langle g\rangle$, a product of two cyclic
groups. Then $G^{\prime}$ is abelian by an Ito’s theorem in [13]. Thus
$G^{\prime}$ acts semiregularly on $V(\Gamma)$, from which we have
$G^{\prime}\cap\langle a\rangle=G\bigcap G_{\gamma_{11}}=1$. Furthermore, by
[5, Corollary C] we know that $G^{\prime}/(G^{\prime}\cap\langle a\rangle)$ is
isomorphic to a subgroup of $\langle b\rangle$, which implies that
$G^{\prime}$ is cyclic. Set $c=a^{\frac{p^{e}(p-1)}{2}}b$. Then
$c^{2}=a^{\frac{p^{e}(p-1)}{2}}ba^{\frac{p^{e}(p-1)}{2}}b=[a^{\frac{p^{e}(p-1)}{2}},b]\in
G^{\prime}$ and hence $c\in G^{\prime}$. Since $G^{\prime}$ is cyclic,
$\langle c\rangle\lhd G$ and thus $\langle c\rangle\langle a\rangle\leq G$.
From $G=\langle a,b\rangle=\langle a,c\rangle$, we get $G=\langle
c\rangle\langle a\rangle$. In particular, $G^{\prime}=\langle c\rangle.$
Set $c^{a}=c^{r}$. Since $\langle a\rangle$ is core-free,
$c^{r^{i}}=c^{a^{i}}\neq c$ for any $i\not\equiv 0(\hbox{\rm mod
}p^{e}(p-1))$. Therefore, $\mathbb{Z}_{p^{e+1}}^{*}=\langle r\rangle$. Take
two such $r$ and $r^{\prime}$ and denote the corresponding groups by $G(r)$
and $G(r^{\prime})$. Set $r=r^{\prime s}$ for some integer $s$. Then the
mapping $\sigma:a\to a^{s}$, $c\to c$ gives an isomorphism from $G(r)$ to
$G(r^{\prime})$. Therefore, $r$ can be chosen to be any given generator of
$\mathbb{Z}_{p^{e+1}}^{*}.$
Now $G$ satisfies all the defining relations of $G_{1}(p,e)$ (take $r=2$ if
$p=3$). A direct checking shows that $|G_{1}(p,e)|=|G|$ and so $G\cong
G_{1}(p,e)$.
(2) Determination the map $\mathcal{M}$. By the above proof, we know that
${\cal M}$ is isomorphic to one of the maps
$\mathcal{M}_{1}(p,e,j)=\mathcal{M}\big{(}G_{1}(p,e);a^{j},b\big{)}~{}\mbox{where}~{}j\in\mathbb{Z}_{p^{e}(p-1)}^{*}.$
By Lemma 2.1, all of the maps $\mathcal{M}_{1}(p,e,j)$ are regular embeddings
of $K_{p[p^{e}]}$. Suppose that for two parameters $j_{1}$ and $j_{2}$,
$\mathcal{M}_{1}(p,e,j_{1})\cong\mathcal{M}_{1}(p,e,j_{2})$. Then there exists
an automorphism $\sigma$ of $G_{1}(p,e)$ such that
$\sigma(a^{j_{1}})=a^{j_{2}}$ and $\sigma(b)=b$. If follows that
$\sigma(c)=\sigma(a^{\frac{p^{e}(p-1)}{2}}b)=\sigma(a^{j_{1}\frac{p^{e}(p-1)}{2}}b)=a^{j_{2}\frac{p^{e}(p-1)}{2}}b=a^{\frac{p^{e}(p-1)}{2}}b=c,$
and hence
$c^{r^{j_{2}}}=c^{a^{j_{2}}}=\big{(}\sigma(c)\big{)}^{\sigma(a^{j_{1}})}=\sigma(c^{a^{j_{1}}})=\sigma(c^{r^{j_{1}}})=c^{r^{j_{1}}}.$
Therefore $r^{j_{1}}\equiv r^{j_{2}}\pmod{p^{e+1}}$ and then $j_{1}\equiv
j_{2}\pmod{p^{e}(p-1)}$. Thus the maps $\mathcal{M}_{1}(p,e,j)$ are uniquely
determined by the parameter $j$. ∎
###### Remark 3.2.
In Theorem 3.1, $H=\langle x,w\rangle,$ where $x=a^{p-1}$ and $w=c^{p}$. Now
$H=\langle x,w\mid x^{p^{e}}=w^{p^{e}}=1,w^{x}=w^{r^{p-1}}\rangle.$
Since $\mathbb{Z}_{p^{e+1}}^{*}=\langle r\rangle$, $r^{p-1}$ is of order
$p^{e}$ in $\mathbb{Z}_{p^{e+1}}$. It is well known that the subgroup of order
$p^{e}$ of $\mathbb{Z}_{p^{e+1}}^{*}$ is $\\{1+pk\bigm{|}0\leq k\leq
p^{e}-1\\}$. Therefore, $p\bigm{|}\bigm{|}(r^{p-1}-1).$ Hence, if $e\geq 2$,
then $H$ is nonableian and $f=1$.
###### Lemma 3.3.
Let $p\geq 5$. Then $\mathcal{M}_{1}(p,e,j)$ are chiral maps of type
$\\{p^{e}(p-1),p^{e}(p-1)\\}$ if $p\equiv 1\pmod{4}$; and
$\\{\frac{p^{e}(p-1)}{2},p^{e}(p-1)\\}$ if $p\equiv 3\pmod{4}$.
###### Proof.
Since $j\not\equiv-j\pmod{p^{e}(p-1)}$ for all
$j\in\mathbb{Z}_{p^{e}(p-1)}^{*}$, we have $\mathcal{M}_{1}(p,e,j)$ is not
isomorphic to $\mathcal{M}_{1}(p,e,-j)$ and hence $\mathcal{M}_{1}(p,e,j)$ are
chiral maps.
Set $l=j-\frac{p^{e}(p-1)}{2}$. Since $p\geq 5$, we get $p\nmid r^{l}-1$. Then
for any integer $i$,
$(a^{j}b)^{i}=(a^{l}c)^{i}=a^{li}c^{1+r^{l}+\cdots+r^{(i-1)l}}=a^{li}c^{\frac{r^{li}-1}{r^{l}-1}}.$
Noting that $l$ is odd if $p\equiv 1\pmod{4}$ and even if $p\equiv 3\pmod{4}$,
we have that the order of $a^{j}b$ is $p^{e}(p-1)$ if $p\equiv 1\pmod{4}$ and
$\frac{p^{e}(p-1)}{2}$ if $p\equiv 3\pmod{4}$. It follows that the maps
$\mathcal{M}_{1}(p,e,j)$ has the type $\\{p^{e}(p-1),p^{e}(p-1)\\}$ if
$p\equiv 1\pmod{4}$ and $\\{\frac{p^{e}(p-1)}{2},p^{e}(p-1)\\}$ if $p\equiv
3\pmod{4}$. ∎
### 3.2 $\hbox{\rm Exp}(P)=p^{e}$
In this subsection, we discuss the case $\hbox{\rm Exp}(P)=p^{e}$. Before
giving the main results, we prove two lemmas for later use.
###### Lemma 3.4.
$N:=\langle x^{p^{e-f}},z\rangle\unlhd G$.
###### Proof.
Set $N=\langle x^{p^{e-f}},z\rangle$. Since $H^{\prime}=\langle
z^{p^{f}}\rangle\leq N$ and $H/H^{\prime}=\langle
x,z\rangle/H^{\prime}\cong\mathbb{Z}_{p^{e}}\times\mathbb{Z}_{p^{f}}$, we have
$N/H^{\prime}=\\{gH^{\prime}\in
H/H^{\prime}\mid(gH^{\prime})^{p^{f}}=H^{\prime}\\}$, which is characteristic
in $H/H^{\prime}$ and hence is normal in $G/H^{\prime}$. It follows that
$N\unlhd G$. ∎
###### Lemma 3.5.
Suppose that $x^{g}=x^{i}z^{j}$ and $z^{g}=x^{k}z^{l}$ for some $g\in
P\setminus H$. Then
1. (1)
$p^{e-f}\mid k$, $p^{e-f}|(i-1)$, $p\nmid j$ and $l\equiv 1\pmod{p}$;
2. (2)
one can set $x^{g}=xz$ by reselecting $z$.
###### Proof.
(1) By lemma 3.4, we have $p^{e-f}\mid k$ and hence $p\nmid l$. It follows
that $x^{kq}=x^{k}$ and $x^{k}\in Z(H)$. Then
$\left\\{\begin{array}[]{l}(z^{g})^{q}=(x^{k}z^{l})^{q}=x^{kq}z^{lq}=x^{k}z^{lq};\\\
(z^{q})^{g}=(z^{x})^{g}=(z^{g})^{x^{g}}=(x^{k}z^{l})^{x^{i}z^{j}}=x^{k}(z^{l})^{x^{i}}=x^{k}z^{lq^{i}}.\end{array}\right.$
Therefore we have $x^{k}z^{lq}=x^{k}z^{lq^{i}}$ and then $lq\equiv
lq^{i}\pmod{p^{e}}$. Noting that $q=1+p^{f}$ and $p\nmid l$, we have $p\nmid
lq$ and hence $q^{i-1}\equiv 1\pmod{p^{e}}$. By [16, Lemma 6], we get
$p^{e-f}|(i-1)$.
Write $\overline{G}=G/\Phi(H)$. Then
$\overline{x}^{\overline{g}}=\overline{x}^{i-1}\overline{x}\overline{z}^{j}=\overline{x}\overline{z}^{j}$,
$\overline{z}^{\overline{g}}=\overline{x}^{k}\overline{z}^{l}=\overline{z}^{l}$.
Since $x$ fixes the vertex $\gamma_{11}$ and $g$ moves away any part of
$\Gamma$, we have $\overline{x}^{\overline{g}}\neq\overline{x}$. It follows
that $\overline{g}$ can be represented on
$\overline{H}\cong\mathbb{Z}_{p}^{2}$ as a matrix
$\left(\begin{array}[]{lr}1&0\\\ j&l\\\ \end{array}\right)\in\hbox{\rm
GL}(2,p)$ with respect to the basis $\\{\overline{x},\overline{z}\\}$. Since
the order of $\overline{g}$ is $p$, we have $l\equiv 1\pmod{p}$ and $p\nmid
j$.
(2) Since $x^{g}=x(x^{i-1}z^{j})$ and
$(x^{i-1}z^{j})^{x}=x^{i-1}(z^{j})^{x}=x^{i-1}z^{jq}=(x^{i-1}z^{j})^{q}$, one
may get the desired conclusion by replacing $z$ by $x^{i-1}z^{j}$. ∎
###### Theorem 3.6.
Suppose that $H$ is non-abelian and $\hbox{\rm Exp}(P)=p^{e}$. Then $p=3$ and
$f=e-1$.
###### Proof.
Set $c=a^{\frac{p^{e}(p-1)}{2}}b$, $x^{c^{k}}=x^{u_{k}}z^{v_{k}}$ and
$z^{c^{k}}=x^{s_{k}}z^{t_{k}}$ for $k\geq 1$. By lemma 3.5, one can set
$u_{1}=v_{1}=1$, $s_{1}=s$ and $t_{1}=t$ where $p^{e-f}|s$ and $t\equiv
1\pmod{p}$. Particularly, $x^{s}\in\mathrm{Z}(H)$. Now we prove the theorem by
the following three steps.
(1) Show that
${\footnotesize\left(\begin{array}[]{cc}u_{k}&s_{k}\\\ v_{k}&t_{k}\\\
\end{array}\right)\equiv\left(\begin{array}[]{cc}1&s\\\ 1&t\\\
\end{array}\right)^{k}}\pmod{p^{e}}.$ (3)
We proceed the proof by induction on $k$. The assertion is trivial if $k=1$.
Let $k\geq 2$. Then we have
$(xz)^{u_{k-1}}=x^{u_{k-1}}z^{(q^{u_{k-1}}-1)/(q-1)}\quad\mbox{and}\quad(x^{s}z^{t})^{v_{k-1}}=x^{sv_{k-1}}z^{tv_{k-1}}.$
By Lemma 3.5., we have that $p^{e-f}$ divide both $u_{k-1}-1$ and $s_{k-1}$.
Then by Lemma 2.2., we get
$(q^{u_{k-1}}-1)/(q-1)\equiv
u_{k-1}\pmod{p^{e}}\quad\mbox{and}\quad(q^{s_{k-1}}-1)/(q-1)\equiv
s_{k-1}\pmod{p^{e}}.$
Therefore,
$\begin{array}[]{lll}x^{c^{k}}&=&(x^{u_{k-1}}z^{v_{k-1}})^{c}=(x^{c})^{u_{k-1}}(z^{c})^{v_{k-1}}=(xz)^{u_{k-1}}(x^{s}z^{t})^{v_{k-1}}\\\
&=&x^{u_{k-1}}z^{(q^{u_{k-1}}-1)/(q-1)}x^{sv_{k-1}}z^{tv_{k-1}}=x^{u_{k-1}+sv_{k-1}}z^{(q^{u_{k-1}}-1)/(q-1)+tv_{k-1}}\\\
&=&x^{u_{k-1}+sv_{k-1}}z^{u_{k-1}+tv_{k-1}}.\end{array}$
Similarly, we get $z^{c^{k}}=x^{s_{k-1}+st_{k-1}}z^{s_{k-1}+tt_{k-1}}$. It
follows that
$\left(\begin{array}[]{cc}u_{k}&s_{k}\\\ v_{k}&t_{k}\\\
\end{array}\right)\equiv\left(\begin{array}[]{cc}1&s\\\ 1&t\\\
\end{array}\right)\left(\begin{array}[]{cc}u_{k-1}&s_{k-1}\\\
v_{k-1}&t_{k-1}\\\ \end{array}\right)\pmod{p^{e}}.$
Then we get the conclusion by employing the inductive hypothesis.
(2) Show that $p\|v_{p}$ if $p\geq 5$.
Suppose that $p\geq 5$. Set $A=\left(\begin{array}[]{cc}0&s\\\ 1&t-1\\\
\end{array}\right)$. Noting that $p$ divide both $s$ and $t-1$. We have
$A^{2}\equiv 0\pmod{p}\quad\quad\mbox{and}\quad\quad A^{4}\equiv
0\pmod{p^{2}}.$
By Eq.(3), we have
$\left(\begin{array}[]{cc}u_{p}&s_{p}\\\ v_{p}&t_{p}\\\
\end{array}\right)=(E+A)^{p}\equiv E+pA=\left(\begin{array}[]{cc}1&ps\\\
p&1+p(t-1)\\\ \end{array}\right)\pmod{p^{2}},$
from which we have $p\|v_{p}$.
(3) Show that $p=3$ and $f=e-1.$
Set $c^{p}=x^{i}z^{j}$. Since $\hbox{\rm Exp}(P)=p^{e}$, we have that $p$
divides both $i$ and $j$. Then we have
$x^{c^{p}}=x^{x^{i}z^{j}}=x^{z^{j}}=xz^{-jq}z^{j}=xz^{-jp^{f}}\quad\mbox{and}\quad
z^{c^{p}}=z^{x^{i}z^{j}}=z^{x^{i}}=z^{q^{i}}.$
It follows that
$u_{p}\equiv 1,~{}v_{p}\equiv-jp^{f},~{}s_{p}\equiv 0,~{}t_{p}\equiv
q^{i}\pmod{p^{e}}.$ (4)
Since $p\mid j$, we have $p^{2}|v_{p}$. Combining with the result of _Step 2_
, we get $p=3$. It can be straightforward to calculate that
$\left(\begin{array}[]{cc}1&s\\\ 1&t\\\
\end{array}\right)^{3}\equiv\left(\begin{array}[]{cc}1+s(t+2)&s(1+t+t^{2}+s)\\\
1+t+t^{2}+s&s+2st+t^{3}\\\ \end{array}\right)\pmod{3^{e}}.$ (5)
By Eq.(4) and Eq.(5), we have
$\left(\begin{array}[]{cc}1+s(t+2)&s(1+t+t^{2}+s)\\\
1+t+t^{2}+s&s+2st+t^{3}\\\
\end{array}\right)\equiv\left(\begin{array}[]{cc}1&0\\\
-3^{f}j&(1+3^{f})^{i}\\\ \end{array}\right)\pmod{3^{e}}.$
Particularly,
$1+t+t^{2}+s\equiv-3^{f}j\pmod{3^{e}}$ (6)
Since $3\mid(t-1)$ and $3\mid j$, we have $3\|(1+t+t^{2})$ and
$9\mid(-3^{f}j)$. Then by Eq.(6), we get $3\|s$. Recalling that $3^{e-f}|s$,
we have $f=e-1$. ∎
###### Theorem 3.7.
Suppose that $p\geq 5$, $H$ is abelian and $\hbox{\rm Exp}(P)=p^{e}$. Then
${\cal M}$ is isomorphic to one of the maps ${\cal M}_{2}(p,j)$ which are
uniquely determined by the parameter $j\in\mathbb{Z}_{p-1}^{*}$. Moreover, all
of the maps ${\cal M}_{2}(p,j)$ are orientable regular embeddings of
$K_{p[p]}$ and they are chiral maps with the type $\\{p(p-1),p(p-1)\\}$ if
$p\equiv 1\pmod{4}$ and $\\{\frac{p(p-1)}{2},p(p-1)\\}$ if $p\equiv
3\pmod{4}$.
###### Proof.
Suppose that $p\geq 5$, $H$ is abelian and $\hbox{\rm Exp}(P)=p^{e}$. By the
classification of orientably-regular embeddings of $K_{p[p]}$ in [8], it
suffices to show that $e=1$.
Recalling the notations: $\langle a\rangle=G_{\gamma_{11}}$, $b$ reverses the
arc $(\gamma_{11},\gamma_{21})$, $x=a^{p-1}$, $y=x^{b}$ and $H=\langle
x,y\rangle$. Considering the conjugacy action of $a$ on $H$, we have $x^{a}=x$
and set $y^{a}=x^{s}y^{t}$ for integers $s$ and $t$. Then
$y=y^{x}=y^{a^{p-1}}=x^{s(1+t+\cdots+t^{p-2})}y^{t^{p-1}},$
and hence
$s(1+t+\cdots+t^{p-2})\equiv
0\pmod{p^{e}}~{}~{}~{}\mbox{and}~{}~{}~{}t^{p-1}\equiv 1\pmod{p^{e}}.$
For any $i\in[p-2]$, noting that $a^{i}$ moves away the block $\Delta_{2}$ and
$H_{\gamma_{21}}=\langle(a^{p-1})^{b}\rangle=\langle y\rangle$, we have
$\langle y^{a^{i}}\rangle\bigcap\langle y\rangle=1$. Therefore, $t$ is of
order $p-1$ modulo $p^{e}$. It follows that
$t^{\frac{p-1}{2}}\equiv-1\pmod{p^{e}}$ and then
$(t-1)\big{(}1+t+\cdots+t^{\frac{p^{e}(p-1)}{2}-1}\big{)}=t^{\frac{p^{e}(p-1)}{2}}-1\equiv-2\pmod{p^{e}}.$
Therefore $p\nmid(t-1)$ and hence
$1+t+\cdots+t^{\frac{p^{e}(p-1)}{2}-1}\equiv\frac{2}{1-t}\pmod{p^{e}}.$
Let $c=a^{\frac{p^{e}(p-1)}{2}}b$ and $v=\frac{2s}{1-t}$. Then
$x^{c}=y~{}\mbox{and}~{}y^{c}=\Big{(}x^{s\big{(}1+t+\cdots+t^{\frac{p^{e}(p-1)}{2}-1}\big{)}}y^{t^{\frac{p^{e}(p-1)}{2}}}\Big{)}^{b}=(x^{v}y^{-1})^{b}=x^{-1}y^{v}.$
(7)
The conjugacy action of $G$ on $H$ gives a unfaithful homomorphism $\pi$ from
$G$ to $\hbox{\rm Aut}(H)$. By Eq.(7), $\pi(c)$ is represented as a matrix
$\mathrm{C}=\left(\begin{array}[]{cc}0&-1\\\ 1&v\\\
\end{array}\right)\in\hbox{\rm GL}(2,\mathbb{Z}_{p^{e}})$
with respect to two generators $x$ and $y$. Set $\overline{G}=G/H$. Then
$\overline{G}=\langle\overline{a},\overline{c}\rangle\cong\hbox{\rm
AGL}(1,p)$. Since the product of two different involutions in $\hbox{\rm
AGL}(1,p)$ must be of order $p$, the order of $\overline{c}$ is $p$ and hence
$c^{p}\in H$. Therefore we have
$\mathrm{C}^{p}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\pmod{p^{e}}.$ (8)
It follows that $\mathrm{C}^{p}=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)\pmod{p}$. Noting that any matrix of order $p$ in $\hbox{\rm
GL}(2,p)$ has the eigenvalue 1, one gets $v\equiv 2\pmod{p}$. Set
$v-2=rp,~{}\mathrm{A}=\left(\begin{array}[]{cc}0&-1\\\ 1&2\\\
\end{array}\right)~{}\mbox{and}~{}\mathrm{B}=\left(\begin{array}[]{cc}0&0\\\
0&1\\\ \end{array}\right).$
Then
$\mathrm{C}^{p}=\big{(}\mathrm{A}+rp\mathrm{B}\big{)}^{p}\equiv\mathrm{A}^{p}+rp(\mathrm{A}^{p-1}\mathrm{B}+\mathrm{A}^{p-2}\mathrm{B}\mathrm{A}+\cdots+\mathrm{B}\mathrm{A}^{p-1})\pmod{p^{2}}.$
It can be straightforward to check that
$\mathrm{A}^{k}=\left(\begin{array}[]{cc}1-k&-k\\\ k&k+1\\\
\end{array}\right)$ for all $k\geq 1$. Therefore
$\displaystyle\mathrm{C}^{p}$ $\displaystyle\equiv$
$\displaystyle\mathrm{A}^{p}+rp\sum_{i+j=p-1}\mathrm{A}^{i}\mathrm{B}\mathrm{A}^{j}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}1-p&-p\\\ p&p+1\\\
\end{array}\right)+rp\sum_{i+j=p-1}\left(\begin{array}[]{cc}1-i&-i\\\ i&i+1\\\
\end{array}\right)\left(\begin{array}[]{cc}0&0\\\ 0&1\\\
\end{array}\right)\left(\begin{array}[]{cc}1-j&-j\\\ j&j+1\\\
\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1-p&-p\\\ p&p+1\\\
\end{array}\right)+rp\sum_{i=0}^{p-1}\left(\begin{array}[]{cc}i^{2}-(p-1)i&i^{2}-pi\\\
-i^{2}+(p-2)i+p-1&-i^{2}+(p-1)i+p\\\ \end{array}\right)$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}1-p&-p\\\ p&p+1\\\
\end{array}\right)+rp\sum_{i=0}^{p-1}\left(\begin{array}[]{cc}i^{2}+i&i^{2}\\\
-i^{2}-2i-1&-i^{2}-i\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1-p&-p\\\ p&p+1\\\
\end{array}\right)+rp\left(\begin{array}[]{cc}\frac{p(p-1)(2p-1)}{6}+\frac{p(p-1)}{2}&\frac{p(p-1)(2p-1)}{6}\\\
-\frac{p(p-1)(2p-1)}{6}-p^{2}&-\frac{p(p-1)(2p-1)}{6}-\frac{p(p-1)}{2}\\\
\end{array}\right)$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}1-p&-p\\\ p&p+1\\\ \end{array}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right)+\left(\begin{array}[]{cc}-p&-p\\\ p&p\\\
\end{array}\right)\pmod{p^{2}}.$
Combining with Eq.(8), we get $e=1$. ∎
## 4 Regular embeddings of $K_{3[n]}$
In this section, we assume that $m=3$ and $n=3^{e}k\geq 2$ where $3\nmid k$.
Set $\Gamma=K_{3[n]}$, with the vertex set
$V(\Gamma)=\Delta_{1}\bigcup\Delta_{2}\bigcup\Delta_{3}~{}~{}\mbox{where}~{}~{}\Delta_{i}=\\{\gamma_{i1},\gamma_{i2},\cdots,\gamma_{in}\\}$
and two vertices are adjacent if and only if they are in different
$\Delta_{i}$. Let ${\cal M}$ be an orientably-regular embedding of $\Gamma$
and $G=\hbox{\rm Aut}^{+}({\cal M})=\langle a,b\rangle$ where $\langle
a\rangle=G_{\gamma_{11}}$ and
$(\gamma_{11},\gamma_{21})^{b}=(\gamma_{21},\gamma_{11})$. Set $a^{2}=x$,
$y=x^{b}$ and $H=\langle x,y\rangle$. Then $\hbox{\rm Aut}_{0}^{+}({\cal
M})=H$. By Proposition 1.1, $H=Q\times K$ where $Q$ is a $3$-group and $K$ is
an abelian $3^{\prime}$\- group. Let $P$ be a Sylow 3-subgroup of $G$ and we
divide the discussions into two subsections according to $\hbox{\rm
Exp}(P)=3^{e+1}$ or $\hbox{\rm Exp}(P)=3^{e}$.
### 4.1 $\hbox{\rm Exp}(P)=3^{e+1}$
###### Theorem 4.1.
Suppose that $\hbox{\rm Exp}(P)=3^{e+1}$. Then ${\cal M}\cong{\cal
M}_{3}(k,e,j)$ where $j\in\mathbb{Z}_{2k\cdot 3^{e}}^{*}$ and ${\cal
M}_{3}(k,e,j_{1})\cong{\cal M}_{3}(k,e,j_{2})$ if and only if $j_{1}\equiv
j_{2}\pmod{2\cdot 3^{e}}$. Moreover, all the maps ${\cal M}_{3}(k,e,j)$ are
chiral regular embeddings of $K_{3[n]}$ with the type $\\{3^{e+1},2\cdot
3^{e}k\\}$ and the number of such maps is $2\cdot 3^{e-1}$ up to isomorphism.
###### Proof.
Recall that
$\displaystyle G_{3}(k,e)$ $\displaystyle=$ $\displaystyle\langle a,b\mid
a^{2\cdot 3^{e}k}=b^{2}=1,c=a^{3^{e}}b,a^{2\cdot
3^{e}}=x_{1},x_{1}^{b}=y_{1},[x_{1},y_{1}]=1,c^{3^{e+1}}=1,$
$\displaystyle\hskip
28.45274pty_{1}^{a}=x_{1}^{-1}y_{1}^{-1},c^{a}=c^{2}x_{1}^{u}y_{1}^{\frac{1-3^{e}}{2}u}\rangle,$
where $u3^{e}\equiv 1\pmod{k}$ if $k>1$ (note that $c^{a}=c^{2}$ if $k=1$).
Now we divide the proof into three steps.
(1) Show that $G\cong G_{3}(k,e)$.
Let $x_{1}=a^{2\cdot 3^{e}}$, $y_{1}=x_{1}^{b}$ and $c=a^{3^{e}}b$. Then
$K=\langle x_{1},y_{1}\rangle\cong\mathbb{Z}_{k}\times\mathbb{Z}_{k}$.
Set $\widetilde{G}=G/Q$ and let $\widetilde{\mathcal{M}}$ be the quotient map
of $\mathcal{M}$ induced by $Q$. Then $\hbox{\rm
Aut}^{+}(\widetilde{\mathcal{M}})\cong\widetilde{G}$ and
$\widetilde{\mathcal{M}}$ is an orientably-regular embedding of $K_{3[k]}$. By
the classification of orientably-regular embeddings of $K_{3[k]}$ for $k$
coprime to 3 (see [25, Lemma 5.2]), we have
$\widetilde{G}=\langle\widetilde{a},\widetilde{b}\mid\widetilde{a}^{2k}=\widetilde{b}^{2}=(\widetilde{a}\widetilde{b})^{3}=\widetilde{1},\widetilde{a}^{2}=\widetilde{x},\widetilde{x}^{\widetilde{b}}=\widetilde{y},[\widetilde{x},\widetilde{y}]=\widetilde{1},\widetilde{y}^{\widetilde{a}}=\widetilde{x}^{-1}\widetilde{y}^{-1}\rangle.$
(15)
Thus we get
$\widetilde{y_{1}}^{\widetilde{a}}=\widetilde{(y^{3^{e}})}^{\widetilde{a}}=\widetilde{x^{-3^{e}}}\widetilde{y^{-3^{e}}}=\widetilde{x}_{1}^{-1}\widetilde{y}_{1}^{-1}$
and then $y_{1}^{a}y_{1}x_{1}\in Q\cap K=1$, that is
$y_{1}^{a}=x_{1}^{-1}y_{1}^{-1}$.
Set $\overline{G}=G/K$ and let $\overline{{\cal M}}$ be the quotient map of
$\mathcal{M}$ induced by $K$. Then $\hbox{\rm Aut}(\overline{{\cal
M}})=\overline{G}.$ Noting that Theorem 3.1 holds for $p=3$, we have
$\overline{G}^{\prime}=\langle\overline{c}\rangle\quad\mbox{and}\quad\overline{c}^{\overline{a}}=\overline{c}^{2}.$
(16)
Now we show two facts:
Fact 1: $\langle K,c\rangle=K\rtimes\langle c\rangle\unlhd G$ with $|\langle
c\rangle|=3^{e+1},~{}x_{1}^{c}=y_{1},~{}y_{1}^{c}=x_{1}^{-1}y_{1}^{-1}~{}\mbox{and}~{}[c^{3},x_{1}]=[c^{3},y_{1}]=1$.
Since $\langle\overline{c}\rangle=\overline{G}^{\prime}\unlhd\overline{G}$ and
$K\unlhd G$, we have $\langle K,c\rangle\unlhd G$. Write $g=ab$. By Eq.(15),
we get $\widetilde{g}^{3}=1$. Noting that
$\langle\widetilde{a}^{2}\rangle=\langle\widetilde{x}\rangle=\langle\widetilde{x_{1}}\rangle$,
we get $\widetilde{a}^{3^{e}-1}\in\langle\widetilde{x_{1}}\rangle$. Let
$\widetilde{a}^{3^{e}-1}=\widetilde{x_{1}}^{i}$ for some integer $i$. Then
$\tilde{c}=\widetilde{a^{3^{e}}b}=\widetilde{a}^{3^{e}-1}\widetilde{a}\widetilde{b}=\widetilde{x_{1}}^{i}\widetilde{g}$.
Since
$\widetilde{x_{1}}^{\widetilde{g}}=\widetilde{x_{1}}^{\widetilde{a}\widetilde{b}}=\widetilde{x_{1}}^{\widetilde{b}}=\widetilde{y_{1}}$
and
$\widetilde{y_{1}}^{\widetilde{g}}=\widetilde{y_{1}}^{\widetilde{a}\widetilde{b}}=(\widetilde{x_{1}}^{-1}\widetilde{y_{1}}^{-1})^{\widetilde{b}}=\widetilde{x_{1}}^{-1}\widetilde{y_{1}}^{-1}$,
we have
$(\widetilde{c})^{3}=\widetilde{g}^{3}(\widetilde{x_{1}}^{i})^{\widetilde{g}^{3}}(\widetilde{x_{1}}^{i})^{\widetilde{g}^{2}}(\widetilde{x_{1}}^{i})^{\widetilde{g}}=\widetilde{x_{1}}^{i}\widetilde{x_{1}}^{-i}\widetilde{y_{1}}^{-i}\widetilde{y_{1}}^{i}=1$
and hence $c^{3}\in Q$. Then $|\langle c\rangle|=3^{e+1}$, since $\hbox{\rm
Exp}(Q)=3^{e}$ and the order of $\overline{c}$ is $3^{e+1}$ in $\overline{G}$.
Noting that $y_{1}^{a^{i}}=x_{1}^{-1}y_{1}^{-1}$ for odd $i$ and $y_{1}$ for
even $i$, we get
$x_{1}^{c}=x_{1}^{a^{3^{e}}b}=x_{1}^{b}=y_{1},\,y_{1}^{c}=y_{1}^{a^{3^{e}}b}=(x_{1}^{-1}y_{1}^{-1})^{b}=x_{1}^{-1}y_{1}^{-1}~{}\mbox{and}~{}[c^{3},x_{1}]=[c^{3},y_{1}]=1.$
Since $\gcd\\{|K|,|\langle c\rangle|\\}=1$ and $K\unlhd G$, we have $\langle
K,c\rangle=\langle K\rangle\rtimes\langle c\rangle$.
Fact 2: $G=\langle K,c\rangle\langle a\rangle$ with
$c^{a}=c^{2}x^{u}y^{\frac{1-3^{e}}{2}u}$ where $u3^{e}\equiv 1\pmod{k}$.
Since $G=\langle a,b\rangle=\langle a,c\rangle$, we have $G=\langle
K,c\rangle\langle a\rangle$. By Eq.(16), we can set
$c^{a}=c^{2}x_{1}^{u}y_{1}^{v}$ for some integers $u$ and $v$ for $k\geq 2$.
The remaining is to show $u3^{e}\equiv 1\pmod{k}$ and
$v\equiv\frac{1-3^{e}}{2}u\pmod{k}$.
One can deduce the following formulas by induction on $i$:
$(c^{2}x_{1}^{u}y_{1}^{v})^{i}=\left\\{\begin{array}[]{ll}c^{2i},&~{}i\equiv
0\pmod{3}\\\ c^{2i}x_{1}^{u}y_{1}^{v},&~{}i\equiv 1\pmod{3}\\\
c^{2i}x_{1}^{v}y_{1}^{v-u},&~{}i\equiv 2\pmod{3}\end{array}\right.$ (17)
and
$c^{a^{2i}}=c^{4^{i}}x_{1}^{iu}y_{1}^{-iu}.$ (18)
Then we have
$c^{a^{2\cdot
3^{e}}}=c^{4^{3^{e}}}x_{1}^{3^{e}u}y_{1}^{-3^{e}u}=cx_{1}^{3^{e}u}y_{1}^{-3^{e}u}.$
On the other hand, $c^{a^{2\cdot
3^{e}}}=c^{x_{1}}=c(x_{1}^{-1})^{c}x_{1}=cx_{1}y_{1}^{-1}$. Therefore
$u3^{e}\equiv 1\pmod{k}$.
By Eq.(18), we have
$c^{a^{3^{e}-1}}=c^{2^{3^{e}-1}}x_{1}^{\frac{3^{e}-1}{2}u}y_{1}^{\frac{1-3^{e}}{2}u}.$
(19)
Note that $2^{3^{e}}\equiv 2\pmod{3}$ and $2^{3^{e}-1}\equiv 1\pmod{3}$. Then
by Eq.(19) and Eq.(17), we have
$c^{a^{3^{e}}}=(c^{2^{3^{e}-1}}x_{1}^{\frac{3^{e}-1}{2}u}y_{1}^{\frac{1-3^{e}}{2}u})^{a}=(c^{2}x_{1}^{u}y_{1}^{v})^{2^{3^{e}-1}}x_{1}^{(3^{e}-1)u}y_{1}^{\frac{3^{e}-1}{2}u}=c^{2^{3^{e}}}x_{1}y_{1}^{\frac{3^{e}-1}{2}u+v}$
(20)
and
$(c^{2^{3^{e}}}x_{1}y_{1}^{\frac{3^{e}-1}{2}u+v})^{2^{3^{e}}}=c^{2^{2\cdot
3^{e}}}x_{1}^{\frac{3^{e}-1}{2}u+v}y_{1}^{-\frac{3^{e}+1}{2}u+v}=cx_{1}^{\frac{3^{e}-1}{2}u+v}y_{1}^{-\frac{3^{e}+1}{2}u+v}.$
(21)
Since $c=a^{3^{e}}b$, we get $c^{b}=c^{a^{3^{e}}}$. Then by Eq.(20) and
Eq.(21), we have
$c=c^{c}=c^{a^{3^{e}}b}=(c^{2^{3^{e}}}x_{1}y_{1}^{\frac{3^{e}-1}{2}u+v})^{b}=(c^{b})^{2^{3^{e}}}x_{1}^{\frac{3^{e}-1}{2}u+v}y_{1}=cx_{1}^{(3^{e}-1)u+2v}y_{1}^{\frac{3^{e}-1}{2}u+v}.$
Therefore $x_{1}^{(3^{e}-1)u+2v}y_{1}^{\frac{3^{e}-1}{2}u+v}=1$ and hence
$v\equiv\frac{1-3^{e}}{2}u~{}\pmod{k}$.
From the _Fact 1_ and _Fact 2_ , $G$ satisfies all the relations of
$G_{3}(k,e)$ and then $G$ is an homomorphic image of $G_{3}(k,e)$. A checking
shows that $|G_{3}(k,e)|=|G|$. Therefore $G\cong G_{3}(k,e)$.
(2) Determination of $\mathcal{M}$.
Recall that ${\cal M}_{3}(k,e,j)=\mathcal{M}\big{(}G_{3}(k,e);a^{j},b\big{)}$
where $j\in\mathbb{Z}_{2n}^{*}$. Since $G\cong G_{3}(k,e)$, we have
$\mathcal{M}$ is isomorphic to one of the maps $\mathcal{M}_{3}(k,e,j)$. If
${\cal M}_{3}(k,e,j_{1})\cong{\cal M}_{3}(k,e,j_{2})$ for two parameters
$j_{1},j_{2}\in\mathbb{Z}_{2n}^{*}$, then there exists an automorphism $\psi$
of $\hbox{\rm Aut}\big{(}G_{3}(k,e)\big{)}$ such that
$\psi(a^{j_{1}})=a^{j_{2}}$ and $\psi(b)=b$. Clearly, $\psi$ induces an
automorphism $\overline{\psi}$ of $\overline{G}$ such that
$\overline{\psi}(\overline{a}^{j_{1}})=\overline{a}^{j_{2}}$ and
$\overline{\psi}(\overline{b})=\overline{b}$. By Proposition 3.1, we get
$j_{1}\equiv j_{2}\pmod{2\cdot 3^{e}}$.
Conversely, suppose that $j_{1}\equiv j_{2}~{}\pmod{2\cdot 3^{e}}$ for
$j_{1},j_{2}\in\mathbb{Z}_{2k\cdot 3^{e}}$. Then $j=j_{2}j_{1}^{-1}\equiv
1\pmod{2\cdot 3^{e}}$. It suffices to show that the mapping $\psi:~{}a\mapsto
a^{j},~{}b\mapsto b$ can be extended to an automorphism of $G_{3}(k,e)$. That
is, $\psi$ can be extended to a bijection preserving all the defining
relations of $G_{3}(k,e)$. By the presentation of $G_{3}(k,e)$, we can set
$\psi(x_{1})=x_{1}^{j},\,\psi(y_{1})=y_{1}^{j}\quad\mbox{and}\quad\psi(c)=(a^{j})^{3^{e}}b.$
Then the four formulas
$\psi(a)^{2\cdot
3^{e}k}=\psi(b)^{2}=1,\,[\psi(x_{1}),\,\psi(y_{1})]=1,\,\psi(x_{1})^{\psi(a)}=\psi(x_{1})\,\psi(y_{1})^{\psi(a)}=\psi(x_{1})^{-1}\psi(y_{1})^{-1}$
clearly hold. Since
$\psi(c)=(a^{j})^{3^{e}}b=a^{2\cdot
3^{e}\frac{j-1}{2}}a^{3^{e}}b=x_{1}^{\frac{j-1}{2}}c,$
$(x_{1}^{\frac{j-1}{2}}c)^{3}=x_{1}^{\frac{j-1}{2}}c^{2}(x_{1}^{\frac{j-1}{2}})^{c}x_{1}^{\frac{j-1}{2}}c=x_{1}^{\frac{j-1}{2}}c^{3}(y_{1}^{\frac{j-1}{2}}x_{1}^{\frac{j-1}{2}})^{c}=x_{1}^{\frac{j-1}{2}}c^{3}x_{1}^{-\frac{j-1}{2}}=c^{3},$
we know that $\psi(c)$ and $c$ have the same order. Set $j=2\cdot 3^{e}i+1$.
Then $a^{j}=x_{1}^{i}a$ and hence
$c^{a^{j}}=c^{x_{1}^{i}a}=\big{(}c(x_{1}^{-i})^{c}x_{1}^{i}\big{)}^{a}=(cy_{1}^{-i}x_{1}^{i})^{a}=c^{2}x_{1}^{u}y_{1}^{\frac{1-3^{e}}{2}u}x_{1}^{2i}y_{1}^{i}=c^{2}x_{1}^{u+2i}y_{1}^{\frac{1-3^{e}}{2}u+i}.$
Noting that
$\psi(c)^{2}=(x_{1}^{\frac{j-1}{2}}c)^{2}=x_{1}^{\frac{j-1}{2}}cy_{1}^{\frac{j-1}{2}}$,
we have
$\psi(c)^{a^{j}}=(x_{1}^{\frac{j-1}{2}}c)^{a^{j}}=x_{1}^{\frac{j-1}{2}}c^{a^{j}}=x_{1}^{\frac{j-1}{2}}c^{2}x_{1}^{u+2i}y_{1}^{\frac{1-3^{e}}{2}u+i}=\psi(c)^{2}x_{1}^{u+2i}y_{1}^{\frac{1-3^{e}}{2}u-\frac{j-1}{2}+i}.$
(22)
Since $j=2\cdot 3^{e}i+1$, $3^{e}u\equiv 1~{}\pmod{k}$, we have
$u+2i\equiv u+2i\cdot 3^{e}u\equiv(1+2i\cdot 3^{e})u\equiv uj\pmod{k},$
and
$\frac{1-3^{e}}{2}u-\frac{j-1}{2}+i\equiv\frac{1-3^{e}-3^{e}j+3^{e}+2\cdot
3^{e}i}{2}u\equiv j\frac{1-3^{e}}{2}u\pmod{k}.$
Then by Eq.(22), we have
$\psi(c)^{a^{j}}=\psi(c)^{2}(x_{1}^{j})^{u}(y_{1}^{j})^{\frac{1-3^{e}}{2}u}$.
Thus, $\psi$ can be indeed extended to a bijection preserving all the defining
relations of $G_{3}(k,e)$. In summary,
$\mathcal{M}_{4}(k,e,j_{1})\cong\mathcal{M}_{4}(k,e,j_{2})$ if and only if
$j_{1}\equiv j_{2}~{}\pmod{2\cdot 3^{e}}$.
(3) Determination of the number and type of the resulting maps.
By Lemma 2.1, all of the maps $\mathcal{M}_{3}(k,e,j)$ for
$j\in\mathbb{Z}_{2n}^{*}$ are orientably-regular embeddings of $K_{2[n]}$. By
(2), ${\cal M}_{3}(k,e,j_{1})\cong{\cal M}_{3}(k,e,j_{2})$ if and only if
$j_{1}\equiv j_{2}\pmod{2\cdot 3^{e}}$. Noting that for
$j\in\mathbb{Z}_{2n}^{*}$ with $(j,2\cdot 3^{e})=1$, the set $\\{j+i\cdot
2\cdot 3^{e}\bigm{|}0\leq i\leq k-1\\}$ contains at least one number which is
coprime to $2\cdot 3^{e}k$, then we have that the number of maps in this
family is $\phi(2\cdot 3^{e})=2\cdot 3^{e-1}$. With the same arguments as in
_Fact 1_ of (1), one may check $|\langle a^{j}b\rangle|=3^{e+1}$. Thus, all of
the resulting maps have type $\\{3^{e+1},2\cdot 3^{e}k\\}$. Clearly, they are
all chiral. ∎
### 4.2 $\hbox{\rm Exp}(P)=3^{e}$
The following theorem quoted from [25, Lemma 5.2] gives a determination of the
case when $H$ is abelian.
###### Theorem 4.2.
If $H$ is abelian, then
$\mathcal{M}\cong\mathcal{M}_{3}(k,e,0,l,j)$
where $(l,j)=(0,1)$ if $e=0$ and $(l,j)=(0,1)$, $(1,1)$ or $(1,-1)$ if $e\geq
1$. The resulting maps $\mathcal{M}_{3}(k,e,0,0,1)$ and
$\mathcal{M}_{3}(k,e,0,1,\pm 1)$ have type $\\{3,2n\\}$ and $\\{9,2n\\}$
respectively, and all of them are orientably-regular embeddings of $K_{3[n]}$.
If $H$ is nonabelian, then we have the following theorem.
###### Theorem 4.3.
Suppose that $H$ is nonabelian and $\hbox{\rm Exp}(P)=3^{e}$. Then ${\cal M}$
is isomorphic to one of the maps ${\cal M}_{4}(k,e,1,l,j)$, where $l=0,\pm 1$
and $j=\pm 1$. Moreover, all of the maps ${\cal M}_{4}(k,e,1,l,j)$ are chiral
regular embeddings of $K_{3[n]}$ with the type $\\{3,2n\\}$ if $l=0$ and
$\\{9,2n)\\}$ if $l=\pm 1$.
###### Proof.
We divide the proof into two steps.
(1) Show that $G$ is isomorphic to one of the following groups
$\displaystyle G_{4}(k,e,1,l)$ $\displaystyle=$ $\displaystyle\langle a,b\mid
a^{2n}=b^{2}=1,a^{2}=x,x^{b}=y,[x,y]=x^{\frac{n}{3}}y^{-\frac{n}{3}},y^{a}=x^{-1}y^{-1},$
$\displaystyle\hskip
25.60747pt(ab)^{3}=x^{\frac{ln}{3}}y^{-\frac{ln}{3}}\rangle,$
where $l=0,\pm 1$.
Set $\overline{G}=G/K$ and let $\overline{{\cal M}}$ be the quotient map of
$\mathcal{M}$ induced by $K$. Then $\mathcal{M}$ is an orientably-regular
embedding of $K_{3[3^{e}]}$ and $\hbox{\rm Aut}(\overline{{\cal
M}})=\overline{G}$. Clearly, $\hbox{\rm Exp}(\overline{P})=\hbox{\rm
Exp}(P)=3^{e}$ and $\overline{H}\cong Q$. By Theorem 3.7, we have that
$\mathrm{Z}(Q)=\langle g^{3}\mid g\in
Q\rangle\cong\mathbb{Z}_{3^{e-1}}\times\mathbb{Z}_{3^{e-1}}$ and
$Q^{\prime}\cong\mathbb{Z}_{3}$. Since $H=Q\times K$ and $K$ is abelian, we
have $\mathrm{Z}(H)=\langle x^{3},y^{3}\rangle$ and
$H^{\prime}=\langle[x,y]\rangle\cong\mathbb{Z}_{3}$. Set
$[x,y]=x^{\frac{n}{3}i}y^{\frac{n}{3}j}$ where $i,j=0\;\mbox{or}\;\pm 1$. Then
from
$y^{-\frac{n}{3}j}x^{-\frac{n}{3}i}=[x,y^{-1}]=[y,x]=[x,y]^{b}=(x^{\frac{n}{3}i}y^{\frac{n}{3}j})^{b}=y^{\frac{n}{3}i}x^{\frac{n}{3}j},$
we get $i=-j$ and hence $[x,y]=x^{\pm\frac{n}{3}}y^{\mp\frac{n}{3}}$. Noting
that these two cases can be interconvertible by replacing $a$ by $a^{-1}$, we
can set $[x,y]=x^{\frac{n}{3}}y^{-\frac{n}{3}}$ without loss of any
generalities. Therefore, $H$ has a presentation
$H=\langle x,y\mid
x^{n}=y^{n}=[x^{3},y]=[x,y^{3}]=1,~{}[x,y]=x^{\frac{n}{3}}y^{-\frac{n}{3}}\rangle.$
Set $[x,y]=w$. One can check that the multiplications and powers in $H$ are
given by
$(x^{i}y^{l})(x^{r}y^{d})=x^{i+r}y^{l+d}w^{-ld},\quad(x^{i}y^{l})^{r}=x^{ri}y^{rl}w^{-\frac{r(r-1)}{2}il}.$
(23)
Since $G$ is an extension of $H$ by $G/H\cong S_{3}$ with $a^{2}=x$, $b^{2}=1$
and $x^{b}=y$, it follows that $G$ can be determined by the following
relations
$y^{a}=x^{u}y^{v},~{}~{}(ab)^{3}=x^{s}y^{t},$
where $u$, $v$, $s$ and $t$ are undetermined parameters. Set $c=ab$. Then
$x^{c}=x^{b}=y,~{}~{}x^{c^{2}}=y^{c}=(x^{u}y^{v})^{b}=y^{u}x^{v}~{}\mbox{and}~{}x^{c^{3}}=(y^{u}x^{v})^{c}=(y^{u}x^{v})^{u}y^{v}.$
By Eq.(23), we have
$(y^{u}x^{v})^{u}=(x^{v}y^{u}w^{-uv})^{u}=x^{uv}y^{u^{2}}w^{-\frac{u(u-1)}{2}uv-u^{2}v}=x^{uv}y^{u^{2}}w^{-\frac{u(u+1)}{2}uv}$
and hence
$x^{c^{3}}=x^{uv}y^{u^{2}+v}w^{-\frac{u(u+1)}{2}uv}=x^{uv-\frac{nu(u+1)}{6}uv}y^{u^{2}+v+\frac{nu(u+1)}{6}uv}.$
On the other hand, since $\hbox{\rm Exp}(P)=3^{e}$, we know that 3 divides
both $s$ and $t$. It follows that $c^{3}\in\mathrm{Z}(H)$ and hence
$x^{c^{3}}=x$. Therefore
$\left\\{\begin{gathered}uv-\frac{nu(u+1)}{6}uv\equiv 1\pmod{n},\\\
u^{2}+v+\frac{nu(u+1)}{6}uv\equiv 0\pmod{n}.\end{gathered}\right.$ (24)
Since
$y^{x}=y^{a^{2}}=(x^{u}y^{v})^{a}=x^{u}(x^{u}y^{v})^{v}=x^{u+uv}y^{v^{2}}w^{-\frac{v(v-1)}{2}uv}=x^{u+uv-\frac{nv(v-1)}{6}uv}y^{v^{2}+\frac{nv(v-1)}{6}uv}$
and
$y^{x}=[x,y^{-1}]y=yw^{-1}=x^{-\frac{n}{3}}y^{1+\frac{n}{3}},$
we have
$\left\\{\begin{gathered}u+uv-\frac{nv(v-1)}{6}uv\equiv-\frac{n}{3}\pmod{n},\\\
v^{2}+\frac{nv(v-1)}{6}uv\equiv 1+\frac{n}{3}\pmod{n}.\end{gathered}\right.$
(25)
By solving the equations (24) and (25), one can obtain $u\equiv
v\equiv-1\pmod{n}$, that is $y^{a}=x^{-1}y^{-1}$. Since
$c^{3}=(ab)^{3}=x^{s}y^{t}$ and $3$ divides both $s$ and $t$, we have
$x^{s}y^{t}=(x^{s}y^{t})^{ab}=[x^{s}(x^{-1}y^{-1})^{t}]^{b}=(x^{s-t}y^{-t})^{b}=x^{-t}y^{s-t}$
and hence
$s\equiv-t(\hbox{\rm mod }{n}),\,t\equiv s-t(\hbox{\rm mod }n),$
that is $3s\equiv 3t\equiv 0\pmod{n}$. It follows that
$(ab)^{3}=x^{-t}y^{t}=x^{\frac{ln}{3}}y^{-\frac{ln}{3}}$, where
$l=0~{}\mbox{or}~{}\pm 1$. Now we have proved that $G$ satisfies all the
defining relations of $G_{4}(k,e,1,l)$. Checking directly, one has
$|G_{4}(k,e,1,l)|=|G|$. Therefore $G\cong G_{4}(k,e,1,l)$.
(2) Determination the map $\mathcal{M}$.
Take two parameters $j_{1},\,j_{2}\in\mathbb{Z}_{2n}^{*}$. Then one may verify
that the mapping $a^{j_{1}}\mapsto a^{j_{2}},\,b\mapsto b$ can be extended to
an automorphism of $G_{4}(k,e,1,l)$ if and only if $j_{1}\equiv
j_{1}\pmod{3}$. Therefore, for given $k,e,l$, $\mathcal{M}$ is isomorphic to
one of the maps $\mathcal{M}_{4}(k,e,1,l,j)$ where $j=\pm 1$.
The remaining is to show different $l$ give nonisomorphic maps. Suppose that
for $l_{1}\neq l_{2}$, $\mathcal{M}_{4}(k,e,1,l_{1},j_{1})$ is isomorphic to
$\mathcal{M}_{4}(k,e,1,l_{2},j_{2})$. Then there exists an isomorphism $\psi$
from $G_{4}(k,e,1,l_{1})$ to $G_{4}(k,e,1,l_{2})$. Set
$\displaystyle G_{4}(k,e,,l_{1})$ $\displaystyle=$ $\displaystyle\langle
a^{\prime},b^{\prime}\mid a^{\prime 3^{e}}=b^{\prime 2}=1,a^{\prime
2}=x^{\prime},x^{\prime b}=y^{\prime},[x^{\prime},y^{\prime}]=x^{\prime
3^{e-1}}y^{\prime-3^{e-1}},$ $\displaystyle\hskip 28.45274pty^{\prime
a}=x^{\prime-1}y^{\prime-1},(a^{\prime}b^{\prime})^{3}=x^{\prime
l_{1}\frac{n}{3}}y^{\prime-\l_{1}\frac{n}{3}}\rangle,$
and
$\displaystyle G_{4}(k,e,1,l_{2})$ $\displaystyle=$ $\displaystyle\langle
a,b\mid a^{3^{e}}=b^{2}=1,a^{2}=x,x^{b}=y,[x,y]=x^{3^{e-1}}y^{-3^{e-1}},$
$\displaystyle\hskip
22.76219pty^{a}=x^{-1}y^{-1},(ab)^{3}=x^{l_{2}\frac{n}{3}}y^{-l_{2}\frac{n}{3}}\rangle.$
Since for any given $k,e,l$, we have two maps $\mathcal{M}_{4}(k,e,1,l,j)$
where $j=\pm 1$, we may assume that $\psi(a^{\prime})=a^{j}$ where $j=\pm 1$
and $\psi(b^{\prime})=b$. Then from
$(\psi(a^{\prime})\phi(b^{\prime}))^{3}=\phi(x^{\prime})^{l_{1}\frac{n}{3}}\psi(y^{\prime})^{-l_{1}\frac{n}{3}}$,
we get $(a^{j}b)^{3}=x^{jl_{1}\frac{n}{3}}y^{-jl_{1}\frac{n}{3}}$. However,
one can check that this equation does not hold provided $l_{1}\neq l_{2}$.
By Lemma 2.1, all of the maps $\mathcal{M}_{4}(k,e,1,l,j)$ are orientably-
regular embeddings of $K_{3[n]}$. Clearly, all of these maps are chiral. By a
simple calculation, we have that the order of $a^{\pm 1}b$ is $3$ if $l=0$ and
$9$ if $l=\pm 1$. Therefore the resulting maps have type $\\{3,2n\\}$ if $l=0$
and $\\{9,2n\\}$ if $l=\pm 1$. ∎
## 5 Proof of Theorem 1.2
Let $\mathcal{M}$ be an orientably-regular embedding of $\Gamma=K_{m[n]}$
where $m\geq 3$ and $n\geq 2$, and let $\hbox{\rm Aut}^{+}_{0}(\mathcal{M})$
be the normal subgroup of $\hbox{\rm Aut}^{+}(\mathcal{M})$ consisting of
automorphisms preserving each part setwise. By Proposition 1.1, either
$\Gamma=K_{p[p^{e}]}$ where $p\geq 5$ is prime, or $\Gamma=K_{3[n]}$. For the
case $\Gamma=K_{p[p^{e}]}$ where $p\geq 5$, we get ${\cal M}\cong{\cal
M}_{1}(p,e,j)$ or ${\cal M}_{2}(p,j)$ in Section 3. For the case
$\Gamma=K_{3[n]}$, we get ${\cal M}\cong\mathcal{M}_{3}(k,e,j)$ or
$\mathcal{M}_{4}(k,e,i,l,j)$ in Section 4. As proved in Section 3 and Section
4, all of the resulting maps are indeed the orientably-regular embedding of
$\Gamma=K_{m[n]}$ and such maps are unique determined by the given parameters.
Finally, Table 1 and Table 2 directly can be obtained from Theorem 3.1,
Theorem 3.7, Theorem 4.1 and Theorem 4.3. This finish the proof of Theorem
1.2.∎
## References
* [1] N. L. Biggs, Classification of complete maps on orientable surfaces, _Rend Math_ 4(6)(1971), 132–138.
* [2] N. L. Biggs, Algebraic Graph Theory, second ed., Cambridge University Press, Cambridge, 1993.
* [3] H. S .M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, fourth ed., Springer, Berlin, 1984.
* [4] M.D.E. Conder, Regular maps and hypermaps of Euler characteristic $-1$ to $-200$, _J. Combin. Theory Ser. B_ 99 (2009) 455-459, with associated lists of computational data available at http://www.math.auckland.ac.nz/ conder/hypermaps.html.
* [5] M. Conder, I. M. Isaacs, Derived subgroups of products of an abelian and a cyclicsubgroup, _J. London Math. Soc._ 69(2004), 333–348.
* [6] S. F. Du, G. A. Jones, J. H. Kwak, R. Nedela and M. $\rm\check{S}koviera$, Regular embeddings of $K_{n,n}$ where n is a power of 2. I: Metacyclic case, _European J. Combin._ 28(2007) 1595–1609.
* [7] S. F. Du, G.A. Jones, J.H. Kwak, R. Nedela and M. $\rm\check{S}koviera$, Regular embeddings of $K_{n,n}$ where n is a power of 2. II: The non-metacyclic case, _European J. Combin._ 31(2010) 1946–1956.
* [8] S. F. Du, J.H. Kwak, R. Nedela, Regular maps with $pq$ vertices, _J. Algebraic Combin._ 19(2004), 123–141.
* [9] S. F. Du, J. H. Kwak, R. Nedela, Regular embeddings of complete multipartite graphs, _European J. Combin._ 26(2005), 505–519.
* [10] A. Gardiner, R. Nedela, J.Širáň, and M. Škoviera, Characterization of graphs which underlie regular maps on closed surfaces, _J Lond Math Soc_ 59(1999), 100–108.
* [11] B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, _Math. Z._ 58(1953), 243-264.
* [12] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
* [13] N. Ito, Über das Produkt von zwei abelschen Gruppen, _Math. Z._ 62(1955) 400–401.
* [14] L. D. James and G. A. Jones, Regular orientable imbeddings of complete graphs, _J Combin Theory Ser B_ 39(1985), 353–367.
* [15] G. A. Jones, Regular embeddings of complete bipartite graphs: classification and enumeration, _Proc. London Math. Soc._ 101(2010), 427–453.
* [16] G. A. Jones, R. Nedela and M. Škoviera, Regular embeddings of $K_{n,n}$ where n is an odd prime power, _European J. Combin._ 28(2007), 1863–1875.
* [17] G. A. Jones, R. Nedela and M. Škoviera, Complete bipartite graphs with a unique regular embedding, _J. Combin. Theory Ser. B_ 98(2008), 241–248.
* [18] J. H. Kwak and Y.S. Kwon, Regular orientable embeddings of complete bipartite graphs, _J. Graph Theory_ 50(2005), 105–122.
* [19] J. H. Kwak and Y.S. Kwon, Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs, _Discrete Math._ 308(2008) 2156–2166.
* [20] J. H. Kwak and Y. S. Kwon, Classification of nonorientable regular embeddings of complete bipartite graphs, _J. Combin. Theory, Ser. B_ 101(2011) 191–205.
* [21] R. Nedela, M. Škoviera, Exponents of orientable maps, _Proc. London Math. Soc._ 75(1997) 1–31.
* [22] R. Nedela, M. Škoviera, and A. Zlatoš, Regular embeddings of complete bipartite graphs, _Discrete Math._ 258(2002) 379–381.
* [23] H. Wielandt, Uber das produkt von paarweise abelschen gruppen, _Math. Z._ 62(1955), 1–7.
* [24] S. E. Wilson, Cantankerous maps and rotary embeddings of $K_{n}$, _J. Combin. Theory Ser. B_ 47(1989), 262–273.
* [25] J. Y. Zhang, S. F. Du, On the Orientable Regular Embeddings of Complete Multipartite Graphs, _European J. Combin._ , 33(2012) 1303–1312.
|
arxiv-papers
| 2012-02-09T13:22:33 |
2024-09-04T02:49:27.291474
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shaofei Du and Junyang Zhang",
"submitter": "Shaofei Du",
"url": "https://arxiv.org/abs/1202.1974"
}
|
1202.2004
|
# A New Type of Cipher
Fabio F. G. Buono
buonof@cli.di.unipi.it
(5-1-2012)
###### Abstract
We will define a new type of cipher that doesn’t use neither an easy to
calcualate and hard to invert matematical function like RSA nor a classical
mono or polyalphabetic cipher.
Our cipher will use an intrinsic and not syntactic property of a sequence. The
core idea of this cipher is a process that is close to changing the
measurement units of the input sequence.
###### Contents
1. 1 Introduction
2. 2 The Idea
3. 3 The cipher
## 1 Introduction
As known a cipher is an algorithm that takes as input some binary sequence and
gives back a ciphered sequence as output. This process is supposed to be
simple to compute but difficult to reverse unless the key is known (this is
known as a one way trapdoor mechanism).
The problem of most known ciphers (except the one-time-pad [2][3][7][8]), is
that their ciphertexts preserve some kind of structure which can be used from
some cryptanalyst to obtain the plaintext they originated from.
Many kind of cryptoanalisis use this strategy. Therefor we shall eliminate
this problem. To be uneffected by this pathology we shall mold our cipher
aroud a intrinsect non syntactic property of the information sequences, i.e.
its measurabilty as we will explain.
We are going to define a process (the measurement) that will be something like
a input sequence “weighing“, the ciphertext spawned by our cipher isn’t a
simple encoding in the classical meaning, but an allocation function of its
value, called from now $\nabla$.
I m going to explain in this article, some possible implementations of this
idea. These implementations will help on clarifying how we use $\nabla$ in our
process.
## 2 The Idea
The property we shall be using in our cipher, is a redefinition of a measure
of the input sequence. To give an idea of what we mean by a measure of
sequence we start with a simple example. The following base two sequence
$101000001$
is written base ten, giving exponents in reverse sequence, as shown in the
following example:
$2^{0}+2^{2}+2^{8}=261.$
Now we define $\nabla$ as a set of positive integer such that every element of
set will be smaller than previous and such that last element will be 1.
We call measure by nabla, a set of integer values. The first one is the result
of integer division between decimal value of input sequence and the value of
the first element of $\nabla$. Next value of the set is the result of integer
division between decimal value of the previous element of set ( first one in
this istance ) multiplied by the value of the element of $\nabla$ previously
used, and so on.
Now, we will be able to define a first cipher pseudo-code.
## 3 The cipher
To show a first example of how the cipher works, we can propose the pseudocode
below, there we take in input a binary sequence and translate it in its
decimal value. after that we take in input an array of integral value as
$\nabla$ and encode it. The output will be an array of measures by $\nabla$.
* Function Cript: begin /* plaintext input as decimal value */
|
arxiv-papers
| 2012-02-09T14:46:47 |
2024-09-04T02:49:27.301824
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabio F. G. Buono",
"submitter": "Fabio Francesco Gabriele Buono",
"url": "https://arxiv.org/abs/1202.2004"
}
|
1202.2158
|
# The Impact of Visual Appearance on User Response in Online Display
Advertising
Javad Azimi
azimi@eecs.oregonstate.edu
Oregon State University Ruofei Zhang
rzhang @yahoo-inc.com
Yahoo! Labs, Silicon Valley Yang Zhou
yangzhou @yahoo-inc.com
Yahoo! Labs, Silicon Valley Vidhya Navalpakkam
nvidhya@yahoo-inc.com
Yahoo! Labs, Silicon Valley Jianchang Mao
jmao@yahoo-inc.com
Yahoo! Labs, Silicon Valley Xiaoli Fern
xfern@eecs.oregonstate.edu
Oregon State University
###### Abstract
Display advertising has been a significant source of revenue for publishers
and ad networks in online advertising ecosystem. One of the main goals in
display advertising is to maximize user response rate for advertising
campaigns, such as click through rates (CTR) or conversion rates. Although in
the online advertising industry we believe that the visual appearance of ads
(creatives) matters for propensity of user response, there is no published
work so far to address this topic via a systematic data-driven approach. In
this paper we quantitatively study the relationship between the visual
appearance and performance of creatives using large scale data in the world’s
largest display ads exchange system, RightMedia. We designed a set of $43$
visual features, some of which are novel and some are inspired by related
work. We extracted these features from real creatives served on RightMedia. We
also designed and conducted a series of experiments to evaluate the
effectiveness of visual features for CTR prediction, ranking and performance
classification. Based on the evaluation results, we selected a subset of
features that have the most important impact on CTR. We believe that the
findings presented in this paper will be very useful for the online
advertising industry in designing high-performance creatives. It also provides
the research community with the first ever data set, initial insights into
visual appearance’s effect on user response propensity, and evaluation
benchmarks for further study.
## 1 Introduction
The Internet revolution has transformed how people experience information,
media and advertising. Web advertising, although nonexisting twenty years ago,
has become a vital component of the modern Internet, where advertisements are
delivered from advertisers to users through different online channels. Recent
trends have shown that an increasingly large share of advertisers’ budgets are
devoted to the online world, and online advertising spending has greatly
outpaced some of the traditional advertising media, such as radio and
magazine. Display advertising is one type of online advertising which,
together with search advertising, contributes the majority of the revenue for
many large Internet companies. In display advertising, display ad instances
are shown to the user on webpages in different formats such as image, flash,
and video. Each display ad instance is called a creative. By showing the
creatives, advertisers aim to either promote brand awareness among users
(brand advertising) or receive desirable responses from users (performance
advertising), such as the action of purchasing, clicking or signing up for a
promotion list from the advertiser’s website. In performance advertising, the
advertiser strives to optimize their ad’s performance metrics such as the
effective cost per click (eCPC) or effective cost per action (eCPA), which in
turn relates to maximizing the user response rate on the creatives as measured
by click through rates (CTR) or conversion rates (CVR). There are several
factors that greatly influence the user response rate of display advertising
campaigns: 1) the position of the ads on the webpage; 2) the relevancy of the
ads to the online users, which is generally captured by the targeting profiles
of the advertising campaigns; 3) the relevancy of the ads to the webpage
content and 4) the quality and visual appearance of the creatives.
