Special Tensor Research Articles

PSEUDO-RIEMANNIAN SPACES WITH A SPECIAL RIEMANN TENSOR

The paper considers pseudo-Riemannian spaces, the Riemann tensor of which has a special structure. The structure of the Riemann tensor is given as a combination of special symmetric and obliquely symmetric tensors. Tensors are selected so that the results can be applied in the theory of geodetic mappings, the theory of holomorphic-projective mappings of Kähler spaces, as well as other problems arising in differential geometry and its application in general relativity, mechanics and other fields. Through the internal objects of pseudo-Riemannian space, others are determined, which are studied depending on what problems are solved in the study of pseudo-Riemannian spaces. By imposing algebraic or differential constraints on internal objects, we obtain special spaces. In particular, if constraints are imposed on the metric we will have equidistant spaces. If on the Ricci tensor, we obtain spaces that allow φ (Ric)-vector fields, and if on the Einstein tensor, we have almost Einstein spaces. The paper studies pseudo-Riemannian spaces with a special structure of the curvature tensor, which were introduced into consideration in I. Mulin paper. Note that in his work these spaces were studied only with the requirement of positive definiteness of the metric. The proposed approach to the specialization of pseudo-Riemannian spaces is interesting by combining algebraic requirements for the Riemann tensor with differential requirements for its components. In this paper, the research is conducted in tensor form, without restrictions on the sign of the metric. Depending on the structure of the Riemann tensor, there are three special types of pseudo-Riemannian spaces. The properties which, if necessary, satisfy the Richie tensors of pseudoriman space and the tensors which determine the structure of the curvature tensor are studied. In all cases, it is proved that special tensors satisfy the commutation conditions together with the Ricci tensor. The importance and usefulness of such conditions for the study of pseudo-Riemannian spaces is widely known. Obviously, the results can be extended to Einstein tensors. Proven theorems allow us to effectively investigate spaces with constraints on the Ricci tensor.

Decomposition of third-order constitutive tensors

Third-order tensors are widely used as a mathematical tool for modeling the physical properties of media in solid-state physics. In most cases, they arise as constitutive tensors of proportionality between basic physical quantities. The constitutive tensor can be considered the complete set of physical parameters of a medium. The algebraic features of the constitutive tensor can be used as a tool for proper identification of natural materials, such as crystals, and for designing artificial nanomaterials with prescribed properties. In this paper, we study the algebraic properties of a general asymmetric third-order tensor relative to its invariant decomposition. In correspondence with different groups acting on the basic vector space, we present the hierarchy of different types of tensor decomposition into invariant subtensors. In particular, we discuss the problem of non-uniqueness and reducibility of high-order tensor decomposition. For a general asymmetric third-order tensor, these features are described explicitly. In the case of special tensors with a prescribed symmetry, the decomposition is demonstrated to be irreducible and unique. We present the explicit results for two physically interesting models: the piezoelectric tensor as an example of pair symmetry and the Hall tensor as an example of pair skew-symmetry.

Fully coupled functional equations for the quark sector of QCD

We present a comprehensive study of the quark sector of $2+1$ flavour QCD, based on a self-consistent treatment of the coupled system of Schwinger-Dyson equations for the quark propagator and the full quark-gluon vertex. The individual form factors of the quark-gluon vertex are expressed in a special tensor basis obtained from a set of gauge-invariant operators. The sole external ingredient used as input to our equations is the Landau gauge gluon propagator with $2+1$ dynamical quark flavours, obtained from studies with Schwinger-Dyson equations, the functional renormalisation group approach, and large volume lattice simulations. The appropriate renormalisation procedure required in order to self-consistently accommodate external inputs stemming from other functional approaches or the lattice is discussed in detail, and the value of the gauge coupling is accurately determined at two vastly separated renormalisation group scales. Our analysis establishes a clear hierarchy among the vertex form factors. We identify only three dominant ones, in agreement with previous results. The components of the quark propagator obtained from our approach are in excellent agreement with the results from Schwinger-Dyson equations, the functional renormalisation group, and lattice QCD simulation, a simple benchmark observable being the chiral condensate in the chiral limit, which is computed as $(245\,\textrm{MeV})^3$. The present approach has a wide range of applications, including the self-consistent computation of bound-state properties and finite temperature and density physics, which are briefly discussed.

