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Journal articles on the topic 'Real symmetric tensors'

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Author: Grafiati

Published: 10 March 2023

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1

Li, Min, Haifeng Sang, Panpan Liu, and Guorui Huang. "Practical Criteria for H-Tensors and Their Application." Symmetry 14, no. 1 (January 13, 2022): 155. http://dx.doi.org/10.3390/sym14010155.

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Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method.

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2

Gong, Wenbin, and Yaqiang Wang. "Some new criteria for judging $ \mathcal{H} $-tensors and their applications." AIMS Mathematics 8, no. 4 (2023): 7606–17. http://dx.doi.org/10.3934/math.2023381.

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<abstract><p>$ \mathcal{H} $-tensors play a key role in identifying the positive definiteness of even-order real symmetric tensors. Some criteria have been given since it is difficult to judge whether a given tensor is an $ \mathcal{H} $-tensor, and their range of judgment has been limited. In this paper, some new criteria, from an increasing constant $ k $ to scale the elements of a given tensor can expand the range of judgment, are obtained. Moreover, as an application of those new criteria, some sufficient conditions for judging positive definiteness of even-order real symmetric tensors are proposed. In addition, some numerical examples are presented to illustrate those new results.</p></abstract>

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3

Cui, Chun-Feng, Yu-Hong Dai, and Jiawang Nie. "All Real Eigenvalues of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 35, no. 4 (January 2014): 1582–601. http://dx.doi.org/10.1137/140962292.

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4

VALERO, CARLOS. "MORSE THEORY FOR EIGENVALUE FUNCTIONS OF SYMMETRIC TENSORS." Journal of Topology and Analysis 01, no. 04 (December 2009): 417–29. http://dx.doi.org/10.1142/s1793525309000199.

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Given a symmetric tensor on a real vector bundle of dimension two, we construct a space where this tensor corresponds to a scalar function. We prove that under certain regularity conditions such a space and the corresponding scalar function are smooth. We study the topology of this space for the case of surfaces and produce a version of Morse inequalities for symmetric tensors. We apply our results to the geometry of surfaces.

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5

Chang, K. C., Kelly Pearson, and Tan Zhang. "On eigenvalue problems of real symmetric tensors." Journal of Mathematical Analysis and Applications 350, no. 1 (February 2009): 416–22. http://dx.doi.org/10.1016/j.jmaa.2008.09.067.

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6

Zhao, Jianxing, and Caili Sang. "Two new eigenvalue localization sets for tensors and theirs applications." Open Mathematics 15, no. 1 (October 9, 2017): 1267–76. http://dx.doi.org/10.1515/math-2017-0106.

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Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324) and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50). As an application, a weaker checkable sufficient condition for the positive (semi-)definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1), 187-198). As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

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7

Ballico, Edoardo, and Alessandra Bernardi. "Real and Complex Rank for Real Symmetric Tensors with Low Ranks." Algebra 2013 (March 21, 2013): 1–5. http://dx.doi.org/10.1155/2013/794054.

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We study the case of a real homogeneous polynomial whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that if the sum of the complex and the real ranks of is at most , then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.

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8

Friedland, Shmuel. "Best rank one approximation of real symmetric tensors can be chosen symmetric." Frontiers of Mathematics in China 8, no. 1 (December 6, 2012): 19–40. http://dx.doi.org/10.1007/s11464-012-0262-x.

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9

Wang, Gang, Linxuan Sun, and Yiju Wang. "Sharp Z-eigenvalue inclusion set-based method for testing the positive definiteness of multivariate homogeneous forms." Filomat 34, no. 9 (2020): 3131–39. http://dx.doi.org/10.2298/fil2009131w.

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In this paper, we establish a sharp Z-eigenvalue inclusion set for even-order real tensors by Z-identity tensor and prove that new Z-eigenvalue inclusion set is sharper than existing results. We propose some sufficient conditions for testing the positive definiteness of multivariate homogeneous forms via new Z-eigenvalue inclusion set. Further, we establish upper bounds on the Z-spectral radius of weakly symmetric nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms. The given numerical experiments show the validity of our results.

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10

Pearson, Kelly Jeanne, and Tan Zhang. "The nonexistence of rank4IP tensors in signature(1,3)." International Journal of Mathematics and Mathematical Sciences 31, no. 5 (2002): 259–69. http://dx.doi.org/10.1155/s0161171202108106.

