Let A = (aij)n×n be a nonnegative, symmetric, irreducible and invertible matrix. We prove the existence and uniqueness of radial solutions to the following Liouville system with singularity:  ∆ui + ∑n j=1 aij |x| βj ej = 0, R2, i = 1, ..., n ∫ R2 |x| βieui(x)dx <∞, i = 1, ..., n where β1, ..., βn are constants greater than −2. If all βis are negative we prove that all solutions are radial and...

The linear system of equations Ax = b where A = [aij ] in Cn.n is a crispsingular matrix and the right-hand side is a fuzzy vector is called a singularfuzzy linear system of equations. In this paper, solving singular fuzzy linearsystems of equations using generalized inverses such as Drazin inverse andpseudo-inverse are investigated.

Let G be a finite, undirected, and simple graph. If {v1, · · · , vn} is the set of vertices of G, then the adjacency matrix A(G) = [aij ] is an n-by-n matrix where aij = 1 if vi and vj are adjacent and aij = 0 otherwise. The energy of a graph, E(G), is defined as the sum of the absolute values of eigenvalues of A(G). The concept of energy originates in chemistry and was first defined by I. Gutm...

We consider compact intervals [a, a] := {x ∈ IR : a ≤ x ≤ a} and denote the set of all such intervals by IR. We also write [a] instead of [a, a]. Furthermore, we consider matrices with an interval in each of its elements; i.e., [A,A] = ([aij ]) = ([aij , aij ]). We also write [A,A] := {A ∈ IRn×n : A ≤ A ≤ A}. By IRn×n we denote the set of all these so-called interval matrices. We also write [A]...

Let G = (V, E) be a multigraph with no loops on the vertex set V = {1, 2, . . . , n}. Define S+(G) as the set of symmetric positive semidefinite matrices A = [aij ] with aij 6= 0, i 6= j, if ij ∈ E(G) is a single edge and aij = 0, i 6= j, if ij / ∈ E(G). Let M+(G) denote the maximum multiplicity of zero as an eigenvalue of A ∈ S+(G) and mr+(G) = |G|−M+(G) denote the minimum semidefinite rank of...

We prove that there is a universal constant C > 0 with the following property. Suppose that n ∈ N and that A = (aij) ∈ Mn(R) is a symmetric stochastic matrix. Denote the second-largest eigenvalue of A by λ2(A). Then for any finite-dimensional normed space (X, ∥ · ∥) we have ∀x1, . . . , xn ∈ X, dim(X) > 1 2 exp ( C 1− λ2(A) √ n ( ∑n i=1 ∑n j=1 ∥xi − xj∥ 2 ∑n i=1 ∑n j=1 aij∥xi − xj∥ ) 1 2 )

We establish a characterization of almost P-matrices via sign non-reversal property. In this we are inspired by the analogous results for N-matrices. Next, interval hull two m×n matrices A=(aij) and B=(bij), denoted I(A,B), is collection all C∈Rn×n such that each cij convex combination aij bij. Using property, identify finite subset I(A,B) determines if in N-matrices/almost P-matrices. This pro...

Let A = (aij) be the generic n×n circulant matrix given by aij = xi+j , with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n...

We denote vectors in bold small letters, for example v. If v is a vector of dimension d, then vk, k ∈ {1, . . . , d} is the kth component of v, and vI , I ⊆ {1, . . . , d} denotes the vector with components vk, k ∈ I (with the vk’s arranged in the same order as in v). Similarly, matrices will be written in bold capital letters, for example A. If A is an m × n matrix, then Aij represents the ijt...