The purpose of this book is to give a comprehensive introduction to the study of non-linear operator theory in probabilistic metric spaces. This book is introduced as a survey of the latest and new results on the following topics: Basic theory of probabilistic metric spaces; Fixed point theorems for single-valued and multi-valued mappings in probabilistic metric spaces; Ekeland's variational principle and Caristi's fixed point theorem in probabilistic metric spaces; Coincidence point theorems, minimisation and fixed degree theorems in probabilistic metric spaces; Probabilistic contractors, accretive mappings and topological degree in probabilistic normed spaces; Nonlinear semigroups and differential equations in probabilistic metric spaces; KKM theorems, minimax theorems and variational inequalities.

Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory. Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces. In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces. Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.

Random Operator Theory provides a comprehensive discussion of the random norm of random bounded linear operators, also providing important random norms as random norms of differentiation operators and integral operators. After providing the basic definition of random norm of random bounded linear operators, the book then delves into the study of random operator theory, with final sections discussing the concept of random Banach algebras and its applications. - Explores random differentiation and random integral equations - Delves into the study of random operator theory - Discusses the concept of random Banach algebras and its applications

In this book, generally speaking, some properties of bitopological spaces generated by certain non-symmetric functions are studied. These functions, called "probabilistic quasi-pseudo-metrics" and "fuzzy quasi-pseudo-metrics", are generalisations of classical quasi-pseudo metrics. For the sake of completeness as well as for convenience and easy comparison, most of the introductory paragraphs are mainly devoted to fundamental notions and results from the classical -- deterministic or symmetric -- theory.

Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory. Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces. In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces. Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.

TheInternationalConferenceonRoughSetandKnowledgeTechnology(RSKT) has been held every year since 2006. RSKT serves as a major forum that brings researchers and industry practitioners together to discuss and deliberate on fundamental issues of knowledge processing and management and knowled- intensive practical solutions in the current knowledge age. Experts from around the world meet to present state-of-the-art scienti?c results, to nurture academic and industrial interaction, and to promote collaborative research in rough sets and knowledge technology. The ?rst RSKT was held in Chongqing, China, f- lowed by RSKT 2007 in Toronto, Canada, RSKT 2008 in Chengdu, China and RSKT 2009 in Gold Coast, Australia. RSKT 2010, the 5th in the series, was held in Beijing, China, October 15–17, 2010. This volume contains 98 papers selected for presentation at RSKT 2010. Following the success of the previous conferences, RSKT 2010 continued the tradition of a very rigorous reviewing process. Every submission was reviewed byatleasttworeviewers.Moreover,RSKT2010invitedseveralareachairsto- pervise the review process of every submission. Most submissions were reviewed by three experts. The Program Committee members were deeply involved in a highly engaging selection process with discussions among reviewers and area chairs. When necessary, additional expert reviews were sought. As a result, only top-quality papers were chosen for presentation at the conference, including 49 regular papers (acceptance rate of 28%) and 25 short papers (acceptance rate of 14.3%). We would like to thank all the authors for contributing their best papers. Without their support, this conference would not have been possible.