The problem of predicting the user response rate for online ads, especially
CTR, has been studied by several researchers in the last few years. One major
research focus has been in predicting clicks by studying the relationship
between CTR and the aforementioned ad factors (and their combinations). For
example in [2], the authors considered the ad’s relevancy to the content of
the webpage in predicting CTR. They show that improving the ad’s content
relevancy is more efficient than considering the content of ads by themselves
[25]. Although it is generally believed that visually appealing ads can
perform better in attracting online users, as a result of which advertisers
always care about the creative designs, there is no, to the best of our
knowledge, published work so far to quantitatively study the effect of visual
appearance of creatives on campaign performance in online display advertising.
This motivates us to investigate the correlation between the visual features
of the creative and CTR, regardless of other ad factors, and to predict
creative performance based on its visual appearance alone.
Our proposed approach consists of two main steps, 1) feature extraction and 2)
correlation investigation. We first extract some informative visual features
from the creatives. We introduce $43$ visual features classified into three
categories, 1) _global features_ which characterize the overall properties of
a given creative, 2) _local features_ representing the properties of specific
parts within a given creative and 3) _advanced features_ which are a group of
features developed based on more complicated algorithms such as the number of
faces and number of characters in a creative. We then develop three regression
approaches to predict the CTR based on these features. The study is conducted
using real creatives and their performance data from the world’s largest
display ads exchange system, RightMedia. Based on the weights of developed
features, we further select a subset of features that have high impact on the
creative’s CTR. The benefit of this work is three-fold. First, our findings on
the visual features and their relationship to CTR can provide useful
recommendations to designers on what features to consider while designing
creatives, and/or can help in automated creative generation. Second, the
visual features and the regression methods developed here can be used in
addition to the traditionally investigated ad factors (such as ad relevancy,
position etc.) for improving CTR prediction in online ads selection. Third, it
provides the research community with the first ever data set, initial insights
into the effect of visual appearance on user response propensity, and
evaluation benchmarks for further study.
The paper is organized as follows. Section 2 introduces the related work. We
introduce the visual features in Section 3. The regression and feature
selection results for CTR prediction are presented in Section 4, followed by
our conclusion in Section 5.
## 2 Background and Related Work
The relationship between various print ad characteristics and measures of
advertising effectiveness has been studied by advertising researchers for
almost a century. A wide variety of characteristics have been investigated.
These characteristics are roughly in two categories: mechanical and content-
based. The mechanical characteristics include ad size, number of colors,
proportional of illustrations to copy, the absence of borders, and type size.
The content factors include message appeal like status, quality, fear and
fantasy, attention-getting techniques like free offers, presence of women, and
psycholinguistic variables like product or personal reference in headline,
interrogative or imperative headline, visual rhetorics, among others. See [20]
for summaries.
Even though the online advertising has taken a large market share of the
advertising industry, and the whole industry is steadily and continuously
shifting to the online domain, study on the effectiveness of the counterpart
of print ads online, generally called display ads, is limited. We list the
studies of several factors below.
Some existing studies try to investigate the effect of several different
factors on the performance of display advertising campaigns. These factors
include targeting and obtrusiveness [9], advertisement size (large vs. small)
and ad exposure format (intrusive vs. voluntary) [4], cognitive impact from ad
size and animation [18], emotional appeal and incentive offering in the ads
[8], repetition of varied execution vs. single execution [30].
To the best of our knowledge, we are not aware of any study on the
relationship between the visual appearance and the performance of creatives in
online display ads. We try to tackle this problem by first defining a set of
visual features and then evaluating their effects on ad performance,
specifically CTR in our experiments, from the actively served ad campaigns on
the world’s largest ad exchange system, RightMedia. Below, we present some
previous computational studies on image properties which provide us
inspiration in designing our visual features.
There are several studies that try to investigate a specific property of
images (photos or paintings) using computational approaches. Such properties
include quality and aesthetic in photos [19, 15, 29, 7] or in paintings [17],
saliency [14], composition [10, 24], color harmony [5] and memorability [13].
Initial work on image quality evaluation concentrated on evaluating and
reconstructing low graded, compressed or degraded images by simple noise model
[6, 1]. However, in most of the beauty evaluation work, including this paper,
we assume that high quality images are available and we are interested in
evaluating the visual aesthetic of images based on visual features.
Recently some researchers tried to evaluate the beauty of an image based on
its visual features. In [15] the authors aim to classify the pictures into
professional and snapshot photos using some basic features including spatial
distribution of edges, color distribution and hue count, etc. In [7] the
authors introduced a regression based approach for rating photos based on
their beauty, using features such as average pixel density, colorfulness,
saturation hue, and the rule of thirds. In addition to these studies, in [19]
the author proposed an approach to classifying images into high and low
quality. The main idea comes from the fact that a professional photographer
makes the background blurry and the subject distinguishable in the image. By
separating the blurry part of the image from the subject, they design a set of
well-motivated features from both the subject and the whole image such as the
clarity contrast of the subject, lighting, simplicity, color harmony and
composition geometry. They show that the combination of these features can
provide a promising performance. All of the above work tries to extract visual
features from photos. Recently Li et al. [17] tried to extract some features
from paintings to evaluate their beauty and classify them into high and low
quality. They introduced a set of global and local features, $40$ in total, to
capture the painting properties such as the brightness contrast between
segments, the brightness contrast across the whole image and the average
saturation for the largest segment of the image.
Computational approaches have also been used to investigate other visual
properties of an image. In [13] the authors studied what properties of images
make them more memorable. They found that statistical properties of an image
such as mean hue, mean saturation, intensity mean, intensity variance,
intensity skewness and number of objects do not have any non-trivial
correlation with memorability in their generated data set. However, they found
that if they label the objects and scenes in the images, they can find a non-
trivial and interesting correlation between images and their memorability. For
example, their results show that the attendance of human being, close up
objects and human scale objects in an image improve its memorability more than
natural scene. This result is not possible to be applied to our work since it
requires large amounts of supervision to tag different parts of the images.
However, we evaluate the impact of the number of human faces in an image in
our work.
Color harmonization is another approach for making an image more appealing. In
[5] the authors proposed to harmonize the colors in a given image using
harmonization templates from [31, 28], which include $8$ different harmonized
color templates. We also used color harmony models to evaluate the hue
distribution of an image in our experiments.
In summary, existing work in related areas has focused primarily on properties
of an image, photo or painting. In contrast, we examine creatives in online
display ads, which contain both graphical features and text. In addition, some
of the existing approaches require significant amount of supervision in their
feature extraction step, which is not possible in large scale applications
where we need to learn from large data sets with minimum amount of
supervision. Finally, we would also like to extract a set of features that are
visually understandable and can be practically controlled to guide the human
designers or automatic creative generators (like in smart ads) to produce
high-performance creatives. These objectives make our problem novel and
interesting for the online advertising industry.
## 3 Feature Extraction
In this section we introduce a set of $43$ different visual features. We
categorize the developed features into three different sets, 1) global
features, 2) local features and 3) advanced features. A complete list of the
features can be found in Table 3. Below we describe the detailed definition of
the proposed features in each category.
In the following sections we use $I$ to indicate an image and use $|I|$ to
indicate the size of the image measured by the number of pixels. We use
variable $x$ to denote an arbitrary pixel when we do not care about its
location in the image. Otherwise we use $(i,j)$ to denote the pixel in the
$i$-th row and $j$-th column in the image.
### 3.1 Global Features
Global features are a set of features which represent the overall properties
of the whole image. We describe the details of $19$ different global features
in this section.
#### 3.1.1 Gray Level Features
We describe 3 features extracted from the gray level histogram of the image,
namely the gray level contrast $f_{1}$, number of dominant gray level bins
$f_{2}$, and the standard deviation of the gray level values among all pixels
$f_{3}$.
The gray level contrast is the width of the middle $95\%$ mass in the gray
level histogram [15]. From the original gray level histogram, we prune the
extreme $2.5\%$ from the 0 side and $2.5\%$ from the 255 side. Gray level
contrast feature $f_{1}$ is calculated as the width of the remaining
histogram.
We count the number of dominant bins in the gray level histogram as our second
feature. Suppose the set $G=\\{g_{0},g_{1},\cdots,g_{255}\\}$ indicates the
set of $256$ bins in the gray level histogram such that $g_{i}$ is the number
of pixel in $i$-th bins. We define the number of dominant gray level bins as
$f_{2}=\sum_{k=0}^{255}\textbf{1}(g_{k}\geq c_{1}\max_{i}g_{i})$, where
$\textbf{1}(\cdot)$ is the indicator function and $c_{1}$ is a threshold value
which is set to be $0.01$ in this paper. 111This parameter, and similar ones
in the rest of the paper, is set inspired by related works such as [19].
The last gray level feature, $f_{3}$, is defined as the standard deviation of
gray level values of all pixels in the image. It is used to capture the
variance of the gray level distribution.
#### 3.1.2 Color Distribution
To avoid distraction from objects in the background, professional
photographers tend to keep the background simple. In [19], the authors use the
color distribution of the background to measure this simplicity. We use a
similar approach to measure the simplicity of color distribution in the image.
For a given image, we quantize each RGB channel into $8$ values, creating a
histogram $H_{rgb}=\\{h_{0},h_{1},\cdots,h_{511}\\}$ of 512 bins, where
$h_{i}$ indicates the number of pixels in $i$-th bin. We define feature
$f_{4}$ to indicate the number of dominant colors as
$f_{4}=\sum_{k=0}^{512}\textbf{1}(h_{k}\geq c_{2}\max_{i}h_{i})$ where
$c_{2}=0.01$ is the threshold parameter. We also calculate the size of the
dominant bin relative to the image size as $f_{5}=\frac{\max_{i}h_{i}}{|I|}$.
This feature indicates the extent to which one of $512$ colors is dominant in
the image.
By replacing the RGB color map with HSV (Hue, Saturation, Value) color map and
using the above methods in calculating features $f_{4}$ and $f_{5}$, we obtain
two other features $f_{6}$ and $f_{7}$.
#### 3.1.3 Model-Based Color Harmony
The concept of color harmony in this paper is based on $8$ different harmonic
color distributions (illustrated in Figure 1) that are based on the hue of the
HSV color wheel [31]. These distributions are called _i, V, L, I, T, Y, X, N_.
Note that each distribution can be rotated by $0\leq\alpha\leq 360$ degrees.
The specific size of color harmony distributions are set as follows: the large
sectors of types $V,Y$ and $X$ are $26\%$ of the disk ($93.6^{\circ}$); the
small sectors of types $i,L,I$ and $Y$ are $5\%$ of the disk ($18^{\circ}$);
the largest sector of type $L$ is $22\%$ of the disk ($79.2\%$); the sector of
type $T$ is $50\%$ of the disk ($180^{\circ}$). The angle between the centers
of the two sectors is $180^{\circ}$ for $I$, $X$, $Y$, and $90^{\circ}$ for
$L$.
Figure 1: Color harmony models
Let us define the set of $8$ distributions as
$\mathcal{D}=\\{d^{1},d^{2},\cdot\cdot\cdot,d^{8}\\}$. We say
$\phi(d^{i}_{\alpha},x)$ indicates the hue of the closest point in the $i$-th
distribution to $x$ after $\alpha$ degree rotation, where $x$ is any arbitrary
pixel in the image. We compute the distance between the hue distribution of
our image $I$ and the distribution $d^{i}\in\mathcal{D}$ as:
$\gamma(I,d^{i})=\mathop{\mathrm{argmin}}_{\alpha}\frac{1}{|I|}\sum_{x\in
I}\parallel hue(x)-\phi(d^{i}_{\alpha},x)\parallel\cdot sat(x),$ (1)
where $hue(x)$ and $sat(x)$ indicate the hue and saturation at pixel $x$, and
$\|\cdot\|$ denotes the arc-length distance. We are interested in the best
fitting model $d^{*}$ which has the least $\gamma(\cdot)$ value,
$d^{*}=\mathop{\mathrm{argmin}}_{d^{i}}\gamma(I,d^{i})$. We define feature
$f_{8}=\gamma(I,d^{*})$. Intuitively, it tells us how different is the hue
distribution of image $I$ from the best fitting model of color harmony.
Some models are superset of other models in Figure 1 concluding that the
$\gamma(\cdot)$ value of some smaller models are higher than some larger
models given any image $I$, e.g.
$\gamma(I,d_{i})\geq\gamma(I,d_{V})\geq\gamma(I,d_{T})$. Therefore, if an
image hue distribution fits into some small models, type $i,V,L,I$, it fits
into larger models as well. This can emphasize the color harmony property of
the images which can fit into a few models rather than just one model. We
consider this property as one potential positive property of the image. To
quantify this property, we introduce a new feature, $f_{9}$, which indicates
the average color harmony deviation from the best two fitted models given an
image $I$. In general, in addition to the deviation from the best fitted model
illustrated by feature $f_{8}$, we consider the deviation from the second best
fitted model as well, and the average of these two deviations is returned as
$f_{9}$. Clearly, for the images fitting into small color harmony models, we
will have $f_{8}$ and $f_{9}$ very close to each other. However, for the
images which fit into the largest model, we will have $f_{9}$ considerably
larger than $f_{8}$. We believe these two numerical features can represent the
color harmony property of an image appropriately.
#### 3.1.4 Color Coherence
We extract a set of features based on the color coherence of pixels resulting
in connected coherent components [23]. A connected coherent component in an
image is defined as:
* •
A set of pixels that fall into the same bin in the histogram.
* •
For any two pixels $p_{i}$ and $p_{j}$ in a connected coherent component
$P=\\{p_{1},p_{2},\cdots,p_{m}\\}$ of $m$ pixels, there is a path of
sequential pixels, $p_{i},p_{i+1},\cdot\cdot\cdot,p_{j}$. Two sequential
pixels in a path must be one of the $8$ neighborhoods of each other.
* •
The size of the connected coherent component is larger than a predefined
threshold $c_{4}$. In our experiment we set $c_{4}=0.01|I|$.
We denote the set of connected coherent components and their color index as
$\mathcal{P}=\\{(P_{1},h_{1}),(P_{2},h_{2}),\cdot\cdot\cdot(P_{n},h_{n})\\}$,
where $P_{i}$ is the set of pixels in the $i$-th component, and $h_{i}$ is its
corresponding color in the HSV color histogram with $512$ bins. We use
$|P_{i}|$ to denote the number of pixels in $P_{i}$. We extract the following
features based on the above definition:
* •
$f_{10}=n$, which indicates the number of connected coherent components in the
image.
* •
$f_{11}=\frac{\max_{i}|P_{i}|}{|I|}$, which indicates the size of the largest
component relative to the whole image.
* •
$f_{12}=\frac{\max\limits_{j,j\neq\arg\max\limits_{i}|P_{i}|}|P_{j}|}{|I|}$,
representing the size of the second largest connected coherent component
relative to the whole image.
* •
$f_{13}=\mathrm{rank}(h_{i}),i=\arg\max_{j}|P_{j}|$, indicating the rank of
the bin, considering the bin size in descending order, associated with the
largest connected coherent component in the image. For example, the value of
this feature is $1$ if the bin associated with the largest coherent component,
$\arg\max_{j}|P_{j}|$, is the largest bin in the color histogram as well;
$\max_{i}|h_{i}|$ where $h_{i}$ is the size of the $i-{th}$ bin in the color
histogram. This feature indicates how the colors are disperse in the image. We
expect to have value $f_{13}=1$ if the colors in the images are not very
randomly distributed. It means the pixel with the same colors are mostly
connected together.
* •
$f_{14}=\mathrm{rank}(h_{i}),i=\arg\max_{j,j\neq\arg\max_{k}|P_{k}|}|P_{j}|$,
similar to $f_{13}$, it shows the bin rank, considering the bin size in
descending order, of the second largest connected coherent component in the
image.
#### 3.1.5 Hue Distribution
In this section we introduce three features based on the hue in HSV color
space. We quantize hues in an image in a similar way as in [17] by eliminating
the pixels with saturation and value less than $0.2$. This will eliminate all
the pixels with white or black colors. Then we calculate the hue histogram of
remaining pixels with $20$ different bins, $18^{\circ}$ for each bin, which
results in $\mathcal{H}_{hue}=\\{h_{1},h_{2},\cdot\cdot\cdot,h_{20}\\}$ where
$h_{i}$ indicates the set of pixels in $i$-th bin. We then extract the
following features:
* •
$f_{15}=\sum_{i=1}^{20}\textbf{1}(|h_{i}|\geq c_{5}|I|)$ where $c_{5}=0.01$ in
our experiments. This feature indicates the number of dominant hues in an
image.
* •
$f_{16}=\max_{i,j}\parallel|h_{i}|-|h_{j}|\parallel$ where $|h_{i}|\geq
c_{5}|I|$, and $\|\cdot\|$ is the arc length. This feature indicates the
largest contrast between two dominant hues in the image.
* •
$f_{17}=\mathrm{std}(\Phi)$ where $\Phi=\\{\cup_{i\in I}\;\parallel
h_{i}(i)-0\parallel\\}$ and $\parallel.\parallel$ is the arc length value.
This feature indicates the standard deviation of all pixel’s hues distance
from the origin $0$. It simply can determine how much the hue colors in an
images has been distributed from each other.
#### 3.1.6 Lightness Features
We use the lightness $L$ in the HSL color space to calculate feature $f_{18}$
and $f_{19}$. In the HSL color space, $L$ value is small when the color is
white and is large when the color is black. The $L$ value in HSL color space
can be calculated as follows:
$L(x)=\frac{\max\left(r(x),g(x),b(x)\right)+\min\left(r(x),g(x),b(x)\right)}{2},$
(2)
where $r(x),g(x),b(x)$ denotes the R, G, B values of pixel $x$ in RGB color
space. We calculate two lightness features as:
* •
$f_{18}=\frac{1}{|I|}\sum_{x\in I}L(x)$, the average lightness of pixels in
the image.
* •
$f_{19}=\mathrm{std}(L(\cdot))$, the standard deviation of lightness of all
pixels in the image.
### 3.2 Local Features
Local features represent a set of features extracted from specific parts of
the image rather than the whole image. We apply the normalized cut
segmentation method [26] to partition the image into $5$ smaller segments. Let
$\mathcal{S}=\\{S_{1},S_{2},\cdots,S_{5}\\}$ indicate the set of $5$ different
segments where $S_{i}$ is the set of pixels in segment $i$. Note that a
segment is considered as noise and is dropped if it is smaller than $5\%$ of
the image. We develop the following features based on the segmentation result.
#### 3.2.1 Segment Size
Two features are extracted from segment size as follows:
* •
$f_{20}=\frac{\max_{i}|S_{i}|}{|I|}$, indicating the size of the largest
segment relative to the whole image.
* •
$f_{21}=\frac{1}{|I|}\max_{i,j}\big{|}|S_{i}|-|S_{j}|\big{|}$, indicating the
contrast among the segmentation sizes of the image.
#### 3.2.2 Segment Hues
Similar to section 3.1.5, we generate the hue histogram of each segment. We
define the set of hue histograms of all $5$ segments as
$\mathcal{H}^{s}_{hue}=\\{h_{1,1},h_{1,2},\cdot\cdot\cdot,h_{1,20},h_{2,1},\cdot\cdot\cdot,h_{5,20}\\}$
where $h_{i,j}$ indicates the set of pixels that fall in the $j$-th bin of
$i$-th segment. Then we extract five features to capture different hue
properties. Below we describe the formal definition of developed features:
* •
$f_{22}=\sum_{j=1}^{20}\textbf{1}(|h_{i,j}\geq c_{6}|I|)$ where
$i=\arg\max_{i}S_{i}$ and $c_{6}=0.01$. This feature denotes the number of
image-wide dominant hues in the largest segment. In general, we would like to
have most of the image hues in the largest segment.
* •
$f_{23}=\sum_{j=1}^{20}\textbf{1}(|h_{i,j}\geq c_{6}|S_{i}|)$ where
$i=\arg\max_{i}S_{i}$. This feature denotes the number of segment-wide
dominant hues in the largest segment.
* •
$f_{24}=\max\limits_{i}q_{i}$ where
$q_{i}=\sum_{j=1}^{20}\textbf{1}(|h_{i,j}\geq c_{6}|S_{i}|)$ is the number of
dominant hues in $i-{th}$ segment. This feature essentially denotes the
largest number of dominant hues in one segment. We would like to have the same
value as $f_{23}$ for this feature illustrating that the largest segment has
the largest number of dominant colors.
* •
$f_{25}=\max\limits_{i,j}|q_{i}-q_{j}|$. This feature denotes the contrast of
the number of dominant hues among the segments. We usually do not like to have
lots of different hues in one segment and a few hues in another segment in an
image. We expect to have unappealing images with large value for $f_{25}$.
* •
$f_{26}=\max\limits_{j,k}\|h_{i,j}-h_{i,k}\|$ where $|h_{i,j}|,|h_{i,k}|\geq
c_{6}|S_{i}|$, $i=\arg\max_{i}|S_{i}|$ and $\|\cdot\|$ is the arc length
distance. This feature captures the contrast of number of pixels among the hue
bins in the largest segment. In general, we expect to have an appealing image
with one bin dominating the largest segment in addition to a few more small
bins. This makes the contrast value very large.
* •
$f_{27}=\mathrm{std}(T(\cdot))$ where $T(i)=\max_{j,k}$,
$|h_{i,j}|,|h_{i,k}|\geq c_{6}|S_{i}|$. This feature returns the standard
deviation of contrast among the segments. If we have different hue contrasts
among different segments, this feature will achieve a significant value.
#### 3.2.3 Segment Color Harmony
Two features are extracted based on the largest segment color harmony. Feature
$f_{28}$ is the minimum deviation from the best fitted color harmony model for
the largest segment, and feature $f_{29}$ is the average deviation of the best
two fitted color harmony models for the largest segment. The details of color
harmony models have been introduced in section 3.1.3.
#### 3.2.4 Segment Lightness
Three segment lightness features are extracted using similar method as in
section 3.1.6:
* •
$f_{30}$: average lightness in the largest segment.
* •
$f_{31}$: standard deviation of average lightness among the segments.
* •
$f_{32}$: contrast of average lightness among the segments.
### 3.3 Advanced features
In this section we develop a set of features based on more complicated
algorithms. Most of the advanced features are based on the saliency map of the
image which determines the visually salient areas in the image that are more
likely to be noticed by the humans. We also extract two additional features
related to the number of characters and number of faces in an image. Below we
describe the details of these features.
#### 3.3.1 Saliency Features
Saliency computation is a well known phenomenon in human vision where
attention tends to be drawn to interesting parts of an image that appear
visually different from the rest of the image (e.g., a red coke can in a green
background appears salient and is immediately noticed, while the same coke can
in an orange-reddish background is not salient and less likely to be noticed).
We compute saliency according to the algorithm described in [12]. Figure 2
shows the saliency output of the algorithm presented in [12] for a sample
creative. The areas with higher lightness in the saliency map indicate more
salient part of the image.
Figure 2: The saliency map of an image. Left: original image. Right: saliency
map.
The saliency algorithm returns a matrix $\tau$ (also referred to as saliency
map) where $\tau(i,j)$ represents the saliency value of pixel $(i,j)$. We also
extract a binary image based on the saliency map, by setting a threshold
$\alpha$ to the saliency map where the pixels with saliency value larger than
$\alpha$ are set to $1$ and the rest of the pixels are set to $0$. Similar to
[12], the parameter $\alpha$ is set as $\alpha=3\bar{\tau}$ where
$\bar{\tau}=1/n\sum_{i,j}\tau(i,j)$ is the average saliency value in the
image. After this binarization, we have some connected components with value
1. These components indicate saliency areas, and the other parts of the image
are considered as background. Then we extract the following features based on
the saliency results, saliency map and binary saliency map.
* •
$f_{33}$: background size. Salient objects usually appear in the foreground
and not in the background. Therefore we return the size of the background as a
function of image size which is calculated as:
$f_{33}=\frac{\sum_{i,j}\textbf{1}(\tau(i,j)<\alpha)}{|I|}.$
* •
$f_{34}$: number of connected components in the binary map.
* •
$f_{35}$: size of the largest components in the binary saliency map relative
to the whole image.
* •
$f_{36}$: average saliency value of the largest component in the binary
saliency map.
* •
$f_{37}$: number of connected components in the image background. In some
images, the saliency areas can divide the background into several disconnected
segments. Usually it is not desirable to have multiple background components.
* •
$f_{38}$: size of the largest connected component in the background relative
to the whole image. If the number of connected components in the background is
equal to one, then this feature has the same value as $f_{33}$.
* •
$f_{39}$: distance between connected components. Let the set
$\mathcal{C}=\\{c_{1},c_{2},\cdots,c_{n}\\}$, $c_{i}=(x_{i},y_{i})$ indicates
the set of $n$ different points such that each $c_{i}$ indicates a pixel
corresponding to the center of mass of the $i$-th saliency area. To make the
rest of the computation scale independent from the image size, we update the
properties of each point $c_{i}$ as $s_{i}=(x_{i}/I_{x},y_{i}/I_{y})$ such
that $I_{x}$ and $I_{y}$ are the horizontal and vertical size of the image.
Then we build up a complete weighted graph given the set $\mathcal{C}$ such
that the weight $w_{i,j}$ between two vertices $c_{i},c_{j}$ is calculated as
$w_{i,j}=\|s_{i}-s_{j}\|_{2}$. Then we return the summation of all edge
weights as the distance between connected components.
* •
$f_{40}$: distance from the rule of third points. Professional photographers
usually locate their main object in one of the four interest points based on
the rule of third. The four interest points in rule of third is the
intersection of two vertical and two horizontal lines dividing the image into
$9$ equal segments. Figure 3 shows the four interested points based on rule of
third. This is an important feature in photo beauty evaluation [17],
motivating us to investigate its effect in creative performance. We define
this feature as the minimum distance from the center of mass of the largest
saliency area to one of the four interest points based on rule of third.
Figure 3: The four interested points based on rule of third.
* •
$f_{41}$: distance from the center of image. This feature is the distance of
saliency components to the center of image which is the most focused part of
an image. The overall distance from the centers of all connected components to
the center of image is returned as feature $f_{41}$. Note that for both
features $f_{40}$ and $f_{41}$, we normalize the position of each pixel
similar as feature $f_{39}$.
#### 3.3.2 Number of Characters
We consider the number of characters in an image as feature $f_{42}$. We tried
a number of OCR toolbox and one of them provides us with appropriate results
considering the number of characters in ads[22]. Note that we are interested
in the number of characters in the image regardless of its meaning. To
evaluate the accuracy of the OCR toolbox, we counted the true number of
characters in $100$ random images and compared it to the returned number of
characters from the OCR toolbox. We found strong linear correlation of $0.80$,
suggesting that our toolbox is reasonably accurate in evaluating the number of
characters in images. Note that extracting the exact text from ad creatives is
challenging as they often appear in different fonts, sizes and orientations.
#### 3.3.3 Number of Faces
The last feature, $f_{43}$, captures the effect of the human face appearance
on creative performance. In [13] the authors concluded that the human
appearance in an image could make the image more memorable. This motivates us
to test whether face appearance affects creative performance. We count the
number of faces in an image using an available toolbox [16]. Our toolbox is
reasonably accurate and has a correlation more than $0.9$ with the true number
of faces in images in our experiments with a sample size of $100$.
## 4 Experimental Results
In this section we present the algorithms and experiments we designed to
evaluate the relationship between visual features and the performance of
creatives in online display advertising.
### 4.1 Data Set
We extracted creatives of advertising campaigns from the world’s largest
online advertising exchange system, RightMedia. We filtered out animated
creatives because our features are designed for static images. We also
calculated the average CTR of these creatives from online serving history log
during a two-month period.
As discussed in Section 1, the performance of creatives is determined by many
factors. One important factor is the ad position in the webpage. Generally the
available position of a creative on a webpage is determined by the creative’s
size. To remove the impact on performance introduced by ad position (and
size), we create two different data sets, each of which consists of creatives
with the same size. The first data set, ID$2$, consists of $6272$ creatives
with size $250\times 300$ pixels, and the second data set, ID$6$, includes
$3888$ images with $90\times 730$ pixels. All of the creatives have a minimum
of $100K$ impressions guaranteeing that their CTRs have converged to their
true values. The CTR distribution of each data set is shown in Figure 4.
(a) Data set ID2
(b) Data set ID6
Figure 4: The CTR distribution of two data sets.
We further created two sub-categories from data set ID$2$: “dating” with $927$
images and “traveling” with $599$ images. Since there are not many images in
these two categories, we consider the images with a minimum of $20k$ and $10k$
impressions for “dating” and “traveling” respectively.
### 4.2 Learning Methods
The main goal of this work is to study the relationship between the
performance of creatives and their visual features. In the first step we try
to predict CTR from visual features using regression methods. We used three
different regression algorithms to predict CTR, 1) Linear Regression (LR), 2)
Support Vector Regression with RBF kernel(SVR), and 3) Constrained Lasso
(C-Lasso) which is a modification to Lasso [27].
We used LIBSVM [3] to implement the SVR and performed cross validation to
determine the parameters of the model. We describe our constrained Lasso
optimization approach as follows. Suppose we have a set of $n$ creatives at
disposal and the visual features of these creatives are represented as a
matrix $A\in\textbf{R}^{d\times n}$ such that
$A=(\mathbf{a}_{1},\mathbf{a}_{2},\cdots,\mathbf{a}_{n})$ where
$\mathbf{a}_{k}\in\textbf{R}^{d}$ is a column vector representing the $d$
dimensional visual features of creative $k$. In our experiment $d=43$. The CTR
values of the $n$ creatives are represented as a vector
$\mathbf{y}=({y_{1},\cdots,y_{n}})^{\top}\in\textbf{R}^{n}$ where each $y_{k}$
is the CTR of the $k$-th creative. We bound the CTR of each creative by
$y_{min}\leq y_{i}\leq y_{\max}$ where $y_{min}$ and $y_{max}$ can be obtained
from online serving history log. To predict CTR of the creatives, we try to
solve the following optimization problem:
$\displaystyle\min_{w}$
$\displaystyle\|A^{\top}\mathbf{w}-\mathbf{y}\|_{F}^{2}+\lambda\|\mathbf{w}\|_{1}$
(3) $\displaystyle s.t.$ $\displaystyle y_{min}\leq A^{\top}\mathbf{w}\leq
y_{max}$
where $\|\cdot\|_{F}^{2}$ is Frobinius-2 norm and $\|\cdot\|_{1}$ is
$\ell_{1}$ norm, also called lasso. We call the above optimization problem as
constrained Lasso (C-Lasso) and we used [11] to find the solution of this
optimization problem. Note that the proposed C-Lasso approach performs better
than Lasso in our application.
### 4.3 Evaluation
In this section we present different evaluation methods to analyze the
efficacy of the developed visual features in predicting the performance of
creatives.
#### 4.3.1 CTR Prediction
| | |
---|---|---|---
ID$2$ | ID$6$ | ID$2$-Dating | ID$2$-Traveling, IMP$\geq 10k$
Figure 5: The amount of preserved ranking for each method.
To evaluate the CTR prediction accuracy of the algorithms, we run each
algorithm for $200$ independent runs where in each run $80\%$ of each data set
is selected randomly for training and $20\%$ for testing. The accuracy
evaluation results are reported over the prediction of the test data. Mean
Squared Error (MSE) is used to measure the prediction accuracy for each
algorithm as follows:
$MSE=\frac{1}{n}\sum_{k=1}^{n}|y_{k}-\hat{y}_{k}|^{2}$ (4)
where $n$ is the number of test samples, $y_{k}$ is the true CTR of the $k$-th
creative calculated from history log, and $\hat{y}_{k}$ is the predicted CTR.
To meaningfully interpret the MSE value, we introduce two baseline approaches,
_Random_ and _Constant Mean_(CM) policy.
Table 1: The prediction accuracy of each method against _Random_ policy. Data set | Samples | CM | LR | C-Lasso | SVR
---|---|---|---|---|---
ID$2$ | $6272$ | $1.71$ | $2.28$ | $2.22$ | 3.27
ID$6$ | $3888$ | $1.75$ | $2.27$ | $2.14$ | 2.77
ID$2$-Dating | $927$ | $1.79$ | $2.65$ | $2.58$ | 2.79
ID$2$-Traveling | $599$ | $1.68$ | $2.13$ | $2.03$ | 2.26
The _Random_ policy simply samples from the CTR distribution of the training
data to predict the CTR of each testing creative, while the CM policy assigns
a constant value, $c_{m}$, to all ads where $c_{m}$ is the mean CTR of the
training data. Table 1 shows the average results over $200$ independent runs
for each algorithm. Each entry is the MSE value of the random policy divided
by MSE value of each algorithm. Results show that we can perform up to $3.27$
times better than _Random_ policy in predicting the CTR from visual features
only. All learners perform consistently better than baseline CM as well. This
result demonstrates the non-trivial impact of visual appearance of the
creative on its advertising performance.
#### 4.3.2 CTR Ranking
We introduce a ranking criterion to investigate the ability of using visual
features to rank the creatives by their CTRs. Given a test set of creatives,
suppose $c_{1}^{-},c_{2}^{-},\cdots,c_{k}^{-}$ represent the $k$ images with
the lowest CTR values and $c_{1}^{+},c_{2}^{+},\cdots,c_{k}^{+}$ represent the
$k$ images with the highest CTR. Therefore we have $k^{2}$ pairs
$(c_{i}^{-},c_{j}^{+})$ such that $ctr(c_{i}^{-})\leq ctr(c_{j}^{+})$ for
$i,j\in\\{1,\cdots,k\\}$. We wish to know whether our prediction of CTR using
visual features preserves the ranking of pairs $(c_{i}^{-},c_{j}^{+})$. To
test this, we change the value of $k$ as a function of test data size. We then
measure the percentage of match between the predicted ranking of creatives,
and the truly observed ranking in the test data. The results over $200$
independent runs are shown in figure 5 for different data sets. The $x-$axis
indicates the value of $\delta$ such that $k=\delta n$ for
$\delta=0.02,0.04,0.06,\cdots,0.50$ where $n$ is the number of creatives in
the data set, and $y-$axis represents the percentage of correctly ranked
pairs.
Results show that SVR consistently outperforms other learners. As we increase
the size of $k$, the percentage of correctly ranked predictions decreases for
all learning algorithms. This is as expected, since differentiating the images
of creatives which have CTRs close to the mean of the CTR distribution, using
visual features only, is very difficult even for a human. Interestingly, the
results show that by just using visual features, we can preserve more than
$90\%$ of the ranking for data set ID$2$ (for $\delta=0.1$). This number
remains high at $75\%$ when we consider all top-half images against low-half
images for all data sets ($\delta=0.5$). This is an encouraging result that
demonstrates the utility of visual features in predicting the ranking of CTR.
#### 4.3.3 CTR Classification
Previous studies in beauty evaluation [7, 15, 17] mostly try to classify the
images into high and low quality category rather than assigning scores to
their beauty based on visual features. Similarly, we evaluate the performance
of classifying the creatives into high (+1) and low (-1) CTR category using
visual features only. We use support vector machine with RBF kernel as our
classifier. Similar to the previous section, we randomly separate $80\%$ of
data as training and use the rest as testing data. Then, we train our
classifier on creatives that belong to the top and bottom $30\%$ in CTR. In
fact, we are disregarding $40\%$ of data that are close to the training data
CTR mean, $\mu_{t}$, to reduce the noise for the classifier. Similar to the
ranking experiments, we filter our test set by focusing on the $k$ creatives
with highest CTR values (labeled as positive) and the $k$ creatives with the
lowest CTR values (labeled as negative), where $k=\delta n$ is varied by
changing $\delta$. We obtain the classification accuracy by comparing the
predicted classes to the true classes obtained from real CTR values. Figure 6
demonstrates the average classification accuracy over $200$ independent runs
where each run uses randomly selected training and testing data. The $x$-axis
indicates the value of $\delta$ and $y$-axis represents the classification
accuracy for each data set given a fixed value of $\delta$. As seen in the
figure, using visual features yields a classification accuracy of $70\%$ when
$\delta=0.5$. Together with the previous results on predicting and ranking
CTR, these results show the efficacy of using visual features of creatives in
predicting CTR.
---
Figure 6: The classification accuracy for each data set.
### 4.4 Feature Selection
The above analysis shows that visual features are useful in predicting the
performance of creatives in online advertising. A natural question is to
identify the visual features that have strong impact on ad performance. Such
information could be very useful in many areas. For example, human graphic
designers may use this information to guide their design of high-performance
creatives. Smart ads system may use this information to dynamically generate
creatives that are more appealing to online users. Ad exchange system may use
this information to determine which creative will win in the auction
marketplace for each advertising opportunity. In this section we conduct a
series of experiments to select such important visual features.
We first calculate the Linear Correlation (LC) and Mutual Information (MI)
between all features and CTR in each data set. Mutual information can provide
us with the information of non-linear correlation between features. Note that,
to calculate the mutual information between any pair of features $(X,Y)$, we
discretized each feature and CTR values into $50$ equal intervals.The results
are shown in table 3. The top $5$ features in each data set with highest
absolute values are highlighted in bold. The table shows that there is no
feature with high linear correlation or mutual information except $f_{12}$ in
data set ID$6$. Thus we use Forward Feature Selection (FFS) to select the top
$k$ features.
Before running FFS, we first cluster the features based on the Normalized
Mutual Information (NMI) of all feature pairs. We discretize each feature into
$50$ equal intervals, and calculate NMI as follows:
$NMI(X;Y)=\frac{I(X;Y)}{\sqrt{H(X)H(Y)}},$ (5)
where $H(X)$ is the entropy of random variable $X$. Then we cluster the
features using the average linkage algorithm [21]. Two clusters are merged
into one if their average NMI is at least $0.2$. This results in $20$ clusters
for data set ID$2$ and $21$ clusters for data set ID$6$. The resulting
clusters are shown in Table 3. In the table, $S_{i}$ represents a set of
features in cluster $i$. We now apply a simple change to the FFS algorithm to
select the top $k$ clusters rather than features. After selecting a feature by
FFS, all the correlated features that belong to the same cluster are removed
from the next steps of FFS. The selected top $k=10$ clusters are shown in
table 2. Note that clustering the features in the above manner helps select
different features (or feature sets) that are less correlated with each other.
For example, all color harmony features are in the same cluster $S_{4}$.
Therefore by selecting one of the features from this cluster, we indicate the
importance of color harmony in CTR, and by removing the highly correlated
features at each step in FFS, we can guarantee to select a set of features
which are less correlated with each other. Below we investigate some of the
selected clusters that are common to both data sets.
Table 2: The top 10 selected clusters by FFS. Data Set | Selected Clusters
---|---
ID$2$ | $S_{5}$, $S_{1}$, $S_{19}$, $S_{17}$, $S_{13}$, $S_{18}$, $S_{20}$, $S_{10}$, $S_{11}$, $S_{9}$
ID$6$ | $S_{1}$, $S_{2}$, $S_{20}$, $S_{5}$, $S_{17}$, $S_{14}$, $S_{9}$, $S_{13}$, $S_{4}$, $S_{18}$
Table 2 shows that $S_{1}$ is the best feature set (or cluster) for data set
ID$6$ and the second best set for data set ID$2$ which illustrates the
importance of set $S_{1}$. $S_{1}$ consists of the gray level features $f_{1}$
and $f_{2}$ of the image. The scatter plot of both features in data set ID$2$
is shown in Figure7 (the scatter plot in data set ID$6$ is similar). Figure 7
shows that for creatives with small value in both features, high CTR value is
unlikely, and creatives with high CTR values should have high values in these
two features. This is consistent with the intuition that creatives with higher
contrast should perform better. Note that having high values in these two
features does not guarantee a high CTR value.
|
---|---
Figure 7: The scatter plot of $f_{1}$ and $f_{2}$ against CTR.
$S_{5}$ is the best feature set for data set ID$2$ and the fourth for ID6. It
only includes $f_{10}$ which is the number of connected coherent components.
The scatter plot of $f_{10}$ in both data sets are shown in figure 8. The
scatter plot shows that creatives with more than $15$ connected coherent
components in data set ID$6$ and more than $20$ in data set ID$2$ are unlikely
to achieve a CTR higher than $0.01$. In other words, this suggests that
cluttered creatives containing many objects tend to have lower CTR.
|
---|---
Figure 8: The scatter plot of $f_{10}$.
The number of characters, $S_{19}$ in data set ID$2$ and $S_{20}$ in data set
ID$6$, is interestingly the third important feature set in both data sets.
Figure 9 shows the scatter plot of the number of characters in both data sets.
It can be seen that the creatives with higher number of characters are
unlikely to achieve high CTR values in both data sets, once again suggesting
that textual clutter is undesirable.
|
---|---
ID$2$ | ID$6$
Figure 9: The scatter plot of number of characters.
The next selected categories is $S_{17}$ which is the $4$-th selected category
in ID$2$ and the $5$-th in ID$6$. $S_{17}$ represents the number of connected
components in saliency binary map, distance between salient components,
distance of saliency areas from the center of image and rule of third closest
point. This indicates the importance of saliency features as well as
considering professional photography rules such as the rule of third in
designing ads. Intuitively, a small number of salient components, closer to
the center of the creative, and consistent with the rule of third are
desirable features in a creative. Finally, $S_{13}$, which contains features
describing the number of hues and the contrast of hues in the largest segment
of the image, is the $5$-th important common category considering both data
sets. Note that the scatter plot of the last $2$ selected categories have been
omitted due to space limit. In summary, our top $5$ selected categories
include the features from all proposed feature categories, global, local and
advanced features, indicating the importance of each of them in predicting the
creatives CTR.
Table 3: Complete list of visual features. Feature | Short Description | LC ID$2$ | MI ID$2$ | LC ID$6$ | MI ID$6$ | C-ID$2$ | C-ID$6$
---|---|---|---|---|---|---|---
$f_{1}$ | gray level contrast | -0.048 | 0.063 | -0.102 | 0.098 | $S_{1}$ | $S_{1}$
$f_{2}$ | number of dominant bins in gray level histogram | -0.107 | $0.071$ | -0.137 | 0.112 | $S_{1}$ | $S_{1}$
$f_{3}$ | standard deviation of gray level images | -0.063 | 0.061 | -0.016 | 0.082 | $S_{2}$ | $S_{2}$
$f_{4}$ | number of dominant bins in RGB histogram | -0.083 | 0.067 | -0.074 | 0.078 | $S_{3}$ | $S_{3}$
$f_{5}$ | size of the dominant bin in RGB histogram | 0.078 | 0.093 | 0.064 | 0.104 | $S_{3}$ | $S_{3}$
$f_{6}$ | number of dominant bins in HSV histogram | -0.083 | 0.073 | -0.078 | 0.092 | $S_{3}$ | $S_{3}$
$f_{7}$ | size of the dominant bin in HSV histogram | 0.119 | 0.096 | 0.085 | 0.094 | $S_{3}$ | $S_{3}$
$f_{8}$ | deviation from the best color harmony model | -0.062 | 0.039 | -0.040 | 0.052 | $S_{4}$ | $S_{4}$
$f_{9}$ | average deviation from the best two color harmony models | -0.069 | 0.047 | -0.046 | 0.063 | $S_{4}$ | $S_{4}$
$f_{10}$ | number of connected coherent components | 0.094 | 0.062 | -0.004 | 0.064 | $S_{5}$ | $S_{5}$
$f_{11}$ | size of the largest connected coherent component | 0.102 | 0.090 | 0.069 | 0.099 | $S_{3}$ | $S_{3}$
$f_{12}$ | size of the second largest connected coherent component | 0.085 | 0.068 | 0.315 | 0.122 | $S_{6}$ | $S_{6}$
$f_{13}$ | color size rank of the largest connected coherent component | -0.002 | 0.010 | -0.012 | 0.014 | $S_{7}$ | $S_{7}$
$f_{14}$ | color size rank of the second largest connected coherent component | -0.046 | 0.020 | -0.039 | 0.022 | $S_{8}$ | $S_{8}$
$f_{15}$ | number of dominant hues | -0.102 | 0.036 | -0.119 | 0.045 | $S_{9}$ | $S_{9}$
$f_{16}$ | contrast of dominant hues | -0.015 | 0.027 | -0.015 | 0.046 | $S_{9}$ | $S_{9}$
$f_{17}$ | standard deviation of hues | 0.010 | 0.084 | -0.039 | 0.102 | $S_{10}$ | $S_{10}$
$f_{18}$ | average lightness | 0.031 | 0.072 | 0.035 | 0.099 | $S_{11}$ | $S_{11}$
$f_{19}$ | standard deviation of lightness | -0.065 | 0.063 | -0.086 | 0.084 | $S_{2}$ | $S_{2}$
$f_{20}$ | size of the Largest Segments (LS) | -0.052 | 0.058 | -0.080 | 0.081 | $S_{12}$ | $S_{12}$
$f_{21}$ | segments size contrast | -0.044 | 0.054 | -0.122 | 0.092 | $S_{12}$ | $S_{12}$
$f_{22}$ | number of image dominant hues in the LS | -0.104 | 0.044 | -0.068 | 0.029 | $S_{13}$ | $S_{13}$
$f_{23}$ | number of dominant hues in the LS | -0.102 | 0.032 | -0.080 | 0.036 | $S_{13}$ | $S_{13}$
$f_{24}$ | largest number of dominant hues in one segment | -0.079 | 0.031 | -0.125 | 0.045 | $S_{14}$ | $S_{14}$
$f_{25}$ | contrast of hues number among segments | -0.006 | 0.027 | -0.098 | 0.033 | $S_{14}$ | $S_{14}$
$f_{26}$ | contrast of hues in the LS | -0.120 | 0.032 | -0.085 | 0.036 | $S_{13}$ | $S_{13}$
$f_{27}$ | standard deviation of hues contrast among segments | -0.056 | 0.072 | -0.073 | 0.098 | $S_{10}$ | $S_{10}$
$f_{28}$ | deviation from the best color harmony model for LS | -0.047 | 0.030 | -0.034 | 0.042 | $S_{4}$ | $S_{4}$
$f_{29}$ | average deviation from the best two color harmony models for LS | -0.055 | 0.037 | -0.049 | 0.053 | $S_{4}$ | $S_{4}$
$f_{30}$ | average lightness in LS | 0.027 | 0.075 | 0.065 | 0.096 | $S_{11}$ | $S_{11}$
$f_{31}$ | standard deviation of average lightness among the segments | 0.080 | 0.076 | -0.049 | 0.091 | $S_{15}$ | $S_{15}$
$f_{32}$ | contrast of average lightness among the segments | 0.075 | 0.067 | -0.027 | 0.081 | $S_{15}$ | $S_{15}$
$f_{33}$ | background size in Saliency Map (SM) | -0.165 | 0.082 | -0.175 | 0.106 | $S_{16}$ | $S_{16}$
$f_{34}$ | number of connected components in SM | -0.111 | 0.029 | -0.062 | 0.088 | $S_{17}$ | $S_{17}$
$f_{35}$ | size of the largest connected components in SM | 0.132 | 0.075 | 0.036 | 0.073 | $S_{16}$ | $S_{18}$
$f_{36}$ | average saliency weight of largest connected component | 0.165 | 0.075 | 0.071 | 0.080 | $S_{16}$ | $S_{18}$
$f_{37}$ | number of connected components in image background SM | -0.012 | 0.015 | -0.035 | 0.038 | $S_{18}$ | $S_{19}$
$f_{38}$ | size of the largest connected component of background in SM | -0.163 | 0.091 | -0.145 | 0.052 | $S_{16}$ | $S_{16}$
$f_{39}$ | distance between connected components in SM | 0.117 | 0.056 | -0.027 | 0.072 | $S_{17}$ | $S_{17}$
$f_{40}$ | distance from rule of third points in SM | 0.126 | 0.063 | 0.010 | 0.086 | $S_{17}$ | $S_{17}$
$f_{41}$ | distance from center of image in SM | 0.091 | 0.065 | 0.011 | 0.087 | $S_{17}$ | $S_{17}$
$f_{42}$ | number of characters (OCR) | -0.082 | 0.057 | -0.017 | 0.071 | $S_{19}$ | $S_{20}$
$f_{43}$ | number of faces in the image | -0.035 | 0.008 | -0.024 | 0.011 | $S_{20}$ | $S_{21}$
## 5 Conclusion
In this paper we investigated the relationship between the user response rate
and the visual appearance of creatives in online display advertising. To the
best of our knowledge, this is the first work in this area. We designed $43$
visual features for our experiments. We extracted the features from large
scale data produced by the world’s largest ad exchange system. We tested the
utility of visual features in CTR prediction, ranking and classification. The
experimental results demonstrate that our proposed framework is able to
outperform baseline consistently, indicating the efficacy of visual features
in predicting CTR. We also performed feature selection to select the top
visual feature categories that have strongest importance for increasing CTR.
The findings from this work will be useful for ads selection and developing
visually appealing creatives with higher user response propensity in online
display advertising.
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|
arxiv-papers
| 2012-02-10T00:26:22 |
2024-09-04T02:49:27.310706
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Javad Azimi, Ruofei Zhang, Yang Zhou, Vidhya Navalpakkam, Jianchang\n Mao, Xiaoli Fern",
"submitter": "Javad Azimi",
"url": "https://arxiv.org/abs/1202.2158"
}
|
1202.2379
|
# A revised asteroid polarization-albedo relationship using WISE/NEOWISE data
Joseph R. Masiero11affiliation: Jet Propulsion Laboratory/California Institute
of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109, USA,
Joseph.Masiero@jpl.nasa.gov , A. K. Mainzer11affiliation: Jet Propulsion
Laboratory/California Institute of Technology, 4800 Oak Grove Dr., MS 321-520,
Pasadena, CA 91109, USA, Joseph.Masiero@jpl.nasa.gov , T. Grav22affiliation:
Planetary Science Institute, 1700 East Fort Lowell, Suite 106, Tucson, AZ
85719-2395 , J. M. Bauer11affiliation: Jet Propulsion Laboratory/California
Institute of Technology, 4800 Oak Grove Dr., MS 321-520, Pasadena, CA 91109,
USA, Joseph.Masiero@jpl.nasa.gov 33affiliation: Infrared Processing and
Analysis Center, California Institute of Technology, Pasadena, CA 91125 USA ,
E. L. Wright44affiliation: UCLA Astronomy, PO Box 91547, Los Angeles, CA
90095-1547 USA , R. S. McMillan55affiliation: Lunar and Planetary Laboratory,
University of Arizona, 1629 East University Blvd, Kuiper Space Science Bldg.
#92, Tucson, AZ 85721-0092 USA , D. J. Tholen66affiliation: Institute for
Astronomy, University of Hawaii, Honolulu, HI 96822 USA , A. W.
Blain77affiliation: Department of Physics and Astronomy, University of
Leicester, University Road, Leicester, LE1 7RH, United Kingdom
###### Abstract
We present a reanalysis of the relationship between asteroid albedo and
polarization properties using the albedos derived from the Wide-field Infrared
Survey Explorer. We find that the function that best describes this relation
is a three-dimensional linear fit in the space of $\log$ (albedo)-$\log$
(polarization slope)-$\log$ (minimum polarization). When projected to two
dimensions the parameters of the fit are consistent with those found in
previous work. We also define $p^{\star}$ as the quantity of maximal
polarization variation when compared with albedo and present the best fitting
albedo-$p^{\star}$ relation. Some asteroid taxonomic types stand out in this
three-dimensional space, notably the E, B, and M Tholen types, while others
cluster in clumps coincident with the S- and C-complex bodies. We note that
both low albedo and small ($D<30~{}$km) asteroids are under-represented in the
polarimetric sample, and we encourage future polarimetric surveys to focus on
these bodies.
## 1 Introduction
As light scatters off the surface of atmosphereless bodies, it is instilled
with a small linear polarization. The degree of linear polarization of the
scattered light measured by the observer is a function of the phase angle of
observation and the composition and structure of the surface, in particular
the interrelated parameters of albedo, index of refraction, and space between
scattering elements (e.g. Muinonen, 1989; Shkuratov et al., 1994). Early work
quantified the relation between phase angle (the angle between the direction
to the sun and the observer as seen from the target, $\alpha$) and
polarization (Dollfus & Zellner, 1979) and this effect can be used in parallel
with the magnitude-phase effect to probe the scattering physics of
atmosphereless surfaces (Muinonen et al., 2002, 2009).
As expected from classical scattering models, the light reflected from a
surface is polarized perpendicular to the scattering plane for large phase
angles, which is referred to as a positive polarization. For small phase
angles, however, light acquires a polarization in the scattering plane due to
an increase in the dominance of second-order scattering. This case is referred
to as negative polarization, as it is perpendicular to the positive case and
thus carries a negative sign when the polarization coordinate system is
rotated to account for the viewing geometry. The angle where the phase curve
transitions from positive to negative is referred to as the inversion angle
($\alpha_{0}$). By definition the value of the polarization must go to zero at
$\alpha=0^{\circ}$, though some work has suggested that surfaces may have a
secondary trough at very small angles related to the optical opposition effect
(Rosenbush et al., 1997). Cellino et al. (2005a) find no evidence for a
polarimetric opposition effect in their sample, though high albedo objects are
not represented there.
From the studies of the scattering properties of the lunar surface, a
relationship was found between albedo and the parameters used to describe the
polarimetric-phase effect of the lunar regolith (Bowell et al., 1973) that was
then extended to asteroids (Zellner et al., 1974; Cellino et al., 1999), of
the form:
$\displaystyle\log p_{V}=C_{1}\log h+C_{2}$ (1) $\displaystyle\log
p_{V}=C_{3}\log P_{min}+C_{4}$ (2)
where $p_{V}$ is the geometric albedo, $h$ is the linear slope of the phase
curve at the inversion angle, and $P_{min}$ is the value of the largest
negative polarization (i.e. the depth of the negative trough), usually
expressed as an absolute value. We show an illustration of these parameters
and two typical polarization-phase curves in Figure 1. We note that the
polarization shown in this figure is the $P_{r}$ value that has been rotated
to account for viewing geometry, such that $P_{r}>0$ is the amplitude of
polarization perpendicular to the scattering plane and $P_{r}<0$ is the
amplitude parallel to the scattering plane. A polarization component $\pm
45^{\circ}$ from the scattering plane is typically not observed at any phase
angle for asteroids, and thus is ignored in this diagram.
Cellino et al. (1999) present the most recent best-fitting values for the
constants in the above equations: $C_{1}=-1.118\pm 0.071$, $C_{2}=-1.779\pm
0.062$, $C_{3}=-1.357\pm 0.140$, and $C_{4}=-0.858\pm 0.030$. In this work, we
revise the best-fitting values for these constants in light of new albedo data
from the Wide-field Infrared Survey Explorer (WISE, Wright et al., 2010) and
the planetary science extension NEOWISE (Mainzer et al., 2011a). The aim of
this work is two-fold: firstly, while WISE provides us with albedos for a
large fraction of the known asteroids, calibration of this relationship will
allow it to be applied to objects that were not observed by WISE; secondly,
the behavior of the polarization of asteroid surfaces helps us determine the
surface mineralogy and this relationship represents a critical component in
this determination. Through application of both thermal infrared and
polarimetric data we can gain a broader understanding of the behavior of
asteroids across the Solar system.
Figure 1: Illustration of the typical polarization-phase behavior for two
different types of asteroids, with the inversion angle ($\alpha_{0}$), the
minimum polarization ($P_{min}$), and the slope of polarization at the
inversion angle ($h$) labeled for the dashed blue curve. Example curves for a
generic S-type asteroid (solid red line) and a generic C-type (dashed blue
line) are shown.
## 2 Data
We draw our list of polarimetric properties for asteroids from a range of
sources. The dominant contributor is the Astronomical Polarimetric Database
presented in the Planetary Data System (PDS) (Lupishko & Vasilyev, 2008) which
was a compilation of the polarimetric properties of individual asteroids in
the literature up to the data of publication. We also incorporate values for
$h$, $P_{min}$, and/or $\alpha_{0}$ for asteroids given by: Cellino et al.
(1999, 2005a, 2005b); Fornasier et al. (2006); Gil-Hutton (2007a); Gil-Hutton
et al. (2007b, 2008); Masiero & Cellino (2009); Belskaya et al. (2010). We
note that as these data are drawn from a range of different instruments,
uncertainties in the absolute calibration may result in a larger scatter than
is actually present. A comprehensive survey of polarimetric properties of a
large number of objects conducted with a single instrument would reduce this
possible source of error, and so is strongly encouraged.
Determination of $h$, $P_{min}$, and $\alpha_{0}$ all require polarimetric
measurements spanning a range of phase angles. The inversion angle can
typically be determined to a reasonable level of accuracy with a few bounding
measurements at $\alpha\sim 20^{\circ}$. Polarimetric slope is more difficult
to determine, especially for objects located farther from the sun that are
rarely observable at phase angles much beyond the inversion angle (e.g.
objects that do not come within $\sim 2.9~{}$AU of the sun can never be
observed at phase angles $\alpha>20^{\circ}$). Careful timing of observations
can ensure adequate phase coverage that will allow for an accurate
determination of the slope. The depth of minimum polarization is often the
most difficult parameter to determine for some objects, as it requires
observing at small phase angles that are not frequently available for
asteroids in the inner Main Belt, Hungaria, Mars Crosser, and NEO populations.
Additionally, determining this value requires evenly spaced observations over
the full branch of negative polarization, rather than just a few bounding
measurements as required for both $\alpha_{0}$ and $h$. As such, relative
errors on $P_{min}$ tend to be larger than measured for the other polarimetric
parameters. Where errors on polarimetric parameters were not given by the
source, we assume values based on the errors on the published data in those
sources.
The albedos we use for this work are drawn from the values derived for Main
Belt asteroids (MBAs) published in Masiero et al. (2011). For objects in the
NEO or Mars Crossing populations, we draw albedos from the appropriate lists
(Mainzer et al., 2011e, f) which were derived using a method identical to that
used for the MBAs. All of the objects with both defined polarimetric phase
curves as well as WISE-determined albedos had low identifying numbers,
implying that they were some of the first objects discovered, and thus likely
to preferentially sample the largest minor bodies of the solar system. These
large asteroids were more likely to have been seen in multiple bands by WISE
which allows for fitting of the beaming parameter. Mainzer et al. (2011b) show
that in cases such as this the error on albedo as an absolute measurement is
$\sim 20\%$ of the measured albedo value, however internal comparisons are
better than this limit.
A primary concern in any analysis of albedos derived from infrared-determined
diameters is the quality of the optical measurements used. We draw our $H$
magnitudes from the Minor Planet Center’s orbital element catalog
(MPCORB111http://www.minorplanetcenter.net/iau/MPCORB.html), as discussed in
Masiero et al. (2011). While other studies have found an offset between
measured magnitudes and those predicted from the $H$ absolute magnitude value,
with objects being fainter than expected (e.g. Jurić et al., 2002; Parker et
al., 2008), Mainzer et al. (2011b) find that, in general, no offset
corrections to $H$ are required for the most recent releases of the
magnitudes. An exception to this result has been found for some objects with
unusually high albedos in Masiero et al. (2011) and Mainzer et al. (2011e);
many of these objects are coincident in orbital-element-space with the
Hungarias and the Vesta family. Harris et al. (1989) found that the commonly
assumed value of $G=0.15$ is inappropriate for these types of high-albedo
objects, and a value of $G\sim 0.4$ may be more appropriate. Revising $G$ to
this value would result in an offset of up to $\sim 0.3~{}$mag in the $H$
magnitude depending on the initial $H$ fit, however this correction is not
required for most objects. In the past, photometric measurements for many
asteroids that contributed to the $H$ magnitudes in the MPCORB catalog were
acquired with unfiltered CCDs. New, filtered observations and refined handling
of previous photometry have largely mitigated the effect of unfiltered
measurements on $H$ values. (T. Spahr, 2012, private communication).
Mainzer et al. (2011c) present a comparison of the WISE albedos to the IRAS
albedos and find a good match for most objects, though some scatter is seen,
especially at the smallest sizes where the IRAS signal-to-noise ratios were
poor compared to WISE. This albedo error assumes moderate-to-low light curve
amplitudes and well characterized $H$ and $G$ values. This error will result
in an uncertainty in the offsets of the linear fits (i.e. $C_{2}$ and $C_{4}$
in Equations 1 and 2) though the slopes should be unaffected. We include this
error in our fits below. We note that recently Muinonen et al. (2010) have
introduced a three-parameter photometric system ($H$,$G_{1}$,$G_{2}$) to
better characterize the behavior of the photometric phase effect which may
reduce some of these errors, but we note that this system requires accurate
photometry over a large phase window, which is not available for many
asteroids.
We show the polarimetric and albedo data used for this work in Table
LABEL:tab.data (ellipses indicate an unmeasured polarimetric property). When
these two data sets are combined, we have $65$ objects with measured albedos
and polarimetric slopes, and $112$ with albedos and minimum polarization
values. This result is a improvement compared to the data set presented by
Cellino et al. (1999), who performed a similar analysis using IRAS albedos of
$37$ objects for the slope-albedo fit and $16$ for the minimum polarization-
albedo fit. We note that due to the brightness requirements of most
polarimeters and the polarimetric survey strategies employed, these lists are
dominated by the largest known asteroids. Approximately half our sample have
sizes over $100~{}$km, and three-quarters are larger than $50~{}$km. Thus
while the largest asteroids are well sampled, there is a distinct lack of
small bodies in these lists. We also note that despite the fact that low-
albedo objects dominate the Main Belt population (Masiero et al., 2011), they
are under-represented in the polarimetric surveys (see below). As WISE is
sensitive to thermal infrared light, the detection probability for asteroids
is effectively unbiased with respect to the albedos of the objects observed
(Mainzer et al., 2011e), and thus the distribution of albedos seen with WISE
is a more accurate representation of the true population than is the
distribution seen for optically-selected samples. Extending polarimetric
coverage to both smaller sizes and low albedo objects through a large-scale
campaign is critical to extending and generalizing the trends seen here and in
previous work.
