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Journal articles on the topic 'Symmetric tensor'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

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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2

Comon, Pierre, Gene Golub, Lek-Heng Lim, and Bernard Mourrain. "Symmetric Tensors and Symmetric Tensor Rank." SIAM Journal on Matrix Analysis and Applications 30, no. 3 (January 2008): 1254–79. http://dx.doi.org/10.1137/060661569.

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3

Dresselhaus, M. S., and G. Dresselhaus. "Note on sufficient symmetry conditions for isotropy of the elastic moduli tensor." Journal of Materials Research 6, no. 5 (May 1991): 1114–18. http://dx.doi.org/10.1557/jmr.1991.1114.

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Group theoretical methods are used to obtain the form of the elastic moduli matrices and the number of independent parameters for various symmetries. Particular attention is given to symmetry groups for which 3D and 2D isotropy is found for the stress-strain tensor relation. The number of independent parameters is given by the number of times the fully symmetric representation is contained in the direct product of the irreducible representations for two symmetrical second rank tensors. The basis functions for the lower symmetry groups are found from the compatibility relations and are explicitly related to the elastic moduli. These types of symmetry arguments should be generally useful in treating the elastic properties of solids and composites.

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4

Perras, Frédéric A., and David L. Bryce. "A ZORA-DFT and NLMO study of the one-bond fluorine–X indirect nuclear spin-spin coupling tensors for various VSEPR geometries." Canadian Journal of Chemistry 89, no. 7 (July 2011): 789–802. http://dx.doi.org/10.1139/v10-172.

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Zeroth-order regular approximation (ZORA) density functional theory (DFT) calculations of one-bond X–19F indirect nuclear spin-spin coupling (J) tensors were performed on a series of fluorine-containing compounds covering several valence shell electron pair repulsion (VSEPR) theory geometries for which J, by symmetry, is not required to be axially symmetric. The calculations show that the antisymmetric components of J are only of the same order of magnitude as the principal components of the symmetric J-coupling tensor for a few geometries, and that in cases of approximate axial symmetry along the bond, J remains nearly axially symmetric with its unique component along the bond. In general, different species having the same nominal geometry tend to have similar tensor orientations, magnitudes of anisotropy of J relative to the isotropic coupling constant, as well as the same dominant contributions from the different coupling mechanisms. Structures are also systematically modified to determine how the tensor components depend on geometrical parameters. The isotropic coupling constants are subsequently interpreted using a natural localized molecular orbital (NLMO) approach. Our results could prove to be useful for future experimental characterizations of J tensors in systems having symmetry properties that do not force J to be axially symmetric or coincident with the dipolar coupling tensor.

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5

Zlatanovic, Milan, and Svetislav Mincic. "Identities for curvature tensors in generalized Finsler space." Filomat 23, no. 2 (2009): 34–42. http://dx.doi.org/10.2298/fil0902034z.

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In the some previous works we have obtained several curvature tensors in the generalized Finsler space GFN (the space with non-symmetric basic tensor and non-symmetric connection in Rund's sence). In this work we study identities for the mentioned tensors (the antisymmetriy with respect of two indices, the cyclic symmetry, the symmetry with respect of pairs of indices).

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6

Bóna, Andrej. "Symmetry characterization and measurement errors of elasticity tensors." GEOPHYSICS 74, no. 5 (September 2009): WB75—WB78. http://dx.doi.org/10.1190/1.3184013.

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It is often desirable to approximate a full anisotropic tensor, given by 21 independent parameters, by one with a higher symmetry. If one considers measurement errors of an elasticity tensor, the standard approaches of finding the best approximation by a higher symmetric tensor do not produce the most likely tensor. To find such a tensor, I replace the distance metric used in previous studies with one based on probability distribution functions of the errors of the measured quantities. In the case of normally distributed errors, the most likely tensor with higher symmetries coincides with the closest higher symmetric tensor, using a deviation-scaled Euclidean metric.

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7

Suh, Young Jin, Carlo Alberto Mantica, Uday Chand De, and Prajjwal Pal. "Pseudo B-symmetric manifolds." International Journal of Geometric Methods in Modern Physics 14, no. 09 (August 2, 2017): 1750119. http://dx.doi.org/10.1142/s0219887817501195.

