# Colloquium Series

## Department of Mathematics Colloquium

## FALL 2014

## Fractals in Chaotic Transport

### Dr. John Delos

### Thursday, September 25th, 2014

12:20 - 1:20 PM, Luter 372

**Abstract**: Chaotic transport, and the escape of trajectories from defined regions of phase space, has been an important topic in dynamics for many years, because it describes phenomena that occur in many branches of physics. For example, some meterorites that fell on Antarctica are believed to have come from Mars; how they escaped from Mars' gravitational field is a problem in the theory of chaotic transport. At a smaller scale, one of the important topics in nanophysics is transport of cold atoms through a junction. At the molecular level, we may think about the breakup of a temporarily bound complex, such as a He atom weakly bound to an I2 molecule. At the atomic level, the ionization of an excited hydrogen atom in applied electric and magnetic fields is an ideal candidate for the laboratory study of chaotic transport. We examine the time required for the electron to escape as a function of its initial direction. This escape-time plot has fractal structure. In this talk, we describe new methods for describing, analyzing and observing this fractal.

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## Interesting Series Involving

## Sums of the Reciprocal of the

## Binomial Coefficients

### Matt McCarthy

Christopher Newport University

### Thursday, October 9

### 12:20 - 1:20 PM, Luter 372

**Abstract**: We consider a set of infinite series involving the sum of the reciprocals of the binomial coefficients and try to determine their exact values by means of generating functions. Via specification, differentiation, and integration of these generating functions, we obtain a wide class of "interesting series" in terms of well-known constants.

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## Four Colors Suffice: an Introduction to the Four-Color Theorem

Dr. Paul Wrayno

Christopher Newport University

Thursday, October 23, 2014

12:20 - 1:20 PM, Luter 372

**Abstract**: Many problems in combinatorics and graph theory are easily posed, but far less easily solved. The Four Color Theorem, or as it was long known, the Four Color Problem, is such a problem. Originally posed to Augustus De Morgan by a student in 1852, the problem has seen many claimed proofs over the years, one of which was even accepted by the mathematical community for over a decade. The first correct proof by Appel and Haken was published in 1977 and required computer assistance, provoking a great deal of controversy. Some improvements have been found to this proof, but all current results remain reliant on computer assistance.

Briefly stated, the Four Color Problem is this:

Can every map be colored with at most four colors in such a way that neighboring countries are colored differently?

In this talk, we will discuss some of the history of the problem, and then prove the simpler six and five color theorems. We will also discuss some of the implication of this proof and the role chance circumstances can play in major discoveries. A secondary goal of this talk is to introduce the basics of graph theory, so no background knowledge is required.

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## A Group Theory Approach to Rubik's Cube

## Dr. Matthew Comer

Christopher Newport University

### Thursday, November 20, 2014

### 12:20 - 1:20 PM, Luter 372

**Abstract**: In 1974, Erno Rubik created the first prototype of a toy that would soon captivate mathematicians, engineers, and puzzle-solvers of all ages. The "Magic Cube," renamed "Rubik's Cube" a few years later, was itself a 3x3x3 cube of smaller cubes (or "cubelets"). Each of the six faces of the Magic Cube could be rotated in increments of 90 degrees, thereby permuting some of the cubelets with each rotation.

Since its creation and spread of influence in the 1970's, the art of "Cubing" has been well-studied by puzzle-solvers from all walks of life. In this talk, we will develop a mathematical approach to understanding Rubik's Cube from the perspective of group theory (specifically, symmetric groups), which will enable us to discuss some interesting results and properties of the Cube.

While this talk may be of particular interest to students who have taken MATH 370, we will not assume any experience with groups, and all necessary definitions will be introduced during the talk. A few cubes will be available at the talk for tangible demonstrations.

## Department of Mathematics Colloquium

** SPRING 2014**

### **Model Theory for Metric Spaces:**

** Ultraproducts and Stability**

### Dr. Alexander Berenstein* *

* **Universidad de los Andes*

* Bogota, Colombia*

##### **12:20 -1:20 p.m.**

** Thursday, January 9, 2014**

** Luter 372**

**___________________________________________________________________________________________________**

** Independence and Linearity:**

** Model Theoretic Perspectives**

** **

** Dr. Yevgeniy Vasilyev**

** ****Memorial University of Newfoundland**** **

**Memorial University of Newfoundland**

** 12:20 -1:20 p.m.**

** Thursday, January 23rd, 2014**

** Luter 372**

**Abstract:** Linear independence and algebraic independence are fundamental concepts used to describe linear or polynomial relationships between mathematical objects. Model theory, a branch of mathematical logic studying mathematical structures from the point of view of a formal language, provides a framework in which linear independence and algebraic independence are viewed as special cases of a more general notion. Using this approach, we can define analogues of the basic concepts of linear algebra, such as span, basis and dimension, in an arbitrary mathematical structure satisfying certain “geometric” requirements.

