Sixth-order Tensor Research Articles

Simultaneous Channel Estimation and Localization of Terahertz Massive MIMO Systems via Bayesian Tensor Decomposition

In this letter, a Bayesian tensor-based scheme is proposed for simultaneous channel estimation and localization (SCEAL) in wideband terahertz (THz) massive multiple-input multiple-output (MIMO) systems with hybrid analog-digital architectures. Considering the beam squint effect, we construct the received training signal into a sixth-order tensor model, including the angle-of-arrival (AOA)/angle-of-departure (AOD), time delay, and path gains. Then, a real-valued Bayesian inference algorithm is developed to fit the constructed tensor model for SCEAL. The proposed algorithm achieves SCEAL with and without line-of-sight (LOS) path, and offers higher accuracy compared to the state-of-the-art algorithms. Simulations verify the effectiveness of the proposed algorithm.

Eigenvector-based centralities for multilayer temporal networks under the framework of tensor computation

In recent years, many centralities have been developed to identify the key nodes in multilayer and temporal networks. Among these centrality measures, eigenvector-based centralities are very efficient ranking algorithms. In the real world, some complex systems have multilayer structure and edges are dynamic, i.e., they appear and disappear over time, referred to as the multilayer temporal networks. Moreover, some eigenvector-based centralities have been extended to rank the nodes in multilayer temporal networks. However, these existing eigenvector-based centralities ignore the inter-layer interactions between different time stamps and may get incorrect ranking results of nodes. In order to better describe the multilayer and temporal features of networks, in this paper, we construct a sixth-order tensor to represent multilayer temporal networks. In particular, cosine similarity is used to depict the relationships of inter-layer. Then, we propose multilayer temporal eigenvector and PageRank centralities based on the sixth-order tensor and cosine similarity for ranking the nodes, layers and time stamps in networks, referred to as MTEIGBC and MTPRBC centralities. Furthermore, the existence and uniqueness of our centrality measures can be guaranteed under very reasonable conditions. Finally, numerical experiments on three networks are performed to demonstrate the effectiveness and superiority of our proposed ranking methods.

Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application

<abstract><p>An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor is presented. As an application, a sufficient condition for the positive definiteness of a sixth-order real symmetric tensor (also a homogeneous polynomial form) is obtained, which is used to judge the asymptotically stability of time-invariant polynomial systems.</p></abstract>

HITS centrality based on inter-layer similarity for multilayer temporal networks

In this paper, we propose a novel HITS centrality based on inter-layer similarity for multilayer temporal networks, referred to as the MT-HITS centrality. We firstly introduce a sixth-order tensor to represent multilayer temporal networks and define an inter-layer similarity index to measure the strength of the interactions between different layers. Then, based on the established tensor, we construct a set of tensor equations to define centrality vectors of nodes, layers, and time stamps, respectively. These centrality vectors can be regarded as a Perron eigenvector of a multi-homogeneous map. Under some reasonable conditions, the existence and uniqueness of our proposed centrality measure are guaranteed by existing results. To demonstrate the effectiveness and superiority of our proposed method, we test it on three networks, and then we apply it for link prediction in temporal networks.

Co-Ranking for nodes, layers and timestamps in multilayer temporal networks

Understanding the structure of multilayer temporal networks requires the evaluation of nodes importance, the relationship between them and the timestamps simultaneously. In this paper,we propose a parameters-free centrality algorithm referred to as Co-Rank. The proposed algorithm uses a sixth-order tensor to describe the multilayer temporal network which considers the inter-layer connections between the adjacent timestamps across different layers. After describing the multilayer temporal network, the next step is to build and solve a set of tensor equations following the mutual relationships to get the centrality. The existence of the centrality metric is formally proven, and the convergence of the Co-Rank is also shown so that it can be effectively applied for the ranking. The results of experiments on synthetic and real-world networks show the effectiveness of our proposed algorithm.

Mesoscopic continuum theory for liquid crystals

A summary of the mesoscopic theory of liquid crystals is presented. Mesoscopic field quantities are defined on the enlarged space of position, time and orientation. An orientation distribution function is defined as the fraction of the mesoscopic mass density over the total mass density. For the mesoscopic field quantities of mass, momentum, angular momentum and energy balance equations on the enlarged domain are formulated. They allow the derivation of a differential equation for the orientation distribution function. Measurable quantities are macroscopic ones. They are obtained from the mesoscopic quantities by averaging with the orientation distribution function. In addition to the usual macroscopic fields, alignment tensors are obtained by the averaging procedure. These tensors of successive order are a measure of the orientational order, present on the mesoscopic level. For these alignment tensors equations of motion are derived here. They form a set of coupled equations for infinitely many alignment tensors, and a closure is needed. At the lowest level, only the alignment tensor of second order is an independent variable, and the fourth order tensor is substituted by an algebraic closure relation. At this level of approximation Landau theory of phase transitions can be recovered in case of vanishing flow field. In a refined approach, the second and the fourth order alignment tensor are considered as independent variables, and the sixth order tensor is eliminated by an algebraic closure relation. This refined alignment tensor dynamics is considered in detail in the case of a uniaxial orientation distribution function.

