Beta ensembles on the real line with three classical weights (Gaussian, Laguerre and Jacobi) are now realized as eigenvalues of certain tridiagonal random matrices. The paper deals beta Jacobi ensembles, type weight. Making use matrix model, we show that in regime where $\beta N \to const \in [0, \infty)$, $N$ system size, empirical distribution converges weakly to a limiting measure which belo...

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the ‘holes’ in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two differe...

Introduction Non-invasive Fluorescent Reflectance Imaging (FRI) is used for accessing physiological and molecular processes in biological media. The aim of this article is to separate the overlapping emission spectra of quantum dots within tissue-equivalent phantom using SVD, Jacobi SVD, and NMF methods in the FRI mode. Materials and Methods In this article, a tissue-like phantom and an optical...

We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the th...

We study the dispersion of a particle whose motion dynamics can be described by forced velocity jump process. To investigate large deviations results, we Kolmogorov forward equation this process in hyperbolic scaling (t,x,v)→(t/ε,x/ε,v) and then, perform Hopf–Cole transform which gives us kinetic on potential. prove convergence potential as ε→0 to solution Hamilton–Jacobi equation. The hamilton...

This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For first class subject a constraint on Stiefel manifold, we reformulate it as an optimization problem unitary group, which makes possible apply (Jacobi-G) algorithm. Then, if subproblem can always be r...

In this paper, we define the Schrödinger-Weil representation for the Jacobi group and construct covariant maps for the Schrödinger-Weil representation. Using these covariant maps, we construct Jacobi forms with respect to an arithmetic subgroup of the Jacobi group.