Topological Minors of Cover Graphs and Dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that ‐free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.     

Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra

Aristotle contended that (regular) tetrahedra tile space, an opinion that remained widespread until it was observed that non-overlapping tetrahedra cannot subtend a solid angle of 4π around a point if this point lies on a tetrahedron edge. From this 15th century argument, we can deduce that tetrahedra do not tile space but, more than 500 years later, we are unaware of any known non-trivial upper bound to the packing density of tetrahedra. In this article, we calculate such a bound. To this end, we show the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is not fully covered by the packing. The bound on the amount of space that is not covered in each sphere is obtained in a recursive way by building on the solid angle argument. The argument can be readily modified to apply to other polyhedra. The resulting lower bound on the fraction of empty space in a packing of regular tetrahedra is 2.6…×10⁻²⁵ and reaches 1.4…×10⁻¹² for regular octahedra.     

On the Dimension of Posets with Cover Graphs of Treewidth 2

In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth 3. In this paper we focus on the boundary case of treewidth 2. It was recently shown that the dimension is bounded if the cover graph is outerplanar (Felsner, Trotter, and Wiechert) or if it has pathwidth 2 (Biró, Keller, and Young). This can be interpreted as evidence that the dimension should be bounded more generally when the cover graph has treewidth 2. We show that it is indeed the case: Every such poset has dimension at most 1276.     

Partially distributed outer approximation

Journal of Global Optimization - This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming...     

Über die Synthese von 7-Methoxycarbonyl-5-phenyl-1H-thieno[2,3-e]1,4-diazepin-2(3H)-on

The preparation of 7-methoxycarbonyl-5-phenyl-1H-thieno[2,3-e]1,4-diazepin-2(3H)-one (8) and its 1-methyl-derivative (10) is described. Hydrolysation of these two products yielded neither of the two possible acids3 respectively13.     

Synthesis and evaluation of caged Garcinia xanthones

Inspired by the combination of unique structure and potent bioactivities exhibited by several family members of the caged Garcinia xanthones, we developed a synthesis of simplified analogues that maintain the overall caged motif. The caged structure of these compounds was constructed via a site-selective Claisen/Diels-Alder reaction cascade. We found that the fully substituted caged structure, in which are included the C18 and C23 geminal methyl groups, is necessary to maintain bioactivity. Analogue had comparable activity to the natural products of this family, such as gambogic acid. These compounds exhibit cytotoxicity in a variety of tumor cell lines at low micromolar concentrations and were found to induce apoptosis in HUVE cells. In addition, studies with HL-60 and HL-60/ADR cells indicate that these compounds are not affected by the mechanisms of multidrug resistance, conferred by P glycoprotein expression, typical of relapsed cancers and thus represent a new and potent pharmacophore.     

Über die Synthese von 4-Phenyl-[1,4]-diazepino[6,7,1-kl]phenthiazin-1(2H)-on

The preparation of 4-phenyl-[1.4]-diazepino[6.7.1-kl]phenothiazin-1(2H)-one (5) is described and some of its properties are compared with those of 1.4-benzodiazepines.     

Dead-time free measurement of dipole-dipole interactions between electron spins

A four-pulse version of the pulse double electron-electron resonance (DEER) experiment is presented, which is designed for the determination of interradical distances on a nanoscopic length-scale. With the new pulse sequence electron-electron couplings can be studied without dead-time artifacts, so that even broad distributions of electron-electron distances can be characterized. A version of the experiment that uses a pulse train in the detection period exhibits improved signal-to-noise ratio. Tests on two nitroxide biradicals with known length indicate that the accessible range of distances extends from about 1.5 to 8 nm. The four-pulse DEER spectra of an ionic spin probe in an ionomer exhibit features due to probe molecules situated both on the same and on different ion clusters. The former feature provides information on the cluster size and is inaccessible with previous methods.