Conductivity tensor in semiconductor with the presence of a laser field and an electromagnetic wave field for the case of scattering on ionized impurities

There are a lot of studies on semiconductors and in semiconductors also appear many dynamic effects, from the research results that have many applications in science and technology. Conductivity tensor is a familiar concept in physics in general and semiconductors in particular, but research on scattering on ionized impurities is still limited. This paper presents the influence of laser field and electromagnetic waves on conductivity tensor for the case of scattering on ionized impurities in semiconductor. The particle system is placed in an electromagnetic wave field and a laser field. Using quantum kinetic equations for electrons in semiconductor under the action of an electromagnetic wave field, a laser field, the conductivity tensor is calculated with case of scattering on ionized impurities. Conductivity tensor depends on the frequency of the laser field, the laser field amplitude, the electromagnetic wave frequency and typical parameters for semiconductor system. With different frequency bands, the dependence of tensor conductivity on the frequency is different, and the special conductivity tensor is strongly dependent on the amplitude of the laser field but depends very little on its frequency. The results will be plotted and discussed with GaAs semiconductor case and compared with studies for different scattering cases.

Gravitational waves on Earth and their warming effects

The gravity or reactive bundle energy is the outlet of the morphogenetic impact, known as “BIG BANG”, creating a bounded ordered/structured universe along with the solar system, including the EARTH-world with its human race. Post-impact, the huge kinetic energy is spread into stellar bodies associated with the light flux under strong mutual connections or gravitational bundle. Einstein’s general relativity theory including the gravitational field can be expressed under a condensed tensor formulation as E  R − Rg =  T where E defines the geometry via a curved space-time structure (R) over the gravity field (1/2Rg), embedded in a matter distribution T The fundamental (ten non-linear partial differential) equations of the gravitational field are a kind of the space-time machine using the curvature of a four-dimensional space-time to engender the gravity field carrying away material structures. Gravity according to the curved space-time theory is not seen as a gravitational force, but it manifests itself in the relativistic form of the space-time curvature needing the constancy of the light speed. But the constant light velocity makes the tidal wave/pulsating energy, a characteristic of solar energy, impossible. The Einstein’s field equation, expressed in terms of tensor formulation along with the constant light speed postulate, needs two special space-time tensors (curvature and torsion) in 4 dimensions, where for the simplicity the torsion/twist tensor is less well approximated (Bianchi identity) leading to a constant/frozen gravity (twist-free gravity).The non-zero torsion tensor plays a significant physical role in the planetary dynamics as a finest gear of a planet, where its spinning rotation is directly connected to the own work and space-time structure (or clock), controlled by light fluctuations (or tidal effect of gravity). The spin correction of Einstein’s gravitational field refers to the curvature-torsion effect coupled with fluctuating light speed. The mutual curvature-torsion bundle self-sustained by the quantum fluctuations of light speed engenders helical gravitational wave fields of a quantum nature where bodies orbit freely in the light speed field (cosmic wind). In contrast to the Einstein’s field equation describing a gravitational frozen field, a quantum tidal gravity model is proposed in the paper.

Microphone Array Speech Separation Algorithm Based on TC-ResNet

Traditional separation methods have limited ability to handle the speech separation problem in high reverberant and low signal-to-noise ratio (SNR) environments, and thus achieve unsatisfactory results. In this study, a convolutional neural network with temporal convolution and residual network (TC-ResNet) is proposed to realize speech separation in a complex acoustic environment. A simplified steered-response power phase transform, denoted as GSRP-PHAT, is employed to reduce the computational cost. The extracted features are reshaped to a special tensor as the system inputs and implements temporal convolution, which not only enlarges the receptive field of the convolution layer but also significantly reduces the network computational cost. Residual blocks are used to combine multiresolution features and accelerate the training procedure. A modified ideal ratio mask is applied as the training target. Simulation results demonstrate that the proposed microphone array speech separation algorithm based on TC-ResNet achieves a better performance in terms of distortion ratio, source-to-interference ratio, and short-time objective intelligibility in low SNR and high reverberant environments, particularly in untrained situations. This indicates that the proposed method has generalization to untrained conditions.

