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University of Alaska Fairbanks
Department of Mathematical Sciences

2004-2005 Colloquium Series

Colloquiums are held on Thursdays at 1:00 in Chapman 106.

Next Colloquium

Schedule

Abstracts

Existence (or Not?) of a Solution For Some Two-Point Boundary Value Problems in Ordinary Differential Equations

Marvin C. Papenfuss
Department of Mathematical Sciences
University of Alaska Fairbanks

The many areas of research in ordinary differential equations are often grouped into the three general categories of existence, uniqueness, and solution techniques. Although the general second order scalar ordinary differential equations may appear to be quite innocent, with these three general questions completely answered, this is far from true.

In this talk we look at certain classes of two-point boundary value problems for which we investigate the existence and uniqueness questions, and primarily the existence question. In the first half of the talk we will consider several global existence and uniqueness results, where various types of growth assumptions are made for the differential equation. Some of these are shown to be best possible results for that class of problems. A'priori bounds for the solution are then presented.

In the second half of the talk we present a curious existence and uniqueness result from the literature, which then motivates the question of non-existence of a solution. A'priori bounds are used to obtain some local assumptions that give sufficient conditions as well as some that give necessary conditions for existence of a solution. The open question that remains is whether or not both a necessary and sufficient condition for existence can be found for a particular class of problems. Examples will be given showing that existence is denied for some time intervals and assured for other time intervals. Those in between give us an uncertainty interval, which we would like to make as small as possible.

I encourage students who have had at least a first course in differential equations to attend. Although specialized results will be presented, the underlying concepts are quite elementary.

Two Approaches To The Problem Of The Controllability of Non-Autonomic Equations

Igor S. Filippov and Victor S. Mikhaylov
Department of Mathematical Sciences
University of Alaska Fairbanks

Controllability of an Elastic Ring
Igor S. Filippov

We consider a flexible ring as the simplest model of a blood vessel. Specifically, we study the exact controllability problem for a ring under stretching tension that varies slowly in time. We are looking for a couple of forces g(x) f1(t) and g(x) f2(t), which drive the state solution to rest and explain why one force would not be enough. The controllability problem is reduced to a moment problem for the controlling forces f1(t) and f2(t). We describe the set of initial data which may be driven to rest by a control forces. The description is obtained in terms of the Fourier coefficients of the initial data.

Controllability of an Elastic String
Victor S. Mikhaylov

We consider the problem of controllability of a string under the axial tension that depends on time. Mathematically this problem means the controllability of the wave equation in the case when density of the string depends on the point of the string and on time. It occurred that the most used way of solving such kind of problems, i.e. Fourier method, yields additional restrictions on the tension. To avoid this, we apply the method of characteristics (D. Russell, 72). This method reduces the second order hyperbolic equation to the system of ordinary differential equations, given on the characteristics of the original string equation. The method gives also the possibility of the numerical simulation of the control.

Wiener-Hopf Operators in Signal Processing

Anna S. Bulanova
Department of Mathematical Sciences
University of Alaska Fairbanks

We address the problem of construction of sampling and interpolation sets for multi-band signals, that is signals whose spectra lie in a finite union of disjoint intervals. This problem and its solution are of fundamental significance in signal processing and have numerous applications in physics, communications, and defense.

Almost all processing of signals of all kinds is ultimately done in the so-called digital layer. On the other hand the transmission and reception process of almost all signals requires the signals to be analog. The key link between the analogue and digital layers is the analog-to-digital converter. This samples analog (continuous) data to produce a discrete sequence of values. In order to minimize the amount of data produced by this process, it is important not to sample the analog data at higher rates than are required to extract all of the significant information from the signals.

Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence and prove the invertibility of this operator.

Boundary Inverse Problems in Glaciology

Valeriy Groshev and Elchin Jafarov
Department of Mathematical Sciences
University of Alaska Fairbanks

The basal boundary of glaciers and ice sheets are generally not observationally accessible. This poses a major problem for glaciers modelling studies, because the basal boundary condition is an essential part of a well-posed problem. The surface, however, is accessible to ground based, airborne, and satellite measurements. It is now possible to measure surface topography, ice thickness, and surface velocities over large areas. This sets up a classic inverse problem with too many boundary conditions at the top and not enough at the bottom. Our objective is to develop a new theoretical and numerical approach to solving this boundary inverse problem. The application of inverse methods will lead to derived basal velocity fields, also over large areas. This will be a major improvement in the understanding of the subglacial environment, and provides a major step towards predictive glacier models. Such models are important in the study of glacier-climate interaction, dating of ice cores, and assessment of natural hazards.

