Flat surfaces,Bratteli diagrams and unique ergodicity à la Masur

Recalling the construction of a flat surface from a Bratteli diagram, this paper considers the dynamics of the shift map on the space of all bi-infinite Bratteli diagrams as the renormalizing dynamics on a moduli space of flat surfaces of finite area. A criterion of unique ergodicity similar to that of Masur’s for flat surface holds: if there is a subsequence of the renormalizing dynamical system which has a good accumulation point, the translation flow or Bratteli–Vershik transformation is uniquely ergodic. Related questions are explored.     

Tightness games with bounded finite selections

For a topological space X and a point x ∈ X, consider the following game—related to the property of X being countably tight at x. In each inning n ∈ ω, the first player chooses a set A _n that clusters at x, and then the second player picks a point a _n ∈ A _n ; the second player is the winner if and only if \(x \in \overline {\left\{ {{a_n}:n \in \omega } \right\}} \).In this work, we study variations of this game in which the second player is allowed to choose finitely many points per inning rather than one, but in which the number of points they are allowed to choose in each inning has been fixed in advance. Surprisingly, if the number of points allowed per inning is the same throughout the play, then all of the games obtained in this fashion are distinct. We also show that a new game is obtained if the number of points the second player is allowed to pick increases at each inning.     

Constraining a possible time variation of the gravitational constant through "gravitochemical heating" of neutron stars

A hypothetical time variation of the gravitational constant G would cause neutron star matter to depart from beta equilibrium, due to the changing hydrostatic equilibrium. This induces nonequilibrium beta processes, which release energy that is invested partly in neutrino emission and partly in internal heating. Eventually, the star arrives at a stationary state in which the temperature remains nearly constant, as the forcing through the change of G is balanced by the ongoing reactions. Using the surface temperature of the nearest millisecond pulsar, PSR J0437-4715, inferred from ultraviolet observations, we estimate two upper limits for this variation: (1) |.G/G|< 2 x 10(-10) yr(-1), if direct Urca reactions are allowed, and (2) |.G/G|< 4 x 10(-12) yr(-1), considering only modified Urca reactions. The latter is among the most restrictive obtained by other methods.     

On elliptic problems involving critical Hardy-Sobolev exponents and sign-changing function

In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving the critical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nehari manifold and by exploring some properties of the best Hardy-Sobolev constant together with an approach developed by Brezis and Nirenberg.     

Uniform continuity on unbounded intervals

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives:

1. To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous;

2. To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions.

Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity.     

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to _ω1${\omega }_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “_ℵ1${\aleph }_1$-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.     

Supramolecular packing of alkyl substituted Janus face all-cis 2,3,4,5,6-pentafluorocyclohexyl motifs

This study uses X-ray crystallography, theory and Langmuir isotherm analysis to explore the conformations and molecular packing of alkyl all-cis 2,3,4,5,6-pentafluorocyclohexyl motifs, which are prepared by direct aryl hydrogenations from alkyl- or vinyl-pentafluoroaryl benzenes. Favoured conformations retain the more polar triaxial C–F bond arrangement of the all-cis 2,3,4,5,6-pentafluorocyclohexyl ring systems with the alkyl substituent adopting an equatorial orientation, and accommodating strong supramolecular interactions between rings. Langmuir isotherm analysis on a water subphase of a long chain fatty acid and alcohol carrying terminal all-cis 2,3,4,5,6-pentafluorocyclohexyl rings do not show any indication of monolayer assembly relative to their cyclohexane analogues, instead the molecules appear to aggregate and form higher molecular assemblies prior to compression. The study indicates the power and potential of this ring system as a motif for ordering supramolecular assembly.

Theory and Langmuir isotherm analysis was used to explore the conformations and molecular packing of alkyl all-cis 2,3,4,5,6-pentafluorocyclohexyl motifs, which are prepared by direct aryl hydrogenations from alkyl- or vinyl-pentafluoroaryl benzenes.