Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

Exploratory analysis of human urine by LC–ESI-TOF MS after high intake of olive oil: understanding the metabolism of polyphenols

Olive oil polyphenols have important biological properties which closely depend on their bioavailability; it is, therefore, essential to understand how polyphenols are absorbed, metabolized, and eliminated from the body. An analytical method based on rapid-resolution liquid chromatography (RRLC) coupled with mass spectrometric detection with a time-of-flight analyzer (RRLC–ESI-TOF MS) has been developed for analysis of the main olive oil phenolic compounds and their metabolites in human urine. Urine samples from ten healthy volunteers were collected before and 2, 4, and 6 h after intake of 50 mL extra-virgin olive oil. The proposed method includes liquid–liquid extraction with ethyl acetate, which provides extraction recoveries of the phenolic compounds studied between 35 and 75% from spiked urine samples. Good repeatability was obtained—the relative standard deviations (RSDs) of peak areas in intra-day and inter-day studies were 4.3 and 6.5%, respectively. Statistical studies enabled us to discriminate between urine samples before and after intake, and facilitated the search for m/z values enabling this discrimination. Based on the very accurate mass information and the isotopic pattern provided by the TOF MS analyzer, together with other available information, ten of these biomarkers and more than 50 metabolites, obtained through phase I and phase II biotransformation reactions, were tentatively identified. Additionally, kinetic studies were conducted on the metabolites identified as possible biomarkers; for most of the compounds concentrations were maximum in the first two hours.     

Regularity of solutions to the polyharmonic equation in general domains

The present paper establishes boundedness of $[m-\frac{n}{2}+\frac{1}{2} ]$ derivatives for the solutions to the polyharmonic equation of order 2m in arbitrary bounded open sets of ${\mathbb{R}}^{n}$ , 2≤n≤2m+1, without any restrictions on the geometry of the underlying domain. It is shown that this result is sharp and cannot be improved in general domains. Moreover, it is accompanied by sharp estimates on the polyharmonic Green function.     

Layer potentials and boundary value problems for Laplacian in Lipschitz domains with data in quasi-Banach Besov spaces

We study the Dirichlet and Neumann boundary value problems for the Laplacian in a Lipschitz domain ${\Omega}$ , with boundary data in the Besov space ${B_{s}^{p,p} (\partial\Omega).}$ The novelty is to identify a way of measuring smoothness for the solution u that allows us to consider the case p < 1. This is accomplished by using a Besov-based nontangential maximal function in place of the classical one. This builds on the works of Jerison and Kenig [14], where the case p > 1 was treated.     

Boundedness of the Hessian of a Biharmonic Function in a Convex Domain

We consider the Dirichlet problem for the biharmonic equation on an arbitrary convex domain and prove that the second derivatives of the variational solution are bounded in all dimensions.     

Capillary electrophoresis-time of flight-mass spectrometry using noncovalently bilayer-coated capillaries for the analysis of amino acids in human urine

A capillary electrophoresis-time of flight-mass spectrometry (CE-TOF-MS) method for the analysis of amino acids in human urine was developed. Capillaries noncovalently coated with a bilayer of Polybrene (PB) and poly(vinyl sulfonate) (PVS) provided a considerable EOF at low pH, thus facilitating the fast separation of amino acids using a BGE of 1 M formic acid (pH 1.8). The PB-PVS coating proved to be very consistent yielding stable CE-MS patterns of amino acids in urine with favorable migration time repeatability (RSDs <2%). The relatively low sample loading capacity of CE was circumvented by an in-capillary preconcentration step based on pH-mediated stacking allowing 100-nL sample injection (i.e. ca. 4% of capillary volume). As a result, LODs for amino acids were down to 20 nM while achieving satisfactory separation efficiencies. Preliminary validation of the method with urine samples showed good linear responses for the amino acids (R(2) >0.99), and RSDs for peak areas were <10%. Special attention was paid to the influence of matrix effects on the quantification of amino acids. The magnitude of ion suppression by the matrix was similar for different urine samples. The CE-TOF-MS method was used for the analysis of urine samples of patients with urinary tract infection (UTI). Concentrations of a subset of amino acids were determined and compared with concentrations in urine of healthy controls. Furthermore, partial least squares-discriminant analysis (PLS-DA) of the CE-TOF-MS dataset in the 50-450 m/z region showed a distinctive grouping of the UTI samples and the control samples. Examination of score and loadings plot revealed a number of compounds, including phenylalanine, to be responsible for grouping of the samples. Thus, the CE-TOF-MS method shows good potential for the screening of body fluids based on the analysis of endogenous low-molecular weight metabolites such as amino acids and related compounds.     

Carbocyanines — derivatives of nitrogen-containing heterocycles with bridging groups in the chromophore

Tetramethylene-2,2′-bis(1,3,3-trimethyl-(3H)-indolium),-3-ethylbenzothiazolium,-1-methyl-quinolinium, and-1-methyl-and-1-butylbenz[c,d]indolium salts were synthesized starting from them trimethinecyanines containing an ethylene bridging group in the α, α′-positions of the chromophore have been prepared. These compounds are converted to their analogs containing a vinylene group by the action of benzoquinones. In the case of thiacarbocyanines the effect of the indicated bridging groups on the conformation and protonation route of the carbocyanines has been studied. The chromaticity of the dyes has been investigated. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 94–102, January, 2008.