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Homomorphism Tensors and Linear Equations [article]

Martin Grohe, Gaurav Rattan, Tim Seppelt
2022 arXiv   pre-print
The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational  ...  In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs.  ...  Homomorphism Tensors and Homomorphism Tensor Maps.  ... 
arXiv:2111.11313v2 fatcat:7xwmantsajayxivhhel5avf2we

Page 109 of Mathematical Reviews Vol. 7, Issue 3 [page]

1946 Mathematical Reviews  
amounts e¢; (when all ¢; are zero we have the special case of linear equations).  ...  Thus the theory of functors is also applicable to abstract sets, linear spaces, and so on. In the case of groups, mappings are understood to be homomorphisms.  ... 

Tensors as module homomorphisms over group rings [article]

Carmeliza Navasca, Michael Opperman, Timothy Penderghest, Christino Tamon
2010 arXiv   pre-print
Braman [B08] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars.  ...  Our generalization allows for any abelian group to replace the cyclic group, any commutative ring with identity to replace the field of real numbers, and an arbitrary order tensor to replace third-order  ...  Moreover, any third-order tensor is a linear transformation over this module and any linear transformation over the module can be represented as a third-order tensor.  ... 
arXiv:1005.1894v1 fatcat:l7kawcrtszezhalmpca4o5fytu

Page 674 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
Barvinok (S-RIT) 15 LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY 93b:15034 Li, Jian Xin A new algorithm for the greatest solution of fuzzy bilinear equations.  ...  Fuzzy Sets and Systems 46 (1992), no. 2, 193-210. From the summary: “The equations A,;x = A2x are called fuzzy bilinear equations. In this paper, we define the extended fuzzy bilinear equations x!  ... 

Algebraic Reductions of Knowledge [article]

Abhiram Kothapalli, Bryan Parno
2022 IACR Cryptology ePrint Archive  
We construct and study the tensor reduction, which is capable of reducing any homomorphic statement composed via the tensor product, and provides knowledge soundness unconditionally when working over vector  ...  We additionally show that tensor reductions can be employed to construct reductions of knowledge with logarithmic communication for familiar linear algebraic statements, and in turn, these can be composed  ...  Equation (18), the prover and verifier run a reduction of knowledge for linear forms.  ... 
dblp:journals/iacr/KothapalliP22 fatcat:b2msxt5nb5ck7my4l7qz5pvuoe

Conformal linear gravity in de Sitter space

M. V. Takook, M. R. Tanhayi, S. Fatemi
2010 Journal of Mathematical Physics  
It has been shown that the theory of linear conformal quantum gravity must include a tensor field of rank-3 and mixed symmetry [1].  ...  Then, in order to relate this field with the symmetric tensor field of rank-2, _αβ related to graviton, we will define homomorphisms between them.  ...  As a future work, it may be possible to find a CI gravitational field which in its linear approximation gives this linear physical equation.  ... 
doi:10.1063/1.3321581 fatcat:u3bugpjlyzhafowwj2ukgqzltq

Differential Operators Preserving Relations of Automorphy

R. C. Gunning
1963 Transactions of the American Mathematical Society  
A holomorphic differential %^-homomorphism 2 of order 1 is a dè-homomorphism if and only if: formula (2.10) of Theorem 1, the right-hand side R of equation (4.7) can be written as Vij(zaczbd) -(2.10)  ...  ) in [6]), and solving the resulting set of linear equations in c,c1,c2,c3.Since this is a straightforward calculation, the details will be omitted, except for the following observation.  ... 
doi:10.2307/1993611 fatcat:mnuqryo7ubczrk7z2amryx5e4i

Differential operators preserving relations of automorphy

R. C. Gunning
1963 Transactions of the American Mathematical Society  
A holomorphic differential %^-homomorphism 2 of order 1 is a dè-homomorphism if and only if: formula (2.10) of Theorem 1, the right-hand side R of equation (4.7) can be written as Vij(zaczbd) -(2.10)  ...  ) in [6]), and solving the resulting set of linear equations in c,c1,c2,c3.Since this is a straightforward calculation, the details will be omitted, except for the following observation.  ... 
doi:10.1090/s0002-9947-1963-0168803-0 fatcat:eu73qyzi25ei7fxivceycktwiy

