Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for every finite n>0$n>0$ for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ  -uniform families in which ρ≥_?ω(ν)$\rho \ge {\mathrm{?}}_{\omega }(\nu )$. The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.     

Discrete approximations for complex Kac–Moody groups

We construct a map from the classifying space of a discrete Kac–Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac–Moody group and prove that it is a homology equivalence at primes q different from p. This generalizes a classical result of Quillen, Friedlander and Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac–Moody groups compatible with the Frobenius homomorphism. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac–Moody groups.     

Renormalization and conjugacy of piecewise linear Lorenz maps

We investigate the uniform piecewise linearizing question for a family of Lorenz maps. Let f be a piecewise linear Lorenz map with different slopes and positive topological entropy, we show that f is conjugate to a linear mod one transformation and the conjugacy admits a dichotomy: it is either bi-Lipschitz or singular depending on whether f is renormalizable or not. f is renormalizable if and only if its rotation interval degenerates to be a rational point. Furthermore, if the endpoints are periodic points with the same rotation number, then the conjugacy is quasisymmetric.     

Typical curvature behaviour of bodies of constant width

It is known that an n-dimensional convex body, which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical n-dimensional convex body of constant width 1 (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to 1. (In contrast, note that a ball of width 1 has radius 1/2, hence all its curvatures are equal to 2.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.     

Dimension gap under Sobolev mappings

We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Hölder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.     

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rⁿ${\mathbb{R}}^n$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.     

Gas‐phase elimination kinetics of selected aliphatic α,β‐unsaturated aldehydes catalyzed by hydrogen chloride

The gas‐phase elimination of 2‐methyl‐2‐propenal catalyzed by HCl yields propene and CO gas, while E‐2‐pentenal with the same catalyst gives butene and CO gas. The kinetics determinations were carried out in a static system with the reaction vessels deactivated with allyl bromide and the presence of the free radical inhibitor toluene. Temperature and pressure ranges were 350.0–410.0 °C and 34–76 Torr. The elimination reactions are homogeneous and unimolecular, and follow a first‐order rate law. The rate coefficients for the reactions are expressible by the following Arrhenius equations: Data from the kinetic and thermodynamic parameters of these catalyzed elimination reactions implies a mechanism of a concerted five‐membered cyclic transition state structure for the formation of the corresponding olefin and carbon monoxide. Copyright © 2015 John Wiley & Sons, Ltd.