In this paper, we introduce a new concept of generalized matrix rings and build up the general theory of radicals for g.m.rings. Meantime, we obtain r̄b(A) = g.m.rb(A) = ∑ {rb(Aij) | i, j ∈ I} = rb(A)

Proof. We prove the statement by induction on n. n = 1 case can be easily verified. Let’s assume the theorem is true for n − 1. We take an orthonormal basis [e1, · · · , en] for W and write Lei = ∑n j=1 ajiej. Since L is symmetric, we have aji = aij or A = [aij] is a symmetric matrix. From the lemma, any root of the characteristic polynomial of A is real. Take λ1 to be a root, then A−λ1I is sin...

Corrigendum: Characterizing Si:P quantum dot qubits with spin resonance techniques Yu Wang, Chin-Yi Chen, Gerhard Klimeck, Michelle Y. Simmons & Rajib Rahman Scientific Reports 6:31830; doi: 10.1038/srep31830; published online 23 August 2016; updated on 30 November 2016 This Article contains errors in Figure 4 where the hyperfine values for the donor separations along the [110] direction were c...

THE quadratic assignement problem (QAP) can be described as an optimizing the assignment a set of facilities to a set of locations with given costs between the locations and given flows between the facilities in order to minimize the sum of the product between flows and distances. A concrete example, would be a hospital planning optimization the location of various medical facilities in order t...

where Aij = a(i, j) and the domain of the function is just the cross-product of the two index sets. Such a structure is called a matrix. The values Aij are called the elements or entries of the matrix. A sequence of elements with the same first index is called a row of the matrix; similarly, a sequence of elements with the same second index is called a column. The dimension of the matrix specif...

This paper presents the results of reliability analyses load-carrying capacity circular concrete filled steel tubes (CFST) using different codes. Four parameters that govern CFST were considered as variables. These variables include yield strength steel, concrete, diameter CFST, and thickness tube. The Monte Carlo technique was used for simulation. Simulations conducted, obtained capacities ana...

Combinatorial matrix theory, which involves connections between linear algebra, graph theory, and combinatorics, is a vital area and dynamic area of research, with applications to fields such as biology, chemistry, economics, and computer engineering. One area generating considerable interest recently is the study of the minimum rank of matrices associated with graphs. Let F be any field. For a...

In this paper we study a game problem, called bin packing game with an interest matrix, which is a generalization of all the currently known bin packing games. In this game, there are some items with positive sizes and identical bins with unit capacity as in the classical bin packing problem; additionally we are given an interest matrix with rational entries, whose element aij stands for how mu...