On smoothing,regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems

Traditionally, most stochastic approximation (SA) schemes for stochastic variational inequality (SVI) problems have required the underlying mapping to be either strongly monotone or monotone and Lipschitz continuous. In contrast, we consider SVIs with merely monotone and non-Lipschitzian maps. We develop a regularized smoothed SA (RSSA) scheme wherein the stepsize, smoothing, and regularization parameters are reduced after every iteration at a prescribed rate. Under suitable assumptions on the sequences, we show that the algorithm generates iterates that converge to the least norm solution in an almost sure sense, extending the results in Koshal et al. (IEEE Trans Autom Control 58(3):594–609, 2013) to the non-Lipschitzian regime. Additionally, we provide rate estimates that relate iterates to their counterparts derived from a smoothed Tikhonov trajectory associated with a deterministic problem. To derive non-asymptotic rate statements, we develop a variant of the RSSA scheme, denoted by aRSSA\(_r\), in which we employ a weighted iterate-averaging, parameterized by a scalar r where \(r = 1\) provides us with the standard averaging scheme. The main contributions are threefold: (i) when \(r<1\) and the parameter sequences are chosen appropriately, we show that the averaged sequence converges to the least norm solution almost surely and a suitably defined gap function diminishes at an approximate rate \(\mathcal{O}({1}\slash {\root 6 \of {k}})\) after k steps; (ii) when \(r<1\), and smoothing and regularization are suppressed, the gap function admits the rate \(\mathcal{O}({1}\slash {\sqrt{k}})\), thus improving the rate \(\mathcal{O}(\ln (k)/\sqrt{k})\) under standard averaging; and (iii) we develop a window-based variant of this scheme that also displays the optimal rate for \(r < 1\). Notably, we prove the superiority of the scheme with \(r < 1\) with its counterpart with \(r=1\) in terms of the constant factor of the error bound when the size of the averaging window is sufficiently large. We present the performance of the developed schemes on a stochastic Nash–Cournot game with merely monotone and non-Lipschitzian maps.     

An efficient synthesis of dexlansoprazole employing asymmetric oxidation strategy

An alternative and scalable synthesis of dexlansoprazole ((R)-(+)-1); the (R)-enantiomer of Lansoprazole with an enantiomeric excess of >99.8% is presented.     

Brain delivery of transferrin coupled indinavir submicron lipid emulsions--pharmacokinetics and tissue distribution

Indinavir, an antiretroviral protease inhibitor used in treatment of HIV infection has limited penetration into brain due to efflux of P-glycoprotein. The aim of this work was to develop transferrin coupled submicron lipid emulsions (SLEs) containing indinavir for delivery to brain. Stearylamine containing SLEs were prepared, characterized, tested for stability and optimized formulation (SLE-4) was developed. Transferrin was coupled to get SLE-6 by water soluble EDC method and purified by gel filtration. The coupled transferrin was quantified by modified Bradford dye assay method. The fluorescent dye (DiD oil) incorporated SLEs were used to check the brain specific delivery of SLEs. The in vivo pharmacokinetic and tissue distribution were conducted in mice. During pharmacokinetic studies, there was no significant difference in the serum levels of indinavir from SLE-1, SLE-4 and SLE-6 formulations at all time points. In tissue distribution studies the therapeutic availability (TA) of indinavir in brain from SLE-6 was 4.69, 3.1 and 1.7 times higher than drug solution, SLE-1 and SLE-4 respectively whereas, the TA of indinavir from SLE-4 was 2.76 and 1.82 times the drug solution and SLE-1. The brain to serum ratios with SLE-6 were above one indicates the brain specific delivery. The brain delivery of indinavir was improved with transferrin ligand attachment to SLEs by receptor mediated transcytosis.     

Some polishing experiments on copper in a hyperbolic cell

Some polishing experiments have been carried out on copper anodes in a hyperbolic cell designed by Gilmont and Walton, using orthophosphoric acid as the electrolyte. The results obtained have been compared to those obtained in similar experiments in a Hull cell. It has been found that very similar bands of different reflectivity and polishes are found to form in both the cells. These bands shift with time and a study of such displacements has been made. The results are briefly discussed.     

A previously unrecognised hydronium di-cation in the crystal structure of a cucurbituril derivative

Examination of the crystal structure of the adduct of hydronium tetrachloroferrate dichloride with cucurbituril resulted in the discovery that the hydronium cation present therein is not the well known H7O3+ species; rather, it is an unprecedented cyclic species of composition (H14O6)2+, templated by the formation of five hydrogen bonds to cucurbituril.     

Spin state selective coherence transfer: a method for discrimination and complete analyses of the overlapped and unresolved 1H NMR spectra of enantiomers

In general, the proton NMR spectra of chiral molecules aligned in the chiral liquid crystalline media are broad and featureless. The analyses of such intricate NMR spectra and their routine use for spectral discrimination of R and S optical enantiomers are hindered. A method is developed in the present study which involves spin state selective two dimensional correlation of higher quantum coherence to its single quantum coherence of a chemically isolated group of coupled protons. This enables the spin state selective detection of proton single quantum transitions based on the spin states of the passive nuclei. The technique provides the relative signs and magnitudes of the couplings by overcoming the problems of enantiomer discrimination, spectral complexity and poor resolution, permitting the complete analyses of the otherwise broad and featureless spectra. A non-selective 180 degrees pulse in the middle of MQ dimension retains all the remote passive couplings. This accompanied by spin selective MQ-SQ conversion leads to spin state selective coherence transfer. The removal of field inhomogeneity contributes to dramatically enhanced resolution. The difference in the cumulative additive values of chemical shift anisotropies and the passive couplings, between the enantiomers, achieved by detecting Nth quantum coherence of N magnetically equivalent spins provides enhanced separation of enantiomer peaks. The developed methodology has been demonstrated on four different chiral molecules with varied number of interacting spins, each having a chiral centre.     

Correlation between photophysical properties and lasing performances of Rhodamine-19 in three types of sol–gel glass hosts

Rhodamine-19 (Rh-19) incorporated three types of sol–gel samples (mentioned as dope route-I, dope route-II and dip method) were prepared by using two different methods of dye impregnation. The photophysical properties of all the three types of Rh-19 incorporated sol–gel solids were studied by using the UV–Visible absorption and the fluorescence spectroscopy. A single photon counting technique was used to estimate the lifetime of fluorescent species. From the study of fluorescence spectra, the coexistence of dimers (fluorescent and non-fluorescent both) and monomer of Rh-19 was observed. The photophysical properties of Rh-19 were found to be the best in dip sample, medium in dope route-II sample and low in dope route-I sample. After studying photophysical properties, the samples were subjected to laser study under nitrogen laser pumping at 337.1 nm at the rate of 1.5 Hz in transverse dye laser cavity. The highest laser efficiency and photostability of the dye were observed in dip sample, medium in dope route-II sample and small in dope route-I sample. A comparison between photophysical properties and laser performance of these materials showed a very good correlation.     

Tris-hydroxymethylaminomethane (THAM): An efficient organocatalyst in diversity-oriented and environmentally benign synthesis of spirochromenes

Tris-hydroxymethylaminomethane has been demonstrated to be an efficient organocatalyst in diversity-oriented synthesis of medicinally prevalent spirochromenes by one-pot, three-component reactions between isatins, malononitrile, and enolizable CH acids like dimedone, 4-hydroxycoumarin, 4-hydroxy-N-methylquinolin-2-one, or in situ generated 2-methylpyrazolon-2-one. Biodegradability and extremely low cost of the catalyst are the noteworthy features of this chromatography-free protocol.