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24
Asymmetric multiple description lattice vector quantizers
 IEEE Trans. Inf. Theory
, 2002
"... Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, a ..."
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Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The highrate asymptotic performance of the quantizer is also studied. We evaluate the ratedistortion performance of the quantizer and compare it to known informationtheoretic bounds. The highrate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, highrate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.
Experimental study of energyminimizing point configurations on spheres
, 2006
"... Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as eviden ..."
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Cited by 18 (6 self)
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Abstract. In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima. [T]he problem of finding the configurations of stable equilibrium for a number of equal particles acting on each other according to some law of force...is of great interest in connexion with the relation between the properties of an element and its atomic weight. Unfortunately the equations which determine the stability of such a collection of particles increase so rapidly in complexity with the number of particles that a general mathematical investigation is scarcely possible.
LowDimensional Lattices VII: Coordination Sequences
 Proc. Royal Soc. A453
, 1996
"... The coordination sequence fS(n)g of a lattice or net gives the number of nodes that are n bonds away from a given node. S(1) is the familiar coordination number. Extending work of O'Keeffe and others, we give explicit formulae for the coordination sequences of the root lattices A d , D d , E 6 ..."
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The coordination sequence fS(n)g of a lattice or net gives the number of nodes that are n bonds away from a given node. S(1) is the familiar coordination number. Extending work of O'Keeffe and others, we give explicit formulae for the coordination sequences of the root lattices A d , D d , E 6 , E 7 , E 8 and their duals. Proofs are given for many of the formulae, and for the fact that in every case S(n) is a polynomial in n, although some of the individual formulae are conjectural. In the majority of cases the set of nodes that are at most n bonds away from a given node form a polytopal cluster whose shape is the same as that of the contact polytope for the lattice. It is also shown that among all the Barlow packings in three dimensions the hexagonal close packing has the greatest coordination sequence, and the facecentered cubic lattice the smallest, as conjectured by O'Keeffe. 1. Introduction The coordination sequence of an infinite vertextransitive graph G is the sequence fS...
Complexity and algorithms for computing Voronoi cells of lattices
, 2009
"... In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #Phard problem. On the other hand, we describe an algorithm for this problem which is especially suited for lowdimens ..."
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In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #Phard problem. On the other hand, we describe an algorithm for this problem which is especially suited for lowdimensional (say dimensions at most 12) and for highlysymmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.
The Sphere Packing Problem
, 1998
"... . A brief report on recent work on the spherepacking problem. 1991 Mathematics Subject Classification: 52C17 Keywords and Phrases: Sphere packings; lattices; quadratic forms; geometry of numbers 1 Introduction The sphere packing problem has its roots in geometry and number theory (it is part of Hi ..."
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Cited by 16 (0 self)
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. A brief report on recent work on the spherepacking problem. 1991 Mathematics Subject Classification: 52C17 Keywords and Phrases: Sphere packings; lattices; quadratic forms; geometry of numbers 1 Introduction The sphere packing problem has its roots in geometry and number theory (it is part of Hilbert's 18th problem), but is also a fundamental question in information theory. The connection is via the sampling theorem. As Shannon observes in his classic 1948 paper [37] (which ushered in the age of digital communication), if f is a signal of bandwidth W hertz, with almost all its energy concentrated in an interval of T secs, then f is accurately represented by a vector of 2WT samples, which may be regarded as the coordinates of a single point in R n , n = 2WT . Nearly equal signals are represented by neighboring points, so to keep the signals distinct, Shannon represents them by ndimensional `billiard balls', and is therefore led to ask: what is the best way to pack `billiard balls...
Hermitian vector bundles and extension groups on arithmetic schemes II. THE ARITHMETIC ATIYAH EXTENSION
, 2008
"... In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a ..."
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In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element — the arithmetic Atiyah class — in a suitable arithmetic extension group. Namely, if E is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class bat X/Z(E) lies in the group d Ext 1 X(E, E ⊗ Ω1), and is an obstruction to the algebraicity over X of the X/Z unitary connection on the vector bundle EC over the complex manifold X(C) that is compatible with its holomorphic structure. In the first sections of this article, we develop the basic properties of the arithmetic Atiyah class which can be used to define characteristic classes in arithmetic Hodge cohomology. Then we study the vanishing of the first Chern class ĉH 1 (L) of a hermitian line bundle L in the arithmetic Hodge cohomology group d Ext 1
Hypermetrics in Geometry of Numbers
, 1993
"... . A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connection ..."
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Cited by 11 (4 self)
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. A finite semimetric d on a set X is hypermetric if it satisfies the inequality P i;j2X b i b j d ij 0 for all b 2 Z X with P i2X b i = 1. Hypermetricity turns out to be the appropriate notion for describing the metric structure of holes in lattices. We survey hypermetrics, their connections with lattices and applications. 2 M. Deza, V.P. Grishukhin and M. Laurent Contents 1 Introduction 2 Preliminaries 2.1 Distance spaces Metric notions Operations on distance spaces Preliminary results on distance spaces 2.2 Lattices and Lpolytopes Lattices Lpolytopes Lpolytopes and Voronoi polytopes Lattices and positive quadratic forms Lpolytopes and empty ellipsoids Basic facts on Lpolytopes Construction of Lpolytopes Lpolytopes in dimension k 4 2.3 Finiteness of the number of types of Lpolytopes in given dimension 3 Hypermetrics and Lpolytopes 3.1 The connection between hypermetrics and Lpolytopes 3.2 Polyhedrality of the hypermetric cone 3.3 Lpolytopes in root lattic...
Classification of integrable discrete equations of octahedron type
 INT. MATH. RES. NOT. IMRN
, 2010
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The affine Weyl group symmetry of Desargues maps and of the noncommutative HirotaMiwa system
 Phys. Lett. A
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