Journal articles: 'Rank of symmetric tensors'

1

Ballico, E. "Gaps in the pairs (border rank, symmetric rank) for symmetric tensors." Sarajevo Journal of Mathematics 9, no. 2 (November 2013): 169–81. http://dx.doi.org/10.5644/sjm.09.2.02.

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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

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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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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5

Bernardi, Alessandra, Alessandro Gimigliano, and Monica Idà. "Computing symmetric rank for symmetric tensors." Journal of Symbolic Computation 46, no. 1 (January 2011): 34–53. http://dx.doi.org/10.1016/j.jsc.2010.08.001.

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6

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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7

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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8

Friedland, Shmuel. "Remarks on the Symmetric Rank of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 37, no. 1 (January 2016): 320–37. http://dx.doi.org/10.1137/15m1022653.

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9

Zhang, Xinzhen, Zheng-Hai Huang, and Liqun Qi. "Comon's Conjecture, Rank Decomposition, and Symmetric Rank Decomposition of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 37, no. 4 (January 2016): 1719–28. http://dx.doi.org/10.1137/141001470.

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10

Wen, Jie, Qin Ni, and Wenhuan Zhu. "Rank-r decomposition of symmetric tensors." Frontiers of Mathematics in China 12, no. 6 (May 5, 2017): 1339–55. http://dx.doi.org/10.1007/s11464-017-0632-5.

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11

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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12

Bozorgmanesh, Hassan, and Anthony Chronopoulos. "On rank decomposition and semi-symmetric rank decomposition of semi-symmetric tensors." Computational Mathematics and Computer Modeling with Applications (CMCMA) 1, no. 1 (January 1, 2022): 37–47. http://dx.doi.org/10.52547/cmcma.1.1.37.

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13

Wu, Leqin, Xin Liu, and Zaiwen Wen. "Symmetric rank-1 approximation of symmetric high-order tensors." Optimization Methods and Software 35, no. 2 (October 21, 2019): 416–38. http://dx.doi.org/10.1080/10556788.2019.1678034.

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14

BERGQVIST, G. "CAUSAL TENSORS AND SIMPLE FORMS." International Journal of Modern Physics A 17, no. 20 (August 10, 2002): 2748. http://dx.doi.org/10.1142/s0217751x02011758.

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A rank-r tensor on a Lorentzian manifold of dimension N is causal if the contraction with r arbitrary causal future-directed vectors is non-negative. General superenergy tensors1, such as the Bel and Bel-Robinson tensors, are examples of even ranked causal tensors1,2, and may therefore be useful when defining norms for geometric evolution equations3. We here show that any symmetric rank-2 causal tensor (energy-momentum tensors satisfying the dominant energy condition) can be written as a sum of at most N superenergy tensors of simple forms. If N=4 this can be expressed in an elegant way as the sum of four spinors squared. Since, for arbitrary N, the superenergy of any simple form is a self-map of the cone (its square is proportional to the metric) this leads to new representations and classifications of all conformal Lorentz transformations and to generalisations of the classical four dimensional Rainich-Misner-Wheeler (RMW) theory of determining the space-time physics from its geometry4. For non-simple forms more complicated equations are satisfied by the superenergy tensors, but also in this case we are able to generalise the RMW theory, and as an example the complete algebraic RMW theory in five dimensions is obtained5.

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15

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

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

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16

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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17

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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18

Mourrain, Bernard, and Alessandro Oneto. "On minimal decompositions of low rank symmetric tensors." Linear Algebra and its Applications 607 (December 2020): 347–77. http://dx.doi.org/10.1016/j.laa.2020.06.029.

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19

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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20

Ostrosablin, N. I. "Functional relation between two symmetric second-rank tensors." Journal of Applied Mechanics and Technical Physics 48, no. 5 (September 2007): 734–36. http://dx.doi.org/10.1007/s10808-007-0094-8.

