Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.     

The Ulam number of infinite graphs

The following is a conjecture of Ulam: In any partition of the integer lattice on the plane into uniformly bounded sets, there exists a set that is adjacent to at least six other sets. Two sets are adjacent if each contain a vertex of the same unit square. This problem is generalized as follows. Given any uniformly bounded partitionP of the vertex set of an infinite graphG with finite maximum degree, letP (G) denote the graph obtained by letting each set of the partition be a vertex ofP (G) where two vertices ofP (G) are adjacent if and only if the corresponding sets have an edge between them. The Ulam number ofG is defined as the minimum of the maximum degree ofP (G) where the minimum is taken over all uniformly bounded partitionsP. We have characterized the graphs with Ulam number 0, 1, and 2. Restricting the partitions of the vertex set to connected subsets, we obtain the connected Ulam number ofG. We have evaluated the connected Ulam numbers for several infinite graphs. For instance we have shown that the connected Ulam number is 4 ifG is an infinite grid graph. We have settled the Ulam conjecture for the connected case by proving that the connected Ulam number is 6 for an infinite triangular grid graph. The general Ulam conjecture is equivalent to proving that the Ulam number of the infinite triangular grid graph equals 6. We also describe some interesting geometric consequences of the Ulam number, mainly concerning good drawings of infinite graphs.     

An improved algorithm to locate critical points in a 3D scalar field as implemented in the program MORPHY

A new algorithm for location of the critical points in general scalar fields is described. The new method has been developed as part of an on-going process to exploit the topologic analysis of general 3D scalar fields. Part of this process involves the use of topologic information to seed the critical point search algorithm. The continuing move away from topologic studies of just the electron density requires more general algorithms and the ability to easily "plug in" new functions, for example, the Laplacian of the electron density ( triangle down (2)rho), the Electron Localisation Function (ELF), the Localised Orbital Locator (LOL), the Lennard-Jones function (LJF), as well as any new functions that may be proposed in the future. Another important aspect of the current algorithm is the retention of all possible intermediate information, for example, the paths describing the connectivity of critical points, as well as an ability to restart searches, something that becomes increasingly important when analysing larger systems. This new algorithm represents a core part of a new local version of the MORPHY code. We distinguish nine universal types of gradient paths.     

Chemoenzymatic synthesis of sialyl oligosaccharides with sialidases employing transglycosylation methodology

A series of sialyloligosaccharides was synthesized using the transglycolytic activity of the sialidases from Vibrio cholerae, Clostridium perfringens, Salmonella typhimurium, and Newcastle disease virus. According to their hydrolytic activities the sialidases from V. cholerae and C. perfringens catalyze preferentially the formation of sialyl alpha(2-6)-linkages whereas the sialidases from S.typhimurium and Newcastle disease virus show a distinct preference for alpha(2-3) directed sialylations. Using combined chemical and enzymatic methodologies structures such as T-(Thomsen-Friedenreich) antigen [beta-D-Gal-(1-3)-alpha-D-GalNAc-OThr], Tn-(Thomsen nouveau) antigen (alpha-D-GalNAc-OThr) and beta-D-Gal-(1-4)-alpha-D-2-deoxy-Gal-OMe were sialylated in alpha(2-3)- and alpha(2-6)-positions regioselectively or in high regioisomeric excess and purified by simple isolation procedures. Depending on the enzyme source and acceptor structure yields for transsialylation varied between 10 and 30%.     

H2/CO2 separations in multicomponent metal-adeninate MOFs with multiple chemically distinct pore environments

Metal–organic frameworks constructed from multiple (≥3) components often exhibit dramatically increased structural complexity compared to their 2 component (1 metal, 1 linker) counterparts, such as multiple chemically unique pore environments and a plurality of diverse molecular diffusion pathways. This inherent complexity can be advantageous for gas separation applications. Here, we report two isoreticular multicomponent MOFs, bMOF-200 (4 components; Cu, Zn, adeninate, pyrazolate) and bMOF-201 (3 components; Zn, adeninate, pyrazolate). We describe their structures, which contain 3 unique interconnected pore environments, and we use Kohn–Sham density functional theory (DFT) along with the climbing image nudged elastic band (CI-NEB) method to predict potential H₂/CO₂ separation ability of bMOF-200. We examine the H₂/CO₂ separation performance using both column breakthrough and membrane permeation studies. bMOF-200 membranes exhibit a H₂/CO₂ separation factor of 7.9. The pore space of bMOF-201 is significantly different than bMOF-200, and one molecular diffusion pathway is occluded by coordinating charge-balancing formate and acetate anions. A consequence of this structural difference is reduced permeability to both H₂ and CO₂ and a significantly improved H₂/CO₂ separation factor of 22.2 compared to bMOF-200, which makes bMOF-201 membranes competitive with some of the best performing MOF membranes in terms of H₂/CO₂ separations.

Tailorable multicomponent MOFs and MOF membranes for efficient H₂/CO₂ separation.     

Synthesis of fluorescently labeled UDP-GlcNAc analogues and their evaluation as chitin synthase substrates

[structure: see text] Chitin synthase (CS) polymerizes UDP-GlcNAc to form chitin (poly-beta(1,4)-GlcNAc), a key component of fungal cell wall biosynthesis. Little is known about the substrate specificity of chitin synthase or the scope of substrate modification the enzyme will tolerate. Following a previous report suggesting that 6-O-dansyl GlcNAc is biosynthetically incorporated into chitin, we became interested in developing an assay for CS activity based on incorporation of a fluorescent substrate. We describe the synthesis of two fluorescent UDP-GlcNAc analogues and their evaluation as chitin synthase substrates.     

Beitrag zum Typenproblem der Riemannschen Flächen

Ohne Zusammenfassung Für das Zustandekommen der vorliegenden Abhandlungs bich ich in erster Linie den Herren Prof.R. Nevanlinna und Dr.H. Wittich für die liebenswürdige Unterstützung, die sie mir w?hrend der Ausarbeitung zuteilkommen lie?en, zu tiefstem Dank verpflichtet. Meine Arbeit bei Herrn Prof.R. Nevanlinna wurde durch dieJubil?umsstiftung der Universit?t Zürich erm?glicht, der mein weiterer Dank gilt.