Quarter Plane Research Articles

Rényi entropy of the totally asymmetric exclusion process

The Rényi entropy is a generalisation of the Shannon entropy that is sensitive to the fine details of a probability distribution. We present results for the Rényi entropy of the totally asymmetric exclusion process (TASEP). We calculate explicitly an entropy whereby the squares of configuration probabilities are summed, using the matrix product formalism to map the problem to one involving a six direction lattice walk in the upper quarter plane. We derive the generating function across the whole phase diagram, using an obstinate kernel method. This gives the leading behaviour of the Rényi entropy and corrections in all phases of the TASEP. The leading behaviour is given by the result for a Bernoulli measure and we conjecture that this holds for all Rényi entropies. Within the maximal current phase the correction to the leading behaviour is logarithmic in the system size. Finally, we remark upon a special property of equilibrium systems whereby discontinuities in the Rényi entropy arise away from phase transitions, which we refer to as secondary transitions. We find no such secondary transition for this nonequilibrium system, supporting the notion that these are specific to equilibrium cases.

On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

SummaryMatrix equations of the kind A1X2+A0X+A−1=X, where both the matrix coefficients and the unknown are semi‐infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are quasi‐Toeplitz matrices, are encountered in certain quasi‐birth–death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi‐infinite quasi‐Toeplitz matrix arithmetic. In particular, we show that the algorithm of cyclic reduction can be effectively applied and can approximate the infinite‐dimensional solutions with quadratic convergence at a cost that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi‐birth–death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.

A combinatorial understanding of lattice path asymptotics

We provide a combinatorial derivation of the exponential growth constant for counting sequences of lattice path models restricted to the quarter plane. The values arise as bounds from analysis of related half planes models. We give explicit formulas, and the bounds are provably tight. The strategy is easily generalizable to cones in higher dimensions, and has implications for random generation.

Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane

The focus of the current paper is the higher order nonlinear dispersive equation which models unidirectional propagation of small amplitude long waves in dispersive media. The specific interest is in the initial-boundary value problem where spatial variable lies in \begin{document}$\mathbb R^+,$\end{document} namely, quarter plane problem. With proper requirement on initial and boundary condition, we show local and global well posedness.

Histopathology Image Analysis and Classification Using ARMA Models: Application to Brain Cancer Detection

Objective: To design ARMA model for the analysis of histopathology images. Keywords: Autoregressive model and moving average (ARMA), least square (LS), quarter plane (QP), markov random field model (MRF), radial basis function (RBF), support vector machine (SVM).

Weighted lattice walks and universality classes

In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

Brownian motion in [Formula: see text] with covariance matrix Σ and drift μ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in [Formula: see text] is found and its main term is identified depending on parameters (Σ, μ, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in [Formula: see text] with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on [Formula: see text] with reflections on the axes.

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s>−2. Copyright © 2017 John Wiley & Sons, Ltd.

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0, t) in the sense that u(0, t) tends to a periodic function g̃0(t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data ux (0, t) and uxx (0, t) are asymptotically t-periodic with period τ which tend to the functions g̃1(t) and g̃2(t) as t → ∞, respectively, then the periodic functions g̃0(t) and g̃0(t) can be uniquely determined in terms of the function g̃1(t). Furthermore, we characterize the Fourier coefficients of g̃1(t) and g̃2(t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.

Asymptotic expansion of stationary distribution for reflected Brownian motion in the quarter plane via analytic approach

Brownian motion in R 2 + with covariance matrix $\Sigma$ and drift $\mu$ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters ($\Sigma$, $\mu$, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on the axes.

Tutte’s invariant approach for Brownian motion reflected in the quadrant

We consider a Brownian motion with negative drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte’s invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or −1. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N2, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna (2010) identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers (2009). We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss’ hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19×4 combinatorially meaningful specializations only four are algebraic functions.

Analysis of an Offloading Scheme for Data Centers in the Framework of Fog Computing

In the context of fog computing, we consider a simple case where data centers are installed at the edge of the network and assume that if a request arrives at an overloaded data center, then it is forwarded to a neighboring data center with some probability. Data centers are assumed to have a large number of servers, and traffic at some of them is assumed to cause saturation. In this case, the other data centers may help to cope with this saturation regime by accepting some of the rejected requests. Our aim is to qualitatively estimate the gain achieved via cooperation between neighboring data centers. After proving some convergence results related to the scaling limits of loss systems for the process describing the number of free servers at both data centers, we show that the performance of the system can be expressed in terms of the invariant distribution of a random walk in the quarter plane. By using and developing existing results in the technical literature, explicit formulas for the blocking rates of such a system are derived.

PAPR Reduction via Constellation Extension in OFDM Systems Using Generalized Benders Decomposition and Branch-and-Bound Techniques

In this paper, a novel constellation extension (CE)-based approach is presented to address the high peak-to-average power ratio (PAPR) problem at the transmitter side, which is an important drawback of orthogonal frequency-division multiplexing (OFDM) systems. This new proposal is formulated as a mixed-integer nonlinear programming optimization problem, which employs generalized Benders decomposition (GBD) and branch-and-bound (BB) methods to determine the most adequate extension factor and the optimum set of input symbols to be extended within a proper quarter plane of the constellation. The optimum technique based on GBD, which is denoted as GBD for constellation extension (GBDCE), provides a bound with relevant improvement in terms of PAPR reduction compared with other CE techniques, although it may exhibit slow convergence. To avoid excessive processing time in practical systems, the suboptimum BB for constellation extension (BBCE) scheme is proposed. Simulation results show that BBCE achieves a significant PAPR reduction, providing a good tradeoff between complexity and performance. We also show that the BBCE scheme performs satisfactorily in terms of power spectral density and bit error rate in the presence of a nonlinear high-power amplifier.

