## Colloquium Abstracts

**Wednesday, Sept 17, 2003**

3:10pm, 107 DPC

**Prof. Bob Franzosa**, *Dept. of Mathematics and Statistics, University of Maine
*What is Applied Topology?

An introduction to the basic concepts of topology, focusing on their role in applied problems. Some applications will be briefly discussed, including applications in molecular biology, geographic information systems, and computer graphics.

**Thursday, Sept 18, 2003**

2:10pm, 421 Neville

**Xue Li**, *Dept. of Mathematics and Statistics, University of Maine
*Statistical Inference for the Common Mean of Two Independent Log-Normal Distributions

The statistical analysis for comparing the means of two independent lognormal distributions is often of interest in biomedical research. For example, in bio-availability studies, the relative potency of a new drug to that of a standard one is expressed in terms of the ratio of means, and analysts often need to construct a confidence interval for the ratio or to test the null hypothesis that the ratio is one, i.e., the mean outcome of the two products are the same.

In this connection, Zhou et al. (1997) proposed likelihood-based procedures for comparing the means of two independent lognormal populations.

In this talk, we shall study the large sample procedures for comparing the means of two lognormal populations. In addition to Zhou’s test, we develop a score test and compare it with Zhou’s test in terms of type I error rates and powers.

Assuming that the means of two lognormal distributions are the same, we shall develop procedures for estimating the common mean of two independent lognormal distributions. Confidence intervals for the common mean will be constructed and their performance will be examined by simulation studies in terms of the coverage probabilities and their average length.

Illustrative examples are also provided.

**Thursday, Oct 16, 2003**

2:10pm, 421 Neville

**Xue Li**, *Dept. of Mathematics and Statistics, University of Maine
*Statistical Inference for the Common Mean and Estimation of R=P(X < Y) of Two Independent Log-Normal Distributions

The statistical analysis for comparing the means of two independent lognormal distributions is often of interest in biomedical research. For example, in bio-availability studies, the relative potency of a new drug to that of a standard one is expressed in terms of the ratio of means, and analysts often need to construct a confidence interval for the ratio or to test the null hypothesis that the ratio is one, i.e., the mean outcome of the two products are the same.

In this connection, Zhou et al. (1997) proposed likelihood-based procedures for comparing the means of two independent lognormal populations.

In this talk, we shall study the large sample procedures for comparing the means of two lognormal populations. In addition to Zhou’s test, we develop a score test and compare it with Zhou’s test in terms of type I error rates and powers.

Assuming that the means of two lognormal distributions are the same, we shall develop procedures for estimating the reliability function R = P(X < Y), where X and Y come from independent lognormal populations with the common mean. Confidence intervals for R will be constructed and their performance will be examined, by simulation studies, in terms of the coverage probabilities and their average length. We also perform simulation studies to compare the confidence intervals with the assumption of equal means to those without the assumption of equal means. In addition, simulation studies are carried out to investigate the distribution of .

Examples are also provided to illustrate the procedure.

**Thursday, Oct 23, 2003**

2:10pm, 100 Neville

**Dr. Naitee Ting**, *Associate Director, Biostatistics Group Pfizer Global Research and Development New London, Connecticut
*Statistical Applications in the Drug Development Process

Most drugs start out as a simple chemical compound. Chemists and biologists study a certain disease process, and synthesize many compounds in the laboratory with the hope that some of these compounds can help patients with this disease. If a compound demonstrates good efficacy and low toxicity, this compound will progress into animal testing (non-clinical studies) and human testing (clinical studies). Statistics is an important tool in the pharmaceutical research and development. It is useful in both non-clinical and clinical drug development processes.

Dr. Naitee Ting is currently an Associate Director in the Biostatistics group of Pfizer Global Research and Development at New London, CT. Naitee received his Ph.D. in 1987 from Colorado State University (major in Statistics). He has an M.S. degree from Mississippi State University (1979, Statistics) and a B.S. degree from College of Chinese Culture (1976, Forestry). He has been with Pfizer since 1987.

