Starting from first principles, this book covers all of the foundational material needed to develop a clear understanding of the Mathematica language, with a practical emphasis on solving problems. ...Concrete examples throughout the text demonstrate how Mathematica can be used to solve problems in science, engineering, economics/finance, computational linguistics, geoscience, bioinformatics, and a range of other fields.
The book will appeal to students, researchers and programmers wishing to further their understanding of Mathematica. Designed to suit users of any ability, it assumes no formal knowledge of programming so it is ideal for self-study. Over 290 exercises are provided to challenge the reader's understanding of the material covered and these provide ample opportunity to practice using the language. Mathematica notebooks containing examples, programs and solutions to exercises are available from www.cambridge.org/wellin.
Mathematica is today's most advanced technical computing system, featuring a rich programming environment, two-and three-dimensional graphics capabilities and hundreds of sophisticated, powerful ...programming and mathematical functions using state-of-the-art algorithms. Combined with a user-friendly interface and a complete mathematical typesetting system, Mathematica offers an intuitive, easy-to-handle environment of great power and utility."The Mathematica GuideBook for Numerics" (text and code fully tailored for Mathematica 5.1) concentrates on Mathematica's numerical mathematics capabilities. The available types of arithmetic (machine, high-precision, and interval) are introduced, discussed, and put to use. Fundamental numerical operations, such as compiling programs, fast Fourier transforms, minimization, numerical solution of equations, ordinary/partial differential equations are analyzed in detail and are applied to a large number of examples in the main text and solutions to the exercises.
We have developed a Mathematica program package SpaceGroupIrep which is a database and tool set for irreducible representations (IRs) of space group in BC convention, i.e. the convention used in the ...famous book “The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell. Using this package, elements of any space group, little group, Herring little group, or central extension of little co-group can be easily obtained. This package can give not only little-group (LG) IRs for any k-point but also space-group (SG) IRs for any k-stars in intuitive table form, and both single-valued and double-valued IRs are supported. This package can calculate the decomposition of the direct product of SG IRs for any two k-stars. This package can determine the LG IRs of Bloch states in energy bands in BC convention and this works for any input primitive cell thanks to its ability to convert any input cell to a cell in BC convention. This package can also provide the correspondence of k-points and LG IR labels between BCS (Bilbao Crystallographic Server) and BC conventions. In a word, the package SpaceGroupIrep is very useful for both study and research, e.g. for analyzing band topology or determining selection rules.
Program title:SpaceGroupIrep
CPC Library link to program files:https://doi.org/10.17632/3vm4g32t4d.1
Developer's repository link:https://github.com/goodluck1982/SpaceGroupIrep
Licensing provisions: GNU General Public Licence 3.0
Programming language: Mathematica
External routines/libraries used:spglib (http://spglib.github.io/spglib)
Nature of problem: Space groups and their representations are important mathematical language to describe symmetry in crystals. The book—“The mathematical theory of symmetry in solids” by C.J. Bradley & A.P. Cracknell (called the BC book)—is highly influential because it contains not only systematic theory but also detailed complete data of space groups and their representations. The package SpaceGroupIrep digitizes these data in the BC book and provides tens of functions to manipulate them, such as obtaining group elements and calculating their multiplications, identifying k-points, showing the character table of any little group, determining the little-group (LG) irreducible representations (IRs) of energy bands, and calculating the direct product of space-group (SG) IRs. This package is a useful database and tool set for space groups and their representations in BC convention.
Solution method: The direct data in the BC book is used to calculate the LG IRs for standard k-points defined in the book. For a non-standard k-point, we first relate it to a standard k-point by an element which makes the space group self-conjugate and then calculate the LG IRs through the element. SG IRs are obtained by calculating the induced representations of the corresponding LG IRs. The full-group method based on double coset is used to calculate the direct products of SG IRs. In addition, an external package spglib is utilized to help convert any input cell to a cell in BC convention.
•Overview: The analytical derivation are carried out for fifth-order asymptotic geometric aberrations of electron lenses.•Core findings: All analytical expressions are found out for fifth-order ...asymptotic geometric aberrations of electron lenses.•Essence of the work: To analytically derive the expressions of fifth-order asymptotic geometric aberrations of electron lenses.•Distinctive highlight: The derived formulae are numerically cross-validated with the differential algebraic method.
In this article the analytical expressions of the fifth-order asymptotic geometric aberrations of electron lenses are derived by Mathematica. The process of the derivation is analogous to the method described in “Principles of Electron Optics” by P.W. Hawkes and E. Kasper. All the analytical formulae for asymptotic aberration coefficients in polynomials in the reciprocal magnification are numerically cross-validated with the differential algebraic (DA) method. The results indicate that the derived formulae are doubtless correct.
FormTracer. A mathematica tracing package using FORM Cyrol, Anton K.; Mitter, Mario; Strodthoff, Nils
Computer physics communications,
October 2017, 2017-10-00, 2017-10-01, Letnik:
219, Številka:
C
Journal Article
Recenzirano
Odprti dostop
We present FormTracer, a high-performance, general purpose, easy-to-use Mathematica tracing package which uses FORM. It supports arbitrary space and spinor dimensions as well as an arbitrary number ...of simple compact Lie groups. While keeping the usability of the Mathematica interface, it relies on the efficiency of FORM. An additional performance gain is achieved by a decomposition algorithm that avoids redundant traces in the product tensors spaces. FormTracer supports a wide range of syntaxes which endows it with a high flexibility. Mathematica notebooks that automatically install the package and guide the user through performing standard traces in space–time, spinor and gauge-group spaces are provided.
