The dynamics of flows in spatial networks, such as the pressure-driven blood flow in the human arterial network or the flow of cars in a traffic network, is most suitably described by PDE-based 'macroscopic' models. To cope with the computational complexity, often simplified models are employed, including at the level of individual particle tracking, usually called 'microscopic' models. Here we describe a mathematical model for blood flow in vascular networks, and compare numerical solutions of the underlying system of PDEs with those of a simplified models, based on pulse-tracking arguments (mesoscopic models). We then use these models to study flow optimization task, for variable size and/or topology of the network. Physiologically realistic control mechanisms are tested in the context of these simplified models.
September 18, 2014
The ubiquity of the Fibonacci Sequence: It comes up in the study of Leavitt path algebras too!
The majority of this talk should be quite accessible to math majors, to graduate students, even to math faculty: indeed, to anyone who has heard of the Fibonacci sequence ... Since its origin (more than eight centuries ago) as a puzzle about the number of rabbits in a (fantasmagorically expanding) colony, the Fibonacci Sequence 1,1,2,3,5,8,13,... has arguably become the most well-known of numerical lists, due in part to its simple recursion formula, as well as to the numerous connections it enjoys with many branches of mathematics and science. Since its origin (less than ten years ago), the study of Leavitt path algebras (a type of algebraic object which arises from directed graphs) has been the focus of much research energy throughout the mathematical world (well, at least throughout the ring-theory world), especially here at UCCS. In this talk we'll show how Fibonacci's sequence is naturally connected to data associated with the Leavitt path algebras of a natural collection of directed graphs. No prior knowledge about Leavitt path algebras will be required. [But in fact we will show how to compute the Grothendieck group $K_0(L(E))$ of the Leavitt path algebra $L(E)$ for a directed graph $E$, by considering only elementary-level properties of the graph. Those properties will lead us directly to Fibonacci. Plenty of easy-to-see examples will be given.] This is joint work with Gonzalo Aranda Pino of the University of Malaga (Spain). Many of you have met Gonzalo: he is a very frequent visitor to UCCS.
College of Charleston
October 2, 2014
UC 116 A
Integrable Curve Flows: the solitary travels of a vortex filament
The Vortex Filament Equation, describing the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable curve dynamics. Its connection with the cubing focusing Nonlinear Schrodinger equation through the well-known Hasimoto map allows the use of many of the tools of soliton theory to study properties of its solutions. I will discuss the construction of knotted solutions, their dynamics, and their stability properties.
University of Dayton
October 9, 2014
The center of rings associated to directed graphs
In 2005 Abrams and Aranda Pino began a program studying rings constructed from directed graphs. These rings, called Leavitt Path algebras, generalized the rings without invariant basis number introduced by Leavitt in the 1950's. Leavitt path algebras are the algebraic analogues of the graph C*-algebras and have provided a bridge for communication between ring theorists and operator algebraists. Many of the properties of Leavitt path algebras can be inferred from properties of the graph, and for this reason provide a convenient way to construct examples of algebras with a particular set of attributes. In this talk we will explore how central elements of the algebra can be read from the graph.
Jason P. Bell
University of Waterloo
October 23, 2014
Game theory and the mathematics of altruism
Game theory is a branch of mathematics that deals with strategy and decision making and is applied in economics, computer science, biology, and many other disciplines as well. We will discuss some of the basic points of game theory and discuss the so-called iterated prisoner dilemma, a game that is of central importance in the study of cooperation between individuals. We will then describe various strategies to this game and explain why altruism is something that can evolve naturally.
Myopic Models of Population Dynamics on Infinite Networks
Population models In mathematical biology often use equations blending diffusion (for movement) with local descriptions of population growth and multispecies interactions (reaction diffusion models). A modern problem is how to make sense of such models on gigantic networks such as the human population or the World Wide Web. One approach is to work in a space of functions which 'look flat' at 'infinity'. A correct formulation of this idea supports a theory of reaction-diffusion models on infinite networks where the network is compactified by adding points at infinity, diffusive effects vanish at infinity, and finite dimensional approximations can be described http://cmes.uccs.edu/temp/colloquium110614.mov.
November 20, 2014
Nonlinear magnetization dynamics in nanoparticles and thin films
Even the simplest magnetic system can undergo unusual nonlinear dynamics. In this talk I will discuss two magnetic systems that display unexpected nonlinear phenomena. Firstly, the magnetization dynamics in a nanoparticle will be detailed. It is found that the transient dynamics in this system can be made to persist for extremely long times when the nanoparticle is driven by oscillating magnetic fields at a very particular frequency and strength.  Secondly, thin magnetic films will be discussed and a perturbative expansion of nonlinear dynamic terms will be presented. In thin films, the threshold above which the system is driven nonlinear depends sensitively on the thickness of the film.  Connections to experiments will briefly be mentioned.  M.G. Phelps, K.L. Livesey and R.E. Camley, in preparation (2014).  K.L. Livesey, M.P. Kostylev and R.L. Stamps, Phys. Rev. B. 75, 174427 (2007).
