Past

We derive a two-dimensional compressible elasticity model for thin elastic sheets as a Gamma-limit of a fully three-dimensional incompressible theory. The energy density of the reduced problem is obtained in two

steps: first one optimizes locally over out-of-plane deformations, then one passes to the quasiconvex envelope of the resulting energy density. This work extends the results by LeDret and Raoult on smooth and finite-valued energies to the case incompressible materials. The main difficulty in this extension is the construction of a recovery sequence which satisfies the nonlinear constraint of incompressibility pointwise everywhere.

This is joint work with Sergio Conti.

1. It can generate flexible credit spread curves.
2. It leads to flexible implied volatility curves, thus providing a link between credit spread and implied volatility.
3. It implies that high tech firms tend to have very little debts.
4. It yields analytical solutions for debt and equity values.

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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.