Related papers: Norms on Complex Matrices Induced by Random Vector…

Using standard methods (due to Janson, Stein-Chen, and Talagrand) from probabilistic combinatorics, we explore the following general theme: As one progresses from each member of a family of objects ${\cal A}$ being "covered" by at most one…

We construct a family of groups from suitable higher rank graphs which are analogues of the finite symmetric groups. We introduce homological invariants showing that many of our groups are, for example, not isomorphic to $nV$, when $n \geq…

Using the character expansion method, we generalize several well-known integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and…

Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the…

Random matrices in the large N expansion and the so-called double scaling limit can be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has generated a tremendous expansion of random matrix…

We introduce an algorithm that conjectures the structure of a permutation class in the form of a disjoint cover of "rules"; similar to generalized grid classes. The cover is usually easily verified by a human and translated into an…

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…

Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class…

We study the topology of real polynomial maps $\mathbb{R}^{4n} \longrightarrow \mathbb{R}^{4}$ expressed in terms of bicomplex variables and their conjugates, which we refer to as bicomplex mixed polynomials. We introduce the notion of…

Probabilistic models learned as density estimators can be exploited in representation learning beside being toolboxes used to answer inference queries only. However, how to extract useful representations highly depends on the particular…

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on non zero complex parameters. The method we use is via…

We consider linear recurrences with polynomial coefficients of Poincar\'e type and with a unique simple dominant eigenvalue. We give an algorithm that proves or disproves positivity of solutions provided the initial conditions satisfy a…

We survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner random matrix ensembles, focusing in particular on the Four Moment Theorem and its applications.

We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary…

We derive a general large deviation principle for a canonical sequence of probability measures, having its origins in random matrix theory, on unbounded sets $K$ of ${\bf C}$ with weakly admissible external fields $Q$ and very general…

For an arbitrary evolution family, we consider the notion of a polynomial dichotomy with respect to a family of norms and characterize it in terms of the admissibility property, that is, the existence of a unique bounded solution for each…

The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…

This paper develops a polynomial normal transformation model, whereby various non-normal probability distributions can be simulated by the standard normal distribution. Two methods are presented to determine the coefficients of polynomial…