Synthesis of 21-nor-22-oxa-1α,25-dihydroxyvitamin D3 derivatives in quest of a drug with low calcemic activity

Two 21-nor-22-oxa-1α,25-dihydroxyvitamin D₃ derivatives have been synthesized in quest of a drug with lower calcemic activity than Maxacalcitol, 22-oxa-1α,25-dihydroxyvitamin D₃, being used as antihyperparathyroidism and antipsoriatic drug. Of two 21-nor products obtained, the product carrying one carbon elongated side chain with diethylcarbinol moiety has been found to exhibit comparable differentiation-inducing activity to Maxacalcitol with much lower exhibition of calcemic activity.     

Strukturelle Abwandlungen anN-Acetylneuraminsäure, 3

From methyl-5-acetylamino-7,8-anhydro-4,9-O-bis-(t-butyldimethylsilyl)-3,5-dideoxy--D-glycero-D-galacto-2-nonulopyranosidonic acid methylester (1) the derivatives1 a and1 b were obtained by removing the 9-O-(t-butyldimethylsilyl)group withBu ₄NF, followed by acetylation. Treatment of1 b with 80% acetic acid and acetanhydride/pyridine yields the 8-epi-N-acetylneuraminic acid derivative2 a and the 7-epi-N-acetylneuraminic acid derivative3 a in a ratio of 3:1 (Scheme 1). The structure elucidation of2 b was achieved by converting2 b via the 4,9-bis-O-(tBDMSi)-8-O-tosyl-derivative2 d into the epoxide1 (Scheme 2). Using the same sequence the epoxides4 and5 were transformed into theN-acetylneuraminic acid derivative6 a and the 7,8-bis-epi-N-acetylneuraminic acid derivative7 a (Scheme 3). After treatment with sodium hydroxide and 0.025m HCl and Dowex 50 H⁺ the 8-epi-, 7-epi- and 7,8-bis-epi-N-acetylneuraminic acids2,3, and7 were obtained. These three compounds were tested withCMP-N-acetylneuraminic acid synthetase.

Herrn KollegenK. Schlögl mit den besten Wünschen zum 60. Geburtstag.     

Cyclotomic units over finite fields

Let ζ be a primitivesp-th root of unity for a primep>2, and consider the group Ω(ζ) of cyclotomic units in the ringR(ζ)=ℒ[ζ+ζ^-1]. This paper deals with the image of Ω(ζ) in the unit group ofR(ζ)/qR(ζ), whereq is a prime ≠p. In particular, it obtains criteria for this image to be essentially everything, and a lower bound on the density of primesp (withq fixed) for which it cannot be. These results have a direct bearing on previous work about units in integral group rings for cyclic groups of orderpq. Work supported in part by an operating grant from NSERC (Canada).     

Cyclic caps in PG(3,q)

This article investigates cyclic completek-caps in PG(3,q). Namely, the different types of completek-capsK in PG(3,q) stabilized by a cyclic projective groupG of orderk, acting regularly on the points ofK, are determined. We show that in PG(3,q),q even, the elliptic quadric is the only cyclic completek-cap. Forq odd, it is shown that besides the elliptic quadric, there also exist cyclick-caps containingk/2 points of two disjoint elliptic quadrics or two disjoint hyperbolic quadrics and that there exist cyclick-caps stabilized by a transitive cyclic groupG fixing precisely one point and one plane of PG(3,q). Concrete examples of such caps, found using AXIOM and CAYLEY, are presented.     

Packing random intervals

Summary Letn random intervalsI ₁, ...,I _n be chosen by selecting endpoints independently from the uniform distribution on [0.1]. Apacking is a pairwise disjoint subset of the intervals; itswasted space is the Lebesgue measure of the points of [0,1] not covered by the packing. In any set of intervals the packing with least wasted space is computationally easy to find; but its expected wasted space in the random case is not obvious. We show that with high probability for largen, this best packing has wasted space . It turns out that if the endpoints 0 and 1 are identified, so that the problem is now one of packing random arcs in a unit-circumference circle, then optimal wasted space is reduced toO(1/n). Interestingly, there is a striking difference between thesizes of the best packings: about logn intervals in the unit interval case, but usually only one or two arcs in the circle case.     

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h ₁,h ₂) ·x = h ₁ xh ₂ ⁻¹ · LetP(G, H) denote the group ofH ¹-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H ⁰([0, 1], g) isometrically byg * u = gug ⁻¹ −g′g ⁻¹. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn.     

Maximal estimates on measure spaces for weak-type multipliers

We describe sufficient conditions for transferring from locally compact abelian groups to measure spaces the weak-type bounds of maximal operators defined by multipliers of weak type. This leads to homomorphism theorems for maximal multiplier operators. Communicated by Guido Weiss     

BV periodic solutions of an evolution problem associated with continuous moving convex sets

This paper is concerned with BV periodic solutions for multivalued perturbations of an evolution equation governed by the sweeping process (or Moreau's process). The perturbed equation has the form –DuN _C (t)(u(t))+F(t,u(t)), whereC is a closed convex valued continuousT-periodic multifunction from [0,T] to ^d ,N ^C (t)(u(t)) is the normal cone ofC(t) atu(t),F: [0,T]× ^{d d} is a compact convex valued multifunction and Du is the differential measure of the periodic BV solutionu. Several existence results for this differential inclusion are stated under various assumptions on the perturbationF.     

On Lipschitz property of solutions of functional equations

In this work it is proved that under certain conditions the continuous vector-valued solutionsf of the functional equation

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.