We develop a method that allows us to construct families of orthogonal matrix polynomials of size N ×N satisfying second order difference equations with polynomial coefficients. The existence (and properties) of these orthogonal families strongly depends on the non commutativity of the matrix product, the existence of singular matrices and the matrix size N .

We show that the variance of the number of simultaneous zeros ofm i.i.d. Gaussian random polynomials of degree N in an open set U ⊂ C with smooth boundary is asymptotic to N νmm Vol(∂U), where νmm is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on C. Our res...

In this paper, we generalize sone well-known commutativity theorems for associative rings as follows: Let ’, > 1. ,,, .,, and be fixed nou-ncgative integers such that s ik m1, or i/= n1, and let R be a ring xvith unity satisfying the polynomial identity y*[x’,y] [x,y’]x for all y R. Sul,lose that (i) R has Q(z) (that is n[x,y] 0 implies [z,y] 0); (ii) the set of d] nilpotent ,,lem,’nts of R is ...

Numerous scientific texts insist on safe topical application of specific herbal products like those containing Boswellia oleogumresin, but adverse effects arising during clinical trials of herbal medicines give the opportunity to clarify the blind spots in the area of drug safety. A 36-year-old woman without allergic background had two burn wounds because of hot oil. Her hand's wound w...

We prove that the existence of a homogeneous invariant of degree n for a representation of a semi-simple Lie group guarantees the existence of non-trivial solutions of D α = 0: these correspond to the maximum value of the square of the invariant divided by the norm of the representation to the n th power.

Let K be a totally real cyclic number field of degree n that is the product of two distinct primes and such that the class number of the n-th cyclotomic field equals 1. We derive certain necessary and sufficient conditions for the existence of a Minkowski unit for K.