Symmetric tensors and symmetric tensor rank

... has been used to fit data arrays to a multilinear model [36, 50]. Yet, the minimization of this matching error is an ill-posed problem in general, since the set of symmetric tensors of symmetric rank not more than r is not closed, unless r = 1 (see Sections 6 and 8) — a fact that parallels the illpo ...

... has been used to fit data arrays to a multilinear model [36, 50]. Yet, the minimization of this matching error is an ill-posed problem in general, since the set of symmetric tensors of symmetric rank not more than r is not closed, unless r = 1 (see Sections 6 and 8) — a fact that parallels the illpo ...

Contents - Harvard Math Department

... Definition 1.4 For S ⊂ R a multiplicative set, the localization S −1 R is a commutative ring as above. In fact, it is an R-algebra; there is a natural map φ : R → S −1 R sending r → r/1. We can, in fact, describe φ : R → S −1 R by a universal property. Note that for each s ∈ S, φ(s) is invertible. T ...

... Definition 1.4 For S ⊂ R a multiplicative set, the localization S −1 R is a commutative ring as above. In fact, it is an R-algebra; there is a natural map φ : R → S −1 R sending r → r/1. We can, in fact, describe φ : R → S −1 R by a universal property. Note that for each s ∈ S, φ(s) is invertible. T ...

EXTERIOR POWERS 1. Introduction Let R be a commutative ring

... Example 2.5. The function C × C → R given by (z, w) 7→ Im(zw) is skew-symmetric and alternating. Example 2.6. Let R contain Z/2Z, so −1 = 1 in R. The multiplication map R × R → R is symmetric and skew-symmetric, but not alternating. In Definition 2.1, the variables are indexed in the order from 1 to ...

... Example 2.5. The function C × C → R given by (z, w) 7→ Im(zw) is skew-symmetric and alternating. Example 2.6. Let R contain Z/2Z, so −1 = 1 in R. The multiplication map R × R → R is symmetric and skew-symmetric, but not alternating. In Definition 2.1, the variables are indexed in the order from 1 to ...

On zero product determined algebras

... Yu’s study of idempotent preserving bilinear maps and the notion of an idempotent elements determined algebra.[8] In this paper, we provide several main results that compliment the existing body of research. We reformulate the definition of a zero product determined algebra in terms of tensor produc ...

... Yu’s study of idempotent preserving bilinear maps and the notion of an idempotent elements determined algebra.[8] In this paper, we provide several main results that compliment the existing body of research. We reformulate the definition of a zero product determined algebra in terms of tensor produc ...

Linear Algebra I

... idea to split the vector space V into subspaces on which the endomorphism can be more easily described. In order to make this precise, we introduce the notion of direct sum of linear subspaces of V . 4.1. Lemma and Definition. Let V be a vector space, Ui ⊂ V (for i ∈ I) linear subspaces. Then the fo ...

... idea to split the vector space V into subspaces on which the endomorphism can be more easily described. In order to make this precise, we introduce the notion of direct sum of linear subspaces of V . 4.1. Lemma and Definition. Let V be a vector space, Ui ⊂ V (for i ∈ I) linear subspaces. Then the fo ...

Fast structured matrix computations: tensor rank and Cohn Umans method

... for a variety of operations involving structured matrices, as measured by the bilinear complexity of the problem. Roughly speaking, the bilinear complexity of an algorithm for a problem that can be cast as the evaluation of a bilinear map is the number of multiplications required in the algorithm; t ...

... for a variety of operations involving structured matrices, as measured by the bilinear complexity of the problem. Roughly speaking, the bilinear complexity of an algorithm for a problem that can be cast as the evaluation of a bilinear map is the number of multiplications required in the algorithm; t ...

Frobenius algebras and monoidal categories

... bicategories. In Vect k - Mod the pseudomonoids include monoidal k-linear categories such as Vect k itself. The Frobenius requirement is related to the notion of star-autonomy due to Michael Barr. Every rigid (autonomous, compact) monoidal category is star-autonomous. In particular, Vect k is Froben ...

... bicategories. In Vect k - Mod the pseudomonoids include monoidal k-linear categories such as Vect k itself. The Frobenius requirement is related to the notion of star-autonomy due to Michael Barr. Every rigid (autonomous, compact) monoidal category is star-autonomous. In particular, Vect k is Froben ...

Interpretations and Representations of Classical Tensors

... A/I is said to be a quotient algebra. [0A ] is obviously the zero element in A/I, while [1A ] is the unit element. For any set S ⊂ A there is a unique smallest ideal I(S) ⊃ S, and a corresponding quotient algebra A/I(S). Alternatively, we can define a quotient algebra by specifying a set R of relati ...

... A/I is said to be a quotient algebra. [0A ] is obviously the zero element in A/I, while [1A ] is the unit element. For any set S ⊂ A there is a unique smallest ideal I(S) ⊃ S, and a corresponding quotient algebra A/I(S). Alternatively, we can define a quotient algebra by specifying a set R of relati ...

