Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces

The probabilistic version of the classical Banach Contraction Principle was proved in 1972 by Sehgal and Bharucha-Reid [V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on PM spaces. Math. Syst. Theory 6, 97–102]. Their fixed point theorem is further generalized by many authors. In the intervening years many others have proved the probabilistic versions of the other known metric fixed point theorems. However, the problem to prove the probabilistic versions of the very important generalization of the Banach Contraction Principle, obtained in 1969 by Boyd and Wong [D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969) 458–464], who proved the fixed point theorem for a self-mapping of a metric space, satisfying the very general nonlinear contractive condition, is unsolved in these days. Similarly, as in the metric space case, to prove a fixed point theorem for a mapping, satisfying the general probabilistic nonlinear contractive condition, it was necessary to find a new approach, substantially different from the previous technique for cases where a mapping satisfies the probabilistic linear contraction condition, introduced by Sehgal and Bharucha-Reid and further used by many authors. So, the problem to obtain a truthful probabilistic version of the Banach fixed point principle for general nonlinear contractions existed unsolved for over 35 years. We have solved this problem in this paper.     

Absorption and Fluorescence Characteristics of Some 2-Aryl-and 2-Heteroaryl-benzothiazoles in Different Solvents and at Various pH Values

The parameters characterizing the absorption and fluorescence of seven 2-substituted benzothiazoles in different solvents and at various acid/base concentrations were studied. It was found that both the kind of substituent and the solvent polarity influenced the changes of the spectral properties of the examined molecules. 2-Aryl substituted benzothiazoles exhibit larger Stokes shifts than 2-heteroaryl derivatives. The linear relationship between solvent polarity and pH values against Stokes shift and fluorescence sensitivity, respectively, was observed. The mostly fluorescent compound was 2-(2-benzofuryl)benzothiazole.     

Synthesis,characterization and solubility studies of four new highly water soluble 1,3,5-triaza-7-phosphaadamantane (PTA) salts and their gold(I) complexes

The synthesis of new water soluble N-alkylated derivatives of 1,3,5-triaza-7-phosphaadamantane is presented. The compounds were characterized by means of NMR and IR spectroscopy, mass spectrometry, high resolution mass spectrometry, elemental analysis and X-ray diffraction analysis. The water solubility of these compounds was found to be up to an astonishing 1450 g/L. Several different reactions were performed utilizing these highly interesting compounds as starting materials. It was shown that the substitution of the counter ion can be performed easily. Also transformations at the PTA framework were possible, as shown by an ester cleavage example. To prove that the resulting PTA derivatives are competent as ligands for transition metals, gold(I) complexes were synthesized, using Au(tht)Cl as the metal source. The resulting gold(I) complexes were characterized by ¹H NMR and IR spectroscopy, mass and high resolution mass spectrometry or elemental analysis.     

Supramolecular phenomena in organic redox films at electrodes: Part I. The methylene blue/leucomethylene blue redox couple at the platinum electrode

It is possible to generate conductive multilayer structures by reduction of methylene blue cations at the interface: clean surface of a platinum electrode/1 M aqueous electrolyte solution.The characteristics of the multilayer ordered phase generated in fluoride and nitrate solutions are presented and the mechanism of a fast charge transfer between electrode/organic film and organic film/aqueous solution interfaces is discussed. The conductivity of the film is interpreted by postulating formation of a mixed valence structure with the generation of a cation radical intermediate which is favoured in the solid state at characteristic potentials.     

Interlayer inclusion of tetracyanonickelate anion and water molecules: The crystal and molecular structure of [-ZN(en)2-μ-(NC)2-NI-μ -(CN)2-ZN(en) -] n 2n+ ·n[NI(CN)4]2− · 3n H2O

The crystal structure of [Zn(en)₂- (NC)₂Ni(CN)₂ Zn(en)][Ni(CN)₄]· 3 H₂O consists of infinite positively charged wave-shaped layers of composition [Zn(en)₂-(NC)₂Ni(CN)₂- Zn(en)] _n ²ⁿ⁺ with the [Ni(CN)₄]^2– anions and water molecules included between them. Both Ni atoms exhibit square-planar coordination. The chelate bonded en (=ethylenediamine) and N-bonded cyano ligands around two independent zinc atoms form a deformed tetrahedron and a deformed octahedron, respectively. Yellow needles of the complex belong to the orthorhombic space groupPbcm witha = 6.977(1),b = 25.407(4),c = 14.876(2)Å,Z = 4,D _m = 1.74(1) g cm^–3 andD _c = 1.739 g cm^–3. The structure was refined toR = 6.31 %.     

