Topological sweep of the complete graph

We present a novel, simple and easily implementable algorithm to report all intersections in an embedding of a complete graph. For graphs with N vertices and complexity K measured as the number of segments of the embedding, the running time of the algorithm is Θ(K+NM), where M is the maximum number of edges cut by any vertical line. Our algorithm handles degeneracies, such as vertical edges or multiply intersecting edges, without requiring numerical perturbations to achieve general position.The algorithm is based on the sweep line technique, one of the most fundamental techniques in computational geometry, where an imaginary line passes through a given set of geometric objects, usually from left to right. The algorithm sweeps the graph using a topological line, borrowing the concept of horizon trees from the topological sweep method [H. Edelsbrunner, L.J. Guibas, Topologically sweeping an arrangement, J. Comput. Syst. Sci. 38 (1989) 165-194; J. Comput. Syst. Sci. 42 (1991) 249-251 (corrigendum)].The novelty in our approach is to control the topological line through the use of the moving wall that separates at any time the graph into two regions: the region of known structure, in front of the moving wall, and the region that may contain intersections generated by edges-that have not yet been registered in the sweep process-behind the wall.Our method has applications to graph drawing and for depth-based statistical analysis, for computing the simplicial depth median for a set of N data points [G. Aloupis, S. Langerman, M. Soss, G. Toussaint, Algorithms for bivariate medians and a Fermat-Torricelli problem for lines, Comp. Geom. Theory Appl. 26 (1) (2003) 69-79].We present the algorithm, its analysis, experimental results and extension of the method to general graphs.     

Regioselectivity of enzymatic and photochemical single electron transfer promoted carbon-carbon bond fragmentation reactions of tetrameric lignin model compounds

New types of tetrameric lignin model compounds, which contain the common β-O-4 and β-1 structural subunits found in natural lignins, have been prepared and carbon-carbon bond fragmentation reactions of their cation radicals, formed by photochemical (9,10-dicyanoanthracene) and enzymatic (lignin peroxidase) SET-promoted methods, have been explored. The results show that cation radical intermediates generated from the tetrameric model compounds undergo highly regioselective C-C bond cleavage in their β-1 subunits. The outcomes of these processes suggest that, independent of positive charge and odd-electron distributions, cation radicals of lignins formed by SET to excited states of sensitizers or heme-iron centers in enzymes degrade selectively through bond cleavage reactions in β-1 vs β-O-4 moieties. In addition, the findings made in the enzymatic studies demonstrate that the sterically large tetrameric lignin model compounds undergo lignin peroxidase-catalyzed cleavage via a mechanism involving preliminary formation of an enzyme-substrate complex.     

Molecular weight and polydispersity estimation of adsorbing polymer brushes by atomic force microscopy

We have estimated the molecular weight, Mn, and polydispersity, PDI, of densely grafted poly(N-isopropylacrylamide) (PNIPAM) brushes using a novel atomic force microscopy (AFM) approach. When compression of a polymer brush induced adsorption of multiple chains to an AFM tip, the resulting decompression force profile exhibited a maximum attractive force at a separation, Lm, that decayed to zero with increasing tip-sample separation. We have found that the separation Lm approximates the average contour length, Lc, determined by gel permeation chromatography (GPC). The detection of a decaying attractive force at separations larger than Lc suggests that chains of above average length sequentially break free from the tip as they are stretched away from the grafting surface. The shape of the decompression profile in this region approximately paralleled the cumulative weight fraction of the grafted chains determined by GPC. The fraction of chains of a given molecular weight determined from a single force curve fit a log-normal distribution, having a standard deviation that provided an estimate of the PDI. We have characterized two PNIPAM brushes by this AFM technique as well as by GPC coupled to a multiangle laser light-scattering detector (MALLS). The values obtained by AFM-(1) Mn,AFM = (3.8+/-0.5) x 10(4), PDI,(AFM) = 1.3+/-0.1 and (2) Mn,AFM = (9.4+/-1.4) x 10(4), PDI,(AFM) = 1.3+/-0.1-agreed quite well with the corresponding GPC/MALLS values of (1) Mn,GPC = 4.77 x 10(4), PDI,GPC = 1.33 and (2) Mn,GPC = 9.49 x 10(4), PDI = 1.35. This technique requires only a single force curve to obtain a statistical distribution of contour lengths and provides a novel method for estimating the Mn and PDI of appropriate uniformly grafted dense polymer layers.     

A REVIEW OF POLAR STRATOSPHERIC CLOUD FORMATION

Liquid and solid particles in polar stratospheric clouds （PSCs） have been known to play a crucial role in the chemical loss of stratospheric ozone over the Antarctic and Arctic regions in late winter and early spring. The stratospheric aerosol and cloud particles provide the sites where fast heterogeneous chemical reactions convert inactive halogen reservoir species into potential ozone destroying radicals. The sedimentation of nitric acid-containing PSC particles irreversibly removes HNO3 gas （denitrification） from the lower stratosphere, which slows the return of chlorine to its inactive forms, resulting in more severe stratospheric ozone destruction. Although these clouds have been investigated extensively during the past decade using in situ field observation, laboratory experiment and modeling studies, the detailed microphysics processes under cold stratospheric conditions are still uncertain. This paper reviews the recent advances in our understanding of PSCs.     

Co-ordinating Two Searchers for an Object Hidden on an Interval

Two unit-speed searchers starting at 0 seek a stationary target hidden according to a known bounded symmetric distribution. The objective is to minimize the expected time for the searchers to return to 0 after one of them has found the target. We find a general optimality condition and use it to solve the problem when the target has a uniform, triangular, or truncated, exponential distribution. The problem has applications to parallel processing and to the optimal choice of drilling depths in the search for an underground mineral.     

A concise synthetic approach to the sorbicillactones: total synthesis of sorbicillactone A and 9-epi-sorbicillactone A

A concise (12 step) total synthesis of sorbicillactone A and 9-epi-sorbicillactone A is reported. Unlike typical routes to the sorbicillinoids, this strategy does not start from sorbicillin and allows for the production of the bicyclic core on a multigram scale. The intramolecular conjugate addition of a tethered malonate serves as an effective means of introducing the lactone ring and provides a synthetic handle for installing the amide nitrogen.     

A unique three-dimensional coordination cluster based on a silver carbene complex

In an attempt to perform a simple anion-exchange reaction on a pincer-carbene-ligated nickel complex using AgNO(3), we instead obtained an unexpected three-dimensional (3D) Ag(7) cluster containing a [Ag(6)] core in a twisted-bowtie geometry. The reverse-transmetalation reaction by which the carbene is transferred from nickel to silver is virtually unprecedented. The CNC pincer-carbene ligands exhibit unusual bridging modes of ligand bonding for all three donor atoms. Another unique feature is that the final structure exhibits a 3D structure brought about by the connection of two-dimensional layers of the [Ag(6)] core via a seventh Ag ion.