# Abstracts for the Second Annual Regional Workshop in Mathematics & Statistics

As the contents of this page are of a past conference, no attempt to update the page has been made in order to preserve the historical context.

#### ABSTRACTS FOR SOME FRIDAY TALKS

KATHERINE KIME, Control of Differential Equations: When we encounter equations we often focus on solving them as given. Control theory considers differential equations from a different viewpoint: Can one make a "good choice" of some term in the equation (the control) so that the resulting solution behaves in a desired fashion? The differential equations usually model physical systems, both macroscopic --waves, beams, vibrating structures--and microscopic --atoms and molecules. We will discuss basic ideas of control theory and examples of research interest.

GLENN LEDDER, A Mathematical Modeling Problem for a Mission to Mars: Mathematical modeling is the process of creating a mathematical world based on a real situation in the hope that the mathematical behavior resembles the real behavior. The models are used to try to obtain qualitative understanding and/or quantitative predictions for applications in science and engineering. The speaker will construct, simplify, and analyze a model from mechanics that can be used to solve a specific design problem connected with a mission to bring rock samples back from Mars. This example serves to illustrate some basic themes of mathematical modeling, in particular the idea of a model as a map from a space of input data to a space of output data.

JOHN MEAKIN, Free Groups and their Subgroups: Group theory is one of the most central areas of modern mathematics, with close connections to many other fields. In particular, groups provide an essential tool to study the notion of symmetry (in mathematics and in nature). It is common to think of a group as being "generated" by a small subset of the elements of the group, usually subject to some constraints on the generators. Free groups are "freely" generated by some set of elements, subject to no constraints on these generators (other than those imposed by the fact that they must generate a group). In this talk I will explain, via lots of examples, what free groups are, and I will illustrate a very simple method of associating free groups with finite graphs. Many central ideas and unsolved problems about free groups and their subgroupsmay be easily understood this way.

DAN NETTLETON, Searching for Statistical Associations Between Genes and Quantitative Traits: Quantitative trait loci (QTL) are the genes that affect the quantitative characteristics of plants and animals. Examples include the genes affecting fruit weight of tomatoes, body fat of pigs, yield of wheat, and IQ of humans. I will explain,through examples, how statistical methods can be used to locate QTL in experimental plants and animals.

#### ABSTRACTS FOR SOME SATURDAY TALKS

ELVAN AKIN, Boundary Value Problems for a Differential Equation on a Measure Chain: I will prove existence and uniquness theorems for solution of the boundary value problem. In one of my results we use upper and lower solutions.

RICHARD AVERY, Existence of Solutions to a Discrete Fourth Order Lidstone Boundary Value Problem: For the fourth order discrete Lidstone boundary value problem, δ4 x(t-2) = f(δ2 x(t-1),x(t)) a+2 \leq t \leq; b+2, x(a) = \Delta^2 x(a) = 0 = x(b+4) = \Delta^2 x(b+2)$, where$f : R \times R \rightarrow [0,\infty)$, growth conditions are imposed on f which yield the existence of at least three positive solutions. JEAN-CAMILLE BIRGET, Groups and Computational Complexity: Presentations of groups, by generators and relators, go back to the 19th century,and can be considered as the earliest general models of computation. Group presentations are the principal tool for describing groups in topology, and they important role in other areas of mathematics. Therefore, the connection between computing and presentations is important: it connects computing with some older fields of mathematics. The main problem for a given presentation is the word problem: Given to strings x and y over the set of generators, can y be obtained from x by repeated application of relators of the given presentation? This is similar to the acceptance problem for programs, and it was proved in the 1950s that the two are equivalent; so the word problem for some finitely presented groups is undecidable. The Higman embedding theorem (1960), and the Boone-Higmantheorem (1974) made the connection between computing and presentations very clear. Until recently, no connection between computational complexity and presentations was known. With co-authors (M. Sapir, A. Olshanskii, E. Rips), I found a characterization of group presentations with reachability problem in NP: a group presentation of a group G has its reachability problem in NP iff the group G can be embedded into a group with finite presentation and with polynomial "Dehn function". The connection between presentations and most other complexity classes is still open. MARK BRITTENHAM, Knots and Crayons: The goal of much of (algebraic) topology is to find (easily computed) ways of assigning algebraic objects to topological objects, as a way to distinguish spaces from one another. In the case of knotted circles in space, one way to do this is to assign colors to pieces of the knot according to a certain set of rules, and ask for how many colors can we satisfy the rules? The resulting lists of numbers can be used to distinguish knots. The talk will explore this procedure of `coloring knots', accompanied by lots of colored chalk. LARRY BROWN, How to Compare Two Subspaces of Hilbert Space: Hilbert space is the infinite dimensional version of Euclidean space and subspaces are analogous to lines, planes, etc. through the origin. Two lines in a plane are compared by measuring the angle between them. One can compare two subspaces of higher dimensional Euclidean spaces or Hilbert space by angles also, though the story is more complicated. RANDY CRIST, Local Maps: An introduction to local maps and some recent results about local maps on matrix algebras and operator algebras. GAVIN CROSS, Creating Statistical Applications for the Web: Teaching at a small liberal arts institution can present unique challenges. This is an example of how one isolated statistician found creative ways to introduce concepts and get students interested in probability & statistics, while keeping himself amused. STEVEN R. DUNBAR, The Track of a Bicycle Back Tire and Riccati Equations: A bicycle goes down the road, the front tire setting the path, possibly weaving back and forth or just following a prescribed motion, the back tire following in a similar path. What is the path of the back tire? A related question is: If the front tire travels some distance, how much less does the back tire travel? The path of the back tire satisfies a Ricatti differential equation. Certain front tire paths admit exact solutions and other interetsting front tire paths admit good approximate solutions. In both cases, I derive an expression for the back tire pathlength as a fraction of the front tire path length. LEON HALL, Roads, Wheels, and Fourier Series: Given a continuous, periodic, negative function$y = f(x)$(the road), there exists a polar function$r = g(\theta)$(the wheel) and a point initially at the origin (the axle) such that the wheel will roll on the road without slipping and the axle moves along the$x\$-axis. However, the wheel does not close up to form a topological disk unless an additional constraint, called the {\it closed wheel condition} holds. Further, even if a road does satisfy the closed wheel condition, the Fourier approximations to the road do not, in general, inherit this property. In this talk, we will present a method of constructing Fourier-like approximations to a road which do inherit the closed wheel property.

