A stochastic algorithm using one sample point per iteration and diminishing stepsizes

A stochastic algorithm for finding stationary points of real-valued functions defined on a Euclidean space is analyzed. It is based on the Robbins-Monro stochastic approximation procedure. Gradient evaluations are done by means of Monte Carlo simulations. At each iteratex _i , one sample point is drawn from an underlying probability space, based on which the gradient is approximated. The descent direction is against the approximation of the gradient, and the stepsize is 1/i. It is shown that, under broad conditions, w.p.1 if the sequence of iteratesx ₁,x ₂,...generated by the algorithm is bounded, then all of its accumulation points are stationary.     

On the optimal allocation of two active redundancies in a two-component series system

Components C₁ and C₂ form a series system. Suppose we can allocate the spare R₁ in parallel with C₁ and the spare R₂ in parallel with C₂, or otherwise, allocate R₁ with C₂ and R₂ with C₁. In this paper, we compare these two options using hazard rate ordering.     

General strategy for a short and efficient synthesis of 3-hydroxy-4-methylprolines (HMP)

The non-proteinogenic amino acid 3-hydroxy-4-methylproline (HMP) is an active constituent of some potent antimicrobials including echinocandins, nostopeptins, pneumocandins, sporiofungin and mulundocandins. A synthesis has been achieved in 10 steps with 29% overall yield; the Evans’ aldol reaction using Crimmins’ modified method was pivotal to the success of the strategy.     

Remote stereocontrol by the sulfinyl group: Mukaiyama aldol reactions of (S)-2-[2-(p-tolylsulfinyl)phenyl]acetaldehyde in the asymmetric synthesis of β-hydroxyacids and 1,3-diols

(S)-2-[2-(p-Tolylsulfinyl)phenyl]acetaldehyde reacts with different O-silylated ketenethioacetals in the presence of Yb(OTf)₃ yielding β-hydroxythioesters in high yields and diastereoselectivities. The obtained compounds were readily transformed into β-hydroxyacids and their corresponding diols. These Mukaiyama aldol reactions are a direct evidence of the ability of the sulfinyl group to control 1,5- and 1,6-asymmetric induction processes.     

Stochastic techniques for global optimization: A survey of recent advances

In this paper stochastic algorithms for global optimization are reviewed. After a brief introduction on random-search techniques, a more detailed analysis is carried out on the application of simulated annealing to continuous global optimization. The aim of such an analysis is mainly that of presenting recent papers on the subject, which have received only scarce attention in the most recent published surveys. Finally a very brief presentation of clustering techniques is given.     

Fast and efficient preparation of Baylis-Hillman-derived (E)-allylic azides and related compounds in aqueous medium

A practical access to alkyl- and aryl-substituted (E)-2-(azidomethyl)alkenoates and related azido compounds from the corresponding allylic bromides in aqueous acetone is described. An alternative method to obtain the starting bromides based on heterogeneous catalysis under mild conditions was also investigated.     

Monte Carlo study of interfacial silicon suboxide layers and oxidation kinetics

A simple simulation scheme that simultaneously describes the growth kinetics of SiO₂ films at the nanometer scale and the SiO_x/Si interface dynamics (its extent, and spatial/temporal evolution) is presented. The simulation successfully applies to experimental data in the region above and below 10 nm, reproduces the Deal and Grove linear-parabolic law and the oxide growth rate enhancement in the very thin film regime (the so-called anomalous region). According to the simulation, the oxidation is governed mainly by two processes: (a) the formation of a transition suboxide layer and (b) its subsequent drift towards the silicon bulk. We found that it is the superposition of these two processes that produces the crossover from the anomalous oxidation region behavior to the linear-parabolic law.     

A priori tests of a stochastic mode reduction strategy

Several a priori tests of a systematic stochastic mode reduction procedure recently devised by the authors [Proc. Natl. Acad. Sci. 96 (1999) 14687; Commun. Pure Appl. Math. 54 (2001) 891] are developed here. In this procedure, reduced stochastic equations for a smaller collections of resolved variables are derived systematically for complex nonlinear systems with many degrees of freedom and a large collection of unresolved variables. While the above approach is mathematically rigorous in the limit when the ratio of correlation times between the resolved and the unresolved variables is arbitrary small, it is shown here on a systematic hierarchy of models that this ratio can be surprisingly big. Typically, the systematic reduced stochastic modeling yields quantitatively realistic dynamics for ratios as large as 1/2. The examples studied here vary from instructive stochastic triad models to prototype complex systems with many degrees of freedom utilizing the truncated Burgers–Hopf equations as a nonlinear heat bath. Systematic quantitative tests for the stochastic modeling procedure are developed here which involve the stationary distribution and the two-time correlations for the second and fourth moments including the resolved variables and the energy in the resolved variables. In an important illustrative example presented here, the nonlinear original system involves 102 degrees of freedom and the reduced stochastic model predicted by the theory for two resolved variables involves both nonlinear interaction and multiplicative noises. Even for large value of the correlation time ratio of the order of 1/2, the reduced stochastic model with two degrees of freedom captures the essentially nonlinear and non-Gaussian statistics of the original nonlinear systems with 102 modes extremely well. Furthermore, it is shown here that the standard regression fitting of the second-order correlations alone fails to reproduce the nonlinear stochastic dynamics in this example.     

Cosmic dust and our origins

The small solid particles in the space between the stars provide the surfaces for the production of many simple and complex molecules. Processes involving the effects of ultraviolet irradiation of the thin (hundredth micron) mantles are shown to produce a wide range of molecules and ions also seen in comets. Some of the more complex ones inferred from laboratory experiments are expected to play an important role in the origin of life. An outline of the chemical evolution of interstellar dust as observed and as studied in the laboratory is presented. Observations of comets are shown to provide substantial evidence for their being fluffy aggregates of interstellar dust as it was in the protosolar nebula, i.e. the interstellar cloud which collapsed to form the solar system. The theory that comets may have brought the progenitors of life to the earth is summarized.