On characterizing collections arising from N-gons in the plane

Consider a labeled convex n-gon in the plane and a point x in general position in the interior. We associate with this a collection (x) of 3-element sets, namely those 3-element subsets of the vertices which contain x in their interior. We investigate the problem of characterizing those collections of subsets which can be represented in this matter. We give a complete characterization of this and a related problem.School of Mathematics, University of Minnesota, Minneapolis, MN 55455. This work was supported in part by NSF grant DMS-8700995.Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England. Work done while Hill Visiting Professor at the University of Minnesota.     

Jacobi elliptic functions and change of variable in a convolution

We solve the functional equation

Radical concentration measurements in hydrocarbon diffusion flames

Absolute OH· concentrations and relative H-atom and O-atom profiles have been measured in a laminar, co-flowing methane/air diffusion flame burning at atmospheric pressure. Laser absorption and laser-induced fluorescence methods were used to probe the A ² X ²II_itransition in OH·. The maximum OH· concentration is found to be 1.8±0.2 ×10¹⁶ cm^–3 (mole fraction =5.0×10^–3) at a temperature of 2080 K, which is twice the value calculated assuming local total equilibrium but less than half that predicted from partial equilibrium (O₂+H₂ 2OH·). Multiphoton ionization (2+1 process at 243 nm) has been used to detect H atoms, while laser-induced fluorescence at 845 nm excited by two-photon absorption at 226 nm was employed to observe O atoms. In both cases it was found that low photon intensities (2×10⁸W/cm²) and a retroreflected beam, Doppler-free geometry was required in order to avoid the photolytic production of the species of interest. For all of the concentration profile data it is necessary to correct the raw signals for variations in collisional quenching. In the case of the multiphoton ionization measurements the variation in electron detection sensitivity as a function of flame position must be accounted for as well. Establishing absolute H-atom and O-atom concentrations is discussed in terms of partial equilibrium considerations and detailed flame structure calculations.Formerly the National Bureau of Standards     

Wiener-Hopf factorisation of Brownian motion

Summary We study how Brownian motion behaves under time change by a fluctuating additive functionalA _t , in particular letting be the first passage time ofA _t to zero we computeP _–x[B dy] explicitly in certain cases. The calculation is not an easy one, our method uses the Désiré André relation for the overshoot of a Lévy process and depends on some elliptic function identities. This paper only considers the one boundary case whereA _t is increasing (resp. decreasing) on the positive (resp. negative) half line.     

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereX _t is a Brownian motion andG is the generator of a (C ₀) contraction semigroupe ^tG.     

Crystal and molecular structure of 1R,3S,5S,7R,10R-3,6,6,10,14-Pentamethyltricyclo[10.3.0.05,7]pentadeca-11,14-diene-1, 10-dihydroxy-2,13-dione (Japodagrol), C20H28O4

A new cytotoxic macrocyclic diterpenoid named Japodagrol, C₂₀H₂₈O₄, has been isolated from the ornamental plantJatropha podagrica, family Euphorbiaceae, grown in Nigeria. The structure of the title compound, C₂₀H₂₈O₄, was solved by X-ray analysis.M _r =332.44, monoclinic space groupC2,a=23.285(4),b=6.5105(12),c=12.505(3) Å,=98.504(17),V=1874.8 ų,Z=4,D _c =1.178 Mg/m³. CuK radiation (graphite crystal monochromator, =1.54184 A), (CuK)=6.13 cm^–1,F(000)=720,T=290 K. Final conventionalR factor=0.035,R _w =0.038 for 3349 observed reflections. The structure was solved with the programDirdif. The 5-membered ring is closed to a half-chair form. The compound contains inter- and intramolecular hydrogen bonds.     

Acid/Base-treated activated carbons: characterization of functional groups and metal adsorptive properties

Surface modification of activated carbons by various physicochemical methods directs an attractive approach for improvement of heavy metal uptake from aqueous solutions. Activated carbons were modified with HCl and HNO3 optionally followed by NaOH. The effects of surface modifications on the properties of the carbons were studied by the specific surface area, carbon pH, and total acidity capacity as well as by scanning electron microscopy, X-ray photoelectron spectroscopy, and Fourier transform infrared spectroscopy. The modifications bring about substantial variation in the chemical properties whereas the physical properties remain nearly unchanged. NaOH causes an increase in the content of hydroxyl groups, while the HCl treatment results in an increase in the amount of single-bonded oxygen functional groups such as phenols, ethers, and lactones. The HNO3 modification generates a large number of surface functional groups such as carbonyl, carboxyl, and nitrate groups. The HNO3 modification significantly increases the copper adsorption, while the HCl treatment slightly reduces the copper uptake. Most of the copper ions are adsorbed rapidly in the first 2 h; the adsorption equilibrium is established in around 8 h. An intraparticle diffusion model successfully describes the kinetics of copper adsorption onto the carbons.     

N-α-Benzyloxyacetyl derivatives of (S)-4-benzyl-5,5-dimethyloxazolidin-2-one for the asymmetric synthesis of differentially protected α,β-dihydroxyaldehydes

α-Dibenzylamino- and α-benzyloxy- derivatives of N-acetyl-(S)-4-benzyl-5,5-dimethyloxazolidin-2-one readily undergo highly stereoselective boron mediated syn-aldol reactions with a range of aromatic and aliphatic aldehydes, generating the syn-aldol products in good to excellent yields as single diastereoisomers after purification. In the α-dibenzylamino series, deprotection of the functionalised aldol fragments to the corresponding α-amino-β-hydroxy methyl ester or α-amino-β-hydroxyaldehyde proved problematic, with a range of N- and O-protecting groups giving mixtures of products arising from endocyclic and exocyclic cleavage pathways. However, in the α-benzyloxy series, O-silyl protection of the aldol products, and subsequent DIBAL reduction gives stereoselectively the corresponding N-1′-hydroxyalkyloxazolidin-2-ones, which undergo base promoted fragmentation to the desired highly functionalised and differentially protected α,β-dihydroxyaldehydes in good yields and without loss of stereochemical integrity.