Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

From probability to sequences and back

This is a survey covering sequential structures and their applications to the foundations of probability theory. Sequential convergence, convergence groups and the extension of sequentially continuous maps belong to general topology and Trieste for long has been a center of sequential topology. We begin with some personal reflections, con- tinue with topological problems motivated by the extension of probability measures, and close with some recent results related to the categorical foundations of probability theory

Fuzzification of crisp domains

summary:The present paper is devoted to the transition from crisp domains of probability to fuzzy domains of probability. First, we start with a simple transportation problem and present its solution. The solution has a probabilistic interpretation and it illustrates the transition from classical random variables to fuzzy random variables in the sense of Gudder and Bugajski. Second, we analyse the process of fuzzification of classical crisp domains of probability within the category $ID$ of $D$-posets of fuzzy sets and put into perspective our earlier results concerning categorical aspects of fuzzification. For example, we show that (within $ID$) all nontrivial probability measures have genuine fuzzy quality and we extend the corresponding fuzzification functor to an epireflector. Third, we extend the results to simplex-valued probability domains. In particular, we describe the transition from crisp simplex-valued domains to fuzzy simplex-valued domains via a “simplex” modification of the fuzzification functor. Both, the fuzzy probability and the simplex-valued fuzzy probability is in a sense minimal extension of the corresponding crisp probability theory which covers some quantum phenomenon

First-order limits, an analytical perspective

In this paper we present a novel approach to graph (and structural) limits based on model theory and analysis. The role of Stone and Gelfand dualities is displayed prominently and leads to a general theory, which we believe is naturally emerging. This approach covers all the particular examples of structural convergence and it put the whole in new context. As an application, it leads to new intermediate examples of structural convergence and to a "grand conjecture" dealing with sparse graphs. We survey the recent developments

Approximation Theory and Related Applications

In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics

Set-Valued Analysis

This Special Issue contains eight original papers with a high impact in various domains of set-valued analysis. Set-valued analysis has made remarkable progress in the last 70 years, enriching itself continuously with new concepts, important results, and special applications. Different problems arising in the theory of control, economics, game theory, decision making, nonlinear programming, biomathematics, and statistics have strengthened the theoretical base and the specific techniques of set-valued analysis. The consistency of its theoretical approach and the multitude of its applications have transformed set-valued analysis into a reference field of modern mathematics, which attracts an impressive number of researchers