DH 3rd floor SR

The Hopfield model took his name and its popularity within the theory

of formal neural networks. It was introduced in 1982 to describe and

implement associative memories. In fact, the mathematical model was

already defined, and studied in a simple form by Pastur and Figotin in

an attempt to describe spin-glasses, which are magnetic materials with

singular behaviour at low temperature. This model indeed shows a very

complex structure if considered in a slightly different regime than

the one they studied. In the present talk we will focus on the

fluctuations of the free energy in the high-temperature phase. No

prior knowledge of Statistical mechanics is required to follow the

talk.

Joe Doob, who died recently aged 94, was the last survivor of the

founding fathers of probability. Doob was best known for his work on

martingales, and for his classic book, Stochastic Processes (1953).

The talk will combine an appreciation of Doob's work and legacy with

reminiscences of Doob the man. (I was fortunate to be a colleague of

Doob from 1975-6, and to get to know him well during that year.)

Following Doob's passing, the mantle of greatest living probabilist

descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick

man.

We are interested in a microscopic stochastic description of a

population of discrete individuals characterized by one adaptive

trait. The population is modeled as a stochastic point process whose

generator captures the probabilistic dynamics over continuous time of

birth, mutation and death, as influenced by each individual's trait

values, and interactions between individuals. An offspring usually

inherits the trait values of her progenitor, except when a mutation

causes the offspring to take an instantaneous mutation step at birth

to new trait values. Once this point process is in place, the quest

for tractable approximations can follow different mathematical paths,

which differ in the normalization they assume (taking limit on

population size , rescaling time) and in the nature of the

corresponding approximation models: integro or integro-differential

equations, superprocesses. In particular cases, we consider the long

time behaviour for the stochastic or deterministic models.

The role of electron emission (either thermionic, secondary or

photoelectric) in charging an object immersed in a plasma is

investigated, both theoretically and numerically.

In fact, recent work [1] has shown how electron emission can

fundamentally affect the shielding potential around the object. In

particular, depending on the physical parameters of the system (that

were chosen such to correspond to common experimental conditions), the

shielding potential can develop an attractive potential well.

The conditions for the formation of the well will be reviewed, based

on a theoretical model of electron emission from the

grain. Furthermore, simulations will be presented regarding specific

laboratory, space and astrophysical applications.

[1] G.L. Delzanno, G. Lapenta, M. Rosenberg, Phys. Rev.

Lett., 92, 035002 (2004).

Time delays are associated with rock fracture and earthquakes. The delay associated with the initiation of a single fracture can be attributed to stress corrosion and a critical stress intensity factor [1]. Usually, however, the fracture of a brittle material, such as rock, results from the coalescence and growth of micro cracks. Another example of time delays in rock is the systematic delay before the occurrence of earthquake aftershocks. There is also a systematic time delay associated with rate-and-state friction. One important question is whether these time delays are related. Another important question is whether the time delays are thermally activated. In many cases systematic scaling laws apply to the time delays. An example is Omori92s law for the temporal decay of after shock activity. Experiments on the fracture of fiber board panels, subjected instantaneously to a load show a systematic power-law decrease in the delay time to failure as a function of the difference between the applied stress and a yield stress [2,3]. These experiments also show a power-law increase in energy associated with acoustic emissions prior to rupture. The frequency-strength statistics of the acoustic emissions also satisfy the power-law Gutenberg-Richter scaling. Damage mechanics and dynamic fibre-bundle models provide an empirical basis for understanding the systematic time delays in rock fracture and seismicity [4-7]. We show that these approachesgive identical results when applied to fracture, and explain the scaling obtained in the fibre board experiments. These approaches also give Omori92s type law. The question of precursory activation prior to rock bursts and earthquakes is also discussed. [1] Freund, L. B. 1990. Dynamic Fracture Mechanics, Cambridge University Press, Cambridge.20
[2] Guarino, A., Garcimartin, A., and Ciliberto, S. 1998. An experimental test of the critical behaviour of fracture precursors. Eur. Phys. J.; B6:13-24.20
[3] Guarino, A., Ciliberto, S., and Garcimartin, A. 1999. Failure time and micro crack nucleation. Europhys. Lett.; 47: 456.20
[4] Kachanov, L. M. 1986. Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, Netherlands.20
[5] Krajcinovic, D. 1996. Damage Mechanics, Elsevier, Amsterdam.20
[6] Turcotte, D. L., Newman, W. I., and Shcherbakov, R. 2002. Micro- and macroscopic models of rock fracture, Geophys. J. Int.; 152: 718-728.
[7] Shcherbakov, R. and Turcotte, D. L. 2003. Damage and self-similarity in fracture. Theor. and Appl. Fracture Mech.; 39: 245-258.

In this talk we discuss the analytic approximation to the loss

distribution of large conditionally independent heterogeneous portfolios. The

loss distribution is approximated by the expectation of some normal

distributions, which provides good overall approximation as well as tail

approximation. The computation is simple and fast as only numerical

integration is needed. The analytic approximation provides an excellent

alternative to some well-known approximation methods. We illustrate these

points with examples, including a bond portfolio with correlated default risk

and interest rate risk. We give an analytic expression for the expected

shortfall and show that VaR and CVaR can be easily computed by solving a

linear programming problem where VaR is the optimal solution and CVaR is the

optimal value.

Complete stochastic volatility models provide prices and

hedges. There are a number of complete models which jointly model an

underlying and one or more vanilla options written on it (for example

see Lyons, Schonbucher, Babbar and Davis). However, any consistent

model describing the volatility of options requires a complex

dependence of the volatility of the option on its strike. To date we

do not have a clear approach to selecting a model for the volatility

of these options

A version of Lyons theory of rough path calculus which applies to a

subclass of rough paths for which more geometric interpretations are

valid will be presented. Application will be made to the Brownian and

to the (fractional) support theorem.

The convergence to stationarity of many finite ergodic Markov

chains presents a sharp cut-off: there is a time T such that before

time T the chain is far from its equilibrium and, after time T,

equilibrium is essentially reached. We will discuss precise

definitions of the cut-off phenomenon, examples, and some partial

results and conjectures.