The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the ?triangular norms? introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.

The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the “triangular norms” introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.

These four volumes (CCIS 297, 298, 299, 300) constitute the proceedings of the 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, held in Catania, Italy, in July 2012. The 258 revised full papers presented together with six invited talks were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections on fuzzy machine learning and on-line modeling; computing with words and decision making; soft computing in computer vision; rough sets and complex data analysis: theory and applications; intelligent databases and information system; information fusion systems; philosophical and methodological aspects of soft computing; basic issues in rough sets; 40th anniversary of the measures of fuziness; SPS11 uncertainty in profiling systems and applications; handling uncertainty with copulas; formal methods to deal with uncertainty of many-valued events; linguistic summarization and description of data; fuzzy implications: theory and applications; sensing and data mining for teaching and learning; theory and applications of intuitionistic fuzzy sets; approximate aspects of data mining and database analytics; fuzzy numbers and their applications; information processing and management of uncertainty in knowledge-based systems; aggregation functions; imprecise probabilities; probabilistic graphical models with imprecision: theory and applications; belief function theory: basics and/or applications; fuzzy uncertainty in economics and business; new trends in De Finetti's approach; fuzzy measures and integrals; multicriteria decision making; uncertainty in privacy and security; uncertainty in the spirit of Pietro Benvenuti; coopetition; game theory; probabilistic approach.

This volume collects the extended abstracts of 45 contributions of participants to the Seventh International Summer School on Aggregation Operators (AGOP 2013), held at Pamplona in July, 16-20, 2013. These contributions cover a very broad range, from the purely theoretical ones to those with a more applied focus. Moreover, the summaries of the plenary talks and tutorials given at the same workshop are included. Together they provide a good overview of recent trends in research in aggregation functions which can be of interest to both researchers in Physics or Mathematics working on the theoretical basis of aggregation functions, and to engineers who require them for applications.

This book offers a timely report on key theories and applications of soft-computing. Written in honour of Professor Gaspar Mayor on his 70th birthday, it primarily focuses on areas related to his research, including fuzzy binary operators, aggregation functions, multi-distances, and fuzzy consensus/decision models. It also discusses a number of interesting applications such as the implementation of fuzzy mathematical morphology based on Mayor-Torrens t-norms. Importantly, the different chapters, authored by leading experts, present novel results and offer new perspectives on different aspects of Mayor’s research. The book also includes an overview of evolutionary fuzzy systems, a topic that is not one of Mayor’s main areas of interest, and a final chapter written by the Spanish pioneer in fuzzy logic, Professor E. Trillas. Computer and decision scientists, knowledge engineers and mathematicians alike will find here an authoritative overview of key soft-computing concepts and techniques.

These three volumes (CCIS 442, 443, 444) constitute the proceedings of the 15th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2014, held in Montpellier, France, July 15-19, 2014. The 180 revised full papers presented together with five invited talks were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections on uncertainty and imprecision on the web of data; decision support and uncertainty management in agri-environment; fuzzy implications; clustering; fuzzy measures and integrals; non-classical logics; data analysis; real-world applications; aggregation; probabilistic networks; recommendation systems and social networks; fuzzy systems; fuzzy logic in boolean framework; management of uncertainty in social networks; from different to same, from imitation to analogy; soft computing and sensory analysis; database systems; fuzzy set theory; measurement and sensory information; aggregation; formal methods for vagueness and uncertainty in a many-valued realm; graduality; preferences; uncertainty management in machine learning; philosophy and history of soft computing; soft computing and sensory analysis; similarity analysis; fuzzy logic, formal concept analysis and rough set; intelligent databases and information systems; theory of evidence; aggregation functions; big data - the role of fuzzy methods; imprecise probabilities: from foundations to applications; multinomial logistic regression on Markov chains for crop rotation modelling; intelligent measurement and control for nonlinear systems.

This carefully edited book comprises the papers from EUROFUSE 2011 Workshop on Fuzzy Methods for Knowledge-based Systems. EUROFUSE was established in 1998 as the EURO (the Association of European Operational Research Societies) Working Group on Fuzzy Sets, as a successor of the former European Chapter of IFSA (the International Fuzzy Systems Association). The present EUROFUSE 2011 workshop is held at Régua, Portugal, 21-23 September, in the World Heritage Site of the Douro Wine Region and is organised by Pedro Melo-Pinto and Pedro Couto. The workshop is devoted to Fuzzy Methods for Knowledge-based Systems. And has the goal to bring together researchers and practitioners developing and applying fuzzy techniques in preference modelling and decision making in an informal atmosphere. EUROFUSE 2011 has three distinguished invited speakers: Francisco Herrera (Granada, Spain), Radko Mesiar (Bratislava, Slovakia) and Jose Luis Garcia Lapresta (Valladolid, Spain). Next to the invited speakers, the three day program consists of 37 lectures. In total, there are 70 participants from 10 countries. This edited volume contains the final revised manuscripts on the basis of which the program was put together.

This book discusses heuristic methods – methods lacking a solid theoretical justification – which are ubiquitous in numerous application areas, and explains techniques that can make heuristic methods more reliable. Focusing on algebraic techniques, i.e., those that use only a few specific features of a situation, it describes various state-of-the-art applications, ranging from fuzzy methods for dealing with imprecision to general optimization methods and quantum-based methods for analyzing economic phenomena. The book also includes recent results from leading researchers, which could (and hopefully will) provide the basis for future applications. As such, it is a valuable resource for mathematicians interested in potential applications of their algebraic results and ideas, as well as for application specialists wanting to discover how algebraic techniques can help in their domains.

This book is a printed edition of the Special Issue "Fuzzy Mathematics" that was published in Mathematics

This is a succinct guide to the application and modelling of dependence models or copulas in the financial markets. First applied to credit risk modelling, copulas are now widely used across a range of derivatives transactions, asset pricing techniques and risk models and are a core part of the financial engineer's toolkit.