143,009 research outputs found
Perfect Sets and -Ideals
A square-free monomial ideal is called an {\it -ideal}, if both
and have the same
-vector, where (,
respectively) is the facet (Stanley-Reisner, respectively) complex related to
. In this paper, we introduce and study perfect subsets of and use
them to characterize the -ideals of degree . We give a decomposition of
by taking advantage of a correspondence between graphs and sets of
square-free monomials of degree , and then give a formula for counting the
number of -ideals of degree , where is the set of -ideals of
degree 2 in . We also consider the relation between an
-ideal and an unmixed monomial ideal.Comment: 15 page
A Comparison of Quantum Oracles
A standard quantum oracle for a general function is
defined to act on two input states and return two outputs, with inputs
and () returning outputs and
. However, if is known to be a one-to-one function, a
simpler oracle, , which returns given , can also be
defined. We consider the relative strengths of these oracles. We define a
simple promise problem which minimal quantum oracles can solve exponentially
faster than classical oracles, via an algorithm which cannot be naively adapted
to standard quantum oracles. We show that can be constructed by invoking
and once each, while invocations of
and/or are required to construct .Comment: 4 pages, 1 figure; Final version, with an extended discussion of
oracle inverses. To appear in Phys Rev
Tree decompositions with small cost
The f-cost of a tree decomposition ({Xi | i e I}, T = (I;F))
for a function f : N -> R+ is defined as EieI f(|Xi|). This measure
associates with the running time or memory use of some algorithms
that use the tree decomposition. In this paper we investigate the
problem to find tree decompositions of minimum f-cost.
A function f : N -> R+ is fast, if for every i e N: f(i+1) => 2*f(i).
We show that for fast functions f, every graph G has a tree decomposition
of minimum f-cost that corresponds to a minimal triangulation
of G; if f is not fast, this does not hold. We give polynomial time
algorithms for the problem, assuming f is a fast function, for graphs
that has a polynomial number of minimal separators, for graphs of
treewidth at most two, and for cographs, and show that the problem
is NP-hard for bipartite graphs and for cobipartite graphs.
We also discuss results for a weighted variant of the problem derived
of an application from probabilistic networks
A limit theorem for the six-length of random functional graphs with a fixed degree sequence
We obtain results on the limiting distribution of the six-length of a random
functional graph, also called a functional digraph or random mapping, with
given in-degree sequence. The six-length of a vertex is defined from
the associated mapping, , to be the maximum such that the
elements are all distinct. This has relevance to
the study of algorithms for integer factorisation
Renormalization of the Fayet-Iliopoulos Term in Softly Broken SUSY Gauge Theories
It is shown that renormalization of the Fayet-Iliopoulos term in a softly
broken SUSY gauge theory, in full analogy with all the other soft terms
renormalizations, is completely defined in a rigid or an unbroken theory.
However, contrary to the other soft renormalizations, there is no simple
differential operator that acts on the renormalization functions of a rigid
theory and allows one to get the renormalization of the F-I term. One needs an
analysis of the superfield diagrams and some additional diagram calculations in
components. The method is illustrated by the four loop calculation of some part
of renormalization proportional to the soft scalar masses and the soft triple
couplings.Comment: Latex2e, 14 pages, uses axodraw.sty. References adde
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