We extend the results in [5] to non-compactly supported perturbations for a class of symmetric first order systems. The purpose of this note is to extend the results obtained in [5] to non-compactly supported perturbations. Consider in R, n ≥ 3 odd, a first order matrix-valued differential operator of the form ∑n j=1A 0 jDxj , A 0 j being constant Hermitian d× d matrices, and denote by G0 its s...

The sign-rank of a real matrix M is the least rank of a matrix R in which every entry has the same sign as the corresponding entry of M.We determine the sign-rank of every matrix of the form M = [ D(|x ∧ y|) ]x,y, where D : {0, 1, . . . , n} → {−1,+1} is given and x and y range over {0, 1}n . Specifically, we prove that the sign-rank of M equals 22̃(k), where k is the number of times D changes s...

The following notational conventions and terminology will be in force. Inequalities for vectors are understood component-wise. Given two matrices M and N with the same number of columns, the notation col(M, N) denotes the matrix obtained by stacking M over N. Let M be a matrix. The sub-matrix M JK of M is the matrix whose entries lie in the rows of M indexed by the set J and the columns indexed...

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form Lu(x) = − ∑ aij∂iju+ PV ∫ Rn (u(x)− u(x+ y))K(y)dy. These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case aij ≡ 0 and for ...

Let X be a projective manifold of dimension n, a symmetric differential of degree m is a section of the m-th symmetric power of the sheaf of holomorphic 1-forms, SΩX . A symmetric differential w of degree m is of rank 1 if it can be locally written in the form w|U = fμ, where μ ∈ H(U,ΩX) and f ∈ O(U). Symmetric differentials of degree 1, i.e. holomorphic 1-forms, are trivially of rank 1. Holomo...

In this manuscript we will present a new fast technique for solving the generalized eigenvalue problem T x = λSx, in which both matrices T and S are symmetric tridiagonal matrices and the matrix S is assumed to be positive definite.1 A method for computing the eigenvalues is translating it to a standard eigenvalue problem of the following form: L−1T L−T (LT x) = λ(LT x), where S = LLT is the Ch...

An integral symmetric matrix S = (sij) ∈ M sym m (Z) with sii ∈ 2Z gives rise to an integral quadratic form q(x) = 12 xSx on Z. If S is positive definite, the number r(q, t) of solutions x ∈ Z of the equation q(x) = t is finite, and it is one of the classical tasks of number theory to study the qualitative question which numbers t are represented by q or the quantitative problem to determine th...

We address the problem of learning a symmetric positive definite matrix. The central issue is to design parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: On-line learning with a simple square loss and finding a symmetric positiv...

We examine the problem of approximating a positive, semidefinite matrix Σ by a dyad xxT , with a penalty on the cardinality of the vector x. This problem arises in the sparse principal component analysis problem, where a decomposition of Σ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, th...

in this paper we study the impact of minkowski metric matrix on a projection in theminkowski space m along with their basic algebraic and geometric properties.the relationbetween the m-projections and the minkowski inverse of a matrix a in the minkowski spacemis derived. in the remaining portion commutativity of minkowski inverse in minkowski spacem is analyzed in terms of m-projections as an a...