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Journal articles on the topic 'Rank-one tensors'

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Author: Grafiati

Published: 10 March 2023

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1

POPA, FLORIAN CATALIN, and OVIDIU TINTAREANU-MIRCEA. "IRREDUCIBLE KILLING TENSORS FROM THIRD RANK KILLING–YANO TENSORS." Modern Physics Letters A 22, no. 18 (June 14, 2007): 1309–17. http://dx.doi.org/10.1142/s0217732307023559.

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We investigate higher rank Killing–Yano tensors showing that third rank Killing–Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric from two-rank Killing–Yano tensors to obtain a reducible Killing tensor and from third-rank Killing–Yano tensors, we obtain three Killing tensors, one reducible and two irreducible.

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2

Tyrtyshnikov, Eugene E. "Tensor decompositions and rank increment conjecture." Russian Journal of Numerical Analysis and Mathematical Modelling 35, no. 4 (August 26, 2020): 239–46. http://dx.doi.org/10.1515/rnam-2020-0020.

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AbstractSome properties of tensor ranks and the non-closeness issue of sets with given restrictions on the rank of tensors entering those sets are studied. It is proved that the rank of the d-dimensional Laplacian equals d. The following conjecture is formulated: for any tensor of non-maximal rank there exists a nonzero decomposable tensor (tensor of rank 1) such that the rank increases by one after adding this tensor. In the general case, it is proved that this property holds algebraically almost everywhere for complex tensors of fixed size whose rank is strictly less than the generic rank.

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3

Zhang, Tong, and Gene H. Golub. "Rank-One Approximation to High Order Tensors." SIAM Journal on Matrix Analysis and Applications 23, no. 2 (January 2001): 534–50. http://dx.doi.org/10.1137/s0895479899352045.

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4

Hu, Shenglong, Defeng Sun, and Kim-Chuan Toh. "Best Nonnegative Rank-One Approximations of Tensors." SIAM Journal on Matrix Analysis and Applications 40, no. 4 (January 2019): 1527–54. http://dx.doi.org/10.1137/18m1224064.

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5

Bachmayr, Markus, Wolfgang Dahmen, Ronald DeVore, and Lars Grasedyck. "Approximation of High-Dimensional Rank One Tensors." Constructive Approximation 39, no. 2 (November 12, 2013): 385–95. http://dx.doi.org/10.1007/s00365-013-9219-x.

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6

Friedland, S., V. Mehrmann, R. Pajarola, and S. K. Suter. "On best rank one approximation of tensors." Numerical Linear Algebra with Applications 20, no. 6 (March 19, 2013): 942–55. http://dx.doi.org/10.1002/nla.1878.

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7

Breiding, Paul, and Nick Vannieuwenhoven. "On the average condition number of tensor rank decompositions." IMA Journal of Numerical Analysis 40, no. 3 (June 20, 2019): 1908–36. http://dx.doi.org/10.1093/imanum/drz026.

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Abstract We compute the expected value of powers of the geometric condition number of random tensor rank decompositions. It is shown in particular that the expected value of the condition number of $n_1\times n_2 \times 2$ tensors with a random rank-$r$ decomposition, given by factor matrices with independent and identically distributed standard normal entries, is infinite. This entails that it is expected and probable that such a rank-$r$ decomposition is sensitive to perturbations of the tensor. Moreover, it provides concrete further evidence that tensor decomposition can be a challenging problem, also from the numerical point of view. On the other hand, we provide strong theoretical and empirical evidence that tensors of size $n_1~\times ~n_2~\times ~n_3$ with all $n_1,n_2,n_3 \geqslant 3$ have a finite average condition number. This suggests that there exists a gap in the expected sensitivity of tensors between those of format $n_1\times n_2 \times 2$ and other order-3 tensors. To establish these results we show that a natural weighted distance from a tensor rank decomposition to the locus of ill-posed decompositions with an infinite geometric condition number is bounded from below by the inverse of this condition number. That is, we prove one inequality towards a so-called condition number theorem for the tensor rank decomposition.

