BMO Solvability and the <Emphasis Type=Italic>A</Emphasis><Subscript>∞</Subscript> Condition for Elliptic Operators

We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L ^p solvability for all p>p ₀.     

How tightly can you fold a sphere?

Consider a compact, connected Lie group G acting isometrically on a sphere Sn of radius 1. The quotient of Sn by this group action, Sn/G, has a natural metric on it, and so we may ask what are its diameter and q-extents. These values have been computed for cohomogeneity one actions on spheres. In this paper, we compute the diameters, extents, and several q-extents of cohomogeneity two orbit spaces resulting from such actions, and we also obtain results about the q-extents of Euclidean disks. Additionally, via a simple geometric criterion, we can identify which of these actions give rise to a decomposition of the sphere as a union of disk bundles. In addition, as a service to the reader, we give a complete breakdown of all the isotropy subgroups resulting from cohomogeneity one and two actions.     

BMO from dyadic BMO on the bidisc

We generalize to the bidisc a theorem of Garnett and Jones relatingthe space BMO of functions of bounded mean oscillation to itsmartingale counterpart, dyadic BMO. Namely, translation-averagesof suitable families of dyadic BMO functions belong to BMO.As a corollary, we deduce a biparameter version of a theoremof Burgess Davis connecting the Hardy space H¹ to martingaleH¹. We also prove the analogs of the theorem of Garnett andJones in the one-parameter and biparameter VMO spaces of functionsof vanishing mean oscillation.     

Translation ovoids over skew fields

We extend Tits' characterization of the translation ovoids in PG(n, F) to the case that F is a skew field, and give examples. We show that, for n3, a translation ovoid in PG(n, F) is (p, q) transitive for some (every) {p, q}F is communtative and is a hyperquadric.Dedicated to Professor Dr Helmut R. Salzmann on the occasion of his 60th birthday     

Semigroups with the ideal retraction property

Abstract. In this paper we investigate the structure of semigroups with the ideal retraction property i.e., semigroups which are not simple and have the property that each ideal is a homomorphic retract of the semigroup. We present examples to show that the ideal retraction property is neither hereditary nor productive. That this property is preserved by homomorphisms is established for some classes of semigroups, but the general question remains open. The classes of semigroups investigated in this paper are separative semigroups, ideal semigroups, semilattices, cyclic semigroups, nil semigroups, and Clifford semigroups. It is established that a semigroup with zero 0 which is expressible as a direct sum of each ideal and a dual ideal (complement with 0 adjoined) has the ideal retraction property. The converse holds for ideal semigroups, and an example is presented which demonstrates that the converse does not hold in general.     

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ³. In ℝn,n≥4, the Agmon-Miranda maximum principle andL ^p-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theL ^p Dirichlet problem for Δ ^m fails to be solvable forp>2(n−1)/(n−3). Supported in part by the NSF.     

Guanidinium-Type resonance stabilization and its biological implications. I. the guanidine and extended-guanidine series

An unusual type of π-electron delocalization in Y-shaped molecules related to guanidine and its protonated form, the guanidinium ion, has been studied by ab initio methods at the STO -3G and 3-21G levels. Results are reported for tautomeric, rotameric, and protonated forms of the oxygen-substituted guanidine series (urea, carbamic, and carbonic acids); “extended-guanidine” (aminomethylene guanidine) including pseudocyclic forms; and simple ring systems in which the extended-guanidine group is incorporated (3-amino-1,2,4-triazole, 2,4-diaminopyrimidine). Both the guanidine and guanidinium type stabilizations have been characterized in terms of a number of structural and energetic parameters: degree of single/double bond character from bond lengths and π-bond orders, electron distributions, and protonation energies. The major finding is that the structural and energetic properties of the isolated extended-guanidinium group resemble those of the group when incorporated within 6-membered heterocyclic or heterobicyclic rings, although the details vary with the nature of the ring and possibility of reinforcement or interference with the substructure resonance from overall ring delocalization. The implications for stabilization of the protonated forms of some biologically important pteridines is discussed.     

A diterpene related to cladiellin from a pacific soft coral

A new diterpene $\text{2}$ from a soft coral collected at Majuro Atoll has been related to cladiellin ($\text{1}$.     

Terminal acetylene-silver trifluoromethanesulfonate π complexes. Synthesis, stoichiometry, and activation of the terminal carbon---hydrogen bond

Terminal acetylene---silver(I) trifluoromethanesulfonate π complexes have been isolated. They are of 1 : 1 stoichiometry in the solid state and solution with all carbon---carbon triple bonds coordinated to silver ion. Their stability constants as measured by proton NMR are strongly solvent dependent and decrease in the order 1,5-hexadiyne> 1,7-octadiyne> 1-hexyne, 3-methyl-1-pentyne> 3,3-dimethyl-1-butyne. IR and NMR analysis indicates weakening and polarization of the terminal carbon---hydrogen bond upon silver ion coordination. In solution terminal acetylenes π coordinated to silver ion undergo deuterium exchange approximately 10⁵ times faster than uncoordinated acetylenes. The mechanism of deuterium exchange does not involve silver acetylides as intermediates.