Partitioning the triangles of the cross polytope into surfaces

We present a constructive proof that there exists a decomposition of the 2-skeleton of the k-dimensional cross polytope $\beta^k$ into closed surfaces of genus $g \leq 1$, each with a transitive automorphism group given by the vertex transitive $\mathbb{Z}_{2k}$-action on $\beta^k$. Furthermore we show that for each $k \equiv 1,5(6)$ the 2-skeleton of the (k-1)-simplex is a union of highly symmetric tori and M\"obius strips.Comment: 13 pages, 1 figure. Minor update. Journal-ref: Beitr. Algebra Geom. / Contributions to Algebra and Geometry, 53(2):473-486, 201

Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure

Crystal structure of the Escherichia coli RNA degradosome component enolase.

The crystal structure of Escherichia coli enolase (EC 4.2.1.11, phosphopyruvate hydratase), which is a component of the RNA degradosome, has been determined at 2.5 Å. There are four molecules in the asymmetric unit of the C2 cell, and in one of the molecules, flexible loops close onto the active site. This closure mimics the conformation of the substrate-bound intermediate. A comparison of the structure of the E. coli enolase with the eukaryotic enolase structures available (lobster and yeast) indicates a high degree of conservation of the hydrophobic core and the subunit interface of this homodimeric enzyme. The dimer interface is enriched in charged residues compared with other protein homodimers, which may explain our observations from analytical ultracentrifugation that dimerisation is affected by ionic strength. The putative role of enolase in the RNA degradosome is discussed; although it was not possible to ascribe a specific role to it, a structural role is possible.http://dx.doi.org

A Closed Contour of Integration in Regge Calculus

The analytic structure of the Regge action on a cone in $d$ dimensions over a boundary of arbitrary topology is determined in simplicial minisuperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone vertex, and a single boundary edge length to all 1-simplices lying on the boundary. The Regge action is analyzed in the space of complex edge lengths, and it is shown that there are three finite branch points in this complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes of the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value which depends on the topology of the bounding universe. For values less than the critical value the wave function exhibits exponential behaviour. It is shown that the critical value is positive for spherical topology in arbitrary dimensions. In three dimensions we compute the critical value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums.Comment: 16 pages, Latex, To appear in Gen. Rel. Gra

Addition of cetuximab to first-line chemotherapy in patients with advanced non-small-cell lung cancer: a cost-utility analysis

Background: Adding cetuximab to standard chemotherapy results in a moderate increase of overall survival in patients with advanced non-small-cell lung cancer (NSCLC), but the cost-effectiveness is unknown. Materials and methods: A Markov model was constructed based on the results of the First-Line ErbituX in lung cancer randomized trial, adding cetuximab to cisplatin-vinorelbine first-line chemotherapy in patients with advanced NSCLC. The primary outcome was the incremental cost-effectiveness ratio (ICER) of adding cetuximab, expressed as cost per quality-adjusted life year (QALY) gained, and relative to a willingness-to-pay threshold of €60 000/QALY. The impact of cetuximab intermittent dosing schedules on the ICER was also evaluated. Results: Adding cetuximab to standard chemotherapy leads to a gain of 0.07 QALYs per patient at an additional cost of €26 088. The ICER for adding cetuximab to chemotherapy was €376 205 per QALY gained. Intermittent cetuximab dosing schedules resulted in ICERs per QALY gained between €31 300 and €83 100, under the assumption of equal efficacy. Conclusions: From a health economic perspective, the addition of cetuximab to standard first-line chemotherapy in patients with epidermal growth factor receptor-expressing advanced NSCLC cannot be recommended to date, due to a high ICER compared with other health care interventions. Treatment schedules resulting in more favorable cost-utility ratios should be evaluate

Triangulations and Severi varieties

We consider the problem of constructing triangulations of projective planes over Hurwitz algebras with minimal numbers of vertices. We observe that the numbers of faces of each dimension must be equal to the dimensions of certain representations of the automorphism groups of the corresponding Severi varieties. We construct a complex involving these representations, which should be considered as a geometric version of the (putative) triangulations

Phase transitions and configuration space topology

Equilibrium phase transitions may be defined as nonanalytic points of thermodynamic functions, e.g., of the canonical free energy. Given a certain physical system, it is of interest to understand which properties of the system account for the presence of a phase transition, and an understanding of these properties may lead to a deeper understanding of the physical phenomenon. One possible approach of this issue, reviewed and discussed in the present paper, is the study of topology changes in configuration space which, remarkably, are found to be related to equilibrium phase transitions in classical statistical mechanical systems. For the study of configuration space topology, one considers the subsets M_v, consisting of all points from configuration space with a potential energy per particle equal to or less than a given v. For finite systems, topology changes of M_v are intimately related to nonanalytic points of the microcanonical entropy (which, as a surprise to many, do exist). In the thermodynamic limit, a more complex relation between nonanalytic points of thermodynamic functions (i.e., phase transitions) and topology changes is observed. For some class of short-range systems, a topology change of the M_v at v=v_t was proved to be necessary for a phase transition to take place at a potential energy v_t. In contrast, phase transitions in systems with long-range interactions or in systems with non-confining potentials need not be accompanied by such a topology change. Instead, for such systems the nonanalytic point in a thermodynamic function is found to have some maximization procedure at its origin. These results may foster insight into the mechanisms which lead to the occurrence of a phase transition, and thus may help to explore the origin of this physical phenomenon.Comment: 22 pages, 6 figure