In this talk, I will define the notions of “geometric structure” and “pregeometry”, and discuss the question of distinguishing between linear (“vector space-like”) and non-linear geometric structures.

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## Riemann Zeta Function at Integers

Hongwei Chen, Ph.D.

*Christopher Newport University*

12:20 - 1:20 PM

Thursday, February 6, 2014

Luter 372

**Abstract**: Riemann zeta function is one of the most important and fascinating functions in mathematics. This talk is in two parts: the first part is about the history and significance of the Riemann zeta function, including the Riemann hypothesis; the second part is about my recent work on the values of the subseries of the Riemann zeta function at integers. Our results illuminate the similarities between the even and odd cases, and may give some insight into why the odd case is much more difficult.

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### An Introduction to Saturation Problems in Graph Theory

### Dr. Paul Wrayno

*Christopher Newport University*

12:20 - 1:20 PM

Thursday, February 20

Luter 372

**Abstract**: Broadly speaking, saturation problems are concerned with finding and analyzing graphs that "almost" have a particular desired or forbidden property. The most well studied saturation problems are extremal problems where one asks "How large/full can a graph be while avoiding this property?" Another type of saturation question asks "How small/sparse can a graph be while still being saturated?" A third type of saturation question asks "For what sizes is there a saturated graph?" The first two question types give upper and lower bounds for these sizes, but it is interesting to know whether there is a saturated graph of each size between the upper and lower bounds, and whether and where gaps occur in that interval.

More formally, a graph G is F-saturated if G does not contain a copy of F, but with the addition of any new edge it does. A graph is an extremal graph if it is an F-Saturated graph of maximum size and its size is the extremal number. The saturation number in contrast is the minimum size of any F-saturated graph.

This talk will provide an introduction to basic properties of mathematical graphs, extremal and saturation numbers and present a sampling of saturation results. If time allows, Dr. Wrayno will discuss his own work on a saturation problem.

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### Competition and Community Composition

### in streams and rivers

**Dr. Olga Vasilyeva**

#### Department of Mathematics

#### *Christopher Newport University*

12:20 - 1:20 PM

Tuesday, March 18, 2014

Luter Hall 372

**Abstract**: The flow speed in stream and rivers can change due to natural causes or human activities. Any change in flow speed can impact biological communities in streams and rivers that are shaped not only by their biotic interactions but also by their response to water flow. In this talk, we will discuss some transport-based modelling approaches describing the population dynamics of two or more competing species in stream ecosystems. We show that alterations of flow speed can influence the outcome of competition and thereby change community composition. Our analysis shows that at relatively high flow speed, each species' intrinsic growth rate is the crucial factor that determines the outcome of competition. At low flow speeds, in contrast, the strength of interspecific competition determines community composition. This is a joint work with Frithjof Lutscher, University of Ottawa, Canada.

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Two predator-prey models

with a Holling-type I functional response that has a

sharp corner in its graph

### Department of Mathematics

*Colgate University*

**Abstract**: Mathematical biology is one of the fastest growing interdisciplinary areas in applied mathematics. In particular, the subject of mathematical ecology is one of the oldest in mathematical biology and recently grabs a lot of attention of mathematicians and biologists. Among many themes in mathematical ecology, the study of the dynamics of predator-prey interaction is a popular subject. In my talk, I will present the dynamics of a laissez-faire and a Leslie predator-prey model with a Holling type I functional response, which is a monotonically increasing function that tends toward and reaches a maximum per capita consumption rate as the prey density increases. I study local and global stability of the equilibrium where the predator and prey coexist. For the Leslie-type model, I use a generalized Jacobian of Clarke to determine how eigenvalues jump at the corner of the functional response. In my numerical results, I show that both models can possess two limit cycles that surround a stable coexistence equilibrium and that the cycles arise through a global cyclic-fold bifurcation. To smooth out the sharp corner of the type I functional response, I introduce an arctangent approximation. I analyze the same models with an arctan functional response and show both models possess Hopf, cyclic-fold, and Bautin bifurcations.

