Related papers: Adding and multiplying random matrices: a generali…
Combinatorial transition matrices arise frequently in the theory of symmetric functions and their generalizations. The entries of such matrices often count signed, weighted combinatorial structures such as semistandard tableaux, rim-hook…
We review elementary properties of random matrices and discuss widely used mathematical methods for both hermitian and nonhermitian random matrix ensembles. Applications to a wide range of physics problems are summarized. This paper…
We consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability…
We describe a natural generalization of irreducibility in order lattices with arbitrary metrics. We analyse the special cases of valuation metrics and more general metrics for lattices. This article is mainly based on a part of the author's…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most…
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…
The popular generalized additive model framework is extended to allow both the mean curves and the response distribution to be nonparametric. The approach is demonstrated to be a flexible yet parsimonious tool for data analysis in its own…
The $\tau$-function theory of Painlev\'e systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlev\'e equations. The…
We present a general scheme for the construction of new eficient generalized Schultz iterative methods for computing the inverse matrix. These methods have the form $$ X_{k+1} = X_k(a_0^{(k)}I+a_1^{(k)}AX_k),\quad k\in\mathbb{N}, $$ where…
Let $R$ be a finite local ring. We prove a quantitative universality statement for the cokernel of random matrices with i.i.d. entries valued in $R$. Rather than use the moment method, we use the Lindeberg replacement technique. This…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
We consider certain scalar product of symmetric functions which is parameterized by a function $r$ and an integer $n$. One the one hand we have a fermionic representation of this scalar product. On the other hand we get a representation of…
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve…
This article handles in a short manner a few Laplace transform pairs and some extensions to the basic equations are developed. They can be applied to a wide variety of functions in order to find the Laplace transform or its inverse when…
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of…