Order Symmetric Tensors Research Articles

Quantifying low rank approximations of third order symmetric tensors

Quantifying low rank approximations of third order symmetric tensors

Schouten–Codazzi gravity

Abstract We propose a new phenomenological second order gravity theory to be denoted as ''Schouten-Codazzi' Gravity'' (SCG), as it is based on Schouten and Codazzi tensors. The theory is related, but is clearly distinct from Cotton Gravity. By assuming as source the energy momentum of General Relativity, we form a second order system with its geometric sector given by the sum of the Schouten tensor and a generic second order symmetric tensor complying with the following properties: (i) it must satisfy the Codazzi differential condition and (ii) it must be concomitant with the invariant characterization based on the algebraic structure of curvature tensors for specific spacetimes or classes of spacetimes. We derive and briefly discuss the properties of SCG solutions for static spherical symmetry (vacuum and perfect fluid), FLRW models and spherical dust fluids. While we do recognize that SCG is ``work in progress'' in an incipient stage that still requires significant theoretical development, we believe that the theory provides valuable guidelines in the search for alternatives to General Relativity

A note on the paper "An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator"

A note on the paper "An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator"

Classification of second orders symmetric tensors on manifolds through an associated fourth order tensor

For a manifold M admitting a metric 1 and given a second order symmetric tensor T on M one can construct from 1 and (the trace-free part of) T a fourth order tensor E on M which is related in a one-to-one way with T and from which T may be readily obtained algebraically. In the case when dimM = 4 this leads to an interesting relationship between the Jordan-Segre algebraic classification of T, viewed as a linear map on the tangent space to M with respect to 1, and the Jordan-Segre classification of E, viewed as a linear map on the 6?dimensional vector space of 2?forms to itself (with respect to the usual metric on 2?forms). This paper explores this relationship for each of the three possible signatures for 1.

Perfect Fluid Spacetimes and Gradient Solitons

In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by p=frac{3-n}{n-1} sigma , or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.

RotEqNet: Rotation-equivariant network for fluid systems with symmetric high-order tensors

In the recent application of scientific modeling, machine learning models are largely applied to facilitate computational simulations of fluid systems. Rotation symmetry is a general property for most symmetric fluid systems. However, in general, current machine learning methods have no theoretical guarantee of Rotation symmetry. By observing an important property of contraction and rotation operation on high order symmetric tensors, we prove that the rotation operation is preserved via tensor contraction. Based on this theoretical justification, in this paper, we introduce Rotation-Equivariant Network (RotEqNet) to guarantee the property of rotation-equivariance for high order tensors in fluid systems. We implement RotEqNet and evaluate our claims with four case studies on various fluid systems. The property of error reduction and rotation-equivariance is verified in these case studies. Results are showing the high superiority of RotEqNet compared to traditional machine learning methods.

An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

This article offers a study of the Calderón type inverse problem of determining up to second order coefficients of higher order elliptic operators. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain, by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain.

Copositivity for 3rd-Order Symmetric Tensors and Applications

The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter.

The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors

Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).

Computing tensor Z-eigenvalues via shifted inverse power method

The positive definiteness of an even degree homogeneous polynomial plays an important role in the stability study of nonlinear autonomous systems via Lyapunov’s direct method in automatic control, the detection of P or P0 tensor in tensor complementarity problems and spectral hypergraph theory, and more. Owing to the positive definiteness of an even degree homogeneous polynomial is equivalent to that of an even order symmetric tensor. In this paper, we propose a shifted inverse power method for computing tensor Z-eigenpairs, which can be viewed as a generalization of the inverse power method for matrices case. We also formulate it as a fixed point iteration form, and reveal that the relationship between the fixed points and the Z-eigenvectors of symmetric tensors. The advantages of the proposed method are simple operations and readily comprehensible convergence analysis. An efficient initialization strategy is also developed, which makes the proposed method converges to a better solution compared to not using the initialization strategy. Finally, we present applications of the proposed method in nonlinear autonomous systems, the detection of P or P0 tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.

Reprint of: Tykhonov well-posedness of a rate-type viscoplastic constitutive law

We consider a rate-type constitutitve law given by an implicit nonlinear differential equation in the space of second order symmetric tensors on Rd, in which the unknowns are the stress and the linearized strain fields. We list the assumptions on the constitutive functions then we state and prove its well-posedness with respect to two different Tykhonov triples. We use these well-posedness properties in order to deduce two convergence results. Finally, we provide the mechanical interpretation of these results as well as some concluding remarks.

An iterative scheme for identifying the positive semi-definiteness of even-order real symmetric H-tensor

H-tensor plays an important role in identifying the positive (semi-) definiteness of even-order real symmetric tensor. In this paper, we propose an algorithm for identifying general H-tensor and prove the algorithm will terminate within finite iterative steps. As application, we give an iterative method for identifying positive semi-definiteness of the even order symmetric tensor. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

Positive definiteness for 4th order symmetric tensors and applications

In particle physics, the vacuum stability of scalar potentials is to check positive definiteness (or copositivity) of its coupling tensors, and such a coupling tensor is a 4th order and symmetric tensor. In this paper, we mainly discuss precise expressions of positive definiteness of 4th order tensors. More specifically, two analytically sufficient conditions of positive definiteness for 4th order 2 dimensional symmetric tensors are given by reducing orders of tensors, and applying these conclusions, some sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors are derived. We also present several other sufficient conditions for the positive definiteness of 4th order 3 dimensional symmetric tensors. Finally, we test and verify the vacuum stability of general scalar potentials of two real singlet scalar fields and the Higgs boson by using these results.

