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Статті в журналах з теми "Fourth-order Tensors"

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Автор: Grafiati

Опубліковано: 6 вересня 2023

Оновлено: 7 вересня 2023

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1

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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Анотація:

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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2

Betten, Josef. "Irreducible invariants of fourth-order tensors." Mathematical Modelling 8 (1987): 29–33. http://dx.doi.org/10.1016/0270-0255(87)90535-5.

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3

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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Анотація:

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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4

He, Jun, Yanmin Liu, Junkang Tian, and Zhuanzhou Zhang. "New Sufficient Condition for the Positive Definiteness of Fourth Order Tensors." Mathematics 6, no. 12 (December 5, 2018): 303. http://dx.doi.org/10.3390/math6120303.

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In this paper, we give a new Z-eigenvalue localization set for Z-eigenvalues of structured fourth order tensors. As applications, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative fourth order tensors is obtained and a new Z-eigenvalue based sufficient condition for the positive definiteness of fourth order tensors is also presented. Finally, numerical examples are given to verify the efficiency of our results.

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5

Desmorat, Boris, and Rodrigue Desmorat. "Tensorial Polar Decomposition of 2D fourth-order tensors." Comptes Rendus Mécanique 343, no. 9 (September 2015): 471–75. http://dx.doi.org/10.1016/j.crme.2015.07.002.

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6

Zhao, Jianxing. "E-eigenvalue Localization Sets for Fourth-Order Tensors." Bulletin of the Malaysian Mathematical Sciences Society 43, no. 2 (April 27, 2019): 1685–707. http://dx.doi.org/10.1007/s40840-019-00768-y.

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7

Ishida, Akio, Takumi Noda, Jun Murakami, Naoki Yamamoto, and Chiharu Okuma. "Calculation of Fourth-Order Tensor Product Expansion by Power Method and Comparison of it with Higher-Order Singular Value Decomposition." Applied Mechanics and Materials 444-445 (October 2013): 703–11. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.703.

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Анотація:

Higher-order singular value decomposition (HOSVD) is known as an effective technique to reduce the dimension of multidimensional data. We have proposed a method to perform third-order tensor product expansion (3OTPE) by using the power method for the same purpose as HOSVD, and showed that our method had a better accuracy property than HOSVD, and furthermore, required fewer computation time than that. Since our method could not be applied to the tensors of fourth-order (or more) in spite of having those useful properties, we extend our algorithm of 3OTPE calculation to forth-order tensors in this paper. The results of newly developed method are compared to those obtained by HOSVD. We show that the results follow the same trend as the case of 3OTPE.

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8

Papenfuss, Christina. "Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure." Journal of Modeling and Simulation of Materials 5, no. 1 (December 31, 2022): 39–52. http://dx.doi.org/10.21467/jmsm.5.1.39-52.

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A closure relation expresses the fourth order orientation tensor as a function of the second order one. Two well-known closure relations, the hybrid closure and the maximum entropy closure, are compared in the case of a rotation symmetric orientation distribution function. The maximum entropy closure predicts a positive fourth order parameter in the whole range of the second order parameter, whereas the hybrid closure results in negative fourth order parameters for small values of the second order one. For the maximum entropy closure quadratic fit polynomials are presented. For a general distribution without rotation symmetry, the expression for the entropy is exploited to derive an explicit form for the maximum entropy distribution. Lowest order approximation of this distribution function leads to simple closure forms for the fourth order alignment tensor and also for higher order alignment tensors.

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9

Xiao, B., and J. Feng. "Higher order elastic tensors of crystal structure under non-linear deformation." Journal of Micromechanics and Molecular Physics 04, no. 04 (December 2019): 1950007. http://dx.doi.org/10.1142/s2424913019500073.

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The higher-order elastic tensors can be used to characterize the linear and non-linear mechanical properties of crystals at ultra-high pressures. Besides the widely studied second-order elastic constants, the third- and fourth-order elastic constants are sixth and eighth tensors, respectively. The independent tensor components of them are completely determined by the symmetry of the crystal. Using the relations between elastic constants and sound velocity in solid, the independent elastic constants can be measured experimentally. The anisotropy in elasticity of crystal structures is directly determined by the independent elastic constants.

