02/15/2012 - 1:15pm

02/15/2012 - 2:15pm

Speaker:

Pablo Amster (University of Buenos Aires)

Abstract:

We give a proof of Brouwer's fixed point theorem and related results

by means of basic calculus. More specifically, if the mapping is ,

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

equation.

Where:

RN 103

01/18/2012 - 1:15pm

01/18/2012 - 2:15pm

Speaker:

David Uminsky (UCLA)

Abstract:

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.

Where:

RN 103

02/08/2012 - 1:15pm

02/08/2012 - 2:15pm

Speaker:

Nicholas Pippenger, HMC

Abstract:

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 , and that the variance is (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).

Where:

RN 103

12/09/2011 - 1:15pm

12/09/2011 - 2:15pm

Speaker:

Blake Hunter, UCLA

Abstract:

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

reviews.

Where:

Davidson

11/16/2011 - 1:15pm

11/16/2011 - 1:25pm

Speaker:

Erin Byrne, HMC

Abstract:

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.

Where:

Davidson, CMC

Misc. Information:

TBA

04/11/2012 - 1:15pm

04/11/2012 - 2:15pm

Speaker:

Henry Schellhorn, CGU

Abstract:

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.

Where:

RN 103

11/09/2011 - 1:15pm

11/09/2011 - 2:15pm

Speaker:

Mario Martelli, CGU

Abstract:

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 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 , the results of this paper are more general than those obtained previously.

Where:

Davidson, CMC

12/02/2011 - 3:00pm

12/02/2011 - 4:00pm

Speaker:

Eun Heui Kim, Cal State University Long Beach

Abstract:

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.

Where:

Davidson

02/22/2012 - 1:15pm

02/22/2012 - 2:15pm

Speaker:

Adolfo Rumbos, Pomona College

Abstract:

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

where is a continuous map between Banach spaces, and ,

is a Fredholm operator of index , and . 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.

Where:

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)

01/25/2012 - 1:15pm

01/25/2012 - 2:15pm

Speaker:

Ami Radunskaya, Pomona College

Abstract:

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).

Where:

RN 103

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