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21

de Gracia, G. B., and G. P. de Brito. "Simple prescription for computing the nonrelativistic interparticle potential energy related to dual models." International Journal of Modern Physics A 31, no. 12 (April 28, 2016): 1650070. http://dx.doi.org/10.1142/s0217751x16500706.

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Following a procedure recently utilized by Accioly et al. to obtain the D-dimensional interparticle potential energy for electromagnetic models in the nonrelativistic limit, and relaxing the condition assumed by the authors concerning the conservation of the external current, the prescription found out by them is generalized so that dual models can also be contemplated. Specific models in which the interaction is mediated by a spin-0 particle described first by a vector field and then by a higher-derivative vector field, are analyzed. Systems mediated by spin-1 particles described, respectively, by symmetric rank-2 tensors, symmetric rank-2 tensors augmented by higher derivatives and antisymmetric rank-2 tensors, are considered as well.

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22

Ballico, Edoardo, Alessandra Bernardi, Matthias Christandl, and Fulvio Gesmundo. "On the partially symmetric rank of tensor products of $W$-states and other symmetric tensors." Rendiconti Lincei - Matematica e Applicazioni 30, no. 1 (April 1, 2019): 93–124. http://dx.doi.org/10.4171/rlm/837.

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23

Bernardi, Alessandra, Enrico Carlini, Maria Catalisano, Alessandro Gimigliano, and Alessandro Oneto. "The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition." Mathematics 6, no. 12 (December 8, 2018): 314. http://dx.doi.org/10.3390/math6120314.

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We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety X. The case we concentrate on is when X is a Veronese variety, a Grassmannian or a Segre variety. Not only these varieties are among the ones that have been most classically studied, but a strong motivation in taking them into consideration is the fact that they parameterize, respectively, symmetric, skew-symmetric and general tensors, which are decomposable, and their secant varieties give a stratification of tensors via tensor rank. We collect here most of the known results and the open problems on this fascinating subject.

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24

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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25

Rodríguez, Jorge Tomás. "On the rank and the approximation of symmetric tensors." Linear Algebra and its Applications 628 (November 2021): 72–102. http://dx.doi.org/10.1016/j.laa.2021.07.002.

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26

Xu, Peiliang. "Spectral theory of constrained second-rank symmetric random tensors." Geophysical Journal International 138, no. 1 (July 1999): 1–24. http://dx.doi.org/10.1046/j.1365-246x.1999.00807.x.

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27

Chiantini, Luca, Giorgio Ottaviani, and Nick Vannieuwenhoven. "On generic identifiability of symmetric tensors of subgeneric rank." Transactions of the American Mathematical Society 369, no. 6 (November 8, 2016): 4021–42. http://dx.doi.org/10.1090/tran/6762.

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28

Baker, Mark Robert, and Julia Bruce-Robertson. "Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities." Canadian Journal of Physics 99, no. 9 (September 2021): 764–71. http://dx.doi.org/10.1139/cjp-2020-0623.

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Curvature tensors of higher-spin gauge theories have been known for some time. In the past, they were postulated using a generalization of the symmetry properties of the Riemann tensor (curl on each index of a totally symmetric rank-n field for each spin-n). For this reason they are sometimes referred to as the generalized “Riemann” tensors. In this article, a method for deriving these curvature tensors from first principles is presented; the derivation is completed without any a priori knowledge of the existence of the Riemann tensors or the curvature tensors of higher-spin gauge theories. To perform this derivation, a recently developed procedure for deriving exactly gauge invariant Lagrangian densities from quadratic combinations of N order of derivatives and M rank of tensor potential is applied to the N = M = n case under the spin-n gauge transformations. This procedure uniquely yields the Lagrangian for classical electrodynamics in the N = M = 1 case and the Lagrangian for higher derivative gravity (“Riemann” and “Ricci” squared terms) in the N = M = 2 case. It is proven here by direct calculation for the N = M = 3 case that the unique solution to this procedure is the spin-3 curvature tensor and its contractions. The spin-4 curvature tensor is also uniquely derived for the N = M = 4 case. In other words, it is proven here that, for the most general linear combination of scalars built from N derivatives and M rank of tensor potential, up to N = M = 4, there exists a unique solution to the resulting system of linear equations as the contracted spin-n curvature tensors. Conjectures regarding the solutions to the higher spin-n N = M = n are discussed.