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10

Desmorat, R., N. Auffray, B. Desmorat, B. Kolev, and M. Olive. "Generic separating sets for three-dimensional elasticity tensors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20190056. http://dx.doi.org/10.1098/rspa.2019.0056.

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We define a generic separating set of invariant functions (a.k.a. a weak functional basis ) for tensors. We then produce two generic separating sets of polynomial invariants for three-dimensional elasticity tensors, one consisting of 19 polynomials and one consisting of 21 polynomials (but easier to compute), and a generic separating set of 18 rational invariants. As a by-product, a new integrity basis for the fourth-order harmonic tensor is provided.

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11

Kintzel, O., and Y. Başar. "Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis." ZAMM 86, no. 4 (April 3, 2006): 291–311. http://dx.doi.org/10.1002/zamm.200410242.

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12

Voyiadjis, G. Z., and T. Park. "Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor." Journal of Applied Mechanics 64, no. 1 (March 1, 1997): 106–10. http://dx.doi.org/10.1115/1.2787259.

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Анотація:

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

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13

Milton, Graeme W., and Andrej V. Cherkaev. "Which Elasticity Tensors are Realizable?" Journal of Engineering Materials and Technology 117, no. 4 (October 1, 1995): 483–93. http://dx.doi.org/10.1115/1.2804743.

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Анотація:

It is shown that any given positive definite fourth order tensor satisfying the usual symmetries of elasticity tensors can be realized as the effective elasticity tensor of a two-phase composite comprised of a sufficiently compliant isotropic phase and a sufficiently rigid isotropic phase configured in an suitable microstructure. The building blocks for constructing this composite are what we call extremal materials. These are composites of the two phases which are extremely stiff to a set of arbitrary given stresses and, at the same time, are extremely compliant to any orthogonal stress. An appropriately chosen subset of the extremal materials are layered together to form the composite with elasticity tensor matching the given tensor.

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14

Kintzel, O. "Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part II: Tensor analysis on manifolds." ZAMM 86, no. 4 (April 3, 2006): 312–34. http://dx.doi.org/10.1002/zamm.200410243.

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15

Le Quang, H., and Q. C. He. "The number and types of all possible rotational symmetries for flexoelectric tensors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2132 (March 2, 2011): 2369–86. http://dx.doi.org/10.1098/rspa.2010.0521.

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Flexoelectricity is due to the electric polarization generated by a non-zero strain gradient in a dielectric material without or with centrosymmetric microstructure. It is characterized by a fourth-order tensor, referred to as flexoelectric tensor, which relates the electric polarization vector to the gradient of the second-order strain tensor. This paper solves the fundamental problem of determining the number and types of all possible rotational symmetries for flexoelectric tensors and specifies the number of independent material parameters contained in a flexoelectric tensor belonging to a given symmetry class. These results are useful and even indispensable for experimentally identifying or theoretically/numerically estimating the flexoelectric coefficients of a dielectric material.

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16

Le Quang, Hung, Qi-Chang He, and Nicolas Auffray. "Classification of first strain-gradient elasticity tensors by symmetry planes." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2251 (July 2021): 20210165. http://dx.doi.org/10.1098/rspa.2021.0165.

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First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

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17

Che, Haitao, Haibin Chen, and Yiju Wang. "On the M-eigenvalue estimation of fourth-order partially symmetric tensors." Journal of Industrial & Management Optimization 16, no. 1 (2020): 309–24. http://dx.doi.org/10.3934/jimo.2018153.

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18

Moakher, M. "Fourth-order cartesian tensors: old and new facts, notions and applications." Quarterly Journal of Mechanics and Applied Mathematics 61, no. 2 (January 23, 2008): 181–203. http://dx.doi.org/10.1093/qjmam/hbm027.

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19

Kaushik, Sumit, and Jan Slovák. "HARDI Segmentation via Fourth-Order Tensors and Anisotropy Preserving Similarity Measures." Journal of Mathematical Imaging and Vision 61, no. 8 (August 23, 2019): 1221–34. http://dx.doi.org/10.1007/s10851-019-00897-w.

