Biquadratic tensors,biquadratic decompositions,and norms of biquadratic tensors

Biquadratic tensors play a central role in many areas of science.Examples include elastic tensor and Eshelby tensor in solid mechanics,and Riemannian curvature tensor in relativity theory.The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor,respectively.The tensor product operation is closed for biquadratic tensors.All of these motivate us to study biquadratic tensors,biquadratic decomposition,and norms of biquadratic tensors.We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure.Then,either the number of variables is reduced,or the feasible region can be reduced.We show constructively that for a biquadratic tensor,a biquadratic rank-one decomposition always exists,and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition.We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor.Finally,we define invertible biquadratic tensors,and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse,and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor,and the spectral norm of its inverse.     

Coherent sequences of polynomial ideals on Banach spaces

This article deals with the relationship between an operator ideal and its natural polynomial extensions. We define the concept of coherent sequence of polynomial ideals and also the notion of compatibility between polynomial and operator ideals. We study the stability of these properties for maximal and minimal hulls, adjoint and composition ideals. We also relate these concepts with conditions on the underlying tensor norms (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Index Theory For Quantum Dynamical Semigroups

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

     

Simultaneous state‐time approximation of the chemical master equation using tensor product formats

We apply the novel tensor product formats (tensor train, quantized TT [QTT], and QTT‐Tucker) to the solution of d‐dimensional chemical master equations for gene regulating networks (signaling cascades, toggle switches, and phage‐ λ). For some important cases, for example, signaling cascade models, we prove analytical tensor product representations of the system operator. The quantized tensor representations (QTT, QTT‐Tucker) are employed in both state space and time, and the global state‐time (d + 1)‐dimensional system is solved in the tensor product form by the alternating minimal energy iteration, the ALS‐type algorithm. This approach leads to the logarithmic dependence of the computational complexity on the volume of the state space. We investigate the proposed technique numerically and compare it with the direct chemical master equation solution and some previously known approximate schemes, where possible. We observe that the newer tensor methods demonstrate a good potential in simulation of relevant biological systems. Copyright © 2014 John Wiley & Sons, Ltd.     

An infinite dimensional functional calculus in Banach algebras with a modulus

The presence of a modulus function on a Banach algebra gives rise to tensor product norms analogous to the greatest cross‐norm and allows us to extend the Waelbroeck functional calculus. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust tensor train component analysis

Robust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low-rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor ${?}_1$ norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush-Kuhn-Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real-world datasets are presented to demonstrate the effectiveness of the proposed approach.     

High-order copositive tensors and its applications

With the coming of the big data era, high-order high-dimensional structured tensors received much attentions of researchers" in recent years, and now they are developed into a new research branch in mathematics named multilinear algebra. As a special kind of structured tensor, the copositive tensor receives a special concern due to its wide applications in vacuum stability of a general scalar potential, polynomial optimization, tensor complementarity problem and tensor eigenvalue complementarity problem. In this review, we will give a simple survey on recent advances of high-order copositive tensors and its applications. Some potential research directions in the future are also listed in the paper.     

TENSOR SUM AND DYNAMICAL SYSTEMS

In this paper we introduce the concept of tensor sum semigroups. Also we have given the examples of tensor sum operators which induce dynamical system on weighted locally convex function spaces.     

Upper bounds for eigenvalues of Cauchy-Hankel tensors

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m－1)-homogeneous operator from l¹ into l^p (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

The Generalized Order Tensor Complementarity Problems

The main propose of this paper is devoted to studying the solvability of the generalized order tensor complementarity problem. We define two problems: the generalized order tensor complementarity problem and the vertical tensor complementarity problem and show that the former is equivalent to the latter. Using the degree theory, we present a comprehensive analysis of existence, uniqueness and stability of the solution set of a given generalized order tensor complementarity problem.     

Tensor sum and dynamical systems

In this paper we introduce the concept of tensor sum semigroups. Also we have given the examples of tensor sum operators which induce dynamical system on weighted locally convex function spaces.     