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LetVbe a real vector space of dimension4with a nondegenerate symmetric bilinear form of signature(1,3). We show that there exists no algebraic curvature tensorRonVso that its associated skew-symmetric operatorR(⋅)has rank4and constant eigenvalues on the Grassmannian of nondegenerate2-planes inV.

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11

Neto, E. Lucena, A. L. Carvalho Neto, and P. I. B. Queiroz. "Eigenvalue Finding of Three-Dimensional Second-Order Real Symmetric Tensors." International Journal of Mechanical Engineering Education 40, no. 3 (July 2012): 172–81. http://dx.doi.org/10.7227/ijmee.40.3.2.

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12

Jaffe, Ariel, Roi Weiss, and Boaz Nadler. "Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 39, no. 3 (January 2018): 1071–94. http://dx.doi.org/10.1137/17m1133312.

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13

Wang, Feng, and Deshu Sun. "New iterative codes for 𝓗-tensors and an application." Open Mathematics 14, no. 1 (January 1, 2016): 212–20. http://dx.doi.org/10.1515/math-2016-0022.

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AbstractNew iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.

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14

Gama, Sílvio M. A., Roman Chertovskih, and Vladislav Zheligovsky. "Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation." Fluids 4, no. 2 (June 14, 2019): 110. http://dx.doi.org/10.3390/fluids4020110.

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We present examples of Padé approximations of the α -effect and eddy viscosity/diffusivity tensors in various flows. Expressions for the tensors derived in the framework of the standard multiscale formalism are employed. Algebraically, the simplest case is that of a two-dimensional parity-invariant six-fold rotation-symmetric flow where eddy viscosity is negative, indicating intervals of large-scale instability of the flow. Turning to the kinematic dynamo problem for three-dimensional flows of an incompressible fluid, we explore the application of Padé approximants for the computation of tensors of magnetic α -effect and, for parity-invariant flows, of magnetic eddy diffusivity. We construct Padé approximants of the tensors expanded in power series in the inverse molecular diffusivity 1 / η around 1 / η = 0 . This yields the values of the dominant growth rate to satisfactory accuracy for η , several dozen times smaller than the threshold, above which the power series is convergent. We do computations in Fortran in the standard “double” (real*8) and extended “quadruple” (real*16) precision, and perform symbolic calculations in Mathematica.

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15

Han, Lixing. "An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors." Numerical Algebra, Control and Optimization 3, no. 3 (July 2013): 583–99. http://dx.doi.org/10.3934/naco.2013.3.583.

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16

Jin, Hongwei, M. Rajesh Kannan, and Minru Bai. "Lower and upper bounds for H-eigenvalues of even order real symmetric tensors." Linear and Multilinear Algebra 65, no. 7 (October 11, 2016): 1402–16. http://dx.doi.org/10.1080/03081087.2016.1242112.

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17

Materassi, Massimo, Giulia Marcucci, and Claudio Conti. "Metriplectic Structure of a Radiation–Matter-Interaction Toy Model." Entropy 24, no. 4 (April 4, 2022): 506. http://dx.doi.org/10.3390/e24040506.

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A dynamical system defined by a metriplectic structure is a dissipative model characterized by a specific pair of tensors, which defines a Leibniz bracket; and a free energy, formed by a “Hamiltonian” and an entropy, playing the role of dynamics generator. Generally, these tensors are a Poisson bracket tensor, describing the Hamiltonian part of the dynamics, and a symmetric metric tensor, that models purely dissipative dynamics. In this paper, the metriplectic system describing a simplified two-photon absorption by a two-level atom is disclosed. The Hamiltonian component is sufficient to describe the free electromagnetic radiation. The metric component encodes the radiation–matter coupling, driving the system to an asymptotically stable state in which the excited level of the atom is populated due to absorption, and the radiation has disappeared. First, a description of the system is used, based on the real–imaginary decomposition of the electromagnetic field phasor; then, the whole metriplectic system is re-written in terms of the phase–amplitude pair, named Madelung variables. This work is intended as a first result to pave the way for applying the metriplectic formalism to many other irreversible processes in nonlinear optics.

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18

Murakami, S. "Mechanical Modeling of Material Damage." Journal of Applied Mechanics 55, no. 2 (June 1, 1988): 280–86. http://dx.doi.org/10.1115/1.3173673.