Table 1: Compiled asteroid albedos and polarimetric properties Asteroid | $p_{V}$ | $\alpha_{0}$ | $h$ | $P_{min}$
---|---|---|---|---
2 | $0.142\pm 0.018$ | $18.1\pm 0.1$ | $0.228\pm 0.003$ | $1.38\pm 0.05$
5 | $0.245\pm 0.051$ | $19.1\pm 0.1$ | $0.096\pm 0.050$ | $0.70\pm 0.05$
6 | $0.269\pm 0.049$ | $20.8\pm 0.2$ | $0.091\pm 0.050$ | $0.80\pm 0.05$
8 | $0.261\pm 0.048$ | $20.0\pm 0.1$ | $0.104\pm 0.003$ | $0.68\pm 0.05$
9 | $0.134\pm 0.016$ | $21.8\pm 0.1$ | $0.102\pm 0.003$ | $0.74\pm 0.05$
10 | $0.058\pm 0.005$ | $...$ | $...$ | $1.50\pm 0.05$
11 | $0.158\pm 0.036$ | $18.9\pm 0.2$ | $0.124\pm 0.003$ | $0.73\pm 0.05$
12 | $0.140\pm 0.014$ | $20.8\pm 0.2$ | $0.121\pm 0.003$ | $0.73\pm 0.01$
13 | $0.069\pm 0.022$ | $21.7\pm 0.5$ | $0.257\pm 0.003$ | $2.10\pm 0.05$
14 | $0.221\pm 0.022$ | $20.5\pm 0.2$ | $0.105\pm 0.003$ | $0.82\pm 0.10$
15 | $0.206\pm 0.055$ | $20.6\pm 0.2$ | $0.087\pm 0.005$ | $0.72\pm 0.02$
17 | $0.160\pm 0.009$ | $...$ | $0.131\pm 0.003$ | $0.74\pm 0.05$
18 | $0.221\pm 0.082$ | $21.6\pm 0.1$ | $0.101\pm 0.003$ | $0.87\pm 0.05$
19 | $0.050\pm 0.020$ | $21.7\pm 0.2$ | $0.305\pm 0.003$ | $1.72\pm 0.05$
22 | $0.169\pm 0.061$ | $...$ | $...$ | $0.83\pm 0.04$
24 | $0.064\pm 0.016$ | $...$ | $0.191\pm 0.003$ | $1.63\pm 0.10$
27 | $0.201\pm 0.058$ | $...$ | $0.099\pm 0.003$ | $0.70\pm 0.05$
29 | $0.157\pm 0.035$ | $22.0\pm 0.2$ | $0.098\pm 0.003$ | $0.88\pm 0.10$
30 | $0.171\pm 0.034$ | $19.8\pm 0.5$ | $0.104\pm 0.003$ | $0.78\pm 0.05$
31 | $0.045\pm 0.044$ | $...$ | $...$ | $1.32\pm 0.10$
32 | $0.230\pm 0.065$ | $...$ | $...$ | $0.63\pm 0.05$
39 | $0.245\pm 0.056$ | $21.0\pm 0.1$ | $0.090\pm 0.003$ | $0.79\pm 0.05$
40 | $0.195\pm 0.019$ | $20.8\pm 0.2$ | $0.100\pm 0.003$ | $0.85\pm 0.05$
46 | $0.052\pm 0.011$ | $...$ | $...$ | $1.54\pm 0.10$
47 | $0.067\pm 0.009$ | $...$ | $0.204\pm 0.003$ | $1.44\pm 0.05$
51 | $0.100\pm 0.026$ | $...$ | $0.292\pm 0.050$ | $1.86\pm 0.05$
54 | $0.049\pm 0.008$ | $22.2\pm 0.5$ | $0.357\pm 0.050$ | $1.95\pm 0.05$
56 | $0.050\pm 0.006$ | $19.7\pm 0.2$ | $0.318\pm 0.003$ | $1.47\pm 0.05$
57 | $0.182\pm 0.047$ | $...$ | $...$ | $0.71\pm 0.10$
58 | $0.059\pm 0.005$ | $...$ | $...$ | $1.70\pm 0.10$
63 | $0.159\pm 0.028$ | $19.8\pm 0.1$ | $0.102\pm 0.003$ | $0.70\pm 0.05$
64 | $0.676\pm 0.223$ | $18.2\pm 0.2$ | $0.036\pm 0.001$ | $0.32\pm 0.05$
68 | $0.207\pm 0.025$ | $...$ | $...$ | $0.68\pm 0.05$
70 | $0.040\pm 0.009$ | $...$ | $...$ | $1.83\pm 0.05$
71 | $0.248\pm 0.035$ | $...$ | $0.061\pm 0.005$ | $0.61\pm 0.02$
73 | $0.186\pm 0.018$ | $...$ | $...$ | $0.76\pm 0.06$
75 | $0.098\pm 0.014$ | $20.2\pm 0.1$ | $0.103\pm 0.017$ | $...$
77 | $0.153\pm 0.046$ | $...$ | $...$ | $1.25\pm 0.14$
80 | $0.182\pm 0.026$ | $...$ | $...$ | $0.75\pm 0.05$
83 | $0.086\pm 0.021$ | $...$ | $...$ | $1.47\pm 0.10$
84 | $0.053\pm 0.017$ | $20.3\pm 0.5$ | $0.306\pm 0.050$ | $1.49\pm 0.05$
85 | $0.063\pm 0.025$ | $...$ | $...$ | $1.36\pm 0.10$
89 | $0.185\pm 0.034$ | $...$ | $0.119\pm 0.050$ | $0.90\pm 0.05$
95 | $0.056\pm 0.009$ | $...$ | $...$ | $1.78\pm 0.05$
97 | $0.206\pm 0.046$ | $22.1\pm 0.1$ | $0.174\pm 0.018$ | $...$
113 | $0.223\pm 0.031$ | $...$ | $0.081\pm 0.005$ | $...$
114 | $0.088\pm 0.010$ | $...$ | $...$ | $1.24\pm 0.10$
115 | $0.654\pm 0.124$ | $...$ | $...$ | $0.71\pm 0.05$
118 | $0.139\pm 0.031$ | $...$ | $...$ | $0.80\pm 0.12$
121 | $0.077\pm 0.010$ | $...$ | $...$ | $1.72\pm 0.05$
125 | $0.115\pm 0.027$ | $...$ | $0.145\pm 0.033$ | $0.83\pm 0.02$
129 | $0.157\pm 0.026$ | $...$ | $...$ | $0.90\pm 0.05$
131 | $0.164\pm 0.011$ | $...$ | $0.208\pm 0.059$ | $...$
132 | $0.120\pm 0.008$ | $...$ | $0.146\pm 0.006$ | $1.13\pm 0.03$
135 | $0.152\pm 0.050$ | $...$ | $...$ | $1.06\pm 0.10$
138 | $0.161\pm 0.028$ | $...$ | $0.103\pm 0.020$ | $...$
139 | $0.045\pm 0.023$ | $...$ | $0.262\pm 0.050$ | $1.31\pm 0.05$
141 | $0.049\pm 0.010$ | $20.6\pm 0.5$ | $0.330\pm 0.050$ | $1.78\pm 0.05$
145 | $0.043\pm 0.004$ | $...$ | $...$ | $1.86\pm 0.05$
153 | $0.046\pm 0.008$ | $...$ | $...$ | $1.05\pm 0.05$
182 | $0.210\pm 0.059$ | $...$ | $...$ | $0.64\pm 0.05$
184 | $0.107\pm 0.019$ | $...$ | $...$ | $0.93\pm 0.06$
188 | $0.157\pm 0.055$ | $...$ | $0.140\pm 0.015$ | $...$
189 | $0.199\pm 0.024$ | $...$ | $...$ | $1.26\pm 0.10$
192 | $0.288\pm 0.040$ | $19.8\pm 0.1$ | $0.084\pm 0.003$ | $0.75\pm 0.05$
197 | $0.239\pm 0.026$ | $...$ | $...$ | $0.79\pm 0.08$
201 | $0.097\pm 0.006$ | $...$ | $...$ | $1.00\pm 0.05$
204 | $0.163\pm 0.044$ | $...$ | $...$ | $0.83\pm 0.12$
216 | $0.111\pm 0.034$ | $...$ | $...$ | $1.27\pm 0.05$
217 | $0.044\pm 0.005$ | $...$ | $...$ | $0.82\pm 0.05$
230 | $0.171\pm 0.076$ | $20.6\pm 0.2$ | $0.122\pm 0.003$ | $0.94\pm 0.05$
234 | $0.151\pm 0.034$ | $27.0\pm 2.0$ | $...$ | $1.60\pm 0.20$
250 | $0.112\pm 0.021$ | $...$ | $...$ | $0.88\pm 0.08$
259 | $0.042\pm 0.005$ | $...$ | $...$ | $1.25\pm 0.05$
270 | $0.254\pm 0.043$ | $...$ | $...$ | $0.65\pm 0.05$
305 | $0.182\pm 0.028$ | $...$ | $...$ | $0.64\pm 0.10$
306 | $0.174\pm 0.060$ | $...$ | $...$ | $0.66\pm 0.10$
324 | $0.063\pm 0.012$ | $20.0\pm 0.1$ | $0.278\pm 0.003$ | $1.46\pm 0.05$
334 | $0.051\pm 0.016$ | $...$ | $...$ | $1.32\pm 0.05$
338 | $0.165\pm 0.028$ | $...$ | $...$ | $0.98\pm 0.10$
345 | $0.059\pm 0.012$ | $...$ | $...$ | $1.55\pm 0.05$
347 | $0.213\pm 0.041$ | $22.6\pm 0.1$ | $0.113\pm 0.003$ | $0.78\pm 0.03$
349 | $0.153\pm 0.018$ | $...$ | $...$ | $0.39\pm 0.05$
351 | $0.171\pm 0.046$ | $...$ | $...$ | $0.74\pm 0.09$
354 | $0.173\pm 0.032$ | $...$ | $...$ | $0.51\pm 0.10$
356 | $0.053\pm 0.015$ | $...$ | $...$ | $1.50\pm 0.10$
377 | $0.056\pm 0.025$ | $19.8\pm 0.2$ | $0.206\pm 0.005$ | $1.76\pm 0.04$
384 | $0.190\pm 0.040$ | $...$ | $...$ | $0.94\pm 0.35$
396 | $0.139\pm 0.025$ | $...$ | $...$ | $1.34\pm 0.09$
409 | $0.050\pm 0.010$ | $19.9\pm 0.2$ | $0.191\pm 0.005$ | $...$
410 | $0.043\pm 0.007$ | $...$ | $0.313\pm 0.050$ | $1.94\pm 0.05$
415 | $0.086\pm 0.009$ | $...$ | $...$ | $1.28\pm 0.10$
423 | $0.066\pm 0.005$ | $...$ | $...$ | $1.40\pm 0.05$
441 | $0.139\pm 0.026$ | $...$ | $...$ | $1.41\pm 0.12$
451 | $0.069\pm 0.006$ | $...$ | $...$ | $1.62\pm 0.05$
466 | $0.086\pm 0.009$ | $...$ | $...$ | $1.60\pm 0.10$
511 | $0.073\pm 0.006$ | $19.4\pm 0.1$ | $0.277\pm 0.003$ | $1.69\pm 0.05$
532 | $0.202\pm 0.039$ | $...$ | $0.122\pm 0.003$ | $0.78\pm 0.05$
550 | $0.137\pm 0.024$ | $...$ | $0.157\pm 0.005$ | $...$
558 | $0.117\pm 0.010$ | $...$ | $...$ | $0.75\pm 0.06$
584 | $0.244\pm 0.060$ | $19.1\pm 0.1$ | $0.108\pm 0.003$ | $0.64\pm 0.05$
600 | $0.177\pm 0.036$ | $...$ | $...$ | $0.43\pm 0.20$
602 | $0.052\pm 0.007$ | $...$ | $...$ | $1.76\pm 0.10$
624 | $0.077\pm 0.020$ | $...$ | $...$ | $1.30\pm 0.05$
625 | $0.197\pm 0.058$ | $...$ | $0.070\pm 0.003$ | $...$
654 | $0.043\pm 0.011$ | $20.5\pm 0.5$ | $0.280\pm 0.050$ | $1.46\pm 0.10$
662 | $0.193\pm 0.028$ | $...$ | $...$ | $1.32\pm 0.33$
674 | $0.206\pm 0.033$ | $...$ | $...$ | $0.81\pm 0.10$
704 | $0.076\pm 0.010$ | $15.7\pm 0.1$ | $0.305\pm 0.003$ | $1.45\pm 0.10$
737 | $0.136\pm 0.043$ | $...$ | $...$ | $0.84\pm 0.05$
787 | $0.120\pm 0.022$ | $...$ | $0.087\pm 0.003$ | $...$
796 | $0.205\pm 0.041$ | $...$ | $0.124\pm 0.011$ | $0.98\pm 0.02$
849 | $0.115\pm 0.016$ | $...$ | $...$ | $0.95\pm 0.05$
857 | $0.225\pm 0.026$ | $...$ | $...$ | $0.75\pm 0.16$
863 | $0.112\pm 0.016$ | $18.1\pm 0.2$ | $0.052\pm 0.005$ | $0.40\pm 0.10$
887 | $0.230\pm 0.018$ | $...$ | $0.101\pm 0.050$ | $0.76\pm 0.05$
925 | $0.253\pm 0.053$ | $19.6\pm 0.2$ | $0.065\pm 0.005$ | $...$
1036 | $0.212\pm 0.026$ | $20.6\pm 0.2$ | $0.112\pm 0.003$ | $0.84\pm 0.02$
1052 | $0.273\pm 0.074$ | $...$ | $...$ | $0.67\pm 0.05$
1058 | $0.242\pm 0.024$ | $...$ | $...$ | $0.69\pm 0.10$
1105 | $0.102\pm 0.017$ | $...$ | $...$ | $1.20\pm 0.16$
1355 | $0.466\pm 0.082$ | $18.2\pm 0.1$ | $0.083\pm 0.020$ | $...$
1627 | $0.153\pm 0.046$ | $...$ | $0.131\pm 0.003$ | $...$
1672 | $0.094\pm 0.016$ | $...$ | $0.131\pm 0.003$ | $...$
1685 | $0.292\pm 0.127$ | $...$ | $0.099\pm 0.003$ | $...$
2131 | $0.198\pm 0.034$ | $...$ | $...$ | $0.86\pm 0.15$
2577 | $0.377\pm 0.062$ | $20.9\pm 0.1$ | $0.124\pm 0.044$ | $...$
3169 | $0.423\pm 0.067$ | $19.6\pm 0.1$ | $0.276\pm 0.018$ | $...$
6249 | $0.878\pm 0.140$ | $22.4\pm 0.1$ | $0.164\pm 0.035$ | $...$
6911 | $0.454\pm 0.083$ | $...$ | $...$ | $0.83\pm 0.16$
Table 1: (continued)
## 3 Revised Polarimetric-Albedo Relationship
In Figures 2-4 we compare the measured WISE albedo to the slope of the
polarization beyond the inversion angle, the depth of the negative branch of
polarization, and the inversion angle, respectively, for all objects with
recorded values for these parameters. We distinguish objects in the Hungaria
region and in the NEO population as red squares and cyan triangles,
respectively. While the NEOs appear consistent with the MBAs, the Hungaria
objects deviate from the general trend significantly. As discussed in Masiero
et al. (2011) the albedos for these objects are suspect: large deviations in
the magnitude-phase slope parameter from the assumed $G=0.15$ used for most
asteroids can result in incorrect $H$ values, and thus poorly constrained
albedos (Harris et al., 1989). Alternatively, large-amplitude long-period
light curves may also corrupt the $H$ values calculated from optical
photometry. We are currently working on a program to better constrain the
photometric parameters and albedos of these objects, but for the following
discussion we will disregard the Hungaria asteroids.
Figure 2: Albedo vs. slope of the polarization phase curve beyond the
inversion angle. Objects located in the Hungaria region are noted as red
squares, objects in the NEO population as cyan triangles. The green dashed
line shows the best fit found for Equation 1 with our data. Figure 3: The same
as Figure 2, but for albedo vs. depth of the minimum polarization branch. The
green dashed line shows the best fit found for Equation 2 with our data.
Figure 4: The same as Figure 2, but for albedo vs. inversion angle.
We see no overall trend between albedo and inversion angle in our data. The
object with the anomalously high inversion angle is (234) Barbara, the
principal member of the “Barbarian” group of objects with strange polarimetric
properties (Cellino et al., 2006). The object with an inversion angle well
below the general trend is (704) Interamnia: some F-class objects like
Interamnia have previously been shown to display unusually small inversion
angles (Belskaya et al., 2005).
We see the expected general trends when looking at slope and Pmin, with low
albedo objects showing steeper slopes and deeper troughs. We note, however,
that the lowest albedo objects, those with $p_{V}<0.04$, have almost no
representation in the polarimetric sample despite representing over $10\%$ of
the total population of MBAs observed by WISE, even before correcting the
population for objects without optical followup (and thus without measured
albedos). We therefore cannot comment on the reliability of the polarization-
phase relation at the lowest albedos. A campaign of polarimetric observations
of low albedo asteroids is critical to test these relations at their low
albedo extreme.
We find that the optimal description of the relation between albedo and
polarimetric parameters is a linear fit in the three dimensional space of
$\log p_{V}$-$\log h$-$\log P_{min}$. We use only those objects where both
polarimetric parameters are measured to an accuracy of $20\%$ or better,
leaving us with $41$ objects in our high-confidence sample. We perform
orthogonal distance regression on the three dimensional data, using the
associated errors on each measurement to determine the best fitting linear
parameters as well as each parameter’s error. We then reduce the best-fit
parameters back to two-dimensional projections, which result in the following
constant parameters for the relationships in Equations 1 and 2:
$C_{1}=-1.207\pm 0.067$ $C_{2}=-1.892\pm 0.141$ $C_{3}=-1.579\pm 0.084$
$C_{4}=-0.880\pm 0.106$
These projected fits are shown as green dashed lines in Figures 2 and 3. With
the exception of $C_{3}$, these parameters are all within one-sigma of the
values found by Cellino et al. (1999), and all are within 1.5 sigma. As the
WISE albedos for the largest asteroids have been shown to be generally
consistent with the IRAS values (Mainzer et al., 2011c) and all of the objects
used here are in the size range sampled by IRAS, this agreement was not
unexpected. Of the objects in our high-confidence polarimetric sample only $6$
were not observed by IRAS, however the WISE albedos are all derived from a
minimum of $5$ observations (and an average of $>10$) spread over time and
thus are less sensitive to rotation effects. As the WISE data cover MBAs down
to a few kilometers and NEOs to much smaller sizes, future polarimetric
surveys focusing on smaller asteroids will allow this relationship to be
tested over a more extensive size range.
We can also project our fit of the polarimetric properties onto an axis of
maximal variation. We define $p^{\star}$ as the quantity of maximum
polarimetric variation and find a best-fitting transform of:
$p^{\star}=(0.79\pm 0.02)\log h+(0.61\pm 0.03)\log P_{min}$
Using our data we find a best-fitting relation between $p^{\star}$ and albedo
of:
$\log p_{V}=(-1.58\pm 0.09)+(-1.04\pm 0.04)p^{\star}$
We show $p^{\star}$ compared to albedo for all of the high-confidence objects
with both measured $h$ and $P_{min}$, along with this fit, in Figure 5. We
label (64) Angelina as “E”, (2) Pallas as “B”, and (132) Aethra as “M”
following their Tholen taxonomic classifications (Neese, 2010). The clusters
of objects with Tholen S and C taxonomic classifications are labeled as such,
and include other objects within those taxonomic complexes (e.g. F- and
D-types are included in the C-complex) . We indicate (71) Niobe with “n” in
this plot; though it has a Tholen class of S it is distinct enough from the
general cluster to warrant mention. Additional polarimetric and spectroscopic
followup of this object will help determine why its properties differ from the
general S complex. We focus on Tholen classifications here as this taxonomic
system shows the greatest distinction in albedo between the different types
(Mainzer et al., 2011d).
Figure 5: The maximal variation of the polarimetric properties $p^{\star}$ as
defined in the text compared to the measured albedo. The best-fitting
relationship projected from the three-dimensional fit is shown as the dashed
line. Text labels denote specific asteroids or groupings as discussed in the
text.
In addition to the lack of the lowest albedo objects in the polarimetric
sample, we observe an over-representation of high albedo objects compared to
the distribution for all similarly sized MBAs. Figure 6 shows the distribution
of albedos of all objects in the sample with high-confidence polarimetric
properties used to derive the linear three dimensional fit (the smallest of
which is $D\sim 35~{}$km), as well as all MBAs larger than $30~{}$km in
diameter that were observed by WISE. The difference in these two distributions
can be traced to the optical selection bias in the acquisition and measurement
of the polarimetric properties of asteroids: very high signal-to-noise levels
are needed to reach the polarimetric sensitivities that allow for accurate
measurement of $P_{min}$ and $h$. Thus, even though at a given phase angle low
albedo objects will show larger degrees of polarization, the reduction in
photons received from these sources make these measurements less precise.
Focusing future polarimetric surveys on low albedo asteroids will help make
this sample more representative of the true distribution of MBAs.
Figure 6: Normalized albedo distribution of the high-confidence polarimetric
sample (solid) and all MBAs larger than $30~{}$km that were detected during
the fully cryogenic portion of the NEOWISE survey (dotted).
## 4 Conclusions
Using the newly available albedos from the WISE space telescope thermal
infrared survey data, we have fit the relationships between albedo, the slope
of the polarimetric-phase curve beyond the inversion angle, and the maximum
depth of the negative polarization trough. We restrict ourselves to objects
with well characterized polarimetric properties (i.e. relative errors
$<20\%$). Due to the selection of the polarimetrically observed objects this
results in our sample consisting of only objects larger than $D>30~{}$km, with
nearly three-quarters having $D>50~{}$km.
We find that the function that best describes the albedo and polarimetry is a
three dimensional linear fit in $\log p_{V}$-$\log h$-$\log P_{min}$ space.
Orthogonal distance regression allows us to find the best fitting parameters
while accounting for measurement error on all parameters. When the best fit
line is projected to two dimensions, we find the resultant fit parameters are
all within $1.5$ sigma of those found by Cellino et al. (1999). We also define
a new polarimetric quantity $p^{\star}$ that describes the maximum variation
in polarimetric properties when compared with albedo.
We observe distinct separation of some taxonomic classes in $p^{\star}$ space.
In particular E-type, B-type, and some M-type asteroids are far removed from
the clumps that trace the more generic S- and C-complex objects. Asteroid (71)
Niobe also holds a distinct location in this space despite its S-type
classification under the Tholen system, and warrants further study. We note
that the principal component (PC) analysis of Niobe from the Eight Color
Asteroid Survey indicates that it is on the edge of the S-complex (Tholen,
1984) and it has a PC4 in the bottom two percent of all asteroids in that
survey (one of only 3 S-type or probable S-type objects with PC4 that low
Zellner et al., 2009).
Finally, despite the prevalence of low albedo asteroids seen throughout the
Main Belt (Masiero et al., 2011) we find that they are under-represented in
the polarimetric sample. Notably, roughly $10\%$ of MBAs have albedos
$p_{V}<0.04$, but there are no objects in our polarimetric sample with albedos
this low. We recommend that future surveys focus on measuring polarization-
phase curves for low albedo asteroids to properly sample this population.
## Acknowledgments
The authors wish to thank referee Alberto Cellino for his helpful review of
this paper. J.R.M. was supported by an appointment to the NASA Postdoctoral
Program at JPL, administered by Oak Ridge Associated Universities through a
contract with NASA. This publication makes use of data products from the Wide-
field Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration. This publication also makes use of data products from NEOWISE,
which is a project of the Jet Propulsion Laboratory/California Institute of
Technology, funded by the Planetary Science Division of the National
Aeronautics and Space Administration. This research has made use of the
NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
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|
arxiv-papers
| 2012-02-10T22:53:43 |
2024-09-04T02:49:27.325990
|
{
"license": "Public Domain",
"authors": "Joseph R. Masiero, A. K. Mainzer, T. Grav, J. M. Bauer, E. L. Wright,\n R. S. McMillan, D. J. Tholen, and A. W. Blain",
"submitter": "Joseph Masiero",
"url": "https://arxiv.org/abs/1202.2379"
}
|
1202.2435
|
arxiv-papers
| 2012-02-11T11:55:34 |
2024-09-04T02:49:27.339815
|
{
"license": "Public Domain",
"authors": "Lu Hoang Chinh",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1202.2435"
}
|
|
1202.2436
|
# Solutions to degenerate complex Hessian equations
Lu Hoang Chinh
###### Abstract.
Let $(X,\omega)$ be an $n$-dimensional compact Kähler manifold. We study
degenerate complex Hessian equations of the form
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n}.$ Under
some natural conditions on $F$, this equation has a unique continuous
solution. When $(X,\omega)$ is rational homogeneous we further show that the
solution is Hölder continuous.
## 1\. Introduction
Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$. Fix an
integer $m$ between $1$ and $n$, and let $d,d^{c}$ denote the usual real
differential operators
$d:=\partial+\bar{\partial},d^{c}=\frac{\sqrt{-1}}{2\pi}(\bar{\partial}-\partial)$
so that $dd^{c}=\frac{i}{\pi}\partial\bar{\partial}.$ We are studying
degenerate complex Hessian equations of the form
(1.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n},$
where the density $F:X\times\mathbb{R}\rightarrow\mathbb{R}^{+}$ satisfies
some natural integrability conditions (see Theorem A below).
The case $m=1$ corresponds to the Laplace equation and the case $m=n$
corresponds to degenerate complex Monge-Ampère equations which have been
studied intensively in recent years (see [Bl03, Bl05, Bl12, BGZ08, BK07,
EGZ09, GKZ08, GZ05, GZ07, Kol98, Kol02, Kol03, Kol05]). So, equation (1.1) is
a generalization of both Laplace and Monge-Ampère equations.
The non degenerate complex Hessian equation on compact Kähler manifold, where
$F(x,\varphi)=f(x)$, with $0<f\in\mathcal{C}^{\infty}(X)$, has been studied
recently in [H09, HMW10, Jb10, DK12]. In [H09] and [Jb10], the authors
independently solved this equation with a strong additional hypothesis,
assuming $(X,\omega)$ has non negative holomorphic bisectional curvature.
Later on, in [HMW10] an a priori $\mathcal{C}^{2}$ estimate was obtained
without curvature assumption. Recently, using this estimate and a blowing up
analysis suggested in [HMW10], Dinew and Kolodziej solved the equation in full
generality.
Following Blocki [Bl05] we develop a potential theory for the complex Hessian
equation on compact Kähler manifold. We define the class of
$(\omega,m)$-subharmonic functions which is a generalization of the class of
$\omega$-plurisubharmonic functions when $m=n.$ The definition of the complex
Hessian operator on bounded $(\omega,m)$-subharmonic functions is delicate due
to difficulties in regularization process.
To go around this difficulty, we introduce a capacity and use it to define the
concept of quasi-uniform convergence. This allows us to define a suitable
class of bounded and quasi-continuous $(\omega,m)$-subharmonic functions on
which the complex Hessian operator is well defined and continuous under quasi-
uniform convergence. We show that this definition coincides with the
definition in the spirit of Bedford and Taylor method for the complex Monge-
Ampère operator. A comparison principle and convergence results for this
operator are also established.
With these potential tools in hand, we then consider the degenerate complex
Hessian equation (1.1). The first main result of this paper is the following:
Theorem A. Let $(X,\omega)$ be a $n$-dimensional compact Kähler manifold. Fix
$1\leq m\leq n$. Let $F:X\times\mathbb{R}\rightarrow[0,+\infty)$ be a function
satisfying the following conditions:
(F1) for all $x\in X$, $t\mapsto F(x,t)$ is non-decreasing and continuous,
(F2) for any fixed $t\in\mathbb{R}$, there exists $p>n/m$ such that the
function $x\mapsto F(x,t)$ belongs to $L^{p}(X)$,
(F3) there exists $t_{0}\in\mathbb{R}$ such that
$\int_{X}F(.,t_{0})\omega^{n}=\int_{X}\omega^{n}.$
Then there exists a function
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ , unique up to an
additive constant, such that
$(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=F(x,\varphi)\omega^{n}.$
Moreover if $\forall x\in X,\ t\mapsto F(x,t)$ is increasing, then the
solution is unique.
Note that the condition (F3) is automatically satisfied if $F(.,-\infty)=0$
and $F(.,+\infty)=+\infty.$ An important particular case is the exponential
function $F(x,t)=f(x)e^{t}.$
A particular case of this result has been obtained in [DK11]. The key point in
their proof is a domination between volume and capacity. Our main result is
proved using this technique and the recent result in the smooth case [DK12].
When $(X,\omega)$ is rational homogeneous with $\omega$ being invariant under
the Lie group action, we can easily regularize $(\omega,m)$-subharmonic.
Adapting the techniques in [EGZ09] we obtain Hölder continuity of the
solution:
Theorem B. Under the same assumption as in Theorem A, assume further that
$(X,\omega)$ is rational homogeneous and $\omega$ is invariant under the Lie
group action. Then the unique solution is Hölder continuous with exponent
$0<\gamma<\frac{2(mp-n)}{mnp+2mp-2n}$.
When $m=n$ we get the same exponent $\gamma$ as in [EGZ09].
Acknowledgement. The paper is part of my Ph.D Thesis. I would like to express
my deep gratitude to my advisor, Professor Ahmed Zeriahi, for sacrificing his
very valuable time for me. I wish to express my sincere gratitude to Professor
Vincent Guedj for his very useful suggestions and discussions to improve the
paper. I also wish to say a special word of thanks to Professor Sébastien
Boucksom for his kind invitation to IMJ and useful discussions. This paper
owes much to their help and constant encouragement.
## 2\. Preliminaries
In this section we introduce the notion of $(\omega,m)$-subharmonic functions
following Blocki’s ideas [Bl05] (see also [DK11]). Using classical techniques
for plurisubharmonic functions we obtain similar results.
### 2.1. Elementary symmetric functions
First, we recall some basic properties of elementary symmetric functions (see
[Bl05], [CW01], [Ga59]). We use the notations in [Bl05]. Let $S_{k}$,
$k=1,...,n$ be the $k$-elementary symmetric function, that is, for
$\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}$,
$S_{k}(\lambda)=\sum_{1\leq i_{1}<i_{2}<...<i_{k}\leq
n}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{k}}.$
Let $\Gamma_{k}$ denote the closure of the connected component of
$\\{S_{k}(\lambda)>0\\}$ containing $(1,...,1).$ It is easy to show that
$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ \mid\
S_{k}(\lambda_{1}+t,...,\lambda_{n}+t)\geq 0,\ \forall t\geq 0\big{\\}}.$
and hence
$\Gamma_{k}=\big{\\{}\lambda\in\mathbb{R}^{n}\ \mid\ S_{j}(\lambda)\geq 0,\ \
\forall 1\leq j\leq k\big{\\}}.$
We have an obvious inclusion $\Gamma_{n}\subset...\subset\Gamma_{1}.$
By Gårding [Ga59] the set $\Gamma_{k}$ is a convex cone in $\mathbb{R}^{n}$
and $S_{k}^{1/k}$ is concave on $\Gamma_{k}.$ Let $\mathcal{H}$ denote the
vector space (over $\mathbb{R}$) of complex hermitian $n\times n$ matrices.
For $A\in\mathcal{H}$ we set
$\widetilde{S}_{k}(A)=S_{k}(\lambda(A)),$
where $\lambda(A)\in\mathbb{R}^{n}$ are is the vector of eigenvalues of $A.$
The function $\widetilde{S}_{k}$ can also be defined as the sum of all
principal minors of order $k$,
$\widetilde{S}_{k}(A)=\sum_{|I|=k}A_{II}.$
From the latter we see that $\widetilde{S}_{k}$ is a homogeneous polynomial of
order $k$ on $\mathcal{H}$ which is hyperbolic with respect to the identity
matrix $I$ (that is for every $A\in\widetilde{S}$ the equation
$\widetilde{S}_{k}(A+tI)=0$ has $n$ real roots; see [Ga59]). As in [Ga59] (see
also [Bl05]), the cone
$\widetilde{\Gamma}_{k}:=\big{\\{}A\in\mathcal{H}\ \mid\
\widetilde{S}_{k}(A+tI)\geq 0,\forall t\geq 0\big{\\}}=\\{A\in\mathcal{H}\
\mid\ \lambda(A)\in\Gamma_{k}\\}$
is convex and the function $\widetilde{S}_{k}^{1/k}$ is concave on
$\widetilde{\Gamma}_{k}.$
### 2.2. $\omega$-subharmonic functions
In this section, we consider $\Omega\subset X$ an open subset contained in a
local chart.
###### Definition 2.1.
A function $u\in L^{1}(\Omega)$ is called weakly $\omega$-subharmonic if
$dd^{c}u\wedge\omega^{n-1}\geq 0,$
in the weak sense of currents.
Thanks to Littman [Lit63] we have the following approximation properties.
###### Proposition 2.2.
Let $u$ be a weakly $\omega$-subharmonic function in $\Omega$. Then there
exists a one parameter family of functions $u_{h}$ with the following
properties: for every compact subset $\Omega^{\prime}\subset\Omega$
a) $u_{h}$ is smooth in $\Omega^{\prime}$ for $h$ sufficiently large,
b) $dd^{c}u_{h}\wedge\omega^{n-1}\geq 0$ in $\Omega^{\prime},$
c) $u_{h}$ is non increasing with increasing $h,$ and
$\lim_{h\to\infty}u_{h}(x)=u(x)$ almost every where in $\Omega^{\prime},$
d) $u_{h}$ is given explicitly as
$u_{h}(y)=\int_{\Omega}K_{h}(x,y)u(x)dx,$
where $K_{h}$ is a smooth non negative function and
$\int_{\Omega}K_{h}(x,y)dy\to 1,$
uniformly in $x\in\Omega^{\prime}.$
###### Definition 2.3.
A function $u$ is called $\omega$-subharmonic if it is weakly
$\omega$-subharmonic and for every $\Omega^{\prime}\Subset\Omega$,
$\lim_{h\to\infty}u_{h}(x)=u(x),\forall x\in\Omega^{\prime},$ where $u_{h}$ is
constructed as in Proposition 2.2.
###### Remark 2.4.
Any continuous weakly $\omega$-subharmonic function is $\omega$-subharmonic.
If $(u_{j})$ is a sequence of continuous $\omega$-subharmonic functions
decreasing to $u$ and if $u\neq-\infty$ then $u$ is $\omega$-subharmonic.
If $u$ is weakly $\omega$-subharmonic then the pointwise limit of $(u_{h})$ is
a $\omega$-subharmonic function.
Let $(u_{j})$ be a sequence of $\omega$-subharmonic functions and $(u_{j})$ is
uniformly bounded from above. Then $u:=(\limsup_{j}u_{j})^{\star}$ is
$\omega$-subharmonic. Where for a function $v$, $v^{\star}$ denotes the upper
semicontinuous regularization of $v.$
The following Hartogs lemma can be proved in the same way as in the case of
subharmonic functions.
###### Lemma 2.5.
Let $u_{t}(x),t>0$ be a family of non positive $\omega$-subharmonic functions
in $\Omega$ and $u_{t}$ is uniformly bounded in $L^{1}_{loc}(\Omega)$. Suppose
that for compact subset $K$ in $\Omega$ there exists a constant $C$ such that
$v(x)=[\limsup_{t\to+\infty}u_{t}(x)]^{\star}\leq C$ on $K.$ Then for every
$\epsilon>0,$ there exists $T_{\epsilon}$ such that $u_{t}(x)\leq C+\epsilon$
for $t\geq T_{\epsilon}$ and $x\in K.$
### 2.3. $(\omega,m)$-subharmonic functions
We associate real (1,1)-forms $\alpha$ in $\mathbb{C}^{n}$ with hermitian
matrices $[a_{j\bar{k}}]$ by
$\alpha=\frac{i}{\pi}\sum_{j,k}a_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$
Then the canonical Kähler form $\beta$ is associated with the identity matrix
$I.$ It is easy to see that
$\binom{n}{k}\alpha^{k}\wedge\beta^{n-k}=\widetilde{S}_{k}(A)\beta^{n}.$
###### Definition 2.6.
Let $\alpha$ be a real $(1,1)$-form on $X$. We say that $\alpha$ is
$(\omega,m)$-positive at a given point $P\in X$ if at this point we have
$\alpha^{k}\wedge\omega^{n-k}\geq 0,\ \ \forall k=1,...,m.$
We say that $\alpha$ is $(\omega,m)$-positive if it is $(\omega,m)$-positive
at any point of $X.$
###### Remark 2.7.
Locally at $P\in X$ with local coordinates $z_{1},...,z_{n}$, we have
$\alpha=\frac{i}{\pi}\sum_{j,k}\alpha_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}},$
and
$\omega=\frac{i}{\pi}\sum_{j,k}g_{j\bar{k}}dz_{j}\wedge d\bar{z_{k}}.$
Then $\alpha$ is $(\omega,m)$-positive at $P$ if and only if the eigenvalues
$\lambda(g^{-1}\alpha)=(\lambda_{1},...,\lambda_{n})$ of the matrix
$\alpha_{j\bar{k}}(P)$ with respect to the matrix $g_{j\bar{k}}(P)$ is in
$\Gamma_{m}$. These eigenvalues are independent of any choice of local
coordinates.
We can show easily the following result:
###### Proposition 2.8.
Let $\alpha\in\Lambda^{1,1}(X)$ be a real (1,1)-form on $X.$ Then $\alpha$ is
$(\omega,m)$-positive if and only if
$\alpha\wedge\beta_{1}\wedge...\wedge\beta_{m-1}\wedge\omega^{n-m}\geq 0,$
for all $(\omega,m)$-positive forms $\beta_{1},...,\beta_{m-1}.$
###### Definition 2.9.
A current $T$ of bidegree $(p,p)$ is said to be $(\omega,m)$-positive if
$\alpha_{1}\wedge...\wedge\alpha_{n-p}\wedge T\geq 0,$
for all smooth $(\omega,m)$-positive (1,1)-forms $\alpha_{i}.$
Following Blocki [Bl05] we can define $(\omega,m)$-subharmonicity for non-
smooth functions.
###### Definition 2.10.
A function $\varphi:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$(\omega,m)$-subharmonic if the following conditions hold
(i) In any local chart $\Omega,$ given $\rho$ the local potential of $\omega$
and set $u:=\rho+\varphi$, then $u$ is $\omega$-subharmonic,
(ii) for every smooth $(\omega,m)$-positive forms $\beta_{1},...,\beta_{m-1}$
we have, in the weak sense of distributions,
$(\omega+dd^{c}\varphi)\wedge\beta_{1}\wedge...\wedge\beta_{m-1}\wedge\omega^{n-m}\geq
0.$
Let $SH_{m}(X,\omega)$ be the set of all $(\omega,m)$-subharmonic functions on
$X.$ Observe that, by definition, any $\varphi\in SH_{m}(X,\omega)$ is upper
semicontinuous.
The following properties of $(\omega,m)$-subharmonic functions are easy to
show.
###### Proposition 2.11.
(i) A $\mathcal{C}^{2}$ function $\varphi$ is $(\omega,m)$-subharmonic if and
only if the form $(\omega+dd^{c}\varphi)$ is $(\omega,m)$-positive or
equivalently
$(\omega+dd^{c}\varphi)\wedge(\omega+dd^{c}u_{1})\wedge...\wedge(\omega+dd^{c}u_{m-1})\wedge\omega^{n-m}\geq
0,$
for all $\mathcal{C}^{2}$ $(\omega,m)$-subharmonic functions
$u_{1},...,u_{m-1}.$
(ii) If $\varphi,\psi\in SH_{m}(X,\omega)$ then $\max(\varphi,\psi)\in
SH_{m}(X,\omega).$
(ii) If $\varphi,\psi\in SH_{m}(X,\omega)$ and $\lambda\in[0,1]$ then
$\lambda\varphi+(1-\lambda)\psi\in SH_{m}(X,\omega).$
(iii) If $(\varphi_{j})\subset SH_{m}(X,\omega)$ is uniformly bounded from
above then $(\limsup_{j}\varphi_{j})^{\star}\in SH_{m}(X,\omega).$
Thanks to Hartogs Lemma 2.5, the following proposition can be proved in the
same way as in the case of $\omega$-plurisubharmonic function (see [GZ05]).
###### Proposition 2.12.
Let $(\varphi_{j})$ be a sequence of functions in $SH_{m}(X,\omega)$.
(i) If $(\varphi_{j})$ is uniformly bounded from above on $X$, then either
$\varphi_{j}$ converges uniformly to $-\infty$ or the sequence $(\varphi_{j})$
is relatively compact in $L^{1}(X).$
(ii) If $\varphi_{j}\rightarrow\varphi$ in $L^{1}(X)$ then
$sup_{X}\varphi=\lim_{j}sup_{X}\varphi_{j}.$
The compactness result can be deduced easily from Proposition 2.12.
###### Lemma 2.13.
There exists a constant $C_{0}>0$ such that for all $\varphi\in
SH_{m}(X,\omega)$ satisfying $\sup_{X}\varphi=0$ we have
$\int_{X}\varphi\omega^{n}\geq-C_{0}.$
It then follows that
$\mathcal{C}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\ \sup_{X}\varphi\leq
0;\int_{X}\varphi\omega^{n}\geq-C_{0}\\}$
is a convex compact subset of $L^{1}(X).$
### 2.4. Non degenerate complex Hessian equations
We summarize here some recent results on the non degenerate complex Hessian
equation on compact Kähler manifolds,
(2.1) $(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}=f\omega^{n},$
where $0<f$ is smooth such that
(2.2) $\int_{X}f\omega^{n}=\int_{X}\omega^{n}.$
The following existence result was solved by Dinew and Kolodziej:
###### Theorem 2.14.
[DK12] If $(X,\omega)$ is a compact Kähler manifold and
$0<f\in\mathcal{C}^{\infty}(X)$ satisfies (2.2) then equation (2.1) has a
unique (up to an additive constant) smooth solution.
This result was known to hold when $(X,\omega)$ has non negative holomorphic
bisectional curvature [H09, Jb10].
The complex Hessian equation in domains of $\mathbb{C}^{n}$, i.e. equations of
the form
$(dd^{c}u)^{m}\wedge\beta^{n-m}=f\beta^{n},$
where $\beta$ is the canonical Kähler form in $\mathbb{C}^{n},$ was considered
by Li [Li04] and Blocki [Bl05]. Existence and uniqueness of smooth solution to
the Dirichlet problem in smoothly bounded domains with $(m-1)$\- pseudoconvex
boundary was obtained in [Li04]. In [Bl05], a potential theory for
$m$-subharmonic functions was developed and the corresponding degenerate
Dirichlet problem was solved. Recently, Sadullaev and Abdullaev studied
capacities and polar sets for $m$-subharmonic functions [SA12]. Note that the
corresponding problem when $\beta$ is not the euclidean Kähler form is fully
open.
It is important to mention that the study of real Hessian equations is a
classical subject which has been developed previously in many papers, for
example [CNS85, CW01, ITW04, Kr95, La02, Tr95, TW99, Ur01, W09].
## 3\. Complex Hessian operators.
One of the key points in pluripotential theory is the smooth approximation
which holds for quasi plurisubharmonic functions ([BK07], [De92]). Such a
result for $(\omega,m)$-subharmonic functions seems to be very difficult. To
overcome this difficulty we work in an (a priori) restrictive class which is
defined by means of uniform convergence with respect to capacity.
### 3.1. Capacity
###### Definition 3.1.
Let $E\in X$ be a Borel subset. We define the inner $(\omega,m)$-capacity of
$E$ by
$\text{cap}_{\omega,m}(E):=\sup\Big{\\{}\int_{E}\omega_{\varphi}^{m}\wedge\omega^{n-m}\
\mid\ \varphi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X),0\leq\varphi\leq
1\Big{\\}}.$
The outer $(\omega,m)$-capacity of $E$ is defined to be
$\text{Cap}_{\omega,m}(E):=\inf\Big{\\{}\text{cap}_{\omega,m}(U)\ \mid\
E\subset U,\ \ U\ \ \text{is an open subset of X}\Big{\\}}.$
It follows directly from the definition that $\text{Cap}_{\omega,m}$ is
monotone and $\sigma$-sub-additive.
Observe that if $\varphi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$,
$0\leq\varphi\leq M$ then, for any Borel subset $E\subset X,$
(3.1) $\int_{E}\omega_{\varphi}^{m}\wedge\omega^{n-m}\leq
M^{m}\text{cap}_{\omega,m}(E).$
###### Definition 3.2.
A sequence $(\varphi_{j})$ converges in $\text{cap}_{\omega,m}$ to $\varphi$
if for any $\delta>0$ we have
$\lim_{j\to\infty}\text{cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$
###### Definition 3.3.
A sequence of functions $(\varphi_{j})$ converges quasi-uniformly to $\varphi$
on $X$ (w.r.t $\text{Cap}_{\omega,m}$) if for every $\epsilon>0$ there exists
an open subset $U\subset X$ such that $\text{Cap}_{\omega,m}(U)\leq\epsilon$
and $\varphi_{j}$ converges uniformly to $\varphi$ in $X\setminus U.$
This convergence is almost equivalent to the convergence in capacity as the
following result shows
###### Proposition 3.4.
(i) If $\varphi_{j}$ converges quasi-uniformly to $\varphi$, then for each
$\delta>0$,
$\lim_{j\to\infty}\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$
(ii) Conversely, assume that $(\varphi_{j})$ is a sequence of functions and
$\varphi$ is a function such that, for every $\delta>0,$
$\lim_{j\to\infty}\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>\delta)=0.$
Then there exists a subsequence $(\varphi_{j_{k}})$ converging quasi-uniformly
to $\varphi$.
###### Proof.
The first part is obvious, so we only prove the second part. We can find a
subsequence (and for convenience we still denote it by $(\varphi_{j})$) such
that
$\text{Cap}_{\omega,m}(|\varphi_{j}-\varphi|>1/j)\leq 2^{-j},\ \forall j.$
For each $j$, let $U_{j}$ be an open subset of $X$ such that
$(|\varphi_{j}-\varphi|>1/j)\subset U_{j}$ and
$\text{cap}_{\omega,m}(U_{j})\leq 2^{-j+1}.$ Then for each $\epsilon>0$, we
can find $k\in\mathbb{N}$ such that $\cup_{j\geq k}U_{j}$ is the open subset
of $\text{Cap}_{\omega,m}$ less than $\epsilon$ and $\varphi_{j}$ converges
uniformly to $\varphi$ on its complement. ∎
###### Definition 3.5.
We denote $\mathcal{P}_{m}(X,\omega)$ the set of all functions $\varphi\in
SH_{m}(X,\omega)$ such that there exists a sequence of $\mathcal{C}^{2}$,
$(\omega,m)$-subharmonic functions $(\varphi_{j})$ converging quasi-uniformly
to $\varphi$ on $X$. Equivalently, we can replace quasi-uniform convergence by
convergence in Capacity thanks to Proposition 3.4.
###### Proposition 3.6.
(i) Any $\varphi\in\mathcal{P}_{m}(X,\omega)$ is quasi continuous, that means,
for any $\epsilon>0$ there exists an open subset $U\subset X$ of
$\text{Cap}_{\omega,m}$ less than $\epsilon$ such that $\varphi$ is continuous
on $X\setminus U$.
(ii) If $\varphi_{j}\downarrow\varphi$ in $\mathcal{P}_{m}(X,\omega)$ then
$(\varphi_{j})$ converges quasi-uniformly to $\varphi.$
###### Proof.
The first statement follows directly from the definition. From (i) , for each
$\epsilon>0$, there exists an open subset $U$ of $\text{cap}_{\omega,m}$ less
than $\epsilon$ such that $\varphi_{j},\varphi$ are continuous on $X\setminus
U$ which is compact. By Dini’s Theorem, $\varphi_{j}$ converges uniformly to
$\varphi$ on $X\setminus U.$ ∎
We have obvious inclusions
$SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)\subset\mathcal{P}_{m}(X,\omega)\subset
SH_{m}(X,\omega),$
and
$PSH(X,\omega)\subset\mathcal{P}_{m}(X,\omega).$
###### Remark 3.7.
Quasi-uniform convergence implies convergence point wise outside a subset of
$\text{Cap}_{\omega,m}$ zero. Moreover, if $\varphi_{j}$ is uniformly bounded
and converges quasi-uniformly to $\varphi$, then we have convergence in
$L^{p}$ for every $p>0$. Indeed, for any $\epsilon>0$ and an open subset $U$
as in definition 3.3, we have
$\displaystyle\int_{X}|\varphi_{j}-\varphi|^{p}\omega^{n}$ $\displaystyle\leq$
$\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+\int_{U}|\varphi_{j}-\varphi|^{p}\omega^{n}$
$\displaystyle\leq$ $\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+\sup_{X,j}|\varphi_{j}-\varphi|^{p}.\text{cap}_{\omega,m}(U)$
$\displaystyle\leq$ $\displaystyle\int_{X\setminus
U}|\varphi_{j}-\varphi|^{p}\omega^{n}+C\epsilon.$
Taking the limsup over $j$ and then letting $\epsilon\to 0$ we obtain
$\limsup_{j}\|\varphi_{j}-\varphi\|_{p}=0.$
###### Lemma 3.8.
If $\varphi,\psi$ belong to the class $\mathcal{P}_{m}(X,\omega)$ then so does
$\max(\varphi,\psi).$
###### Proof.
Let $(\varphi_{j}),(\psi_{j})$ be uniformly bounded sequences of functions in
$SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ converging quasi-uniformly to
$\varphi,\psi$ respectively. Set
$u:=\max(\varphi,\psi);\ u_{j}:=\max(\varphi_{j},\psi_{j});\
v_{j}:=\frac{1}{j}\log(e^{j\varphi_{j}}+e^{j\psi_{j}}).$
For each $\epsilon>0$ there exists an open subset $U$ of
$\text{cap}_{\omega,m}$ less than $\epsilon$ and $\varphi_{j},\psi_{j}$
converges uniformly on $X\setminus U$ to $\varphi,\psi$ respectively. Since
$u_{j}\leq v_{j}\leq\log(2)/j+u_{j}$ and $u_{j}$ converges uniformly to $u$ on
$X\setminus U$ we deduce that $v_{j}$ converges uniformly to $u$ on
$X\setminus U.$ ∎
### 3.2. Hessian measure
In this section we define complex Hessian measure for functions in
$SH_{m}(X,\omega)$ which can be approximated in $\text{Cap}_{\omega,m}$ by
$\mathcal{C}^{2}$-functions in $SH_{m}(X,\omega)$. In particular, for
functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ this notion of
Hessian measure can be defined by Bedford-Taylor’s method.
###### Theorem 3.9.
Let $\varphi\in SH_{m}(X,\omega)$ such that there exists a uniformly bounded
sequence $(\varphi_{j})$ of $\mathcal{C}^{2}$ $(\omega,m)$-subharmonic
functions converging in $\text{cap}_{\omega,m}$ to $\varphi.$ Then the
sequence of measures
$H_{m}(\varphi_{j}):=(\omega+dd^{c}\varphi_{j})^{m}\wedge\omega^{n-m}$
converges (weakly in the sense of measures) to a positive Radon measure $\mu.$
Moreover, the measure $\mu$ does not depend on the choice of the approximating
sequence $(\varphi_{j}).$ We define the Hessian measure of $\varphi$ to be
$H_{m}(\varphi):=\mu.$
###### Proof.
Since all the measures $H_{m}(\varphi_{j})$ have uniformly bounded mass (which
is $\int_{X}\omega^{n}$), they stay in a weakly compact subset. It suffices to
show that all accumulation points of this sequence are just the same. To do
this it is enough to show that for every test function
$\chi\in\mathcal{C}^{\infty}(X),$
$\lim_{j,k\to\infty}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]=0.$
By integration by part formula we have
$\displaystyle\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]$
$\displaystyle=$ $\displaystyle\int_{X}\chi
dd^{c}(\varphi_{j}-\varphi_{k})\wedge T$ $\displaystyle=$
$\displaystyle\int_{X}(\varphi_{j}-\varphi_{k})dd^{c}\chi\wedge T,$
where
$T=\sum_{l=0}^{m-1}(\omega+dd^{c}\varphi_{j})^{l}\wedge(\omega+dd^{c}\varphi_{k})^{m-1-l}\wedge\omega^{n-m}.$
Fix $\epsilon>0$, and set
$U=U(j,k,\epsilon)=\\{|\varphi_{j}-\varphi_{k}|\geq\epsilon\\}.$ By $C$ we
will denote a constant that does not depend on $j,k,\epsilon.$ Then by (3.2)
and (3.1) there exists $C>0$ such that
$\displaystyle\Big{|}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]\Big{|}$
$\displaystyle\leq$
$\displaystyle\int_{U}|\varphi_{j}-\varphi_{k}|C\omega\wedge
T+\int_{X\setminus U}|\varphi_{j}-\varphi_{k}|C\omega\wedge T$
$\displaystyle\leq$ $\displaystyle
C\text{cap}_{\omega,m}(U)+C\epsilon\sup_{X\setminus
U}|\varphi_{j}-\varphi_{k}|\int_{X\setminus U}\omega\wedge T.$
Now, it follows that
$\limsup_{j,k\to\infty}\Big{|}\int_{X}\chi[H_{m}(\varphi_{j})-H_{m}(\varphi_{k})]\Big{|}\leq
C\epsilon.$
The result follows by letting $\epsilon\downarrow 0.$ For the independence in
the choice of the sequence it is enough to repeat the above arguments. ∎
###### Lemma 3.10.
Let $U\subset X$ be an open subset and $\varphi$ be a bounded function in
$\mathcal{P}_{m}(X,\omega).$ Then
$\int_{U}H_{m}(\varphi)\leq 2(\sup_{X}|\varphi|+1)\text{Cap}_{\omega,m}(U).$
###### Proof.
Let $\varphi_{j}$ be a sequence of $\mathcal{C}^{2}$ functions in
$SH_{m}(X,\omega)$ converging quasi uniformly to $\varphi.$ We can assume that
$-\sup_{X}|\varphi|-1\leq\varphi_{j}\leq\sup_{X}|\varphi|+1,\ \forall j.$
Then
$\int_{U}H_{m}(\varphi)\leq\liminf_{j\to+\infty}\int_{U}H_{m}(\varphi_{j})\leq
2(\sup_{X}|\varphi|+1)\text{Cap}_{\omega,m}(U).$
∎
For functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ we can also
define the Hessian measure in a weak sense following Bedford-Taylor method.
###### Lemma 3.11.
Let $\varphi_{1},\varphi_{2}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$.
Then the current
$\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}$ is well
defined in the weak sense (Bedford-Taylor), symmetric and
$(\omega,m)$-positive. Then we can define inductively the Hessian measure of
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$,
$H_{m}(\varphi):=(\omega+dd^{c}\varphi)^{m}\wedge\omega^{n-m}.$
Moreover, this definition coincides with the one in Theorem 3.9.
###### Proof.
It follows from definition of $(\omega,m)$-subharmonic functions that
$T_{1}=(\omega+dd^{c}\varphi_{1})\wedge\omega^{n-m}$ is a
$(\omega,m)$-positive current. If $\varphi_{2}\in\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$ then $dd^{c}\varphi_{2}\wedge T_{1}$ is the current defined by
$dd^{c}\varphi_{2}\wedge T_{1}=dd^{c}(\varphi_{2}T_{1}).$
We denote by
$T_{2}=\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}.$
Since $\varphi_{1},\varphi_{2}$ are in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$, there exist uniformly bounded sequences
$(\varphi_{1}^{j}),(\varphi_{2}^{j})$ in
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{2}(X)$ converging quasi-uniformly
to $\varphi_{1},\varphi_{2}$ respectively. The sequence of currents
$T_{2}^{j}=\omega_{\varphi_{1}^{j}}\wedge\omega_{\varphi_{2}^{j}}\wedge\omega^{n-m}$
converges to $T_{2}$ and hence $T_{2}$ is $(\omega,m)$\- positive and
$\omega_{\varphi_{1}}\wedge\omega_{\varphi_{2}}\wedge\omega^{n-m}=\omega_{\varphi_{2}}\wedge\omega_{\varphi_{1}}\wedge\omega^{n-m}.$
To prove that $T_{2}^{j}$ converges to $T_{2}$, let us choose some test form
$\chi$ and prove the following convergence
(3.3) $\lim_{j\to\infty}\int_{X}\chi\wedge
dd^{c}(\varphi_{2}^{j}-\varphi_{2})\wedge T_{1}=0.$
We have
$\displaystyle\Big{|}\int_{X}\chi\wedge
dd^{c}(\varphi_{2}^{j}-\varphi_{2})\wedge T_{1}\Big{|}$ $\displaystyle=$
$\displaystyle\Big{|}\int_{X}(\varphi_{2}^{j}-\varphi_{2})dd^{c}\chi\wedge
T_{1}\Big{|}$ $\displaystyle\leq$ $\displaystyle
C\int_{X}|\varphi_{2}^{j}-\varphi_{2}|\omega_{\varphi_{1}}\wedge\omega^{n-1},$
where the constant $C$ depends only on $\chi,\omega.$ Now (3.3) follows from
the last inequality in view of
$\int_{U}\omega_{\varphi_{1}}\wedge\omega^{n-1}\leq
C\text{cap}_{\omega,m}(U),$
for every open subset $U\subset X.$ ∎
We can prove inductively that the current
$T_{k}=\omega_{\varphi_{1}}\wedge...\wedge\omega_{\varphi_{k}}\wedge\omega^{n-m}$
is well-defined, symmetric, $(\omega,m)$-positive, for each $k\leq m$ and
$\varphi_{i}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$. The Hessian
measure of $\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ is defined
in this way
$H_{m}(\varphi)=\omega_{\varphi}\wedge...\wedge\omega_{\varphi}\wedge\omega^{n-m}.$
Now, given $\varphi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, it is easy
to see that the Hessian measure of $\varphi$ defined by the above construction
coincides with the Hessian measure $H_{m}(\varphi)$ defined in Theorem 3.9.
### 3.3. Some Convergence results
In this section we state some convergence results and the comparison principle
for functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$. The proofs are
nearly the same as for the Monge-Ampère operator and hence are omitted.
###### Proposition 3.12.
Let $(\varphi^{1}_{j}),...,(\varphi^{m}_{j})$ be uniformly bounded sequence of
functions in $\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ converging quasi-
uniformly to $\varphi^{1},...,\varphi^{m}$ respectively. Assume that $(f_{j})$
is a uniformly bounded sequence of quasi continuous functions converging quasi
uniformly to $\varphi.$ Then we have the weak convergence of measures
$f_{j}\omega_{\varphi_{j}^{1}}\wedge...\wedge\omega_{\varphi_{j}^{m}}\wedge\omega^{n-m}\rightharpoonup
f\omega_{\varphi^{1}}\wedge...\wedge\omega_{\varphi^{m}}\wedge\omega^{n-m}.$
###### Proof.
Thanks to Lemma 3.10 we can follow the lines in [Kol05]. ∎
The integration by parts formula is valid for $\mathcal{C}^{2}$ functions (by
Stokes). By Proposition 3.12 we see that it is also valid for functions in
$\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$
###### Theorem 3.13 (Integration by parts).
Let $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ and $T$ be a
current of the form
$T=\omega_{\varphi_{1}}\wedge...\wedge\omega_{\varphi_{m-1}}\wedge\omega^{n-m},$
with $\varphi_{i}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$ Then
$\int_{X}\varphi dd^{c}\psi\wedge T=\int_{X}\psi dd^{c}\varphi\wedge T.$
The maximum principle for functions in $\mathcal{P}_{m}(X,\omega)$ can be
proved by the same way as in the classical case.
###### Theorem 3.14 (Maximum principle).
If $\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$ then
$1{\hskip-2.5pt}\hbox{{I}}_{\\{\varphi>\psi\\}}H_{m}(\max(\varphi,\psi))=1{\hskip-2.5pt}\hbox{{I}}_{\\{\varphi>\psi\\}}H_{m}(\varphi).$
From Theorem 3.14 we easily get
###### Corollary 3.15 (Comparison principle).
If $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$ then
$\int_{(\varphi>\psi)}H_{m}(\varphi)\leq\int_{(\varphi>\psi)}H_{m}(\psi).$
###### Lemma 3.16.
Let $\varphi,\psi$ be two non positive functions in
$\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X).$ If $s>0$ and $0<t<1$ then we
have
(3.4)
$t^{m}\text{Cap}_{\omega,m}(\varphi-\psi<-t-s)\leq(1+M)^{m}\int_{(\varphi-\psi<-s)}H_{m}(\varphi),$
where $M=\|\psi\|_{L^{\infty}(X)}.$
###### Proof.
We can assume that $\psi$ is continuous on $X$. For the general case we can
approximate $\psi$ quasi-uniformly by sequence of $\mathcal{C}^{2}$ functions
in $SH_{m}(X,\omega)$. In (3.4) We can replace $\text{Cap}_{\omega,m}$ by
$\text{cap}_{\omega,m}$ since they coincide on open sets. Now, it suffices to
repeat the arguments in [EGZ09].
∎
###### Proposition 3.17 (Chern-Levine-Nirenberg inequality).
Let $T$ be any current of the form
$T=\omega_{u_{1}}\wedge...\wedge\omega_{u_{m-1}}\wedge\omega^{n-m}$ with
$u_{1},...,u_{m-1}\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X),$ and
$\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)\cap
L^{\infty}(X)$. Then
(3.5) $\int_{X}|\psi|\omega_{\varphi}\wedge
T\leq\int_{X}|\psi|T\wedge\omega+\Big{(}2|\sup_{X}\psi|+\sup_{X}\varphi-\inf_{X}\varphi\Big{)}\int_{X}\omega^{n}.$
###### Proof.
The proof is nearly the same as in [GZ05] and is omitted. ∎
Applying (3.5) for $T_{i}=\omega_{\varphi}^{i}\wedge\omega^{n-m+i}$ for
$i=m-1,...,0$ we obtain
###### Corollary 3.18.
Let $\varphi,\psi$ be two functions in $\mathcal{P}_{m}(X,\omega)$ such that
$0\leq\varphi\leq 1$. Then
$\int_{X}|\psi|H_{m}(\varphi)\leq\int_{X}|\psi|\omega^{n}+m\Big{(}2|\sup_{X}\psi|+1\Big{)}\int_{X}\omega^{n}.$
Applying Corollary 3.18 we obtain:
###### Corollary 3.19.
There exists a constant $C>0$ such that for all
$\psi\in\mathcal{P}_{m}(X,\omega)$ satisfying $\sup_{X}\psi=-1$ and for every
$t>0$ we have
$\text{Cap}_{\omega,m}(\psi<-t)\leq C/t.$
We end this section by showing that the class $\mathcal{P}_{m}(X,\omega)$ is
stable under decreasing sequences.
###### Proposition 3.20.
Let $(\varphi_{j})$ be a decreasing sequence of functions in
$\mathcal{P}_{m}(X,\omega)$ converging to $\varphi\not\equiv-\infty.$ Then
$\varphi_{j}$ converges to $\varphi$ quasi-uniformly. In particular,
$\varphi\in\mathcal{P}_{m}(X,\omega).$
###### Proof.
It is easy to see that $\varphi$ is $(\omega,m)$-subharmonic. It suffices to
show that there exists a subsequence of $(\varphi_{j})$ converging quasi-
uniformly to $\varphi.$
In view of Corollary 3.19 we can assume that $\varphi$ is bounded.
Fix $k\in\mathbb{N}.$ For each $j>k\in\mathbb{N},$ by applying Lemma 3.16 with
$\varphi=\varphi_{j},\psi=\varphi_{k},s=t$ we obtain
$\displaystyle t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\varphi_{k}<-2t)$
$\displaystyle\leq$
$\displaystyle(1+M)^{m}\int_{(\varphi_{j}-\varphi_{k}<-t)}H_{m}(\varphi_{j})$
$\displaystyle\leq$
$\displaystyle\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi_{j})H_{m}(\varphi_{j}).$
After extracting a subsequence if necessary we can assume that
$H_{m}(\varphi_{j})\rightharpoonup\mu$ in the weak sense of measures.We apply
Lemma 3.21 below to get
$\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})=\int_{X}\varphi
d\mu.$
From the quasi-continuity of the functions $\varphi_{j},j\in\mathbb{N}$ and
the $\sigma$-subadditivity of $\text{Cap}_{\omega,m}$ we deduce that for each
fixed $\epsilon>0$ there exists an open subset $U$ such that
$\text{Cap}_{\omega,m}(U)<\epsilon$ and there exists a subsequence
$(\tilde{\varphi}_{j})$ of continuous functions on $X$ such that for any $j,$
$\varphi_{j}=\tilde{\varphi}_{j}$ on $X\setminus U.$
From basic properties of $\text{Cap}_{\omega,m}$ we have
$\displaystyle
t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\tilde{\varphi}_{k}<-2t)$
$\displaystyle\leq$ $\displaystyle
t^{m}\text{Cap}_{\omega,m}(\varphi_{j}-\varphi_{k}<-2t)+t^{m}\epsilon$
$\displaystyle\leq$
$\displaystyle\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi_{j})H_{m}(\varphi_{j})+t^{m}\epsilon.$
Recall that $\text{cap}_{\omega,m}$ is continuous under increasing sequence.
Note also that $\text{Cap}_{\omega,m}$ and $\text{cap}_{\omega,m}$ coincide on
open sets. By taking the limit when $j\to+\infty$ in (3.3), we obtain
$t^{m}\text{Cap}_{\omega,m}(\varphi-\tilde{\varphi}_{k}<-2t)\leq\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi)d\mu+t^{m}\epsilon.$
It follows that
$t^{m}\text{Cap}_{\omega,m}(\varphi-\varphi_{k}<-2t)\leq\frac{(1+M)^{m}}{t}\int_{X}(\varphi_{k}-\varphi)d\mu+2t^{m}\epsilon,$
and hence,
$\lim_{k\to+\infty}\text{Cap}_{\omega,m}(\varphi-\varphi_{k}<-2t)=0.$
Now, by Proposition 3.4 there exists a subsequence of $(\varphi_{j})$
converging quasi-uniformly to $\varphi$. To complete the proof it remains to
prove the following lemma. ∎
###### Lemma 3.21.
Assume that $(\varphi_{j})$ is a sequence in $\mathcal{P}_{m}(X,\omega)$
decreasing to $\varphi\in L^{\infty}(X).$ If $H_{m}(\varphi_{j})$ converges
weakly to $\mu$ in the sense of measures then
$\lim_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})=\int_{X}\varphi
d\mu.$
###### Proof.
We prove this lemma by induction. It obviously holds when $m=1.$ Remark also
that
$\limsup_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\leq\int_{X}\varphi
d\mu.$
Thus, it suffices to prove that
$\liminf_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\geq\int_{X}\varphi
d\mu.$
Fix $k\in\mathbb{N}$. For each $j>k$, By integration by parts we get
$\displaystyle\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-H_{m}(\varphi_{j})]$
$\displaystyle=$
$\displaystyle\int_{X}\varphi_{k}dd^{c}(\varphi_{k}-\varphi_{j})\wedge
T\wedge\omega^{n-m}$ $\displaystyle=$
$\displaystyle\int_{X}(\varphi_{k}-\varphi_{j})dd^{c}\varphi_{k}\wedge
T\wedge\omega^{n-m}$ $\displaystyle\geq$
$\displaystyle-\int_{X}(\varphi_{k}-\varphi_{j})T\wedge\omega^{n-m+1},$
where
$T=\sum_{p=0}^{m-1}(\omega+dd^{c}\varphi_{k})^{p}\wedge(\omega+dd^{c}\varphi_{j})^{m-1-p}.$
By setting $\psi_{j}=\frac{\varphi_{j}+\varphi_{k}}{2},$ we get
$T\wedge\omega^{n-m+1}\leq
2^{m-1}(\omega+dd^{c}\psi_{j})^{m-1}\wedge\omega^{n-m+1}.$
As a consequence, (3.3) yields
(3.9)
$\displaystyle\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-H_{m}(\varphi_{j})]$
$\displaystyle\geq$
$\displaystyle-2^{m}\int_{X}(\varphi_{k}-\psi_{j})H_{m-1}(\psi_{j})$
After extracting a subsequence if necessary, we can assume that
$H_{m-1}(\psi_{j})\rightharpoonup\nu$ in the weak sense of measures. By
letting $j\to+\infty$ in (3.9), the induction hypothesis gives us
$\int_{X}\varphi_{k}[H_{m}(\varphi_{k})-\mu]\geq-2^{m-1}\int_{X}(\varphi_{k}-\varphi)d\nu.$
We then infer that
$\liminf_{j\to+\infty}\int_{X}\varphi_{j}H_{m}(\varphi_{j})\geq\int_{X}\varphi
d\mu,$
and the result follows. ∎
## 4\. Stability results
In this section we use the volume-capacity estimate in [DK11] and mimic the
arguments in [EGZ09] to prove stability results for the complex Hessian
equation.
Using Blocki’s technique [Bl03] we obtain the following stability results.
###### Theorem 4.1.
Let $\varphi,\psi\in SH_{m}(X,\omega)\cap\mathcal{C}^{2}(X,\omega)$, $r\geq
2$, and set $\rho=\varphi-\psi$. Then
$\int_{X}|\rho|^{r-2}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C\Big{(}\int_{X}|\rho|^{r-2}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$
where $C$ is a positive constant depending only on $n,m,r$, and upper bounds
of $\|\varphi\|_{L^{\infty}(X)}$, $\|\psi\|_{L^{\infty}(X)}$, and
$\int_{X}\omega^{n}$.
From Theorem 4.1 and Corollary 3.2 we thus get
###### Corollary 4.2.
Let $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, and set
$\rho=\varphi-\psi$. Then
$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C\Big{(}\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$
where $C$ is a positive constant depending only on $n,m$, and upper bounds of
$\|\varphi\|_{L^{\infty}(X)}$, $\|\psi\|_{L^{\infty}(X)}$, and
$\int_{X}\omega^{n}$.
Corollary 4.2 is useful to prove uniqueness results as we will see in the
proof of Theorem A.