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8

Grasedyck, Lars, and Wolfgang Hackbusch. "An Introduction to Hierarchical (H-) Rank and TT-Rank of Tensors with Examples." Computational Methods in Applied Mathematics 11, no. 3 (2011): 291–304. http://dx.doi.org/10.2478/cmam-2011-0016.

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Abstract We review two similar concepts of hierarchical rank of tensors (which extend the matrix rank to higher order tensors): the TT-rank and the H-rank (hierarchical or H-Tucker rank). Based on this notion of rank, one can define a data-sparse representation of tensors involving O(dnk + dk^3) data for order d tensors with mode sizes n and rank k. Simple examples underline the differences and similarities between the different formats and ranks. Finally, we derive rank bounds for tensors in one of the formats based on the ranks in the other format.

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9

Krieg, David, and Daniel Rudolf. "Recovery algorithms for high-dimensional rank one tensors." Journal of Approximation Theory 237 (January 2019): 17–29. http://dx.doi.org/10.1016/j.jat.2018.08.002.

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10

Milošević, Ivanka. "Second-rank tensors for quasi-one-dimensional systems." Physics Letters A 204, no. 1 (August 1995): 63–66. http://dx.doi.org/10.1016/0375-9601(95)00412-v.

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11

O'Hara, Michael J. "On the perturbation of rank-one symmetric tensors." Numerical Linear Algebra with Applications 21, no. 1 (July 31, 2012): 1–12. http://dx.doi.org/10.1002/nla.1851.

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12

Ballico, Edoardo. "Ranks with Respect to a Projective Variety and a Cost-Function." AppliedMath 2, no. 3 (August 2, 2022): 457–65. http://dx.doi.org/10.3390/appliedmath2030026.

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Let X⊂Pr be an integral and non-degenerate variety. A “cost-function” (for the Zariski topology, the semialgebraic one, or the Euclidean one) is a semicontinuous function w:=[1,+∞)∪+∞ such that w(a)=1 for a non-empty open subset of X. For any q∈Pr, the rank rX,w(q) of q with respect to (X,w) is the minimum of all ∑a∈Sw(a), where S is a finite subset of X spanning q. We have rX,w(q)<+∞ for all q. We discuss this definition and classify extremal cases of pairs (X,q). We give upper bounds for all rX,w(q) (twice the generic rank) not depending on w. This notion is the generalization of the case in which the cost-function w is the constant function 1. In this case, the rank is a well-studied notion that covers the tensor rank of tensors of arbitrary formats (PARAFAC or CP decomposition) and the additive decomposition of forms. We also adapt to cost-functions the rank 1 decomposition of real tensors in which we allow pairs of complex conjugate rank 1 tensors.

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13

Khoromskij, B. N. "Structured Rank-(r1, . . . , rd) Decomposition of Function-related Tensors in R_D." Computational Methods in Applied Mathematics 6, no. 2 (2006): 194–220. http://dx.doi.org/10.2478/cmam-2006-0010.

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AbstractThe structured tensor-product approximation of multidimensional nonlocal operators by a two-level rank-(r1, . . . , rd) decomposition of related higher-order tensors is proposed and analysed. In this approach, the construction of the desired approximant to a target tensor is a reminiscence of the Tucker-type model, where the canonical components are represented in a fixed (uniform) basis, while the core tensor is given in the canonical format. As an alternative, the multilevel nested canonical decomposition is presented. The complexity analysis of the corresponding multilinear algebra shows an almost linear cost in the one-dimensional problem size. The existence of a low Kronecker rank two-level representation is proven for a class of function-related tensors.

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14

Li, Zhening, Yuji Nakatsukasa, Tasuku Soma, and André Uschmajew. "On Orthogonal Tensors and Best Rank-One Approximation Ratio." SIAM Journal on Matrix Analysis and Applications 39, no. 1 (January 2018): 400–425. http://dx.doi.org/10.1137/17m1144349.