Complete symmetry classification and compact matrix representations for 3D strain gradient elasticity

Strain Gradient Elasticity (SGE) is now often used in mechanics and physics, owing to its capability to model some non-classical phenomena, such as size effects, in materials and structures. However, certain fundamental questions about it have not yet received complete responses. In its linear setting, the constitutive law of SGE is characterized by a fourth-order elasticity tensor, a fifth-order one and a sixth-order one. Even if the matrix representations for the 3D fourth- and sixth-order elasticity tensors are available for all possible symmetry classes, the counterparts for the 3D fifth-order tensor, whose presence is unavoidable for materials with non-centrosymmetric microstructure, are still lacking. In addition, although the symmetry classes for each of the fourth-, fifth- and sixth-order elasticity tensor spaces are known, the symmetry classes for these tensor spaces as a whole have never been reported and clarified in the literature. The present work solves these two fundamental problems preventing the full understanding and exploitation of the linear constitutive law of SGE. Precisely, the matrix representations of the fifth-order tensor for all of its 29 symmetry classes are provided in a compact and well-structured way. Further, linear SGE is shown to possess 48 symmetry classes, and for each of these symmetry classes, the matrix representations of its fourth-, fifth- and sixth-order tensors are now available.

Exact solution for size-dependent elastic response in laminated beams considering generalized first strain gradient elasticity

In this paper, exact solutions for the static bending response of laminated composite beams have been derived considering the effect of strain gradient elasticity. Separate solutions for the laminate comprising of isotropic and orthotropic laminae have been presented here. The sixth-order tensor for the higher order elastic coefficients has been evaluated using the generalized first strain gradient elasticity model with three independent material length constants. The significance of the gradient elasticity in low-dimensional structures has been established employing numerical examples of micro and nano-beams, and comparing the current results with the classical elasticity results. A parametric study over the variation of the axial, transverse Cauchy and physical stresses across the thickness of the beam has been conducted. Also, the discontinuity of the inter-laminar transverse Cauchy stresses and higher order stresses across the thickness has been studied in the current framework. The exact solutions developed in this paper may be used as benchmark results for validating further research involving the strain gradient elastic response of low-dimensional laminated composite beams.

Crack opening displacements under remote stress gradient: Derivation with a canonical basis of sixth order tensors

In this paper, we derive the crack opening displacement of a penny-shaped crack embedded in an infinite isotropic elastic medium and subjected to a remote constant stress gradient. The solution is derived by taking advantage of the solution of the equivalent ellipsoidal inclusion problem subjected to a linear polarization. The case of the penny-shaped crack is thereafter investigated by considering the case of a spheroidal cavity which has one principal axis infinitesimally small compared to both others. The derivation of the explicit solution for the inhomogeneity subjected to a remote stress gradient raises the problem of the inversion of a sixth order tensor. For the problem having a symmetry axis (this including the case of penny shaped crack), this problem can be tackled by using a decomposition on the canonical basis for transversely isotropic sixth order tensors.

Algebra of Transversely Isotropic Sixth Order Tensors and Solution to Higher Order Inhomogeneity Problems

In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

On the inversion of non symmetric sixth-order isotropic tensors and conditions of positiveness of third-order tensor valued quadratic functions

Abstract In the present paper we propose new results concerning linear tensorial algebra for third-order and non symmetric isotropic sixth-order tensors in the most general case (i.e. having not the major and minor symmetries). Such tensors are used, for instance, in the theory of microstructured elastic media. A formalism based on an irreducible basis for isotropic sixth-order tensors is introduced, which is useful for performing the classical tensorial operations. Specially, a condensed expression for the product between two isotropic sixth-order tensors is provided, which allows the obtaining of a condition on these tensors for being invertible and a closed form expression of the inverse of such a tensor. Finally, the condition of positiveness of third-order tensor-valued quadratic functions is derived. For instance, such conditions are required for computing the elastic energy of microstructured media.

Conditions for Strong Ellipticity of Anisotropic Elastic Materials

In this paper, we derive necessary and sufficient conditions for the strong ellipticity condition of anisotropic elastic materials. We first observe that the strong ellipticity condition holds if and only if a second order tensor function is positive definite for any unit vectors. Then we further link this condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors and a sixth-order tensor. In particular, we consider conditions of strong ellipticity of the rhombic classes, for which we need to check the copositivity of three second-order tensors and the positive definiteness of a sixth-order tensor. A direct method is presented to verify our conditions.

