Abstract:
In CAGD/CAD research and education, users are involved with development of mathematical algorithms and followed by the analysis of the resultant algorithm. This process involves geometric display which can only be carried out with high end graphics display. There are many approaches practiced and one of the so-called easiest approaches is by using C/C++ programming language and OpenGL application program interface, API. There are practitioners uses C/C++ programming language to develop the algorithms and finally utilize AutoCAD for graphics display. On the other hand, high end CAD users manage to use Auto Lisp as their programming language in AutoCAD. Nevertheless, these traditional ways are definitely time consuming. This paper introduces an alternative method whereby the practitioners may maximize scientific computation programs, SCPs: Mathematica and MATLAB in the context of CAGD/CAD for research and education.

Abstract:
We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

Abstract:
Symbolic integration is an important module of a typical Computer Algebra System. As for now, Mathematica, Matlab, Maple and Sage are all mainstream CAS. They share the same framework for symbolic integration at some points. In this book first we review the state of the art in the field of CAS. Then we focus on typical frameworks of the current symbolic integration systems and summarize the main mathematical theories behind these frameworks. Based on the open-source computer algebra system maTHmU developed by our team in our university, we propose a potential framework to improve the performance of the current symbolic integration system.

Abstract:
We present a \emph{Mathematica} notebook allowing for the symbolic calculation of the $3\times3$ dielectric tensor of a electron-beam plasma system in the fluid approximation. Calculation is detailed for a cold relativistic electron beam entering a cold magnetized plasma, and for arbitrarily oriented wave vectors. We show how one can elaborate on this example to account for temperatures, arbitrarily oriented magnetic field or a different kind of plasma.

Abstract:
In many-particle problems involving interacting fermions or bosons, the most natural language for expressing the Hamiltonian, the observables, and the basis states is the language of the second-quantization operators. It thus appears advantageous to write numerical computer codes which allow the user to define the problem and the quantities of interest directly in terms of operator strings, rather than in some low-level programming language. Here I describe a Mathematica package which provides a flexible framework for performing the required translations between several different representations of operator expressions: condensed notation using pure ASCII character strings, traditional notation ("pretty printing"), internal Mathematica representation using nested lists (used for automatic symbolic manipulations), and various higher-level ("macro") expressions. The package consists of a collection of transformation rules that define the algebra of operators and a comprehensive library of utility functions. While the emphasis is given on the problems from solid-state and atomic physics, the package can be easily adapted to any given problem involving non-commuting operators. It can be used for educational and demonstration purposes, but also for direct calculations of problems of moderate size.

Abstract:
In this paper, the author present reliable symbolic algorithms for solving a general bordered tridiagonal linear system. The first algorithm is based on the LU decomposition of the coefficient matrix and the computational cost of it is O(n). The second is based on The Sherman-Morrison-Woodbury formula. The algorithms are implementable to the Computer Algebra System (CAS) such as MAPLE, MATLAB and MATHEMATICA. Three examples are presented for the sake of illustration.

Abstract:
In this paper, some unsolved problems of the Mathematica software package are documented. At first, it is shown, using a number of examples, that the processing (simplification) of rational-fractional expressions involving powers in the general form, has been implemented in Maple more carefully than in Mathematica. Then, an error in Mathematica is demonstrated, leading to incorrect results at a change of a function’s body. For each example of symbolic or symbolic-numeric computations, alternate routes for solving the emerged problem are proposed (where possible). An added problem, related to Mathematica’s computing speed is documented, employing examples related to two-dimensional gas dynamics problems. It is shown that Mathematica computes rather slowly (about one thousand times slower) compared to a Fortran code.

Abstract:
functionalObjects.h allows the C++ programmer performing common mathematical calculations to use a more symbolic syntax rather than an algorithmic syntax. This is not as ambitious as a symbolic manipulation program such as Mathematica; it is more like having the ability to drop a very simple Mathematica statement into a C++ program.

Abstract:
The theme of symbolic computation in algebraic categories has become of utmost importance in the last decade since it enables the automatic modeling of modern algebra theories. On this theoretical background, the present paper reveals the utility of the parameterized categorical approach by deriving a multivariate polynomial category (over various coefficient domains), which is used by our Mathematica implementation of Buchberger's algorithms for determining the Groebner basis. These implementations are designed according to domain and category parameterization principles underlining their advantages: operation protection, inheritance, generality, easy extendibility. In particular, such an extension of Mathematica, a widely used symbolic computation system, with a new type system has a certain practical importance. The approach we propose for Mathematica is inspired from D. Gruntz and M. Monagan's work in Gauss, for Maple.

Abstract:
The current paper is mainly devoted to construct a generalized symbolic Thomas algorithm that will never fail. Two new efficient and reliable computational algorithms are given. The algorithms are suited for implementation using computer algebra systems (CAS) such as Mathematica, Macsyma and Maple. Some illustrative examples are given.