The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

Ditopic hexaazamacrocycles containing pyridine: synthesis, protonation and complexation studies

Structural studies of metal complexes of five ditopic hexaazamacrocycles containing two pyridine rings ([n]py2N4 n= 18, 20, 22, 24 and 26) have been carried out. The synthesis of macrocycles [22]- to [26]-py2N4 are also reported. The protonation constants of the last three compounds and the stability constants of their complexes with Ni2+, Cu2+, Zn2+, and Pb2+ were determined at 25 degrees C in 0.10 mol dm(-3) KNO3 in aqueous solution. Our results with [22]py2N4 show significant differences from those described previously, while [24]py2N4 has not been studied before and [26]py2N4 is a new compound. Mononuclear and dinuclear complexes of the divalent metal ions studied with [22]- to [26]-py2N4 were found in solution. The stability constants for the ML complexes of the three ligands follow the Irving-Williams order: NiL2+ < CuL2+ > ZnL2+ > PbL2+, however for the dinuclear complexes the values for Pb2+ complexes are higher than the corresponding values for the Ni2+ and the Zn2+ complexes. The X-ray single crystal structures of the supramolecular aggregates [Cu2([20]py2N4)(H2O)4][Cu(H2O)6](SO4)3 x 3H2O and [Cu(2)([20]py(2)N4)(CH3CN)4][Ni([20]py2N4)]2(ClO4)8 x H2O, which are composed of homodinuclear [Cu2([20]py2N4])(H2O)4]4+ and [Cu2([20]py2N4])(CH3CN))4]4+, and mononuclear species, [Cu(H2O)6]2+ and [Ni([20]py2N4)]2+, respectively, assembled by an extensive network of hydrogen bonds, are also reported. In both homodinuclear complexes the copper centres are located at the end of the macrocycle and display distorted square pyramidal coordination environments with the basal plane defined by three consecutive nitrogen donors and one solvent molecule, water in and acetonitrile in . The macrocycle adopts a concertina-type conformation leading to the formation of macrocyclic cavities with the two copper centres separated by intramolecular distances of 5.526(1) and 5.508(7) A in 1a and 2a, respectively. The mononuclear complex [Ni([20]py2N4])]2+ displays a distorted octahedral co-ordination environment with the macrocycle wrapping the metal centre in a helical shape. EPR spectroscopy of the copper complexes indicated the presence of mono- and dinuclear species.     

Ruspolia hypercrateriformisM.R.: Isolierung und Strukturaufklärung von neuen Pyrrolidin-Alkaloiden. 169. Mitt. über organische Naturstoffe

Ruspolia hypercrateriformis M.R. : Isolation and Structure Elucidation of New Pyrrolidine Alkaloids Three new pyrrolidine alkaloids have been isolated from Ruspolia hypercrateriformis, which belongs to the plant family of Acanthaceae. The structure of the alkaloids ruspolinone ( 1 ), norruspolinone ( 2 ) and norruspoline ( 3 ) (Scheme 1) has been elucidated by means of spectroscopic data of the pure compounds and their derivatives, by chemical transformation fo 2 to 1 by methylation, by trans-formation of 3 and 2 to identical hydrogenation products, and by comparison of degradation products with synthetically prepared model compounds.     

Isothermal and nonisothermal gravimetry of nontronite dehydroxylation

A pure typical nontronite from Czechoslovakia (Sampor) was analysed by the title techniques in the range 570–1070 K. The isothermal dehydroxylation of nontronite, recorded at 630–730 K, was described reasonably, in the decomposition range =0.05–0.95, by diffusioncontrolled mechanismsD ₃ andD ₄. Application of the current solid-state reaction equations to the non-isothermal curve gave very poor results, except in the limited decomposition range =0.10 to 0.55. A unimolecular mechanism equation (Fm ₁) and also a second-order (SO) mechanism gave the best linearization of the curve. The activation energies estimated from the isothermal (D ₃,D ₄) and non-isothermal (F ₁) experiments were 125 and 151 kJ·mol^–1, respectively. Reduced time plots indicate the probable presence of a sequence of different mechanism for both techniques.

