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Mathematics --- Wiskunde. --- Mathematics. --- Mathematical Sciences --- Applied Mathematics --- analysis --- convexity --- algebra --- topology --- geometry --- operator theory --- Math --- Science
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Among the participants discussing recent trends in their respective fields and in areas of common interest in these proceedings are such world-famous geometers as H.S.M. Coxeter, L. Danzer, D.G. Larman and J.M. Wills, and equally famous graph-theorists B. Bollobás, P. Erdös and F. Harary. In addition to new results in both geometry and graph theory, this work includes articles involving both of these two fields, for instance ``Convexity, Graph Theory and Non-Negative Matrices'', ``Weakly Saturated Graphs are Rigid'', and many more. The volume covers a broad spectrum of topics in graph theory,
Discrete mathematics --- Graph theory --- Convex domains --- CONVEX DOMAINS --- Congresses --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Graph theory - Congresses --- CONVEX DOMAINS - Congresses
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Generalized convexity conditions play a major role in many modern mechanical applications. They serve as the basis for existence proofs and allow for the design of advanced algorithms. Moreover, understanding these convexity conditions helps in deriving reliable mechanical models. The book summarizes the well established as well as the newest results in the field of poly-, quasi and rank-one convexity. Special emphasis is put on the construction of anisotropic polyconvex energy functions with applications to biomechanics and thin shells. In addition, phase transitions with interfacial energy and the relaxation of nematic elastomers are discussed.
Convex domains. --- Mechanics, Applied -- Mathematical models. --- Mechanics, Applied --- Convex domains --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Civil Engineering --- Mathematical models --- Mathematical models. --- Convex regions --- Convexity --- Engineering. --- Mechanics. --- Mechanics, Applied. --- Theoretical and Applied Mechanics. --- Calculus of variations --- Convex geometry --- Point set theory --- Mechanics, applied. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Rank-one convexity
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All the existing books in Infinite Dimensional Complex Analysis focus on the problems of locally convex spaces. However, the theory without convexity condition is covered for the first time in this book. This shows that we are really working with a new, important and interesting field. Theory of functions and nonlinear analysis problems are widespread in the mathematical modeling of real world systems in a very broad range of applications. During the past three decades many new results from the author have helped to solve multiextreme problems arising from important situations, non-c
Holomorphic functions. --- Functional analysis. --- Convexity spaces. --- Convex surfaces. --- Complexes. --- Linear complexes --- Algebras, Linear --- Coordinates --- Geometry --- Line geometry --- Transformations (Mathematics) --- Convex areas --- Convex domains --- Surfaces --- Spaces, Convexity --- Convex sets --- Vector spaces --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functions, Holomorphic --- Functions of several complex variables
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In this monograph the author presents the theory of duality for nonconvex approximation in normed linear spaces and nonconvex global optimization in locally convex spaces. Key topics include: * duality for worst approximation (i.e., the maximization of the distance of an element to a convex set) * duality for reverse convex best approximation (i.e., the minimization of the distance of an element to the complement of a convex set) * duality for convex maximization (i.e., the maximization of a convex function on a convex set) * duality for reverse convex minimization (i.e., the minimization of a convex function on the complement of a convex set) * duality for d.c. optimization (i.e., optimization problems involving differences of convex functions). Detailed proofs of results are given, along with varied illustrations. While many of the results have been published in mathematical journals, this is the first time these results appear in book form. In addition, unpublished results and new proofs are provided. This monograph should be of great interest to experts in this and related fields. Ivan Singer is a Research Professor at the Simion Stoilow Institute of Mathematics in Bucharest, and a Member of the Romanian Academy. He is one of the pioneers of approximation theory in normed linear spaces, and of generalizations of approximation theory to optimization theory. He has been a Visiting Professor at several universities in the U.S.A., Great Britain, Germany, Holland, Italy, and other countries, and was the principal speaker at an N. S. F. Regional Conference at Kent State University. He is one of the editors of the journals Numerical Functional Analysis and Optimization (since its inception in 1979), Optimization, and Revue d'analyse num'erique et de th'eorie de l'approximation. His previous books include Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces (Springer 1970), The Theory of Best Approximation and Functional Analysis (SIAM 1974), Bases in Banach Spaces I, II (Springer, 1970, 1981), and Abstract Convex Analysis (Wiley-Interscience, 1997).
