Tensors: Geometry and Applications
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Asymptotic Behavior of the Dimension of the Chow Variety
ASYMPTOTIC BEHAVIOR OF THE DIMENSION OF THE CHOW VARIETY BRIAN LEHMANN Abstract. First, we calculate the dimension of the Chow variety of degree d cycles on projective space. We show that the component of maximal dimension usually parametrizes degenerate cycles, confirming a conjecture of Eisenbud and Harris. Second, for a numerical class α on an arbitrary variety, we study how the dimension of the components of the Chow variety parametrizing cycles of class mα grows as we increase m. We show that when the maximal growth rate is achieved, α is represented by cycles that are \degenerate" in a precise sense. 1. Introduction Let X be an integral projective variety over an algebraically closed field. Fix a numerical cycle class α on X and let Chow(X; α) denote the com- ponents of Chow(X) which parametrize cycles of class α. Our goal is to study the dimension of Chow(X; α) and its relationship with the geometry and positivity of the cycles representing α. In general one expects that the maximal components of Chow should parametrize subvarieties that are \de- generate" in some sense, and our results verify this principle in a general setting. We first consider the most important example: projective space. [EH92] n analyzes the dimension of the Chow variety of curves on P . Let ` denote n the class of a line in P . Then for d > 1, d2 + 3d dim Chow( n; d`) = max 2d(n − 1); + 3(n − 2) : P 2 n The first number is the dimension of the space of unions of d lines on P , and the second is the dimension of the space of degree d planar curves. -
Families of Cycles and the Chow Scheme
Families of cycles and the Chow scheme DAVID RYDH Doctoral Thesis Stockholm, Sweden 2008 TRITA-MAT-08-MA-06 ISSN 1401-2278 KTH Matematik ISRN KTH/MAT/DA 08/05-SE SE-100 44 Stockholm ISBN 978-91-7178-999-0 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i matematik måndagen den 11 augusti 2008 klockan 13.00 i Nya kollegiesalen, F3, Kungl Tek- niska högskolan, Lindstedtsvägen 26, Stockholm. © David Rydh, maj 2008 Tryck: Universitetsservice US AB iii Abstract The objects studied in this thesis are families of cycles on schemes. A space — the Chow variety — parameterizing effective equidimensional cycles was constructed by Chow and van der Waerden in the first half of the twentieth century. Even though cycles are simple objects, the Chow variety is a rather intractable object. In particular, a good func- torial description of this space is missing. Consequently, descriptions of the corresponding families and the infinitesimal structure are incomplete. Moreover, the Chow variety is not intrinsic but has the unpleasant property that it depends on a given projective embedding. A main objective of this thesis is to construct a closely related space which has a good functorial description. This is partly accomplished in the last paper. The first three papers are concerned with families of zero-cycles. In the first paper, a functor parameterizing zero-cycles is defined and it is shown that this functor is represented by a scheme — the scheme of divided powers. This scheme is closely related to the symmetric product. -
Generic Properties of Symmetric Tensors
2006 – 1/48 – P.Comon Generic properties of Symmetric Tensors Pierre COMON I3S - CNRS other contributors: Bernard MOURRAIN INRIA institute Lek-Heng LIM, Stanford University I3S 2006 – 2/48 – P.Comon Tensors & Arrays Definitions Table T = {Tij..k} Order d of T def= # of its ways = # of its indices def Dimension n` = range of the `th index T is Square when all dimensions n` = n are equal T is Symmetric when it is square and when its entries do not change by any permutation of indices I3S 2006 – 3/48 – P.Comon Tensors & Arrays Properties Outer (tensor) product C = A ◦ B: Cij..` ab..c = Aij..` Bab..c Example 1 outer product between 2 vectors: u ◦ v = u vT Multilinearity. An order-3 tensor T is transformed by the multi-linear map {A, B, C} into a tensor T 0: 0 X Tijk = AiaBjbCkcTabc abc Similarly: at any order d. I3S 2006 – 4/48 – P.Comon Tensors & Arrays Example Example 2 Take 1 v = −1 Then 1 −1 −1 1 v◦3 = −1 1 1 −1 This is a “rank-1” symmetric tensor I3S 2006 – 5/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... I3S 2006 – 6/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity I3S 2006 – 7/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... CanD/PARAFAC: = + ... + . ... ... ... ... ... ... ... ... PARAFAC cannot be used when: • Lack of diversity • Proportional slices I3S 2006 – 8/48 – P.Comon Usefulness of symmetric arrays CanD/PARAFAC vs ICA . -
