Applied Math Seminar

In the beginning was Green...

02/15/2012 - 1:15pm
02/15/2012 - 2:15pm
Pablo Amster (University of Buenos Aires)

We give a proof of Brouwer's fixed point theorem and related results
by means of basic calculus. More specifically, if the mapping is $ C^2 $,
our proof relies only on Green's Theorem.
Then, we prove the existence of solutions for some boundary value problems.
In particular, we shall give an elementary proof of a theorem by Lazer on
the existence of periodic solutions for a second order differential

RN 103

From swarming and self-assembly to vortex interaction: Applications of nonlocal PDE

01/18/2012 - 1:15pm
01/18/2012 - 2:15pm
David Uminsky (UCLA)

I will survey results in applications of a large class of nonlocal PDEs. The problems are of active scalar type and the applications are dictated by the nonlocal interaction kernel. We will begin by considering nonlocal kernels with a gradient, attraction-repulsion structure which arise in minimal models of biological swarming and self-assembly. We use tools from dynamical systems and analysis to develop the mathematical theory for predicting which patterns will arise. We also discuss solving the inverse problem of designing kernels for a given ground state structure. We conclude by turning to kernels which are incompressible in nature and discuss the development of a new convergent, higher order deformable vortex method for simulating fluids.

RN 103

Self-Intersections of Two-Dimensional Equilateral Random Walks

02/08/2012 - 1:15pm
02/08/2012 - 2:15pm
Nicholas Pippenger, HMC

We consider the classical "drunkard's walk" as a polygonal path of n steps in the plane, in which each side has unit length and the angles between successive sides are independent random variables uniformly distributed on the circle. How may self-intersections does such a walk have? We show that the mean number of self-intersections is $ (2/\pi^2) n log n+O(n) $, and that the variance is $ O(n^2) $ (which implies that the distribution is concentrated around its mean). We also obtain analogous results for closed polygons (walks conditioned to return to their origin after n steps), and discuss its possible extensions to other models.

This work was done jointly with Max Kutler and Maggie Rogers (both HMC Math '11).

RN 103

Social Network Clustering of Sparse Data

12/09/2011 - 1:15pm
12/09/2011 - 2:15pm
Blake Hunter, UCLA

Trillions of devices around the world are continuously
producing exabytes of data every day. The impact of search engines
has been enormous, but there is also a parallel development in the
applications of these methods to other related problems concerning the
extraction of knowledge from large datasets. Data mining is the
mathematics, methodologies and procedures used to extract knowledge
from large datasets. While this includes topics related to search
engines it is mainly devoted to the more general problem of finding
features and structure in a dataset. There are many active scientific
fields, including pure and applied mathematics, statistics, computer
science and engineering with numerous applications such as finance,
the social sciences, and the humanities. Spectral embedding uses
eigenfunctions of a Laplace operator (or related graph affinity
matrix) for extracting the underlying global structure of a dataset.
This talk will give an introduction to spectral embeddings.
Applications presented will include clustering LA street gang members
based on sparse observations of where and who they are seen with, and
automatically topic detection of Twitter tweets and Amazon product


The post-fragmentation density function for bacterial aggregates

11/16/2011 - 1:15pm
11/16/2011 - 1:25pm
Erin Byrne, HMC

Multicellular communities are a dominant, if not the predominant, form
of bacterial growth. Growing affixed to a surface, they are termed
biofilms. When growing freely suspended in aqueous environments, they
are usually referred to as flocs. Flocculated growth is important in
conditions as varied as bloodstream infections (where flocs can be
seen under the microscope) to algal blooms (where they can be seen
from low earth orbit). Understanding the distribution of floc sizes in
a disperse collection of bacterial colonies is a significant
experimental and theoretical challenge. One analytical approach is the
application of the Smoluchowski coagulation equations, a group of PDEs
that track the evolution of a particle size distribution over time.

The equations are characterized by kernels describing the result of
floc collisions as well as hydrodynamic-mediated fragmentation into
daughter aggregates. The post-fragmentation probability density of
daughter flocs is one of the least well-understood aspects of modeling
flocculation. A wide variety of functional forms have been used over
the years for describing fragmentation, and few have had experimental
data to aid in its construction. In this talk, we discuss the use of
3D positional data of Klebsiella pneumoniae bacterial flocs in
suspension, along with the knowledge of hydrodynamic properties of a
laminar flow field, to construct a probability density function of
floc volumes after a fragmentation event. Computational results are
provided which predict that the primary fragmentation mechanism for
medium to large flocs is erosion, as opposed to the binary
fragmentation mechanism (i.e. a fragmentation that results in two
similarly-sized daughter flocs) that has traditionally been assumed.

