Photochemistry of nonconjugated bichromophoric systems: Photoreductive polyrecombination of diaryl ketones

The irradiation of nonconjugated bichromophoric systems, diaryl ketones, with ultraviolet light in the presence of hydrogen donors, such as isopropanol or bisbenzhydrol, yields high polymers. Two series of arylketones were investigated, namely bisbenzophenones and bisketotriazoles. The cage reaction, which arises from the cross combination between the acetone ketyl and bisbenzophenone ketyl radicals, was estimated to be very small but may constitute a chain termination reaction in the photopolymerization in isopropanol. The photopolymerization of bisdiketone–dihydrol systems was found to yield polymers with much higher molecular weights than those obtained from the photopolymerization of bisdiketone–isopropanol systems. The photolysis of some binary mixtures of aromatic diketones in the presence of isopropanol yielded statistical copolypinacols. In some favorable cases, the NMR spectrum of the copolymer obtained showed three distinct linkages and from the infrared spectrum intensities the copolymers composition and the number average length of the sequences may be determined.     

Chiral (iminophosphoranyl)ferrocenes: highly efficient ligands for rhodium- and iridium-catalyzed enantioselective hydrogenation of unfunctionalized olefins

A series of chiral (iminophosphoranyl)ferrocenes (1-3) are highly efficient ligands for Rh- and Ir-catalyzed hydrogenation of a number of unfunctionalized olefins; almost perfect enantiomeric excesses (up to 99% ee) have been achieved under mild reaction conditions.     

Guiding electron emissions by excess negative charges in multiply charged molecular anions

Using photoelectron imaging, we show the effects of excess negative charges on the directions of outgoing electrons in multiply charged anions. Photoemissions are observed to occur either in a perpendicular or parallel direction, depending on the molecular configurations and origins of the detached electrons. Detachment of the π electrons from biphenyl-disulfonate dianions is shown to occur in a perpendicular direction due to the Coulomb repulsion from the two terminal charges, whereas detachment from the sulfonate groups in linear aliphatic disulfonates occurs in parallel directions.     

A general transformation for theta series associated with the quadratic form $$x^2+ky^2$$

Using elementary techniques, we prove a general transformation for theta series associated with the quadratic form \(x^2+ky^2.\) The transformation is then applied to establish several infinite families of identities involving theta series whose Fourier coefficients are interlinked.     

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

In this paper, we consider the following nonlinear Kirchhoff wave equation

$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0

(1)

where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)₁ is studied.

     

Strong Convergence to Solutions for a Class of Variational Inequalities in Banach Spaces by Implicit Iteration Methods

In this paper, in order to solve a variational inequality problem over the set of common fixed points of an infinite family of nonexpansive mappings on a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, we introduce two new implicit iteration methods. Their strong convergence is proved, by using new V-mappings instead of W-ones.     

Regularity Index of S + 2 Fat Points not on a Linear (S − 1)-Space

We will give a formula to compute the regularity index of s + 2 fat points not lying on a linear (s ? 1)-space in ? ⁿ , s ≤ n (Theorem 3.4). Our result generalizes a formula to compute the regularity index of fat points in general position in ? ⁿ ([3 Catalisano , M. V. , Trung , N. V. , Valla , G. ( 1993 ). A sharp bound for the regularity index of fat points in general position . Proc. Amer. Math. Soc. 118 : 717 – 724 .[Crossref], [Web of Science ®] , [Google Scholar]], Corollary 8). Our result also shows that the Segre bound is attained by s + 2 points not lying on a linear (s ? 1)-space.