###### Definition 4.3.
Let $\alpha>0,A>0.$ A Borel measure $\mu$ on $X$ satisfies condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$ if for all Borel subsets $K$ of $X,$
$\mu(K)\leq A\text{Cap}_{\omega,m}(K)^{1+\alpha}.$
###### Proposition 4.4.
Let $\mu$ be a Borel measure satisfying condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$. Suppose that
$\varphi\in\mathcal{P}_{m}(X,\omega)$ solves $H_{m}(\varphi)=\mu$, and
$\sup_{X}\varphi=-1.$ Then there exists a constant $C=C(\alpha,A,\omega,n,m)$
such that
$\sup_{X}|\varphi|\leq C.$
Sketch of proof. Set
$f(s):=[\text{Cap}_{\omega,m}(\varphi<-s)]^{1/m}.$
Observe that $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is right continuous,
decreasing with $\lim_{+\infty}f=0.$ Since $\mu$ satisfies condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$, it follows from Lemma 3.16 applied to the
function $\psi\equiv 0$ that $f$ satisfies the condition in Lemma 2.4 in
[EGZ09]. Moreover it follows from Corollary 3.19 that
$f(s)\leq C_{1}s^{-1/m},$
for some constant $C_{1}$ which only depends on $\omega.$ Thus, by following
the lines in [EGZ09], page 615, we have the desired uniform estimate.
###### Theorem 4.5.
Suppose that $\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$
satisfy
$\sup_{X}\varphi=\sup_{X}\psi=-1.$
Assume that $H_{m}(\varphi)$, $H_{m}(\psi)$ satisfy condition
$\mathcal{Q}_{m}(\alpha,A,\omega)$ for some $\alpha,A>0.$ Then there exists
$C=C(\alpha,A,\omega,\|\varphi\|_{L^{\infty}(X)},\|\psi\|_{L^{\infty}(X)})>0$
such that, for any $\epsilon>0,$
$\sup_{X}(\psi-\varphi)\leq\epsilon+C[\text{Cap}_{\omega,m}(\varphi-\psi<-\epsilon)]^{\alpha/m}.$
###### Proof.
The same as in [EGZ09], Proposition 2.6. ∎
The following Proposition is due to Kolodziej and Dinew [DK11, Propsition
2.1]. We include here a slightly different proof.
###### Proposition 4.6.
[DK11] Let $1<p<\frac{n}{n-m}.$ There exists a constant $C=C(p,\omega)$ such
that for every Borel subset $K$ of $X$, we have
$V(K)\leq C\text{Cap}_{\omega,m}(K)^{p},$
where $V(K):=\int_{K}\omega^{n}.$
###### Proof.
Fix an open subset $U$ such that $K\subset U.$ Solve the complex Monge-Ampère
equation to find $u\geq 0$ such that $\omega_{u}^{n}=f\omega^{n}$ on $X$, with
$f=V(U)^{-1}\chi_{U}.$ From [BGZ08], Corollary 3.2, the solution $u$ is
continuous and moreover, for each $r>1$,
$\sup_{X}u\leq C\|f\|_{r}^{1/n},$
where the constant $C=C(r,\omega)$ does not depend on $K.$ The inequality
between mixed complex Monge-Ampère measures [Di09] tells us that
$\omega_{u}^{m}\wedge\omega^{n-m}\geq f^{m/n}\omega^{n}.$
Thus since $u\in\mathcal{P}_{m}(X,\omega)\cap L^{\infty}(X)$, we obtain
$\text{Cap}_{\omega,m}(U)\geq(\sup_{X}u)^{-m}\int_{U}H_{m}(u)\geq(\sup_{X}u)^{-m}\int_{U}f^{m/n}\omega^{n}\geq
C^{-m}V(U)^{1-\frac{m}{rn}}.$
Thus, for every $r>1$, there exists a constant $C$ not depending on $K$ such
that $V(K)\leq C.\text{Cap}_{\omega,m}(K)^{\frac{nr}{nr-m}}$. The proof is
complete. ∎
As a consequence of Proposition 4.6 we have some examples of measures
satisfying condition $\mathcal{Q}_{m}(\alpha,A,\omega).$
###### Lemma 4.7.
Assume $\mu=f\omega^{n}$ is a Borel measure with $0\leq f\in L^{p}(X)$ for
some $p>n/m$. Then for any $0<\alpha<\frac{mp-n}{(n-m)p},$ there exists
$A_{\alpha}>0$ such that $\mu$ satisfies
$\mathcal{Q}_{m}(\alpha,A_{\alpha},\omega).$
The following stability theorem was established in [EGZ09] for the Monge-
Ampère equation.
###### Theorem 4.8.
Assume $H_{m}(\varphi)=f\omega^{n},\ H_{m}(\psi)=g\omega^{n},$ where
$\varphi,\psi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and $f,g\in
L^{p}(X)$ with $p>n/m.$ Fix $r>0$. Then if $\gamma$ small enough such that
$\frac{\gamma mq}{r-\gamma(r+mq)}<\frac{mp-n}{(n-m)p}$, we have
$\|\varphi-\psi\|_{L^{\infty}(X)}\leq C\|\varphi-\psi\|_{L^{r}(X)}^{\gamma},$
where $q=\frac{p}{p-1}$ denotes the conjugate exponent of $p,$ and the
constant $C$ depends only on $n,m,p,r$ and upper bounds of $\|f\|_{p},\
\|g\|_{p}$.
###### Proof.
Fix $\epsilon>0,$ and $\alpha>0$ to be chosen later. It follows from Theorem
4.5 and Proposition 4.4 that
$\|\varphi-\psi\|_{L^{\infty}(X)}\leq\epsilon+C_{1}[\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)]^{\alpha/m}.$
Applying Lemma 3.16 we see that
$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq\frac{C_{2}}{\epsilon^{m+r/q}}\int_{X}|\varphi-\psi|^{r/q}(f+g)\omega^{n}.$
It follows thus from Hölder’s inequality that
$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq\frac{C_{3}\|f+g\|_{p}}{\epsilon^{m+r/q}}\|\varphi-\psi\|_{L^{r}}^{r/q}.$
Choose $\epsilon:=\|\varphi-\psi\|_{L^{r}}^{\gamma}.$ Then
$\text{Cap}_{\omega,m}(|\varphi-\psi|>\epsilon)\leq
C_{4}[\|\varphi-\psi\|_{L^{r}}]^{r/q-\gamma(m+r/q)}.$
We infer that
$\|\varphi-\psi\|_{L^{\infty}(X)}\leq\|\varphi-\psi\|_{L^{r}(X)}^{\gamma}+C_{5}\|\varphi-\psi\|_{L^{r}(X)}^{\gamma^{\prime}},$
where $\gamma^{\prime}=\frac{\alpha}{m}[r/q-\gamma(m+r/q)].$ We finally choose
$\alpha$ so that $\gamma=\gamma^{\prime}$: this yields the desired estimate. ∎
## 5\. Proof of the main results
### 5.1. Proof of Theorem A
We first prove the uniqueness. Suppose that $\varphi$ and $\psi$ are two
continuous solutions of (1.1). Set $\rho:=\varphi-\psi.$ It follows from
Corollary 4.2 that
$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}\leq
C.\Big{(}\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))\Big{)}^{2^{1-m}},$
where $C$ is a positive constant. Since $F$ is non decreasing in the second
variable, it follows from Stokes formula that
$0\leq\int_{X}\rho(H_{m}(\psi)-H_{m}(\varphi))=\int_{X}(\varphi-\psi)(F(.,\psi)-F(.,\varphi))\omega^{n}\leq
0.$
Thus,
$\int_{X}d\rho\wedge d^{c}\rho\wedge\omega^{n-1}=0,$
which implies that $\rho$ is constant. If moreover $t\mapsto F(x,t)$ is
increasing for every $x\in X$, it is easy to see that $\rho=0.$
To prove the existence, we consider three cases.
Case 1: $F$ does not depend on the second variable, $F(x,t)=f(x),\forall x,t$.
Take a sequence of smooth strictly positive functions $(f_{j})$ converging to
$f$ in $L^{p}(X).$ We can assume that
$\int_{X}f_{j}\omega^{n}=\int_{X}\omega^{n}$, for every $j$. We use the
existence result in Theorem 2.14 to produce a sequence of smooth solutions
$(\varphi_{j})$ normalized by $\sup_{X}\varphi_{j}=0,\forall j.$ By passing to
a subsequence we can assume that $(\varphi_{j})$ converges in $L^{1}(X).$
Since $\|f_{j}\|_{p}$ is uniformly bounded, by Lemma 4.7 we can find
$\alpha,A$ which do not depend on $j$ such that all the measures
$f_{j}\omega^{n}$ satisfy condition $\mathcal{Q}_{m}(\alpha,A,\omega).$ By
Proposition 4.4, the sequence $(\varphi_{j})$ is uniformly bounded. Now it
follows from Theorem 4.8 that $\varphi_{j}$ converges uniformly to a
continuous function $\varphi\in\mathcal{P}_{m}(X,\omega)$ which solves
equation $H_{m}(\varphi)=f\omega^{n}$.
In the next two cases we will use the Schauder fixed point Theorem.
Case 2: There exists $t_{1}\in\mathbb{R}$ such that
$\int_{X}F(x,t_{1})\omega^{n}>\int_{X}F(x,t_{0})\omega^{n}$.
We set
$\mathcal{C}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\
\int_{X}\varphi\omega^{n}\geq-C_{0};\ \sup_{X}\varphi\leq 0\\},$
where $C_{0}$ is the constant introduced in Lemma 2.13. It follows that
$\mathcal{C}$ is a compact convex subset of $L^{1}(X).$
Take $\psi\in\mathcal{C},$ we use the result in case 1 to find
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ such that
$\sup_{X}\varphi=0$ and
$H_{m}(\varphi)=F(.,\psi+c_{\psi})\omega^{n},$
where $c_{\psi}\geq t_{0}$ is a constant such that
(5.1) $\int_{X}F(.,\psi+c_{\psi})\omega^{n}=\int_{X}\omega^{n}.$
This can be done because $F$ satisfies conditions (F2) and (F3). Indeed, by
Fatou’s Lemma we have
$\liminf_{t\to+\infty}\int_{X}F(.,\psi+t)\omega^{n}\geq\int_{X}F(.,t_{1})\omega^{n}>\int_{X}\omega^{n}.$
Moreover
$\int_{X}F(.,\psi+t_{0})\omega^{n}\leq\int_{X}F(.,t_{0})=\int_{X}\omega^{n}$.
Thus by continuity of $t\mapsto\int_{X}F(.,\psi+t)\omega^{n}$ we can find
$c_{\psi}$ satisfying (5.1). Observe that $\varphi$ is well-defined and does
not depend on $c_{\psi}.$ Indeed, assume that $c_{1},c_{2}$ are two constants
such that
$\int_{X}F(.,\psi+c_{1})\omega^{n}=\int_{X}F(.,\psi+c_{2})\omega^{n}=\int_{X}\omega^{n},$
and $\varphi_{1},\varphi_{2}$ are two continuous functions in
$\mathcal{P}_{m}(X,\omega)$ such that
$H_{m}(\varphi_{1})=F(.,\psi+c_{1}),\ \ H_{m}(\varphi_{2})=F(.,\psi+c_{2}).$
Since $t\mapsto F(x,t)$ is non decreasing for every $x\in X,$ we have
$F(.,\psi+c_{1})=F(.,\psi+c_{2})$ almost every where on $X$. Thus by the
uniqueness result above, $\varphi_{1}=\varphi_{2}+c$ for some constant $c$
which must be $0$ by the normalization. Then we define the map
$\Phi:\mathcal{C}\rightarrow\mathcal{C},\ \psi\mapsto\varphi.$
Now we prove that $\Phi$ is continuous on $\mathcal{C}.$ Suppose that
$(\psi_{j})$ is a sequence in $\mathcal{C}$ converging to $\psi\in\mathcal{C}$
in $L^{1}(X)$ and let $\varphi_{j}=\Phi(\psi_{j})$. We set
$c_{j}:=c_{\psi_{j}}$ and prove that $(c_{j})$ is uniformly bounded. Suppose
in the contrary that $c_{j}\uparrow+\infty.$ By subtracting a subsequence if
necessary we can assume that $\psi_{j}\to\psi$ almost everywhere in $X$. Then
by Fatou’s lemma we have
$\int_{X}\omega^{n}=\lim_{j\to+\infty}\int_{X}F(.,\psi_{j}+c_{j})\omega^{n}\geq\int_{X}F(.,t_{1})\omega^{n},$
which is impossible. Therefore the sequence $(c_{j})$ is bounded. This implies
that the sequence $(F(.,\psi_{j}+c_{j}))_{j}$ is bounded in $L^{p}(X),$ for
some $p>n/m$ which does not depend on $j.$ To prove the continuity of $\Phi$
it suffices to show that any cluster point of $(\varphi_{j})$ satisfies
$\Phi(\psi)=\varphi.$ Suppose that $\varphi_{j}\to\varphi$ in $L^{1}(X)$. It
follows from Theorem 4.8 that the sequence $(\varphi_{j})$ is Cauchy in
$\mathcal{C}^{0}(X)$. Thus $(\varphi_{j})$ converges to $\varphi$ in
$\mathcal{C}^{0}(X)$ and
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$. By subtracting a
subsequence if necessary we can assume that $\psi_{j}\to\psi$ almost
everywhere on $X$ and $c_{j}\to c.$ Since $t\mapsto F(x,t)$ is continuous we
see that $F(.,\psi_{j}+c_{j})\to F(.,\psi+c)$ almost everywhere. Thus
$H_{m}(\varphi)=F(.,\psi+c)$ which means $\Phi(\psi)=\varphi$ and hence $\Phi$
is continuous on $\mathcal{C}.$
By the Schauder fixed point Theorem, it follows that $\Phi$ has a fixed point
in $\mathcal{C}$, say $\varphi$. By definition of $\Phi$, the function
$\varphi$ must be in the class
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and we have
$H_{m}(\varphi)=F(.,\varphi+c_{\varphi})\omega^{n}.$
The function $\varphi+c_{\varphi}$ is the required solution.
Case 3: $\int_{X}F(.,t)\omega^{n}=\int_{X}F(.,t_{0})\omega^{n},\forall t\geq
t_{0}.$ In this case we have $F(x,t)=F(x,t_{0})$ for all $t\geq t_{0}$ and for
almost $x\in X.$ Thus, for every $t\geq t_{0},$
$\|F(.,t_{0})\|_{L^{p}(X)}=\|F(.,t)\|_{L^{p}(X)}.$
From Proposition 4.4 we can find a positive constant $C_{1}$ such that for any
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ satisfying
$\sup_{X}\varphi=0$ and
$H_{m}(\varphi)=f\omega^{n},$
with $\|f\|_{p}\leq\|F(.,t_{0})\|_{p}$ then
$\varphi\geq-C_{1}.$
We set
$\mathcal{C}^{\prime}:=\\{\varphi\in SH_{m}(X,\omega)\ \mid\
-C_{1}\leq\varphi\leq 0\\}.$
Then $\mathcal{C}^{\prime}$ is a compact convex subset of $L^{1}(X).$
Take $\psi\in\mathcal{C}^{\prime},$ we use the result in case 1 to find
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ such that
$\sup_{X}\varphi=0$ and
$H_{m}(\varphi)=F(.,\psi+c_{\psi})\omega^{n},$
where $t_{0}\leq c_{\psi}\leq t_{0}+C_{1}$ is a constant such that
$\int_{X}F(.,\psi+c_{\psi})\omega^{n}=\int_{X}\omega^{n}.$
This can be done because $F$ satisfies the condition (F2) and (F3) . Indeed,
$\int_{X}F(.,\psi+t_{0})\omega^{n}\leq\int_{X}\omega^{n}\leq\int_{X}F(.,\psi+t_{0}+C_{1})\omega^{n}.$
Thus by continuity we can find $c_{\psi}$ as above. As in case 2, $\varphi$ is
well-defined and does not depend on the choice of $c_{\psi}$. By the choice of
$C_{1}$, we see that $\varphi\in\mathcal{C}^{\prime}.$ So, we can define a map
$\Phi:\mathcal{C}^{\prime}\rightarrow\mathcal{C}^{\prime}$ by setting
$\Phi(\psi)=\varphi.$
Now we prove that $\Phi$ is continuous on $\mathcal{C}^{\prime}.$ Suppose that
$(\psi_{j})$ is a sequence in $\mathcal{C}^{\prime}$ converging to
$\psi\in\mathcal{C}^{\prime}$ in $L^{1}(X)$ and let
$\varphi_{j}=\Phi(\psi_{j})$. We set $c_{j}:=c_{\psi_{j}}$. For each
$j\in\mathbb{N}$,
$\int_{X}[F(.,\psi_{j}+c_{j})]^{p}\omega^{n}\leq\int_{X}[F(.,c_{j})]^{p}\omega^{n}=\int_{X}[F(.,t_{0})]^{p}\omega^{n}.$
Therefore, the sequence $(F(.,\psi_{j}+c_{j}))_{j}$ is bounded in $L^{p}(X).$
As in case 2, we can assume that $\varphi_{j}\to\varphi$ in $L^{1}(X)$. It
follows from Theorem 4.8 that the sequence $(\varphi_{j})$ is Cauchy in
$\mathcal{C}^{0}(X)$. Thus $(\varphi_{j})$ converges to $\varphi$ in
$\mathcal{C}^{0}(X)$ and
$\varphi\in\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$. By subtracting a
subsequence if necessary we can assume that $\psi_{j}\to\psi$ in $L^{1}(X)$
and $c_{j}\to c.$ Then $H_{m}(\varphi)=F(.,\psi+c)$ and $\Phi(\psi)=\varphi$
which implies that $\Psi$ is continuous on $\mathcal{C}^{\prime}.$
By the Schauder fixed point Theorem, it follows that $\Phi$ has a fixed point
in $\mathcal{C}^{\prime}$, say $\varphi$. By definition of $\Phi$, the
function $\varphi$ must be in the class
$\mathcal{P}_{m}(X,\omega)\cap\mathcal{C}^{0}(X)$ and we have
$H_{m}(\varphi)=F(.,\varphi+c_{\varphi})\omega^{n}.$
The function $\varphi+c_{\varphi}$ is the required solution.
### 5.2. Proof of Theorem B
In this section we consider a special class of compact Kähler manifolds. We
assume that $(X,\omega)$ is a rational homogeneous manifold. That means
$X=G/H$, where $G$ is a complex semi-simple algebraic group and $H$ is a
parabolic subgroup. Let $K$ be a maximal compact subgroup of $G.$ Then $K$
acts transitively on $X.$ We assume moreover that $\omega$ is fixed by action
of $K.$ In this case we can regularize singular $(\omega,m)$-subharmonic
functions by using the group action which preserves the metric.
Let $\varphi$ be a continuous $(\omega,m)$-subharmonic function on $X.$ We
consider the following regularizing sequence
$\varphi_{\epsilon}(x):=\int_{K}\varphi(g^{-1}.x)\chi_{\epsilon}(g)dg,$
where $dg$ is the Haar measure on $K$ and $\chi_{\epsilon}$ are cut-off
functions whose supports decreases to $\\{e\\}$ (the identity of $K$), and
$\int_{K}\chi_{\epsilon}(g)dg=1,\forall\epsilon>0.$
It follows from [G99], [Hu94] that $\varphi_{\epsilon}$ is smooth for every
$\epsilon>0.$
###### Theorem 5.1.
Let $\varphi$ be a continuous $(\omega,m)$-subharmonic function on $X.$ Then
for each $\epsilon>0$, $\varphi_{\epsilon}$ is smooth $(\omega,m)$-subharmonic
and
$\lim_{\epsilon\to 0}\varphi_{\epsilon}=\varphi$
uniformly on $X.$
###### Proof.
The uniform convergence always holds for continuous functions. Let us show the
second assertion. Let $\alpha_{1},...,\alpha_{m-1}$ be $(\omega,m)$-positive
closed (1,1)-forms on $X$, and denote (for short)
$\alpha=\alpha_{1}\wedge...\wedge\alpha_{m-1}.$ Let $\mathcal{L}_{g}$ denote
the left action of $g\in K$, i.e.
$\mathcal{L}_{g}(x)=g.x,\ \ x\in X.$
Then $\mathcal{L}^{*}_{g}\alpha_{j}$ is also $(\omega,m)$-positive for every
$j,$ since $\mathcal{L}^{*}_{g^{-1}}\omega=\omega,$ and
$\mathcal{L}^{*}_{g^{-1}}(\mathcal{L}^{*}_{g}\alpha^{k}\wedge\omega^{n-k})=\alpha^{k}\wedge\omega^{n-k}.$
Fix a positive test function $\psi.$ We have
$\displaystyle\int_{X}\psi(\omega+dd^{c}\varphi_{\epsilon})\wedge\alpha\wedge\omega^{n-m}=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{X}\varphi_{\epsilon}dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{X}\Big{(}\int_{K}\mathcal{L}^{*}_{g}\varphi\chi_{\epsilon}(g)dg\Big{)}dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{X}\psi\alpha\wedge\omega^{n-m+1}+\int_{K}\Big{(}\int_{X}\mathcal{L}^{*}_{g}\varphi
dd^{c}\psi\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi(\omega+dd^{c}\mathcal{L}^{*}_{g}\varphi)\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi(\omega+\mathcal{L}^{*}_{g}dd^{c}\varphi)\wedge\alpha\wedge\omega^{n-m}\Big{)}\chi_{\epsilon}(g)dg$
$\displaystyle\,\,\,\,\,\,\,\,=\int_{K}\Big{(}\int_{X}\psi\mathcal{L}^{*}_{g}\big{[}(\omega+dd^{c}\varphi)\wedge\mathcal{L}^{*}_{g^{-1}}\alpha\wedge\omega^{n-m}\big{]}\Big{)}\chi_{\epsilon}(g)dg\geq
0.$
∎
###### Remark 5.2.
Thanks to Theorem 5.1, every continuous $(\omega,m)$-subharmonic function
belongs to $\mathcal{P}_{m}(X,\omega).$
Proof of Theorem B. Let $\varphi$ be the unique continuous solution to (1.1).
For $h\in K$, let $\varphi_{h}(x):=\varphi(h.x),\ x\in X.$ If $u$ is smooth
then
$\|u_{h}-u\|^{2}_{L^{2}}\leq
Cdist^{2}(h,e)\int_{X}(-u)dd^{c}u\wedge\omega^{n-1},$
where $C$ is some universal constant. Then, it follows from the approximation
theorem (Theorem 5.1) that
$\|\varphi_{h}-\varphi\|_{L^{2}(X)}\leq Cdist(h,e).$
For fixed $h\in K$, observe that $\varphi_{h}$ is $(\omega,m)$-subharmonic and
satisfies
$H_{m}(\varphi_{h})=F(h.x,\varphi(h.x))\omega^{n}.$
Thus, by applying Theorem 4.8 with $r=2$ we obtain
$\|\varphi_{h}-\varphi\|_{L^{\infty}}\leq
C^{\prime}.\|\varphi_{h}-\varphi\|_{L^{2}(X)}^{\gamma},$
where $0<\gamma<\frac{2(mp-n)}{mnp+2mp-2n}$ is a given constant and
$C^{\prime}>0$ is another constant which does not depend on $h.$ We thus get
$\|\varphi_{h}-\varphi\|_{L^{\infty}(X)}\leq C.C^{\prime}dist(h,e)^{\gamma},\
\forall h\in K.$
This yields the $\gamma$-Hölder continuity of $\varphi$ (see [EGZ09]).
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Lu Hoang Chinh
Laboratoire Emile Picard
UMR 5580, Université Paul Sabatier
118 route de Narbonne
31062 TOULOUSE Cedex 04 (FRANCE)
lu@math.univ-toulouse.fr
|
arxiv-papers
| 2012-02-11T12:00:41 |
2024-09-04T02:49:27.345070
|
{
"license": "Public Domain",
"authors": "Lu Hoang Chinh",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1202.2436"
}
|
1202.2538
|
# Dynamic Control of Collapse in a Vortex Airy Beam
Rui-Pin Chen1,2, Khian-Hooi Chew3, Sailing He1,4 1Centre for Optical and
Electromagnetic Research, State Key Laboratory of Modern Optical
Instrumentations, JORCEP [KTH-ZJU-LU Joint Research Center of Photonics],
Zhejiang University (ZJU), Hangzhou 310058, China
2School of Sciences, Zhejiang A $\&$ F University, Lin’an 311300, Zhejiang
Province, China
3Department of Physics, Faculty of Science, University of Malaya, Kuala Lumpur
50603, Malaysia
4Department of Electromagnetic Engineering, School of Electrical Engineering,
Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden
###### Abstract
We study the self-focusing dynamics and collapse of vortex Airy optical beams
in a Kerr medium. The collapse is suppressed compared to a non- vortex Airy
beam in a Kerr medium as a result of the existence of vortex fields. The
locations of collapse depend sensitively on the initial power, vortex order,
and modulation parameters. Unlike the collapses reported before for any beam,
the collapse may occur in a position where the initial field is nearly zero
while no collapse appears in the region where the initial field is mainly
distributed. This study sheds light on how to control and manipulate the
location of collapse based on the initial power, vortex order and modulation
parameter.
###### pacs:
42.65.Jx; 42.65.Sf
Nonlinear wave collapse has been investigated in many areas of physics,
including optics, fluidics, plasma physics, and Bose-Einstein condensates [1].
In nonlinear optics, collapses of beams with various spatial distributions
have been studied [1-3]. One of the challenges facing this field of interest
is the possibility of controlling and manipulating the collapse dynamics.
Recently, the Airy beam has attracted considerable attention after its
experimental generation by Siviloglou et al. [4] due to its intriguing
properties and potential applications such as weak-diffraction, transverse
acceleration [4-6], self-healing [7], and sorting microscopic particles [8].
The evolution characteristics of an Airy beam in nonlinear medium have been
studied [9-17], such as plasma channel generation [13], laser filamentation
[14], supercontinuum and solitary wave generation [15-17]. Vortices have been
the subject of many studies and appear in many branches of physics [18]. A
vortex Airy beam is formed by superposition of an Airy beam and a vortex
optical field. Recently, interesting propagation dynamics and non-classicality
of a vortex Airy beam have been reported [19, 20]. The nonlinear dynamics of a
vortex Airy beam in a Kerr medium is also expected to give some interesting
and nontrivial properties.
In this letter, we investigate the spatial collapse dynamics of a vortex Airy
beam. The coupling between the vortex and Airy beam strongly affects the
nonlinear dynamic properties of the vortex and Airy beam in the Kerr medium.
The diffraction against self-focusing of a vortex Airy beam is effectively
enhanced compared to a non-vortex Airy beam. In addition, the vortex of beam
tends to suppress the collapse in a Kerr medium. For a non-vortex Airy beam in
a Kerr medium, partial collapse can occur in the beam’s center [12]. However,
the collapse of the vortex Airy beam never occurs at the beam’s center. This
is because the center of the vortex is located at the center of the beam. Our
findings reveal that the position of the partial collapse and the propagation
distance for the appearance of partial collapse are dependent on the initial
powers, vortex orders, and modulation parameters. The partial collapse may
occur in the main lobe, side lobes, or outermost lobes, depending on the
strength of initial powers, the order of vortex and the modulation parameters.
The collapse can occur in the position where the initial field is almost zero
while no collapse appears in the position where the field originally exists
with a appropriate power. We further show that the initial power, vortex order
and modulation parameters can be exploited to control and manipulate the
position of collapse dynamics in a vortex Airy beam. Even if the initial power
is ultrahigh, the partial collapse occurs separately at the side lobes and the
beam still propagates along a similar accelerating curve trajectory as that of
an Airy beam in free space. Since the field distribution of a vortex Airy beam
is modulated by some exponential factors, we also study the effect of these
exponential factors on the nonlinear evolution of the beam in a Kerr medium.
Finally, the evolution and collapse of a quasi-one-dimensional vortex Airy
beam in a Kerr medium are analyzed as a limiting case. Our study shows the
possibility of controlling and manipulating the collapse, especially the
position of collapse in the vortex Airy beam, by choosing the initial powers,
vortex orders or modulation parameters. Analogous to optics, the results can
be extended to manage the collapse of other nonlinear waves such as fluid and
matter waves.
The moments approach analysis. The propagation of a light beam in a Kerr
medium is described in the paraxial approximation by the following nonlinear
Schrodinger (NLS) equation:
$\nabla_{\bot}^{2}E-2ik\frac{{\partial E}}{{\partial
z}}+\frac{{2n_{2}k^{2}}}{{n_{0}}}\left|E\right|^{2}E=0,$ (1)
where $k$ is the linear wave number, $x$ and $y$ are the transverse
coordinates, $z$ is the longitudinal coordinate, $n_{0}$ is the linear
refraction index of the medium, $n_{2}$ is the third order nonlinear
coefficient. Due to the complexity of the evolution of a vortex Airy beams in
a Kerr medium, we apply the approach of moments [21] to study the nonlinear
dynamics by analyzing the evolution of several integral quantities derived
from the NLS. These quantities are defined as
$\displaystyle I_{1}(z)$ $\displaystyle=$
$\displaystyle\iint_{s}{\left|E\right|^{2}dxdy},$ (2a) $\displaystyle
I_{2}(z)$ $\displaystyle=$
$\displaystyle\iint_{s}{(x^{2}+y^{2})\left|E\right|^{2}dxdy},$ (2b)
$\displaystyle I_{3}(z)$ $\displaystyle=$
$\displaystyle\frac{i}{k}\iint_{s}{\left[{x\left({E\frac{{\partial
E^{*}}}{{\partial x}}-E^{*}\frac{{\partial E}}{{\partial
x}}}\right)+y\left({E\frac{{\partial E^{*}}}{{\partial
y}}-E^{*}\frac{{\partial E}}{{\partial y}}}\right)}\right]}dxdy,$ (2c)
$\displaystyle I_{4}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{{2k^{2}}}\iint_{s}{\left({\left|{\frac{{\partial
E}}{{\partial x}}}\right|^{2}+\left|{\frac{{\partial E}}{{\partial
y}}}\right|^{2}-\frac{{k^{2}n_{2}}}{{n_{0}}}\left|E\right|^{4}}\right)dxdy}.$
(2d)
These quantities are associated with the beam power $I_{1}$, beam width
$I_{2}$, momentum $I_{3}$, and Hamiltonian $I_{4}$; and satisfy a closed set
of coupled ordinary differential equations [21]: $dI_{1}(z)/dz=0$,
$dI_{2}(z)/dz=I_{3}(z)$, $dI_{3}(z)/dz=4I_{4}(z)$, $dI_{4}(z)/dz=0$, and the
important invariant under evolution, $Q=2I_{4}I_{2}-I_{3}^{2}/4$, Therefore,
the following Ermakov-Pinney equation describing the dynamics of the scaled
beam width can be obtained:
$\frac{{d^{2}I_{2}^{1/2}(z)}}{{dz^{2}}}=\frac{Q}{{I_{2}^{3/2}(z)}}.$ (3)
For a vortex Airy beam, an initial field distribution can be described by
[19]:
$E(x,y;z=0)=A_{0}Ai(x/x_{0})\exp(a_{x}x/x_{0})Ai(y/x_{0})\exp(a_{y}y/x_{0})(x+iy)^{m},$
(4)
(a)
(b)
Figure 1: (Color online) The critical powers of vortex Airy beams for
different $a_{x}$ and $a_{y}$. (a) $m=1$; (b) $m=2$.
where $A_{0}$ is the amplitude of the complex amplitude $E(x,y,z=0)$, $x_{0}$
is an arbitrary transverse scale. Exponential factors $a_{x}$ and $a_{y}$ are
positive in order to ensure containment of the infinite Airy tail in the $-x$
and $-y$ directions, respectively. The azimuthal index $m$ represents the
order of vortex. The general solution to Eq. (3) with the vortex Airy beam as
an initial field distribution can be given as
$I_{2}(z)=I_{2}(z=0)+\frac{Q}{{I_{2}(z=0)}}z^{2},$ (5)
Eq. (5) describes the variation of the scaled beam width of the vortex Airy
beam in a Kerr medium. When $Q=0$, the rms beam width remains constant as
recognized from Eq. (5) and the critical value, $I_{1}^{cr}$, is derived from
$2I_{4}I_{2}-I_{3}^{2}/4=0$ and Eq. (2). The corresponding power can be found
through $P_{cr}=n_{0}c\varepsilon_{0}I_{1}^{cr}/2$ [21]. The expression for
$P_{cr}$ is not presented here due to its lengthy expression.
Figure 1 shows the denary logarithmic vertical scale of the ratio of the
critical power of a vortex Airy beam (with $m=1,2$) to the critical power
$P_{cr}^{G}=\pi c\varepsilon_{0}n_{0}^{2}/(n_{2}k^{2})$ of a Gaussian beam ,
where $c$ is the speed of light in vacuum and $\varepsilon_{0}$ is the
permittivity of free space. In Fig. 1 one sees that the value of the critical
power of the vortex Airy beam depends mainly on the beam profile of the
transverse distribution (besides the nonlinear parameters of the medium). The
critical power of the vortex Airy beam increases as the the order of vortex
modes increases, but decreases as the modulation parameters $a_{x}$ and
$a_{y}$ increase. If the initial power exceeds the critical power $P_{cr}$,
the rms beam width goes to zero at a finite propagation distance, as predicted
by the approach of moments. Obviously, the critical power is the upper bound
for the collapse power [22, 23].
Evolution and collapse of vortex Airy beams in Kerr media. Let us examine the
evolution of a vortex Airy beam in a Kerr medium. In the numerical
calculation, we take $\lambda=0.53\mu m$, $x_{0}=100\mu m$ and
$z_{0}=kx_{0}^{2}/2=6cm$. The intensity distributions of the 1st and 2nd
vortex Airy beams with $P_{in}=5P_{cr}^{G}$ and $P_{in}=15P_{cr}^{G}$ in the
focusing nonlinear medium at different propagation distances are shown in Fig.
2, where hereafter the intensity distribution is normalized with respect to
its initial peak intensity. One sees that the beams with different initial
powers in the Kerr medium propagate along the same accelerating curve
trajectory as an Airy beam in free space, as expected. The collapse will not
occur during the propagation for these cases but the evolution of intensity
distributions of the beam, however, depends sensitively on the order of vortex
and the initial power.
Figure 2: (Color online) (Color online) The intensity distribution of vortex
Airy beam $(a_{x}=a_{y}=0.1)$ at different propagation distances with initial
powers in focusing Kerr medium. Upper row:the $m=1$,
$P_{in}=5P_{cr}^{G}=0.0023P_{cr}$; and lower: $m=2$,
$P_{in}=15P_{cr}^{G}=0.0024P_{cr}$. Figure 3: (Color online) The intensity
distribution of the vortex Airy beam $(a_{x}=a_{y}=0.1)$ in focusing medium.
(a) $m=1$,$P_{in}=15P_{cr}^{G}=0.0069P_{cr}$, $z=5z_{0}$; (b)$m=1$,
$P_{in}=15P_{cr}^{G}=0.0069P_{cr}$, $z=6z_{0}$; (c)
$m=2$,$P_{in}=50P_{cr}^{G}=0.0079P_{cr}$, $z=7z_{0}$; (d)
$m=2$,$P_{in}=50P_{cr}^{G}=0.0079P_{cr}$, $z=7.3z_{0}$; Figure 4: (Color
online) The intensity distribution of the vortex Airy beam $(a_{x}=a_{y}=0.1)$
in focusing medium. (a) $m=1$,$P_{in}=50P_{cr}^{G}=0.023P_{cr}$, $z=0.7z_{0}$;
(b) $m=1$,$P_{in}=50P_{cr}^{G}=0.023P_{cr}$,$z=0.8z_{0}$. (c)
$m=1$,$P_{in}=2500P_{cr}^{G}=1.15P_{cr}$, $z=0.02z_{0}$; (d)
$m=1$,$P_{in}=2500P_{cr}^{G}=1.15P_{cr}$,$z=0.03z_{0}$;
(a)
(b)
Figure 5: (Color online) The peak intensity of vortex Airy beam
$a_{x}=a_{y}=0.1$ as a function of the propagation distance with different
initial powers (a) $m=1$; (b) $m=2$.
As the initial power increases, more energy accumulates at various locations
of the beam, leading to partial collapse of the beam. If the initial power is
$P_{in}=15P_{cr}^{G}$, the 1st order vortex Airy beam collapses at the
outermost main lobe of the beam, as illustrated in Fig. 3 (upper). No collapse
of vortex Airy beam with the order of $m=2$ is expected, if
$P_{in}=15P_{cr}^{G}$ (see Fig. 2 (lower)). The 2nd order vortex Airy beam,
however, is expected to collapse at $P_{in}=50P_{cr}^{G}$, as shown in Fig.
3(c) and 3(d). In this case, the partial collapse will occur separately in two
lobes in the beam (but, not in the outermost lobes). Numerical simulations
indicate that the actual collapse power for the 1st order vortex Airy beam
with $a_{x}=a_{y}=0.1$ is $P_{in}=0.0063P_{cr}=14P^{G}_{cr}$ and that for the
2nd order vortex Airy beam with $a_{x}=a_{y}=0.1$ is
$P_{in}=0.0058P_{cr}=37P^{G}_{cr}$. It is interesting to see that the 1st
order vortex Airy beam collapses at the outermost lobes, if
$P_{in}=50P_{cr}^{G}$ (see Fig. 4(a) and (b)). A stronger initial power
increases the number of lobes that collapse at the outermost region, and
decreases the propagation distances of beam (e.g. Fig. 4(c) and 4(d) with
$P_{in}=1.15P_{cr}=2500P^{G}_{cr}$). In Fig. 5, we show the normalized peak
intensities as a function of the propagation distance with different initial
powers. Although the approach of moments predicts that the rms beam width will
be broadened when $P_{in}<P_{cr}$, the results from the variation of peak
intensities suggest a deformation and redistribution of the vortex Airy beams
during the propagation. The intensity at some lobes will dominate and
eventually collapse while the rms beam width increases or remains constant,
such as for the cases $P_{in}=15P_{cr}^{G}$ and $50P_{cr}^{G}$ for the 1st
order vortex Airy beam and $P_{in}=50P_{cr}^{G}$ for the 2nd order vortex Airy
beam. The results indicate that a higher initial power leads a shorter
propagation distance before collapse. When the initial power begin to exceed
the collapse power, the flow of transverse energy of Airy beam and vortex
leads to a collapse occurring at the outermost main lobe of the beam. It is
interesting to see that the energy accumulates to the position where the field
is almost zero and finally a collapse occurs. On the other hand, no collapse
occurs at a position where initial field mainly exists. In all the studies
reported in the literature (for either vortex or non-vortex beams), the field
never collapses at a location where the initial field is almost zero
[3,22,23]. In the present study, however, a unusual collapse is observed for
the present vortex Airy beam. The vortex Airy beam may collapse at a location
where the initial field is nearly zero, e.g., the collapse point in Fig. 3(b)
(cf. the zero initial field at this point, marked as point A on the first
subfigure of Fig.2). During the propagation, the energy accumulates at this
collapse position. The higher the initial power value, the closer the collapse
position to the beam’s center and the shorter propagation distance before
collapse. These results, therefore, provide useful information on how to
spatially manipulate a collapse in experiment. If the initial power increases
further to a certain threshold value, the beam collapses at many side lobes
simultaneously, but not in the outermost lobes, see e.g. the collapse position
in Fig. 3(d) (cf. the initial field at these points around point A in Fig. 2
with $z=0$). In this case, we see that the position of collapse, the number of
collapse lobes, and the propagation distance (before collapse) also depend on
the initial power. As the initial power reaches a considerably high value, the
collapse will appear at many outermost lobes with a very short propagation
distance with a higher initial power and a larger number of collapsed lobes.
Even if the approach of moments predicts that the rms beam width will go to be
zero at some propagation distance when the initial power exceeds the critical
power $P_{cr}$, the vortex Airy beam collapses separately at the outermost
lobes as shown in Fig. 4(c) and 4(d).
Quasi-one-dimensional vortex Airy beam. We now investigate the effect of
modulation parameters on the nonlinear evolution of the vortex Airy beam. When
$a_{x}=1.5$ and $a_{y}=0.05$, the beam becomes a quasi-one-dimensional vortex
Airy beam, as shown in Fig. 6. Under a low initial power of
$P_{in}=1.5P_{cr}^{G}$, the beam propagates along the accelerating trajectory
with a feature similar to the propagation behavior of an Airy beam in free
space, the field distribution is evidently different with that of the beam in
free space, and the beam does not collapse as shown in Fig. 6(a). By
increasing the initial power from $P_{in}=1.5P_{cr}^{G}$ to $5.5P_{cr}^{G}$,
the off-center part of the beam partially collapses at certain propagation
distance although the propagation trajectory is the same as that of
$P_{in}=1.5P_{cr}^{G}$, (compared with the zero initial field of this collapse
point to the point A on first subfigure in Fig. 6(b)). On the other hand, no
collapse occurs at the position where the field originally exists as shown in
Fig. 6(b). The underlying physics are similar to the collapse behavior of a
vortex Airy beam.
Figure 6: (Color online) The intensity distribution of a vortex Airy beam
($a_{x}=1.5,a_{y}=0.05$) at different propagation distances with initial
powers (a)$P_{in}=1.5P_{cr}^{G}=0.00043P_{cr}$; (b)
$P_{in}=5.5P_{cr}^{G}=0.00157P_{cr}$. The thin red line in the first plot
indicates the position of the longitudinal cross-section.
In conclusion, we have systematically studied the collapse dynamics of vortex
Airy optical fields in a Kerr medium. The collapse of a vortex Airy beam
requires a higher energy than that of a non-vortex beam. Although vortex Airy
beams with different initial powers propagate along a similar acceleration
curve trajectory in the Kerr medium as a non-vortex Airy beam propagates in
free space, the evolution and appearance of the collapse in a vortex Airy beam
exhibits unusual features. The collapse can occur in the position where the
initial field is almost zero. Our study shows that the collapse can occur at
different locations other than the central point, and the locations can be
controlled through choosing appropriate initial powers, the order of vortex,
and the modulation parameters. This work is supported in part by the State Key
Program for Basic Research of China under Grant No.2006CB921805, the Science
and Technology Department of Zhejiang Province (2010R50007), and the Key
Project of the Education Commission of Zhejiang Province of China
No.Z201120128.
## References
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|
arxiv-papers
| 2012-02-12T15:50:18 |
2024-09-04T02:49:27.363069
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rui-Pin Chen, Khian-Hooi Chew, Sailing He",
"submitter": "Ruipin Chen",
"url": "https://arxiv.org/abs/1202.2538"
}
|
1202.2736
|
# Function call overhead benchmarks
with MATLAB, Octave, Python, Cython and C
André Gaul111 gaul@math.tu-berlin.de, http://www.math.tu-berlin.de/?78347.
This work is licensed under the creative commons license CC-BY-3.0. All used
source code files are published under GPL 3.0 in the git repository
https://bitbucket.org/andrenarchy/funcall.
## 1 Background
In many applications a function has to be called very often inside a loop. One
such application in numerical analysis is the finite element method (FEM) for
the approximate solution of a partial differential equation (PDE). For example
we would like to approximate the solution $u:\Omega\rightarrow\mathbb{R}$ of
the PDE
$\displaystyle-\nabla\cdot(a\nabla u)+b\cdot\nabla u+cu$
$\displaystyle=f\qquad\text{in}\quad\Omega$ $\displaystyle u$
$\displaystyle=g\qquad\text{on}\quad\partial\Omega$
where $\Omega\subseteq\mathbb{R}^{d}$ is a $d$-dimensional domain with
boundary $\partial\Omega$ and $a,c,f:\Omega\rightarrow\mathbb{R}$,
$b:\Omega\rightarrow\mathbb{R}^{d}$ and
$g:\partial\Omega\rightarrow\mathbb{R}$ are given functions with special
properties that will not be discussed here. In FEM a domain is discretized
into a mesh by splitting the domain into “simple” geometric shapes (intervals,
triangles, tetrahedrons, …). Along with special functions (usually piecewise
polynomials) these shapes are called _elements_. A system of linear algebraic
equations $Ax=b$ is obtained by computing integrals on each element and
sorting them into a large (and usually sparse) “system” matrix $A$ and right
hand side (RHS) $b$.
From a computational point of view this can be achieved by writing a function
getElementIntegrals(...) that computes all necessary integrals on one element
and is then called within a loop for each element of the mesh. A corresponding
Python code could look like this:
⬇
for el in mesh:
elMat, elRHS = getElementIntegrals(el,a,b,c,f,g)
# process entries of elMat and elRHS by sorting them
# into a "big" matrix and right hand side.
Now imagine that the mesh is very fine, i.e. the number of elements in the
mesh is large. For example a uniform tetrahedral discretization of the unit
cube $[0,1]^{3}$ with grid size $h=1/n$ ($n\in\mathbb{N})$ results in a mesh
consisting of $6n^{3}$ tetrahedral elements. The function getElementIntegrals
is thus called $6n^{3}$ times.
Especially in interpreted programming languages like MATLAB, Octave or Python
a function call may be very time-consuming. By _function call_ we mean the
setup needed for the function to start executing the actual function code in
the function’s body and cleaning it up afterwards. This includes possible
copying of memory and dynamic type checking for the parameters passed to and
returned from the function.
In the above setting the functions a, b, c, f, g and even getElementIntegrals
can often be evaluated in a fast way. However, the function call itself may
exhibit an overhead that consumes far more time than the actual function code.
In this report we present results of function call benchmarks for programming
languages or interpreters often used in numerical analysis: MATLAB/Octave,
Python and C. While C is a compiled language and optimizations are possible
even on a very low level, MATLAB and the free software alternative Octave are
interpreted languages and mainly draw on automated optimization and low-level
improvements are usually only possible by switching to plain C, C++ or Fortran
with so-called mex-files. In contrast, in the interpreted language Python
time-critical parts can be compiled with Cython – the C-Extensions for Python
[1]. Cython’s syntax is very similar to Python’s and introduces static typing
as well as the ability to call C code easily.
Here we present a benchmark that helps to identify and quantify optimization
potentials with respect to time consumption caused by function calls in the
mentioned languages. Section 2 describes the setup of the benchmark and
Section 3 presents and discusses the results.
## 2 Benchmark setup
The situation outlined in Section 1 can be boiled down to a function that is
called in a loop very often. For benchmarking the overhead of the function
call itself it is reasonable to make the function body as simple as possible.
Therefore, we will use a function that accepts one double precision floating-
point number and returns its square. We ran separate tests for several types
of function definitions that are available in the used programming languages.
The different options are enumerated for later reference.
MATLAB and Octave
1. Option 1.
1. (a)
The called function defined in an external .m-file:
⬇
function result = fun_external(a)
result = a*a;
end
The loop is defined in a separate file:
⬇
function result = loop_external(n)
result = zeros(n,1);
for i=1:n
result(i) = fun_external(i);
end
end
2. (b)
Both functions from 1(a) are placed in the same file consecutively.
3. (c)
A nested function definition in the body of the calling function is used, that
is the called function is placed _inside_ the body of the loop function:
⬇
function result = loop_nested(n)
result = zeros(n,1);
for i=1:n
result(i) = fun_nested(i);
end
function ret = fun_nested(a)
ret = a*a;
end
end
4. (d)
Anonymous function definition in the body of the calling function:
⬇
fun_anonymous = @(a) (a*a);
Python
Python is an interpreted language but can be tuned by writing time-critical
parts in Cython [1]. With Cython one can blend Python code and C code easily.
We take a closer look at the following options for implementing the loop and
the called function:
1. Option 1.
The called function can be implemented
1. (a)
together with the loop function in the same .py-file or
2. (b)
in a separate .py-file and imported in the .py-file implementing the loop.
2. Option 2.
The loop can be implemented with a numpy array [3] in
1. (a)
plain Python:
⬇
def loop(n):
result = numpy.empty(n)
for i in xrange(0,n):
result[i] = fun_samefile(i+1)
return result
or in
2. (b)
Cython where the code still is plain Python code as in 2(a) or in
3. (c)
Cython enriched with static typing:
⬇
def loop(n):
cdef numpy.ndarray[numpy.double_t] result = \
numpy.empty(n)
cdef int i
for i in xrange(0,n):
result[i] = fun_samefile(i+1)
return result
Note that the result array and the i variable are now typed which allows
Cython to address the elements of the numpy array in the loop efficiently.
3. Option 3.
Similarly, the called function can be implemented in
1. (a)
plain Python:
⬇
def fun(a):
return a**2
or in
2. (b)
Cython where the code still is plain Python code as in 3(a) or in
3. (c)
Cython enriched with static typing:
⬇
cpdef double fun(double a):
return a**2
If the function is imported in the .py-file running the loop then an
additional .pxd-file with the corresponding function declaration should be
provided. A .pxd-files works like a C header file and in our case simply
contains the line
⬇
cpdef double fun(double a)
Several combinations are not possible and are thus omitted. For example,
option 1 1(a) with option 2 2(a) and option 3 3(b) are impossible because both
the loop and the called function are compiled with Cython if they are defined
in the same file).
C
1. Option 1.
1. (a)
The function is defined in the same .c-file as the loop and compiled with the
options -O3 -fomit-frame-pointer. The function code is
⬇
double fun(double a) {
return a*a;
}
while the loop code is
⬇
double* loop(int n) {
double* result =
(double*) malloc(sizeof(double)*n);
for (int i=0; i<n; i++)
result[i] = fun(i);
free(result);
return result;
}
2. (b)
The function is compiled in a shared library (.so-file) which is then
dynamically linked to the compiled loop function. The compiler options are the
same as for 1(a).
For further details we refer to the source code [2].
## 3 Benchmark results
In this section we present results of the benchmark setup described in Section
2 conducted with the languages/interpreters
* •
MATLAB 2011b
* •
Octave 3.2.4
* •
Python 2.7.2
* •
Cython 0.14.1 (C-Extensions for Python)
* •
C with GCC 4.6.1.
All experiments have been carried out on a Intel Core i5 M540 CPU running at
2.53 GHz with Ubuntu 11.10. We computed $a^{2}$ for $a=1,\ldots,10^{7}$ with
all possible variations of implementations with the options presented in 2. By
using this test setup we wish to identify and quantify possibilities for
optimization with respect to time consumption caused by function calls. The
experiment was repeated 10 times and the arithmetic mean of the measured
timings are presented in Table 1.
The files used for the experiments are published [2] under GPL3 so further
results can be produced with later versions of the above software and on
different hardware.
Rank | Time in s | Language | Variant
---|---|---|---
| | | Option 1 | Option 2 | Option 3
1 | 0.055 | C | 1(a) | – | –
2 | 0.076 | C | 1(b) | – | –
3 | 0.077 | Python/Cython | 1(b) | 2(c) | 3(c)
4 | 0.077 | Python/Cython | 1(a) | 2(c) | 3(c)
5 | 1.283 | Python/Cython | 1(a) | 2(c) | 3(b)
6 | 1.323 | Python/Cython | 1(a) | 2(b) | 3(c)
7 | 1.337 | Python/Cython | 1(b) | 2(b) | 3(c)
8 | 1.598 | Python/Cython | 1(b) | 2(c) | 3(b)
9 | 2.124 | Python/Cython | 1(a) | 2(b) | 3(b)
10 | 2.298 | Python/Cython | 1(b) | 2(a) | 3(c)
11 | 2.426 | Python/Cython | 1(b) | 2(a) | 3(b)
12 | 2.553 | Python/Cython | 1(b) | 2(c) | 3(a)
13 | 2.862 | Python/Cython | 1(b) | 2(b) | 3(b)
14 | 2.941 | Python | 1(a) | 2(a) | 3(a)
15 | 2.973 | MATLAB | 1(b) | – | –
16 | 3.018 | MATLAB | 1(a) | – | –
17 | 3.359 | Python/Cython | 1(b) | 2(b) | 3(a)
18 | 3.715 | Python | 1(b) | 2(a) | 3(a)
19 | 4.181 | MATLAB | 1(c) | – | –
20 | 6.590 | MATLAB | 1(d) | – | –
21 | 112.154 | Octave | 1(d) | – | –
22 | 133.725 | Octave | 1(c) | – | –
23 | 138.603 | Octave | 1(b) | – | –
24 | 152.452 | Octave | 1(a) | – | –
Table 1: Execution time in seconds for $10^{7}$ function calls. The variants
are described in Section 2.
Unsurprisingly, both C implementations with enabled compiler optimizations are
the fastest implementations in this benchmark. The fact that Octave is slower
with function calls than MATLAB is also well-known. More interesting is the
observation that Python and MATLAB approximately consume the same amount of
time when no optimizations are used in Python. However, we can see in the
first lines of Table 1 that Python can be tuned with the C-Extensions Cython
such that the execution time reaches the one of the dynamically linked C
implementation, which is about 40 times faster than the plain Python or MATLAB
implementation.
The possibility to write performance-critical parts in the Python-like Cython
syntax is a clear advantage over MATLAB and Octave because currently an
optimization of function calls in MATLAB can only be achieved by writing .mex-
files that require a complete rewrite of the code in another language and are
often hard to handle – especially if several versions of MATLAB and thus the
mex-API are used. However, we want to point out that in principle the
performance of the C variants can be achieved with mex-based implementations
by calling C code in the mex-files.
The Cython approach requires less effort since the Cython syntax is very
similar to the plain Python syntax. Thus optimizations can be implemented
easily with Cython where they are needed while maintaining the full
flexibility of Python.
## References
* [1] R. Bradshaw, S. Behnel, D. S. Seljebotn and G. Ewing “The Cython compiler”, 2012 URL: http://cython.org
* [2] André Gaul “Funcall git repository”, 2012 URL: https://bitbucket.org/andrenarchy/funcall
* [3] Eric Jones, Travis Oliphant and Pearu Peterson “SciPy: Open source scientific tools for Python”, 2001– URL: http://www.scipy.org/
|
arxiv-papers
| 2012-02-13T14:14:00 |
2024-09-04T02:49:27.375640
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andr\\'e Gaul",
"submitter": "Andr\\'e Gaul",
"url": "https://arxiv.org/abs/1202.2736"
}
|
1202.2747
|
# Viewing the CPT Invariance as a Basic Postulate in Physics
Guang-jiong Ni a,b pdx01018@pdx.edu a Department of Physics, Portland State
University, Portland, OR97207, U. S. A.
b Department of Physics, Fudan University, Shanghai, 200433, China Suqing
Chen b suqing_chen@yahoo.com b Department of Physics, Fudan University,
Shanghai, 200433, China Jianjun Xu b xujj@fudan.edu.cn b Department of
Physics, Fudan University, Shanghai, 200433, China
###### Abstract
The $CPT$ invariance has been firmly established via experimental tests. Its
theoretical implication and derivation at two levels of both relativistic
quantum mechanics ($RQM$) and quantum field theory ($QFT$) are discussed.
Being a basic symmetry, the $CPT$ invariance can be expressed as ${\cal
P}{\cal T}={\cal C}$, where ${\cal P}{\cal T}$ represent the ”strong
reflection”, i.e., the (newly defined) space-time inversion (${\bf x}\to-{\bf
x},t\to-t$), invented by Lüders and Pauli in the proof of the $CPT$ theorem
and ${\cal C}$ the new particle-antiparticle transformation proposed by Lee
and Wu. Actually, the renamed $CPT$ invariance, ${\cal P}{\cal T}={\cal C}$,
could be viewed as a basic postulate being injected implicitly into the theory
since the nonrelativistic quantum mechanics ($NRQM$) was combined with the
theory of special relativity ($SR$) to become $RQM$ and then the $QFT$. The
Klein-Gordon ($KG$) equation is highlighted to become a self-consistent theory
in $RQM$, based on two sets of wavefunctions ($WFs$) and momentum-energy
operators for particle and antiparticle respectively, together with the above
postulate. Hence the Klein paradox for both $KG$ equation and Dirac equation
can be solved without resorting to the ”hole theory”.
Keywords: CPT invariance, Antiparticle, Quantum mechanics, Quantum field
theory
PACS: 03.65.-w; 03.65.Ta; 03.65.Ud; 11.10.-z
## I I. Introduction
The famous paper by Einstein, Podolsky and Rosen (EPR 1 (1)) in 1935 is not
easy to read, at least to us. And the deep implication of a remarkable $EPR$
experiment by CPLEAR collaboration on $K^{0}-\bar{K}^{0}$ system in 1998 2 (2)
is not easy to explore either. We discuss these two papers in parallel in
section II before we will be able to extract some new conception about the
antiparticle’s wavefunction ($WF$) and momentum-energy operators. Then in
section III the Klein-Gordon ($KG$) equation is highlighted to become a self-
consistent theory in $RQM$ based on two sets of wavefunctions ($WFs$) and
momentum-energy operators for particle and antiparticle respectively together
with a symmetry under the (newly defined) space-time inversion ${\cal P}{\cal
T}$ (${\bf x}\to-{\bf x},t\to-t$) or mass inversion ($m\to-m$) which is
inspired by the Feshbach-Villars dissociation of KG equation (1958, 25 (25)).
The above symmetry is further discussed and is identified with the ”strong
reflection” invariance invented by Lüders and Pauli in their proof of the
$CPT$ theorem (1954-1957, 31 (31, 32, 33)) in QFT as well as the new
definition (${\cal C}$) for particle-antiparticle transformation proposed by
Lee and Wu (1965, 21 (21)). All discussions are combined to become a renamed
symmetry for the $CPT$ invariance as ${\cal P}{\cal T}={\cal C}$. In section
IV, the Dirac equation is discussed. In section V, we discuss $QFT$. Section
VI is a brief summary and discussion. The Klein paradox is solved in the
Appendix for both $KG$ equation and Dirac equation without resorting to the
”hole theory”.
## II II. What the $K^{0}\bar{K}^{0}$ correlation experimental data are
telling?
To our knowledge, beginning from Bohm 3 (3) and Bell 4 (4), physicists
gradually turned their research of EPR paradox onto the entangled state
composed of electrons, especially photons with spin and achieved fruitful
results. However, as pointed out by Guan (1935-2007), EPR’s paper 1 (1) is
focused on two spinless particles and Guan found that there is a commutation
relation hiding in such a system as follows 5 (5):
Consider two particles in one dimensional space with positions
$x_{i}\,(i=1,2)$ and momentum operators
$\hat{p}_{i}=-i\hbar\frac{\partial}{\partial x_{i}}$. Then a commutation
relation arises as
$[x_{1}-x_{2},\hat{p}_{1}+\hat{p}_{2}]=0$ (2.1)
According to QM’s principle, there may be a kind of common eigenstate having
eigenvalues of these two commutative (i.e., compatible)observables like:
$p_{1}+p_{2}=0,\;(p_{2}=-p_{1})\quad\text{and}\quad(x_{1}-x_{2})=D$ (2.2)
with $D$ being their distance. The existence of such kind of eigenstate
described by Eq.(2.2) puzzled Guan, he asked: ”How can such kind of quantum
state be realized?” A discussion between Guan and one of present authors (Ni)
in 1998 led to a paper 6 (6).
Here we are going to discuss further, showing that the correlation experiment
on a $K^{0}\bar{K}^{0}$ system (which just realized an entangled state
composed of two spinless particles) in 1998 by CPLEAR collaboration 2 (2)
actually revealed some important features of QM and then answered the puzzle
raised by EPR in a surprising way. First, besides Eq.(2.1), let us consider
another three commutation relations simultaneously:
$[t_{1}+t_{2},\hat{E}_{1}-\hat{E}_{2}]=0$ (2.3)
$[x_{1}+x_{2},\hat{p}_{1}-\hat{p}_{2}]=0$ (2.4)
$[t_{1}-t_{2},\hat{E}_{1}+\hat{E}_{2}]=0$ (2.5)
($E_{i}=i\hbar\frac{\partial}{\partial t_{i}}$ with $t_{i}$ being the time
during which the i’th particle is detected). In accordance with Ref.2 (2), we
also focus on back-to-back events. The evolution of $K^{0}\bar{K}^{0}$’s
wavefunction (WF) will be considered in three inertial frames: The center-of-
mass system $S$ is at rest in laboratory with its origin $x=0$ located at the
apparatus’ center, where the antiprotons’ beam is stopped inside a hydrogen
gas target to create $K^{0}\bar{K}^{0}$ pairs by $p\bar{p}$ annihilation. The
$K^{0}\bar{K}^{0}$ pairs are detected by a cylindrical tracking detector
located inside a solenoid providing a magnetic field parallel to the
antiprotons’ beam. For back-to-back events, the space-time coordinates in
Eqs.(2.1)-(2.5) refer to particles moving to the right ($x_{1}>0$) and left
($x_{2}<0$) respectively. Second, we take an inertial system $S^{\prime}$ with
its origin located at particle 1 (i.e., $x^{\prime}_{1}=0$). $S^{\prime}$ is
moving in a uniform velocity $v$ with respect to $S$. (For Kaon’s momentum of
$800\,MeV/c,\;\beta=v/c=0.849$). Another $S^{\prime\prime}$ system is chosen
with its origin located at particle 2 ($x^{\prime\prime}_{2}=0$).
$S^{\prime\prime}$ is moving in a velocity ($-v$) with respect to $S$. Thus we
have Lorentz transformation among the space-time coordinates being
$\left\\{\begin{array}[]{ll}x^{\prime}=\dfrac{x-vt}{\sqrt{1-\beta^{2}}},&\\\\[14.22636pt]
t^{\prime}=\dfrac{t-vx/c^{2}}{\sqrt{1-\beta^{2}}},&\end{array}\right.\qquad\left\\{\begin{array}[]{ll}x^{\prime\prime}=\dfrac{x+vt}{\sqrt{1-\beta^{2}}},&\\\\[14.22636pt]
t^{\prime\prime}=\dfrac{t+vx/c^{2}}{\sqrt{1-\beta^{2}}},&\end{array}\right.$
(2.6)
Here $t^{\prime}_{1}$ and $t^{\prime\prime}_{2}$ correspond to the proper time
$t_{a}$ and $t_{b}$ in Ref.2 (2) respectively. The common time origin
$t=t^{\prime}=t^{\prime\prime}=0$ is adopted.
A $K^{0}\bar{K}^{0}$ pair, created in a $J^{PC}=1^{--}$ antisymmetric state,
can be described by a two-body WF depending on time as (2 (2), see also 7 (7,
8))
$\begin{array}[]{l}|\Psi(0,0)\rangle^{(antisym)}=\dfrac{1}{\sqrt{2}}\left[|K^{0}(0)\rangle_{a}|\bar{K}^{0}(0)\rangle_{b}-|\bar{K}^{0}(0)\rangle_{a}|K^{0}(0)\rangle_{b}\right]\\\\[14.22636pt]
|\Psi(t_{a},t_{b})\rangle^{(antisym)}=\dfrac{1}{\sqrt{2}}\left[|K_{S}(0)\rangle_{a}|K_{L}(0)\rangle_{b}e^{-i(\alpha_{S}t_{a}+\alpha_{L}t_{b})}-|K_{L}(0)\rangle_{a}|K_{S}(0)\rangle_{b}e^{-i(\alpha_{L}t_{a}+\alpha_{S}t_{b})}\right]\end{array}$
(2.7)
with
$|K_{S}\rangle=\dfrac{1}{\sqrt{2}}[|K^{0}\rangle-|\bar{K}^{0}\rangle],\;|K_{L}\rangle=\dfrac{1}{\sqrt{2}}[|K^{0}\rangle+|\bar{K}^{0}\rangle]$
(2.8)
where the CP violation has been neglected and
$\alpha_{S,L}=m_{S,L}-i\gamma_{S,L}/2$, $m_{S,L}$ and $\gamma_{S,L}$ being the
$K_{S,L}$ masses and decay widths, respectively. From Eq.(2.7), the
intensities of events with like-strangeness ($K^{0}K^{0}$ or
$\bar{K}^{0}\bar{K}^{0}$) and unlike-strangeness ($K^{0}\bar{K}^{0}$ or
$\bar{K}^{0}K^{0}$) can be evaluated as
$I_{like}^{(antisy)}(t_{a},t_{b})=\dfrac{1}{8}e^{-2\gamma\tilde{t}}\left\\{e^{-\gamma_{S}|t_{a}-t_{b}|}+e^{-\gamma_{L}|t_{a}-t_{b}|}-2e^{-\gamma|t_{a}-t_{b}|}\cos[\Delta
m(t_{a}-t_{b})]\right\\}$ (2.9)
$I_{unlike}^{(antisy)}(t_{a},t_{b})=\dfrac{1}{8}e^{-2\gamma\tilde{t}}\left\\{e^{-\gamma_{S}|t_{a}-t_{b}|}+e^{-\gamma_{L}|t_{a}-t_{b}|}+2e^{-\gamma|t_{a}-t_{b}|}\cos[\Delta
m(t_{a}-t_{b})]\right\\}$ (2.10)
where $\Delta m=m_{L}-m_{S},\;\gamma=(\gamma_{S}+\gamma_{L})/2$ and
$\tilde{t}=t_{a}\,(\text{for}\;t_{a}<t_{b})$ or
$\tilde{t}=t_{b}\,(\text{for}\;t_{a}>t_{b})$.
Similarly, for $K^{0}\bar{K}^{0}$ created in a $J^{PC}=0^{++}$ or $2^{++}$
symmetric state as:
$\begin{array}[]{l}|\Psi(0,0)\rangle^{(sym)}=\dfrac{1}{\sqrt{2}}\left[|K^{0}(0)\rangle_{a}|\bar{K}^{0}(0)\rangle_{b}+|\bar{K}^{0}(0)\rangle_{a}|K^{0}(0)\rangle_{b}\right]\\\\[14.22636pt]
|\Psi(t_{a},t_{b})\rangle^{(sym)}=\dfrac{1}{\sqrt{2}}\left[|K_{L}(0)\rangle_{a}|K_{L}(0)\rangle_{b}e^{-i(\alpha_{L}t_{a}+\alpha_{L}t_{b})}-|K_{S}(0)\rangle_{a}|K_{S}(0)\rangle_{b}e^{-i(\alpha_{S}t_{a}+\alpha_{S}t_{b})}\right]\end{array}$
(2.11)
the predicted intensities read
$\begin{array}[]{l}I_{like}^{(sym)}(t_{a},t_{b})=\dfrac{1}{8}\left\\{e^{-\gamma_{S}(t_{a}+t_{b})}+e^{-\gamma_{L}(t_{a}+t_{b})}-2e^{-\gamma(t_{a}+t_{b})}\cos[\Delta
m(t_{a}+t_{b})]\right\\}\\\\[14.22636pt]
I_{unlike}^{(sym)}(t_{a},t_{b})=\dfrac{1}{8}\left\\{e^{-\gamma_{S}(t_{a}+t_{b})}+e^{-\gamma_{L}(t_{a}+t_{b})}+2e^{-\gamma(t_{a}+t_{b})}\cos[\Delta
m(t_{a}+t_{b})]\right\\}\end{array}$ (2.12)
The experiment 2 (2) reveals that the $K^{0}\bar{K}^{0}$ pairs are mainly
created in the antisymmetric state shown by Eqs.(2.9)-(2.10) while the
contribution in a symmetric state shown by Eqs.(2.11)-(2.12) accounts for
$7.4\%$.
