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...tion and illustrate some related results by numerically computed eigenfunctions. 1. Introduction The geometrical structure of Laplacian eigenfunctions has been thoroughly investigated (see the review =-=[1]-=- and references therein). When a domain can be seen as a union of two (or many) subdomains with narrow connections, some lowfrequency eigenfunctions can be found localized (or trapped) in one subdomai...

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...an trajectories that hit or crossed the boundary ∂Ω prior to time t. The diffusion propagator can be expressed in terms of the (L2-normalized) Laplace operator eigenfunctions um(r) and eigenvalues λm =-=[1, 2, 26]-=- Gt(r0, r) = ∞∑ m=1 e−Dλmtum(r0)u∗m(r), (2) January 29, 2014 13:52 BC9104 – First-Passage Phenomena and their Applications 23˙grebenkov˙revised page 575 Efficient Monte Carlo methods 575 where asteris...

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... spectrum is the infinite set 0≤λ1≤· · ·≤λn≤· · · , λn→+∞, of eigenvalues for the Dirichlet problem:{ ∆Ωf + λf = 0 in Ω f = 0 on ∂Ω. (2) Denoting by φn the associated L2(Ω)-orthonormal eigenfunctions, let KΩ(t,x,y) ,∑ n≥1 exp{−λnt}φn(x)φn(y) the heat kernel. The heat kernel is the fundamental solution to the heat equation ∂tKΩ(t,x,y) = ∆ΩKΩ(t,x,y) over Ω, withKΩ(0,x,y)≡ δx(y) and Dirichlet boundary conditions. It is at the basis of a variety of point matching techniques in the computer vision and computer graphics literature, as KΩ(t,x,x) encodes information about the local neighbourhood of x [10]. Similarly the trace of the heat kernel a.k.a. the heat-trace, ZΩ(t) , ∫ Ω KΩ(t,x,x)dx = ∑+∞ n=1 e −λnt, summarizes information about local and global invariants of the shape [12]. It provides a convenient link between the intrinsic geometry of the object and the spectrum. For two shapes Ωλ and Ωξ with respective spectrum λ and ξ, let ∆nλ,ξ quantify the influence of the change in the nth eigenmode on the heat-trace: ∆nλ,ξ , ∫ +∞ 0 |e−λnt − e−ξnt |dt = |λn − ξn| λnξn . (3) The pseudo-metric WESD is obtained by summing the contributions of all modes: ρp(Ωλ,Ωξ) , [ +∞∑ n=1 (∆nλ,ξ) p ]1/p = [ +∞∑...

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... eigenfunctions. In general, this new problem is even more time-consuming than the original one. However, for a special class of “symmetric” domains, the Laplacian eigenfunctions are known explicitly =-=[92]-=- that tremendously speeds up computations and greatly improves the accuracy. Examples of such domains are intervals/rectangles/rectangular parallelepipeds, equilateral triangles, disks/cylinders, sphe...

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...odic orbits in a spiral-shaped billiards. Bies and co-workers presented a brief survey on scarring and symmetric effects of eigenfunctions in a stadium billiard [24]. Other references can be found in =-=[95]-=-. It is worth mentioning that there are many physical experiments related to whispering gallery and bouncing ball modes. For instance, Wiersig investigated the structure of whispering gallery modes in...

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.... We remark that there is a vast literature on the use of the Laplacian operator for extracting various types of geometric information on static manifolds and we refer the reader to the recent survey =-=[15]-=-, with over 500 references. We now proceed through a few simple domains to illustrate the relationship between the solution to the isoperimetric problems and the eigenvalues and eigenfunctions of the ...

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...idge (cf. [18]). The Laplace operator’s eigenvalues and eigenfunctions depend on the geometry of the region as well as on the imposed boundary conditions, but are invariant under rotations and shifts =-=[19]-=-. Moreover, when the domain is scaled by a factor c > 0 the eigenvalues are rescaled by 1/c2, with associated eigenfunctions ψk(x/c) for x ∈ cD. In the case of Dirichlet- or Robin boundary conditions,...