Special Tensor Research Articles

Pseudosymmetric normal paracontact metric space forms admitting (α, β)− type almost η−Ricci-Yamabe solitons

In this paper, we have considered normal paracontact metric space forms ad- mitting (α, β) −type almost η−Ricci-Yamabe solitons by means of some curvature ten- sors. Ricci pseudosymmetry concepts of normal paracontact metric space forms admit- ting (α, β) −type almost η−Ricci-Yamabe soliton have introduced according to choos- ing of some special curvature tensors such as Riemann, concircular, projective, W1 curvature tensor. After that, according to choosing of the curvature tensors, necessary conditions are given for normal paracontact metric space form admitting (α, β) −type almost η−Ricci-Yamabe soliton to be Ricci semisymmetric. Then some characteriza- tions are obtained and some classifications are made under the some conditions.

On some important characterizations of Lorentz para-Kenmotsu manifolds on some special curvature tensors

In this paper, some properties of Lorentz para-Kenmotsu manifolds are studied using specified curvature tensors. The Lorentz para-Kenmotsu manifold is investigated in terms of the curvature tensors [Formula: see text] and [Formula: see text]. Initially, the tensor-based characterization of semisymmetric Lorentz para-Kenmotsu manifolds is studied. Subsequently, we consider the Lorentzian para-Kenmotsu manifold, which admits almost [Formula: see text]-Ricci solitons via these curvature tensors. According to the [Formula: see text] and [Formula: see text] curvature tensors, Ricci pseudosymmetry notions of Lorentzian para-Kenmotsu manifolds accepting [Formula: see text]-Ricci soliton have been developed. Following that, required conditions for the Lorentzian para-Kenmotsu manifold, admitting [Formula: see text]-Ricci soliton to be Ricci semisymmetric, are presented based on the curvature tensors chosen. Further, various characterizations are provided, and classifications are made under certain conditions. Finally, the characterizations of the invariant submanifolds of Lorentz para-Kenmotsu manifold on the [Formula: see text] and [Formula: see text] curvature tensors are investigated. We obtained the necessary and sufficient conditions for an invariant submanifold of a para-Kenmotsu to be [Formula: see text] and [Formula: see text] pseudoparallel.

Inequalities for eigenvalues of operators in divergence form on Riemannian manifolds isometrically immersed in Euclidean space

In this paper, we compute universal inequalities of eigenvalues of a large class of second-order elliptic differential operators in divergence form, that includes, e.g., the Laplace-Beltrami and Cheng-Yau operators, on a bounded domain in a complete Riemannian manifolds isometrically immersed in Euclidean space. A key step in order to obtain the sequence of our estimates is to get the right Yang-type first inequality. We also prove some inequalities for manifolds supporting some special functions and tensors.

Tensor Decompositions and Their Properties

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor decomposition, especially (1, σ), σ=1,2,3 tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition.

$\left(LCS\right)_{n}-$Manifolds Admitting Almost $\eta-$Ricci Solitons on Some Special Curvature Tensors

In this paper, we consider $\left(LCS\right)_{n}$ manifold admitting almost $\eta-$Ricci solitons by means of curvature tensors. Ricci pseudosymmetry concepts of $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as pseudo-projective, $W_{1}$, $W_{1}^{\ast}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are searched for $\left(LCS\right)_{n}$ manifold admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.

On the Almost $\eta-$Ricci Solitons on Pseudosymmetric Lorentz Generalized Sasakian Space Forms

In this paper, we consider Lorentz generalized Sasakian space forms admitting almost $\eta-$Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of \ Lorentz generalized Sasakian space forms admitting $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, $\mathcal{M-}$projective, $W_{1}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz generalized Sasakian space form admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made.

CP decomposition for tensors via alternating least squares with QR decomposition

AbstractThe CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low‐rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill‐conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP‐ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP‐ALS subproblems efficiently, have the same complexity as the standard CP‐ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill‐conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with nonsmooth data.