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A systematic theory to describe the anisotropic damage states of materials and a consistent definition of effective stress tensors are developed within the framework of continuum damage mechanics. By introducing a fictitious undamaged configuration, mechanically equivalent to the real damaged configuration, the classical creep damage theory is extended to the general three-dimensional states of material damage; it is shown that the damage state can be described in terms of a symmetric second rank tensor. The physical implications, mathematical restrictions, and the limitations of this damage tensor, as well as the effects of finite deformation on the damage state, are discussed in some detail. The notion of the fictitious undamaged configuration is then applied also to the definition of effective stresses. Finally, the extension of the effective stresses incorporating the effects of crack closure is discussed. The resulting effective stress tensor is employed to analyze the stress-path dependence of the elastic behavior of a cracked elastic-brittle material.

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19

Ballico, Edoardo. "ON THE TYPICAL RANK OF REAL POLYNOMIALS (OR SYMMETRIC TENSORS) WITH A FIXED BORDER RANK." Acta Mathematica Vietnamica 39, no. 3 (August 8, 2014): 367–78. http://dx.doi.org/10.1007/s40306-014-0068-x.

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20

Elsheikh, Safa, Andrew Fish, and Diwei Zhou. "Exploiting Spatial Information to Enhance DTI Segmentations via Spatial Fuzzy c-Means with Covariance Matrix Data and Non-Euclidean Metrics." Applied Sciences 11, no. 15 (July 29, 2021): 7003. http://dx.doi.org/10.3390/app11157003.

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A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods.

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21

Watanabe, Atsuo, Kunio Hayakawa, and Shinichiro Fujikawa. "An Anisotropic Damage Model for Prediction of Ductile Fracture during Cold-Forging." Metals 12, no. 11 (October 27, 2022): 1823. http://dx.doi.org/10.3390/met12111823.

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Researchers have formulated equations of ductile fracture to simulate and predict defects in cold-forged parts, e.g., the Cockcroft–Latham criterion. However, these equations are not applicable to certain cases of fracture in forged products. This study formulates a new equation for predicting ductile fractures with better prediction accuracy than the convention by which the cost for trial-and-error design can be reduced. The equation is expressed as a second-rank symmetric tensor, which is the inner product of the stress and strain-increment tensors. The theoretical efficacy of the equation in predicting ductile fractures is verified via a uniaxial tensile test. The practicability of the equation is confirmed by applying it to the simulations of two real cold-forged components: a cold-forged hollow shaft and a flanged shaft. For the hollow shaft, the equation predicts the position where the ductile fracture would initiate, which—to the best of the authors’ knowledge—is unique to this study. For the flanged shaft, the equation predicts the occurrence of diagonal cracks due to different lubrication conditions.

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22

SEGAL, ARKADY Y. "POINT PARTICLE–SYMMETRIC TENSORS INTERACTION AND GENERALIZED GAUGE PRINCIPLE." International Journal of Modern Physics A 18, no. 27 (October 30, 2003): 5021–38. http://dx.doi.org/10.1142/s0217751x03015842.

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The model of a point particle in the background of external symmetric tensor fields is analyzed from the higher spin theory perspective. It is proposed that the gauge transformations of the infinite collection of symmetric tensor fields may be read off from the covariance properties of the point particle action w.r.t. general canonical transformations. The gauge group turns out to be a semidirect product of all phase space canonical transformations to an Abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. A general configuration of external fields includes rank-0,1,2 symmetric tensors, so the whole system may be truncated to ordinary particle in Einstein–Maxwell backgrounds by switching off the higher-rank symmetric tensors. When otherwise all the higher rank tensors are switched on, the full gauge group provides a huge gauge symmetry acting on the whole infinite collection of symmetric tensors. We analyze this gauge symmetry and show that the symmetric tensors which couple to the point particle should not be interpreted as Fronsdal gauge fields, but rather as gauge fields of some conformal higher spin theories. It is shown that the Fronsdal fields system possesses twice as many symmetric tensor fields as is contained in the general background of the point particle. Besides, the particle action in general backgrounds is shown to reproduce De Wit–Freedman point particle–symmetric tensors first order interaction suggested many years ago, and extends their result to all orders in interaction, while the generalized equivalence principle completes the first order covariance transformations found in their paper, in all orders.