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15

Gao, Tong, Hao Chen, and Wen Chen. "MCMS-STM: An Extension of Support Tensor Machine for Multiclass Multiscale Object Recognition in Remote Sensing Images." Remote Sensing 14, no. 1 (January 2, 2022): 196. http://dx.doi.org/10.3390/rs14010196.

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The support tensor machine (STM) extended from support vector machine (SVM) can maintain the inherent information of remote sensing image (RSI) represented as tensor and obtain effective recognition results using a few training samples. However, the conventional STM is binary and fails to handle multiclass classification directly. In addition, the existing STMs cannot process objects with different sizes represented as multiscale tensors and have to resize object slices to a fixed size, causing excessive background interferences or loss of object’s scale information. Therefore, the multiclass multiscale support tensor machine (MCMS-STM) is proposed to recognize effectively multiclass objects with different sizes in RSIs. To achieve multiclass classification, by embedding one-versus-rest and one-versus-one mechanisms, multiple hyperplanes described by rank-R tensors are built simultaneously instead of single hyperplane described by rank-1 tensor in STM to separate input with different classes. To handle multiscale objects, multiple slices of different sizes are extracted to cover the object with an unknown class and expressed as multiscale tensors. Then, M-dimensional hyperplanes are established to project the input of multiscale tensors into class space. To ensure an efficient training of MCMS-STM, a decomposition algorithm is presented to break the complex dual problem of MCMS-STM into a series of analytic sub-optimizations. Using publicly available RSIs, the experimental results demonstrate that the MCMS-STM achieves 89.5% and 91.4% accuracy for classifying airplanes and ships with different classes and sizes, which outperforms typical SVM and STM methods.

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16

Zhu, Yada, Jingrui He, and Rick Lawrence. "Hierarchical Modeling with Tensor Inputs." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (September 20, 2021): 1233–39. http://dx.doi.org/10.1609/aaai.v26i1.8283.

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In many real applications, the input data are naturally expressed as tensors, such as virtual metrology in semiconductor manufacturing, face recognition and gait recognition in computer vision, etc. In this paper, we propose a general optimization framework for dealing with tensor inputs. Most existing methods for supervised tensor learning use only rank-one weight tensors in the linear model and cannot readily incorporate domain knowledge. In our framework, we obtain the weight tensor in a hierarchical way — we first approximate it by a low-rank tensor, and then estimate the low-rank approximation using the prior knowledge from various sources, e.g., different domain experts. This is motivated by wafer quality prediction in semiconductor manufacturing. Furthermore, we propose an effective algorithm named H-MOTE for solving this framework, which is guaranteed to converge. The time complexity of H-MOTE is linear with respect to the number of examples as well as the size of the weight tensor. Experimental results show the superiority of H-MOTE over state-of-the-art techniques on both synthetic and real data sets.

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17

Silber, Gerhard, Uwe Janoske, Mansour Alizadeh, and Guenther Benderoth. "An Application of a Gradient Theory With Dissipative Boundary Conditions to Fully Developed Turbulent Flows." Journal of Fluids Engineering 129, no. 5 (December 22, 2006): 643–51. http://dx.doi.org/10.1115/1.2720476.

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The paper presents a complete gradient theory of grade two, including new dissipative boundary conditions based on an axiomatic conception of a nonlocal continuum theory for materials of grade n. The total stress tensor of rank two in the equation of linear momentum contains two higher stress tensors of rank two and three. In the case of isotropic materials, both the tensors of rank two and three are tensor valued functions of the second order strain rate tensor and its first gradient. So the vector valued differential equation of motion is of order four, where the necessary dissipative boundary conditions are generated by using porosity tensors. An application to hydrodynamic turbulence by a linear theory is shown, whereby fully developed steady turbulent channel flows with fixed walls and one moving wall are also examined. The velocity distribution parameters are identified by a numerical optimization algorithm, using experimental data of velocity profiles of channel flow with fixed walls from the literature. These profiles were compared with others given in the literature. With these derived parameters, the predicted velocity gradient of a channel flow agrees well with data from the literature. In addition all simulations were successfully carried out using the finite difference method.

