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Although it is well known that regular hexagons tessellate the plane by reflection, the tessellation is not strict, because the lines that contain the edges of the hexagon cut through the interior of the reflected copies.
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The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, Bérard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace...
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Context 1
... are not aware of the term "strict tessellation" in the literature, but it might be known under a different name. An example of a strict tessellation of the plane is given in Figure 3; a tessellation of the plane which is not strict is given in Figure 4. Definition 2. A polytope ∈ ℘ n strictly tessellates R n if 1. R n = j ∈Z j , such that each j is isometric to , and j ∩ k = ∅ for any j = k. ...
Citations
... In Section 2, we construct the exact wave kernel and derive a generalized Poisson summation formula on strictly tessellating polytopes. We then use this Poisson summation formula to prove Theorems 1.4 and 1.5 in Section 3. The proof of Theorems 1.1 and 1.2 are given in Section 4. In Appendix A, we show that the definition of strictly tessellating polytopes presented here is equivalent to [22,Definition 2]. In Appendix B, we collect some well-known facts about the wave equation that are used throughout the proofs. ...
... Remark. The term strictly tessellates is adopted from [22]. Note, however, that the definition given here is different from the one in [22,Definition 2]. ...
... The term strictly tessellates is adopted from [22]. Note, however, that the definition given here is different from the one in [22,Definition 2]. The definition above is more convenient for our purposes, as can be seen from the proof of Lemma 2.3 below. ...
We derive a two-term asymptotic expansion for the exchange energy of the free electron gas on strictly tessellating polytopes and fundamental domains of lattices in the thermodynamic limit. This expansion comprises a bulk (volume-dependent) term, the celebrated Dirac exchange, and a novel surface correction stemming from a boundary layer and finite-size effects. Furthermore, we derive analogous two-term asymptotic expansions for semi-local density functionals. By matching the coefficients of these asymptotic expansions, we obtain an integral constraint for semi-local approximations of the exchange energy used in density functional theory.
... The term 'Robin boundary condition' came much later, see[6] for a historical discussion.4 The only polygonal domains where all Dirichlet or Neumann eigenfunctions are trigonometric are rectangles, and the equilateral, hemi-equilateral and right isosceles triangles[17], see[20] for a higher dimensional version. ...
The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin-Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.
... where ∂ ∂n is the derivative in the outward pointing normal direction, and σ > 0 is the Robin parameter (which we take to be constant). Lamé only determined the Robin eigenfunctions possessing 120 • rotational symmetry, and it is only in 2004 that McCartin [14,16] completely determined the eigenproblem, showing that all of the eigenfunctions are trigonometric polynomials 2 , and that the eigenvalues are determined by a system of transcendental coupled secular equations as follows: Define auxiliary parameters L ∈ (−π/2, 0], M, N ∈ [0, π/2), which are required to satisfy the coupled system of equations The corresponding Robin eigenvalues are (1.2) Λ m,n (σ) = 4π 2 27r 2 (µ 2 + ν 2 + µν) where µ = 2M − N − L π + m, ν = 2N − L − M π + n. 2 The only polygonal domains where all Dirichlet or Neumann eigenfunctions are trigonometric are rectangles, and the equilateral, hemi-equilateral and right isosceles triangles [15], see [18] for a higher dimensional version Note that there is a systematic multiplicity of order 2 coming from the symmetry Λ m,n = Λ n,m . ...
The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lam\'e in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin-Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.
