# Fall 2006

### Stabilizing Heegaard splittings of graph manifolds

#### Prof. Ryan Derby-Talbot

Dept. of Mathematics, University of Texas

#### Friday, 18 August at 2:30 in AB 635

**Abstract:** A useful way of studying 3 dimensional manifolds is to decompose them into two easily understood pieces via a Heegaard splitting. While this technique is quite old, there are fundamental questions about Heegaard splittings which remain largely unanswered. For example, is is unknown how quickly two Heegaard splittings of a given 3-manifold can be made the same via a certain basic move called stabilization. Using lots of pictures and intuition, we will investigate this problem for a class of 3-manifolds called graph manifolds.

### Ladies and Gentlemen...Introducing the Department of Mathematics & Statistics!

#### Thursday, 5 Oct at 2:30 in AB 635

In this special colloquium the Department will "introduce" itself, specifically to current and prospective graduate students. After some introductory comments, each of the departments' research "groups" will make a short presentation, in which they will describe (in very general terms) the research that their constituent members do. Graduate students will introduce themselves, and tell about the research areas in which they are interested. We enthusiastically invite any graduate student, prospective graduate student, or anyone else interested in finding out more about the math/stat department here at UNR.

### Asymptotics of Exceedance with Stationary Stable Steps Generated by Dissipative Flows

#### Tuncay Alparslan, Professor

#### Department of Mathematics and Statistics, University of Nevada

#### Thursday, 19 Oct at 2:30 in AB 635

**Abstract:** We analyze the exceedance probability of a high threshold for a random walk with a negative deterministic drift, where the steps constitute a stationary symmetric stable process. This is a popular problem in many fields including insurance, finance, and queuing theory among others. We refer to ergodic theory to split the step process into two independent processes. We focus on the processes generated by dissipative flows, which are known to have a mixed moving average representation, and we restrict our attention to regular moving averages with non-negative kernels. The cases of discrete-time and continuous-time steps are treated separately.

Department meeting with President Glick, no colloquium

Thursday, 26 October at 2:30 in AB 635

### An Election Model

#### Alexander Matros, Professor

#### Department of Economics, University of Pittsburgh

#### Tuesday, 31 October at 2:30 in AB 635

**Absatract:** We consider the following game. K players compete in N simultaneous contests. Each player i has a limited resource X(i) and must decide how to allocate it to N contests. In each contest, if player i allocates more resources than player j, player i has a higher chance to win the contest than player j. We find a unique Nash equilibrium in pure strategies. Each player competes in all contests in the equilibrium. If individual resources are private information, there exists a monotonic symmetric Bayesian equilibrium where each player competes in all contests. We analyze two extensions of the model: (1) players want to win two out of three states (majority game); and (2) the limited resources are costly. In both extensions, a unique Nash equilibrium is described. In extension (1), a player has the same chance to win each state in the equilibrium. In extension (2), the equilibrium spending depends on the number of unrestricted players, the total resources of the restricted players, and the total prize value of all contests. Applications for elections, R&D, arms races, military conflicts, simultaneous rent-seeking activities are discussed.

### Relationships between braid length and the number of braid strands

#### Cornelia Van Cott, Professor

#### Department of Mathematics, Indiana University

#### Friday, 3 November at 2:00 in AB 201

**Abstract:** Knot theory and braid theory were first connected in 1923 when Alexander proved that every knot or link can be deformed into a closed braid. The complicating twist is that this braid is not unique. In fact, every knot is isotopic to infinitely many distinct closed braids. We will study this infinite set of braids associated to a knot, focusing our study on braids which have a minimal number of crossings.

### The Laplace distribution and generalizations: Fundamental properties, applications, and recent developments

#### Tomasz Kozubowski, Professor

#### Department of Mathematics and Statistics, University of Nevada

#### Thursday, 9 November at 2:30

**Abstract:** In his memoir in 1774, P.S. Laplace introduced an error distribution that now bears his name. Since then, for many years the popularity of the Laplace distribution in stochastic modeling has been by far less than that of its four-years-older sibling the second law of Laplace, better known as the Gaussian (normal) distribution. It is only in recent years that this distribution, together with its various generalizations, has been revived, and is now being used in a variety of fields, including archaeology, biology, biostatistics, climatology, economics, environmental science, finance, geosciences, and physics. In this talk, we will review fundamental properties of the Laplace and related distributions, discuss their applications, and present some recent developments in this area.

