Related papers: Tensor Calculus
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on…
These are the lecture notes for a short course on tensor categories. The coverage in these notes is relatively non-technical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
We present the basic concepts of tensor products of vector spaces, emphasizing linear algebraic and combinatorial techniques as needed for applied areas of research. The topics include (1) Introduction; (2) Basic multilinear algebra; (3)…
This note is a sequel to the previous series "Tensor Track I-III". Assuming some familiarity with the tensor track approach to quantum gravity, we provide a brief introduction to the developments of the last two years and to their…
The goal of this note is to provide a recursive algorithm that allows one to calculate the expansion of the metric tensor up to the desired order in Riemann normal coordinates. We test our expressions up to fourth order and predict results…
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also…
We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed…
An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces. This method of differentiation is known, but not explained well, in the literature.…
Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
These notes concern linear transformations on R^n and C^n, exponentials of linear transformations, and some related geometric questions.
In this paper, we analyze the existing rules for constructing derivatives of the scalar and tensor functions of the tensor argument with respect to the tensor argument and the theoretical positions underlying the construction of these…
In a previous paper a new approach has been introduced for computing, recursively and numerically, one-loop tensor integrals. Here we describe a few modifications of the original method that allow a more efficient numerical implementation…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
This is a self-contained set of lecture notes covering various aspects of the theory of open quantum system, at a level appropriate for a one-semester graduate course. The main emphasis is on completely positive maps and master equations,…
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…
These notes, connected to a "potpourri" topics class currently underway, discuss some basic topics in analysis and connections with other areas of mathematics.
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of…
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples