Tensor, Exterior and Symmetric Algebras
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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. -
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.. -
28. Exterior Powers
28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1. -
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. -
CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V
CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n. -
1.3 Cartesian Tensors a Second-Order Cartesian Tensor Is Defined As A
1.3 Cartesian tensors A second-order Cartesian tensor is defined as a linear combination of dyadic products as, T Tijee i j . (1.3.1) The coefficients Tij are the components of T . A tensor exists independent of any coordinate system. The tensor will have different components in different coordinate systems. The tensor T has components Tij with respect to basis {ei} and components Tij with respect to basis {e i}, i.e., T T e e T e e . (1.3.2) pq p q ij i j From (1.3.2) and (1.2.4.6), Tpq ep eq TpqQipQjqei e j Tij e i e j . (1.3.3) Tij QipQjqTpq . (1.3.4) Similarly from (1.3.2) and (1.2.4.6) Tij e i e j Tij QipQjqep eq Tpqe p eq , (1.3.5) Tpq QipQjqTij . (1.3.6) Equations (1.3.4) and (1.3.6) are the transformation rules for changing second order tensor components under change of basis. In general Cartesian tensors of higher order can be expressed as T T e e ... e , (1.3.7) ij ...n i j n and the components transform according to Tijk ... QipQjqQkr ...Tpqr... , Tpqr ... QipQjqQkr ...Tijk ... (1.3.8) The tensor product S T of a CT(m) S and a CT(n) T is a CT(m+n) such that S T S T e e e e e e . i1i2 im j1j 2 jn i1 i2 im j1 j2 j n 1.3.1 Contraction T Consider the components i1i2 ip iq in of a CT(n). -
Tensor-Spinor Theory of Gravitation in General Even Space-Time Dimensions
Physics Letters B 817 (2021) 136288 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Tensor-spinor theory of gravitation in general even space-time dimensions ∗ Hitoshi Nishino a, ,1, Subhash Rajpoot b a Department of Physics, College of Natural Sciences and Mathematics, California State University, 2345 E. San Ramon Avenue, M/S ST90, Fresno, CA 93740, United States of America b Department of Physics & Astronomy, California State University, 1250 Bellflower Boulevard, Long Beach, CA 90840, United States of America a r t i c l e i n f o a b s t r a c t Article history: We present a purely tensor-spinor theory of gravity in arbitrary even D = 2n space-time dimensions. Received 18 March 2021 This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in Accepted 9 April 2021 4D to general even dimensions with the signature (2n − 1, 1). In the original BM-theory in D = (3, 1), Available online 21 April 2021 the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a Editor: N. Lambert m lagrangian free of both the fundamental metric gμν and the vierbein eμ . We first improve the original Keywords: BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance = =− = Bars-MacDowell theory under the nilpotent local fermionic symmetry: δψ Dμ and δ χ . We next generalize it to D Vector-spinor (2n − 1, 1), following the same principle based on a lagrangian free of fundamental metric or vielbein Tensors-spinors rs − now with the field content (ψμ1···μn−1 , ωμ , χμ1···μn−2 ), where ψμ1···μn−1 (or χμ1···μn−2 ) is a (n 1) (or Metric-less formulation (n − 2)) rank tensor-spinor. -
Mathematics 552 Algebra II Spring 2017 the Exterior Algebra of K
Mathematics 552 Algebra II Spring 2017 The exterior algebra of K-modules and the Cauchy-Binet formula Let K be a commutative ring, and let M be a K-module. Recall that the tensor ⊗i algebra T (M) = ⊕i=0M is an associative K-algebra and that if M is free of finite rank ⊗k m with basis v1; : : : ; vm then T (M) is a graded free K module with basis for M given ⊗k by the collection of vi1 ⊗ vi2 ⊗ · · · vik for 1 ≤ it ≤ m. Hence M is a free K-module of rank mk. We checked in class that the tensor algebra was functorial, in that when f : M ! N is a K-module homomorphism, then the induced map f ⊗i : M ⊗i ! N ⊗i gives a homomorphism of K-algebras T (f): T (M) ! T (N) and that this gives a functor from K-modules to K-algebras. The ideal I ⊂ T (M) generated by elements of the form x ⊗ x for x 2 M enables us to construct an algebra Λ(M) = T (M)=I. Since (v+w)⊗(v+w) = v⊗v+v⊗w+w⊗v+w⊗w we have that (modulo I) v ⊗ w = −w ⊗ v. The image of M ⊗i in Λ(M) is denoted Λi(M), the i-th exterior power of M. The product in Λ(M) derived from the product on T (M) is usually called the wedge product : v ^ w is the image of v ⊗ w in T (M)=I. Since making tensor algebra of modules is functorial, so is making the exterior algebra. -
Tensor Algebra
TENSOR ALGEBRA Continuum Mechanics Course (MMC) - ETSECCPB - UPC Introduction to Tensors Tensor Algebra 2 Introduction SCALAR , , ... v VECTOR vf, , ... MATRIX σε,,... ? C,... 3 Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”. The components of a tensor will depend on the reference system chosen and will vary with it. 4 Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. a Scalar: zero dimension 3.14 1.2 v 0.3 a , a Vector: 1 dimension i 0.8 0.1 0 1.3 2nd order: 2 dimensions A, A E 02.40.5 ij rd A , A 3 order: 3 dimensions 1.3 0.5 5.8 A , A 4th order … 5 Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: eeˆˆ12,, ee ˆˆ 23 ee ˆˆ 31 Where: eeeˆˆˆ1231, 1, 1 Note that 1 if ij eeˆˆi j ij 0 if ij 6 Cylindrical Coordinate System x3 xr1 cos x(,rz , ) xr2 sin xz3 eeeˆˆˆr cosθθ 12 sin eeeˆˆˆsinθθ cos x2 12 eeˆˆz 3 x1 7 Spherical Coordinate System x3 xr1 sin cos xrxr, , 2 sin sin xr3 cos ˆˆˆˆ x2 eeeer sinθφ sin 123sin θ cos φ cos θ eeeˆˆˆ cosφφ 12sin x1 eeeeˆˆˆˆφ cosθφ sin 123cos θ cos φ sin θ 8 Indicial or (Index) Notation Tensor Algebra 9 Tensor Bases – VECTOR A vector v can be written as a unique linear combination of the three vector basis eˆ for i 1, 2, 3 . -