Multilinear Algebra and Applications July 15, 2014
Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3. -
On Manifolds of Tensors of Fixed Tt-Rank
ON MANIFOLDS OF TENSORS OF FIXED TT-RANK SEBASTIAN HOLTZ, THORSTEN ROHWEDDER, AND REINHOLD SCHNEIDER Abstract. Recently, the format of TT tensors [19, 38, 34, 39] has turned out to be a promising new format for the approximation of solutions of high dimensional problems. In this paper, we prove some new results for the TT representation of a tensor U ∈ Rn1×...×nd and for the manifold of tensors of TT-rank r. As a first result, we prove that the TT (or compression) ranks ri of a tensor U are unique and equal to the respective seperation ranks of U if the components of the TT decomposition are required to fulfil a certain maximal rank condition. We then show that d the set T of TT tensors of fixed rank r forms an embedded manifold in Rn , therefore preserving the essential theoretical properties of the Tucker format, but often showing an improved scaling behaviour. Extending a similar approach for matrices [7], we introduce certain gauge conditions to obtain a unique representation of the tangent space TU T of T and deduce a local parametrization of the TT manifold. The parametrisation of TU T is often crucial for an algorithmic treatment of high-dimensional time-dependent PDEs and minimisation problems [33]. We conclude with remarks on those applications and present some numerical examples. 1. Introduction The treatment of high-dimensional problems, typically of problems involving quantities from Rd for larger dimensions d, is still a challenging task for numerical approxima- tion. This is owed to the principal problem that classical approaches for their treatment normally scale exponentially in the dimension d in both needed storage and computa- tional time and thus quickly become computationally infeasable for sensible discretiza- tions of problems of interest. -
Parallel Spinors and Connections with Skew-Symmetric Torsion in String Theory*
ASIAN J. MATH. © 2002 International Press Vol. 6, No. 2, pp. 303-336, June 2002 005 PARALLEL SPINORS AND CONNECTIONS WITH SKEW-SYMMETRIC TORSION IN STRING THEORY* THOMAS FRIEDRICHt AND STEFAN IVANOV* Abstract. We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the V- parallel spinors. In particular, we obtain partial solutions of the type // string equations in dimension n = 5, 6 and 7. 1. Introduction. Linear connections preserving a Riemannian metric with totally skew-symmetric torsion recently became a subject of interest in theoretical and mathematical physics. For example, the target space of supersymmetric sigma models with Wess-Zumino term carries a geometry of a metric connection with skew-symmetric torsion [23, 34, 35] (see also [42] and references therein). In supergravity theories, the geometry of the moduli space of a class of black holes is carried out by a metric connection with skew-symmetric torsion [27]. The geometry of NS-5 brane solutions of type II supergravity theories is generated by a metric connection with skew-symmetric torsion [44, 45, 43]. The existence of parallel spinors with respect to a metric connection with skew-symmetric torsion on a Riemannian spin manifold is of importance in string theory, since they are associated with some string solitons (BPS solitons) [43]. Supergravity solutions that preserve some of the supersymmetry of the underlying theory have found many applications in the exploration of perturbative and non-perturbative properties of string theory. -
Causalx: Causal Explanations and Block Multilinear Factor Analysis
