10,150 research outputs found
Parametric Polyhedra with at least Lattice Points: Their Semigroup Structure and the k-Frobenius Problem
Given an integral matrix , the well-studied affine semigroup
\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be
stratified by the number of lattice points inside the parametric polyhedra
. Such families of parametric polyhedra appear in
many areas of combinatorics, convex geometry, algebra and number theory. The
key themes of this paper are: (1) A structure theory that characterizes
precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{
Sg}(A) such that has at least solutions. We
demonstrate that this set is finitely generated, it is a union of translated
copies of a semigroup which can be computed explicitly via Hilbert bases
computations. Related results can be derived for those right-hand-side vectors
for which has exactly solutions or fewer
than solutions. (2) A computational complexity theory. We show that, when
, are fixed natural numbers, one can compute in polynomial time an
encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function,
using a short sum of rational functions. As a consequence, one can identify all
right-hand-side vectors of bounded norm that have at least solutions. (3)
Applications and computation for the -Frobenius numbers. Using Generating
functions we prove that for fixed the -Frobenius number can be
computed in polynomial time. This generalizes a well-known result for by
R. Kannan. Using some adaptation of dynamic programming we show some practical
computations of -Frobenius numbers and their relatives
Chaotic and fractal properties of deterministic diffusion-reaction processes
We study the consequences of deterministic chaos for diffusion-controlled
reaction. As an example, we analyze a diffusive-reactive deterministic
multibaker and a parameter-dependent variation of it. We construct the
diffusive and the reactive modes of the models as eigenstates of the
Frobenius-Perron operator. The associated eigenvalues provide the dispersion
relations of diffusion and reaction and, hence, they determine the reaction
rate. For the simplest model we show explicitly that the reaction rate behaves
as phenomenologically expected for one-dimensional diffusion-controlled
reaction. Under parametric variation, we find that both the diffusion
coefficient and the reaction rate have fractal-like dependences on the system
parameter.Comment: 14 pages (revtex), 12 figures (postscript), to appear in CHAO
Measures of Variability for Bayesian Network Graphical Structures
The structure of a Bayesian network includes a great deal of information
about the probability distribution of the data, which is uniquely identified
given some general distributional assumptions. Therefore it's important to
study its variability, which can be used to compare the performance of
different learning algorithms and to measure the strength of any arbitrary
subset of arcs.
In this paper we will introduce some descriptive statistics and the
corresponding parametric and Monte Carlo tests on the undirected graph
underlying the structure of a Bayesian network, modeled as a multivariate
Bernoulli random variable. A simple numeric example and the comparison of the
performance of some structure learning algorithm on small samples will then
illustrate their use.Comment: 19 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:0909.168
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