In this paper, we focus on developing a high efficient algorithm for solving d-dimension time-fractional diffusion equation (TFDE). For TFDE, the initial function or source term is usually not smooth, which can lead to the low regularity of exact solution. And such low regularity has a marked impact on the convergence rate of numerical method. In order to improve the convergence rate of the algorithm, we introduce the space-time sparse grid (STSG) method to solve TFDE. In our study, we employ the sine basis and the linear element basis for spatial discretization and temporal discretization, respectively. The sine basis can be divided into several levels, and the linear element basis can lead to the hierarchical basis. Then, the STSG can be constructed through a special tensor product of the spatial multilevel basis and the temporal hierarchical basis. Under certain conditions, the function approximation on standard STSG can achieve the accuracy order with degrees of freedom (DOF) for and DOF for , where J denotes the maximal level of sine coefficients. However, if the solution changes very rapidly at the initial moment, the standard STSG method may reduce accuracy or even fail to converge. To overcome this, we integrate the full grid into the STSG, and obtain the modified STSG. Finally, we obtain the fully discrete scheme of STSG method for solving TFDE. The great advantage of the modified STSG method can be shown in the comparative numerical experiment.

Special core tensors of multi-qubit states and the concurrency of three lines

Special core tensors of multi-qubit states and the concurrency of three lines

Semi-symmetric almost C(α)-manifold on some curvature tensors

In this article, semi-symmetry of almost C(α)-manifold is investigated on some special curvature tensors. First, the behavior of the almost C(α)-manifold is investigated when the special curvature tensors discussed are flat. Then, for these special curvature tensors, the behavior of the manifold in the semi-symmetric condition is observed and for some special curvature tensors, important properties such as the semi-symmetric almost C(α)-manifold being Einstein and η-Einstein manifold are obtained.

Almost $\eta-$Ricci Solitons on Pseudosymmetric Lorentz Sasakian Space Forms

In this paper, we consider pseudosymmetric Lorentz Sasakian space forms admitting almost $\eta-$Ricci solitons in some curvature tensors. Ricci pseudosymmetry concepts of Lorentz Sasakian space forms admits $\eta-$Ricci soliton have introduced according to the choice of some special curvature tensors such as Riemann, concircular, projective, $\mathcal{M-}$projective, $W_{1}$ and $W_{2}.$ Then, again according to the choice of the curvature tensor, necessary conditions are given for Lorentz Sasakian space form admits $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are obtained and some classifications have made under the some conditions.

ECBC: Efficient Convolution via Blocked Columnizing.

Direct convolution methods are now drawing increasing attention as they eliminate the additional storage demand required by indirect convolution algorithms (i.e., the transformed matrix generated by the im2col convolution algorithm). Nevertheless, the direct methods require special input-output tensor formatting, leading to extra time and memory consumption to get the desired data layout. In this article, we show that indirect convolution, if implemented properly, is able to achieve high computation performance with the help of highly optimized subroutines in matrix multiplication while avoid incurring substantial memory overhead. The proposed algorithm is called efficient convolution via blocked columnizing (ECBC). Inspired by the im2col convolution algorithm and the block algorithm of general matrix-to-matrix multiplication, we propose to conduct the convolution computation blockwisely. As a result, the tensor-to-matrix transformation process (e.g., the im2col operation) can also be done in a blockwise manner so that it only requires a small block of memory as small as the data block. Extensive experiments on various platforms and networks validate the effectiveness of ECBC, as well as the superiority of our proposed method against a set of widely used industrial-level convolution algorithms.

The symmetric rank and decomposition of m-order n-dimensional (n = 2,3,4) symmetric tensors over the binary field

Firstly, we propose an algebraic method to compute the symmetric rank and symmetric rank decomposition for symmetric tensors over the binary field, when the order is at least the dimension minus one. Then we discuss Comon's conjecture and rank decomposition for low dimension symmetric tensors. For 2×2×⋯×2 symmetric tensors over the binary field, we show Comon's conjecture is true except for one special tensor (actually matrix), and the rank decomposition problem is solved when the order is greater than or equal to 4. For 3×3×⋯×3 symmetric tensors over the binary field, we prove Comon's conjecture is true when the order is greater than or equal to 4, and obtain the unique rank decomposition when the order is greater than or equal to 6. Finally, for 4×4×⋯×4 symmetric tensors over the binary field, we show Comon's conjecture is true and obtain the unique rank decomposition when the order is greater than or equal to 8.

Special holomorphic tensors on orthogonal modular varieties and applications to the Lang conjecture

We introduce a method to construct special holomorphic tensors on orthogonal modular varieties from scalar-valued modular forms, and give applications to the Lang conjecture on the birational type of subvarieties of orthogonal modular varieties.

