Sensitivity of Solutions to a Parametric Generalized Equation

In the present paper we prove that, under some suitable conditions on multifunctions $C\!: {I\!\!R}^l\rightrightarrows {I\!\!R}^n$ , $F\!:{I\!\!R}^d\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , and $K\!:{I\!\!R}^l\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , the generalized perturbation multifunction $G\!: {I\!\!R}^d\times{I\!\!R}^l\times {I\!\!R}^m\rightrightarrows {I\!\!R}^n$ , of the form $$G(\mu,\lambda,\nu)=\{x\in C(\lambda)\ |\ \nu\in F(\mu,x)+K(\lambda,x)\},$$ is proto-differentiable at (μ, λ, ν) relative to x?∈?G(μ, λ, ν). Moreover, in a special case, where K(λ, x) is a normal cone to C(λ) at x, we also provide sufficient conditions for G(·) to be single-valued on a neighborhood of (μ, λ, ν) and semi-differentiable at (μ, λ, ν).     

A One-Pot-Three-Step Route to Triazolotriazepinoindazolones from Oxazolino-2H-indazoles

A one-pot-three-step method has been developed for the conversion of oxazolino-2H-indazoles into triazolotriazepinoindazolones with three points of diversity. Step one of this process involves a propargyl bromide-initiated ring opening of the oxazolino-2H-indazole (available by the Davis-Beirut reaction) to give an N(1)-(propargyl)-N(2)-(2-bromoethyl)-disubstituted indazolone, which then undergoes -CH(2)Br → -CH(2)N(3) displacement (step two) followed by an uncatalyzed intramolecular azide-alkyne 1,3-dipolar cycloaddition (step three) to form the target heterocycle. Employing 7-bromooxazolino-2H-indazole allows for further diversification through, for example, palladium-catalyzed coupling chemistry, as reported here.     

Neutral gold complexes with tridentate SNS thiosemicarbazide ligands

Na[AuCl(4)]·2H(2)O reacts with tridentate thiosemicarbazide ligands, H(2)L1, derived from N-[N',N'-dialkylamino(thiocarbonyl)]benzimidoyl chloride and thiosemicarbazides under formation of air-stable, green [AuCl(L1)] complexes. The organic ligands coordinate in a planar SNS coordination mode. Small amounts of gold(I) complexes of the composition [AuCl(L3)] are formed as side-products, where L3 is an S-bonded 5-diethylamino-3-phenyl-1-thiocarbamoyl-1,2,4-triazole. The formation of the triazole L3 can be explained by the oxidation of H(2)L1 to an intermediate thiatriazine L2 by Au(3+), followed by a desulfurization reaction with ring contraction. The chloro ligands in the [AuCl(L1)] complexes can readily be replaced by other monoanionic ligands such as SCN(-) or CN(-) giving [Au(SCN)(L1)] or [Au(CN)(L1)] complexes. The complexes described in this paper represent the first examples of fully characterized neutral Gold(III) thiosemicarbazone complexes. All the [AuCl(L1)] compounds present a remarkable cell growth inhibition against human MCF-7 breast cancer cells. However, systematic variation of the alkyl groups in the N(4)-position of the thiosemicarbazone building blocks as well as the replacement of the chloride by thiocyanate ligands do not considerably influence the biological activity. On the other hand, the reduction of Au(III) to Au(I) leads to a considerable decrease of the cytotoxicity.     

A metallocene-pyrrolidinopyridine nucleophilic catalyst for asymmetric synthesis

[reaction: see text] A highly active chiral 4-aminopyridine nucleophilic catalyst, available in three steps from (S,S)-hexane-2,5-diol, was applied to the asymmetric Steglich rearrangement of O-aceylated azlactones (1 mol % loading, up to 76% ee).     

A Hausdorff‐like moment problem and the inversion of the Laplace transform

We consider the problem of finding u ∈ L ²(I ), I = (0, 1), satisfying ∫_I u (x )x dx = μ _k , where k = 0, 1, 2, …, (α _k ) is a sequence of distinct real numbers greater than –1/2, and μ = (μ _kl ) is a given bounded sequence of real numbers. This is an ill‐posed problem. We shall regularize the problem by finite moments and then, apply the result to reconstruct a function on (0, +∞) from a sequence of values of its Laplace transforms. Error estimates are given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetic Kronig-Penney-type graphene superlattices: finite energy Dirac points with anisotropic velocity renormalization

We study the energy band structure of magnetic graphene superlattices with delta-function magnetic barriers and zero average magnetic field. The dispersion relation obtained using the T-matrix approach shows the emergence of an infinite number of Dirac-like points at finite energies, while the original Dirac point is still located at the same place as that for pristine graphene. The carrier group velocity at the original Dirac point is isotropically renormalized, but at finite energy Dirac points it is generally anisotropic. An asymmetry in the width between the wells and the barriers of the periodic potential induces a shift of the original Dirac point in the zero-energy plane, keeping the velocity renormalization isotropic.     

Proof of Modulational Instability of Stokes Waves in Deep Water

It is proven that small-amplitude steady periodic water waves with infinite depth are unstable with respect to long-wave perturbations. This modulational instability was first observed more than half a century ago by Benjamin and Feir. It has been proven rigorously only in the case of finite depth. We provide a completely different and self-contained approach to prove the spectral modulational instability for water waves in both the finite and infinite depth cases. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

Molecular analysis of the interface of poly(ether-ether-ketone) and the basal plane of graphite single crystals

The interfacial interaction of the basal plane of graphite single crystals and poly(aryl-ether-etherketone) (PEEK) was investigated using Fourier transform infrared reflection-absorption spectroscopy (FT-IR-RAS). PEEK melt-quenched on the basal plane of a graphite single crystal does not show preferential orientation. However, upon annealing, the polymer chains crystallize perpendicularly to the basal plane of the graphite single crystal. The rate of this surface-induced orientation during crystallization is found to be dependent on the film thickness. These results were verified by using model compounds of PEEK and by calculations based on optical theories.