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Partial symplectic conditional and joint probability representations of quantum mechanics are considered. The correspondence rules for most interesting physical operators are found and the expressions of the dual symbols of operators are…
We compute the limiting statistical distribution of the eigenvalues of sequences of matrices whose entries satisfy what we call a vanishing mean variation condition and are $\mu$-distributed for some probability measure. As an application…
We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…
Optimization of complex functions, such as the output of computer simulators, is a difficult task that has received much attention in the literature. A less studied problem is that of optimization under unknown constraints, i.e., when the…
We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic…
The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on…
A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the…
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and…
There are many advantages to use probability method for nonlinear system identification, such as the noises and outliers in the data set do not affect the probability models significantly; the input features can be extracted in probability…
We show that every n-by-n matrix is generically a product of [n/2] + 1 Toeplitz matrices and always a product of at most 2n+5 Toeplitz matrices. The same result holds true if the word "Toeplitz" is replaced by "Hankel", and the generic…
Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In…
We consider the sensitivity of real zeros of structured polynomial systems to perturbations of their coefficients. In particular, we provide explicit estimates for condition numbers of structured random real polynomial systems, and extend…
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
Bayesian properties of the signed root likelihood ratio statistic are analysed. Conditions for first-order probability matching are derived by the examination of the Bayesian posterior and frequentist means of this statistic. Second-order…
This paper considers estimating a covariance matrix of $p$ variables from $n$ observations by either banding or tapering the sample covariance matrix, or estimating a banded version of the inverse of the covariance. We show that these…
We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of…
The article presents a systematic study of the problem of conditioning a Gaussian random variable $\xi$ on nonlinear observations of the form $F \circ \phi(\xi)$ where $\phi: \mathcal{X} \to \mathbb{R}^N$ is a bounded linear operator and…
The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing…
Maximum likelihood learning with exponential families leads to moment-matching of the sufficient statistics, a classic result. This can be generalized to conditional exponential families and/or when there are hidden data. This document…
We calculate the eigenvalues of a class of random matrices, namely the randomly segmented tridiagonal quasi-Toeplitz (rstq-T) matrix, in exact closed-form. The contexts under which these matrices arise are ubiquitous in physics. In our…