Naitee published articles in Technometrics, Drug Information Journal, Journal of Statistical Planning and Inference, Journal of Biopharmaceutical Statistics, Biometrical Journal, Statistics and Probability Letters, Statistics in Medicine and Journal of Statistical Computation and Simulation. He also published a few book chapters. Naitee served various roles in American Statistical Association (ASA): Representative at Council of Chapters for the Connecticut Chapter (1997-1999), President of the Connecticut Chapter (1991-1992), and Vice President of the Connecticut Chapter (1990-1991). He also served as the Executive Director of the International Chinese Statistical Association (ICSA) between 1998 and 2000.

**Thursday, Nov 6, 2003**

3:30pm, 115 DPC

**Dr. Linda Rottman**, *Onward Program, University of Maine
*Brain Science and Learning Algebra

New Technologies have enabled brain researchers to better understand the biological processes that occur in the brain as people acquire new learning. Could it be that helping students understand these processes — how their brain works — would improve their ability to learn and remember new material in their mathematics courses? Linda Rottmann has been teaching “brain science” to her developmental math students at UMaine for the past few years. In this workshop, she will share her experiences and approaches to creating a brain friendly learning environment for students. She will also provide you with an opportunity to explore how you might use these concepts with your own students.

**Wednesday, Dec 10, 2003**

3:10pm, 421 Neville

**Xue Li**, *Dept. of Mathematics and Statistics, University of Maine
*Statistical Inference for the Common Mean of Two Independent Log-Normal Distributions and Some Applications in Reliability

The statistical analysis for comparing the means of two independent log-normal distributions is often of interest in biomedical research. For examples, in bio-availability studies, the relative potency of a new drug to that of a standard one is expressed in terms of the ratio of means, and analysts often need to construct a confidence interval for the ratio or to test the null hypothesis that the ratio is one, i.e., the mean outcome of the two products are the same, (Berger and Hsu, 1996; Chow and Liu, 2000).

In this connection, Zhou et al. (1997) proposed likelihood-based procedures for comparing the means of two independent log-normal populations. One is a Z test and the other is a nonparametric bootstrap approach. However, Wu et al. (2002) pointed out that the Z test does not perform well in a range of small sample settings.

In this thesis, we shall study the large sample as well as small sample procedures for comparing the means of two log-normal populations. In addition to Zhou’s test, we develop a Rao’s score test for testing the equality of the means of two log-normal populations and compare its performance with that of Zhou’stest. Assuming that the means of two log-normal distributions are the same, we shall also:

- Develop procedures for estimating the common mean of two independent log-normal distributions. Confidence intervals for the common mean will be constructed and their performance will be examined by simulation studies in terms of the coverage probabilities and their average length.
- Estimate R = P(X < Y), where X and Y come from independent log-normal distributions with equal means. The problem originated in estimating the reliability in the stress-strength model where X is the stress and Y is the strength, see Gupta et al. (1999). The performance of the confidence intervals developed will be examined by simulation studies in terms of the coverage probability and the average length of the confidence intervals.

**Thursday, Dec 11, 2003**

2:30pm, 108 Neville

**Carrie Diaz Eaton**, *Dept. of Mathematics and Statistics, University of Maine
*Investigating the Dynamics of Mechanosensory Processing in the Cricket Cercal Sensory System

We are interested in the representation and processing of information about the dynamics of air current stimuli in the primary sensory interneurons of the cricket cercal system. Past modeling studies have used two ion channels, a transient sodium channel and delayed-rectifier type potassium channel, to model the spike-producing mechanism in these cells. In this work, we develop channel models based on physiology data obtained by Kloppenburg and Horner (1998), use the channels to construct a model neuron, and provide an analysis of the underlying mathematical structure. We also examine the frequency sensitivity of the model neuron and its dependence on channel dynamics. This study provides a first step toward developing more accurate models of primary sensory interneurons.

**Thursday, Jan 29, 2004**

2:10pm, 100 Neville

**Andrew Knightly**, *Dept. of Mathematics, University of Rochester
*Modular Forms and Galois Representations

Modular forms are certain holomorphic functions on the unit disk in the complex plane. For some modular forms, the coefficients of the Taylor series about 0 hold important arithmetic information. I will discuss in broad terms the various connections these coefficients hold to Number Theory, Representation Theory, and Algebraic Geometry as unified by the conjectures of Langlands. In this context, I will discuss my recent work on the trace formula for Hecke operators (an analytic approach), and on special cases of Tate’s conjecture (which is a p-adic analog of the Hodge conjecture).

