Abstract We prove a very general fixed point theorem in the space of functions taking values random normed (RN-space). Next, we show several its consequences and, among others, present applications it proving Ulam stability results for inhomogeneous linear functional equation with variables class f mapping vector X into an RN-space. Particular cases are instance equations Cauchy, Jensen, Jordan...

In this paper we propose a new geometric interpretation for General Relativity (GR). It has always been presumed that the gravitational field is described in GR by a LeviCivita connection. We suggest that this may not necessarily be the case. We show that in the presence of an arbitrary affine connection, the gravitational field is described as nonmetricity of the affine connection. An affine c...

We find and study four kinds of 6-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of types B (1) 6 , D (1) 6 and D (2) 7 . We also give an explicit description of a confluence to the Noumi-Yamada system of type A (1) 5 . 0. Introduction In 1912, considering the significant problem of searching for higher order analogues of the Painlevé equations, Garnier discover...

We give new formulations of the solutions of the field equations of the affine Toda and conformal affine Toda theories on a cylinder and two-dimensional Minkowski space-time. These solutions are parameterised in terms of initial data and the resulting covariant phase spaces are diffeomorphic to the Hamiltonian ones. We derive the fundamental Poisson brackets of the parameters of the solutions a...

We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the...

This paper deals with the optimal control techniques for a class of affine switched systems. The affine structure of the state equations makes it possible to establish a continuity property (with respect to the control parameter) for the above systems. Under some additional assumptions, the optimal control problem (OCP) associated with the affine dynamic models corresponds to a standard convex ...

The construction of generalized Backlund transformation for the $A_n$ Affine Toda hierarchy is proposed in terms gauge acting on zero curvature representation. Such based upon graded structure underlying affine algebra which induces a classification transformations. Moreover, explicit examples $su(3)$ and $su(4)$ lead to uncover interesting composition properties various types universality char...

An overview is given on recent developments in the affine Weyl group approach to Painlevé equations and discrete Painlevé equations, based on the joint work with Y. Yamada and K. Kajiwara. 2000 Mathematics Subject Classification: 34M55, 39A12, 37K35.