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20

Moreno, Rodrigo, Örjan Smedby, and Dieter H. Pahr. "Prediction of apparent trabecular bone stiffness through fourth-order fabric tensors." Biomechanics and Modeling in Mechanobiology 15, no. 4 (September 4, 2015): 831–44. http://dx.doi.org/10.1007/s10237-015-0726-5.

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21

Yao, Yuyan, and Gang Wang. "Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors." Mathematical Foundations of Computing 5, no. 1 (2022): 33. http://dx.doi.org/10.3934/mfc.2021018.

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Анотація:

<p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum <inline-formula><tex-math id="M2">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.</p>

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22

Ahmad Mirshafeazadeh, Mir, and Behroz Bidabad. "On generalized quasi-Einstein manifolds." Advances in Pure and Applied Mathematics 10, no. 3 (July 1, 2019): 193–202. http://dx.doi.org/10.1515/apam-2017-0112.

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Abstract We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with {(n-1)} -dimensional Einstein fibers in suitable cases.

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23

Jog, C. S. "A Concise Proof of the Representation Theorem for Fourth-Order Isotropic Tensors." Journal of Elasticity 85, no. 2 (August 29, 2006): 119–24. http://dx.doi.org/10.1007/s10659-006-9074-0.

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24

Helnwein, Peter. "Some remarks on the compressed matrix representation of symmetric second-order and fourth-order tensors." Computer Methods in Applied Mechanics and Engineering 190, no. 22-23 (February 2001): 2753–70. http://dx.doi.org/10.1016/s0045-7825(00)00263-2.

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25

Qasem, Fahmi Yaseen Abdo, та Wafa'a Hadi Ali Hadi. "On a generalized βK-Birecurrent Finsler space". University of Aden Journal of Natural and Applied Sciences 22, № 1 (30 квітня 2018): 167–73. http://dx.doi.org/10.47372/uajnas.2018.n1.a14.

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Анотація:

In the present paper, we introduceda Finsler space whose Cartan's fourth curvature tensor \(K_{jkh}^i\) satisfies the condition \(B_n B_m K_{jkh}^i=a_{mn} K_{jkh}^i+b_{mn} (δ_k^i g_{jh}-δ_h^i g_{jk} )- 2y^r μ_n B_r (δ_k^i C_{jhm}-δ_h^i C_{jkm})\), where Bn Bm are Berwald's covariant differential operator of the second order with respect to xm and xn, successively, Br is Berwald's covariant differential operator of the first order with respect to xr, amn and bmn are non-zero covariant tensors field of second order called recurrence tensorsfield and μn is non-zero covariant vector field, such space is called as a generalized βK-birecurrent space. The aim of this paper is to prove that thecurvature tensor \(H_{jkh}^i\) satisfies the generalized birecurrence property. We proved that Ricci tensors Hjk, Kjk, the curvature vector Hk and the curvature scalarHof such space are non-vanishing andunder certain conditions, a generalized βK-birecurrent space becomes Landsberg space. Also, some conditions have been pointed out which reduce a generalized βK-birecurrent space Fn (n>2) into Finsler space of curvature scalar.

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26

Wang, Xuezhong. "Best Rank-One Approximation of Fourth-Order Partially Symmetric Tensors by Neural Network." Numerical Mathematics: Theory, Methods and Applications 11, no. 4 (June 2018): 673–700. http://dx.doi.org/10.4208/nmtma.2018.s01.

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27

Itskov, Mikhail. "On the theory of fourth-order tensors and their applications in computational mechanics." Computer Methods in Applied Mechanics and Engineering 189, no. 2 (September 2000): 419–38. http://dx.doi.org/10.1016/s0045-7825(99)00472-7.

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28

Li, Suhua, and Yaotang Li. "Checkable Criteria for the M-Positive Definiteness of Fourth-Order Partially Symmetric Tensors." Bulletin of the Iranian Mathematical Society 46, no. 5 (December 20, 2019): 1455–63. http://dx.doi.org/10.1007/s41980-019-00335-y.

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29

He, Jun, Yanmin Liu, and Guangjun Xu. "Z-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Tensors." Bulletin of the Malaysian Mathematical Sciences Society 43, no. 2 (January 22, 2019): 1069–93. http://dx.doi.org/10.1007/s40840-019-00727-7.

