Asymptotic behaviour of certain sets of prime ideals

Let be ideals of the commutative ring , let be a Noetherian -module and let be a submodule of ; also let be an Artinian -module and let be a submodule of . It is shown that, whenever is a sequence of -tuples of non-negative integers which is non-decreasing in the sense that for all and all , then Ass is independent of for all large , and also Att is independent of for all large . These results are proved without any regularity conditions on the ideals , and so (a special case of) the first answers in the affirmative a question raised by S. McAdam.

     

The circle problem on surfaces of variable negative curvature

In this note we study the problem of orbit counting for certain groups of isometries of simply connected surfaces with possibly variable negative curvature. We show that ifN(t) denotes the orbit counting function for a convex co-compact group of isometries then for some constantsC, h>0,N(t) Ce ^ht , ast +.     

Direct evaluation of DLVO theory for predicting long-range forces between a yeast cell and a surface

Using a combined gradient optical trap and evanescent wave light-scattering force-measurement technique, long-range colloidal forces were measured between a single Candida albicans yeast cell and a flat, bare glass surface in electrolyte concentrations ranging from 0.1 to 100 mM NaCl. The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory was compared to experimentally measured equilibrium force curves and found to provide a close approximation to the decay length of the measured forces for electrolyte concentrations up to about 0.23 mM NaCl. At higher electrolyte concentrations (>/=0.5 mM NaCl), decay lengths of force curves in experimental measurements were consistently longer than Debye lengths calculated from the electrolyte concentrations. In electrolyte concentrations of 10 and 100 mM NaCl, most cells attached rapidly, which prevented measurements of long-range forces. The small fraction of cells remaining unattached in these higher electrolyte concentrations displayed purely repulsive forces. These results show that the DLVO theory accurately describes cell-surface interactions when the Debye length is in the range of 20-30 nm but underpredicts the decay length of the interactions at higher electrolyte concentrations.     

Molybdenum alkyl- and aryl-thiolates

The reaction between [(η⁵-C₅H₅)MoH(CO)₃] and disulphides gives dimeric or trimeric complexes depending upon the conditions. The syntheses of the novel trinuclear molybdenum carbonyl complex [{Mo(η⁵-C₅H₅)(SR)(μ-CO)(CO)}₃] (R = Me), and dinuclear compounds [Mo₂(η⁵-C₅H₅)(μ-SR)₃(CO)₄] (R = Me) and [Mo₂(η⁵-C₅H₅)₂(SR)₂(CO)₂(μ-SR)(μ-Br)] (R = Me or Ph) are reported.     

Fast atom bombardment mass spectrometry of non-volatile organometallic compounds. Rhodium,iridium and platinum complexes containing a cumulene ligand

Fast atom bombardment (FAB) mass spectrometry has been used to examine a series of rhodium, iridium and platinum organimetallic complexes, in which a cumulene ligand is attached to the metal in either σ-or π-bonding fashion. The most intense ion formed in the rhodium and platinum series is the metal-bis(triphenylphosphine) ion, while the [Ir(P(C₆H₅)₃)₂CO]⁺ ion is most intense for the iridium series. The platinum complexes show the most intense molecular ion peaks (up to 35% relative intensity), while the rhodium complexes show the least intense molecular ion peaks. The primary fragmentations of all these complexes occur at the metal-ligand bonds. The cumulenic ligand is lost as an impact unit in all cases. The FAB mass spectra of Rh(P(C₆H₅)₃)₃Cl (Wilkinson's catalyst), Ir(P(C₆H₅)₃)₂COCl (Vaska's compound), Rh(P(C₆H₅)₃)₂COCl and Pt(P(C₆H₅)₃)₂(C₂H₄)–synthetic precursors or related compounds to the organometallic complexes examined here–are included for comparison.     

Particle spin from representations of the diffeomorphism group

The semidirect productI∧K of Schwartz' spaceI of functions on ?³ with the groupK of diffeomorphisms of ?³ provides a model for quantum theory based on local currents. Certain unitary representations of ?^K are induced by representations of \(\overline {{\text{SL(3, }}\mathbb{R}{\text{)}}}\) . From the local currents in these representations, we construct the generators of local rigid rotations, with respect to which the Hilbert space decomposes into invariant subspaces of fixed spin carrying representations of local SU(2). The physical interpretation of this procedure is discussed.     

NMR paramagnetic relaxation due to the S=5/2 complex, Fe(III)-(tetra-p-sulfonatophenyl)porphyrin: central role of the tetragonal fourth-order zero-field splitting interaction

The metalloporphyrins, Me-TSPP [Me=Cr(III), Mn(III), Mn(II), Fe(III), and TSPP=meso-(tetra-p-sulfonatophenyl)porphyrin], which possess electron spins S=3/2, 2, 5/2, and 5/2, respectively, comprise an important series of model systems for mechanistic studies of NMR paramagnetic relaxation enhancement (NMR-PRE). For these S>1/2 spin systems, the NMR-PRE depends critically on the detailed form of the zero-field splitting (zfs) tensor. We report the results of experimental and theoretical studies of the NMR relaxation mechanism associated with Fe(III)-TSPP, a spin 5/2 complex for which the overall zfs is relatively large (D approximately = 10 cm(-1)). A comparison of experimental data with spin dynamics simulations shows that the primary determinant of the shape of the magnetic relaxation dispersion profile of the water proton R1 is the tetragonal fourth-order component of the zfs tensor. The relaxation mechanism, which has not previously been described, is a consequence of zfs-induced mixing of the spin eigenfunctions of adjacent Kramers doublets. We have also investigated the magnetic-field dependence of electron-spin relaxation for S=5/2 in the presence of a large zfs, such as occurs in Fe(III)-TSPP. Calculations show that field dependence of this kind is suppressed in the vicinity of the zfs limit, in agreement with observation.