Robust angle estimation algorithm for MIMO radar based on tensor analysis and reduced-dimension subspace reconstruction

A robust angle estimation algorithm based on tensor analysis and reduced-dimension subspace reconstruction is proposed for the noncircular signals with unknown mutual coupling in MIMO radar. The proposed algorithm constructs a special tensor analysis to capture the non-circular characteristics and multi-dimensional structure of non-circular signals, and can eliminate the influence of mutual coupling in the tensor domain. The signal subspace is obtained by using high-order singular value decomposition (HOSVD), and the direction-of-departure (DOD) of the target is obtained by combining the real value subspace estimation of direction of arrival (DOA). In addition, according to the working principle and echo model of bistatic MIMO radar, combined with array signal processing, the angle estimation algorithm of reduced-dimension subspace reconstruction is proposed, which can effectively solve the angle estimation problem of bistatic MIMO radar, reduce the calculation cost, improve the robustness of the algorithm, and solve the problem that angle matching cannot be realized after dimension reduction. The simulation results show that the algorithm proposed in this paper has good performance in recognition.

Existence and uniqueness of positive solution for [formula omitted]-tensor equations

The analysis of the existence of solutions for the systems of polynomial equations is much more difficult to handle in general. Nonetheless, we can still confirm the existence of some particular solutions, especially for some special structured tensor equations. In this paper, we theoretically prove that a system of H+-tensor equations with the positive right-hand side has a unique positive solution by proposing a positive increasing map. We also prove that this system also has a negative solution. Numerical examples are presented to verify our results are true.

An iterative scheme for testing the positive definiteness of multivariate homogeneous forms

ABSTRACTA positive-definite homogeneous multivariate form plays a critical role in the medical imaging and automatic control, and the definiteness of this form can be identified by a special structure tensor. In this paper, we first state the equivalence between the positive-definite multivariate form and the corresponding tensor and account for the links between the positive-definite tensor with a strong H-tensor. Then based on weak reducibility, some criteria were provided to identify strong H-tensors. Furthermore, with these relations, we establish an iterative scheme to identify the positive-definite multivariate homogeneous form and prove it is theoretically valid. Numerical experiments were given to illustrate the practicality of the scheme.

Tensor-Based Angle Estimation Approach for Strictly Noncircular Sources with Unknown Mutual Coupling in Bistatic MIMO Radar

In the paper, the estimation of joint direction-of-departure (DOD) and direction-of-arrival (DOA) for strictly noncircular targets in multiple-input multiple-output (MIMO) radar with unknown mutual coupling is considered, and a tensor-based angle estimation method is proposed. In the proposed method, making use of the banded symmetric Toeplitz structure of the mutual coupling matrix, the influence of the unknown mutual coupling is removed in the tensor domain. Then, a special enhancement tensor is formulated to capture both the noncircularity and inherent multidimensional structure of strictly noncircular signals. After that, the higher-order singular value decomposition (HOSVD) technology is applied for estimating the tensor-based signal subspace. Finally, the direction-of-departure (DOD) and direction-of-arrival (DOA) estimation is obtained by utilizing the rotational invariance technique. Due to the use of both noncircularity and multidimensional structure of the detected signal, the algorithm in this paper has better angle estimation performance than other subspace-based algorithms. The experiment results verify that the method proposed has better angle estimation performance.

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.

One Pass Thalamic and Subthalamic Stimulation for Patients with Tremor-Dominant Idiopathic Parkinson Syndrome (OPINION): Protocol for a Randomized, Active-Controlled, Double-Blinded Pilot Trial.