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30

Yang, Qingzhi, Yujin Paek, and Wei Mei. "Further Investigation of Positive Semi-definiteness of Fourth-order Cauchy and Hilbert Tensors." Frontiers of Mathematics 18, no. 4 (July 22, 2023): 935–52. http://dx.doi.org/10.1007/s11464-020-0180-2.

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31

Fahrenthold, E. P., and A. Wu. "Bond Graph Modeling of Continuous Solids in Finite Strain Elastic-Plastic Deformation." Journal of Dynamic Systems, Measurement, and Control 110, no. 3 (September 1, 1988): 284–87. http://dx.doi.org/10.1115/1.3152683.

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Анотація:

The physical systems modeling theory of bond graphs may be employed to represent the hysteretic deformation of continuous solids infinite strain elastic-plastic deformation. Second and fourth order tensors represent stress power conjugate variables and transformer or gyrator moduli, respectively, with corresponding inner products generalizing the power flow and modulation calculations of scalar and vector bond graphs. Capacitors and resistors of a tensor type suitably represent strain energy storage, associated flow rules in plasticity, and other familiar concepts. An important application of this simulation technique arises in constitutive modeling studies of nonlinear materials.

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32

CAPRASSE, H., J. DEMARET, K. GATERMANN, and H. MELENK. "POWER-LAW TYPE SOLUTIONS OF FOURTH-ORDER GRAVITY FOR MULTIDIMENSIONAL BIANCHI I UNIVERSES." International Journal of Modern Physics C 02, no. 02 (June 1991): 601–11. http://dx.doi.org/10.1142/s0129183191000901.

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This paper is devoted to the application of computer algebra to the study of solutions of the field equations derived from a non-linear Lagrangian, as suggested by recently proposed unified theories. More precisely, we restrict ourselves to the most general quadratic Lagrangian, i.e. containing quadratic contributions in the different curvature tensors exclusively. The corresponding field equations are then fourth-order in the metric tensor components. The cosmological models studied are the simplest ones in the class of spatially homogeneous but anisotropic models, i.e. Bianchi I models. For these models, we consider only power-law type solutions of the field equations. All the solutions of the associated system of algebraic equations are found, using computer algebra, from a search of its Groebner bases. While, in space dimension d=3, the Einsteinian-Kasner metric is still the most general power-law type solution, for d>3, no solution, other than the Minkowski space-time, is common to the three systems of equations corresponding to the three contributions to the Lagrangian density. In the case of a pure Riemann-squared contribution to the Lagrangian (suggested by a recent calculation of the effective action for the heterotic string), the possibility exists to realize a splitting of the d-dimensional space into a (d−3)-dimensional internal space and a physical 3-dimensional space, the latter expanding in time as a power bigger than 2 (about 4.5 when d=9).

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33

JOSEPH, K. BABU, and M. SABIR. "REFORMULATION OF EINSTEIN GRAVITY AS A FLAT SPACE GAUGE THEORY." Modern Physics Letters A 03, no. 05 (April 1988): 497–509. http://dx.doi.org/10.1142/s021773238800060x.

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Based on an algebraic decomposition of a fourth rank tensor in terms of second rank tensors we suggest a reformulation of Einstein’s gravitational theory as a flat space gauge theory. This has been done by associating a curved manifold with a flat space U(2)×U(2) gauge theory. It is shown that while, in order to reproduce Einstein field equations one has to use a non-Yang-Mills action, the linearized equations follow from a Yang-Mills action. A relation between the metric and gauge fields is obtained. The consistency of the postulates is also verified.

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34

Hunana, P., T. Passot, E. Khomenko, D. Martínez-Gómez, M. Collados, A. Tenerani, G. P. Zank, Y. Maneva, M. L. Goldstein, and G. M. Webb. "Generalized Fluid Models of the Braginskii Type." Astrophysical Journal Supplement Series 260, no. 2 (June 1, 2022): 26. http://dx.doi.org/10.3847/1538-4365/ac5044.

