Symmetric Tensor Research Articles

Higher order distance-like functions and Sobolev spaces

On non-compact Riemannian manifolds, we construct distance-like functions with derivatives controlled up to some order k assuming bounds on the growth of the derivatives of the curvature up to order k−2 and on the decay of the injectivity radius. This construction extends previously known results in various directions, permitting to obtain consequences which are (in a sense) sharp. As a first main application, we give refined conditions guaranteeing the density of compactly supported smooth functions in the Sobolev space Wk,p on the manifold. Contrary to all previously known results this can be obtained also on manifolds with possibly unbounded geometry. In the particular case p=2, making use of the Weitzenböck formula for a Lichnerowicz Laplacian acting on k-covariant totally symmetric tensor fields, we can weaken the assumptions needed to obtain the density property, avoiding any condition on the highest order derivatives of the curvature. Distance-like functions are also used to obtain new disturbed Sobolev inequalities, disturbed Lp-Calderón-Zygmund inequalities and the full Omori-Yau maximum principle for the Hessian under weak assumptions.

Revisiting the Reynolds-averaged Navier–Stokes equations

Abstract This study revisits the Reynolds-averaged Navier–Stokes (RANS) equations and finds that the existing literature is erroneous regarding the primary unknowns and the number of independent unknowns in the RANS. The literature claims that the Reynolds stress tensor has six independent unknowns, but in fact the six unknowns can be reduced to three that are functions of the three velocity fluctuation components, because the Reynolds stress tensor is simply an integration of a second-order dyadic tensor of flow velocity fluctuations rather than a general symmetric tensor. This difficult situation is resolved by returning to the time of Reynolds in 1895 and revisiting Reynolds’ averaging formulation of turbulence. The study of turbulence modeling could focus on the velocity fluctuations instead of the Reynolds stress. An advantage of modeling the velocity fluctuations is, from both physical and experimental perspectives, that the velocity fluctuation components are observable whereas the Reynolds stress tensor is not.

Symmetry-Consistent Expansion of Interaction Kernels Between Rigid Molecules

We discuss the expansion of interaction kernels between anisotropic rigid molecules. The expansion decouples the correlated orientational variables so that it can be utilized to derive macroscopic models. Symmetries of two types are considered. First, we examine the symmetry of the interacting cluster, including the translation and rotation of the whole cluster, and label permutation within the cluster. The expansion is expressed by symmetric traceless tensors, and the linearly independent terms are identified. Then, we study the molecular symmetry characterized by a point group in $O(3)$. The proper rotations determine what symmetric traceless tensors can appear. The improper rotations decompose these tensors into two subspaces and determine how the tensors in the two subspaces are coupled. For each point group, we identify the two subspaces, so that the expansion consistent with the point group is established.

Kaehlerian Manifolds on Einstein-Recurrent curvature tensor

Bochner (1949) obtained some results on curvature and betti numbers. Hasegawa (1974) studied on H projective recurrent Kaehlerian manifolds and Bochner recurrent Kaehlerian manifolds. Also, Negi (2016) obtained Some Recurrence Properties in Kaehlerian, Einstein Kaehlerian and Tachibana Spaces. After that, Negi et. al. (2019) studied on Projective recurrent and symmetric tensor in almost Kaehlerian spaces. Again, Negi and Sulochana (2021) have studied on conformal symmetric tensor of kaehlerian manifolds. In this paper, the author calculated some properties of Kaehlerian Manifolds on Einstein-Recurrent curvature tensor and computes the relations between special type curvature tensors

The Jordan algebras of Riemann, Weyl and curvature compatible tensors

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatible.

On generalized W2-curvature tensor of para-Kenmotsu manifolds

The object of the present paper is to generalizeW2-curvature tensor of para-Kenmotsu manifold with the help of a new generalized (0,2) symmetric tensor Z introduced by Mantica and Suh [11]. Various geometric properties of generalized W2-curvature tensor of para-Kenmotsu manifold have been studied. It is shown that a generalized W2?-symmetric para-Kenmotsu manifold is an Einstein manifold.

