CycloSiFA: The Next Generation of Silicon-Based Fluoride Acceptors for Positron Emission Tomography (PET)

The ring-opening Si-fluorination of a variety of azasilole derivatives cyclo-1-(iPr₂Si)−4-X−C₆H₃−2-CH₂NR ( 4 : R=2,6-iPr₂C₆H₃, X=H; 4 a : R=2,4,6-Me₃C₆H₂, X=H; 9 : R=2,6-iPr₂C₆H₃, X=tBuMe₂SiO; 10 : R=2,6-iPr₂C₆H₃, X=OH; 13 : R=2,6-iPr₂C₆H₃, X=HCCCH₂O; 22 : R=2,6-iPr₂C₆H₃, X=tBuMe₂SiCH₂O) with different ¹⁹F-fluoride sources was studied, optimized and the experience gained was used in a translational approach to create a straightforward ¹⁸F-labelling protocol for the azasilole derivatives [¹⁸F] 6 and [¹⁸F] 14 . The latter constitutes a potential clickable CycloSiFA prosthetic group which might be used in PET tracer development using Cu-catalysed triazole formation. Based on our findings, CycloSiFA has the potential to become a new entry into non-canonical labelling methodologies for radioactive PET tracer development.     

Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family R_t(z) = (1 + (4/27)z³/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rⁿ())_n. In particular, it is shown thatif escapes (that is, Rⁿ() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs.     

On index formulas for manifolds with metric horns

In this paper was discuss the index problem for geometric differential operators (Spin–Dirac opreatorm Gauβ–Bonnet operator, Singnature operator) on manifolds with metric horns. On singular manifolds these operators in general do not unique closed extensions. But there always exist two extremal extensions D$sub:min$esub:and D$sub:max$esub: We describe the quotient D (D$sub:max$esub:)/D(D$sub:esub:) explicitely in geometric resp.topological terms of the base manifolds of the metric horns.

We derive index formulas for the3 Spin–Dirac and GauβBonnet operator. For the Singnature operator we present a partial result.     

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).     

Dedekind-Endlichkeit und Wohlordenbarkeit

Weak forms of the axiom of choice are defined, their set theoretical strength is investigated and as an application topological wellordering theorems are derived.     

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Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

Arrow's Theorem,Weglorz' Models and the Axiom of Choice

Applying Weglorz' mode s of set theory without the axiom of choice, we investigate Arrow‐type social we fare functions for infinite societies with restricted coalition algebras. We show that there is a reasonable, nondictatorial social welfare function satisfying “finite discrimination”, if and only if in Weglorz' mode there is a free ultrafilter on a set representing the individuals.