Conjugate points and closed geodesic arcs on convex surfaces

This paper discusses conjugate points on the geodesics of convex surfaces. It establishes their relationship with the cut locus. It shows the possibility of having many geodesics with conjugate points at very large distances from each other. It also shows that on many surfaces there are arbitrarily many closed geodesic arcs originating and ending at a common point. To achieve these goals, Baire category methods are employed.     

Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

Renormalization of the inverse square potential

The quantum-mechanical D-dimensional inverse square potential is analyzed using field-theoretic renormalization techniques. A solution is presented for both the bound-state and scattering sectors of the theory using cutoff and dimensional regularization. In the renormalized version of the theory, there is a strong-coupling regime where quantum-mechanical breaking of scale symmetry takes place through dimensional transmutation, with the creation of a single bound state and of an energy-dependent s-wave scattering matrix element.     

Reduction in principal fiber bundles: Covariant Euler-Poincaré equations

Let be a principal -bundle, and let be a -invariant Lagrangian density. We obtain the Euler-Poincaré equations for the reduced Lagrangian defined on , the bundle of connections on .

     

Ionic liquids with perfluorinated alkoxyaluminates

Ionic liquids with the weakly coordinating perfluoroalkoxyaluminate anions, Al(ORF)4-, are obtained by a metathesis reaction of LiAl(ORF)4 and [bmim]Cl or [bmpyr]Cl in high yields. The compounds [bmim][Al(hfip)4] and [bmpyr][Al(hfip)4] have surprisingly low viscosities at elevated temperatures (60 degrees C) and an exceptionally high electrochemical window, which might render them useful for certain applications.     

Gauge Theories Labelled by Three-Manifolds

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional ${\mathcal{N} = 2}$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that ${S^{3}_{b}}$ partition functions of two mirror 3d ${\mathcal{N} = 2}$ gauge theories are equal. Three-dimensional ${\mathcal{N} = 2}$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional ${\mathcal{N} = 2}$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.     

Dissipative Stochastic Evolution Equations Driven by General Gaussian and Non-Gaussian Noise

We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers examples of Gaussian processes, like fractional and bifractional Brownian motion and also non Gaussian examples like the Hermite process. We give an application of our results to the study of the stochastic version of a common model of potential spread in a dendritic tree. Our investigation is specially motivated by possibility to introduce long-range dependence in time of the stochastic perturbation.     

Integration of individual TiO2 nanotube on the chip: Nanodevice for hydrogen sensing

Titania (TiO₂) exists in several phases possessing different physical properties. In view of this fact, we report on three types of hydrogen sensors based on individual TiO₂ nanotubes (NTs) with three different structures consisting of amorphous, anatase or anatase/rutile mixed phases. Different phases of the NTs were produced by controlling the temperature of post‐anodization thermal treatment. Integration of individual TiO₂ nanotubes on the chip was performed by employing metal deposition function in the focused ion beam (FIB/SEM) instrument. Gas response was studied for devices made from an as‐grown individual nanotube with an amorphous structure, as well as from thermally annealed individual nanotubes exhibiting anatase crystalline phase or anatase/rutile heterogeneous structure. Based on electrical measurements using two Pt complex contacts deposited on a single TiO₂ nanotube, we show that an individual NT with an anatase/rutile crystal structure annealed at 650 °C has a higher gas response to hydrogen at room temperature than samples annealed at 450 °C and as‐grown. The obtained results demonstrate that the structural properties of the TiO₂ NTs make them a viable new gas sensing nanomaterial at room temperature. (© 2015 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)