__Seminar Description__: The seminar will include lectures, a computer lab and research projects.
During the lectures, the students will learn background material on differential equations, diffusion, solution by the method of characteristics
and aspects of fluid dynamics. The computer lab will prepare the students
to write computer programs to solve the necessary equations and to modify other programs written in advance by the seminar leader or associate.
In addition, students will learn the software MATLAB and possibly some basic FORTRAN. Four groups of three students will work on different
assigned research projects on topics related to biological fluid motion.

Some of the material will be taken from notes developed for a
first-year graduate course in Applied Mathematics at Tulane University and also from standard textbooks in partial differential equations and
numerical analysis, such as *Partial Differential Equations * by Lawrence C. Evans and
*Introduction to Theoretical and Computational Fluid Dynamic* by Costas Pozrikidis.

__Sample Projects__: The goal of the projects is to give the students a taste of research
in computational mathematics. Some tentative projects include: (1) the analysis and evaluation of three
related deterministic diffusion methods, (2) the investigation of the swimming speed of
"tubular" organisms as a function of their thickness, (3) the extension of an existing method based on forces
to a method for problems in which the fluid motion is driven by torques, (4) a comparison of different types of flows around solid bodies using
particle methods.

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**Integrals:
An Introduction to Research in Mathematics**

** Seminar Leader/Research Advisor: ****Victor
H. Moll**, Tulane University

Click here for a PDF file containing the
description below.

__Required
Background:__
Integral Calculus (usually second semester calculus) and basic familiarity
with a computational language like *Mathematica*
of Maple. A basic course in
number theory would be helpful, but not essential.

__Seminar and Research Description:__** **The evaluation of integrals is one of the topics that
every undergraduate calculus course covers in some detail. The goal of this project
is to employ integrals as a bridge to more
sophisticated mathematics. The
central question to be explored is this: if somebody gives you an integral
to evaluate, can you do it? If
you can, what did you learn from the process and the answer? If
you do not succeed, is it because it cannot be done or you have seen the
right method? What does it
really mean that an integral cannot be evaluated?

I
hope that the examples below illustrate these points.

**1.**
Consider the simplest of all examples:

^{
}

The
proof of this is elementary: it is equivalent to the fact that

The
proof of this simple fact will remind you of the binomial theorem, and it
should inspire you to think about the definition of .
What does
mean? Or even worse, what
is ?

The
definite integral analog to the integral above is

^{
}

Now
if you differentiate this with respect to the parameter *n* you get

^{}

^{ }**2.**
During the last twenty years there has been an interesting development of *symbolic languages*.
These
are languages that allow you to compute a large number of integrals. Another interesting use of these languages is to create a large
amount of data that will help you generate interesting conjectures. We illustrate the idea with a proof of **Wallis'
formula**:

^{}

The
first approach is by **blind evaluation**. This simply means to ask the
machine to do the problem for you. If
you ask *Mathematica* for the value
of ,
it will tell you that

^{}

It
is unclear what is going one here. What is the **gamma function** that appears in the answer?
Is there a better way to find this integral? What is doing
in the answer?

If
you ask for specifics, then things get better:

^{}

Now
you see that the answer is a rational multiple of *p*,
the question is how to guess the exact form of it. One the methods that will be discussed in SIMU 2002 is that
of **recurrences**. For example, integration by parts will show you that

^{}

From
here we can guess the answer and **prove
it**!

**3.**
During
SIMU 2000 we discussed the following problem.
It is not hard to show that

^{
}

where

^{}

is
a polynomial in a with
rational coefficients. A
sequence of numbers is called **unimodal**
if they increase up to a point and then they decrease. We had observed that is,
for fixed *m*,
a unimodal sequence. A new proof by one the SIMU 2000 groups became a paper in the *Electronic
Journal of Combinatorics*. The
stronger fact that
are **log
concave**:
is
still an open question.

**4.**
Consider the iterated indefinite integrals of ln(1 + *x*):

^{}

Define
the
function that appears at the *n*th step. Then it is not hard to
show that

^{}

where
and are
polynomials in *x*.
It is easy to figure out the
value

^{}

The
value of is
very mysterious. Write

^{}

where
has
integer coefficients and

^{}

It
turns out that

^{}

^{Why? }

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