Related papers: Iterated Differential Forms I: Tensors
We present a tensor calculus for exceptional generalised geometry. Expressions for connections, torsion and curvature are given a unified formulation for different exceptional groups E_n(n). We then consider "tensor gauge fields" coupled to…
There is a need in general relativity for a consistent and useful mathematical theory defining the multiplication of tensor distributions in a geometric (diffeomorphism invariant) way. Significant progress has been made through the concept…
Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$…
The concept of a crossed tensor product of algebras is studied from a few points of views. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative…
A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
In this paper we introduce a class of mathematical objects called \emph{extensors} and develop some aspects of their theory with considerable detail. We give special names to several particular but important cases of extensors. The…
In this paper we propose a general spectral theory for tensors. Our proposed factorization decomposes a tensor into a product of orthogonal and scaling tensors. At the same time, our factorization yields an expansion of a tensor as a…
We investigate the algebraic complexity of tensor calulus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture VP, the class of polynomial families efficiently computable…
Random tensors are the natural generalization of random matrices to higher order objects. They provide generating functions for random geometries and, assuming some familiarity with random matrix theory and quantum field theory, we discuss…
We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the…
Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…
High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional…
We prove that any derived equivalence between derived discrete algebras is standard, i.e.\ is isomorphic to the derived tensor product by a two-sided tilting complex.
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. This extends the familiar notion that matrices form the set of all linear…
These notes are the second part of the tensor calculus documents which started with the previous set of introductory. In the present text, we continue the discussion of selected topics of the subject at a higher level expanding, when…
Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups…
Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its…
We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…