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Previous
Talks
May 3, 2011
Savanna Modeling and Species Invasion: Two Dynamical
Systems
Christopher
Strickland
In this talk, I summarize my research to date on
modeling savanna ecosystems and on modeling species invasion. First, I will motivate and
describe current efforts to model savanna dynamics while showing some details of a model
currently under development by ecologists at CSU. I will then describe my contribution to
this model, as well as my plans for further research this summer in Australia. Next, I will
motivate and describe current efforts to model the spread of an invasive species, building up
to our own model which is under development. In doing so, I will discuss relevant nonstandard
diffusion models, including some new formulations that are particular to our problem. This
talk will be non-technical, and hopefully will serve to describe several possible areas of
mathematical research which are of current relevance to ecologists.
April 19, 2011
A Stochastic Model of Programmed
Ribosomal Frameshifting in Viral Protein Synthesis
Brenae Bailey
The University of Arizona
Many viruses can produce different proteins from
the same RNA sequence by encoding them in overlapping genes. The mechanism that causes the
ribosomes of infected cells to decode both genes is called programmed ribosomal frameshifting
(PRF). Although PRF has beenrecognized for 25 years, the mechanism is not well understood. I will
present a model that treats RNA translation into proteins as a stochastic process in which the
transition probabilities are based on the free energies of local molecular interactions. The model
can reproduce observed translation rates and frameshift efficiencies, and can be used to predict
the effects of mutations in the viral RNA sequence on the frameshift efficiency.
March 9
2011
Bargaining over Productivity and Wages when Technical Change is
Induced: Implications for Growth, Distribution, and
Employment
Daniele
Tavani
In a simple one-sector economy
operating at full capacity, workers and firms bargain a la [Nash (1950)] over wages and
productivity gains taking into account the trade-o s faced by fi rms in choosing factor-augmenting
technologies. The aggregate environment resulting from optimal decision rules on wages,
productivity gains, savings and investment, is described by a two-dimensional dynamical system in
the employment rate and output/capital ratio. The economy converges cyclically to a long-run
equilibrium involving a Harrod-neutral pro file of technical change, a constant rate of employment
of labor, and constant input shares. The type of oscillations predicted by the model is
qualitatively consistent with the available data on the United States (1963-2003), replicates the
dynamics found in earlier models of growth cycles such as [Goodwin (1967)] [Shah and Desai (1981)],
[van der Ploeg (1987)], and is veri fied numerically in simulations. Institutional change, as
captured by variations in workers' bargaining power, has a positive effect on the rate of growth of
output per worker but a negative effect on employment. Economic policy can also aff ect the growth
and distribution pattern through changes in unemployment compensations, which also have a positive
impact on labor productivity growth but a negative impact on employment.
Feb 15
2010
Security from
Chaos
Francis Motta
A degree-three polynomial with a single parameter
is constructed and used to construct a discrete dynamical system that is topologically conjugate to
the shift map on three indices. For each positive integer greater than 1, a piecewise continuous
function on [0,1] with a single parameter is constructed and a topological conjugacy is
demonstrated between the shift map on n symbols and this function. Using the correspondance for n =
10, a theoretical algorithm is developed to construct dynamic passwords which depend on time. The
chaotic nature of the shift map and its sensititvity to initial conditions suggests a robust and
difficult-to-break dynamic password system.
Feb 22 2011
Roundtable Disscusion:
Experimental Mathematics Laboratory
Friday, Dec. 3
2010
Storing Cycles in Continuous Asymmetric
Hopfield-type Networks
Chuan Zhang
This report investigates conditions for cycles
of binary state vectors which can be stored in an asymmetric Hopfield-type neural network, and
analyzes bifurcations in the networks of a ring of neurons with nearest neighbor coupling. In a
simplified case, a sufficient condition for admissible cycles is proposed and proved. Using a set
of admissible cycles of the binary vectors, networks of a ring of neurons with nearest neighbor
coupling are constructed using the pseudoinverse learning rule. Bifurcations in these networks are
analyzed systematically, and the results show that networks of odd and even numbers have different
bifurcation structures.
Friday, Dec. 10 2010.
Chemical Oscillators
Melody
Dodd
The study of oscillating chemical reactions is an
active area of research in both chemistry and mathematics. These reactions oscillate as they
progress, producing cyclic color changes; the reaction mechanism can be modeled with a system
of ordinary differential equations. This presentation includes an overview of the history and
mathematics of chemical oscillators and a live demonstration of one such
reaction.