Tensor Framework and Combined Symmetry for Hypertext Mining

We have made a case here for utilizing tensor framework for hypertext mining. Tensor is a generalization of vector and tensor framework discussed here is a generalization of vector space model which is widely used in the information retrieval and web mining literature. Most hypertext documents have an inherent internal tag structure and external link structure that render the desirable use of multidimensional representations such as those offered by tensor objects. We have focused on the advantages of Tensor Space Model, in which documents are represented using sixth-order tensors. We have exploited the local-structure and neighborhood recommendation encapsulated by the proposed representation. We have defined a similarity measure for tensor objects corresponding to hypertext documents, and evaluated the proposed measure for mining tasks. The superior performance of the proposed methodology for clustering and classification tasks of hypertext documents have been demonstrated here. The experiment using different types of similarity measure in the different components of hypertext documents provides the main advantage of the proposed model. It has been shown theoretically that, the computational complexity of an algorithm performing on tensor framework using tensor similarity measure as distance is at most the computational complexity of the same algorithmperforming on vector space model using vector similarity measure as distance.

Sixth-order Fitted Closures for Short-fiber Reinforced Polymer Composites

Orientation tensors are widely used in short-fiber reinforced injection molding simulations of polymer composite products. In these computations, the evolution of an orientation tensor is given as a function of higher-order tensors, necessitating the use of a closure approximation. Current fourth-order fitted closures enable accurate and efficient computation of second-order tensor evolutions, but have been shown to be limited in their ability to accurately represent the fiber orientation distribution function. This article presents two sixth-order fitted closures that may be used to improve the accuracy of fiber orientation distribution function reconstructions. These new closures are written in terms of the second-order orientation tensor where it is assumed that the orthogonal planes of material symmetry of the sixth-order tensor are defined by the principal directions of the second-order tensor. Sixth-order tensor components are fit to either eigenvalues or invariants of the second-order tensor over numerous flow conditions and interaction coefficients. Example calculations illustrate that the new closures are capable of surpassing accuracy limits experienced when using any fourth-order closure.

An invariant based fitted closure of the sixth-order orientation tensor for modeling short-fiber suspensions

Orientation tensors are commonly used in short-fiber reinforced injection molding simulations of industrial polymer composite products. The evolution equation for each even-order orientation tensor is written in terms of the next higher even-order orientation tensor necessitating the use of a closure. It has been shown that current fourth-order closures approach the fourth-order truncation limit when representing the fiber orientation distribution function so that an increase in accuracy necessitates the development of a robust sixth-order closure. This paper presents a new fitted sixth-order closure formed from a general expression for a fully symmetric sixth-order tensor written as a function of a fourth-order orientation tensor. The components of this sixth-order closure are fit to a linear polynomial of the fourth-order orientation tensor invariants whose coefficients are computed by fitting the sixth-order components obtained from the closure to those computed from distribution function simulations for a variety of flow fields and interaction coefficients. The fitted sixth-order closure is shown to more accurately predict the second-order orientation tensor than simulations that employ existing fourth-order and sixth-order closures. Additionally, it is shown that the sixth-order closure more accurately represents the distribution function of fibers than current closure methods.

(3+1)-dimensional optical soliton dragging logic.

A review of the varieties of optical solitons and their possible interactions, combined with the requirements for a robust digital logic gate motivate the use of (3+1)-dimensional optical solitons (light bullets) as information carriers and soliton dragging gates as switches. Soliton dragging is the asymmetric interaction between two initially overlapping, orthogonally polarized solitons propagating at different angles so that a weak signal soliton can drag a strong pump out of a spatial aperture, thereby implementing a phase-insensitive, high-contrast, logical switch with gain. Light bullets may be an ideal choice for use in these soliton dragging gates but are unstable in Kerr media, but stable (for sufficient pulse energy) in materials with physically reasonable saturating or negative ${\mathit{n}}_{4}$${\mathit{I}}^{2}$ nonlinearities. An efficient technique for the propagation of spherically symmetric (3+1)-dimensional field envelopes is developed and used to verify the theoretical stability predictions. A split-step numerical algorithm that models the propagation and phase-independent interaction of arbitrary (3+1)-dimensional, vector e.m. fields in anisotropic media with up to sixth-order tensor nonlinearities is developed and used to demonstrate the features of the gates. NOT and single-stage, two- and four-input NOR light-bullet dragging logic gates are simulated and their performance over a range of operating parameters is presented. It is shown that, with material parameters in the range of those currently available from highly nonlinear organic crystals, high-contrast, all-optical, soliton logic gates with a clock rate greater than 1 THz, latency of a few picoseconds, and switching energy of 25 pJ may be possible.

High-order matter tensors in symmetrical systems

The schemes of components of general fifth and sixth order tensors of polar and axial nature are derived for all the crystallographic groups by using the method of direct inspection for triclinic, monoclinic, rhombic, tetragonal and cubic groups, and elementary group theory for trigonal and hexagonal groups.