First eigenvalue of the Laplacian of a geodesic ball and area-based symmetrization of its metric tensor

Given a Riemmanian manifold, we provide a new method to compute a sharp upper bound for the first eigenvalue of the Laplacian for the Dirichlet problem on a geodesic ball of radius less than the injectivity radius of the manifold. This upper bound is obtained by transforming the metric tensor into a rotationally symmetric metric tensor that preserves the area of the geodesic spheres. The provided upper bound can be computed using only the area function of the geodesic spheres contained in the geodesic ball and it is sharp in the sense that the first eigenvalue of geodesic ball coincides with our upper bound if and only if the mean curvature pointed inward of each geodesic sphere is a radial function.

Wave optics and weak gravitational lensing of lights from spherically symmetric static scalar vector tensor Brans Dicke wormholes

This paper has three parts. In first step we use modified scalar tensor vector Brans Dicke gravity \cite{GH0} to obtain metric solution of a spherically symmetric static black hole via perturbation method which asymptotically behaves as modified Schwarzschild de Sitter black hole in weak field limits. Corrections on the line element with respect to the point mass Schwarzschild de Sitter black hole are two parts: power law function for black hole mass instead of point mass and additional logarithmic metric potential which they are produced from effects of interacting timelike dynamical vector fields. Second part of the paper is dedicated to review Fresnel-Kirchhoff diffraction theory for massless scalar field instead of the vector electromagnetic waves. In third part of the work we consider the obtained black hole metric to be lens and we study weak gravitational lensing of moving light originated from point star. Production of stationary images is investigated via both interference of waves and geometric optics approaches. At last we compare results of these approaches and give some outlooks.

Damage and Fracture Analysis under Multi-axial Stress State Based on Damage Mechanics

In various press forming and deep drawing processes, it is important to anticipate the forming limit in multi-axial stress state. In the uniaxial tension test, it is necessary to take stress triaxiality into account for accurate prediction of failure strain, because after the onset of the necking, the constricted portion, which is actually subjected to loading, is in the multiaxial stress state due to the change in local cross-sectional area. Recently, several damage and fracture models have been proposed that focus on the effect of stress triaxiality on the forming limit, while most of them are phenomenological models. In this study, the constitutive equation and damage evolution equation, which are based on a second-order symmetric damage tensor within the framework of the continuum damage mechanics, are constructed as a function of the stress triaxiality to analyze damage and fracture of materials in multiaxial stress state. Then, FEM analysis of an example problem was carried out to validate this material model.

Feature Curves and Surfaces of 3D Asymmetric Tensor Fields.

3D asymmetric tensor fields have found many applications in science and engineering domains, such as fluid dynamics and solid mechanics. 3D asymmetric tensors can have complex eigenvalues, which makes their analysis and visualization more challenging than 3D symmetric tensors. Existing research in tensor field visualization focuses on 2D asymmetric tensor fields and 3D symmetric tensor fields. In this paper, we address the analysis and visualization of 3D asymmetric tensor fields. We introduce six topological surfaces and one topological curve, which lead to an eigenvalue space based on the tensor mode that we define. In addition, we identify several non-topological feature surfaces that are nonetheless physically important. Included in our analysis are the realizations that triple degenerate tensors are structurally stable and form curves, unlike the case for 3D symmetric tensors fields. Furthermore, there are two different ways of measuring the relative strengths of rotation and angular deformation in the tensor fields, unlike the case for 2D asymmetric tensor fields. We extract these feature surfaces using the A-patches algorithm. However, since three of our feature surfaces are quadratic, we develop a method to extract quadratic surfaces at any given accuracy. To facilitate the analysis of eigenvector fields, we visualize a hyperstreamline as a tree stem with the other two eigenvectors represented as thorns in the real domain or the dual-eigenvectors as leaves in the complex domain. To demonstrate the effectiveness of our analysis and visualization, we apply our approach to datasets from solid mechanics and fluid dynamics.