Valeriy Groshev will talk about an approach to solving inverse problem using the method of successive iterations suggested by V. Kozlov and V. Maz'ya. Elchin Jafarov will talk about formulation of the boundary inverse model and optimal control methods for solving it.

Sampling and Interpolation of Band-Limited Signals

Sergei Avdonin
Department of Mathematical Sciences
University of Alaska Fairbanks

From our perspective, a signal is a function s(t), say, in L2(R); in engineering terms this is a finite energy signal. The spectrum of a signal is the set of points &omega where its Fourier transform does not vanish. This talk is about sequences {tn} of real numbers with the property that knowledge of a signal s at these points, that is, {s(tn)} tells us everything about the signal. In order that this problem makes sense it is necessary to assume that the spectrum of the signal is constrained in some way. Classically such constraints have been that the spectrum lies in a finite interval. Such a signal is said to be band-limited. Recently there has been considerable interest in signals whose spectra lie in finite unions of such intervals (multi-band signals). We shall describe the main features of the classical theory and then discuss some of the more recent work.

Inverse Scattering Methods in Nonlinear PDE's: A Topic Prompted by the Recent Tsunami Disaster

Alexei Rybkin
Department of Mathematical Sciences
University of Alaska Fairbanks

An essential element in the study of Applied Mathematics (understood broadly) is to explain physical phenomena by mathematical models, frequently leading to nonlinear systems. The central theme is to understand (by approximation, numerical and exact methods) solutions to these underlying partial differential equations (PDE's) and their properties. An important method used to solve certain nonlinear PDE's is the so called Inverse Scattering Transform (IST), a remarkable marriage of two seemingly unrelated areas: nonlinear PDE's and the (essentially linear) direct/inverse scattering theory for Schrodinger type equations. The IST is conceptually analogous to the Fourier Transform; IST employs methods of direct and inverse scattering (techniques originally developed by physicists and mathematicians studying quantum mechanics). The IST allows one to construct general solutions to certain initial-boundary value problems that arise in a variety of physical problems such as nonlinear optics, water waves (including tsunamis), plasma physics, lattice vibrations, and relativity.

Historically, the IST method was discovered first for the Korteweg - de Vries (KdV) equation in the 60's and since then has been adapted to many other nonlinear PDE's (too many (and too difficult to spell!) to be listed here). We concentrate on the IST in the context of the KdV equation. The main nonlinear feature of the KdV equation is a special class of solutions referred to as solitons, which are extremely stable localized waves important in physical applications. A simple example of a soliton is a tsunami wave.

IST methods are based on sophisticated mathematics beyond the standard graduate curriculum. The talk, however, will focus on main ideas while leaving out technicalities.

Modeling Molecular Evolution: Markov models of DNA mutation on trees

Elizabeth S. Allman
Department of Mathematics
University of Southern Maine

The problem of tracing evolutionary relationships between species has long interested scientists. In recent years, as DNA sequences have become readily available, researchers in the areas of biology, mathematics, statistics, and computer science have been working to develop new techniques of phylogenetic tree construction using this data source.

In a model-based method of phylogeny reconstruction, one assumes that a particular probabilistic model governs the mutation process. Statistical techniques such as maximum likelihood can then reconstruct phylogenetic relationships fairly reliably. For any model-based method of phylogenetic inference to be accurate, however, it is essential that the model `fits' the data. One way to begin to address the question of model fit is by identifying the polynomial relationships that patterns in the sequence data must satisfy. Such constraints are known as phylogenetic invariants.

This talk will give a brief overview of models of molecular mutation, highlighting a mixture model known as the general Markov plus invariable sites model (GM+I). In the latter part of the talk, we explain our work as a problem in `algebraic statistics', where we exploit the observation that a model of molecular evolution gives rise to a parameterization of an affine algebraic variety. Finally, we explain how this viewpoint has led to results including a proof of the identifiability of parameters for the GM+I model (which is necessary to establish the consistency of maximum likelihood methods) and suggest possible improvements to estimates of parameters of interest to biologists.

