4,519 research outputs found
Some hard families of parameterised counting problems
We consider parameterised subgraph-counting problems of the following form:
given a graph G, how many k-tuples of its vertices have a given property? A
number of such problems are known to be #W[1]-complete; here we substantially
generalise some of these existing results by proving hardness for two large
families of such problems. We demonstrate that it is #W[1]-hard to count the
number of k-vertex subgraphs having any property where the number of distinct
edge-densities of labelled subgraphs that satisfy the property is o(k^2). In
the special case that the property in question depends only on the number of
edges in the subgraph, we give a strengthening of this result which leads to
our second family of hard problems.Comment: A few more minor changes. This version to appear in the ACM
Transactions on Computation Theor
Survey on the Tukey theory of ultrafilters
This article surveys results regarding the Tukey theory of ultrafilters on
countable base sets. The driving forces for this investigation are Isbell's
Problem and the question of how closely related the Rudin-Keisler and Tukey
reducibilities are. We review work on the possible structures of cofinal types
and conditions which guarantee that an ultrafilter is below the Tukey maximum.
The known canonical forms for cofinal maps on ultrafilters are reviewed, as
well as their applications to finding which structures embed into the Tukey
types of ultrafilters. With the addition of some Ramsey theory, fine analyses
of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page
Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds
We prove an existence theorem for gauge invariant -normal neighborhoods
of the reduction loci in the space of oriented connections on a
fixed Hermitian 2-bundle . We use this to obtain results on the topology of
the moduli space of (non-necessarily irreducible) oriented
connections, and to study the Donaldson -classes globally around the
reduction loci. In this part of the article we use essentially the concept of
harmonic section in a sphere bundle with respect to an Euclidean connection.
Second, we concentrate on moduli spaces of instantons on definite 4-manifolds
with arbitrary first Betti number. We prove strong generic regularity results
which imply (for bundles with "odd" first Chern class) the existence of a
connected, dense open set of "good" metrics for which all the reductions in the
Uhlenbeck compactification of the moduli space are simultaneously regular.
These results can be used to define new Donaldson type invariants for definite
4-manifolds. The idea behind this construction is to notice that, for a good
metric , the geometry of the instanton moduli spaces around the reduction
loci is always the same, independently of the choice of . The connectedness
of the space of good metrics is important, in order to prove that no
wall-crossing phenomena (jumps of invariants) occur. Moreover, we notice that,
for low instanton numbers, the corresponding moduli spaces are a priori compact
and contain no reductions at all so, in these cases, the existence of
well-defined Donaldson type invariants is obvious. The natural question is to
decide whether these new Donaldson type invariants yield essentially new
differential topological information on the base manifold have, or have a
purely topological nature.Comment: LaTeX, 45 page
Complexity of distances: Theory of generalized analytic equivalence relations
We generalize the notion of analytic/Borel equivalence relations, orbit
equivalence relations, and Borel reductions between them to their continuous
and quantitative counterparts: analytic/Borel pseudometrics, orbit
pseudometrics, and Borel reductions between them. We motivate these concepts on
examples and we set some basic general theory. We illustrate the new notion of
reduction by showing that the Gromov-Hausdorff distance maintains the same
complexity if it is defined on the class of all Polish metric spaces, spaces
bounded from below, from above, and from both below and above. Then we show
that is not reducible to equivalences induced by orbit pseudometrics,
generalizing the seminal result of Kechris and Louveau. We answer in negative a
question of Ben-Yaacov, Doucha, Nies, and Tsankov on whether balls in the
Gromov-Hausdorff and Kadets distances are Borel. In appendix, we provide new
methods using games showing that the distance-zero classes in certain
pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and
Tsankov.
There is a complementary paper of the authors where reductions between the
most common pseudometrics from functional analysis and metric geometry are
provided.Comment: Based on the feedback we received, we decided to split the original
version into two parts. The new version is now the first part of this spli
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