Perfect Lattices In Euclidean Spaces
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| Author | : Jacques Martinet |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 535 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 3662051672 |
Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3. This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property. Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290. Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices.
| Author | : Boris Nikolaevich Delone |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 220 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 9780821831472 |
This collection of papers honors the 100th anniversary of the birth of Boris Nikolaevich Delone, whose mathematical interests centered on the geometry of positive quadratic forms. After an initial paper presenting an account of Delone's life, including his scientific work, the book centers on discrete geometry and combinatorics. The book presents new methods that permit a description of the structure of some $L$-bodies and $L$-partitionings and that, in many cases, provide a definitive description. Also studied are combinatorial-topological problems arising in the statistical Ising model, the disposition of finite point sets in convex bodies of high dimension under certain conditions, and investigations of regular partitionings of spaces of constant curvature.
| Author | : Achill Schurmann |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 183 |
| Release | : 2009 |
| Genre | : Mathematics |
| ISBN | : 082184735X |
"Starting from classical arithmetical questions on quadratic forms, this book takes the reader step by step through the connections with lattice sphere packing and covering problems. As a model for polyhedral reduction theories of positive definite quadratic forms, Minkowski's classical theory is presented, including an application to multidimensional continued fraction expansions. The reduction theories of Voronoi are described in great detail, including full proofs, new views, and generalizations that cannot be found elsewhere. Based on Voronoi's second reduction theory, the local analysis of sphere coverings and several of its applications are presented. These include the classification of totally real thin number fields, connections to the Minkowski conjecture, and the discovery of new, sometimes surprising, properties of exceptional structures such as the Leech lattice or the root lattices." "Throughout this book, special attention is paid to algorithms and computability, allowing computer-assisted treatments. Although dealing with relatively classical topics that have been worked on extensively by numerous authors, this book is exemplary in showing how computers may help to gain new insights."--BOOK JACKET.
| Author | : Wai Kiu Chan |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 259 |
| Release | : 2013 |
| Genre | : Mathematics |
| ISBN | : 0821883186 |
This volume contains the proceedings of the International Workshop on Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms. The articles cover the arithmetic theory of quadratic forms and lattices, as well as the effective Diophantine analysis with height functions.
| Author | : Ricardo Baeza |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 424 |
| Release | : 2009-08-14 |
| Genre | : Mathematics |
| ISBN | : 0821846485 |
This volume presents a collection of articles that are based on talks delivered at the International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms held in Frutillar, Chile in December 2007. The theory of quadratic forms is closely connected with a broad spectrum of areas in algebra and number theory. The articles in this volume deal mainly with questions from the algebraic, geometric, arithmetic, and analytic theory of quadratic forms, and related questions in algebraic group theory and algebraic geometry.
| Author | : Guillaume Hanrot |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 407 |
| Release | : 2010-07-07 |
| Genre | : Computers |
| ISBN | : 3642145175 |
This book constitutes the refereed proceedings of the 9th International Algorithmic Number Theory Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The papers are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography.
| Author | : Murray R. Bremner |
| Publisher | : CRC Press |
| Total Pages | : 336 |
| Release | : 2011-08-12 |
| Genre | : Computers |
| ISBN | : 1439807027 |
First developed in the early 1980s by Lenstra, Lenstra, and Lovász, the LLL algorithm was originally used to provide a polynomial-time algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.
| Author | : Matthias Beck |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 202 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0821841734 |
"The AMS-IMS-SIAM Joint Summer Research Conference "Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics" was held in Snowbird, Utah in June 2006. This proceedings volume contains research and survey articles originating from the conference. The volume is a cross section of recent advances connected to lattice-point questions. Similar to the talks given at the conference, topics range from commutative algebra to optimization, from discrete geometry to statistics, from mirror symmetry to geometry of numbers. The book is suitable for researchers and graduate students interested in combinatorial aspects of the above fields." -- Back cover.
| Author | : Larry J. Gerstein |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 274 |
| Release | : 2008 |
| Genre | : Mathematics |
| ISBN | : 0821844652 |
The arithmetic theory of quadratic forms is a rich branch of number theory that has had important applications to several areas of pure mathematics--particularly group theory and topology--as well as to cryptography and coding theory. This book is a self-contained introduction to quadratic forms that is based on graduate courses the author has taught many times. It leads the reader from foundation material up to topics of current research interest--with special attention to the theory over the integers and over polynomial rings in one variable over a field--and requires only a basic background in linear and abstract algebra as a prerequisite. Whenever possible, concrete constructions are chosen over more abstract arguments. The book includes many exercises and explicit examples, and it is appropriate as a textbook for graduate courses or for independent study. To facilitate further study, a guide to the extensive literature on quadratic forms is provided.
| Author | : Alf J. van der Poorten |
| Publisher | : Springer |
| Total Pages | : 463 |
| Release | : 2008-05-07 |
| Genre | : Computers |
| ISBN | : 3540794565 |
This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008. The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.