**Monday, Feb 2, 2004**

2:10pm, 227 Neville

**Dr. Rachel Weir**, *Dept. of Mathematics, University of Virginia
*Invariant Subspaces and Extremal Functions in Weighted Bergman Spaces

For 0 < p < infty and -1 < alpha < infty, the weighted Bergman space A_alpha^p consists of all functions f, analytic in the unit disk in the complex plane, for which |f|^p w_{alpha} is area-integrable, where w_{alpha}(z) = (alpha+1)(1-|z|^2)^{alpha}. We will discuss the subspaces of A_{alpha}^p which are invariant under multiplication by z, and properties of their canonical extremal functions.

For -1 < alpha leq 0 and 0 < p < infty, these canonical extremal functions are known to act as contractive zero-divisors in the weighted Bergman space A_{alpha}^p. We will show that for 0 < alpha leq 1 and 0 < p < infty, the analogous extremal functions to not have any extra zeros in the unit disk and, hence, have the potential to act as zero-divisors. As a corollary, we find that certain families of hypergeometric functions either have no zeros in the unit disk or have no zeros in a half-plane.

**Thursday, Feb 5, 2004**

2:10pm, 100 Neville

**Dr. Laura Ghezzi**, *Dept. of Mathematics, University of Missouri-Colombia
*Valuations in Algebraic Geometry

This is joint work with S.D. Cutkosky.

Let *k* be a field of characteristic zero, *K* an algebraic function field over *k*, and *V* a *k*-valuation ring of *K*. Zariski’s theorem of local uniformization shows that there exist algebraic regular local rings *Ri* with quotient field *K*which are dominated by *V*, and such that the direct limit union *Ri* = *V*.

After giving the necessary background and definitions we discuss generalizations of Zariski’s theorem and we give examples of valuations that arise naturally in Algebraic Geometry.

**Monday, Feb 9, 2004**

3:00pm, 105 DPC

**Dr. James Barta**, *Dept. of Elementary Education, Utah State University
*Singing Blocks, Talking Shapes, and Dancing Numbers: Mathematical Connections for Native American Students

Through my research I have come to understand that while cultural connections are important aspects of the educational process, we must also consider the “world views” of those we teach. I intend to discuss my understanding of indigenous world view and the effects on how we teach and how some Native students learn. Admittedly, I have only begun to probe what this really means in terms of pedagogy and curriculum but I am excited to share my current insights and experiences and hope through them we can enhance the study through collaborative efforts.

**Monday, Feb 9, 2004**

2:10pm, 227 Neville

**Artem Zvavitch**, *Dept. of Mathematics, University of Missouri-Colombia
*Convex Bodies and the Fourier Transform

The study of geometric properties of bodies using information about sections and projections of these bodies has important applications to many areas of mathematics and science. A new approach to projections and sections of convex bodies, based on methods of Fourier analysis, has recently been developed.

The crucial role in the Fourier approach to sections belongs to a certain formula connecting the volume of sections with the Fourier transform of powers of the Minkowski functional. In this talk we present an analog of this formula for the case of projections, which expresses the volume of projections in terms of the Fourier transform of the curvature function. We will also show some generalizations of Fourier analytic technique to the case of Gaussian measures, giving a solution of Busemann-Petty problem for Gaussian measures.

