7,733 research outputs found

### Separability and the genus of a partial dual

Partial duality generalizes the fundamental concept of the geometric dual of
an embedded graph. A partial dual is obtained by forming the geometric dual
with respect to only a subset of edges. While geometric duality preserves the
genus of an embedded graph, partial duality does not. Here we are interested in
the problem of determining which edge sets of an embedded graph give rise to a
partial dual of a given genus. This problem turns out to be intimately
connected to the separability of the embedded graph. We determine how
separability is related to the genus of a partial dual. We use this to
characterize partial duals of graphs embedded in the plane, and in the real
projective plane, in terms of a particular type of separation of an embedded
graph. These characterizations are then used to determine a local move relating
all partially dual graphs in the plane and in the real projective plane

### Integration and conjugacy in knot theory

This thesis consists of three self-contained chapters. The first two concern
quantum invariants of links and three manifolds and the third contains results
on the word problem for link groups.
In chapter 1 we relate the tree part of the Aarhus integral to the
mu-invariants of string-links in homology balls thus generalizing results of
Habegger and Masbaum.
There is a folklore result in physics saying that the Feynman integration of
an exponential is itself an exponential. In chapter 2 we state and prove an
exact formulation of this statement in the language which is used in the theory
of finite type invariants.
The final chapter is concerned with properties of link groups. In particular
we study the relationship between known solutions from small cancellation
theory and normal surface theory for the word and conjugacy problems of the
groups of (prime) alternating links. We show that two of the algorithms in the
literature for solving the word problem, each using one of the two approaches,
are the same. Then, by considering small cancellation methods, we give a normal
surface solution to the conjugacy problem of these link groups and characterize
the conjugacy classes. Finally as an application of the small cancellation
properties of link groups we give a new proof that alternating links are
non-trivial.Comment: University of Warwick Ph.D. thesi

### A permanent formula for the Jones polynomial

The permanent of a square matrix is defined in a way similar to the
determinant, but without using signs. The exact computation of the permanent is
hard, but there are Monte-Carlo algorithms that can estimate general
permanents. Given a planar diagram of a link L with $n$ crossings, we define a
7n by 7n matrix whose permanent equals to the Jones polynomial of L. This
result accompanied with recent work of Freedman, Kitaev, Larson and Wang
provides a Monte-Carlo algorithm to any decision problem belonging to the class
BQP, i.e. such that it can be computed with bounded error in polynomial time
using quantum resources.Comment: To appear in Advances in Applied Mathematic

### Comparison of models of CO2-laser impedance fluctuations

There is a large opto-galvanic effect (OGE) in CO2-N2-He laser mixtures and this is exploited in laser frequency and power stabilisation systems. Two substantially different theories have been advanced to explain the effect. The two models are compared and it is concluded that the multi-step ionisation model is not adequate to describe the OGE in CO2 lasers, but the temperature perturbation or discharge cooling model describes the phenomenon with considerable precision

### Bipartite partial duals and circuits in medial graphs

It is well known that a plane graph is Eulerian if and only if its geometric
dual is bipartite. We extend this result to partial duals of plane graphs. We
then characterize all bipartite partial duals of a plane graph in terms of
oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric

### NSW recorded crime statistics 2012

This report finds that in New South Wales, the surge in incidents involving the discharge of a firearm into premises which began in February 2010 has been reversed.
In February 2010, incidents of discharge firearm into premises were running at an average of five a month. They peaked in August 2012 at an average rate of more than 11 per month. By December 2012, they were back down to around 6.7 a month. Other types of shooting incident are also down from their peaks between 2001 and 2003.
This report presents further detail on crime trends across New South Wales.
It finds that between January 2011 and December 2012:
Fraud increased by 14.6 per cent;
Assault - Non-domestic related fell by 5.7 per cent;
Break and enter (non-dwelling) fell by 4.9 per cent; and
Motor vehicle theft fell by 7.0 per cent
The remaining major categories of crime remained stable across the state as a whole.
 

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