Stationary strategies in topological games

We prove general theorems on the existence of stationery strategies (i.e., strategies depending only on the opponent's last move) in certain infinite positional games of perfect information and we derive some consequences for various topological games.     

Stationary measures and rectifiability

For integers , we consider -valued Radon measures on an open set which satisfy

Strong uniform convergence of composition sequences of probability measures on locally compact topological semigroups

Weak convergence of convolution products of non-identical probability measures on topological semigroups is investigated. Firstly, the dependence relationship between the semigroupN of measures and suppN is studied; secondly, a necessary and sufficient condition that insures the strong uniform composition convergence of sequence μ_n is presented; thirdly, a more extensive class of semigroups is found, on which μ_n must be of strong uniform composition convergence, if it is of composition convergence.     

Homotopy simplicial faces and the homology of realizations of simplicial topological spaces

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of A _∞-algebras and D _∞-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.     

Fuzzy probability measures

In [4] Höhle has defined fuzzy measures on G-fuzzy sets [2] where G stands for a regular Boolean algebra. Consequently, since the unit interval is not complemented, fuzzy sets in the sense of Zadeh [8] do not fit in this framework in a straightforward manner. It is the purpose of this paper to continue the work started in [5] which deals with [0,1]-fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space [5]. We give necessary and sufficient conditions for such a measure to be a classical integral as in [9] in the case the space is generated. A counterexample in the general case is also presented. Finally it is shown that a fuzzy probability measure is always an integral (if the space is generated) if we replace the operations ∧ and ∨ by the t-norm T_o and its dual S₀ (see [6]).     

Banach-valued probability measures

We introduce and study the notion of Banach-valued probability measures on a compact semitopological semigroup. In particular, we prove that these measures are nontrivial idempotents and convolution is separately continuous. We give an example of a Banach-valued measure where the support may not be simple; though for any idempotent measure the support is a closed simple subsemigroup.     

Determinantal probability measures

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.     

Applications of realizations (aka linearizations) to free probability

We show how the combination of new “linearization” ideas in free probability theory with the powerful “realization” machinery – developed over the last 50 years in fields including systems engineering and automata theory – allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II₁ von Neumann algebras, with a precise control of the domains of the rational expressions.The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.     

Super-measures and finitely defined topological measures

We introduce a class of set-functions on the set of natural numbers, which are called super-measures. Super-measures are then utilized to characterize a certain class of topological measures (previously called quasi-measures, see below) which arises naturally. The members of this class of topological measures are called finitely defined, and are shown to be dense in the set of all topological measures. This revised version was published online in June 2006 with corrections to the Cover Date.     

q-Functions and extreme topological measures

q-Functions provide a method for constructing topological measures. We give necessary and sufficient conditions for a composition of a q-function and a topological measure to be a topological measure. Regular and extreme step q-functions are characterized by certain regions in Rn. Then extreme q-functions are used to study extreme topological measures. For example, we prove (under some assumptions on the underlying set) that given n, there are different types of extreme topological measures with values 0,1/n,…,1. In contrast, in the case of measures the only extreme points are {0,1}-valued, i.e., point masses.     

Stationary probability distributions for multiresponse linear learning models

A necessary and sufficient condition is given for the existence of stationary probability distributions of a non-Markovian model with linear transition rule. Similar to the Markovian case, stationary probability distributions are characterized as eigenvectors of nonnegative matrices. The model studied includes as special cases the Markovian model as well as the linear learning model and has applications in psychological and biological research, in control theory, and in adaption theory.     

Stationary measures for randomly chosen maps

A Markov process on a compact metric space,X is given by random transformations.S is a finite set of continuous transformations ofX to itself. A random evolution onX is defined by lettingx inX evolve toT(x) forT inS with probability that depends onx andT but is independent of any other past measurable events. This type of model is often called a place dependent iterated function system. The transformations are assumed to have either monotone or contractive properties. Theorems are given to describe the number and types of ergodic invariant measures. Special emphasis is given to learning models and self-reinforcing random walks.Supported in part by AFOSR Grant No. 91-0215, the Alexander von Humboldt Foundation and SFB 170, University of Göttingen.     

The shape of data and probability measures

We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only quantifies global variation about the mean. Empirical forms of localized covariance previously have been used in data analysis and visualization, for example, in local principal component analysis, but we develop a framework for the systematic treatment of theoretical questions and mathematical analysis of computational models. We prove strong stability theorems with respect to the Wasserstein distance between probability measures, obtain consistency results for estimators, as well as bounds on the rate of convergence of empirical CTFs. These results show that CTFs are robust to sampling, noise and outliers. We provide numerous illustrations of how CTFs let us extract shape from data and also apply CTFs to manifold clustering, the problem of categorizing data points according to their noisy membership in a collection of possibly intersecting smooth submanifolds of Euclidean space. We prove that the proposed manifold clustering method is stable and carry out several experiments to illustrate the method.