The Singular Set of Minima of Integral Functionals

In this paper we provide upper bounds for the Hausdorff dimension of the singular set of minima of general variational integrals where F is suitably convex with respect to Dv and Hölder continuous with respect to (x,v). In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than n, where .     

Online stochastic optimization under time constraints

This paper considers online stochastic combinatorial optimization problems where uncertainties, i.e., which requests come and when, are characterized by distributions that can be sampled and where time constraints severely limit the number of offline optimizations which can be performed at decision time and/or in between decisions. It proposes online stochastic algorithms that combine the frameworks of online and stochastic optimization. Online stochastic algorithms differ from traditional a priori methods such as stochastic programming and Markov Decision Processes by focusing on the instance data that is revealed over time. The paper proposes three main algorithms: expectation E, consensus C, and regret R. They all make online decisions by approximating, for each decision, the solution to a multi-stage stochastic program using an exterior sampling method and a polynomial number of samples. The algorithms were evaluated experimentally and theoretically. The experimental results were obtained on three applications of different nature: packet scheduling, multiple vehicle routing with time windows, and multiple vehicle dispatching. The theoretical results show that, under assumptions which seem to hold on these, and other, applications, algorithm E has an expected constant loss compared to the offline optimal solution. Algorithm R reduces the number of optimizations by a factor |R|, where R is the number of requests, and has an expected ρ(1+o(1)) loss when the regret gives a ρ-approximation to the offline problem.     

Tragoponol, a dimeric dihydroisocoumarin from Tragopogon porrifolius L.

A phytochemical re-investigation of Tragopogon porrifolius L. (Asteraceae) yielded (7S,15S)-2,4,12-trihydroxy-7-(4-hydroxyphenyl)-10-methoxy-15-(4-methoxyphenyl)-7,8,15,16-tetrahydrodibenzo[c,i][1,7]dioxacyclododecine-5,13-dione, named tragoponol, a dimeric dihydroisocoumarin.The compound, which represents the first of its kind, is comprised of the open-chained forms of two different mono-methoxylated dihydroisocoumarin moieties, scorzocreticin and hongkongenin, which are connected via two ester bonds to form a macrolide with two lactone moieties featuring a 12-membered ring. The structure of the nearly symmetrical compound was established by HR mass spectrometry, CD measurements, and extensive 1D and 2D NMR experiments.     

Formation of surface-bound acyl groups by reaction of acyl halides on ge(100)-2x1

We have investigated the reaction of a series of acyl halides, including acetyl chloride, acetyl bromide, acetyl-d3 chloride, benzoyl chloride, and pivaloyl chloride, on Ge(100)-2x1 with multiple internal reflection infrared (MIR-IR) spectroscopy, X-ray photoelectron spectroscopy (XPS), and density functional theory (DFT). Infrared spectra following saturation exposures of acetyl chloride and acetyl bromide to Ge(100)-2x1 at 310 K are nearly identical, both exhibiting strong nu(C=O) stretching peaks near 1685 cm-1 and no vibrational modes in the nu(Ge-H) region. These data provide strong evidence for the presence of a surface-bound acetyl group on Ge(100)-2x1, which results from a C-X dissociation reaction (where X=Cl, Br). For acetyl chloride, DFT calculations predict that the barrier to C-Cl dissociation is only 1 kcal/mol above a chlorine-bound precursor state and is considerably smaller than barriers leading to the [2+2] C=O cycloaddition and alpha-CH dissociation products. In addition to the C-X dissociation product, both infrared and photoelectron results point to the presence of a second structure for acetyl halides where the oxygen of the surface-bound acetyl group donates charge to a nearby surface atom. This interaction is not observed for benzoyl chloride and pivaloyl chloride.     