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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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30

SEGAL, ARKADY Y. "POINT PARTICLE IN GENERAL BACKGROUND FIELDS AND FREE GAUGE THEORIES OF TRACELESS SYMMETRIC TENSORS." International Journal of Modern Physics A 18, no. 27 (October 30, 2003): 4999–5019. http://dx.doi.org/10.1142/s0217751x03015830.

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Point particle may interact with traceless symmetric tensors of arbitrary rank. Free gauge theories of traceless symmetric tensors are constructed, which provides a possibility for a new type of interactions, when particles exchange by those gauge fields. The gauge theories are parametrized by the particle's mass m and otherwise are unique for each rank s. For m=0, they are local gauge models with actions of 2s th order in derivatives, known in d=4 as "pure spin," or "conformal higher spin" actions by Fradkin and Tseytlin. For m≠0, each rank-s model undergoes a unique nonlocal deformation which entangles fields of all ranks, starting from s. There exists a nonlocal transform which maps m≠0 theories onto m=0 ones, however, this map degenerates at some m≠0 fields whose polarizations are determined by zeros of Bessel functions. Conformal covariance properties of the m=0 models are analyzed, the space of gauge fields is shown to admit an action of an infinite-dimensional "conformal higher spin" Lie algebra which leaves gauge transformations intact.

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31

TINTAREANU-MIRCEA, OVIDIU. "f-SYMBOLS, KILLING TENSORS AND CONSERVED BEL-TYPE CURRENTS." Modern Physics Letters A 26, no. 05 (February 20, 2011): 337–49. http://dx.doi.org/10.1142/s0217732311034888.

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In the framework of the General Relativity we show that from three generalizations of Killing vector fields, namely f-symbols, symmetric Stäckel–Killing and antisymmetric Killing–Yano tensors, some conserved currents can be obtained through adequate contractions of the above-mentioned objects with rank-four tensors having the properties of Bel or Bel–Robinson tensors in Einstein spaces.

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32

Ceruti, Gianluca, and Christian Lubich. "Time integration of symmetric and anti-symmetric low-rank matrices and Tucker tensors." BIT Numerical Mathematics 60, no. 3 (January 28, 2020): 591–614. http://dx.doi.org/10.1007/s10543-019-00799-8.

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33

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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34

Ballico, E. "Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support." ISRN Geometry 2013 (December 29, 2013): 1–4. http://dx.doi.org/10.1155/2013/704072.

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Let νd:ℙm→ℙn, n:=(n+dn)-1, denote the degree d Veronese embedding of ℙm. For any P∈ℙn, let sr(P) be the minimal cardinality of S⊂νd(ℙm) such that P∈〈S〉. Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯(S) powers Ld with L a linear form (or as a sum of ♯(S) d-powers of vectors). Here we fix an integral variety T⊊ℙm and P∈〈νd(T)〉 and study a similar decomposition with S⊈T and ♯(S) minimal. For instance, if T is a linear subspace, then we prove that ♯(S)≥♯(S∩T)+d+1 and classify all (S,P) such that ♯(S)-♯(S∩T)≤2d-1.

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35

Leineweber, Andreas. "Anisotropic diffraction-line broadening due to microstrain distribution: parametrization opportunities." Journal of Applied Crystallography 39, no. 4 (July 15, 2006): 509–18. http://dx.doi.org/10.1107/s0021889806019546.

