Strictly semi-positive tensors and the boundedness of tensor complementarity problems

In this paper, we present the boundedness of solution set of tensor complementarity problem defined by a strictly semi-positive tensor. For strictly semi-positive tensor, we prove that all \(H^+(Z^+)\)-eigenvalues of each principal sub-tensor are positive. We define two new constants associated with \(H^+(Z^+)\)-eigenvalues of a strictly semi-positive tensor. With the help of these two constants, we establish upper bounds of an important quantity whose positivity is a necessary and sufficient condition for a general tensor to be a strictly semi-positive tensor. The monotonicity and boundedness of such a quantity are established too.     

On the free automata and tensor product

In this paper we shall introduce the algebraic structure of a tensor product for arbitrarily given automata, giving a defintion of the tensor product for automata. We introduce and study that for any setX there always exists a free automaton onX. The existence of a tensor product for automata will be investigated in the same way like modules do.     

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Leibniz cohomology and the calculus of variations

A geometric interpretation of the Leibniz coboundary is given in terms of the calculus of variations. For a differentiable manifold M, Leibniz cohomology generalizes de Rham cohomology by including all tensors as cochains. When applied to two-tensors, the conditions for the vanishing of a Leibniz cochain are related to the necessary conditions to achieve an extreme value of the integral of the tensor over an immersed surface. A local formula for the coboundary of any tensor is given in terms of a coordinate chart, and the Leibniz coboundary of the Riemann curvature tensor is computed in terms of the derivative of sectional curvature.     

Some geometric properties of the Bakry-Émery-Ricci tensor

The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Émery tensor. We show that the Bakry-Émery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Émery tensor and measured Gromov-Hausdorff limits.     

Tensor inversion and its application to the tensor equations with Einstein product

Recently, the inverse of an even-order square tensor has been put forward in [Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl. 2013;34(2):542–570] by means of the tensor group consisting of even-order square tensors equipped with the Einstein product. In this paper, several necessary and sufficient conditions for the invertibility of a tensor are obtained, and some approaches for calculating the inverse (if it exists) are proposed. Furthermore, the Cramer's rule and the elimination method for solving the tensor equations with the Einstein product are derived. In addition, the tensor eigenvalue problem mentioned in [Qi L-Q. Theory of tensors (hypermatrices). Hong Kong: Department of Applied Mathematics, The Hong Kong Polytechnic University; 2014] can also be addressed by using the elimination method mentioned above. By the way, the LU decomposition and the Schur decomposition of matrices are extended to tensor case. Numerical examples are provided to illustrate the main results.     

Critical metrics of the Schouten functional

Given a Riemannian metric on a compact smooth manifold, we consider its Schouten tensor, which is a tensor field of type (0, 2) arising in the remainder of the Weyl part in the standard decomposition of the curvature tensor of the metric. We study extremal properties of the Schouten functional, defined to be the scaling-invariant L ²-norm of the Schouten tensor. It is proved, for instance, that space form metrics are characterized as critical points of the Schouten functional among conformally flat metrics.     

An Iterative Method for Finding the Least Solution to the Tensor Complementarity Problem

In this paper, we are concerned with finding the least solution to the tensor complementarity problem. When the involved tensor is strongly monotone, we present a way to estimate the nonzero elements of the solution in a successive manner. The procedure for identifying the nonzero elements of the solution gives rise to an iterative method of solving the tensor complementarity problem. In each iteration, we obtain an iterate by solving a lower-dimensional tensor equation. After finitely many iterations, the method terminates with a solution to the problem. Moreover, the sequence generated by the method is monotonically convergent to the least solution to the problem. We then extend this idea for general case and propose a sequential mathematical programming method for finding the least solution to the problem. Since the least solution to the tensor complementarity problem is the sparsest solution to the problem, the method can be regarded as an extension of a recent result by Luo et al. (Optim Lett 11:471–482, 2017). Our limited numerical results show that the method can be used to solve the tensor complementarity problem efficiently.     

On the tensor product of well generated dg categories

We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying a universal property. The resulting monoidal structure is symmetric and closed with respect to the cocontinuous RHom of dg categories (in the sense of Toën [32]). We give a construction of the tensor product in terms of localisations of dg derived categories, making use of the enhanced derived Gabriel-Popescu theorem [27]. Given a regular cardinal α, we define and construct a tensor product of homotopically α-cocomplete dg categories and prove that the well generated tensor product of α-continuous derived dg categories (in the sense of [27]) is the α-continuous dg derived category of the homotopically α-cocomplete tensor product. In particular, this shows that the tensor product of well generated dg categories preserves α-compactness.     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005     

On algorithms for and computing with the tensor ring decomposition

Tensor decompositions such as the canonical format and the tensor train format have been widely utilized to reduce storage costs and operational complexities for high‐dimensional data, achieving linear scaling with the input dimension instead of exponential scaling. In this paper, we investigate even lower storage‐cost representations in the tensor ring format, which is an extension of the tensor train format with variable end‐ranks. Firstly, we introduce two algorithms for converting a tensor in full format to tensor ring format with low storage cost. Secondly, we detail a rounding operation for tensor rings and show how this requires new definitions of common linear algebra operations in the format to obtain storage‐cost savings. Lastly, we introduce algorithms for transforming the graph structure of graph‐based tensor formats, with orders of magnitude lower complexity than existing literature. The efficiency of all algorithms is demonstrated on a number of numerical examples, and in certain cases, we demonstrate significantly higher compression ratios when compared to previous approaches to using the tensor ring format.     