What we learn from Ref.2 (2) in combination with Eqs.(2.1)-(2.5) are as
follows:
(a) Because only back-to-back events are involved in the $S$ system, we denote
three commutative operators as: the ”distance” operator
$\hat{D}=x_{1}-x_{2}=v(t_{1}+t_{2})$, $\hat{A}=\hat{p}_{1}+\hat{p}_{2}$ and
$\hat{B}=\hat{E}_{1}-\hat{E}_{2}$, Eqs.(2.1) and (2.3) read
$[\hat{D},\hat{A}]=0,\;[\hat{D},\hat{B}]=0,\;[\hat{A},\hat{B}]=0$ (2.13)
So they may have a kind of common eigenstate during the measurement composed
of $K^{0}K^{0}$ and projected from the symmetric state shown by Eq.(2.11). It
is assigned by a continuous eigenvalue $D_{j}=v(t_{1}+t_{2})$ (with continuous
index $j$) of operator $\hat{D}$ acting on the WF,
$\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$, as111The WF reads
approximately as: $\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})\sim
e^{i(p_{1}x_{1}-E_{1}t_{1})}e^{i(p_{2}x_{2}-E_{2}t_{2})}$ $None$ which can be
calculated from $\langle K^{0}K^{0}|\Psi(t_{a},t_{b})\rangle^{sym}$ with two
terms. The squares of WF’s amplitude reproduces the
$I_{like}^{(sym)}(t_{a},t_{b})$ in Eq.(2.12).
$\hat{D}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=D_{j}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=v(t_{1}+t_{2})\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$
$None$
$\hat{A}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=A^{like}_{j}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=(p_{1}+p_{2})\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$
(2.15)
$\hat{B}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=B^{like}_{j}\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})=(E_{1}-E_{2})\Psi^{sym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$
(2.16)
where the lowest eigenvalue of $\hat{A}$ is
$A^{like}_{j}=p_{1}+p_{2}=0,\,(p_{2}=-p_{1})$, and that of $\hat{B}$ is
$B^{like}_{j}=E_{1}-E_{2}=0,\,(E_{2}=E_{1})$ respectively. These eigenstates
of like-strangeness events predicted by Eq.(2.11) are really observed in the
experiment 2 (2) (these eigenstates of $K^{0}K^{0}$ were overlooked in the
Ref.6 (6)).
(b) The more interesting case occurs for $K^{0}\bar{K}^{0}$ pair created in
the antisymmetric state with intensity given by Eq.(2.10) being a function of
$(t_{a}-t_{b})$ (not $(t_{a}+t_{b})$ as shown by Eq.(12) for symmetric states)
which is proportional to $(t_{1}-t_{2})$ in the $S$ system. In the EPR limit
$t_{1}=t_{2}$, $K^{0}\bar{K}^{0}$ events dominate whereas like-strangeness
events are strongly suppressed as shown by Eq.(2.9) (see Fig.1 in 2 (2)). So
the experimental facts remind us of the possibility that $K^{0}\bar{K}^{0}$
events may be related to common lowest (zero) eigenvalues of some commutative
operators (just like what happened in Eqs.(2.15) and (2.16) for operators
$\hat{A}$ and $\hat{B}$ (which are applied to symmetric states (due to
$\hat{D}=x_{1}-x_{2}=v(t_{1}+t_{2})$) but are not suitable for antisymmetric
states), there are another three operators shown by Eqs.(2.4) and (2.5) being:
the operator of ”flight-path difference” $\hat{F}=x_{1}+x_{2}=v(t_{1}-t_{2})$,
$\hat{M}=\hat{p}_{1}-\hat{p}_{2}$ and $\hat{G}=\hat{E}_{1}+\hat{E}_{2}$ with
commutation relations as:
$[\hat{F},\hat{M}]=0,\;[\hat{F},\hat{G}]=0,\;[\hat{M},\hat{G}]=0$ (2.17)
which are just suitable for antisymmetric states. For $K^{0}\bar{K}^{0}$ back-
to-back events, assume that one of two particles, say 2, is an antiparticle
with its momentum and energy operators being
$\hat{p}_{x}^{c}=i\hbar\dfrac{\partial}{\partial
x},\;\hat{E}^{c}=-i\hbar\dfrac{\partial}{\partial t}$ (2.18)
(the superscript $c$ means ”antiparticle”) just opposite in the sign to that
for a particle. For instance, a freely moving particle’s WF reads111Please see
the derivation of Eqs.(2.19) and (2.20) from the quantum field theory (QFT) at
Eqs.(5-7)-(5-9).:
$\psi(x,t)\sim\exp\left[\frac{i}{\hbar}(px-Et)\right]$ (2.19)
whereas
$\psi_{c}(x,t)\sim\exp\left[-\frac{i}{\hbar}(p_{c}x-E_{c}t)\right]$ (2.20)
for its antiparticle with $p_{c}$ and $E_{c}\,(>0)$ being momentum and energy
of the antiparticle in accordance with Eq.(2.18). If using Eqs.(2.18)-(2.20),
we find
$\hat{F}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=F^{unlike}_{k}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=v(t_{1}-t_{2})\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$
(2.21)
with continuous index $k$ referring to continuous eigenvalues
$F_{k}=v(t_{1}-t_{2})$. Here, the WF in space-time of this system during
measurement reads approximately:
$\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})\sim
e^{i(p_{1}x_{1}-E_{1}t_{1})}e^{-i(p_{2}^{c}x_{2}-E_{2}^{c}t_{2})}$ (2.22)
with antiparticle 2 moving opposite to particle 1 and $p_{2}^{c}=-p_{1}$.
Now we use $\hat{M}(=\hat{p}_{1}-\hat{p}_{2})=\hat{p}_{1}+\hat{p}^{c}_{2}$ on
$K^{0}\bar{K}^{0}$ system, yielding
$\hat{M}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=M^{unlike}_{k}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=(p_{1}+p^{c}_{2})\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$
(2.23)
Similarly, we have
$\hat{G}(=\hat{E}_{1}+\hat{E}_{2})=\hat{E}_{1}-\hat{E}^{c}_{2}$ and find
$\hat{G}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=G^{unlike}_{k}\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})=(E_{1}-E^{c}_{2})\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$
(2.24)
Hence we see that once Eqs.(2.18) and (2.20) are accepted, the WFs
$\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$ show up in
experiments as the only WFs with strongest intensity at the EPR limit
($t_{1}=t_{2}$) corresponding to their three eigenvalues being all zero:
$F_{k}=M^{unlike}_{k}=G^{unlike}_{k}=0$ and they won’t change even when
accelerator’s energies are going up.
If using Eq.(2.18), the eigenvalues of $\hat{A}$ and $\hat{B}$ for the WF
$\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$ are
$A^{unlike}_{j}=p_{1}-p^{c}_{2}=2p_{1}$ and
$B^{unlike}_{j}=E_{1}+E^{c}_{2}=2E_{1}$ respectively, while that of $\hat{M}$
and $\hat{G}$ for the WF
$\Psi^{antisym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$ are
$M^{like}_{k}=p_{1}-p_{2}=2p_{1}$ and $G^{like}_{k}=E_{1}+E_{2}=2E_{1}$,
respectively, those eigenvalues are much higher than zero and going up with
the accelerator’s energy.
Something is very interesting here: If we deny Eq.(2.18) but insist on unified
operators $\hat{p}$ and $\hat{E}$ for both particle and antiparticle, there
would be no difference in eigenvalues between like-strangeness events and
unlike-strangeness ones. For example, the $M^{unlike}_{k}$ and
$G^{unlike}_{k}$ would be $2p_{1}$ and $2E_{1}$ too (instead of ”0” as in
Eqs.(2.23) and (3.26)). This would mean that three commutative operators
$\hat{F},\hat{M}$ and $\hat{G}$ are not enough to distinguish the WF
$\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$ from the WF
$\Psi^{antisym}_{K^{0}K^{0}}(x_{1},t_{1};x_{2},t_{2})$ even they behave so
differently as shown by Eqs.(2.9) and (2.10)), especially at the EPR limit
($t_{1}=t_{2}$).
Eq.(2.18) together with the identification of WF
$\Psi^{antisym}_{K^{0}\bar{K}^{0}}(x_{1},t_{1};x_{2},t_{2})$ by three zero
eigenvalues implies that the difference of a particle from its antiparticle is
not something hiding in the ”intrinsic space” like opposite charge (for
electron and positron) or opposite strangeness (for $K^{0}$ and $\bar{K}^{0}$)
but can be displayed in their WFs evolving in space-time at the level of QM.
To our knowledge, Eq.(2.18) can be found at a page note of a paper by
Konopinski and Mahmaud in 1953 9 (9), also appears in Refs.6 (6, 10, 11, 12,
13). 111Eqs.(2.7)-(2.12), the WFs and predicted intensities for
$K^{0}-\bar{K}^{0}$ system, are derived from QFT (see Appendix I in Ref1 (1)).
On the other hand, our discussions about commutation relations,
Eqs.(2.1)-(2.5), and their eigenvalues of measurements, Eqs.(2.13)-(2.24), are
at the level of RQM, where two particles’ WFs with their space-time
coordinates are taken into account. Here we use Eqs.(2.7)-(2.12) only to
divide commutative operators into two groups, one for symmetric states, one
for antisymmetric states. So the EPR problem is discussed at the level of RQM.
## III III. How to Make Klein-Gordon Equation a Self-Consistent Theory in RQM
?
Let us begin with the energy conservation law for a particle in classical
mechanics:
$E=\frac{1}{2m}{\bf p}^{2}+V({\bf x})$ (3.1)
Consider the rule promoting observables into operators:
$E\to\hat{E}=i\hbar\frac{\partial}{\partial t},\quad{\bf p}\to\hat{\bf
p}=-i\hbar\nabla$ (3.2)
and let Eq.(3.1) act on a wavefunction (WF) $\psi({\bf x},t)$, the Schrödinger
equation
$i\hbar\frac{\partial}{\partial t}\psi({\bf
x},t)=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi({\bf x},t)+V({\bf x})\psi({\bf
x},t)$ (3.3)
follows immediately. In mid 1920’s, considering the kinematical relation for a
particle in the theory of special relativity (SR):
$(E-V)^{2}=c^{2}{\bf p}^{2}+m^{2}c^{4}$ (3.4)
and using Eq.(3.2) again, the Klein-Gordon (KG) equation was established as:
$(i\hbar\frac{\partial}{\partial t}-V)^{2}\psi({\bf
x},t)=-c^{2}\hbar^{2}\nabla^{2}\psi({\bf x},t)+m^{2}c^{4}\psi({\bf x},t)$
(3.5)
For a free KG particle, its plane-wave solution reads:
$\psi({\bf x},t)\sim\exp[\frac{i}{\hbar}({\bf p}\cdot{\bf x}-Et)]$ (3.6)
However, two difficulties arose:
(a) The energy $E$ in Eq.(3.6) has two eigenvalues:
$E=\pm\sqrt{c^{2}{\bf p}^{2}+m^{2}c^{4}}$ (3.7)
What the ”negative energy” means?
(b)The continuity equation is derived from Eq.(3.5) as
$\frac{\partial\rho}{\partial t}+\nabla\cdot{\bf j}=0$ (3.8)
where
$\rho=\frac{i\hbar}{2mc^{2}}(\psi^{*}\frac{\partial}{\partial
t}\psi-\psi\frac{\partial}{\partial t}\psi^{*})-\frac{1}{mc^{2}}V\psi^{*}\psi$
(3.9)
and
${\bf j}=\frac{i\hbar}{2m}(\psi\nabla\psi^{*}-\psi^{*}\nabla\psi)$ (3.10)
are the ”probability density” and ”probability current density” respectively.
While the latter is the same as that derived from Eq.(3.3), Eq.(3.9) seems not
positive definite and dramatically different from $\rho=\psi^{*}\psi$ in
Eq.(3.3).
In 1958, Feshbach and Villars 25 (25) recast Eq.(3.5) into two coupled
Schrödinger-like equations as:111Interestingly, if ignoring the coupling
between $\phi$ and $\chi$ and $V=0$ in Eq.(3.11), they satisfy respectively
the ”two equations” written down by Schrödinger in his 6th paper in 1926
(Annaten der Physik (4) V.81, 1926, p.104)when he invented NRQM in the form of
wave mechanics.
$\left\\{\begin{array}[]{ll}\left(i\hbar\dfrac{\partial}{\partial
t}-V\right)\phi=mc^{2}\phi-\dfrac{\hbar^{2}}{2m}\nabla^{2}(\phi+\chi)\\\\[11.38109pt]
\left(i\hbar\dfrac{\partial}{\partial
t}-V\right)\chi=-mc^{2}\chi+\dfrac{\hbar^{2}}{2m}\nabla^{2}(\phi+\chi)\end{array}\right.$
(3.11)
where
$\left\\{\begin{array}[]{ll}\phi=\dfrac{1}{2}\left[\left(1-\dfrac{1}{mc^{2}}V\right)\psi+\dfrac{i\hbar}{mc^{2}}\dot{\psi}\right]\\\\[11.38109pt]
\chi=\dfrac{1}{2}\left[\left(1+\dfrac{1}{mc^{2}}V\right)\psi-\dfrac{i\hbar}{mc^{2}}\dot{\psi}\right]\end{array}\right.$
(3.12)
($\dot{\psi}=\frac{\partial\psi}{\partial t}$). Interestingly, while
$\psi=\phi+\chi$, the ”probability density”, Eq.(3.9) can be recast into a
difference between two positive-definite densities 6 (6, 8):
$\rho=\phi^{*}\phi-\chi^{*}\chi$ (3.13)
For simplicity, consider a free KG particle ($V=0$) with WF Eq.(3.6). Then
$\left\\{\begin{array}[]{ll}\phi=\dfrac{1}{2}\left(1+\dfrac{E}{mc^{2}}\right)\psi&\\\
\chi=\dfrac{1}{2}\left(1-\dfrac{E}{mc^{2}}\right)\psi&\end{array}\right.$
(3.14)
So $|\phi|>|\chi|$ and $\rho>0$ for $E>0$ case. But for a negative-energy
($E<0$) particle, we would have $\rho_{E<0}=|\phi|^{2}-|\chi|^{2}<0$. Thus we
see that the difficulty of negative probability density is intimately related
to that of negative-energy. The later difficulty is actually solved in the
last section by regarding the negative-energy WF of a particle directly as the
positive-energy WF of its antiparticle and introducing operators as
$\hat{E}_{c}=-i\hbar\dfrac{\partial}{\partial t},\quad\hat{\bf
p}_{c}=i\hbar\nabla$ (3.15)
when these two operators act on the antiparticle’s WF
$\psi_{c}({\bf x},t)\sim\exp\left[-\dfrac{i}{\hbar}({\bf p}_{c}\cdot{\bf
x}-E_{c}t)\right],\quad(E_{c}>0)$ (3.16)
The antiparticle’s energy $E_{c}(>0)$ and momentum ${\bf p}_{c}$ will be
picked up. If substituting $\psi_{c}$, Eq.(3.16), into Eq.(3.12) to replace
$\psi$, we obtain (after adding subscript ”c” for antiparticle,
$\psi_{c}=\phi_{c}+\chi_{c}$ and $V=0$ again)
$\left\\{\begin{array}[]{ll}\phi_{c}=\dfrac{1}{2}\left(1-\dfrac{E_{c}}{mc^{2}}\right)\psi_{c}&\\\
\chi_{c}=\dfrac{1}{2}\left(1+\dfrac{E_{c}}{mc^{2}}\right)\psi_{c}&\end{array}\right.$
(3.17)
Now $|\phi_{c}|<|\chi_{c}|$. To set a positive-definite probability density
$\rho_{c}$ for describing the antiparticle, we need
$\rho_{c}=|\chi_{c}|^{2}-|\phi_{c}|^{2}>0$ (3.18)
instead of Eq.(3.13) for particle. However, if we directly add Eq.(3.18) for
antiparticle, this would not be a good theory. So we should have a unified
prescription to get everything right for KG equation. Inspecting Eq.(3.11)
carefully, we do find that it is invariant under a (newly defined) space-time
inversion (${\bf x}\to-{\bf x},t\to-t$), i.e., there is a hidden symmetry as
follows:
$\left\\{\begin{array}[]{l}V({\bf x},t)\to-V({\bf x},t)=V_{c}({\bf x},t),\\\
\psi({\bf x},t)\to\psi(-{\bf x},-t)=\psi_{c}({\bf x},t),\\\ \phi({\bf
x},t)\to\phi(-{\bf x},-t)=\chi_{c}({\bf x},t),\\\ \chi({\bf x},t)\to\chi(-{\bf
x},-t)=\phi_{c}({\bf x},t)\end{array}\right.$ (3.19)
For example, space-time inversion will bring Eq.(3.2) for particle into
Eq.(3.15) for antiparticle. Meanwhile, performing Eq.(3.19) on Eq.(3.12), we
find $\chi_{c}(\phi_{c})$ satisfying the same equation of $\chi(\phi)$ ,
Eq.(3.11), and they read:
$\left\\{\begin{array}[]{l}\chi_{c}=\dfrac{1}{2}\left[\left(1+\dfrac{1}{mc^{2}}V\right)\psi_{c}-\dfrac{i\hbar}{mc^{2}}\dot{\psi}_{c}\right]\\\\[11.38109pt]
\phi_{c}=\dfrac{1}{2}\left[\left(1-\dfrac{1}{mc^{2}}V\right)\psi_{c}+\dfrac{i\hbar}{mc^{2}}\dot{\psi}_{c}\right]\end{array}\right.$
(3.20)
which is in conformity with the transformation of $\rho$, Eq.(3.9), as
expected:
$\rho\to\rho_{c}=\frac{i\hbar}{2mc^{2}}(\psi_{c}\dot{\psi}_{c}^{*}-\psi_{c}^{*}\dot{\psi}_{c})+\frac{1}{mc^{2}}V\psi_{c}^{*}\psi_{c}=\chi_{c}^{*}\chi_{c}-\phi_{c}^{*}\phi_{c}>0$
(3.21)
Similarly, we have
${\bf j}\to{\bf
j}_{c}=\frac{i\hbar}{2m}(\psi^{*}_{c}\nabla{\psi}_{c}-\psi_{c}\nabla\psi^{*}_{c})$
(3.22)
which, for $V=0$ case, means
${\bf j}=\dfrac{{\bf p}}{m}|\psi|^{2}\to{\bf j}_{c}=\dfrac{{\bf
p}_{c}}{m}|\psi_{c}|^{2},\;\;(V=0)$ (3.23)
with ${\bf j}_{c}$ along the direction of ${\bf p}_{c}$ as expected. Eq.(3.8)
remains valid after the above transformation too. Thus we see that both the
”probability density” $\rho$ for a particle and $\rho_{c}$ for an antiparticle
are positive definite before they can be normalized as expected:
$\int\rho d^{3}x=\int\rho_{c}d^{3}x=1$ (3.24)
Hence we see that the space-time inversion Eq.(3.19) reflects the underlying
symmetry between particle and antiparticle and overcomes two difficulties of
KG equation simultaneously in an elegant way.
Here, we would like to introduce a ”mass inversion” which can realize the same
symmetry of Eq.(3.18)as follows:
$\left\\{\begin{array}[]{l}m\to-m\\\ V({\bf x},t)\to V({\bf x},t),\\\
\psi({\bf x},t)\to\psi_{c}({\bf x},t),\\\ \phi({\bf x},t)\to\chi_{c}({\bf
x},t),\\\ \chi({\bf x},t)\to\phi_{c}({\bf x},t)\end{array}\right.$ (3.25)
Notice that, when $m\to-m$, we have $p\to-p_{c}$ and $E\to-E_{c}$ for a free
particle transforming into its antiparticle because the momentum and energy in
their WFs are proportional to the mass in SR. 111Here $m$ always refers to the
”rest mass” for a particle or its antiparticle, see the excellent paper by
Okun in Ref.23 (23).
The reason why $V\to-V$ in the space-time inversion Eq.(3.19) whereas $V\to V$
in the mass inversion Eq.(3.25) can be seen from the classical equation: The
Lorentz force ${\bf F}$ on a particle exerted by an external potential $\Phi$
reads: ${\bf F}=-\nabla V=-\nabla(q\Phi)=m{\bf a}$. As the acceleration ${\bf
a}$ of particle will change to $-{\bf a}$ for its antiparticle, there are two
alternative explanations: either due to the inversion of charge $q\to-q$
(i.e., $V\to-V$ but keeping $m$ unchanged) or due to the inversion of mass
$m\to-m$ (but keeping $V$ unchanged).
The reason why Feshbach-Villars’ dissociation of KG equation, Eq.(3.11), is so
important is because they unveiled a new point of view for us to see a
particle as follows:
For a free KG particle (say, $K^{-}$ meson) moving at a high speed $v$, its WF
$\psi\sim e^{-iEt}\,(E>0)$ is always composed of two fields, $\phi$ and
$\chi$, in confrontation as shown by Eqs.(3.11) and (3.12). Calculations (as
can be seen from Eq.(3.14)) show that: as long as $E>0$, then $|\phi|>|\chi|$
and $\rho>0$, so $\phi$ dominates $\chi$ and $K^{-}$ remains as a particle.
However, the amplitude of $\chi$ increases with the increase of particle’s
energy $E$: when $v\to 0,E\to m,|\chi|\to 0$, but when
$E\to\infty,|\chi|\to|\phi|$, the ratio between them reads:
$|\chi|/|\phi|=[1-(1-v^{2}/c^{2})^{1/2}]/[1+(1-v^{2}/c^{2})^{1/2}]$. What does
this mean? It seems to us that while $\phi$ (hidden in $\psi$) characterizes
the particle’s property, $\chi$ represents the hidden ”antiparticle (say,
$K^{+}$) field” in the WF of this $K^{-}$ particle. Indeed, in Eqs.(3.6) and
(3.16), the WFs of particle (dominated by $\phi$) and antiparticle (dominated
by $\chi_{c}$), their phase variations with respect to space-time (keeping the
same values of momentum and energy) are just in opposite directions, meaning
that the intrinsic tendencies of space-time evolution of $\phi$ and $\chi$ are
also in opposite directions essentially (see Eq.(3.11) with $V=0$). Hence,
even the $\chi$ is in a subordinate position in a particle, it still strives
to display itself as follows: On one hand, $\chi$ holds $\phi$ back from going
forward in space, so the particle’s velocity $v$ has an upper limit value $c$
when $|\chi|$ approaches $|\phi|$. And during the ”boosting” process of a
particle’s wave-packet, it shows the Lorentz contraction due to the
entanglement between $\phi$ and $\chi$ (see calculation shown in Fig.9.5.1 of
Ref.11 (11)). On the other hand, a clock attached to the particle will show
the time dilatation effect in SR with the increase of velocity $v$. This is
because, in some sense, the ”intrinsic clock” of $\phi(\chi)$ is running
clockwise (anticlockwise). With the enhancement of $\chi$, the particle’s
clock, though still runs clockwise, tends to stop. Therefore, it seems to us
that all SR effects of a particle could be calculated and understood by the
existence and enhancement of ”hidden antiparticle field” $\chi$ inside (for
detail, please see 11 (11)).
Now we are going to prove that in RQM the antiparticle’s WF $\psi_{c}$
obtained from the (newly defined) space-time inversion is coinciding with the
CPT transformation of particle’s WF $\psi$, but not that from $C\psi$.
As is well known, these three discrete transformations of $C,P$ and $T$ in RQM
were defined for spinless particles separately as follows:
(a) Space-inversion ($P$):
The sign change of space coordinates (${\bf x}\to-{\bf x}$) in the wave
function ($WF$) of $QM$ may lead to two eigenstates:
$\psi_{\pm}({\bf x},t)\to P\psi_{\pm}({\bf x},t)=\psi_{\pm}(-{\bf
x},t)=\pm\psi_{\pm}({\bf x},t)$ (3.26)
with eigenvalues $1$ or $-1$ being the even or odd parity.
(b) Time reversal ($T$):
The so-called $T$ transformation is actually not a ”time reversal” but a
”reversal of motion” 11 (11, 16), which implies an antiunitary operator acting
on the WF:
$\psi({\bf x},t)\to T\psi({\bf x},t)=\psi^{*}({\bf x},-t)$ (3.27)
(c) Charge conjugation transformation($C$):
The $C$ transformation brings a particle (with charge $q$) into its
antiparticle (with charge $-q$) and implies a complex conjugation on the $WF$:
$\psi({\bf x},t)\to C\psi({\bf x},t)=\psi^{*}({\bf x},t)$ (3.28)
Note that the $WF$ $\psi^{*}$ implies a negative-energy particle. Usually, one
has to resort to so-called ”hole theory” for electron — the vacuum is fully
filled with infinite negative-energy electrons and a ”hole” created in the
”sea” would correspond to a positron15 (15, 17). But how could the ”hole
theory” be applied to the boson particle? No one knows.
Fortunately, at the level of RQM, if one performs the product operator CPT on
a particle’s WF $\psi({\bf x},t)$, two operations of complex conjugation in
the C and T will cancel each other, yielding 7 (7, 17)
$\psi({\bf x},t)\to CPT\psi({\bf x},t)=\psi_{CPT}({\bf x},t)=\psi(-{\bf
x},-t)$ (3.29)
On the right-side-hand (RHS), the $\psi_{CPT}({\bf x},t)$ is just the
antiparticle’s WF obtained from the (newly defined) space-time inversion
(${\bf x}\to-{\bf x},t\to-t$), Eq.(3.16).
One may ask: Is the $C\psi=\psi^{*}$ in Eq.(3.28) gives the same space-time
evolution behavior of EQ.(3.29) or Eq.(3.16)? The answer is ”yes” and ”no”. We
say ”yes” because it seems formally correct but ”no” because it is essentially
incorrect — given one set of operators like Eq.(3.2) for both $\psi$ and
$C\psi=\psi^{*}$, then the $C\psi$ is a negative energy WF, which cannot be
accepted in physics. Moreover, as we will see in the next section, for Dirac
equation, while the particle’s WF $\psi({\bf x},t)$ and its antiparticle’s
$\psi_{c}({\bf x},t)=\psi_{CPT}({\bf x},t)\sim\psi(-{\bf x},-t)$ describes the
same energy and momentum, they must have opposite helicities. Given the
definition of C, $C\psi=\psi^{*}$ is bound to fail in this aspect too.
Above discussions are strictly at the level of RQM. We have shown that the KG
equation is really a self-sufficient and simplest model in RQM as long as the
(newly defined) space-time inversion is introduced. As we will continue to do
so in next sections, let us write down it in a compact equation as a renamed
CPT invariance:
${\cal P}{\cal T}={\cal C}$ (3.30)
Here, ${\cal P}{\cal T}$ represent (${\bf x}\to-{\bf x},t\to-t$), i.e., the
”strong reflection” invented by Lüders and Pauli (1954-1957) in their proof of
CPT theorem at the level of QFT, and ${\cal C}$, the newly defined particle-
antiparticle transformation operator, whose definition is precisely contained
in Eq.(3.30).
In 1965, Lee and Wu proposed that21 (21):
$|\bar{a}\rangle=CPT|a\rangle$ (3.31)
where $|a\rangle$ and $|\bar{a}\rangle$ represent a particle and its
antiparticle. To our understanding, the physical essence of Eq. (3.31) is just
that claimed by Eq. (3.30). What we add in Eq.(3.30) here is: It can be
derived at the level of RQM much easier than that at the level of QFT, as long
as we admit that ”performing the operation of complex conjugation twice means
doing nothing at all”, so what left after CPT product operation in RQM is
merely the space-time inversion operation, ${\bf x}\to-{\bf x},t\to-t$. And we
should accept Eq.(3.15) for antiparticle (so $\psi(-{\bf x},-t)=\psi_{c}({\bf
x},t)$) and the underlying symmetry Eq.(3.19) to reach Eq.(3.30). The ${\cal
C}$ operator in Eq.(3.30) shows that a particle’s ”intrinsic property” is
intimately linked to the space-time.
Based on the ideas and method in this section, the Klein paradox for KG
equation can be solved as discussed in the Appendix A.
## IV IV. Dirac Equation as Coupled equations of two-component Spinors
Let us turn to the Dirac equation describing an electron
$\left(i\hbar\dfrac{\partial}{\partial t}-V\right)\psi=H\psi=(-i\hbar
c{\boldsymbol{\alpha}}\cdot\nabla+\beta mc^{2})\psi$ (4.1)
with ${\boldsymbol{\alpha}}$ and $\beta$ being $4\times 4$ matrices, the WF
$\psi$ is a four-component spinor
$\psi=\begin{pmatrix}\phi\\\ \chi\end{pmatrix}$ (4.2)
Usually, the two-component spinors $\phi$ and $\chi$ are called ”positive” and
”negative” energy components. In our point of view, they are the hiding
”particle” and ”antiparticle” fields in a particle (electron) respectively (11
(11), see below). Substitution of Eq.(4.2) into Eq.(4.1) leads to
$\left\\{\begin{array}[]{l}\left(i\hbar\dfrac{\partial}{\partial
t}-V\right)\phi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\chi+mc^{2}\phi\\\\[8.53581pt]
\left(i\hbar\dfrac{\partial}{\partial t}-V\right)\chi=-i\hbar
c{\boldsymbol{\sigma}}\cdot\nabla\phi-mc^{2}\chi\end{array}\right.$ (4.3)
(${\boldsymbol{\sigma}}$ are Pauli matrices). Eq.(4.3) is invariant under the
space-time inversion $({\bf x}\to-{\bf x},t\to-t)$ with
$\left\\{\begin{array}[]{l}\phi({\bf x},t)\to\phi(-{\bf x},-t)=\chi_{c}({\bf
x},t),\;\chi({\bf x},t)\to\chi(-{\bf x},-t)=\phi_{c}({\bf x},t)\\\ V({\bf
x},t)\to-V({\bf x},t)=V_{c}({\bf x},t)\end{array}\right.$ (4.4)
again, showing that Dirac equation is in conformity with the underlying
symmetry Eq.(3.30). Note that under the space-time inversion, the
${\boldsymbol{\sigma}}$ remain unchanged (However, see Eqs.(4.9)-(4.11)
below). Alternatively, Eq.(4.3) also remains invariant under a mass inversion
as
$m\to-m,\;\phi({\bf x},t)\to\chi_{c}({\bf x},t),\;\chi({\bf
x},t)\to\phi_{c}({\bf x},t),\;V\to V$ (4.5)
In either case of Eq.(4.4) or (4.5), we have222The reason why we use
$\psi^{\prime}_{c}$ instead of $\psi_{c}$ will be clear in Eqs.(4.12)-(4.15).
Actually, we emphasize Dirac equation as a coupling equation of two two-
component spinors, Eq.(4.3), rather than merely a four-component spinor
equation.
$\psi({\bf x},t)=\begin{pmatrix}\phi({\bf x},t)\\\ \chi({\bf
x},t)\end{pmatrix}\to\begin{pmatrix}\chi_{c}({\bf x},t)\\\ \phi_{c}({\bf
x},t)\end{pmatrix}=\psi^{\prime}_{c}({\bf x},t)$ (4.6)
For concreteness, we consider a free electron moving along the $z$ axis with
momentum $p=p_{z}>0$ and having a helicity $h={\boldsymbol{\sigma}}\cdot{\bf
p}/|{\bf p}|=1$, its WF reads:
$\psi(z,t)\sim\begin{pmatrix}\phi\\\ \chi\end{pmatrix}\sim\begin{pmatrix}1\\\
0\\\ \frac{p}{E+m}\\\ 0\end{pmatrix}\exp[i(pz-Et)]$ (4.7)
with $|\phi|>|\chi|$. Under a space-time inversion ($z\to-z,t\to-t,p\to
p_{c},E\to E_{c}$) or mass inversion ($m\to-m,p\to-p_{c},E\to-E_{c}$), it
transforms into a WF for positron (moving along $z$ axis)
$\psi^{\prime}_{c}(z,t)\sim\begin{pmatrix}\chi_{c}\\\
\phi_{c}\end{pmatrix}\sim\begin{pmatrix}1\\\ 0\\\ \frac{p_{c}}{E_{c}+m}\\\
0\end{pmatrix}\exp[-i(p_{c}z-E_{c}t)]$ (4.8)
with $|\chi_{c}|>|\phi_{c}|,\;(p_{c}>0,E_{c}>0)$. However, the positron’s
helicity becomes $h_{c}={\boldsymbol{\sigma}}_{c}\cdot{\bf p}_{c}/|{\bf
p}_{c}|=-1$. This is because the total angular momentum operator for an
electron reads
$\hat{\bf J}=\hat{\bf L}+\frac{\hbar}{2}{\boldsymbol{\sigma}}$ (4.9)
Under a space-time inversion, the orbital angular momentum operator transforms
as
$\hat{\bf L}={\bf r}\times\hat{\bf p}={\bf r}\times(-i\hbar\nabla)\to{-\bf
r}\times(i\hbar\nabla)=-{\bf r}\times\hat{\bf p}_{c}=-\hat{\bf L}_{c}$ (4.10)
To get $\hat{\bf j}\to-\hat{\bf j}_{c}$ with $\hat{\bf
j}_{c}=\hat{L}_{c}+\frac{\hbar}{2}\hat{\boldsymbol{\sigma}}_{c}$, we should
have
$\hat{\boldsymbol{\sigma}}_{c}=-\hat{\boldsymbol{\sigma}}$ (4.11)
Hence the values of matrix element for positron’s spin operator
${\boldsymbol{\sigma}}_{c}$ is just the negative to that for
${\boldsymbol{\sigma}}$ in the same matrix representation.
Notice that Eq.(4.7) describes an electron with positive helicity, i.e.,
${\boldsymbol{\Sigma}}\cdot\hat{{\bf p}}\psi=p_{z}\psi=p\psi$
222${\boldsymbol{\Sigma}}=\begin{pmatrix}{\boldsymbol{\sigma}}&0\\\
0&{\boldsymbol{\sigma}}\end{pmatrix},\;{\boldsymbol{\Sigma}}_{c}=\begin{pmatrix}{\boldsymbol{\sigma}}_{c}&0\\\
0&{\boldsymbol{\sigma}}_{c}\end{pmatrix}$. Under a space-time inversion, it
transforms into $(-{\boldsymbol{\Sigma}}_{c})\cdot\hat{{\bf
p}_{c}}\psi^{\prime}_{c}=\Sigma_{z}(i\hbar\frac{\partial}{\partial
z})\psi^{\prime}_{c}=p_{c}\psi^{\prime}_{c}$ in Eq.(4.8), i.e.,
${\boldsymbol{\Sigma}}_{c}\cdot\hat{{\bf
p}_{c}}\psi^{\prime}_{c}=-p_{c}\psi^{\prime}_{c}$, meaning that Eq.(4.8)
describes a positron with negative helicity.
Dirac equation is usually written in a covariant form as (Pauli metric is
used:
$x_{4}=ict,\gamma_{k}=-i\beta\alpha_{k},\gamma_{4}=\beta,\gamma_{5}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=-\begin{pmatrix}0&I\\\
I&0\end{pmatrix}$, see 15 (15)):
$(\gamma_{\mu}\partial_{\mu}+m)\psi=0$ (4.12)
Under a space-time (or mass) inversion, it turns into an equation for
antiparticle:
$(-\gamma_{\mu}\partial_{\mu}+m)\psi^{\prime}_{c}=0$ (4.13)
with an example of $\psi^{\prime}_{c}$ shown in Eq.(4.8). Let us perform a
representation transformation:
$\psi^{\prime}_{c}\to\psi_{c}=(-\gamma_{5})\psi^{\prime}_{c}=\begin{pmatrix}\phi_{c}\\\
\chi_{c}\end{pmatrix}$ (4.14)
and arrive at
$(\gamma_{\mu}\partial_{\mu}+m)\psi_{c}=0$ (4.15)
due to $\\{\gamma_{5},\gamma_{\mu}\\}=0$. Since $\psi_{c}$ and
$\psi^{\prime}_{c}$ are essentially the same in physics, (this is obviously
seen from its resolved form, Eq.(4.3)), it is merely a trivial thing to change
the position of $\chi_{c}$ in the 4-component spinor (lower in Eq.(4.14) and
upper in Eq.(4.8)). What important is $|\chi_{c}|>|\phi_{c}|$ for
characterizing an antiparticle versus $|\phi|>|\chi|$ for a particle.
Therefore, if a particle with energy $E$ runs into a potential barrier
$V=V_{0}>E+m$, its kinetic energy becomes negative, and its WF’s third
component in Eq.(4.7) suddenly turns into
$\frac{p^{\prime}}{E-V_{0}+m}=\frac{-p^{\prime}}{V_{0}-E-m},(p^{\prime}=\sqrt{(E-V_{0})^{2}-m^{2}})$,
whose absolute magnitude is larger than that of the first component. This
means that it is an antiparticle’s WF satisfying Eq.(4.15) (with
$E_{c}=V_{0}-E(>m)$ and $|\chi_{c}|>|\phi_{c}|$) and will be crucial for the
explanation of Klein paradox in Dirac equation as shown in the Appendix A.
However, we need to discuss the ”probability density” $\rho$ and ”probability
current density” ${\bf j}$ for a Dirac particle versus $\rho_{c}$ and ${\bf
j}_{c}$ for its antiparticle. Different from that in KG equation, now we have
$\rho=\psi^{\dagger}\psi=\phi^{\dagger}\phi+\chi^{\dagger}\chi\to\rho_{c}=\psi_{c}^{\dagger}\psi_{c}=\chi_{c}^{\dagger}\chi_{c}+\phi_{c}^{\dagger}\phi_{c}$
(4.16)
which is positive definite for either particle or antiparticle. On the other
hand, we have
${\bf
j}=c\psi^{\dagger}{\boldsymbol{\alpha}}\psi=c(\phi^{\dagger}{\boldsymbol{\sigma}}\chi+\chi^{\dagger}{\boldsymbol{\sigma}}\phi)\to{\bf
j}_{c}=c\psi_{c}^{\dagger}{\boldsymbol{\alpha}}\psi_{c}=c(\chi_{c}^{\dagger}{\boldsymbol{\sigma}}\phi_{c}+\phi_{c}^{\dagger}{\boldsymbol{\sigma}}\chi_{c})$
(4.17)
(we prefer to keep ${\boldsymbol{\sigma}}$ rather than
${\boldsymbol{\sigma}}_{c}$ for antiparticle). For Eqs.(4.7), (4.8) and
(4.14), we find ($c=\hbar=1$)
$j_{z}\sim\dfrac{2p}{E+m}>0\to
j_{z}^{c}\sim\dfrac{2p_{c}}{E_{c}+m}>0\quad(V=0)$ (4.18)
which means that the probability current is always along the momentum’s
direction for either a particle or antiparticle.
Above discussions at RQM level may be summarized as follows: The first symptom
for the appearance of an antiparticle is: If we perform an energy operator (
$E=i\hbar\partial/\partial t$) on a WF and find a negative energy ($E<0$) or a
negative kinetic energy ($E-V<0$), we’d better to doubt the WF being a
description of antiparticle and use the operators for antiparticle, Eq.(3.15).
Then for further confirmation, two more criterions for $\rho$ and ${\bf j}$
are needed (see Appendix).
In hindsight, for a linear equation in RQM, either KG or Dirac equation, the
emergence of both positive and negative energy ($E$) WFs is inevitable and
natural. From mathematical point of view, the set of WFs cannot be complete if
without taking the negative energy solutions into account. And physicists
believe that these negative-energy solutions might be relevant to
antiparticles. However, we physicists admit that both a rest particle’s energy
$E=mc^{2}$ and a rest antiparticle’s energy $E_{c}=m_{c}c^{2}=mc^{2}$ are
positive, as verified by the experiments of pair-creation process $\gamma\to
e^{+}+e^{-}$. The above contradiction constructs so-called ”negative-energy
paradox” in RQM. For Dirac particle, majority (not all) of physicists accept
the ”hole theory” to explain the ”paradox”. But for KG particle, no such kind
of ”hole theory” can be acceptable. It was this ”negative-energy paradox” and
”Klein paradox” as well as the four ”commutation relations”, Eqs.(2.1)-(2.5),
hidden in the two-particle system discussed by EPR 1 (1) gradually prompted us
to realize that the root cause of difficulty in RQM lies in an a priori notion
— only one kind of WF with one set of operators (like Eq.(3.2)) can be
acceptable in QM, either for NRQM or RQM.
Once getting rid of the constraint in the above notion and introducing two
sets of WFs and operators for particle and antiparticle respectively, we are
able to see that many difficulties in RQM disappear immediately. What we
emphasize in section III and IV is: The CPT invariance in RQM, i.e., the
invariance of (newly defined) space-time inversion Eq.(3.30) (which dictates
Eq.(3.15), $\rho_{c}$ and ${\bf j}_{c}$ for antiparticle) is capable of
helping the RQM to become complete and more useful in applications (see
Appendix A).
## V V. Why QFT is Correct from Scratch?
In QFT, Lüders and Pauli proved the CPT theorem (Refs.31 (31, 32, 33, 34)),
claiming that ”a wide class of QFTs which are invariant under the proper
Lorentz group is also invariant with respect to the product of $T,C$ and $P$”.
The proof of CPT theorem contains a crucial step being the construction of so-
called ”strong reflection”, consisting in a reflection of space and time about
some arbitrarily chosen origin, i.e., ${\bf r}\to-{\bf r},t\to-t$.
Pauli first proposed and explained the strong reflection in 33 (33) as
follows: When the space-time coordinates change their sign, every particle
transforms into its antiparticle simultaneously. The physical sense of the
strong reflection is the substitution of every emission (absorption) operator
of a particle by the corresponding absorption (emission) operator of its
antiparticle. And there is no need to reverse the sign of the electric charge
when the sign of space-time coordinates is reversed.
After combining with other necessary postulates, Lüders and Pauli proved that
the Hamiltonian density of QFT (constructed from fields of spin zero, one-half
and one by local interactions which are invariant under the proper Lorentz
group) is invariant with respect to the strong reflection, ${\cal H}({\bf
x},t)\to{\cal H}(-{\bf x},-t)$. The strong reflection, after combining with
the Hermitian conjugation (H.C.), turns to be identical with the product of
$T,C$ and $P$ operations in QFT, thus completing the proof of CPT theorem 32
(32).
Hence we can rename the CPT invariance and write down Eq.(3.30) again, but at
the level of QFT:
${\cal P}{\cal T}={\cal C}$ (5.1)
It’s time to look at the CPT theorem upside down as follows:
For a theorem, either in mathematics or in physics, its consequences are
already hidden in its premises (which are essentially beyond the proof of
theorem itself). Evidently, the premise of CPT theorem is QFT, but the premise
of QFT had not been unveiled explicitly until 1954-1957. The great merit of
Lüders and Pauli is: they discovered the premise of QFT is just the invariance
of the ”strong reflection”(in combination with the H.C.), or equivalently, the
CPT invariance. In other words, why CPT theorem looks so unique is just
because its consequence proves its premise exactly.
We are encouraged to say so because the validity of CPT invariance, i.e., the
”strong reflection”, has also been proved at the level of RQM in last two
sections. Below, we highlight Pauli-Lüders’ idea to show that the ”field
operator” in QFT is defined precisely in accordance with the CPT invariance,
i.e., Eq.(5.1). What we need is some supplement added in the Fock space
according to Pauli’s idea:
Let us look at the free ”field operator” of charged scalar boson field, i.e.,
the complex KG field and its hermitian conjugate being defined as
($\hbar=c=1$)
$\left\\{\begin{array}[]{ll}\hat{\psi}({\bf x},t)=\sum\limits_{\bf
p}\dfrac{1}{\sqrt{2V\omega_{\bf p}}}\left\\{\hat{a}_{\bf p}\exp[i({\bf
p}\cdot{\bf x}-Et)]+\hat{b}^{\dagger}_{\bf p}\exp[-i({\bf p}\cdot{\bf
x}-Et)]\right\\}\\\\[11.38109pt] \hat{\psi}^{\dagger}({\bf
x},t)=\sum\limits_{\bf p}\dfrac{1}{\sqrt{2V\omega_{\bf
p}}}\left\\{\hat{a}^{\dagger}_{\bf p}\exp[-i({\bf p}\cdot{\bf
x}-Et)]+\hat{b}_{\bf p}\exp[i({\bf p}\cdot{\bf
x}-Et)]\right\\}\end{array}\right.$ (5.2)
As Eq.(5.1) works at the level of QFT, we expect that Eq.(5.2) remains
invariant under the operation ${\cal P}{\cal T}={\cal C}$ in the sense of
$\begin{array}[]{l}\hat{\psi}({\bf x},t)\to{\cal P}{\cal T}\hat{\psi}({\bf
x},t)({\cal P}{\cal T})^{-1}=\hat{\psi}(-{\bf x},-t)=\hat{\psi}({\bf x},t),\\\
\hat{\psi}^{\dagger}({\bf x},t)\to{\cal P}{\cal T}\hat{\psi}^{\dagger}({\bf
x},t)({\cal P}{\cal T})^{-1}=\hat{\psi}^{\dagger}(-{\bf
x},-t)=\hat{\psi}^{\dagger}({\bf x},t)\end{array}$ (5.3)
Indeed, it does, as long as a transformation of operators in Fock space is
expressed by 11 (11, 28):
$\hat{a}_{\bf p}\leftrightarrows\hat{b}^{\dagger}_{\bf p}$ (5.4)
simultaneously together with the inversion of space-time coordinates (${\bf
x}\to-{\bf x},t\to-t$) in c-numbers WFs in Eq.(5.2). Eqs.(5.2)-(5.4) imply
that under the space-time inversion, the process of a particle’s annihilation
transformations into that of its antiparticle’s creation (or vice versa), an
ansatz could be understood as a necessary implementation of Eq.(5.1) at the
level of QFT and further reflects Pauli’s idea that the space-time inversion
is indivisible from the transformation between particle and antiparticle.
111Unlike in RQM, the ”mass inversion” is not applicable in QFT.
The commutation relation between $\hat{a}_{\bf p}$ and $\hat{a}^{\dagger}_{\bf
p}$ is assumed as usual:
$[\hat{a}_{\bf p},\hat{a}^{\dagger}_{\bf p^{\prime}}]=\delta_{\bf
pp^{\prime}},\quad[\hat{a}_{\bf p},\hat{b}_{\bf p^{\prime}}]=0$ (5.5)
Then, performing Eq.(5.1) on Eq.(5.5), we arrive at:
$[\hat{b}_{\bf p},\hat{b}^{\dagger}_{\bf p^{\prime}}]=\delta_{\bf
pp^{\prime}},\quad[\hat{a}^{\dagger}_{\bf p},\hat{b}^{\dagger}_{\bf
p^{\prime}}]=0$ (5.6)
where Eq.(5.4) has been used and one more rule is added as follows32 (32): The
order of an operator product in Fock space has to be reversed under the space-
time inversion, e.g., $({\cal P}{\cal T})\hat{A}\hat{B}({\cal P}{\cal
T})^{-1}=({\cal P}{\cal T})\hat{B}({\cal P}{\cal T})^{-1}({\cal P}{\cal
T})\hat{A}({\cal P}{\cal T})^{-1}$. So is the order of a process occurred in a
many-particle system under the operation Eq.(5.1). This is a necessary
postulate (together with the Hermiticity of the Hamiltonian density) for QFT
being capable of dealing with real problems successfully.
As QFT is such a successful theory, we may expect that the WFs for a particle
and its antiparticle derived from the QFT will also be identified with that in
the RQM.
In QFT, the WF of a single particle should be defined rigorously as the
nondiagonal matrix element of the relevant field operator between the vacuum
state and one-particle state. For instance, assume Eq.(5.2) to be the ”field
operator of $K^{-}$ meson field”, then the WF of a freely moving $K^{-}$ meson
(with momentum ${\bf p}_{1}$) is given by
$\psi_{K^{-}}({\bf x},t)=\langle 0|\hat{\psi}({\bf x},t)|K^{-},{\bf
p}_{1}\rangle=\langle 0|\hat{\psi}({\bf x},t)\hat{a}^{\dagger}_{{\bf
p}_{1}}|0\rangle=\dfrac{1}{\sqrt{2V\omega_{{\bf p}_{1}}}}e^{i({\bf
p}_{1}\cdot{\bf x}-E_{1}t)}$ (5.7)
whereas the hermitian (i.e., complex) conjugation of a $K^{+}$ meson’s WF is
given by
$\psi^{*}_{K^{+}}({\bf x},t)=\langle 0|\hat{\psi}^{\dagger}({\bf
x},t)|K^{+},{\bf p}_{1}\rangle=\langle 0|\hat{\psi}^{\dagger}({\bf
x},t)\hat{b}^{\dagger}_{{\bf p}_{1}}|0\rangle=\dfrac{1}{\sqrt{2V\omega_{{\bf
p}_{1}}}}e^{i({\bf p}_{1}\cdot{\bf x}-E_{1}t)}$ (5.8)
which leads to $K^{+}$’s WF being
$\psi_{K^{+}}({\bf x},t)=\dfrac{1}{\sqrt{2V\omega_{{\bf p}_{1}}}}e^{-i({\bf
p}_{1}\cdot{\bf x}-E_{1}t)}$ (5.9)
as expected.
Similarly, the ”field operator” for Dirac field is constructed in the
following form [see 14 (14, 15). However, instead of spin $s$ (projection
along $z$ axis in space) usually used, here $h$, the helicity, is used in the
expansion]:
$\left\\{\begin{array}[]{ll}\hat{\psi}({\bf
x},t)=\dfrac{1}{\sqrt{V}}\sum\limits_{\bf p}\sum\limits_{h=\pm
1}\sqrt{\dfrac{m}{E}}\left[\hat{a}_{\bf p}^{(h)}u^{(h)}({\bf p})e^{i({\bf
p}\cdot{\bf x}-Et)}+\hat{b}^{(h){\dagger}}_{\bf p}v^{(h)}({\bf p})e^{-i({\bf
p}\cdot{\bf x}-Et)}\right]\\\\[11.38109pt] \hat{\psi}^{\dagger}({\bf
x},t)=\dfrac{1}{\sqrt{V}}\sum\limits_{\bf p}\sum\limits_{h=\pm
1}\sqrt{\dfrac{m}{E}}\left[\hat{a}_{\bf p}^{(h){\dagger}}u^{(h){\dagger}}({\bf
p})e^{-i({\bf p}\cdot{\bf x}-Et)}+\hat{b}^{(h)}_{\bf p}v^{(h){\dagger}}({\bf
p})e^{i({\bf p}\cdot{\bf x}-Et)}\right]\end{array}\right.$ (5.10)
Here for arbitrary momentum ${\bf p}$ (with direction denoted by angles
$\theta$ and $\phi$ in spherical coordinates) and energy $E=\sqrt{{\bf
p}^{2}+m^{2}}>0$, the spinors attached to particle’s annihilation operators
$\hat{a}_{\bf p}^{(h)}$ are
$u^{(1)}({\bf p})=\sqrt{\dfrac{E+m}{2m}}\begin{pmatrix}\phi_{0}^{(1)}({\bf
p})\\\ \frac{|{\bf p}|}{E+m}\phi_{0}^{(1)}({\bf p})\end{pmatrix},\quad
u^{(-1)}({\bf p})=\sqrt{\dfrac{E+m}{2m}}\begin{pmatrix}\phi_{0}^{(-1)}({\bf
p})\\\ \frac{-|{\bf p}|}{E+m}\phi_{0}^{(-1)}({\bf p})\end{pmatrix}$ (5.11)
$\phi_{0}^{(1)}({\bf p})=\begin{pmatrix}\cos\theta/2\\\
e^{i\phi}\sin\theta/2\end{pmatrix},\;\phi_{0}^{(-1)}({\bf
p})=\begin{pmatrix}\sin\theta/2\\\
-e^{i\phi}\cos\theta/2\end{pmatrix},\;u^{(h){\dagger}}({\bf
p})u^{(h^{\prime})}({\bf p})=\dfrac{E}{m}\delta_{hh^{\prime}}$ (5.12)
$\phi_{0}^{(h)}(-{\bf p})=\phi_{0}^{(-h)}({\bf p},\quad\gamma_{4}u^{(h)}(-{\bf
p})=u^{(-h)}({\bf p})$ (5.13)
while that attached to antiparticle’s creation operators $\hat{b}_{\bf
p}^{(h){\dagger}}$ are
$v^{(h)}({\bf p})=(-\gamma_{5})u^{(-h)}({\bf p}),\quad\gamma_{4}v^{(h)}(-{\bf
p})=-v^{(-h)}({\bf p})$ (5.14) $v^{(h){\dagger}}({\bf p})v^{(h^{\prime})}({\bf
p})=\dfrac{E}{m}\delta_{hh^{\prime}},\;v^{(h^{\prime}){\dagger}}(-{\bf
p})u^{(h)}({\bf p})=u^{(h^{\prime}){\dagger}}(-{\bf p})v^{(h)}({\bf p})=0$
(5.15)
Like Eq.(5.4) for KG field, an ansatz is added:
$\hat{a}_{\bf p}^{(h)}\rightleftarrows\hat{b}_{\bf
p}^{(-h){\dagger}},\quad\hat{a}_{\bf
p}^{(h){\dagger}}\rightleftarrows\hat{b}_{\bf p}^{(-h)}\quad({\bf x}\to-{\bf
x},t\to-t)$ (5.16)
with $h\rightleftarrows(-h)$ under the space-time inversion in complying with
Eqs.(4.7)-(4.11) and the discussion after them.
Different from that for KG field, now the operators $\hat{a}_{\bf p}^{(h)}$
and $\hat{b}_{\bf p}^{(h)}$ etc.are assumed to obey anticommutation relations:
$\\{\hat{a}^{(h)}_{\bf p},\hat{a}^{(h^{\prime}){\dagger}}_{{\bf
p}^{\prime}}\\}=\\{\hat{b}^{(h)}_{\bf p},\hat{b}^{(h^{\prime}){\dagger}}_{{\bf
p}^{\prime}}\\}=\delta_{{\bf p}{\bf
p}^{\prime}}\delta_{hh^{\prime}},\\{\hat{a}^{(h)}_{\bf
p},\hat{b}^{(h^{\prime})}_{{\bf p}^{\prime}}\\}=0,\;{\it etc.}$ (5.17)
which, like Eqs.(5.5)-(5.6) for KG field, remain invariant under the operation
of Eq.(5.1).
However, for Dirac field operator, we should define:
$\begin{array}[]{l}\hat{\psi}({\bf x},t)\to{\cal P}{\cal T}\hat{\psi}({\bf
x},t)({\cal P}{\cal T})^{-1}=-\gamma_{5}\hat{\psi}(-{\bf
x},-t)=\hat{\psi}({\bf x},t)\\\\[11.38109pt] \hat{\psi}^{\dagger}({\bf
x},t)\to{\cal P}{\cal T}\hat{\psi}^{\dagger}({\bf x},t)({\cal P}{\cal
T})^{-1}=\hat{\psi}^{\dagger}(-{\bf
x},-t)(-\gamma_{5})=\hat{\psi}^{\dagger}({\bf x},t)\end{array}$ (5.18)
for keeping their invariance in the 4-component spinor form rigorously. Thus
$\hat{\psi}(-{\bf x},-t)=-\gamma_{5}\hat{\psi}({\bf
x},t),\quad\hat{\psi}^{\dagger}(-{\bf x},-t)=\hat{\psi}^{\dagger}({\bf
x},t)(-\gamma_{5})$ (5.19)
which are useful for proving the ”spin-statistics connection” by ${\cal
P}{\cal T}$ invariance.
Another rule is: One should always take the normal ordering when dealing with
quadratic forms like $\hat{\bar{\psi}}(x)\psi(x)$ etc.
Now, like Eqs.(5.7)-(5.9) for KG particles’ WFs, for Dirac field, we have the
WF of an electron being
$\psi_{e^{-}}({\bf x},t)=\langle 0|\hat{\psi}({\bf x},t)|e^{-},{\bf
p}_{1},h_{1}\rangle=\langle 0|\hat{\psi}({\bf x},t)\hat{a}_{\bf
p_{1}}^{(h_{1}){\dagger}}|0\rangle=\dfrac{1}{\sqrt{V}}\sqrt{\dfrac{m}{E_{1}}}u^{(h_{1})}({\bf
p}_{1})e^{i({\bf p}_{1}\cdot{\bf x}-E_{1}t)}$ (5.20)
but the hermitian conjugate of a positron’s WF is given by
$\psi_{e^{+}}^{\dagger}({\bf x},t)=\langle 0|\hat{\psi}^{\dagger}({\bf
x},t)|e^{+},{\bf p}_{c},h_{c}\rangle=\langle 0|\hat{\psi}^{\dagger}({\bf
x},t)\hat{b}_{\bf
p_{c}}^{(h_{c}){\dagger}}|0\rangle=\dfrac{1}{\sqrt{V}}\sqrt{\dfrac{m}{E_{c}}}v^{(h_{c}){\dagger}}({\bf
p}_{c})e^{i({\bf p}_{c}\cdot{\bf x}-E_{c}t)}$ (5.21)
which leads to positron’s WF being
$\psi_{e^{+}}({\bf
x},t)=\dfrac{1}{\sqrt{V}}\sqrt{\dfrac{m}{E_{c}}}v^{(h_{c})}({\bf
p}_{c})e^{-i({\bf p}_{c}\cdot{\bf x}-E_{c}t)}$ (5.22)
as expected. To our surprise, we couldn’t find such simple derivations like
Eqs.(5.7)-(5.9) and Eqs.(5.20)-(5.22) in existing textbooks. 111Notice that in
Eq.(5.21), the existence of spinor $v^{(h_{c}){\dagger}}({\bf p}_{c})$ makes
the claim for ”hermitian conjugate” an indisputable one.
Hence the historical merit of Pauli and Lüders could be highlighted as
follows: On one hand, what they did is actually to correct a systematic error
(”only one set of WF and operators is allowed”) in RQM via the approach of
QFT. On the other hand, they unveiled the underlying symmetry of QFT being the
invariance of the Hamiltonian density ${\hat{\cal H}}({\bf x},t)$ under an
operation of ”strong reflection”, i.e.,
${\hat{\cal H}}({\bf x},t)\to{\cal P}{\cal T}{\hat{\cal H}}({\bf x},t)({\cal
P}{\cal T})^{-1}={\hat{\cal H}}(-{\bf x},-t)={\hat{\cal H}}({\bf x},t)$ (5.23)
as well as that under a H.C.:
${\hat{\cal H}}({\bf x},t)\to{\hat{\cal H}}^{\dagger}({\bf x},t)={\hat{\cal
H}}({\bf x},t)$ (5.24)
The validity of both Eqs.(5.23) and (5.24) has been verified experimentally
since the discovery of parity violation and the establishment (and
development) of standard model in particle physics till today.
## VI VI. Summary and Discussion
1\. Since the CPT theorem was proved in the framework of QFT, the CPT
invariance was often discussed at the level of QFT. By contrast, its
discussion at the level of RQM was, to our knowledge, rarely seen. The reason
might be as follows: After performing the CPT transformation on a particle’s
WF as Eq.(3.29) at the level RQM, one encountered a WF with ”negative energy”
inevitably. And together with the ”negative probability density”, no concensus
could be reached on its explanation within the physics community. In our
opinion, the RQM cannot be considered complete until two sets of WFs and
operators are introduced for particle versus antiparticle respectively and
they are linked with the (newly defined) space-time inversion, Eq.(3.19).
Alternatively, the symmetry between particle and antiparticle can also be
realized by the mass inversion ($m\to-m$) as shown in Eq.(3.25) at the level
of RQM (but not for QFT). The fact that the symmetry ${\cal P}{\cal T}={\cal
C}$ exists in both RQM and QFT strongly hints that it is a basic postulate in
physics rather than merely a consequence of the CPT theorem.
2\. Evidently, the renamed CPT invariance, Eq.(5.1), discovered by Lüders and
Pauli is already there in the present framework of QFT. However, one thing is
important here: We should assign the helicity ($h$) rather than the spin
projection (in space) ($s$) to particle’s state in the expansion of field
operator, Eq.(5.10), as required by the Eq.(5.1) with Eq.(5.16). The helicity
of a particle is just opposite to that of its antiparticle after the operation
Eq.(5.1).
Therefore, the experimental tests for the CPT invariance should include not
only the equal mass and lifetime of particle versus antiparticle, but also the
following fact: A particle and its antiparticle with opposite helicities must
coexist in nature with no exception. A prominent example is the neutrino — A
neutrino $\nu_{L}$ (antineutrino $\bar{\nu}_{R}$) is permanently left-handed
(right-handed) polarized whereas the fact that no $\nu_{R}$ exists in nature
must means no $\bar{\nu}_{L}$ as well (as verified by the GGS experiment 24
(24)).
3\. Despite of Eqs.(3.30) and (5.1) being always valid, we wish to stress that
all discussions about C, P, CP and T symmetries remain meaningful regardless
of being conserved or not. In particular, T always means the ”reversal of
motion” at the level of either QM or QFT even though it is always an
antiunitary operator.
4\. In hindsight, after learning Feshbach-Villars dissociation of KG equation
Eqs.(3.11)-(3.12), we realized that the postulate, i.e., the renamed CPT
invariance Eqs.(3.30) and (5.1) could reflect some intrinsic property of the
theory of SR established by Einstein in 1905.
Actually, there are two Lorentz invariants in the kinematics of SR:
$c^{2}(t_{1}-t_{2})^{2}-({\bf x}_{1}-{\bf
x}_{2})^{2}=c^{2}(t^{\prime}_{1}-t^{\prime}_{2})^{2}-({\bf
x^{\prime}}_{1}-{\bf x^{\prime}}_{2})^{2}=const$ (6.1) $E^{2}-c^{2}{\bf
p}^{2}=E^{\prime 2}-c^{2}{\bf p^{\prime}}^{2}=m^{2}c^{4}$ (6.2)
It seems quite clear that Eq.(6.1) is invariant under the space-time inversion
(${\bf x}\to-{\bf x},t\to-t$) and Eq.(6.2) remains invariant under the mass
inversion ($m\to-m$). We believe that these two discrete symmetries are deeply
rooted at the SR’s dynamics via its combination with QM and developing into
RQM and QFT — the particle and its antiparticle are treated on equal footing
and linked by the symmetry ${\cal P}{\cal T}={\cal C}$ essentially.
Hence, the invariance of a theory (either in RQM or in classical physics)
under the space-time inversion or the mass inversion in one coordinate frame
could be used as a tool to find or test a new equation for it being
relativistic or not11 (11, 12, 13, 35, 38, 39, 40).
## Appendix A: Klein Paradox for Klein-Gordon Equation and Dirac Equation
We will discuss the Klein paradox for both KG equation and Dirac equation
based on the basic postulate, Eq.(3.30), at the level of QM.
### AI: Klein Paradox for KG Equation
Consider that a KG particle moves along $z$ axis in one-dimensional space and
hits a step potential
$V(z)=\left\\{\begin{array}[]{ll}0,&\hbox{$z<0$;}\\\
V_{0},&\hbox{$z>0$.}\end{array}\right.$ $None$
Its incident WF with momentum $p\,(>0)$ and energy $E\,(>0)$ reads
$\psi_{i}=a\exp[i(pz-Et)],\quad(z<0)$ $None$
If $E=\sqrt{p^{2}+m^{2}}<V_{0}$, we expect that the particle wave will be
partly reflected at $z=0$ with WF $\psi_{r}$ and another transmitted wave
$\psi_{t}$ emerged at $z>0$:
$\psi_{r}=b\exp[i(-pz-Et)],\quad(z<0)$ $None$
$\psi_{t}=b^{\prime}\exp[i(p^{\prime}z-Et)],\quad(z>0)$ $None$
with $p^{\prime 2}=(E-V_{0})^{2}-m^{2}$. See Fig.1(a).
Figure 1: Klein paradox: (a) If $V_{0}>E+m$, there will be a wave $\psi_{t}$
at $z>0$.
(b) Just look at $z>0$ region, making a shift
$V(z)\to\tilde{V}(z)=V(z)-V_{0},\,E\to E^{\prime}=E-V_{0}<-m$.
(c) An antiparticle (at $z>0$) appears with its energy $E_{c}=|E^{\prime}|>m$
and the potential is $V_{c}(z)=-\tilde{V}(z)$
Two continuity conditions for WFs and their space derivatives at the boundary
$z=0$ give two simple equations
$\left\\{\begin{array}[]{l}a+b=b^{\prime}\\\
(a-b)p=b^{\prime}p^{\prime}\end{array}\right.$ $None$
The Klein paradox happens when $V_{0}>E+m$ because the momentum
$p^{\prime}=\pm\sqrt{(V_{0}-E)^{2}-m^{2}}$ is real again and the reflectivity
$R$ of incident wave reads
$R=\left|\dfrac{b}{a}\right|^{2}=\left|\dfrac{p-p^{\prime}}{p+p^{\prime}}\right|^{2},\;\left\\{\begin{array}[]{l}R<1,\quad\text{if}\;p^{\prime}>0\\\
R>1,\quad\text{if}\;p^{\prime}<0\end{array}\right.$ $None$
(See Ref.6 (6) or §9.4 in Ref.11 (11), where discussions were not complete and
need to be complemented and corrected here). Because the kinetic energy
$E^{\prime}$ at $z>0$ is negative: $E^{\prime}=E-V_{0}<0$, what does it mean?
Does the particle still remain as a particle?
