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A third-order three-dimensional symmetric traceless tensor, called the octupolar tensor, has been introduced to study tetrahedratic nematic phases in liquid crystals. The octupolar potential, a scalar-valued function generated on the unit sphere by that tensor, should ideally have four maxima capturing the most probable molecular orientations (on the vertices of a tetrahedron), but it was recently found to possess an equally generic variant with three maxima instead of four. It was also shown that the irreducible admissible region for the octupolar tensor in a three-dimensional parameter space is bounded by a dome-shaped surface, beneath which is a separatrix surface connecting the two generic octupolar states. The latter surface, which was obtained through numerical continuation, may be physically interpreted as marking a possible intra-octupolar transition. In this paper, by using the resultant theory of algebraic geometry and the E-characteristic polynomial of spectral theory of tensors, we give a closed-form, algebraic expression for both the dome-shaped surface and the separatrix surface. This turns the envisaged intra-octupolar transition into a quantitative, possibly observable prediction. Some other properties of octupolar tensors are also studied.

Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order threedimensional symmetric and traceless tensor has four invariants with degrees two, four, six and ten respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no polynomial syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.Key words. minimal integrity basis, irreducible function basis, symmetric and traceless tensor, syzygy. Nomenclature D a third order three-dimensional symmetric and traceless tensor with components D ijk T(m, n) the space of real tensors of order m and dimension n S(m, n) the subspace of symmetric tensors St(m, n) the subspace of symmetric and traceless tensors O(n) the orthogonal group of dimension n SO(n) the special orthogonal group of dimension n Gl(n, R) the general linear group of real matrices m n = m! n!(m−n)! the binomial coefficient for m ≥ n ≥ 0The next theorem claims that there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 , where {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of D.Theorem 4.1. Let {J 2 , J 4 , J 6 , J 10 } be an arbitrary minimal integrity basis of a third order three-dimensional symmetric and traceless tensor D. Then there is no syzygy relation among four invariants J 2 , J 4 , J 6 and J 10 .

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In signal processing, data analysis and scientific computing, one often encounters the problem of decomposing a tensor into a sum of contributions. To solve such problems, both the search direction and the step size are two crucial elements in numerical algorithms, such as alternating least squares algorithm (ALS). Owing to the nonlinearity of the problem, the often used linear search direction is not always powerful enough. In this paper, we propose two higher-order search directions. The first one, geometric search direction, is constructed via a combination of two successive linear directions. The second one, algebraic search direction, is constructed via a quadratic approximation of three successive iterates. Then, in an enhanced line search along these directions, the optimal complex step size contains two arguments: modulus and phase. A current strategy is ELSCS that finds these two arguments alternately. So it may suffer from a local optimum. We broach a direct method, which determines these two arguments simultaneously, so as to obtain the global optimum. Finally, numerical comparisons on various search direction and step size schemes are reported in the context of blind separation-equalization of convolutive DS-CDMA mixtures. The results show that the new search directions have greatly improve the efficiency of ALS and the new step size strategy is competitive.

Abstract. The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H-and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Lojasiewicz property, we prove that the sequence of iterates generated by CEST converges to an eigenvector of the tensor. When CEST is started from multiple randomly initial points, the resulting best eigenvalue could touch the extreme eigenvalue with a high probability. Finally, numerical experiments on small hypergraphs show that CEST is efficient and promising. Moreover, CEST is capable of computing eigenvalues of tensors corresponding to a hypergraph with millions of vertices. Recently, spectral hypergraph theory is proposed to explore connections between the geometry of a uniform hypergraph and H-and Z-eigenvalues of some related symmetric tensors. Cooper and Dutle [13] proposed in 2012 the concept of adjacency tensor for a uniform hypergraph. Two years later, Qi [49] gave definitions of Laplacian and signless Laplacian tensors associated with a hypergraph. When an even-uniform hypergraph is connected, the largest H-eigenvalues of the Laplacian and signless Laplacian tensors are equivalent if and only if the hypergraph is odd-bipartite [28]. This result gives a certification to check whether a connected even-uniform hypergraph is odd-bipartite or not.