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23

Ma, Rongsheng, and Donghe Pei. "The ∗-Ricci Operator on Hopf Real Hypersurfaces in the Complex Quadric." Mathematics 11, no. 1 (December 26, 2022): 90. http://dx.doi.org/10.3390/math11010090.

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We study the ∗-Ricci operator on Hopf real hypersurfaces in the complex quadric. We prove that for Hopf real hypersurfaces in the complex quadric, the ∗-Ricci tensor is symmetric if and only if the unit normal vector field is singular. In the following, we obtain that if the ∗-Ricci tensor of Hopf real hypersurfaces in the complex quadric is symmetric, then the ∗-Ricci operator is both Reeb-flow-invariant and Reeb-parallel. As the correspondence to the semi-symmetric Ricci tensor, we give a classification of real hypersurfaces in the complex quadric with the semi-symmetric ∗-Ricci tensor.

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24

Sharma, Ramesh. "Second order parallel tensor in real and complex space forms." International Journal of Mathematics and Mathematical Sciences 12, no. 4 (1989): 787–90. http://dx.doi.org/10.1155/s0161171289000967.

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Levy's theorem ‘A second order parallel symmetric non-singular tensor in a real space form is proportional to the metric tensor’ has been generalized by showing that it holds even if one assumes the second order tensor to be parallel (not necessarily symmetric and non-singular) in a real space form of dimension greater than two. Analogous result has been established for a complex space form.It has been shown that an affine Killing vector field in a non-flat complex space form is Killing and analytic.

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25

Kaparulin, Dmitry S. "Conservation Laws and Stability of Field Theories of Derived Type." Symmetry 11, no. 5 (May 7, 2019): 642. http://dx.doi.org/10.3390/sym11050642.

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We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time gives rise to the series of symmetries of action functional. If the wave operator is given by n-th-order polynomial then this series includes n independent entries, which can be explicitly constructed. The Noether theorem is then used to construct an n-parameter set of second-rank conserved tensors. The canonical energy-momentum tensor is included in the series, while the other entries define independent integrals of motion. The Lagrange anchor concept is applied to connect the general conserved tensor in the series with the original space-time translation symmetry. This result is interpreted as existence of multiple energy-momentum tensors in the class of derived systems. To study stability we seek for bounded-conserved quantities that are connected with the time translations. We observe that the derived theory is stable if its wave operator is defined by a polynomial with simple and real roots. The general constructions are illustrated by the examples of the Pais–Uhlenbeck oscillator, higher-derivative scalar field, and extended Chern–Simons theory.

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26

SASAKURA, NAOKI. "A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY." International Journal of Modern Physics A 25, no. 23 (September 20, 2010): 4475–92. http://dx.doi.org/10.1142/s0217751x10050433.

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Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

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27

Breiding, Paul. "How many eigenvalues of a random symmetric tensor are real?" Transactions of the American Mathematical Society 372, no. 11 (September 6, 2019): 7857–87. http://dx.doi.org/10.1090/tran/7910.

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28

Vanchurin, Vitaly. "Covariant information theory and emergent gravity." International Journal of Modern Physics A 33, no. 34 (December 10, 2018): 1845019. http://dx.doi.org/10.1142/s0217751x18450197.

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Informational dependence between statistical or quantum subsystems can be described with Fisher information matrix or Fubini-Study metric obtained from variations/shifts of the sample/configuration space coordinates. Using these (noncovariant) objects as macroscopic constraints, we consider statistical ensembles over the space of classical probability distributions (i.e. in statistical space) or quantum wave functions (i.e. in Hilbert space). The ensembles are covariantized using dual field theories with either complex scalar field (identified with complex wave functions) or real scalar field (identified with square roots of probabilities). We construct space–time ensembles for which an approximate Schrodinger dynamics is satisfied by the dual field (which we call infoton due to its informational origin) and argue that a full space–time covariance on the field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy certain conservation equations. Then we switch to a thermodynamic description of the quantum/statistical systems and argue that the (inverse of) space–time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In (local) equilibrium, the entropy production vanishes, and the metric is not dynamical, but away from the equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described approximately by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium, these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry, the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein–Hilbert term. This proves that general relativity is equivalent to a theory of nonequilibrium (thermo)dynamics of the metric, but the theory is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.