MIKE IRA, A Block-Size Bound for Steiner 6-wise Balanced Designs: A t-wise balanced design (tBD) of type t-( v, K , lambda) is a pair (X, B) where X is a v-element set of points and B is a collection of subsets of X called blocks with the property that the size of every block is in K and every t-element subset of X is contained in exactly lambda blocks. If K is a set of positive integers strictly between t and v then we say the tBD is proper. If B is a block in a proper Steiner 6-wise balanced design then |B| <= v/2.

KARL KATTCHEE, Integer Programming and Direct Sum Decompositions: The integer programming problem is a restriction of the linear programming problem: Only solutions with nonnegative integer entries are considered. In algebraic terms, the restriction is from a vector space setting to a monoid setting. An interesting connection with decomposition of finitely-generated modules over a local ring is noted, and an algebraic solution of the integer programming problem is sketched.

DAVID LOGAN, A Model of Cavern Formation: The speaker will present a simple mathematical model of how chemical reactions and transport in porous media can change the porosity of the domain and result in cavern formation or ore deposition.

VALENTIN MATACHE, The Numerical Range or the Field of Values of a Linear Operator: Introduced by Hausdorff and Toeplitz in the early decades of this century, the notion of numerical range designated the immage of the unit sphere under the quadratic form associated to a matrix. Its convexity made the object of an early theorem by the aforementioned mathematicians. This talk will start by comparing the relatively monotonous situation of the numerical ranges of 2 by 2 matrices with the diversity of shapes one gets by upgrading from 2 by 2 to 3 by 3. Stepping from finite to infinite-dimensional spaces the shapes of numerical ranges of the operators acting on some function spaces by composition with a fixed mapping will be described.

LANCE NIELSEN, A Weak Convergence Problem: We prove weak convergence of two sequences of measures related to Lebesgue measure on [0,1]. These sequences will then be applied to the formation of the exponential of the sum of two bounded self-adjoint operators on a Hilbert space.

DAVE SKOUG, Irrational Numbers and the Function sin(1/x): Let r be a nonzero rational number. 1) Is sin(1/r) always an irrational number? 2) For any trig function f(x), is f(r) always an irrational number?

ROGER WIEGAND, Factoring Algebraic Numbers into Factors of Low Degree: Let K be the field Q(sqrt(2),i). Most elements of K have degree 4 over Q. Lots of elements, e.g. sqrt(2) + i, even though they have degree 4, can be factored as products of elements of degree 2. But not all of them: for example 1 + sqrt(2) + i cannot be factored in this way. Suppose now that L is a Galois extension of Q of degree n. (That is, L is obtained by adjoining to Q all the roots of some polynomial over Q.) When is it the case that every element of L can be factored as a product of elements whose degrees are smaller than n? Surprisingly, the answer is "almost always". The key thing to study is the Galois group of the extension, and there is a complete classification of those groups for which there exist elements that cannot be factored in this way. The special linear groups play a special role in the answer.

LINDA YOUNG, Estimating Animal Populations Using Capture-Recapture Techniques: An activity simulating a capture-recapture study of fish in a pond will be used to develop the Lincoln-Petersen index. Insight into the wide application of capture-recapture techniques will be provided. The assumptions inherent in the Lincoln-Petersen index and extensions to this model will be explored.