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Author: Jacques Martinet
Publisher: Springer Science & Business Media
ISBN: 3662051672
Category : Mathematics
Languages : en
Pages : 535
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Book Description
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: Jacques Martinet
Publisher: Springer Science & Business Media
ISBN: 3662051672
Category : Mathematics
Languages : en
Pages : 535
Get Book Here
Book Description
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: Heinz-Georg Quebbemann
Publisher:
ISBN:
Category :
Languages : de
Pages : 10
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Book Description
Author: H.-G. Quebbemann
Publisher:
ISBN:
Category :
Languages : en
Pages : 10
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Book Description
Author: Maria Teider
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
Author: Gaëtan Chenevier
Publisher: Springer
ISBN: 3319958917
Category : Mathematics
Languages : en
Pages : 417
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Book Description
This book includes a self-contained approach of the general theory of quadratic forms and integral Euclidean lattices, as well as a presentation of the theory of automorphic forms and Langlands' conjectures, ranging from the first definitions to the recent and deep classification results due to James Arthur. Its connecting thread is a question about lattices of rank 24: the problem of p-neighborhoods between Niemeier lattices. This question, whose expression is quite elementary, is in fact very natural from the automorphic point of view, and turns out to be surprisingly intriguing. We explain how the new advances in the Langlands program mentioned above pave the way for a solution. This study proves to be very rich, leading us to classical themes such as theta series, Siegel modular forms, the triality principle, L-functions and congruences between Galois representations. This monograph is intended for any mathematician with an interest in Euclidean lattices, automorphic forms or number theory. A large part of it is meant to be accessible to non-specialists.
Author: Fumitomo Maeda
Publisher: Springer Science & Business Media
ISBN: 3642462480
Category : Mathematics
Languages : en
Pages : 204
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Book Description
Of central importance in this book is the concept of modularity in lattices. A lattice is said to be modular if every pair of its elements is a modular pair. The properties of modular lattices have been carefully investigated by numerous mathematicians, including 1. von Neumann who introduced the important study of continuous geometry. Continu ous geometry is a generalization of projective geometry; the latter is atomistic and discrete dimensional while the former may include a continuous dimensional part. Meanwhile there are many non-modular lattices. Among these there exist some lattices wherein modularity is symmetric, that is, if a pair (a,b) is modular then so is (b,a). These lattices are said to be M-sym metric, and their study forms an extension of the theory of modular lattices. An important example of an M-symmetric lattice arises from affine geometry. Here the lattice of affine sets is upper continuous, atomistic, and has the covering property. Such a lattice, called a matroid lattice, can be shown to be M-symmetric. We have a deep theory of parallelism in an affine matroid lattice, a special kind of matroid lattice. Further more we can show that this lattice has a modular extension.
Author: Wai Kiu Chan
Publisher: American Mathematical Soc.
ISBN: 0821883186
Category : Mathematics
Languages : en
Pages : 259
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Book Description
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: Peter Jipsen
Publisher: Springer
ISBN: 3540475141
Category : Mathematics
Languages : en
Pages : 171
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Book Description
The study of lattice varieties is a field that has experienced rapid growth in the last 30 years, but many of the interesting and deep results discovered in that period have so far only appeared in research papers. The aim of this monograph is to present the main results about modular and nonmodular varieties, equational bases and the amalgamation property in a uniform way. The first chapter covers preliminaries that make the material accessible to anyone who has had an introductory course in universal algebra. Each subsequent chapter begins with a short historical introduction which sites the original references and then presents the results with complete proofs (in nearly all cases). Numerous diagrams illustrate the beauty of lattice theory and aid in the visualization of many proofs. An extensive index and bibliography also make the monograph a useful reference work.
Author: Ralph S. Freese
Publisher: American Mathematical Soc.
ISBN: 0821821814
Category : Mathematics
Languages : en
Pages : 103
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Book Description
A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let [capital script]M [infinity symbol] [over][subscript italic]n denote the lattice variety generated by all modular lattices of width not exceeding [subscript italic]n. [capital script]M [infinity symbol] [over]1 and [capital script]M [infinity symbol] [over]2 are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that [capital script]M [infinity symbol] [over]3 is also finitely based. On the other hand, K. Baker has shown that [capital script]M [infinity symbol] [over][subscript italic]n is not finitely based for 5 [less than or equal to symbol] [italic]n [less than] [lowercase Greek]Omega. This paper settles the finite bases problem for [capital script]M [infinity symbol] [over]4.
Author: Myung-Hwan Kim
Publisher: American Mathematical Soc.
ISBN: 0821819496
Category : Mathematics
Languages : en
Pages : 314
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Book Description
This volume presents the proceedings of an international conference held at Seoul National University (Korea). Talks covered recent developments in diverse areas related to the theory of integral quadratic forms and hermitian forms, local densities, linear relations and congruences of theta series, zeta functions of prehomogeneous vector spaces, lattices with maximal finite matrix groups, globally irreducible lattices, Mordell-Weil lattices, and more. Articles in the volume represent expository lectures by leading experts on recent developments in the field. The book offers a comprehensive introduction to the current state of knowledge in the arithmetic theory of quadratic forms and provides active directions of research with new results. Topics addressed in the volume emphasize connections with related fields, such as group theory, arithmetic geometry, analytic number theory, and modular forms. The book is an excellent introductory guide for students as well as a rich reference source for researchers.