Quarter Plane ARMA Model for Analysis and Classification of Histopathology Images: Application to Cancer Detection

Objective: Cancer diagnostic using clinical pathology have been proved as a standard method in which histologist/ pathologist examines biopsy sample for cell morphology and tissue distribution. Pathologist detects random growth and random placements in tissue samples. These diagnostics are very subjective and based on experience/knowledge base of pathologists. This work presents the use of 2D Autoregressive And Moving Average (ARMA) model in computer assisted automatic cancer detection. Analysis: ARMA model parameters have been considered for representing entire histopathology image. These features have further used for analysis and classification. Parameter estimation has been carried out by Yule walker Least Square (LS) method. Histology images have been classified into healthy and malignant images according to ARMA parameters. K- Fole cross validation has been performed with Linear Kernel support vector machine classifier for classification. Findings: As an outcomes of this experimentation, it is proven that ARMA model parameters works as an excellent discriminating features. These ARMA features are capable of extracting hidden information of the underlying cancer decease. This study also presents the role of neighborhood pixel in image analysis and classification. Improvement: This work have described innovative way of using ARMA features in histopathology imagery and can be implemented in computer assisted diagnosis.

A human proof of Gessel’s lattice path conjecture

Gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, again using computer algebra tools, that the complete generating function of Gessel walks is algebraic. In this article we propose the first "human proofs" of these results. They are derived from a new expression for the generating function of Gessel walks in terms of Weierstrass zeta functions.

Global well-posedness and singularity propagation for the BBM-BBM system on a quarter plane

Nonlinear, dispersive wave equations arise as models of various physical phenomena. A major preoccupation on the mathematical side of the study of such equations has been to settle the fundamental issues of local and global well-posedness in Hadamard's classical sense. The development so far has been mostly for the initial-value problem for single equations. However, systems of such equations have also received consideration, and there is now theory for pure initial-value problems where data are given on the entire space or on the torus. Here, consideration is given to non-homogeneous initial-boundary-value problems for a class of BBM-type systems having the form \begin{equation*} \begin{aligned} u_t + u_x -u_{xxt} + P(u,v)_x \, = \, 0, \\ v_t + v_x - v_{xxt} + Q(u,v)_x \, = \, 0, \end{aligned} \end{equation*} where $P$ and $Q$ are homogeneous, quadratic polynomials, $u$ and $v$ are real-valued functions of a spatial variable $x$ and a temporal variable $t$, and subscripts connote partial differentiation. Local in time well-posedness is established in the quarter plane $\{(x,t): x \geq 0, \, t \geq 0 \}$. Under certain restrictions on the coefficients of the nonlinearities $P$ and $Q$, global well posedness is also shown to obtain.

Phase Characteristics for the Stability of 2-D Quarter-Plane Recursive Digital All-Pass Filters

In this brief, the monotone phase-response property of a 2-D causal digital all-pass filter (DAF) with real coefficients or complex coefficients in the quarter plane (QP) support region is presented. Regarding the circumstance of real coefficients, we also prove that the previously proposed stability criterion on the viewpoint of unwrapped phase by Pei and Shyu is necessary and sufficient for 2-D separable DAFs but is only sufficient for QP DAFs. The resultant property possesses the advantage of increasing the freedom of phase design over the previously proposed one. A remarkable application of the presented property is choosing an appropriate specification for the desired phase response of a 2-D DAF design. A design example is given to show this application, and the stability is also inspected by the 2-D spectral factorization algorithm and the root maps for confirmation.

Image modeling based on a 2-D stochastic subspace system identification algorithm

Fitting a causal dynamic model to an image is a fundamental problem in image processing, pattern recognition, and computer vision. In image restoration, for instance, the goal is to recover an estimate of the true image, preferably in the form of a parametric model, given an image that has been degraded by a combination of blur and additive white Gaussian noise. In texture analysis, on the other hand, a model of a particular texture image can serve as a tool for simulating texture patterns. Finally, in image enhancement one computes a model of the true image and the residuals between the image and the modeled image can be interpreted as the result of applying a de-noising filter. There are numerous other applications within the field of image processing that require a causal dynamic model. Such is the case in scene analysis, machined parts inspection, and biometric analysis, to name only a few. There are many types of causal dynamic models that have been proposed in the literature, among which the autoregressive moving average and state-space models (i.e., Kalman filter) are the most commonly used. In this paper we introduce a 2-D stochastic state-space system identification algorithm for fitting a quarter plane causal dynamic Roesser model to an image. The algorithm constructs a causal, recursive, and separable-in-denominator 2-D Kalman filter model. The algorithm is tested with three real images and the quality of the estimated images are assessed.

A corrective solution for finding the effects of edge-rounding on complete contact between elastically similar bodies. Part I: Contact law and normal contact considerations

The solution is found for the contact pressure and contact law when a rounded quarter plane is pressed onto an elastically similar half-plane, for the case when the interfacial coefficient of friction is sufficient to inhibit all slip away from the contact edge. The far field is specified by stress intensity factors as if the contacting pair were a monolithic wedge of internal angle 270 degrees. The solution found is used to investigate corner rounding for a simple contact between a square block and a half-plane, as an illustration of its use.