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18

Mu, Cun, Daniel Hsu, and Donald Goldfarb. "Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 36, no. 4 (January 2015): 1638–59. http://dx.doi.org/10.1137/15m1010890.

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19

Novak, Erich, and Daniel Rudolf. "Tractability of the Approximation of High-Dimensional Rank One Tensors." Constructive Approximation 43, no. 1 (March 19, 2015): 1–13. http://dx.doi.org/10.1007/s00365-015-9282-6.

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20

Gibiansky, L. V. "Effective moduli of plane polycrystals with a rank-one local compliance tensor." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 6 (1998): 1325–54. http://dx.doi.org/10.1017/s0308210500027359.

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Recently, Grabovsky and Milton have studied effective properties of plane polycrystals made of a crystal with a rank-one local compliance tensor and a positive definite ‘weak’ direction. This paper continues their investigation. By using several complementary approaches, it is proved that the effective compliance tensor of such a polycrystal has exactly one weak direction. The new proofs clarify the link of the degenerate polycrystal problem with previously obtained results on the polycrystal properties. Investigation of the phase boundary conditions has allowed us to find the L-closure set of the effective properties of all polycrystals that can be constructed by sequential laminations of the original degenerate crystals. Bounds on the G-closure, i.e. a set of the effective properties of all the polycrystals that can be built from the given set of degenerate crystals, are found. They are given by a polygonal set in the plane of two linear invariants of the effective compliance tensors. The effective moduli of the polycrystals made from crystals with non-definite ‘weak’ directions are also studied.

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21

Hardhienata, Hendradi, Tony Ibnu Sumaryada, Benedikt Pesendorfer, and Adalberto Alejo-Molina. "Bond Model of Second- and Third-Harmonic Generation in Body- and Face-Centered Crystal Structures." Advances in Materials Science and Engineering 2018 (September 23, 2018): 1–11. http://dx.doi.org/10.1155/2018/7153247.

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In this work, we describe the third- and fourth-rank tensors of body- and face-centered cubic systems and derive the s- and p-polarized SHG far field using the simplified bond-hyperpolarizability model. We also briefly discuss bulk nonlinear sources in such structures: quadrupole contribution, spatial dispersion, electric-field second-harmonic generation, and third-harmonic generation, deriving the corresponding fourth rank tensor. We show that all the third- and fourth-rank tensorial elements require only one independent fitting parameter.

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22

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

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

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23

Chen, Yannan. "Successive unconstrained dual optimization method for rank-one approximation to tensors." Journal of Applied Mathematics and Computing 38, no. 1-2 (November 17, 2010): 9–23. http://dx.doi.org/10.1007/s12190-010-0459-7.

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24

Chang, Jingya, Wenyu Sun, and Yannan Chen. "A modified Newton’s method for best rank-one approximation to tensors." Applied Mathematics and Computation 216, no. 6 (May 2010): 1859–67. http://dx.doi.org/10.1016/j.amc.2009.12.019.

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25

BĂLEANU, D., and S. BAŞKAL. "DUAL METRICS FOR A CLASS OF RADIATIVE SPACE–TIMES." Modern Physics Letters A 16, no. 03 (January 30, 2001): 135–42. http://dx.doi.org/10.1142/s0217732301003218.

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Second-rank nondegenerate Killing tensors for some subclasses of space–times admitting parallel null one-planes are investigated. Lichnérowicz radiation conditions are imposed to provide a physical meaning to space–times whose metrics are described through their associated second-rank Killing tensors. Conditions under which the dual space–times retain the same physical properties are presented.

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26

Exl, Lukas, Claas Abert, Norbert J. Mauser, Thomas Schrefl, Hans Peter Stimming, and Dieter Suess. "FFT-based Kronecker product approximation to micromagnetic long-range interactions." Mathematical Models and Methods in Applied Sciences 24, no. 09 (May 20, 2014): 1877–901. http://dx.doi.org/10.1142/s0218202514500109.

