22,779 research outputs found
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Spin reorientation in TlFe1.6Se2 with complete vacancy ordering
The relationship between vacancy ordering and magnetism in TlFe1.6Se2 has
been investigated via single crystal neutron diffraction, nuclear forward
scattering, and transmission electron microscopy. The examination of chemically
and structurally homogenous crystals allows the true ground state to be
revealed, which is characterized by Fe moments lying in the ab-plane below
100K. This is in sharp contrast to crystals containing regions of order and
disorder, where a competition between c-axis and ab-plane orientations of the
moments is observed. The properties of partially-disordered TlFe1.6Se2 are
therefore not associated with solely the ordered or disordered regions. This
contrasts the viewpoint that phase separation results in independent physical
properties in intercalated iron selenides, suggesting a coupling between
ordered and disordered regions may play an important role in the
superconducting analogues.Comment: Minor changes; updated references and funding acknowledgemen
A self-dual poset on objects counted by the Catalan numbers and a type-B analogue
We introduce two partially ordered sets, and , of the same
cardinalities as the type-A and type-B noncrossing partition lattices. The
ground sets of and are subsets of the symmetric and the
hyperoctahedral groups, consisting of permutations which avoid certain
patterns. The order relation is given by (strict) containment of the descent
sets. In each case, by means of an explicit order-preserving bijection, we show
that the poset of restricted permutations is an extension of the refinement
order on noncrossing partitions. Several structural properties of these
permutation posets follow, including self-duality and the strong Sperner
property. We also discuss posets and similarly associated with
noncrossing partitions, defined by means of the excedence sets of suitable
pattern-avoiding subsets of the symmetric and hyperoctahedral groups.Comment: 15 pages, 2 figure
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
GCD matrices, posets, and nonintersecting paths
We show that with any finite partially ordered set one can associate a matrix
whose determinant factors nicely. As corollaries, we obtain a number of results
in the literature about GCD matrices and their relatives. Our main theorem is
proved combinatorially using nonintersecting paths in a directed graph.Comment: 10 pages, see related papers at http://www.math.msu.edu/~saga
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