The purpose of this work is a numerical verification of weak (or wave) turbulence theory. We made a simulation of the deep water surface gravity waves dynamics in the framework of the primordial dynamical equations. An effective numerical method conserving some of the properties of Hamilonian system was developed. We compare the results with the predictions following from the Hasselmann kinetic equation for waves.

Solitons in fiber optical systems with dispersion management have been studied intensively and put to use by the communication industry. In fall of 2005, experimental group in University of Rostock, Germany, showed that bound pairs of dispersion managed solitons may form a bound state — bisoliton — and propagate in such a way over long distances. In the limit of strong dispersion map NLSE is reduced to an integral equation. The naive iterative procedure is shown to be unstable. The method to stabilize the iterations is developed. Using this method we have found parametric bifurcation of bisoliton solutions.

A protein fold is usually represented by the position of their C_α. Some of these folds can be obtained accurately by experimental procedure such as X-ray crystallography. Here we introduce a continuous representation of protein folds that can be used to gain some insight on the geometry of proteins and consider large deformations and transformations of protein folds. The basic idea is to represent a protein fold as a sequence of helices passing through the C_α's. This approach is particularly well-suited to study either proteins with repetitive sequences or coiled-coils. In this talk I will explore the geometry and mechanics of this representation and show how it can be used to relate existing proteins through evolutionary paths or build proto-proteins which are possible candidates for protein design. I will also study the mechanics of fibrous proteins such as collagen and keratin which are made of helical proteins wound together to form coiled-coils. These superstructures have themselves a handedness dictated by the position of residues, external loadings, and their folding. I will revisit and generalize classical results by Crick to understand the chirality and mechanics of these structures. This is joint work with Andrew Hausrath and Sébastien Neukirch.

Some materials with crystal lattices of high symmetry (the “austenite” phase) at high temperature undergo phase transitions upon cooling and are left with lattices with fewer symmetries (the “martensite” phase). Applying a “lost” symmetry operation will then yield a distinct lattice, a “variant” of the low-symmetry lattice. When a body of such material is cooled, its high-symmetry lattice may be converted to different variants of the low-symmetry lattice in different regions. It is this phenomenon that gives rise to the “shape-memory” effect, in which a deformed solid appears to remember its undeformed shape. Until 1987, there was only a phenomenological theory of how arrangements of different variants of the low-symmetry phase arise. In 1987 a mathematical theory used continuum mechanics, crystal lattice theory, abstract analysis, and group theory to explain such formations as minimizers of the material's free energy. In this talk I will introduce some of the far-flung ideas that have been cobbled together (continuum mechanics shouldn't be applicable to crystal lattices, should it?) to create a field of study that now engages researchers in physics, engineering, and both analytical and numerical mathematics.

In this seminar, I will discuss computational modeling of an inter-protein electron transfer (ET) reaction, with emphasis on the mathematical and physical models employed in the modeling procedure. Interprotein ET reactions play an important role in biological energy conversion processes. One of these reactions, the ET between cytochrome c2 (cyt) and reaction center (RC) from photosynthetic bacteria, is the focus of this theoretical study. The proposed mechanism of cyt/RC binding involves an encounter complex (EC) stabilized by electrostatic interactions, followed by a transition state (TS), leading to the bound complex (BC) active in ET. The present work is a computational analysis to determine the ensemble configurations of the TS and EC and the molecular detail of the interprotein ET reactions. The EC ensemble was obtained by calculating the electrostatic interactions for a wide range of cyt/RC configurations. The TS ensemble was obtained by comparing the experimental data of the changes in the TS energies due to different mutations with the simulated differences in the electrostatic energies. The resulting TS ensemble is close to the average position of the EC ensemble, with strong electrostatic interactions between cyt and the RC surface. The similarity between the structures of the EC, TS, and BC can account for the rapid association. The changes in the ET rate constant during the association process were calculated as the cyt moved from the EC to the BC. The ET rate increased smoothly as the cyt approached from the EC to the BC, with a tunneling decay factor of 1.1 Å⁻¹. This relatively efficient coupling between redox centers is due to the ability of interfacial water molecules to form multiple strong hydrogen bonding pathways connecting tunneling pathways between the two proteins.