On universal solution to reflection equation [article]

J. Donin, P. P. Kulish, A. I. Mudrov
2002 arXiv   pre-print
We introduce universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices  ...  We show that the braided bialgebra structure of ^* is naturally described by means of twisted tensor powers of and their module algebras.  ...  It will satisfy equations (25) and (26) in B ⊗ B ⊗B * .  ... 
arXiv:math/0210242v1 fatcat:d4vxv57dfbgbnmgjuu7ha7thqm

Contents [chapter]

2022 Abstract Algebra  
of equations by radicals | 280 Group actions | 86 Orbits and stabilizers | 89 12.5 Roots of Polynomials and Discriminants | 288 5.3 13 Applications to the structure of groups | 93 Tensor Products | 292  ...  8.2 Linear independence, basis and dimension | 148 8.3 Linear mappings | 158 8.4 Eigenvalues and eigenvectors | 167 9 Introduction to Modules | 180 9.1 Elements of module theory | 180 9.2 Modules over  ... 
doi:10.1515/9783110691160-toc fatcat:7wqnwsd6dned7jpyqnoo5fuxv4

Sofic Lie Algebras [article]

Cameron Cinel
2022 arXiv   pre-print
We introduce and study soficity for Lie algebras, modelled after linear soficity in associative algebras.  ...  Finally, we give explicit families of almost representations for the Witt and Virasoro algebras.  ...  The lemma show that SR(L) = Θ∈S ker Θ where S = Θ : L → ω gl n k (F )/ρ ω | Θ is a Lie algebra homomorphism Tensor each component of Ψ m with an appropriately sized identity matrix gives us maps Ψm :  ... 
arXiv:2202.13025v3 fatcat:2ww65fdvzfev3hpblmzlynntbe

WIGNER–ECKART THEOREM FOR TENSOR OPERATORS OF HOPF ALGEBRAS

MAREK MOZRZYMAS
2005 International Journal of Geometric Methods in Modern Physics (IJGMMP)  
We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible.  ...  Two classes of tensor operators for the Hopf algebra U_t(su(2)) are considered. The reduced matrix elements for the class of irreducible tensor operators are calculated.  ...  The linear mapping f ∈ Hom(V, W ) is a homomorphism of representations if and only if f is an invariant element of the representation (Hom(V, W ), δ).  ... 
doi:10.1142/s0219887805000594 fatcat:ghjkfx446zdflecr4z6yomb7iq

A Conformal Wave Equation

O. Veblen
1935 Proceedings of the National Academy of Sciences of the United States of America  
The wave equation which we are seeking should be a linear combination of these covariant derivatives with appropriate matrices as coefficients.  ...  Thus, when dealing with a homology theory involving a direct homomorphism sequence, the integers form the natural coefficient group; and, when dealing with a homology theory involving an inverse homomorphism  ... 
doi:10.1073/pnas.21.7.484 pmid:16588007 pmcid:PMC1076633 fatcat:er6gsrjxyngm7kw4auzxouvb3e

Page 5086 of Mathematical Reviews Vol. , Issue 82m [page]

1982 Mathematical Reviews  
The main results concern relations between solutions of a pair of tensor products of systems of linear homogeneous inequalities and solutions of tensor factor systems of matrix equations and vice versa  ...  matrices A and B to establish some relationships between tensor factor systems of linear homogeneous inequalities and tensor products of matrices.  ... 

Multilinear Algebras and Tensors with Vector Subbundle of Manifolds

Khondokar M Ahmed, Saraban Tahora
2015 Dhaka University Journal of Science  
A Theorem 1.32. is established by using sections and fibrewise orthogonal sections of an application of Gran-Schmidt.  ...  In the present paper some aspects of tensor algebra, tensor product, exterior algebra, symmetric algebra, module of section, graded algebra, vector subbundle are studied.  ...  Finally, this induced homomorphism makes a covariant function from the category of −modules −linear maps to the category of graded algebras over and graded algebra homomorphisms. Definition 1.16.  ... 
doi:10.3329/dujs.v62i1.21957 fatcat:xed2xez5jzc7llmldny6r7fhvm
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