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Author : Chuanming Zong
Publisher : Cambridge University Press
Page : 196 pages
File Size : 52,6 Mb
Release : 2006-02-02
Category : Mathematics
ISBN : 0521855357
Analysis, Algebra, Combinatorics, Graph Theory, Hyperbolic Geometry, Number Theory.
Author : Chuanming Zong
Publisher : Cambridge University Press
Page : 162 pages
File Size : 54,9 Mb
Release : 2006-02-02
Category : Mathematics
ISBN : 9781139448642
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 : Károly Bezdek
Publisher : Springer Science & Business Media
Page : 166 pages
File Size : 43,8 Mb
Release : 2010-06-23
Category : Mathematics
ISBN : 9781441906007
Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics. In this way, the works of a number of outstanding mathema- cians including H. S. M. Coxeter (Canada), C. A. Rogers (United Kingdom), and L. Fejes-T oth (Hungary) led to the new and fast developing eld called discrete geometry. One can brie y describe this branch of geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. This, as a classical core part, also includes the theory of polytopes and tilings in addition to the theory of packing and covering. D- crete geometry is driven by problems often featuring a very clear visual and applied character. The solutions use a variety of methods of modern mat- matics, including convex and combinatorial geometry, coding theory, calculus of variations, di erential geometry, group theory, and topology, as well as geometric analysis and number theory.
Author : Daniel Hug,Wolfgang Weil
Publisher : Springer Nature
Page : 287 pages
File Size : 43,6 Mb
Release : 2020-08-27
Category : Mathematics
ISBN : 9783030501808
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 : 48,8 Mb
Release : 2014
Category : Mathematics
ISBN : 9781107601017
A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.
Author : Barry Simon
Publisher : Cambridge University Press
Page : 357 pages
File Size : 47,7 Mb
Release : 2011-05-19
Category : Mathematics
ISBN : 9781139497596
Convexity is important in theoretical aspects of mathematics and also for economists and physicists. In this monograph the author provides a comprehensive insight into convex sets and functions including the infinite-dimensional case and emphasizing the analytic point of view. Chapter one introduces the reader to the basic definitions and ideas that play central roles throughout the book. The rest of the book is divided into four parts: convexity and topology on infinite-dimensional spaces; Loewner's theorem; extreme points of convex sets and related issues, including the Krein–Milman theorem and Choquet theory; and a discussion of convexity and inequalities. The connections between disparate topics are clearly explained, giving the reader a thorough understanding of how convexity is useful as an analytic tool. A final chapter overviews the subject's history and explores further some of the themes mentioned earlier. This is an excellent resource for anyone interested in this central topic.
Author : Sinai Robins
Publisher : American Mathematical Society
Page : 352 pages
File Size : 42,5 Mb
Release : 2024-04-24
Category : Mathematics
ISBN : 9781470470333
This book offers a gentle introduction to the geometry of numbers from a modern Fourier-analytic point of view. One of the main themes is the transfer of geometric knowledge of a polytope to analytic knowledge of its Fourier transform. The Fourier transform preserves all of the information of a polytope, and turns its geometry into analysis. The approach is unique, and streamlines this emerging field by presenting new simple proofs of some basic results of the field. In addition, each chapter is fitted with many exercises, some of which have solutions and hints in an appendix. Thus, an individual learner will have an easier time absorbing the material on their own, or as part of a class. Overall, this book provides an introduction appropriate for an advanced undergraduate, a beginning graduate student, or researcher interested in exploring this important expanding field.
Author : Cho-Ho Chu
Publisher : Cambridge University Press
Page : 273 pages
File Size : 44,8 Mb
Release : 2011-11-17
Category : Mathematics
ISBN : 9781139505437
Jordan theory has developed rapidly in the last three decades, but very few books describe its diverse applications. Here, the author discusses some recent advances of Jordan theory in differential geometry, complex and functional analysis, with the aid of numerous examples and concise historical notes. These include: the connection between Jordan and Lie theory via the Tits–Kantor–Koecher construction of Lie algebras; a Jordan algebraic approach to infinite dimensional symmetric manifolds including Riemannian symmetric spaces; the one-to-one correspondence between bounded symmetric domains and JB*-triples; and applications of Jordan methods in complex function theory. The basic structures and some functional analytic properties of JB*-triples are also discussed. The book is a convenient reference for experts in complex geometry or functional analysis, as well as an introduction to these areas for beginning researchers. The recent applications of Jordan theory discussed in the book should also appeal to algebraists.
Author : Kalyan B. Sinha,Debashish Goswami
Publisher : Cambridge University Press
Page : 301 pages
File Size : 48,5 Mb
Release : 2007-01-25
Category : Mathematics
ISBN : 9781139461696
The classical theory of stochastic processes has important applications arising from the need to describe irreversible evolutions in classical mechanics; analogously quantum stochastic processes can be used to model the dynamics of irreversible quantum systems. Noncommutative, i.e. quantum, geometry provides a framework in which quantum stochastic structures can be explored. This book is the first to describe how these two mathematical constructions are related. In particular, key ideas of semigroups and complete positivity are combined to yield quantum dynamical semigroups (QDS). Sinha and Goswami also develop a general theory of Evans-Hudson dilation for both bounded and unbounded coefficients. The unique features of the book, including the interaction of QDS and quantum stochastic calculus with noncommutative geometry and a thorough discussion of this calculus with unbounded coefficients, will make it of interest to graduate students and researchers in functional analysis, probability and mathematical physics.
Author : Sergei Kuksin,Armen Shirikyan
Publisher : Cambridge University Press
Page : 337 pages
File Size : 48,7 Mb
Release : 2012-09-20
Category : Mathematics
ISBN : 9781139576956
This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D Navier–Stokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) – proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.
Author : Horst Osswald
Publisher : Cambridge University Press
Page : 429 pages
File Size : 45,6 Mb
Release : 2012-03
Category : Mathematics
ISBN : 9781107016149
After functional, measure and stochastic analysis prerequisites, the author covers chaos decomposition, Skorohod integral processes, Malliavin derivative and Girsanov transformations.
Author : Bas Lemmens,Roger Nussbaum
Publisher : Cambridge University Press
Page : 337 pages
File Size : 42,9 Mb
Release : 2012-05-03
Category : Mathematics
ISBN : 9780521898812
Guides the reader through the nonlinear Perron-Frobenius theory, introducing them to recent developments and challenging open problems.
Author : Ivan Nourdin,Giovanni Peccati
Publisher : Cambridge University Press
Page : 255 pages
File Size : 40,6 Mb
Release : 2012-05-10
Category : Mathematics
ISBN : 9781107017771
This book shows how quantitative central limit theorems can be deduced by combining two powerful probabilistic techniques: Stein's method and Malliavin calculus.
Author : Yann Bugeaud
Publisher : Cambridge University Press
Page : 317 pages
File Size : 41,6 Mb
Release : 2012-07-05
Category : Mathematics
ISBN : 9780521111690
A treatment of cutting-edge research on the distribution modulo one of sequences and related topics, much of it from the last decade. There are numerous exercises to aid student understanding of the topic, and researchers will appreciate the notes at the end of each chapter, extensive references and open problems.
Author : A. Ivić
Publisher : Cambridge University Press
Page : 265 pages
File Size : 49,6 Mb
Release : 2013
Category : Mathematics
ISBN : 9781107028838
A comprehensive account of Hardy's Z-function, one of the most important functions of analytic number theory.
