On Shellsort and the frobenius problem

A boundO(N ^1+1/k ) for the running time of Shellsort, withO(logN) passes, is proved very simply by application of a Frobenius basis withk elements.     

Zur Berechnung der Geschossabweichung unter dem Einfluss eines Seitenwindes

Summary The first order correction which should be made to thez-coordinate of the standard trajectory of a projectile to account for a cross-wind is given by (1). It is shown that this formula is a very good approximation also for angles of departure near 90°.     

Hard-sphere binary-collision operators

The time displacement operator is described for a system of hard-sphere particles. We show how to avoid needing a representation for this operator in unphysical regions of phase space, and how to construct a useful representation in terms of binary collision operators in the physical region. The various binary collision operators used for hard-sphere systems are derived for the case of a system of two spheres, and the results are generalized toN-particle systems.Dedicated to Prof. E. G. D. Cohen on the occasion of his 65th birthday.     

Triplet state excitation of alkali molecules on helium droplets: experiments and theory

In this paper, we discuss the electronic structure of alkali dimer molecules in 3Pig states on the surface of a helium droplet. The perturbation due to the droplet will in general not satisfy rotational symmetry around the internuclear axis of the diatom and thus, in addition to a broadening and blue shift, will cause a splitting of electronic levels that are degenerate in the free molecules. We propose a model based on general symmetry arguments and on a small number of physically reasonable parameters. We demonstrate that such a model accounts for the essential features of laser-induced fluorescence (LIF) and magnetic circular dichroism (MCD) spectra of the (1)3Pig-a3Sigma+ transition of Rb2 and K2. Furthermore the MCD spectra, analyzed according to the approach of Langford and Williamson [J. Phys. Chem. A 1998, 102, 2415], allow a determination of the populations of Zeeman sublevels in the ground state and thus a measurement of the surface temperature of the droplet. The latter agrees with the accepted temperature, 0.37 K, measured in the interior of a droplet.     

The renaissance of high-energy CID for structural elucidation of complex lipids: MALDI-TOF/RTOF-MS of alkali cationized triacylglycerols

Triacylglycerols were analyzed as cationized species (Li⁺, Na⁺, K⁺) by high-energy CID at 20 keV collisions utilizing MALDI-TOF/RTOF mass spectrometry. Precursor ions, based on [M+Li]⁺-adduct ions exhibited incomplete fragmentation in the high and low m/z region whereas [M+K]⁺-adducts did not show useful fragmentation. Only sodiated precursor ions yielded product ion spectra with structurally diagnostic product ions across the whole m/z range. The high m/z region of the CID spectra is dominated by abundant charge-remote fragmentation of the fatty acid substituents. In favorable cases also positions of double bonds or of hydroxy groups of the fatty acid alkyl chains could be determined. A-type product ions represent the end products of these charge-remote fragmentations. B- and C-type product ions yield the fatty acid composition of individual triacylglycerol species based on loss of either one neutral fatty acid or one sodium carboxylate residue, respectively. Product ions allowing fatty acid substituent positional determination were present in the low m/z range enabling identification of either the sn-1/sn-3 substituents (E-, F-, and G-type ions) or the sn-2 substituent (J-type ion). These findings were demonstrated with synthetic triacylglycerols and plant oils such as cocoa butter, olive oil, and castor bean oil. Typical features of 20 keV CID spectra of sodiated triacylglycerols obtained by MALDI-TOF/RTOF MS were an even distribution of product ions over the entire m/z range and a mass accuracy of ±0.1 to 0.2 u. One limitation of the application of this technique is mainly the insufficient precursor ion gating after MS1 (gating window at 4 u) of species separated by 2 u.     

Pre-organization of the core structure of E-selectin antagonists

A new class of N-acetyl-D-glucosamine (GlcNAc) mimics for E-selectin antagonists was designed and synthesized. The mimic consists of a cyclohexane ring substituted with alkyl substituents adjacent to the linking position of the fucose moiety. Incorporation into E-selectin antagonists led to the test compounds 8 and the 2'-benzoylated analogues 21, which exhibit affinities in the low micromolar range. By using saturation transfer difference (STD)-NMR it could be shown that the increase in affinity does not result from an additional hydrophobic contact of the alkyl substituent with the target protein E-selectin, but rather from a steric effect stabilizing the antagonist in its bioactive conformation. The loss of affinity found for antagonists 10 and 35 containing a methyl substituent in a remote position (and therefore unable to support to the stabilization of the core) further supports this hypothesis. Finally, when a GlcNAc mimetic containing two methyl substituents (52 and 53) was used, in which one methyl was positioned adjacent to the fucose linking position and the other was in a remote position, the affinity was regained.