On an identity by Chaundy and Bullard. I

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.Comment: 20 pages; added in v3: more references to earlier occurrences of the identity and its multivariable analogue, combinatorial proof of the identity and extension to noninteger m,n, proof of multivariable identity by splitting up Dirichlet's multivariable beta integra

Anti-phospholipid-antibodies in patients with relapsing polychondritis

Relapsing polychondritis (RP) is an extremly rare multisystemic disease thought to be of autoimmune origin. In order to assess if RP is associated with anti-phospholipid antibodies (aPL), clinical data and sera of 21 patients with RP were collected in a multicentre study. Concentration of anti-cardiolipin antibodies (aCL) (IgG-, IgM-and IgA-isotypes), anti-phosphatidylserine-antibodies (aPS) (IgG-and IgM-isotypes) and anti-β-2-glycoprotein I-antibodies (aβ2 GPI) were measured by ELISA. In eight patients aCL were found to be elevated. One patient had elevated aPS. No patient had elevated aβ2 GPI. No patient had clinical signs and symptoms of a aPL syndrome. Interestingly, the two RP patients with the highest aPL had concomitant systemic lupus erythematosus (SLE). Therefore the presence of elevated aPL in RP is probably more closely related to an associated SLE than to RP itself. There is no convincing evidence that aPL are associated with RP

Spatial Light Modulators for the Manipulation of Individual Atoms

We propose a novel dipole trapping scheme using spatial light modulators (SLM) for the manipulation of individual atoms. The scheme uses a high numerical aperture microscope to map the intensity distribution of a SLM onto a cloud of cold atoms. The regions of high intensity act as optical dipole force traps. With a SLM fast enough to modify the trapping potential in real time, this technique is well suited for the controlled addressing and manipulation of arbitrarily selected atoms.Comment: 9 pages, 5 figure

Summation and transformation formulas for elliptic hypergeometric series

Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.Comment: 21 pages, AMS-LaTe

Structural studies of cesium, lithium/cesium and sodium/cesium bis(trimethylsilyl)amide (HMDS) complexes

Reacting cesium fluoride with an equimolar n-hexane solution of lithium bis(trimethylsilyl)amide (LiHMDS), allows the isolation of CsHMDS (1) in 80% yield (after sublimation). This preparative route to 1 negates the need for pyrophoric Cs metal or organocesium reagents in its synthesis. If a 2:1 LiHMDS:CsF ratio is employed the heterobimetallic polymer [LiCs(HMDS)2]∞ 2 was isolated (57% yield). By combining equimolar quantities of NaHMDS and CsHMDS in hexane/toluene [NaCs(HMDS)2(toluene)]∞ 3 was isolated (62% yield). Attempts to prepare the corresponding potassium-cesium amide failed, and instead yielded the known monometallic polymer [Cs(HMDS)(toluene)]∞ 4. With the aim of expanding the structural diversity of Cs(HMDS) species, 1 was reacted with several different Lewis basic donor molecules of varying denticity; namely, (R,R)-N,N,N′,N′-tetramethylcyclohexane-1,2-diamine [(R,R)-TMCDA] and N,N,Nʹ,Nʹ-tetramethylethylenediamine (TMEDA), N,N,Nʹ,Nʹʹ,Nʹʹ-pentamethyldiethylenetriamine (PMDETA), tris[2-(dimethylamino)ethyl]amine (Me6-TREN) and tris[2-(2-methoxyethoxy)ethyl]amine (TMEEA). These reactions yielded dimeric [Cs(HMDS)·donor]2 5-7 [where donor is (R,R)-TMCDA, TMEDA and PMDETA respectively], the tetranuclear ‘open’-dimer [{Me6-TREN·Cs(HMDS)}2{Cs(HMDS)}2] 8 and the monomeric Cs(HMDS)·TMEEA 9. Complexes 2,3 and 5-9 were characterized by X-ray crystallography and in solution by multinuclear NMR spectroscopy

An examination of the precipitation delivery mechanisms for Dolleman Island, eastern Antarctic Peninsula

Copyright @ 2004 Wiley-BlackwellThe variability of size and source of significant precipitation events were studied at an Antarctic ice core drilling site: Dolleman Island (DI), located on the eastern coast of the Antarctic Peninsula. Significant precipitation events that occur at DI were temporally located in the European Centre for Medium-Range Weather Forecasting (ECMWF) reanalysis data set, ERA-40. The annual and summer precipitation totals from ERA-40 at DI both show significant increases over the reanalysis period. Three-dimensional backwards air parcel trajectories were then run for 5 d using the ECMWF ERA-15 wind fields. Cluster analyses were performed on two sets of these backwards trajectories: all days in the range 1979–1992 (the climatological time-scale) and a subset of days when a significant precipitation event occurred. The principal air mass sources and delivery mechanisms were found to be the Weddell Sea via lee cyclogenesis, the South Atlantic when there was a weak circumpolar trough (CPT) and the South Pacific when the CPT was deep. The occurrence of precipitation bearing air masses arriving via a strong CPT was found to have a significant correlation with the southern annular mode (SAM); however, the arrival of air masses from the same region over the climatological time-scale showed no such correlation. Despite the dominance in both groups of back trajectories of the westerly circulation around Antarctica, some other key patterns were identified. Most notably there was a higher frequency of lee cyclogenesis events in the significant precipitation trajectories compared to the climatological time-scale. There was also a tendency for precipitation trajectories to come from more northerly latitudes, mostly from 50–70°S. The El Niño Southern Oscillation (ENSO) was found to have a strong influence on the mechanism by which the precipitation was delivered; the frequency of occurrence of precipitation from the east (west) of DI increased during El Niño (La Niña) events

Quantum Logic for Trapped Atoms via Molecular Hyperfine Interactions

We study the deterministic entanglement of a pair of neutral atoms trapped in an optical lattice by coupling to excited-state molecular hyperfine potentials. Information can be encoded in the ground-state hyperfine levels and processed by bringing atoms together pair-wise to perform quantum logical operations through induced electric dipole-dipole interactions. The possibility of executing both diagonal and exchange type entangling gates is demonstrated for two three-level atoms and a figure of merit is derived for the fidelity of entanglement. The fidelity for executing a CPHASE gate is calculated for two 87Rb atoms, including hyperfine structure and finite atomic localization. The main source of decoherence is spontaneous emission, which can be minimized for interaction times fast compared to the scattering rate and for sufficiently separated atomic wavepackets. Additionally, coherent couplings to states outside the logical basis can be constrained by the state dependent trapping potential.Comment: Submitted to Physical Review