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In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

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8

Huang, Zheng-Hai, and Liqun Qi. "Stationary Probability Vectors of Higher-Order Two-Dimensional Symmetric Transition Probability Tensors." Asia-Pacific Journal of Operational Research 37, no. 04 (July 24, 2020): 2040019. http://dx.doi.org/10.1142/s0217595920400199.

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In this paper, we investigate stationary probability vectors of higher-order two-dimensional symmetric transition probability tensors. We show that there are two special symmetric transition probability tensors of order [Formula: see text] dimension 2, which have and only have two stationary probability vectors; and any other symmetric transition probability tensor of order [Formula: see text] dimension 2 has a unique stationary probability vector. As a byproduct, we obtain that any symmetric transition probability tensor of order [Formula: see text] dimension 2 has a unique positive stationary probability vector, and that any symmetric irreducible transition probability tensor of order [Formula: see text] dimension 2 has a unique stationary probability vector.

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9

BERGMAN, JONAS, S. BRIAN EDGAR, and MAGNUS HERBERTHSON. "THE BACH TENSOR AND OTHER DIVERGENCE-FREE TENSORS." International Journal of Geometric Methods in Modern Physics 02, no. 01 (February 2005): 13–21. http://dx.doi.org/10.1142/s0219887805000442.

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In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

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10

Olive, M., B. Desmorat, B. Kolev, and R. Desmorat. "Reduced algebraic conditions for plane or axial tensorial symmetries." Mathematics and Mechanics of Solids 25, no. 12 (June 15, 2020): 2155–77. http://dx.doi.org/10.1177/1081286520920691.

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In this article, we formulate necessary and sufficient polynomial equations for the existence of a symmetry plane or an order-two axial symmetry for a totally symmetric tensor of order [Formula: see text]. These conditions are effective and of degree [Formula: see text] (the tensor’s order) in the components of the normal to the plane (or the direction of the axial symmetry). These results are then extended to obtain necessary and sufficient polynomial conditions for the existence of such symmetries for an elasticity tensor, a piezo-electricity tensor or a piezo-magnetism pseudo-tensor.

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11

Cvetkovic, Milica, and Milan Zlatanovic. "New Cartan’s tensors and pseudotensors in a generalized Finsler space." Filomat 28, no. 1 (2014): 107–17. http://dx.doi.org/10.2298/fil1401107c.

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In this work we defined a generalized Finsler space (GFN) as 2N-dimensional differentiable manifold with a non-symmetric basic tensor gij(x,x?), which applies that gij_?|m(x,x?)=0; ?=1,2. Based on non-symmetry of basic tensor, we obtained ten Ricci type identities, comparing to two kinds of covariant derivative of a tensor in Rund?s sense. There appear two new curvature tensors and fifteen magnitudes, we called ?curvature pseudotensors?.

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12

Leng, Kuang-dai, and Qiang Yang. "Fabric Tensor Characterization of Tensor-Valued Directional Data: Solution, Accuracy, and Symmetrization." Journal of Applied Mathematics 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/516060.

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Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame-indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor-valued orientation distribution functions (ODFs). Two characterizations of the tensor-valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respectively. The later proves to be nonconvergent and less accurate but still an available solution for where fabric tensors are required in full symmetry. Analytic solutions of the two types of fabric tensors characterizing centrosymmetric and anticentrosymmetric tensor-valued ODFs are presented in terms of orthogonal irreducible decompositions in both two- and three-dimensional (2D and 3D) spaces. Accuracy analysis is performed on normally distributed random ODFs to evaluate the approximation quality of the two characterizations, where fabric tensors of higher orders are employed. It is shown that the fitness is dominated by the dispersion degree of the original ODFs rather than the orders of fabric tensors. One application of tensor-valued ODF and fabric tensor in continuum damage mechanics is presented.