An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms

Abstract A positive definite homogeneous multivariate form plays an important role in the field of optimization, and positive definiteness of the form can be identified by a special structured tensor. In this paper, based on the equivalence between the form and the corresponding tensor, and the links of the positive definiteness of a tensor with ℋ-tensor, we propose an ℋ-tensor-based criterion for identifying the positive definiteness of multivariate homogeneous forms. Some numerical examples are provided to illustrate the efficiency and validity of our results.

Six-Dimensional Manifold with Symmetric Signature in a Unified Theory of Gravity and Electromagnetism

A six dimensional manifold of symmetric signature (3,3) is proposed as a space structure for building combined theory of gravity and electromagnetism. Special metric tensor is proposed, yielding the space which combines the properties of Riemann, Weyl and Finsler spaces. Geodesic line equations are constructed where coefficients can be divided into depending on the metric tensor (relating to the gravitational interaction) and depending on the vector field (relating to the electromagnetic interaction). If there is no gravity, the geodesics turn into the equations of charge motion in the electromagnetic field. Furthermore, symmetric six-dimensional electrodynamics can be reduced to traditional four-dimensional Maxwell system, where two additional time dimensions are compactified. A purely geometrical interpretation of the concept of electromagnetic field and point electric charge is proposed.

ST-SVD factorization and s-diagonal tensors

A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f-diagonal tensor an s-diagonal tensor. An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values. Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode of the considered tensor is $2, 3$ and $4$, we present direct sufficient and necessary conditions which do not involve such a complex number.

On the geodesic mappings of pseudo-Riemannian spaces with special supplementary tensor

В роботі досліджуються два псевдоріманових простори, які мають спільні геодезичні лінії. Вимагається виконання умов алгебраїчного та диференціального характеру на тензор Рімана одного з них. А операція опускання індексів та обчислення коваріантної похідної здійснюється відносно метрики та об'єктів зв'язності іншого простору. Для досліджень використовується спеціальний допоміжний тензор. Доведено, що виконання додаткових умов приводить до просторів, що не допускають нетривіальних геодезичних відображень, або простори належать до еквідістантних просторів. Використовуються тензорні методи без обмежень на знак метрики.

A Rayleigh quotient‐gradient neural network method for computing 𝒵‐eigenpairs of general tensors

AbstractRecently, Zhao, Zheng, Liang and Xu (A locally and cubically convergent algorithm for computing ‐eigenpairs of symmetric tensors. Numer Linear Algebra Appl, 2020, 27:e2284) studied on an efficient method for computing ‐eigenpairs of symmetric tensors. Whereas, symmetric tensors are just special tensors. This article is concerned with the computation of ‐eigenpairs of general real tensors. We propose a Rayleigh quotient‐gradient neural network model (RGNN) for computing ‐eigenpairs of a general real tensor and the Euler‐type difference rule is used to discretize RGNN model. Theoretical analysis of the convergence for RGNN model is provided. Numerical experiments show that our method can capture all ‐eigenpairs for some small‐scale general tensors and enjoys efficient computation for large‐scale tensors.

Exact Closed-Form Solutions of the Motion in Non-Inertial Reference Frames, Using the Properties of Lie Groups SO3 and SE3

The paper offers a general symbolic method to study the motion in a non-inertial reference frame. In order to achieve this, we use the algebraic and geometric properties of the Lie group of special orthogonal tensors, SO3, and the Lie group of the rigid body displacements, SE3. We obtain a simplified form of the initial value problem that models the non-inertial motion using a tensor instrument introduced in this paper. Thus, the study of the motion in a non-inertial reference frame is transferred into the study of a classical motion in an inertial reference frame. The applications of this method refer to solving the relative motion problem and deriving the straightforward solution to classical theoretical mechanics problems. The motion in a uniform gravitational force field in a rotating reference frame, the motion of a charged particle in non-stationary electric and magnetic fields, the exact solution of the relative rigid body motion in the non-inertial reference frame are studied. Using this symbolic method in studying the motion in a non-inertial reference frame reduces the number of computations. In addition, it offers, in some essential particular cases, exact closed-form coordinate-free analytical solutions.