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

Probability Theory Compatible with the New Conception of Modern Thermodynamics. Economics and Crisis of Debts

We show that G\"odel's negative results concerning arithmetic, which date back to the 1930s, and the ancient "sand pile" paradox (known also as "sorites paradox") pose the questions of the use of fuzzy sets and of the effect of a measuring device on the experiment. The consideration of these facts led, in thermodynamics, to a new one-parameter family of ideal gases. In turn, this leads to a new approach to probability theory (including the new notion of independent events). As applied to economics, this gives the correction, based on Friedman's rule, to Irving Fisher's "Main Law of Economics" and enables us to consider the theory of debt crisis.Comment: 48p., 14 figs., 82 refs.; more precise mathematical explanations are added. arXiv admin note: significant text overlap with arXiv:1111.610

Tropicalizing the simplex algorithm

We develop a tropical analog of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in O(n(m+n)) time, where m is the number of constraints and n is the dimension.Comment: v1: 35 pages, 7 figures, 4 algorithms; v2: improved presentation, 39 pages, 9 figures, 4 algorithm

A Fuzzy Social Network Analysis Method and a Case Study on Tianya

Social networking service (SNS) has become online service platforms that focus on facilitating the building of social networks among people who share interests, activities, backgrounds, or real-life connections and has had a rapid development in China in the past few years. This paper aims to develop a fuzzy social network service analysis method, which combines graph theory with related fuzzy approach, to analyze the social network structural features and the distribution characteristics of interpersonal nodes in SNS community. A case study on a very famous Chinese tourism BBS-Tianya-is conducted to illustrate and validate the proposed approach. The research findings are as follows: (1) The attraction degrees of various areas in the forum are significantly different; (2) interpersonal nodes in the forum are concentrated relatively; (3) the fuzzy out-degrees and the fuzzy in-degrees of interpersonal nodes in the forum conflict each other; and (4) the distribution of interpersonal nodes is influenced by geographical relations. These findings can directly support social network service management and particularly tourism online service developments. © Springer-Verlag Berlin Heidelberg 2014

The core of games on ordered structures and graphs

In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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