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We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

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27

Litvin, Daniel B. "Axial point groups: rank 1, 2, 3 and 4 property tensor tables." Acta Crystallographica Section A Foundations and Advances 71, no. 3 (March 26, 2015): 346–49. http://dx.doi.org/10.1107/s2053273315002740.

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The form of a physical property tensor of a quasi-one-dimensional material such as a nanotube or a polymer is determined from the material's axial point group. Tables of the form of rank 1, 2, 3 and 4 property tensors are presented for a wide variety of magnetic and non-magnetic tensor types invariant under each point group in all 31 infinite series of axial point groups. An application of these tables is given in the prediction of the net polarization and magnetic-field-induced polarization in a one-dimensional longitudinal conical magnetic structure in multiferroic hexaferrites.

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28

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

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29

Wu, Fengsheng, Chaoqian Li, and Yaotang Li. "Algorithms for Structure Preserving Best Rank-one Approximations of Partially Symmetric Tensors." Frontiers of Mathematics 18, no. 1 (January 2023): 123–52. http://dx.doi.org/10.1007/s11464-021-0088-5.

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30

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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31

Ouyang, Zhiyuan, Liqi Zhang, Huazhong Wang, and Kai Yang. "High-Dimensional Seismic Data Reconstruction Based on Linear Radon Transform–Constrained Tensor CANDECOM/PARAFAC Decomposition." Remote Sensing 14, no. 24 (December 11, 2022): 6275. http://dx.doi.org/10.3390/rs14246275.

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Random noise and missing seismic traces are common in field seismic data, which seriously affects the subsequent seismic processing flow. The complete noise-free high-dimensional seismic dataset in the frequency–space (f-x) domain under the local linear assumption are regarded as a low-rank tensor, and each high dimensional seismic dataset containing only one linear event is a rank-1 tensor. The tensor CANDECOM/PARAFAC decomposition (CPD) method estimates complete noise-free seismic signals by characterizing high-dimensional seismic signals as the sum of several rank-1 tensors. In order to improve the stability and effect of the tensor CPD algorithm, this paper proposes a linear Radon transform–constrained tensor CPD method (RCPD) by using the sparsity of factor matrix in the Radon domain after high-dimensional seismic signal tensor CPD and uses alternating direction multiplier method (ADMM) to solve the established optimization problem. This proposed method is an essential realization of the high-dimensional linear Radon transform, and the results of synthetic and field data reconstruction prove the effectiveness of the proposed method.

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32

Brovko, George L. "Tensors in Newtonian Physics and the Foundations of Classical Continuum Mechanics." Mathematical and Computational Applications 24, no. 3 (September 3, 2019): 79. http://dx.doi.org/10.3390/mca24030079.

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In the Newtonian approach to mechanics, the concepts of objective tensors of various ranks and types are introduced. The tough classification of objective tensors is given, including tensors of material and spatial types. The diagrams are constructed for non-degenerate (“analogous”) relations between tensors of one and the same (any) rank, and of various types of objectivity. Mappings expressing dependence between objective tensor processes of various ranks and types are considered. The fundamental concept of frame-independence of such mappings is introduced as being inherent to constitutive relations of various physical and mechanical properties in the Newtonian approach. The criteria are established for such frame-independence. The mathematical restrictions imposed on the frame-independent mappings by the objectivity types of connected tensors are simultaneously revealed. The absence of such restrictions is established exclusively for mappings and equations linking tensors of material types. Using this, a generalizing concept of objective differentiation of tensor processes in time, and a new concept of objective integration, are introduced. The axiomatic construction of the generalized theory of stress and strain tensors in continuum mechanics is given, which leads to the emergence of continuum classes and families of new tensor measures. The axioms are proposed and a variant of the general theory of constitutive relations of mechanical properties of continuous media is constructed, generalizing the known approaches by Ilyushin and Noll, taking into account the possible presence of internal kinematic constraints and internal body-forces in the body. The concepts of the process image and the properties of the five-dimensional Ilyushin’s isotropy are generalized on the range of finite strains.