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Анотація:

Abstract Several generalizations of the well-known fluid model of Braginskii (1965) are considered. We use the Landau collisional operator and the moment method of Grad. We focus on the 21-moment model that is analogous to the Braginskii model, and we also consider a 22-moment model. Both models are formulated for general multispecies plasmas with arbitrary masses and temperatures, where all of the fluid moments are described by their evolution equations. The 21-moment model contains two “heat flux vectors” (third- and fifth-order moments) and two “viscosity tensors” (second- and fourth-order moments). The Braginskii model is then obtained as a particular case of a one ion–electron plasma with similar temperatures, with decoupled heat fluxes and viscosity tensors expressed in a quasistatic approximation. We provide all of the numerical values of the Braginskii model in a fully analytic form (together with the fourth- and fifth-order moments). For multispecies plasmas, the model makes the calculation of the transport coefficients straightforward. Formulation in fluid moments (instead of Hermite moments) is also suitable for implementation into existing numerical codes. It is emphasized that it is the quasistatic approximation that makes some Braginskii coefficients divergent in a weakly collisional regime. Importantly, we show that the heat fluxes and viscosity tensors are coupled even in the linear approximation, and that the fully contracted (scalar) perturbations of the fourth-order moment, which are accounted for in the 22-moment model, modify the energy exchange rates. We also provide several appendices, which can be useful as a guide for deriving the Braginskii model with the moment method of Grad.

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35

Soto-Campos, Carlos A., and Susana Valdez-Alvarado. "Noncommutative Reissner–Nordstrøm black hole." Canadian Journal of Physics 96, no. 12 (December 2018): 1259–65. http://dx.doi.org/10.1139/cjp-2017-0599.

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Анотація:

In this work we construct a deformed embedding of the Reissner–Nordstrøm (R-N) space–time within the framework of a noncommutative Riemannian geometry. We provide noncommutative corrections to the usual Riemannian expressions for the metric and curvature tensors. For the case of the metric tensor, the expression obtained possesses terms that are valid to all orders in the deformation parameter. Then we calculate the correction to the area of the event horizon of the corresponding noncommutative R-N black hole, obtaining an expression for the area of the black hole, which is correct up to fourth-order terms in the deformation parameter. Finally we include some comments on the noncommutative version on one of the second-order scalar invariants of the Riemann tensor, the so-called Kretschmann invariant, a quantity that is regularly used to extend gravity to the quantum level.

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36

Sun, Linxuan, Gang Wang, and Lixia Liu. "Further Study on Z-Eigenvalue Localization Set and Positive Definiteness of Fourth-Order Tensors." Bulletin of the Malaysian Mathematical Sciences Society 44, no. 1 (May 3, 2020): 105–29. http://dx.doi.org/10.1007/s40840-020-00939-2.

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37

Itskov, M., and Y. Basar. "On the theory of fourth-order tensors and their application in large strain elasticity." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 80, S2 (2000): 523–24. http://dx.doi.org/10.1002/zamm.200008014132.

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38

Zhang, Peng, Huize Ren, Xiaobin Dong, Xuechao Wang, Mengxue Liu, Ying Zhang, Yufang Zhang, Jiuming Huang, Shuheng Dong, and Ruiming Xiao. "Understanding and Applications of Tensors in Ecosystem Services: A Case Study of the Manas River Basin." Land 12, no. 2 (February 10, 2023): 454. http://dx.doi.org/10.3390/land12020454.

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Анотація:

Ecosystem services (ESs) are a multiple whole composed of multiple services and their multiple relations, which can be expressed as tensors (multiple functions of multiple vectors). This study attempts to introduce the concept and method of tensors into ES research to solve problems caused by the multiplicity of ESs, such as multiple descriptions and perceptions of ESs, repeated calculation of ES values, and cascading relationships with the social economy. Taking the Manas River Basin composite ecosystem as an example, we constructed five different types of ES tensors based on different understandings and applications: (1) As multiple vectors, three eigenvectors were extracted from the ES state tensor (ESST), including farmland service (FS), vegetation service (VS) and water service (WS). According to the ES response tensor (ESRT), an increase in FS may lead to a decline in overall services. (2) As multiple functions, the ES value (ESV) of the basin was measured by the ESV metric tensor (ESVMT), with a gross value of 14.8 billion USD and a net value of 10.17 billion USD. From different stakeholders perceptions constructed by the ecosystem services to human well-beings (ES-HW) tensor, the human well-being values (HWV) were ranked as citizens > farmers ≈ herdsmen > public. (3) The HWV output efficiency of different LULC per unit of water use was calculated by a fourth-order mixed tensor constructed by water–LULC–ES–HW multiple cascading relations. Among them, the HWV efficiency of water areas and wetlands was the highest, but the area was the smallest. Cultivated land and unused land had the lowest HWV efficiency and largest area. In general, the ES tensor is the extension and integration of the ES scalars/indicators to the ES vectors/bundles, which can provide tools for the integral expression, objective measurement and multiple perceptions of ESs.

