Caloric Measure Null Sets

We give a systematic treatment of caloric measure null sets on the essential boundary $\partial_eE$ of an arbitrary open set $E$ in ${\bf R}$. We discuss two characterisations of such sets and present some basic properties. We investigate the dependence of caloric measure null sets on the open set $E$. Thus, if $D$ is an open subset of $E$ and $Z\subseteq\partial_eE\cap\partial_eD$, we show that $Z$ is caloric measure null for $D$ if it is caloric measure null for $E$. We also give conditions on $E$ and $Z$ which imply that the reverse implication is true. We know from \cite{watson2011} that any polar subset of $\partial_eD$ is caloric measure null for $D$, but the reverse implication is not generally true. In our final result we show that, for subsets of a certain component of $\partial_eD$, caloric measure null sets are necessarily polar.

Miracles and Violations of Laws of Nature

The aim of this article is to spell out the relationship between miracles and violations of laws of nature. I argue that the former do not necessarily entail the latter, even in the case of the type of miraculous event which cannot be brought about by natural operations alone. The idea that they do is based on a deterministic assumption which is too often overlooked. The article also explores the reverse implication, i.e. the question whether violations of laws of nature entail miracles. It turns out that there are conceptual difficulties in defining what sort of events would qualify as such violations in the first place, but that a more general notion of God’s action contravening nature is viable. However, there are theological reasons against the assumption that God ever acts in this way.

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

Cyclical monotonicity and the ergodic theorem

It is well known that optimal transport plans are cyclically monotone. The reverse implication that cyclically monotone transport plans are optimal needs some assumptions and the proof is non-trivial even if the costs are given by the squared euclidean distance on ${ \mathbb{R} }^{n} $. We establish this result as a corollary to the ergodic theorem.

Logical Distinction Between Diagrammatic and Cohen–Lyndon Asphericity

We resolve the status of the implication CLA ⇒ DA in the negative by showing that the two-dimensional model of the group presentation (a, b : a, b-1 aba-1 b-1) is DA but not CLA. This settles a question that has been addressed in [2, 7, 10, 11]. In 1941, Whitehead posed the question whether asphericity is a hereditary property for two-dimensional CW complexes. This question remains unanswered. Out of its study developed the formulation of several combinatorial properties for group presentations that are sufficient (but not necessary) for asphericity of the associated two-dimensional model. The logical relationships between these flavors of asphericity are just partially understood. In this article we show that two of these flavors of asphericity are in fact distinct. As a consequence, all of the known flavors are distinct. An argument of Lyndon and Schupp [5, III Property 10.6] shows that if a two-dimensional CW complex K is Cohen–Lyndon aspherical (CLA), then K is also diagrammatically aspherical (DA). The status of the reverse implication had been open prior to this writing. We will present an alternative proof for the implication CLA ⇒ DA and demonstrate that every spherical picture over the presentation (a, b : a, b-2 aba-1) can be reduced without insertions of dipoles, thus concluding that the presentation is DA. A straightforward argument shows that the presentation cannot be CLA. Our main tool is the theory of spherical pictures and picture moves of which we will give a short survey.

A RECYCLED CHARACTERIZATION OF KNEADING SEQUENCES

For any infinite sequence E on two symbols one can define two sequences of positive integers S(E) (the splitting times) and T(E) (the cosplitting times), which each describe the self-replicative structure of E. If E is the kneading sequence of a unimodal map, it is known that S(E) and T(E) carry a lot of information on the dynamics, and that they are disjoint. We show the reverse implication: A nonperiodic sequence E is the kneading sequence of some unimodal map if the sequences S(E) and T(E) are disjoint.

Random set and coverage measure

It is well known that a random set determines its random coverage measure. The paper gives a necessary and sufficient condition for the reverse implication. An equivalent formulation of the condition constitutes a first step in the search for a way to recognize a random measure as being the random coverage measure of a random set.

Random set and coverage measure

It is well known that a random set determines its random coverage measure. The paper gives a necessary and sufficient condition for the reverse implication. An equivalent formulation of the condition constitutes a first step in the search for a way to recognize a random measure as being the random coverage measure of a random set.

Maximum idempotents in naturally ordered regular semigroups

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense thatNo assumption is made about the reverse implication.