The main contribution is showing that the known convergence properties of the Hopfield model can be reduced to a very simple case, for which we have an elementary proof. The convergence properties of the Hopfield model are dependent on the structure of the interconnections matrix W and the method by which the nodes are updated. Three cases are known: (1) convergence to a stable state when opera...

We consider a supercritical branching process (Zn) in a random environment ξ. Let W be the limit of the normalized population size Wn = Zn/E[Zn|ξ]. We first show a necessary and sufficient condition for the quenched L (p > 1) convergence of (Wn), which completes the known result for the annealed L convergence. We then show that the convergence rate is exponential, and we find the maximal value ...

We describe a method of random generation of the integrals A 1;2 (t; t + h) = Z t+h t Z s t dw 1 (r)dw 2 (s) ? Z t+h t Z s t dw 2 (r)dw 1 (s) together with the increments w 1 (t+h)?w 1 (t) and w 2 (t+h)?w 2 (t) of a two-dimensional Brownian path (w 1 (t);w 2 (t)). The method chosen is based on Marsaglia's `rectangle-wedge-tail' method, gen-eralised to higher dimensions. The motivation is the ne...

The double Fourier series of functions of the generalized bounded variation class {n/ ln(n + 1)}∗BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class Λ∗BV, defined here, have quadrant limits at every point and, for f ∈ Λ∗BV, there exist at most countable sets P and Q such that, for x / ∈ P and y / ∈ Q, f is conti...

Hongmin Ren College of Information and Engineering, Hangzhou Polytechnic, Hangzhou 311402, Zhejiang, PR China Email:[email protected] Abstract.We present new sufficient semilocal convergence conditions for the Newton-Kantorovich method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Examples are given to show that our results apply but earlier on...

Suppose that A E (]J',n is a block p-cyclic consistently ordered matrix and let Band Sw denote , respectively, the black Jacobi and the block Symmetric Successive Overrelaxation (SSOR) iteration matrices associated with A. Neumaier a.nd Varga found (in the (p(IBI),w)-plane) the exact convergence and divergence domains of the SSOR method for the class of H-matrice8. Hadjidimos and Neumann applie...

W is the category of archimedean l-groups with distinguished weak order unit, with l-group homomorphisms which preserve unit. This category includes all rings of continuous functions C(X) and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in W. There is a rich theor...

For a simple class of Lagrangians and variational integrators, derived by time discretization of the action functional, we establish (i) the -convergence of the discrete action sum to the action functional; (ii) the relation between -convergence and weak∗ convergence of the discrete trajectories in W 1,∞(R;Rn); and (iii) the relation between -convergence and the convergence of the Fourier trans...

We study homogenization by Γ-convergence, with respect to the L1-strong convergence, of periodic multiple integrals in W 1,∞ when the integrand can take infinite values outside of a convex bounded open set of matrices.

In this paper, we mainly study the convergence properties of stochastic gradient descent (SGD) as described in Needell et al. [2]. The function to be minimized with SGD is assumed to be strongly convex. Also, its gradients are assumed to be Lipschitz continuous. First, we discuss the superior bound on convergence (of standard SGD) obtained by Needell et al. [2] as opposed to the previous work o...