40,846 research outputs found
On the complement of the Richardson orbit
We consider parabolic subgroups of a general algebraic group over an
algebraically closed field whose Levi part has exactly factors. By a
classical theorem of Richardson, the nilradical of a parabolic subgroup has
an open dense -orbit. In the complement to this dense orbit, there are
infinitely many orbits as soon as the number of factors in the Levi part is
. In this paper, we describe the irreducible components of the
complement. In particular, we show that there are at most irreducible
components.Comment: 15 page
The t-stability number of a random graph
Given a graph G = (V,E), a vertex subset S is called t-stable (or
t-dependent) if the subgraph G[S] induced on S has maximum degree at most t.
The t-stability number of G is the maximum order of a t-stable set in G. We
investigate the typical values that this parameter takes on a random graph on n
vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed
non-negative integer t, we show that, with probability tending to 1 as n grows,
the t-stability number takes on at most two values which we identify as
functions of t, p and n. The main tool we use is an asymptotic expression for
the expected number of t-stable sets of order k. We derive this expression by
performing a precise count of the number of graphs on k vertices that have
maximum degree at most k. Using the above results, we also obtain asymptotic
bounds on the t-improper chromatic number of a random graph (this is the
generalisation of the chromatic number, where we partition of the vertex set of
the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version
apart from formatting and a minor amendment to Lemma 8 (and its proof on p.
21
Colouring exact distance graphs of chordal graphs
For a graph and positive integer , the exact distance- graph
is the graph with vertex set and with an edge between
vertices and if and only if and have distance . Recently,
there has been an effort to obtain bounds on the chromatic number
of exact distance- graphs for from certain
classes of graphs. In particular, if a graph has tree-width , it has
been shown that for odd ,
and for even . We
show that if is chordal and has tree-width , then for odd , and for even .
If we could show that for every graph of tree-width there is a
chordal graph of tree-width which contains as an isometric subgraph
(i.e., a distance preserving subgraph), then our results would extend to all
graphs of tree-width . While we cannot do this, we show that for every graph
of genus there is a graph which is a triangulation of genus and
contains as an isometric subgraph.Comment: 11 pages, 2 figures. Versions 2 and 3 include minor changes, which
arise from reviewers' comment
The largest fragment of a homogeneous fragmentation process
We show that in homogeneous fragmentation processes the largest fragment at
time has size where is the L\'evy exponent of the
fragmentation process, and is the unique solution of the equation
. We argue that this result is in line
with predictions arising from the classification of homogeneous fragmentation
processes as logarithmically correlated random fields.Comment: 20 page
Generalized vector space partitions
A vector space partition in is a set of
subspaces such that every -dimensional subspace of is
contained in exactly one element of . Replacing "every point" by
"every -dimensional subspace", we generalize this notion to vector space
-partitions and study their properties. There is a close connection to
subspace codes and some problems are even interesting and unsolved for the set
case .Comment: 12 pages, typos correcte
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