**Wednesday, Mar 24, 2004**

3:10pm, 421 Neville

**Dr. Sylvia Valdes-Leon**, *Dept. of Mathematics, University of Southern Maine
*Strongly Associate Rings

Let *R* be a commutative ring with identity. For *a,b* in *R* we define *a* and *b*to be *associates*, denoted *a* ~ *b*, if *a/b* and *b/a*, to be *strong associates*, denoted *a* approx *b*, if *a = ub* for some unit *u* of *R*, and to be *very strong associates*, denoted *a* cong *b*, if *a* ~ *b* and either *a = b = 0* or *a = rb*implies that *r* is a unit. Certainly *a* cong *b* Longrightarrow *a* approx *b*Longrightarrow *a* ~ *b*. Here we study commutative rings *R*, called *strongly associative rings*, with the property that for *a, b* in *R*, *a* ~ *b* implies *a*approx *b* and commutative rings *R*, called *presimplifiable rings*, with the property that for *a, b* in *R*, *a* ~ *b* (or *a* approx *b*) implies that *a* cong *b*. We also study whether or not this property of a commutative ring *R* being*strongly associate* is inherited by the ring of polynomials over *R*, denoted by *R[x]*, and the power series ring over *R*, denoted by *R[[x]]*.

**Thursday, Mar 25, 2004**

3:10pm, 421 Neville

**Suzhong Tian**, *Dept. of Mathematics and Statistics, University of Maine
*Statistical Inference for the Risk Ratio in 2 X 2 Binomial Trials with Structural Zero

In some statistical analyses, researchers may encounter the problem of analyzing correlated 2×2 table with a structural zero in one of the off diagonal cells. Structural zeros arise in situation where it is theoretically impossible for a particular cell to be observed. For instance, Agresti (1990) provided an example involving a sample of 156 calves born in Okeechobee County, Florida. Calves are first classified according to whether they get a pneumonia infection within certain time. They are then classified again according to whether they get a secondary infection within a period after the first infection clears up. Because subjects cannot, by definition, have a secondary infection without first having a primary infection, a structural void in the cell of the summary table that corresponds with no primary infection and has secondary infection is introduced. For discussion of this phenomenon, see Tang and Tang (2002), and Liu (1998).

The rate ratio (RR) between the secondary infection, given the primary infection, and the primary infection may be a useful measure of change in the pneumonia infection rates of the primary infection and the secondary infection.

In this talk, we will investigate the large sample confidence interval of RR. Example, from the literature, will be provided to illustrate the confidence intervals. Simulation studies will be carried out to examine the performance of these procedures.

**Thursday, Apr 1, 2004**

2:10pm, 421 Neville

**Anna Kettermann**, *Dept. of Mathematics and Statistics, University of Maine
*Estimation of Standardized Mortality Ratio in Geographic Epidemiology

The analysis of the geographic variation of disease and its representation on a map is an important topic in epidemiological research and in public health in general. Identification of spatial heterogeneity of relative risk using morbidity and mortality data is required. Frequently, interest is also in the analysis of space data with respect to time, where physically data are used which are aggregated in certain time windows like 5 or 10 years. The occurrence measure of interest is usually the standardized mortality (morbidity) ratio (SMR). It is well known that disease maps in space or in space in time not solely be based upon the crude SMR but rather some smoothed version of it. This fact has led to a tremendous amount of theoretical developments in spatial methodology, in particular in the area of hierarchical modeling in connection with fully Bayesian estimation techniques like Markov chain and Monte Carlo.

In this talk we will focus on the developments that avoid the pitfalls of the crude SMR. More specifically, we shall propose several estimates of the mean and variance of the heterogeneity. It turns out that some of these estimators are biased. An adjustment is made to make these estimators unbiased. A simple procedure to evaluate the heterogeneity is also explored.

**Thursday, Apr 8, 2004**

3:10pm, 421 Neville Hall

**Thomas Enkosky**, *Dept. of Mathematics and Statistics, University of Maine
*An Algorithm to Determine the Monomials in a Polynomial Ideal

An algorithm modeled after Buchberger’s algorithm, will be presented. Given a homogeneous ideal in a polynomial ring over a field the algorithm will first determine whether or not there are monomials in the ideal, and then it will give the generating set for the subideal generated by all monomials. An example will be given to show that the number of generators of the monomial subideal can be arbitrarily large, regardless of the number of generators of the ideal. Some preliminary information about Gröbner bases will be reviewed.