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33

Пенкин, Юрий Михайлович, and Алина Александровна Федосеева. "СТРУКТУРНЫЕ КОНЕЧНЫЕ АВТОМАТЫ В ВИДЕ ТЕНЗОРОВ ТРЕТЬЕГО РАНГА ТИПА СУДОКУ." RADIOELECTRONIC AND COMPUTER SYSTEMS, no. 4 (December 25, 2019): 79–87. http://dx.doi.org/10.32620/reks.2019.4.09.

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The construction concept and general principles of the operation of a new kind of finite state machine are presented, for which the input and output elements are panels of square matrices, and the transitions between their states are determined by numerical tensors of the third rank. In this case, the structure of the tensors is specified in the form of cubic grids, in whose cells' natural numbers are located according to the principle of Sudoku construction. An algorithm for constructing such tensors of arbitrary size is indicated. The structures of tensors constructed using ranked sets of natural numbers are defined as standard. It is shown that the possibility of determining Sudoku type tensors using a one-dimensional parameter is basic for the manifestation of their functional self-similarity. The property of additive conservation of the structure of numerical tensors of the third rank to the requirements of Sudoku is formulated as a theorem. It is proved that the tensor obtained by summing an arbitrary tensor structure and a constant, taking into account the introduced cyclic ranking rule, satisfies the general requirements of Sudoku. The problems of abstract and structural synthesis of finite state machine based on the analyzed tensor structures are considered. In this case, the task of abstract synthesis has traditionally been defined as the creation of a mathematical model of an automaton, and the task of structural synthesis is just the development of its functional logical scheme. Based on the ambiguity of the function of the output of the finite state machine, the possibility of the simultaneous use of several different output alphabets is substantiated. The modes of functioning of the minimal finite state machine of the proposed type are described by the example of an initial state machine with a distinguished standard initial state. In the general case, it is shown that the finite state machines defined on the group of these requirements can be attributed to generalized first-order finite-state machines (or Mealy machines) with a multi-valued output. The features of network applications of structural automata are presented. Variants of possible applications of the considered finite state machines are analyzed.

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34

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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35

Che, Maolin, Andrzej Cichocki, and Yimin Wei. "Neural networks for computing best rank-one approximations of tensors and its applications." Neurocomputing 267 (December 2017): 114–33. http://dx.doi.org/10.1016/j.neucom.2017.04.058.

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36

Felício Fuck, Rodrigo, and Ilya Tsvankin. "Analysis of the symmetry of a stressed medium using nonlinear elasticity." GEOPHYSICS 74, no. 5 (September 2009): WB79—WB87. http://dx.doi.org/10.1190/1.3157251.

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Velocity variations caused by subsurface stress changes play an important role in monitoring compacting reservoirs and in several other applications of seismic methods. A general way to describe stress- or strain-induced velocity fields is by employing the theory of nonlinear elasticity, which operates with third-order elastic (TOE) tensors. These sixth-rank strain-sensitivity tensors, however, are difficult to manipulate because of the large number of terms involved in the algebraic operations. Thus, even evaluation of the anisotropic symmetry of a medium under stress/strain proves to be a challenging task. We employ a matrix representation of TOE tensors that allows computation of strain-related stiffness perturbations from a linear combination of [Formula: see text] matrices scaled by the components of the strain tensor. In addition to streamlining the numerical algorithm, this approach helps to predict strain-induced symmetry using relatively straightforward algebraic considerations. For example, our analysis shows that a transversely isotropic (TI) medium acquires orthorhombic symmetry if one of the principal directions of the strain tensor is aligned with the symmetry axis. Otherwise, the strained TI medium can become monoclinic or even triclinic.

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37

SAVVIDY, GEORGE. "PARTICLE SPECTRUM OF NON-ABELIAN TENSOR GAUGE FIELDS." Modern Physics Letters A 25, no. 14 (May 10, 2010): 1137–61. http://dx.doi.org/10.1142/s0217732310033001.

