## Abstracts

Note: This page contains LaTeX markup to display mathematical notation.

Friday, June 28, 2013. **Thesis defense.** (Advisor: George Markowsky)**Joshua Case**, *UMaine Mathematics MA candidate***“Necklaces and Bracelets: Enumeration, Algebraic Properties, and TheirRelationship to Music Theory”**

2:00 – 2:50 pm, 115 DPC.

A necklace is a set consisting of a string and all of its rotations while a bracelet is a set consisting of a string and its rotations and reversals. Using musical chords as our motivation, we investigate methods for counting necklaces and bracelets. Additionally, we discuss the enumeration of Lyndon words, which are aperiodic strings that are *lexically-least* (least in dictionary order) in their necklace equivalence class. We investigate the algebraic properties of Lyndon words, including a result that proves that every q-ary string can be expressed as a non-increasing (lexically-speaking) product of Lyndon words.

Wednesday, May 15, 2013. **Thesis defense.** (Advisor: Ramesh Gupta)**Jie Huang**, *UMaine Mathematics MA candidate***“Analysis of Survival Data by a Weibull-Generalized Poisson Distribution”**

11:00 – 11:50 pm, 421 Neville Hall.

In life testing and survival analysis, sometimes the components are arranged in

series or parallel system and the number of components is initially unknown. Thus, the number of components is considered as random with an appropriate probability mass function. More specifically, the problem arises in cancer clinical trials where the number (Z) of metastasis- competent cells (clonogens) is unknown and the event occurs as soon as one of the clonogens metastasizes. In damage models, the number of cracks is unknown and the system fails as soon as the first failure occurs. In this connection, several distributions of Z have been considered in the literature, including the Poisson distribution, the logarithmic series distribution, the COM-Poisson distribution and the power series distribution, with exponential as the baseline distribution. More recently, Gupta et al. [1] proposed a model with the generalized Poisson distribution as the distribution of Z. With exponential as the baseline distribution, the resulting model has decreasing failure rate. In this thesis, we will model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model and increasing, decreasing, bathtub and upside bathtub failure rate. The maximum likelihood estimation of the parameters will be studied and the results will be compared to the existing models, especially the exponential generalized Poisson distribution which has been studied by Gupta et al. [1].

Tuesday, May 14, 2013. **Graduate seminar.****Adam Duncan**, *UMaine Mathematics MA student***“Wavelet-Based Edge Detection and Segmentation of Chromosome Territories in 3D Images of Cell Nuclei”**

2:10 – 3:00 pm, 421 Neville Hall.

We develop a novel tool for detecting the bounding surfaces of bright regions in 3D images. Using a vector of wavelets given by partial derivatives of isotropic Gaussian functions, we obtain smoothed estimates of the image gradient vector at each point. On the resulting gradient image, we extract triangle mesh surfaces where the gradient magnitude is locally maximum in the direction of the gradient vector. This process is done with Gaussian functions over a range of scales to determine the amount of smoothing which gives the most detail without being overwhelmed by image noise and error due to discretization. We then compute the topologically invariant Euler characteristic of each component of the mesh to determine which components are closed surfaces without holes. For each such surface, we find the scale at which the average gradient is maximized without compromising the surface topology. Finally, we compute various geometric properties of the resulting surfaces. This entire process is performed on simulated images to test its accuracy, and then on images of cell nuclei to extract information about chromosome territories in those images. The resulting segmentation allows us to more easily visualize the 3D data and objectively compute geometric properties of different chromosomes in the same nucleus. Finding the distance between chromosomes and their relative radial distance is a problem of interest in biomedical fields and we present these techniques as an interdisciplinary research tool.

Friday, April 26, 2013. **Thesis defense.** (Advisor: Andrew Knightly)**Carl Ragsdale**, *UMaine Mathematics MA candidate***“Matrix Coeﬃcients of Certain Supercuspidal Representations”**

2:10 – 3:00 pm, 204 Neville Hall.

In the past 60 years, an important problem in algebraic number theory has been to understand the group , where denotes the ﬁeld of -adic numbers. One way to study this group is through its complex representations. An easier related problem is to study the irreducible representations of the finite group . We will provide a complete classiﬁcation of irreducible representations of . After this is done, we will discuss how to obtain supercuspidal representations of from representations of . The main result of this thesis is the computation of matrix coeﬃcients of supercuspidal representations that can be constructed in this way.