**Friday, Apr 9, 2004**

2:10pm, 211 Little Hall

**Dr. Sat Gupta**, *Dept. of Mathematics, University of Southern Maine
*Use of Randomized Response Techniques in Circumventing Social Desirability Response Bias In Personal Interview Surveys

Social desirability is a major source of bias and non-response in personal interview surveys involving sensitive questions. This response bias may reflect impression management concerns and the desire to present oneself in a more conventional and socially acceptable manner. Psychologists have used various approaches to circumventing this bias. These include a Bogus Pipeline (BPL) technique and a particular type of Randomized Response Technique (RRT). In this talk, we present various RRT approaches; their mathematical properties and their validation using simulated data. We also compare some of the RRT approaches to the BPL approach using survey data obtained from students enrolled in mathematics and psychology courses at USM. Survey results indicate that a “partial RRT approach” works better than the “full RRT approach” that is used by psychologists. We also find that the “partial RRT approach” works at least as well as the BPL approach and is much more user friendly than the BPL approach.

Dr. Gupta holds MS and Ph. D. degrees in mathematics from University of Delhi (1972,1977) and a Ph. D. degree in statistics from Colorado State University (1987). He has held teaching positions at University of Delhi (1976-1982) and at the University of Southern Maine (1986- Present). Currently, he is a Professor of Statistics at the University of Southern Maine.

Dr. Gupta has published in several areas including, Sampling Methods, Time Series Forecasting, Biostatistics, Education and Coding Theory. He has consulted extensively with biotechnology companies and other businesses, and has testified as a statistical expert in several major legal cases.

**Thursday, Apr 15, 2004**

3:10pm, 421 Neville Hall

**Richard W. Beveridge**, *Dept. of Mathematics and Statistics, University of Maine
*Developing a Survey to Assess Beliefs About Mathematics

Thesis defense; Advisor: John Donovan

The thesis considers the development of an instrument, the Mathematical Disposition Survey (MDS), to assess respondents beliefs about mathematics, and ways in which the resulting data may be analyzed. This presentation will discuss theoretical conceptualizations of beliefs in general and beliefs about mathematics in particular, as well as the difference between mathematical beliefs and mathematical knowledge. The survey from which the MDS was adapted will be discussed as will four other survey instruments developed for similar purposes. A method of analysis calculating the mean distance from optimal answer will be introduced and compared to the method of analysis performed by the researchers at the University of Maryland who developed the survey from which the MDS was adapted. A third method of analysis based on the method of analysis performed by the University of Maryland research team will also be introduced and a very high positive correlation will be shown between the result of this method of analysis and the Mean Distance from Optimal calculation.

**Friday, Apr 23, 2004**

2:10pm, 421 Neville Hall

**Carrie Eaton**, *Dept. of Mathematics and Statistics, University of Maine
*Adaptive Landscapes in Evolutionary Modeling

I will provide an introduction to evolutionary genetics and Darwinian fitness to describe the adaptive landscape as a visual, mathematical and heuristic tool for discussing evolutionary dynamics. We start with a 2-dimensional overdominance model and perform stability analysis. Then we explore the construction and behavior of a 3-dimensional introduced in by Gavrilets in Nature, (2000). We will also discuss results from a later model by Gavrilets and Waxman, with two interesting behavioral regimes: Buridan’s ass and sympatric speciation.

**Thursday, May 6, 2004**

3:10pm, 421 Neville Hall

**Thomas Enkosky**, *Dept. of Mathematics and Statistics, University of Maine
*Groebner Bases And An Algorithm To Find The Monomials Of An Ideal

Thesis defense; Advisor: Henrik Bresinsky

This thesis gives a background information on algebra and Groebner bases to solve the following problem: Given an ideal *I* in the polynomial ring *k[x1 , x2 , … , xn]*, what monomials, if any, are in the ideal. This thesis shows that there is no loss in generality in assuming that the ideal is homogeneous. Because the ideal is homogeneous we can use the properties of colon ideals and Groebner basis to produce a term *X* such that there are monomials in *I* if and only if *X* is in *I*. We then present an algorithm, modeled after Buchberger’s algorithm, that gives all the generators of the monomial subideal.

