# Fall 2011

### Using Boolean Networks to Model Gene Interactions

#### Karen Schlauch and Bryson Wheeler

#### Biochemistry and Molecular Biology

University of Nevada, Reno

#### Thursday, December 8, 2011, 2:30-3:45pm, AB 102

**Abstract:** The analysis of large amounts of microarray data is a significant challenge for the researcher. The parallel assay of thousands of data points, not all of which are independent, across a number of temporal states, provides an interesting platform for statistical analyses and the construction of models. Complex networks are often used to model genetic regulatory systems or protein interaction networks, both represented by these types of data.

As a first approximation, Boolean networks can provide useful models of the underlying regulatory networks of a collection of genes (or proteins) over time (e.g., a pathway). Gene signals can be typically characterized as being on or off, lending perfectly to the Boolean model. Interactions among individual genes are inferred by examining gene expression levels measured over a series of temporal states. Often, more than one regulatory model may exist to represent the genes' interactions.

In this approach, we generate all possible Boolean network models that correctly represent the system under observation. Using published experimental data about the genes within the given network, as well as simple entropy measures, many of these network models can be quickly eliminated. The resulting network space includes biologically meaningful models that characterize possible structure and interaction patterns of the regulatory system.

### Inequalites: Theory of Majorization and its Applications

#### Ingram Olkin

#### Department of Statistics, Stanford University

#### Thursday, December 1, 2011, 2:30-3:45pm, AB 102

**Abstract:** There are many theories of "equations": linear equations, differential equations, functional equations, and more, However, there is no central theory of "inequations" There are several general themes that lead to many inequalities. One such theme is convexity. Another theme is majorization, which is a particular partial order. What us important in this context is that the partial order have lots of examples, and that the order-preserving functions be a rich class. In this case majorization arises in many fields: in mathematics:geometry, numerical analysis, graph theory; in other fields: physics, chemistry, political science, economics. In this talk we describe the origins of majorization and many examples of majorization and its consequences.

### Montesinos knots and the slice-ribbon conjecture

#### Ana Garcia Lecuona, Visiting Post-Doc

#### Penn State

#### Thursday, November 17, 2011, 2:30-3:45pm, AB 102

**Abstract:** The slice-ribbon conjecture states that a knot in the three sphere is the boundary of an embedded disc in the four ball if and only if it bounds a disc in the sphere which has only ribbon singularities. This conjecture was proposed by Fox in the early 70s. There doesn't seem to be any conceptual reason for it to be true, but large families of knots (i.e. pretzel knots, two bridge knots) satisfy it. In this colloquium we will show different ways to approach this conjecture with special emphasis in the case of Montesinos knots.

### Numerical Laplace Transform Inversion Methods with Selected Applications

#### Patrick Kano, Ph.D.

#### Applied Mathematics, U of Arizona

#### Cofounder, Acunum Algorithms and Simulations, LLC

#### Friday, November 4, 2011, 2:30-3:45pm, DMS 102

**Abstract:** Mathematical methods based on the use of the Laplace transform are a standard component of undergraduate engineering, mathematics, and physics education. Outside of the classroom however, real world problems often yield Laplace space solutions which are too complex to be analytically inverted to expressions in physically meaningful variables. A robust numerical inversion approach is thus desirable for these nontrivial cases. In this talk, I will present a few of the more common methods that have been developed to compute an approximate inverse. I will also discuss the inherent difficulties in performing numerical Laplace transform inversion. Finally, I will show through a selection of applications that these numerical inversion methods can be utilized to efficiently produce accurate results.

### A Gentle Introduction to Stable Distributions

#### John Nolan

#### American University, Washington, D.C.

#### Thursday, November 3, 2011, 2:30-3:45pm, AB 102

**Abstract:** Stable distributions are a class of heavy tailed probability distributions that generalize the Gaussian distribution and that can be used to model a variety of problems. An overview of univariate stable laws is given, with emphasis on the practical aspects of working with stable distributions. Then a range of statistical applications will be explored. If there is time, a brief introduction to multivariate stable distributions will be given.