Purification method for recombinant proteins based on a fusion between the target protein and the C-terminus of calmodulin

Calmodulin (CaM) was used as an affinity tail to facilitate the purification of the green fluorescent protein (GFP), which was used as a model target protein. The protein GFP was fused to the C-terminus of CaM, and a factor Xa cleavage site was introduced between the two proteins. A CaM-GFP fusion protein was expressed in E. coli and purified on a phenothiazine-derivatized silica column. CaM binds to the phenothiazine on the column in a Ca(2+)-dependent fashion and it was, therefore, used as an affinity tail for the purification of GFP. The fusion protein bound to the affinity column was then subjected to a proteolytic digestion with factor Xa. Pure GFP was eluted with a Ca(2+)-containing buffer, while CaM was eluted later with a buffer containing the Ca(2+)-chelating agent EGTA. The purity of the isolated GFP was verified by SDS-PAGE, and the fluorescence properties of the purified GFP were characterized.     

Syntheses and Reactions of 1′,2′-Unsaturated 2′,3′-Seconucleoside Analogues

The 1′,2′-unsaturated 2′,3′-secoadenosine and 2′,3′-secouridine analogues were synthesized by the regioselective elimination of the corresponding 2′,3′-ditosylates, 2 and 18 , respectively, under basic conditions. The observed regioselectivity may be explained by the higher acidity and, hence, preferential elimination of the anomeric H–C(1′) in comparison to H? C(4′). The retained (tol-4-yl)sulfonyloxy group at C(3′) of 3 allowed the preparation of the 3′-azido, 3′-chloro, and 3′-hydroxy derivatives 5–7 by nucleophilic substitution. ZnBr₂ in dry CH₂Cl₂ was found to be successful in the removal (85%) of the trityl group without any cleavage of the acid-sensitive, ketene-derived N,O-ketal function. In the uridine series, base-promoted regioselective elimination (→ 19 ), nucleophilic displacement of the tosyl group by azide (→ 20 ), and debenzylation of the protected N(3)-imide function gave 1′,2′-unsaturated 5′-O-trityl-3′-azido-secouridine derivative 21 . The same compound was also obtained by the elimination performed on 2,2′-anhydro-3′-azido-3′-azido-3′-deoxy-5′-O-2′,3′-secouridine ( 22 ) that reacted with KO(t-Bu) under opening of the oxazole ring and double-bond formation at C(1′).     

Copper(I) and silver(I) complexes of 1,5-methylene- and diethylmethylene-bridged bis(oxazoline) ligands. In situ Cu(II)-catalyzed oxidation of methylene bridge

Silver(I) and copper(I) complexes of C₂-symmetric bis(oxazoline) ligands were studied by UV, NMR, IR, EPR and ES-MS spectroscopies. The stability constants of the Ag-1a and Ag-1b complexes with 1:1 and 1:2 stoichiometries in acetonitrile were determined by NMR spectrometric titrations. The evidence of tetrahedral coordination for complex (Ag(1a)₂(⁺ was obtained from the complexation induced shifts (CIS) and NOEs. Mass spectra revealed the Cu(II) mediated oxidation of methylene bridge in copper complexes of 1a and 1b, which was in accordance with the UV, NMR, IR and EPR findings. The efficiency of Cu(I) complexes of methylene-bridged 1,5-bis(oxazoline)s 1 as chiral catalysts in stereoselective cyclopropanation of styrene with ethyl diazoacetate, was compared to that of the dialkylmethylene-bridged 1,5-bis(oxazoline)s 2.     

Synthesis, Structure, and Antimicrobial Activity of Complexes of Pt(II), Pd(II), and Ni(II) with the Condensation Product of 2-(Diphenylphosphino)benzaldehyde and Semioxamazide

Summary. Complexes of Pt(II), Pd(II), and Ni(II) with the condensation derivative of 2-(diphenylphosphino)benzaldehyde and semioxamazide were synthesized, characterized, and their antimicrobial activity was evaluated. The ligand and the complexes were characterized by spectroscopic methods with the particular accent on NMR spectral analysis. For the palladium(II) complex, the crystal structure was determined by X-ray analysis. In all the complexes the ligand is coordinated as a tridentate via a P, N, O donor set. The Pd(II) and Pt(II) complexes have a square planar geometry, whereas the geometry of the Ni(II) complex is tetrahedral. The ligand showed antibacterial and antifungal activity, which was enhanced upon complexation.