A NOTE ON THE UNIQUENESS OF SOLUTIONS FOR THE YAMABE PROBLEM

Using recent results on the compactness of the space of solutions of the Yamabe problem, we show that in conformal classes of metrics near the class of a nondegenerate solution which is unique (up to scaling) the Yamabe problem has a unique solution as well. This provides examples of a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

Existence of Solutions for Abstract Non-Autonomous Neutral Differential Equations

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous neutral functional differential equations. An application to partial neutral differential equations is considered.

EXISTENCE OF SOLUTIONS FOR A FRACTIONAL NEUTRAL INTEGRO-DIFFERENTIAL EQUATION WITH UNBOUNDED DELAY

In this article, we study the existence of mild solutions for fractional neutral integro-differential equations with infinite delay.

STUDY BY MASS SPECTROMETRY OF SOLUTIONS OF [HYDROXY(TOSYLOXY)IODO]BENZENE: PROPOSED DISPROPORTIONATION MECHANISMS

STUDY BY MASS SPECTROMETRY OF SOLUTIONS OF [HYDROXY(TOSYLOXY)IODO]BENZENE: PROPOSED DISPROPORTIONATION MECHANISMS. Solutions of [hydroxy(tosyloxy)iodo]benzene (HTIB or Koser's reagent) in acetonitrile were analyzed using high resolution electrospray ionization mass spectrometry (ESI-MS) and electrospray ionization tandem mass spectrometry (ESI-MS/MS) under different conditions. Several species were characterized in these analyses. Based on these data, mechanisms were proposed for the disproportionation of the iodine(III) compounds in iodine(V) and iodine(I) species.

Multivariate spectroscopic methods for the analysis of solutions

In this thesis some multivariate spectroscopic methods for the analysis of solutions are proposed. Spectroscopy and multivariate data analysis form a powerful combination for obtaining both quantitative and qualitative information and it is shown how spectroscopic techniques in combination with chemometric data evaluation can be used to obtain rapid, simple and efficient analytical methods. These spectroscopic methods consisting of spectroscopic analysis, a high level of automation and chemometric data evaluation can lead to analytical methods with a high analytical capacity, and for these methods, the term high-capacity analysis (HCA) is suggested. It is further shown how chemometric evaluation of the multivariate data in chromatographic analyses decreases the need for baseline separation. The thesis is based on six papers and the chemometric tools used are experimental design, principal component analysis (PCA), soft independent modelling of class analogy (SIMCA), partial least squares regression (PLS) and parallel factor analysis (PARAFAC). The analytical techniques utilised are scanning ultraviolet-visible (UV-Vis) spectroscopy, diode array detection (DAD) used in non-column chromatographic diode array UV spectroscopy, high-performance liquid chromatography with diode array detection (HPLC-DAD) and fluorescence spectroscopy. The methods proposed are exemplified in the analysis of pharmaceutical solutions and serum proteins. In Paper I a method is proposed for the determination of the content and identity of the active compound in pharmaceutical solutions by means of UV-Vis spectroscopy, orthogonal signal correction and multivariate calibration with PLS and SIMCA classification. Paper II proposes a new method for the rapid determination of pharmaceutical solutions by the use of non-column chromatographic diode array UV spectroscopy, i.e. a conventional HPLC-DAD system without any chromatographic column connected. In Paper III an investigation is made of the ability of a control sample, of known content and identity to diagnose and correct errors in multivariate predictions something that together with use of multivariate residuals can make it possible to use the same calibration model over time. In Paper IV a method is proposed for simultaneous determination of serum proteins with fluorescence spectroscopy and multivariate calibration. Paper V proposes a method for the determination of chromatographic peak purity by means of PCA of HPLC-DAD data. In Paper VI PARAFAC is applied for the decomposition of DAD data of some partially separated peaks into the pure chromatographic, spectral and concentration profiles.

Spectral and variational characterizations of solutions to semilinear eigenvalue problems