To appear: Proc. of the 2020 25th International Conference on Pattern Recognition (ICPR 2020) Milan, Italy, Jan. 10-15, 2021. CausalX: Causal eXplanations and Block Multilinear Factor Analysis M. Alex O. Vasilescu1;2 Eric Kim2;1 Xiao S. Zeng2 [email protected] [email protected] [email protected] 1Tensor Vision Technologies, Los Angeles, California 2Department of Computer Science,University of California, Los Angeles Abstract—By adhering to the dictum, “No causation without I. INTRODUCTION:PROBLEM DEFINITION manipulation (treatment, intervention)”, cause and effect data Developing causal explanations for correct results or for failures analysis represents changes in observed data in terms of changes from mathematical equations and data is important in developing in the causal factors. When causal factors are not amenable for a trustworthy artificial intelligence, and retaining public trust. active manipulation in the real world due to current technological limitations or ethical considerations, a counterfactual approach Causal explanations are germane to the “right to an explanation” performs an intervention on the model of data formation. In the statute [15], [13] i.e., to data driven decisions, such as those case of object representation or activity (temporal object) rep- that rely on images. Computer graphics and computer vision resentation, varying object parts is generally unfeasible whether problems, also known as forward and inverse imaging problems, they be spatial and/or temporal. Multilinear algebra, the algebra have been cast as causal inference questions [40], [42] consistent of higher order tensors, is a suitable and transparent framework for disentangling the causal factors of data formation. Learning a with Donald Rubin’s quantitative definition of causality, where part-based intrinsic causal factor representations in a multilinear “A causes B” means “the effect of A is B”, a measurable framework requires applying a set of interventions on a part- and experimentally repeatable quantity [14], [17]. -
Edinburgh Research Explorer
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Edinburgh Research Explorer Edinburgh Research Explorer A Holant Dichotomy: Is the FKT Algorithm Universal? Citation for published version: Cai, J-Y, Fu, Z, Guo, H & Williams, T 2015, A Holant Dichotomy: Is the FKT Algorithm Universal? in IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015. IEEE, pp. 1259-1276, Foundations of Computer Science 2015, Berkley, United States, 17/10/15. DOI: 10.1109/FOCS.2015.81 Digital Object Identifier (DOI): 10.1109/FOCS.2015.81 Link: Link to publication record in Edinburgh Research Explorer Document Version: Peer reviewed version Published In: IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015 General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 05. Apr. 2019 A Holant Dichotomy: Is the FKT Algorithm Universal? Jin-Yi Cai∗ Zhiguo Fu∗y Heng Guo∗z Tyson Williams∗x [email protected] [email protected] [email protected] [email protected] Abstract We prove a complexity dichotomy for complex-weighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. -
Algebraic Cycles, Chow Varieties, and Lawson Homology Compositio Mathematica, Tome 77, No 1 (1991), P
COMPOSITIO MATHEMATICA ERIC M. FRIEDLANDER Algebraic cycles, Chow varieties, and Lawson homology Compositio Mathematica, tome 77, no 1 (1991), p. 55-93 <http://www.numdam.org/item?id=CM_1991__77_1_55_0> © Foundation Compositio Mathematica, 1991, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 77: 55-93,55 1991. (Ç) 1991 Kluwer Academic Publishers. Printed in the Netherlands. Algebraic cycles, Chow varieties, and Lawson homology ERIC M. FRIEDLANDER* Department of Mathematics, Northwestern University, Evanston, Il. 60208, U.S.A. Received 22 August 1989; accepted in revised form 14 February 1990 Following the foundamental work of H. Blaine Lawson [ 19], [20], we introduce new invariants for projective algebraic varieties which we call Lawson ho- mology groups. These groups are a hybrid of algebraic geometry and algebraic topology: the 1-adic Lawson homology group LrH2,+i(X, Zi) of a projective variety X for a given prime1 invertible in OX can be naively viewed as the group of homotopy classes of S’*-parametrized families of r-dimensional algebraic cycles on X. Lawson homology groups are covariantly functorial, as homology groups should be, and admit Galois actions. If i = 0, then LrH 2r+i(X, Zl) is the group of algebraic equivalence classes of r-cycles; if r = 0, then LrH2r+i(X , Zl) is 1-adic etale homology. -
Matrices and Tensors