Program Title: FormTracer
Program Files doi:http://dx.doi.org/10.17632/7rd29h4p3m.1
Licensing provisions: GPLv3
Programming language: Mathematica and FORM
Nature of problem: Efficiently compute traces of large expressions
Solution method: The expression to be traced is decomposed into its subspaces by a recursive Mathematica expansion algorithm. The result is subsequently translated to a FORM script that takes the traces. After FORM is executed, the final result is either imported into Mathematica or exported as optimized C/C++/Fortran code.
Unusual features: The outstanding features of FormTracer are the simple interface, the capability to efficiently handle an arbitrary number of Lie groups in addition to Dirac and Lorentz tensors, and a customizable input-syntax.
We present LieART 2.0 which contains substantial extensions to the Mathematica application LieART (LieAlgebras and Representation Theory) for computations frequently encountered in Lie algebras and ...representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. The basic procedure is unchanged—it computes root systems of Lie algebras, weight systems and several other properties of irreducible representations, but new features and procedures have been included to allow the extensions to be seamless. The new version of LieART continues to be user friendly. New extended tables of properties, tensor products and branching rules of irreducible representations are included in the supplementary material for use without Mathematica software. LieART 2.0 now includes the branching rules to special subalgebras for all classical and exceptional Lie algebras up to and including rank 15.
Program Title: LieART 2.0
CPC Library link to program files:http://dx.doi.org/10.17632/8vm7j67bwt.1
Licensing provisions: GNU Lesser General Public License
Programming language: Mathematica
External routines/libraries: Wolfram Mathematica 8–12
Nature of problem: The use of Lie algebras and their representations is widespread in physics, especially in particle physics. The description of nature in terms of gauge theories requires the assignment of fields to representations of compact Lie groups and their Lie algebras. Mass and interaction terms in the Lagrangian give rise to the need for computing tensor products of representations of Lie algebras. The mechanism of spontaneous symmetry breaking leads to the application of subalgebra decomposition. This computer code was designed for the purpose of Grand Unified Theory (GUT) model building, (where compact Lie groups beyond the U(1), SU(2) and SU(3) of the Standard Model of particle physics are needed), but it has found use in a variety of other applications. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU(N), SO(N) and Sp(2N) and all the exceptional groups E6, E7, E8, F4 and G2. This includes both regular and irregular (special) subgroup decomposition of all Lie groups up through rank 15, and many more.
Solution method: LieART generates the weight system of an irreducible representation (irrep) of a Lie algebra by exploiting the Weyl reflection groups, which is inherent in all simple Lie algebras. Tensor products are computed by the application of Klimyk’s formula, except for SU(N)’s, where the Young-tableaux algorithm is used. Subalgebra decomposition of SU(N)’s are performed by projection matrices, which are generated from an algorithm to determine maximal subalgebras as originally developed by Dynkin 1,2. We generate projection matrices by the Dynkin procedure, i.e., removing dots from the Dynkin or extended Dynkin diagram, for regular subalgebras, and we implement explicit projection matrices for special subalgebras.
Restrictions: Internally irreps are represented by their unique Dynkin label. LieART’s default behavior in TraditionalForm is to print the dimensional name, which is the labeling preferred by physicist. Most Lie algebras can have more than one irrep of the same dimension and different irreps with the same dimension are usually distinguished by one or more primes (e.g. 175 and 175′ of A4). To determine the need for one or more primes of an irrep a brute-force loop over other irreps must be performed to search for irreps with the same dimensionality. Since Lie algebras have an infinite number of irreps, this loop must be cut off, which is done by limiting the maximum Dynkin digit in the loop. In rare cases for irreps of high dimensionality in high-rank algebras the used cutoff is too low and the assignment of primes is incorrect. However, this only affects the display of the irrep. All computations involving this irrep are correct, since the internal unique representation of Dynkin labels is used.
We present a Mathematica program package MagneticTB, which can generate the tight-binding model for arbitrary magnetic space group. The only input parameters in MagneticTB are the (magnetic) space ...group number and the orbital information in each Wyckoff position. Some useful functions including getting the matrix expression for symmetry operators, manipulating the energy band structure by parameters, and interfacing with other software are also developed. MagneticTB can help to investigate the physical properties in both magnetic and non-magnetic system, especially for topological properties.
Program Title: MagneticTB
CPC Library link to program files:https://doi.org/10.17632/kws9xbvz3y.1
Developer's repository link:https://github.com/zhangzeyingvv/MagneticTB
Licensing provisions: GNU General Public Licence 3.0
Programming language: Mathematica
External routines/libraries: ISOTROPY (iso.byu.edu)
Nature of problem: Construct the symmetry adapted tight-binding model for the system with arbitrary magnetic space group.
Solution method: The symmetry-adapted tight-binding model in this code is based on the Bloch theorem and group representation theory.
Additional comments including restrictions and unusual features: This code supports the spinless system and spinful system with collinear or non-collinear magnetization configuration.
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
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