Nonlinear random ergodic theorems for affine operators

Let (Ω,ß,μ) be a finite measure space and let (S,F,ν) be another probability measure space on which a measure preserving transformation φ is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly F-measurable family of affine operators, where B is a reflexive Banach space, is a strongly F-measurable family of linear contractions on L₁(Ω,B) as well as on L_∞(Ω,B) and ξ is a function in (I−T)Lp(S×Ω,B) (1?p<∞) with the operator T defined by Tf(s,ω)=[Tsfφs](ω) which denotes the F⊗ß-measurable version of Tsfφs(ω). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.

     

On the normalized eigenvalue problems for nonlinear elliptic operators II

This paper continues our previous research on the following form of normalized eigenvalue problem

     

Generalized bi-quasi-variational inequalities for quasi-pseudo-monotone type II operators on compact sets

In this paper, the authors prove some existence results of solutions for a new class of generalized bi-quasi-variational inequalities (GBQVI) for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining these results on GBQVI for quasi-pseudo-monotone type II and strongly quasi-pseudo-monotone type II operators, we shall use Chowdhury and Tan’s generalized version [3] of Ky Fan’s minimax inequality [7] as the main tool.     

Abel wave operators. II. Wave operators for functions of operators

It is shown that in a scattering theory with averaging the invariance principle of M. S. Birman and T. Kato (in its strong form) is always valid.     

Generalized affine chain geometries

The affine chain geometry over a group with a partial fibration into subgroups and a certain involution is introduced. This concept generalizes the affine trace of the chain geometry over an associative algebra. We study the geometric properties of these geometries and give examples.Dedicated to Professor Helmut Mäurer on the occasion of his 60th birthday     

Positive linear extension operators for spaces of affine functions

The following result is proved: letE be anF-space (that is, the space of all continuous affine functions defined on a compact universal cap van shing at zero) and letMχE be anM-ideal. Then, ifE/M is a π₁ with positive defining projections, then there is a positive linear operator ϱ:E/M→E of norm one such that ϱ lifts the canonical mapE→E/M. In the proof, which heavily depends on work of Ando, we study ensor products of certain convex cones with compact bases, and we calculate the norm of a positive linear operator defined on a finite dimensional space with range in aF-space. Various corollaries are deduced for split faces of compact convex sets and for morphisms ofC ^*-algebras.     

Generalized hermitian operators

In this paper, the concept of generalized hermitian operators defined on a complex Hilbert space is introduced. It is shown that the spectrums and the Fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis. Some other results on generalized hermitian operators are obtained.     

Generalized fractional operators for nonstandard Lagrangians

In this note, we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems, and the corresponding Euler–Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized sum of operators

For a semibounded below self-adjoint operatorA in a Hilbert spaceH and a singular operatorV acting in theA-scale of Hilbert spaces, the notion of generalized sumA?V is introduced. Conditions are found forA?V to be self-adjoint in ?. In particular, it is shown that if a symmetric operatorV is semibounded or has a spectral gap, then there exists an α such that the generalized sumA?αV is a self-adjoint operator inH. For a symmetric restrictionA = A‖D, D C D(A), with deficiency indices (1, 1), it is proved that each self-adjoint extension à of A admits representation as a generalized sum Ã=A?V.     

Generalized G-convergence for quasilinear elliptic differential operators

Generalized G-convergence for a quasilinear elliptic differential equation is defined and studied. The equation describes heat conduction in the cores of large electric transformers. The coefficients of the equation depend on temperature and the corresponding differential operator is neither potential nor monotone. A theory which generalizes the classical G-convergence is proposed. The theory is applied to the homogenization of the quasilinear elliptic differential equation with periodic coefficients.     

Time averaging for nonlinear parabolic operators

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 11, pp. 1483–1487, November, 1989.