Brown Bag Seminar
Archive

Fall 2007

August

08/24
Paul Dostert
Post-Doc, Pogram in Applied Mathematics
University of Arizona

: Applications of Sparse Grid Collocation to Uncertainty Quantification Problems in Porous Media Flows

Many porous media processes are affected by heterogeneities at various length scales as well as uncertainties. We present an approach for sampling permeability (conductivity) conditioned to an integrated response. The permeability fields can be characterized using the Karhunen-Loeve expansion which results in a parameterization of the uncertainty. In order to reduce the computational effort in sampling the permeability, we consider the use of sparse grid collocation methods in uncertainty space. Numerical results for uncertainty quantification applied to both two-phase immiscible flow and Richards' equation are presented.

Additional food will be provided by SIAM. SIAM organizational meeting will follow the Brown Bag. Bring your ideas!
08/31
Joe McMahon
Graduate Student, Program in Applied Mathematics
University of Arizona

A Continuum Model of a Growing Membrane

In 1978 two mathematicians started from the three-dimensional theory of elastic solids and derived a simplified set of equations describing the shape of a very thin elastic membrane in equilibrium. One of the creators later showed that, for a cylindrically symmetric membrane with a simple type of strain-energy function, no equilibrium configuration can be cavitated, or have a rip in it.

For a particular model of an elastic membrane with growth, however, I have numerical results that suggest that some equilibrium configurations can be cavitated.

I will present the rudiments of traditional elasticity theory as well a qualitative and physical introduction to the incorporation of growth into this formalism. I will review the results of the cylindrically symmetric membrane model without and with growth.

September

09/07
Brenae Bailey
Graduate Student, Program in Applied Mathematics
University of Arizona

Orbital Evolution of Centaurs

The Centaurs are a transient population of small bodies in the outer solar system which are thought to be a dynamical class intermediate between the Kuiper Belt and the Jupiter family comets. They suffer frequent close encounters with the giant planets and their orbits are strongly chaotic. We are investigating the chaotic behavior of these orbits. Our numerical analysis of the orbital chaos in these orbits shows two types of behavior: (1) intermittent resonance sticking characterized by sudden jumps from one mean motion resonance to another, which may have characteristics similar to Levy flights, and (2) random walks resulting in anomalous diffusion, in which the mean square deviation of the semimajor axis grows as t^H where t is time and H is different than 1/2. These results will constrain the possible origins and eventual fates of the Centaurs and will be applicable to the study of chaotic orbital evolution of other small body populations in the solar system.

09/14
Mark Robertson-Tessi
Graduate Student, Program in Applied Mathematics
University of Arizona

Mathematical Modeling of Tumor Growth

A model for tumor progression under chemotheraputic treatment, including interactions with the immune system, will be presented. Some of the difficulties and open questions will be discussed.

09/28
Serina Diniega
Graduate Student, Program in Applied Mathematics
University of Arizona

See dune, see dune run (into another dune!)

Dunes (those sandy structures seen in all desert movies and along many beaches) can be found wherever there is a supply of particles -- like sand or ice grains, and a wind to move them. This type of landform has been imaged on Earth, Mars, and even Titan (a moon of Saturn). This talk will present my current work towards understanding the evolution of a simplified two-dimensional dune. Additionally, investigations regarding different interactions between dunes will be discussed. Currently, the math tools used include dimensional analysis, Fourier-based stability analysis of coupled partial differential equations, reduced model approximation, and numerical simulation. This talk should be accessible to all graduate students and to undergraduates who have seen this type of analysis before, as it covers a simple application of material presented during the first year Applied Math core.

October

10/05
Dr. Moshe Dror
MIS, Eller College of Management
University of Arizona

Cooperative Inventory Games

In this talk we will examine a family of cooperative games termed Inventory Games. Leaving the noncooperative games to others, we focus on the cooperative facets of inventory games. Following a short motivational cooperative games examples the basic cooperative game terminology is introduced. Inventory games are naturally divided into deterministic inventory management and stochastic inventory environment. In the deterministic setting, of primary interest are the Economic Order Quantity (EOQ) like policies with the option for consolidated ordering. The main questions that require resolution are that of fair cost allocation. In the stochastic setting we examine Newsvendor like centralization games and their extensions.

