162 research outputs found
Totally free expansions of matroids
The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science
Confinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix
representable over a sub-partial field P' of P. We say that B confines M to P'
if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled
P'-matrix. We show that, under some conditions on the partial fields, on M, and
on B, verifying whether B confines M to P' amounts to a finite check. A
corollary of this result is Whittle's Stabilizer Theorem.
A combination of the Confinement Theorem and the Lift Theorem from
arXiv:0804.3263 leads to a short proof of Whittle's characterization of the
matroids representable over GF(3) and other fields.
We also use a combination of the Confinement Theorem and the Lift Theorem to
prove a characterization, in terms of representability over partial fields, of
the 3-connected matroids that have k inequivalent representations over GF(5),
for k = 1, ..., 6.
Additionally we give, for a fixed matroid M, an algebraic construction of a
partial field P_M and a representation A over P_M such that every
representation of M over a partial field P is equal to f(A) for some
homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic
analog of the theory of free expansions by Geelen et al.Comment: 45 page
Dimension, matroids, and dense pairs of first-order structures
A structure M is pregeometric if the algebraic closure is a pregeometry in
all M' elementarily equivalent to M. We define a generalisation: structures
with an existential matroid. The main examples are superstable groups of U-rank
a power of omega and d-minimal expansion of fields. Ultraproducts of
pregeometric structures expanding a field, while not pregeometric in general,
do have an unique existential matroid.
Generalising previous results by van den Dries, we define dense elementary
pairs of structures expanding a field and with an existential matroid, and we
show that the corresponding theories have natural completions, whose models
also have a unique existential matroid. We extend the above result to dense
tuples of structures.Comment: Version 2.8. 61 page
The Number of Nowhere-Zero Flows on Graphs and Signed Graphs
A nowhere-zero -flow on a graph is a mapping from the edges of
to the set \{\pm1, \pm2, ..., \pm(k-1)\} \subset \bbZ such that, in
any fixed orientation of , at each node the sum of the labels over the
edges pointing towards the node equals the sum over the edges pointing away
from the node. We show that the existence of an \emph{integral flow polynomial}
that counts nowhere-zero -flows on a graph, due to Kochol, is a consequence
of a general theory of inside-out polytopes. The same holds for flows on signed
graphs. We develop these theories, as well as the related counting theory of
nowhere-zero flows on a signed graph with values in an abelian group of odd
order. Our results are of two kinds: polynomiality or quasipolynomiality of the
flow counting functions, and reciprocity laws that interpret the evaluations of
the flow polynomials at negative integers in terms of the combinatorics of the
graph.Comment: 17 pages, to appear in J. Combinatorial Th. Ser.
Confinement of matroid representations to subsets of partial fields
Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representation matrix A of M has a submatrix B, A is a scaled P'-matrix. We show that, under some conditions on the partial fields, on M, and on B, verifying whether B confines M to P' amounts to a finite check. A corollary of this result is Whittle's Stabilizer Theorem.
A combination of the Confinement Theorem and the Lift Theorem from arXiv:0804.3263 leads to a short proof of Whittle's characterization of the matroids representable over GF(3) and other fields.
We also use a combination of the Confinement Theorem and the Lift Theorem to prove a characterization, in terms of representability over partial fields, of the 3-connected matroids that have k inequivalent representations over GF(5), for k = 1, ..., 6.
Additionally we give, for a fixed matroid M, an algebraic construction of a partial field P_M and a representation A over P_M such that every representation of M over a partial field P is equal to f(A) for some homomorphism f:P_M->P. Using the Confinement Theorem we prove an algebraic analog of the theory of free expansions by Geelen et al
General duality for abelian-group-valued statistical-mechanics models
We introduce a general class of statistical-mechanics models, taking values
in an abelian group, which includes examples of both spin and gauge models,
both ordered and disordered. The model is described by a set of ``variables''
and a set of ``interactions''. A Gibbs factor is associated to each variable
and to each interaction. We introduce a duality transformation for systems in
this class. The duality exchanges the abelian group with its dual, the Gibbs
factors with their Fourier transforms, and the interactions with the variables.
High (low) couplings in the interaction terms are mapped into low (high)
couplings in the one-body terms. The idea is that our class of systems extends
the one for which the classical procedure 'a la Kramers and Wannier holds, up
to include randomness into the pattern of interaction. We introduce and study
some physical examples: a random Gaussian Model, a random Potts-like model, and
a random variant of discrete scalar QED. We shortly describe the consequence of
duality for each example.Comment: 26 pages, 2 Postscript figure
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