Tuesday, April 23, 2013. **Thesis defense.** (Advisor: Andre Khalil)**Kendra Mooers**, *UMaine Mathematics MA candidate***“Characterization of Mammographic Breast Lesions and Their Microenvironment: An Application of a Wavelet-Based Multifractal Formalism”**

11:10 – 12:00 pm, 165 Barrows Hall. (Refreshments at 11:00 am)

The Wavelet-Transform Modulus Maxima (WTMM) method has been implemented into nearly all fields of applied sciences. In this adaption of the 2D WTMM method, the continuous wavelet transform is the mathematical microscope used to characterize the fractal geometry of clusters of microcalcifications (MC) in human breast tissue and to determine the roughness of the background tissue seen in mammograms. The WTMM method yields the so-called singularity spectrum, D(h), i.e. the fractal dimension D, of points having a Holder exponent of h. The MC are seen as Dirac singularities by the WTMM method, therefore having Holder exponent value of h = -1. This allows the MC with h ~ -1 to be abstracted from the background tissue which has h ~ 0.30 for fatty breast tissue and h ~ 0.65 for dense tissue. Thus, the WTMM method is used to perform a segmentation of the breast tumors based on the strength of the singularities composing the mammogram images, and to simultaneously quantify their fractal dimension. After analyzing several hundred images from a digital databank of mammograms with known radiologist diagnostics, the fractal dimensions of benign and malignant breast lesions are significantly different, with benign having an integer dimension corresponding to a non-invasive Euclidean object and cancer having a non-integer dimension, representing an invasive structure. In addition, the microenvironments characterized by the roughness of the tissue in which the lesions are embedded are different for benign and malignant tumors, and provides an insight into the onset and development of breast cancer.

Wednesday, April 17, 2013. **Graduate Seminar.****Carl Ragsdale**, *UMaine Mathematics MA student***“An introduction to complex representations of finite groups”**

2:10 – 3:00 pm, 421 Neville Hall.

Representation theory allows us to apply results from linear algebra to the study of general groups. Given a group G, a complex representation of G is a pair (ρ, V), where V is a complex vector space, and

ρ : G −→ GL(V)

is a group homomorphism. This seminar talk will look at examples of

complex representations of ﬁnite groups before discussing some of the key tools that are used to study such representations, including direct sums of representations, complete reducibility, intertwining operators, Schur’s Lemma, and Frobenius Reciprocity.

Wednesday, February 20, 2013.**Prof. Eva Curry**, *Acadia University (and UMaine Math BA Alum)***“Multidimensional Radix Representations and Number Systems”**

3:10 – 4:00 pm, 100 Neville Hall.

Digital (also known as radix) representations for integers and real numbers, such as base 10 or base 2, can be generalized to higher dimensions. The “base” (radix) in this case is a square matrix with integer entries, and the digits are a set of vectors with integer entries that represent cosets of the integer lattice under multiplication by the base matrix. The first part of this talk will give some examples and introduce some of the number theoretic questions that can be studied for this type of representation.

In the regular one-dimensional case, the set of “decimals” in standard base 10, base 2, or similar radix representation is the unit interval. The set of “decimals” in a multidimensional radix representation can be an interesting fractal set, however. The second part of this talk will give some examples and introduce some of the topological questions that can be studied for multidimensional radix representations.

ALSO:

At noon on Wednesday, we will have pizza in room 421 Neville. Eva will be available to chat with students and faculty about being a math major at UM and then going on to get a Ph.D. in mathematics and ultimately a mathematics faculty position.

Thursday, November 29, 2012. **Pizza Pi.****Dr. Banu Baydil**, *UMaine Mathematics***“Modeling Multi-cellular Structures”**

12:30 – 1:20 pm, 208 Neville Hall.

Recently, there is growing interest in mathematical modeling of tissue at different scales both from a data-driven modeling perspective and from a first principles based modeling perspective. This talk will present an introduction to mathematical modeling of multi-cellular structures.

Wednesday, November 28, 2012. **Graduate Seminar.****Jie Huang**, *UMaine Mathematics MA student***“A Versatile Model to Analyze Life time Data”**

3:10 pm, 421 Neville Hall. Refreshments will be served at 3pm.

In life testing and survival analysis, the components are arranged in series or

parallel system and the number of components is, initially unknown. Thus, the

number of components is considered as random with an appropriate probability mass function. More specifically, the problem arises in cancer clinical trials where the number (N) of metastasis-competent cells (clonogens) is unknown and the event occurs as soon as one of the clonogens metastasizes. In damage models, the number of cracks is unknown and the system fails as soon as the first failure occurs. In this connection, several distributions of N have been considered in the literature, including the Poisson distribution, the Logarithmic series distribution, the COM-Poisson distribution and the power series distribution, with exponential as the baseline ditribution. More recently, Gupta et al. (2012) proposed a model with the generalized Poisson distribution as the distribution of N.

