Vieta–Fibonacci wavelets: Application in solving fractional pantograph equations

In this paper, the Vieta–Fibonacci wavelets as a new family of orthonormal wavelets are generated. An operational matrix concerning fractional integration of these wavelets is extracted. A numerical scheme is established based on these wavelets and their fractional integral matrix together with the collocation technique to solve fractional pantograph equations. The presented method reduces solving the problem under study into solving a system of algebraic equations. Several examples are provided to show the accuracy of the method.     

Green Strategies for the Preparation of Enantiomeric 5–8-Membered Carbocyclic β-Amino Acid Derivatives through CALB-Catalyzed Hydrolysis

Candida antarctica lipase B-catalyzed hydrolysis of carbocyclic 5–8-membered cis β-amino esters was carried out in green organic media, under solvent-free and ball-milling conditions. In accordance with the high enantioselectivity factor (E > 200) observed in organic media, the preparative-scale resolutions of β-amino esters were performed in tBuOMe at 65 °C. The unreacted β-amino ester enantiomers (1R,2S) and product β-amino acid enantiomers (1S,2R) were obtained with modest to excellent enantiomeric excess (ee) values (ee_s > 62% and ee_p > 96%) and in good chemical yields (>25%) in one or two steps. The enantiomers were easily separated by organic solvent/H₂O extraction.     

Morphology and Mechanical Properties of Polyamide 6/Acrylonitrile-Butadiene-Styrene Blends Containing Carbon Nanotubes

The blends of polyamide 6/acrylonitrile-butadiene-styrene (PA6/ABS), with added styrene-maleic acid copolymer (SMA) compatibilizer, were prepared through melt mixing in an internal mixer. The effects of blend composition and various process conditions, as well as the addition of multi-wall carbon nanotubes (MWCNTs) to the blends, on the morphology and mechanical properties were investigated. The morphology of the blends and blend nanocomposites were observed by scanning electron microscopy (SEM) and analyzed using an image analysis technique. The mechanical behavior of the blends was investigated by tensile and also impact testing. The results showed that the blend composition as well as the processing conditions significantly affected the morphology and mechanical properties of the PA6/ABS blends. Among the various compositions, the blend with 36?wt.% of ABS and 4?wt.% of SMA compatibilizer exhibited the best mechanical properties. Comparing various speeds and times of mixing, it was found that less mixing speed and longer mixing times resulted in the favorable morphology and conditions for achievement of the desired toughness for the polyamide 6. By adding different amounts of MWCNTs to the blends, it was found that the presence of the carbon nanotubes changed the viscosity of the resulting nanocomposite and thus changed the morphology. These nanocomposites also showed an improvement in mechanical properties. The MWCNTs acted as a second compatibilizer, resulting in a synergistic effect on the mechanical properties of the PA6/ABS blend nanocomposites.     

A characterization of nonlinear Hölder seminorm preserving bijections

For a compact metric space (X, d) and \(\alpha \in (0,1)\), let \(\mathrm{Lip}^\alpha (X)\) be the linear space of all complex-valued functions f on X satisfying

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and \(\mathrm{lip}^\alpha (X)\) be the subspace of \(\mathrm{Lip}^\alpha (X)\) consisting of functions f with \(\lim \frac{f(x)-f(y)}{d^\alpha (x,y)} =0\) as \(d(x,y) \rightarrow 0\). In this paper, we give a characterization of a bijective map \(T:\mathrm{lip}^\alpha (X)\longrightarrow \mathrm{lip}^\alpha (Y)\), not necessarily linear, which is an isometry with respect to the Hölder seminorm \(L(\cdot )\). It is shown that there exist \(K_0>0\), a surjective map \(\Psi : Y \longrightarrow X\) with \(d^\alpha (y,z)= K_0 \, d^\alpha (\Psi (y),\Psi (z))\) for all \(y,z\in Y\), and a function \(\Lambda : \mathrm{lip}^\alpha (X) \longrightarrow {\mathbb {C}}\) (which is linear or real-linear if T is so) such that either

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0\, f(\Psi (y))+\Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y) \end{aligned}$$

or

$$\begin{aligned} Tf(y)= T0(y)+\overline{\tau } K_0 \,\overline{f(\Psi (y))}+ \Lambda (f)\quad (f\in \mathrm{lip}^\alpha (X), y\in Y), \end{aligned}$$

where \(\tau =e^{i\theta }\) for some \(\theta \in [0,\pi )\).

     

An Efficient One-Pot Synthesis of 2,3-Dialkyl-1,1-Diethyl-1-(Acetylamino)-3-(1,1,1-Triphenylphosphanilidene)-1,1,2,3-Propanetetracarboxylates and Their Temperature-Dependent NMR Spectra

Abstract

Stable crystalline phosphorus ylides of 1,1-diethyl 2,3-dimethyl 1-(acetylamino)-3-(1,1,1-triphenylphosphanilidine)-1,1,2,3-propanetetracarboxylates were obtained in excellent yields from the 1:1:1 addition reaction between triphenylphosphine and dialkyl acetylenedicarboxylates in the presence of diethyl acetamidomalonate as a CH-acid. These stable ylides exist in solution as a mixture of two geometrical isomers (E and Z) as a result of restricted rotation around the carbon–carbon partial double bond resulting from conjugation of the ylide moiety with the adjacent carbonyl group on the nuclear magnetic resonance (NMR) time scale at ambient temperature. The dynamic effects in the ylide moieties were investigated by ¹H NMR spectra.

Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file.     

Synthesis of polyfunctionalized alkenes and α,β-unsaturated γ-lactams from the reaction of alkyl propiolates and CH-acids such as diethyl acetamidomalonate and ethyl acetamidocyanoacetate in the presence of triphenylphosphine

Dialkyl 1-(Z)-2-(acetylamino)-2-butenedioates, 1,1-diethyl-3-alkyl (E)-(1-acetylamino)-2-propene-1,1,3-tricarboxylate, 5-ethyl-1-methyl (E)-4-(acetylamino)-4-cyano-2-pentenedioate and ethyl 1-acetyl-2-cyano-5-oxo-2,5-dihydro-1H-pyrrole-2-carboxylate were obtained from the three-component reactions between alkyl propiolates and CH-acids such as diethyl acetamidomalonate and ethyl acetamidocyanoacetate in the presence of triphenylphosphine at room temperature in dry dichloromethane. Correspondence: Sakineh Asghari, Chemistry Department, Mazandaran University, P. O. Box 453, Babolsar, Iran.     

Application of Mie theory and fractal models to determine the optical and surface roughness of Ag–Cu thin films

Thin films of Ag/Cu were deposited by reactive DC magnetron sputtering on (001)-oriented Si and glass substrates for various deposition times (4–24 min). These films were characterized by atomic force microscopy (AFM), and a power law scaling was performed on the obtained micrographs to investigate the self-affine nature of the sample morphology, which is indicative of a fractal structure. We applied the Higuchi’s algorithm to the AFM data to determine the fractal dimension of each sample, and the Hurst exponents were computed. The deposition time dependences of these parameters and the grain size distributions estimated from the UV–visible spectra using the Mie theory, allowed us to describe a particle formation mechanism during the deposition process, in which the length of continuous paths of conductive particles increases as the deposition time is increased. In agreement with this explanation, the electrical resistance decreased with the increment of the deposition time.