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39

EGGERS, H. C., and P. LIPA. "QUANTITATIVE CHARACTERISATION OF CORRELATION FUNCTION SHAPES WITH A MULTIVARIATE EDGEWORTH EXPANSION." International Journal of Modern Physics E 16, no. 10 (November 2007): 3205–23. http://dx.doi.org/10.1142/s0218301307009208.

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Second-order HBT correlation functions can be expanded in terms of correlated gaussian ellipsoids and their derivatives. The resulting multidimensional Edgeworth expansion in terms of cumulants and hermite tensors contains 15 fourth-order and 28 sixth-order cumulants which act as shape parameters. Off-diagonal terms dominate both the character and magnitude of shape changes. We show how cumulants can be measured directly and so the procedure has no need for fitting.

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40

Chen, Yannan, Liqun Qi, and Qun Wang. "Positive semi-definiteness and sum-of-squares property of fourth order four dimensional Hankel tensors." Journal of Computational and Applied Mathematics 302 (August 2016): 356–68. http://dx.doi.org/10.1016/j.cam.2016.02.019.

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41

Voyiadjis, George Z., and Peter I. Kattan. "Finite Strain Plasticity and Damage in Constitutive Modeling of Metals With Spin Tensors." Applied Mechanics Reviews 45, no. 3S (March 1, 1992): S95—S109. http://dx.doi.org/10.1115/1.3121396.

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The analysis of damage and plastic deformation in metals is very important towards the full understanding of the various damage mechanisms in these materials. A coupled theory of damage mechanics and finite strain plasticity is proposed. The theory is based on a sound mathematical and mechanical background and is thermodynamically consistent. It is formulated using spatial coordinates utilizing a von Mises type yield criterion with both isotropic and kinematic hardening. The derivation is based on the concept of effective stress that was originally proposed by Kachanov [1] for the case of uniaxial tension. The plasticity model is first formulated in a fictitious undamaged configuration of the body. Then certain transformation equations are derived to transform this model into a damage-plasticity model in the damaged configuration of the body. Certain assumptions are made in order to make this transformation possible. These assumptions include small elastic strains and the hypothesis of elastic energy equivalence of Ref 17. The corotational stress rate equations are also discussed since they are used extensively in the constitutive relations. Therefore, the use of spin tensors is also discussed since they play a major role in the definition of the corotational rates. In addition, a modified spin tensor is proposed to be used in the coupled model. Furthermore, the nature of the fourth-rank damage effect tensor is discussed for a general state of deformation and damage. Also, the explicit matrix representation of this tensor is rigorously derived and can be used in future applications to solve plane stress and plane strain problems involving damage. Finally, the problem of finite simple shear is investigated using the proposed model. The resulting equations are solved using a Runge-Kutta-Verner fifth order and sixth order method. The stress-strain curves are obtained for a certain expression of the modified spin tensor and are compared with other spin tensors. Also, the evolution of the backstress and damage variables is presented. The results obtained compare favorably with previous results.

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42

Che, Haitao, Haibin Chen, and Guanglu Zhou. "New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors." Journal of Industrial & Management Optimization 13, no. 5 (2017): 0. http://dx.doi.org/10.3934/jimo.2020139.

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43

Kreimer, Nadia, Aaron Stanton, and Mauricio D. Sacchi. "Tensor completion based on nuclear norm minimization for 5D seismic data reconstruction." GEOPHYSICS 78, no. 6 (November 1, 2013): V273—V284. http://dx.doi.org/10.1190/geo2013-0022.1.