Lecture 21: Symmetric Products and Algebras
LECTURE 21: SYMMETRIC PRODUCTS AND ALGEBRAS Symmetric Products The last few lectures have focused on alternating multilinear functions. This one will focus on symmetric multilinear functions. Recall that a multilinear function f : U ×m ! V is symmetric if f(~v1; : : : ;~vi; : : : ;~vj; : : : ;~vm) = f(~v1; : : : ;~vj; : : : ;~vi; : : : ;~vm) for any i and j, and for any vectors ~vk. Just as with the exterior product, we can get the universal object associated to symmetric multilinear functions by forming various quotients of the tensor powers. Definition 1. The mth symmetric power of V , denoted Sm(V ), is the quotient of V ⊗m by the subspace generated by ~v1 ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vm − ~v1 ⊗ · · · ⊗ ~vj ⊗ · · · ⊗ ~vi ⊗ · · · ⊗ ~vm where i and j and the vectors ~vk are arbitrary. Let Sym(U ×m;V ) denote the set of symmetric multilinear functions U ×m to V . The following is immediate from our construction. Lemma 1. We have an natural bijection Sym(U ×m;V ) =∼ L(Sm(U);V ): n We will denote the image of ~v1 ⊗: : :~vm in S (V ) by ~v1 ·····~vm, the usual notation for multiplication to remind us that the terms here can be rearranged. Unlike with the exterior product, it is easy to determine a basis for the symmetric powers. Theorem 1. Let f~v1; : : : ;~vmg be a basis for V . Then _ f~vi1 · :::~vin ji1 ≤ · · · ≤ ing is a basis for Sn(V ). Before we prove this, we note a key difference between this and the exterior product. For the exterior product, we have strict inequalities. For the symmetric product, we have non-strict inequalities. -
Left-Symmetric Algebras of Derivations of Free Algebras
LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS OF FREE ALGEBRAS Ualbai Umirbaev1 Abstract. A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of deriva- tions are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric al- gebra of derivations of a free nonassociative symmetric m-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described. Mathematics Subject Classification (2010): Primary 17D25, 17A42, 14R15; Sec- ondary 17A36, 17A50. Key words: left-symmetric algebras, free algebras, derivations, Jacobian matrices. 1. Introduction If A is an arbitrary algebra over a field k, then the set DerkA of all k-linear derivations of A forms a Lie algebra. If A is a free algebra, then it is possible to define a multiplication · on DerkA such that it becomes a left-symmetric algebra and its commutator algebra becomes the Lie algebra DerkA of all derivations of A. The language of the left-symmetric algebras of derivations is more convenient to describe some combinatorial properties of derivations. Recall that an algebra B over k is called left-symmetric [4] if B satisfies the identity (1) (xy)z − x(yz)=(yx)z − y(xz). This means that the associator (x, y, z) := (xy)z −x(yz) is symmetric with respect to two left arguments, i.e., (x, y, z)=(y, x, z). -
Low-Level Image Processing with the Structure Multivector
Low-Level Image Processing with the Structure Multivector Michael Felsberg Bericht Nr. 0202 Institut f¨ur Informatik und Praktische Mathematik der Christian-Albrechts-Universitat¨ zu Kiel Olshausenstr. 40 D – 24098 Kiel e-mail: [email protected] 12. Marz¨ 2002 Dieser Bericht enthalt¨ die Dissertation des Verfassers 1. Gutachter Prof. G. Sommer (Kiel) 2. Gutachter Prof. U. Heute (Kiel) 3. Gutachter Prof. J. J. Koenderink (Utrecht) Datum der mundlichen¨ Prufung:¨ 12.2.2002 To Regina ABSTRACT The present thesis deals with two-dimensional signal processing for computer vi- sion. The main topic is the development of a sophisticated generalization of the one-dimensional analytic signal to two dimensions. Motivated by the fundamental property of the latter, the invariance – equivariance constraint, and by its relation to complex analysis and potential theory, a two-dimensional approach is derived. This method is called the monogenic signal and it is based on the Riesz transform instead of the Hilbert transform. By means of this linear approach it is possible to estimate the local orientation and the local phase of signals which are projections of one-dimensional functions to two dimensions. For general two-dimensional signals, however, the monogenic signal has to be further extended, yielding the structure multivector. The latter approach combines the ideas of the structure tensor and the quaternionic analytic signal. A rich feature set can be extracted from the structure multivector, which contains measures for local amplitudes, the local anisotropy, the local orientation, and two local phases. Both, the monogenic signal and the struc- ture multivector are combined with an appropriate scale-space approach, resulting in generalized quadrature filters.