CU Boulder App Math Dept
December 4, 2014
Experiments on Solitons, Dispersive Shock Waves, and Their Interactions
A soliton is a localized traveling wave solution to a special class of partial differential equations (integrable equations). A defining property of solitons is their interaction behavior. In his seminal work of 1968 introducing a notion of integrability (the Lax pair), Peter Lax also proved that the Korteweg-de Vries (KdV) equation admits two soliton solutions whose interaction behavior is quite remarkable. Two solitons interact elastically, i.e., each soliton maintains the same speed and shape post-interaction as they had pre-interaction. Moreover, Lax classified the interaction geometry into three categories depending on the soliton amplitude ratio. This talk will present a physical medium (corn syrup and water) modeled by the KdV equation in the weakly nonlinear regime that supports approximate solitons. Numerical analysis and laboratory experiments will be used to show that the three Lax categories persist into the strongly nonlinear regime, beyond the applicability of the KdV model. Additionally, a wavetrain of solitons called a dispersive shock wave in this medium will be described and investigated using a nonlinear wave averaging technique (Whitham theory) and experiment. Interactions of dispersive shock waves and solitons reveal remarkable behavior including soliton refraction, soliton absorption, and two-phase dynamics.
Over the last fifteen years, A unified approach has recently been developed to solve boundary value problems (BVPs) for integrable nonlinear partial differential equations (PDEs). The approach is a generalization of the inverse scattering transform (IST), which was originally introduced in the 1970's to solve initial value problems for such PDEs. Interestingly, this approach also provides a novel and powerful way to solve BVPs for linear PDEs. This talk will discuss the application of this method for linear PDEs. Specifically, we will look in detail at the solution of BVPs on the half line (0<x<infty) for linear evolution PDEs in 1 spatial and 1 temporal dimension. Time permitting, two-point BVPs, multi-dimensional PDEs and BVPs for linear elliptic PDEs will also be discussed.
A classical theorem of Shoda from 1936 says that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or stated another way, that evaluating the polynomial f(x,y)=xy-yx on matrices over K gives precisely all the matrices having trace 0. I will describe various attempts over the years to generalize this result.
Large equilibrium configurations of two-dimensional fluid vortices
The point-vortex equations, a discretization of the Euler equations, describe the motion of collections of two-dimensional fluid vortices. The poles and zeros of rational solutions to the Painleve II equation describe equilibrium configurations of vortices of the same strength and mixed rotation directions. There is an infinite sequence of such rational solutions with an increasing number of poles and zeros. In joint work with P. Miller (Michigan), we compute detailed asymptotic behavior of these rational functions with error estimates. Our results include the limiting density of vortices for these configurations. We will also describe how knowledge of the asymptotic behavior of the rational Painleve II functions is useful in understanding critical phenomena in the solution of nonlinear wave equations.
Solving non-linear dispersive equations by the method of inverse scattering:
The celebrated Korteweg-de Vries (KdV) equation and the nonlinear Schrodinger (NLS) equations are partial differential equation that describe the motion of weakly nonlinear long waves in a narrow channel. They predict "solitary waves" which do not disperse, which have been observed in nature, and used in many applications. In this lecture we'll talk about the "KdV miracle" of complete integrability that explains the solitary waves and establishes a remarkable connection between these equation and quantum mechanics. We will also discuss work in progress involving generalizations of the KdV and NLS equations to two space dimensions that describe surface waves and, like their one-dimensional counterpart, are completely integrable.
Dispersive shock waves and shallow ocean-wave line-soliton interactions:
Many physical phenomena are understood and modeled with nonlinear partial differential equations (PDEs). A special subclass of these nonlinear PDEs has stable localized waves -- called solitons -- with important applications in engineering and physics. I'll talk about two such applications: dispersive shock waves and shallow ocean-wave line-soliton interactions.
Dispersive shock waves (DSWs) occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg de Vries (KdV) equation is the universal model for phenomena with weak dispersion and weak quadratic nonlinearity. I'll show that the long-time asymptotic solution of the KdV equation for general step-like data is a single-phase DSW; the boundary data determine its form and the initial data determine its position. I find this asymptotic solution using the inverse scattering transform (IST) and matched-asymptotic expansions.
Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these nonlinear interactions look like an X or a Y or an H from above; much less frequently, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. I'll show photographs and videos of such interactions, which occur every day,close to low tide, on two flat beaches that are about 2,000 km apart. These interactions are related to the analytic, soliton solutions of the Kadomtsev Petviashvili equation, which extended the KdV equation to include transverse effects. On a much larger scale, tsunami waves can merge in similar ways.
Otterbein University-Western Ohio
November 7, 2013
Weight Problems in Harmonic Analysis, Especially the Fourier Transform:
Three important operators in harmonic analysis include the maximal operator, the singular integral operator, and the Fourier transform. A recurring problem in studying these operators is measuring the ''size" of an output function given some knowledge of the size of the input function--that is, finding the mapping properties of the operator. A further complication is introduced by using weighted measures of size. Determining whether an operator maps a weighted space into another weighted space is sometimes referred to as a ''weight problem" for the operator.
The weight problem is completely solved for the maximal operator,mostly solved for the singular integral operator, but unsolved for the Fourier transform. This is peculiar, since the Fourier transform is, in fact, the most widely used and oldest of the operators. In this talk I will review weight problems, their solutions, and focus especially on recent progress on the weight problem for the Fourier transform.
Large Deviations in The Reinforced Random Walk Model on Trees
In this talk, we consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. However,the lower tail is in polynomial decay for the linearly reinforced random walk model.