On cohomology in symmetric tensor categories in prime characteristic

We describe graded commutative Gorenstein algebras ${\mathcal E}_n(p)$ over a field of characteristic $p$, and we conjecture that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)\cong{\mathcal E}_{n}(p)$, where $\mathsf{Ver}_{p^{n+1}}$ are the new symmetric tensor categories recently constructed in \cite{Benson/Etingof:2019a,Benson/Etingof/Ostrik,Coulembier}. We investigate the combinatorics of these algebras, and the relationship with Minc's partition function, as well as possible actions of the Steenrod operations on them. Evidence for the conjecture includes a large number of computations for small values of $n$. We also provide some theoretical evidence. Namely, we use a Koszul construction to identify a homogeneous system of parameters in ${\mathcal E}_n(p)$ with a homogeneous system of parameters in $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$. These parameters have degrees $2^i-1$ if $p=2$ and $2(p^i-1)$ if $p$ is odd, for $1\le i \le n$. This at least shows that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{p^{n+1}}}(1,1)$ is a finitely generated graded commutative algebra with the same Krull dimension as ${\mathcal E}_n(p)$. For $p=2$ we also show that $\mathrm{Ext}^\bullet_{\mathsf{Ver}_{2^{n+1}}}(1,1)$ has the expected rank $2^{n(n-1)/2}$ as a module over the subalgebra of parameters.

Nonlinear Filtering Based on Third-Degree Approximation Via Sequential Homotopy

To better describe statistical characteristics of asymmetric distributions often encountered in nonlinear systems, this paper aims at providing effective third-degree moment approximation by a new deterministic sample, named tau-point set. The set is not limited to a particular structure such as in the unscented transformation. It is computationally stable and can be asymmetric, which makes its moment approximation better than some state-of-the-art methods. The construction of a tau-point set is formulated first as a constrained third-order symmetric tensor decomposition problem. Inspired by the idea of homotopy, a simple problem with an easy (or known) solution is proposed, and then a novel approach initialized adaptively in filtering is developed for solving the decomposition problem by tracing analytic zero paths emanating from a solution of the simple problem. The tau points are used to approximate the moments in the linear minimum mean square error (LMMSE) estimation framework. Thus a new filter is developed and demonstrated to outperform some popular nonlinear filters through examples.

On rank decomposition and semi-symmetric rank decomposition of semi-symmetric tensors

A tensor is called semi-symmetric if all modes but one, are symmetric. In this paper, we study the CP decomposition of semi-symmetric tensors or higher-order individual difference scaling (INDSCAL). Comon's conjecture states that for any symmetric tensor, the CP rank and symmetric CP rank are equal, while it is known that Comon's conjecture is not true in the general case but it is proved under several assumptions in the literature. In the paper, Comon's conjecture is extended for semi-symmetric CP decomposition and CP decomposition of semi-symmetric tensors under suitable assumptions. Specially, we show that if a semi-symmetric tensor has a CP rank smaller or equal to its order, or when the semi-symmetric CP rank is less than/or equal to the dimension, then the semi-symmetric CP rank is equal to the CP rank.

