Quarter Plane Research Articles

Discrete analytic functions, structured matrices and a new family of moment problems

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An important role is played by a Krein space realization result of Dijksma, Langer and de Snoo for functions analytic in a neighborhood of the origin. A key observation is that, using a simple Moebius transform, one can reduce the study of discrete analytic functions in the upper right quadrant to problems of function theory in the open unit disk. As an example, we associate to each finite positive measure on [0,2π] a discrete analytic function on the right-upper quarter plane with values on the lattice defining a positive definite function. Emphasis is put on the rational case, both when an underlying Carathéodory function is rational and when, in the positive case, the spectral function is rational. The rational case and the general case are linked via the existence of a unitary dilation, possibly in a Krein space.

Method of reflections for the Klein–Gordon equation

Using the method of reflections, the solutions of the first and second mixed problem for the homogenous Klein–Gordon equation in a quarter plane and of the first mixed problem for the homogenous Klein–Gordon equation in a halfstrip are written out in an explicit analytical form. The Cauchy conditions of these problems are inhomogeneous, but the Dirichlet boundary condition (or the Neumann boundary condition) is homogeneous. Conditions are formulated, under which the solutions to these problems are classical.

Constructing discrete harmonic functions in wedges

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

Lower regularity solutions of the non-homogeneous boundary-value problem for a higher order Boussinesq equation in a quarter plane

We continue to study the initial–boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: utt−uxx+βuxxxx−uxxxxxx+(u2)xx=0,x,t∈R+,u(x,0)=φ(x),ut(x,0)=ψ′′(x),u(0,t)=h1(t),uxx(0,t)=h2(t),uxxxx(0,t)=h3(t),where β=±1. We show that the problem is locally analytically well-posed in the space Hs(R+) for any s>−34 with the initial-value data (φ,ψ)∈Hs(R+)×Hs−1(R+)and the boundary-value data (h1,h2,h3)∈Hs+13(R+)×Hs−13(R+)×Hs−33(R+).

Length derivative of the generating function of walks confined in the quarter plane

Length derivative of the generating function of walks confined in the quarter plane

Classical solution of the mixed problem for a nonlinear equation

The first mixed problem for a nonlinear equation is considered in the quarter plane. The Cauchy conditions are set at the bottom of the boundary. The Dirichlet condition is set on the left part of the boundary. The solution is constructed using the method of characteristics in an implicit analytical form as a solution of the integral equation. The solvability of these integral equations, the smoothness of the solutions, and their dependence on the initial data are investigated. The uniqueness is proved and the conditions are established, under which there exists a piecewise smooth and classical solution of the first mixed problem.

Hopf dreams and diagonal harmonics

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday–Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on ν $\nu$ -trees related to ν $\nu$ -Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics.

Walks Obeying Two-Step Rules on the Square Lattice: Full, Half and Quarter Planes

We consider walks on the edges of the square lattice $\mathbb Z^2$ which obey two-step rules, which allow (or forbid) steps in a given direction to be followed by steps in another direction. We classify these rules according to a number of criteria, and show how these properties affect their generating functions, asymptotic enumerations and limiting shapes, on the full lattice as well as the upper half plane.
 For walks in the quarter plane, we only make a few tentative first steps. We propose candidates for the group of a model, analogous to the group of a regular short-step quarter plane model, and investigate which models have finite versus infinite groups. We demonstrate that the orbit sum method used to solve a number of the original models can be made to work for some models here, producing a D-finite solution. We also generate short series for all models and guess differential or algebraic equations where possible. In doing so, we find that there are possibilities here which do not occur for the regular short-step models, including cases with algebraic or D-finite generating functions but infinite groups, as well as models with non-D-finite generating functions but finite groups.

A Unified Singular Solution of Laplace’s Equation with Neumann and Dirichlet Boundary Conditions

Laplace’s equation is one of the important equations in studying applied mathematics and engineering problems including the study of temperature distribution of steady-state heat conduction or the concentration distribution of steady-state diffusion problems. In this study, the analytical method has been applied to solve the Laplace's equation in a two-dimensional domain. For the specified Neumann or Dirichlet boundary conditions, the analytical solution of temperature distribution in the quarter-plane can be found by several methods including the Fourier transform method, similarity method, and the method of Green’s function with images. For different boundary conditions, the solution of temperature distribution of the Laplace’s equation will be in a totally different form. Nevertheless, the merit of this research is that the solutions of steady-state temperature distribution in the quarter plane with Neumann and Dirichlet boundary conditions are unified under the singular similarity solution with source type singularity. With the typical benchmarked examples for finding the temperature distribution by the numerical integral method, it is shown that Gibbs phenomenon behaves at a jump discontinuity, where serious oscillation result was found especially near the singular points of the boundary. In addition, the temperature distribution in the domain can be easily calculated without oscillation phenomenon near the singular points from the similarity solutions.

