Strange Phenomena In Convex And Discrete Geometry

Author : Chuanming Zong
Publisher : Springer Science & Business Media
Page : 167 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461384816

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Strange Phenomena in Convex and Discrete Geometry by Chuanming Zong Pdf

Convex and discrete geometry is one of the most intuitive subjects in mathematics. One can explain many of its problems, even the most difficult - such as the sphere-packing problem (what is the densest possible arrangement of spheres in an n-dimensional space?) and the Borsuk problem (is it possible to partition any bounded set in an n-dimensional space into n+1 subsets, each of which is strictly smaller in "extent" than the full set?) - in terms that a layman can understand; and one can reasonably make conjectures about their solutions with little training in mathematics.

Author : Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher : CRC Press
Page : 1557 pages
File Size : 51,8 Mb
Release : 2004-04-13
Category : Mathematics
ISBN : 9781420035315

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Handbook of Discrete and Computational Geometry, Second Edition by Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman Pdf

While high-quality books and journals in this field continue to proliferate, none has yet come close to matching the Handbook of Discrete and Computational Geometry, which in its first edition, quickly became the definitive reference work in its field. But with the rapid growth of the discipline and the many advances made over the past seven years, it's time to bring this standard-setting reference up to date. Editors Jacob E. Goodman and Joseph O'Rourke reassembled their stellar panel of contributors, added manymore, and together thoroughly revised their work to make the most important results and methods, both classic and cutting-edge, accessible in one convenient volume. Now over more then 1500 pages, the Handbook of Discrete and Computational Geometry, Second Edition once again provides unparalleled, authoritative coverage of theory, methods, and applications. Highlights of the Second Edition: Thirteen new chapters: Five on applications and others on collision detection, nearest neighbors in high-dimensional spaces, curve and surface reconstruction, embeddings of finite metric spaces, polygonal linkages, the discrepancy method, and geometric graph theory Thorough revisions of all remaining chapters Extended coverage of computational geometry software, now comprising two chapters: one on the LEDA and CGAL libraries, the other on additional software Two indices: An Index of Defined Terms and an Index of Cited Authors Greatly expanded bibliographies

Author : Chuanming Zong
Publisher : Cambridge University Press
Page : 162 pages
File Size : 46,6 Mb
Release : 2006-02-02
Category : Mathematics
ISBN : 9781139448642

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The Cube-A Window to Convex and Discrete Geometry by Chuanming Zong Pdf

This tract has two purposes: to show what is known about the n-dimensional unit cubes and to demonstrate how Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory, can be applied to the study of them. The unit cubes, from any point of view, are among the most important and fascinating objects in an n-dimensional Euclidean space. However, our knowledge about them is still quite limited and many basic problems remain unsolved. In this Tract eight topics about the unit cubes are introduced: cross sections, projections, inscribed simplices, triangulations, 0/1 polytopes, Minkowski's conjecture, Furtwangler's conjecture, and Keller's conjecture. In particular the author demonstrates how deep analysis like log concave measure and the Brascamp-Lieb inequality can deal with the cross section problem, how Hyperbolic Geometry helps with the triangulation problem, how group rings can deal with Minkowski's conjecture and Furtwangler's conjecture, and how Graph Theory handles Keller's conjecture.

Author : Daniel Hug,Wolfgang Weil
Publisher : Springer Nature
Page : 287 pages
File Size : 53,7 Mb
Release : 2020-08-27
Category : Mathematics
ISBN : 9783030501808

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Lectures on Convex Geometry by Daniel Hug,Wolfgang Weil Pdf

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Author : Rolf Schneider
Publisher : Cambridge University Press
Page : 759 pages
File Size : 42,7 Mb
Release : 2014
Category : Mathematics
ISBN : 9781107601017

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Convex Bodies: The Brunn–Minkowski Theory by Rolf Schneider Pdf

A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

Author : Gergely Ambrus,Imre Bárány,Károly J. Böröczky,Gábor Fejes Tóth,János Pach
Publisher : Springer
Page : 458 pages
File Size : 41,9 Mb
Release : 2018-11-03
Category : Mathematics
ISBN : 9783662574133

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New Trends in Intuitive Geometry by Gergely Ambrus,Imre Bárány,Károly J. Böröczky,Gábor Fejes Tóth,János Pach Pdf

This volume contains 17 surveys that cover many recent developments in Discrete Geometry and related fields. Besides presenting the state-of-the-art of classical research subjects like packing and covering, it also offers an introduction to new topological, algebraic and computational methods in this very active research field. The readers will find a variety of modern topics and many fascinating open problems that may serve as starting points for research.