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13

MANTICA, CARLO ALBERTO, and YOUNG JIN SUH. "PSEUDO-Q-SYMMETRIC RIEMANNIAN MANIFOLDS." International Journal of Geometric Methods in Modern Physics 10, no. 05 (April 3, 2013): 1350013. http://dx.doi.org/10.1142/s0219887813500138.

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In this paper, we introduce a new kind of tensor whose trace is the well-known Z tensor defined by the present authors. This is named Q tensor: the displayed properties of such tensor are investigated. A new kind of Riemannian manifold that embraces both pseudo-symmetric manifolds ( PS )n and pseudo-concircular symmetric manifolds [Formula: see text] is defined. This is named pseudo-Q-symmetric and denoted with ( PQS )n. Various properties of such an n-dimensional manifold are studied: the case in which the associated covector takes the concircular form is of particular importance resulting in a pseudo-symmetric manifold in the sense of Deszcz [On pseudo-symmetric spaces, Bull. Soc. Math. Belgian Ser. A44 (1992) 1–34]. It turns out that in this case the Ricci tensor is Weyl compatible, a concept enlarging the classical Derdzinski–Shen theorem about Codazzi tensors. Moreover, it is shown that a conformally flat ( PQS )n manifold admits a proper concircular vector and the local form of the metric tensor is given. The last section is devoted to the study of ( PQS )n space-time manifolds; in particular we take into consideration perfect fluid space-times and provide a state equation. The consequences of the Weyl compatibility on the electric and magnetic part of the Weyl tensor are pointed out. Finally a ( PQS )n scalar field space-time is considered, and interesting properties are pointed out.

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14

Brachat, Jerome, Pierre Comon, Bernard Mourrain, and Elias Tsigaridas. "Symmetric tensor decomposition." Linear Algebra and its Applications 433, no. 11-12 (December 2010): 1851–72. http://dx.doi.org/10.1016/j.laa.2010.06.046.

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15

Casarotti, Alex, Alex Massarenti, and Massimiliano Mella. "On Comon’s and Strassen’s Conjectures." Mathematics 6, no. 11 (October 25, 2018): 217. http://dx.doi.org/10.3390/math6110217.

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Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.

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16

Tsvetkov, S. V. "Non-Linear Constitutive Equations for Transversely Isotropic Materials Belonging to the С∞ and С∞h Symmetry Groups." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 84 (June 2019): 46–59. http://dx.doi.org/10.18698/1812-3368-2019-3-46-59.

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Transversely isotropic materials feature infinite-order symmetry axes. Depending on which other symmetry elements are found in the material structure, five symmetry groups may be distinguished among transversely isotropic materials. We consider constitutive equations for these materials. These equations connect two symmetric second-order tensors. Two types of constitutive equations describe the properties of these five material groups. We derived constitutive equations for materials belonging to the C∞ and C∞h symmetry groups in the tensor function form. To do this, we used corollaries of Curie's Symmetry Principle. This makes it possible to obtain a fully irreducible form of the tensor function.

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17

MANTICA, CARLO ALBERTO, and YOUNG JIN SUH. "PSEUDO Z SYMMETRIC RIEMANNIAN MANIFOLDS WITH HARMONIC CURVATURE TENSORS." International Journal of Geometric Methods in Modern Physics 09, no. 01 (February 2012): 1250004. http://dx.doi.org/10.1142/s0219887812500041.

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In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

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18

He, Q. C. "A Remarkable Tensor in Plane Linear Elasticity." Journal of Applied Mechanics 64, no. 3 (September 1, 1997): 704–7. http://dx.doi.org/10.1115/1.2788952.

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It is shown that any two-dimensional elastic tensor can be orthogonally and uniquely decomposed into a symmetric tensor and an antisymmetric tensor. To within a scalar multiplier, the latter turns out to be equal to the right-angle rotation on the space of two-dimensional second-order symmetric tensors. On the basis of these facts, several useful results are derived for the traction boundary value problem of plane linear elasticity.

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19

Obster, Dennis, and Naoki Sasakura. "Counting Tensor Rank Decompositions." Universe 7, no. 8 (August 15, 2021): 302. http://dx.doi.org/10.3390/universe7080302.

