On Eutactic Forms

Ash, Avner

doi:10.4153/cjm-1977-101-2

Cited by 28 publications

(25 citation statements)

References 6 publications

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“…In view of the results on spherical designs by Goethals and Seidel [17], Bachoc and Venkov [3] and others, proposition (1) shows that many special lattices in the literature are strongly eutactic, or all their layers (with obvious normalization) are strongly eutactic.…”

Section: Minimum Properties Of ζmentioning

confidence: 98%

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Application of an idea of Voronoĭ to lattice zeta functions

Gruber

2012

Proc. Steklov Inst. Math.

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A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoȋ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoȋ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.

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Section: Minimum Properties Of ζmentioning

confidence: 98%

“…Recent contributions are due to Sarnak and Strömbergsson [34] and Coulangeon [11]. Ash [1] showed that the density of lattice packings of balls is a Morse function. Bergé and Martinet [7] studied duality problems for the density.…”

Section: Introductionmentioning

confidence: 99%

Application of an idea of Voronoĭ to lattice zeta functions

Gruber

2012

Proc. Steklov Inst. Math.

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“…The best packing, in fact, is both a perfect and eutactic lattice (we give the characterization of these lattices later) and both the number of perfect 13,14 and that of eutactic lattices is finite 15 (hence the intersection is). This algorithm has been applied to dimensions d ≤ 8 to systematically find all such lattices [16][17][18][19][20][21][22] .…”

Section: Introductionmentioning

confidence: 99%

Random perfect lattices and the sphere packing problem

Andreanov

Scardicchio

2012

Phys. Rev. E

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Motivated by the search for best lattice sphere packings in Euclidean spaces of large dimensions we study randomly generated perfect lattices in moderately large dimensions (up to d = 19 included). Perfect lattices are relevant in the solution of the problem of lattice sphere packing, because the best lattice packing is a perfect lattice and because they can be generated easily by an algorithm. Their number however grows super-exponentially with the dimension so to get an idea of their properties we propose to study a randomized version of the algorithm and to define a random ensemble with an effective temperature in a way reminiscent of a Montecarlo simulation. We therefore study the distribution of packing fractions and kissing numbers of these ensembles and show how as the temperature is decreased the best know packers are easily recovered. We find that, even at infinite temperature, the typical perfect lattices are considerably denser than known families (like A d and D d ) and we propose two hypotheses between which we cannot distinguish in this paper: one in which they improve Minkowsky's bound φ ∼ 2 −(0.84±0.06)d , and a competitor, in which their packing fraction decreases super-exponentially, namely φ ∼ d −ad but with a very small coefficient a = 0.06 ± 0.04. We also find properties of the random walk which are suggestive of a glassy system already for moderately small dimensions. We also analyze local structure of network of perfect lattices conjecturing that this is a scale-free network in all dimensions with constant scaling exponent 2.6 ± 0.1.

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“…They provide perfect analogues to classical problems of lattice sphere packings. An important fact (see section 5) is also that the function syst which associates to every surface the length of its systole is a topological Morse function on Teichmüller space, equivariant with respect to the mapping class group ( [90]), again in analogy to the packing function for lattice sphere packings (Ash [4]). …”

Section: Introductionmentioning

confidence: 99%

Untitled

1998

Bull. Amer. Math. Soc.

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Abstract. The paper presents a survey on recent results on the geometry of Riemann surfaces showing that the study of closed geodesics provides a link between different aspects of Riemann surface theory such as hyperbolic geometry, topology, spectral theory, and the theory of arithmetic Fuchsian groups. Of particular interest are the systoles, the shortest closed geodesics of a surface; their study leads to the hyperbolic geometry of numbers with close analogues to classical sphere packing problems.

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On Eutactic Forms

Cited by 28 publications

References 6 publications

Application of an idea of Voronoĭ to lattice zeta functions

Application of an idea of Voronoĭ to lattice zeta functions

Random perfect lattices and the sphere packing problem

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