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# The degree of the E-characteristic polynomial of an even order tensor

Department of Mathematics, National University of Defense Technology, Changsha, Hunan 410073, PR China; Department of Mathematics, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong; Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong; School of Operations Research and Management Sciences, Qufu Normal University, Rizhao, Shandong 276800, PR China

Journal of Mathematical Analysis and Applications (Impact Factor: 1.05). 01/2007; DOI: 10.1016/j.jmaa.2006.07.064 -
##### Article: Space tensor conic programming

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**ABSTRACT:**Space tensors appear in physics and mechanics. Mathematically, they are tensors in the three-dimensional Euclidean space. In the research area of diffusion magnetic resonance imaging, convex optimization problems are formed where higher order positive semi-definite space tensors are involved. In this short paper, we investigate these problems from the viewpoint of conic linear programming (CLP). We characterize the dual cone of the positive semi-definite space tensor cone, and study the CLP formulation and the duality of positive semi-definite space tensor conic programming.Computational Optimization and Applications 10/2014; · 1.28 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The quantum eigenvalue problem arises in the study of the geometric measure of the quantum entanglement. In this paper, we convert the quantum eigenvalue problem to the Z-eigenvalue problem of a real symmetric tensor. In this way, the theory and algorithms for Z-eigenvalues can be applied to the quantum eigenvalue problem. In particular, this gives an upper bound for the number of quantum eigenvalues. We show that the quantum eigenvalues appear in pairs, i.e., if a real number $\lambda$ is a quantum eigenvalue of a square symmetric tensor $\Psi$, then $-\lambda$ is also a quantum eigenvalue of $\Psi$. When $\Psi$ is real, we show that the entanglement eigenvalue of $\Psi$ is always greater than or equal to the Z-spectral radius of $\Psi$, and that in several cases the equality holds. We also show that the ratio between the entanglement eigenvalue and the Z-spectral radius of a real symmetric tensor is bounded above in a real symmetric tensor space of fixed order and dimension.05/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.Frontiers of Mathematics in China 12/2013; 8(1). · 0.32 Impact Factor

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