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Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

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20

Sutcliffe, S. "Spectral Decomposition of the Elasticity Tensor." Journal of Applied Mechanics 59, no. 4 (December 1, 1992): 762–73. http://dx.doi.org/10.1115/1.2894040.

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The elasticity tensor in anisotropic elasticity can be regarded as a symmetric linear transformation on the nine-dimensional space of second-order tensors. This allows the elasticity tensor to be expressed in terms of its spectral decomposition. The structures of the spectral decompositions are determined by the sets of invariant subspaces that are consistent with material symmetry. Eigenvalues always depend on the values of the elastic constants, but the eigenvectors are, in part, independent of these values. The structures of the spectral decompositions are presented for the classical symmetry groups of crystallography, and numerical results are presented for representative materials in each group. Spectral forms for the equilibrium equations, the acoustic tensor, and the stored energy function are also derived.

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21

Stankovic, Mica, Milan Zlatanovic, and Nenad Vesic. "Some properties of ET-projective tensors obtained from Weyl projective tensor." Filomat 29, no. 3 (2015): 573–84. http://dx.doi.org/10.2298/fil1503573s.

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Vanishing of linearly independent curvature tensors of a non-symmetric affine connection space as functions of vanished curvature tensor of the associated space of this one are analyzed in the first part of this paper. Projective curvature tensors of a non-symmetric affine connection space are expressed as functions of the affine connection coefficients and Weyl projective tensor of the corresponding associated affine connection space in the second part of this paper.

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22

Surana, Karan S., and Stephen W. Long. "Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality." Entropy 22, no. 4 (April 14, 2020): 443. http://dx.doi.org/10.3390/e22040443.

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This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders n ε as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders n Θ . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

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23

CLARK, T. E., C. H. LEE, and S. T. LOVE. "SUPERSYMMETRIC TENSOR GAUGE THEORIES." Modern Physics Letters A 04, no. 14 (July 20, 1989): 1343–53. http://dx.doi.org/10.1142/s0217732389001532.

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The supersymmetric extensions of anti-symmetric tensor gauge theories and their associated tensor gauge symmetry transformations are constructed. The classical equivalence between such supersymmetric tensor gauge theories and supersymmetric non-linear sigma models is established. The global symmetry of the supersymmetric tensor gauge theory is gauged and the locally invariant action is obtained. The supercurrent on the Kähler manifold is found in terms of the supersymmetric tensor gauge field.

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24

Bernard, Guy M., Klaus Eichele, Gang Wu, Christopher W. Kirby, and Roderick E. Wasylishen. "Nuclear magnetic shielding tensors for the carbon, nitrogen, and selenium nuclei of selenocyanates - a combined experimental and theoretical approach." Canadian Journal of Chemistry 78, no. 5 (May 1, 2000): 614–25. http://dx.doi.org/10.1139/v00-046.

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The principal components of the carbon, nitrogen, and selenium chemical shift (CS) tensors for several solid selenocyanate salts have been determined by NMR measurements on stationary or slow magic-angle-spinning powder samples. Within experimental error, all three CS tensors are axially symmetric, consistent with the expected linear geometry of these anions. The spans (Ω) of the carbon and selenium CS tensors for the selenocyanate anion (SeCN-) are approximately 300 and 800 ppm, respectively, much less than the corresponding values for carbon diselenide (CSe2). This difference is a consequence of the difference in the CS tensor components perpendicular to the C infiniti symmetry axes in these systems. Ab initio calculations show that the orbital symmetries of these compounds are a significant factor in the shielding. For CSe2, efficient mixing of the σ and π orbitals results in a large paramagnetic contribution to the total shielding of the chemical shielding tensor components perpendicular to the molecular axis. Such mixing is less efficient for the SeCN-, resulting in a smaller paramagnetic contribution and hence in greater shielding in directions perpendicular to the molecular axis.Key words: selenocyanates, solid-state NMR, carbon shielding tensors, nitrogen shielding tensors, selenium shielding tensors, ab initio calculations.

