Iteration of certain exponential-like meromorphic functions

The dynamics of functions \(f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0\) is studied showing that there exists \(\lambda ^* > 0\) such that the Julia set of \(f_\lambda \) is disconnected for \(0< \lambda < \lambda ^*\) whereas it is the whole Riemann sphere for \(\lambda > \lambda ^*\). Further, for \(0< \lambda < \lambda ^*\), the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of \(\infty \) is shown to be disconnected for \(0<\lambda < \lambda ^*\) whereas it is connected for \(\lambda > \lambda ^*\). For complex \(\lambda \), it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions \(E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}\), which we call exponential-like, are studied as a generalization of \(f(z)=\frac{\mathrm{e}^{z}}{z+1}\) which is nothing but \(E_1(z)\). This name is justified by showing that \(E_n\) has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over \(\infty \) and both are direct. Non-existence of Herman rings are proved for \(\lambda E_n \).     

Synthesis and Characterization of Some BODIPY‐based Substituted Salicylaldimine Schiff Bases

Salicylaldimine Schiff bases represent an important class of hetero‐polydentate ligands capable of forming mononuclear, binuclear, and polynuclear complexes with transition and non‐transition metals. In this report, we developed an easy synthesis of BODIPY‐based salicylaldimine Schiff bases and synthesized five new derivatives. These were characterized by elemental analysis, infrared, UV‐Vis, nuclear magnetic resonance spectroscopy, and X‐ray crystallography. Finally, one of the Schiff bases was reacted with BF₃·OEt₂ to synthesize corresponding bis‐BF₂ boron complex. The photophysical and electrochemical properties of the Schiff bases and the boron complex were evaluated and rationalized by theoretical calculations. The bis‐BF₂ boron complex showed excited state charge redistribution, thus could be useful as sensitizers for designing new dye‐sensitized solar cells.     

Extending the applicability of the inexact Newton-HSS method for solving large systems of nonlinear equations

Numerical Algorithms - We revisit the study of the semi-local convergence of the inexact Newton-HSS method (INHSS) introduced by Amiri et al. (2018), for solving large systems of nonlinear...     

β-Functionalized ethylchalcogenolate complexes of lead(II): Synthesis, structures and their conversion into lead chalcogenide nanoparticles

The reactions of Pb(OAc)₂·3H₂O with NaER yield homoleptic colorless to yellow complexes of composition, [Pb(ER)₂] (ER = SCH₂CH₂NMe₂ (1), SeCH₂CH₂NMe₂ (2) and SeCH₂CH₂COOMe (3)). These complexes were characterized by elemental analyses, UV–Vis and ¹H NMR data. Molecular structures of [Pb(ECH₂CH₂NMe₂)₂] (E = S or Se) have been established by single crystal X-ray diffraction analyses. These molecules have a distorted trigonal bipyramidal configuration around lead with the nitrogen atoms of the chelating chalcogenolate ligands occupying the axial positions. Thermal behavior of these complexes was studied by TG analysis. Pyrolysis in a furnace or in HDA (hexadecylamine) gave PbE nanoparticles which were characterized by UV–Vis, photoluminescence, XRD, EDAX, and TEM measurements.     

Group 12 metal monoselenocarboxylates: synthesis, characterization, structure and their transformation to metal selenide (MSe; M = Zn, Cd, Hg) nanoparticles

Reactions of [MCl2(tmeda)] with potassium salts of monoselenocarboxylic acids gave complexes of the general formula [M(SeCOR)2(tmeda)] (M = Zn, Cd; R = Ph, Tol; Tol = C6H4-p-CH3; tmeda = Me2NCH2CH2NMe2). The analogous mercury complexes were unstable at room temperature and afforded HgSe nanoparticles during the course of reaction. All the complexes were characterized by elemental analysis, IR, UV-vis, NMR (1H, 13C, 77Se, 113Cd) data. The X-ray structural analysis of [Cd(SeCOPh)2(tmeda)] revealed that the complex is a discrete monomer having an approximate tetrahedral coordination environment around the central metal atom with monodentate (Se-bonded) selenocarboxylates. Thermal behavior of these complexes was studied by TG analysis. Pyrolysis in a furnace or in HDA (hexadecylamine) gave MSe nanoparticles, which were characterized by XRD, EDAX, SEM and absorption spectroscopy.     

A Hybrid Open-Framework Aluminum Phosphate-Oxalate Possessing Large Circular 12-Membered Channels

A new aluminum phosphate-oxalate, I, [N₂C₄H₁₂]Al₂(PO₄) (HPO₄)(C₂O₄)]H₂O, has been synthesized hydrothermally in the presence of structure-directing amines. The hybrid structure comprises a vertex-linkage of AlO₆ octahedra, PO₄ tetrahedra, and C₂O₄ units leading to three-dimensional connectivity. The connectivity between AlO₆ and PO₄ units are such that it forms double-six rings that are connected to each other via the oxalate units, thereby leading to the formation of a large circular 12-membered channel of width 9 Å along the c axis. The structure-directing amine along with one water molecule is situated within this channel. The connectivity also forms two different types of 8-membered channels along the a and b axes. The three-dimensional structure of I, is very similar to the naturally occurring aluminosilicate zeolite, gmelinite. Crystal data for I are: monoclinic, space group Pccm, a=9.992(1), b=11.644(1), c=12.231(1) Å, V=1423.0(2), M=438.9, Z=4, R_F=0.07.     

Copper(I) 2-pyridyl selenolates and tellurolates: synthesis, structures and their utility as molecular precursors for the preparation of copper chalcogenide nanocrystals and thin films

The complexes, [Cu{EC(5)H(3)(R-3)N}](4) (E/R = Se/Me or Te/R; R = H or Me) were isolated by the reaction between CuCl and NaEC(5)H(3)(R-3)N and were characterized by elemental analyses, uv-vis and NMR ((1)H, (13)C) spectroscopy. The crystal structures of [Cu{SeC(5)H(3)(Me-3)N}](4) and [Cu(TeC(5)H(4)N)](4) revealed that the molecules are tetrameric in which each copper atom lies at the vertex of the tetrahedron and each face of the tetrahedron is capped by the bridging pyridylchalcogenolate ligand. Thermal behavior of these complexes was studied by thermogravimetric analysis. Depending on reaction conditions, thermolysis gave both stoichiometric and non-stoichiometric copper chalcogenides, which were characterized by XRD, EDX, SEM, TEM and SAED techniques. These precursors were used for the preparation of nanocrystals and for deposition of thin films of copper chalcogenides by AACVD (Aerosol Assisted Chemical Vapor Deposition).