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We review the non-Abelian tensor gauge field theory and analyze its free field equations for lower rank gauge fields when the interaction coupling constant tends to zero. The free field equations are written in terms of the first-order derivatives of extended field strength tensors similar to the electrodynamics and non-Abelian gauge theories. We determine the particle content of the free field equations and count the propagating modes which they describe. In four-dimensional spacetime the rank-2 gauge field describes propagating modes of helicity two and zero. We show that the rank-3 gauge field describes propagating modes of helicity-three and a doublet of helicity-one gauge bosons. Only four-dimensional spacetime is physically acceptable, because in five- and higher-dimensional spacetime the equation has solutions with negative norm states. We discuss the structure of the particle spectrum for higher rank gauge fields.

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38

ANTONIADIS, IGNATIOS, and GEORGE SAVVIDY. "EXTENSION OF CHERN–SIMONS FORMS AND NEW GAUGE ANOMALIES." International Journal of Modern Physics A 29, no. 03n04 (February 10, 2014): 1450027. http://dx.doi.org/10.1142/s0217751x14500274.

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We present a general analysis of gauge invariant, exact and metric independent forms which can be constructed using higher-rank field-strength tensors. The integrals of these forms over the corresponding space–time coordinates provides new topological Lagrangians. With these Lagrangians one can define gauge field theories which generalize the Chern–Simons quantum field theory. We also present explicit expressions for the potential gauge anomalies associated with the tensor gauge fields and classify all possible anomalies that can appear in lower dimensions.

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Bykov, Dmitri. "Ricci-flat metrics and Killing-Yano tensors." EPJ Web of Conferences 191 (2018): 06010. http://dx.doi.org/10.1051/epjconf/201819106010.

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We consider the problem of constructing Ricci-flat metrics on the total space of the canonical bundle over the del Pezzo surface of rank one. We analyze the so-called ‘orthotoric metric’ and its first-order deformation, whose existence is compatible with the Calabi-Yau theorem.

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Wang, Gang, Xiaoxuan Yang, Wei Shao, and Qiuling Hou. "Further Study on C-Eigenvalue Inclusion Intervals for Piezoelectric Tensors." Axioms 11, no. 6 (May 26, 2022): 250. http://dx.doi.org/10.3390/axioms11060250.

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The C-eigenpair of piezoelectric tensors finds applications in the area of the piezoelectric effect and converse piezoelectric effect. In this paper, we provide some characterizations of C-eigenvectors by exploring the structure of piezoelectric tensors, and establish sharp C-eigenvalue inclusion intervals via Cauchy–Schwartz inequality. Further, we propose the lower and upper bounds of the largest C-eigenvalue and evaluate the efficiency of the best rank-one approximation of piezoelectric tensors. Numerical examples are proposed to verify the efficiency of the obtained results.

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Wang, Xuezhong, Maolin Che, and Yimin Wei. "Partial orthogonal rank-one decomposition of complex symmetric tensors based on the Takagi factorization." Journal of Computational and Applied Mathematics 332 (April 2018): 56–71. http://dx.doi.org/10.1016/j.cam.2017.09.050.

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Fu, Tao-Ran, and Jin-Yan Fan. "Successive Partial-Symmetric Rank-One Algorithms for Almost Unitarily Decomposable Conjugate Partial-Symmetric Tensors." Journal of the Operations Research Society of China 7, no. 1 (February 9, 2018): 147–67. http://dx.doi.org/10.1007/s40305-018-0194-6.

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Yang, Yuning, Qingzhi Yang, and Liqun Qi. "Properties and methods for finding the best rank-one approximation to higher-order tensors." Computational Optimization and Applications 58, no. 1 (November 5, 2013): 105–32. http://dx.doi.org/10.1007/s10589-013-9617-9.

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Li, Shuang, and Qiuwei Li. "Local and Global Convergence of General Burer-Monteiro Tensor Optimizations." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 9 (June 28, 2022): 10266–74. http://dx.doi.org/10.1609/aaai.v36i9.21267.

