Self assembly is the idea of creating a system whose component parts spontaneously assemble into a structure of interest. In this talk I will outline our research program aimed at creating self-assembled structures out of very small spheres, that bind to each other on sticking. The talk will focus on

(i) some fundamental mathematical questions in finite sphere packings (e.g. how do the number of rigid packings grow with N, the number of spheres);

(ii) algorithms for self assembly (e.g. suppose the spheres are not identical, so that every sphere does not stick to every other; how to design the system to promote particular structures);

(iii) physical questions (e.g. what is the probability that a given packing with N particles forms for a system of colloidal nanospheres); and

(iv) comparisons with experiments on colloidal nanospheres.

Many-body scattering problem is solved asymptotically when the size of the particles tends to zero and the number of the particles tends to infinity.
A method is given for calculation of the number of small particles and their boundary impedances such that embedding of these particles in a bounded domain, filled with known material, results in creating a new material with a desired refraction coefficient.
iThe new material may be created so that it has negative refraction, that is, the group velocity in this material is directed opposite to the phase velocity.
Another possible application consists of creating the new material with some desired wave-focusing properties. For example, one can create a new material which scatters plane wave mostly in a fixed given solid angle. In this application it is assumed that the incident plane wave has a fixed frequency and a fixed incident direction.
An inverse scattering problem with scattering data given at a fixed wave number and at a fixed incident direction is formulated and solved. Acoustic and electromagnetic (EM) wave scattering problems are discussed.

A.G.Ramm's vita, list of publications and some papers can be printed from the Internet address http://www.math.ksu.edu/~ramm

Turbulence is a state of flows which is characterized by a combination of chaotic and random behaviours affecting a very large range of scales. It is governed by Navier-Stokes equations and corresponds to their solutions in the limit where the fluid viscosity becomes negligible, the nonlinearity dominant and the turbulent dissipation constant. In this regime one observes that fluctuations tend to self-organize into coherent structures which seem to have their own dynamics.

A prominent tool for multiscale decomposition are wavelets. A wavelet is a well localized oscillating smooth function, e.g. a wave packet, which is translated and dilated. The wavelet transform decomposes a flow field into scale-space contributions from which it can be reconstructed.

We will show how the wavelet transform can decompose turbulent flows into coherent and incoherent contributions presenting different statistical and dynamical properties. We will then propose a new way to analyze and predict the evolution of turbulent flows.

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The presentation will use different results obtained in collaboration with:

Kai Schneider (Universite de Provence, Marseille, France),
Naoya Okamoto, Katsunori Yoshimatsu and Yukio Kaneda (Nagoya University, Japan)

Related publications can be downloaded from the web page
http://wavelets.ens.fr

The Schwarzchild, respectively the Kerr space-times are solutions for the vacuum Einstein equation which model a spherically symmetric, respectively a rotating black hole. In this talk I will discuss the decay properties of solutions to the linear wave equation on
such backgrounds.