A general analysis of the limiting distribution of neural network functions is performed, with emphasis on non-Gaussian limits. We show that with i.i.d. symmetric stable output weights, and more generally with weights distributed from the normal domain of attraction of a stable variable, that the neural functions converge in distribution to stable processes. Conditions are also investigated und...

A weighted U -statistic based on a random sample X1, . . . ,Xn has the form Un = ∑1≤i,j≤n wi−jK(Xi,Xj ), where K is a fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U -statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a nonlinear time serie...

The completely 2, symmetric S-matrix defined by Belavin is shown to satisfy the Yang-Baxter equations. In the projective space of Boltzmann weights, the curves on which there exist commuting transfer matrices are shown to be embedded elliptic curves. Explicit polynomial equations for these curves are given. For n = 2 these results reduce to the results of Baxter for the symmetric eight-vertex m...

In this paper, we develop new results on consensus for continuous-time multi-agent systems. We introduce the assumptions of persistent connectivity of the interaction graph and of slow divergence of reciprocal interaction weights. Persistent connectivity can be considered as the counterpart of the notion of ultimate connectivity used in discrete-time consensus protocols. Slow divergence of reci...

Models of associative memory usually have full connectivity or if diluted, random symmetric connectivity. In contrast, biological neural systems have predominantly local, non-symmetric connectivity. Here we investigate sparse networks of threshold units, trained with the perceptron learning rule. The units are given position and are arranged in a ring. The connectivity graph varies between bein...

A weighted U -statistic based on a random sample X1, . . . , Xn has the form Un = ∑ 1≤i,j≤n wi−jK(Xi, Xj) where K is fixed symmetric measurable function and the wi are symmetric weights. A large class of statistics can be expressed as weighted U -statistics or variations thereof. This paper establishes the asymptotic normality of Un when the sample observations come from a non-linear time serie...

In the architecture of cellular neural networks (CNN), connections among cells are built on linear coupling laws. These laws are characterized by the so-called templates which express the local interaction weights among cells. Recently, the complete stability for CNN has been extended from symmetric connections to cycle-symmetric connections. In this presentation, we investigate a class of temp...

We study the existence of unitriangular basic sets for symmetric group which behave nicely with respect to Mullineux involution. Such give a natural labelling modular irreducible representations. show that, any odd prime p, p-blocks weight 2 have stable we describe by studying combinatorics partitions in these blocks.