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Various generalizations of the classical concept of a convex function have been introduced, especially during the second half of the 20th century. Generalized convex functions are the many nonconvex functions which share at least one of the valuable properties of convex functions. Apart from their theoretical interest, they are often more suitable than convex functions to describe real-word problems in disciplines such as economics, engineering, management science, probability theory and in other applied sciences. More recently, generalized monotone maps which are closely related to generalized convex functions have also been studied extensively. While initial efforts to generalize convexity and monotonicity were limited to only a few research centers, today there are numerous researchers throughout the world and in various disciplines engaged in theoretical and applied studies of generalized convexity/monotonicity (see http://www.genconv.org). The Handbook offers a systematic and thorough exposition of the theory and applications of the various aspects of generalized convexity and generalized monotonicity. It is aimed at the non-expert, for whom it provides a detailed introduction, as well as at the expert who seeks to learn about the latest developments and references in his research area. Results in this fast growing field are contained in a large number of scientific papers which appeared in a variety of professional journals, partially due to the interdisciplinary nature of the subject matter. Each of its fourteen chapters is written by leading experts of the respective research area starting from the very basics and moving on to the state of the art of the subject. Each chapter is complemented by a comprehensive bibliography which will assist the non-expert and expert alike.

Convex functions. --- Monotonic functions. --- Functions, Monotonic --- Functions of real variables --- Functions, Convex --- Convex functions --- Monotonic functions --- Mathematics. --- Real Functions. --- Game Theory, Economics, Social and Behav. Sciences. --- Operations Research, Management Science. --- Math --- Science --- Functions of real variables. --- Game theory. --- Operations research. --- Management science. --- Games, Theory of --- Theory of games --- Mathematical models --- Mathematics --- Real variables --- Functions of complex variables --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory

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In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than using a “big convexification” of the graph of the multifunction and the “minimax technique”for proving the existence of linear functionals satisfying certain conditions, the Fitzpatrick function is used. The journey begins with a generalization of the Hahn-Banach theorem uniting classical functional analysis, minimax theory, Lagrange multiplier theory and convex analysis and culminates in a survey of current results on monotone multifunctions on a Banach space. The first two chapters are aimed at students interested in the development of the basic theorems of functional analysis, which leads painlessly to the theory of minimax theorems, convex Lagrange multiplier theory and convex analysis. The remaining five chapters are useful for those who wish to learn about the current research on monotone multifunctions on (possibly non reflexive) Banach space.

Monotone operators. --- Monotonic functions. --- Banach spaces. --- Opérateurs monotones --- Fonctions monotones --- Banach, Espaces de --- Monotone operators --- Monotonic functions --- Banach spaces --- Duality theory (Mathematics) --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Maxima and minima. --- Opérateurs monotones --- EPUB-LIV-FT SPRINGER-B --- Functions, Monotonic --- Minima --- Mathematics. --- Functional analysis. --- Operator theory. --- Calculus of variations. --- Functional Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Operator Theory. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science --- Functions of real variables --- Operator theory --- Algebra --- Mathematical analysis --- Topology --- Functions of complex variables --- Generalized spaces --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis

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This book provides an in depth discussion of Loewner’s theorem on the characterization of matrix monotone functions. The author refers to the book as a ‘love poem,’ one that highlights a unique mix of algebra and analysis and touches on numerous methods and results. The book details many different topics from analysis, operator theory and algebra, such as divided differences, convexity, positive definiteness, integral representations of function classes, Pick interpolation, rational approximation, orthogonal polynomials, continued fractions, and more. Most applications of Loewner’s theorem involve the easy half of the theorem. A great number of interesting techniques in analysis are the bases for a proof of the hard half. Centered on one theorem, eleven proofs are discussed, both for the study of their own approach to the proof and as a starting point for discussing a variety of tools in analysis. Historical background and inclusion of pictures of some of the main figures who have developed the subject, adds another depth of perspective. The presentation is suitable for detailed study, for quick review or reference to the various methods that are presented. The book is also suitable for independent study. The volume will be of interest to research mathematicians, physicists, and graduate students working in matrix theory and approximation, as well as to analysts and mathematical physicists.

Monotonic functions. --- Matrices. --- Mathematics. --- Matrix theory. --- Operator theory. --- Group theory. --- Real Functions. --- Linear and Multilinear Algebras, Matrix Theory. --- Operator Theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Functional analysis --- Math --- Science --- Functions of real variables. --- Algebra. --- Mathematics --- Mathematical analysis --- Real variables --- Functions of complex variables

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This book should be accessible for graduate or advanced undergraduate students both in mathematics and computer science. We have designed this book especially

Algebraic logic. --- Monotonic functions. --- Fixed point theory. --- Lattice theory. --- Machine theory. --- Abstract automata --- Abstract machines --- Automata --- Mathematical machine theory --- Algorithms --- Logic, Symbolic and mathematical --- Recursive functions --- Robotics --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Fixed point theorems (Topology) --- Nonlinear operators --- Coincidence theory (Mathematics) --- Functions, Monotonic --- Functions of real variables