As discussed in section IV, for a KG particle (or its antiparticle), two
criterions must be held: its probability density $\rho$ (or $\rho_{c}$) must
be positive and its probability current density ${\bf j}$ (or ${\bf j}_{c}$)
must be in the same direction of its momentum ${\bf p}$ (or ${\bf p}_{c}$).
See Fig.1(b), after making a shift in the energy scale, i.e., basing on the
new vacuum at $z>0$ region, we redefine a WF $\tilde{\psi}_{t}$ (which is
actually the WF in the ”interaction picture”,
$\tilde{\psi}_{t}=\psi_{t}e^{iV_{0}t}\,(z>0)$)
$\psi_{t}\to\tilde{\psi}_{t}=b^{\prime}\exp[i(p^{\prime}z-E^{\prime}t)],\quad(z>0)$
$None$
($E^{\prime}=E-V_{0}<0$). From now on we will replace KG WF $\tilde{\psi}_{t}$
by $\tilde{\phi}_{t}$ and $\tilde{\chi}_{t}$ according to Eq.(3.12), if
$\tilde{\psi}_{t}$ still describes a ”particle”, whose probability density
$\rho_{t}$ should be evaluated by Eq.(3.13) with $V\to\tilde{V}(z)=0\,(z>0)$
yielding:
$\rho_{t}=|\tilde{\phi}_{t}|^{2}-|\tilde{\chi}_{t}|^{2}=\dfrac{E^{\prime}}{m}|b^{\prime}|^{2}<0,\quad(z>0)$
$None$
And its probability current density $j_{t}$ should be given by Eq.(3.10),
yielding:
$j_{t}=\dfrac{p^{\prime}}{m}|b^{\prime}|^{2},\quad(z>0)$ $None$
Eq.(A.8) is certainly not allowed. So to consider a ”particle” with momentum
$p^{\prime}>0$ moving to the right makes no sense. Instead, we should consider
$p^{\prime}<0$ (which also makes no sense for a particle due to the boundary
condition) and regard $\tilde{\psi}_{t}$ as an antiparticle’s WF by rewriting
it as:
$\tilde{\psi}_{t}=\psi_{c}=b^{\prime}\exp[-i(p_{c}z-E_{c}t)],\quad(z>0)$
$None$
Now using Eq.(2.18) we see that Eq.(A.10) does describe an antiparticle with
momentum $p_{c}=-p^{\prime}=|p^{\prime}|=\sqrt{E_{c}^{2}-m^{2}}>0$ and energy
$E_{c}=|E^{\prime}|=V_{0}-E>0$. In the mean time, from the antiparticle’s
point of view (i.e., with $E_{c}>m$), the potential becomes
$V_{c}(z)=-\tilde{V}(z)$ (comparing Eq.(3.12) with Eq.(3.17)) as shown by
Fig.1(c).
It is easy to see from Eqs.(3.16),(3.18) and (A.10) that111We had discarded
the solution of $p^{\prime}>0$ in Eqs.(A.8)-(A.9) as a particle. However, if
we consider $p^{\prime}=-p_{c}>0$ for an antiparticle, then similar to
Eqs.(A.10)-(A.11), we would get $\rho_{t}^{c}>0$ but both $j_{t}^{c}$ and
$p_{c}$ are negative, meaning that the antiparticle is coming from $z=\infty$,
not in accordance with our boundary condition. So the case of $p^{\prime}>0$
should be abandoned either as a particle or as an antiparticle.
$\left\\{\begin{array}[]{l}\rho_{t}^{c}=|\tilde{\chi}_{t}^{c}|^{2}-|\tilde{\phi}_{t}^{c}|^{2}=\dfrac{E_{c}}{m}|b^{\prime}|^{2}>0,\\\\[11.38109pt]
j_{t}^{c}=\dfrac{p_{c}}{m}|b^{\prime}|^{2}\end{array}\right.\quad(z>0)$ $None$
So the reflectivity, Eq.(A.6), should be fixed as:
$R_{KG}=\left|\dfrac{b}{a}\right|^{2}=\left|\dfrac{p+p_{c}}{p-p_{c}}\right|^{2}=\left(\dfrac{1+\gamma^{\prime}}{1-\gamma^{\prime}}\right)^{2},\;\gamma^{\prime}=\dfrac{p_{c}}{p}>0$
$None$
And the transmission coefficient can also be predicted as:
$T_{KG}=\dfrac{j_{t}^{c}}{j_{i}}=\dfrac{p_{c}}{p}\left|\dfrac{b^{\prime}}{a}\right|^{2}=\dfrac{p_{c}}{p}\left|1+\dfrac{b}{a}\right|^{2}=\dfrac{4pp_{c}}{(p-p_{c})^{2}}=\dfrac{4\gamma^{\prime}}{(1-\gamma^{\prime})^{2}}$
$None$ $R_{KG}-T_{KG}=1$ $None$
The variation of $T_{KG}$ seems very interesting:
$T_{KG}=\left\\{\begin{array}[]{l}0,\;\gamma^{\prime}\to 0\quad(p_{c}\to
0,E_{c}\to m)\\\\[8.53581pt] \infty,\;\gamma^{\prime}\to
1\quad(p_{c}=p,E_{c}=E=V_{0}/2)\\\\[8.53581pt]
0,\;\gamma^{\prime}\to\infty\quad(p_{c}\to\infty,E_{c}=V_{0}-E\to\infty)\\\\[8.53581pt]
0,\;\gamma^{\prime}\to\infty\quad(p\to 0,E\to m)\end{array}\right.$ $None$
Above equations show us that the incident KG particle triggers a process of
”pair creation” occurring at $z=0$, creating new particles moving to the left
side (to join the reflected incident particle) so enhancing the reflectivity
$R_{KG}>1$ and new antiparticles (with equal number of new particles) moving
to the right.
To our understanding, this is not a stationary state problem for a single
particle, but a nonstationary creation process of many particle-antiparticle
system. It is amazing to see the Klein paradox in KG equation being capable of
giving some prediction for such kind of process at the level of RQM. Further
investigations are needed both theoretically and experimentally. 222We find
from the Google search that R. G. Winter in 1958 had examined the Klein
paradox for KG equation at the QM level and reached basically the same result
as ours. So he was the first author dealing with this problem. Regrettably, it
seems that his paper had never been published on some journal.
### AII: Klein Paradox for Dirac Equation
Beginning from Klein 27 (27), many authors e.g.Greiner et al.36 (36, 37), have
studied this topic. We will join them by using the similar approach like that
for KG equation discussed above.
Based on similar picture shown in Fig.1, now we have three Dirac WFs under the
condition $V_{0}>E+m$:
$\psi_{i}=a\begin{pmatrix}1\\\ 0\\\ \frac{p}{E+m}\\\ 0\end{pmatrix}e^{i(pz-
Et)},\psi_{r}=b\begin{pmatrix}1\\\ 0\\\ \frac{-p}{E+m}\\\
0\end{pmatrix}e^{i(-pz-Et)}\quad(z<0)$ $None$
$\psi_{t}=b^{\prime}\begin{pmatrix}1\\\ 0\\\ \frac{p^{\prime}}{E-V_{0}+m}\\\
0\end{pmatrix}e^{i(p^{\prime}z-Et)}=b^{\prime}\begin{pmatrix}1\\\ 0\\\
\frac{-p^{\prime}}{V_{0}-E-m}\\\
0\end{pmatrix}e^{i(p^{\prime}z-Et)}=\begin{pmatrix}\phi_{t}\\\
\chi_{t}\end{pmatrix}\quad(z>0)$ $None$
where $p^{\prime}=\pm\sqrt{(V_{0}-E)^{2}-m^{2}}$. Unlike Eq.(A.8) for KG
equation, the probability density for Dirac WF $\psi_{t}$ is positive definite
(see Eq.(4.16))
$\rho_{t}=\psi^{\dagger}_{t}\psi_{t}=\phi^{\dagger}_{t}\phi_{t}+\chi^{\dagger}_{t}\chi_{t}$
$None$
Hence we will rely on two criterions: First, the probability current density
and momentum must be in the same direction for either a particle or
antiparticle. For $\psi_{i}$ and $\psi_{r}$, their probability current density
are ($c=1$)
$\begin{array}[]{l}j_{i}=\psi^{\dagger}_{i}\alpha_{z}\psi_{i}=\phi_{i}^{\dagger}\sigma_{z}\chi_{i}+\chi^{\dagger}_{i}\sigma_{z}\phi_{i}=\dfrac{2p}{E+m}|a|^{2}>0\\\
j_{r}=\psi^{\dagger}_{r}\alpha_{z}\psi_{r}=\dfrac{-2p}{E+m}|b|^{2}<0\end{array}\quad(z<0)$
$None$
as expected. However, for $\psi_{t}$, we meet difficulty similar to that in
Eq.(A.9)
$j_{t}=\psi^{\dagger}_{t}\alpha_{z}\psi_{t}=\dfrac{-2p^{\prime}}{V_{0}-E-m}|b^{\prime}|^{2}\quad(z>0)$
$None$
the direction of $j_{t}$ is always opposite to that of $p^{\prime}$! The
second criterion is: while $|\phi|>|\chi|$ for particle, we must have
$|\chi_{c}|>|\phi_{c}|$ for antiparticle. Now in $\psi_{i}$ (or $\psi_{r}$),
$|\phi_{i}|>|\chi_{i}|$ (or $|\phi_{r}|>|\chi_{r}|$), but the situation in
$\psi_{t}$ is dramatically changed, the existence of $V_{0}$ renders
$|\chi_{t}|>|\phi_{t}|$!
The above two criterions, together with the experience in KG equation, prompt
us to choose $p^{\prime}<0$ and regard $\psi_{t}$ as an antiparticle’s WF. So
we rewrite:
$\psi_{t}=\psi_{t}^{c}e^{-iV_{0}t}$ $None$
$\psi_{t}^{c}=b^{\prime}\begin{pmatrix}1\\\ 0\\\ \frac{p_{c}}{E_{c}-m}\\\
0\end{pmatrix}e^{-i(p_{c}z-E_{c}t)}=\begin{pmatrix}\phi_{t}^{c}\\\
\chi_{t}^{c}\end{pmatrix},\tilde{\psi}^{c}_{t}=b^{\prime}_{c}\begin{pmatrix}1\\\
0\\\ \frac{p_{c}}{E_{c}+m}\\\
0\end{pmatrix}e^{-i(p_{c}z-E_{c}t)}=\begin{pmatrix}\chi_{t}^{c}\\\
\phi_{t}^{c}\end{pmatrix}\quad(z<0)$ $None$
where $\tilde{\psi}^{c}_{t}=(-\gamma^{5})\psi_{t}^{c}$ (with new normalization
constant $b^{\prime}_{c}$ replacing $b^{\prime}$) describes an antiparticle
with momentum $p_{c}=|p^{\prime}|=-p^{\prime}=\sqrt{E_{c}^{2}-m^{2}}>0$,
energy $E_{c}=V_{0}-E>0$ and $|\chi_{t}^{c}|>|\phi_{t}^{c}|$. Using Eq.(4.17)
we find
$j_{t}^{c}=\dfrac{2p_{c}}{E_{c}+m}|b^{\prime}_{c}|^{2}>0,\quad(z>0)$ $None$
as expected. Now it is easy to match Dirac WFs at the boundary $z=0$,
($\psi_{i}+\psi_{r})|_{z=0}=\tilde{\psi}_{t}^{c}|_{z=0}$, yielding111Eq.(A.23)
means that the large (small) component of spinor is connected with large
(small) component at both sides of $z=0$. However, if instead of
$\tilde{\psi}^{c}_{t}$, the $\psi_{t}^{c}$ is used directly with its first
(small) component being connected with the first (large) components of
$\psi_{i}$ and $\psi_{r}$, it would lead to a different expression of
Eq.(A.27):
$\gamma\to\tilde{\gamma}=\sqrt{\frac{(E_{c}-m)(E-m)}{(E+m)(E_{c}+m)}}$, which
is just the $1/\gamma$ ($\gamma$ and $1/\gamma$ make no difference in the
result of, say, Eqs.(A.24) and (A.25)) defined by Eq.(8) on page 266 of Ref.36
(36) (see Eq.(A31) below) or that by Eq.(5.36) in Ref.37 (37)
$\left\\{\begin{array}[]{l}a+b=b^{\prime}_{c}\\\
\dfrac{(a-b)p}{E+m}=\dfrac{b^{\prime}_{c}p_{c}}{E_{c}+m}\end{array}\right.\to\left\\{\begin{array}[]{l}\dfrac{b}{a}=\dfrac{\xi-\eta}{\xi+\eta}\\\
\dfrac{b^{\prime}_{c}}{a}=1+\dfrac{b}{a}=\dfrac{2\xi}{\xi+\eta}\end{array}\right.$
$None$
where $\xi=p(E_{c}+m)>0,\eta=p_{c}(E+m)>0$. The reflectivity $R_{D}$ and
transmission coefficient $T_{D}$ follow from Eq.(A.19) and (A.22) as:
$R_{D}=\dfrac{|j_{r}|}{j_{i}}=\left|\dfrac{b}{a}\right|^{2}=\left(\dfrac{1-\gamma}{1+\gamma}\right)^{2}$
$None$
$T_{D}=\dfrac{j_{t}^{c}}{j_{i}}=\left|\dfrac{b^{\prime}_{c}}{a}\right|^{2}\dfrac{p_{c}(E+m)}{p(E_{c}+m)}=\dfrac{4\gamma}{(1+\gamma)^{2}}$
$None$ $R_{D}+T_{D}=1$ $None$
where
$\gamma=\dfrac{\eta}{\xi}=\sqrt{\dfrac{(E_{c}-m)(E+m)}{(E-m)(E_{c}+m)}}\geq
0\;(E_{c}=V_{0}-E\geq m)$ $None$
and
$T_{D}=\left\\{\begin{array}[]{l}0,\;\gamma\to 0\quad(p_{c}\to 0,E_{c}\to
m)\\\ 1,\;\gamma=1\quad(p_{c}=p,E_{c}=E=V_{0}/2)\;(\text{resonant\
transmission})\\\
\frac{2p}{E+p},\;\gamma\to\sqrt{\frac{E+m}{E-m}}\quad(E_{c}=V_{0}-E\to\infty)\\\
0,\;\gamma\to\infty\quad(p\to 0,E\to m)\end{array}\right.$ $None$
The variation of $T_{D}$ bears some resemblance to Eq.(A.15) for KG equation
but shows striking difference due to sharp contrast between Eqs.(A.24)-(A.28)
and Eqs.(A.12)-(A.15).
To our understanding, in the above Klein paradox for Dirac equation, there is
no ”pair creation” process occurring at the boundary $z=0$. The paradox just
amounts to a steady transmission of particle’s wave $\psi_{i}$ into a high
potential barrier $V_{0}>E+m$ at $z>0$ region where $\psi_{t}$ shows up as an
antiparticle’s WF propagating to the right. In some sense, the existence of a
potential barrier $V_{0}$ plays a ”magic” role of transforming the particle
into its antiparticle. Because the probability densities of both particle and
antiparticle are positive definite, the total probability can be normalzed
over the entire space like that for one particle case:
$\int_{-\infty}^{\infty}[\rho(z)\Theta(-z)+\rho_{c}(z)\Theta(z)]dz=1$ $None$
($\Theta(z)$ is the Heaviside function) and the probability current density
remains continuous at the boundary $z=0$. In other words, the continuity
equation holds in the whole space just like what happens in a one-particle
stationary state.
It is interesting to compare our result with that in Refs.36 (36) and 37 (37).
In Ref.36 (36), Eqs.(13.24)-(13.28) are essentially the same as ours. But the
argument there for choosing $\bar{p}<0$ in Eq.(13.23) is based on the
criterion of the group velocity $v_{gr}$ being positive (for the transmitted
wave packet moving toward $z=\infty$). And the $v_{gr}$ is stemming from
Eq.(13.16) which is essentially the probability current density in our
Eq.(A.19) or (A.20).
However, the author in Ref.36 (36) also considered the other choice
$\bar{p}>0$ in the example (p.265-267 in 36 (36)) based on the hole theory,
ending up with the prediction as:
$R=\left(\dfrac{1+\gamma}{1-\gamma}\right)^{2},\;T=\dfrac{4\gamma}{(1-\gamma)^{2}},\;R-T=1$
$None$
where
$\gamma=\dfrac{p_{2}}{p_{1}}\dfrac{E+m}{V_{0}-E-m}=\sqrt{\dfrac{(V_{0}-E+m)(E+m)}{(V_{0}-E-m)(E-m)}}$
$None$
The argument for the validity of his Eqs.(A.30)-(A.31) is based on the hole
theory (see also section 5.2 in Ref.37 (37)), saying that once $V_{0}>E+m$,
there would be an overlap between the occupied negative continuum for $z>0$
and the empty positive continuum for $z<0$, providing a mechanism for
electron-positron pair creation if the ”hole” at $z>0$ can be identified with
a positron. We doubt the ”hole” theory seriously because there are only two
electrons (with opposite spin orientations) staying at each energy level in
the negative continuum. So it seems that there is no abundant source for
electrons and ”holes” to account for the huge value of $T>1$ in Eq.(A.30).
Fortunately, we learn from section 10.7 in Ref.37 (37) that if the Klein
paradox in Dirac equation is treated at the level of QFT, their result turns
out to be the same form as our Eqs.(A.24)-(A.28), rather than
Eqs.(A.30)-(A.31).
## Acknowledgements
We thank E. Bodegom, T. Chang, Y. X. Chen, T. P. Cheng, X. X. Dai, G.
Tananbaum, V. Dvoeglaznov, Y. Q. Gu, F. Han, J. Jiao, A. Kellerbauer, A.
Khalil, R. Konenkamp, D. X. Kong, J. S. Leung, P. T. Leung, Q. G. Lin, S. Y.
Lou, D. Lu, Z. Q. Ma, D. Mitchell, E. J. Sanchez, Z. Y. Shen, Z. Q. Shi, P.
Smejtek, X. T. Song, R. K. Su, Y. S. Wang, Z. M. Xu, X. Xue, J. Yan, F. J.
Yang, J. F. Yang, R. H. Yu, Y. D. Zhang and W. M. Zhou for encouragement,
collaborations and helpful discussions.
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|
arxiv-papers
| 2012-02-13T14:39:23 |
2024-09-04T02:49:27.382723
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Guang-jiong Ni, Suqing Chen and Jianjun Xu",
"submitter": "Jianjun Xu",
"url": "https://arxiv.org/abs/1202.2747"
}
|
1202.2771
|
11institutetext: Microsoft Research New England, One Memorial Drive,
Cambridge, MA 02142 11email: {borgs,jchayes}@microsoft.com 22institutetext:
Computer and Information Science Department, University of Pennsylvania,
3330 Walnut Street, Philadelphia, PA 19104
22email: brautbar@cis.upenn.edu 33institutetext: Computer Science Department,
University of Southern California,
941 Bloom Walk, Los Angeles, CA 90089
33email: shanghua@usc.edu
# A Sublinear Time Algorithm for PageRank Computations
Christian Borgs 11 Michael Brautbar 22 Jennifer Chayes 11 Shang-Hua Teng 33
###### Abstract
In a network, identifying all vertices whose PageRank is more than a given
threshold value $\Delta$ is a basic problem that has arisen in Web and social
network analyses. In this paper, we develop a nearly optimal, sublinear time,
randomized algorithm for a close variant of this problem. When given a
directed network $G=(V,E)$, a threshold value $\Delta$, and a positive
constant $c>3$, with probability $1-o(1)$, our algorithm will return a subset
$S\subseteq V$ with the property that $S$ contains all vertices of PageRank at
least $\Delta$ and no vertex with PageRank less than $\Delta/c$. The running
time of our algorithm is always $\tilde{O}(\frac{n}{\Delta})$. In addition,
our algorithm can be efficiently implemented in various network access models
including the Jump and Crawl query model recently studied by [6], making it
suitable for dealing with large social and information networks.
As part of our analysis, we show that any algorithm for solving this problem
must have expected time complexity of ${\Omega}(\frac{n}{\Delta})$. Thus, our
algorithm is optimal up to logarithmic factors. Our algorithm (for identifying
vertices with significant PageRank) applies a multi-scale sampling scheme that
uses a fast personalized PageRank estimator as its main subroutine. For that,
we develop a new local randomized algorithm for approximating personalized
PageRank which is more robust than the earlier ones developed by Jeh and Widom
[9] and by Andersen, Chung, and Lang [2].
## 1 Introduction
A basic problem in network analysis is to identify the set of its vertices
that are “significant.” For example, the significant nodes in the web graph
defined by a query could provide the authoritative content in web search; they
could be the critical proteins in a protein interaction network; and they
could be the set of people (in a social network) most effective to seed the
influence for online advertising. As the networks become larger, we need more
efficient algorithms to identify these “significant” nodes.
### 1.1 Identifying Nodes with Significant PageRanks: Our Results
The meaning of ’significant’ vertices depend on the semantics of the network
and the applications. In this paper, we focus on a particular measure of
significance — the PageRanks of the vertices. PageRank was introduced by Page
and Brin in their seminal work for ranking webpages [11]. Mathematically, the
PageRank (with restart constant, also known as the teleportation constant,
$\alpha$) of a web-page is proportional to the the probability that the page
is visited by a random surfer who explores the web using the following simple
random walk: at each step, with probability $(1-\alpha)$ go to a random
webpage linked to from the current page, and with probability $\alpha$,
restarts the process from a randomly chosen page. For reasons to be cleared
shortly, we consider a normalization of the PageRank so that the sum of the
PageRank values over all vertices is equal to $n$, the number of vertices in
the network. In other words, suppose $\text{PageRank}(u)$ denote the PageRank
of vertex $u$ in the network $G=(V,E)$. Then,
$\sum_{u\in V}\text{PageRank}(u)=n.$
PageRank has been used by the Google search engine and has found applications
in wide range of data analysis problems [7, 4]. In this context, the problem
of identifying “significant” vertices could be illustrated by the following
search problem: Let Top PageRanks denote the problem of identifying all
vertices whose PageRanks in a network $G=(V,E)$ are more than a given
threshold value $1\leq\Delta\leq|V|$.
In this paper, we consider for the following close variant of Top PageRanks:
> Significant PageRanks: Given a network $G=(V,E)$, a threshold value
> $1\leq\Delta\leq|V|$ and a positive constant $c>1$, compute, with success
> probability $1-o(1)$, a subset $S\subseteq V$ with the property that $S$
> contains all vertices of PageRank at least $\Delta$ and no vertex with
> PageRank less than $\Delta/c$.
We develop a nearly optimal, sublinear time randomized algorithm for
Significant PageRanks for any fixed $c>3$. The running time of our algorithm
is always $\tilde{O}(\frac{n}{\Delta})$. We show that any algorithm for
Significant PageRanks must have time complexity of
${\Omega}(\frac{n}{\Delta})$. Thus, our algorithm is optimal up to logarithmic
factors. Our Significant PageRanks algorithm applies a multi-scale sampling
scheme that uses a fast personalized PageRank estimator (see below) as its
main subroutine.
### 1.2 Matrix Sampling and Personalized PageRank Approximation
While the PageRank of a vertex captures the importance of the vertex
collectively assigned by all vertices in the network, as pointed out by
Haveliwala [8], one can use the distributions of the following random walk to
define the pairwise contributions of significances: Given a teleportation
probability $\alpha$ and a starting vertex $u$ in a network $G=(V,E)$, at each
step, with probability $(1-\alpha)$ go to a random neighboring vertex, and
with probability $\alpha$, restarts the process from $u$. For $v\in V$, the
probability that $v$ is visited by this random process, denoted by
$\text{PersonalizedPageRank}_{u}(v)$, is the $u$’s personal PageRank
contribution of significance to $v$. It is not hard to verify that
$\forall u\in V$, $\displaystyle\sum_{v\in
V}\text{PersonalizedPageRank}_{u}(v)=1;\mbox{\ and }$ $\forall v\in V$,
$\displaystyle\text{PageRank}(v)=\sum_{u\in
V}\text{PersonalizedPageRank}_{u}(v).$
Personalized PageRanks has been widely used to describe personalized behavior
of web-users [11] as well as for designing good network clustering
techniques[2]. As a result, fast algorithms for computing or approximating
personalized PageRank can be very useful. One can approximate PageRanks and
personalized PageRanks by the power method [4], which involves costly matrix
multiplications for large scale networks. Applying effective truncation, Jeh
and Widom [9] and Andersen, Chung, and Lang [2] developed personalized
PageRank approximation algorithms that can find an $\epsilon$-additive
approximation in time proportional to the product of $\epsilon^{-1}$ and the
maximum in-degree in the graph.
Our sublinear-time algorithm for Significant PageRanks also requires fast
subroutines for estimating personalized PageRanks. It uses a multi-scale
sampling approach by selecting a set of precision parameters
$\\{\epsilon_{1},...,\epsilon_{h}\\}$ where $h$ depends on $n$ and $\Delta$,
$\epsilon_{i}=1/2^{i}$. Then, for each $i$ in range $1\leq i\leq h$, it
computes the $\epsilon_{i}$-precise personalized PageRanks defined by a sample
of $\tilde{O}({\epsilon_{i}n}/{\Delta})$ vertices. For networks with constant
maximum degrees, we can simply use the Jeh-Widom or Andersen-Chung-Lang
personalized PageRank approximation algorithms in our multi-scale sampling
scheme. However, for networks such as web graphs and social networks that may
have nodes with large degrees, these two earlier algorithms are not robust
enough for our purpose.
We develop a new local algorithm for approximating personalized PageRank that
satisfies the desirable robust property that the multi-scale sample scheme
requires. Given $\rho,\epsilon>0$ and a starting vertex $u$ in a network
$G=(V,E)$, our algorithm estimates each entry in the personalized PageRank
vector,
$\text{PersonalizedPageRank}_{u}(.)$
defined by $u$ to a multiplicative factor of at most $(1+\rho)$ plus an
additive precision error of at most $\epsilon$ 111Formally, estimated value
$\hat{val}$ of $val$ would have the property that
$(1-\rho)\cdot\text{val}-\epsilon\leq\hat{\text{val}}\leq(1+\rho)\cdot{\text{val}}+\epsilon$..
The time complexity of our algorithm is
$O(\frac{\log(|V|)\log(\epsilon^{-1})}{\epsilon\rho^{2}})$. Our algorithm
requires a careful simulation of random walks from the starting node $u$ to
ensure that its complexity does not depends on the degree of any node.
Our algorithms can be efficiently implemented in various network querying
models assuming no direct global access to the network. In particular, our
algorithms can be efficiently implemented in the Jump and Crawl query model
[6], making the algorithm suitable for processing large social and information
networks.
In particular, our sublinear algorithm for Significant PageRanks could be used
in Web search engines, which often need to build a core of web-pages to be
later used for web-search. It is desirable that pages in the core have high
PageRank values. These search engines usually apply crawling to discover new
significant pages and add them to the core. The property that our sublinear-
time algorithms have a natural implementation in the Jump and Crawl model may
make them useful in a search engine for selecting pages with high PageRank
values to update the current core by using them to replace the existing core
pages that have relatively low PageRank values. We anticipate that our
algorithm for Significant PageRanks will be useful for many other network
analysis tasks.
### 1.3 Additional Related Work
For personalized PageRanks approximation, in addition to the work of [9, 5, 2,
11, 4], Andersen et al [1] developed a ’backward’ version of the local
algorithm of [2]. Their algorithm finds all nodes that contribute at least
some fixed fraction $\rho$ to a page’s PageRank in time $O(d_{\text{max-
out}}/{\rho})$ where $d_{\text{max-out}}$ is the maximum out-degree in the
network. This algorithm can be used to provide some reliable estimate to a
node’s PageRank. For example, for a given $k$, in time $\tilde{\Theta}(k)$ it
can bound the total contribution from the $k$ highest contributors to the
node’s PageRank. However, for networks with large $d_{\text{max-out}}$, its
complexity may not be sublinear.
As suggested in [1], one can view the entire set of personalized PageRanks
(defined by all vertices in a network) as an $|V|\times|V|$ matrix, which is
referred to as the PageRank matrix of the graph. In the PageRank matrix, each
row represents the personalized PageRanks from a particular vertex, and each
column represents the contributions to its PageRank from all vertices in the
network. Note that the sum of each row is 1 and the sum of the $u^{th}$ column
is the PageRank of $u$.
In light of this, the problem of Significant PageRanks can be viewed as a
matrix sparsification or matrix approximation problem. There has been a large
body of work of finding a low complexity approximation to a matrix that
preserves some of its properties. Perhaps the most relevant one to our goal is
a low rank matrix approximation under the $l_{2}$ matrix norm.
All current methods for finding such low rank approximations runs in time at
least linear in the size of the input matrix. See [10] for a survey of recent
results.
Next, a linear time Monte Carlo based method to estimate PageRank of all nodes
is devised in [3]. The method is based on running constant number of random
walks from each of the nodes in the network.
Last, in the context of sublinear time graph algorithms, our research is
related to the work of [12], in which sublinear time algorithms are presented
for estimating several quantities. Our implementation of the Jump and Crawl
query model can be viewed as a stringent type of the adjacency-list graph
model used in [12].
### 1.4 Organization
Section $2$ contains the needed definitions and notations. Section $3$
presents our multi-scale sampling algorithm for Significant PageRanks. In
section $4$ we provide a lower bound construction for Significant PageRanks.
In Section $5$ we give a robust local algorithm for approximating personalized
PageRank vectors.
## 2 Preliminaries
We consider a network which is defined as a directed graph $G=(V,E)$ with $n$
nodes and $m$ edges. Usually, a network is massive. Our algorithms access a
network using a rather natural implementation of the Jump-and-Crawl query
model of [6] developed for processing large social and information networks.
The Jump and Crawl model is concerned with informational complexity of nodes,
where each node access reveal its full list of adjacent neighbors at no extra
cost. Our algorithms shall be designed to work under the following compelling
implementation of the Jump-and-Crawl query model. We allow two types of
queries:
* •
Jump: A call to the Jump query needs no input and returns a uniformly at
random node from the network.
* •
RandomCrawl: A call to the RandomCrawl query requires a vertex $v$ as input.
RandomCrawl(v) returns a uniformly at random out-neighbor of $v$.
Note for example, that the random surfer procedure used in the definition of
PageRank is itself a natural algorithm under our implementation of the Jump-
and-Crawl query model.
We now move to define personalized PageRank as well as PageRank.
Mathematically, the personalized PageRank vector of a node $v$ is the
stationary point of the following equation:
$\text{PersonalizedPageRank}_{v}(\cdot)=\alpha\textbf{1}_{v}+(1-\alpha)\text{PersonalizedPageRank}_{v}(\cdot)\cdot
D^{-1}A,$
where $\alpha$ is the teleportation probability, $A$ is the adjacency matrix
of the directed network $G=(V,E)$ so $A(i,j)=1$ iff $(i,j)\in E$. In this
notation, $D$ is a diagonal matrix with $d_{out}(v)$ at entry $(v,v)$ and
$\textbf{1}_{v}$ is the indicator vector of $v$. We will follow the standard
[4] by assuming that each node has at least one out-link 222Otherwise, as
commonly done [4], consider that node as having out links into all nodes in
the network..
Then, one can define the RageRank vector as
$\text{PageRank}(\cdot)=\sum_{v\in V}\text{PersonalizedPageRank}_{v}(\cdot)$
Note that in this definition, the sum of the all PageRank values is equal to
$n$.
Following [1], we define a matrix PPR (short for personalized PageRank) to be
the $n\times n$ matrix, whose $v^{th}$ row is
$\text{PersonalizedPageRank}_{v}(\cdot)$.
Unless stated otherwise, for any $x$, $\log(x)$ would mean $\log_{2}(x)$.
## 3 Multi-scale Matrix Sampling and Approximation of PageRank
In this section, we present our nearly optimal, sublinear time algorithm for
Significant PageRanks. Recall that
> Significant PageRanks: Given a network $G=(V,E)$, a threshold value
> $1\leq\Delta\leq|V|$ and a positive constant $c>1$, compute, with
> probability $1-o(1)$, a subset $S\subseteq V$ with the property that $S$
> contains all vertices of PageRank at least $\Delta$ and no vertex with
> PageRank less than $\Delta/c$.
Note that the PageRank value of each vertex is at least $\alpha$ and at most
$n$. Instrumental to our algorithm, we present a multi-scale algorithm for
sampling the PageRank matrix PPR that achieves, for any fixed $c>3$, the
following goals: The algorithm makes $\tilde{O}(\frac{n}{\Delta})$ total
queries and updates, and with high probability,
1. 1.
For each vertex with PageRank value at least $\Delta$, the sum of the sampled
entries of of the column corresponding to the vertex will provide a quality
estimate to the PageRank value of that vertex.
2. 2.
The algorithm does not return any vertex whose PageRank value is less than
$\Delta/c$.
In our algorithm, we will use a new local algorithm ApproxRow for personalized
PageRank approximation. Algorithm ApproxRow takes three input parameters:
$v\in V$, an additive error factor $\epsilon\in(0,1)$ and a multiplicative
factor $\rho\in(0,1)$. It returns an approximation to
$\text{PersonalizedPageRank}_{v}(\cdot)$ such that for every
$\text{PPR}(v,j)>\epsilon$, it returns a non-negative estimated value between
$(1-\rho)\text{PPR}(v,j)-\epsilon$ to $(1+\rho)\text{PPR}(v,j)+\epsilon$. The
running time of ApproxRow is essentially
$O\large(\frac{\log(n)\log(\epsilon^{-1})}{\epsilon\rho^{2}}\large)$.
ApproxRow and its analysis will be presented in Section 5.
We start with some high-level idea of our multi-scale sampling algorithm. To
assist our exposition, we will present our algorithm and its analysis for
$c=6$. Both are easily extended to any other constant value $c>3$. Our
algorithm will use $O(\log n)$ precision scales: $\epsilon_{t}=2^{-t}$ for
$0\leq t\leq\log(\frac{4n}{\Delta})$. We conceptually divide each column of
the PPR matrix into chunks, where the chunk corresponding to $\epsilon_{t}$
contains its entries with values between $\epsilon_{t}$ to $2\epsilon_{t}$.
Thus, we ignore all entries in the PPR matrix column of value less than
$\frac{\Delta}{4n}$, the finest scale. Note that entries with value at most
$\frac{\Delta}{4n}$ can contribute to at most a quarter to the PageRank of a
vertex whose PageRank value is least $\Delta$.
If the sum of a chunk’s entries is at least $\Delta/(2\log(n))$, we will refer
to it as a heavy chunk. The central idea of our algorithm is to efficiently
generate robust estimates of the sums for all heavy chunks, as we shall show
that it is also sufficient to only provide estimates to heavy chunks.
As the entries in each chunk are within a factor of 2 of each other, we then
reduce the task of estimating the sum in a chunk to the problem of
approximately counting the size of the chunk. Then conceptually, we estimate
the size of each heavy chunk at scale $\epsilon_{t}$ by taking
$\tilde{O}(\epsilon_{t}4n/\Delta)$ random entries from its column and counting
the numbers of samples in this chunk. The challenge we need to overcome is to
efficiently sample all heavy chunks at a scale simultaneously.
This is where we will use our local PageRank approximation algorithm
ApproxRow, which in $O(\frac{\log(n)\log(\epsilon^{-1})}{\epsilon})$ time when
given a vertex $v$, returns robust estimates to all entries of values at least
$\epsilon$ in $v$’s row in the PPR matrix. To achieve $\tilde{O}(n/\Delta)$
queries and running time, we call ApproxRow
$\tilde{\Theta}(\frac{n}{\Delta}\epsilon_{t})$ times at scale $\epsilon_{t}$,
and we will show that it is sufficient to sample this much (or little).
In the last step of the algorithm, for each node $j$, we will simply sum up
over all scales $\epsilon_{t}$, its estimated values weighted by a normalizing
factor $\frac{\Delta}{\epsilon_{t}2\log^{2}(n)}$. Then the algorithm will
output only those $j$’s where the sum is at least $\frac{\Delta}{4}$ and their
estimated PageRank values.
A detailed pseudo-code of our algorithm, ApproximatePageRank, is given below.
Algorithm 1 ApproximatePageRank
0: PageRank threshold $\Delta$, a network $G=(V,E)$ on $n$ nodes accessible
only by Jump and RandomCrawl queries. // First-Part //
1: Initialize a binary search tree, ChunkTree, indexed lexicographically by a
two-tuple key $(\text{nodeID},\epsilon)$.
2: for $t=0$ to $\log(\frac{n}{4\Delta})$ do
3: Set the additive error $\epsilon_{t}=2^{-t}$.
4: for $(\frac{n}{\Delta}\epsilon_{t}4\log^{2}(n))$ times do
5: Jump to a random node, call it $v$.
6: Call $\text{list}=\text{ApproxRow}(v,\frac{\epsilon_{t}}{2},\frac{1}{2})$
and update the chunk size estimate affiliated vertices in list as the
following:
7: for each pair $(\text{nodeID},\epsilon_{t})$ in the list do
8: if there exists an entry $e$ with key $(\text{nodeID},\epsilon_{t})$ in
ChunkTree then
9: Update entry $e$’s value by adding 1 to its current value.
10: else
11: Create an entry in ChunkTree with key $(\text{nodeID},\epsilon_{t})$ and
value 1.
12: end if
13: end for
14: end for (at scale $\epsilon_{t}$).
15: end for (for all scales) // Second-Part //
16: Initialize a final tree, called TreeofPageRankValues, indexed by key
$(\text{nodeID})$.
17: for all elements (chunks) in ChunkTree that all belong to same node $i$
(namely, have $i$ as the first part of their key) do
18: if chunk has value, val, at least $\frac{1}{2}\log(n)$ then
19: Let $\epsilon$ be the second part of the chunk’s key.
20: Add $\frac{\Delta}{2\epsilon\log^{2}(n)}$ to the entry indexed by $(i)$ in
TreeofPageRankValues.
21: end if
22: Output all elements in TreeofPageRankValues with at least ${\Delta}/{4}$
23: end for
In the proofs for the following two theorems, we will analyze the performance
of this algorithm. Note that we will ignore the dependence of the running time
on $\alpha$ as for all standard PageRank computations, it is taken to be a
fixed constant independent of input size [4].
###### Theorem 3.1 (Complexity of ApproximatePageRank)
The runtime of algorithm ApproximatePageRank is upper bounded by
$\tilde{O}({n}/{\Delta})$.
###### Proof
The algorithm uses $O(\log(n/\Delta))$ scales. In First-Part of the algorithm,
for scale $\epsilon_{t}$, it makes $\frac{n}{\Delta}\epsilon_{t}4\log^{2}(n)$
Jump queries and for each query it runs
$\text{ApproxRow}(v,\epsilon_{t}/2,1/2)$, where $v$ represents the random
vertex returned by the query. ApproxRow then has a runtime of
$O(\frac{\log(n)\log(\epsilon_{t}^{-1})}{\epsilon_{t}})$. Thus, the total
runtime complexity is $\tilde{O}(\frac{n}{\Delta})$ as the finest scale is
${\Delta}/{n}$ and there are at most $\log n$ scales. In addition to the time
spent on querying the network, the algorithm takes $\Theta(\log(n))$ per step
overhead for each access/update in its data structure.
In Second-Part of the algorithm, it makes no new queries. As there are only
$\tilde{O}({n}/{\Delta})$ items in the data structure ChunkTree and then
TreeofPageRankValues, the complexity of this summation part is
$\tilde{O}({n}/{\Delta})$. The last step of outputting all nodes in the tree
with value bigger than a threshold can easily be done in linear time in the
size of the tree, which is $\tilde{O}({n}/{\Delta})$.
###### Theorem 3.2 (Correctness of ApproximatePageRank)
Given $\Delta$ and constant $c>3$, ApproximatePageRank outputs, with
probability $1-o(1)$, all nodes with PageRank at least $\Delta$ but no node
with PageRank smaller than333Again for exposition, we present our algorithm
and its analysis for $c=6$. We later show that the theorem on a slightly
modified algorithm holds for any constant $c>3$. ${\Delta}/{c}$.
###### Proof
For $v\in V$, let $(p_{1}^{v},p_{2}^{v},\ldots,p_{n}^{v})$ be $v$’s column in
the PPR matrix. Let $\text{ChunkSet}{(v,\epsilon)}=\\{i:\epsilon\leq
p_{i}^{v}<2\epsilon\\}$,
$\text{ChunkSize}{(v,\epsilon)}=|\text{ChunkSet}{(v,\epsilon)}|$, and
$\text{ChunkSum}{(v,\epsilon)}=\sum_{i=1}^{n}{\\{p_{i}^{v}:\epsilon\leq
p_{i}^{v}<2\epsilon\\}}.$
Recall a chunk is heavy is its chunksum ${\Delta}/{\log(n)}$. We now prove
that at the end of First-Part in Algorithm ApproximatePageRank, all heavy
chunks are well approximated.
To focus on the essence of the proof for multi-scale matrix sampling, we first
assume that all the values returned by ApproxRow are exact (with no error at
all). We call this assumption, the perfect row approximation assumption. We
will later show that when removing this assumption the approximation scheme
would only be affected by a multiplicative factor of three, namely the
effective value of $c$ in Significant PageRanks would be one third its value
under perfect row approximations.
###### Lemma 1 (key lemma)
Let $\epsilon_{t}=2^{-t}$, for $1\leq t\leq\frac{n}{4\Delta}$. The following
holds with probability $1-o(1)$:
* •
If $\text{ChunkSum}{(v,\epsilon_{t})}\geq\frac{\Delta}{2\log(n)}$ then at the
end of First-Part in the algorithm the entry in ChunkTree with key
$(v,\epsilon_{t})$, namely the algorithm’s approximation of
$\text{ChunkSize}{(v,\epsilon_{t})}$, is at least $\log n/2$ and is between
$\frac{\text{ChunkSize}{(v,\epsilon_{t})}}{\Delta}\cdot\epsilon_{t}2\log^{2}(n)$
to
$\frac{\text{ChunkSize}{(v,\epsilon_{t})}}{\Delta}\cdot\epsilon_{t}8\log^{2}(n)$.
* •
If $\text{ChunkSum}{(v,\epsilon_{t})}\leq\frac{\Delta}{4\log(n)}$ then at the
end of First-Part in the algorithm the entry in ChunkTree with key
$(v,\epsilon_{t})$, namely the algorithm’s approximation of
$\text{ChunkSize}{(v,\epsilon_{t})}$, is smaller than $\frac{\log(n)}{2}$.
###### Proof
Note that
$\frac{1}{2\epsilon_{t}}\text{ChunkSum}{(v,\epsilon_{t})}\leq\text{ChunkSize}{(v,\epsilon_{t})}\leq\frac{1}{\epsilon_{t}}\text{ChunkSum}{(v,\epsilon_{t})}$.
So, if $\text{ChunkSum}{(v,\epsilon_{t})}\geq\frac{\Delta}{2\log(n)}$ then
${\text{ChunkSize}{(v,\epsilon_{t})}}/n\geq{\Delta}/{(4\epsilon_{t}n\log n)}$.
Thus, when sampling $4\epsilon_{t}n\log^{2}(n)/\Delta$ random rows (as in line
$5$ of the algorithm), the expected number of entries in the chunk that
ApproxRow discovers is at least
${\text{ChunkSize}{(v,\epsilon_{t})}}\epsilon_{t}4\log^{2}(n)/{\Delta}\geq\log(n)$.
By a standard multiplicative Chernoff bound (see appendix), with probability
$1-o(1)$, after multiplying the count by
${\Delta}/{(2\epsilon_{t}\log^{2}n)}$, we can approximate
$\text{ChunkSize}{(v,\epsilon_{t})}$ within a multiplicative factor of 2\.
Moreover, if $\text{ChunkSum}{(v,\epsilon_{t})}\leq\frac{\Delta}{4\log(n)}$
then, its estimated value is at most twice its value, namely smaller than
$\frac{\log(n)}{2}$.
###### Lemma 2
The following holds with probability $1-o(1)$ under the perfect row
approximation assumption:
* •
If $\text{PageRank}(v)\geq\Delta$, then the algorithm will output $v$ and will
estimate its PageRank value to a value between $\text{PageRank}(v)/4$ to
$2\text{PageRank}(v)$.
* •
If $\text{PageRank}(v)<\Delta/8$, then the algorithm will not output $v$.
* •
If $\Delta/8\leq\text{PageRank}(v)<\Delta$, then the algorithm might output
$v$. If $v$ is outputted, then its estimated PageRank value is between
$\text{PageRank}(v)/16$ to $2\text{PageRank}(v)$.
###### Proof
By lemma 1, that the sums of each heavy chunk are well estimated to within a
multiplicative factor of 2.
Since there are at most $\log n$ chunks in column, the contribution from all
non-heavy chunks is at most $\log n(\Delta/(2\log n))=\Delta/2$. Thus, if
$v$’s PageRank is at least $\Delta$, then the contribution from its heavy
chunks is at least ${\Delta}/2$. Consequently, and the algorithm’s
approximation to $v$’s PageRank will be at least $\Delta/4$ and at most
$2\Delta$, and this vertex will be outputted.
We can similarly establish the other two cases as stated in the lemma.
We now turn to discuss the effect of having only approximate values computed
in ApproxRow calls on the guarantees of ApproxmatePageRank.
###### Lemma 3
Given parameters $0<\epsilon,\rho<1$, removing the perfect row approximation
assumption changes the approximation constant $c$ by at most $3$ times its
value as well as changes the estimated PageRank values computed by
ApproximatePageRank to be at most three times their value.
###### Proof
The PPR matrix is effectively computed using calls to ApproxRow by the
algorithm.
Given $\epsilon>0$, consider an element $\epsilon\leq PPR(v,j)\leq 2\epsilon$
for some nodes $v,j$. There are two sources for having this element estimator
differ from it real value. First, ApproxRow (with parameters
$\epsilon=\epsilon/2$ and $\rho=1/2$) computes approximate values so the
estimated value is between $(1-\rho-1/4)$ its real value to $(1+\rho+1/4)$ (we
could put the additive $\epsilon/4$ error in the multiplicative approximation
factor since $\epsilon\leq PPR(v,j)\leq 2\epsilon$). In particular, in the
algorithm we pick $\rho=1/2$. However, one could replace both $\rho$ and
$\epsilon$ with smaller values to get an approximation that gets closer to the
true value: Replacing $\rho$ by $k_{1}\rho$ and $\epsilon$ by $k_{2}\epsilon$
for any integral $k_{1},k_{2}$ would increase the total runtime by only a
factor of $k_{1}^{2}k_{2}\frac{\log{(k_{2})}}{\log(\epsilon^{-1})}$. The
second source why the estimator differs from its true value is double
counting. An element with a true value between $\epsilon/2$ to $\epsilon$ as
well as one with a true value between $2\epsilon$ to $4\epsilon$ could appear
in a realization as an element with a value between $\epsilon$ to $2\epsilon$.
However (by applying Chernoff bound), elements with true value smaller than
$\epsilon/2$ as well as those with value bigger than $4\epsilon$ would not
appear as such. Thus due to double counting the sum of elements in each column
can be at most three times its real value. If we denote the PageRank of node
$j$ by $\Delta(j)$ and the value it gets from the realized column values by
$\Delta^{\prime}(j)$ then,
$(1-k_{1}\rho-k_{2}/2)\Delta(j)\leq\Delta^{\prime}(j)\leq
3(1+k_{1}\rho+k_{2}/2)\Delta(j).$
In particular, algorithm ApproximatePageRank uses $\rho=\frac{1}{2}$,
$k_{1}=1$ and $k_{2}=\frac{1}{2}$ which gives
$\frac{1}{4}\Delta(j)\leq\Delta^{\prime}(j)<6\Delta(j).$
This ends the proof of Theorem 3.2.
## 4 Lower Bound Construction for PageRank Approximations
We now turn to prove a corresponding lower bound for PageRank approximations.
The lower bound construction will show that, any algorithm, making less than
$\Omega(\frac{n}{\Delta})$ Jump and Crawl queries, will fail, with constant
probability, to find any node with PageRank at least $\Delta$ on the graph.
This holds true for any type of implementation of a Crawl query (including the
RandomCrawl one). Given positive integers $n$ and $\Delta<\frac{n}{2}-1$, we
construct an undirected graph on $n$ nodes made of a path subgraph on $n-d-1$
nodes and an isolated star subgraph on $d+1$ nodes, where $d=2\Delta$. See
figure 1 for an illustration. Fix $0<\alpha<1$, the teleportation probability.
By solving the PageRank equations it is not hard to check that each node on
the path subgraph has PageRank value of $1$, the hub of the subgraph has
PageRank $\frac{d}{2}+\frac{1}{2(1-\alpha)}$ and each leaf of the star
subgraph has PageRank of
$\frac{1}{d}(d+1-\frac{d}{2}-\frac{1}{2(1-\alpha)})\leq\frac{1}{2}+\frac{1}{d}$.
As $\Delta=\frac{d}{2}$, the only node with PageRank at least $\Delta$ is the
hub of the star subgraph. However, for any $\epsilon>0$, in order to find any
node that belongs to the star subgraph one needs to make, with probability at
least $1-\frac{1}{e}-\epsilon$, at least
$\frac{n}{d}=\Omega(\frac{n}{\Delta})$ Jump queries.
Figure 1: An example illustrating the path-star graph of the lower bound
construction for PageRank computations.
## 5 Local Robust Computation of Personalized PageRank
We now describe a method, ApproxRow, based only on local computations that
approximates a node’s personalized PageRank vector. The pseudo-code is given
on the next page.
###### Theorem 5.1 (Complexity of ApproxRow)
For any node $v$ and values $0<\epsilon,\rho<1$, the runtime of
$\text{ApproxRow}(v,\epsilon,\rho)$ is upper bounded by
$O\large(\frac{\log(n)\log(\epsilon^{-1})}{\epsilon\rho^{2}\log_{2}(\frac{1}{1-\alpha})}\large)$.
###### Proof
The algorithm performs $\frac{1}{\epsilon\rho^{2}}\cdot 16\log(n)$ rounds
where at each round it simulates a random walk with termination probability of
$\alpha$ for at most $length$ steps. Each step is simulated by taking a Jump
(’termination’ step) with probability $\alpha$ and taking a RandomCrawl step
with probability $1-\alpha$. Thus the total number of queries used is
$\frac{16\log(n)}{\epsilon\rho^{2}}\cdot\log_{\frac{1}{(1-\alpha)}}(\frac{3}{\epsilon})=O\large(\frac{\log(n)\log(\epsilon^{-1})}{\epsilon\rho^{2}\log_{2}(\frac{1}{1-\alpha})}\large)$.
Algorithm 2 ApproxRow
0: A node $v$ in $G=(V,E)$, additive error parameter $0<\epsilon<1$,
multiplicative approximation parameter $0<\rho<1$, teleportation probability
$0<\alpha<1$.
1: Initialize a binary search tree NodeCountTree where the key is a node’s
identity.
2: Set $length=\log_{\frac{1}{(1-\alpha)}}(\frac{4}{\epsilon})$.
3: Set $r=\frac{1}{\epsilon\rho^{2}}\cdot 16\log(n)$.
4: for $r$ times do
5: Run one realization of a random walk with restart probability $\alpha$: the
walk starts at $v$ and at each time makes, with probability $\alpha$ a
’termination’ step by returning to $v$ and terminating, and with probability
$1-\alpha$ a RandomCrawl step. The walk is artificially stopped after $length$
steps if it has not terminated already.
6: if the walk visited a node $u$ just before making a termination step then
7: Add $1$ to the count stored at $u$’s entry in NodeCountTree.
8: end if
9: Output all nodes in NodeCountTree together with their average count (over
the $r$ rounds).
10: end for
###### Theorem 5.2 (Correctness of ApproxRow)
For any node $v$ and values $0<\epsilon,\rho<1$, with probability of at least
$1-\Theta(\frac{1}{n^{2}})$, $\text{ApproxRow}(v,\epsilon,\rho)$ computes a
list $l$ with the following properties:
* •
Every node $j$ that is outputted in the list $l$ has an estimated value which
is non-negative and lies between $(1-\rho)\text{PPR}(v,j)-\frac{\epsilon}{4}$
to $(1+\rho)\text{PPR}(v,j)$.
* •
Every node not in the list $l$ has $\text{PPR}(v,j)\leq\epsilon/2$.
###### Proof
We start with an observation. The personalized PageRank contribution from a
node $v$ to node $j$ is exactly the probability that a random walk that starts
at $v$, and at each time step terminates with probability $\alpha$, and with
probability $1-\alpha$ moves to a random out-link of the node it is currently
at, was at node $j$ one step before termination. Define $1_{v}$ to be the
indicator vector of $v$. The proof of the observation follows from a series of
algebraic manipulations on the definition of the PersonalizedPageRankVector of
$v$:
$\text{PersonalizedPageRank}_{v}(\cdot)=\alpha\textbf{1}_{v}+(1-\alpha)\text{PersonalizedPageRank}_{v}(\cdot)\cdot
D^{-1}A.$
Solving the system gives
$\text{PersonalizedPageRank}_{v}(\cdot)=\alpha\textbf{1}_{v}(I-(1-\alpha)D^{-1}A)^{-1}=\alpha\textbf{1}_{v}\sum_{i=0}^{\infty}{((1-\alpha)D^{-1}A)^{i}}$.
This last equation makes the observation clear.
Given a node $j$, denote by $p_{k}(v,j)$ the contribution to $v$’s
Personalized PageRank vector from walks that are of length at most $k$. By the
above observation,
$p_{k}(v,j)=\alpha\textbf{1}_{v}\sum_{i=0}^{k}{((1-\alpha)D^{-1}A)^{i}}$.
We ask how much is contributed to $j$’s entry in the Personalized PageRank
vector of $v$ from walks of length bigger or equal to $k$. The contribution is
at most $(1-\alpha)^{k}$ since the walk needs to survive at least $k$
consecutive steps. Taking $(1-\alpha)^{k}\leq\frac{\epsilon}{4}$ will
guarantee that at most $\frac{\epsilon}{4}$ is lost by only considering walks
of length smaller than $k$, namely: $\text{PPR}(v,j)-\frac{\epsilon}{4}\leq
p_{k}(v,j)\leq\text{PPR}(v,j)$.
For that it suffices to take
$k=\log_{\frac{1}{(1-\alpha)}}{(\frac{4}{\epsilon})}$. This is exactly
$length$, the length of each walk the algorithm simulates, is set to.
Next, the algorithm provide a estimate of $p_{k}(v,j)$ by realizing walks of
length at most $k$. The algorithm does so by taking the average count over
$\frac{1}{\epsilon\rho^{2}}\cdot 16\log(n)$ trials. Denote the algorithm’s
output by $\hat{p}_{k}(v,j)$. Then, if $p_{k}(v,j)\geq\frac{\epsilon}{4}$, by
the multiplicative Chernoff bound,
$Pr(\hat{p}_{k}(v,j)>(1+\rho)p_{k}(v,j))\leq\exp(-2\log(n))$ and
$Pr(\hat{p}_{k}(v,j)<(1-\rho)p_{k}(v,j))\leq\exp(-2\log(n))$.
We can conclude that
$(1-\rho)(\text{PPR}(v,j)-\frac{\epsilon}{4})\leq\hat{p}_{k}(v,j)\leq(1+\rho)\text{PPR}(v,j)$.
In particular, nodes with $\text{PPR}(v,j)>\epsilon$ will be estimated to a
positive value and outputted as claimed.
Similarly, if $p_{k}(v,j)\leq\frac{\epsilon}{4}$ then by the multiplicative
Chernoff bound,
$Pr(\hat{p}_{k}(v,j)>\frac{\epsilon}{2})\leq\exp(-2\log(n))$. In this case we
conclude that $\hat{p_{k}}(v,j)\leq\frac{\epsilon}{2}$. Also,
$\text{PPR}(v,j)\leq\frac{\epsilon}{4}+\frac{\epsilon}{4}\leq\frac{\epsilon}{2}$
so $|\text{PPR}(v,j)-\hat{p}_{k}(v,j)|\leq\frac{\epsilon}{2}$.
## 6 Acknowledgments
We thank Brendan Lucier, Elchanan Mossel and Eugene Vorobeychik for their
suggestions and the anonymous reviewers for their helpful comments.
## References
* [1] Reid Andersen, Christian Borgs, Jennifer T. Chayes, John E. Hopcroft, Vahab S. Mirrokni, and Shang-Hua Teng. Local computation of pagerank contributions. Internet Mathematics, 5(1):23–45, 2008.
* [2] Reid Andersen, Fan R. K. Chung, and Kevin J. Lang. Local graph partitioning using pagerank vectors. In FOCS, pages 475–486, 2006.
* [3] K. Avrachenkov, N. Litvak, D. Nemirovsky, and N. Osipova. Monte carlo methods in pagerank computation: When one iteration is sufficient. SIAM Journal on Numerical Analysis, 45, 2007.
* [4] Pavel Berkhin. Survey: A survey on pagerank computing. Internet Mathematics, 2(1), 2005.
* [5] Pavel Berkhin. Bookmark-coloring approach to personalized pagerank computing. Internet Mathematics, 3(1), 2006.
* [6] Mickey Brautbar and Michael Kearns. Local algorithms for finding interesting individuals in large networks. In ICS, pages 188–199, 2010.
* [7] Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual web search engine. Computer Networks, 30(1-7):107–117, 1998.
* [8] T.H Haveliwala. Topic-sensitive pagerank: A context-sensitive ranking algorithm for web search. In Trans. Knowl. Data Eng, volume 15(4), pages 784––796, 2003\.
* [9] Glen Jeh and Jennifer Widom. Scaling personalized web search. In WWW, pages 271–279, 2003.
* [10] Ravindran Kannan. Spectral methods for matrices and tensors. In STOC, pages 1–12, 2010.
* [11] Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The pagerank citation ranking: Bringing order to the web. Stanford University 1998.
* [12] Ronitt Rubinfeld and Asaf Shapira. Sublinear time algorithms. SIAM Journal on Discrete Math, 25:1562–1588, 2011.
## Appendix: Concentration Bounds
###### Lemma 4 (multiplicative Chernoff bound)
Let $X_{i}$ be i.i.d. Bernoulli random variables with expectation $\mu$ each.
Define $X=\sum_{i=1}^{n}{X_{i}}$. Then,
* •
For $0<\lambda<1,Pr[X<(1-\lambda)\mu n]<\exp(-\mu n\lambda^{2}/2)$.
* •
For $0<\lambda<1,Pr[X>(1+\lambda)\mu n]<\exp(-\mu n\lambda^{2}/4)$.
* •
For $\lambda\geq 1,Pr[X>(1+\lambda)\mu n]<\exp(-\mu n\lambda/2)$.
|
arxiv-papers
| 2012-02-13T15:49:20 |
2024-09-04T02:49:27.395394
|
{
"license": "Public Domain",
"authors": "Christian Borgs, Michael Brautbar, Jennifer Chayes and Shang-Hua Teng",
"submitter": "Michael Brautbar",
"url": "https://arxiv.org/abs/1202.2771"
}
|
1202.2907
|
# The weight Enumerator of some irreducible cyclic codes
Yun Song , Zhihui Li ∗ College of Mathematics and Information Science, Shaanxi
Normal University, Xi’an, 710062, P. R. China
###### Abstract
Irreducible cyclic codes are one of the largest known classes of block codes
which have been investigated for a long time. However, their weight
distributions are known only for a few cases. In this paper, a class of
irreducible cyclic codes are studied and their weight distributions are
determined. Moreover, all codewords of some irreducible cyclic codes are
obtained through programming in order to explain their distributions. The
number of distinct nonzero weights in these codes dealt with in this paper
varies among 1,2,3,6,8.
###### keywords:
Gaussian periods; Period polynomial; Irreducible cyclic codes; Linear codes;
Weight distributions; Weight enumerator
## 1 Introduction
Let $q=p^{s}$ and $r=q^{m},$ where $s$ and $m$ are positive integers, and $p$
a prime. A linear $[n,k,d]$ code $C$ over $F_{q}$ is a $k$-dimensional
subspace of $F_{q}^{n}$ with minimum (Hamming) distance $d.$ Let $A_{i}$
denote the number of codewords in a code $C$ with Hamming weight $i.$ The
weight enumerator of $C$ is defined by
$1+A_{1}x+A_{2}x^{2}+\cdots+A_{n}x^{n}.\ \ \ \ \ \ \ (1)$
The sequence$(1,A_{1},\ldots,A_{n})$ is called the weight distribution of the
code.
Let $N>1$ be an integer dividing $r-1$, and put $n=\frac{(r-1)}{N}.$ Let
$\alpha$ be a primitive element of $F_{q^{m}}$ and $\theta=\alpha^{N}.$ The
set
$\textit{C(r,N)}=\\{(\textrm{Tr}_{r/q}(\beta),\textrm{Tr}_{r/q}(\beta\theta),\cdots,\textrm{Tr}_{r/q}(\beta\theta^{n-1}))|\beta\in
F_{r}\\}\ \ \ \ \ \ \ \ \ \ \ \ (2)$
is called an irreducible cyclic code (ICC) over $F_{q},$ where Tr is the trace
function from $F_{r}$ onto $F_{q}.$
The determination of the weight enumerator of ICC is a fascinating problem,
which provides some valuable informations about the performance for the code.
Important contributions in this direction can be found in [1-3]. McEliece
showed that the weights of an irreducible cyclic codes can be expressed as a
linear combination of Gauss sums via the Fourier transform[4]. Schmidt and
White gave a characterization of ICC with at most two weights[5]. Langevin
investigated a class of ICC with at most three weights[6]. However, in
general, determining the weight distribution of ICC is quite complicated. The
following special cases were studied in the literature.
$\bullet$ When $N|(q^{j}+1)$ for some $j$ being a divisor of $m/2,$ then codes
have two weights. These codes were discussed in [7].
$\bullet$ When $N=2,$ the weight distribution of $C(r,n)$ was founded by
Baumert and McEliece[7].
$\bullet$ When $N$ is a prime with $N\equiv 3(\rm{mod}\ 4)$ and
$ord_{q}(N)=(N-1)/2,$ the weight distribution was studied by Baumert and
Mykkeltveit[8].
$\bullet$ When $2\leq N\leq 4$, the number of distinct nonzero weights in the
ICC dealt with in [9] varies between one and four.
The objective of this paper is to determine the weight enumerator of a class
of ICC for all $N$ with $5\leq N\leq 8.$ The number of distinct nonzero
weights in these codes dealt with in this paper varies among 1,2,3,6,8.
## 2 Cyclotomic classes and period polynomials
To study the weight enumerator of the irreducible cyclic codes, we need to
introduce cyclotomic classes and period polynomials[10].
###### Definition 2.1
Let $r-1=nN$ for two positive integers $n>1$ and $N>1$, and let $\alpha$ be a
fixed primitive element of $F_{r}.$ Define
$C_{i}^{(N,r)}=\alpha^{i}\langle\alpha^{N}\rangle$ for $i=0,1,\ldots,N-1,$
where $\langle\alpha^{N}\rangle$ denotes the subgroup of $F_{r}^{*}$generated
by $\langle\alpha^{N}\rangle$. The cosets $C_{i}^{(N,r)}$ are called the
cyclotomic classes of order in $F_{r}$ .
###### Definition 2.2
The Gaussian periods are defined by
$\eta_{i}^{(N,r)}=\sum_{x\in C_{i}^{(N,r)}}\chi(x),\ \ \ \ \ \ \
i=0,1,\ldots,N-1,$
where $\chi$ is the canonical additive character of $F_{r}.$
To determine the weight distribution of the ICC, we need to compute the
Gaussian periods.Unfortunally, Definition 2.2 is not enough to determine the
values of the Gaussian periods $\eta_{i}.$ To this end, we need to introduce
the period polynomials.
###### Definition 2.3
The period polynomials are defined by
$\psi_{(N,r)}(X)=\prod_{i=0}^{N-1}(X-\eta_{i}^{(N,r)}).\ \ \ \ \ \ \ \ (3)$
It is known that $\psi_{(N,r)}(X)$ is a polynomial with integer coefficients
$[10].$ We will need the following four lemmas whose proofs can be found in
$[10-11].$
###### Lemma 2.4
Let $N=5$. We have the following results on the factorization of
$\psi_{(5,r)}(X).$
(i) If $p\equiv 4(\rm{mod}5),$ then
$\psi_{(5,r)}(X)=\\{\begin{array}[]{c}5^{-5}(5X+1+4\sqrt{r})(5X+1-\sqrt{r})^{4},\
\ \ \ \ ifsm/2\ even,\\\ 5^{-5}(5X+1-4\sqrt{r})(5X+1+\sqrt{r})^{4},\ \ \ \
ifsm/2\ odd.\\\ \end{array}$
(ii) If $p\equiv 2(\rm{mod}\ 5)$ or $p\equiv 3(\rm{mod}\ 5)$, then
$\psi_{(5,r)}(X)=\\{\begin{array}[]{c}5^{-5}(5X+1+4\sqrt{r})(5X+1-\sqrt{r})^{4},\
\ \ \ \ ifsm/4\ even,\\\ 5^{-5}(5X+1-4\sqrt{r})(5X+1+\sqrt{r})^{4},\ \ \ \
ifsm/4\ odd.\\\ \end{array}$
(iii) If $p\equiv 1(\rm{mod}\ 5),$ and $5\nmid ms$, then $\psi_{(5,r)}(X)$ is
irreducible over the rationals.
###### Lemma 2.5
Let $N=6.$ We have the following results on the factorization of
$\psi_{(6,r)}(X).$
(i) If $p\equiv 5(\rm{mod}\ 6)$, and $ms$ is even, then
$\psi_{(6,r)}(X)=\\{\begin{array}[]{c}6^{-6}(6X+1+5\sqrt{r})(6X+1-\sqrt{r})^{5},\
\ \ \ \ ifsm/2\ even,\\\ 6^{-6}(6X+1-5\sqrt{r})(6X+1+\sqrt{r})^{5},\ \ \ \
ifsm/2\ odd.\\\ \end{array}$
(ii) If $p\equiv 1(\rm{mod}\ 6)$,
(a) If $3\nmid ms$, and $ms$ is odd, then $\psi_{(6,r)}(X).$is irreducible
over the rationals.