Matrix Rank, Tensor Rank, and the Algebraic Statistics of Molecular Evolution

John Rhodes
Department of Mathematics
Bates College

The notion of the matrix rank is fundamental in linear algebra and can be easily computed either algorithmically (via Gaussian elimination or SVD) or algebraically (via the vanishing of minors). In addition, an elementary observation shows that rank plays a role in describing what types of 2-dimensional tables might arise from some simple statistical models.

When these statistical models are generalized only slightly, the tables become 3- or more dimensional, and we must deal with the tensor rank of higher dimensional arrays. While the concept is an old one, determining tensor rank poses much more difficulty than matrix rank, whether approached algorithmically or algebraically. Many basic questions are still unanswered.

After developing the rank concept as it relates to statistical models, this talk will give recent results that indicate the key role it plays in molecular phylogenetics: Standard statistical models of the evolution of DNA sequences along trees predict high-dimensional data tables that can be characterized in terms of ranks of `flattened' tables of dimension < 3.

In addition, the viewpoint of statistical models, and specifically phylogenetic ones, has led to new insights in algebraic geometry: We provide construction of new, explicit polynomials that vanish on all tensors of certain ranks and size which are thus higher-dimensional analogues of matrix minors.

A Sightseeing Tour of the Einstein Constraint Equations

David Maxwell
Department of Mathematical Sciences
University of Alaska Fairbanks

The Einstein constraint equations are a system of geometric partial differential equations arising in general relativity. These equations bring together researchers in differential geometry, PDE theory, and physics. Although they have been studied since the early days of general relativity, they remain a rich and active research field. In this talk I will discuss some recent progress made by myself and others in well-posedness questions, boundary value problems, and gluing constructions for the constraint equations. The goal is to visit these topics in breadth and to discuss some open problems along the way.

0 for April, or, Are Batting Slumps Inevitable?

E. Lee May, Jr.
Department of Mathematics
Salisbury University

There was no easy explanation for [Arizona Diamondbacks Center Fielder Steve] Finley's wicked slump, which had him hitless from the fourth game of the season through the 11th game, 0-for-25, in all.

It's hard to imagine that Ripken would be playing this well after he started the season in a 1-for-19 funk and continued to struggle well into June.

Aug. 11, 1950 - Stuck in a 4-for-38 slump and batting just .279 on the year, DiMaggio is benched for the first time in his career.

[Seattle Mariners] Designated hitter Edgar Martinez has been slumping, at least by his standards, and his average has slipped to .299--23 points below his career average.

Fans of baseball are accustomed to statements such as these. Remarks about slumps are heard almost every day of the season, from April through October. But wait! A "good" hitter is one who, over the entire season, amasses, on the average, three hits in every ten official at-bats. A "very good" hitter collects one hit in every three at-bats over the long haul. This means that, even for the very good hitter, two of every three of his official at-bats are going to end in his failing to "hit 'em where they ain't," to quote early-twentieth-century ballplayer "Wee Willie" Keeler. That's twice as many outs as hits which must occur in his total number of at-bats. Should we be surprised when "slumps," even ones as pronounced as those described above, occur?

This talk will explore the issue of batting slumps in baseball with the tools of arithmetic, probability, statistics, and computing. It will present a definition of "batting slump" and, using that definition, discuss the likelihood that a given hitter will experience a slump in a given season and examine the devastation which a slump wreaks in that season.

The talk is aimed at a broad audience. Anyone who has ever enjoyed playing or watching baseball, as well as anyone who has taken a course in introductory statistics or computing, might find something of interest in it. Some of the questions raised in the talk might constitute suitable topics for either graduate or undergraduate research.

Transcendence of Values and Periods of Special Functions

Paula Cohen
Department of Mathematics
Texas A&M University

Hermite's proof in 1873 of the transcendence of e began a new era in number theory. Within a decade, Lindemann proved the transcendence of &pi. These results concern the transcendence of values and periods of the exponential function, which is singly periodic. Inspired by Hilbert's 7th problem, Siegel (1932) and Schneider (1937) obtained the first significant results about the transcendence of values and periods of doubly periodic functions and of values of modular functions at certain algebraic points. Siegel formulated similar problems for G-functions, a particular case of which is the classical hypergeometric function. The modern development of this circle of ideas, including our own contribution, is the focus of our lecture. The lecture will be self-contained and accessible to a general audience.

