Common best proximity points: global minimal solutions

Let us suppose that A and B are nonempty subsets of a metric space. Let S:A?B and T:A?B be nonself-mappings. Considering the fact S and T are nonself-mappings, it is feasible that the equations Sx=x and Tx=x have no common solution, designated as a common fixed point of the mappings S and T. Eventually, when the equations have no common solution, one contemplates to figure out an element x that is in close proximity to Sx and Tx in the sense that d(x,Sx) and d(x,Tx) are minimum. In fact, common best proximity point theorems scrutinize the existence of such optimal approximate solutions, known as common best proximity points, to the equations Sx=x and Tx=x in the event that the equations have no common solution. Further, one can perceive that the real-valued functions x?d(x,Sx) and x?d(x,Tx) estimate the magnitude of the error involved for any common approximate solution of the equations Sx=x and Tx=x. In light of the fact that the distance between x and Sx, and the distance between x and Tx are at least the distance between A and B for all x in A, a common best proximity point theorem ascertains global minimum of both functions x?d(x,Sx) and x?d(x,Tx) by limiting a common approximate solution of the equations Sx=x and Tx=x to fulfil the requirement that d(x,Sx)=d(A,B) and d(x,Tx)=d(A,B). This article discusses a common best proximity point theorem for a pair of nonself-mappings, one of which dominates the other proximally, thereby yielding common optimal approximate solutions of some fixed point equations when there is no common solution.     

Synthesis and Antioxidant Activity of Bis‐oxazolyl/thiazolyl/imidazolyl 1,3,4‐Oxadiazoles and 1,3,4‐Thiadiazoles

A variety of symmetrical trisheterocycles bis‐oxazolyl/thiazolyl/imidazolylacetamido‐sulfonylmethyl 1,3,4‐oxadiazoles and 1,3,4‐thiadiazoles was prepared, and their antioxidant activity was studied. The compounds bisoxazolyl oxadiazole and bisoxazolyl thiadiazole having methyl substituent on the aromatic ring exhibited greater antioxidant activity than the standard ascorbic acid.     

Spectroscopic studies of atmospheric interest on NO and NO

In recent years, high-resolution photoelectron spectroscopy and ab initio calculations have considerably revised and enlarged the understanding of the electronic structure of the NO and NO⁺ molecules. The experimental potential energy curves for the different electronic states of atmospheric interest molecules like NO and NO⁺ are constructed by using the Rydberg-Klein-Rees method as modified by Vanderslice et al. The ground state dissociation energies are determined by curve fitting technique using the five parameter Hulburt-Hirschfelder (H-H) function. The estimated dissociation energies are 6.381 and 10.693 eV for NO and NO⁺, respectively. These values are in good agreement with the literature values. The r-centroids and Franck-Condon factors (FC Factors) for the band system of B²Π_r-X²Π of NO and a³Σ⁺-X¹Σ⁺, A¹Π-X¹Σ⁺ of NO⁺ molecules have been calculated employing an approximate analytical methods of Jarmain and Fraser, and Nicholls and Jarmain. The absence of the bands in these systems is explained.     

Electrical conduction in copper-poly(vinyl-alcohol) composites

The temperature dependence of d.c. conductivity was investigated in composites of copper (Cu) and poly(vinyl-alcohol) (PVA) prepared by compression of a mixture. A minimum appeared in the conductivity vs reciprocal temperature curves, which is discussed as a result of the competition between hopping and thermally activated modes of conduction. It was also found that the conduction of Cu-PVA composites obeys a percolation model. Dependences of reverse current vs voltage were investigated and discussed in terms of a clean-up effect of charge carriers. Quantitative analysis was carried out to determine the activation energy, drift mobility and carrier concentration.     

Rapid Determination of Fingolimod Hydrochloride-Related Substances and Degradation Products in API and Pharmaceutical Dosage Forms by Use of a Stability-Indicating UPLC Method

Ultra-performance liquid chromatography coupled with photodiode-array detection has been used to develop a simple, sensitive, and reproducible reversed-phase method for quantitative determination of fingolimod hydrochloride and all possible process-related impurities. Chromatographic separation was achieved on a Waters Acquity BEH C18 (100 mm × 2.1 mm, 1.7 µm) column. The mobile phase was a gradient prepared from potassium dihydrogen phosphate (20 mM) containing 0.1 % (v/v) triethylamine and adjusted to pH 6.5 with trifluoroacetic acid (component A) and 85:15 (v/v) acetonitrile–water (component B); the gradient program (time (min)/% B) was: 0.01/20, 2.0/20, 6.0/75, 9.0/90, 12.0/90, 14.0/20, 16.0/20; the run time was 16 min and fingolimod hydrochloride and its six impurities were well separated. Eluting compounds were monitored at 220 nm. The method was validated for precision, specificity, linearity, limit of detection, limit of quantification, accuracy, and robustness in accordance with International Conference on Harmonization guidelines. Fingolimod hydrochloride was subjected to oxidative, acid, base, hydrolytic, thermal, and photolytic stress, and analysis was conducted to determine the amounts of related impurities formed.

     

Practical Synthesis of 4-Benzylidene-2-phenyl-5(4H)-oxazolones

A simple and alternative method has been developed for the synthesis of 4-benzylidene-2-phenyl-5(4H)-oxazolones via reactions of hippuric acid with various aldehydes in the presence of 2-chloro-4,6-dimethoxy-1,3,5-triazine/N-methylmorpholine at 75 °C.

Supplemental materials are available for this article. Go to the publisher's online edition of Synthetic Communications® to view the free supplemental file.     

Real-time monitoring of protein complexes reveals their quaternary organization and dynamics

The dynamics of protein complexes are crucial for their function yet are challenging to study. Here, we present a nanoelectrospray (nESI) mass spectrometry (MS) approach capable of simultaneously providing structural and dynamical information for protein complexes. We investigate the properties of two small heat shock proteins (sHSPs) and find that these proteins exist as dodecamers composed of dimeric building blocks. Moreover, we show that these proteins exchange dimers on the timescale of minutes, with the rate of exchange being strongly temperature dependent. Because these proteins are expressed in the same cellular compartment, we anticipate that this dynamical behavior is crucial to their function in vivo. Furthermore, we propose that the approach used here is applicable to a range of nonequilibrium systems and is capable of providing both structural and dynamical information necessary for functional genomics.