A series of systems of nonlinear equations with affine Weyl group symmetry of type A (1) l is studied. This series gives a generalization of Painlevé equations P IV and PV to higher orders. Introduction In this paper we propose a series of systems of nonlinear differential equations which have symmetry under the affine Weyl groups of type A (1) l (l ≥ 2). These systems are thought of as higher ...

Suppose that attached to each site z ∈ Z is a coin with bias θ(z), and only finitely many of these coins have non-zero bias. Allow a simple random walker to generate observations by tossing, at each move, the coin attached to its current position. Then we can determine the biases {θ(z)}z∈Z, using only the outcomes of these coin tosses and no information about the path of the random walker, up t...

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of stable manifold zero solution. In particular, for $n$ dimensional case, solution globally asymptotically if and only this $n.$ Fixed $n,$ let $X$ be random variable that assigns to each system index, $p_k$ with $k=0,1,\ldots,n,$ denote probabilities $P(X=k)$. paper we obtain eith...

In recognition of the fundamental relation between regular arrays and systems of affine recurrence equations, the ALPHA language was developed as the basis of a computer aided design methodology for regular array architectures. ALPHA is used to initially specify algorithms at a very high algorithmic level. Regular array architecures can then be derived from the algorithmic specification using a...

An important question about a ne skeletons is the existence of di erential equations that are related to the A ne Distance and the Area Distance (hence to a ne skeletons) in the same way the Eikonal equation is related to the Euclidean Distance (and the medial axis). We show a surprisingly simple nonlinear second order PDE of Monge-Ampère type that relates to the a ne skeletons (and extends the...

Let σ be a simple involution of an algebraic semisimple group G and let H be the subgroup of G of points fixed by σ. If the restricted root system is of type A,C or BC and G is simply connected or if the restricted root system is of type B and G is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring k[G/H ] using the standard monomial theory and the Pl...

Recently, Cartwright and Woess [5] provided a detailed analysis of isotropic random walks on the vertices of thick affine buildings of type Ãn. Their results generalise results of Sawyer [18] where homogeneous trees are studied (these are Ã1 buildings), and Lindlbauer and Voit [9], where Ã2 buildings are studied. In this paper we apply techniques of spherical harmonic analysis to prove a local ...

Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to difference equations by evolution of parameters their linear marginals. On other hand, any equation defines an autoregressive Markov chain (a affine recursion) which can be positive recurrent, null recurrent and transient which, as forward iterations iter...

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. The purpose of this paper is to propose an iterative method for the symmetric affine SOCCP that is based on the idea ofmatrix splitting.Matrix-splittingmethods have originally been developed for the solution of the system of linear eq...

We describe a class of affine toric varieties V that are set-theoretically minimally defined by codimV + 1 binomial equations over fields of any characteristic.