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25

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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26

BALEANU, DUMITRU, and AYŞE KALKANLI KARASU. "LAX TENSORS, KILLING TENSORS AND GEOMETRIC DUALITY." Modern Physics Letters A 14, no. 37 (December 7, 1999): 2587–94. http://dx.doi.org/10.1142/s0217732399002716.

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The solution of the Lax tensor equations in the case Lαβγ= -Lβαγ was analyzed. The Lax tensors on the dual metrics were investigated. We classified all two-dimensional metrics having the symmetric Lax tensor Lαβγ. The Lax tensors of the flat space, Rindler system and its dual were found.

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27

MANSSUR, L. R. U., R. PORTUGAL, and B. F. SVAITER. "GROUP-THEORETIC APPROACH FOR SYMBOLIC TENSOR MANIPULATION." International Journal of Modern Physics C 13, no. 07 (September 2002): 859–79. http://dx.doi.org/10.1142/s0129183102004571.

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We describe how Computational Group Theory provides tools for manipulating indexed objects (tensors, spinors, and so on) in explicit index notation. In particular, we present algorithms that put tensors with free and dummy indices obeying permutation symmetries into the canonical form. The method is based on algorithms for determining canonical double coset representatives of subgroups of the symmetric group. The complexity of the algorithms is polynomial on the number of indices in practical applications and allows one to address the simplification of tensor expressions with hundreds of indices, going beyond what is needed in practical applications.

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28

Carando, Daniel, and Daniel Galicer. "Natural symmetric tensor norms." Journal of Mathematical Analysis and Applications 387, no. 2 (March 2012): 568–81. http://dx.doi.org/10.1016/j.jmaa.2011.09.027.

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29

Nie, Jiawang. "Symmetric Tensor Nuclear Norms." SIAM Journal on Applied Algebra and Geometry 1, no. 1 (January 2017): 599–625. http://dx.doi.org/10.1137/16m1083384.

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30

Ballico, E. "Tensor ranks and symmetric tensor ranks are the same for points with low symmetric tensor rank." Archiv der Mathematik 96, no. 6 (May 15, 2011): 531–34. http://dx.doi.org/10.1007/s00013-011-0274-x.

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31

Biswas, Indranil, and Niels Leth Gammelgaard. "Vassiliev invariants from symmetric spaces." Journal of Knot Theory and Its Ramifications 25, no. 10 (September 2016): 1650055. http://dx.doi.org/10.1142/s0218216516500553.

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We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

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32

Ali, Musavvir, Naeem Pundeer, and Young Suh. "Proper semiconformal symmetries of spacetimes with divergence-free semiconformal curvature tensor." Filomat 33, no. 16 (2019): 5191–98. http://dx.doi.org/10.2298/fil1916191a.

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In the present paper, the symmetries admitted by semiconformal curvature tensor in semiconformally symmetric spacetime have been studied and we show that a four-dimensional spacetime admitting a proper semiconformal symmetry is semiconformally flat or of the Petrov type N. It is also shown that a four-dimensional spacetime with divergence-free semiconformal curvature tensor admitting a proper semiconformal symmetry is locally of the Petrov type O or has four distinct principal null directions. In both the cases, we found that if the spacetime admits an infinitesimal semiconformal Killing vector field then the scalar curvature of the spacetime vanishes.

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33

De Paris, Alessandro. "Seeking for the Maximum Symmetric Rank." Mathematics 6, no. 11 (November 12, 2018): 247. http://dx.doi.org/10.3390/math6110247.

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We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.

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34

Tang, Changxin, Wei Wan, Lei Zhang, and Wennan Zou. "The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor." Materials 14, no. 18 (September 17, 2021): 5388. http://dx.doi.org/10.3390/ma14185388.

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The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

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35

Bazˇant, Zdeneˇk P. "Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate." Journal of Engineering Materials and Technology 120, no. 2 (April 1, 1998): 131–36. http://dx.doi.org/10.1115/1.2807001.

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It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.

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36

De, Uday, Young Suh, Sudhakar Chaubey, and Sameh Shenawy. "On pseudo H-symmetric Lorentzian manifolds with applications to relativity." Filomat 34, no. 10 (2020): 3287–97. http://dx.doi.org/10.2298/fil2010287d.

