Locally minimal Epstein zeta functions

Fields, K. L.

doi:10.1112/s002557930000989x

Cited by 11 publications

(11 citation statements)

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“…Remark 3.2 (Properties of universal minimizers -Parametrization). Since any universally minimal lattice in L d (M, λ) is also locally minimal for L → ζ L (s), it follows by [34,Cor. 2] that there is only a finite number of such lattices in R d and, by [34,Cor.…”

Section: Universal Minimality and Consequencesmentioning

confidence: 91%

“…Since any universally minimal lattice in L d (M, λ) is also locally minimal for L → ζ L (s), it follows by [34,Cor. 2] that there is only a finite number of such lattices in R d and, by [34,Cor. 1] that any such lattice has d linearly independent minimum vectors.…”

Section: Universal Minimality and Consequencesmentioning

confidence: 91%

“…To finish this section, we give the definition (see also [34]) of a type of lattices that plays a crucial role in our theoretical results. Definition 2.4 (Strongly eutactic lattices).…”

Section: Name Notation and Definitionmentioning

confidence: 99%

“…When restricted to simple periodic configurations, the number of degrees of freedom of the system can be reduced, leading to more general optimality results that give important clues concerning the possible ground states for the corresponding general crystallization problem. Rigorous optimization of crystal's shapes under constraints has been performed for inverse power laws [18,23,27,30,31,34,56],…”

Section: Introductionmentioning

confidence: 99%

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On energy ground states among crystal lattice structures with prescribed bonds

Bétermin

2021

Preprint

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We consider pairwise interaction energies and we investigate their minimizers among lattices with prescribed minimal vectors (length and coordination number), i.e. the one corresponding to the crystal's bonds. In particular, we show the universal minimality -i.e. the optimality for all completely monotone interaction potentials -of strongly eutactic lattices among these structures. This gives new optimality results for the square, triangular, simple cubic (SC), face-centred-cubic (FCC) and body-centred-cubic (BCC) lattices in dimensions 2 and 3 when points are interacting through completely monotone potentials. We also show the universal maximality of the triangular and FCC lattices among all lattices with prescribed bonds. Furthermore, we apply our results to Lennard-Jones type potentials, showing the minimality of any universal minimizer (resp. maximizer) for small (resp. large) bond lengths, where the ranges of optimality are easily computable. Finally, a numerical investigation is presented where a phase transition of type "square-rhombic-triangular" (resp. "SC-rhombic-BCC-rhombic-FCC") in dimension d = 2 (resp. d = 3) among lattices with more than 4 (resp. 6) bonds is observed.

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Section: Universal Minimality and Consequencesmentioning

confidence: 91%

Section: Universal Minimality and Consequencesmentioning

confidence: 91%

“…To finish this section, we give the definition (see also [34]) of a type of lattices that plays a crucial role in our theoretical results. Definition 2.4 (Strongly eutactic lattices).…”

Section: Name Notation and Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On energy ground states among crystal lattice structures with prescribed bonds

Bétermin

2021

Preprint

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“…For example, this function is the simplest case of the zeta functions studied in Minakshisundaram and Pleijel [470] in order to obtain information about the eigenvalues of the Laplacian on compact differentiable manifolds (see also Singer [603]). And minima of Epstein zeta functions for fixed s have been investigated in order to find the best lattice to use in numerical integration among other things (see Delone and Ryskov [128], Fields [179], as well as Sarnak and Strömbergsson [561]). Applications of Epstein zeta functions to quantum-statistical mechanics are considered by Hurt and Hermann [310,Chap.…”

Section: Crystallographymentioning

confidence: 99%

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Terras

2013

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Whenever there is a large earthquake the Earth vibrates for days afterwards. The vibrations consist of the superposition of the elastic-gravitational normal modes of the Earth that are excited by the earthquake.-From F. Gilbert [212, p. 107].A (surface or Laplace) spherical harmonic is an eigenfunction of the Laplacian on the sphere. These are the analogues of exponentials for Fourier analysis on the sphere. Laplace and Legendre introduced these functions in order to study gravitational theory in the 1780s. Spherical harmonics are necessary for the analysis of any phenomena with spherical symmetry; e.g., earthquakes, the hydrogen atom, and the solar corona. Some of these topics will be discussed later in this section.Since our treatment of harmonic analysis on the sphere is rather condensed, the reader may want to consult some of the following references for more information: Before discussing spherical harmonics, we need to understand something about the geometry of the sphere. This symmetric space is closely related to the orthogonal group O(n) of real n × n matrices U such that t UU = I, where t U denotes the transpose of U and I denotes the n × n identity matrix. The special orthogonal SO(n), is the subgroup of matrices U in O(n) such that the determinant of U is one. You can regard U in SO(n − 1) as an element of SO(n) by formingThe group O(n) is a compact group and the sphere is a compact symmetric space, making some of the analysis on it somewhat easier.Exercise 2.1.1 (The Sphere as a Quotient or Homogeneous Space). Consider the sphere S n−1 = {x ∈ R n | x = 1}. Show that S n−1 can be identified with the quotient space SO(n)/SO(n − 1), using the preceding identification of SO(n − 1) as a subgroup of SO(n). Hint.Map the coset gSO(n − 1) to the vector ge n for g in SO(n − 1) and e n = t (0,... ,0, 1).You may also want to read the discussion of the topology of spheres and orthogonal groups in Chevalley [85,. In particular, the fundamental (or Poincaré) group of SO(2) is isomorphic to Z, while that of SO(n), n ≥ 3, has order 2.From now on we shall consider only the sphere S 2 . See the references for the general case. Now the sphere S 2 is a differentiable manifold. This means that locally it looks like two-dimensional Euclidean space. To make this precise, we use the usual angular coordinates (θ , ϕ), 0 < ϕ < 2π, 0 < θ < π, pictured in Fig. 2.1, to parameterize S 2 except for the semi-circle C through the poles and (1, 0, 0). A similar coordinate patch can be constructed to cover the rest of the sphere. The equations for the rectangular coordinates (x, y, z) of a point on S 2 in terms of the angular coordinates are x = sin θ cos ϕ y = sin θ sin ϕ z = cos θ ⎫ ⎬ ⎭ (2.1)

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Flat Space: Fourier Analysis on $${\mathbb{R}}^{m}$$

Terras

2013

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

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Locally minimal Epstein zeta functions

Cited by 11 publications

References 5 publications

On energy ground states among crystal lattice structures with prescribed bonds

On energy ground states among crystal lattice structures with prescribed bonds

Harmonic Analysis on Symmetric Spaces—Euclidean Space, the Sphere, and the Poincaré Upper Half-Plane

Flat Space: Fourier Analysis on $${\mathbb{R}}^{m}$$

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