Groups with exponents I. Fundamentals of the theory and tensor

... commutative. Put f~ = r162 and f2 = r162 Since fi is identical on Ai(G) and Ai(G) B-generates Gi, we have fi = id; i.e., r and r are mutually inverse B-homomorphisms. [] In applications # will mostly be an embedding of rings. However, even in this case, the homomorphism A : G ---* G B'" is not alway ...

... commutative. Put f~ = r162 and f2 = r162 Since fi is identical on Ai(G) and Ai(G) B-generates Gi, we have fi = id; i.e., r and r are mutually inverse B-homomorphisms. [] In applications # will mostly be an embedding of rings. However, even in this case, the homomorphism A : G ---* G B'" is not alway ...

Tensors and hypermatrices

... useful device. First, it gives us a concrete way to think about tensors, one that allows a parallel to the usual matrix theory. Second, a hypermatrix is what we often get in practice: As soon as measurements are performed in some units, bases are chosen implicitly, and the values of the measurements ...

... useful device. First, it gives us a concrete way to think about tensors, one that allows a parallel to the usual matrix theory. Second, a hypermatrix is what we often get in practice: As soon as measurements are performed in some units, bases are chosen implicitly, and the values of the measurements ...

Square Deal: Lower Bounds and Improved Relaxations for Tensor

... ranktc (X 0 )} is empty.2 The recovery performance of (2.1) depends heavily on the properties of G. Suppose (2.1) fails to recover X 0 ∈ Tr . Then there exists another X 0 ∈ Tr such that G[X 0 ] = G[X 0 ]. To guarantee that (2.1) recovers any X 0 ∈ Tr , a necessary and sufficient condition is that G ...

... ranktc (X 0 )} is empty.2 The recovery performance of (2.1) depends heavily on the properties of G. Suppose (2.1) fails to recover X 0 ∈ Tr . Then there exists another X 0 ∈ Tr such that G[X 0 ] = G[X 0 ]. To guarantee that (2.1) recovers any X 0 ∈ Tr , a necessary and sufficient condition is that G ...

Chapter 22 Tensor Algebras, Symmetric Algebras and Exterior

... isomorphism between T1 and T2 . Now that we have shown that tensor products are unique up to isomorphism, we give a construction that produces one. Theorem 22.5 Given n ≥ 2 vector spaces E1 , . . . , En , a tensor product (E1 ⊗ · · · ⊗ En , ϕ) for E1 , . . . , En can be constructed. Furthermore, den ...

... isomorphism between T1 and T2 . Now that we have shown that tensor products are unique up to isomorphism, we give a construction that produces one. Theorem 22.5 Given n ≥ 2 vector spaces E1 , . . . , En , a tensor product (E1 ⊗ · · · ⊗ En , ϕ) for E1 , . . . , En can be constructed. Furthermore, den ...

TENSOR PRODUCTS II 1. Introduction Continuing our study of

... makes no sense in I ⊗2 since 1 is not an element of I.) To show two tensors are not equal, the best approach is to construct a linear map from the tensor product space that has different values at the two tensors. The function I × I → A given by (f, g) 7→ fX (0, 0)gY (0, 0), where fX and gY are part ...

... makes no sense in I ⊗2 since 1 is not an element of I.) To show two tensors are not equal, the best approach is to construct a linear map from the tensor product space that has different values at the two tensors. The function I × I → A given by (f, g) 7→ fX (0, 0)gY (0, 0), where fX and gY are part ...

MULTILINEAR ALGEBRA: THE EXTERIOR PRODUCT This writeup

... The argument for the remaining elementary divisors is more of the same, going up to F ∧m = Ae1 ∧ · · · ∧ em ⊕ · · · and S ∧m = a1 · · · am e1 ∧ · · · ∧ em . Note that m itself is described intrinsically as the highest exponent of a nonzero exterior power of S. Now we can see that the ingredients of ...

... The argument for the remaining elementary divisors is more of the same, going up to F ∧m = Ae1 ∧ · · · ∧ em ⊕ · · · and S ∧m = a1 · · · am e1 ∧ · · · ∧ em . Note that m itself is described intrinsically as the highest exponent of a nonzero exterior power of S. Now we can see that the ingredients of ...

Explicit tensors - Computational Complexity

... so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and rational numbers by tuples of integers represented in binary. The actual encoding does not matter as long as it is “reasonably nice”, that is, all operatio ...

... so we need to be able to encode the field elements by {0, 1}-strings. For instance, elements from finite fields can be represented in binary and rational numbers by tuples of integers represented in binary. The actual encoding does not matter as long as it is “reasonably nice”, that is, all operatio ...