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29

Jankun-Kelly, T. J., Y. S. Lanka, and J. E. Swan. "An Evaluation of Glyph Perception for Real Symmetric Traceless Tensor Properties." Computer Graphics Forum 29, no. 3 (August 12, 2010): 1133–42. http://dx.doi.org/10.1111/j.1467-8659.2009.01711.x.

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30

Jiang, Bo, Zhening Li, and Shuzhong Zhang. "Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations." SIAM Journal on Matrix Analysis and Applications 37, no. 1 (January 2016): 381–408. http://dx.doi.org/10.1137/141002256.

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31

BYTSENKO, A. A., V. S. MENDES, and A. C. TORT. "THERMODYNAMICS OF ABELIAN GAUGE FIELDS IN REAL HYPERBOLIC SPACES." International Journal of Modern Physics A 20, no. 13 (May 20, 2005): 2847–57. http://dx.doi.org/10.1142/s0217751x05020823.

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We work with N-dimensional compact real hyperbolic space XΓ with universal covering M and fundamental group Γ. Therefore, M is the symmetric space G/K, where G = SO 1(N, 1) and K = SO (N) is a maximal compact subgroup of G. We regard Γ as a discrete subgroup of G acting isometrically on M, and we take XΓ to be the quotient space by that action: XΓ = Γ∖M = Γ∖G/K. The natural Riemannian structure on M (therefore on X) induced by the Killing form of G gives rise to a connection p-form Laplacian 𝔏p on the quotient vector bundle (associated with an irreducible representation of K). We study gauge theories based on Abelian p-forms on the real compact hyperbolic manifold XΓ. The spectral zeta function related to the operator 𝔏p, considering only the coexact part of the p-forms and corresponding to the physical degrees of freedom, can be represented by the inverse Mellin transform of the heat kernel. The explicit thermodynamic functions related to skew-symmetric tensor fields are obtained by using the zeta-function regularization and the trace tensor kernel formula (which includes the identity and hyperbolic orbital integrals). Thermodynamic quantities in the high and low temperature expansions are calculated and new entropy/energy ratios established.

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32

Yao, Tinglan. "An optimal $ Z $-eigenvalue inclusion interval for a sixth-order tensor and its an application." AIMS Mathematics 7, no. 1 (2021): 967–85. http://dx.doi.org/10.3934/math.2022058.

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<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>

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33

Grigoryev, Sergey, and Arkadiy Leonov. "On several static cylindrically symmetric solutions of the Einstein equations." Modern Physics Letters A 31, no. 11 (April 10, 2016): 1650068. http://dx.doi.org/10.1142/s0217732316500681.

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We study the Einstein equations in the static cylindrically symmetric case with the stress–energy tensor of the form [Formula: see text], where [Formula: see text] is an unknown function and [Formula: see text], [Formula: see text], [Formula: see text] are arbitrary real constants ([Formula: see text] is assumed to be nonzero). The stress–energy tensor of this form includes as special cases several well-known solutions, such as the perfect fluid solution with the barotropic equation of state, the solution with the static electric field and the solution with the massless scalar field. We solve the Einstein equations with this stress–energy tensor and study some properties of the obtained metric.

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34

Decu, S., M. Petrovic-Torgasev, A. Sebekovic, and L. Verstraelen. "On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds." Tamkang Journal of Mathematics 41, no. 2 (June 30, 2010): 109–16. http://dx.doi.org/10.5556/j.tkjm.41.2010.662.

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In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.

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35

Ballico, E. "An upper bound for the real tensor rank and the real symmetric tensor rank in terms of the complex ranks." Linear and Multilinear Algebra 62, no. 11 (September 24, 2013): 1546–52. http://dx.doi.org/10.1080/03081087.2013.839671.

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36

Berezovskii, Volodymyr, Irena Hinterleitner, and Josef Mikes. "Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds." Filomat 32, no. 2 (2018): 379–85. http://dx.doi.org/10.2298/fil1802379b.

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In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than n(n+1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto symmetric manifolds.

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37

Deszyński, Krzysztof. "A theorem on polynomial lorentz structures." Glasgow Mathematical Journal 28, no. 2 (July 1986): 229–35. http://dx.doi.org/10.1017/s001708950000656x.

