The mimetic finite difference (MFD) method mimics fundamental properties of mathematical and physical systems including conservation laws, symmetry and positivity of solutions, duality and ...self-adjointness of differential operators, and exact mathematical identities of the vector and tensor calculus. This article is the first comprehensive review of the 50-year long history of the mimetic methodology and describes in a systematic way the major mimetic ideas and their relevance to academic and real-life problems. The supporting applications include diffusion, electromagnetics, fluid flow, and Lagrangian hydrodynamics problems. The article provides enough details to build various discrete operators on unstructured polygonal and polyhedral meshes and summarizes the major convergence results for the mimetic approximations. Most of these theoretical results, which are presented here as lemmas, propositions and theorems, are either original or an extension of existing results to a more general formulation using polyhedral meshes. Finally, flexibility and extensibility of the mimetic methodology are shown by deriving higher-order approximations, enforcing discrete maximum principles for diffusion problems, and ensuring the numerical stability for saddle-point systems.
We present the tensor computer algebra package xTras, which provides functions and methods frequently needed when doing (classical) field theory. Amongst others, it can compute contractions, make ...Ansätze, and solve tensorial equations. It is built upon the tensor computer algebra system xAct, a collection of packages for Mathematica.
Program title: xTras
Catalogue identifier: AESH_v1_0
Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESH_v1_0.html
Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland
Licensing provisions: GNU General Public License, version 3
No. of lines in distributed program, including test data, etc.: 155879
No. of bytes in distributed program, including test data, etc.: 565389
Distribution format: tar.gz
Programming language: Mathematica.
Computer: Any computer running Mathematica 6 or newer.
Operating system: Linux, Unix, Windows, OS X.
RAM: 100 Mb
Classification: 5.
External routines: xACT (www.xact.es)
Subprograms used:Cat IdTitleReferenceAEBH_v1_0xPermCPC 179 (2008) 597ADZK_v2_0Invar Tensor Package 2.0CPC 179 (2008) 586Nature of problem:
Common problems in classical field theory: making Ansätze, computing contractions, solving tensorial equations, etc.
Solution method:
Various (group theory, brute-force, built-in Mathematica functions, etc.)
Running time:
1–60 s
Pauli first noticed the hidden SO(4) symmetry for the hydrogen atom in the early stages of quantum mechanics 1. Starting from that symmetry, one can recover the spectrum of a spinless hydrogen atom ...and the degeneracy of its states without explicitly solving Schrödinger's equation 2, 3. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem is well known 4, 5, its solution involves several steps of manipulating expressions with tensorial quantum operators, including simplifying them by taking into account a combination of commutator rules and Einstein's sum rule for repeated indices. Therefore, it is an excellent model to test the current status of CAS concerning this kind of quantum-and-tensor-algebra computations and to showcase the CAS technique. Generally speaking, when capable, CAS can significantly help with manipulations that, like non-commutative tensor calculus subject to algebra rules, are tedious, time-consuming and error-prone. The presentation also shows two alternative patterns of computer algebra steps that can be used for systematically tackling more complicated symbolic problems of this kind.
Program Title: Maple specific worksheet
CPC Library link to program files:https://doi.org/10.17632/knbckjrwfc.1
Developer's repository link:https://www.maplesoft.com/applications/view.aspx?SID=154764
Licensing provisions: GPLv3
Programming language: Maple (Maplesoft)
Nature of problem: Provide a general framework to handle Quantum Mechanics' non-commutative algebra systematically. Non-commutative tensor calculus, subject to algebra rules, is tedious, time-consuming and error-prone. This Maple worksheet provides a general framework to implement, test and address a wide range of non-commutative problems commonly found in advanced quantum mechanics. It can be customized and adapted to potentially more complicated algebra problems.
Solution method: Make use of Maple's Physics package. First, configure the problem parameters: space dimension, tensor objects and non-commutative quantities. Then provide the most basic commutation rules, such as position, linear and angular momentum commutators. From there, using the Simplify command and a short set of other key commands, demonstrate further commutation rules until the complete target algebra is reconstructed. One should notice that the Physics package commands employed here take into account custom algebra rules, the sum rule for repeated indices, and use tensor-simplification algorithms.
Additional comments including restrictions and unusual features: This article mirrors a Maple worksheet. As such, it is not an usual code intended to make a computation for a given set of parameters. It is rather a general template or guideline, useful to extend the present problem with new computations or, more generally, widely open to tackle new quantum mechanics problem involving tensor and commutator heavy calculus.