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In this paper, we introduce a new type of curvature tensor named H-curvature tensor of type (1, 3) which is a linear combination of conformal and projective curvature tensors. First we deduce some basic geometric properties of H-curvature tensor. It is shown that a H-flat Lorentzian manifold is an almost product manifold. Then we study pseudo H-symmetric manifolds (PHS)n (n > 3) which recovers some known structures on Lorentzian manifolds. Also, we provide several interesting results. Among others, we prove that if an Einstein (PHS)n is a pseudosymmetric (PS)n, then the scalar curvature of the manifold vanishes and conversely. Moreover, we deal with pseudo H-symmetric perfect fluid spacetimes and obtain several interesting results. Also, we present some results of the spacetime satisfying divergence free H-curvature tensor. Finally, we construct a non-trivial Lorentzian metric of (PHS)4.

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37

Liu, Qilong, and Yaotang Li. "Bounds for the Z-eigenpair of general nonnegative tensors." Open Mathematics 14, no. 1 (January 1, 2016): 181–94. http://dx.doi.org/10.1515/math-2016-0017.

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AbstractIn this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.

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38

Mantica, Carlo Alberto, and Young Jin Suh. "On weakly conformally symmetric pseudo-Riemannian manifolds." Reviews in Mathematical Physics 29, no. 03 (March 14, 2017): 1750007. http://dx.doi.org/10.1142/s0129055x17500076.

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In this paper, we study the properties of weakly conformally symmetric pseudo- Riemannian manifolds focusing particularly on the [Formula: see text]-dimensional Lorentzian case. First, we provide a new proof of an important result found in literature; then several new others are stated. We provide a decomposition for the conformal curvature tensor in [Formula: see text]. Moreover, some important identities involving two particular covectors are stated; for example, it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further, we study weakly conformally symmetric [Formula: see text]-dimensional Lorentzian manifolds (space-times): it is proven that one of the previously defined co-vectors is null and unique up to a scaling. Moreover, it is shown that under certain conditions, the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is stated that such space-time is of Petrov type N with respect to the same vector.

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39

Petrović, Miloš Z., Mića S. Stanković, and Patrik Peška. "On Conformal and Concircular Diffeomorphisms of Eisenhart’s Generalized Riemannian Spaces." Mathematics 7, no. 7 (July 15, 2019): 626. http://dx.doi.org/10.3390/math7070626.

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We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart’s generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart’s generalized Riemannian spaces. Finally, we describe some properties of covariant derivatives of tensors analogous to Yano’s tensor of concircular curvature in Eisenhart symmetric spaces of various kinds.

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40

Rigelesaiyin, Ji, Adrian Diaz, Weixuan Li, Liming Xiong, and Youping Chen. "Asymmetry of the atomic-level stress tensor in homogeneous and inhomogeneous materials." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2217 (September 2018): 20180155. http://dx.doi.org/10.1098/rspa.2018.0155.

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The stress tensor is described as a symmetric tensor in all classical continuum mechanics theories and in most existing statistical mechanics formulations. In this work, we examine the theoretical origins of the symmetry of the stress tensor and identify the assumptions and misinterpretations that lead to its symmetric property. We then make a direct measurement of the stress tensor in molecular dynamics simulations of four different material systems using the physical definition of stress as force per unit area acting on surface elements. Simulation results demonstrate that the stress tensor is asymmetric near dislocation cores, phase boundaries, holes and even in homogeneous material under a shear loading. In addition, the atomic virial stress and Hardy stress formulae are shown to significantly underestimate the stress tensor in regions of stress concentration.

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41

MURTHY, GARIMELLA RAMA. "MULTI/INFINITE DIMENSIONAL NEURAL NETWORKS, MULTI/INFINITE DIMENSIONAL LOGIC THEORY." International Journal of Neural Systems 15, no. 03 (June 2005): 223–35. http://dx.doi.org/10.1142/s0129065705000190.