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Let M be a differentiable manifold of dimension m. A tensor field f of type (1, 1) on M is called a polynomial structure on M if it satisfies the equation:where a1, a2, …, an are real numbers and I denotes the identity tensor of type (1, 1).We shall suppose that for any x ∈ Mis the minimal polynomial of the endomorphism fx: TxM → TxM.We shall call the triple (M, f, g) a polynomial Lorentz structure if f is a polynomial structure on M, g is a symmetric and nondegenerate tensor field of type (0, 2) of signaturesuch that g (fX, fY) = g(X, Y) for any vector fields X, Y tangent to M. The tensor field g is a (generalized) Lorentz metric.

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38

Wang, Peitao, Zhaoshui He, Jun Lu, Beihai Tan, YuLei Bai, Ji Tan, Taiheng Liu, and Zhijie Lin. "An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation." Symmetry 12, no. 7 (July 17, 2020): 1187. http://dx.doi.org/10.3390/sym12071187.

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Symmetric nonnegative matrix factorization (SNMF) approximates a symmetric nonnegative matrix by the product of a nonnegative low-rank matrix and its transpose. SNMF has been successfully used in many real-world applications such as clustering. In this paper, we propose an accelerated variant of the multiplicative update (MU) algorithm of He et al. designed to solve the SNMF problem. The accelerated algorithm is derived by using the extrapolation scheme of Nesterov and a restart strategy. The extrapolation scheme plays a leading role in accelerating the MU algorithm of He et al. and the restart strategy ensures that the objective function of SNMF is monotonically decreasing. We apply the accelerated algorithm to clustering problems and symmetric nonnegative tensor factorization (SNTF). The experiment results on both synthetic and real-world data show that it is more than four times faster than the MU algorithm of He et al. and performs favorably compared to recent state-of-the-art algorithms.

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Liu, Jianlin, Gaofeng Cao, and Jing Sun. "Towards a Unified Route in Mechanics Based on the Second-Order Real Symmetric Tensor." International Journal of Mechanical Engineering Education 42, no. 2 (April 2014): 166–74. http://dx.doi.org/10.7227/ijmee.0011.

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Jankun-Kelly, T. j., and Ketan Mehta. "Superellipsoid-based, Real Symmetric Traceless Tensor Glyphs Motivated by Nematic Liquid Crystal Alignment Visualization." IEEE Transactions on Visualization and Computer Graphics 12, no. 5 (September 2006): 1197–204. http://dx.doi.org/10.1109/tvcg.2006.181.

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Liu, Xianyong, and Lizhuang Ma. "Topological analysis for 3D real, symmetric second-order tensor fields using Deviatoric Eigenvalue Wheel." Computers & Graphics 54 (February 2016): 28–37. http://dx.doi.org/10.1016/j.cag.2015.07.009.

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Azami, Shahroud. "The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow." Mathematica Slovaca 72, no. 6 (December 1, 2022): 1585–96. http://dx.doi.org/10.1515/ms-2022-0109.

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Abstract Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow $\begin{array}{} \displaystyle \frac{\partial g(x,t)}{\partial t}=2h(x,t), \end{array}$ where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential $$\begin{array}{} \displaystyle \partial_{t}u(x,t)=\Delta_{m}u(x,t)-\mathcal{R}(x,t)u(x,t) -au(x,t)\log u(x,t),\quad(x,t)\in M\times [0,T], \end{array}$$ along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.

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Clozel, Laurent, and Jack A. Thorne. "Level-raising and symmetric power functoriality, I." Compositio Mathematica 150, no. 5 (March 26, 2014): 729–48. http://dx.doi.org/10.1112/s0010437x13007653.

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AbstractAs the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. This is known for $n=2,3$ and $4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume $F$ totally real, and the initial representation $\pi $ of classical type.

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Srivastava, Ankit. "Causality and passivity in elastodynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2180 (August 2015): 20150256. http://dx.doi.org/10.1098/rspa.2015.0256.

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What are the constraints placed on the constitutive tensors of elastodynamics by the requirements that the linear elastodynamic system under consideration be both causal (effects succeed causes) and passive (system does not produce energy)? The analogous question has been tackled in other areas but in the case of elastodynamics its treatment is complicated by the higher order tensorial nature of its constitutive relations. In this paper, we clarify the effect of these constraints on highly general forms of the elastodynamic constitutive relations. We show that the satisfaction of passivity (and causality) directly requires that the hermitian parts of the transforms (Fourier and Laplace) of the time derivatives of the constitutive tensors be positive semi-definite. Additionally, the conditions require that the non-hermitian parts of the Fourier transforms of the constitutive tensors be positive semi-definite for positive values of frequency. When major symmetries are assumed these definiteness relations apply simply to the real and imaginary parts of the relevant tensors. For diagonal and one-dimensional problems, these positive semi-definiteness relationships reduce to simple inequality relations over the real and imaginary parts, as they should. Finally, we extend the results to highly general constitutive relations which include the Willis inhomogeneous relations as a special case.