Ideals of spaces of degenerate matrices Vill, Julian; Michałek, Mateusz; Taveira Blomenhofer, Alexander
Linear algebra and its applications,
09/2022, Letnik:
648
Journal Article
Recenzirano
Odprti dostop
The variety Singn,m consists of all tuples X=(X1,…,Xm) of n×n matrices such that every linear combination of X1,…,Xm is singular. Equivalently, X∈Singn,m if and only if det(λ1X1+…+λmXm)=0 for all ...λ1,…,λm∈Q. Makam and Wigderson 12 asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Singn,m lies in the ideal generated by the polynomials det(λ1X1+…+λmXm). We answer this question in the negative by determining the vanishing ideal of Sing2,m for all m∈N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m≥n2−n+1, we prove there are equations vanishing on Singn,m that are not in the ideal generated by polynomials of type det(λ1X1+…+λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.
This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order ...tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.
Least-mean-squares (LMS) algorithms constitute a prevalent approach to implement the linear adaptive filters whose coefficients can be updated sample by sample so as to track time-varying dynamics. ...As the memory and computational complexities required for the realization of LMS filters are very low, they have been widely adopted in many real-time signal processing applications. The input of any conventional LMS filter has to be a sequence of scalar samples (one-dimensional time series), whereas such assumption is too restrictive nowadays for multi-channel (high-dimensional) signals and multi-relational data in the rise of a big-data era. It is crucial to deal with high-dimensional data-arrays, a.k.a. tensors, to manifest the variety and complex interrelations of data. Owing to lack of a sufficient mathematical framework to govern relevant tensor operations, the general tensor LMS filter, whose input is allowed to be an arbitrary tensor, has never been established for realization to the best of our knowledge. In this work, we will dedicate a new mathematical framework for tensors to establish the general tensor least-mean-squares (TLMS) filter theory and propose two novel TLMS algorithms with update rules based on stochastic gradient-descent and Newton's methods, respectively. Furthermore, as we establish the tensor calculus theory, the performance evaluation on convergence-rate and misadjustment for our proposed TLMS filters can be conducted. Finally, the memory and computational complexities of the new TLMS algorithms are also studied in this paper.
Одним из основных подходов к обработке, анализу и визуализации геофизических данных является применение геоинформационных систем и технологий, что обусловлено их геопространственной привязкой. Вместе ...с тем, сложность представления геофизических данных связана с их комплексной структурой, предполагающей множество составляющих, которые имеют одну и ту же геопространственную привязку. Яркими примерами данных такой структуры и формата являются гравитационные и геомагнитные поля, которые в общем случае задаются трех и четырехкомпонентными векторами с разнонаправленными осями координат. При этом на сегодняшний день отсутствуют решения, позволяющие визуализировать указанные данные в комплексе, не декомпозируя их на отдельные скалярные значения, которые, в свою очередь, могут быть представлены в виде одного или многих пространственных слоев. В этой связи в работе предложена концепция, использующая элементы тензорного исчисления для обработки, хранения и визуализации информации такого формата. Формализован механизм тензорного представления компонент поля с возможностью его комбинирования с другими данными такого же формата, с одной стороны, и свертки при сочетании с данными более низкого ранга. На примере гибридной реляционно-иерархической модели данных предложен механизм хранения информации по тензорным полям, предусматривающий возможность описания и применения инструкций по трансформации при переходе между различными системами координат. В работе рассматривается применение подхода при переходе от декартовой к сферической системе координат при представлении параметров геомагнитного поля. Для комплексной визуализации параметров тензорного поля предложен подход, основанный на применении тензорных глифов. В качестве последних при этом используются суперэллипсы с осями, соответствующими рангу тензора. При этом атрибутивные значения предлагается визуализировать относительно осей графического примитива таким образом, что распределение данных может быть задано посредством варьирования градиента монохромного представления параметра вдоль оси. Работоспособность концепции была исследована в ходе сравнительного анализа тензорного подхода с решениями, основанными на скалярной декомпозиции соответствующих комплексных значений с последующим их представлением в виде одного или многих пространственных слоев. Проведенный анализ показал, что применение предложенного подхода позволит в значительной степени повысить наглядность формируемого геопространственного изображения без необходимости сложного перекрывания пространственных слоев.
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