Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ?⁺ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > m _c = max {(N? 2)/N,0}, in which the limits of classical solutions are also continuous in ? ^N as extended functions with values in ?₊∪{∞}. We introduce a precise class of extended continuous solutions ? _c and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ? _c has an initial trace in ?⁺, and (iii) that the solutions in ? _c are limits of classical solutions. Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈??, t>0. Well-posedness is established for this problem for m _c < m > 1 when ? is any open subset of ? ^N and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=f ^q posed in any open set ?. For 1 > q > N/(N? 2)₊ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense. As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦m _c , where the limits of smooth solutions need not be extended-continuously.     

Simple approach for charge renormalization in highly charged macroions

We revisit the notion of renormalized charge, which is a concept of central importance in the field of highly charged colloidal or polyelectrolyte solutions. Working at the level of a linear Debye-Hückel-like theory only, we propose a versatile method to predict the saturated amount of charge renormalization, which is, however, a nonlinear effect arising at strong electrostatic coupling. The results are successfully tested against nonlinear Poisson-Boltzmann theory for polyions of various shapes (planar, cylindrical, and spherical), both in the infinite dilution limit or in confined geometry, with or without added electrolyte. Our approach, accurate for monovalent microions in solvents such as water, is finally confronted against experimental results on charged colloids and B-DNA solutions.     

Investigating nonlinear dynamics from time series: The influence of symmetries and the choice of observables

When a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators. (c) 2002 American Institute of Physics.     

Microelectrode study of single cavitational bubbles induced by 500 kHz ultrasound

Insight is gained into about the processes governing cavitational activity and acoustic streaming induced by high frequency (500 kHz) ultrasound by the use of microelectrodes with short time resolution electrochemical equipment to allow monitoring of the activity of single cavitating bubbles. Current transients are interpreted as showing the flux of solution towards the electrode surface due to microstreaming. In order to explain the current amplitude, a simplified model is produced. Important parameters such as bubble size and shape on the surface as well as the boundary layer thickness for microstreaming are taken into account. This model leads to the amplitude of the oscillations of the cavitating bubble. Introducing realistic bubble sizes, this amplitude is found to be in the order of 1 micron. The conclusions arising from this work allow a further interpretation of previous observations at millimeter scale electrodes.     

Combined protocol for the analysis of polycyclic aromatic hydrocarbons (PAHs) and polychlorobiphenyls (PCBx) from sediments using focussed microwave assisted (FMW) extraction at atmospheric pressure

A microwave-assisted protocol has been developed using focussed microwaves at atmospheric pressure for the extraction of PAHs and PCBs from sediments. It combines extraction and purification assisted by microwaves. This protocol has been applied to the quantification of 12 individual PAHs, 8 individual PCBs and 8 PCB coeluted mixtures from two Standard Reference Materials and one natural sediment. The results for both classes of compounds (PAHs and PCBs) are good in terms of recoveries, which are always greater than 70% and in most cases around 100%. The reproducibility is also good with coefficients of variation below 10% in most cases. This protocol has the great advantage of saving time, the time dedicated to the preparation / extraction assisted by microwaves being reduced to less than half an hour.     

Chromium (II) chloride : a new reagent for cross-aldol reactions

The proposed cross aldolization of α-bromoketones with aldehydes in presence of Chromium (II) chloride is an excellent stereospecific reaction. One exception to the stereospecific orientation is reported.     

Robust discretizations versus increase of the time step for the Lorenz system

When continuous systems are discretized, their solutions depend on the time step chosen a priori. Such solutions are not necessarily spurious in the sense that they can still correspond to a solution of the differential equations but with a displacement in the parameter space. Consequently, it is of great interest to obtain discrete equations which are robust even when the discretization time step is large. In this paper, different discretizations of the Lorenz system are discussed versus the values of the discretization time step. It is shown that the sets of difference equations proposed are more robust versus increases of the time step than conventional discretizations built with standard schemes such as the forward Euler, backward Euler, or centered finite difference schemes. The nonstandard schemes used here are Mickens' scheme and Monaco and Normand-Cyrot's scheme.     

Synthesis of the C12-C19 fragment of (+)-peloruside A through a diastereomer-discriminating RCM reaction

[reaction: see text]. A short and efficient asymmetric synthesis of the C12-C19 fragment of the cytotoxic macrolide (+)-peloruside A has been achieved via a highly diastereomer-discriminating RCM of alpha-branched but-3-enoate ester of a methallylic alcohol derived from hydrolytically resolved (S)-(-)-propylene oxide.     

Fusion of conventional ultrasound imaging and acousto-optic sensing by use of a standard pulsed-ultrasound scanner

Acousto-optic sensing (AOS) is a dual-wave sensing technique based on the ultrasound modulation of diffuse light in a turbid medium. We experimentally demonstrate the feasibility of combining AOS and conventional ultrasound imaging by use of a commercially available pulsed-ultrasound scanner coupled with a photorefractive crystal-based optical interferometry system. Optically absorbing targets embedded in highly diffusive phantoms (mus'= 10 cm(-1)) are imaged through a thickness of 27 mm with millimeter resolution. The acousto-optic images are intrinsically coregistered with the ultrasound images.