Research Project 1: Numerical differentiation and integration
Research Project 2: Solving nonlinear equations using Newton’s method
Boundary layer problems for differential equations
Research Project: Boundary layer resolving spectral method
Consider the differential equation
on the interval with .
There has been substantial progress in the development of numerical methods for the solution of such problems in the last few decades.
A small region near the ends of the interval where the slope of the solution curve is changing most rapidly is called the boundary layer.
A collocation method is a method for the numerical solution of differential equations is based on choosing a collection of candidate solutions, usually polynomials up to a certain degree, and a number of points in the domain (called collocation points), and to select that solution which satisfies the given equation at the collocation points.
- T.-W. Tee and L. N. Trefethen, “A rational spectral collocation method with adaptively transformed Chebyshev grid points“, SIAM J. Sci. Comp., Vol. 28 (2006), No. 5, pp. 1798-1811.
- Michael S. Floater & Kai Hormann, “Barycentric rational interpolation with no poles and high rates of approximation“, Numerische Mathematik, 2007
- John T. Anderson Jr, ” Ludwig Prandtl’s boundary layer“, Physics Today, pp. 42-48, December 2005
Computational Number Theory
Adviser: Professor Gary Davis
An excellent – and free – reference for background to research projects in this area is Victor Shoup’s book: A Computational Introduction to Number Theory and Algebra. Carl Pomerance has an excellent, but more advanced, review of big research problems in computational number theory.
Research project 1: Searching for Catalan pseudoprimes
The Catalan numbers arise in a wide variety of counting problems. The nth Catalan number Cn turns out to be , where is the middle binomial number: the number of ways of choosing n objects from 2n.
In 2008, Christian Aebi and Grant Cairns discovered a connection between Catalan numbers and prime numbers:
Theorem: If p is an odd prime then (-1)(p-1)/2 C(p-1)/2 = 2 (mod p)
[Christian Aebi &Grant Cairns, Catalan Numbers, Primes and Twin Primes, Elemente der Mathematik 63 (2008), no. 4, 153–164]
This says that for prime numbers the remainder when (-1)(p-1)/2 C(p-1)/2 is divided by p is 2.
This test allows us to tell which numbers p are not prime numbers. However, Aebi & Cairns theorem does not say that if (-1)(p-1)/2 C(p-1)/2 = 2 (mod p) then p is prime. Indeed there are counterexamples to this: p = .
Numbers p >2 for which p is not prime but (-1)(p-1)/2 C(p-1)/2 = 2 (mod p) are called Catalan pseudoprimes by Aebi & Cairns.
So 5907 is a Catalan pseudoprime.
Are there others?
Yes: 1194649 = 10932 and 12327121 = 35112 are both Catalan pseudoprimes.
These are the only Catalan pseudoprimes Aebi & Cairns could find.
Through a computational search they showed that there are no Catalan pseudoprimes less than 1010 of the form pq, where p, q are distinct primes.
The aim of this project is to search for Catalan pseudoprimes, or to show that there are no more.
Reference: Catalan pseudoprimes
Research project 2: Computational evidence for an inequality It is widely believed, but still not known, if the fractional part of , namely , satisfies the inequality: for all
If this inequality is true then a famous problem of number theory – Waring’s problem – is known to be true.
The inequality seems innocent enough, yet no-one to date has been able to prove it is true.
The aim of this project is to computationally quantify how close the inequality above comes to failing, and, if possible, to find a counterexample.
Research project 3: Zeroes in the digits of powers of 2
It is suspected that 86 is the last n for which does not have a (decimal) digit equal to 0.
Showing that always has a digit equal to 0 for seems to be beyond the tools of present day mathematics.
However, we can statistically investigate a stronger question: how many zeroes are there in the digits of ?
A simple model is that should have about of its digits equal to 0, at least for large n.
The number of digits of is about for large n, so we suspect that should have about of its digits equal to 0 for large n.
Computations indicate that the number of zeroes in is where the “error” is fairly well behaved.
The aim of this project is to get tight bounds on the “error” and to potentially prove the estimate for the number of zeroes in .
Reference: How many zeros are there in 2^n?
Discrete Dynamical Systems
Research Project 1: Random Fibonacci-type sequences
Adviser: Professor Alfa Heryudono
The Fibonacci sequence is defined recursively by:
It is known – and not hard to show – that the terms increase exponentially at the rate of where .
If we modify the recursive definition of the Fibonacci sequence to: where each sign is chosen independently and randomly at each iteration with equal probabilities for + and for -, then these new sequences (there are many, dependent on how the + and – signs pan out) also increase exponentially in size – i.e. increases exponentially – at a rate where , with probability 1 (i.e. “almost all” of them do so).
- Viswanath, D. (1999), ”Random Fibonacci sequences and the number 1.13198824”, Mathematics of Computation 69 (231): 1131-1155.
- Ivars Peterson, “Fibonacci at random: Uncovering a new mathematical constant”, Science News, June 12, 1999, Vol. 155 No. 24 p. 376
Now consider the random sequence where the sign + or − is chosen independently with equal probability at random at every iteration.
It turns out that these sequences decay exponentially as with probability 1 for and growexponentially for where is known as the Embree-Trefethen constant. Reference:
- Embree, M. and Trefethen, L. N. (1999), ”Growth and decay of random Fibonacci sequences”, Proceedings of the Royal Society 455 (455): 2471-2485.
This project is concerned with modifying recurrence formula by choosing + and – signs, as above, according to different probability distributions, and studying the Lyapunov constant which measures the growth of solutions .
Random Fibonacci-type sequences are important in many fields such as dynamical systems, statistical mechanics, spectral theory, and condensed matter physics. We will pick up some applications based on students’ interests.
- Eran Makover and Jeffrey McGowan, “An elementary proof that random Fibonacci sequences grow exponentially“, 2008
- Divakar Viswanath, “Lyapunov exponents from random Fibonacci sequences to the Lorenz equations”, Ph.D. thesis, August 1998
- Benoit Rittaud, “On the average growth of random Fibonacci sequences“, Journal of Integer Sequences, Vol. 10 (2007)
- Elise Janvresse, Benoît Rittaud & Thierry De La Rue, “Growth rate for the expected value of a generalized random Fibonacci sequence“, Journal of Physics A, Mathematical and Theoretical, 42 (2009)
- Elise Janvresse, Benoît Rittaud & Thierry De La Rue, “Almost-sure Growth Rate of Generalized Random Fibonacci sequences“, Annales de l’IHP – Probabilités et Statistiques 46, 1 (2010), 135-158
- Elise Janvresse, Benoît Rittaud & Thierry De La Rue, “How do random Fibonacci sequences grow?“, Probability Theory and Related Fields Volume 142, 3-4 (2008) 619-648
Research Proejct 2: Integer values of recurrences
Adviser: To be determined
Victor Moll writes:
“The recurrence comes from a simple finite sum of values of the arctangent function. Starting at you will see that . We have conjectured that this never happens again.
The paper “Arithmetical properties of a sequence arising from an arctangent sum” (see below) contains some dynamical systems that needs to be explored.” The aim of his project is to explore this conjecture computationally.
In particular, what is the behavior of the sequence of fractional parts of the defined in the Amdeberhan et al(2007) article?
A plot of shows two values of for which is small, namely
However, a check reveals that and .
A plot of reveals spots, increasingly rare, where
References & readings
- Tewodros Amdeberhan, Luis A. Medina & Victor H. Moll(2007) Arithmetical properties of a sequence arising from an arctangent sum. J. Number Theory (2007)(arithmetical _ properties _ of _a_ sequence _ arising _ from _ an _ arctangent _ sum)