Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.Comment: v4: Minor error in Lemma 5.5 fixed, Section 6.6 added, minor improvements. 44 pages, 14 figure

A topological classification of convex bodies

The shape of homogeneous, generic, smooth convex bodies as described by the Euclidean distance with nondegenerate critical points, measured from the center of mass represents a rather restricted class M_C of Morse-Smale functions on S^2. Here we show that even M_C exhibits the complexity known for general Morse-Smale functions on S^2 by exhausting all combinatorial possibilities: every 2-colored quadrangulation of the sphere is isomorphic to a suitably represented Morse-Smale complex associated with a function in M_C (and vice versa). We prove our claim by an inductive algorithm, starting from the path graph P_2 and generating convex bodies corresponding to quadrangulations with increasing number of vertices by performing each combinatorially possible vertex splitting by a convexity-preserving local manipulation of the surface. Since convex bodies carrying Morse-Smale complexes isomorphic to P_2 exist, this algorithm not only proves our claim but also generalizes the known classification scheme in [36]. Our expansion algorithm is essentially the dual procedure to the algorithm presented by Edelsbrunner et al. in [21], producing a hierarchy of increasingly coarse Morse-Smale complexes. We point out applications to pebble shapes.Comment: 25 pages, 10 figure

There is no triangulation of the torus with vertex degrees 5, 6, ..., 6, 7 and related results: Geometric proofs for combinatorial theorems

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be bicolored. Similar statements hold for 4,8-triangulations and 2,6-quadrangulations. We prove these results, of which the first two are known and the others seem to be new, as corollaries of a theorem on the holonomy group of a euclidean cone metric on the torus with just two cone points. We provide two proofs of this theorem: One argument is metric in nature, the other relies on the induced conformal structure and proceeds by invoking the residue theorem. Similar methods can be used to prove a theorem of Dress on infinite triangulations of the plane with exactly two irregular vertices. The non-existence results for torus decompositions provide infinite families of graphs which cannot be embedded in the torus.Comment: 14 pages, 11 figures, only minor changes from first version, to appear in Geometriae Dedicat

Structural basis for the assembly and quinone transport mechanisms of the dimeric photosynthetic RC-LH1 supercomplex

The reaction center (RC) and light-harvesting complex 1 (LH1) form a RC–LH1 core supercomplex that is vital for the primary reactions of photosynthesis in purple phototrophic bacteria. Some species possess the dimeric RC–LH1 complex with a transmembrane polypeptide PufX, representing the largest photosynthetic complex in anoxygenic phototrophs. However, the details of the architecture and assembly mechanism of the RC–LH1 dimer are unclear. Here we report seven cryo-electron microscopy (cryo-EM) structures of RC–LH1 supercomplexes from Rhodobacter sphaeroides. Our structures reveal that two PufX polypeptides are positioned in the center of the S-shaped RC–LH1 dimer, interlocking association between the components and mediating RC–LH1 dimerization. Moreover, we identify another transmembrane peptide, designated PufY, which is located between the RC and LH1 subunits near the LH1 opening. PufY binds a quinone molecule and prevents LH1 subunits from completely encircling the RC, creating a channel for quinone/quinol exchange. Genetic mutagenesis, cryo-EM structures, and computational simulations provide a mechanistic understanding of the assembly and electron transport pathways of the RC–LH1 dimer and elucidate the roles of individual components in ensuring the structural and functional integrity of the photosynthetic supercomplex

Structural basis for the assembly and electron transport mechanisms of the dimeric photosynthetic RC–LH1 supercomplex

AbstractThe reaction center (RC) and light-harvesting complex 1 (LH1) form a RC–LH1 core supercomplex that is vital for the primary reactions of photosynthesis in purple photosynthetic bacteria. Some species possess the dimeric RC–LH1 complex with an additional polypeptide PufX, representing the largest photosynthetic complex in anoxygenic phototrophs. However, the details of the architecture and assembly mechanism of the RC–LH1 dimer are unclear. Here we report seven cryo-electron microscopy (cryo-EM) structures of RC–LH1 supercomplexes from Rhodobacter sphaeroides. Our structures reveal that two PufX polypeptides are positioned in the center of the S-shaped RC–LH1 dimer, interlocking association between the components and mediating RC–LH1 dimerization. Moreover, we identify a new transmembrane peptide, designated PufY, which is located between the RC and LH1 subunits near the LH1 opening. PufY binds a quinone molecule and prevents LH1 subunits from completely encircling the RC, creating a channel for quinone/quinol exchange. Genetic mutagenesis, cryo-EM structures, and computational simulations enable a mechanistic understanding of the assembly and electron transport pathways of the RC–LH1 dimer and elucidate the roles of individual components in ensuring the structural and functional integrity of the photosynthetic supercomplex.</jats:p

Cuinse2 Homojunction Diode Fabricated by Phosphorus OP9

Homojunction diodes were fabricated by doping of phosphorus to n-type Cu-In-Se thin films. The junction prepared by P implantation at the energy of 50 keV with the dose of 1 x 10(15) ions/cm 2 showed a rectification ratio of more than 100. Conduction in Cu-In-Se thin films, whose crystal structure is of the chalcopyrite type, changes from n- to p-type in such a way that group V elements (N, P, Sb, or Bi) substitute for Se in the film

Cuinse2 Homojunction Diode Fabricated by Phosphorus OP9

Homojunction diodes were fabricated by doping of phosphorus to n-type Cu-In-Se thin films. The junction prepared by P implantation at the energy of 50 keV with the dose of 1 x 10(15) ions/cm 2 showed a rectification ratio of more than 100. Conduction in Cu-In-Se thin films, whose crystal structure is of the chalcopyrite type, changes from n- to p-type in such a way that group V elements (N, P, Sb, or Bi) substitute for Se in the film

Homojunction Diode of Cuinse2 Thin-Film Fabricated by Nitrogen Implantation

Rectifying homojunction have been fabricated in polycrystalline CuInSe2 thin film. The p-n junction diode was obtained by short annealing in nitrogen atmosphere at 450-degrees-C following the ion implantation of nitrogen with the energy and the dose of 50 keV and 1 x 10(15) cm-2, respectively. The properties of the near surface region in the films implanted have been studied by the Raman scattering spectroscopy. The secondary ion mass spectroscopy depth profile of the nitrogens in the CuInSe2 film and the capacitive-voltage characteristics of the rectifying diode have been measured to characterize the junction properties. The photovoltaic characteristics in AM 1.0, 100 mW/cm2 are shown with the efficiency of 0.35%