Related papers: Tensor computations in computer algebra systems
We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable…
We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor…
In the present paper, we describe some experiences in using programming, commands and graphical interfaces based on computer algebra systems, as tools for learning Physics and Mathematics.
Efficient simulation of quantum computers is essential for the development and validation of near-term quantum devices and the research on quantum algorithms. Up to date, two main approaches to simulation were in use, based on either full…
GPL Maxima is an open-source computer algebra system based on DOE-MACSYMA. GPL Maxima included two tensor manipulation packages from DOE-MACSYMA, but these were in various states of disrepair. One of the two packages, CTENSOR, implemented…
Computer Algebra Systems (CASs) like Cadabra Software play a prominent role in a wide range of research activities in physics and related fields. We show how Cadabra language is easily implemented in the well established Python programming…
We use computational algorithms recently developed by us to study completely four index divergence free quadratic in Riemann tensor polynomials in GR. Some results are new and some other reproduce and/or correct known ones. The algorithms…
The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $\F$ is at most $k$ over the algebraic closure of $\F$, where $k$ is a given positive integer. We estimate the arithmetic…
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…
Quantum computers are expected to enable fast solving of large-scale combinatorial optimization problems. However, their limitations in fidelity and the number of qubits prevent them from handling real-world problems. Recently, a…
While Spectral Methods have long been used for Principal Component Analysis, this survey focusses on work over the last 15 years with three salient features: (i) Spectral methods are useful not only for numerical problems, but also discrete…
Tensor methods have become a promising tool to solve high-dimensional problems in the big data era. By exploiting possible low-rank tensor factorization, many high-dimensional model-based or data-driven problems can be solved to facilitate…
A method of fast linear transform algorithm synthesis for an arbitrary tensor, matrix, or vector is proposed. The method is based on factorization of a tensor and using the factors for building computational structures performing fast…
Tensor algebra is essential for data-intensive workloads in various computational domains. Computational scientists face a trade-off between the specialization degree provided by dense tensor algebra and the algorithmic efficiency that…
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…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…
Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks…
Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees,…
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…
A systematic method is presented for the construction and classification of algebras of gauge transformations for arbitrary high rank tensor gauge fields. For every tensor gauge field of a given rank, the gauge transformation will be…