(b) If $ms\equiv 0(\rm{mod}\ 6)$, then
$\psi_{(6,r)}(X)=6^{-6}[(6X+1)+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(u+\overline{u})+r^{\frac{1}{3}}(u_{1}+\overline{u}_{1})+(-1)^{1+\frac{kms}{2}}\sqrt{r}]\times$
$[(6X+1)+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(v+\overline{v})+r^{\frac{1}{3}}(v_{1}+\overline{v}_{1})+(-1)^{2+\frac{kms}{2}}\sqrt{r}]\times$
$\\{(6X+1)+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}[(v+\overline{v})-(u+\overline{u})]-r^{\frac{1}{3}}[(v_{1}+\overline{v}_{1})+(u_{1}+\overline{u}_{1}]$
$+(-1)^{3+\frac{kms}{2}}\sqrt{r}\\}\times[(6X+1)-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(u+\overline{u})+r^{\frac{1}{3}}(u_{1}+\overline{u}_{1})+$
$(-1)^{4+\frac{kms}{2}}\sqrt{r}]\times[(6X+1)-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(v+\overline{v})+r^{\frac{1}{3}}(v_{1}+\overline{v}_{1})+(-1)^{5+\frac{kms}{2}}\sqrt{r}]$
$\times\\{(6X+1)-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}[(v+\overline{v})-(u+\overline{u})]-r^{\frac{1}{3}}[(v_{1}+\overline{v}_{1})+(u_{1}+$
$\overline{u}_{1}]+(-1)^{6+\frac{kms}{2}}\sqrt{r}\\}.$
(c) If $ms\equiv 3(\rm{mod}\ 6)$, then
$\psi_{(6,r)}(X)=6^{-6}[(6X+1)^{2}-2r^{\frac{1}{3}}(u_{1}+\overline{u}_{1})(6X+1)+p^{\frac{2}{3}ms}(u_{1}+\overline{u}_{1})^{2}-(-1)^{k}r^{\frac{1}{3}}(u+$
$\overline{u}-r^{\frac{1}{3}})^{2}]\times[(6X+1)^{2}-2r^{\frac{1}{3}}(v_{1}+\overline{v}_{1})(6X+1)+p^{\frac{2}{3}ms}(v_{1}+\overline{v}_{1})^{2}-$
$(-1)^{k}r^{\frac{1}{3}}(v+\overline{v}-r^{\frac{1}{3}})^{2}]\times\\{(6X+1)^{2}+2r^{\frac{1}{3}}[(v_{1}+\overline{v}_{1})+(u_{1}+\overline{u}_{1})](6X$
$+1)+p^{\frac{2}{3}ms}[(v_{1}+\overline{v}_{1})+(u_{1}+\overline{u}_{1})]^{2}-(-1)^{k}r^{\frac{1}{3}}[(v+\overline{v})-(u+\overline{u})-r^{\frac{1}{3}})^{2}],$
where $|u|^{2}=|v_{1}|^{2}p^{\frac{1}{3}ms}$,
$|v|^{2}=|u_{1}|^{2}p^{\frac{1}{3}ms}$. In addition, $u$ and $v$ are given by
$(u-v)(\overline{u}-\overline{v})=p^{\frac{2}{3}ms},$ $u_{1}$ and $v_{1}$ are
given by $(u_{1}+v_{1})(\overline{u}_{1}+\overline{v}_{1})=p^{\frac{1}{3}ms},$
and $p=6k+1.$
###### Lemma 2.6
Let $N=7$.We have the following results on the factorization of
$\psi_{(7,r)}(X).$
(i) If $p\equiv 6(\rm{mod}\ 7),$ then
$\psi_{(7,r)}(X)=\\{\begin{array}[]{c}7^{-7}(7X+1+6\sqrt{r})(7X+1-\sqrt{r})^{6},\
\ \ \ \ ifsm/2\ even,\\\ 7^{-7}(7X+1-6\sqrt{r})(7X+1+\sqrt{r})^{6},\ \ \ \
ifsm/2\ odd.\\\ \end{array}$
(ii) If $p\equiv 3(\rm{mod}\ 7)$ or $p\equiv 5(\rm{mod}\ 7)$, then
$\psi_{(7,r)}(X)=\\{\begin{array}[]{c}7^{-7}(7X+1+6\sqrt{r})(7X+1-\sqrt{r})^{6},\
\ \ \ \ ifsm/6\ even,\\\ 7^{-7}(7X+1-6\sqrt{r})(7X+1+\sqrt{r})^{6},\ \ \ \
ifsm/6\ odd.\\\ \end{array}$
(iii) If $p\equiv 1(\rm{mod}\ 7)$, and $7\nmid ms$, then $\psi_{(7,r)}(X)$ is
irreducible over the rationals.
###### Lemma 2.7
Let $N=8$. We have the following results on the factorization of
$\psi_{(8,r)}(X).$
(i) If $p\equiv 7(\rm{mod}\ 8)$, and $ms$ is even, then
$\psi_{(8,r)}(X)=\\{\begin{array}[]{c}8^{-8}(8X+1+7\sqrt{r})(8X+1-\sqrt{r})^{7},\
\ \ \ \ ifsm/2\ even,\\\ 8^{-8}(8X+1-7\sqrt{r})(8X+1+\sqrt{r})^{7},\ \ \ \
ifsm/2\ odd.\\\ \end{array}$
(ii) If $p\equiv 5(\rm{mod}\ 8)$,
(a) If $ms\equiv 0(\rm{mod}\ 8)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)-\sqrt{r}+8r^{\frac{1}{4}}cd]^{2}\times[(8X+1)+3\sqrt{r}-4r^{\frac{1}{4}}c^{2}+8r^{\frac{3}{8}}d]\times$
$[(8X+1)-\sqrt{r}-8r^{\frac{1}{4}}cd]^{2}\times[(8X+1)-\sqrt{r}+4r^{\frac{1}{4}}c^{2}-4r^{\frac{3}{8}}c]\times[(8X+$
$1)+3\sqrt{r}-4r^{\frac{1}{4}}c^{2}-8r^{\frac{3}{8}}d]\times[(8X+1)-\sqrt{r}+4r^{\frac{1}{4}}c^{2}+4r^{\frac{3}{8}}c],$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv
1(\rm{mod}\ 4),and\ \rm{gcd}(c,p)=1.$
(b) If $ms\equiv 4(\rm{mod}\ 8)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)-\sqrt{r}+8r^{\frac{1}{4}}cd]^{2}\times\\{[(8X+1)+3\sqrt{r}-4r^{\frac{1}{4}}c^{2}]^{2}-64r^{\frac{3}{4}}d^{2}\\}$
$\times[(8X+1)-\sqrt{r}-8r^{\frac{1}{4}}cd]^{2}\times\\{[(8X+1)+4r^{\frac{1}{4}}c^{2}-\sqrt{r}]^{2}-16r^{\frac{1}{4}}c^{2}\\},$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv\pm
1(\rm{mod}\ 4),and\ \rm{gcd}(c,p)=1.$
(c) If $ms\equiv 2(\rm{mod}\ 4)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)^{2}-2\sqrt{r}(8X+1)+r-16\sqrt{r}z^{2}]^{2}\times[(8X+1)^{4}+4\sqrt{r}(8X$
$+1)^{3}+2\sqrt{r}(8X+1)^{2}(11\sqrt{r}-4w^{2})+4r(8X+1)(9\sqrt{r}-20w^{2})+$
$r(9\sqrt{r}-4w^{2})^{2}],$
where $w$ and $z$ are given by $w^{2}+4z^{2}=\sqrt{r},w\equiv 1(\rm{mod}\
4),and\ \rm{gcd}(w,p)=1.$
(iii) If $p\equiv 3(\rm{mod}\ 8)$,
(a) If $ms\equiv 0(\rm{mod}\ 4)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)-\sqrt{r}]^{2}\times[(8X+1)^{2}-2\sqrt{r}(8X+1)+r-16\sqrt{r}t^{2}]^{2}\times$
$[(8X+1)^{2}+6\sqrt{r}(8X+1)+9r-16\sqrt{r}l^{2}],$
where $l$ and $t$ are given by $l^{2}+2t^{2}=\sqrt{r},l\equiv-1(\rm{mod}\
8),and\ \rm{gcd}(l,p)=1.$
(b) If $ms\equiv 2(\rm{mod}\ 4)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)^{2}+2\sqrt{r}(8X+1)+r+16\sqrt{r}t^{2}]^{2}\times[(8X+1)-3\sqrt{r}]^{2}$
$\times[(8X+1)^{2}+2\sqrt{r}(8X+1)+r+16\sqrt{r}l^{2}],$
where $l$ and $t$ are given by $l^{2}+2t^{2}=\sqrt{r},l\equiv\pm 1(\rm{mod}\
4),and\ \rm{gcd}(l,p)=1.$
(iv) If $p\equiv 1(\rm{mod}\ 8)$,
(a) If $ms$ is odd, then $\psi_{(8,r)}(X).$is irreducible over the rationals.
(b) If $ms\equiv 0(\rm{mod}\ 8)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)+\sqrt{r}+8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(2c-4d)t]\times[(8X+1)+\sqrt{r}-$
$4r^{\frac{1}{4}}c^{2}+8r^{\frac{1}{8}}dl]\times[(8X+1)+\sqrt{r}-8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(2c+4d)t]\times[(8X+$
$1)-3\sqrt{r}+4r^{\frac{1}{4}}c^{2}-4r^{\frac{1}{8}}cl]\times[(8X+1)+\sqrt{r}+8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(4d-2c)t]$
$[(8X+1)+\sqrt{r}-4r^{\frac{1}{4}}c^{2}-8r^{\frac{1}{8}}dl]\times[(8X+1)+\sqrt{r}-8r^{\frac{1}{4}}cd-2r^{\frac{1}{8}}(2c+4d)t]$
$\times[(8X+1)-3\sqrt{r}+4r^{\frac{1}{4}}c^{2}+4r^{\frac{1}{8}}cl],$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv
1(\rm{mod}\ 8),and\ \rm{gcd}(c,p)=1;$ $l$
and $t$ are given by $l^{2}+2t^{2}=\sqrt{r},l\equiv 1(\rm{mod}\ 8),and\
\rm{gcd}(l,p)=1.$
(c) If $ms\equiv 4(\rm{mod}\ 8)$, then
$\psi_{(8,r)}(X)=8^{-8}[(8X+1)^{2}+2r^{\frac{1}{4}}(r^{\frac{1}{4}}+8cd)(8X+1)+\sqrt{r}(r^{\frac{1}{4}}+8cd)^{2}-4r^{\frac{1}{4}}(2c-$
$4d)^{2}t^{2}]\times[(8X+1)^{2}+2r^{\frac{1}{4}}(3r^{\frac{1}{4}}-4c^{2})(8X+1)+\sqrt{r}(3r^{\frac{1}{4}}-4c^{2})^{2}-64r^{\frac{1}{4}}d^{2}l^{2}]$
$\times[(8X+1)^{2}+2r^{\frac{1}{4}}(r^{\frac{1}{4}}-8cd)(8X+1)+\sqrt{r}(r^{\frac{1}{4}}-8cd)^{2}-4r^{\frac{1}{4}}(2c+4d)^{2}t^{2}]\times$
$[(8X+1)^{2}+2r^{\frac{1}{4}}(4c^{2}-r^{\frac{1}{4}})(8X+1)+\sqrt{r}(4c^{2}-r^{\frac{1}{4}})^{2}-16r^{\frac{1}{4}}c^{2}l^{2},$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv
1(\rm{mod}\ 4),and\ \rm{gcd}(c,p)=1;$ $l$
and $t$ are given by $l^{2}+2t^{2}=\sqrt{r},l\equiv 1(\rm{mod}\ 8),and\
\rm{gcd}(l,p)=1.$
(d) If $ms\equiv 2(\rm{mod}\ 4)$, then
$\psi_{(8,r)}(X)=8^{-8}\\{(8X+1)^{4}-4\sqrt{r}(8X+1)^{3}-2\sqrt{r}(8X+1)^{2}(16z^{2}+16t^{2}-3\sqrt{r})-$
$4\sqrt{r}(8X+1)[r-\sqrt{r}(16z^{2}+16t^{2})+128t^{2}z^{2}]+r(\sqrt{r}+16z^{2}-16t^{2})^{2}-$
$64\sqrt{r}(\sqrt{r}-16t^{2})^{2}z^{2}\\}\times(8X+1)^{4}+4\sqrt{r}(8X+1)^{3}-2\sqrt{r}(8X+1)^{2}$
$(4w^{2}+8l^{2}-3\sqrt{r})+4\sqrt{r}(8X+1)[r-\sqrt{r}(4w^{2}+8l^{2})+16w^{2}l^{2}]+r(\sqrt{r}$
$+4w^{2}-8l^{2})^{2}-4\sqrt{r}(2\sqrt{r}-4l^{2})^{2}w^{2},$
where $w$ and $z$ are given by $w^{2}+4z^{2}=\sqrt{r},w\equiv 1(\rm{mod}\
4),and\ \rm{gcd}(w,p)=1;$
$l$ and $t$ are given by $l^{2}+2t^{2}=\sqrt{r},l\equiv-1(\rm{mod}\ 4),and\
\rm{gcd}(l,p)=1.$
## 3 The weight enumerator of some classes of irreducible cyclic codes
Recall that $q=p^{s}$and $r=q^{m},$where $s$ and $m$ are positive integers.
$r-1=nN$ throughout this paper. Let $Z(r,a)$ denote the number of solutions
$x\in F_{r}$ of the equation $Tr_{r/q}(ax^{N})=0.$ We have then
$Z(r,a)=\frac{1}{q}\sum_{x\in F_{r}}\sum_{\chi\in
F_{q}^{\wedge}}\chi(Tr_{r/q}(ax^{N})\overline{\chi(0)})$
$=\frac{1}{q}\sum_{x\in F_{r}}\sum_{y\in F_{q}}\chi_{y}(Tr_{r/q}(ax^{N}))$
$=\frac{1}{q}\sum_{x\in F_{r}}\sum_{y\in F_{q}}\chi_{1}(yTr_{r/q}(ax^{N}))$
$=\frac{1}{q}\sum_{x\in F_{r}}\sum_{y\in F_{q}}\chi_{1}(Tr_{r/q}(yax^{N}))$
$=\frac{1}{q}\sum_{x\in F_{r}}\sum_{y\in F_{q}}\chi_{r/p}(yax^{N})\ \ \ \ \ \
$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{1}{q}[r+q-1+\sum_{x\in
F_{r^{*}}}\sum_{y\in F_{q^{*}}}\chi_{1}(Tr_{r/q}(yax^{N}))]$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=\frac{1}{q}[r+q-1+N\sum_{y\in F_{q^{*}}}\sum_{x\in
C_{0}^{(N,r)}}\chi_{1}(Tr_{r/q}(yax^{N}))].\ \ \ \ \ \ (4)$
Hence, for any $\beta\in F_{r}^{*}$ the Hamming weight of the codeword
$\textbf{c}(\beta)=({\textrm{Tr}_{r/q}(\beta),\textrm{Tr}_{r/q}(\beta\theta),\cdots,\textrm{Tr}_{r/q}(\beta\theta^{n-1})})$
in the ICC of (2) is equal to $n-\frac{Z(r,\beta)-1}{N}.$
###### Theorem 3.1
([9]) If $q\equiv 1(\rm{mod}\ N)$ and $\rm{gcd}(m,N)=1,$ then the set $C(r,N)$
in (2) is a $[(q^{m}-1)/N,m,q^{m-1}(q-1)/N]$ code with the only nonzero weight
$q^{m-1}(q-1)/N.$
A. The weight enumerator in the Case $N=5$
###### Lemma 3.2
Let $N=5$. If $q\equiv 1(\rm{mod}\ 5),$ then
$(r-1)/(q-1)\rm{mod}\ N=m\ \rm{mod}\ 5.$
###### Proof 1
Note that $(r-1)/(q-1)\rm{mod}\ 5=(q^{m-1}+q^{m-2}+\ldots+q+1)\rm{mod}\ 5,$
and $q\equiv 1(\rm{mod}\ 5),$ The conclusion then follows. $\Box$
###### Theorem 3.3
Let $N=5$ and $q\equiv 1(\rm{mod}\ 5).$ If $5\nmid m,$ then the set $C(r,5)$
is a $[(q^{m}-1)/5,m,q^{m-1}(q-1)/5]$ code with the only nonzero weight
$q^{m-1}(q-1)/5.$
###### Proof 2
Since $\rm{gcd}(m,5)=1.$ The conclusion then follows from Theorem 3.1. $\Box$
###### Example 3.4
Let $q=11$ and let $m=2.$ Then the set $C(r,5)$ in (2) is a $[24,2,22]$ code
with the only nonzero weight $22.$
###### Theorem 3.5
Let $q\equiv 1(\rm{mod}\ 5),$ $p\equiv 4(\rm{mod}\ 5).$ and $m\equiv
0(\rm{mod}\ 5).$ Let $r-1=nN$, where$N=5.$
(i) If $sm\equiv 0(\rm{mod}\ 4),$ then $C(r,5)$ is an
$[(r-1)/5,m,(q-1)(r-\sqrt{r})/5q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{4(r-1)}{5}x^{\frac{(q-1)(r-\sqrt{r})}{5q}}+\frac{(r-1)}{5}x^{\frac{(q-1)(r+4\sqrt{r})}{5q}}.$
(ii) If $sm\equiv 2(\rm{mod}\ 4),$ then $C(r,5)$ is an
$[(r-1)/5,m,(q-1)(r-4\sqrt{r})/5q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{4(r-1)}{5}x^{\frac{(q-1)(r+\sqrt{r})}{5q}}+\frac{(r-1)}{5}x^{\frac{(q-1)(r-4\sqrt{r})}{5q}}.$
###### Proof 3
If $q\equiv 1(\rm{mod}\ 5),$ $m\equiv 0(\rm{mod}\ 5),$ then by Lemma
3.2,$(r-1)/(q-1)\rm{mod}\ 5=0.$ Let $\alpha$ be a fixed primitive element of
$F_{r}.$ Note that every element of $F_{q}^{*}$ is the form
$\alpha^{i(r-1)/(q-1)}$ for some integer $i$. Hence, $F_{q}^{*}\subset
C_{0}^{(5,r)}.$ It follows from Lemma 2.4 and (3) that the Gaussian period
$\eta_{0}^{(5,r)}=\frac{-1-4\sqrt{r}}{5},$
$\eta_{j}^{(5,r)}=\frac{-1+\sqrt{r}}{5}$ $(1\leq j\leq 4).$ Note that (4)
becomes
$Z(r,a)=\frac{1}{q}[r+q-1+N\sum_{y\in F_{q}^{*}}\sum_{x\in\in
C_{0}^{(N,r)}}\chi(yax)]$ $=\frac{1}{q}[r+q-1+5(q-1)\eta_{i}^{(5,r)}],$
where $a\in C_{i}^{(5,r)}$ for some $i$. Hence, the Hamming weight of the
codeword $\textbf{c}(\alpha)$ is
$n-\frac{Z(r,a)-1}{N}=\frac{r-Z(r,a)}{N}=\frac{(q-1)(r-1-5\eta_{i}^{(5,r)}}{5q}).$
Then the weight enumerator can be determined in terms of (1).
The proof for the case that $sm\equiv 2(\rm{mod}\ 4)$ is the same as that of
part(i). $\Box$
###### Theorem 3.6
Let $q\equiv 1(\rm{mod}\ 5),$ $p\equiv 2(\rm{mod}\ 5)$ or $p\equiv 3(\rm{mod}\
5).$ and $m\equiv 0(\rm{mod}\ 5).$ Let $r-1=nN$, where $N=5.$
(i) If $sm\equiv 0(\rm{mod}\ 8),$ then $C(r,5)$ is an
$[(r-1)/5,m,(q-1)(r-\sqrt{r})/5q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{4(r-1)}{5}x^{\frac{(q-1)(r-\sqrt{r})}{5q}}+\frac{(r-1)}{5}x^{\frac{(q-1)(r+4\sqrt{r})}{5q}}.$
(ii) If $sm\equiv 4(\rm{mod}\ 8),$ then $C(r,5)$ is an
$[(r-1)/5,m,(q-1)(r-4\sqrt{r})/5q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{4(r-1)}{5}x^{\frac{(q-1)(r+\sqrt{r})}{5q}}+\frac{(r-1)}{5}x^{\frac{(q-1)(r-4\sqrt{r})}{5q}}.$
The proof of Theorem 3.6 is similar to that of the Theorem 3.5.
B. The weight enumerator in the Case $N=6$
###### Lemma 3.7
Let $N=6$. If $q\equiv 1(\rm{mod}\ 6),$ then
$(r-1)/(q-1)\rm{mod}\ N=m\ \rm{mod}\ 6.$
###### Theorem 3.8
Let $N=6$ and $q\equiv 1(\rm{mod}\ 6).$ If $m$ is odd and $3\nmid m$, then the
set $C(r,6)$ is a $[(q^{m}-1)/6,m,q^{m-1}(q-1)/6]$ code with the only nonzero
weight $q^{m-1}(q-1)/6.$
###### Example 3.9
Let $q=7$ and $m=5.$ Then the set $C(r,6)$ in (2) is a $[2801,5,2401]$ code
with the only nonzero weight $2401.$
###### Theorem 3.10
Let $q\equiv 1(\rm{mod}\ 6),$ $p\equiv 5(\rm{mod}\ 6),$ and let $s$ be even.
(i) If $m\equiv 0(\rm{mod}\ 6),$ then $C(r,6)$ is an
$[(r-1)/6,m,(q-1)(r-\sqrt{r})/6q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{5(r-1)}{6}x^{\frac{(q-1)(r-\sqrt{r})}{6q}}+\frac{(r-1)}{6}x^{\frac{(q-1)(r+5\sqrt{r})}{6q}}.$
(ii) If $m\equiv 2(\rm{mod}\ 6),$ then $C(r,6)$ is an
$[(r-1)/6,m,(q-1)(r-\sqrt{r})/6q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{6q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{6q}}.$
###### Proof 4
(i) Since $q\equiv 1(\rm{mod}\ 6),$ $m\equiv 0(\rm{mod}\ 6),$ then by Lemma
3.6,$(r-1)/(q-1)\rm{mod}\ 6=0.$ Then similar to the proof of Theorem 3.5.
(ii) Since $q\equiv 1(\rm{mod}\ 6),$ $m\equiv 2(\rm{mod}\ 6),$ then by Lemma
3.7,$(r-1)/(q-1)\rm{mod}\ 6=2.$ Hence, $\rm{gcd}(6,r-1)/(q-1)mod\rm{mod}\
6)=2.$ From[12], If $r$ be an even power of an odd prime, then
$\chi(C_{0}^{(2,r)})=\frac{-1\pm\sqrt{r}}{2},\ \ \ \
\chi(C_{1}^{(2,r)})=\frac{-1\mp\sqrt{r}}{2}.$
It then follows from (4) that
$Z(r,a)=\frac{1}{q}[r+q-1+6\sum_{y\in F_{q}^{*}}\sum_{x\in\in
C_{0}^{(6,r)}}\chi(yax)]$ $\ \ \ \ \ \ \ \ \ \ \ \
=\frac{1}{q}[r+q-1+6\times\frac{q-1}{3}\sum_{x\in C_{0}^{(2,r)}}\chi(ax)]$ $\
\ \ \ \ \ =\frac{1}{q}[r+q-1+(q-1)(-1\pm\sqrt{r})].$
Hence, the Hamming weight of the codeword $c(a)$ is
$n-\frac{Z(r,a)-1}{N}=\frac{r-Z(r,A)}{N}=\frac{(q-1)(r\pm\sqrt{r})}{6q}.$
This is a complete proof for the second part. $\Box$
###### Example 3.11
Let $q=5^{2}$ and $m=2.$ Then the set $(r,5)$ in (2) is a $[2801,5,2401]$ code
over $F_{25}$ with the weight distribution $1+312x^{96}+312x^{104}.$
###### Theorem 3.12
Let $q\equiv 1(\rm{mod}\ 6),$ $p\equiv 1(\rm{mod}\ 6).$ .
(i) If $m\equiv 0(\rm{mod}\ 6),$ then $C(r,6)$ is an $[(r-1)/6,m]$ code over
$F_{q}$ with the weight enumerator
$1+\frac{r-1}{6}x^{\frac{(q-1)[r+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(u+\overline{u})+r^{\frac{1}{3}}(u_{1}+\overline{u}_{1})+(-1)^{1+\frac{kms}{2}}\sqrt{r}]}{6q}}+$
$\frac{r-1}{6}x^{\frac{(q-1)[r+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(v+\overline{v})+r^{\frac{1}{3}}(v_{1}+\overline{v}_{1})+(-1)^{2+\frac{kms}{2}}\sqrt{r}]}{6q}}+\
\ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{r-1}{6}x^{\frac{(q-1)\\{r+(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}[(v+\overline{v})-(u+\overline{u})]-r^{\frac{1}{3}}[(v_{1}+\overline{v}_{1})+(u_{1}+\overline{u}_{1})]+(-1)^{3+\frac{kms}{2}}\sqrt{r}\\}}{6q}}+$
$\frac{r-1}{6}x^{\frac{(q-1)[r-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(u+\overline{u})+r^{\frac{1}{3}}(u_{1}+\overline{u}_{1})+(-1)^{4+\frac{kms}{2}}\sqrt{r}}{6q}]}+$
$\frac{r-1}{6}x^{\frac{(q-1)[r-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}(v+\overline{v})+r^{\frac{1}{3}}(v_{1}+\overline{v}_{1})+(-1)^{5+\frac{kms}{2}}\sqrt{r}]}{6q}}+$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\frac{r-1}{6}x^{\frac{(q-1)\\{r-(-1)^{\frac{kms}{2}}r^{\frac{1}{6}}[(v+\overline{v})-(u+\overline{u})]-r^{\frac{1}{3}}[(v_{1}+\overline{v}_{1})+(u_{1}+\overline{u}_{1}]+(-1)^{6+\frac{kms}{2}}\sqrt{r}\\}}{6q}},$
where $|u|^{2}=|v_{1}|^{2}p^{\frac{1}{3}ms}$,
$|v|^{2}=|u_{1}|^{2}p^{\frac{1}{3}ms}$. In addition, $u$ and $v$ are given by
$(u-v)(\overline{u}-\overline{v})=p^{\frac{2}{3}ms},$ $u_{1}$ and $v_{1}$ are
given by $(u_{1}+v_{1})(\overline{u}_{1}+\overline{v}_{1})=p^{\frac{1}{3}ms},$
and $p=6k+1.$
(ii) If $m\equiv 2(\rm{mod}\ 6),$ then $C(r,6)$ is an
$[(r-1)/6,m,(q-1)(r-\sqrt{r})/6q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{6q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{6q}}.$
The proof of Theorem 3.12 is similar to that of the Theorem 3.10.
###### Example 3.13
Let $q=7$ and $m=2.$ Then the set $C(r,6)$ in (2) is a $[8,2,6]$ code over
$F_{7}$ with the weight distribution $1+24x^{6}+24x^{8}.$ We can then obtain
nonzero codewords of $C(r,6)$ in this example through programming, as follows
:
$\displaystyle(4,4,5,2,3,3,2,5),\ \ (6,3,6,0,1,4,1,0),\ \ (4,6,6,4,3,1,1,3),\
\ (0,2,1,2,0,5,6,5),$ $\displaystyle(1,6,2,2,6,1,5,5),\ \ (4,0,3,5,3,0,4,2),\
\ (4,5,2,3,3,2,5,4),\ \ (3,6,0,1,4,1,0,6),$ $\displaystyle(6,6,4,3,1,1,3,4),\
\ (2,1,2,0,5,6,5,0),\ \ (6,2,2,6,1,5,5,1),\ \ (0,3,5,3,0,4,2,4),$
$\displaystyle(5,2,3,3,2,5,4,4),\ \ (6,0,1,4,1,0,6,3),\ \ (6,4,3,1,1,3,4,6),\
\ (1,2,0,5,6,5,0,2),$ $\displaystyle(2,2,6,1,5,5,1,6),\ \ (3,5,3,0,4,2,4,0),\
\ (2,3,3,2,5,4,4,5),\ \ (0,1,4,1,0,6,3,6),$ $\displaystyle(4,3,1,1,3,4,6,6),\
\ (2,0,5,6,5,0,2,1),\ \ (2,6,1,5,5,1,6,2),\ \ (5,3,0,4,2,4,0,3),$
$\displaystyle(3,3,2,5,4,4,5,2),\ \ (1,4,1,0,6,3,6,0),\ \ (3,1,1,3,4,6,6,4),\
\ (0,5,6,5,0,2,1,2),$ $\displaystyle(6,1,5,5,1,6,2,2),\ \ (3,0,4,2,4,0,3,5),\
\ (3,2,5,4,4,5,2,3),\ \ (4,1,0,6,3,6,0,1),$ $\displaystyle(1,1,3,4,6,6,4,3),\
\ (5,6,5,0,2,1,2,0),\ \ (1,5,5,1,6,2,2,6),\ \ (0,4,2,4,0,3,5,3),$
$\displaystyle(2,5,4,4,5,2,3,3),\ \ (1,0,6,3,6,0,1,4),\ \ (1,3,4,6,6,4,3,1),\
\ (6,5,0,2,1,2,0,5),$ $\displaystyle(5,5,1,6,2,2,6,1),\ \ (4,2,4,0,3,5,3,0),\
\ (5,4,4,5,2,3,3,2),\ \ (0,6,3,6,0,1,4,1),$ $\displaystyle(3,4,6,6,4,3,1,1),\
\ (5,0,2,1,2,0,5,6),\ \ (5,1,6,2,2,6,1,5),\ \ (2,4,0,3,5,3,0,4),$
it is obvious that the number of nonzero codewords with Hamming weight either
6 or 8 is $24.$
C. The weight enumerator in the Case $N=7$
###### Lemma 3.14
Let $N=7$. If $q\equiv 1(\rm{mod}\ 7),$ then
$(r-1)/(q-1)\rm{mod}\ N=m\ \rm{mod}\ 7.$
###### Theorem 3.15
Let $N=7$, $q\equiv 1(\rm{mod}\ 7),$ and $7\nmid m.$ Then the set $C(r,7)$ is
a $[(q^{m}-1)/7,m,q^{m-1}(q-1)/7]$ code with the only nonzero weight
$q^{m-1}(q-1)/7.$
###### Example 3.16
Let $q=29$ and $m=2.$ Then the set $C(r,7)$ in (2) is a $[120,2,116]$ code
with the only nonzero weight 116.
###### Theorem 3.17
Let $q\equiv 1(\rm{mod}\ 7),$ $p\equiv 6(\rm{mod}\ 7),$ and $m\equiv
0(\rm{mod}\ 7).$ Let $r-1=nN$, where $N=7.$
(i) If $sm\equiv 0(\rm{mod}\ 4),$ then $C(r,7)$ is an
$[(r-1)/7,m,(q-1)(r-\sqrt{r})/7q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{6(r-1)}{7}x^{\frac{(q-1)(r-\sqrt{r})}{7q}}+\frac{(r-1)}{7}x^{\frac{(q-1)(r+6\sqrt{r})}{7q}}.$
(ii) If $sm\equiv 2(\rm{mod}\ 4),$ then $C(r,7)$ is an
$[(r-1)/7,m,(q-1)(r-6\sqrt{r})/7q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{6(r-1)}{7}x^{\frac{(q-1)(r+\sqrt{r})}{7q}}+\frac{(r-1)}{7}x^{\frac{(q-1)(r-6\sqrt{r})}{7q}}.$
###### Theorem 3.18
Let $q\equiv 1(\rm{mod}\ 7),$ $p\equiv 3(\rm{mod}\ 7)$ or $p\equiv 5(\rm{mod}\
7)$ and $m\equiv 0(\rm{mod}\ 7).$ Let $r-1=nN$, where $N=7.$
(i) If $sm\equiv 0(\rm{mod}\ 12),$ then $C(r,7)$ is an
$[(r-1)/7,m,(q-1)(r-\sqrt{r})/7q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{6(r-1)}{7}x^{\frac{(q-1)(r-\sqrt{r})}{7q}}+\frac{(r-1)}{7}x^{\frac{(q-1)(r+6\sqrt{r})}{7q}}.$
(ii) If $sm\equiv 6(\rm{mod}\ 12),$ then $C(r,7)$ is an
$[(r-1)/7,m,(q-1)(r-6\sqrt{r})/7q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{6(r-1)}{7}x^{\frac{(q-1)(r+\sqrt{r})}{7q}}+\frac{(r-1)}{7}x^{\frac{(q-1)(r-6\sqrt{r})}{7q}}.$
The proof of Theorem 3.17, 3.18 is similar to that of the Theorem 3.5.
From[13], We have the following theorem.
###### Theorem 3.19
Let $N=7$, $r=q^{3m},$ $p=q$ and $a=\frac{\omega_{p}(n)}{p-1}.$ Let
$n=n_{0}+n_{1}p+\cdots$ be the expansion of $n$ in the base $p$ and let
$\omega_{p}(n)=n_{0}+n_{1}+\cdots.$ Let $c_{m},d_{m}$ be the unique positive
integers prime to $p$ that satisfy the diophantine equation
$c_{m}^{2}+Nd_{m}^{2}=4p^{m(3-2a)}.$ Then $C(r,7)$ of length $n_{m}$ have
three weights with the weight distribution
$1+n_{m}x^{\frac{(p-1)(2r\pm 3p^{ma}c_{m})}{7p}}+3n_{m}x^{\frac{(p-1)[2r\pm
p^{ma}(7d_{m}-c_{m})]}{14p}}+3n_{m}x^{\frac{(p-1)[2r\mp
p^{ma}(7d_{m}+c_{m})]}{14p}}.\ \ \ \ \ \ (5)$
###### Example 3.20
Let $p=2$ and let $m=2.$ Then $n_{2}=9,a=d_{2}=1,c_{2}=3.$ It then follows
from (2) that $C(r,7)$ in (2) is a $[9,2,2]$ code over $F_{2}$ with the weight
distribution $1+9x^{2}+27x^{4}+27x^{6}.$ We can then obtain all codewords of
$C(r,7)$ in this example through programming, as follows :
$\displaystyle(1,1,0,0,1,0,1,0,0),(1,0,1,1,0,0,0,0,1),(1,1,1,1,0,1,0,1,0),(1,0,0,0,1,0,1,1,0),$
$\displaystyle(1,1,1,1,1,1,0,0,0),(1,1,1,0,1,1,1,0,0),(0,0,1,0,0,1,0,0,0),(1,0,0,1,0,1,0,0,1),$
$\displaystyle(0,1,1,0,0,0,0,1,1),(1,1,1,0,1,0,1,0,1),(0,0,0,1,0,1,1,0,1),(1,1,1,1,1,0,0,0,1),$
$\displaystyle(1,1,0,1,1,1,0,0,1),(1,1,0,1,1,0,0,0,0),(0,1,1,1,1,0,1,0,1),(0,1,0,0,0,1,0,1,1),$
$\displaystyle(0,1,0,0,1,0,0,0,0),(0,0,1,0,1,0,0,1,1),(1,1,0,0,0,0,1,1,0),(1,1,0,1,0,1,0,1,1),$
$\displaystyle(0,0,1,0,1,1,0,1,0),(1,1,1,1,0,0,0,1,1),(1,0,1,1,1,0,0,1,1),(1,0,0,1,0,0,0,0,0),$
$\displaystyle(0,1,0,1,0,0,1,1,0),(1,0,0,0,0,1,1,0,1),(1,0,1,0,1,0,1,1,1),(0,1,0,1,1,0,1,0,0),$
$\displaystyle(1,1,1,0,0,0,1,1,1),(0,1,1,1,0,0,1,1,1),(0,0,1,0,0,0,0,0,1),(1,0,1,0,0,1,1,0,0),$
$\displaystyle(0,0,0,0,1,1,0,1,1),(0,1,0,1,0,1,1,1,1),(1,0,1,1,0,1,0,0,0),(1,1,0,0,0,1,1,1,1),$
$\displaystyle(1,1,1,0,0,1,1,1,0),(0,1,0,0,0,0,0,1,0),(0,1,0,0,1,1,0,0,1),(0,0,0,1,1,0,1,1,0),$
$\displaystyle(1,0,1,0,1,1,1,1,0),(0,1,1,0,1,0,0,0,1),(1,0,0,0,1,1,1,1,1),(1,1,0,0,1,1,1,0,1),$
$\displaystyle(1,0,0,0,0,0,1,0,0),(1,0,0,1,1,0,0,1,0),(0,0,1,1,0,1,1,0,0),(0,1,0,1,1,1,1,0,1),$
$\displaystyle(1,1,0,1,0,0,0,1,0),(0,0,0,1,1,1,1,1,1),(1,0,0,1,1,1,0,1,1),(0,0,0,0,0,1,0,0,1),$
$\displaystyle(0,0,1,1,0,0,1,0,1),(0,1,1,0,1,1,0,0,0),(1,0,1,1,1,1,0,1,0),(1,0,1,0,0,0,1,0,1),$
$\displaystyle(0,0,1,1,1,1,1,1,0),(0,0,1,1,1,0,1,1,1),(0,0,0,0,1,0,0,1,0),(0,1,1,0,0,1,0,1,0),$
$\displaystyle(0,1,1,1,1,1,1,0,0),(0,1,1,1,0,1,1,1,0),(0,0,0,1,0,0,1,0,0),(0,0,0,0,0,0,0,0,0),$
it is obvious that the number of nonzero codewords with Hamming weight $2,4\
or\ 6\ is\ 9,27\ \\\ or\ 27,$ respectively.
D. The weight enumerator in the Case $N=8$
###### Lemma 3.21
Let $N=8$. If $q\equiv 1(\rm{mod}8),$ then
$(r-1)/(q-1)\rm{mod}\ N=m\ \rm{mod}\ 8.$
###### Theorem 3.22
Let $N=8$ and $q\equiv 1(\rm{mod}\ 8).$ If $m$ is odd , then the set $C(r,8)$
is a $[(q^{m}-1)/8,m,q^{m-1}(q-1)/8]$ code with the only nonzero weight
$q^{m-1}(q-1)/8.$
###### Example 3.23
Let $q=17$ and let $m=3.$ Then the set $C(r,8)$ in (2) is a $[614,3,578]$ code
with the only nonzero weight 578.
###### Theorem 3.24
Let $q\equiv 1(\rm{mod}\ 8),$ $p\equiv 7(\rm{mod}\ 8),$ and $s$ be even.
(i) If $m\equiv 0(\rm{mod}\ 8),$ then $C(r,8)$ is an
$[(r-1)/8,m,(q-1)(r-\sqrt{r})/8q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{7(r-1)}{8}x^{\frac{(q-1)(r-\sqrt{r})}{8q}}+\frac{(r-1)}{8}x^{\frac{(q-1)(r+7\sqrt{r})}{8q}}.$
(ii) If $m\equiv 2(\rm{mod}\ 8),$ then $C(r,8)$ is an
$[(r-1)/6,m,(q-1)(r-\sqrt{r})/8q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{8q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{8q}}.$
The proof of Theorem 3.24 is similar to that of the Theorem 3.10.
###### Theorem 3.25
Let $q\equiv 1(\rm{mod}\ 8),$ $p\equiv 5(\rm{mod}\ 8),$ and let $s$ be even.
(i) If $m\equiv 0(\rm{mod}\ 8),$ then $C(r,8)$ is an $[(r-1)/8,m]$ code over
$F_{q}$ with the weight enumerator
$1+\frac{r-1}{4}x^{\frac{(q-1)(r-\sqrt{r}+8r^{\frac{1}{4}}cd)}{8q}}+\frac{r-1}{4}x^{\frac{(q-1)(r-\sqrt{r}-8r^{\frac{1}{4}}cd)}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)(r+3\sqrt{r}-4r^{\frac{1}{4}}c^{2}+8r^{\frac{3}{8}}d)}{8q}}+$
$\frac{r-1}{8}x^{\frac{(q-1)(r-\sqrt{r}+4r^{\frac{1}{4}}c^{2}-4r^{\frac{3}{8}}c)}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)(r+3\sqrt{r}-4r^{\frac{1}{4}}c^{2}-8r^{\frac{3}{8}}d)}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)(r-\sqrt{r}+4r^{\frac{1}{4}}c^{2}+4r^{\frac{3}{8}}c)}{8q}},$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv
1(\rm{mod}\ 4),and\ \rm{gcd}(c,p)=1.$
(ii) If $m\equiv 2(\rm{mod}\ 8),$ then $C(r,8)$ is an
$[(r-1)/8,m,(q-1)(r-\sqrt{r})/8q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{8q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{8q}}.$
The proof of Theorem 3.25 is similar to that of the Theorem 3.10.
###### Theorem 3.26
Let $q\equiv 1(\rm{mod}\ 8),$ $p\equiv 3(\rm{mod}\ 8).$ and let $s$ be even.
If $m\equiv 2(\rm{mod}\ 8),$ then $C(r,8)$ is an
$[(r-1)/8,m,(q-1)(r-\sqrt{r})/8q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{8q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{8q}}.$
The proof of Theorem 3.26 is similar to that of the Theorem 3.10(ii).
###### Example 3.27
Let $q=3^{2}$ and let $m=2.$ Then $C(r,8)$ in (2) is a $[10,2,8]$ code over
$F_{9}$ with the weight distribution $1+40x^{8}+40x^{10}.$
###### Theorem 3.28
Let $q\equiv 1(\rm{mod}\ 8),$ $p\equiv 1(\rm{mod}\ 8).$
(i) If $m\equiv 0(\rm{mod}\ 8),$ then $C(r,8)$ is an $[(r-1)/8,m]$ code over
$F_{q}$ with the weight enumerator
$1+\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}+8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(2c-4d)t]}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}-4r^{\frac{1}{4}}c^{2}+8r^{\frac{1}{8}}dl]}{8q}}+$
$\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}-8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(2c+4d)t]}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)[r-3\sqrt{r}+4r^{\frac{1}{4}}c^{2}-4r^{\frac{1}{8}}cl]}{8q}}+$
$\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}+8r^{\frac{1}{4}}cd+2r^{\frac{1}{8}}(4d-2c)t]}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}-4r^{\frac{1}{4}}c^{2}-8r^{\frac{1}{8}}dl]}{8q}}+$
$\frac{r-1}{8}x^{\frac{(q-1)[r+\sqrt{r}-8r^{\frac{1}{4}}cd-2r^{\frac{1}{8}}(2c+4d)t]}{8q}}+\frac{r-1}{8}x^{\frac{(q-1)[r-3\sqrt{r}+4r^{\frac{1}{4}}c^{2}+4r^{\frac{1}{8}}cl]}{8q}},$
where $c$ and $d$ are given by $c^{2}+4d^{2}=r^{\frac{1}{4}},c\equiv
1(\rm{mod}\ 8),and\rm{gcd}(c,p)=1;$ $l$ and $t$ are given by
$l^{2}+2t^{2}=\sqrt{r},l\equiv 1(\rm{mod}\ 8),and\ \rm{gcd}(l,p)=1.$
(ii) If $m\equiv 2(\rm{mod}\ 8),$ then $C(r,8)$ is an
$[(r-1)/8,m,(q-1)(r-\sqrt{r})/8q]$ code over $F_{q}$ with the weight
enumerator
$1+\frac{r-1}{2}x^{\frac{(q-1)(r-\sqrt{r})}{8q}}+\frac{r-1}{2}x^{\frac{(q-1)(r+\sqrt{r})}{8q}}.$
The proof of Theorem 3.28 is similar to that of the Theorem 3.10.
###### Example 3.29
Let $q=17$ and let $m=2.$ Then $C(r,8)$ in (2) is a $[36,2,32]$ code over
$F_{17}$ with the weight distribution $1+144x^{32}+144x^{36}.$
## 4 Conclusions
In this paper, we studied the weight enumerator of irreducible cyclic codes
$C(r,N)$ for all $5\leq N\leq 8,$ and gave some specific examples.
Furthermore, all codewords of $C(r,N)$ in certain examples can be obtained
through programming, which is beneficial to verify the weight distributions of
codes. Due to space limitation and the complexity of the weight enumerator of
codes, it is hard to describe the weight distributions in more general cases.
We shall work on this subject in a future work.
## 5 Acknowledgements
This work was supported by the National Natural Science Foundation of China
(Grant No. 60873119).
## 6 References
## References
* [1] C. Ding, D. Feng , C. Ma, et al.. The weight enumerator of a class of cyclic codes[J]. IEEE Trans. Inform. Theory, 2011,57(1): 397-402.
* [2] J.Yuan, C.Carlet, and C.Ding. The weight distribution of a class of linear codes from perfect nonlinear functions[J]. IEEE Trans. Inform. Theory, 2006,52(2): 712-717.
* [3] Z.H. Li, T. Xue, and H. Lai. Secret sharing schemes from binary linear codes[J]. Information Science, 2011,180(22): 4412-4419.
* [4] R.J.McEliece and H. Rumsey Jr.. Euler produces, cyclotomy, and coding[J]. Number Theroy,1972, 4:302-311.
* [5] B.Schmidt and C.White. All two-weight irreducible cyclic codes? [J]. Finite Fields Appl., 2002, 8:1-17.
* [6] P.Langevin. A new class of two weight codes[J].. Finite Fields Appl., 1996,181-187.
* [7] L.D.Baumert and R.J.McEliece. Weights of irreducible cyclic codes[J]. Inf.Contr., 1972,20(2):158-175.
* [8] L.D.Baumert and J.Mykkeltveit. Weight distributions of some irreducible cyclic codes[J]. DSN Progr. Rep., 1973,16:128-131.
* [9] C. Ding . The weight distribution of some irreducible cyclic codes[J]. IEEE Trans. Inform. Theory, 2009, 55 (3): 955-960.
* [10] S.Gurak. Period polynomials for of fixed small degree[J]. Number Theroy, 2004,127-145.
* [11] A.Hoshi. Explicit lifts of quintic Jacobi sums and period polynomials for [J]. Proc. Japan Acad., 2006, 82(7):87-92.
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Zhi Hui Li
College of Mathematics and Information Science
Shaanxi Normal University
Xi’an 710062, P. R. China
e-mail:snnulzh@yahoo.com.cn
|
arxiv-papers
| 2012-02-14T01:42:20 |
2024-09-04T02:49:27.407085
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yun Song, Zhihui Li",
"submitter": "Yun Song",
"url": "https://arxiv.org/abs/1202.2907"
}
|
1202.3095
|
# Viscous and resistive accretion flows with radially self-similar and
outflows
Kazem Faghei and Azam Mollatayefeh
School of Physics, Damghan University, Damghan, Iran
E-mail: kfaghei@du.ac.irE-mail: azam.molatayefe@yahoo.com
###### Abstract
The existence of outflow in accretion flows is confirmed by observations and
magnetohydrodynamics (MHD) simulations. In this paper, we study outflows of
accretion flows in the presence of resistivity and toroidal magnetic field.
The mechanism of energy dissipation in the flow is assumed to be the viscosity
and the magnetic diffusivity due to turbulence in the accretion flow. It is
also assumed that the magnetic diffusivity and the kinematic viscosity are not
constant and vary by position and $\alpha$-prescription is used for them. The
influence of outflow emanating from accretion disc is considered as a sink for
mass, angular momentum and energy. The self-similar method is used to solve
the integrated equations that govern the behavior of the accretion flow in the
presence of outflow. The solutions represent the disc which rotates faster and
becomes cooler for stronger outflows. Moreover, by adding the magnetic
diffusivity, the surface density and rotational velocity decrease, while the
radial velocity and temperature increase. The study of present model with the
magnitude of magnetic field implies that the disc rotates and accretes faster
and becomes hotter, while the surface density decreases. The disc thickness
increases by adding the magnetic field or resistivity, while it becomes
thinner for more losses of mass and energy due to the outflows.
###### keywords:
accretion, accretion discs, magnetohydrodynamics: MHD
## 1 Introduction
Accretion disc is an important physical object in astrophysics. The standard
model of thin accretion disc is widely regarded as a successful model for
explaining the observational features in active galactic nuclei (AGN) and
X-ray binaries (Shakura & Sunyaev 1973). However, the standard disk model
cannot explain the spectral energy distributions (SED) of many sources, such
as Sgr A∗. To understand such systems, the model of advection dominated
accretion flow (ADAF) is introduced (Ichimaru 1977; Narayan & Yi 1994). In
this accretion flows, the energy released due to dissipation processes is
retained in the fluid rather than being radiated away. The models of ADAF
place an intermediate position between the spherically symmetric accretion
flow of non-rotating fluid (Bondi 1952) and the cool, thin disc of classical
accretion disc theory (e. g. Pringle 1981).
The observational evidence of accretion flows imply that outflow is an
important property in these systems. For example, comparison of the accretion
luminosities of neutron stars and white dwarfs in quiescence with similar
binary companions implies that the accretion rate onto white dwarfs is larger
by three orders of magnitude than that on the surface of neutron star (Loeb et
al. 2001). This indicates significant outflows in accretion flows. Moreover,
outflows seem to be common in the nuclei of galaxies. Marrone et al. (2006)
suggested the accretion rate of Sgr A∗ at small radii, much smaller than the
Bondi radius, must be low, below $10^{-7}M_{\odot}yr^{-1}$. While, Baganoff et
al. (2003) estimated the hot plasma surrounding Sgr A∗ should supply the
accretion rate of $10^{-6}M_{\odot}yr^{-1}$ at the Bondi radius. This
significant difference between the inner and outer mass accretion rates
indicates mass loss from the accretion flow due to the outflow (Kawabata &
Mineshige 2009). Thus, the accretion discs in the presence of outflow have
been studied by several authors (Knigge 1999; Fukue 2002, 2004; Shadmehri
2008; Xie & Yuan 2008; Kawabata & Mineshige 2009; Bu et al. 2009; Li & Cao
2009; Abbassi et al. 2010).
Knigge (1999) derived the radial distribution of dissipation rate and
effective temperature across a Keplerian, steady-state, mass losing accretion
disc, using a simple, parametric approach that is sufficiently general to be
applicable to many types of dynamical disc-wind models. Fukue (2002) examined
a hydrodynamical wind, which emanates from an accretion disc and is driven by
thermal and radiation pressure, under an one-dimensional approximation along
supposed streamlines. In another study, Fukue (2004) studied a supercritical
accretion regime, where the mass accretion rate was regulated just at the
critical rate with the help of wind mass-loss. He derived a critical radius
that outside of it, the disc is in a radiation-pressure dominated standard
state, while inside this radius the disc is in a critical state, where the
excess mass is expelled by wind, and the accretion rate is kept being just at
the critical rate at any radius inside of critical radius. Shadmehri (2008)
studied the effects of thermal conduction and outflows on ADAFs. He found that
in comparison to accretion flows without winds, the disc which rotates faster
and becomes cooler because of the angular momentum and energy flux which are
taking away by the winds. Xie & Yuan (2008), based on one-and-a-half-
dimensional description of the accretion flow, considered the interchange of
mass, radial and azimuthal momentum, and the energy between the outflow and
inflow. Bu et al. (2009) presented the self-similar solutions for ADAFs with
outflows and ordered magnetic fields. They assumed the magnetic field has a
strong toroidal component and a vertical component in addition to a stochastic
component. They found that the dynamical properties of ADAFs can be
significantly changed in the presence of ordered magnetic fields and outflows.
Abbassi et al. (2010) examined the effects of a hydrodynamical wind on ADAFs
in the presence of a toroidal magnetic field under a self-similar treatment.
Their results implied that in the presence of the wind, the disc temperature
decreases due to energy flux, which is taken away by winds and the accretion
velocity enhances.
The importance of magnetic diffusivity has been studied in several accreting
systems, such as the protostellar discs (Stone et al. 2000; Fleming & Stone
2003), discs in dwarf nova systems (Gammie & Menou 1998), the discs around
black holes (Kudoh & Kaburaki 1996), and Galactic centre (Melia & Kowalenkov
2001; Kaburaki et al. 2010). Moreover, two and three-dimensional MHD
simulations have shown that resistive dissipation is one of the crucial
processes that determines the saturation amplitude of the magnetorotational
instability (MRI). As, the linear growth rate of MRI can be reduced
significantly due to the suppression by ohmic dissipation (Sano et al. 1998;
Fleming et al. 2000; Masada & Sano 2008). Moreover, from comparison of ideal
and resistive MHD simulations, it seems the magnetic diffusivity may play an
important role in astrophysical outflows (Fendt & Čemeljić 2002; Čemeljić et
al. 2008).
As mentioned, semi-analytical studies of ADAFs with outflow are typically
related to systems without magnetic diffusivity. While, non-ideal MHD
simulations imply that the resistivity plays an important role on outflows (e.
g. Fendt & Čemeljić 2002; Čemeljić et al. 2008). Akizuki & Fukue (2006)
proposed a self-similar solution for ADAFs with a highly ionized gas. Thus,
they assumed that the plasma resistivity is zero, and only viscosity is due to
turbulence and dissipation in the disc. Also, they ignored the effects of
outflow/wind in their model. In this paper, we want to explore the effects of
the resistivity on magnetized ADAFs in the presence of outflows. Thus, we
adopt the presented solutions by Knigge (1999), Akizuki & Fukue (2006), and
Shadmehri (2008). The paper is organized as follow. In section 2, the basic
equations of constructing a model for ADAF in the presence of toroidal
magnetic field, resistivity and outflows will be defined. In section 3, self-
similar method for solving equations, which govern the behavior of the
accreting gas will be used. Results of the present model are brought in
sections 4 and 5. The summary of the model will appear in section 6.
## 2 Basic Equations
We consider a non-ideal magnetohydrodynamics of steady, axisymmetric, viscous,
accreting and rotating flow in presence of a purely toroidal magnetic field.
We use a cylindrical coordinate ($r$, $\phi$, $z$) centred on the accreting
object. We ignore the general-relativistic effects and use Newtonian gravity.
Under these assumptions, the continuity equation is
$\frac{\partial}{\partial r}(r\Sigma
v_{r})+\frac{1}{2\pi}\frac{\partial\dot{M}_{w}}{\partial r}=0,$ (1)
where $v_{r}$ is the radial velocity, $\Sigma=2\rho H$ is the surface density
of the disc, $\rho$ and $H$ being the mid-plane density and half-thickness of
the disc, respectively, and $\dot{M}_{w}$ is the mass loss rate by
outflow/wind. The half-thickness of the disc is given by
$H=c_{s}\sqrt{1+\Pi}/\Omega_{K}$, where $\Omega_{K}$ is the Keplerian angular
velocity, $c_{s}$ is the sound speed, $\Pi$ is defined below in equation (9).
It will be reduced to its traditional form of $H=c_{s}/\Omega_{K}$ in absence
of the toroidal component of magnetic field ($B_{\varphi}$; equation 8). The
sound speed is defined as $c_{s}=(p_{gas}/\rho)^{1/2}$, where $p_{gas}$ is the
gas pressure. The cumulative mass-loss rate from the disc can be written as
$\dot{M}_{w}(r)=\int^{r}_{{r_{in}}}4\,\pi\,r^{\prime}\,\dot{m}_{w}(r^{\prime})\,dr^{\prime},$
(2)
where $r_{in}$ denotes the radius at the inner edge of the disc and
$\dot{m}_{w}(r)$ is the mass-loss rate per unit area from each disc face.
Since the mass accretion rate is $\dot{M}=-2\pi r\Sigma v_{r}$, from the
equations (1) and (2), we can write
$\frac{\partial\dot{M}}{\partial r}=\frac{\partial\dot{M}_{w}}{\partial r}.$
(3)
Above equation implies that the mass accretion rate varies by radius due to
outflow. Thus, we exploit a power-law dependence for mass accretion rate as
follows (e.g. Blandford & Begelman 1999)
$\dot{M}(r)=\dot{M}(R)\left(\frac{r}{R}\right)^{s},$ (4)
where $R$ is the radius at the outer edge of the disc, $\dot{M}(R)$ is the
mass accretion rate at $R$, and $s$ is a free parameter, which for a disc
without outflow/wind, $s=0$ and in the presence of the outflow/wind, $s>0$
(e.g. Fukue 2004). The observed broad-band spectra of Sgr A∗ and soft X-ray
transients can also be fitted by ADAF models with moderate outflows, $s\sim
0.3$ – $0.4$, if the direct heating of electrons in ADAFs is efficient
(Quataert & Narayan 1999; Yuan et al. 2003). Equations (1)-(4) imply that
$\dot{m}_{w}(r)=s\,\frac{\dot{M}(R)}{4\pi
R^{2}}\left(\frac{r}{R}\right)^{s-2}.$ (5)
The radial equation of momentum is
$v_{r}\frac{dv_{r}}{dr}=r\,(\Omega^{2}-\Omega_{K}^{2})-\frac{1}{\Sigma}\frac{d}{dr}(\Sigma
c_{s}^{2})-\frac{c_{A}^{2}}{r}-\frac{1}{2\Sigma}\frac{d}{dr}(\Sigma
c_{A}^{2})$ (6)
where $\Omega$ is the angular velocity of the flow and $c_{A}$ is Alfven
speed, which is defined as $c_{A}^{2}\equiv
B_{\varphi}^{2}/(4\pi\rho)=2p_{mag}/\rho$, $p_{mag}$ being the magnetic
pressure.
The angular momentum transfer equation with consideration of the outflow/wind
can be written as (e.g. Knigge 1999)
$\Sigma
v_{r}\frac{d}{dr}(r^{2}\Omega)=\frac{1}{r}\frac{d}{dr}\left(r^{3}\nu\Sigma\frac{d\Omega}{dr}\right)-\frac{(lr)^{2}\Omega}{2\pi
r}\frac{d\dot{M}_{w}}{dr}$ (7)
where the two terms on the right-hand side of above equation describe the
effects of viscous torques due to shear ($\nu$, which is the effective
kinematic viscosity) and the angular momentum sink provided by the outflow.
Here, it will be assumed that matter outflowed at radius $r$ on the disc
carries away specific angular momentum $(lr)^{2}\Omega$. Thus, $l=0$
corresponds to a non-rotating disc wind and $l=1$ to outflowing material that
carries away the specific angular momentum it had at the point of outflow
(Knigge 1999).
The hydrostatic balance in the vertical direction is integrated to
$\frac{GM}{r^{3}}H^{2}=c_{s}^{2}\left[1+\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2}\right]=(1+\Pi)c_{s}^{2}.$
(8)
Here, we introduce the parameter $\Pi$ by
$\Pi=\frac{p_{mag}}{p_{gas}}=\frac{1}{2}\left(\frac{c_{A}}{c_{s}}\right)^{2},$
(9)
which is the degree of magnetic pressure to the gas pressure. Since we will
apply a steady self-similar method to solve system equation, this parameter
will be constant throughout the disc. Really, this parameter is a function of
position and time (Machida et al. 2006; Oda et al. 2007; Khesali & Faghei
2008, 2009). Studies of hot accretion flows represent the typical value of
$\Pi$ lies in the range $0.01$-$1$ (De Villiers et al. 2003; Beckwith et al.
2008), but here we also consider the magnetically dominated case ($\Pi>1$).
Because, MHD simulation by Machida et al. (2006) shows as thermal instability
grows in an accretion flow, the magnetic pressure exceeds the gas pressure due
to the disc shrink in the vertical direction and conservation of the toroidal
magnetic flux. This will result in large $\Pi$ and forms a magnetically
dominated accretion flow (Oda et al. 2007).
We assume both of the viscosity and the diffusivity are due to turbulence in
the disc, so that it is reasonable to use these parameters in analogy to the
$\alpha$-prescription of Shakura & Sunyaev (1973) for the turbulent
$\nu=P_{m}\eta=\alpha c_{s}H,$ (10)
where $P_{m}$ is the magnetic Prandtl number of the turbulence assumed a
constant of order of unity, $\eta$ is the magnetic diffusivity, and $\alpha$
is a free parameter less than unity.
Here, we can write the energy equation considering energy balance in the
system. We will assume the energy released due to viscous and resistive
dissipations can be balanced by the advection cooling and energy loss of
outflow (e.g. Shadmehri 2008). Thus,
$\displaystyle\frac{\Sigma
v_{r}}{\gamma-1}\frac{dc_{s}^{2}}{dr}-2Hv_{r}c_{s}^{2}\frac{d\rho}{dr}=$
$\displaystyle\frac{\alpha\sqrt{1+\Pi}fc_{s}^{2}}{\Omega_{K}}\left[\Sigma
r^{2}\left(\frac{d\Omega}{dr}\right)^{2}+\frac{H}{2\pi
P_{m}}\left(\frac{1}{r}\frac{d}{dr}(rB_{\varphi})\right)^{2}\right]$
$\displaystyle-\frac{1}{2}\xi\dot{m}_{w}(r)v_{K}^{2}(r),$ (11)
where $f$ is a constant less than unity and is called advection parameter. The
parameter $f$ measures how much the flow is advection-dominated (Narayan & Yi
1994). The first two terms on the right-hand side of above equation represent
the energy generated due to viscous and resistive dissipation, respectively.
The resistive dissipation is derived by $(4\pi/c^{2})\eta\textbf{J}^{2}$,
where $\textbf{J}=(c/4\pi)\nabla\times\textbf{B}$ is the current density.
Moreover, the last term on the right-hand side of energy equation is the
energy loss due to wind or outflow. Depending on the energy loss mechanism,
dimensionless parameter $\xi$ may change. We consider it as a free parameter
of our model so that larger $\xi$ corresponds to more energy extraction from
the disc due to the outflows (Knigge 1999; Shadmehri 2008).
The creation/escape rate of the magnetic field can be described by dynamo and
diffusion. We define the advection rate of the toroidal magnetic field as (Oda
et al. 2007)
$\dot{\Phi}=\int v_{r}B_{\varphi}dz,$ (12)
which is used instead of the induction equation. Since we study a steady-state
accreting system, the above quantity will be constant in absence of the dynamo
and the diffusion effects. This quantity can vary with radius due to the
presence of the dynamo/diffusion effect (Machida et al. 2006; Oda et al.
2007). In the present model, we expect the magnetic flux advection rate varies
with radius due to the presence of resistivity. We will consider this property
in next section.
## 3 Self-Similar Solutions
We seek self-similar solutions in the following form (e.g. Akizuki & Fukue
2006; Shadmehri 2008):
$\Sigma(r)=c_{0}\Sigma_{0}\left(\frac{r}{R}\right)^{s-\frac{1}{2}}$ (13)
$v_{r}(r)=-c_{1}\sqrt{\frac{GM}{R}}\left(\frac{r}{R}\right)^{-\frac{1}{2}}$
(14)
$\Omega(r)=c_{2}\sqrt{\frac{GM}{R^{3}}}\left(\frac{r}{R}\right)^{-\frac{3}{2}}$
(15)
$c_{s}^{2}(r)=c_{3}\left(\frac{GM}{R}\right)\left(\frac{r}{R}\right)^{-1}$
(16) $c_{A}^{2}(r)=\frac{B_{\varphi}^{2}}{4\pi\rho}=2\Pi
c_{3}\left(\frac{GM}{R}\right)\left(\frac{r}{R}\right)^{-1}$ (17)
where $\Sigma_{0}$ and $R$ are exploited to write the equations in the non-
dimensional forms. Substituting the above solutions in the continuity, radial
momentum, angular momentum, hydrostatic, and energy equations [(1),(6)-(8),
and (11)], we can obtain the following relations:
$c_{0}c_{1}=\dot{m}$ (18)
$-\frac{1}{2}c_{1}^{2}+\left[(s+\frac{1}{2})\Pi+s-\frac{3}{2}\right]c_{3}-c_{2}^{2}+1=0$
(19) $-\frac{1}{2}c_{0}\left[c_{1}-3\alpha
c_{3}(s+\frac{1}{2})\sqrt{1+\Pi}\right]+l^{2}s\dot{m}=0$ (20)
$\frac{H}{r}=\sqrt{(1+\Pi)c_{3}}$ (21) $\displaystyle\alpha\Pi
f(\gamma-1)\left(s-\frac{1}{2}\right)^{2}c_{0}c_{3}^{2}+P_{m}c_{0}c_{3}\times$
$\displaystyle\left\\{\frac{9}{2}\alpha
f(\gamma-1)c_{2}^{2}-\frac{2c_{1}}{\sqrt{1+\Pi}}\left[\left(s-\frac{3}{2}\right)\gamma-\left(s-\frac{5}{2}\right)\right]\right\\}-$
$\displaystyle\frac{2\,s\,\dot{m}\,\xi\,P_{m}(\gamma-1)}{\sqrt{1+\Pi}}=0$ (22)
where $\dot{m}$ is non-dimensional mass accretion rate and is defined
$\dot{m}=\frac{\dot{M}(R)}{2\pi\Sigma_{0}\sqrt{GMR}}.$ (23)
Using the equations (18)-(22), we obtain a quadratic equation for the
coefficient of $c_{3}$:
$\displaystyle-\frac{81}{16}\alpha^{2}(1+\Pi)\left[\frac{1+2s}{2l^{2}s-1}\right]^{2}c_{3}^{2}+$
$\displaystyle\Big{\\{}\frac{3}{(\gamma-1)f}\left[\frac{1+2s}{2l^{2}s-1}\right]\left[\left(s-\frac{3}{2}\right)\gamma-\left(s-\frac{5}{2}\right)\right]+$
$\displaystyle\frac{9}{2}\left[\left(s+\frac{1}{2}\right)\Pi+\left(s-\frac{3}{2}\right)\right]+\frac{\Pi}{P_{m}}\left(s-\frac{1}{2}\right)^{2}\Big{\\}}c_{3}+$
$\displaystyle\frac{9}{2}\left[1+\frac{2s\xi}{3f}\left(\frac{1+2s}{2l^{2}s-1}\right)\right]=0,$
(24)
and the rest of the coefficients are
$c_{0}=-\frac{2}{3}\frac{\dot{m}}{\alpha\sqrt{1+\Pi}}\left[\frac{2l^{2}s-1}{1+2s}\right]c_{3}^{-1},$
(25)
$c_{1}=-\frac{3}{2}\alpha\sqrt{1+\Pi}\left[\frac{1+2s}{2l^{2}s-1}\right]c_{3},$
(26)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}c_{2}^{2}=1-\frac{9}{8}\alpha^{2}(1+\Pi)\left[\frac{1+2s}{2l^{2}s-1}\right]^{2}c_{3}^{2}+$
$\displaystyle\left[(s+\frac{1}{2})\Pi+s-\frac{3}{2}\right]c_{3}.$ (27)
Without mass outflows, resistivity, and toroidal magnetic field, i.e.
