in this paper we will prove that if $l$ is a continuous symmetric n-linear form on a hilbert spaceand $widehat{l}$ is the associated continuous n-homogeneous polynomial, then $||l||=||widehat{l}||$.for the proof we are using a classical generalized  inequality due to s. bernstein for entire functions of exponential type. furthermore we study the case that if x is a banach space then we have tha...

This paper provides a simple approach for solving a semidefinite program, SDP. As is common with many other approaches, we apply a primal-dual method that uses the perturbed optimality equations for SDP, Fμ(X, y, Z) = 0, where X, Z are n × n symmetric matrices and y ∈ Rm. However, as in Kruk et al [19], we look at this as an overdetermined system of nonlinear (bilinear) equations on vectors X, ...

In this paper, we consider the following type of non-local (pseudo-differential) operators L on Rd: Lu(x) = 1 2 d ∑ i,j=1 ∂ ∂xi ( aij(x) ∂ ∂xj ) + lim ε↓0 ∫ {y∈Rd: |y−x|>ε} (u(y) − u(x))J(x, y)dy, where A(x) = (aij(x))1≤i,j≤d is a measurable d × d matrix-valued function on Rd that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial nonnegative kernel on Rd × Rd satisfy...

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$‎, ‎to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$‎. ‎If there exists an $ntimes n$ symmetric doubly stochastic ...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In this paper we describe block algorithms for the reduction of a real symmetric matrix to tridiagonal form and for the reduction of a general real matrix to either bidiagonal or Hessenberg form using Householder transformations. The approach is to aggregate the transformations and to apply them in a blocked fashion, thus achieving algorithms that are rich in matrix-matrix operations. These red...

In the case of a linear constant coefficient differential equation, & = Ax, where x is a (complex) n-vector and A is a (complex) nXn matrix, it is well known when all solutions are bounded; namely, if all eigenvalues of A are purely imaginary and all elementary divisions of A are simple. This condition is equivalent to the Jordan normal form, / , of A being (Hermitian) skew symmetric. That is i...

Two vectors a and b are called perpendicular if a'Gb = 0. Two subspaces yt* and 9f** are perpendicular if x'Gy = 0 for all xC9i*, yC9t**Obviously, these relations are symmetric. The vectors perpendicular to a given vector respectively to a given wz-space form an (n — 1)space, respectively in — ra)-space. We call the matrix T orthogonal if it leaves the expression x'Gy unchanged for all x and y....

The speed-gradient variational principle (SG-principle) for nonstationary far from equilibrium systems is formulated and illustrated by examples. The SG-model of transient (relaxation) dynamics for systems of a finite number of particles based on maximum entropy principle is derived. It has the form dX(t)/dt = A lnX(t), where X(t) is the vector of the cell populations, A is a symmetric matrix w...