BackgroundBesides fluctuations, therapy refractory tremor is one of the main indications of deep brain stimulation (DBS) in patients with idiopathic Parkinson syndrome (IPS). Although thalamic DBS (ventral intermediate nucleus [Vim] of thalamus) has been shown to reduce tremor in 85-95% of patients, bradykinesia and rigidity often are not well controlled. The dentato-rubro-thalamic tract (DRT) that can directly be targeted with special diffusion tensor magnetic resonance imaging sequences has been shown as an efficient target for thalamic DBS. The subthalamic nucleus (STN) is typically chosen in younger patients as the target for dopamine-responsive motor symptoms. This study investigates a one-path thalamic (Vim/DRT) and subthalamic implantation of DBS electrodes and possibly a combined stimulation strategy for both target regions.ObjectiveThis study investigates a one path thalamic (Vim/DRT) and subthalamic implantation of DBS electrodes and a possibly combined stimulation strategy for both target regions.MethodsThis is a randomized, active-controlled, double-blinded (patient- and observer-blinded), monocentric trial with three treatments, three periods and six treatment sequences allocated according to a Williams design. Eighteen patients will undergo one-path thalamic (Vim/DRT) and STN implantation of DBS electrodes. After one month, a double-blinded and randomly-assigned stimulation of the thalamic target (Vim/DRT), the STN and a combined stimulation of both target regions will be performed for a period of three months each. The primary objective is to assess the quality of life obtained by the Parkinson’s Disease Questionnaire (39 items) for each stimulation modality. Secondary objectives include tremor reduction (obtained by the Fahn-Tolosa-Marin tremor rating scale, video recordings, the Unified Parkinson’s disease rating scale, and by tremor analysis), psychiatric assessment of patients, and to assess the safety of intervention.ResultsAt the moment, the recruitment is stopped and 12 patients have been randomized and treated. A futility analysis is being carried out by means of a conditional power analysis.ConclusionsThe approach of the OPINION trial planned to make, for the first time, a direct comparison of the different stimulation conditions (Vim/DRT, compared to STN, compared to Vim/DRT+STN) in a homogeneous patient population and, furthermore, will allow for intraindividual comparison of each condition with the “quality of life” outcome parameter. We hypothesize that the combined stimulation of the STN and the thalamic (Vim/DRT) target will be superior with respect to the patients’ quality of life as compared to the singular stimulation of the individual target regions. If this holds true, this work might change the standardized treatment described in the previous section.Trial RegistrationClinicalTrials.gov: NCT02288468; https://clinicaltrials.gov/ct2/show/NCT02288468 (Archived by WebCite at http://www.webcitation.org/6wlKnt2pJ); and German Clinical Trials Register: DRKS00007526; https://www.drks.de/drks_ web/navigate.do?navigationId=trial.HTML&TRIAL_ID=DRKS00007526 (Archived by WebCite at http://www.webcitation.org/6wlKyXZZL).

Multiscale Tensor Dictionary Learning Approach for Multispectral Image Denoising

Taking advantage of different sensitivities of each component/material in an object to different bands, multispectral image (MSI) is obtained by shooting the object from multiple bands individually or simultaneously. Including these different complementary information with the space-time correlation, the MSI can describe the object more clearly and comprehensively. In practice, however, it is unavoidable that MSIs are corrupted by noise. In order to solve the denoising problem of MSIs, a framework is proposed to suppress noise by learning multiscale sparse representations of MSIs with an overcomplete tensor dictionary. In our method, the tensor patches are extracted from an image tensor, and a tensor-based dictionary is trained using a special tensor decomposition, in which each atom is a rank-one tensor. The so-called multiscale learned representation is obtained based on an efficient quadtree decomposition of the trained tensor dictionary. Experimental results on numerical simulations and real MSIs demonstrate that multiscale tensor dictionary gets better indexes in terms of PSNR and SSIM compared with single-scale tensor dictionary and other related competing methods. At the same time, from the perspective of visual quality, our method restores more image details than other methods.

Real eigenvalues of nonsymmetric tensors

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the contrast, every nonsymmetric tensor has finitely many H-eigenvalues. We propose Lasserre type semidefinite relaxation methods for computing such eigenvalues. For every nonsymmetric tensor that has finitely many real Z-eigenvalues, we can compute all of them; each of them can be computed by solving a finite sequence of semidefinite relaxations. For every nonsymmetric tensor, we can compute all its real H-eigenvalues; each of them can be computed by solving a finite sequence of semidefinite relaxations. Various examples are demonstrated.