### Energy exchange, localization, and transfer in finite oscillatory chains: Weak coupling approximation

#### Leonid I. Manevitch

#### Institute of Chemical Physics

Moscow, Russia

#### Thursday, September 22, 2011, 2:30-3:45pm, PE 208

**Abstract:** I will present a general approach to non-stationary dynamics of weakly coupled nonlinear oscillators. Two significant limiting cases can be distinguished. (i) Infinite (or very long) oscillatory chains can be considered in continuum approximation. Hence, they present appropriate objects for application of the ideas and methods of classical nonlinear field theory. Depending on the initial conditions, both wave-like (normal vibrations and waves) and particle-like (solitons) excitations may take place. However, for (ii) relatively short chains a different approach is required. The reason is that the formation of localized excitations and irreversible energy transfer is preceded by the stage of intensive energy exchange between groups of particles ("effective particles"). Maximal possible energy exchange occurs on the Limiting Phase Trajectory, which is a novel concept alternative to the quantum stationary state and the classical nonlinear normal mode. We obtain the description of the short chain in terms of "effective particles" when we reduce the oscillatory chain (in a certain frequency range) to a system of weakly coupled oscillators. Mathematically, the latter is similar to a multi-level quantum system. Then we determine the threshold for the transition from thestate of intensive energy exchange to the state of energy localization on an effective particle (or, possibly, energy transfer along the chain). Special attention will be paid to classical linear and nonlinear systems with variable parameters, whose mathematical description is similar to that for a quantumsystem in an external field. An outstanding example of such a quantum system is given by the Landau-Zener Tunneling. The very possibility of a unified description of quantum and classical systems clearly shows the asymptotic nature of the wave-particle duality.

Download a PowerPoint presentation of the talk here.

### High-Order Staggered Finite Difference Methods for Maxwell's Equations in Dispersive Media

#### Vrushali Bokil

#### Department of Mathematics, Oregon State University

#### Thursday, September 15, 2011, 2:30-3:45pm, AB 102

**Abstract:** We consider high order (in space) staggered finite difference schemes for Maxwell's equations coupled with a Debye or Lorentz polarization model. A novel expansion of the symbol of arbitrary (even) order finite difference approximations of the first order spatial derivative operator allows us to derive a concise formula for the numerical dispersion relation for all (even) order schemes applied to each model, including the limiting (infinite order) case. We further derive a closed-form analytical stability condition for these schemes as a function of the order of the method. Using representative numerical values for the physical parameters, we validate the stability criterion while quantifying numerical dissipation. Lastly, we demonstrate the effect that the spatial discretization order, and the corresponding stability constraint, has on the dispersion error.

### An Open Courseware for Numerical Methods

#### Autar K Kaw

#### Department of Mechanical Engineering

University of South Florida

#### Tuesday, September 6, 2011, 2:30-3:45pm, AB 102

**Abstract:** Funded by National Science Foundation, since 2001, an innovative open courseware (http://numericalmethods.eng.usf.edu) has been developed for a comprehensive undergraduate course in Numerical Methods. The topics include 1) Introduction to Scientific Computing, 2) Differentiation, 3) Nonlinear Equations, 4) Simultaneous Linear Equations, 5) Interpolation, 6) Regression, 7) Integration, 8) Ordinary Differential Equations, 9) Partial DifferentialEquations, 10) Optimization, and 11) Fast Fourier Transforms.

The open courseware resources enhance instructor preparation and development as well as the student educational experience by facilitating a hybrid educational approach to theteaching of Numerical Methods, a pivotal STEM course, via a) customized textbooks, b) adapted course websites, c) social networking via blogs and YouTube, d) YouTube and iTunes digital audiovisual lectures, e) concept inventory, f) self-assessment of the level of learning via online multiple-choice question tests and algorithm-based unlimited attempt quizzes, g) worksheets in a computational system of choice, and h) real-life applications based on the choice of one's STEM major.

The popularity of the open courseware is unprecedented. If you conduct a web search for "numerical methods", you will find that the courseware is ranked #2 on Google, #4 on Yahoo, and #4 on Bing. In 2010, there were

- 1,016,206 page views (330,597 visits) to http://numericalmethods.eng.usf.edu,
- 532,675 views of the audiovisual lectures on YouTube (http://youtube.com/numericalmethodsguy), and
- 79,034 visits to the "Numerical Methods Guy" blog (http://autarkaw.wordpress.com).

In this talk, the speaker will discuss the development, refinement, and assessment process of the open courseware. The assessment results will include those of comparing several instructional modalities, measuring student learning, effect of collecting homework for a grade, using online quizzes as a substitute for grading homework, and interpreting summative ratings of the courseware, student satisfaction, and Google analytics.

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