20,805 research outputs found
A Lexicographic Product Cancellation Property for Digraphs
There are four prominent product graphs in graph theory: Cartesian, strong, direct, and lexicographic. Of these four product graphs, the lexicographic product graph is the least studied. Lexicographic products are not commutative but still have some interesting properties. This paper begins with basic definitions of graph theory, including the definition of a graph, that are needed to understand theorems and proofs that come later. The paper then discusses the lexicographic product of digraphs, denoted , for some digraphs and . The paper concludes by proving a cancellation property for the lexicographic product of digraphs , , , and : if and , then . It also proves additional cancellation properties for lexicographic product digraphs and the author hopes the final result will provide further insight into tournaments
Final state interaction in the production of heavy unstable particles
We make an attempt to discuss in detail the effects originating from the
final state interaction in the processes involving production of unstable
elementary particles and their subsequent decay. Two complementary scenarios
are considered: the single resonance production and the production of two
resonances. We argue that part of the corrections due to the final state
interaction can be connected with the Coulomb phases of the involved charge
particles; the presence of the unstable particle in the problem makes the
Coulomb phase ``visible''. It is shown how corrections due to the final state
interaction disappear when one proceeds to the total cross-sections. We derive
one-loop non-factorizable radiative corrections to the lowest order matrix
element of both single and double resonance production. We discuss how the
infrared limit of the theories with the unstable particles is modified. In
conclusion we briefly discuss our results in the context of the forthcoming
experiments on the and the production at LEP and NLC.Comment: 33 pages, latex, 6 figures (added), version accepted for publication
in Nuc. Phys. B, substantial revisio
Factors of disconnected graphs and polynomials with nonnegative integer coefficients
We investigate the uniqueness of factorisation of possibly disconnected
finite graphs with respect to the Cartesian, the strong and the direct product.
It is proved that if a graph has connected components, where is prime,
or , and satisfies some additional conditions, it factors uniquely
under the given products. If, on the contrary, or 10, all cases of
nonunique factorisation are described precisely.Comment: 14 page
Finite index subgroups without unique product in graphical small cancellation groups
We construct torsion-free hyperbolic groups without unique product whose
subgroups up to some given finite index are themselves non-unique product
groups. This is achieved by generalising a construction of Comerford to
graphical small cancellation presentations, showing that for every subgroup
of a graphical small cancellation group there exists a free group such that
admits a graphical small cancellation presentation.Comment: 8 pages, 1 figur
Every group is the outer automorphism group of an HNN-extension of a fixed triangle group
Fix an equilateral triangle group
with arbitrary. Our main result is: for every presentation
of every countable group there exists an HNN-extension
of such that . We construct the HNN-extensions explicitly, and examples are given. The
class of groups constructed have nice categorical and residual properties. In
order to prove our main result we give a method for recognising malnormal
subgroups of small cancellation groups, and we introduce the concept of
"malcharacteristic" subgroups.Comment: 39 pages. Final version, to appear in Advances in Mathematic
Rips construction without unique product
Given a finitely presented group we produce a short exact sequence such that is a torsion-free
Gromov hyperbolic group without the unique product property and is without
the unique product property and has Kazhdan's Property (T). Varying we
show a wide diversity of concrete examples of Gromov hyperbolic groups without
the unique product property. As an application, we obtain Tarski monster groups
without the unique product property.Comment: 22 page
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