Reflected Brownian motion in the quarter plane: An equivalence based on time reversal

We consider a semimartingale reflected Brownian motion (SRBM) Z whose state space is the non-negative quarter plane; the apparently more general case of SRBM in a convex wedge can be transformed to the quarter plane by a simple change of variable. The data of the stochastic process Z are a drift vector μ, a covariance matrix Σ, and a 2 × 2 reflection matrix R whose columns are the directions of reflection on the two axes. We consider only the case where R has non-positive off-diagonal elements, that is, the direction of reflection is either normal or toward the origin from each axis.Under that restriction, we define a dual RBM Zˆ that is constructed using the same data (μ,Σ,R) but with certain sign reversals. Using a time reversal argument, we show the following: to find the distribution of the random two-vector Z(t) at an arbitrary time t, assuming that Z(0)=0, it suffices to find the probability that the dual RBM Zˆ, starting from an arbitrary point z in the quarter plane, reaches the origin before time t. Letting t→∞ and assuming that μ and R jointly satisfy the known condition for positive recurrence of Z, we then have the following: to determine the stationary distribution of Z, it suffices to determine the probability that the dual RBM Zˆ, starting from an arbitrary point z, ever reaches the origin.

Promotion of Kreweras words

Kreweras words are words consisting of nmathrm {A}’s, nmathrm {B}’s, and nmathrm {C}’s in which every prefix has at least as many mathrm {A}’s as mathrm {B}’s and at least as many mathrm {A}’s as mathrm {C}’s. Equivalently, a Kreweras word is a linear extension of the poset mathsf{V}times [n]. Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3n applications of promotion on a Kreweras word merely swaps the mathrm {B}’s and mathrm {C}’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s mathfrak {sl}_3-webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.

Boundary value problems for one pseudohyperbolic equation in a quarter plane

Boundary value problems for one pseudohyperbolic equation in a quarter plane

Mating of Discrete Trees and Walks in the Quarter-Plane

We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.

On differentially algebraic generating series for walks in the quarter plane

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell–Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.

On a Method for Solving Boundary Value Problems of the Dynamical Theory of Elasticity in a Quarter Plane

On a Method for Solving Boundary Value Problems of the Dynamical Theory of Elasticity in a Quarter Plane

On Green\u2019s function of Cauchy\u2013Dirichlet problem for hyperbolic equation in a quarter plane

The definition of a Green’s function of a Cauchy–Dirichlet problem for the hyperbolic equation in a quarter plane is given. Its existence and uniqueness have been proven. Representation of the Green’s function is given. It is shown that the Green’s function can be represented by the Riemann–Green function.

An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane

An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane

On the Nature of Four Models of Symmetric Walks Avoiding a Quadrant

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.

On partially homogeneous nearest-neighbour random walks in the quarter plane and their application in the analysis of two-dimensional queues with limited state-dependency

This work deals with the stationary analysis of two-dimensional partially homogeneous nearest-neighbour random walks. Such random walks are characterized by the fact that the one-step transition probabilities are functions of the state-space. We show that stationary behaviour is investigated by solving a finite system of linear equations, two matrix functional equations, and a functional equation with the aid of the theory of Riemann (–Hilbert) boundary value problems. This work is strongly motivated by emerging applications in flow level performance of wireless networks that give rise in queueing models with scalable service capacity, as well as in queue-based random access protocols, where the network’s parameters are functions of the queue lengths. A simple numerical illustration, along with some details on the numerical implementation are also presented.

Переходные вероятности марковского процесса на отрезке, лежащем в четверти плоскости

The paper considers a quadratic birth-death Markov process. The points on a line segment located within a quarter-plane represent the states of the random process. We designate the set of vectors that have integer non-negative coordinates as our quarter plane. The process is defined by infinitesimal characteristics, or transition probability densities. These characteristics are determined by a quadratic function of the coordinates at the segment points with integer coordinates. The boundary points of the segment are absorbing; at these points, the random process stops. We investigated a critical case when process jumps are equally probable at the moment of exiting a point. We derived expressions describing transition probabilities of the Markov process as a spectral series. We used a two-dimensional exponential generating function of transition probabilities and a two-dimensional generating function of transition probabilities. The first and second systems of ordinary differential Kolmogorov equations for Markov process transition probabilities are reduced to second-order mixed type partial differential equations for a double generating function. We solve the resulting system of linear equations using separation of variables. The spectrum obtained is discrete. The eigen-functions are expressed in terms of hypergeometric functions. The particular solution constructed is a Fourier series, whose coefficients are derived by means of expo-nential expansion. We employed sums of functional series known in the theory of special functions to construct the exponential expansion required