Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

Full-size image

and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.     

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

Water-assisted self-assembly of harmonic single and triple helices in a polymeric coordination complex

A novel two-dimensional coordination polymer containing infinite, coherently pitched single and triple helical motifs is formed by the self-assembly of Cd2+, succinate, water and a bipyridyl ligand.     

Direct optical emission spectroscopy of liquid analytes using an electrolyte as a cathode discharge source (ELCAD) integrated on a micro-fluidic chip

Atomic emission detection of metallic species in aqueous solutions has been performed using a miniaturised plasma created within a planar, glass micro-fluidic chip. Detection was achieved using an Electrolyte as a Cathode Discharge source (ELCAD) in which the sample solution itself is used as the cathode for the discharge. To realise the ELCAD technique within a micro-fluidic device, a parallel liquid-gas flow was set up in a micro-channel and a glow discharge ignited between the flowing liquid sample surface and a metal wire anode. The detection of copper and sodium was achieved, using atmospheric pressure air as a carrier gas, by observation of atomic emission lines of copper at 324 nm, 327 nm, 511 nm, 515 nm and 522 nm and an atomic emission line of sodium at 589 nm using a commercially available miniaturised spectrometer. A total electrical power of less than 70 mW was required to sustain the discharge. A semi-quantitative, absolute detection limit of 17 nmol s(-1) was obtained for sodium with a sample flow rate of 100 microL min(-1) and an integration time of 100 ms in air at atmospheric pressure. The volume required for such detection is approximately 170 nL. Further analysis was performed with an Echelle spectrometer using both argon and air as a carrier gas. The geometry and flow rates used demonstrate the feasibility of integrating such micro-plasmas into other micro-fluidic devices, such as miniaturised CE devices, as a method of detection. The potential for using such micro-plasmas within highly portable miniaturised systems and mu-TAS devices is presented and discussed.     

Functional polymers and sequential copolymers by phase transfer catalysis. XXVIII. Synthesis and characterization of alternating block copolymers and polyformals of polyisobutylene and aromatic polyether sulfone

Alternating—i.e., -(A-B)_n- type—block copolymers of polyisobutylene (PIB) and aromatic polyether sulfone (PSU) have been prepared by phase transfer catalyzed Williamson polyetherification of α,ω-di(phenol)PIB with α,ω-di(chloroallyl)- or -(bromobenzyl)PSU. Block copolymers of the two prepolymers were also synthesized by the phase transfer catalyzed polyetherification of methylene chloride with α,ω-di(phenol)PIB and α,ω-di(phenol)PSU (bisphenol-A-terminated PSU). This method leads to -[(A)_x-(B)_y]_n- block copolymers with formal linkages between segments. At sufficiently high segment lengths, both types of block copolymers exhibit two distinct T_gs, indicating phase separation into rubbery PIB and glassy PSU domains.     

Expedient Diels-Alder assembly of 4-aryl-4-phenylsulfonyl cyclohexanones

The efficient preparation of 4-aryl-4-phenylsulfonyl cyclohexanones, containing a quaternary sulfone-bearing carbon centre, is described. Their synthesis proceeds in 38-78% overall yield by way of three steps: (i) sulfinate alkylation; (ii) methylenation; and (iii) regioselective Diels-Alder condensation with 2-trimethylsiloxybutadiene. The scope and limitations of the one-pot Mannich-type methylenation described were examined.     

Synthesis and characterisation of a series of alkylmagnesium amide and related oxygen-contaminated "alkoxy" compounds

Synthesised either by an unusual tert-butyl metathesis between tert-butyllithium and a n,s-butylmagnesium amide or by reaction of an alkyl Grignard reagent and a sodium amide, five tert-butylmagnesium amides, Bu(t)MgDBA (5)(DBA=dibenzylamide), Bu(t)MgDA (6)(DA=diisopropylamide), Bu(t)MgHMDS (7)(HMDS=1,1,1,3,3,3-hexamethyldisilazide), Bu(t)MgTMP (8)(TMP=2,2,6,6-tetramethylpiperidide) and Bu(t)MgNCy2 (9)(cy=cyclohexyl) have been isolated as crystalline solids. All five amides have been characterised by X-ray crystallography and solution NMR spectroscopic studies. The former studies reveal a common dimeric molecular structure with amido bridges in a planar (MgN)2 ring and terminal Bu(t) ligands on the Mg atoms. Also described is the dodecameric primary amide [Bu(n)MgN(H)Dipp]12 (10a) and its monomeric solvate Bu(n)MgN(H)Dipp.TMEDA (10b)(Dipp=2,6-diisopropylphenyl; TMEDA=N,N,N',N'-tetramethylethylenediamine). The crystal structures of the oxo-insertion products Bu(t)MgOBu(t).THF (11), Bu(t)Mg(mu-OBu(t))(mu-TMP)MgTMP (12) and Mg(OBu(n))HMDS.solv [solv=THF (13a) or Et2O (13b)], made fortuitously during the course of this work, are also presented.     

Reactive intermediates in asymmetric cross-coupling catalysed by palladium P-N chelates

The palladium dibromide complexes of (S,R)-(1,1′-bis-diphenylphosphino)-2-ferrocencylthyldimethylamine and (S,R)-(1-diphenylphosphino)-2-ferrocenylethyldimethylamine have been reduced with dilithiocyclooctatetraene to form the corresponding Pd⁰ cyclooctatetraene complexes. Their reactions with E-4-methoxy-2′-bromophenylethene, and then benzylmagnesium chloride at −60 to −30°C, provide information on the structure of intermediates in asymmetric cross-coupling.     

Routine determination of major anions in atmospheric aerosols by capillary electrophoresis

Capillary electrophoresis (CE) with indirect UV detection utilizing a pyromellitate-based electrolyte was used for the routine analysis of major anions in atmospheric aerosols collected on filters with high-volume (Hi-Vol) samplers. The long-term reliability of the CE system was checked over an 8-month period during which over 2900 samples were analyzed. In addition, approximately 1100 samples were analyzed in parallel by ion chromatography (IC). It has been shown that acceptable analytical performance can be routinely obtained. The agreement between the CE and IC results is good, generally better than 20% at concentrations larger than 1 mg l⁻¹.