8,256 research outputs found
Trees, Tight-Spans and Point Configuration
Tight-spans of metrics were first introduced by Isbell in 1964 and
rediscovered and studied by others, most notably by Dress, who gave them this
name. Subsequently, it was found that tight-spans could be defined for more
general maps, such as directed metrics and distances, and more recently for
diversities. In this paper, we show that all of these tight-spans as well as
some related constructions can be defined in terms of point configurations.
This provides a useful way in which to study these objects in a unified and
systematic way. We also show that by using point configurations we can recover
results concerning one-dimensional tight-spans for all of the maps we consider,
as well as extend these and other results to more general maps such as
symmetric and unsymmetric maps.Comment: 21 pages, 2 figure
On morphological hierarchical representations for image processing and spatial data clustering
Hierarchical data representations in the context of classi cation and data
clustering were put forward during the fties. Recently, hierarchical image
representations have gained renewed interest for segmentation purposes. In this
paper, we briefly survey fundamental results on hierarchical clustering and
then detail recent paradigms developed for the hierarchical representation of
images in the framework of mathematical morphology: constrained connectivity
and ultrametric watersheds. Constrained connectivity can be viewed as a way to
constrain an initial hierarchy in such a way that a set of desired constraints
are satis ed. The framework of ultrametric watersheds provides a generic scheme
for computing any hierarchical connected clustering, in particular when such a
hierarchy is constrained. The suitability of this framework for solving
practical problems is illustrated with applications in remote sensing
The dissimilarity map and representation theory of
We give another proof that -dissimilarity vectors of weighted trees are
points on the tropical Grassmanian, as conjectured by Cools, and proved by
Giraldo in response to a question of Sturmfels and Pachter. We accomplish this
by relating -dissimilarity vectors to the representation theory of Comment: 11 pages, 8 figure
The Mathematics of Phylogenomics
The grand challenges in biology today are being shaped by powerful
high-throughput technologies that have revealed the genomes of many organisms,
global expression patterns of genes and detailed information about variation
within populations. We are therefore able to ask, for the first time,
fundamental questions about the evolution of genomes, the structure of genes
and their regulation, and the connections between genotypes and phenotypes of
individuals. The answers to these questions are all predicated on progress in a
variety of computational, statistical, and mathematical fields.
The rapid growth in the characterization of genomes has led to the
advancement of a new discipline called Phylogenomics. This discipline results
from the combination of two major fields in the life sciences: Genomics, i.e.,
the study of the function and structure of genes and genomes; and Molecular
Phylogenetics, i.e., the study of the hierarchical evolutionary relationships
among organisms and their genomes. The objective of this article is to offer
mathematicians a first introduction to this emerging field, and to discuss
specific mathematical problems and developments arising from phylogenomics.Comment: 41 pages, 4 figure
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