67,651 research outputs found
How China’s private sector helped the government fight coronavirus
This article is republished from The Conversation under a Creative Commons Attribution NoDerivatives license: https://creativecommons.org/licenses/by-nd/4.0/. Read the original article: https://theconversation.com/how-chinas-private-sector-helped-the-government-fight-coronavirus-136100
Centrosymmetric, Skew Centrosymmetric and Centrosymmetric Cauchy Tensors
Recently, Zhao and Yang introduced centrosymmetric tensors. In this paper, we
further introduce skew centrosymmetric tensors and centrosymmetric Cauchy
tensors, and discuss properties of these three classes of structured tensors.
Some sufficient and necessary conditions for a tensor to be centrosymmetric or
skew centrosymmetric are given. We show that, a general tensor can always be
expressed as the sum of a centrosymmetric tensor and a skew centrosymmetric
tensor. Some sufficient and necessary conditions for a Cauchy tensor to be
centrosymmetric or skew centrosymmetric are also given. Spectral properties on
H-eigenvalues and H-eigenvectors of centrosymmetric, skew centrosymmetric and
centrosymmetric Cauchy tensors are discussed. Some further questions on these
tensors are raised
On a question of Demailly-Peternell-Schneider
This note aims to give an affirmative answer to an open question posed by
Demailly-Peternell-Schneider [DPS] in 2001 and recently by Peternell [P] again.
Let be a surjective morphism from a log canonical pair (X,D)
onto a -Gorenstein variety . If is nef, we show that
is pseudo-effective.Comment: J. Eur. Math. Soc. (to appear), minor corrections to some misleading
misprint
Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors
and their generating vectors in this paper. Hilbert tensors are symmetric
Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite
if and only if its generating vector is positive. An even order symmetric
Cauchy tensor is positive definite if and only if its generating vector has
positive and mutually distinct entries. This extends Fiedler's result for
symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that
the positive semi-definiteness character of an even order symmetric Cauchy
tensor can be equivalently checked by the monotone increasing property of a
homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial
is strictly monotone increasing in the nonnegative orthant of the Euclidean
space when the even order symmetric Cauchy tensor is positive definite.
Furthermore, we prove that the Hadamard product of two positive semi-definite
(positive definite respectively) symmetric Cauchy tensors is a positive
semi-definite (positive definite respectively) tensor, which can be generalized
to the Hadamard product of finitely many positive semi-definite (positive
definite respectively) symmetric Cauchy tensors. At last, bounds of the largest
H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and
several spectral properties on Z-eigenvalues of odd order symmetric Cauchy
tensors are shown. Further questions on Cauchy tensors are raised
A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem
In this paper, we consider the tensor eigenvalue complementarity problem
which is closely related to the optimality conditions for polynomial
optimization, as well as a class of differential inclusions with nonconvex
processes. By introducing an NCP-function, we reformulate the tensor eigenvalue
complementarity problem as a system of nonlinear equations. We show that this
function is strongly semismooth but not differentiable, in which case the
classical smoothing methods cannot apply. Furthermore, we propose a damped
semismooth Newton method for tensor eigenvalue complementarity problem. A new
procedure to evaluate an element of the generalized Jocobian is given, which
turns out to be an element of the B-subdifferential under mild assumptions. As
a result, the convergence of the damped semismooth Newton method is guaranteed
by existing results. The numerical experiments also show that our method is
efficient and promising
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