7,744 research outputs found
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 and are
bounded Borel sets with the same positive Lebesgue measure whose boundaries
have upper Minkowski dimension less than , then and 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 .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
-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 -sphere. This answers a question of
Wagon. We also construct Borel realizations of abstract systems of congruences
for the action of on .
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
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