852 research outputs found
Local Geometric Invariants of Integrable Evolution Equations
The integrable hierarchy of commuting vector fields for the localized
induction equation of 3D hydrodynamics, and its associated recursion operator,
are used to generate families of integrable evolution equations which preserve
local geometric invariants of the evolving curve or swept-out surface.Comment: 15 pages, AMSTeX file (to appear in Journal of Mathematical Physics
Nested hierarchies in planar graphs
We construct a partial order relation which acts on the set of 3-cliques of a
maximal planar graph G and defines a unique hierarchy. We demonstrate that G is
the union of a set of special subgraphs, named `bubbles', that are themselves
maximal planar graphs. The graph G is retrieved by connecting these bubbles in
a tree structure where neighboring bubbles are joined together by a 3-clique.
Bubbles naturally provide the subdivision of G into communities and the tree
structure defines the hierarchical relations between these communities
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 New Approach for Visualizing UML Class Diagrams
UML diagrams have become increasingly important in the engineering and reengineering processes for software systems. Of particular interest are UML class diagrams whose purpose is to display class hierarchies (generalizations), associations, aggregations, and compositions in one picture. The combination of hierarchical and non-hierarchical relations poses a special challenge to a graph layout tool. Existing layout tools treat hierarchical and non-hierarchical relations either alike or as separate tasks in a two-phase process as in, e.g., cite{See97}. We suggest a new approach for visualizing UML class diagrams leading to a balanced mixture of the following aesthetic criteria: Crossing minimization, bend minimization, uniform direction within each class hierarchy, no nesting of one class hierarchy within another, orthogonal layout, merging of multiple inheritance edges, and good edge labelling. We have realized our approach within the graph drawing library GoVisual. Experiments show the superiority to state-of-the-art and industrial standard layouts
Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths
When can a plane graph with prescribed edge lengths and prescribed angles
(from among \}) be folded flat to lie in an
infinitesimally thin line, without crossings? This problem generalizes the
classic theory of single-vertex flat origami with prescribed mountain-valley
assignment, which corresponds to the case of a cycle graph. We characterize
such flat-foldable plane graphs by two obviously necessary but also sufficient
conditions, proving a conjecture made in 2001: the angles at each vertex should
sum to , and every face of the graph must itself be flat foldable.
This characterization leads to a linear-time algorithm for testing flat
foldability of plane graphs with prescribed edge lengths and angles, and a
polynomial-time algorithm for counting the number of distinct folded states.Comment: 21 pages, 10 figure
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