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A mathematical model of an arbitrary multi-dimensional neural network is developed and a convergence theorem for an arbitrary multi-dimensional neural network represented by a fully symmetric tensor is stated and proved. The input and output signal states of a multi-dimensional neural network/logic gate are related through an energy function, defined over the fully symmetric tensor (representing the connection structure of a multi-dimensional neural network). The inputs and outputs are related such that the minimum/maximum energy states correspond to the output states of the logic gate/neural network realizing a logic function. Similarly, a logic circuit consisting of the interconnection of logic gates, represented by a block symmetric tensor, is associated with a quadratic/higher degree energy function. Infinite dimensional logic theory is discussed through the utilization of infinite dimension/order tensors.

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42

Taşci, A. Yavuz, and F. Özen Zengin. "Z-symmetric manifold admitting concircular Ricci symmetric tensor." Afrika Matematika 31, no. 7-8 (April 18, 2020): 1093–104. http://dx.doi.org/10.1007/s13370-020-00782-5.

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43

Wang, Gang, Linxuan Sun, and Lixia Liu. "M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors." Complexity 2020 (January 8, 2020): 1–8. http://dx.doi.org/10.1155/2020/2474278.

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M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

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44

Velimirovic, Ana, and Milan Zlatanovic. "On semisymmetric connection." Filomat 33, no. 4 (2019): 1179–84. http://dx.doi.org/10.2298/fil1904179v.

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Using the non-symmetry of a connection, it is possible to introduce four types of covariant derivatives. Based on these derivatives, several types of Ricci?s identities and twelve curvature tensors are obtained. Five of them are linearly independent but the other curvature tensors can be expressed as linear combinations of these five linearly independent curvature tensors and the curvature tensor of the corresponding associated symmetric space. The semisymmetric connection is defined and the properties of two of the five independent curvature tensors are analyzed. In the same manner, the properties for three others curvature tensors may be derived.

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45

Theocaris, P. S., and D. P. Sokolis. "Linear elastic eigenstates of the compliance tensor for trigonal crystals." Zeitschrift für Kristallographie - Crystalline Materials 215, no. 1 (January 1, 2000): 1–9. http://dx.doi.org/10.1524/zkri.2000.215.1.01.

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The spectral decomposition of the compliance fourth-rank tensor, representative of a trigonal crystalline or other anisotropic medium, is offered in this paper, and its characteristic values and idempotent fourth-rank tensors are established, with respect to the Cartesian tensor components. Consequently, it is proven that the idempotent tensors serve to analyse the second-rank symmetric tensor space into orthogonal subspaces, resolving the stress and strain tensors for the trigonal medium into their eigentensors, and, finally, decomposing the total elastic strain energy density into distinct, autonomous components. Finally, bounds on the values of the compliance tensor components for the trigonal system, dictated by the classical thermodynamical argument for the elastic potential to be positive definite, are estimated by imposing the characteristic values of the compliance tensor to be strictly positive.

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46

Bergqvist, Göran, and Paul Lankinen. "Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2059 (June 15, 2005): 2181–95. http://dx.doi.org/10.1098/rspa.2004.1411.

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We present a study of Rainich-like conditions for symmetric and trace-free tensors T . For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, T must be the Bel–Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.

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47

Chen, Zhongming, Yannan Chen, Liqun Qi, and Wennan Zou. "Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor." Mathematics and Mechanics of Solids 24, no. 10 (March 8, 2019): 3092–102. http://dx.doi.org/10.1177/1081286519835246.

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The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith–Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

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48

Clark, T. E., C. H. Lee, and S. T. Love. "Anti-symmetric tensor gauge theories." Nuclear Physics B 308, no. 2-3 (October 1988): 379–96. http://dx.doi.org/10.1016/0550-3213(88)90569-x.

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49

Kwieciński, Michal. "Tensor powers of symmetric algebras." Communications in Algebra 24, no. 3 (January 1996): 793–801. http://dx.doi.org/10.1080/00927879608825600.

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50

Mendez, M. "Tensor species and symmetric functions." Proceedings of the National Academy of Sciences 88, no. 21 (November 1, 1991): 9892–94. http://dx.doi.org/10.1073/pnas.88.21.9892.

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