5,281 research outputs found
Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps
This paper presents a formalized proof of a discrete form of the Jordan Curve
Theorem. It is based on a hypermap model of planar subdivisions, formal
specifications and proofs assisted by the Coq system. Fundamental properties
are proven by structural or noetherian induction: Genus Theorem, Euler's
Formula, constructive planarity criteria. A notion of ring of faces is
inductively defined and a Jordan Curve Theorem is stated and proven for any
planar hypermap
A new proof of Vassiliev's conjecture
We give a new proof of Vassiliev's planarity criterion for framed four-valent
graphs (and more generally, *-graphs), which is based on Pontryagin-Kuratowski
theorem.Comment: a planarity criterion for noneven *-graphs is adde
Some Remarks on Non-Planar Diagrams
Two criteria for planarity of a Feynman diagram upon its propagators
(momentum flows) are presented. Instructive Mathematica programs that solve the
problem and examples are provided. A simple geometric argument is used to show
that while one can planarize non-planar graphs by embedding them on
higher-genus surfaces (in the example it is a torus), there is still a problem
with defining appropriate dual variables since the corresponding faces of the
graph are absorbed by torus generators.Comment: Presented by K. Bielas at the International Conference of Theoretical
Physics "Matter To The Deepest", Ustron 201
Gauss paragraphs of classical links and a characterization of virtual link groups
A classical link in 3-space can be represented by a Gauss paragraph encoding
a link diagram in a combinatorial way. A Gauss paragraph may code not a
classical link diagram, but a diagram with virtual crossings. We present a
criterion and a linear algorithm detecting whether a Gauss paragraph encodes a
classical link. We describe Wirtinger presentations realizable by virtual link
groups.Comment: 12 pages, 12 figures, v2: new results have been adde
A Planarity Test via Construction Sequences
Optimal linear-time algorithms for testing the planarity of a graph are
well-known for over 35 years. However, these algorithms are quite involved and
recent publications still try to give simpler linear-time tests. We give a
simple reduction from planarity testing to the problem of computing a certain
construction of a 3-connected graph. The approach is different from previous
planarity tests; as key concept, we maintain a planar embedding that is
3-connected at each point in time. The algorithm runs in linear time and
computes a planar embedding if the input graph is planar and a
Kuratowski-subdivision otherwise
Occam's razor meets WMAP
Using a variety of quantitative implementations of Occam's razor we examine
the low quadrupole, the ``axis of evil'' effect and other detections recently
made appealing to the excellent WMAP data. We find that some razors {\it fully}
demolish the much lauded claims for departures from scale-invariance. They all
reduce to pathetic levels the evidence for a low quadrupole (or any other low
cut-off), both in the first and third year WMAP releases. The ``axis of
evil'' effect is the only anomaly examined here that survives the humiliations
of Occam's razor, and even then in the category of ``strong'' rather than
``decisive'' evidence. Statistical considerations aside, differences between
the various renditions of the datasets remain worrying
- …