Tykhonov well-posedness of a rate-type viscoplastic constitutive law

We consider a rate-type constitutitve law given by an implicit nonlinear differential equation in the space of second order symmetric tensors on Rd, in which the unknowns are the stress and the linearized strain fields. We list the assumptions on the constitutive functions then we state and prove its well-posedness with respect to two different Tykhonov triples. We use these well-posedness properties in order to deduce two convergence results. Finally, we provide the mechanical interpretation of these results as well as some concluding remarks.

On norm compression inequalities for partitioned block tensors

When a tensor is partitioned into subtensors, some tensor norms of these subtensors form a tensor called a norm compression tensor. Norm compression inequalities for tensors focus on the relation of the norm of this compressed tensor to the norm of the original tensor. We prove that for the tensor spectral norm, the norm of the compressed tensor is an upper bound of the norm of the original tensor. This result can be extended to a general class of tensor spectral norms. We discuss various applications of norm compression inequalities for tensors. These inequalities improve many existing bounds of tensor norms in the literature, in particular tightening the general bound of the tensor spectral norm via tensor partitions. We study the extremal ratio between the spectral norm and the Frobenius norm of a tensor space, provide a general way to estimate its upper bound, and in particular, improve the current best upper bound for third order nonnegative tensors and symmetric tensors. We also propose a faster approach to estimate the spectral norm of a large tensor or matrix via sequential norm compression inequalities with theoretical and numerical evidence. For instance, the complexity of our algorithm for the matrix spectral norm is Oleft( n^{2+epsilon }right) where epsilon ranges from 0 to 1 depending on the partition and the estimate ranges correspondingly from some close upper bound to the exact spectral norm.

On approximate diagonalization of third order symmetric tensors by orthogonal transformations

In this paper, we study the approximate orthogonal diagonalization problem of third order symmetric tensors. We define several classes of approximately diagonal tensors, including the ones corresponding to the stationary points of this problem. We study the relationships between these classes, and other well-known objects, such as tensor Z-eigenvalue and Z-eigenvector. We also prove results on convergence of the cyclic Jacobi (or Jacobi CoM2) algorithm.

Inverse Boundary Value Problem of Determining Up to a Second Order Tensor Appear in the Lower Order Perturbation of a Polyharmonic Operator

We consider the following perturbed polyharmonic operator \(\mathcal {L}(x,D)\) of order 2m defined in a bounded domain \(\Omega \subset \mathbb {R}^n, n\ge 3\) with smooth boundary, as \( \mathcal {L}(x,D) \equiv (-\Delta )^m + \sum _{j,k=1}^{n}A_{jk} D_{j}D_{k} + \sum _{j=1}^{n}B_{j} D_{j} + q(x),\) where A is a symmetric 2-tensor field, B and q are vector field and scalar potential respectively. We show that the coefficients \(A=[A_{jk}]\), \(B=(B_j)\) and q can be recovered from the associated Dirichlet-to-Neumann data on the boundary. Note that this result shows an example of determining higher order (2nd order) symmetric tensor field in the class of inverse boundary value problem.

Some curvature properties of paracontact metric manifolds

AbstractThe purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying{\nabla_{\xi}h=0}and such that the sectional curvature of the plane section containing ξ equals a non-zero constantc. Also, we study paracontact metric manifolds satisfying the curvature condition{Q\cdot R=0}, whereQandRare the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed.

A non-singular theory of dislocations in anisotropic crystals

We develop a non-singular theory of three-dimensional dislocation loops in a particular version of Mindlin's anisotropic gradient elasticity with up to six length scale parameters. The theory is systematically developed as a generalization of the classical anisotropic theory in the framework of linearized incompatible elasticity. The non-singular version of all key equations of anisotropic dislocation theory are derived as line integrals, including the Burgers displacement equation with isolated solid angle, the Peach-Koehler stress equation, the Mura-Willis equation for the elastic distortion, and the Peach-Koehler force. The expression for the interaction energy between two dislocation loops as a double line integral is obtained directly, without the use of a stress function. It is shown that all the elastic fields are non-singular, and that they converge to their classical counterparts a few characteristic lengths away from the dislocation core. In practice, the non-singular fields can be obtained from the classical ones by replacing the classical (singular) anisotropic Green's tensor with the non-singular anisotropic Green's tensor derived by Lazar and Po (2015b). The elastic solution is valid for arbitrary anisotropic media. In addition to the classical anisotropic elastic constants, the non-singular Green's tensor depends on a second order symmetric tensor of length scale parameters modeling a weak non-locality, whose structure depends on the specific class of crystal symmetry. The anisotropic Helmholtz operator defined by such tensor admits a Green's function which is used as the spreading function for the Burgers vector density. As a consequence, the Burgers vector density spreads differently in different crystal structures. Two methods are proposed to determine the tensor of length scale parameters, based on independent atomistic calculations of classical and gradient elastic constants. The anisotropic non-singular theory is shown to be in good agreement with molecular statics without fitting parameters, and unlike its singular counterpart, the sign of stress components does not show reversal as the core is approached. Compared to the isotropic solution, the difference in the energy density per unit length between edge and screw dislocations is more pronounced.

Second order parallel tensors on some paracontact manifolds

The object of the present paper is to study the symmetric and skew-symmetric properties of a second order parallel tensor on paracontact metric (k, µ)-spaces and almost β-para-Kenmotsu (k, µ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k, µ)-space M, then either M is locally isometric to a product of a flat n + 1-dimensional manifold and an n-dimensional manifold of constant sectional curvature −4, or the second order parallel tensor is a constant multiple of the associated metric tensor g of M2n+1. If there is a second order parallel tensor on an almost β-para-Kenmotsu (k, µ)-space with k ≠ 0, then it is a constant multiple of the associated metric tensor g of M2n+1.