Narrow Results

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms,

It is well known that the full matrix ring over a skew-field is a simple ring. We generalize this theorem to the case of semirings. We characterize the case when the matrix semiring $M_n(S)$, of all $n\times n$ matrices over a semiring S, is congruence-simple, provided that either S has a multiplicatively absorbing element or S is commutative and additively cancellative.

In this paper, we study representations of $G_n^3$-like groups. The group $G_n^3$ itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of $G_n^3$-like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the $G_n^3$-like group structure.

In this paper, we get several new results on permutation polynomials over finite fields. First, by using the linear translator, we construct permutation polynomials of the forms and . These forms generalize the results obtained by Kyureghyan in 2011. Consequently, we characterize permutation polynomials of the form , which extends a theorem of Charpin and Kyureghyan obtained in 2009.

We show that for n × n nonsingular matrices A₁, A₂, …, A_k (k ≥ 2) over a commutative principal ideal domain R with relatively coprime determinants there exist invertible n × n matrices U, V₁, …, V_k over R such that UA_iV_i = S_Ai are the Smith normal forms of the matrices A_i.