Schmidt Games and Conditions on Resonant Sets

Winning sets of Schmidt's game enjoy a remarkable rigidity. Therefore, this game (and modifications of it) have been applied to many examples of complete metric spaces (X, d) to show that the set of "badly approximable points", with respect to a given collection of resonant sets in X, is a winning set. For these examples, strategies were deduced that are, in most cases, strongly adapted to the specific dynamics and properties of the underlying setting. We introduce a new modification of Schmidt's game which is a combination and generalization of the ones of [18] and [20]. This modification allows us to axiomatize conditions on the collection of resonant sets under which there always exists a winning strategy. Moreover, we discuss properties of winning sets of this modification and verify our conditions for several examples - among them, the set of badly approximable vectors in the Euclidian space and the p-adic integers with weights and, as a main example, the set of geodesic rays in proper geodesic CAT(-1) spaces which avoid a suitable collection of convex subsets.Comment: 30 pages, Comments are welcome

On the logical definability of certain graph and poset languages

We show that it is equivalent, for certain sets of finite graphs, to be definable in CMS (counting monadic second-order logic, a natural extension of monadic second-order logic), and to be recognizable in an algebraic framework induced by the notion of modular decomposition of a finite graph. More precisely, we consider the set $F\_\infty$ of composition operations on graphs which occur in the modular decomposition of finite graphs. If $F$ is a subset of $F\_{\infty}$, we say that a graph is an \calF-graph if it can be decomposed using only operations in $F$. A set of $F$-graphs is recognizable if it is a union of classes in a finite-index equivalence relation which is preserved by the operations in $F$. We show that if $F$ is finite and its elements enjoy only a limited amount of commutativity -- a property which we call weak rigidity, then recognizability is equivalent to CMS-definability. This requirement is weak enough to be satisfied whenever all $F$-graphs are posets, that is, transitive dags. In particular, our result generalizes Kuske's recent result on series-parallel poset languages

Valdez-Jimenez v. Eighth Jud. Dist. Ct.,136 Nev. Adv. Op. 20 (April 9, 2020)

The Court determined what process is constitutionally required when a district court sets bail in an amount that the defendant cannot afford, resulting in pretrial detention. The Court found that bail may only be imposed where it is necessary to reasonably ensure the defendant’s appearance at court proceedings or to reasonably protect the community. If a defendant remains in custody after arrest they are (1) entitled to an individualized hearing, where (2) the State must prove by clear and convincing evidence that bail, rather than less restrictive conditions, is necessary to ensure the defendant’s appearance at future court proceedings or to protect the safety of the community, and (3) the district court must also state its findings and reasons for the bail decision on the record

Jarnik-type Inequalities

It is well known due to Jarnik that the set Bad of badly approximable numbers is of Hausdorff-dimension one. If Bad(c) denotes the subset of x in Bad for which the approximation constant c > c(x), then Jarnik was in fact more precise and gave nontrivial lower and upper bounds of the Hausdorff-dimension of Bad(c) in terms of the parameter c > 0. Our aim is to determine simple conditions on a framework which allow to extend 'Jarnik's inequality' to further examples; among the applications, we discuss the set of badly approximable vectors in with weights and the set of geodesics in the hyperbolic space which avoid a suitable collection of convex sets.Comment: Comments are welcome! Corrections and modifications in new versio

The Earlier Wittgenstein on the Notion of Religious Attitude

I defend a new interpretation of Wittgenstein's notion of religious (or ethical) attitude in the Tractatus, one that rejects three key views from the secondary literature: firstly, the view that, for Wittgenstein, the willing subject is a transcendental condition for the religious attitude; secondly, the view that the religious attitude is an emotive response to the world or something closely modelled on this notion of emotive response; and thirdly, the view that, although the religious and ethical pseudo-propositions of the Tractatus are nonsensical, they nevertheless succeed in expressing the religious attitude endorsed by Wittgenstein. In connection to the first, I argue that the notion of willing subject as transcendental condition is abandoned by Wittgenstein in the Notebooks and is no longer a feature of his position in the Tractatus. In connection to the second, I argue that the religious attitude is dispositional rather than emotive for Wittgenstein: it is a disposition to use signs in a way that demonstrates one's conceptual clarity. Finally, in connection to the third, I argue that the religious or ethical attitude is strongly ineffable in that it cannot be described, expressed or conveyed by language at all.Peer reviewe