Matrix Functional Equation Research Articles

On the real form of real solutions of the matrix functional equation $\Phi(x)\Phi(y)=\Phi(xy)$ for non-singular matrices $\Phi$

On the real form of real solutions of the matrix functional equation $\Phi(x)\Phi(y)=\Phi(xy)$ for non-singular matrices $\Phi$

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.

A circulant functional equation for the additive function and its stability

A general solution of a matrix functional equation involving circulant matrices of the additive function is determined, and its stability is established.

Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.

More, or less, trivial matrix functional equations

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1) case, such as \({\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }\); others—such as \({\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }\)—possess nontrivial solutions only in the matrix case (N > 1), namely their scalar (N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two (N × N)-matrices \({\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}\) satisfy the triplet of functional equations written above—and nontrivial examples of such matrices are exhibited— then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read \({\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }\).

Normal forms of “near similarity” transformations and linear matrix equations

A formal solution to a linear matrix differential equation with irregular singularity t 1−rY ′(t)=A(t)Y(t) , where r∈ Z + and the matrix-valued function A(t) is analytic at t=∞, was obtained via reduction of the coefficient A(t) to its Jordan form. The same approach was also utilized to find formal solutions to difference equations and to singularly perturbed differential equations. The linear change of variables Y=TX, where X is the new unknown matrix, generates the transformation A→T −1AT−t 1−rT −1T ′ . When r>0, this transformation can be considered as a “small perturbation” of the similarity transformation A→T −1AT . Various normal forms of these two transformations could be found in the literature. The emphasis of the present paper is to describe some classes of “near similarity” transformations that have the same normal forms as A→T −1AT . Obtained results are used to construct formal solutions to matrix functional equations and to discretized differential equations.

L-Functions for Jacobi Forms of Arbitrary Degree

A finite number ofL-functions are associated to every Jacobi cusp form of degreen. TheseL-functions are infinite series constructed with the Fourier coefficients of the form and a variables in ℂn. It is proved that eachL-function has an integral representation, admits a holomorphic continuation to the whole space ℂn, and the row vector formed with them satisfies a particular matrix functional equation.

Solutions of Discretized Affine Toda Field Equations forAn(1),Bn(1),Cn(1),Dn(1),An(2)andDn+1(2)

It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X_{n}^{(a)} type affine Toda field equation on discrete space time. We present, for A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}, its solutions in terms of determinants or Pfaffians.

Analytic calculation of conformal partition functions: Tricritical hard squares with fixed boundaries

An analytic method of calculating conformal partition functions of solvable lattice models is presented. Our technique involves the solution of transfer matrix functional equations in the scaling limit, and unifies the work of Albertini et al. on the classification of transfer matrix eigenvalues with that of Khimper and Pearce on finite-size corrections. As an illustration, conformal partition functions of the tricritical hard square model with fixed boundaries are derived. The resulting Virasoro characters arise in their fermionic representation and agree with the Coulomb gas calculations of Saleur and Bauer.

A Converse Theorem for Jacobi Forms

Letf(qτ, qz)=∑n, rc(n, r)qnτqrzbe a power series whose coefficients satisfy a particular periodicity condition depending on the integerrmodulo 2m. We first associate tof(qτ, qz) a 2m-vector-valued functionΛ(f, s) via a generalized Mellin transform. Then we show that the functionΛ(f, s) is entire, bounded on vertical strips and satisfies certain matrix functional equation if, and only if,f(qτ, qz) is the Fourier expansion of a Jacobi cusp form of indexminvariant under the group SL(2, Z)⋉Z2. This is the direct analogue of Hecke's converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, Z)⋉Z2.

A storage model for data communication systems

We consider a storage model where the input and demand are modulated by an underlying Markov chain. Such models arise in data communication systems. The input is a Markov-compound Poisson process and the demand is a Markov linear process. The demand is satisfied if physically possible. We study the properties of the demand and its inverse, which may be viewed as transformed time clocks. We show that the unsatisfied demand is related to the infimum of the net input and that, under suitable conditions, it is an additive functional of the input process. The study of the storage level is based on a detailed analysis of the busy period, using techniques based on infinitesimal generators. The transform of the busy period is the unique solution of a certain matrix-functional equation. Steady state results are also obtained; these are not obvious generalizations of the results for simple storage models. In particular, a generalization of the Pollaczek-Khinchin formula brings new insight.

A Storage Model for Data Communication Systems

We consider a storage model where the input and demand are modulated by an underlying Markov chain. Such models arise in data communication systems. The input is a Markov-compound Poisson process and the demand is a Markov linear process. The demand is satisfied if physically possible. We study the properties of the demand and its inverse, which may be viewed as transformed time clocks. We show that the unsatisfied demand is related to the infimum of the net input and that, under suitable conditions, it is an additive functional of the input process. The study of the storage level is based on a detailed analysis of the busy period, using techniques based on infinitesimal generators. The Laplace transform of the busy period is the unique solution of a certain matrix-functional equation. Steady state results are also obtained; these are not obvious generalizations of the results for simple storage models. In particular, a generalization of the Pollaczek-Khinchin formula brings new insight.

Method for separation of variables for bilinear matrix functional equation and its applications

Necessary and sufficient conditions for the solvability of a bilinear matrix functional equation are presented. The conditions are applied in the construction of the solutions of systems of partial differential equations.

Special Analytical Solutions for Use in Debond Stress Analysis

Solutions are presented for the three-dimensional problems of the debonding of a rigid rod from an incompressible elastic matrix. Particular attention is given to debond initiation at key stress singularities in the perfectly bonded configuration and to the case where the rod base is almost or completely debonded. In each of these cases it is possible by a suitable scaling of the co-ordinates to reduce the problem to one in which a three-dimensional stress field is perturbed by a debond geometry which can be viewed as two dimensional. This two-dimensional problem is then treated by solving a matrix functional equation. The results are compared with finite element results of an experimental configuration of a rod pull-out test.