When partitioning the variation of univariate or multivariate ecological data with respect to several submodels of spatial eigenfunctions (e.g., Moran's eigenvector maps, MEM) acting as explanatory data, a problem occurs: although the submodels are constructed to be orthogonal to one another, the partitioning based on adjusted R2 statistics produces nonzero values in the intersections between s...

In this paper, the Extended-Domain-Eigenfunction-Method (EDEM) is combined with the Level Set Method in a composite numerical scheme for simulating a moving boundary problem. The liquid velocity is obtained by formulating the problem in terms of the EDEM methodology and solved using a least square approach. The propagation of the free surface is effected by a narrow band Level Set Method. The t...

A numerical method is developed for solving the two dimensional Helmholtz equation in a two layer region bounded by a flat top, a flat bottom and a curved interface. A local orthogonal transform is used to flatten the curved interface of the waveguide. The one-way re-formulation based on the Dirichlet-to-Neumann (DtN) map is then used to reduce the boundary value problem to an initial value pro...

An interesting example in the paper of Davies and Simon [5] was that of a horn-shaped domain in R. By horn-shaped we mean domains of the form D = {(x, y) : x > 0, ‖y‖ < f(x)} with f : [0,∞) → (0,∞) a function tending to zero as x tends to infinity. Davies and Simon [5] established sufficiently sharp estimates on the first Dirichlet eigenfunction of ∆d (d-dimensional Laplacian) on D to determine...

Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk D and ∆ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of ∆, equivariant by Γ with real eigenvalue λ = −s(1 − s), where s = 1 2 + it, admits an integral representation by a distribution D f,s (the Helgason distribution) which is equivariant by Γ and supported at infinity ∂D = S 1. The geodesic flo...

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite p and investigate the limit problem as p → ∞.

Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation, but due to the computational demands, applications of such methods to spatiotemporal data have been slow to appear. For even a short video sequence, the set of all pairwise voxel similarities is a huge quantity of data: one second of a 256 384 sequence captured at 30Hz entails on the order ...

We analyze the Ritz–Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, apparently the first truly a priori error estimates th...