In imaging applications it frequently occurs that the subject to be imaged is larger than the field of view of the camera, necessitating the acquisition of many overlapping images to capture the region of interest in its entirety. Once acquired, these overlapping “tile images” must then be blended into a single “mosaic” image of the entire subject, ideally without visible artifacts such as distortion, ghosts, or seams. In practice, manual assembly of a high-quality mosaic is difficult and time-consuming due to variations in lighting conditions, camera orientation, focus, and exposure among the source images.

The talk will describe the algorithms employed by the presenter as part of an ongoing collaboration with The Art Institute of Chicago to develop software for the automatic assembly of mosaics from large sets of digitized infrared reflectographs and x-radiographs of paintings in the Art Institute's collection. Each of the following steps in the assembly process will be described: (1) location and fingerprinting of distinctive features in the source images; (2) detection of shared features among source images; (3) global optimization of the relative scale, location, and orientation of source images within the mosaic; (4) determination of the optimum blending mask across which overlapping images will be blended; and (5) multi-scale blending of source images into the final mosaic.

The talk will present background from the computer vision, computational geometry, and computer graphics fields, and will make extensive use of examples from paintings by Memling, Matisse, and Picasso.

Diffusion-limited aggregation (DLA), that could be thought as discretization of the Laplacian growth problem, may be simulated by conformal mapping models. The advantage of this approach is the potential absence of anisotropy in microscopic rules of growth, so one can control the noise arising from growing the cluster by finite size particles and the anisotropy of the growth rules independently. The method of growing clusters by conformal mapping (that has a lot of resemblance to SLE) could be also generalized to study models where the relative growth speed of tips and fjords of the cluster is deviating from DLA.

The behaviour of dry granular matter, like sand, is a continuing source of fascination. How does a heap of sand support itself? The answer is surprising and subtle. Shaken grains self-organize into spectacular lattice patterns and granular gases cool into galaxy-like clusters. Why? Unlike fluids, dry granular materials often stubbornly refuse to mix when shaken or stirred. Instead, they sort themselves by size or shape. These “segregation” effects are common in many industrial processes involving grains from cake mixes to gunpowder, but are only rather poorly understood. In this talk, I will describe experiments on granular mixtures that segregate when tumbled in a partially filled, horizontal rotating drum. The dynamical evolution of segregation can, under certain conditions, be oscillatory. Continuum models of this process posit two coupled fields which oscillate out of phase with one another. We examined several candidate fields and find that all are in phase, in contradiction to a recent order parameter model. We also studied the axial transport in the tube using narrow pulses as initial conditions. Surprizingly, we find that the process is subdiffusive, rather than diffusive as assumed in the models.

The aim of my research is to develop a simple mathematical model which replicates sand dune morphologies and behaviors, thus helping us understand the effects of the multi-scale physical processes involved. Such a model could be used to study dune fields in complex environments (such as fields climbing a mountain or in a varying wind field, on the Earth or other planets), and to predict relationships between different system parameters.

To help guide this research, I have studied different approaches applied to the study of granular material. This talk will briefly present two approaches to the avalanche problem: the BCRE model — a simple phenomenological (intuitional) model, and the St. Venant equations — a depth-averaged hydrodynamical model. I will then focus on a successful BCRE-type model for barchan (crescent) sand dunes developed by Sauermann, et al. (2001), and discuss its derivation, results, strengths, and weaknesses. Finally, I will outline the current structure and approach of my work.

The disk capacities have increased tremendously over the last couple of years, but are approaching its theoretical limit. Heat-Assisted Magnetic Recording (HAMR) is a promising technology that can take the storage density beyond this limit. Unlike the conventional recording schemes, in HAMR, the magnetic medium is heated by a laser before the data is recorded. The system characteristics depend both on the thermal and the magnetic properties of the medium. This talk will focus on the numerical model used to characterize the process of magnetization in a HAMR system in terms of these properties. Certain unique characteristics of the system investigated using this model will also be discussed.