$s=l=\xi=0$, $P_{m}=\infty$, and $\Pi=0$, above relations reduce to the
standard ADAF solutions (Narayan & Yi 1994). Moreover, in the absence of wind
and resistivity but with toroidal magnetic field, above relations reduce to
result of Akizuki & Fukue (2006). However the present model includes toroidal
magnetic field, outflows, and resistivity.
The studies of resistive and magnetized ADAFs (e. g. Faghei 2011) imply that
the solution for a set of the input parameters reaches to a non-rotating limit
at a specific of $\Pi$ which we call it by $\Pi_{c}$. Assuming $c_{2}=0$ and
$s=l=\xi=0$ (no wind case) in equations of (18)-(22), $\Pi_{c}$ can be written
as
$\Pi_{c}=\frac{18P_{m}}{f}\left[\frac{5/3-\gamma}{\gamma-1}\right].$ (28)
For typical values of adiabatic index and advection parameter in ADAFs,
$\gamma=4/3$ and $f=1$, we can write $\Pi_{c}=18P_{m}$. We cannot extend the
solutions for larger values of $\Pi_{c}$, because the right-hand side of
equation (27) becomes negative and a negative $c_{2}$ is clearly unphysical.
Moreover, $\Pi_{c}=18P_{m}$ implies that the flow can be magnetically
dominated for $P_{m}>1/18\sim 0.056$. It means the flow can be had a strong
magnetic field in the presence of resistivity. This property is in accord with
resistive MHD simulations of Machida et al. (2006).
Now, we can investigate the radial dependence of the magnetic flux advection
rate (equation 12). The self-similar solution for this quantity implies that
$\dot{\Phi}(r)=\dot{\Phi}_{out}~{}(\frac{r}{R})^{(s-3/2)/2},$ (29)
where $\dot{\Phi}_{out}$ is the magnetic flux advection rate at outer edge of
the disc, $R$. Since $s<3/2$, the magnetic flux increases with approaching to
central object and the stronger wind/outflow reduces this increasing. The
radial dependence of $\dot{\Phi}$ is qualitatively consistent with results of
Machida et al. (2006) and Oda et al. (2007).
## 4 Results
Now we can investigate the behavior of the solutions in the presence of the
outflows and resistivity. The effects of outflows and resistivity are studied
via parameters $s$, $l$, $\xi$, and $P_{m}^{-1}$. Here, the inverse of Prandtl
number specifies the resistivity of the fluid. Because,
$\eta\propto\alpha/P_{m}$ and $\alpha$ parameter is $0.1$ in all Figures. The
behavior of physical variables as a function of $P_{m}^{-1}$ are shown in
Figures 1 and 2. The solutions in Figures 1 and 2 represent the radial inflow
speed, $c_{1}$, and sound speed, $c_{3}$, both increase with the magnitude of
resistivity ($P_{m}^{-1}$). These properties are qualitatively consistent with
Faghei (2011). The density profiles, $c_{0}$, show that it decreases by adding
resistivity. It can be due to temperature raise of the flow. The rotational
velocity decreases, $c_{2}$, with the magnitude of resistivity. Because, the
viscous torque increases with the temperature ($\nu\propto c_{s}^{2}\propto
T$). These properties are also consistent with results of Faghei (2011).
Figure 1: Physical quantities of the flow as a function of $P_{m}^{-1}$ for
$\gamma=4/3$, $\alpha=0.1$, $\Pi=0.5$, and $f=l=\xi=1$. The solid, dotted,
dashed, and dot-dashed lines represent $s=0,0.01,0.02$, and $0.03$.
In Figure 1, we also studied the effect of parameter $s$ on physical
variables. The value of $s$ measures the strength of outflow and a larger $s$
denotes a stronger outflow. Figure 1 shows that for non-zero $s$, surface
density is lower than the standard ADAF solution and for stronger outflows
this reduction of surface density is more evidence. We can see that ADAF with
wind rotates more quickly than those without winds and lead to enhance
accretion velocity. The solution shows that temperature decreases for stronger
outflows. On the other hand, outflows play as a cooling agent. These
properties are in accord with results of Shadmehri (2008) and Abbassi et al.
(2010).
In Figure 2, the effect of energy loss due to outflows is studied by $\xi$
parameter. As with the magnitude of $\xi$ parameter, the more energy will
carry by outflows. Due to this energy loss, we expect the temperature should
decrease by adding the $\xi$ parameter. Temperature profiles confirmed this
property. Since the turbulence viscosity is proportional to temperature
($\nu\propto T$), the efficiency of angular momentum transport decreases with
the $\xi$ parameter. Decreasing viscous torque increases the rotational
velocity and decreases the radial infall velocity. These properties are
consistent with previous works (e. g. Shadmehri 2008).
Figure 2: Same as Figure 1, but $s=0.1$, and the solid, dotted, dashed, and
dot-dashed lines represent $\xi=0,0.1,0.2$, and $0.3$.
The physical variables as a function of parameter $\Pi$ and several values of
mass and energy losses are shown in Figures 3 and 4. By adding $\Pi$ which
indicates the role of magnetic filed on the dynamics of accretion discs, we
see the sound speed becomes larger, while surface density decreases. Moreover,
Figure 3 represents that the radial and rotational velocities increase with
the magnitude of $\Pi$. Increase in the radial velocity is due to the magnetic
tension term dominates the magnetic pressure term in the radial momentum
equation, which assists the radial velocity of accretion flows. Moreover,
increase in the rotational velocity is because of that the disc should rotate
faster than the case without the magnetic field which results the magnetic
tension. These properties are qualitatively consistent with the previous works
on magnetized ADAFs (e.g. Akizuki & Fukue 2006; Khesali & Faghei 2009; Abbassi
et al. 2010). Figures 3 and 4 imply that the mass and energy losses due to
wind give the same results by Figures 1 and 2. On the other hand, the effects
of mass and energy losses on the physical variables do not change in low and
high values of magnetic field.
Figures 5 and 6 represent that the disc thickness increases with the magnitude
of the resistivity or the magnetic field. Because the temperature increases by
adding the resistivity or the magnetic field, and from equation (21) we can
see the disc thickness increases with temperature. In Figures 5 and 6, the
disc thickness is also studied for several values of parameters $s$ and $\xi$.
The disc thickness profiles imply that it becomes thinner for the stronger
mass or energy losses due to outflows. Because these parameters reduces the
temperature of the flow.
## 5 The Bernoulli parameter
Here, we exploit Bernoulli parameter ($Be$) to consider the effects of the
resistivity to generate/enhance outflows in magnetized ADAFs. Because, this
parameter measures the likelihood that outflow or wind may originate
spontaneously (Narayan & Yi 1994). An adiabatic flow has a constant $Be$ along
streamlines. If $Be$ is positive for any of accreting gas, then this gas can
potentially reach infinity with a net positive kinetic energy. The Bernoulli
parameter, defined as the sum of the kinetic energy, the enthalpy and the
potential energy of the accretion flow,
$Be=\frac{1}{2}(v^{2}+r^{2}\Omega^{2})+\frac{\gamma}{\gamma-1}c_{s}^{2}-r^{2}\Omega_{K}^{2}.$
(30)
Figure 3: Physical quantities of the flow as a function of $\Pi$ for
$\gamma=4/3$, $\alpha=0.1$, $P_{m}=1$, $f=1$, $l=1$ and $\xi=1$. The solid,
dotted, dashed, and dot-dashed lines represent $s=0,0.01,0.03$, and $0.05$.
Narayan & Yi (1994) showed that Bernoulli parameter is positive in height-
integrated advection dominated flows, and suggested this may explain the
frequency occurrence of outflows and wind in many accretion systems. Using the
self-similar transformations of (13)-(17), the Bernoulli parameter can be
written as
$b=\frac{Be}{v_{K}^{2}}=\frac{1}{2}(c_{1}^{2}+c_{2}^{2})+\frac{\gamma}{\gamma-1}c_{3}-1,$
(31)
where $b$ is the normalized Bernoulli parameter. In Figure 7, the behavior of
this parameter as a function of $P_{m}^{-1}$ is studied for different values
of magnetic field. The $Be$ profiles represent that it is positive and
increases with the magnitude of resistivity or magnetic field. It can be due
to increase of the flow temperature by adding the resistivity or magnetic
field. Moreover, the $Be$ profiles show the magnetic field is more important
in high resistivity. Thus, the outflows can be enhanced in resistive and
magnetized ADAFs.
Figure 4: Same as Figure 3, but $s=0.1$, and the solid, dotted, dashed, and
dot-dashed lines represent $\xi=0,1,2$, and $3$.
## 6 Summary and Discussion
Mass loss appears to be a common phenomenon among accretion systems. The
observational evidences of ADAFs imply that outflow is important in such
system. Moreover, the non-ideal MHD simulation results represent that
resistivity can play an importance role in outflow of accretion discs.
In this paper, the structure of a magnetized ADAF in the presence of
resistivity and outflow is investigated. We assumed that the magnetic field
has a purely toroidal component. The outflow emanating affects on the
equations of continuity, angular momentum and energy, and can therefore act as
a sink for mass, angular momentum and energy. We adopted the presented
solutions by Knigge (1999), Akizuki & Fukue (2006), and Shadmehri (2008).
Thus, we assumed that angular momentum transport is due to viscous turbulence
and the $\alpha$-prescription is used for the kinematic coefficient of
viscosity. We also assumed that the flow does not have a good cooling
efficiency and so a fraction of energy accretes along with matter onto the
central object. To solve the equations that govern the structure behavior of
magnetized ADAF with outflow, we have used a steady self-similar solution.
Figure 5: The ratio of height thickness to radius as a function of
$P_{m}^{-1}$ for different $s$ (left panel), and $\xi$ (right panel). The
input parameters in left and right panels are same as Figures 1 and 2,
respectively.
The present model represented the outflows in accretion flows can improve by
the resistivity and magnetic field. These properties are in accord with
resistive MHD simulations of Fendt & Čemeljić (2002) and Čemeljić et al.
(2008). For example, Fendt & Čemeljić (2002) showed that resistivity can
affect the outflows structure in accretion discs. Moreover, they found that
the outflow velocity increases with the magnitude of the resistivity and
toroidal magnetic field.
In this paper, we assumed a purely toroidal magnetic field that provides
restrictions in the present model. For example, a purely toroidal magnetic
field is not enough to have magnetically driven outflows. Thus, the present
model is unsuitable to use in disc with magnetically driven outflows.
Moreover, we considered the presented model in a height-integrated approach
and applied it to calculate Bernoulli parameter. Narayan & Yi (1995) showed
that Bernoulli parameter in ADAF varies by latitude. As Bernoulli parameter in
equatorial is negative and becomes positive for high latitude. One can
investigate latitudinal behavior of the present model.
## Acknowledgments
We would like to thank referee for his/her invaluable comments and very
careful reading of the manuscript that helped us to improve the initial
version of the paper.
Figure 6: The ratio of height thickness to radius as a function of $\Pi$ for
different values of $s$ (left panel), and $\xi$ (right panel). The input
parameters in left and right panels are same as Figures 3 and 4, respectively.
Figure 7: The dimensionless of Bernoulli constant as a function of
$P_{m}^{-1}$ for different values of $\Pi$. The input parameters are same as
Figure 1, but $s=\xi=l=0$, and the solid, dotted, dashed, and dot-dashed lines
represent $\Pi=0.1,0.3,0.5$, and $0.7$, respectively.
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|
arxiv-papers
| 2012-02-14T17:26:09 |
2024-09-04T02:49:27.420757
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kazem Faghei and Azam Mollatayefeh",
"submitter": "Kazem Faghei",
"url": "https://arxiv.org/abs/1202.3095"
}
|
1202.3357
|
040001 2012 A. Goñi J. Chavez Boggio, Leibniz Institut für Astrophysik
Potsdam, Germany. 040001
We present a high-speed wavelength tunable photonic crystal fiber-based source
capable of generating tunable femtosecond solitons in the infrared region.
Through measurements and numerical simulation, we show that both the
pulsewidth and the spectral width of the output pulses remain nearly constant
over the entire tuning range from $860$ to $1160$ nm. This remarkable behavior
is observed even when pump pulses are heavily chirped ($7400$ fs2), which
allows to avoid bulky compensation optics, or the use of another fiber, for
dispersion compensation usually required by the tuning device.
# High-speed tunable photonic crystal fiber-based femtosecond soliton source
without dispersion pre-compensation
Martín Caldarola [lec] Víctor A. Bettachini E-mail: caldarola@df.uba.ar
[itba] Andrés A. Rieznik [itba] Pablo G. König [itba] Martín E. Masip [lec]
Diego F. Grosz [itba, coni] Andrea V. Bragas[lec, ifiba]
(7 July 2011; 1 February 2012)
††volume: 4
99 lec Laboratorio de Electrónica Cuántica, Departamento de Física,
Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, (C1428EHA)
Buenos Aires, Argentina. itba Instituto Tecnológico de Buenos Aires, Eduardo
Madero 399, (C1106ACD) Buenos Aires, Argentina. coni Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina. ifiba IFIBA, Consejo
Nacional de Investigaciones Científicas y Técnicas, Argentina.
## 1 Introduction
Light sources based on the propagation of solitons in optical fibers have
emerged as a compact solution to the need of a benchtop source of ultra-short
tunable pulses [1, 2, 3]. The soliton formation from femtosecond pulses
launched into an optical fiber is explained in terms of the interplay between
self-phase modulation (SPM) and group-velocity dispersion (GVD) in the
anomalous dispersion regime [4]. The wavelength tunability is a consequence of
the Raman-induced frequency shift (RIFS) produced on the pulse when traveling
through the fiber [5]. The term soliton self-frequency shift (SSFS) [6] was
coined to name this effect widely used to produce tunable femtosecond pulses
in different wavelength ranges, e.g., from $850$ to $1050$ nm [7], from $1050$
to $1690$ nm [8], and from $1566$ to $1775$ nm [1]. In most cases, photonic
crystal fibers (PCF) are used for building these sources since their GVD can
be easily tailored to produce solitons in a desired tuning range [9, 10]. For
a given choice of the PCF, full experimental characterization of the pump and
output pulses, complemented with theoretical predictions, is necessary to
understand how nonlinear effects modify the output soliton.
The wavelength tunability in a PCF-based light source is provided by the
modulation of the pump power injected into the fiber [11, 12, 13, 14]. It is
worth noting that the wavelength choice of the output pulse is done without
moving any mechanical part, which is clearly attractive for all the proposed
and imaginable applications of these soliton sources. Moreover, the wavelength
of the output pulse can be chosen as fast as one can modulate the power of the
pump pulse, as introduced in Ref. [15, 16]. By introducing an acousto-optic
modulator (AOM) in the path of the pump pulse, the output wavelength can be
changed at a speed which is ultimately limited only by the laser repetition
rate. This kind of experimental setup has been presented in some previous
reports [17, 14], with stunning applications as the one presented in Ref.
[18], where a pseudo-CW wideband source for optical coherent tomography is
introduced. However, the need to pre-compress the pump pulse to avoid the
chirp produced by the AOM contrives against the compact and mechanically
robust design of the light source. In this paper, we demonstrate that the PCF-
based source presented here is robust against chirped pump pulses. We present
a complete set of measurements showing that the temporal and spectral
characteristics of the generated solitons in the PCF remain unaltered even
when pump pulses are heavily chirped up to $\sim 7400$ fs2. Results are
presented for the whole range of tunability ($860$ nm to $1160$ nm). We also
present numerical simulations which remarkably fit the experimental data and
help to understand the soliton behavior.
This paper is organized as follows: In section II, we describe the
experimental setup. The numerical simulations are described in section III. In
section IV, we present experimental and numerical results and in section V we
further analyze the results with numerical simulations. Finally, in section
VI, we present our conclusions.
## 2 Experimental Setup
A scheme of the experimental setup is shown in Fig. 1. A Ti:Sa laser (KMLabs)
generates ultrashort transform-limited (TL) pulses of $\mathrm{\Delta t}=31$
fs (FWHM-sech2), $\mathrm{\lambda_{pump}}=830$ nm, with a spectral width
$\mathrm{\Delta\lambda}=23$ nm, and a repetition rate of $94$ MHz.
Figure 1: Experimental setup. (a) Titanium-Sapphire (Ti:Sa) laser, (b) Prism
compressor, (c) Acousto-optic modulator (AOM), (d) Coupling lens, (e) Photonic
crystal fiber (PCF), (f) Collimator objective, (g) Spatial filter, (h) Flipper
mirror, (i) Fast-scan interferometric autocorrelator, (j) Optical spectrum
analyzer (OSA).
The AOM not only allows high speed (up to MHz) and accurate control of the
soliton wavelength, as previously discussed, but also prevents feedback into
the Ti:Sa, replacing the optical isolator required in similar setups [19]. As
the AOM introduces $\sim 56$ mm of SF8 glass path, pump pulses gain a positive
chirp of about $\sim 7400$ fs2, which leads to a time spread by a factor of
$\sim 3$ in them. This can be pre-compensated, for example, by introducing an
optical fiber in the anomalous dispersion regime [20, 8] or a prism compressor
in the well-known configuration presented in [21]. In this work, the chirp was
compensated by a pair of SF18 prisms with an apex separation of $78$ cm.
Additionally, the prism compressor allowed us to up-chirp pump pulses in a
controlled fashion from TL to $\sim 1400$ fs2 by introducing an extra glass
path at the second prism of the arrangement [22]. This full or partial
compensation of the phase distortion introduced by the AOM allowed us to study
the role of different chirp figures in the temporal and spectral
characteristics of the solitons generated in the PCF.
Pump pulses were coupled into a non-polarization-maintaining microstructured
fiber commercially used for supercontinuum generation (Thorlabs,
NL-2.3-790-02). Its main parameters are listed in Table 1 and the dispersion
curve and SEM image are shown in Fig. 2.111Datasheet available in
http://www.thorlabs.com. Upon propagation down the fiber, the spectrum is
highly broadened so a spatial band-pass filter made of a prism and razor
blades, similar to the one presented in [23], allowed to filter the spectral
region of the solitonic branch (see Fig. 3) without adding any extra chirp to
the solitons.
Figure 2: Dispersion curve of the PCF, showing the zero dispersion wavelength (ZDW) at $790$ nm. The inset is the scanning electron microscope image of the PCF core. The curve and image were provided by the manufacturer. L | $75$ cm
---|---
ZDW | $790$ nm
$\beta_{2}$ | $-12.4$ ps2km-1
$\beta_{3}$ | $0.07$ ps3km-1
$\gamma(\omega)$ | $\gamma_{0}(\omega_{0})+(\omega-\omega_{0})\gamma_{1}$
$\gamma_{0}(\omega_{0})$ | $78$ W-1km-1
$\gamma_{1}$ | $\gamma_{0}/\omega_{0}$
$\omega_{0}$ | $2271$ THz
Table 1: PCF parameters relevant to the simulation. Further details can be
found in Ref. [14].
Once the spectral selection was achieved, a flipper mirror directed the
filtered beam for analysis either by the optical spectrum analyzer (OSA) or by
the interferometric autocorrelator. A fast-scan system [24] allows to perform
fast interferometric auto-correlations. Briefly, a platform with a hollow
retroreflector is moved sinusoidally back and forth, with a stepper motor at
11 Hz, to produce and optical delay in one of the arms of a Michelson
interferometer. The autocorrelation signal is recorded by a PMT and averaged
with an oscilloscope.
## 3 Numerical Simulations
In order to further validate experimental results, we simulated the
propagation of femtosecond pulses in the PCF by numerically solving the
generalized nonlinear Schrödinger equation (GNLSE) including dispersive, Kerr,
instantaneous and delayed Raman response, and self-steepening effects [25],
with a conservation quantity error (CQE) adaptive step-size algorithm [26].
The GNLSE reads
$\displaystyle\frac{\partial A}{\partial z}+\beta_{1}\frac{\partial
A}{\partial t}+\operatorname{i}\beta_{2}\frac{\partial^{2}A}{\partial t^{2}}$
(1) $\displaystyle-\beta_{3}\frac{\partial^{3}A}{\partial
t^{3}}+...=\operatorname{i}\gamma(\omega)\left(1+\frac{\operatorname{i}}{\omega_{0}}\frac{\partial}{\partial
t}\right)$
$\displaystyle\times\left(A(z,t)\int_{-\infty}^{\infty}R(t^{\prime})|A(z,t-t^{\prime})|^{2}dt^{\prime}\right),$
$\displaystyle R(t)=\left(1-f_{R}\right)\delta(t)+f_{R}h_{R}(t),$
$\displaystyle h_{R}(t)=\left(f_{a}+f_{c}\right)h_{a}(t)+f_{b}h_{b}(t),$
$\displaystyle
h_{a}(t)=\tau_{1}\left(\tau_{1}^{-2}+\tau_{2}^{-2}\right)\operatorname{e}^{-t/\tau_{2}}\sin{(t/\tau_{1})},$
$\displaystyle
h_{b}(t)=\left[\left(2\tau_{b}-t\right)/\tau_{b}^{2}\right]\operatorname{e}^{-t/\tau_{b}},$
where $A(z,t)$ is the complex envelope of the electric field, $\beta_{n}$ are
the expansion terms for the propagation constant around the carrier frequency
$\omega_{0}$ and $\gamma$ is the nonlinear coefficient. $f_{R}(t)$ represents
the fractional contribution of the delayed Raman effect $h_{R}$. Note that Eq.
(1) adopts a more accurate description of this effect than the one usually
used [4]. In our simulation, we adopted $\tau_{1}=12.2$ fs, $\tau_{2}=32$ fs,
$\tau_{b}=96$ fs, $f_{a}=0.75$, $f_{b}=0.21$, $f_{c}=0.04$, and $f_{R}=0.24$
[27]. The dependence of the fiber non-linear parameter $\gamma$ with the
frequency was modeled as a linear function (see Table 1).
## 4 RESULTS
### 4.1 Transform-limited pump pulses
First, we present the full characterization of the soliton source seeded by TL
pump pulses, in an extended wavelength range if compared with the results
presented in our previous paper [14].
In order to investigate the dependence of the output spectrum with the coupled
power, managed by the AOM, we skipped spectral filtering at first. Fig. 3
shows the measured spectrum at the PCF output as a function of the coupled
power. The infrared solitonic branch appears at $\sim 10$ mW and undergoes
red-shift with increasing power. The maximum wavelength attained is $1130$ nm
at $55$ mW. Spectra in Fig. 3 also shows show that some of the input energy is
converted to visible non-solitonic radiation.
Figure 3: Experimental spectra vs coupled power to the PCF with transform
limited (TL) pump pulses. The color map shows spectral intensity. The maximum
achieved soliton shift, $\lambda_{s}\simeq 1130$ nm, was reached at $55$ W.
The pulsewidth of the filtered soliton as a function of its wavelength,
$\lambda_{s}$, is shown in Fig. 4. The pulsewidth remains constant at $\sim
45$ fs, for the entire tunability range. Numerical simulations are also
plotted in the same figure, showing an excellent agreement with experimental
measurements.
Figure 4: Experimental pulsewidth of the soliton as a function of its
wavelength, pumping the PCF with TL pulses. The results for the three lower
wavelengths were already present in Ref. [14]. Full line: numerical
simulations.
### 4.2 Chirped pump pulses
The effect over the soliton produced by the chirp of pump pulses was studied
systematically by introducing a known amount of extra glass path on the second
prism of the compressor. This scheme allowed to change the GVD of pump pulses
from $0$ to $1400$ fs2. Further chirping was achieved by the complete removal
of the prism compressor, leading to a total amount of positive chirp $\sim
7400$ fs2.
Figure 5: Soliton temporal (a) and spectral width (b) vs. chirp of input pump
pulses . Full line: numerical results. The soliton wavelength is
$\lambda_{s}=1075$ nm.
Figure 5 (a) shows the pulsewidth of solitons with wavelength
$\lambda_{s}=1075$ nm upon variation of pump pulses chirp. Even for a $\sim
7400$ fs2 chirp, the soliton output pulsewidth remained around $45$ fs.
Numerical simulations show very good agreement with these observations, as
they predict nearly constant pulsewidth regardless of the input chirp (full
line in Fig. 5). Measurements and numerical simulations in the spectral domain
also indicate that the bandwidth of the output solitons is almost unaffected
by the pump pulses chirp [see Fig. 5 (b)]. The product $\Delta t\Delta\nu$ was
found to be near $0.315$, as it is expected for transform-limited sech2
pulses.
The effect of this heavy chirping was evident in the auto-correlation traces
of pump pulses, as can be seen by comparing Fig. 6 (a) and (c). However, there
is not a clear difference between traces of the output solitons for the TL (b)
and the highly chirped ($\sim 7400$ fs2) case (d).
Figure 6: Interferometric auto-correlation traces of TL (a) and heavily
chirped, $\sim 7400$ fs2, Ti:Sa pump pulses (c). Interferometric auto-
correlation traces of the output solitons are similar in both cases, unchirped
(b) and heavily chirped (d). Soliton wavelength is $\lambda_{s}\simeq 1075$
nm.
A color map of the spectra as function of the coupled power, for a highly
chirped pump pulse ($\sim 7400$ fs2), is shown in Fig. 7. As in the TL case,
we observe that a solitonic branch is red shifted by increasing the coupled
power. However, in this case, $80$ mW of coupled power is required to produce
a $1160$ nm soliton which represents an increment of about $\sim 45\%$ in
comparison to the TL case.
Figure 7: Spectra vs coupled power to the PCF with highly chirped ($\sim 7400$
fs2) input pulses.
A comparison of the soliton red-shift between TL and chirped pump pulse cases
is presented Fig. 8. The figure shows that more power is always required to
attain the same shift when pump pulses are heavily chirped.
Figure 8: Soliton wavelength shift for chirped (full squares) and TL pump
pulses (empty squares) vs pump pulse power. Dashed and full lines correspond
to numerical results for TL and highly chirped pump pulses, respectively.
Figure 9 (a) shows the soliton pulsewidth as a function of its wavelength,
$\lambda_{s}$, when the pump pulse is heavily chirped ($\sim 7400$ fs2). We
observe an approximately constant output pulsewidth ($\sim 45$ fs) in the
entire tuning range. Furthermore, the $\mathrm{\Delta t\Delta\nu}$ product,
shown in Fig. 9 (b), indicates that the generated pulses can be identified as
fundamental solitons (sech2-like), as in the case of TL pump pulses [14].
Numerical simulations were also performed for this case (full lines in Fig. 9)
showing an excellent agreement with experimental results.
Figure 9: (a) Soliton pulsewidth and (b) the product $\mathrm{\Delta
t\Delta\nu}$ vs wavelength in the case of highly chirped pump pulses
($\sim$7400 fs2).
## 5 DISCUSSION
### 5.1 Fiber soliton self-frequency shift effective length
In order to further analyze soliton formation, we studied the pulse evolution
along the fiber by performing numerical simulations. The spectrum evolution
along the fiber, for a given coupled power, in the TL and the chirped cases
are shown in Fig. 10 (a) and (b), respectively. These simulations show that in
the case of chirped pump pulses ($\sim 7400$ fs2), the spectrum broadening and
the soliton formation take place farther down into the fiber (see Fig. 10), as
compared to the TL case.
The delay in the formation of the soliton can be explained by an interplay of
opposite chirping effects: the positive chirp acquired by traversing the AOM
is compensated as the pulse advances into the PCF, in anomalous propagation,
leading to pulse compression. The PCF itself provides pulse compression in the
first stretch of the fiber previously to the branching of a soliton.
Therefore, the SSFS effective length, i.e., the fiber path where nonlinearity
broadens the spectrum, is longer in the TL case. If the chirp is
overcompensated and a negatively chirped pulse is fed into the fiber, these
pulses would also be compressed within the first stretch of the fiber due to
SPM [28] leading to the same behavior than in the positively chirped case,
resulting in a narrower tunability range.
Once the soliton is formed and the peak power is high enough, intrapulse Raman
scattering red-shifts the soliton as it propagates through the remaining of
the fiber. This spectral shift increases with both fiber length and soliton
peak power [4]. So the fact that the soliton is formed at different lengths
explains the red shifts observed for the same coupled power.
However, as a larger wavelength shift can be achieved with a higher input
power, this shortening in the effective length in the chirped case could be
compensated by coupling more power into the PCF [1]. Another possibility for
compensating this effect on the SSFS is using a longer PCF.
Figure 10: Simulated spectral evolution along the fiber length. Pump pulses
with identical peak power produce more soliton shifting with unchirped (a)
than with heavily chirped ($\sim$7400 fs2) (b) pump pulses.
### 5.2 Fiber power conversion efficiency
Figure 11: Simulation of the spectral evolution along the fiber showing that
the same shifting shown in Fig. 10 (a) for unchirped pump pulses is attained
with chirped ones ($\sim 7400$ fs2) with a higher pump power.
Figure 11 shows a simulation where the same shift wavelength obtained for TL
pump pulses is achieved by increasing the coupled power in the shorter
effective length fiber ($7400$ fs2 chirp). Fission of more than one soliton
branch is visible in this case, as compared to the case of TL pump pulses, for
which only a single soliton branch appears (Fig. 10). Each soliton branch
carries a fundamental soliton ($N=1$) with a peak power $P_{0}$ given by [4]
$N^{2}=1=\frac{\gamma P_{0}T_{0}^{2}}{|\beta_{2}|},$ (2)
As $\gamma$, $\beta_{2}$ and $T_{0}$ (see Figs. 4 and 5) are the same in both,
the TL and chirped cases, the peak power of the solitons is also the same.
The arising of new soliton branches partially accounts for the increased pump
power required in the chirped case (Fig. 11) to attain the same shift. Indeed,
the soliton-pump power ratio is $0.2$ in the chirped case and $0.44$ in the TL
case. This result reveals that the use of the PCF as a compressor decreases
its power conversion efficiency.
On the other hand, it is possible to achieve the same soliton shift as in the
TL case by increasing the fiber length, and keeping the same pump power. In
this case, the power conversion efficiency is even lower, $0.17$, as predicted
by simulations.
## 6 Conclusions
We have presented a high-speed tunable soliton infrared source capable of
generating $\sim 45$ fs transform-limited pulses in the range from $860$ to
$1160$ nm. Both the pulsewidth and the spectral width were shown to remain
constant over the entire tuning range, even when pump pulses were heavily
chirped up to $7400$ fs2. Insensitivity to the chirp of pump pulses points out
to the feasibility of avoiding bulky compensation optics prior to the PCF,
opening up the possibility to build reliable and compact high-speed tunable
femtosecond sources in the near infrared region. A minor drawback of this
source is that either more power needs to be coupled or a longer PCF needs to
be used in order to achieve the same tuning range obtained with transform-
limited pump pulses.
###### Acknowledgements.
This work was supported by ANPCyT PICT 2006-1594, ANPCyT PICT 2006-497 and UBA
Programación Científica 2008-2010, Proyecto N X022.
## References
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* [7] B Washburn, S Ralph, P Lacourt, J Dudley, Tunable near-infrared femtosecond soliton generation in photonic crystal fibers, Electronics Lett. 37, 1510 (2001).
* [8] J Takayanagi, T Sugiura, M Yoshida, N Nishizawa, 1.0-1.7 $\mu$m wavelength-tunable ultrashort-pulse generation using femtosecond yb-doped fiber laser and photonic crystal fiber, IEEE Photon. Technol. Lett. 18, 659 (2006).
* [9] P Russell, Photonic crystal fibers, Science 299, 358 (2003).
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* [12] K S Abedin, F Kubota, Widely tunable femtosecond soliton pulse generation at a 10-ghz repetition rate by use of the soliton self-frequency shift in photonic crystal fiber, Opt. Lett. 28, 1760 (2003).
* [13] N Ishii, C Y Teisset, E E Serebryannikov, T Fuji, T Metzger, F Krausz, A M Zheltikov, Widely tunable soliton frequency shifting of few-cycle laser pulses, Phys. Rev. E 74, 036617 (2006).
* [14] M E Masip, A A Rieznik, P G König, D F Grosz, A V Bragas, O E Martínez, Femtosecond soliton source with fast and broad spectral tunability, Opt. Lett. 34, 842 (2009).
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* [18] K Sumimura, Y Genda, T Ohta, K Itoh, N Nishizawa, Quasi-supercontinuum generation using 1.06 $\mu$m ultrashort-pulse laser system for ultrahigh-resolution optical-coherence tomography. Opt. Lett. 35, 3631 (2010).
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* [20] J Nicholson, A Yablon, P Westbrook, K Feder, M Yan, High power, single mode, all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation, Opt. Express 12, 3025 (2004).
* [21] R L Fork, O E Martínez, J P Gordon, Negative dispersion using pairs of prisms, Opt. Lett. 9, 150 (1984).
* [22] R L Fork, C H B Cruz, P C Becker, C V Shank, Compression of optical pulses to six femtoseconds by using cubic phase compensation, Opt. Lett. 12, 483 (1987).
* [23] J L A Chilla, O E Martinez, Direct determination of the amplitude and the phase of femtosecond light pulses, Opt. Lett. 16, 39 (1991).
* [24] S Costantino, A R Libertun, P D Campo, J R Torga, O E Martínez, Fast scanner with position monitor for large optical delays, Opt. Comm. 198, 287 (2001).
* [25] J Dudley, G Genty, S Coen, Supercontinuum generation in photonic crystal fibers, Rev. Mod. Phys. 78, 1135 (2006).
* [26] A Heidt, Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers, J. Lightwave Technol. 27, 3984 (2009).
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|
arxiv-papers
| 2012-02-15T16:45:30 |
2024-09-04T02:49:27.433534
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mart\\'in Caldarola, V\\'ictor A. Bettachini, Andr\\'es A. Rieznik, Pablo\n G. Konig, Mart\\'in E. Masip, Diego F. Grosz, Andrea V. Bragas",
"submitter": "Mart\\'in Caldarola",
"url": "https://arxiv.org/abs/1202.3357"
}
|
1202.3473
|
11institutetext: Sandia National Laboratories, Livermore, CA 94550
11email: {jairay,apinar,scomand}@sandia.gov
# Are we there yet? When to stop a Markov chain while generating random
graphs††thanks: This work was funded by the Applied Mathematics Program at the
U.S. Department of Energy and performed at Sandia National Laboratories, a
multiprogram laboratory operated by Sandia Corporation, a wholly owned
subsidiary of Lockheed Martin Corporation, for the United States Department of
Energy’s National Nuclear Security Administration under contract DE-
AC04-94AL85000.
Jaideep Ray Ali Pinar C. Seshadhri
###### Abstract
Markov chains are a convenient means of generating realizations of networks,
since they require little more than a procedure for rewiring edges. If a
rewiring procedure exists for generating new graphs with specified statistical
properties, then a Markov chain sampler can generate an ensemble of graphs
with prescribed characteristics. However, successive graphs in a Markov chain
cannot be used when one desires independent draws from the distribution of
graphs; the realizations are correlated. Consequently, one runs a Markov chain
for $N$ iterations before accepting the realization as an independent sample.
In this work, we devise two methods for calculating $N$. They are both based
on the binary “time-series” denoting the occurrence/non-occurrence of edge
$(u,v)$ between vertices $u$ and $v$ in the Markov chain of graphs generated
by the sampler. They differ in their underlying assumptions. We test them on
the generation of graphs with a prescribed joint degree distribution. We find
the $N\propto|E|$, where $|E|$ is the number of edges in the graph. The two
methods are compared by sampling on real, sparse graphs with ${\rm
10^{3}}-{\rm 10^{4}}$ vertices.
###### Keywords:
graph generation, Markov chain Monte Carlo, independent samples
††footnotetext: _Submitted to the 9th Workshop on Algorithms and Models for
the Web Graph, WAW-2012, June 22-23, 2012, Dalhousie University, Halifax, Nova
Scotia, Canada. SAND2012-1169C_
## 1 Introduction
Markov chain Monte Carlo (MCMC) methods are a common means of generating
realizations of graphs which share similar characteristics since they require
nothing more than a procedure that can generate a new graph by “rewiring” the
edges of an existing graph. Much of their use to date has been in generating
graphs with a prescribed degree distribution [1, 2, 3, 4]. Other efforts have
used MCMC to generate graphs with a prescribed joint degree distribution [5].
MCMC methods require a graph to start the chain; thereafter, the “rewiring”
procedure generates new, realizations which preserve certain graph
characteristics. The specific characteristic(s) that are preserved depend
entirely on the “rewiring” procedure.
MCMC methods for generating graphs has two drawbacks. _Initialization bias_
arises from the fact that the starting graph may not even lie in the
population of graphs that we seek to sample, or may be an outlier in that
population. The second issue, _autocorrelation in equilibrium_ , arises from
the fact that successive samples drawn by the MCMC sampler are correlated and
an empirical distribution constructed from them would result in the
statistical error (variance) to be 2$\tau_{int}$larger than the distribution
constructed using independent samples. Here $\tau_{int}$, the integrated
autocorrelation time, is a measure of how slowly correlation in graphical
metrics, calculated from an MCMC series, decays; see Sections 2 and 3 in
Sokal’s lecture notes [6].
Sokal’s method [6] for deciding the “sufficiency” of samples obtained from
MCMC revolve around autocorrelation. The method is general, and was adapted
for use with graphs in [5]. Consider an edge $(u,v)$ between labeled vertices
$u$ and $v$ in the ensemble of graphs generated by the MCMC chain. Denoting
its occurrence/non-occurrence in the chain of graphs by 1/0 gives us a binary
time-series $\\{Z_{t}\\},t=1\ldots T$, with an empirical mean $\mu$. The auto-
correlation, with lag $l$, is given by
$C(l)=(\\{Z_{t}\\}-\mu)(\\{Z_{t+l}\\}-\mu),t=1\ldots T-l$ and the normalized
version of it, $\rho(l)=C(l)/C(0)$ can be used to gauge whether the
autocorrelation in the time-series is observed to be decreasing. In [5], the
authors used this metric, applied to all edges in the graphs that were
sampled, to ensure that their MCMC chain was mixed. One can also set, loosely
speaking (for details, consult [6]), a minimum threshold $\rho_{min}$,
identify the corresponding lag $l_{min}$, and retain every $l_{min}^{th}$
entry in the MCMC chain to serve as independent samples. However, this method
has two practical drawbacks. First the autocorrelation analysis has to be
performed for all the edges (potentially, $|V|^{2}$ in number) that might
appear in the MCMC chain, which quickly becomes prohibitively expensive for
large graphs. Secondly, it requires a user input, $\rho_{min}$, which may have
an arbitrary effect on graphical properties of the ensemble. These
shortcomings motivate our work.
In this paper, we propose two different methods for generating _independent_
graphs using an MCMC method. The first, which we call Method A or “multiple
short runs”, determines the number of iterations $N$ an MCMC method has to be
run to “forget” the initial graph and minimize the initialization bias. The
second approach, Method B or “one long run”, requires $K$ MCMC iterations.
This long run is thinned by a factor $k$ (i.e., every $k^{th}$ MCMC iteration
is preserved) to generate $K/k$ independent samples. Both methods are intended
to be approximate, but simple to evaluate, so that they can be employed in
practice to gauge the “sufficiency” of MCMC iterations. The two methods for
extracting independent graphs are tested on an MCMC chain with the setup
described in [5]. We explore the practical impact of approximations in our
methods. We restrict ourselves to undirected graphs with labeled nodes.
In the next section (Sec. 2) we describe the procedure used to “rewire” a
graph to create a new graph realization with the same joint degree
distribution. In Sec. 3 we describe the two methods for generating independent
samples. In Sec. 4, we test the methods on real sparse graphs. We conclude in
Sec. 5.
## 2 A Markov chain algorithm for sampling graphs with a given joint degree
distribution
Consider an undirected graph $G=(V,E)$, where $|V|=n$ and $|E|=m$. The degree
distribution of the graph is given by the vector $\vec{f}$, where $f(d)$ is
the number of vertices of degree $d$. The _joint degree distribution_ list the
number of edges incident between vertices of specified degrees. Formally, the
$n\times n$ matrix $\mathbf{J}$ denotes the joint degree distribution, where
the entry $J(i,j)$ is the number of edges between vertices of degree $i$ and
degree $j$. Stanton and Pinar [5] studied the problem of generating and random
sampling a graph with a given joint degree distribution. They proposed a
greedy algorithm to construct an instance of a graph with a specified
(feasible) degree distribution, as well as a Markov chain algorithm to
generate random samples of graphs with the same degree distribution.
For the purposes of this paper, we will only focus on the Markov chain
algorithm. The rewiring operation that moves us between the nodes of the
Markov chain is depicted in Fig. 1. At the first step, one picks an edge
$(u_{1},v)$ at random and thereafter, one of the end vertices, e.g., $u_{1}$.
We wish to break $(u_{1},v)$ and connect $u_{1}$ and $v$ to others without
violating the prescribed joint degree distribution. We, therefore, search for
another edge $(u_{2},w)$ where $d_{u_{2}}=d_{u_{1}}$ or $d_{w}=d_{u_{1}}$,
where $d_{p}$ denotes the degree of node $p$. WLOG, let $d_{u_{2}}=d_{u_{1}}$.
Swapping the edges i.e. creating edges $(u_{1},w)$ and $(u_{2},v)$ while
destroying $(u_{1},v)$ and $(u_{2},w)$ leaves the joint degree distribution
unchanged while changing the connectivity pattern of the graph. If the
resulting graph is simple, the graph is retained by the MCMC chain.
Figure 1: The swapping operation for the Markov chain algorithm.
The procedure results in inter-edge correlation. A particular edge is chosen
for swapping, on average, once every $|E|$ Markov chain iterations. In [5],
this procedure was used to generate graphs (of moderate sizes, with $|V|\leq
23,000$), using an MCMC sampler. Autocorrelation analysis showed that the
Markov chain mixes and the autocorrelation decays with edge-dependent rates.
Empirically it was observed that an edge $(u,v)$ de-correlated slowly if
$J(d_{u},d_{v})/(f{d_{u}}f{d_{v}})\approx 0.5$. Empirically, it was observed
that the autocorrelations of the edges decreased very sharply.
## 3 Methods for calculating independence
The algorithm in Sec. 2 has two variants - one where the resulting graph may
not be simple, and another where the graph was always simple (A graph is
simple if it does not have self-loops or parallel edges). For the first
variant, earlier work [1] implies that this chain has a polynomial mixing
time. The second variant is not even known to have any bounds on the mixing
time. As discussed in Sec. 1, one would like to provide a length to run the
MCMC that, although not guaranteeing complete mixing, at least gives some
confidence that the sampled graph is fairly “random.” To that end, we will we
will approximate the behavior of a single edge by a Markov chain. We stress
that we do not give a proof, but only a mathematical argument justifying this.
Below we present two methods for generating independent graph samples.
### 3.1 Method A - “multiple short runs”
We refer to this method as Method A or the “multiple short runs” method. We
generate $M$ graph samples by running $M$ independent Markov chain for $N$
iterations before accepting the resulting graph. All the chains are run from
the same initial graph; however, the state in the random number generator in
each of the $M$ MCMC chains are distinct.
Consider a fixed pair of labeled vertices $\\{u,v\\}$. We will approximate the
occurrence of an edge between $(u,v)$ as a two state Markov chain. Note that
this is an approximation, since these transitions depend on the remaining
graph if $\mathbf{J}$ is to be preserved. Nonetheless, these dependencies
appear to be weak. The first coordinate of the matrix is state $0$ (no edge)
and the second coordinate is $1$, indicating the existence of an edge. The
transition matrix $\mathbf{T}$ for this chain is
$\mathbf{T}_{i,j}=\left(\begin{array}[]{cc}1-\alpha_{i,j}&\alpha_{i,j}\\\
\beta_{i,j}&1-\beta_{i,j}\end{array}\right),$ (1)
where $i=d_{u}$ and $j=d_{v}$ are the degrees of vertices $u$ and $v$.
$\alpha_{i,j}$ and $\beta_{i,j}$ are positive fractions and
$(\alpha_{i,j}+\beta_{i,j})\leq 1$. The eigenvalues of the transition matrix
are 1 and $1-(\alpha_{i,j}+\beta_{i,j})$. Below, we construct a model for
$\alpha_{i,j}$ and $\beta_{i,j}$.
Suppose the state is currently $0$. The state will become $1$ if the edge
$(u,v)$ is swapped in. Let the two edges chosen by the algorithm be $e$ and
$e^{\prime}$, in that order. The edge $(u,v)$ is swapped in if $e$ contains
$u$ and $e^{\prime}$ contains $v$ (or vice versa). Furthermore, the endpoint
$u$ must be chosen, and the other end of $e^{\prime}$ must have degree
$d_{u}$. The probability that $e$ contains $u$ and $u$ is chosen as an
endpoint is exactly $d_{u}/2m$. The probability we choose the edge
$e^{\prime}$ that is incident to $v$ depends on the number of neighbors of $v$
whose degree is $d_{u}$. Clearly, this depends on the graph structure (leading
in a non-Markov probability of this transition). We heuristically guess this
number based on the joint degree distribution. The number of edges from degree
$d_{u}$ to degree $d_{v}$ vertices is $\mathbf{J}(d_{u},d_{v})$. Of these, the
average number of edges incident to a fixed vertex of degree $d_{v}$ is
$\mathbf{J}(d_{u},d_{v})/f_{v}$. We shall approximate the number of edges
incident to $v$ with the other endpoint of degree $d_{u}$ by this quantity.
The total probability is
$\frac{d_{u}}{2m}\times\frac{\mathbf{J}(d_{u},d_{v})}{mf(d_{v})}=\frac{d_{u}\mathbf{J}(d_{u},d_{v})}{2m^{2}f(d_{v})}$
The edge $(u,v)$ is also swapped in when the reverse happens (so we choose $v$
as an endpoint, and an edge incident to $u$ with the other endpoint of degree
$d_{v}$). The total transition probability from $0$ to $1$ is approximated by
$\alpha_{i,j}=\frac{d_{u}\mathbf{J}(d_{u},d_{v})}{2m^{2}f(d_{v})}+\frac{d_{v}\mathbf{J}(d_{u},d_{v})}{2m^{2}f(d_{u})}=\frac{\mathbf{J}(d_{u},d_{v})}{2m^{2}}\left(\frac{d_{u}}{f(d_{v})}+\frac{d_{v}}{f(d_{u})}\right)$
(2)
We now address the transition from $1$ to $0$. Suppose $(u,v)$ is currently an
edge. If the first edge $e$ is chosen to be $(u,v)$, then $(u,v)$ will
definitely be swapped out. The probability of this is $1/m$. If the random
endpoint chosen has degree $d_{u}$ (and is not $u$), then we might choose
$e^{\prime}$ to be $(u,v)$. The total probability of this is
$\frac{(f(d_{u})-1)d_{u}}{2m}\times\frac{1}{m}=\frac{(f(d_{u})-1)d_{u}}{2m^{2}}$
The roles of $u$ and $v$ can also be reversed, so the total transition
probability from $1$ to $0$ is
$\frac{f(d_{u})d_{u}+f(d_{v})d_{v}-d_{u}-d_{v}}{2m^{2}}$
and so
$\beta_{i,j}=\frac{1}{m}+\frac{f(d_{u})d_{u}+f(d_{v})d_{v}-d_{u}-d_{v}}{2m^{2}}$
(3)
We proceed to determining the number of iterations $N$ to run the Markov
chain. We start the Markov chain ${\mathcal{M}}$ with an initial distribution
${\mathbf{v}}$ (which is either $(0,1)$ or $(1,0)$). ${\mathcal{M}}$, which is
represented by $\mathbf{T}_{i,j}$ (Eq. 1), is run for
$N=ln(1/\epsilon)/(\alpha+\beta)$ iterations, $\epsilon>0$. We have dropped
the subscripts $i$ and $j$, since it is implied that this model is being
derived for an edge $(u,v)$ with vertices of degrees $i$ and $j$. After $N$
steps, we realize a 2-state distribution ${\mathbf{p}}=(p_{0},p_{1})$, which
is different from the stationary distribution be ${\mathbf{u}}=(u_{0},u_{1})$.
Denote the unit 2-norm eigenvectors of $\mathbf{T}$, corresponding to the
eigenvalues 1 and $1-(\alpha+\beta)$, as ${\mathbf{e}}_{1}$ and
${\mathbf{e}}_{2}$. The initial state can be expressed as
${\mathbf{v}}=c_{1}{\mathbf{e}}_{1}+c_{2}{\mathbf{e}}_{2}$. After $N$
applications of the transition matrix we get
${\mathbf{p}}=\mathbf{T}^{N}{\mathbf{v}}=c_{1}\mathbf{T}^{N}{\mathbf{e}}_{1}+c_{2}\mathbf{T}^{N}{\mathbf{e}}_{2}=c_{1}{\mathbf{e}}_{1}+c_{2}\left(1-(\alpha+\beta)\right)^{N}{\mathbf{e}}_{2}.$
Since $(1-\\{\alpha+\beta\\})<1$, the second term decays with $N$ and
$c_{1}{\mathbf{e}}_{1}$ is the stationary distribution. We can bound the
decaying term as
$\|(1-(\alpha+\beta))^{N}c_{2}{\mathbf{e}}_{2}\|_{2}=(1-(\alpha+\beta))^{\ln(1/\epsilon)/(\alpha+\beta)}c_{2}\|{\mathbf{e}}_{2}\|_{2}\leq\exp(-\ln(1/\epsilon))=\epsilon$
Hence, $\|{\mathbf{p}}-{\mathbf{u}}\|_{2}\leq\epsilon$, and so each
$|p_{i}-u_{i}|$ is at most $\epsilon$. Further, from Eq. 2 and 3, we see that
$\alpha+\beta\geq 1/m$ (to leading order) and consequently
$N=\frac{\ln(1/\epsilon)}{\alpha+\beta}\leq
m\ln(1/\epsilon)=|E|\ln\left(\frac{1}{\epsilon}\right).$ (4)
### 3.2 Method B - “one long run”
We propose a second method, which we refer to as Method B or “one long run”,
for generating independent graphs. The procedure involves running a Markov
chain for a large number of steps $K$ and thinning it by a factor $k$ i.e.,
preserving every $k^{th}$ instance of the chain. Comparing with the
development in Sec. 3.1, we expect $k\sim N$.
Similar to Method A (Sec. 3.1), this method too begins with the binary time-
series of edge occurrence $\\{Z_{t}\\}$. As observed in [5], the
autocorrelation in $\\{Z_{t}\\}$ decays for all edges. Consequently it is
possible to successively thin the chain $\\{Z_{t}\\}$ (i.e., retain every
$k^{th}$ element to obtain $\\{Z^{k}_{t}\\}$, the $k-$thinned chain) and
compare the likelihoods that the chains were generated by (1) independent
sampling or (2) by a first-order Markov process. When sufficiently thinned,
the independent sampling model is expected to fit the data better. Using this
as the stopping criterion removes an ambiguity (user-specified tolerances). We
will employ a method based on comparison of log-likelihoods of model fit. We
derive these expressions below. While this technique has been applied in other
domains [7, 8], but this paper is the first application of this technique to
graphs.
Consider the chain $\\{Z^{k}_{t}\\}$. We count the number, $x_{ij}$, of the
$(i,j),i,j\in(0,1)$ transitions in it. $x_{ij}$ are used to populate $X$, a
$2\times 2$ contingency table. Dividing each entry by the length of thinned
chain $K/k-1$ provides us with the empirical probabilities $p_{ij}$ of
observing an $(i,j)$ transition in $\\{Z^{k}_{t}\\}$. Let $\widehat{p_{ij}}$
and $\widehat{x_{ij}}=(K/k-1)\widehat{p_{ij}}$ be the predictions of the
probabilities and expected values of the table entries provided by a model. In
such a case, the goodness-of-fit of the model is provided by a likelihood
ratio statistic (called the $G^{2}$-statistic; Chapter 4.2 in [9]) and a
Bayesian Information Criterion (BIC) score
$G^{2}=-2\sum_{i=0}^{i=1}\sum_{i=0}^{i=1}x_{ij}\log\left(\frac{\widehat{x_{ij}}}{x_{ij}}\right),\mbox{\hskip
28.45274pt}BIC=G^{2}+n\log\left(\frac{K}{k}-1,\right)$ (5)
where $n$ is the number of parameters in the model used to fit the table data.
Typically log-linear models are used for the purpose (Chapter 2.2.3 in [9]);
the log-linear models for table entries generated by independent sampling and
a first-order Markov process are
$\log(p_{ij}^{(I)})=u^{(I)}+u^{(I)}_{1,(i)}+u^{(I)}_{2,(j)}\mbox{\hskip
2.84526pt and \hskip
2.84526pt}\log(p_{ij}^{(M)})=u^{(M)}+u^{(M)}_{1,(i)}+u^{(M)}_{2,(j)}+u^{(M)}_{12,(ij)},$
(6)
where superscripts $I,M$ indicate an independent and Markov process
respectively. The maximum likelihood estimates (MLE) of the model parameters
($u^{(W)}_{b,(c)}$) are available in closed form (Chapter 3.1.1 in [9]). They
lead to the model predictions below
$\widehat{x_{ij}^{I}}=\frac{(x_{i+})(x_{+j})}{x_{++}}\mbox{\hskip 5.69054pt
and \hskip 5.69054pt}\widehat{x_{ij}^{M}}=x_{ij},$ (7)
where $x_{i+}$ and $x_{+j}$ are the sums of the table entries in row $i$ and
column $j$ respectively. $x_{++}$ is the sum of all entries (i.e., $K/k-1$,
the number of transitions observed in $\\{Z^{k}_{t}\\}$, or the total number
of data points). We compare the fits of the two models thus:
$\Delta
BIC=BIC^{(I)}-BIC^{(M)}=-2\sum_{i=0}^{i=1}\sum_{i=0}^{i=1}x_{ij}\log\left(\frac{\widehat{x_{ij}^{(I)}}}{x_{ij}}\right)-\log\left(\frac{K}{k}-1\right).$
(8)
Above, we have substituted $\widehat{x_{ij}^{(M)}}=x_{ij}$ and the fact that
the log-linear model for a Markov process has one more parameter than the
independent sampler model. Large BIC values indicate a bad fit. A negative
$\Delta BIC$ indicates that an independent model fits better than a Markov
model.
The procedure for identifying a suitable thinning factor $k$ then reduces to
progressively thinning $\\{Z^{k}_{t}\\}$ till $\Delta BIC$ in Eq. 8 becomes
negative. We search for $k$ in powers of 2\. The value of $k$ so obtained
varies between edges and conservatively, we take the largest $k,k_{*}$.
However, this may be _too_ conservative, i.e, $k_{*}\gg N$, if a few edges are
seen to display a slow autocorrelation decay. If we are interested in certain
global metrics for graph e.g., maximum eigenvalue etc, a few correlated edges
are unlikely to have any substantial effect. Thus, one may be able to thin
with a $k\sim N\ll k_{*}$. We will test this empirically in Sec. 4.
## 4 Tests with real graphs
In this section we first explore the impact of $\epsilon$ (as defined in Sec.
3.1) on the ensemble of graphs generated by a Markov chain, and choose a
$\epsilon$ for further use. Thereafter we compare the graphs generated by the
Methods A and B (Sec. 3.1 and Sec. 3.2) and gauge the impact of choosing a
thinning factor $k<k_{*}$. All tests are done with four real networks - the
neural network of _C. Elegans_ [10] (referred to as “C. Elegans”), the power
grid of the Western states of US [10] (called “Power”), co-authorship graph of
network science researchers [11] (referred to as “Netscience”) and a 75,000
vertex graph of the social network at Epinions.com [12] (“Epinions”). Their
details are in Table 1. The first three were obtained from [13] while the
fourth was downloaded from [14]. All the graphs were converted to undirected
graphs by symmetrizing the edges.
We start the Markov chain using the real networks listed in Table 1. When
comparing ensembles of graphs, we will use the (distributions of) global
clustering coefficient, number of triangles in the graphs, the graph diameter
and the maximum eigenvalue as metrics.
Table 1: Characteristics of the graphs used in this paper. $(|V|,|E|)$ are the numbers of vertices and edges in the graph, $N$ are the number of Markov chain steps used for generating graphs in Sec. 3.1 and $k$ is its equivalent obtained by the method in Sec. 3.2. $K/k_{*}$ are the number of graph samples, obtained by thinning a long run, that were used to generate distributions in the figures. Graph name | $(|V|,|E|)$ | $N/|E|$ | $k_{*}/|E|$ | $K/k_{*}$
---|---|---|---|---
C. Elegans | (297, 4296) | 10 | 13 | 3582
Netscience | (1461, 5484) | 10 | 49 | 737
Power | (4941, 13188) | 10 | 13 | 1214
Epinions | (75879, 405740) | 30 | 720 | various
In Fig. 2 we investigate the impact of $\epsilon$ in Method A (“many short
runs”). We generate 1000 samples by running the Markov chain for
$1|E|,5|E|,10|E|$ and $15|E|$ Markov chain iterations, corresponding to
$\epsilon=0.37,6.7\times 10^{-3},4.5\times 10^{-5}$ and $3.06\times 10^{-7}$.
In Fig. 2, we plot the distributions for the first three graphs (in Table 1)
and find that for all three, $\epsilon<5.0\times 10^{-3}$ lead to
distributions which are very close. We will proceed with $\epsilon=4.5\times
10^{-5}$ i.e., when we use Method A, we will mix the Markov chain $10|E|$
times before extracting a sample.
Figure 2: Plots of the distributions of the global clustering coefficient,
the number of triangles in the graphs, the graph diameter and the max
eigenvalue of the graph Laplacian for “Netscience” (left) and “Power” (right),
evaluated after $1|E|,5|E|,10|E|$ and $15|E|$ iterations of the Markov chain
(green, blue, black and red lines respectively). The corresponding values of
$\epsilon$ are in the legend. We see that the distributions converge at
$\epsilon<1.0^{-5}$.
In Table 1 we see that Method B (“one long run”) method often prescribes a
thinning factor that is larger than the one obtained using Method A (“multiple
short runs”). This large number is often due to the lack of autocorrelation
decay in a few edges. We investigate whether such a lack has a significant
impact on the graphical metrics that we have chosen. In Fig. 3 we plot
distributions of the same metrics for the three graphs. The thinning factors
are in Table 1. We see that the distributions are close, i.e., the existence
of a few edges whose time-series are still correlated do not impact the
metrics of choice. We have repeated these tests with other metrics and the
same result holds true.
Figure 3: Comparison of the distributions of the global clustering
coefficient, the number of triangles in the graphs, the graph diameter and the
max eigenvalue of the graph Laplacian for “C. Elegans” (left), “Netscience”
(middle) and “Power” (right), evaluated using Methods A (“many short runs”)
and B (“one long run”). We see that the distributions are very similar. The
kernel density estimation used to generate the distributions sometimes causes
nonsensical artifacts e.g., a small, but negative clustering coefficient. For
Method A, the Markov chain was run for $10|E|$ iterations. Thinning factors
for Method B are in Table 1.
We now address a large graph (Epinions). Since potentially $|V|^{2}$ distinct
edges might be realized during a Markov chain, it is infeasible to calculate a
thinning factor for all the edges. Consequently, we perform the thinning
analysis for only $0.1|E|$ (40,574) edges, chosen randomly from all the
distinct edges that are realized by the Markov chain. In Fig. 4 we plot the
distribution of $k$ obtained from the 40,574 sampled edges. We see that most
of the $k$ lie between $10|E|$ and $100|E|$; edges with thinning factors
outside that range are about two orders of magnitude less abundant. It is
quite conceivable that there are edges (which were not captured by the sample)
that would prescribe an even higher thinning factor. In order to check whether
these edges have a significant impact on the distribution of graph metrics, we
check their convergence as a function of the thinning factor.
Figure 4: Left: The normalized thinning factor $k/|E|$ for the Epinions graph,
as calculated for the 40,574 sampled edges. We see that the most thinning
factors are lie in ($10|E|,100|E|$). Right: Plot of the graph diameter and
distribution generated using Method A (with $N=30|E|$) and Method B (with $k$
equal to various multiples of $N$). We see that the distributions are very
similar.
We generate separate ensembles of graphs. The reference ensemble is generated
using Method A, with $N=30|E|$. As seen in Table 1, certain edges will still
be correlated ($N=720|E|$ would make them independent). We then use Method B
to generate graph ensembles with thinning factors $k/|E|<720$ which are also
multiples of $N$. In Fig. 4 (right), the diameter distribution obtained with
Method A is compared to that obtained with Method B. While the distributions
are very similar, they do display some small differences. This is surprising
since $N=30|E|$ indicates a minuscule $\epsilon$. In addition, distributions
obtained with $k=5N,9N,$ and $13N$ show some differences between themselves,
indicating that the edges that have not become independent have a small, but
measurable impact on the graph diameter. Further, the distributions using
smaller values of $k$ are marginally wider (have a larger variance),
indicating that they were constructed using samples which were not completely
independent. However, the differences are minute, and for practical purposes
the graph ensemble generated using Method A with $N=30|E|$ is identical to the
one generated using Method B, per our chosen metrics. Consequently, despite
its approximations, the results in Sec. 3.1 furnish a workable estimate of
$N$, if one uses $\epsilon<10^{-5}$. Further, $k_{*}$ is generally too
conservative if our aim is to obtain “converged” distributions of certain
graph metrics. This arises from a few edges that de-correlate slowly, but have
little effect on global graphical metrics due to their rarity.
## 5 Conclusions
We have developed a method that allows one to generate a set of independent
realizations of graphs with a prescribed joint degree distribution. The graphs
are generated using an MCMC approach, employing the algorithm described in [5]
as the “rewiring” mechanism. The graphs so generated are tightly correlated;
our two methods address the question of how one can decorrelate the chain.
The first method, variously called Method A or “multiple short chains”,
involves running the Markov chain for $N$ steps before extracting a graph
realization; the Markov chain is run repeatedly to generate samples. We
developed a model (and a closed-form expression) to estimate $N$ that allows
the Markov chain to converge to its stationary distribution before a graph
realization is extracted from it. This model assumes that edges are
independent. In reality, their behavior is correlated, which leads us to incur
small errors.
The second method, variously called Method B or “one long chain”, is a data
driven method. It uses the time-series of the occurence/non-occurence of edges
in an MCMC run. It does not assume a constant joint degree distribution. It
progressively thins the time-series (by retaining every $k^{th}$ element) and
fits a first-order Markov and an independent sampling model to the data. The
thinning process stops when the independent model has a higher likelihood
(strictly, a lower BIC score) than the Markov process. Since this method is
data-driven and does not require any user-defined tolerances, we use it to
validate Method A. The method is not new, but does not seem to have been used
in the generation of independent graphs.
Comparing the two methods, we find that for practical purposes, the ensembles
generated using Method A are statistically similar to those obtained with
Method B, as gauged by a set of graph metrics. Even at tight tolerance values,
a small number of edges in the graphs generated by Method A remain correlated,
and the metrics’ distributions are slightly wider. This problem is very small
(nearly unmeasureable) in small graphs, but becomes measureable, but still
small, for large graphs.
While this work enables the generation of independent graphs, including large
ones, it poses a number of questions for further investigation. For example,
being able to estimate or bound the difference in the distributions generated
by Methods A and B would be helpful. Further, an intelligent way of
identifying hard-to-decorrelate edges would reduce the computational burden of
checking for the stopping criterion using Method B; currently, we simply use a
random set of edges. Finally, it would be interesting if Method A could be
extended to the generation of independent graphs when some graph property,
other than the joint degree distribution, is held constant. This is currently
being studied.
## References
* [1] Kannan, R., Tetali, P., Vempala, S.: Simple Markov-chain algorithms for generating bipartite graphs and tournaments. Random Struct. Algorithms 14(4) (1999) 293–308
* [2] Jerrum, M., Sinclair, A.: Fast uniform generation of regular graphs. Theor. Comput. Sci. 73(1) (1990) 91–100
* [3] Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4) (2004) 671–697
* [4] Gkantsidis, C., Mihail, M., Zegura, E.W.: The Markov chain simulation method for generating connected power law random graphs. ALENEX (2003) 16–25
* [5] Stanton, I., Pinar, A.: Constructing and sampling graphs with a prescribed joint degree distribution using markov chains. ACM Journal of Experimental Algorithmics to appear.
* [6] Sokal, A.: Monte Carlo methods in statistical mechanics: Foundations and new algorithms (1996)
* [7] Raftery, A., Lewis, S.M.: Implementing MCMC. In Gilks, W.R., Richardson, S., Spiegelhalter, D.J., eds.: Markov Chain Monte Carlo in Practice. Chapman and Hall (1996) 115–130
* [8] Raftery, A.E., Lewis, S.M.: How many iterations in the Gibbs sampler? In Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M., eds.: Bayesian Statistics. Volume 4., Oxford University Press (1992) 765–766
* [9] Bishop, Y.M., Fienberg, S.E., Holland, P.W.: Discrete multivariate analysis: Theory and practice. Springer-Verlag, New York, NY (2007)
* [10] Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393 (1998) 440–442
* [11] Newman, M.E.J.: Finding community structure in networks using the eigenvectors of matrices, 036104. Phys. Rev. E 74 (2006)
* [12] Richardson, M., Agrawal, R., Domingos, P.: Trust management for the semantic Web. In Fensel, D., Sycara, K., Mylopoulos, J., eds.: The Semantic Web - ISWC 2003. Volume 2870 of Lecture Notes in Computer Science. Springer Berlin / Heidelberg (2003) 351–368 10.1007/978-3-540-39718-2_23.
* [13] Newman, M.E.J.: Prof. M. E. J. Newman’s collection of graphs at University of Michigan http://www-personal.umich.edu/m̃ejn/netdata/.
* [14] Stanford Network Analysis Platform Collection of Graphs: The Epinions social network from the Stanford Network Analysis Platform collection http://snap.stanford.edu/data/soc-Epinions1.html.
|
arxiv-papers
| 2012-02-15T23:38:06 |
2024-09-04T02:49:27.443117
|
{
"license": "Public Domain",
"authors": "Jaideep Ray, Ali Pinar and C. Seshadhri",
"submitter": "Jaideep Ray",
"url": "https://arxiv.org/abs/1202.3473"
}
|
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