Folner tilings for actions of amenable groups

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes") with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable.Comment: Minor revisions. Final versio

Optimal Decision Making Through Return On Time Analysis For The Venture Capital Industry

Venture capital firms make capital investments in businesses in return for an equity stake. However finding businesses that deliver the kind of returns venture capital firms seek is not a straight forward task. This undertaking is made all the more difficult when investing in a start up business which has no track record. Once the investment has been made there is no guarantee that the business will survive let alone provide the returns that are sought. This study investigates the investment process and rescuing of failing investments. In addition it looks to understand which of the two activities are more profitable to the venture capital firm. The study is based on primary data gathered through an in depth interview with a single venture capital firm. The surveyed firm specialises in early stage technology investments. The results show that the greatest return on time for the venture capital firm is investing in new businesses. However there is a caveat, the returns are only highest when the business turns out to be a star of the portfolio

Borel asymptotic dimension and hyperfinite equivalence relations

A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of solvable groups including the Baumslag--Solitar group BS(1,2) and the lamplighter group. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov's notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a zero-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Folner tilings, topological dynamics, and $C^*$-algebras

Calculated Vibrational States of Ozone up to Dissociation

A new accurate global potential energy surface for the ground electronic state of ozone [R. Dawes et al., J. Chem. Phys. 139, 201103 (2013)] was published fairly recently. The topography near dissociation differs significantly from previous surfaces, without spurious submerged reefs and corresponding van der Waals wells. This has enabled significantly improved descriptions of scattering processes, capturing the negative temperature dependence and large kinetic isotope effects in exchange reaction rates. The exchange reactivity was found to depend on the character of near-threshold resonances and their overlap with reactant and product wavefunctions, which in turn are sensitive to the potential. Here we present global three-well calculations of all bound vibrational states of three isotopic combinations of ozone (48O3, 16O218O, 16O217O) for J = 0 and J = 1 with a focus on the character and density of highly excited states and discuss their impact on the ozone isotopic anomaly. The calculations were done using a parallel symmetry-adapted Lanczos method with the RV3 code. Some comparisons were made with results obtained with the improved relaxation method implemented in the Heidelberg multi-configuration time-dependent Hartree code

Advanced Booster Liquid Engine Combustion Stability

Combustion instability is a phenomenon in liquid rocket engines caused by complex coupling between the time-varying combustion processes and the fluid dynamics in the combustor. Consequences of the large pressure oscillations associated with combustion instability often cause significant hardware damage and can be catastrophic. The current combustion stability assessment tools are limited by the level of empiricism in many inputs and embedded models. This limited predictive capability creates significant uncertainty in stability assessments. This large uncertainty then increases hardware development costs due to heavy reliance on expensive and time-consuming testing

450 d of Type II SN 2013ej in optical and near-infrared

We present optical and near-infrared photometric and spectroscopic observations of SN 2013ej, in galaxy M74, from 1 to 450 d after the explosion. SN 2013ej is a hydrogen-rich supernova, classified as a Type IIL due to its relatively fast decline following the initial peak. It has a relatively high peak luminosity (absolute magnitude MV = -17.6) but a small 56Ni production of ~0.023 M⊙. Its photospheric evolution is similar to other Type II SNe, with shallow absorption in the Hα profile typical for a Type IIL. During transition to the radioactive decay tail at ~100 d, we find the SN to grow bluer in B - V colour, in contrast to some other Type II supernovae. At late times, the bolometric light curve declined faster than expected from 56Co decay and we observed unusually broad and asymmetric nebular emission lines. Based on comparison of nebular emission lines most sensitive to the progenitor core mass, we find our observations are best matched to synthesized spectral models with a MZAMS = 12-15 M⊙ progenitor. The derived mass range is similar to but not higher than the mass estimated for Type IIP progenitors. This is against the idea that Type IIL are from more massive stars. Observations are consistent with the SN having a progenitor with a relatively low-mass envelop