The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

Evaluating the Prospects for the Development of Own Trademarks of Trade Enterprises

The article presents the results of the research on determining the prospects for the introduction and development of own trademarks (OTM) of the trade enterprise. The strategies of own trademarks of the trade enterprise are provided. The peculiarities of formation and implementation of dumping strategies, replacement of a competitor, brand expansion are specified. To determine the prospects for the introduction and development of own trademarks of the trade enterprise, a methodical approach is substantiated, which is formed using matrix methods, scorecard methods, and additive convolution. The developed methodical approach provides for the implementation of interrelated stages of evaluation of competitive positions of activity in relation to own trademarks compared to other types of current activities, favorable external environment and readiness of the trade enterprise for the implementation of this type of activity. To evaluate the competitive positions of the OTM-related activities in comparison with other types of current activities, the use of the Dibb –Symkin matrix and the modified BCG matrix is proposed; to assess the favorability of the external environment – PEST analysis; to assess the readiness of the trade enterprise for the introduction and development of its own trademarks – the method of scorecard according to the characteristics of organizational, technological and resource aspects of the trade enterprise’s activities regarding OTM. In order to ensure the validity of managerial decisions on optimizing the portfolio of own trademarks, a scientific-methodical approach to the ranking of assortment groups of goods that are part of own trademark is developed. The methodical basis of the developed approach is matrix methods (Dibb–Symkin matrix, modified BCG matrix), coefficient method, expert estimation method and additive convolution. The sequence of determining the competitiveness of the assortment of goods is presented, which involves grouping the assortment according to the indicators of quality, price, latitude of the assortment of goods compared to the range of products on the part of the trademark of competing enterprises and manufacturers.

A Numerical Method for Solving Matrix Coefficient Heat Equations with Interfaces

In this paper, we propose a numerical method for solving the heat equations with interfaces. This method uses the non-traditional finite element method together with finite difference method to get solutions with second-order accuracy. It is capable of dealing with matrix coefficient involving time, and the interfaces under consideration are sharp-edged interfaces instead of smooth interfaces. Modified Euler Method is employed to ensure the accuracy in time. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner. Extensive numerical experiments illustrate the feasibility of the method.

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

It is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.