using the borsuk ulam theorem

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Using The Borsuk Ulam Theorem

Author : Jiri Matousek
ISBN : 9783540766490
Genre : Mathematics
File Size : 39. 60 MB
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To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

The Borsuk Ulam Theorem For Approximable Maps

Author : Zdzisław Dzedzej
ISBN : OCLC:823596239
Genre :
File Size : 48. 24 MB
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A Journey Through Discrete Mathematics

Author : Martin Loebl
ISBN : 9783319444796
Genre : Computers
File Size : 25. 26 MB
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This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka would have approved of. The original research articles, surveys and expository articles, written by leading experts in their respective fields, map Jiří Matoušek’s numerous areas of mathematical interest.

Topological Methods For Variational Problems With Symmetries

Author : Thomas Bartsch
ISBN : 9783540480990
Genre : Mathematics
File Size : 54. 95 MB
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Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in elliptic equations on symmetric domains or in the corresponding semiflows; and when one is looking for "special" solutions of these problems. This book is concerned with Lusternik-Schnirelmann theory and Morse-Conley theory for group invariant functionals. These topological methods are developed in detail with new calculations of the equivariant Lusternik-Schnirelmann category and versions of the Borsuk-Ulam theorem for very general classes of symmetry groups. The Morse-Conley theory is applied to bifurcation problems, in particular to the bifurcation of steady states and hetero-clinic orbits of O(3)-symmetric flows; and to the existence of periodic solutions nearequilibria of symmetric Hamiltonian systems. Some familiarity with the usualminimax theory and basic algebraic topology is assumed.

Proofs From The Book

Author : Martin Aigner
ISBN : 9783662442050
Genre : Mathematics
File Size : 38. 55 MB
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This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises. From the Reviews "... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..." LMS Newsletter, January 1999 "Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011.

A Course In Topological Combinatorics

Author : Mark de Longueville
ISBN : 9781441979100
Genre : Mathematics
File Size : 75. 11 MB
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A Course in Topological Combinatorics is the first undergraduate textbook on the field of topological combinatorics, a subject that has become an active and innovative research area in mathematics over the last thirty years with growing applications in math, computer science, and other applied areas. Topological combinatorics is concerned with solutions to combinatorial problems by applying topological tools. In most cases these solutions are very elegant and the connection between combinatorics and topology often arises as an unexpected surprise. The textbook covers topics such as fair division, graph coloring problems, evasiveness of graph properties, and embedding problems from discrete geometry. The text contains a large number of figures that support the understanding of concepts and proofs. In many cases several alternative proofs for the same result are given, and each chapter ends with a series of exercises. The extensive appendix makes the book completely self-contained. The textbook is well suited for advanced undergraduate or beginning graduate mathematics students. Previous knowledge in topology or graph theory is helpful but not necessary. The text may be used as a basis for a one- or two-semester course as well as a supplementary text for a topology or combinatorics class.

Lectures On Hyperbolic Geometry

Author : Riccardo Benedetti
ISBN : 354055534X
Genre : Mathematics
File Size : 32. 41 MB
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The core of the book is the study of the space of the hyperbolic manifolds endowed with the Chabauty and the geometric topology, and in particular the proof of the hypberbolic surgery theorem in dimension three, based on the representation of three-mainfolds as glued ideal tetrahedra. The development of this main theme requires setting a wide background forming the body of the book: the classical geometry of the hyperbolic space, the Fenchel-Nielsen parametrization of the Teichmüller space, Mostow's rigidity theorem, Margulis' lemma. As a conclusion some features of bounded cohomology, flat fiber bundles and amenable groups are mentioned.

A Version Of The Infinite Dimensional Borsuk Ulam Theorem For Multivalued Maps

Author :
ISBN : OCLC:1051872944
Genre :
File Size : 25. 79 MB
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Abstract: This paper is devoted to the proof of the infinite-dimensional Borsuk-Ulam theorem for odd completely continuous multivalued maps with convex images which are defined on level sets of even functions. The results obtained in the paper are new even for single-valued maps. In the final section some applications of the theorem to analysis and differential equations are discussed. Bibliography: 12 titles.

Combinatorics

Author : Source Wikipedia
ISBN : 1230563512
Genre :
File Size : 40. 94 MB
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 146. Chapters: Factorial, Binomial coefficient, Permutation, Combination, Pigeonhole principle, Catalan's constant, Borsuk-Ulam theorem, Partition, Random permutation statistics, Longest common subsequence problem, Twelvefold way, Permanent is sharp-P-complete, Aanderaa-Karp-Rosenberg conjecture, Butcher group, Oriented matroid, Theory of relations, Permutation pattern, Bent function, Algorithmic Lovasz local lemma, Finite topological space, Enumerations of specific permutation classes, Symbolic combinatorics, Combinatorial number system, Factorial number system, Group testing, Percolation theory, Sparse ruler, Uniform convergence, History of combinatorics, Arrangement of hyperplanes, Longest increasing subsequence, Graph dynamical system, Isolation lemma, Lottery mathematics, Stable roommates problem, Trace monoid, Dividing a circle into areas, Probabilistic method, Sicherman dice, Josephus problem, 3-dimensional matching, No-three-in-line problem, Outline of combinatorics, Inversion, Domino tiling, Set packing, Combinatorics and physics, Sperner's lemma, Partition of a set, Incidence matrix, Index of combinatorics articles, Szemeredi's theorem, Umbral calculus, Natural density, Constraint counting, Multi-index notation, Barycentric-sum problem, Road coloring problem, Difference set, Stars and bars, Sidon sequence, Singmaster's conjecture, Meander, Coupon collector's problem, Rota-Baxter algebra, Combinatorial principles, Small set, Infinitary combinatorics, Laver table, Sequential dynamical system, De Arte Combinatoria, Incidence structure, Method of distinguished element, Dobinski's formula, Pascal's rule, Mnev's universality theorem, All-pairs testing, Combinatorial explosion, Composition, Free convolution, Bruck-Chowla-Ryser theorem, Topological combinatorics, Lubell-Yamamoto-Meshalkin inequality, Block...

Computing Equilibria And Fixed Points

Author : Zaifu Yang
ISBN : 0792383958
Genre : Business & Economics
File Size : 74. 11 MB
Format : PDF, ePub
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Computing Equilibria and Fixed Points is devoted to the computation of equilibria, fixed points and stationary points. This volume is written with three goals in mind: (i) To give a comprehensive introduction to fixed point methods and to the definition and construction of Gröbner bases; (ii) To discuss several interesting applications of these methods in the fields of general equilibrium theory, game theory, mathematical programming, algebra and symbolic computation; (iii) To introduce several advanced fixed point and stationary point theorems. These methods and topics should be of interest not only to economists and game theorists concerned with the computation and existence of equilibrium outcomes in economic models and cooperative and non-cooperative games, but also to applied mathematicians, computer scientists and engineers dealing with models of highly nonlinear systems of equations (or polynomial equations).

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