A ring is feebly nil-clean if for any there exist two orthogonal idempotents and a nilpotent such that . Let be a 2-primal feebly nil-clean ring. We prove that every matrix ring over is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.
We first investigate factorizations of elements of the semigroup of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for , and, given , also provide formulas for , and . As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem...
This paper is concerned with optimal design problems with a special assumption on the coefficients of the state equation. Namely we assume that the variations of these coefficients have a small amplitude. Then, making an asymptotic expansion up to second order with respect to the aspect ratio of the coefficients allows us to greatly simplify the optimal design problem. By using the notion of H-measures we are able to prove general existence theorems for small amplitude optimal design and to provide...