\notices\events\abstracts\differential-equations\woods.shtml
First, I'll give a very brief update on our nonlinear Krylov accelerator for the usual Modified Newton's method. This simple accelerator, which I devised and Neil Carlson implemented as a simple two page Fortran add-on to our implicit stiff ODEs solver, has been robust, simple, cheap, and automatic on all our moving node computations since 1990. I publicize further experience with it here, by us and by others in diverse fields, because it is proving to be of great general usefulness, especially for solving nonlinear evolutionary PDEs or a smooth succession of steady states.
Second, I'll report on some recent work in computerized tomography from X-rays. With colored computer graphics I'll explain how the standard "filtered backprojection" method works for the classical 2D parallel beam problem. Then with that backprojection kernel function H(t) we'll use an integral "change of variables" approach for the 2D fan-beam geometry. Finally, we turn to the tomographic reconstruction of a 3D object f(x,y,z) from a wrapped around cylindical 2D array of detectors opposite a 2D array of sources, such as occurs in PET (positron-emission tomography) or in very-wide-cone-beam tomography with a finely spaced source spiral.
Bacteriophage T4 is a virus that attacks bacteria by a unique mechanism. It
lands on the surface of the bacterium and attaches its baseplate to the cell
wall. Aided by Brownian motion and chemical bonding, its tail fibres stick to
the cell wall, producing a large moment on the baseplate. This triggers an
amazing phase transformation in the tail sheath, of martensitic type, that
causes it to shorten and fatten. The transformation strain is about 50%. With a
thrusting and twisting motion, this transformation drives the stiff inner tail
core through the cell wall of the bacterium. The DNA of the virus then enters
the cell through the hollow tail core, leading to the invasion of the host.
This is a natural machine. As we ponder the possibility of making man-made
machines that can have intimate interactions with natural ones, on the scale of
biochemical processes, it is an interesting prototype. We present a mathematical
theory of the martensitic transformation that occurs in T4 tail sheath.
Following a suggestion of Pauling, we propose a theory of an active protein
sheet with certain local interactions between molecules. The free energy is
found to have a double-well structure. Using the explicit geometry of T4 tail
sheath we introduce constraints to simplify the theory. Configurations
corresponding to the two phases are found and an approximate formula for the
force generated by contraction is given. The predicted behaviour of the sheet is
completely unlike macroscopic sheets. To understand the position of this
bioactuator relative to nonbiological actuators, the forces and energies are
compared with those generated by inorganic actuators, including nonbiological
martensitic transformations. Joint work with Wayne Falk, @email
Wayne Falk and R. D. James, An elasticity theory for self-assembled protein
lattices with application to the martensitic transformation in Bacteriophage T4
tail sheath, preprint.
K. Bhattacharya and R. D. James, The material is the machine, Science 307
(2005), pp. 53-54.
We obtain a new identity giving a quintuple law of overshoot, time of
overshoot, undershoot, last maximum, and time of last maximum of a general Levy
process at ?rst passage. The identity is a simple product of the jump measure
and its ascending and descending bivariate renewal measures. With the help of
this identity, we consider applications for passage problems of stable
processes, recovering and extending results of V. Vigon on the bivariate jump
measure of the ascending ladder process of a general Levy process and present
some new results for asymptotic overshoot distributions for Levy processes with
regularly varying jump measures.
(Parts of this talk are based on joint work with Ron Doney and Claudia
Kluppelberg)
In recent years, the use of random planar maps as discretized random surfaces has received a considerable attention in the physicists community. It is believed that the large-scale properties, or the scaling limit of these objects should not depend on the local properties of these maps, a phenomenon called universality.
By using a bijection due to Bouttier-di Francesco-Guitter between certain classes of planar maps and certain decorated trees, we give instances of such universality
phenomenons when the random maps follow a Boltzmann distribution where each face with degree $2i$ receives a nonnegative weight $q(i)$. For example, we show that under
certain regularity hypothesis for the weight sequence, the radius of the random map conditioned to have $n$ faces scales as $n^{1/4}$, as predicted by physicists and shown in the case of quadrangulations by Chassaing and Schaeffer. Our main tool is a new invariance principle for multitype Galton-Watson trees and discrete snakes.
The aging of spin-glasses has been of much interest in the last decades. Since its explanation in the context of real spin-glass models is out of reach, several effective models were proposed in physics literature. In my talk I will present how aging can be rigorously proved in so called trap models and what is the mechanism leading to it. In particular I will concentrate on conditions leading to the fact that one of usual observables used in trap models converges to arc-sine law for Levy processes.