The Forwarding Indices of Random Graphs

A routing R of a graph G is a set of n(n ? 1) elementary paths R(u, v) specified for all ordered pairs (u, v) of vertices of G. The vertex-forwarding index ξ(G) of G, is defined by Where ξ(G, R) is the maximum number of paths of the routing R passing through any vertex of G and the minimum is taken over all the routings of G. Let G_p denote the random graph on n vertices with edge probability p and let m = np. It is proved among other things that, under natural growth conditions on the function p = p(n), the ratio Tends to 1 in probability as n tends to infinity.     

Circular orbits and relative strains in Schwarzschild space-time

The equation of the relative strain is analyzed in tetrad form with respect to a family of observers moving on spatially circular orbits, in the Schwarzschild space-time. We select a field of tetrads, which we term phase locking frames, and explicitly calculate how, in the equatorial plane, the orbital acceleration, its gradient and the Fermi drag add together to compensate the curvature and assure equilibrium among a set of comoving neighbouring particles. While equilibrium is achieved in the radial and azimuthal directions, in the direction orthogonal to the equatorial plane there is a residue of acceleration which pulls a particle towards that plane leading to a harmonic oscillation with a frequency equal to the proper frequency of the orbital revolution. This measurement, combined with those of the frequency shift of an incoming photon and the frequency of precession of the local compass of inertia, enables one to determine the relativistic ratio 2M/r, whereM is the gravitational mass of the source andr the coordinate radius of the circular orbits.     

Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures shifted by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.This paper is dedicated to Robert A. Minlos on the occasion of his 60th birthday.     

Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors

We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits.     

A multineuron interaction model for neural networks

A multineuron interaction model (RS model) with an energy function given by the product of the squared distances in phase space between the state of the net and the stored patterns is studied in detail within a mean-field approach. Two limits are considered: when the patterns and antipatterns are stored (as in the Hopfield model), PAS case, and when only the patterns are taken into account, OPS case. TheT=0 solutions for the proper memories are exactly obtained for all finite values of, as a consequence of the energy function: whenever one of the overlaps is exactly one the corresponding equations decouple and no configuration average is required. Special interest is focused on the OPS situation, which presents a peculiar phase space topology. On the other hand, the PAS configuration recovers the Hopfield model in the appropriate limit, while keeping associative memory abilities far beyond the critical values of other models when the full Hamiltonian is considered.