The Classical Periods of Abelian Integrals

Marvin Tretkoff
Department of Mathematics
Texas A&M University

The periods of abelian integrals appear in complex function theory, number theory, algebraic geometry and mathematical physics. Beginning with an example from elementary calculus, we will give an informal discussion of abelian integrals and their periods. There will be no rigorous proofs. Towards the end of the lecture, we hope to illustrate our own contribution to the subject by determining the periods associated to the Fermat curve.

Domination in Graphs

or

Ruling the World with Minimum Effort

Glenn G. Chappell
Department of Mathematical Sciences
University of Alaska Fairbanks

How many trusted henchmen does a dictator need to keep everyone in the world in line? What is the minimum number of emergency shelters we have to build so that one can be reached quickly from any part of town? How many guards are required to keep every part of a building under constant observation? These questions are all modeled by a problem in graph theory known as the "domination problem".

A graph is a set of vertices, pairs of which are joined by edges. The domination problem asks for the smallest size of a set of vertices so that every vertex in the graph is either in the set or else joined by an edge to a vertex in the set. (Thus, every vertex in the graph is "dominated" by one in the set.)

We give an introductory discussion of the domination problem, some of its applications, and the fact that it is NP-complete (that is, it belongs to a large class of problems for which efficient algorithmic solutions are unknown, and probably impossible). We conclude with bounds on the ratio between the domination number and other graph parameters that can be computed efficiently (joint work with John Gimbel and Chris Hartman).

The Insolvability of the Fifth Degree Polynomial

Jonathan Wiens
Department of Mathematical Sciences
University of Alaska Fairbanks

Mathematicians have been solving the quadratic (degree 2) polynomial since antiquity. In the early 1500s progress was finally made on solving generic third degree and fourth degree polynomials. Tartaglia, Cardano and Ferrari all made significant contributions and in 1545 Cardano published "Ars Magna, sive de regulis algebraicis" which contained the formulas for both third and fourth degree polynomials. For almost 300 years mathematicians sought a generalization of these formulas to polynomials of degree five and greater. Eventually Ruffini (in 1799) and Abel (in 1824) were able to give proofs (with some gaps, however) that no formula for a quintic was possible. Finally, Galois (in 1831 or 1832) was able to not only rigorously prove that no solution to the quintic was possible, but he was also able to determine precisely which quintics had roots that could be expressed with formulas involving addition, subtraction, multiplication, division and extraction of roots. By doing so, he also founded the Theory of Groups. This talk will give a brief account, in modern terms, of the ideas behind the proof.

The Poincare Conjecture (easiest among seven hard ways to earn your first million?)

Ed Bueler
Department of Mathematical Sciences
University of Alaska Fairbanks

This talk will be aimed at a general mathematical audience, that is, math majors, graduate students, and faculty in all fields.

More than a century ago, at the genesis of topology as a mathematical field, Henri Poincare speculated that a certain way of counting holes which he had largely invented (homology) could distinguish between objects (topological spaces). A few years later he showed that he was wrong.

He tried again. He conjectured that in three dimensions, if one counts no holes by a different method (homotopy; the fundamental group) then the object is in fact (in the category of topology) a three-dimensional sphere. No one has known if he guessed right. Perhaps no one ever will, but there are noises in the mathematical world about a highly credible argument that he was. And that just might earn someone some money...

This talk will describe the conjecture, give some history, and provide a gloss of the new methods for addressing it. At best, this talk will be the mathematical equivalent of a NOVA special---possibly fun but certainly not deep. The speaker is no expert, but hopes to be a competent reporter.

Initial Data for Black Hole Spacetimes

David Maxwell
Department of Mathematical Sciences
University of Alaska Fairbanks

The Einstein constraint equations are PDEs that arise as a compatibility condition on initial data in the Cauchy problem of general relativity. In this talk I will describe the construction of a family of solutions of the constraints that evolve into spacetimes containing black holes. The idea is to work with a manifold with boundary and specify that the boundary be an apparent horizon. I'll show that when certain geometric conditions are satisfied, the constraint equations together with the apparent horizon boundary condition can be reduced to a well-posed elliptic boundary value problem. The talk will be accessible, with much of it devoted to reviewing background material from general relativity and geometric analysis.

Please contact John Gimbel or David Maxwell if you are interested in giving a colloquium talk.

'03 - '04 Colloquium Schedule

doc info last modified Wednesday, 21-Sep-2005 16:38:00 AKDT.