The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

Nonnegative tensor decomposition allows us to analyze data in their ‘native’ form and to present results in the form of the sum of rank-1 tensors that does not nullify any parts of the factors. In this paper, we propose the geometrical structure of a basis vector frame for sum-of-rank-1 type decomposition of real-valued nonnegative tensors. The decomposition we propose reinterprets the orthogonality property of the singularvectors of matrices as a geometric constraint on the rank-1 matrix bases which leads to a geometrically constrained singularvector frame. Relaxing the orthogonality requirement, we developed a set of structured-bases that can be utilized to decompose any tensor into a similar constrained sum-of-rank-1 decomposition. The proposed approach is essentially a reparametrization and gives us an upper bound of the rank for tensors. At first, we describe the general case of tensor decomposition and then extend it to its nonnegative form. At the end of this paper, we show numerical results which conform to the proposed tensor model and utilize it for nonnegative data decomposition.

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l^k,s-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l^k,s-singular values /vectors, some properties of the related l^k,s-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.

Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9(3): 241–250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n(m-1)^n-1. However, there is no known bounds on the maximal number of distinct Heigenvectors in general. We prove that for any m≥2, an m-order 2-dimensional tensor A exists such that A has 2(m - 1) distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Furthermore, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenvectors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenvectors.

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.

We study the real and complex geometric simplicity of nonnegative irreducible tensors. First, we prove some basic conclusions. Based on the conclusions, the real geometric simplicity of the spectral radius of an evenorder nonnegative irreducible tensor is proved. For an odd-order nonnegative irreducible tensor, sufficient conditions are investigated to ensure the spectral radius to be real geometrically simple. Furthermore, the complex geometric simplicity of nonnegative irreducible tensors is also studied.

An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.

Consider the problem of computing the largest eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the largest eigenvalue for any nonnegative tensors.

We study the representations of the restricted two-parameter quantum groups of types B and G. For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive lth root of unity, the factorization of simple ?θy,θz,( )-modules is possible, if and only if (2(y - z), l) = 1 for =??2n+1; (3(y - z), l) = 1 for g= G₂.

In this paper, the Klein-Gordon equation (KGE) with power law nonlinearity will be considered. The bifurcation analysis as well as the ansatz method of integration will be applied to extract soliton and other wave solutions. In particular, the second approach to integration will lead to a singular soliton solution. However, the bifurcation analysis will reveal several other solutions that are of prime importance in relativistic quantum mechanics where this equation appears.

The principal filtration of the infinite-dimensional odd contact Lie superalgebra over a field of characteristic p>2 is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the -1 component with respect to the principal grading. Finally, all the odd contact superalgebras are classified up to isomorphisms.

The base graph of a simple matroid M=(E,?) is the graph G such that V(G)=? and E(G)={BB′:B,B′∈?,|B\B′|=1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connectedsimple matroid M is Z₃-connected if |V (G)|≥5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if |V (G)| = 4. Furthermore, if for every connected component E_i (i≥2) of M, the matroid ase graph G_i of M_i = M|E_i has |V (G_i)|≥5, then G is Z₃-connected which also implies that G admits nowhere-zero 3-flow immediately.

We give a characterization of the λ-central BMO space via the boundedness of commutators of n-dimensional Hardy operators.