With exponential as the baseline distribution, the resulting model has

decreasing failure rate. In this presentation, we have modeled the survival

data with baseline distribution as Weibull and the distribution of N as

generalized Poisson, giving rise to four parameters in the model and increasing, decreasing, bathtub and upside bathtub failure rate. The maximum likelihood estimation of the parameters is studied and the results are compared to the existing models, especially the exponential generalized Poisson distribution of Gupta et al. (2012).

Tuesday, November 7, 2012. **Graduate Seminar.****Rachel Rier**, *UMaine Mathematics MA student***“Moment Differential Equations of a Stochastic Epidemiological Model”**

3:10 pm, 421 Neville Hall. Refreshments will be served.

Compartmental models in epidemiology are commonly used to model the spread of infectious diseases in a population. In this talk we will examine a stochastic SIS (Susceptible-Infected-Susceptible) model and introduce a set of nonlinear moment differential equations which characterize the system. A general formula for the differential equation for any moment is introduced. We will examine ways in which we can “close off” the system of differential equations with moment-closure approximations, so that the system can be numerically solved. Using a similar approach, we can construct a new system of moment differential equations for a household-structured model, in which the population is partitioned into communities with varying levels of interaction.

Tuesday, November 6, 2012. **Colloquium / Pizza Pi.****Prof. Ben Weiss**, *Bates College***“Arithmetic Dynamics and Sarkovskii’s Theorem”**

421 Neville Hall. Pizza at 12:15, lecture from 12:30-1:20pm.

Given a function f(x), and a number A, what does the orbit {A, f(A), f(f(A)), f(f(f(A))), ….} look like? Can it be finite? If so how big can it be? These questions are part of dynamics, which studies functions by analyzing their orbits. The study of orbits has very wide applications to number theory, ergodic theory, and is a beautiful subject in its own right. We’ll discuss how dynamics of polynomials over the integers and rational numbers can tell us about about units and arithmetic properties of these sets. Then we’ll discuss Sarkovskii’s Theorem, which classifies possible orbit sizes of continuous functions over the real numbers, and time permitting will discuss related open problems. This talk will be accessible to all.

Thursday, October 25, 2012. **Pizza Pi Talk.****Prof. Bob Franzosa**, *University of Maine***“The True/False Cards: A Hands-On Deductive Reasoning Computer”**

12:30-1:20 pm, 421 Neville Hall. Pizza will be served.

In the chapter “The Binary System” in *New Mathematical Diversions* from Scientific American, Martin Gardner presents a deck of punched and slotted cards that he uses to demonstrate binary sorting and to solve a logic problem. Building on his approach to the logic problem, I will show how to use the deck for a hands-on approach to basic topics in deductive reasoning, such as truth tables, equivalent statements, and valid arguments.

Tuesday, October 16, 2012. **Graduate Seminar.****Kendra Mooers**, *UMaine Mathematics MA student***“Characterization of Mammographic Breast Lesions and their Microenvironment”**

3:30 pm, 208 Neville Hall. Refreshments will be served.

The Wavelet-Transform Modulus Maxima (WTMM) method has been implemented into nearly all fields of applied sciences. In this adaption of the WTMM method, the continuous wavelet transform is the mathematical microscope used to characterize the fractal geometry of clusters of microcalcifications (MC) in human breast tissue and to determine the roughness of the background tissue seen in mammograms. The WTMM method yields the so-called singularity spectrum, D(h), i.e. the fractal dimension D, of points having a Holder exponent of h. The MC are seen as Dirac singularities by the WTMM method, therefore having Holder exponent value of h = -1. This allows the MC with h ~ -1 to be abstracted from the background tissue which has h ~ 0.30 for fatty breast tissue and h ~ 0.65 for dense tissue. Thus, the WTMM method is used to perform a segmentation of the breast tumors based on the strength of the singularities composing the mammogram images, and to simultaneously quantify their fractal dimension. After analyzing several hundred images from a digital databank of mammograms with known radiologist diagnostics, the fractal dimensions of benign and malignant breast lesions are significantly different, with benign having a dimension close to 1 or 2 and cancer having a dimension close to 1.5. In addition, the microenvironments characterized by the roughness of the tissue in which the lesions are embedded are different for benign and malignant tumors.