Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Examples of Twisted Cyclic Cocycles from Covariant Differential Calculi

For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators for some self-adjoint operator and some *-representation of the underlying *-algebra.     

Polymer chain dynamics under nanoscopic confinements

It is shown that the confinement of polymer melts in nanopores leads to chain dynamics dramatically different from bulk behavior. This so-called corset effect occurs both above and below the critical molecular mass and induces the dynamic features predicted for reptation. A spinodal demixing technique was employed for the preparation of linear poly(ethylene oxide) (PEO) confined to nanoscopic strands that are in turn embedded in a quasi-solid and impenetrable methacrylate matrix. Both the molecular weight of the PEO and the mean diameter of the strands were varied to a certain degree. The chain dynamics of the PEO in the molten state was examined with the aid of field-gradient NMR diffusometry (time scale, 10(-2)-10(0) s) and field-cycling NMR relaxometry (time scale, 10(-9)-10(-4) s). The dominating mechanism for translational displacements probed in the nanoscopic strands by either technique is shown to be reptation. On the time scale of spin-lattice relaxation time measurements, the frequency dependence signature of reptation (i.e., T1 approximately nu(3/4)) showed up in all samples. A "tube" diameter of only 0.6 nm was concluded to be effective on this time scale even when the strand diameter was larger than the radius of gyration of the PEO random coils. This corset effect is traced back to the lack of the local fluctuation capacity of the free volume in nanoscopic confinements. The confinement dimension is estimated at which the crossover from confined to bulk chain dynamics is expected.     

On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy’s uncertainty principle using properties of the prolate spheroidal wave functions. We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on \(L^2(\mathbb {R})\) . A weak version of Hardy’s uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.     

Investigation of the Chemo- and Stereoselectivity of the Ketene-Claisen Rearrangement

A novel type of ketene-Claisen rearrangement in which the precursor of the rearrangement is generated in situ by reaction of optically active allyl thioethers with dichloroketene is described. A characteristic feature of this rearrangement is the excellent chemoselectivity in favor of allyl thioethers vs. allyl ethers, i.e., exclusive chirality transfer of the allylic sulfur moiety is observed with 12, 13 , and 25--27 . The cyclic, optically active allyl thioethers (+)-(R)- 4 and (?)-(S)- 4 and the open-chain allyl thioethers 11--13 rearrange with in situ generated dichloroketene to the optically active thioesters (?)-(S)- 28 , (+)-(R)- 28 , and 31-33 , respectively. A chirality-transfer of > 99% in the cyclic cases (+)-(R)- 4 and (?)-(S)- 4 , and 96--98 % in the open-chain cases 11--13 is observed. Furthermore, the dichloroketene-Claisen rearrangement is characterized by a high asymmetric 1,2-induction. The chiral allylic sulfides 25--27 give the optically active thioesters 36--38 with a 1,2-induction > 99% as determined by NMR-shift experiments.     

Integration of electron density and molecular orbital techniques to reveal questionable bonds: the test case of the direct Fe-Fe bond in Fe2(CO)9

The article illustrates the advantages of partitioning the total electron density rho(rb), its Laplacian (inverted Delta)2 rho(rb), and the energy density H(rb) in terms of orbital components. By calculating the contributions of the mathematically constructed molecular orbitals to the measurable electron density, it is possible to quantify the bonding or antibonding character of each MO. This strategy is exploited to review the controversial existence of direct Fe-Fe bonding in the triply bridged Fe2(CO)9 system. Although the bond is predicted by electron counting rules, the interaction between the two pseudo-octahedral metal centers can be repulsive because of their fully occupied t(2g) sets. Moreover, previous atoms in molecules (AIM) studies failed to show a Fe-Fe bond critical point (bcp). The present electron density orbital partitioning (EDOP) analysis shows that one sigma bonding combination of the t(2g) levels is not totally overcome by the corresponding sigma* MO, which is partially delocalized over the bridging carbonyls. This suggests the existence of some, albeit weak, direct Fe-Fe bonding.