Who did what?
Lindsey Doane is a mathematics major who in Fall, 2014 carried out a research project in MTH 204, Computational Experiments in Mathematics. Her project involved estimating occurrences of Niven Numbers. A Niven number (sometimes known as a Harshad number) is a positive integer exactly divisible by the sum of its (base 10) digits. Examples are 12 and 18. More information at the Online Encyclopedia of Integer Sequences.
No prime number p greater than 10 can be a Niven number because the sum of the digits of p is a number greater than 1 and less than p (and so cannot exactly divide the prime number p).
However, the growth of the Niven numbers is tantalizingly similar to the growth of the prime numbers. The famous prime number theorem says that the number of primes less than or equal to is asymptotically . This means that the ratio of to approaches 1 as .
In a paper in 2001 and a later paper in 2003, Jean-Marie De Koninck, Nicolas Doyon, and later Imre Katai, establish there is a constant such that the number of Niven numbers less than or equal to n is asymptotically .
Lindsey investigated how to better approximate the constant , and also investigated the ratio of the number of Niven numbers up to n, to the number of prime numbers up to n (since they both grow in much the same way, yet are quite different sets of numbers).
Who’s doing what?
Sidafa Conde, a graduate student in the Engineering & Applied Sciences Ph.D. program, specializing in Scientific Computing, is working on numerical solutions of non-linear partial differential equations.
Sidafa said about his experience at U Mass Dartmouth: “I’d say that UMass Dartmouth is truly amazing. Balanced. One-of-a-kind. And actually very affordable. My experience at UMass Dartmouth has been phenomenal.That whole “world class. within reach” is definitely true. The professors care. They’re here not just to teach but to actually show us. It’s awesome. When you walk through the department, they always call you in to have a conversation, even if it’s not about classes. They ask you about your plans for the future. They always encourage you to try new opportunities.”
Who’s doing what?
Bo Dong, an Assistant Professor in the Department of Mathematics, works on finding accurate numerical solutions to certain non-linear differential equations. The differential equations she studies are closely related to the so-called Korteweg-de Vries equation (so called, because the original equation was discovered by Joseph Valentin Boussinesq in 1877 and re-discovered 18 years later by Diederik Korteweg and Gustav de Vries) which provides a mathematical model of waves on shallow water surfaces. A nice historical background to the KdV equation can de found here. Differential equations similar to the KdV equation have many applications in fluid mechanics, nonlinear optics, acoustics, and plasma physics. For example, the KdV equation has been used in the modeling of shallow water waves and the study of Tsunami waves. The study of equations of KdV type has had an enormous impact on the development of modern nonlinear mathematical science: vast areas of mathematics (ordinary differential equations, algebraic geometry, Lie group theory, differential geometry, asymptotics) and theoretical physics (quantum field theory, string and conformal field theory, quantum gravity, classical general relativity) opened up as a consequence of the basic research into the KdV type equations.
Bo uses a method for numerically approximating solutions to KdV type equations known as the discontinuous Galerkin method. (not to be confused with discontinued gherkins!). This method is named, in part, after Boris Galerkin, a Russian mathematician and engineer, who introduced numerical methods, now known by his name, for solving differential equations. The discontinuous Galerkin (DG) method was introduced in 1973 by W.H. Reed & T.R Hill, in the context of modeling neutron transport. A critical feature of the use of DG methods is that KdV type equations may develop discontinuous solutions and at those discontinuities quite complex behavior can occur, making numerical approximation a delicate issue. A very nice account of DG methods is available in: The Development of discontinuous Galerkin methods
Who’s doing what?
Jorge Fernandez is a recent UMass Dartmouth mathematics graduate. Jorge started his studies as a College Now student, and as a Junior and Senior worked on a number of research projects. One of these projects, of his own choice, involved modeling the ingress of hurricanes from water to land. After graduation Jorge applied for an internship at the Hartford, a Fortune 500 insurance company in Hartford, CT. His research work on hurricanes stood him in good stead, and he got the internship that was updated to employment as a data analyst at the end of summer. Jorge commented that he made a fundamental change in his outlook to studies when he started doing research, focusing on learning rather than on grades.
Professor Naiomi Cameron telling math jokes at an Open Math Night, November 22, 2008.
What was up with Pythagoras?