A tighter M-eigenvalue localization set for partially symmetric tensors and its an application

<abstract><p>In this paper, a new M-eigenvalue inclusion set for a partially symmetric tensor is provided. It is proved that the new set is tighter than some existing M-eigenvalue inclusion sets. Based on the obtained results, an upper bound of the largest M-eigenvalue is given and a modified WQZ-algorithm is established which guarantees the generated converges to the largest M-eigenvalue of the tensor faster.</p></abstract>

Electro-magnetic vibration behavior of smart curved micro-/nanoshells

This paper investigates dynamic results of a small-scale-dependent panel subjected to a combination of mechanical, electrical, and magnetic loads using the shear defamation theory. Modified couple stress theory (MCST) is used for accounting small-scale effects in small sizes. Governing equations of motion are derived based on Hamilton’s principle in the cylindrical coordinate system. The micro panel is composed of an elastic core integrated with two piezoelectric/piezomagnetic layers at top and bottom. Microlength scale parameter is included in the couple stress tensor and symmetric curvature tensor to capture size dependency. The analytical solution is applied for the solution of the motion equations. The extended parametric results are provided to show natural frequencies in terms of main significant input parameters such as initial electric/magnetic loads, span angle, length to thickness, radius to thickness, and length to radius ratio. Furthermore, the effect of higher mode numbers along the axial and circumferential directions is studied on the responses. The numerical results indicate that the natural frequencies are increased with an increase in initial magnetic potential and a decrease in initial electric potential.

Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero.
 
 Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂z（λ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.

Riemannian Newton optimization methods for the symmetric tensor approximation problem

The Symmetric Tensor Approximation problem (STA) consists of approximating a symmetric tensor or a homogeneous polynomial by a linear combination of symmetric rank-1 tensors or powers of linear forms of low symmetric rank. We present two new Riemannian Newton-type methods for low rank approximation of symmetric tensor with complex coefficients.The first method uses the parametrization of the set of tensors of rank at most r by weights and unit vectors. Exploiting the properties of the apolar product on homogeneous polynomials combined with efficient tools from complex optimization, we provide an explicit and tractable formulation of the Riemannian gradient and Hessian, leading to Newton iterations with local quadratic convergence. We prove that under some regularity conditions on non-defective tensors in the neighborhood of the initial point, the Newton iteration (completed with a trust-region scheme) is converging to a local minimum.The second method is a Riemannian Gauss–Newton method on the Cartesian product of Veronese manifolds. An explicit orthonormal basis of the tangent space of this Riemannian manifold is described. We deduce the Riemannian gradient and the Gauss–Newton approximation of the Riemannian Hessian. We present a new retraction operator on the Veronese manifold.We analyze the numerical behavior of these methods, with an initial point provided by Simultaneous Matrix Diagonalisation (SMD). Numerical experiments show the good numerical behavior of the two methods in different cases and in comparison with existing state-of-the-art methods.

Finite elements for divdiv conforming symmetric tensors in three dimensions

Finite element spaces on a tetrahedron are constructed for div ⁡ div \operatorname {div}\operatorname {div} -conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the div ⁡ div \operatorname {div}\operatorname {div} Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensor spaces are derived from the polynomial complexes. Second, traces for the div ⁡ div \operatorname {div}\operatorname {div} operator are characterized through a Green’s identity. Besides the normal-normal component, another trace involving combination of first order derivatives of the tensor is continuous across the face. Due to the smoothness of polynomials, the symmetric tensor element is also continuous at vertices, and on the plane orthogonal to each edge. Besides, a finite element for s y m c u r l symcurl -conforming trace-free tensors is constructed following the same approach. Putting all together, a finite element div ⁡ div \operatorname {div}\operatorname {div} complex, as well as the bubble functions complex, in three dimensions is established.

A compendium of sphere path integrals

We study the manifestly covariant and local 1-loop path integrals on Sd+1 for general massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin s ≥ 0 in any dimensions d ≥ 2. After reviewing the cases of massless fields with spin s = 1, 2, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins s ≥ 1. Following the standard procedure of Wick-rotating the negative conformal modes, we find a higher spin analog of Polchinski’s phase for any integer spin s ≥ 2. The derivations for low-spin (s = 0, 1, 2) massive, shift-symmetric and partially massless fields are also carried out explicitly. Finally, we provide general prescriptions for general massive and shift-symmetric fields of arbitrary integer spins and partially massless fields of arbitrary integer spins and depths.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.