Spectral analysis and generation of certain highly oscillatory curves related to chaos,part 2: Calculation aspects

The Fourier spectrum of a curve on a given interval, as represented by the Fourier series, is very reflective of the oscillatory behavior of the curve as the “weights” of the Fourier coefficients can indicate the individual prominence of frequencies of oscillations. But, unfortunately, the calculations of the Fourier coefficients can often get quite cumbersome.     

Umsetzungen von 1-Aminoalkan-1,1-diphosphonsäuren mit salpetriger Säure

Reactions of 1-Aminoalkane-1,1-diphosphonic Acids with Nitrous Acid Compounds of the structure RC(NH₂)(PO₃H₂)₂ 1 give products RC(OH)(PO₃H₂)₂ 2 in high yields by the reaction with nitrous acid. In contrast, when R = Aryl the reaction of 1 often gives in concentrated hydrochloric acid RCCl(PO₃H₂)₂ 3 . Exceptions to the above reaction are described.     

Hyperbranched molecules based on calixarenes

Synthesis of the diamide 2 derived from tris(2-aminoethyl)amine and monocarboxymethylcalix[4]arene provides a starting material useful for the preparation of a variety of hyperbranched molecules. Metal ion binding is one of the potentially useful properties of these new materials.     

Asymptotic Behavior of Positive Solutions of a Nonlinear Combined p-Laplacian Equation

For p > 1, we establish existence and asymptotic behavior of a positive continuous solution to the following boundary value problem $$\left\{\begin{array}{ll}\frac{1}{A} \left( A\Phi _{p}(u^{\prime})\right) ^{\prime}+a_{1}(r)u^{\alpha _{1}}+a_{2}(r)u^{\alpha _{2}}=0, \, {\rm in}\, (0,\infty ),\\ {\rm lim}_{r\rightarrow 0} A\Phi _{p}(u^{\prime})(r)=0, {\rm lim}_{r\rightarrow \infty } u(r)=0,\end{array}\right.$$ where \({\alpha _{1}, \alpha _{2} < p -1, \Phi _{p}(t) = t|t| ^{p-2},A}\) is a positive differentiable function and a ₁, a ₂ are two positive functions in \({C_{\rm loc}^{\gamma}((0, \infty )), 0 < \gamma < 1,}\) satisfying some appropriate assumptions related to Karamata regular variation theory. Also, we obtain an uniqueness result when \({\alpha _{1}, \alpha _{2} \in (1-p,p-1)}\) . Our arguments combine a method of sub and supersolutions with Karamata regular variation theory.     

On a new Kato class and positive solutions of Dirichlet problems for the fractional Laplacian in bounded domains

The purpose of this paper is to extend some results of the potential theory of an elliptic operator to the fractional Laplacian (−Δ)^α/2, 0<α<2, in a bounded C^1,1 domain D in Rⁿ. In particular, we introduce a new Kato class K_α(D) and we exploit the properties of this class to study the existence of positive solutions of some Dirichlet problems for the fractional Laplacian.     

Complex Formation of Some Tetraamido-type Calix[4]arene Derivatives Detected by Vibrational Spectroscopy

Abstract

Dibenzo-14-crown-4 alcohols are shown to aggregate in chloroform by the formation of hydrogen-bonded dimers. Structural variations between crown ether alcohols significantly affect the degree of dimerization. This self-association is of relevance for understanding the complexation and extractive properties of these and related macrocycles.     

Synthesis and cation complexation of calix[4]azacrowns and a spirobiscalix[4]azacrown

The synthesis and complexation properties of three new derivatives, one spirobiscalix[4]azacrown (1) and two calix[4]azacrown (2 and 3), are reported. Complexation studies of the three ligands toward transition and heavy metal cations have been carried out and monitored by UV absorption spectrophotometry in acetonitrile. Mononuclear complexes were detected for all complexes, whereas binuclear species (M₂L) were also formed in the case of ligand 1 with Cu²⁺ and Pb²⁺. The extraction properties of 1, 2 and 3 toward Cu²⁺, Zn²⁺, Pb²⁺, Ag²⁺ and Cd²⁺ are also reported. The results showed that complexation is the main factor affecting extraction with ligands 2 and 3, while with ligand 1 it is not.