Low Tucker rank tensor completion using a symmetric block coordinate descent method

Low Tucker rank tensor completion has wide applications in science and engineering. Many existing approaches dealt with the Tucker rank by unfolding matrix rank. However, unfolding a tensor to a matrix would destroy the data's original multi-way structure, resulting in vital information loss and degraded performance. In this article, we establish a relationship between the Tucker ranks and the ranks of the factor matrices in Tucker decomposition. Then, we reformulate the low Tucker rank tensor completion problem as a multilinear low rank matrix completion problem. For the reformulated problem, a symmetric block coordinate descent method is customized. For each matrix rank minimization subproblem, the classical truncated nuclear norm minimization is adopted. Furthermore, temporal characteristics in image and video data are introduced to such a model, which benefits the performance of the method. Numerical simulations illustrate the efficiency of our proposed models and methods.     

Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition

In this paper, we discuss a fixed point theorem for mappings derived by a pair of mappings satisfying weak(k, k/) contractive type condition on the tensor product spaces. Let X and Y be Banach spaces and T_1 : X γ Y → X and T_2: X γ Y → Y be two operators which satisfy weak(k, k/) contractive type condition. Using T_1 and T_2, we construct an operator T on X γ Y and show that T has a unique fixed point in a closed and bounded subset of X γ Y.We derive an iteration scheme converging to this unique fixed point of T. Conversely, using a weakly contractive mapping T, we construct a pair of mappings(T_1, T_2) satisfying weak(k, k/)contractive type condition on X γ Y and from this pair, we also obtain two self mappings S_1 and S_2 on X and Y respectively with unique fixed points.     

Logarithmic Norms and Nonlinear DAE Stability

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certain unbounded operators, i.e. partial differential algebraic equations. Perturbation bounds are obtained from differential inequalities for any given norm by using the relation between Dini derivatives and semi-inner products. Simple discretizations are also considered.     

Norm estimates for functions of two operators on tensor products of Hilbert spaces

Analytic operator valued functions of two operators on tensor products of Hilbert spaces are considered. A precise norm estimate is established. Applications to operator differential equations are also discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New directions in algebraic dynamical systems

The logarithmic Mahler measure of certain multivariate polynomials occurs frequently as the entropy or the free energy of solvable lattice models (especially dimer models). It is also known that the entropy of an algebraic dynamical system is the logarithmic Mahler measure of the defining polynomial. The connection between the lattice models and the algebraic dynamical systems is still rather mysterious.     

Twisted Alexander norms give lower bounds on the Thurston norm

We introduce twisted Alexander norms of a compact connected orientable 3-manifold with first Betti number greater than one, generalizing norms of McMullen and Turaev. We show that twisted Alexander norms give lower bounds on the Thurston norm of a 3-manifold. Using these we completely determine the Thurston norm of many 3-manifolds which are not determined by norms of McMullen and Turaev.

     

Necessary conditions and a test for the stability of a system of two linear ordinary differential equations of the first order

We use the Riccati equation method for the derivation of necessary conditions and a test for the stability of a system of two linear first-order ordinary differential equations. We consider an example in which our results are compared with the results obtained by the Lyapunov and Bogdanov methods by estimating the norms of solutions via Lozinskii logarithmic norms, and by the freezing method.     

The symmetric Radon-Nikodým property for tensor norms

We introduce the symmetric Radon-Nikodým property (sRN property) for finitely generated s-tensor norms β of order n and prove a Lewis type theorem for s-tensor norms with this property. As a consequence, if β is a projective s-tensor norm with the sRN property, then for every Asplund space E, the canonical mapping is a metric surjection. This can be rephrased as the isometric isomorphism Qmin(E)=Q(E) for some polynomial ideal Q. We also relate the sRN property of an s-tensor norm with the Asplund or Radon-Nikodým properties of different tensor products. As an application, results concerning the ideal of n-homogeneous extendible polynomials are obtained, as well as a new proof of the well-known isometric isomorphism between nuclear and integral polynomials on Asplund spaces. An analogous study is carried out for full tensor products.