The solution U(t) to the linear differential equation dU/dt = h(t)U can be represented by a finite product of exponential operators; In many interesting cases the representation is global. U(t) = exp[g1(t)H1] exp [g2(t)H2] … exp[gn(t)Hn] where gi(t) are scalar functions and Hi are constant operators. The number, n, of terms in this expansion is equal to the dimension of the Lie algebra generated by H(t). Each term in this product has time-independent eigenvectors. Some applications of this solution to physical problems are given.

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To improve the antitumor properties and optimize the pharmaceutical properties including solubility and protein binding of indolin-2-ones, a number of different basic and weakly basic analogues were designed and synthesized. 5-[5-Fluoro-2-oxo-1,2-dihydroindol-(3Z)-ylidenemethyl]-2,4-dimethyl-1H-pyrrole-3-carboxylic acid (2-diethylaminoethyl)amide (12b or SU11248) has been found to show the best overall profile in terms of potency for the VEGF-R2 and PDGF-Rbeta tyrosine kinase at biochemical and cellular levels, solubility, protein binding, and bioavailability. 12b is currently in phase I clinical trials for the treatment of cancers.

We have prepared a novel material, consisting of Pt particles supported on purified singlewalled carbon nanotubes (SWNTs), representing the first reported metal-loaded SWNT material. The material contains 10 wt % Pt on entangled SWNT bundles consisting of 20-100 nanotubes each. The average Pt particle size is 1-2 nm. High-resolution transmission electron microscopy (HRTEM) observations combined with electron energy-loss spectroscopy (EELS) indicate chemical bonding between Pt and the SWNT surfaces. This bonding is accomplished presumably by ion exchange on carboxylic acid sites created on the nanotube surfaces by slow wet oxidation in dilute HNO 3 . In addition, a simple SWNT purification scheme requiring no filtration was developed for the preparation of this material. The purification results in a well-defined structure for the metal support useful for investigating the role of this material as a heterogeneous catalyst and the effects of metal-support interactions (MSI). Preliminary kinetics measurements of the activity of this material for the selective partial hydrogenation of 3-methyl-2-butenal (prenal) to 3-methyl-2-butenol (prenol)sa representative reaction important in the fine chemicals industrysindicates the desired catalytic selectivity above 75°C, demonstrating a utility of this material that can be optimized in future work.

An automated method has been developed to fully characterize the three-dimensional structure of zeolite porous networks. The proposed optimization-based approach starts with the crystallographic coordinates of a structure and identifies all portals, channels, and cages in a unit cell, as well as their connectivity. We apply our algorithms to known zeolites, hypothetical zeolites, and zeolite-like structures and use the characterizations to calculate important quantities such as pore size distribution, accessible volume, surface area, and largest cavity and pore limiting diameters. We aggregate this data over many framework types to gain insights about zeolite selectivity. Finally, we develop a continuous-time Markov chain model to estimate the probability of occupancy of adsorption sites throughout the porous network. ZEOMICS, an online database of structure characterizations and web tool for the automated approach is freely available to the scientific community (http://helios.princeton.edu/zeomics/).

The increasing interest in high energy and high capacity batteries triggers the demand of clarifying the reaction mechanism in battery cathodes during high potential operations. This is critical for further improving the performance of commercially viable Ni rich compounds, however, the mechanism often involves both transition metal and oxygen activities that remain elusive. Here we report a comprehensive study of the both the

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