Tensor convolutions and Hankel tensors

Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n−1). Then A is called a Hankel tensor associated with a vector v ∈ ℝN+1 if Aσ = v k for each k = 0, 1,...,N whenever σ = (i1,..., im) satisfies i1 +· · ·+im = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.

Positive definite and Gram tensor complementarity problems

Given a continuous function and , the non-linear complementarity problem $$\text{ NCP }(g,q)$$ is to find a vector such that $$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$ We say that g has the Globally Uniquely Solvable ( $$\text{ GUS }$$ )-property if $$\text{ NCP }(g,q)$$ has a unique solution for all and C-property if $$\mathrm{NCP}(g,q)$$ has a convex solution set for all . In this paper, we find a class of non-linear functions that have the $$\text{ GUS }$$ -property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.

Tensor complementarity problems: the GUS-property and an algorithm

In this paper, we study the global uniqueness and solvability (GUS-property) of tensor complementarity problems (TCPs) for some special structured tensors. The modulus equation for TCPs is also proposed, and based on this equation, we develop the corresponding nonsmooth Newton’s method, which extends the existing method given in the work of Zheng H and Li W. The modulus-based nonsmooth Newton’s method for solving linear complementarity problems. J Comput Appl Math. 2015;288: 116-126. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.

The Construction of Nonlinear Elasticity Tensors for Crystals and Quasicrystals

The problem of construction of constitutive relations in nonlinear elasticity theory for crystals and quasicrystals is associated with finding the construction of the structure of tensors of elastic modules of 2nd and 3rd orders. In the case of linear constitutive relations, the problem is entirely solved for crystals and some quasicrystals. In the case of nonlinear relations, the quantity of nonzero elastic modules of the 3rd order is known only for some classes of crystals. The problem of construction of nonlinear constitutive relations for quasicrystal materials has not been considered. In this article, representations of linear and nonlinear elastic properties of crystals and quasicrystals in the form of decompositions by special invariant tensor bases are considered. This allows us to concretize constitutive relations for icosahedral and axial quasicrystals. It is shown that axial quasicrystals’ behavior towards elastic properties coincides with transversal-isotropic materials. Icosahedral quasicrystals’ behavior matches with the isotropic materials’ one. On the basis of the obtained decompositions of nonlinear elasticity tensors for axial quasicrystals having a symmetry plane and for graphene films with and without defects, the analysis of their mechanical behavior at some types of loading is fulfilled. It is shown that differences in the behavior of these materials appear only in the second order effects.

Rich gauge structures from a unitary approach of some massless gauge fields of spins one and two

The aim of this paper consists in the investigation of both first- and second-order dynamics for a special massless tensor gauge field of degree [Formula: see text]. Geometrically, this can be interpreted as a bosonic (1-)form-valued [Formula: see text]-form that, in specific space–time dimensions, describes either spin-1 or spin 2 gauge fields. The idea of using multi-forms in describing general tensor gauge fields is not new. It was previously investigated at Lagrangian level[Formula: see text] where was displayed interesting exotic gauge theories. In arbitrary Minkowski space–times, the considered geometric object combines two tensor gauge fields with mixed symmetry namely a [Formula: see text]-form and a massless tensor gauge field with the mixed symmetry [Formula: see text]. Concretely, we first approach the bosonic (1-)form-valued [Formula: see text]-form from the Hamiltonian perspective, Dirac analysis revealing new compelling gauge structures. Second, we construct the Lagrangian first-order formulations for the considered geometric ingredient.

Upper Bounds for the Spectral Radii of Nonnegative Tensors

In this paper, we present several sharper upper bounds for the M-spectral radius and Z-spectral radius based on the eigenvalues of some unfolding matrices of nonnegative tensors. Meanwhile, we show that these bounds could be tight for some special tensors. For a general nonnegative tensor which can be transformed into a matrix, we prove the maximal singular value of this matrix is an upper bound of its Z-eigenvalues. Some examples are provided to show these proposed bounds greatly improve some existing ones.