The perturbation of the quantum Calogero-Moser-Sutherland system and related results

The Hamiltonian of the trigonometric Calogero-Sutherland model coincides with some limit of the Hamiltonian of the elliptic Calogero-Moser model. In other words the elliptic Hamiltonian is a perturbed operator of the trigonometric one. In this article we show the essential self-adjointness of the Hamiltonian of the elliptic Calogero-Moser model and the regularity (convergence) of the perturbation for the arbitrary root system. We also show the holomorphy of the joint eigenfunctions of the commuting Hamiltonians w.r.t the variables (x_1, >..., x_N) for the A_{N-1}-case. As a result, the algebraic calculation of the perturbation is justified.Comment: Revised version, 35 page

Spherical functions on the space of pp-adic unitary hermitian matrices

We investigate the space $X$ of unitary hermitian matrices over \frp-adic fields through spherical functions. First we consider Cartan decomposition of $X$, and give precise representatives for fields with odd residual characteristic, i.e., 2\notin \frp. In the latter half we assume odd residual characteristic, and give explicit formulas of typical spherical functions on $X$, where Hall-Littlewood symmetric polynomials of type $C_n$ appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show the Schwartz space \SKX is a free Hecke algebra \hec-module of rank $2^n$, where $2n$ is the size of matrices in $X$, and give the explicit Plancherel formula on \SKX.Comment: to appear in International Journal of Number Theor

Easy estimation by a new parameterization for the three-parameter lognormal distribution

A new parameterization and algorithm are proposed for seeking the primary relative maximum of the likelihood function in the three-parameter lognormal distribution. The parameterization yields the dimension reduction of the three-parameter estimation problem to a two-parameter estimation problem on the basis of an extended lognormal distribution. The algorithm provides the way of seeking the profile of an object function in the two-parameter estimation problem. It is simple and numerically stable because it is constructed on the basis of the bisection method. The profile clearly and easily shows whether a primary relative maximum exists or not, and also gives a primary relative maximum certainly if it exists.Comment: 13 pages, 3 figure