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Zusammenfassung Mittels der Titeltechniken wurde im Bereich 570–1070 K ein reines typisches Nontronit aus der SSR (Sampor) untersucht. Die isotherme Dehydroxylierung von Nontronit bei 630–730 K konnte im Bereich der Zersetzung von =0,05–0,95 durch die diffusionsbestimmten MechanismenD ₃ undD ₄ befriedigend beschrieben werden. Eine Anwendung der Gleichungen für Feststoffreaktionen auf die nichtisotherme Kurve ergab mit Ausnahme des Bereiches =0,10–0,55 nur sehr unzureichende Ergebnisse. Eine Gleichung für einen monomolekularen (F ₁) Mechanismus sowie für einen mechanismus zweiter Ordnung (SO) ergaben die beste Linearisierung der Kurve. Die aufgrund der isothermen (D ₃,D ₄) und nichtisothermen (F ₁) Experimente geschätzten Aktivierungsenergien betragen 125 bzw. 151 kJ·mol^–1. Es wird angenommen, daß es sich bei beiden Techniken um eine Sequenz verschiedener Mechanismen handelt.

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570–1070 (). , 630–730 =0,05–0,95, - D ₃ D ₄. , =0,10–0,55. (F ₁), . , (D ₃,D ₄) (F ₁) , , , 125 151 ·^–1. .

     

A Water-Soluble Luminescence Oxygen Sensor

We developed a water-soluble luminescent probe for dissolved oxygen. This probe is based on (Ru[dpp(SO₃Na)₂]₃) Cl₂, which is a sulfonated analogue of the well-known oxygen probe (Ru[dpp]₃)Cl₂. The compound dpp is 4,7-diphenyl-1,10-phenanthroline and dpp(SO₃Na)₂ is a disulfonated derivative of the same ligand. In aqueous solution in the absence of oxygen (Ru[dpp(SO₃Na)₂]₃)Cl₂ displays a lifetime of 3.7 μs that decreases to 930 ns on equilibrium with air and 227 ns on equilibrium with 100% oxygen. The Stern–Vohner quenching constant is 11330 M⁻¹. This high oxygen-quenching constant means that the photoluminescence of Ru(dpp[SO₃Na]₂)₃Cl₂ is 10% quenched at an oxygen concentration of 8.8 x 10⁻⁶ M , or equilibration with 5.4 torr of oxygen. The oxygen probe dissolved in water displays minimal interactions with lipid vesicles composed of dipalmityl-L-α-phosphatidyl glycerol but does appear to interact with human serum albumin. The absorption maximum near 480 nm, long lifetime and large Stokes'shift allow this probe to be used with simple instrumentation based on a light-emitting diode light source, allowing low-cost oxygen sensing in aqueous solutions. To the best of our knowledge this is the first practical water-soluble oxygen sensor.     

Über Integralzerlegungen von Darstellungen nilpotenter Liegruppen

For simply connected nilpotent Lie groups we determine explicitly the central decomposition measure of the restriction of a representation onto a subgroup as well as of the induction from a subgroup to the group. For this the Kirillov method is used. Knowing this measure we can show that in the non-abelian case there are non-trivial subgroups such that all representations of these subgroups induce a representation quasi-equivalent to the regular representation. Thus Fell's Frobenius Property (FP) never holds in the non-abelian case-contrary to Fell's conjecture. Finally we prove a weaker Frobenius, Property defined by decomposition measures.     

Light‐Induced Water Splitting Causes High‐Amplitude Oscillation of pH‐Sensitive Layer‐by‐Layer Assemblies on TiO2

We introduce a simple concept of a light induced pH change, followed by high amplitude manipulation of the mechanical properties of an adjacent polymer film. Irradiation of a titania surface is known to cause water splitting, and this can be used to reduce the environmental pH to pH 4. The mechanical modulus of an adjacent pH sensitive polymer film can thus be changed by more than an order of magnitude. The changes can be localized, maintained for hours and repeated without material destruction.     

Induced Cycles in Graphs

The maximum number vertices of a graph G inducing a 2-regular subgraph of G is denoted by \(c_\mathrm{ind}(G)\). We prove that if G is an r-regular graph of order n, then \(c_\mathrm{ind}(G) \ge \frac{n}{2(r-1)} + \frac{1}{(r-1)(r-2)}\) and we prove that if G is a cubic, claw-free graph on order n, then \(c_\mathrm{ind}(G) > \frac{13}{20}n\) and this bound is asymptotically best possible.