Asymptotically radial solutions in expanding annular domains

In this paper we consider the problem

$\left\{\begin{array}{ll}-\Delta u=u^{p}\quad {\rm in}\, \Omega_R,\\ u=0 \quad \quad \quad {\rm on}\, \partial\Omega_R,\quad\quad\quad (0.1)\end{array}\right.$

where p > 1 and Ω _R is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as \({R \longrightarrow \infty}\). The main goal of the paper is to prove, for large R, the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.

     

In Vitro Photodynamic Activity of 5,15-bis(3-Hydroxyphenyl)porphyrin and Its Halogenated Derivatives Against Cancer Cells

5,15-Diarylporphyrins ( 1-5 ) with hydroxyl groups and halogens as substituents were prepared by condensation between unsubstituted dipyrromethane and halogenated m -hydroxybenzaldehydes. Photophysical properties show that the nonhalogenated porphyrin 1 has higher fluorescence yield but lower singlet oxygen formation quantum yield than the halogenated derivatives due to the heavy atom effect. The in vitro activity of these derivatives was tested against WiDr colorectal adenocarcinoma and A375 melanoma cancer cells. All porphyrins present a much higher phototoxicity than Photofrin^® with IC ₅₀ values lower than the 50 n m level for WiDr cells and 25 n m level for A375 cancer cells. The most photoactive compound is the nonhalogenated porphyrin 1 which also presents the highest uptake. Halogenated derivatives present much lower uptakes than 1 . However, their photoactivity is similar to compound 1 showing that their intrinsic photoactivity (ISP) is very high. Iodinated compound 4 presents the highest ISP. The greater ability of these porphyrins to destroy cancer cells could be related to their photophysical and photochemical properties.     

The quality of pH measurements 100 years after its definition

Analytical chemistry is a scientific discipline based on the search of qualitative and quantitative information embodied in unknown samples which are representative of targeted systems. A. Lavoisier measured masses, S. Arrhenius discovered ions, namely hydrogen ions, S. Sørensen introduced pH and the chemical community has been dealing abundantly with them, first in the frame of classical analysis and later by means of instrumental methods of analysis responsible for large amount of data. Although statistical treatment of measurement results became part of the procedures, it took a few decades of thinking and globalization of the economy to realize that more was needed for the sake of evidence of quality; and this was found in the bases of the science of measurement and on the metrological approach. At the dawn of the twenty-first century, the most measured chemical parameter, pH, set the example of such hierarchical targeting of measurement results fit for a certain purpose, which is worth noticing particularly because 100 years have elapsed since its first definition and initial recommendation of the measurement procedure.     

Existence of multiple solutions for a wide class of differential inclusions†

We give a unified approach to study the existence of multiple positive solutions of nonlinear differential inclusions of the form $$-u^{\prime \prime }(t)\in F(t,u(t)),\text{a.e.}t\in (0,1),$$ subject to various nonlocal boundary conditions. We study these problems via a perturbed integral inclusion of the form $u(t)\in Bu(t)+{\int }_0^{1}k(t,s)F(s,u(s))ds$.     

A validated LC–MS/MS method for quantitative determination of L-dopa in Fagioli di Sarconi beans (Phaseolus vulgaris L.)

An analytical method based on ultrasound assisted extraction (UAE) and liquid chromatography coupled to electrospray tandem mass spectrometry (LC–ESI/MS/MS) was validated and applied for determining L-dopa in four ecotypes of Fagioli di Sarconi beans (Phaseolus vulgaris L.), marked with the European label PGI (Protected Geographical Indication). The selectivity of the proposed method was ensured by the specific fragmentation of the analyte. Simple isocratic chromatographic conditions and mass spectrometric detection in multiple reaction monitoring (MRM) acquisition mode were used for sensitive quantification. The LC–ESI/MS/MS method was validated within a linear range of 0.001–5.000 μg/mL. Values of 0.4 and 1.1 ng/mL were obtained for the limits of detection and quantification, respectively. The repeatability, inter-day precision, and recovery values ranges were 0.6%–4.5%, 5.4%–9.9%, and 83%–93%, respectively. Fresh and dried beans, as well as pods, cultivated exclusively with organic methods avoiding any synthetic fertilizers and pesticides were analyzed showing an L-dopa content ranging from 0.020 ± 0.005 to 2.34 ± 0.05 μg/g dry weight.