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Practical linear algebra : a geometry toolbox
\"Practical Linear Algebra covers all the concepts in a traditional undergraduate-level linear algebra course, but with a focus on practical applications. The book develops these fundamental concepts in 2D and 3D with a strong emphasis on geometric understanding before presenting the general (n-dimensional) concept. The book does not employ a theorem/proof structure, and it spends very little time on tedious, by-hand calculations (e.g., reduction to row-echelon form), which in most job applications are performed by products such as Mathematica. Instead the book presents concepts through examples and applications. \"-- Provided by publisher.
Book
SymPy: symbolic computing in Python
SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
Journal Article
Multilinear subspace learning : dimensionality reduction of multidimensional data
\"Due to advances in sensor, storage, and networking technologies, data is being generated on a daily basis at an ever-increasing pace in a wide range of applications, including cloud computing, mobile Internet, and medical imaging. This large multidimensional data requires more efficient dimensionality reduction schemes than the traditional techniques. Addressing this need, multilinear subspace learning (MSL) reduces the dimensionality of big data directly from its natural multidimensional representation, a tensor. Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data gives a comprehensive introduction to both theoretical and practical aspects of MSL for the dimensionality reduction of multidimensional data based on tensors. It covers the fundamentals, algorithms, and applications of MSL. Emphasizing essential concepts and system-level perspectives, the authors provide a foundation for solving many of today's most interesting and challenging problems in big multidimensional data processing. They trace the history of MSL, detail recent advances, and explore future developments and emerging applications.The book follows a unifying MSL framework formulation to systematically derive representative MSL algorithms. It describes various applications of the algorithms, along with their pseudocode. Implementation tips help practitioners in further development, evaluation, and application. The book also provides researchers with useful theoretical information on big multidimensional data in machine learning and pattern recognition. MATLAB source code, data, and other materials are available at www.comp.hkbu.edu.hk/haiping/MSL.html\"-- Provided by publisher
Book
The stability of mathematics students’ beliefs about working with CAS
In Victoria, Australia, senior secondary mathematics students are expected to use technology and thus need to make decisions about using pen-and-paper (P&P) or technology when solving mathematics problems. The predominant technology is a Computer Algebra System (CAS). This study investigated the beliefs about CAS held by twelve Year 11 students as they learnt to use CAS and whether these beliefs were stable over time. These students held a range of beliefs related to the usefulness of CAS, speed of CAS compared to P&P, whether CAS is proper mathematics, choice of CAS or P&P, ease of use, the correctness of answers and solving problems in Mathematical Methods (i.e. the mathematics subject studied). Beliefs are often described as being stable (e.g. McLeod, 1992), but some researchers stress stability needs to be determined empirically rather than being seen as a characteristic of beliefs (e.g. Liljedahl et al., 2012). For this sample of students, stability (rather than instability) is a feature of students’ beliefs about CAS.
Journal Article
Recent developments in using digital technology in mathematics education
In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field.
Journal Article
A space-time calculus based on symmetric 2-spinors
In this paper we present a space-time calculus for symmetric spinors, including a product with a number of index contractions followed by symmetrization. As all operations stay within the class of symmetric spinors, no involved index manipulations are needed. In fact spinor indices are not needed in the formalism. It is also general because any covariant tensor expression in a 4-dimensional Lorentzian spacetime can be translated to this formalism. The computer algebra implementation
SymSpin
as part of
xAct
for
Mathematica
is also presented.
Journal Article
Lower Bounds for the Rank of a Matrix with Zeros and Ones outside the Leading Diagonal
We found a lower bound on the rank of a square matrix where every entry in the leading diagonal is neither zero nor one and every entry outside the leading diagonal is either zero or one. The rank of this matrix is at least half its order. Under an additional condition, the lower bound is higher by one. This condition means that some auxiliary system of linear equations has no binary solution. Some examples are provided that show that the lower bound can be achieved. This lower bound on the matrix rank allows the problem of finding a binary solution to a system of linear equations with a sufficiently large number of linearly independent equations to be reduced to a similar problem in a smaller number of variables. Restrictions on the existence of a large set of solutions are found, each differing from the binary one by the value of one variable. In addition, we discuss the possibility of certifying the absence of a binary solution to a large system of linear algebraic equations. Estimates of the time required for calculating the matrix rank in the SymPy computer algebra system are also provided. It is shown that the rank of a matrix over the field of residues modulo prime number is calculated faster than it generally takes to calculate the rank of a matrix of the same order over the field of rational numbers.
Journal Article
ReLie: A Reduce Program for Lie Group Analysis of Differential Equations
Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions.
Journal Article
Mathematica as Equation Solver in Chemistry
We describe a graphical method for solving a wide range of algebraic equations in one unknown. The method can be used to solve problems in acid/base equilibria, properties of real gases, general equilibria, solubility products and chemical kinetics. The graphical method employs Mathematica software to solve equations by plotting two curves. This is different to current practices which use numerical (iterative) method or complex formulae. The graphical method is easy to apply and requires minimal mathematical manipulations. It introduces students to the use of powerful computer algebra software. The method works not only for polynomial type equations (often encountered in acid/base or general equilibria or real gas equations), but also for non-polynomial equations involving exponential terms. The graphical method implemented by Mathematica represents a \"graphing calculator\". The calculator is a general equilibrium solver for single equilibrium systems and allows students to calculate equilibrium concentrations without recourse to complex analytical formulae or numerical methods. Keywords: teaching physical chemistry or general chemistry, calculator-based learning, computer algebra software.
Journal Article
Semantic preserving bijective mappings for expressions involving special functions between computer algebra systems and document preparation systems
PurposeModern mathematicians and scientists of math-related disciplines often use Document Preparation Systems (DPS) to write and Computer Algebra Systems (CAS) to calculate mathematical expressions. Usually, they translate the expressions manually between DPS and CAS. This process is time-consuming and error-prone. The purpose of this paper is to automate this translation. This paper uses Maple and Mathematica as the CAS, and LaTeX as the DPS.Design/methodology/approachBruce Miller at the National Institute of Standards and Technology (NIST) developed a collection of special LaTeX macros that create links from mathematical symbols to their definitions in the NIST Digital Library of Mathematical Functions (DLMF). The authors are using these macros to perform rule-based translations between the formulae in the DLMF and CAS. Moreover, the authors develop software to ease the creation of new rules and to discover inconsistencies.FindingsThe authors created 396 mappings and translated 58.8 percent of DLMF formulae (2,405 expressions) successfully between Maple and DLMF. For a significant percentage, the special function definitions in Maple and the DLMF were different. An atomic symbol in one system maps to a composite expression in the other system. The translator was also successfully used for automatic verification of mathematical online compendia and CAS. The evaluation techniques discovered two errors in the DLMF and one defect in Maple.Originality/valueThis paper introduces the first translation tool for special functions between LaTeX and CAS. The approach improves error-prone manual translations and can be used to verify mathematical online compendia and CAS.
Journal Article