Shapes of polyhedra, mixed volumes and hyperbolic geometry

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2

Photoemission Beyond the Sudden Approximation

The many-body theory of photoemission in solids is reviewed with emphasis on methods based on response theory. The classification of diagrams into loss and no-loss diagrams is discussed and related to Keldysh path-ordering book-keeping. Some new results on energy losses in valence-electron photoemission from free-electron-like metal surfaces are presented. A way to group diagrams is presented in which spectral intensities acquire a Golden-Rule-like form which guarantees positiveness. This way of regrouping should be useful also in other problems involving spectral intensities, such as the problem of improving the one-electron spectral function away from the quasiparticle peak.Comment: 18 pages, 11 figure

Calorimetric Behavior of Phosphatidylcholine/Phosphatidylethanolamine Bilayers is Compatible with the Superlattice Model

Differential scanning calorimetry was used to study the phase behavior of binary lipid bilayers consisting of phosphatidylcholine (PC) and phosphatidylethanolamine (PE) of varying acyl chain length. A two-state transition model was used to resolve the individual transition components, and the two-state transition enthalpy, the relative enthalpy, and the transition temperature of each component were plotted as a function of composition. Intriguingly, abrupt changes in these thermodynamic parameters were observed at or close to many “critical” XPE values predicted by the superlattice model proposing that phospholipids with different headgroups tend to adopt regular rather than random lateral distributions. Statistical analysis indicated that the agreement between the observed and predicted “critical” compositions is highly significant. Accordingly, these data provide strong evidence that the molecules in PC/PE bilayers tend to adopt regular, superlattice-like lateral arrangements, which could be involved in the regulation of the lipid compositions of biological membranes

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t = −1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added

Collider Implications of Kaluza-Klein Excitations of the Gluons

We consider an asymmetric string compactification scenario in which the SM gauge bosons can propagate into one TeV$^{-1}$-size extra compact dimension. These gauge bosons have associated KK excitations that present additional contributions to the SM processes. We calculate the effects that the KK excitations of the gluons, $g^{\star}$'s, have on multijet final state production in proton-proton collisions at the Large Hadron Collider energy. In the case of dijet final states with very high $p_{{}_T}$, the KK signal due to the exchanges of the $g^{\star}$'s is several factors greater than the SM background for compactification scales as high as about 7 TeV. The high-$p_{{}_T}$ effect is not as dramatic for the direct production of a single on-shell $g^{\star}$, which subsequently decays into $q$-$\bar{q}$ pairs, where the KK signal significantly exceeds the SM three-jet background for compactification scales up to about 3 TeV. We also present our results for the four-jet final state signal from the direct production of two on-shell $g^{\star}$'s.Comment: 33 pages, LaTeX; added Figure 6 showing the dijet mass distribution and corresponding discussion in a paragraph on page 11; some additionaal discussions added; typos corrected; few references adde

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Polydispersity and ordered phases in solutions of rodlike macromolecules

We apply density functional theory to study the influence of polydispersity on the stability of columnar, smectic and solid ordering in the solutions of rodlike macromolecules. For sufficiently large length polydispersity (standard deviation $\sigma>0.25$) a direct first-order nematic-columnar transition is found, while for smaller $\sigma$ there is a continuous nematic-smectic and first-order smectic-columnar transition. For increasing polydispersity the columnar structure is stabilized with respect to solid perturbations. The length distribution of macromolecules changes neither at the nematic-smectic nor at the nematic-columnar transition, but it does change at the smectic-columnar phase transition. We also study the phase behaviour of binary mixtures, in which the nematic-smectic transition is again found to be continuous. Demixing according to rod length in the smectic phase is always preempted by transitions to solid or columnar ordering.Comment: 13 pages (TeX), 2 Postscript figures uuencode

Ergodic infinite group extensions of geodesic flows on translation surfaces

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces

Near-Infrared Spectroscopy of Photodissociation Regions: The Orion Bar and Orion S

To test the PDR model spectra under the physical conditions present in Orion, we have obtained moderate-resolution (R=3000) J-, H-, and K-band long-slit spectra of the Orion bar and Orion S regions These observations provide conclusive evidence for the PDR origin (rather than shocks) of the H_2 emission in each of these regions and demonstrate significant departures in the observed chemical structure from that predicted by models of homogeneous PDRs.Comment: Accepted to Astrophysical Journa