Mathematics & Statistics Department

Lots of manipulations of sounds would benefit from the right analysis of each sound into sinusoidal components with varying amplitudes and frequencies.  Windowed Fourier Transforms (WFT) generate time-frequency distributions ρ(f,t) with lovely mathematical properties, and lots of very effective uses.

Clearly, we should diddle the details of WFT until we get distributions where ρ(f,t) represents as precisely as possible what's happening at frequency f and time t. 

Or not.  I argue that the notational association of WFT data ρ(f,t) with frequency f and time t may blind us to a better use of the information.  Rather than specifying what happens at f and t, ρ(f,t) tells us something about the behavior near frequency f at a time near t.  Techniques from polynomial interpolation, used with an attitude that I got from estimation theory, offer an attractive alternative to distributions.

There is a good baccalaureate thesis in here.

Arrange some vertices in the plane, and some polygon "obstacles", and construct a graph by drawing a straight line edge between vertices whenever it doesn't intersect an obstacle.  Which graphs can be constructed in this way?  Given a graph, how many obstacles do you need?  What if the obstacles are convex polygons?  In this talk we'll answer these questions using some surprising connections to a few different subfields of graph theory.  We'll also pose a number of questions which are still open.  This work was done with undergraduate students in the Willamette Valley REU program.

A spike train metric for estimating the parameters of a model neuron

Gaurav Venkataraman

Symmetric sandpiles and dominoes on a Möbius strip

Julia Porcino, Nick Salter, Tianyuan Xu

Fingerprinting Two-dimensional Mass Spectra using Clustering Algorithms and Decision Trees

Laura Florescu

Using Bayesian Probability to Model Human Perception

David Krueger

Statistical Tests of Equivalence: An Application to cDNA Microarrays

Andrew Winterman

Innocent-sounding questions about numbers and geometry sometimes require serious mathematics to answer. Spectral theory, breaking down complex functions into fundamental “waves,” as a prism breaks down light into its constituent waves of red through violet, is one modern method for dealing with such questions. As a simple example, Fourier series of Bernoulli polynomials compute special values of Riemann’s zeta function, a function related to the distribution of prime numbers. More serious spectral theory, along with standard methods from complex analysis, gives a relationship between the number of lattice points in an expanding region in hyperbolic space and the “automorphic spectrum,” the fundamental waves in this context. As a final example, the recent work of Diaconu, Garrett, and Goldfeld uses spectral theory to produce identities involving moments of L-functions, from which they extract information about the behavior of the L-functions on the critical line, proving a subconvex bound, which, in the absence of the currently unattainable hypotheses of Lindelöf and Riemann, suffices for many number-theoretic applications.

We describe a general first order linear system of partial differential equations and discuss the formulation of an inverse problem for some parameters given measurements of some other quantities.  In many applications a simple constant parameter is not sufficient to represent the dynamics; thus we discuss what is meant by a distribution of parameters.  This approach can be effectively used in uncertainty quantification, but we will apply it in a more straight-forward manner to a particular problem where the resulting system can be efficiently simulated.  Lastly we will present numerical results for a sample inverse problem for a distribution of parameters. Aspects of this research were explored during the Math REU at OSU for the past few years.