A Helly-type theorem for countable intersections of starshaped sets

Let k and d be fixed integers, 0kd, and let be a collection of sets in If every countable subfamily of has a starshaped intersection, then is (nonempty and) starshaped as well. Moreover, if every countable subfamily of has as its intersection a starshaped set whose kernel is at least k-dimensional, then the kernel of is at least k-dimensional, too. Finally, dual statements hold for unions of sets.Received: 3 April 2004     

Illumination by translates of convex sets

Let S be a compact set in the plane. If every three points of S are illuminated clearly by some translate of the compact convex set T, then there is a translate of T which illumines every point of S. Various analogues hold for translates of flats in R ^das well.Supported in part by NSF grant DMS-8705336.     

Influence of alkyl chain length on the structure of dialkyldithiophosphinic Acid self-assembled monolayers on gold

We report the formation and characterization of self-assembled monolayers (SAMs) based on dialkyldithiophosphinic acid adsorbates {[CH(3)(CH(2))(n)](2)P(S)SH (n = 5, 9, 11, 13, 15)} on gold substrates. SAMs were characterized using X-ray photoelectron spectroscopy, reflection-absorption infrared spectroscopy, contact angle measurements, and electrochemical impedance spectroscopy. Data show that there is a roughly 60:40 mixture of bidentate and monodentate adsorbates in each of these SAMs. The presence of monodentate adsorbates is due to the numerous and deep grain boundaries of the underlying gold substrate, which disrupt chelation. Comparing the characterization data of dialkyldithiophosphinic acid SAMs with those of analogous n-alkanethiolate SAMs shows that both SAMs follow a similar trend: The alkyl chains become increasingly organized and crystalline with increasing alkyl chain length. The alkyl groups of dialkyldithiophosphinic acid SAMs, however, are generally less densely packed than those of n-alkanethiolate SAMs. For short alkyl chains (hexyl, decyl, and dodecyl), the significantly lower packing densities cause the alkyl chains to be liquid-like and disorganized. Long-chain dialkyldithiophosphinic acid SAMs are only slightly less crystalline than analogous n-alkanethiolate SAMs.     

Bottom-up view of water network-mediated CO2 reduction using cryogenic cluster ion spectroscopy and direct dynamics simulations

The transition states of a chemical reaction in solution are generally accessed through exchange of thermal energy between the solvent and the reactants. As such, an ensemble of reacting systems approaches the transition state configuration of reactant and surrounding solvent in an incoherent manner that does not lend itself to direct experimental observation. Here we describe how gas-phase cluster chemistry can provide a detailed picture of the microscopic mechanics at play when a network of six water molecules mediates the trapping of a highly reactive "hydrated electron" onto a neutral CO(2) molecule to form a radical anion. The exothermic reaction is triggered from a metastable intermediate by selective excitation of either the reactant CO(2) or the water network, which is evidenced by the evaporative decomposition of the product cluster. Ab initio molecular dynamics simulations of energized CO(2)·(H(2)O)(6)(-) clusters are used to elucidate the nature of the network deformations that mediate intracluster electron capture, thus revealing the detailed solvent fluctuations implicit in the Marcus theory for electron-transfer kinetics in solution.     

Variable time lag and backward ejection in full-dimensional analysis of strong-field double ionization

Ensembles of 400,000 two-electron trajectories in three space dimensions are used with Newtonian equations of motion to track atomic double ionization under very strong laser fields. We report a variable time lag between e-e collision and double ionization, and find that the time lag plays a key role in the emergence directions of the electrons. These are precursors to production of electron momentum distributions showing substantial new agreement with experimental data.     

Raman spectral analysis of renal tissue: a novel application

Renal cell carcinoma (RCC) accounts for 85% of all primary renal cancers. The definitive diagnosis of RCC relies exclusively on the subjective pathological interpretation of the surgical specimen. In this study, we aimed to analyze renal tissue using objective Raman spectroscopy (RS). We obtained 15 pairs of RCC (T) and corresponding normal renal parenchymal tissues (N) from our biobank. There are three subtypes of RCC: clear cell RCC (ccRCC), papillary RCC (pRCC), and chromophobe RCC (cRCC). Five pairs of tissue of each subtype were enrolled. Fresh‐frozen sliced tissues were used for the RS detection. The Raman spectra between T and N were compared and analyzed using partial least squares (PLS) regression. Data for a total of 55 T and 58 N analyzable RS samples were obtained. The spectra were normalized by dividing the intensity of the characteristic peak at 1003 cm⁻¹ using phenylalanine's Raman peak. After further analysis with PLS, the sensitivity and specificity for discriminating T from N were 95% and 93%, respectively. The RCC subtypes can be discriminated at an accuracy of 72% for ccRCC, 88% for cRCC, and 86% for pRCC. This study demonstrates the feasibility of analyzing renal tissue using RS. RS, with its advantages of easy and objective tissue assessment, may be applied to aid intraoperative decision making and pathological tissue assessment. Copyright © 2014 John Wiley & Sons, Ltd.     

Radon and Carathéodory type theorems for staircase convexity

Aequationes mathematicae - We establish the following analogues of the familiar Radon and Carathéodory theorems for staircase convex sets in $$\mathbb {R}^d$$ : Various “best...