A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions

Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λ ∈ V ? ? ⁿ . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions \({\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}\) at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form \(\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} \), z ∈ ?, where P _k and Q _k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P _k and Q _k .     

On exponential stability of second order delay differential equations

We propose a new method for studying stability of second order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.     

Cathodic Stripping Determination of Water in Organic Solvents

A cathodic stripping voltammetry (CSV) method has been developed to determine water content in organic solvents based on electrooxidation of a gold electrode in the presence of water to form gold oxides upon anodic polarizations. Following cathodic potential scan, a sharp reductive stripping peak of gold oxides is generated and the magnitude of the peak current is related to the water content in the organic solvents. The method has been applied for determination of water in four commonly used water miscible organic solvents, acetonitrile, tetrahydrofuran, acetone and glacial acetic acid, with limits of detection of 60, 10, 40 and 10 ppm, respectively.     

Fabrication of a Single Layer and Bilayer Potentiometric Biosensors for Penicillin by Galvanostatic Entrapment of Penicillinase into Polypyrrole Films

A single layer and bilayer potentiometric biosensors for the detection of penicillin have been developed. The favourable conditions that were established for the polypyrrole‐penicillinase ((PPy‐P’nase) single layer biosensor were 0.03 M pyrrole, 50 U/mL P’nase, 0.01 M penicillin, applied current density of 0.9 mA/cm² and a polymerisation time of 40 s. The optimum conditions for the formation of the outer layer of the bilayer were: (a) 0.1 M Py, 19 U/mL P’nase, 0.01 M pen, current density of 0.9 mA/cm² and a polymerisation time of 40 s. The minimum detectable penicillin concentration with the bilayer potentiometric biosensor was 0.3 µM and the linear concentration range was 7.5–146 µM. The average percentage recovery of penicillin that was found in amoxycillin 500 mg was 113±24 %. The determination of penicillin in milk was fraught with problems of non‐specific binding of penicillin to the milk.     

Redox interactions between structurally different alkylresorcinols and iron(III) in aqueous media: frozen-solution 57Fe Mössbauer spectroscopic studies,redox kinetics and quantum chemical evaluation of the alkylresorcinol reactivities

Iron(III)-containing aqueous solutions of 5-methylresorcinol (5-MR), 5-n-propylresorcinol (5-n-PR) and 4-n-hexylresorcinol (4-n-HR) at pH ~ 3 were studied by means of ⁵⁷Fe transmission Mössbauer spectroscopy. Kinetic considerations were applied to the redox reactions. Density Functional Theory (DFT) calculations were performed for the alkylresorcinol (AR) molecules and their non-alkylated analogue (resorcinol). Mössbauer spectra consisted of quadrupole doublets assigned to high-spin Fe(III) and Fe(II) species. From changes in their relative spectral areas, a gradual reduction of Fe(III) by all the ARs studied was observed. However, significant differences were found for the reduction rates among the ARs. The following series of the reduction rates was established by means of Mössbauer spectroscopy: 4-n-HR ? 5-MR > 5-n-PR, supplemented by rate constants calculated using a kinetic model. DFT calculations resulted in the following series: 4-n-HR ? 5-n-PR > 5-MR ? resorcinol (the latter is not oxidised under the conditions applied). The reversed order of the experimentally observed 5-MR and 5-n-PR oxidation rates may be explained in terms of their different kinetic parameters related to their structure.     

Growth Actuated Bending and Twisting of Single Crystals

Crystals of a variety of substances including elements, minerals, simple salts, organic molecular crystals, and high polymers forgo long‐range translational order by twisting and bending as they grow. These deviations have been observed in crystals ranging in size from nanometers to centimeters. How and why so many materials choose dramatic non‐crystallographic distortions is analyzed, with an emphasis on crystal chemistries that give rise to stresses operating either on surfaces of crystallites or within the bulk.     

Synthesis,Structure, and Reactivity of Diazene Adducts: Isolation of iso‐Diazene Stabilized as a Borane Adduct

This work describes the synthesis and full characterization of a series of GaCl₃ and B(C₆F₅)₃ adducts of diazenes R¹?N?N?R² (R¹=R²=Me₃Si, Ph; R¹=Me₃Si, R²=Ph). Trans‐Ph?N?N?Ph forms a stable adduct with GaCl₃, whereas no adduct, but instead a frustrated Lewis acid–base pair is formed with B(C₆F₅)₃. The cis‐Ph?N?N?Ph ? B(C₆F₅)₃ adduct could only be isolated when UV light was used, which triggers the isomerization from trans‐ to cis‐Ph?N?N?Ph, which provides more space for the bulky borane. Treatment of trans‐Ph?N?N?SiMe₃ with GaCl₃ led to the expected trans‐Ph?N?N?SiMe₃ ? GaCl₃ adduct but the reaction with B(C₆F₅)₃ triggered a 1,2‐Me₃Si shift, which resulted in the formation of a highly labile iso‐diazene, Me₃Si(Ph)N?N; stabilized as a B(C₆F₅)₃ adduct. Trans‐Me₃Si?N?N?SiMe₃ forms a labile cis‐Me₃Si?N?N?SiMe₃ ? B(C₆F₅)₃ adduct, which isomerizes to give the transient iso‐diazene species (Me₃Si)₂N?N ? B(C₆F₅)₃ upon heating. Both iso‐diazene species insert easily into one B?C bond of B(C₆F₅)₃ to afford hydrazinoboranes. All new compounds were fully characterized by means of X‐ray crystallography, vibrational spectroscopy, CHN analysis, and NMR spectroscopy. All compounds were further investigated by DFT and the bonding situation was assessed by natural bond orbital (NBO) analysis.