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We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d. 相似文献
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Conditions for strong ellipticity and M-eigenvalues 总被引:1,自引:0,他引:1
The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define
M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of
the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive
definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor
is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order
positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods
for finding M-eigenvalues are presented.
相似文献
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As a global polynomial optimization problem, the best rank-one approximation to higher order tensors has extensive engineering
and statistical applications. Different from traditional optimization solution methods, in this paper, we propose some Z-eigenvalue
methods for solving this problem. We first propose a direct Z-eigenvalue method for this problem when the dimension is two.
In multidimensional case, by a conventional descent optimization method, we may find a local minimizer of this problem. Then,
by using orthogonal transformations, we convert the underlying supersymmetric tensor to a pseudo-canonical form, which has
the same E-eigenvalues and some zero entries. Based upon these, we propose a direct orthogonal transformation Z-eigenvalue
method for this problem in the case of order three and dimension three. In the case of order three and higher dimension, we
propose a heuristic orthogonal transformation Z-eigenvalue method by improving the local minimum with the lower-dimensional
Z-eigenvalue methods, and a heuristic cross-hill Z-eigenvalue method by using the two-dimensional Z-eigenvalue method to find
more local minimizers. Numerical experiments show that our methods are efficient and promising.
This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120). 相似文献
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In this paper, we have proposed an upper bound for the largest Z-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which is called the Brauer upper bound:■where■ As applications, a bound on the Z-spectral radius of uniform hypergraphs is presented. 相似文献
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