15 research outputs found
On the topological classification of binary trees using the Horton-Strahler index
The Horton-Strahler (HS) index has been shown to
be relevant to a number of physical (such at diffusion limited aggregation)
geological (river networks), biological (pulmonary arteries, blood vessels,
various species of trees) and computational (use of registers) applications.
Here we revisit the enumeration problem of the HS index on the rooted,
unlabeled, plane binary set of trees, and enumerate the same index on the
ambilateral set of rooted, plane binary set of trees of leaves. The
ambilateral set is a set of trees whose elements cannot be obtained from each
other via an arbitrary number of reflections with respect to vertical axes
passing through any of the nodes on the tree. For the unlabeled set we give an
alternate derivation to the existing exact solution. Extending this technique
for the ambilateral set, which is described by an infinite series of non-linear
functional equations, we are able to give a double-exponentially converging
approximant to the generating functions in a neighborhood of their convergence
circle, and derive an explicit asymptotic form for the number of such trees.Comment: 14 pages, 7 embedded postscript figures, some minor changes and typos
correcte
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category