Academic literature on the topic 'Real symmetric tensors'

Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

<abstract><p>$ \mathcal{H} $-tensors play a key role in identifying the positive definiteness of even-order real symmetric tensors. Some criteria have been given since it is difficult to judge whether a given tensor is an $ \mathcal{H} $-tensor, and their range of judgment has been limited. In this paper, some new criteria, from an increasing constant $ k $ to scale the elements of a given tensor can expand the range of judgment, are obtained. Moreover, as an application of those new criteria, some sufficient conditions for judging positive definiteness of even-order real symmetric tensors are proposed. In addition, some numerical examples are presented to illustrate those new results.</p></abstract>

Given a symmetric tensor on a real vector bundle of dimension two, we construct a space where this tensor corresponds to a scalar function. We prove that under certain regularity conditions such a space and the corresponding scalar function are smooth. We study the topology of this space for the case of surfaces and produce a version of Morse inequalities for symmetric tensors. We apply our results to the geometry of surfaces.

Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

We study the case of a real homogeneous polynomial whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

LetVbe a real vector space of dimension4with a nondegenerate symmetric bilinear form of signature(1,3). We show that there exists no algebraic curvature tensorRonVso that its associated skew-symmetric operatorR(⋅)has rank4and constant eigenvalues on the Grassmannian of nondegenerate2-planes inV.

I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than 2c+1, if d is odd, and t is greater or equal than max(3,2c+1), if d is even, where c is the number of ovals in the zero locus of f. About binary forms, I prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms. Previously, I worked on the computation of the real ranks of real binary forms of degree four and five with assigned complex rank.

This book is an introduction to the modern methods of the general theory of relativity, tensor calculus, space time geometry, the classical theory of fields, and a variety of theoretical physics oriented topics rarely discussed at the level of the intended reader (mid-college physics major). It does so from the point of view of the so-called principle of covariance; a symmetry that underlies most of physics, including such familiar branches as Newtonian mechanics and electricity and magnetism. The book is written from a minimalist perspective, providing the reader with only the most basic of notions; just enough to be able to read, and hopefully comprehend, modern research papers on these subjects. In addition, it provides a (hopefully short) preparation for the student to be able to conduct research in a variety of topics in theoretical physics; with particular emphasis on physics in curved spacetime backgrounds. The hope is that students with a minimal mathematical background in calculus and only some introductory courses in physics may be able to study this book and benefit from it.

The discrete element modelling allows the study of the mechanical behaviour of a collection of solids, submitted to dynamic excitations. The proposed definition of a stress tensor concerns a cell constituted of its kernel, and its neighbouring solids. We demonstrate that taking into account the inertial effects allows the symmetry of the stress tensor. The relative weight of the centrifugal effects remaining in the final expression of the stress tensor is evaluated in order to determine whether or not it has to be taken into account, depending on the application. The proposed definition allows the definition of the load level on different parts of the discontinuous structure. Applications to real masonry cases are presented.

The analysis is performed on a hydraulic press which is intended for use in the automotive industry and is a part of a production line. The final phase of manufacture of interior and acoustic parts takes place in this press. These interior and acoustic parts are made of sandwich fabric which is inserted into the heated mould of the press and by treatment with a defined pressure (or, more precisely, a defined compression) and temperature, it is formed into its final shape. This press has a frame with four columns and it is not preloaded. Two double acting hydraulic cylinders placed on an upper cross beam exert the compressive force. Due to continuously increasing demands on the accuracy and quality of products not only in the automotive industry, it is necessary to ensure compliance with the accuracy of certain values of machine operation. Especially in this case, the value of accuracy substantially depends on the clamping plates of the press, for which a certain value of flatness is required, both at room temperature and at elevated temperatures. To achieve this accuracy, it is necessary to guarantee sufficient stiffness of the machine to resist the pressing force with the smallest deformation possible. Another crucial factor affecting the accuracy of the machine is heating of the heated clamping plates. Unequal heating of parts of the frame causes additional deformation that has to be quantified and eliminated. The main aim was to verify the design of the press by numerical computation and gather knowledge for modifying the topological design of the press so that it fulfils the required customer parameters of flatness and parallelism for different types of loading. A computational model of the press was created for the numerical solution of a coupled temperature-displacement numerical analysis. The analysis was performed using the finite element method in Abaqus software. The press is symmetrical in two orthogonal planes and the load of the press is considered to be centric. On the basis of these two factors it was possible to carry out the analysis by considering only a quarter of the press. The analysis was used to investigate the effects of static and combined loads from the pressing force and heat on the press. The influence of a cooling circuit located in the press frame for the reduction of frame deformation (and deformation of clamping plates) was investigated. Contacts were defined among individual parts to ensure the computational model had characteristics as close as possible to the real press. The analysis was solved as stationary, on the basis that the cooling of the tool between individual pressing cycles is negligible. The insulating plates are made of a particulate composite material which was considered to have isotropic properties depending on the temperature. For strength evaluation of composite materials all individual components of the stress tensor were examined according to the maximum stress criterion. Hook’s law was considered to be valid for the metallic materials. Von Mises criterion was used to evaluate the strength of the metallic materials. The geometry of the press was discretized using 3D linear thermally coupled brick elements with 8 nodes and full integration (C3D8T). There were approximately 174,000 elements in total. Design procedures for designing a press frame with higher work accuracy (flatness) were proposed with the example of the simplified model of the press table. With these methods it is possible to achieve times higher accuracy than is achieved with conventional method.