abstract

The problem of how one can learn from examples is illustrated on the case of a student perceptron trained by the Hebb rule on examples generated by a teacher perceptron. Two basic quantities are calculated the training error and the generalization error. The obtained results are found to be typical. Other training rules are discussed. For the case of an Ising student with an Ising teacher, the existence of a first order phase transition is shown. Special effects such as dilution, queries, rejection , etc. are discussed and some results for multilayer networks are reviewed. In particular, the properties of a selfsimilar committee machine are derived. Finally, we discuss the statistics of generalization, with a review of the Hoeffding inequality, the Dvoretzky Kiefer Wolfowitz theorem and the Vapnik Chervonenkis theorem.

abstract

Matrix models are a powerful technique to solve some models of statistical mechanics on random planar lattices. A simple introduction is given, to illustrate the basic analytical methods and a few applications.

abstract

We study the transport properties of two models of disordered media. In both cases we consider the motion of a particle which is not in thermal equilibrium with the environment. The first example is a hopping process in a one-dimensional lattice with random spacing, the second one is a random telegraph process in a random potential. We show, by computing the average stationary flux for a finite segment of the system, that the transport is subdiffusive if the temperature of the particle is lower than the temperature of the medium.

abstract

We discuss nonlinear effects which could be the basis for a theory of melting involving processes by which energy, initially evenly distributed in a lattice, can localize itself into large amplitude excitations. In a first part, we show how nonlinearity can affect the equipartition of energy in a lattice, and introduce some properties of nonlinear excitations and solitons. A second part investigates more precisely the question of self-localization of the energy and the third part presents a simple theory of DNA thermal denaturation. We show in particular how the addition of a specific type of nonlinear coupling can cause a sharp “melting” transition in this one-dimensional system.

abstract

A new algorithm which overcomes some specific difficulties arising in modeling of heterogeneous catalytic processes by cellular automata (CA) technique is proposed. The algorithm was tested with scheme introduced by Ziff, Gulari and Barshad and showed a good agreement with their results. The problem of the physical adequacy and interpretation of the algorithm was discussed.

abstract

The nonlinear development of thermal bistability in fluids is investigated. One-dimensional models of interacting fronts, treated with the help of singular perturbation theory yield for exponentially (in their size) long time on the way to phase separation. Spatial and temporal perturbation of the cooling may give rise to complex steady patterns. In two dimensions we perform numerical simulations and find for the purely thermal case and for the hydrodynamic case with open boundaries a qualitatively similar behavior — overall growth of the majority domain as a power of the time. Again, stationary complex pattern may be achieved by locking onto spatio-temporal perturbations. The hydrodynamic case differs from the purely thermal case in the value of the dynamical exponent (the power with which the correlation length grows in time). The probability of occurrence of critical conditions in an observed astrophysical system is assessed using statistical considerations.

abstract

A review of weak localization theory is presented in application to electron transport in semiconductors. While the theory qualitatively works well going into more microscopic level reveals difficulties. The important questions concern relaxing contact potential assumption by using weekly screened potential, effect of impurity correlations, and taking into account spatial distribution of impurities and wave function modulation. The origin of weak-antilocalization in strictly two-dimensional semiconductor systems is analyzed. There are two possible sources of this phenomenon: (1) crystalline: inversion asymmetry term and interface or Rashba term, and (2) resulting from the presence of many subbands. The first mechanism does not seem to agree with experimentally extracted dispersion relations for quasi two-dimensional systems. The second mechanism cannot explain a secondary maximum for magnetoresistance observed in some systems. In addition to need to study interaction effect, further progress can be achieved by using semiclassical theory with the correlated noise.