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 \(\lambda E_n \).     

Stocking and pricing decisions under endogenous demand and reference point effects

In this paper, we study the newsvendor’s pricing and stocking decisions under reference point effects. The demand faced by the newsvendor is endogenous and the customers may also decide to procure the product from an outside option. We characterize the firm’s optimal pricing and stocking decisions. Our analysis reveals a threshold policy on the firm’s ordering and pricing decisions while considering the impact of reference point effects. We also find that as the level of optimism increases, the firm’s optimal ordering level decreases and optimal price increases. We further study the impact of loss aversion on the firm’s ordering and pricing decisions.     

Synthesis,Isolation and X-ray Crystal Structure Characterization of <Emphasis Type="Italic">Trans</Emphasis>-γ-{(C-<Emphasis Type="Italic">meso</Emphasis>-5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane)(diisothiocyanato)} Nickel(II) Complex

Abstract  A new isomer of [Ni(tet-a)]²⁺ (assigned as γ-isomer) (tet-a = C-meso-5,5,7,12,12,14−hexamethyl-1,4,8,11-tetraazacyclotetradecane), which is found in the complex of trans-γ-[Ni(tet-a)(NCS)₂], has been isolated from the reaction of β-[Ni(tet-a)](ClO₄)₂ with NaSCN in a mixture of MeOH–H₂O (1:1, v/v) solution and characterized by X-ray crystallography. The complex crystallizes in orthorhombic Pca2(1) space group with the crystal cell parameters of a = 12.5787(10) ?, b = 11.6834(10) ?, c = 15.4019(12) ?, V = 2263.5(3) ?³, and Z = 4. The crystal structure shows that the complex is noncentrosymmetric. The conformations of the six-asymmetric centers are assigned as 1R,4R,8S,11S,7R,14S or 1S,4S,8R,11R,7S,14R (Trans-III). The coordination polyhedron around the nickel atom is axially elongated octahedron with an N₆ chromophore. The association of the complex units through N(3)–H···S(1) (2.665 ?) hydrogen bonding interactions leads one-dimensional hydrogen bonded network along c-axis. Index Abstract  A new isomer of Ni(tet-a)²⁺, which is assigned as γ-isomer, in the complex of trans-γ-[Ni(tet-a)(NCS)₂] has been synthesized, isolated and characterized by X-ray crystallography. The complex units form a one-dimensional self-assembly hydrogen bonded network.      

Tip‐heating‐assisted Raman spectroscopy at elevated temperatures

We demonstrate tip‐heating‐assisted enhanced Raman spectroscopy to investigate the temperature dependence of the carbon nanotube G‐band with nanoscale resolution. The controllable and nanoscale heat generated at the tip apex was used to thermally perturb and characterize a small volume in a carbon nanotube sample that is precisely positioned underneath the tip. The dependence of tip enhancement with temperature was also experimentally examined, which is in good agreement with the enhancement calculated from the electromagnetic model of isolated spheroids. The technique presented may open up opportunities in the study of controlled heat‐assisted biochemical reactions and physical transformations of nanostructures. It can also be used for thermal characterization of various materials requiring site‐selective and controllable nanoscale heat source and could enable the realization of new photothermal devices. Copyright © 2010 John Wiley & Sons, Ltd.     

Molecular complexity from aromatics. Studies on synthesis of polycyclic system related to crotogoudin

Synthesis of tricyclic systems related to crotogoudin from a simple aromatic precursor has been described. Oxidative dearomatization, tandem retro-Diels–Alder/Diels–Alder reaction and ring-closing metathesis are the key features of our approach.     

Alkali Metal–Promoted Facile Synthesis of Secondary Amines from Imines and Carbodiimides

We present here an efficient method for the hydroboration of aldimines (-C=N-) with pinacolborane (HBpin) using an alkali metal catalyst, potassium benzyl. The reaction was accomplished with unprecedented catalytic efficiency under mild and solvent-free conditions to afford the high yield of the corresponding N-boryl amines up to 97%. Various functionalities on aldimines were incorporated for hydroboration. The corresponding boryl amines were subjected to further hydrolysis to yield the corresponding secondary amines with good yields up to 89%. This protocol for the reaction demonstrates an atom-economic and green method with diverse imines that bears excellent functional group tolerance. Chemoselective reduction of imines was also attained, with good yields of 74–89%. We also propose the most plausible mechanism involving the formation of metal hydride as the active pre-catalyst.     

Reversible Self‐Assembly of Carboxylated Peptide‐Functionalized Gold Nanoparticles Driven by Metal‐Ion Coordination

Carboxylated peptide‐functionalized gold nanoparticles (peptide‐GNPs) self‐assemble into two‐ and three‐dimensional nanostructures in the presence of various heavy metal ions (i.e. Pb²⁺, Cd²⁺, Cu²⁺, and Zn²⁺) in aqueous solution. The assembly process is monitored by following the changes in the surface plasmon resonance (SPR) band of gold nanoparticles in a UV/Vis spectrophotometer, which shows the development of a new SPR band in the higher‐wavelength region. The extent of assembly is dependent on the amount of metal ions present in the medium and also the time of assembly. TEM analysis clearly shows formation of two‐ and three‐dimensional nanostructures. The assembly process is completely reversible by addition of alkaline ethylenediaminetetraacetic acid (EDTA) solution. The driving force for the assembly of peptide‐GNPs is mainly metal ion/carboxylate coordination. The color and spectral changes due to this assembly can be used for detection of these heavy‐metal ions in solution.