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Symbolic computation has been widely applied to Combinatorics, Number Theory, and also other fields. Many reliable and fast algorithms with corresponding implementations now have been established and developed. Using the tool of Experimental Mathematics, especially with the help of mathematical software, in particularly Mathematica, we could visualize the data, manipulate algorithms and implementations. The work presented here, based on symbolic computation, involves the following two parts. The first part introduces a systematic integration method, called the Method of Brackets. It only consists of a small number of simple and direct rules coming from the Schwinger parametrization of Feynman diagrams. Verification of each rule makes this method rigorous. Then it follows a necessary theorem that different series representations of the integrand, though lead to different processes of computations, do not affect the result. Examples of application lead to further discussions on analytic continuation, especially on Pochhammer symbol, divergent series and connection to Mellin transform of the Method of Brackets. In the end, comparison with other integration methods and a Mathematica package manual are presented. The second part provides a symbolic approach on the study of Bernoulli numbers and its generalizations. The Bernoulli symbol $\mathcal{B}$ originally comes from Umbral Calculus, as a formal approach to Sheffer sequences. Recently, a rigorous footing by probabilistic proof makes it also a random variable with its density function a shifted hyperbolic trigonometric function. Such an approach together with general method on random variables gives a variety of results on generalized Bernoulli polynomials, multiple zeta functions, and also other related topics. / Lin Jiu

In recent years, symbolic and constraint-solving techniques have been making major advances and are continually being deployed in new business and engineering applications. A major push behind this trend has been the development and deployment of sophisticated methods that are able to comprehend and evaluate important sub-classes of symbolic problems (such as those in polynomial, linear inequality and finite domains). However, relatively little has been explored in higher-order domains, such as equations with unknown functions. This paper proposes a new symbolic method for solving a class of higher-order equations with an unknown function over the complex domain. Our method exploits the closure property of group structure (for functions) in order to allow an equivalent system of equations to be expressed and solved in the first-order setting. Our work is an initial step towards the relatively unexplored realm of higher-order constraint-solving, in general; and higher-order equational solving, in particular. We shall provide some theoretical background for the proposed method, and also prototype an implementation under Mathematica. We hope that our foray will help open up more sophisticated applications, as well as encourage work towards new methods for solving higher-order constraints. / Singapore-MIT Alliance (SMA)

Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well.

Given a matrix A in F(t)[D;\delta]^{n\times n} over the ring of differential polynomials, we first prove the existence of the Hermite form H of A over this ring. Then we determine degree bounds on U and H such that UA=H. Finally, based on the degree bounds on U and H, we compute the Hermite form H of A by reducing the problem to solving a linear system of equations over F(t). The algorithm requires a polynomial number of operations in F in terms of the input sizes: n, deg_{D} A, and deg_{t} A. When F=Q it requires time polynomial in the bit-length of the rational coefficients as well.

<p> We present a symbolic analysis of a class of while loop programs which can automatically derive a closed-form symbolic expression for the input-output relation embodied in that program.</p> <p> We show that this is especially well-suited to analyzing programs from scientific computation, in particular programs which compute special functions (like Bessel functions) from its Taylor series expansion. Other than making heavy use of algebraic manipulations, as available in any computer algebra system, we also require the use of recurrence relations. It is from these recurrence relations that we derive most of our information.</p> <p> It is important to note that we can often get interesting information about a program (like termination) without requiring closed-form solutions to the recurrences.</p> / Thesis / Master of Science (MSc)

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

The Fourier transform is one of the most useful tools in science and engineering and can be expanded to multi-dimensions and curvilinear coordinates. Multidimensional Fourier transforms are widely used in image processing, tomographic reconstructions and in fact any application that requires a multidimensional convolution. By examining a function in the frequency domain, additional information and insights may be obtained. In this thesis, the development of a symbolic computer algebra toolbox to compute two dimensional Fourier transforms in polar coordinates is discussed. Among the many operations implemented in this toolbox are different types of convolutions and procedures that allow for managing the toolbox effectively. The implementation of the two dimensional Fourier transform in polar coordinates within the toolbox is shown to be a combination of two significantly simpler transforms. The toolbox is also tested throughout the thesis to verify its capabilities.

This thesis presents a new algorithm for computing the Hermite form of a polynomial matrix. Given a nonsingular n by n matrix A filled with degree d polynomials with coefficients from a field, the algorithm computes the Hermite form of A in expected number of field operations similar to that of matrix multiplication. The algorithm is randomized of the Las Vegas type.

Groebner bases have many applications in mathematics, science, and engineering. This dissertation deals with the algorithmic aspects of computing these bases. The dissertation begins with a brief introduction of fundamental concepts about Groebner bases. Following this a discussion of various implementation issues are discussed. Much of the practical difficulties of using Groebner basis algorithms and techniques stems from the high computational complexity. It is shown that the algorithmic complexity of computing a Groebner basis primarily stems from the calculation of normal forms. This is established by studying run profiles of various computations. This leads to two options of making Groebner basis techniques more practical. They are to reduce the complexity by developing new algorithms (heuristics) or reduce running time of normal form calculations by introducing concurrency. The later approach is taken in the remainder of the dissertation where a multithreaded normal form algorithm is presented and discussed. It is shown with a simple example that the new algorithm demonstrates a speedup and scalability. The algorithm also has the advantage of being completion strategy independent. We conclude with an outline of future research involving the new algorithm.