Computational Algebra
Integration of Rational Functions
Integration of Rational Functions
Seminar Leader/Research Advisor
Victor Hugo Moll Becker, Tulane
University
Prerequisites
One year of Calculus. Some familiarity with a
programming language. (We will use either Mathematica or Maple during
SIMU.) The seminar will include lots of numerical and symbolic work, but
the background will be developed at SIMU.
Topic Overview
In spite of the fact that a student learns how to
integrate in the first Calculus course, the problem of integration of special
functions is a source for many interesting mathematical problems. In the special
case of rational functions, that will be the topic of the Seminar, one has the
difficulty that factorization of polynomials of large degree is a difficult
problem. The students might be aware of the fact that there is no formula to
solve a quintic equation, so the method of partial fractions fails.
In the seminar we will discuss a new approach to this
question developed in
joint work with George Boros of the University of New Orleans.
Seminar and Research Topics
(Note that all
expressions are in LaTeX format. Click here
if you would like a PDF version of this description that will have the
mathematical formulas typeset.)
1. Wallis' formula: The expression
$$ \int_{0}^{\infty} \frac{dx}{(x^{2} + 1)^{m+1}} = \frac{\pi}{2^{2m+1}}
\binom{2m}{m}$$
is one of the first formulas of Calculus. We will discuss several proofs
of it
and use it as an introduction to the WZ-method developed by Wilf and
Zeilberger to prove identities on a computer. From here we will discuss
some Number Theory related to binomial coefficients.
2. A quartic integral: The formula
$$\int_{0}^{\infty} \frac{dx}{(x^{4} + 2ax^{2} + 1)^{m+1}} =
\frac{\pi}{2^{3m+3/2} (a+1)^{m+1/2}} \sum_{k=0}^{m} 2^{k}
\binom{2m-2k}{m-k} \binom{m+k}{m} (a+1)^{k}$$
appears to be new. It has some amazing connections, for example it
appears
in the Taylor series of the function $f(c) = \sqrt{a + \sqrt{1+c}}$.
3. A dynamical system: The system
$$ a_{n+1} = \frac{a_{n} b_{n} + 5a_{n} + 5b_{n} + 9}
{(a_{n} + b_{n} + 2)^{4/3}} \;\;\;
b_{n+1} = \frac{a_{n} + b_{n} + 6}
{(a_{n} + b_{n} + 2)^{2/3}} $$
appears in the evaluation of an integral of degree $6$. The iteration of this
system can be used to produce a numerical method to compute integrals. The
long time behaviour of this system seems interesting: if the initial point is
on the first quadrant, then $a_{n}, b_{n} \to 3$, but if you start somewhere
on the third quadrant it seems that one gets chaotical behavior.
4. A dynamical system with number theory flavor: The iteration of
the function
$$ \gamma_{m}(j) = m \left[\frac{j}{2} \right] - \tfrac{1}{2}(m-1)(j-1)
$$
where $m$ is an odd integer and $ 0 \leq j \leq m-2$ appeared in our attempt
to understand the integration of odd rational functions. The set $\{ 0 \leq j
\leq m-2 \}$ is partitioned into orbits when the function $\gamma_{m}$
is iterated. In some cases, there is a unique orbit, for example for $m =
4133$. It turns out that if this happens, then $m$ must be a prime number and
(we have conjectured) that $2$ is a primitive root for $m$. The converse
statement is only a conjecture: if $m$ is prime then either there is a unique
orbit, or there are many orbits but all have the same size. For example,
if $m = 4153$ there are $12$ orbits of length $346$.
Back to the top.
Computational Algebra
Seminar Leader/Research Advisor
Reinhard Laubenbacher, New Mexico State University
Topic Overview
Computational algebra is a very active and rapidly growing field, with many applications to other areas of mathematics, as well as computer science and engineering.
Easily accessible without extensive specialized mathematical expertise, the theory quickly leads to deep research problems as well as realistic applications to other subjects.
It is ideally suited to provide undergraduate students with a genuine research experience in a very limited amount of time.
Seminar Description
The topics of the pre-research seminar have been chosen to expose participants to a variety of pure and applied subjects, unified by the theme of effective computational methods based on polynomial algebra. After presenting the necessary minimum of computational algebra techniques, namely the concept of a Gröbner basis for an ideal in a polynomial ring and the Buchberger algorithm used to compute it, the seminar will move to applications. For the purpose of the seminar, computational algebra will be viewed as a tool, rather than the primary object of study.
Research Projects
The research projects lie in four research areas.
1. Primary decomposition of permanental ideals. This project is based on the recent work by Professor Laubenbacher and Irena Swanson on primary decomposition of permanental ideals. Students will explore different reduced Gröbner bases of permanental ideals obtained with respect to different term orders. Such a study would shed light on permanental ideals as well as general Gröbner bases theory. The project is open-ended as students exhaust the computational resources at their disposal fairly soon and this leads them to increasingly complicated theoretical issues.
2. Applications to the solution of polynomial systems. One of the most important applications of computational algebra techniques is to polynomial systems. Symbolic computation can be used on the one hand as a symbolic preprocessor before applying numerical methods, and on the other as a means to study systems qualitatively. A two-part research project in this area is a study of more advanced methods for solving polynomial systems. Students then apply these methods to interesting polynomial systems arising in signal processing which are difficult to solve reliably with standard numerical methods but which are amenable to a mixture of symbolic and numerical methods.
3. Applications to algebraic geometry. The basic concepts of algebraic geometry will have been introduced as part of the material on polynomial systems. Students will use symbolic computation to define and compute the dimension of an algebraic variety.
4. Applications to cryptography. Several methods have been proposed to use Gröbner bases to design secret codes. The projects will explore these methods and their feasibility.
Back to the top.
