Multi-operator colligations and multivariate characteristic functions

In the spectral theory of non-self-adjoint operators there is a well-known operation of product of operator colligations. Many similar operations appear in the theory of infinite-dimensional groups as multiplications of double cosets. We construct characteristic functions for such double cosets and get semigroups of matrix-valued functions in matrix balls.Comment: 15p

Branching processes with infinite collection of particle types and stochastic fragmentation theory

The stochastic model for the description of the so-called fragmentation process in frameworks of Kolmogorov approach is proposed. It is proved that the branching condition of this process represents the basic equation of the Kolmogorov theor

The structure of one-relator relative presentations and their centres

Suppose that G is a nontrivial torsion-free group and w is a word in the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from w by erasing all letters belonging to G is not a proper power in the free group F(x_1,...,x_n). We show how to reduce the study of the relative presentation \^G= to the case n=1. It turns out that an "n-variable" group \^G can be constructed from similar "one-variable" groups using an explicit construction similar to wreath product. As an illustration, we prove that, for n>1, the centre of \^G is always trivial. For n=1, the centre of \^G is also almost always trivial; there are several exceptions, and all of them are known.Comment: 15 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm . V4: the intoduction is rewritten; Section 1 is extended; a short introduction to Secton 5 is added; some misprints are corrected and some cosmetic improvements are mad

Zero-Forcing Precoding for Frequency Selective MIMO Channels with H∞H^\infty Criterion and Causality Constraint

We consider zero-forcing equalization of frequency selective MIMO channels by causal and linear time-invariant precoders in the presence of intersymbol interference. Our motivation is twofold. First, we are concerned with the optimal performance of causal precoders from a worst case point of view. Therefore we construct an optimal causal precoder, whereas contrary to other works our construction is not limited to finite or rational impulse responses. Moreover we derive a novel numerical approach to computation of the optimal perfomance index achievable by causal precoders for given channels. This quantity is important in the numerical determination of optimal precoders.Comment: Minor Revisions, mainly in introduction and problem statement. Submitted to Signal Processin

On cyclic relatively nonexpansive mappings in generalized semimetric spaces

[EN] In this article, we prove a fixed point theorem for cyclic relatively nonexpansive mappings in the setting of generalized semimetric spaces by using a geometric notion of seminormal structure and then we conclude some results in uniformly convex Banach spaces. We also discuss on the stability of seminormal structure in generalized semimetric spaces.This research was in part supported by a grant from IPM (No. 93470047).Gabeleh, M. (2015). On cyclic relatively nonexpansive mappings in generalized semimetric spaces. Applied General Topology. 16(2):99-108. https://doi.org/10.4995/agt.2015.2988SWORD9910816

Final distribution density of material fragment sizes at slow fragmentation process

The process of slow material fragmentation is studied when the diffusion approximation is applicable. Final distribution density of fragment sizes is calculated in the case of scale-homogeneity of subdivision mechanism.Рассмотрен процесс медленной фрагментации материала в условиях, когда применимо диффузионное приближение. Вычислена финальная плотность распределения размеров фрагментов в случае масштабной однородности механизма дробления.Розглянуто процес повільної фрагментації матеріалу в умовах, коли можна застосувати дiфузiйне наближення. Обчислена фінальна густина розподілу розмiрiв фрагментiв у випадку масштабної однорiдностi механiзму дроблення

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.Comment: 11 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3: revised following referee's comment

On the generalization of the Volterra principle of inversion

In this article a linear operator, K, defined on a Hilbert space equipped with a chain of orthoprojectors is considered. It is proved that if K enjoys a particular property with respect to the chain of orthoprojectors, then the series [summation operator]n = 0[infinity] Kn converges in the uniform operator norm. The proof uses purely algebraic techniques and does not require compactness of K. As such, it is a significant generalization of the well-known Volterra principle of inversion.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22226/1/0000660.pd

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite dimensional Hilbert space into a separable Hilbert space (we call it "operator ball") has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In the appendix we present results of Itai Shafrir about hyperbolic metrics on the operator ball