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.     

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.     

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.     

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.     

Gold nanoshell bioconjugates for molecular imaging in living cells

Advances in scattering-based optical imaging technologies offer a new approach to noninvasive point-of-care detection, diagnosis, and monitoring of cancer. Emerging photonics technologies provide a cost-effective means to image tissue in vivo with high resolution in real time. Advancing the clinical potential of these imaging strategies requires the development of optical contrast agents targeted to specific molecular signatures of disease. We describe the use of a novel class of contrast agents based on nanoshell bioconjugates for molecular imaging in living cells. Nanoshells offer significant advantages over conventional imaging probes including continuous and broad wavelength tunability, far greater scattering and absorption coefficients, increased chemical stability, and improved biocompatibility. We show that nanoshell bioconjugates can be used to effectively target and image human epidermal growth factor receptor 2 (HER2), a clinically relevant biomarker, in live human breast carcinoma cells.     

Phase-sensitive optical low-coherence reflectometry technique applied to the characterization of photonic crystal fiber properties

Localized measurements of group-velocity dispersion and birefringence of photonic crystal fibers are achieved with a phase-sensitive optical low-coherence reflectometry technique. This technique is efficient for fiber samples no longer than 1 m. Theoretical simulations are in good agreement with experimental results. As a result, the stress-induced birefringence proves to be at most 1 order of magnitude below the geometrical-shape birefringence.     

Theory of spin hall conductivity in n-doped GaAs

We develop a theory of extrinsic spin currents in semiconductors, resulting from spin-orbit coupling at charged scatterers, which leads to skew-scattering and side-jump contributions to the spin-Hall conductivity. Applying the theory to bulk n-GaAs, without any free parameters, we find spin currents that are in reasonable agreement with experiments by Kato et al. [Science 306, 1910 (2004)].