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ISBN: 9783764377052 3764377054 3764377062 9786610627769 128062776X Year: 2006 Publisher: Basel : Birkhäuser Basel : Imprint: Birkhäuser,
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Problems linking the shape of a domain or the coefficients of an elliptic operator to the sequence of its eigenvalues are among the most fascinating of mathematical analysis. In this book, we focus on extremal problems. For instance, we look for a domain which minimizes or maximizes a given eigenvalue of the Laplace operator with various boundary conditions and various geometric constraints. We also consider the case of functions of eigenvalues. We investigate similar questions for other elliptic operators, such as the Schrödinger operator, non homogeneous membranes, or the bi-Laplacian, and we look at optimal composites and optimal insulation problems in terms of eigenvalues. Providing also a self-contained presentation of classical isoperimetric inequalities for eigenvalues and 30 open problems, this book will be useful for pure and applied mathematicians, particularly those interested in partial differential equations, the calculus of variations, differential geometry, or spectral theory.
Eigenvalues. --- Maxima and minima. --- Valeurs propres --- Maxima et minima --- Eigenvalues --- Elliptic operators --- Maxima and minima --- Elliptic operators. --- Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- EPUB-LIV-FT SPRINGER-B --- Minima --- Differential operators, Elliptic --- Operators, Elliptic --- Mathematics. --- Operator theory. --- Potential theory (Mathematics). --- Operator Theory. --- Potential Theory. --- Partial differential operators --- Matrices --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functional analysis
ISBN: 9780817635176 0817635173 0817644733 Year: 2006 Publisher: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser,
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Questions of maxima and minima have great practical significance, with applications to physics, engineering, and economics; they have also given rise to theoretical advances, notably in calculus and optimization. Indeed, while most texts view the study of extrema within the context of calculus, this carefully constructed problem book takes a uniquely intuitive approach to the subject: it presents hundreds of extreme-value problems, examples, and solutions primarily through Euclidean geometry. Key features and topics: * Comprehensive selection of problems, including Greek geometry and optics, Newtonian mechanics, isoperimetric problems, and recently solved problems such as Malfatti’s problem * Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning * Presentation and application of classical inequalities, including Cauchy--Schwarz and Minkowski’s Inequality; basic results in calculus, such as the Intermediate Value Theorem; and emphasis on simple but useful geometric concepts, including transformations, convexity, and symmetry * Clear solutions to the problems, often accompanied by figures * Hundreds of exercises of varying difficulty, from straightforward to Olympiad-caliber Written by a team of established mathematicians and professors, this work draws on the authors’ experience in the classroom and as Olympiad coaches. By exposing readers to a wealth of creative problem-solving approaches, the text communicates not only geometry but also algebra, calculus, and topology. Ideal for use at the junior and senior undergraduate level, as well as in enrichment programs and Olympiad training for advanced high school students, this book’s breadth and depth will appeal to a wide audience, from secondary school teachers and pupils to graduate students, professional mathematicians, and puzzle enthusiasts.
Maxima and minima. --- Geometry --- Maxima et minima --- Géométrie --- Problems, exercises, etc. --- Problèmes et exercices --- Electronic books. -- local. --- Geometry -- Problems, exercises, etc. --- Maxima and minima --- Mathematics --- Physical Sciences & Mathematics --- 514.1 --- Minima --- General geometry --- 514.1 General geometry --- Géométrie --- Problèmes et exercices --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Mathematics. --- Algebra. --- Mathematical analysis. --- Analysis (Mathematics). --- Geometry. --- Mathematical optimization. --- Topology. --- Combinatorics. --- Optimization. --- Analysis. --- Global analysis (Mathematics). --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Combinatorics --- Algebra --- Mathematical analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis --- Euclid's Elements --- 517.1 Mathematical analysis
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ISBN: 0444851674 9786612754883 1282754882 0080875270 9780080875279 9780444851673 Year: 1979 Volume: v. 6 Publisher: Amsterdam : New York : North-Holland Pub. Co. ; sole distributors for the U.S.A. and Canada, Elsevier North-Holland,
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Theory of extremal problems
Mathematical optimization --- Maxima and minima --- Calculus of variations --- Extremal problems (Mathematics) --- Optimisation mathématique --- Maxima et minima --- Calcul des variations --- Problèmes extrémaux (Mathématiques) --- 519.85 --- 681.3*F22 --- 681.3*G16 --- Minima --- Mathematics --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Graph theory --- Problems, Extremal (Mathematics) --- Geometric function theory --- Isoperimetrical problems --- Variations, Calculus of --- Mathematical programming --- Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3} --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Extremal problems --- Calculus of variations. --- Mathematical optimization. --- Maxima and minima. --- Extremal problems (Mathematics). --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 681.3*F22 Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3} --- 519.85 Mathematical programming --- Optimisation mathématique --- Problèmes extrémaux (Mathématiques) --- ELSEVIER-B EPUB-LIV-FT
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