The Averaging Method replaces a time-varying perturbation of a differential equation by a time-invariant one, while introducing only a relatively small error. The origin of the method goes back to calculations of stellar orbits; it has many modern applications in both modeling and numerical issues. A variety of mathematical tools have been developed in order to derive accurate estimates of the resulting errors. A new estimation criterion will be offered, which is based on rather "soft" estimates, namely, carrying out a comparison of integrals on a fast time scale. If, in addition, Young measures are employed, the method allows a natural extension to control systems, and carrying out the averaging in an environment with slowly moving averages.

The usual paradigm for encoding signals is based on the Shannon sampling theorem. If the signal is broad-banded then this requires a high sampling rate even though the information content in the signal may be small. Compressed Sensing is an attempt to get out of this dilemma and sample at close to the information rate. The fact that this may be possible is embedded in some old mathematical results in functional analysis, geometry and approximation. This talk will be an excursion into these topics which will focus on the relation between the number of samples we take of a signal and how well we can approximate the signal. It will take place in the discrete setting for vectors in Euclidean space. The talk should be understandable to graduate students and non specialists.

(Co-author: Egemen Kolemen) We give a self-contained modern linear stability analysis of a system of n equal mass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear approximation, compute all of the eigenvalues of the linear stability matrix, and finally derive inequalities that guarantee that none of these eigenvalues have positive real part. In the end, we rederive the result that J.C. Maxwell found for large n in his seminal paper on the nature and stability of Saturn’s rings, which was published 150 years ago. In addition, we identify the exact matrix that defines the linearized system even when n is not large. This matrix is then investigated numerically (by computer) to find stability inequalities. Furthermore, using properties of circulant matrices, the eigenvalues of the large 4n×4n matrix can be computed by solving n quartic equations, which further facilitates the investigation of stability. Finally, we have implemented an n-body simulator and we verify that the threshold mass ratios that we derived mathematically or numerically do indeed identify the threshold between stability and instability. Throughout the paper we consider only the planar n-body problem so that the analysis can be carried out purely in complex notation, which makes the equations and derivations more compact, more elegant and therefore, we hope, more transparent. The result is a fresh analysis that shows that these systems are always unstable for 2 <= n <= 6 and for n > 6 they are stable provided that the central mass is massive enough. We give an explicit formula for this mass-ratio threshold.

The full paper is posted here: orfe.princeton.edu/~rvdb/tex/saturn/ms.pdf (PDF)

Equi-energy sampler: From statistical inference to protein folding

A surge of recent research in machine learning and statistics has developed new techniques for finding patterns of words in document collections using hierarchical probabilistic models. These models are called "topic models" because the word patterns often reflect the underlying topics that are combined to form the documents; however topic models also naturally apply also such data as images and biological sequences.

After reviewing the basics of topic modeling, I will describe two related lines of research in this field, which extend the current state of the art.

First, I will describe probabilistic models designed to capture the dynamics of topics as they evolve over time. Many document collections change over time: scientific articles, emails, and search queries reflect evolving content, and it is important to model the corresponding evolution of the underlying topics.

Second, I will describe a probabilistic topic model which can capture correlations between the hidden topics. Previous models assume that the occurrence of the different topics are independent. In many document collections, however, the presence of a topic may be correlated with the presence of another. For example, a document about sports is more likely to also be about health than international finance.

In addition to giving quantitative, predictive models of a corpus, topic models provide a qualitative window into the structure of a large document collection. This allows a user to explore a corpus in a topic-guided fashion. I will demonstrate the capabilities of these new models on the archives of the journal Science, founded in 1880 by Thomas Edison. The models are built on the noisy text from JSTOR, an online scholarly journal archive, resulting from an optical character recognition engine run over the original bound journals.

(joint work with M. Jordan, A. Ng, and J. Lafferty)

The design and processing of materials with novel physical and mechanical properties requires a fundamental understanding of the connections between processing, microstructure, and properties. For example, mechanical properties in pure metals and alloys can be varied by manipulating the polycrystalline grain size or the size of the compositional domains through heat treatment, while elastic strain provides a way to tune the optical properties of self-assembled quantum dots during growth. In an analogous manner, the biological function of cell membranes is strongly affected by the details of the local "microstructure".

Typically, microstructural evolution takes place across multiple length and time scales, ranging from atomistic to mesoscopic ones. In this talk I will describe our recent efforts in developing physically-based, coarse-grained continuum models, which bridge the atomistic and mesoscopic scales, to elucidate lateral organization and non-equilibrium dynamics of heterogeneous lipid bilayers. In particular, I will focus on spatially organized, dynamic heterogeneities in the local lipid composition ("lipid rafts") which have been implicated in many important cellular processes including signal transduction, membrane trafficking, cytoskeleton organization, and pathogen entry.