Part C4: Tensor product

... (3) list of basic properties (4) distributive property (5) right exactness (6) localization is flat (7) extension of scalars (8) applications 4.1. definition. First I gave the categorical definition and then I gave an explicit construction. 4.1.1. universal condition. Tensor product is usually defin ...

... (3) list of basic properties (4) distributive property (5) right exactness (6) localization is flat (7) extension of scalars (8) applications 4.1. definition. First I gave the categorical definition and then I gave an explicit construction. 4.1.1. universal condition. Tensor product is usually defin ...

Tensor products in the category of topological vector spaces are not

... For example, the tensor products in the class of real Hausdorff locally convex spaces are the projective tensor products, going back to Grothendieck’s memoir [8]. In this case, an explicit description of the locally convex topology (by means of suitable cross-seminorms) is available, and it is well- ...

... For example, the tensor products in the class of real Hausdorff locally convex spaces are the projective tensor products, going back to Grothendieck’s memoir [8]. In this case, an explicit description of the locally convex topology (by means of suitable cross-seminorms) is available, and it is well- ...

Harmonic analysis of dihedral groups

... The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R2 , there are n axes through ...

... The rotations are the symmetries preserving the (cyclic) ordering of vertices. Thus, a rotation g is determined by the image gv, so the subgroup N of rotations has n elements. A reflection is an order-2 symmetry reversing the ordering of vertices. Imbedding the n-gon in R2 , there are n axes through ...

Chapter 3 Cartesian Tensors

... consider the vector relation y = (a . b)x. We have a . b = ai bi , but we cannot write yi = ai bi xi as this would be ambiguous. How can we correct this? Note that a . b = ai b i = aj b j – the suffix we use for the summation is immaterial. (Compare with the use of dummy R∞ R∞ variables in integrat ...

... consider the vector relation y = (a . b)x. We have a . b = ai bi , but we cannot write yi = ai bi xi as this would be ambiguous. How can we correct this? Note that a . b = ai b i = aj b j – the suffix we use for the summation is immaterial. (Compare with the use of dummy R∞ R∞ variables in integrat ...

TENSOR PRODUCTS OF LOCALLY CONVEX ALGEBRAS 124

... Banach algebras with respect to a cross-norm, these authors have given sufficient conditions under which its space of nonzero multiplicative functionals with relative weak* topology can be characterized as the topological direct product of the corresponding spaces associated with the two factors. Gi ...

... Banach algebras with respect to a cross-norm, these authors have given sufficient conditions under which its space of nonzero multiplicative functionals with relative weak* topology can be characterized as the topological direct product of the corresponding spaces associated with the two factors. Gi ...

THE TENSOR PRODUCT OF FUNCTION SEMIMODULES The

... such that µ : SA ⊗ SB → SA×B is not injective (Theorem 3.10). Here, ⊗ denotes the (algebraic) tensor product of S-semimodules satisfying the standard universal property expected of such a product. This tensor product was constructed by Y. Katsov in [Kat97]. The infinite cardinality of A, B has to be ...

... such that µ : SA ⊗ SB → SA×B is not injective (Theorem 3.10). Here, ⊗ denotes the (algebraic) tensor product of S-semimodules satisfying the standard universal property expected of such a product. This tensor product was constructed by Y. Katsov in [Kat97]. The infinite cardinality of A, B has to be ...

Exploring Tensor Rank

... n1 × · · · × nd tensor is simply an n1 × · · · × nd -dimensional array of elements of a field F . We call d the order of the tensor, and (n1 , . . . , nd ) the dimensions of the tensor. The space of n1 × · · · × nd tensors is denoted F n1 ⊗ · · · ⊗ F nd , and is equipped with a natural addition wher ...

... n1 × · · · × nd tensor is simply an n1 × · · · × nd -dimensional array of elements of a field F . We call d the order of the tensor, and (n1 , . . . , nd ) the dimensions of the tensor. The space of n1 × · · · × nd tensors is denoted F n1 ⊗ · · · ⊗ F nd , and is equipped with a natural addition wher ...

An Introduction to Nonlinear Solid Mechanics Marino Arroyo & Anna Pandolfi

... If Q and R are orthogonal, then also the product QR is orthogonal. It is said that the set of orthogonal mappings is a subgroup of GL(n) closed under multiplication, and it is called Orthogonal Group O(n). The subgroup of the orthogonal group that preserves the orientation, i.e. with determinant equ ...

... If Q and R are orthogonal, then also the product QR is orthogonal. It is said that the set of orthogonal mappings is a subgroup of GL(n) closed under multiplication, and it is called Orthogonal Group O(n). The subgroup of the orthogonal group that preserves the orientation, i.e. with determinant equ ...

Introduction to tensor, tensor factorization and its applications

... processing, storage, tensor factorization is often needed. Three-way tensor: ...

... processing, storage, tensor factorization is often needed. Three-way tensor: ...