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Many standard seismic data processing and imaging techniques require regularly sampled data. Prestack seismic data are multidimensional signals that can be represented via low-rank fourth-order tensors in the [Formula: see text] domain. We propose to adopt tensor completion strategies to recover unrecorded observations and to improve the signal-to-noise ratio of prestack seismic volumes. Tensor completion can be posed as an inverse problem and solved by minimizing a convex objective function. The objective function contains two terms: a data misfit and a nuclear norm. The data misfit measures the proximity of the reconstructed seismic data to the observations. The nuclear norm constraints the reconstructed data to be a low-rank tensor. In essence, we solve the prestack seismic reconstruction problem via low-rank tensor completion. The cost function of the problem is minimized using the alternating direction method of multipliers. We present synthetic examples to illustrate the behavior of the algorithm in terms of trade-off parameters that control the quality of the reconstruction. We further illustrate the performance of the algorithm with a land data survey from Alberta, Canada.

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44

Vicente Alvarez, Miguel Angel, Victor Laliena, Florencia Malamud, Javier Campo, and Javier Santisteban. "A novel method to obtain integral parameters of the orientation distribution function of textured polycrystals from wavelength-resolved neutron transmission spectra." Journal of Applied Crystallography 54, no. 3 (May 28, 2021): 903–13. http://dx.doi.org/10.1107/s1600576721003861.

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A novel method to estimate integral parameters of the orientation distribution function (ODF) in textured polycrystals from the wavelength-resolved neutron transmission is presented. It is based on the expression of the total coherent elastic cross section as a function of the Fourier coefficients of the ODF. This method is broken down in detail for obtaining Kearns factors in hexagonal crystals, and other material properties that depend on the average of second- and fourth-rank tensors. The robustness of the method against three situations was analyzed: effects of sample misalignment, of cutoff value l max of the series expansion and of experimental standard deviation. While sample misalignment is shown not to be critical for the determination of Kearns factors and second-order-rank properties, it can be critical for fourth-rank and higher-order tensor properties. The effect of the cutoff value on the method robustness is correlated to the standard deviation of the experimental data. In order to achieve a good estimation of the Fourier coefficients, it is recommended that the experimental standard deviation be around 3–5% of the total scattering cross section of the material for the method to be stable. The method was applied for the determination of Kearns factors from transmission measurements performed at the instrument ENGIN-X (ISIS) on a Zr–2.5 Nb pressure tube along two sample directions and was shown to be able to estimate Kearns factors with an error below 5%.

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45

Itin, Yakov. "Irreducible matrix resolution for symmetry classes of elasticity tensors." Mathematics and Mechanics of Solids 25, no. 10 (April 20, 2020): 1873–95. http://dx.doi.org/10.1177/1081286520913596.

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In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.

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46

Giusti, Sebastián M., Antonio A. Novotny, and Eduardo A. de Souza Neto. "Sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2118 (January 8, 2010): 1703–23. http://dx.doi.org/10.1098/rspa.2009.0499.

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This paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions. The macroscopic response is assumed to be predicted by a well-established multi-scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element (RVE) of material. The proposed formula—a symmetric fourth-order tensor field over the RVE domain—is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the RVE. In the limits, when the inclusion/matrix phase contrast ratio tends to zero and infinity, the sensitivities to the insertion of a hole and a rigid inclusion, respectively, are rigorously obtained. The derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail. The derived fundamental formula is of interest to many areas of applied and computational mechanics. To illustrate its potential applicability, a simple finite element-based example is presented where the topological derivative information is used to automatically generate a bi-material microstructure to meet pre-specified macroscopic properties.

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47

Wang, Yuan, Zhi-feng Cui, and Hong-fei Wang. "Experimental Observables and Macroscopic Susceptibility/Microscopic Polarizability Tensors for Third and Fourth-Order Nonlinear Spectroscopy of Ordered Molecular System." Chinese Journal of Chemical Physics 20, no. 4 (August 2007): 449–60. http://dx.doi.org/10.1088/1674-0068/20/04/449-460.

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48

Shankar, Vijay, Anton Lundberg, Taraka Pamidi, Lars-Olof Landström, and Örjan Johansson. "CFD Analysis of Turbulent Fibre Suspension Flow." Fluids 5, no. 4 (October 8, 2020): 175. http://dx.doi.org/10.3390/fluids5040175.

