3,183 research outputs found
Morphoelastic rods Part 1: A single growing elastic rod
A theory for the dynamics and statics of growing elastic rods is presented. First, a single growing rod is considered and the formalism of three-dimensional multiplicative decomposition of morphoelasticity is used to describe the bulk growth of Kirchhoff elastic rods. Possible constitutive laws for growth are discussed and analysed. Second, a rod constrained or glued to a rigid substrate is considered, with the mismatch between the attachment site and the growing rod inducing stress. This stress can eventually lead to instability, bifurcation, and buckling
Spaces of phylogenetic networks from generalized nearest-neighbor interchange operations
Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are used to represent the evolution of species which have undergone reticulate evolution. In this paper we consider spaces of such networks defined by some novel local operations that we introduce for converting one phylogenetic network into another. These operations are modeled on the well-studied nearest-neighbor interchange (NNI) operations on phylogenetic trees, and lead to natural generalizations of the tree spaces that have been previously associated to such operations. We present several results on spaces of some relatively simple networks, called level-1 networks, including the size of the neighborhood of a fixed network, and bounds on the diameter of the metric defined by taking the smallest number of operations required to convert one network into another.We expect that our results will be useful in the development of methods for systematically searching for optimal phylogenetic networks using, for example, likelihood and Bayesian approaches
Geometric medians in reconciliation spaces
In evolutionary biology, it is common to study how various entities evolve
together, for example, how parasites coevolve with their host, or genes with
their species. Coevolution is commonly modelled by considering certain maps or
reconciliations from one evolutionary tree to another , all of which
induce the same map between the leaf-sets of and (corresponding
to present-day associations). Recently, there has been much interest in
studying spaces of reconciliations, which arise by defining some metric on
the set of all possible reconciliations between and .
In this paper, we study the following question: How do we compute a geometric
median for a given subset of relative to , i.e. an
element such that holds for all
? For a model where so-called host-switches or
transfers are not allowed, and for a commonly used metric called the
edit-distance, we show that although the cardinality of can be
super-exponential, it is still possible to compute a geometric median for a set
in in polynomial time. We expect that this result could
be useful for computing a summary or consensus for a set of reconciliations
(e.g. for a set of suboptimal reconciliations).Comment: 12 pages, 1 figur
Characterizing Block Graphs in Terms of their Vertex-Induced Partitions
Given a finite connected simple graph with vertex set and edge
set , we will show that
the (necessarily unique) smallest block graph with vertex set whose
edge set contains is uniquely determined by the -indexed family of the various partitions
of the set into the set of connected components of the
graph ,
the edge set of this block graph coincides with set of all -subsets
of for which and are, for all , contained
in the same connected component of ,
and an arbitrary -indexed family of
partitions of the set is of the form for some
connected simple graph with vertex set as above if and only if,
for any two distinct elements , the union of the set in
that contains and the set in that contains coincides with
the set , and holds for all .
As well as being of inherent interest to the theory of block graphs, these
facts are also useful in the analysis of compatible decompositions and block
realizations of finite metric spaces
Representing Partitions on Trees
In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set X of species from a multiset Πof partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset ΣΠconsisting of all those bipartitions {A,X − A} with A a part of some partition in Π. The rational behind this approach is that a phylogenetic tree with leaf set X can be uniquely represented by the set of bipartitions of X induced by its edges. Motivated by these considerations, given a multiset Σ of bipartitions corresponding to a phylogenetic tree on X, in this paper we introduce and study the set P(Σ) consisting of those multisets of partitions Πof X with ΣΠ= Σ. More specifically, we characterize when P(Σ) is non-empty, and also identify some partitions in P(Σ) that are of maximum and minimum size. We also show that it is NP-complete to decide when P(Σ) is non-empty in case Σ is an arbitrary multiset of bipartitions of X. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system Πto the multiset ΣΠ, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions
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