Polyhedra as domains of harmonic maps

Certain more or less known properties of polyhedra in combinatorial topology are established. In particular, the concept of normal pseudomanifolds is extended to so-called admissible polyhedra, whereby branching may occur. Admissible Riemannian polyhedra serve as domains of (generalized) harmonic functions or maps.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

Cyclopentadienyl-functionalised polyhedral silsesquioxanes as building blocks for new nanostructured materials

This work is to design new organosilicon precursors, which contain both siloxane and cyclopentadienyl functionalities and use these compounds as building blocks for new nanomaterials. Polyhedral silsesquioxanes, (RSiO_1.5)_n (n = 8 and 10) with cyclopentadienyl (R = -C₅H₅ or -(CH₂)₃-C₅H₅) functionalities have been prepared by hydrolytic condensation of the correspondent silicon organic precursors RSiCl₃ or RSi(OC₂H₅)₃. The products with cyclopentadienyl groups have been used to prepare new cross-linked 3D polymeric materials by Diels-Alder reaction. The compounds have been characterised by multinuclear (¹H and ²⁹Si) NMR, ESI MS and IR spectroscopy, UV/Vis spectrometry, gel permeation chromatography and scanning electron microscopy. The approach above seems to be very promising for the preparation of new nanostructured materials.     

Analysis of alcohol ethoxylates and alkylamine ethoxylates in agricultural soils using pressurised liquid extraction and liquid chromatography–mass spectrometry

Nonionic surfactants e.g. alcohol ethoxylates (AEOs) and alkylamine ethoxylates (ANEOs) are commonly utilised as adjuvants in pesticide formulations to enhance their effectiveness. In this study, analytical methods for AEO and ANEO determination in soil samples using pressurised liquid extraction (PLE) were developed and used in connection with LC–MS. The recovery of the method, which was highly dependent on the soil properties, varied in the range 47–106% for AEO and 27–109% for ANEO. Detection limits (LOD) were 7–13 µg kg^–1 for AEO and 24–43 µg kg^–1 for ANEO. The developed method has been applied to determine AEOs and ANEOs in surface soil samples from fields sprayed with glyphosate herbicides. Tallowalkylamine ethoxylates (an ANEO) were detected in the soil before and after pesticide application, with increasing concentrations after treatment. The highest concentration in the soil samples was observed for the ANEO homologues with the longest ethoxy chains; in the clay soil the concentration decreased with the length of the ethoxy chain. ANEOs added to pesticide formulations as a technical mixture will, as demonstrated in this study, behave as individual homologues, which is reflected in their behaviour in the environment.Abbreviations AEO Alcohol ethoxylates - ANEO Alkylamine ethoxylates - APEO Alkylphenol ethoxylates - APCI Atmospheric pressure chemical ionisation - ASE Accelerated solvent extraction - CEC Cationic exchange capacity - LC–MS Liquid chromatography–mass spectrometry - LOD Limit of detection - MAE Microwave-assisted extraction - PLE Pressurised liquid extraction - SD Standard deviation - SIM Selected-ion monitoring - SPE Solid-phase extraction - TEA Triethylamine     

Studies on organophosphorus compounds—XXVII: Synthesis of thiono-, thiolo- and dithiolactones

Unsubstituted and alkyl- or cyanosubstituted lactones such as dihydro-2(3 H)-furanone, dihydro-5-methyl-2(3 H)-furanone, dihydro-5,5-dimethyl-4-propyl-2(3 H)-furanone and tetrahydro-2,2-dimethyl-5-oxo-3furan-carbonitrile react with the dimer of p-methoxyphenylthionophosphine sulfide, 1, in anhydrous xylene or toluene to give the corresponding thionolactones, 3a-d, in good yields. Dihydro-2(3 H)-thiophenone and 1 produce dihydro-2(3 H)-thiophenthione. Aromatic lactones such as 2 H-1-benzopyran-2-one give the corresponding 2-thione. 1-Oxa-4-thiaspiro[4,5]decan-2-one, when treated with 1 at 120-125°, gave 1,4-dithiaspiro[4,5]decan-2-one and 1,4-dithias-piro[4,5]decan-2-thione. Tetrahydro-5,5-dimethyl-2-oxo-4-propyl-3-furancarboxylic acid ethyl ester reacted with 1 at 110° giving the corresponding 2-thione and 5,5-dimethyl-4-propyl-4,5-dihydrothieno[2,3-c]-1,2-dithiole-3-thione.     

BCS theory for trapped ultracold fermions

We develop an extension of the well-known BCS-theory to systems with trapped fermionic atoms. The theory fully includes the quantized energy levels in the trap. The key ingredient is to model the attractive interaction between two atoms by a pseudo-potential which leads to a well defined scattering problem and consequently to a BCS-theory free of divergences. We present numerical results for the BCS critical temperature and the temperature dependence of the gap. They are used as a test of existing semi-classical approximations. Received 18 December 1998