Author : Silouanos Brazitikos,Apostolos Giannopoulos,Petros Valettas,Beatrice-Helen Vritsiou
Publisher : American Mathematical Soc.
Page : 618 pages
File Size : 41,8 Mb
Release : 2014-04-24
Category : Mathematics
ISBN : 9781470414566

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Geometry of Isotropic Convex Bodies by Silouanos Brazitikos,Apostolos Giannopoulos,Petros Valettas,Beatrice-Helen Vritsiou Pdf

The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

Author : Alexander Kharazishvili
Publisher : CRC Press
Page : 397 pages
File Size : 54,7 Mb
Release : 2024-05-07
Category : Mathematics
ISBN : 9781040014264

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Introduction to Combinatorial Methods in Geometry by Alexander Kharazishvili Pdf

This book offers an introduction to some combinatorial (also, set-theoretical) approaches and methods in geometry of the Euclidean space Rm. The topics discussed in the manuscript are due to the field of combinatorial and convex geometry. The author’s primary intention is to discuss those themes of Euclidean geometry which might be of interest to a sufficiently wide audience of potential readers. Accordingly, the material is explained in a simple and elementary form completely accessible to the college and university students. At the same time, the author reveals profound interactions between various facts and statements from different areas of mathematics: the theory of convex sets, finite and infinite combinatorics, graph theory, measure theory, classical number theory, etc. All chapters (and also the five Appendices) end with a number of exercises. These provide the reader with some additional information about topics considered in the main text of this book. Naturally, the exercises vary in their difficulty. Among them there are almost trivial, standard, nontrivial, rather difficult, and difficult. As a rule, more difficult exercises are marked by asterisks and are provided with necessary hints. The material presented is based on the lecture course given by the author. The choice of material serves to demonstrate the unity of mathematics and variety of unexpected interrelations between distinct mathematical branches.

Author : Joseph O’Rourke
Publisher : Springer Nature
Page : 245 pages
File Size : 54,5 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9783031475115

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Reshaping Convex Polyhedra by Joseph O’Rourke Pdf

Author : Richard A. Holmgren
Publisher : Springer Science & Business Media
Page : 231 pages
File Size : 54,8 Mb
Release : 2012-09-05
Category : Mathematics
ISBN : 9781441987327

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A First Course in Discrete Dynamical Systems by Richard A. Holmgren Pdf

Given the ease with which computers can do iteration it is now possible for almost anyone to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Mathematica programs that illustrate the dynamics are included in an appendix.

Author : Jürgen Jost
Publisher : Springer Science & Business Media
Page : 460 pages
File Size : 51,9 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662223857

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Riemannian Geometry and Geometric Analysis by Jürgen Jost Pdf

FROM REVIEWS OF THE FIRST EDITION "a very readable introduction to Riemannian geometry...it is most welcome...The book is made more interesting by the perspectives in various sections, where the author mentions the history and development of the material and provides the reader with references."-MATHEMATICAL REVIEWS

Author : Marcel Berger
Publisher : Springer Science & Business Media
Page : 840 pages
File Size : 43,5 Mb
Release : 2010-07-23
Category : Mathematics
ISBN : 9783540709978

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Geometry Revealed by Marcel Berger Pdf

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Author : Jong-Shi Pang
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 45,8 Mb
Release : 2012-12-06
Category : Business & Economics
ISBN : 9781461551973

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Computational Optimization by Jong-Shi Pang Pdf

Computational Optimization: A Tribute to Olvi Mangasarian serves as an excellent reference, providing insight into some of the most challenging research issues in the field. This collection of papers covers a wide spectrum of computational optimization topics, representing a blend of familiar nonlinear programming topics and such novel paradigms as semidefinite programming and complementarity-constrained nonlinear programs. Many new results are presented in these papers which are bound to inspire further research and generate new avenues for applications. An informal categorization of the papers includes: Algorithmic advances for special classes of constrained optimization problems Analysis of linear and nonlinear programs Algorithmic advances B- stationary points of mathematical programs with equilibrium constraints Applications of optimization Some mathematical topics Systems of nonlinear equations.

Author : Chuanming Zong
Publisher : Springer Science & Business Media
Page : 245 pages
File Size : 42,5 Mb
Release : 2008-01-20
Category : Mathematics
ISBN : 9780387227801

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Sphere Packings by Chuanming Zong Pdf

Sphere packings is one of the most fascinating and challenging subjects in mathematics. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. This book gives a full account of this fascinating subject, especially its local aspects, discrete aspects, and its proof methods. The book includes both classical and contemporary results and provides a full treatment of the subject.

Author : James Blowey,John P. Coleman,Alan W. Craig
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 47,6 Mb
Release : 2001-08-28
Category : Mathematics
ISBN : 3540418466

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Theory and Numerics of Differential Equations by James Blowey,John P. Coleman,Alan W. Craig Pdf

A compilation of detailed lecture notes on six topics at the forefront of current research in numerical analysis and applied mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to quickly gain an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research is given.