Lattice points in stretched model domains of finite type in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:math>

Guo, Jessica Y; Wang, Weiwei

doi:10.1016/j.jnt.2018.03.009

Cited by 5 publications

(3 citation statements)

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“…More recently these results have been generalized to allow for a shift of the lattice, that is replacing the standard lattice by (N + σ) × (N + τ ), see [39]. Also higher dimensional versions of this problem have been studied by Marshall, and Guo and Wang in [24,42]. A particularly interesting case of the lattice point optimization problem is to consider f (x) = 1 − x.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

Larson

2019

J. Spectr. Theory

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For Ω ⊂ R n , a convex and bounded domain, we study the spectrum of −∆Ω the Dirichlet Laplacian on Ω. For Λ ≥ 0 and γ ≥ 0 let ΩΛ,γ (A) denote any extremal set of the shape optimization problem, where A is an admissible family of convex domains in R n . If γ ≥ 1 and {Λj } j≥1 is a positive sequence tending to infinity we prove that {ΩΛ j ,γ (A)} j≥1 is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences as minimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then ΩΛ,γ converges, up to rotation and translation, to the regular m-gon.2010 Mathematics Subject Classification. 35P15, 47A75, 49Q10.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains

Larson

2019

J. Spectr. Theory

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“…replacing N 2 by (N + σ) × (N + τ ). For work on similar problems in higher dimensions see also [8,18]. However, the results of [3,16,17] all require that the graph of the function f has nonvanishing curvature.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Maximizing Riesz means of anisotropic harmonic oscillators

Larson

2019

Arkiv För Matematik

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We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N − 1 2 ) × (N − 1 2 ) inside triangles with vertices (0, 0), (0, λ √ β) and(λ/ √ β, 0) with respect to β > 0, for fixed λ ≥ 0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N + σ) × (N + τ ), for σ, τ > −1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ = τ = − 1 2 , is a critical case.2010 Mathematics Subject Classification. Primary 35P15. Secondary 11P21, 52C05.

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“…In the authors studied the discrepancy for convex bodies with a finite number of isolated flat points of order at most γ as in Definition . For