Die vorliegende Arbeit befaßt sich mit einer Klasse von nichtlinearen Eigenwertproblemen mit Variationsstrukturin einem reellen Hilbertraum. Die betrachteteEigenwertgleichung ergibt sich demnach als Euler-Lagrange-Gleichung eines stetig differenzierbarenFunktionals, zusätzlich sei der nichtlineare Anteil desProblems als ungerade und definit vorausgesetzt.Die wichtigsten Ergebnisse in diesem abstrakten Rahmen sindKriterien für die Existenz spektral charakterisierterLösungen, d.h. von Lösungen, deren Eigenwert gerade miteinem vorgegeben variationellen Eigenwert eines zugehörigen linearen Problems übereinstimmt. Die Herleitung dieserKriterien basiert auf einer Untersuchung kontinuierlicher Familien selbstadjungierterEigenwertprobleme und erfordert Verallgemeinerungenspektraltheoretischer Konzepte.Neben reinen Existenzsätzen werden auch Beziehungen zwischenspektralen Charakterisierungen und denLjusternik-Schnirelman-Niveaus des Funktionals erörtert.Wir betrachten Anwendungen auf semilineareDifferentialgleichungen (sowieIntegro-Differentialgleichungen) zweiter Ordnung. Diesliefert neue Informationen über die zugehörigenLösungsmengen im Hinblick auf Knoteneigenschaften. Diehergeleiteten Methoden eignen sich besonders für eindimensionale und radialsymmetrische Probleme, während einTeil der Resultate auch ohne Symmetrieforderungen gültigist.

Existence of solutions and asymtptotic limits of the Euler-Poisson and the quantum drift-diffusion systems

My work concerns two different systems of equations used in the mathematical modeling of semiconductors and plasmas: the Euler-Poisson system and the quantum drift-diffusion system. The first is given by the Euler equations for the conservation of mass and momentum, with a Poisson equation for the electrostatic potential. The second one takes into account the physical effects due to the smallness of the devices (quantum effects). It is a simple extension of the classical drift-diffusion model which consists of two continuity equations for the charge densities, with a Poisson equation for the electrostatic potential. Using an asymptotic expansion method, we study (in the steady-state case for a potential flow) the limit to zero of the three physical parameters which arise in the Euler-Poisson system: the electron mass, the relaxation time and the Debye length. For each limit, we prove the existence and uniqueness of profiles to the asymptotic expansion and some error estimates. For a vanishing electron mass or a vanishing relaxation time, this method gives us a new approach in the convergence of the Euler-Poisson system to the incompressible Euler equations. For a vanishing Debye length (also called quasineutral limit), we obtain a new approach in the existence of solutions when boundary layers can appear (i.e. when no compatibility condition is assumed). Moreover, using an iterative method, and a finite volume scheme or a penalized mixed finite volume scheme, we numerically show the smallness condition on the electron mass needed in the existence of solutions to the system, condition which has already been shown in the literature. In the quantum drift-diffusion model for the transient bipolar case in one-space dimension, we show, by using a time discretization and energy estimates, the existence of solutions (for a general doping profile). We also prove rigorously the quasineutral limit (for a vanishing doping profile). Finally, using a new time discretization and an algorithmic construction of entropies, we prove some regularity properties for the solutions of the equation obtained in the quasineutral limit (for a vanishing pressure). This new regularity permits us to prove the positivity of solutions to this equation for at least times large enough.

Regularity of solutions of the p-Laplace equation in the Heisenberg group

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Multiplicity of data in trial reports and the reliability of meta-analyses: empirical study

Objectives To examine the extent of multiplicity of data in trial reports and to assess the impact of multiplicity on meta-analysis results. Design Empirical study on a cohort of Cochrane systematic reviews. Data sources All Cochrane systematic reviews published from issue 3 in 2006 to issue 2 in 2007 that presented a result as a standardised mean difference (SMD). We retrieved trial reports contributing to the first SMD result in each review, and downloaded review protocols. We used these SMDs to identify a specific outcome for each meta-analysis from its protocol. Review methods Reviews were eligible if SMD results were based on two to ten randomised trials and if protocols described the outcome. We excluded reviews if they only presented results of subgroup analyses. Based on review protocols and index outcomes, two observers independently extracted the data necessary to calculate SMDs from the original trial reports for any intervention group, time point, or outcome measure compatible with the protocol. From the extracted data, we used Monte Carlo simulations to calculate all possible SMDs for every meta-analysis. Results We identified 19 eligible meta-analyses (including 83 trials). Published review protocols often lacked information about which data to choose. Twenty-four (29%) trials reported data for multiple intervention groups, 30 (36%) reported data for multiple time points, and 29 (35%) reported the index outcome measured on multiple scales. In 18 meta-analyses, we found multiplicity of data in at least one trial report; the median difference between the smallest and largest SMD results within a meta-analysis was 0.40 standard deviation units (range 0.04 to 0.91). Conclusions Multiplicity of data can affect the findings of systematic reviews and meta-analyses. To reduce the risk of bias, reviews and meta-analyses should comply with prespecified protocols that clearly identify time points, intervention groups, and scales of interest.

On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.