Davidson, CMC
Misc. Information: 


Parabolic Classical and Path-Dependent Partial Differential Equations: Stochastic Methods

04/11/2012 - 1:15pm
04/11/2012 - 2:15pm
Henry Schellhorn, CGU

In the first part of the talk I will review some well-known correspondences between parabolic partial differential equations (PDEs) and stochastic processes, in increasing order of generality. Whenever the solution exists and is unique: (i)  the fundamental solution of the heat equation is a Gaussian density, (ii) the solution of semilinear parabolic PDEs is given by the Feynman-Kac representation, (iii) the solution of quasilinear parabolic PDEs can be represented by the solution of forward-backward stochastic differential equations (FBSDE), while (iv) the solution of the general nonlinear parabolic PDE can be represented as the solution of a second-order FBSDE (2-BSDEs). 2BSDEs were introduced around 2006. The main tool for establishing these correspondences is the chain rule of stochastic calculus, known as Ito’s lemma.

In the second part of the talk, I describe path-dependent PDEs (PPDEs), which generalize classical PDEs. PPDEs correspond to FBSDEs where the terminal condition depends on the path. This is a very common problem in many applications ranging from stochastic control to mathematical finance. One way to view this problem as a PDE is to augment the state space, and describe it as an infinite system of PDEs. Another one, namely the PPDE formulation, was proposed by Dupire, and developed in Cont and Fournie (Journal of Functional Analysis 2011). It consists of introducing a functional derivative, which acts only on the “last coordinate” of the path. As a result a “functional” Ito’s formula can be obtained.

I conclude the talk by showing a new or independently rediscovered result for PPDEs, which generalizes the classical semi-group theory of parabolic PDEs, albeit only for “smooth” problems: the solution of a PPDE can be represented as an exponential of the initial condition. The generator of this exponential is equal to one half times the second-order Malliavin derivative, and thus the exponential can be easily calculated by Dyson series. Dyson series are an important tool in quantum field theory, but they are not so well-known in the area of stochastic processes and PDEs.


RN 103

Longitudinal librations of a satellite

11/09/2011 - 1:15pm
11/09/2011 - 2:15pm
Mario Martelli, CGU

Furi, Martelli and Landsberg gave a theoretical explanation of the chaotic longitudinal librations of Hyperion, a satellite of Saturn. The analysis was made under the simplifying assumption that the spin axis remains perpendicular to the orbit plane. Here, under the same assumption, we investigate the behavior of the longitudinal librations of any satellite. Also we show that they are possibly chaotic depending on two parameters: a constant k related to the principal moments of inertia of the satellite, and the eccentricity e of its orbit. We prove that the plane k-e contains an open region $\Omega$ with the property that the longitudinal librations of any satellite are possibly chaotic if the point (k,e) belongs to this region. Since Hyperion's point is inside $\Omega$, the results of this paper are more general than those obtained previously. 

Davidson, CMC

Transonic problems in multidimensional conservation laws

12/02/2011 - 3:00pm
12/02/2011 - 4:00pm
Eun Heui Kim, Cal State University Long Beach

 In many applications of engineering and physics, the model problems obey the conservation laws and sometime behave in a self-similar manner. A distinctive feature of multidimensional conservation laws in self-similar coordinates is that they change their type, meaning that they are hyperbolic (supersonic) far from the origin, and mixed (subsonic) near the origin. Hence the problem becomes transonic. In this talk, we first discuss the mathematical models and background of the transonic problems. We then discuss the recent development of mathematical analysis and numerical results of transonic problems.




Coincidence Degree Theory and Semilinear Problems at Resonance

02/22/2012 - 1:15pm
02/22/2012 - 2:15pm
Adolfo Rumbos, Pomona College

Coincidence degree theory was developed by Jean Mawhin in the early 1970s as an extension of the Leray-Schauder degree. The theory applies to study of equations of the form

$ Lu =N(u) +f,  $

where $ N\colon X\to Y $ is a continuous map between Banach spaces, $ X $ and $ Y $,

$  L\colon\mbox{Dom}(L)\subset X\to Y  $

is a Fredholm operator of index $ 0 $, and $ f\in Y $. In this talk we outline the coincidence degree theory in the context of its applications to certain semilinear two--point boundary value problems under conditions of resonance.

Roberts Hall North 103, Claremont McKenna College (Roberts Hall North is located northeast of Kravis Center on CMC's campus. Please see attached map for more information)

Modeling drug kinetics in slow-release tablets: mathematical approaches and challenges

01/25/2012 - 1:15pm
01/25/2012 - 2:15pm
Ami Radunskaya, Pomona College

Many drugs are most effective when their concentration in the blood is maintained at a fixed level. To obtain this concentration profile, pharmacists must create sustained-release tablets that deliver the drug as slowly as possible without waste. Matrix tablets made up of drug and polymer are a cost-effective solution, but a theoretical basis for the prediction of release curves is needed. In this talk we will describe several approaches to this problem, and describe some of the questions that still remain.

This work is in collaboration with Peter Hinow (U. of Wisconsin, Milwaukee).

RN 103
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