Friday, Oct 8
2010
Perturbations and Asymptotic Methods:
The Mathematical Nature of the Boundary Layer
Iuliana Oprea
In 1904, Ludwig
Prandtl, a little known physicist at that time, revolutionized fluid dynamic with his idea that the
effects of the friction are experienced only very near an object moving through a fluid. His only
eight pages seminal paper proved to be one of the most important fluid dynamics papers ever
written. We will discuss Prandtl's paradigm example of
the harmonic oscillator to illustrate the fundamental idea of boundary layer theory and its
mathematical nature, as well as to introduce the concept of singularly perturbed problems and basic
math tools to approach them. The mathematical level will not go beyond an undergrad course on
ordinary differential equations.
April 1, 2010
Fractals of the Degree-Two Standard
Family of Circle Maps
Christopher Strickland
In this talk, I seek to characterize the family of
one dimensional fractals formed by the infinite composition and rescaling of the degree-two
standard family. I will first build up some theory on circle maps and their infinite composition
while providing graphical examples from the degree-two standard family. Next, I will motivate the
examination of critical points and their dynamics under infinite composition, which will result in
a bifurcation diagram of critical point values on the circle. Time permitting, I will characterize
the different regions of the bifurcation diagram for qualitative information about the
fractals.
April 22, 2010
New results in nearly continuous Kakutani
equivalence
Bethany Springer
Advisor: Daniel Rudolph
It was previously known that a conjugacy between induced systems on
Polish probability spaces (X,T,\mu) and (Y,S, \nu) on nearly clopen subsets $A \subset X$ and $B
\subset Y$ of same relative measure can be extended to a measurable orbit equivalence between the
systems. This talk gives an algorithm for modifying the conjugacy in order to establish nearly
continuous Kakutani equivalence. We also show that in the case $\mu(A) > \nu(B)$, we can
establish conjugacy between $T$ and an induced system $S_{\bar B}$ where the systems are nearly
uniquely ergodic.
Don't worry, I will also give an introduction to the type of
dynamical systems I work with.
April 29, 2010
Steady State Hopf Mode Interaction in
Anisotropic Systems
Jennifer Maple
We present a
theoretical and numerical analysis of a system of four globally coupled complex Ginzburg-Landau
equations modeling steady oblique-normal Hopf mode interaction observed in experiments in
electroconvection of nematic liquid crystals.
March 4, 2010
Geometric Data Analysis: Techniques and
Iterations
Josh Thompson
We will begin with a discussion of a few common
methods of data analysis, and their motivations. We will end with an investigation into the
dynamics of a not-so-common method, the Multivariate State Estimation Technique. In the middle
hopefully we will learn something, have fun and see some interesting images.
March 11, 2010
Hopf Bifurcation in Anisotropic Reaction
Diffusion Systems Posed in Large Rectangles:
Travis Olson
The oscillatory instability (Hopf bifurcation) for
anisotropic reaction diffusion equations posed in large (but finite) rectangles is investigated.
For the case considered, the solution of the reaction diffusion system is represented in terms of
slowly modulated complex amplitudes of four wave-trains propagating in four oblique directions.
While for the infinitely extended system the modulating amplitudes are independent dynamical
variables, the finite size of the domain leads to relations between them induced by wave
reflections at the boundaries. This leads to a single amplitude equation for a doubly periodic
function that captures all four envelopes in different regions of its fundamental domain. The
amplitude equation is derived by matching an asymptotic bulk solution to an asymptotic boundary
layer solution. Preliminary numerical simulations show that the complexity of the solutions
significantly increases when the rescaled control parameter is increased.
March 25, 2010
Summary of some important types of intermittency:
Yang Zou
Intermittency is usually referred as the
occurrence of a signal alternating randomly between long regular phases and relatively short
irregular bursts. And sometimes it can be the intermittent switching between two or more chaotic
states. In this talk I will summaries some important types of intermittency, which includes
Pomeau-Manneville Type I, II, and III intermittency, Type X intermittency, Type V intermittency,
on-off intermittency, in-out intermittency, and some kinds of crisis-induced intermittency. Their
mechanisms will be briefly described. And some classical examples generating them will be given.
Statistical properties of these intermittencies will also be investigated finally.
Feburary 4, 2010
Patterns and Complexity: An
Introduction
Juliana Oprea
An open problem involving minimal
surfaces and phase separation
Patrick Shipman
Feburary 11, 2010
Tutorial on Symmetries and Differential
Equations
Patrick Shipman
Feburary 18,
2010
Hurricane
Model
Blake Rutherford
Feburary 25,
2010
Bubble
Elasticity
Dan Brake
Fall 09 I at Universiteit Leiden, Holland, I
studied foam elasticity in the Granular and Disordered Media group headed by Martin Van Hecke. We
discovered that near the jamming transition, both in a Couette cell and constrained linearly, a
submerged group of bubbles reacts nonlinearly to an applied compressive force.
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