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We use the Brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings. Also, we provide some finer criteria for the existence of real eigenpairs of two-dimensional real tensors and give a complete classification of the Brouwer degree zero and ±2 maps induced by general third order two-dimensional real tensors.  相似文献   

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We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m-1)-homogeneous operator from l1 into lp (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.  相似文献   

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In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated ?-module M, in terms of a generating set for M.  相似文献   

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Let U be an n-dimensional vector space over an algebraically closed field F. Let U(m) denote the mth symmetric power of U. For each positive integer k≤min{m,n}, let Dk denote the set of all nonzero decomposable elements x1 xm in U(m) such that dim(x1 xm ) = k and Ek denote the set of all decomposable elements x1 xm in U(m) such that dim(x1 xm ) ≤ k. In this paper we first show that Ek is an algebraic variety with Dk as a dense subset and determine the dimension of Ek . We next use these results to study the structure of linear mappings T on Um such that T(Dk ) ? Dk or T(Ek ) ? Ek for some fixed k.  相似文献   

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In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous PerronFrobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A k-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of k-th roots of unity. A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues. In particular, the spectral symmetry(index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be 1-symmetric or 2-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.  相似文献   

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There is a well-developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit property, weak symmetries, and the structure of the nilradical of the isometry group. Also, we give a number of examples of weakly symmetric pseudo-Riemannian manifolds, some mirroring the Riemannian case and some indicating the problems in extending Riemannian results to weakly symmetric pseudo-Riemannian spaces.  相似文献   

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A ring R is defined to be GWS   if abc=0abc=0 implies bac⊆N(R)bacN(R) for a,b,c∈Ra,b,cR, where N(R)N(R) stands for the set of nilpotent elements of R. Since reduced rings and central symmetric rings are GWS, we study sufficient conditions for GWS rings to be reduced and central symmetric. We prove that a ring R is GWS   if and only if the n×nn×n upper triangular matrices ring Un(R,R)Un(R,R) is GWS for any positive integer n. It is proven that GWS rings are directly finite and left min-abel. For a GWS ring R, R is a strongly regular ring if and only if R is a von Neumann regular ring if and only if R is a left SF   ring and J(R)=0J(R)=0; R is an exchange ring if and only if R is a clean ring. Finally, we show that GWS exchange rings have stable range 1 and a GWS semiperiodic ring R   with N(R)≠J(R)N(R)J(R) is commutative.  相似文献   

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The eigenvalues of random symmetric matrices   总被引:1,自引:0,他引:1  
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Let x1,…,xm be linearly independent vectors. We give a necessary and sufficient condition for x11…1xm=y11…1ym to hold. Some consequences of this result are also presented.  相似文献   

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We use variational methods to give a positive answer to a conjecture posed by Liqun Qi [L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput. 40 (2005) 1302-1324] regarding the real eigenvalues of certain higher order tensors.  相似文献   

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In this paper, we shall use the Shannon-Whittacker-Kotelnikov sampling theorem to approximate the eigenvalues of the string, y″(t) + μ2w(t)y(t) = 0, where the weight w(t) ≥ 0 is allowed to vanish on subintervals. After a discussion on the truncation error, numerical results are provided.  相似文献   

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The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.  相似文献   

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An algorithm has been developed for finding the eigenvalues of a symmetric matrixA in a given interval [a, b] and the corresponding eigenvectors using a modification of the method of simultaneous iteration with the same favorable convergence properties. The technique is most suitable for large sparse matrices and can be effectively implemented on a parallel computer such as the ILLIAC IV.  相似文献   

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We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature \((n-1,1)\) and trans-Lorentzian signature \((n-2,2)\).  相似文献   

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We consider d×d tensors A(x) that are symmetric, positive semi-definite, and whose row-divergence vanishes identically. We establish sharp inequalities for the integral of (det?A)1d?1. We apply them to models of compressible inviscid fluids: Euler equations, Euler–Fourier, relativistic Euler, Boltzman, BGK, etc. We deduce an a priori estimate for a new quantity, namely the space–time integral of ρ1np, where ρ is the mass density, p the pressure and n the space dimension. For kinetic models, the corresponding quantity generalizes Bony's functional.  相似文献   

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