A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

Solutions périodiques en temps des équations de Vlasov–Maxwell

We study here the existence of time periodic solution for the Vlasov–Maxwell equations in a three dimensional bounded domain. We assume that the boundary of the domain is strictly star-shaped. We give a priori estimates for the kinetic and electro-magnetic energy, and also for the normal and tangential traces of the electro-magnetic field. This method allows us to treat both classical and relativistic cases. To cite this article: M. Bostan, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Nanomaterials with antioxidant properties,obtained via biotechnology,using the solid state biosynthesis

ABSTRACT

Two nano materials type emulsion (EP and EM), that includes two different bioactive metabolites of Monascus sp. were obtained. Prior obtaining the nanoemulsion, tha enatural extracts, P (Monascus sp.1 – parental strain) and M (Monascus sp.2 - modified strain), were conditioned in order to increase their stability and biodisponibility. The new nanomaterials obtained were tested in terms of physical and chemical characteristics, by dynamic light scattering, infrared spectra and thermal analysis. Biological properties was investigated, by determining the effects of these new bio materials on wound healing and by measuring the antioxidant properties. The results obtained reveal the presence of aggregates with dimensions in the range of 1–100 nm, proving that these bioproducts are nano materials. The infrared spectra indicate the presence of benzene metadisubstituted (bands at 881.25 cm^?1 and 880.4 cm^?1 respectively), which are specific to Monascus metabolites Thermal analysis indicate the similarities between the matrix used in conditioning and the final nano products contains Monascus metabolites. The influence of these nano bio products on cicatrisation process was determined by tests on mices. It was concluded that there is small positive effect of the extracellular EP product. In this case, the bio product increase with 3% the rate of cicatrisation process, in comparison with the witness. All the two nano products show a significant antioxidant properties, with specific quenching rate of 97.7%     

Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).     

Birkhoff Completeness in Institutions

We develop an abstract proof calculus for logics whose sentences are ‘Horn sentences’ of the form: and prove an institutional generalization of Birkhoff completeness theorem. This result is then applied to the particular cases of Horn clauses logic, the ‘Horn fragment’ of preorder algebras, order-sorted algebras and partial algebras and their infinitary variants. the restriction of a logic to Horn sentences     

Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

We consider the viability problem for nonlinear evolutions inclusions of the form u′(t)?∈?Au(t)?+?F(u(t)), where A is an m-dissipative (possible nonlinear and multi-valued) operator acting in a Banach space X, K?X is a nonempty, locally closed set and $F:K{\user1{ \rightsquigarrow }}X$ is with nonempty, convex, closed and bounded values. We define the concept of A-quasi-tangent set to K at a given point ξ?∈?K and we prove a necessary condition for C ⁰-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the necessary condition is also sufficient. We extend the results to the quasi-autonomous case, we deal with the existence of noncontinuable or even global C ⁰-solutions and, as applications, we deduce a comparison result and a sufficient condition for null controllability.     

Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let $K:I\leadsto X$ be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârj? et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set $E\subseteq X$ is right tangent to at if $\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0$ . Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.     

Taylor dispersion analysis discloses the impairment of Aβ peptide aggregation by the presence of a fluorescent tag

The use of fluorescently tagged amyloid peptides, implicated in Alzheimer's disease, to study their aggregation at low concentrations is a common method; however, the fluorescent tag should not introduce a bias in the aggregation process. In this work, native amyloid peptides Aβ(1–40) and Aβ(1–42) and fluorescein-5-isothiocyanate (FITC), tagged ones, were studied using Taylor dispersion analysis coupled with a simultaneous UV and light-emitting diode-induced fluorescence detection, to unravel the effect of FITC on the aggregation process. For that, a total concentration of 100 µM of peptides consisting of a mixture of native and tagged ones (up to 10% in moles) was applied. Results demonstrated that FITC had a strong inhibition effect upon the aggregation behaviour of Aβ(1–42), whereas for Aβ(1–40), only a retardation in kinetics was observed. It was also shown that when mixed solutions of Aβ(1–40) and Aβ(1–42) are used, the Aβ(1–42) alloform was the leading peptide in the aggregation process, and when the latter was tagged, the aggregation kinetics decreased but the lifetime of potentially toxic oligomers was drastically increased. These results confirmed that the hydrophilicity of the N-terminus part of the peptide plays a major role in the aggregation process.