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Анотація:

A new model for turbulent fibre suspension flow is proposed by introducing a model for the fibre orientation distribution function (ODF). The coupling between suspended fibres and the fluid momentum is then introduced through the second and fourth order fibre orientation tensors, respectively. From the modelled ODF, a method to construct explicit expressions for the components of the orientation tensors as functions of the flow field is derived. The implementation of the method provides a fibre model that includes the anisotropic detail of the stresses introduced due to presence of the fibres, while being significantly cheaper than solving the transport of the ODF and computing the orientation tensors from numerical integration in each iteration. The model was validated and trimmed using experimental data from flow over a backwards facing step. The model was then further validated with experimental data from a turbulent fibre suspension channel flow. Simulations were also carried out using a Bingham viscoplastic fluid model for comparison. The ODF model and the Bingham model performed reasonably well for the turbulent flow areas, and the latter model showed to be slightly better given the parameter settings tested in the present study. The ODF model may have good potential, but more rigorous study is needed to fully evaluate the model.

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49

ZUBOV, V. I., J. F. SANCHEZ, N. P. TRETIAKOV, and A. E. YUSEF. "SELF-CONSISTENT THEORY OF ELASTIC PROPERTIES OF STRONGLY ANHARMONIC CRYSTALS I: GENERAL TREATMENT AND COMPARISON WITH COMPUTER SIMULATIONS AND EXPERIMENT FOR FCC CRYSTALS." International Journal of Modern Physics B 09, no. 07 (March 30, 1995): 803–17. http://dx.doi.org/10.1142/s0217979295000318.

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Based on the correlative method of an unsymmetrized self-consistent field,16–23 we have derived expressions for elastic constant tensors of strongly anharmonic crystals of cubic symmetry. Each isothermal elastic constant consists of four terms. The first one is the zeroth approximation containing the main anharmonicity (up to the fourth order). The second term is the quantum correction. It is important at temperatures below the De-bye characteristic temperature. Finally, the third and fourth terms are the perturbation theory corrections which take into account the influence of the correlations in atomic displacements from the lattice points and that of the high-order anharmonicity respectively. These corrections appear to be small up to the melting temperatures. It is sufficient for a personal computer to perform all our calculations with just a little computer time. A comparison with certain Monte Carlo simulations and with experimental data for Ar and Kr is made. For the most part, our results are between. The quasi-harmonic approximation fails at high temperatures, confirming once again the crucial role of strong anharmonicity.

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50

Ginzburg, Irina. "Truncation Errors, Exact And Heuristic Stability Analysis Of Two-Relaxation-Times Lattice Boltzmann Schemes For Anisotropic Advection-Diffusion Equation." Communications in Computational Physics 11, no. 5 (May 2012): 1439–502. http://dx.doi.org/10.4208/cicp.211210.280611a.

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AbstractThis paper establishes relations between the stability and the high-order truncated corrections for modeling of the mass conservation equation with the two-relaxation-times (TRT) collision operator. First we propose a simple method to derive the truncation errors from the exact, central-difference type, recurrence equations of the TRT scheme. They also supply its equivalent three-time-level discretization form. Two different relationships of the two relaxation rates nullify the third (advection) and fourth (pure diffusion) truncation errors, for any linear equilibrium and any velocity set. However, the two relaxation times alone cannot remove the leading-order advection-d if fusion error, because of the intrinsic fourth-order numerical diffusion. The truncation analysis is carefully verified for the evolution of concentration waves with the anisotropic diffusion tensors. The anisotropic equilibrium functions are presented in a simple but general form, suitable for the minimal velocity sets and the d2Q9, d3Q13, d3Q15 and d3Q19 velocity sets. All anisotropic schemes are complemented by their exact necessary von Neumann stability conditions and equivalent finite-difference stencils. The sufficient stability conditions are proposed for the most stable (OTRT) family, which enables modeling at any Peclet numbers with the same velocity amplitude. The heuristic stability analysis of the fourth-order truncated corrections extends the optimal stability to larger relationships of the two relaxation rates, in agreement with the exact (one-dimensional) and numerical (multi-dimensional) stability analysis. A special attention is put on the choice of the equilibrium weights. By combining accuracy and stability predictions, several strategies for selecting the relaxation and free-tunable equilibrium parameters are suggested and applied to the evolution of the Gaussian hill.

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