MSRI-UP 2008: Experimental Mathematics
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The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computational power and accessibility, the view that "real mathematicians do not compute" no longer exists for the current generations. In this program the students will take real advantage of the computational tools that exists in symbolic languages like Mathematica and Maple to investigate interesting problems most of which come from the question of evaluation of definite integrals. As background the applicants must have a solid knowledge of one variable Calculus. Some experience with Discrete Mathematics and Linear Algebra would be helpful but it is not essential. The program will show how computation is used to gain insight and intuition in Mathematics. We will use it to discover new facts, patterns, and relationships. In particular we will show how Analysis, Discrete Mathematics and Computations are just different aspects of the same science: Mathematics. Projects. The first two weeks of the program will be devoted to prepare the students for the most interesting part: the projects. These are mathematical problems for which the instructors and assistants have some ideas on how to solve them, but they are open problems. Our past experience has shown that students will provide unexpected insight into these problems. Here are some examples to show how exciting they could be:
- An elementary calculation gives the integral of 1/(x² + 1)m over the positive real line. Express the integral of 1/(x4 + 2ax² + 1)m as a function of the parameters a and m. The result will involve a polynomial of degree m with rational coefficients. The project consists in exploring the factroization of these coefficients as products of primes. Many beautiful patterns will appear, most of them without a traditional proof. The paper "The 2-adic valuation of a sequence arising from a rational integral" can be downloaded from this website.
- The Stirling numbers S(n, k) count the number of ways to partition n objects into k non-empty parts. These numbers are integers, because they count something. What can you say about the power of 2 that divides them? The paper "The 2adic valuation of Stirling numbers" present interesting conjectures and beautiful pictures.
- The recurrence x[n] = (n + x[n - 1])/(1 - nx[n - 1]) comes from a simple finite sum of values of the arctangent function. Starting at x = 0 you will see that x[n] is an integer for n ≤ 4. We have conjectured that this never happens again. The paper "Arithmetical properties of a sequence arising from an arctangent sum" contains some dynamical systems that needs to be explored.
- Why is it that the sums [graphic missing] are easy to evaluate, but the one with the cubes of binomial coefficients does not appear in elementary texts?
After a numerical calculation, you find that the answer to your problem is
s = 10.56275158164930392825
What is the real answer? We will learn how to figure out that it has be π2 + ln 2. This is a remarkable new insight: from a numerical approximation, we get the exact answer.