Singular fractional differential equations

Fractional differential equations have received increasing attention during recent years since the behavior of many physical systems can be properly described using the fractional order system theory. By fractional analog for Duhamel principle we give the existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions. In order to find the explicit solutions we use integral representation of the solution obtained via Laplace and Fourier transforms in succession and their inverses. We deal with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes, extending the results in viscoelasticity, continuum random walk, seismology, continuum mechanics and many other branches of life and science. The main task is finding existence-uniqueness results like in the case of evolution equations with entire derivatives. By examining the fractional evolution equations it turns out that they lead to till now known results from the evolution equations with entire derivatives in limiting case. They give more, behavior of the solution when order of derivatives are inside the intervals of entire points. In this way we can follow the influence of the operators generated by entire derivative in many fractional time evolution PDEs especially with singular initial data, and non-Lipschitz's nonlinear term. Apart from evolution equations we prove also an existence-uniqueness result for an initial value problem with singularities for linear and nonlinear fractional elliptic equation of Helmholtz type and fractional order α, where 1 < Re(α) ≤ 2, with respect to the one variable from R ₊. As a framework, we employ also Colombeau algebra of generalized functions containing fractional derivatives and operations among them in order to deal with the fractional equations with singularities. We apply the same techniques to the fractional Laplace and Poisson equation linear and nonlinear ones. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal switching for systems governed by nonlinear evolution equations

Optimal switching for a nonlinear evolution equation is considered. A type of approximating problems with value functions {u_u0<u<u₀} is studied. We prove that each u_u is the unique viscositY solution of corresponding Hami1ton-Jacoby system, u_u convergesto the value function u of the original problem as µ goes to 0, and an optimal control is determined.     

A Structure–Activity Study of Nickel NNN Pincer Complexes for Alkyl‐Alkyl Kumada and Suzuki–Miyaura Coupling Reactions

A new series of Ni NNN pincer complexes were synthesized and characterized. The main difference among these complexes is the substituents on the side arm amino group(s). No major structural difference was found except for the C–N–C angle of the various substituents and the ‘pseudo bite angle’ of the complexes. Four new complexes were efficient for the alkyl‐alkyl Kumada reaction of primary alkyl halides, and among them, one complex was also efficient with secondary alkyl halides. The influence of the substituents on the catalytic performance of the Ni complexes in alkyl‐alkyl Kumada and Suzuki–Miyaura cross‐coupling reactions was systematically investigated. No correlation was found between the catalytic activity and the key structural parameters (C–N–C angle and ‘pseudo bite angle’), redox properties or Lewis acidity of the complexes.     

Analysis of quaternary ammonium and phosphonium ionic liquids by reversed-phase high-performance liquid chromatography with charged aerosol detection and unified calibration

Several hydrophobic ionic liquids (ILs) based on long-chain aliphatic ammonium- and phosphonium cations and selected aromatic anions were analyzed by reversed-phase high-performance liquid chromatography (RP-HPLC) employing trifluoroacetic acid as ion-pairing additive to the acetonitrile-containing mobile phase and adopting a step-gradient elution mode. The coupling of charged aerosol detection (CAD) for the non-chromophoric aliphatic cations with diode array detection (DAD) for the aromatic anions allowed their simultaneous analysis in a set of new ILs derived from either tricaprylmethylammonium chloride (Aliquat 336) and trihexyltetradecylphosphonium chloride as precursors. Aliquat 336 is a mix of ammonium cations with distinct aliphatic chain lengths. In the course of the studies it turned out that CAD generates an identical detection response for all the distinct aliphatic cations. Due to lack of single component standards of the individual Aliquat 336 cation species, a unified calibration function was established for the quantitative analysis of the quaternary ammonium cations of the ILs. The developed method was validated according to ICH guidelines, which confirmed the validity of the unified calibration. The application of the method revealed molar ratios of cation to anion close to 1 indicating a quantitative exchange of the chloride ions of the precursors by the various aromatic anions in the course of the synthesis of new ILs. Anomalies of CAD observed for the detection of some aromatic anions (thiosalicylate and benzoate) are discussed.     

Photoluminescence spectroscopy of CdSe nanoparticles embedded in transparent glass

In this paper we present photoluminescence measurements of CdSe nanoparticles embedded in transparent glass. Sample is prepared using an original technique, which combines both heat treatment and ultraviolet laser irradiation. Photoluminescence spectra displayed one main emission band at 2.14 eV. We identify this bands energy as basic interband transition in CdSe nanoparticle. We calculated energy of basic (1s_h–1s_e) transition in spherical CdSe quantum dot (QD), within infinite potential barrier, in effective-mass approximation. On the basis of this model, average radius of synthesized CdSe QDs is about 3 nm, which is in consistence with AFM measurements and UV–VIS absorption measurements.     

Risk premium and fair option prices under stochastic volatility: the HARA solution

We have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For a given HARA utility, the ‘risk premium’, i.e., the ‘market price of volatility risk’ is determined via a solution of a certain nonlinear PDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE and a Black–Scholes type PDE. To cite this article: S.D. Stojanovic, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Physical Science in Bologna

We provide a guide to Bologna, Italy, focusing particularly on sites of interest to physicists. Our first tour is in the city center; it begins in the Piazza Maggiore at the Palazzo d’Accursio, the Basilica di San Petronio, and the Archiginnasio (Old University) and then proceeds to the Two Towers and the Palazzo Poggi, which houses the Astronomical Observatory Museum and other important instrument and art collections; it concludes at the Physics Museum, Department of Physics and Astronomy. Our second tour again begins in the Piazza Maggiore but goes to sites beyond the city center where famous Bolognese physicists and other scientists were born, lived, and are buried. Finally, we point out important museums and other institutions on the outskirts of Bologna.     

Mechanistic Insights into Cu-Catalyzed Asymmetric Aldol Reactions: Chemical and Spectroscopic Evidence for a Metalloenolate Intermediate

In situ IR spectroscopy and transmetalation experiments confirm a postulated catalytic cycle. The metalloenolate 1 describes the active intermediate in the aldol reaction catalyzed by [CuF₂{(S)-tol-binap}] (see reaction scheme). (S)-tol-binap=(S)-(−)-2,2′-bis(di-p-tolylphosphanyl)-1,1′-binaphthyl.     

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.