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Nayebi, Ali, and Hojjatollah Rokhgireh. "Using of Anisotropic Continuum Damage Mechanics to Describe Yield Surface Distortion." Applied Mechanics and Materials 784 (August 2015): 11–18. http://dx.doi.org/10.4028/www.scientific.net/amm.784.11.

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In this paper, yield surface distortion was studied by considering the combination of nonlinear kinematic hardening model of Chaboche and a new anisotropic continuum damage evolution model. The constitutive relations for anisotropic damage of elastoplasic materials were developed based on irreversible thermodynamics. The internal state manifold which consists of internal variables to specify the thermodynamic state of solids was described by a 2nd rank symmetric damage tensor, the kinematic hardening tensor and tensor of movement of damage potential surface. In order to describe the damage state, the fictitious continuum domain was considered and the consistent relations between real and fictitious domains were developed. It was indicated that the combination of the Chaboche’s model and model of anisotropic continuum damage leads to the well description of the subsequent yield surface.

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Micunovic, M. V., C. Albertini, A. Grillo, I. Muha, G. Wittum, and L. Kudrjavceva. "Two dimensional plastic waves in quasi rate independent viscoplastic materials." Theoretical and Applied Mechanics 38, no. 1 (2011): 47–74. http://dx.doi.org/10.2298/tam1101047m.

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The subject of this work is an analysis of the experimental biaxial Hopkinson bar technique when such a device consists of a cruciform tensile specimen surrounded by four very long elastic bars. Unlike commonly applied by-pass analysis which attempts to draw conclusions from behavior of elastic bars we attempt to take into account real plastic waves inside the specimen with few hundreds of reflections. A quasi rate-independent as well as a more general rate-dependent tensor function model for ASME 537 steel are applied. Plastic wave speeds non-existent in traditional elasto-viscoplasticity are analyzed. Some preliminary numerical results for symmetric and non-symmetric loading cases valid for initial and subsequent elastic ranges are given.

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Berezovski, Volodymyr, Josef Mikes, Patrik Peska, and Lenka Rýparová. "On canonical F-planar mappings of spaces with affine connection." Filomat 33, no. 4 (2019): 1273–78. http://dx.doi.org/10.2298/fil1904273b.

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In this paper we study the theory of F-planar mappings of spaces with affine connection. We obtained condition, which preserved the curvature tensor. We also studied canonical F-planar mappings of space with affine connection onto symmetric spaces. In this case, the main equations have the partial differential Cauchy type form in covariant derivatives. We got the set of substantial real parameters on which depends the general solution of that PDE?s system.

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Sharafutdinov, Vladimir A. "Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus." Journal of Mathematics Research 14, no. 1 (December 22, 2021): 1. http://dx.doi.org/10.5539/jmr.v14n1p1.

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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.

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Lv, Changqing, and Changfeng Ma. "An iterative scheme for identifying the positive semi-definiteness of even-order real symmetric H-tensor." Journal of Computational and Applied Mathematics 392 (August 2021): 113498. http://dx.doi.org/10.1016/j.cam.2021.113498.

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Gómez-Ávila, Selim, Lao-Tse López-Lozano, and José Carlos Olvera-Meneses. "Light signals from spin one tensor dark matter." Pädi Boletín Científico de Ciencias Básicas e Ingenierías del ICBI 7, no. 13 (July 5, 2019): 72–75. http://dx.doi.org/10.29057/icbi.v7i13.3634.

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In this work we report on some observables associated with the annihilation of dark matter, modeled as a spin one real field in the antisymmetric tensor representation. This formulation has the interesting feature that stability is achieved without the introductionof ad hoc symmetries, as reported by Cata and Ibarra in 2014. In this model, the coupling to photons ocurrs through a Higgs portal term. The expression for the cross section in the photon scattering is calculated as well as their energy and the gamma ray flux.

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