Classical orthogonal polynomials and matrix polynomials being orthogonal with respect to some Hermitean positive deenite matrix of measures share several properties , e.g., three term recurrencies, Christooel{Darboux formulas; there are connections to the triangular decomposition of the (inverse) moment matrix and to eigenvalue{problems for the banded matrix of recurrence coeecients. Also, a co...

There are many different methods to obtain an orthogonal matrix. Householder discovered an orthogonal matrix, and it can be applied to solve physical problems of reflection. In this classroom note, we can obtain an orthogonal matrix group eAt, where A is antisymmetric with angular vector ω. When t is setting to be t=π/|ω|, eAt is equivalent to a Householder matrix of a reflecting plane with nor...

Dual scattering channel schemes extend the transmission line matrix numerical method (Johns’ TLM algorithm) in two directions. For one point, transmission line links are replaced by abstract scattering channels in terms of paired distributions (characteristic impedances are thus neither needed, nor in general defined, e.g.). In the second place, non-trivial cell interface scattering is admitted...

Semi-supervised clustering is often viewed as using labeled data to aid the clustering process. However, existing algorithms fail to consider dual constraints between data points (e.g. documents) and features (e.g. words). To address this problem, in this paper, we propose a novel semi-supervised document co-clustering model OSS-NMF via orthogonal nonnegative matrix tri-factorization. Our model...

Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable ‘matrix’ quantum mechanics, which is recently proposed by Odake and the author. The (q)Askey-scheme of hypergeometric orthogonal polynomials of a discrete...

Dual scattering channel schemes extend the transmission line matrix numerical method (Johns’ TLM algorithm) in two directions. For one point, transmission line links are replaced by abstract scattering channels in terms of paired distributions (characteristic impedances are thus neither needed, nor in general defined, e.g.). In the second place, non-trivial cell interface scattering is admitted...

A complex square matrix A is called an orthogonal projector if A2 = A = A∗, where A∗ denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish v...