356 research outputs found
Searching for Hyperbolicity
This is an expository paper, based on by a talk given at the AWM Research
Symposium 2017. It is intended as a gentle introduction to geometric group
theory with a focus on the notion of hyperbolicity, a theme that has inspired
the field from its inception to current-day research
Accidental parabolics and relatively hyperbolic groups
By constructing, in the relative case, objects analoguous to Rips and Sela's
canonical representatives, we prove that the set of images by morphisms without
accidental parabolic, of a finitely presented group in a relatively hyperbolic
group, is finite, up to conjugacy.Comment: Revision, 24 pages, 4 figure
A simple proof of the Markoff conjecture for prime powers
We give a simple and independent proof of the result of Jack Button and Paul
Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples
(a,b,c), where a, b, and c are in increasing order, holds whenever is a
prime power.Comment: 5 pages, no figure
Mackintosh lecture: Association and cognition: two processes, one system
This is the author accepted manuscript. The final version is available from SAGE Publications via the DOI in this record.There is another ORE record for this item: http://hdl.handle.net/10871/33264This paper argues that the dual-process position can be a useful first approximation
when studying human mental life, but it cannot be the whole truth. Instead, we argue
that cognition is built on association, in that associative processes provide the
fundamental building blocks that enable propositional thought. One consequence of
this position is to suggest that humans are able to learn associatively in a similar
fashion to a rat or a pigeon, but another is that we must typically suppress the
expression of basic associative learning in favour of rule-based computation. This
stance conceptualizes us as capable of symbolic computation, but acknowledges that,
given certain circumstances, we will learn associatively and, more importantly, be
seen to do so. We present three types of evidence that support this position: The first
is data on human Pavlovian conditioning that directly supports this view. The second
is data taken from task switching experiments that provides convergent evidence for
at least two modes of processing, one of which is automatic and carried out “in the
background”. And the last suggests that when the output of propositional processes is
uncertain, then the influence of associative processes on behaviour can manifest
Geometrical Finiteness, Holography, and the BTZ Black Hole
We show how a theorem of Sullivan provides a precise mathematical statement
of a 3d holographic principle, that is, the hyperbolic structure of a certain
class of 3d manifolds is completely determined in terms of the corresponding
Teichmuller space of the boundary. We explore the consequences of this theorem
in the context of the Euclidean BTZ black hole in three dimensions.Comment: 6 pages, Latex, Version to appear in Physical Review Letter
Qubit-Qutrit Separability-Probability Ratios
Paralleling our recent computationally-intensive (quasi-Monte Carlo) work for
the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to
high numerical accuracy, the formulas of Sommers and Zyczkowski
(quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional
hyperarea of the (separable and nonseparable) N x N density matrices, based on
the Bures (minimal monotone) metric -- and also their analogous formulas
(quant-ph/0302197) for the (non-monotone) Hilbert-Schmidt metric. With the same
seven billion well-distributed (``low-discrepancy'') sample points, we estimate
the unknown volumes and hyperareas based on five additional (monotone) metrics
of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate
all of these seven volume and seven hyperarea (unknown) quantities when
restricted to the separable density matrices. The ratios of separable volumes
(hyperareas) to separable plus nonseparable volumes (hyperareas) yield
estimates of the separability probabilities of generically rank-six (rank-five)
density matrices. The (rank-six) separability probabilities obtained based on
the 35-dimensional volumes appear to be -- independently of the metric (each of
the seven inducing Haar measure) employed -- twice as large as those (rank-five
ones) based on the 34-dimensional hyperareas. Accepting such a relationship, we
fit exact formulas to the estimates of the Bures and Hilbert-Schmidt separable
volumes and hyperareas.(An additional estimate -- 33.9982 -- of the ratio of
the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite
clearly close to integral too.) The doubling relationship also appears to hold
for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit
exact formulas for the Hilbert-Schmidt separable volumes and hyperareas.Comment: 36 pages, 15 figures, 11 tables, final PRA version, new last
paragraph presenting qubit-qutrit probability ratios disaggregated by the two
distinct forms of partial transpositio
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