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Tensor optimization is crucial to massive machine learning and signal processing tasks. In this paper, we consider tensor optimization with a convex and well-conditioned objective function and reformulate it into a nonconvex optimization using the Burer-Monteiro type parameterization. We analyze the local convergence of applying vanilla gradient descent to the factored formulation and establish a local regularity condition under mild assumptions. We also provide a linear convergence analysis of the gradient descent algorithm started in a neighborhood of the true tensor factors. Complementary to the local analysis, this work also characterizes the global geometry of the best rank-one tensor approximation problem and demonstrates that for orthogonally decomposable tensors the problem has no spurious local minima and all saddle points are strict except for the one at zero which is a third-order saddle point.

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Zhu, Hu, Ze Wang, Yu Shi, Yingying Hua, Guoxia Xu, and Lizhen Deng. "Multimodal Fusion Method Based on Self-Attention Mechanism." Wireless Communications and Mobile Computing 2020 (September 23, 2020): 1–8. http://dx.doi.org/10.1155/2020/8843186.

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Multimodal fusion is one of the popular research directions of multimodal research, and it is also an emerging research field of artificial intelligence. Multimodal fusion is aimed at taking advantage of the complementarity of heterogeneous data and providing reliable classification for the model. Multimodal data fusion is to transform data from multiple single-mode representations to a compact multimodal representation. In previous multimodal data fusion studies, most of the research in this field used multimodal representations of tensors. As the input is converted into a tensor, the dimensions and computational complexity increase exponentially. In this paper, we propose a low-rank tensor multimodal fusion method with an attention mechanism, which improves efficiency and reduces computational complexity. We evaluate our model through three multimodal fusion tasks, which are based on a public data set: CMU-MOSI, IEMOCAP, and POM. Our model achieves a good performance while flexibly capturing the global and local connections. Compared with other multimodal fusions represented by tensors, experiments show that our model can achieve better results steadily under a series of attention mechanisms.

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Zheltkov, Dmitry, and Eugene Tyrtyshnikov. "Global optimization based on TT-decomposition." Russian Journal of Numerical Analysis and Mathematical Modelling 35, no. 4 (August 26, 2020): 247–61. http://dx.doi.org/10.1515/rnam-2020-0021.

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AbstractIn contrast to many other heuristic and stochastic methods, the global optimization based on TT-decomposition uses the structure of the optimized functional and hence allows one to obtain the global optimum in some problem faster and more reliable. The method is based on the TT-cross method of interpolation of tensors. In this case, the global optimum can be found in practice even in the case when the approximation of the tensor does not possess a high accuracy. We present a detailed description of the method and its justification for the matrix case and rank-1 approximation.

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Falcó, Antonio, Lucía Hilario, Nicolás Montés, Marta C. Mora, and Enrique Nadal. "Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)." Mathematics 9, no. 1 (December 25, 2020): 34. http://dx.doi.org/10.3390/math9010034.

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A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure.

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BOUCHITTÉ, GUY, WILFRID GANGBO, and PIERRE SEPPECHER. "MICHELL TRUSSES AND LINES OF PRINCIPAL ACTION." Mathematical Models and Methods in Applied Sciences 18, no. 09 (September 2008): 1571–603. http://dx.doi.org/10.1142/s0218202508003133.

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We study the problem of Michell trusses when the system of applied equilibrated forces is a vector measure with compact support. We introduce a class of stress tensors which can be written as a superposition of rank-one tensors carried by curves (lines of principal strains). Optimality conditions are given for such families showing in particular that optimal stress tensors are carried by mutually orthogonal families of curves. The method is illustrated on a specific example where uniqueness can be proved by studying an unusual system of hyperbolic PDEs. The questions we address here are of interest in elasticity theory, optimal designs, as well as in functional analysis.

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Hegland, Markus, and Jochen Garcke. "On the numerical solution of the chemical master equation with sums of rank one tensors." ANZIAM Journal 52 (August 10, 2011): 628. http://dx.doi.org/10.21914/anziamj.v52i0.3895.

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Friedland, Shmuel, and Giorgio Ottaviani. "The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors." Foundations of Computational Mathematics 14, no. 6 (March 27, 2014): 1209–42. http://dx.doi.org/10.1007/s10208-014-9194-z.

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