244 research outputs found
Representations of the homotopy surface category of a simply connected space
We introduce the homotopy surface category of a space which generalizes the
1+1-dimensional cobordism category of circles and surfaces to the situation
where one introduces a background space. We explain how for a simply connected
background space, monoidal functors from this category to vector spaces can be
interpreted in terms of Frobenius algebras with additional structure.Comment: 9 pages, 2 figure
Homotopy Quantum Field Theories and Related Ideas
In this short note we provide a review of some developments in the area of
homotopy quantum field theories, loosely based on a talk given by the second
author at the Xth Oporto Meeting on Geometry, Topology and Physics.Comment: 8 pages, 2 figures; correcte
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
Pattern Avoidance in Poset Permutations
We extend the concept of pattern avoidance in permutations on a totally
ordered set to pattern avoidance in permutations on partially ordered sets. The
number of permutations on that avoid the pattern is denoted
. We extend a proof of Simion and Schmidt to show that for any poset , and we exactly classify the posets for which
equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and
improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated
author email addresse
Spatial Hypersurfaces in Causal Set Cosmology
Within the causal set approach to quantum gravity, a discrete analog of a
spacelike region is a set of unrelated elements, or an antichain. In the
continuum approximation of the theory, a moment-of-time hypersurface is well
represented by an inextendible antichain. We construct a richer structure
corresponding to a thickening of this antichain containing non-trivial
geometric and topological information. We find that covariant observables can
be associated with such thickened antichains and transitions between them, in
classical stochastic growth models of causal sets. This construction highlights
the difference between the covariant measure on causal set cosmology and the
standard sum-over-histories approach: the measure is assigned to completed
histories rather than to histories on a restricted spacetime region. The
resulting re-phrasing of the sum-over-histories may be fruitful in other
approaches to quantum gravity.Comment: Revtex, 12 pages, 2 figure
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
Spacelike distance from discrete causal order
Any discrete approach to quantum gravity must provide some prescription as to
how to deduce continuum properties from the discrete substructure. In the
causal set approach it is straightforward to deduce timelike distances, but
surprisingly difficult to extract spacelike distances, because of the unique
combination of discreteness with local Lorentz invariance in that approach. We
propose a number of methods to overcome this difficulty, one of which
reproduces the spatial distance between two points in a finite region of
Minkowski space. We provide numerical evidence that this definition can be used
to define a `spatial nearest neighbor' relation on a causal set, and conjecture
that this can be exploited to define the length of `continuous curves' in
causal sets which are approximated by curved spacetime. This provides evidence
in support of the ``Hauptvermutung'' of causal sets.Comment: 32 pages, 16 figures, revtex4; journal versio
National health and medical research council statement on electronic cigarettes: 2022 update
Introduction: Electronic cigarette (e-cigarette) use in Australia has rapidly increased since the 2017 National Health and Medical Research Council (NHMRC) Chief Executive Officer (CEO) statement on e-cigarettes. The type of products available and the demographic characteristics of people using these products have changed. New evidence has been published and there is growing concern among public health professionals about the increased use, particularly among young people who do not currently smoke combustible cigarettes. The combination of these issues led NHMRC to review the current evidence and provide an updated statement on e-cigarettes. In this article, we describe the comprehensive process used to review the evidence and develop the 2022 NHMRC CEO statement on electronic cigarettes. Main recommendations: E-cigarettes can be harmful; all e-cigarette users are exposed to chemicals and toxins that have the potential to cause adverse health effects. There are no health benefits of using e-cigarettes if you do not currently smoke tobacco cigarettes. Adolescents are more likely to try e-cigarettes if they are exposed to e-cigarettes on social media. Short term e-cigarette use may help some smokers to quit who have been previously unsuccessful with other smoking cessation aids. There are other proven safe and effective options available to help smokers to quit. Changes in management as a result of this statement: The evidence base for the harms of e-cigarette use has strengthened since the previous NHMRC statement. Significant gaps in the evidence base remain, especially about the longer term health harms of using e-cigarettes and the toxicity of many chemicals in e-cigarettes inhaled as an aerosol
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