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This solutions manual thoroughly goes through the exercises found in Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker. Several solutions are accompanied by detailed illustrations and intuitive explanations. This book will pave the way for students to easily grasp the multitude of solution methods and aspects of convex sets and convex functions. Request Inspection Copy
This introduction to the theory of complex manifolds covers the most important branches and methods in complex analysis of several variables while completely avoiding abstract concepts involving sheaves, coherence, and higher-dimensional cohomology. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Each chapter contains a variety of examples and exercises.
Based on undergraduate teaching to students in computer science, economics and mathematics at Aarhus University, this is an elementary introduction to convex sets and convex functions with emphasis on concrete computations and examples. Starting from linear inequalities and Fourier–Motzkin elimination, the theory is developed by introducing polyhedra, the double description method and the simplex algorithm, closed convex subsets, convex functions of one and several variables ending with a chapter on convex optimization with the Karush–Kuhn–Tucker conditions, duality and an interior point algorithm. Contents:Fourier–Motzkin Elimination Affine SubspacesConvex SubsetsPolyhedraComputations with PolyhedraClosed Convex Subsets and Separating HyperplanesConvex FunctionsDifferentiable Functions of Several VariablesConvex Functions of Several VariablesConvex OptimizationAppendices:AnalysisLinear (In)dependence and the Rank of a Matrix Readership: Undergraduates focusing on convexity and optimization. Keywords:Convex Sets;Covex Functions;Fourier–Motzkin Eliminination;Karush–Kuhn–Tucker Conditions;Quadratic OptimizationKey Features:Emphasis on viewing introductory convexity as a generalization of linear algebra in finding solutions to linear inequalitiesA key point is computation through concrete algorithms like the double description method. This enables students to carry out non-trivial computations alongside the introduction of the mathematical conceptsConvexity is inherently a geometric subject. However, without computational techniques, the teaching of the subject turns easily into a reproduction of abstractions and definitions. The book addresses this issue at a basic levelReviews: “Overall, the author has managed to keep a sound balance between the different approaches to convexity in geometry, analysis, and applied mathematics. The entire presentation is utmost lucid, didactically well-composed, thematically versatile and essentially self-contained. The large number of instructive examples and illustrating figures will certainly help the unexperienced reader grasp the abstract concepts, methods and results, all of which are treated in a mathematically rigorous way. Also, the emphasis on computational, especially algorithmic methods is a particular feature of this fine undergraduate textbook, which will be a great source for students and instructors like-wise … the book under review is an excellent, rather unique primer on convexity in several branches of mathematics.” Zentralblatt MATH “Undergraduate Convexity would make an excellent textbook. An instructor might choose to have students present some of the examples while he or she provides commentary, perhaps alternating coaching and lecturing. A course taught from this book could be a good transition into more abstract mathematics, exposing students to general theory then giving them the familiar comfort of more computational exercises. One could also use the book as a warm-up to a more advanced course in optimization.” MAA Review “The book is didactically written in a pleasant and lively style, with careful motivation of the considered notions, illuminating examples and pictures, and relevant historical remarks. This is a remarkable book, a readable and attractive introduction to the multi-faceted domain of convexity and its applications.” Nicolae Popovici Stud. Univ. Babes-Bolyai Math “Compared to most modern undergraduate math textbooks, this book is unusually thin and portable. It also contains a wealth of material, presented in a concise and delightful way, accompanied by figures, historical references, pointers to further reading, pictures of great mathematicians and snapshots of pages of their groundbreaking papers. There are numerous exercises, both of computational and theoretical nature. If you want to teach an undergraduate convexity course, this looks like an excellent choice for the textbook.” MathSciNet
This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.
A comprehensive introduction to the tools, techniques and applications of convex optimization.
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