10/12
Julia Arciero
Graduate Student, Program in Applied Mathematics
University of Arizona

Blood flow regulation - a matter of supply and demand

The ability of the circulatory system to adequately match blood supply to tissue demand implies the presence of regulatory mechanisms that communicate tissue status to blood vessels. One such mechanism involves red blood cells. Red blood cells have been shown to respond to low tissue oxygen levels by releasing ATP. The ATP triggers a conducted response in which an electrical signal travels upstream along vessel walls and causes arterioles to dilate so that more blood is supplied to the downstream regions of demand. A mathematical model is presented that predicts how blood flow is regulated by the conducted response as oxygen demand increases. The model complements experimental studies and provides a method for assessing the contributions of multiple mechanisms that coordinate changes in blood flow.

10/19
Ben Pittman-Polletta
Graduate Student, Program in Applied Mathematics
University of Arizona

Symmetry-breaking bifurcations in neuroscience

Golubitsky and Stewart have shown that when a dynamical system exhibits symmetry, the stable modes that result from symmetry-breaking bifurcations can be recovered from the structure of the symmetry group. This technique is particularly fruitful when analyzing networks of coupled oscillators. I will develop these techniques, along with some basic group theory. Then, we will take a look at a couple of applications to neuroscience. The first application is to the central pattern generators that control animal locomotion, and the second is to visual hallucinations arising in the cortex.

10/26
Bojan Durickovic
Graduate Student, Program in Applied Mathematics
University of Arizona

Perversions on Elastic Helices

Consider an elastic helix. By applying axial tension and moment, the helical shape is preserved... except if you reach a point when a different configuration is energetically preferable. Such configurations typically contain two helical parts of opposite handedness, joining at a point with zero torsion.

In this talk, I will lay down the basics of the framework for studying elastic helices, and show perversions on a phone cord.

November

11/02
Jared Barber
Graduate Student, Program in Applied Mathematics
University of Arizona

Red blood cell motion through diverging vessel bifurcations

In the circulatory system, red blood cells are not distributed uniformly. In fact, two vessels of the exact same size may contain significantly different percentages of red blood cells. One reason for this nonuniformity is that at diverging vessel bifurcations, where blood flows from one large vessel into two smaller branch vessels, the percentage of total blood entering one of the smaller vessels is not usually proportional to the percentage of red blood cells entering that vessel.

I have used a two-dimensional model to produce trajectories of red blood cells traveling through small diverging vessel bifurcations. This model has allowed us to investigate where red blood cells travel in vessel bifurcations as a function of percentage total blood entering either branch, the angles at which the vessels branch, and the relative size of the branches. Comparing our model predictions with experiments has offered us important insight into red blood cell motion and distribution.

11/16
Joe Stover
Graduate Student, Program in Applied Mathematics
University of Arizona

Advanced Sudoku Methods: Solving the Unsolvable

Sudoku is a logic puzzle which has risen in popularty over the past few years as evidenced by its regular appearance in an increasing number of newspapers. Most puzzles have unique solutions and can be solved using simple logic. Some however fail to be solvable using the simplest of logical rules and require more advanced methods in order to avoid the use of trial and error or backtracking. David Epstein, a computer scientist at UC Irvine, wrote a paper in 2005 involving graph theory and what are known as bilocation/bivalue plots for sudoku puzzles. This analysis has lead to advanced patterns which are recognizable by humans and make harder puzzles solvable. These methods can be used by computers also to search for complex patterns in order to solve puzzles without backtracking. In this talk I will outline some of the most useful advanced methods and give examples on where they are seen; the focus will be on using pencilmarks to look for patterns that allow logical eliminations to solve evil puzzles. You don't need any background in sudoku or mathematics to understand this talk, only logical reasoning is necessary. The goal is that you walk away with the ability to solve the hardest sudoku in the paper 'easily' using pencil marks and bb plot methods.

11/30
David Morales
Graduate Student, Program in Applied Mathematics
University of Arizona

The Computer of Life: a mathematical elucidation

This talk is intended for a general audience that loves to see the beauty and power of mathematics in the most important code that has ever existed; the code for the Computer of Life.