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[en] We study the commutativity of certain class of rings, namely rings with unity 1 and right s-unital rings under each of the following properties [yxm - xn f(y)xp,x]=0, [yxm+xnf(y)xp,x]=0, where f(t) is a polynomial in t2Z[t] varying with pair of ring elements x,y and m,n,p are fixed non-negative integers. Moreover, the results have been extended to the case when m and n depend on the choice of x and y and the ring satisfies the Chacron's Theorem. (author). 14 refs
[en] Decomposition of unipotents gives short polynomial expressions of the conjugates of elementary generators as products of elementaries. It turns out that with some minor twist the decomposition of unipotents can be read backwards to give very short polynomial expressions of the elementary generators themselves in terms of elementary conjugates of an arbitrary matrix and its inverse. For absolute elementary subgroups of classical groups this was recently observed by Raimund Preusser. I discuss various generalizations of these results for exceptional groups, specifically those of types E6 and E7, and also mention further possible generalizations and applications.
[en] Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful framework to meet this challenge. Central factorial numbers play a key role in the theoretical development. Explicit polynomial coefficients for rotations expressed either as exponentials or as rational Cayley transforms are considered here. Structural features of the results are discussed and compared, and large j limits of the coefficients are examined
[en] The Gauss algorithm for diagonalization of constant matrices is modified for application to polynomial matrices. Due to this modification the diagonal elements become pure polynomials rather than rational functions. (author)
[en] Systems of hypercomplex numbers, which had been studied and developed at the end of the last century, are nowadays quite unknown to the scientific community. It is believed that study of their applications ended just before one of the fundamental discoveries of our century, Einstein's equivalence between space and time. Owing to this equivalence, not-defined quadratic forms - which are in a quite strong relationship with hypercomplex numbers possessing divisors of the zero - have got concrete physical meaning. The aim of this work is to study these systems of numbers and to describe them in terms of a familiar mathematical tool, i.e. matrix algebra. Moreover, they will show how hypercomplex numbers possessing divisors of the zero candidate themselves to be the most proper mathematical language for treatment of propagative phenomena
[en] We show that the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) admits a suggestive reformulation through elliptic functions. We also consider one-variable reductions of the dispersionless DKP hierarchy and show that they are described by an elliptic version of the Löwner equation. With a particular choice of the driving function, the latter appears to be closely related to the Painlevé VI equation with a special choice of parameters. (fast track communication)
[en] We prove many congruences for binomial and multinomial coefficients as well as for the coefficients of the Girard-Newton formula in the theory of symmetric functions. These congruences also imply congruences (modulo powers of primes) for the traces of various powers of matrices with integer elements. We thus have an extension of the matrix Fermat theorem similar to Euler's extension of the numerical little Fermat theorem
[en] The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velázquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.