### Nearest-neighbor analysis for marked point processes and aftershock identification problem

#### Ilya Zaliapin, Professor

#### Department of Mathematics and Statistics, University of Nevada

#### Thursday, 16 November at 2:30

**Abstract: **The centennial observations on the world seismicity have revealed a wide variety of clustering phenomena that unfold in the space-time-energy domain and provide the most reliable information about earthquake dynamics. Nevertheless, there is neither a unifying theory nor a convenient statistical apparatus that would naturally account for the different types of seismic clustering. Notably, there is no objective method for aftershock identification, the latter being an important part of seismic hazard analysis and earthquake forecasting. In case of point processes or fields clustering theory is conventionally based on the nearest neighbor concept; yet, such a theory does not transfer naturally to marked pont processes.

In this talk we present a theoretical analysis and new results for the ad hoc seismicity nearest neighbor approach developed by Baiesi and Paczuski (2004). Formally, we define an asymmetric pseudo-distance d in space-time-energy domain such that the nearest-neighbor spanning graph with respect to d becomes a time-oriented tree. We show that under natural assumptions the nearest neighbor distance dnn has Weibull distribution (the same as the Euclidean nearest-neighbor distance for point fields and processes) and demonstrate how the discussed approach can be used to detect earthquake clustering. We apply our analysis to the observed seismicity of California and show that the earthquake clustering part is statistically different from the homogeneous part. This finding serves as a basis for an objective aftershock identification procedure.

### Moodle - an On-line Course Management System

#### Lars Jensen, Professor

#### Departments of Mathematics and Physics, Truckee Meadows Community College

#### Thursday, 30 November at 2:30

**Abstract:** Moodle is on-line learning management system - a software package designed to help educators create quality online courses. Moodle is fast, and easy to use and navigate for the students. It allows instructors to build, manage and organize their classes through an intuitive point-and-click interface. Instructors "own" their courses so courses do not have to be deleted and re-created each semester. College administration can enroll students directly from a database.

Moodle is supported by a large community of developers and users. Through this community, technical support is available 24 hours a day, 7 days a week. Moodle, which is free software (GPL license), has the potential to save institutions tens of thousands of dollars in licensing fees on their current web platform software. This presentation includes an overview of Moodle. Topics included are creating assignments, student communication, user management and authentication, on-line group collaboration, integration with the WeBWorK math homework engine, and other unique course components.

### Scaling limits for workload process and Palm calculus

#### Gennady Samorodnitsky, Professor

#### School of Operations Research and Industrial Engineering, Cornell University

#### Thursday, 7 December at 2:30

**Abstract: **We discuss different scaling behavior of a very general telecommunications workload process. The activities of a telecommunication system are described by a marked point process $((T_n,Z_n))$, where $T_n$ is the arrival time of a packet brought to the system or the starting time of the activity of an individual source and the mark $Z_n$ is the amount of work brought to the system at time $T_n$. This model includes the popular ON/OFF process and the infinite source Poisson model. In addition to the latter models, one can flexibly model dependence of the inter-arrival times $T_n-T_{n-1}$, clustering behavior due to the arrival of an impulse generating a flow of activities but also dependence between the arrival process $(T_n)$ and the marks $(Z_n)$. Similarly to the ON/OFF and infinite source Poisson model one can derive a multitude of scaling limits for the workload process of one source or for the superposition of an increasing number of such sources. The memory in the workload depends on a variety of factors such as the tails of the inter-arrival times or the tails of the distribution of activities initiated at an arrival $T_n$ or the number of activities starting at $T_n$. It turns out that, as in standard results on the scaling behavior of workload processes in telecommunications, Fractional Brownian motion or infinite variance Levy stable motion can occur in the scaling limit. However, the Fractional Brownian motion is a much more robust limit than the stable motion, and many other limits may occur as well. Many of the questions above can be conveniently analyzed using Palm calculus of stationary marked point processes.Joint work with Thomas Mikosch (University of Copenhagen).