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A correlated Gaussian lattice-parameter distribution of an ensemble of crystals, as leading to line broadening in the course of powder diffraction, can be associated with a correlated Gaussian microstrain distribution. The latter can be described in terms of a fourth-rank covariance tensor containing as its 81 componentsEijpq, the variances and the covariances of the nine components ∊ijof the symmetric second-rank strain tensor (formulated with respect to Cartesian coordinates),i.e.Eijpq= 〈∊ij∊pq〉. The restrictions for theEijpqtensor components resulting from assumed crystal class-symmetry invariance are the same as expected for certain fourth-rank property tensors, like compliancy. The parametrization of anisotropic microstrain broadening (e.g.in the course of Rietveld refinement) on the basis of the covariance tensor componentsEijpqhas, in comparison with earlier approaches, the advantage of straightforward recognizability of the case of isotropic microstrain broadening, independently of the actual crystallographic coordinate system.

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36

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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37

Soler, Tomás, and Boudewijn H. W. Gelder. "On covariances of eigenvalues and eigenvectors of second-rank symmetric tensors." Geophysical Journal International 105, no. 2 (May 1991): 537–46. http://dx.doi.org/10.1111/j.1365-246x.1991.tb06732.x.

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38

Ni, Guyan, and Yiju Wang. "On the best rank-1 approximation to higher-order symmetric tensors." Mathematical and Computer Modelling 46, no. 9-10 (November 2007): 1345–52. http://dx.doi.org/10.1016/j.mcm.2007.01.008.

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39

Sam, Steven V. "Ideals of bounded rank symmetric tensors are generated in bounded degree." Inventiones mathematicae 207, no. 1 (May 12, 2016): 1–21. http://dx.doi.org/10.1007/s00222-016-0668-2.

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40

Sam, Steven V. "Syzygies of bounded rank symmetric tensors are generated in bounded degree." Mathematische Annalen 368, no. 3-4 (December 22, 2016): 1095–108. http://dx.doi.org/10.1007/s00208-016-1509-8.

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41

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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42

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

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43

Nie, Jiawang. "Low Rank Symmetric Tensor Approximations." SIAM Journal on Matrix Analysis and Applications 38, no. 4 (January 2017): 1517–40. http://dx.doi.org/10.1137/16m1107528.

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44

Ballico, Edoardo. "Multiple points, scheme rank and symmetric tensor rank." Portugaliae Mathematica 70, no. 3 (2013): 243–50. http://dx.doi.org/10.4171/pm/1933.

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45

Ishteva, Mariya, P. A. Absil, and Paul Van Dooren. "Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors." SIAM Journal on Matrix Analysis and Applications 34, no. 2 (January 2013): 651–72. http://dx.doi.org/10.1137/11085743x.

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46

Lagzdin', A. "Smooth convex limit surfaces in the space of symmetric second-rank tensors." Mechanics of Composite Materials 33, no. 2 (March 1997): 119–27. http://dx.doi.org/10.1007/bf02269597.

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47

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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48

Sam, David D., E. Turan Onat, Pavel I. Etingof, and Brent L. Adams. "Coordinate Free Tensorial Representation of the Orientation Distribution Function With Harmonic Polynomials." Textures and Microstructures 21, no. 4 (January 1, 1993): 233–50. http://dx.doi.org/10.1155/tsm.21.233.

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The crystallite orientation distribution function (CODF) is reviewed in terms of classical spherical function representation and more recent coordinate free tensorial representation (CFTR). A CFTR is a Fourier expansion wherein the coefficients are tensors in the three-dimensional space. The equivalence between homogeneous harmonic polynomials of degree k and symmetric and traceless tensors of rank k allows a realization of these tensors by the method of harmonic polynomials. Such a method provides for the rapid assembly of a tensorial representation from microstructural orientation measurement data. The coefficients are determined to twenty-first order and expanded in the form of a crystallite orientation distribution function, and compared with previous calculations.

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49

Ballico, Edoardo, and Luca Chiantini. "Sets Computing the Symmetric Tensor Rank." Mediterranean Journal of Mathematics 10, no. 2 (July 3, 2012): 643–54. http://dx.doi.org/10.1007/s00009-012-0214-4.

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50

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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Journal articles on the topic 'Rank of symmetric tensors'

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