Convex functions. --- Convex sets. --- Duality theory (Mathematics) --- Approximation theory. --- Convex domains. --- Convexity spaces. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Spaces, Convexity --- Convex sets --- Vector spaces --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Algebra --- Mathematical analysis --- Topology --- Sets, Convex --- Convex domains --- Set theory --- Functions, Convex --- Functions of real variables --- Operator theory. --- Functional analysis. --- Mathematical optimization. --- Mathematics. --- Operator Theory. --- Functional Analysis. --- Optimization. --- Approximations and Expansions. --- Math --- Science --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Functional calculus --- Functional equations --- Integral equations
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This volume presents some of the research topics discussed at the 2014-2015 Annual Thematic Program Discrete Structures: Analysis and Applications at the Institute of Mathematics and its Applications during the Spring 2015 where geometric analysis, convex geometry and concentration phenomena were the focus. Leading experts have written surveys of research problems, making state of the art results more conveniently and widely available. The volume is organized into two parts. Part I contains those contributions that focus primarily on problems motivated by probability theory, while Part II contains those contributions that focus primarily on problems motivated by convex geometry and geometric analysis. This book will be of use to those who research convex geometry, geometric analysis and probability directly or apply such methods in other fields.
Convex domains. --- Probabilities. --- Probability --- Statistical inference --- Convex regions --- Convexity --- Mathematics. --- Convex geometry. --- Discrete geometry. --- Convex and Discrete Geometry. --- Probability Theory and Stochastic Processes. --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Calculus of variations --- Convex geometry --- Point set theory --- Discrete groups. --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry . --- Geometry --- Combinatorial geometry
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This textbook provides a thorough introduction to spectrahedra, which are the solution sets to linear matrix inequalities, emerging in convex and polynomial optimization, analysis, combinatorics, and algebraic geometry. Including a wealth of examples and exercises, this textbook guides the reader in helping to determine the convex sets that can be represented and approximated as spectrahedra and their shadows (projections). Several general results obtained in the last 15 years by a variety of different methods are presented in the book, along with the necessary background from algebra and geometry.
Convex domains. --- Matrix inequalities. --- Inequalities (Mathematics) --- Matrices --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Algebraic geometry. --- Convex geometry. --- Discrete geometry. --- Mathematical optimization. --- Algebraic Geometry. --- Convex and Discrete Geometry. --- Optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Discrete mathematics --- Geometry --- Combinatorial geometry --- Algebraic geometry --- Desigualtats matricials --- Dominis convexos
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This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems.
Mathematics. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Convex geometry. --- Discrete geometry. --- Combinatorics. --- Graph theory. --- Convex and Discrete Geometry. --- Graph Theory. --- Global Analysis and Analysis on Manifolds. --- Graph theory --- Convex domains. --- Convex regions --- Convexity --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- Calculus of variations --- Convex geometry --- Point set theory --- Combinatorial analysis --- Topology --- Discrete groups. --- Global analysis. --- Groups, Discrete --- Infinite groups --- Combinatorics --- Algebra --- Mathematical analysis --- Discrete mathematics --- Convex geometry . --- Geometry, Differential --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Geometry --- Combinatorial geometry
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In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
Convex functions. --- Gamma functions. --- Functions, Convex --- Functions of real variables --- Functions, Gamma --- Transcendental functions --- Difference Equation --- Higher Order Convexity --- Bohr-Mollerup's Theorem --- Principal Indefinite Sums --- Gauss' Limit --- Euler Product Form --- Raabe's Formula --- Binet's Function --- Stirling's Formula --- Euler's Infinite Product --- Euler's Reflection Formula --- Weierstrass' Infinite Product --- Gauss Multiplication Formula --- Euler's Constant --- Gamma Function --- Polygamma Functions --- Hurwitz Zeta Function --- Generalized Stieltjes Constants
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Mathematical optimization --- Convex domains --- Decision making --- Optimisation mathématique --- Algèbres convexes --- Prise de décision --- Mathematical models --- Modèles mathématiques --- Mathematics --- 519.83 --- -Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Theory of games --- -Theory of games --- 519.83 Theory of games --- -Convex regions --- Deciding --- Optimisation mathématique --- Algèbres convexes --- Prise de décision --- Modèles mathématiques --- Fonctions convexes --- Convex functions --- Géometrie convexe --- Microéconomie --- Decision making - Mathematics
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