Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction

Legitimating inaction : differing identity constructions of the Scots language.

The Scots language plays a key role in the political and cultural landscape of contemporary Scotland. From a discourse-historical perspective, this article explores how language ideologies about the Scots language are realized linguistically in a so-called ‘languages strategy’ drafted by the Scottish Executive, and in focus groups consisting of Scottish people. This article shows that although the decline of Scots is said to be a ‘tragedy’, focus group participants seem to reject the notion of Scots as a viable, contemporary language that can be used across a wide range of registers. The policy document also seems to construct Scots in very positive terms, but is shown to be unhelpful or potentially even damaging in the process of changing public attitudes to Scots

Force indeterminacy in the jammed state of hard disks

Granular packings of hard discs are investigated by means of contact dynamics which is an appropriate technique to explore the allowed force-realizations in the space of contact forces. Configurations are generated for given values of the friction coefficient, and then an ensemble of equilibrium forces is found for fixed contacts. We study the force fluctuations within this ensemble. In the limit of zero friction the fluctuations vanish in accordance with the isostaticity of the packing. The magnitude of the fluctuations has a non-monotonous friction dependence. The increase for small friction can be attributed to the opening of the angle of the Coulomb cone, while the decrease as friction increases is due to the reduction of connectivity of the contact-network, leading to local, independent clusters of indeterminacy. We discuss the relevance of indeterminacy to packings of deformable particles and to the mechanical response properties.Comment: 4 pages, 3 figures. Minor changes, journal reference adde