In the presence of a confining potential V, eigenfunctions continuous Schrödinger operator ?? + V decay exponentially with rate governed by part which is above corresponding eigenvalue; this can be quantified method Agmon. Analogous localization properties also established for eigenvectors discrete matrix. This note shows, perhaps surprisingly, that one replace matrix any real symmetric Z-matri...

We discuss the classical efficiency criteria in density estimation and propose some variants. The context is a general density estimation scheme that contains the cases of i.i.d. or dependent random variables, in discrete or continuous time. Unbiased estimation, optimality and asymptotic optimality are considered. An example of a density estimator that satisfies some suggested criteria is given...

We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In trigonometric limit eigenfunctions in question recover previously studied $q$-Racah reduction of Koornwinder-Macdonald polynomials. When inter-particle interaction degenerates to that impenetrable bosons, simplifies terms generalized Schur polynomials on...

We study the n-simplex nonconforming Crouzeix-Raviart element in approximating the n-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are ...

In this paper, we investigate some properties of eigenvalues and eigenfunctions of boundary value problems with separated boundary conditions. Also, we obtain formal series solutions for some partial differential equations associated with the second order differential equation, and study necessary and sufficient conditions for the negative and positive eigenvalues of the boundary value problem....

We show that the equilibrium positions of the Ruijsenaars-Schneider-van Diejen systems with the trigonometric potential are given by the zeros of the Askey-Wilson polynomials with five parameters. The corresponding single particle quantum version, which is a typical example of “discrete” quantum mechanical systems with a q-shift type kinetic term, is shape invariant and the eigenfunctions are t...

well-known seismic design codes have offered an alternative equivalent static procedure for practical purposes instead of verifying design trials with complicated step-y-step dynamic analyses. such a pattern of base-shear distribution over the building height will enforce its special stiffness and strength distribution which is not necessarily best suited for seismic design. the present study, ...

We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t...

Most of the data analysis of microarray gene expression data use a clustering algorithm, as a preprocessing step, in genomic functional analysis for example [6] or as the main discriminating tool, in the tumor classification study [9]. While from the experimenter point of view the simplest clustering method could be the best, it is acknowledged [8] that the reliability of allocation the units t...