**Friday, May 7, 2004**

11:00am, 421 Neville Hall

**Anna Kettermann**, *Dept. of Mathematics and Statistics, University of Maine
*Heterogeneity in Estimation of Standardized Mortality Ratio in Geographic Epidemiology

The analysis of the geographic variation of disease and its representation on a map is an important topic in epidemiological research and in public health in general. Identification of spatial heterogeneity of relative risk using morbidity and mortality data is required. Frequently, interest is also in the analysis of space data with respect to time, where physically data are used which are aggregated in certain time windows like 5 or 10 years. The occurrence measure of interest is usually the standardized mortality (morbidity) ratio (SMR). It is well known that disease maps in space or in space in time not solely be based upon the crude SMR but rather some smoothed version of it. This fact has led to a tremendous amount of theoretical developments in spatial methodology, in particular in the area of hierarchical modeling in connection with fully Bayesian estimation techniques like Markov chain and Monte Carlo.

In this talk we will focus on the developments that avoid the pitfalls of the crude SMR. More specifically, we present a mixture model to evaluate the heterogeneity in estimating SMR. Simulation studies are carried out and the results are analyzed.

**Friday, May 7, 2004**

1:10pm, 421 Neville Hall

**Richard Beveridge**, *Dept. of Mathematics and Statistics, University of Maine
*The Disquisitiones Arithmeticae of C.F. Gauss and the group Z(p)*

Many fundamental ideas of modern algebra were considered by Gauss in his 1801 work Disquisitiones Arithmeticae. The fact that Z(p)* is a cyclic group is easily explained in modern terms by stating that it is the multiplicative group of a finite field. Many of the aspects of modern proofs of this theorem are visible in Gauss’ earlier, more rudimentary work on congruences. I will outline Gauss’ proof that that Z(p)* is cyclic by tracing his demonstration that there exists an element of Z(p)* of degree p -1.

**Thursday, May 20, 2004**

3:10pm, 421 Neville Hall

**Suzhong Tian**, *Dept. of Mathematics and Statistics, University of Maine
*Power of tests on Risk Ratio in 2 X 2 Binomial Trials with Structural Zero

In some statistical analyses, researchers may encounter the problem of analyzing correlated 2×2 tables with a structural zero in one of the off diagonal cells. Structural zeros arise in situation where it is theoretically impossible for a particular cell to be observed. For instance, Agresti (1990) provided an example involving a sample of 156 calves born in Okeechobee County, Florida. Calves are first classified according to whether they get a pneumonia infection within certain time. They are then classified again according to whether they get a secondary infection within a period after the first infection clears up. Because subjects cannot, by definition, have a secondary infection without first having a primary infection, a structural void in the cell of the summary table that corresponds with no primary infection and has secondary infection is introduced. For discussion of this phenomenon, see Tang and Tang (2002), and Liu (1998).

The risk ratio (RR) between the secondary infection, given the primary infection, and the primary infection may be a useful measure of change in the pneumonia infection rates of the primary infection and the secondary infection. In the last talk, we developed and evaluated the large sample confidence intervals of RR. In this presentation, we will investigate the tests about RR and the power of these tests. An example, from the literature, will be provided to illustrate the tests. Simulation studies will be carried out to examine the performance of these tests.

**Friday, June 4, 2004**

11:00am, 421 Neville Hall

**Anna Kettermann**, *Dept. of Mathematics and Statistics, University of Maine
*Estimation of Standardized Mortality Ratio in Geographic Epidemiology

Thesis defense; Advisor: Professor Ramesh C. Gupta

The analysis of the geographic variation of disease and its representation on a map is an important topic in epidemiological research and in public health in general. Identification of spatial heterogeneity of relative risk using morbidity and mortality data is required. Frequently, interest is also in the analysis of space data with respect to time, where physically data are used which are aggregated in certain time windows like 5 or 10 years. The occurrence measure of interest is usually the standardized mortality (morbidity) ratio (SMR). It is well known that disease maps in space or in space in time not solely be based upon the crude SMR but rather some smoothed version of it. This fact has led to a tremendous amount of theoretical developments in spatial methodology, in particular in the area of hierarchical modeling in connection with fully Bayesian estimation techniques like Markov chain and Monte Carlo.

In this talk we will focus on the developments that avoid the pitfalls of the crude SMR. More specifically, we present a mixture model to evaluate the heterogeneity in estimating SMR. Simulation studies are carried out and the results are analyzed.