Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct boundary element method, we numerically obtain the low-lying eigenmodes on a com...

The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain Ω0. If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter –the character...

The Prandtl number dependence of unsteady laminar natural convection along an infinite vertical plate in a thermally stratified fluid is investigated. Flows are induced by an impulsive (step) change in plate temperature and by a suddenly imposed plate heat flux. Analytical solutions of the viscous equations of motion and thermodynamic energy are obtained for Prandtl numbers near unity by the me...

Motivated by practical applications in heat conduction and contaminant transport, we consider mass diffusion across a perturbed interface separating two finite regions of distinct diffusivity. Under the assumption continuity solution diffusive flux at interface, use perturbation theory to develop an asymptotic expansion valid for small perturbations. Each term satisfies initial-boundary value p...

In this paper we consider the Laplace-Dirichlet equation in a polygonal domain perturbed at the small scale ε near a vertex. We assume that this perturbation is self-similar, that is, derives from the same pattern for all relevant values of ε. We construct and validate asymptotic expansions of the solution in powers of ε via two different techniques, namely the method of multiscale expansions a...

We present the first two leading terms of the 1/N (genus) expansion of the free energy for ensembles of normal and complex random matrices. The results are expressed through the support of eigenvalues (assumed to be a connected domain in the complex plane). In particular, the subleading (genus-1) term is given by the regularized determinant of the Laplace operator in the complementary domain wi...

We propose a tensorial expansion of high resolution diffusion imaging (HARDI) data on the unit sphere into a sum of self-similar polynomials, i.e. polynomials that retain their form up to a scaling under the act of lowering resolution via the diffusion semigroup generated by the Laplace-Beltrami operator on the sphere. In this way we arrive at a hierarchy of HARDI degrees of freedom into contra...

Let V ⊂ R be a compact real analytic surface with isolated singularities, and assume its smooth part V0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R . We prove: 1. Each point of V has a neighborhood which is quasi-isometric (naturally and ’almost isometrically’) to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expa...