Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

A NOTE ON OPEN 3-MANIFOLDS SUPPORTING FOLIATIONS BY PLANES

We show that if N, an open connected n-manifold with finitely generated fundamental group, is C-2 foliated by closed planes, then pi(1)(N) is a free group. This implies that if pi(1)(N) has an abelian subgroup of rank greater than one, then F has at least a nonclosed leaf. Next, we show that if N is three dimensional with fundamental group abelian of rank greater than one, then N is homeomorphic to T-2 x R. Furthermore, in this case we give a complete description of the foliation.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, A*) is an attractor-repeller pair for the attractor A of a semigroup {T(t) : t >= 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A*, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas. (C) 2012 Elsevier Ltd. All rights reserved.

Polar foliations and isoparametric maps

A singular Riemannian foliation F on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Sigma, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of H : M -> Sigma, coincide with the level sets of a smooth map H: M -> Sigma, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.

Theoretical study of the absorption and nonradiative deactivation of 1-nitronaphthalene in the low-lying singlet and triplet excited states including methanol and ethanol solvent effects

The photophysics of the 1-nitronaphthalene molecular system, after the absorption transition to the first singlet excited state, is theoretically studied for investigating the ultrafast multiplicity change to the triplet manifold. The consecutive transient absorption spectra experimentally observed in this molecular system are also studied. To identify the electronic states involved in the nonradiative decay, the minimum energy path of the first singlet excited state is obtained using the complete active space self-consistent field//configurational second-order perturbation approach. A near degeneracy region was found between the first singlet and the second triplet excited states with large spin-orbit coupling between them. The intersystem crossing rate was also evaluated. To support the proposed deactivation model the transient absorption spectra observed in the experiments were also considered. For this, computer simulations using sequential quantum mechanic-molecular mechanic methodology was used to consider the solvent effect in the ground and excited states for proper comparison with the experimental results. The absorption transitions from the second triplet excited state in the relaxed geometry permit to describe the transient absorption band experimentally observed around 200 fs after the absorption transition. This indicates that the T-2 electronic state is populated through the intersystem crossing presented here. The two transient absorption bands experimentally observed between 2 and 45 ps after the absorption transition are described here as the T-1 -> T-3 and T-1 -> T-5 transitions, supporting that the intermediate triplet state (T-2) decays by internal conversion to T-1. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4738757]

QUASI-TOPOLOGICAL QUANTUM FIELD THEORIES AND Z(2) LATTICE GAUGE THEORIES

We consider a two-parameter family of Z(2) gauge theories on a lattice discretization T(M) of a three-manifold M and its relation to topological field theories. Familiar models such as the spin-gauge model are curves on a parameter space Gamma. We show that there is a region Gamma(0) subset of Gamma where the partition function and the expectation value h < W-R(gamma)> i of the Wilson loop can be exactly computed. Depending on the point of Gamma(0), the model behaves as topological or quasi-topological. The partition function is, up to a scaling factor, a topological number of M. The Wilson loop on the other hand, does not depend on the topology of gamma. However, for a subset of Gamma(0), < W-R(gamma)> depends on the size of gamma and follows a discrete version of an area law. At the zero temperature limit, the spin-gauge model approaches the topological and the quasi-topological regions depending on the sign of the coupling constant.

An electrochemical gas sensor based on paper supported room temperature ionic liquids

A sensitive and fast-responding membrane-free amperometric gas sensor is described, consisting of a small filter paper foil soaked with a room temperature ionic liquid (RTIL), upon which three electrodes are screen printed with carbon ink, using a suitable mask. It takes advantage of the high electrical conductivity and negligible vapour pressure of RTILs as well as their easy immobilization into a porous and inexpensive supporting material such as paper. Moreover, thanks to a careful control of the preparation procedure, a very close contact between the RTIL and electrode material can be achieved so as to allow gaseous analytes to undergo charge transfer just as soon as they reach the three-phase sites where the electrode material, paper supported RTIL and gas phase meet. Thus, the adverse effect on recorded currents of slow steps such as analyte diffusion and dissolution in a solvent is avoided. To evaluate the performance of this device, it was used as a wall-jet amperometric detector for flow injection analysis of 1-butanethiol vapours, adopted as the model gaseous analyte, present in headspace samples in equilibrium with aqueous solutions at controlled concentrations. With this purpose, the RTIL soaked paper electrochemical detector (RTIL-PED) was assembled by using 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl) imide as the wicking RTIL and printing the working electrode with carbon ink doped with cobalt(II) phthalocyanine, to profit from its ability to electrocatalyze thiol oxidation. The results obtained were quite satisfactory (detection limit: 0.5 mu M; dynamic range: 2-200 mu M, both referring to solution concentrations; correlation coefficient: 0.998; repeatability: +/- 7% RSD; long-term stability: 9%), thus suggesting the possible use of this device for manifold applications.

Magnetic susceptibility study of Ce3+ in PbCeA (A=Te, Se, S)

The magnetic susceptibility of Pb(1-x)Ce(x)A (A=S, Se and Te) crystals with Ce3+ concentrations 0.006 <= x <= 0.036 was investigated in the temperature range from 2 K to 300 K. The magnetic susceptibility data was found to be consistent with a E-2(5/2) lowest manifold for Ce3+ ions with a crystal-field splitting Delta=E(Gamma(8))-E(Gamma(7)) of about 340 K, 440 K and 540 K for Pb1-xCexTe, Pb1-xCexSe, and Pb1-xCexS, respectively. For all the three compounds the doublet Gamma(7) lies below the Gamma(8) quadruplet which confirms the substitution of Pb2+ by Ce3+ ions in the host crystals. The observed values for the crystal-field splitting are in good agreement with the calculated ones based on the point-charge model. Moreover, the effective Lande factors were determined by X-band (similar to 9.5 GHz), electron paramagnetic measurements (EPR) to be g=1.333, 1.364, and 1.402 for Ce ions in PbA, A = S. Se and Te, respectively. The small difference with the predicted Lande factor g of 10/7 for the Gamma(7) (J=5/2) ground state was attributed to crystal-field admixture. (C) 2012 Elsevier B.V. All rights reserved.