APPENDIX MATRICES AND TENSORS A.1. INTRODUCTION AND RATIONALE The purpose of this appendix is to present the notation and most of the mathematical tech- niques that are used in the body of the text. The audience is assumed to have been through sev- eral years of college-level mathematics, which included the differential and integral calculus, differential equations, functions of several variables, partial derivatives, and an introduction to linear algebra. Matrices are reviewed briefly, and determinants, vectors, and tensors of order two are described. The application of this linear algebra to material that appears in under- graduate engineering courses on mechanics is illustrated by discussions of concepts like the area and mass moments of inertia, Mohr’s circles, and the vector cross and triple scalar prod- ucts. The notation, as far as possible, will be a matrix notation that is easily entered into exist- ing symbolic computational programs like Maple, Mathematica, Matlab, and Mathcad. The desire to represent the components of three-dimensional fourth-order tensors that appear in anisotropic elasticity as the components of six-dimensional second-order tensors and thus rep- resent these components in matrices of tensor components in six dimensions leads to the non- traditional part of this appendix. This is also one of the nontraditional aspects in the text of the book, but a minor one. This is described in §A.11, along with the rationale for this approach. A.2. DEFINITION OF SQUARE, COLUMN, AND ROW MATRICES An r-by-c matrix, M, is a rectangular array of numbers consisting of r rows and c columns: ¯MM.. -
Matchgates Revisited
THEORY OF COMPUTING, Volume 10 (7), 2014, pp. 167–197 www.theoryofcomputing.org RESEARCH SURVEY Matchgates Revisited Jin-Yi Cai∗ Aaron Gorenstein Received May 17, 2013; Revised December 17, 2013; Published August 12, 2014 Abstract: We study a collection of concepts and theorems that laid the foundation of matchgate computation. This includes the signature theory of planar matchgates, and the parallel theory of characters of not necessarily planar matchgates. Our aim is to present a unified and, whenever possible, simplified account of this challenging theory. Our results include: (1) A direct proof that the Matchgate Identities (MGI) are necessary and sufficient conditions for matchgate signatures. This proof is self-contained and does not go through the character theory. (2) A proof that the MGI already imply the Parity Condition. (3) A simplified construction of a crossover gadget. This is used in the proof of sufficiency of the MGI for matchgate signatures. This is also used to give a proof of equivalence between the signature theory and the character theory which permits omittable nodes. (4) A direct construction of matchgates realizing all matchgate-realizable symmetric signatures. ACM Classification: F.1.3, F.2.2, G.2.1, G.2.2 AMS Classification: 03D15, 05C70, 68R10 Key words and phrases: complexity theory, matchgates, Pfaffian orientation 1 Introduction Leslie Valiant introduced matchgates in a seminal paper [24]. In that paper he presented a way to encode computation via the Pfaffian and Pfaffian Sum, and showed that a non-trivial, though restricted, fragment of quantum computation can be simulated in classical polynomial time. Underlying this magic is a way to encode certain quantum states by a classical computation of perfect matchings, and to simulate certain ∗Supported by NSF CCF-0914969 and NSF CCF-1217549. -
A Some Basic Rules of Tensor Calculus
A Some Basic Rules of Tensor Calculus The tensor calculus is a powerful tool for the description of the fundamentals in con- tinuum mechanics and the derivation of the governing equations for applied prob- lems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: – the direct (symbolic) notation and – the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three dimensional space. A vector (first rank tensor) a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum of ordered vector pairs A = a b + ... +c d. The scalars, vectors and tensors are handled as invariant (independent⊗ from the choice⊗ of the coordinate system) objects. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. [29, 32, 49, 123, 131, 199, 246, 313, 334] among others. The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi, i = 1, 2, 3 one can write a = aig , A = aibj + ... + cidj g g i i ⊗ j Here the Einstein’s summation convention is used: in one expression the twice re- peated indices are summed up from 1 to 3, e.g. 3 3 k k ik ik a gk ∑ a gk, A bk ∑ A bk ≡ k=1 ≡ k=1 In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.