Continuity of the Solution Map in Quadratic Programs under Linear Perturbations

It is well known that the solution map of a quadratic program where only the linear part of the data is subject to perturbation is an upper Lipschitz multifunction. This paper characterizes the continuity and lower semicontinuity of that solution map.This work was supported by the Brain Korea 21 Project in 2003, the APEC Postdoctoral Fellowships Program, and the KOSEF Brain Pool Program. The authors thank Professor F. Giannessi and two anonymous referees for helpful comments.Communicated by F. Giannessi     

Torus maps and the problem of a one-dimensional optical resonator with a quasiperiodically moving wall

We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d≥2 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. In particular, the Jacobian of this map has (d−1) eigenvalues equal to 1. We present rigorous and numerical results about several dynamical features of such maps. We also show how these dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem—which happens for a Cantor set of positive measure of parameters—the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region—which happens in open sets of parameters inside the gaps of the previous Cantor set—the energy grows exponentially.     

Models of the temporal dynamics of visual processing

Single unit recordings of neurons in primary visual cortex have demonstrated complex temporal patterns in the interspike interval return maps when presented with periodic input. Two models are tested to account for these patterns. An integrate-and-fire model is only able to replicate thein vivo data if its synaptic input is a chaotic function of time (such as a time series derived from the sinusoidally driven Duffing equation). Simpler purely periodic inputs are insufficient to replicate the experimental data. A Hodgkin-Huxley ionic model with a periodic input can replicate some of the features of the neural data, however it seems to be lacking as a complete model. These results indicate that thein vivo dynamics are not a result of the intrinsic properties of the neuron, but arise from a chaotic input to the neuron.     

Geometric Aspects of the Maximum Principle and Lifts over a Bundle Map

A coordinate-free proof of the maximum principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal extremals.     

The invariant densities for maps modeling intermittency

For a one-parameter family of maps modeling intermittency the explicit formula of the invariant density is presented.     

Homoclinic twisting bifurcations and cusp horseshoe maps

Chaotic dynamics arises when the unstable manifold of a hyperbolic equilibrium point changes its twist type along a homoclinic orbit as some generic parameter is varied. Such bifurcation points occur naturally in singularly perturbed systems. Some quotient symbolic systems induced from the Bernoulli symbolic system on two symbols are proved to be characteristic for this new mechanism of chaos generation. Combination of geometrical and analytical methods is proved to be more fruitful.     

Fractal dimensions and homeomorphic conjugacies

We investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes. When the homeomorphic conjugacy is not Lipschitz continuous, we discuss how its singularities can affect the whole set of generalized fractal dimensions. We give applications to homeomorphisms that conjugate critical circle maps with irrational (golden mean) winding numbers. We present numerical studies corroborating the theoretical predictions.     

Suppression of intra-channel four-wave mixing in 40Gbit/s RZ-DQPSK transmission with alternate-polarization

In this paper, we show that the suppression of intra-channel four-wave mixing (IFWM) in 40 Gbit/s RZ-DQPSK transmission system based on different dispersion maps through using alternate polarization of adjacent symbols. The suppression of IFWM results in significant improvement of system performance. Simulation results show that the amplitude fluctuation is efficiently reduced due to the suppression of the IFWM with alternate-polarization.     

Some remarks on the abundance of stable periodic orbits inside homoclinic lobes

We consider a family F_? of area-preserving maps (APMs) with a hyperbolic point H_? whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map (SM) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families F_? with central symmetry and close to an integrable limit, of the relative measure of the set of parameters ? for which F_? has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families F_? of APMs:

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The SM family, and we compare our analytical results with the numerical estimates.

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The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form.

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The conservative Hénon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands.

The results obtained can be seen as the quantitative analogs to those in Simó and Treschev (2008) [9], although here we deal with the a priori stable situation instead.     

Localisations of non prime e-vh-orders

Abstract

If A is an n × n complex matrix and x ∈ ? ⁿ , the conjecture is that if we take the kth power of each component of Ax, the resulting vector belongs to the range of the matrix obtained by taking the kth power of the entries of AA ^?, where A ^? is the adjoint of A. The conjecture is proved here for any k ≥ 2 when we add assumptions of either low dimension (namely, n ≤ 4) or low corank (0, 1, and, with some technical restrictions, 2). This problem arises in the study of the Jacobian Conjecture.