On the Nuclear Norm and the Singular Value Decomposition of Tensors

Finding the rank of a tensor is a problem that has many applications. Unfortunately, it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a systematic study various measures of orthogonality in tensor product spaces and we give a new generalization of the singular value decomposition to higher-order tensors.     

Anisotropic tensor spaces and functions,the mean values of tensor quantities and the limiting criteria

The concept of an anisotropic vector space with a tensor basis which is invariant under a symmetry transformations of a three-dimensional Euclidean vector space is introduced using the example of symmetric second- and fourth-rank Euclidean tensors. In addition to the traditional operation of summation, the operation of multiplication in a fixed tensor basis is introduced for the elements of this space, that is, the axioms of a ring with an identity element and zero divisors, which enable one to carry out algebraic and functional operations. The possibilities of the proposed mathematical procedure are illustrated using examples of anisotropic tensor functions of a tensor argument, by the general solution of the classical problem of calculating the mean value of the tensor of the moduli of elasticity of a single-phase grain-oriented polycrystalline material and the construction of the strength surfaces of anisotropic composite materials.     

On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations

This paper deals with studying some of well‐known iterative methods in their tensor forms to solve a Sylvester tensor equation. More precisely, the tensor form of the Arnoldi process and full orthogonalization method are derived by using a product between two tensors. Then tensor forms of the conjugate gradient and nested conjugate gradient algorithms are also presented. Rough estimation of the required number of operations for the tensor form of the Arnoldi process is obtained, which reveals the advantage of handling the algorithms based on tensor format over their classical forms in general. Some numerical experiments are examined, which confirm the feasibility and applicability of the proposed algorithms in practice. Copyright © 2016 John Wiley & Sons, Ltd.     

Weyl connections and the local sphere theorem for quaternionic contact structures

We apply the theory of Weyl structures for parabolic geometries developed by Čap and Slovák (Math Scand 93(1):53–90, 2003) to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot–Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold, cf. (Geom Appl 28(4):376–394, 2010). In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern–Moser tensor in CR geometry. This tensor was first discovered via different methods by Ivanov and Vasillev (J Math Pures Appl 93:277–307, 2010), and we also get an independent proof of their Local Sphere Theorem. However, as a result of our derivation of this tensor, its fundamental properties—conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure—follow as easy corollaries from the general parabolic theory.     

Control of the elastic properties of a shell working at stability

This examines a shell with elastic properties varying across the coordinates, which are prescribed by means of scalar functions of the invariants of the elasticity tensor. The basis of the arrangement of the tensor for the elasticity consists of q linear-independent tensors of the fourth range (q is the number of linear-independent components of the elasticity tensor) which are obtained by multiplying and turning the first tensor of the surface and the tensor characterizing the class of symmetry of the medium. The invariants of the elasticity tensor present in the stability equation and their derivatives are taken to be the equations and parameters for the state of the system (shell), and the problem is thus reduced to a problem of optimum equations. As an example we shall examine an orthotropic cylindrical shell with a model varying over the length under the action of external pressure.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 93–100, January–February, 1974.     

Tensor norms and operators in the category of Banach spaces

The theory of dual functors in the category of Banach spaces is applied to the study of tensor norms in the sense of Grothendieck. The dual functors of the tensor norms arising from the projective and inductive tensor product as well as from more general tensor norms, such as the norms d_p of Saphar, are identified as various spaces of operators, which include p-integral and absolutely p-summing operators. Properties of these operators are then easily derived by categorical means. Applications of the methods provide simplified proofs of composition theorems and the characterization of dual spaces of type (L).The author acknowledges the hospitality of the University of Massachusetts at Amherst during the year when the first draft of this paper was written as well as support from the Natural Sciences and Engineering Research Council of Canada.     

On exact expressions of the bending tensor in the nonlinear theory of thin shells

Some equivalent exact expressions of the bending tensor in the nonlinear theory of thin shells are reviewed. It is noted that the bending tensor, proposed by Shen et al. (2010) [X.Q. Shen, K.T. Li, Y. Ming “The modified model of Koiter’s type for the nonlinearly elastic shells”, Appl. Math. Model. 34 (2010) 3527-3535] as a third-degree polynomial of displacements, is an approximate expression, not the exact one. Then integrability of the fourth kinematic boundary condition, associated with two different but equivalent exact expressions of the bending tensor, is briefly discussed. Finally, a few modified definitions of the bending tensor proposed in the literature are recalled. Within the first-approximation theory they all lead to energetically equivalent models of elastic shells.