The behavior on fundamental group of a free pro-homotopy equivalence

We study the question: given a morphism $\text{?{(X}{}_{\mathrm{n}}\text{, x}{}_{\mathrm{n}}\text{)}→{(Y}{}_{\mathrm{n}}\text{, y}{}_{\mathrm{<mtext>n</mtext>}}\text{)}}$ in the category pro-(Poi nted. Homotopy) where the domain and range are inverse sequences of well-pointed CW complexes, and given that ? induces an isomorphism {X_n}→{Y_n} in pro-(Homotopy), what additional hypotheses force ? to be an isomorphism in pro-(Pointed Homotopy)? Conjecture. If the dimensions of the Y_n's are bounded, then ? is an isomorphism in pro-(Pointed Homotopy). We first prove the special case of this conjecture in which dim Y_n?d<∞ for all n, and $\text{lim}\text{{H}{}_{\mathrm{d}}\text{Y}{}_{\mathrm{n}}\text{}≠0}$, Y_n being the universal cover of Y_n. Then we deal with the general case: we show that there are certain elements of each π₁Y_n with the properties: (i) these elements commute if and only if ? is an isomorphism in pro-(Pointed Homotopy); (ii) if dim Y_n?d<∞ for all n, then powers of these elements commute. While (i) and (ii) are not incompatible, this result puts severe restrictions on the nature of any possible counter-example to the conjecture.Our method also gives pro-homotopy analogues of the well-known fact that if a K(π, 1) is N-dimensional, then π is torsion-free and contains no abelian subgroup of rank>N. The latter theorems apply to inverse sequences {Y_n} of CW complexes where dim Y_n is finite but not necessarily bounded, hence in particular to infinite-dimensional shape-aspherical compacta.     

Locally optimal rank tests for independence

The problem of testing the hypothesis of independence against multiparametrical set of alternatives is considered. Rank tests, having some locally maximin property are studied and a certain characterization of these tests is given. Finite sample and asymptotic test statistics in a restricted class of tests are derived.     

Synthesis and Photochemical Properties of Luminescent Eu(III) Complexes with Macrocyclic and Macropolycyclic Ligands Incorporating 3,3′-Biisoquinoline-2,2-dioxide Units

Three novel ligands incorporating two, or three 3,3-biisoquinoline-2,2-dioxide units have been synthesized using modified standard methods of cyclization. These ligands formed photoactive mononuclear complexes with a Eu(III) cation. The ligand with two photoactive units displayed a better quantum yield for the Eu(III) emission (0.064) upon UV excitation than the other two complexes incorporating three units, whose luminescence quantum yields have been estimated to be below 0.01. The luminescence of the Eu(III) complex with two sensitizing units was not stable in aqueous solution, whereas with the two other complexes, their luminescence in solution remained for a couple of days.     

On quantum deformations of d=4 conformal algebra

Three classes of classical r-matrices for sl(4, C) algebra are constructed in quasi-Frobenius algebra approach. They satisfy CYBE and are spanned respectively on 8,10,12 generators. The o(4, 2) reality condition can be imposed only on the eight dimensional r-matrices with dimension-full deformation parameters. Contrary to the Poincaré algebra case, it appears that all deformations with a mass-like deformation parameter (-deformations) are described by classical r-matrices satisfying CYBE.     

Theoretische Überlegungen zur Wechselwirkung zwischen verschiedenen Schwingungstypen, 1. Mitt.: Wechselwirkung zwischen Streckschwingungen und Ringwellungsschwingungen (ring-puckering vibrations)

Zusammenfassung Es werden die Folgen der zwei Gleichgewichtszustände bei Ringwellungsschwingungen besprochen. Kopplung zwischen Streckschwingungen mit Deformationsschwingungen einerseits und Ringwellungsschwingungen andererseits muß zu der experimentell beobachteten Abweichung der differentiellen Spektrumsverschiebung führen. Die Frequenzen der Deformationsschwingungen nehmen ab, die der Ringwellungsschwingungen nehmen zu.

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Theoretical discussion of the interaction between different types of vibration, I.: Interaction between stretching and ring-puckering-vibrations

The consequences of the two equilibrium states in ring-puckering vibrations are discussed. Coupling between valence and deformation vibration on the one hand and ring-puckering vibrations on the other has to lead to the experimentally observed irregularity in differential spectrum progression; the deformation frequencies will decrease, and the ring-puckering frequencies will increase.

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Mit 2 Abbildungen     

Spectrophotometric determination of boron in human hair with azomethine H

Summary Investigation of methods of hair sample mineralization was made. The most profitable method of hair mineralization was by combustion in oxygen. Boron was determined spectrophotometrically with azomethine H as a spectrophotometric reagent. Boron in hair was determined in the range of 0–5 g/g.     

Remarks on regular (LF)-spaces

Developing ideas found in a recent paper of Gilsdorf to arbitrary topological vector spaces (tvs) one shows that a Hausdorff (LB) _tv -spaceE is regular provided every null-sequence inE has a series convergent subsequence inE. Received by the editors 1980Mathematics Subject Classification (1985 Revision) 46A12.     

K-theory of twisted Grassmannians

We give a computation of the K-groups of Grassmannians and flag varieties over an arbitrary Noetherian base scheme. We also compute the K-groups of forms of Grassmannians and flag varieties associated to a sheaf of Azumaya algebras. One ingredient in the computation is the extension of the Bott theorem on the cohomology of line bundles on the flag variety (over Q) to a K ₀-Bott theorem valid over arbitrary Noetherian base schemes.Partially supported by the NSF.