Rigorous Solution of the Gardner Problem

 We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between N spins which solve the problem of storing a given set of p random patterns. The Gardner formula for this volume in the limit N, p→∞, p/N→α is proven for all values of α. Besides, we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics. Received: 2 December 2001 / Accepted: 12 June 2002 Published online: 31 January 2003 Communicated by A. Kupiainen     

p-adic dynamic systems

Dynamic systems in non-Archimedean number fields (i.e., fields with non-Archimedean valuations) are studied. Results are obtained for the fields of p-adic numbers and complex p-adic numbers. Simple p-adic dynamic systems have a very rich structure—attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number p plays the role of a parameter of the p-adic dynamic system. Changing p radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear. Alexander von Humboldt Fellowship and SFB 237 Essen-Bochum-Düsseldorf, on leave from Moscow State Institute of Electronic Engineering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 3, pp. 349–365, March, 1998.     

A rigorous study of periodic orbits by means of a computer

We apply a modified version of the method of Sinai and Vul in order to study, by means of a computer, a closed orbit which appears in the five-mode model of bidimensional incompressible fluid on the torus.     

Behavior near the focal points of asymptotic solutions to the Cauchy problem for the linearized shallow water equations with initial localized perturbations

We study the behavior of the wave part of asymptotic solutions to the Cauchy problem for linearized shallow water equations with initial perturbations localized near the origin. The global representation for these solutions based on the generalized Maslov canonical operator was given earlier. The asymptotic solutions are also localized in the neighborhood of certain curves (fronts). The simplification of general formulas and the behavior of asymptotic solutions in a neighborhood of the regular part of fronts was also given earlier. Here the behavior of asymptotic solutions in a neighborhood of the focal point of the fronts is discussed in detail and the proof of formulas announced earlier for the wave equation is given. This paper can be regarded as a continuation of the paper in Russiian Journal of Mathematical Physics 15 (2), 192–221 (2008). In memoriam V.A. Borovikov     

Rigorous results for the free energy in the Hopfield model

We prove that the free energy of the Hopfield model with a finite number of patterns can be represented in terms of an asymptotic series expansion in inverse powers of the neurons number. The series is Borel summable for large temperatures. We also establish mathematically some other interesting properties, partly used before in a seminal paper by Amit, Gutfreund and Sompolinsky.     

Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory

According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of structural self-similarity and stability. They are the algebras of shock waves, narrow solitons, and square-root point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the square-root point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.     

Forecast of the trajectory of the center of typhoons and the Maslov decomposition

We predict the trajectory of the center of a typhoon by using the coordinates of the first three positions of the center. From this information, we obtain the initial distribution of the wind velocity using a neural network trained for solving this inverse problem. We take the wind field at the initial time as the sum of a smooth part and a singular part, according to the Maslov theory. This form of the field ensures the stability and self-similarity of the flow. The trajectory is found by solving the shallow water equations numerically. In some cases, the resulting trajectory approximates the actual trajectory fairly well. This work has been supported by the project DSTN 521241 (Dep. Physics, Univ. La Sapienza, Italy, DSTN)     

Gaussian Packets and Beams with Focal Points in Vector Problems of Plasma Physics

We consider a linearized equation describing plasma motion in a toroidal domain (tokamak) and study the asymptotic forms of steady-state solutions of the Gaussian beam type with a short wave length, which correspond to electric modes. We also study Gaussian wave packets and localized “cigar”-type beams describing the transmission of localized perturbations through the tokamak chamber. We separately consider the case of focal points on a trajectory and the asymptotic forms in a neighborhood of a focal point.