H(n) = uniform bound for the number of limit cycles of (1) . One way to formulate the Hilbert 16th problem is the following: Hilbert 16th Problem (HP). Estimate H(n) for any n ∈ Z+. To prove that H(1) = 0 is an exercise, but to find H(2) is already a difficult unsolved problem (see [DRR,DMR] for work in this direction). Below we discuss two of the most significant branches of research HP has ge...

In the present paper a realization of a classical method for Lyapunov quantities computation in Maple is considered. Key–Words: Lyapunov quantity, focus values, symbolic computation, small-amplitude limit cycles, Maple, Hilbert 16th problem

A continuous time, on-line training algorithm is presented, which learns to classify input patterns P in a Euclidean space R (or a Hilbert space) by attractors of a differential equation dx/dt = F(x) in R (or a Banach space). These nets include simple one-layer adeline-like nets as well as many recurrent nets. The attractors can be quite arbitrary—fixed points, limit cycles, fractals, etc.— but...

The suppression of constant-input limit cycles when designing second-order-state-space digital filters is addressed in this paper. A set of three sections yet proposed in the literature is revisited, focusing on the specific block called type III structure. It is shown that there is an optimum version of that section, what is the main contribution of the paper. The noise performance of the opti...

This paper deals with the problem of generating stable and robust oscillations in triangular nonlinear systems. The proposed method consists of two steps. First, a globally attractive oscillation is generated in a nominal second–order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential...

The aim of the paper is to introduce a two-parameter family of infinitedimensional diffusion processes X related to Pitman’s two-parameter PoissonDirichlet distributions P. The diffusions X are obtained in a scaling limit transition from certain finite Markov chains on partitions of natural numbers. The state space of X is an infinite-dimensional simplex called the Kingman simplex. In the speci...

We consider bosonized QCD2, and prove that after rewriting the theory in terms of gaugeinvariant fields, there exists an integrability condition that is valid for the quantum theory as well. Furthermore, performing a duality-type transformation we obtain an appropriate action for the description of the strong coupling limit, which is still integrable. We also prove that the model displays a com...

In this paper we consider the Hamiltonian of quantum electrodynamics, which describes the system of Dirac fields coupled to quantized radiation fields. We impose ultraviolet cutoffs on the Dirac field and the radiation field, and introduce spatial cutoffs to define the Hamiltonian as a self-adjoint operator on a boson-fermion Fock space. Taking a scaling limit of the Hamiltonian, we derive the ...

In this article, we investigate the question of equivalent keys for two Multivariate Quadratic public key schemes HFE and C∗−− and improve over a previously known result, to appear at PKC 2005. Moreover, we show a new non-trivial extension of these results to the classes HFE, HFEv, HFEv-, and C∗−−, which are cryptographically stronger variants of the original HFE and C∗ / MIA schemes. In partic...

We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We show how these processes arise, in a particular instance, as scaling limits of the REM-like trap model “at low temperature”, and subsequently derive...