1,542 research outputs found
The Arithmetic of Distributions in Free Probability Theory
We give an analytical approach to the definition of additive and
multiplicative free convolutions which is based on the theory of Nevanlinna and
of Schur functions. We consider the set of probability distributions as a
semigroup equipped with the operation of free convolution and prove a
Khintchine type theorem for the factorization of elements of this semigroup. An
element of contains either indecomposable ("prime") factors or it
belongs to a class, say , of distributions without indecomposable factors.
In contrast to the classical convolution semigroup in the free additive and
multiplicative convolution semigroups the class consists of units (i.e.
Dirac measures) only. Furthermore we show that the set of indecomposable
elements is dense in .Comment: 66 pages; latex; 5 figures; corrected version of proofs of Khintchine
type theorems. For details see end of introductio
A fixed point theorem for contractions in modular metric spaces
The notion of a (metric) modular on an arbitrary set and the corresponding
modular space, more general than a metric space, were introduced and studied
recently by the author [V. V. Chistyakov, Metric modulars and their
application, Dokl. Math. 73(1) (2006) 32-35, and Modular metric spaces, I:
Basic concepts, Nonlinear Anal. 72(1) (2010) 1-14]. In this paper we establish
a fixed point theorem for contractive maps in modular spaces. It is related to
contracting rather ``generalized average velocities'' than metric distances,
and the successive approximations of fixed points converge to the fixed points
in a weaker sense as compared to the metric convergence.Comment: 31 pages, LaTeX, uses elsarticle.st
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