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## Numerical Evidence of Reverse Buoyancy Current

## in a Channel

### Dr. James Martin

*Christopher Newport University*

12:20 - 1:20 PM

Thursday, April 3, 2014

Luter 372

**Abstract**: This talk will focus on my numerical calculations of gravity-driven channel flow. Upstream particle seeding is used as one trigger for the starting density gradient in the two included fluids. During the ash fall-out period of volcanic eruption, a spectacular instance of a particle-driven current takes place with the super-heated neighboring air. A more everyday gravity-driven current takes place upon opening your front door and letting the air enter/exit on a cold day. Under certain initial conditions, a time sequence of density contours reveals evidence of the phenomenon known as reverse buoyancy. I will begin this talk, however, with old photographs from Main Steet Library's Virginiana room and other reasons for my ongoing interest in the subject of aerodynamics and fluid flows.

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## Wavelet Sets, Tilings, and Groups

Dr. Gestur Olafsson

*Louisiana State University*

*Thursday, April 17, 2014*

*12:20 - 1:20 *

*Luter 372*

## Department of Mathematics Colloquium

## Fall 2013

### September 26, 12:15 - 1:15, Luter 372

#### Dr. Emily Sprague

Edinboro University

**A mathematical discussion of what makes**

** good music sound “good”**

Abstract: Throughout history, mathematicians have argued whether mathematics drives or describes music. During the twentieth century the pendulum swung in favor of the “intellect over the ear” as musicians famously developed serial composition, then turned to probability theory to eliminate any hint of traditional melodic and harmonic structure. Music theorist Allen Forte, in his book *The Structure of Atonal Music*, proposed a musical "set theory" of pitch-class-set analysis analogous to mathematical set theory in an attempt to provide a method for the analysis of this atonal music.

By the final decades of the last century, music theorist John Clough became interested in extending Forte's atonal methodology to the diatonic system, the scale and interval patterns of our familiar tonal music. Among other goals, he hoped to advance discussion of how to classify with mathematics, music that is pleasing and interesting to the general ear.

This presentation will survey the building blocks, developed by Clough together with mathematician Jack Douthett, of some descriptive set theory which points toward characterizations of scales and intervals of tonal systems. Following this survey, we will review work of Godfried Toussaint who has applied similar techniques to analysis of musical rhythm. In particular, we will illustrate an isomorphism of scale and rhythm and move on to his exploration of the qualities that make a particular rhythm "good."

### October 10, 12:15 - 1:15, Luter 372

#### Dr. Yevgeniy Vasilyev

Christopher Newport University

## Model Theory: the geography of mathematics

#### October 24, 12:15 - 1:15, Luter 372

#### Dr. Paul Wrayno

Christopher Newport University

## Extremal and Saturation Graph Theory

#### November 7, 12:20 - 1:20, Luter 372

#### Dr. Olga Vasilyeva

Christopher Newport University

## Against the flow: population dynamics of stream insects and drift paradox

#### November 21, 12:15 - 1:15, Luter 372

#### TBA

*Biomathematics Semester*

at** **

Christopher Newport University

** **

*Department of Mathematics Colloquium*

** **

Spring 2013 Schedule

- February 14, 12:15 – 1:15, GOSN 202
**MODELING THE INFLAMMATORY RESPONSE**

**Dr. ****Angela Reynolds **Virginia Commonwealth University

- February 28, 12:15 – 1:15, GOSN 202

MATHEMATICAL ANALYSIS OF THE NEURAL CONTROL OF HORMONE SECRETION

- March 21, 12:15 – 1:15, GOSN 202
- March 28, 12:15 – 1:15, GOSN 202
- April 11, 12:15 – 1:15, GOSN 202

**Dr. Natalia Toporikova**

Washington and Lee University

PATIENT-SPECIFIC 3D RECONSTRUCTION OF A STENTED CORONARY ARTERY FOR ANALYSIS

OF CARDIOVASCULAR DISEASE

**Dr. ****Laura Ellwein **Virginia Commonwealth University

PATTERN FORMATION AND CHAOS IN BIOLOGY

**Dr. Heather Hardway**

Christopher Newport University

** **

** **

**Dr. Jing He**

Old Dominion University

## Dr. Heather Hardway

Christopher Newport University

Thursday, September 20, 2012 Gosnold 202

12:20 - 1:20 PM

Reaction-Diffusion Models for Dorsal-Ventral Patterning in Development