12,521 research outputs found
PALP - a User Manual
This article provides a complete user's guide to version 2.1 of the toric
geometry package PALP by Maximilian Kreuzer and others. In particular,
previously undocumented applications such as the program nef.x are discussed in
detail. New features of PALP 2.1 include an extension of the program mori.x
which can now compute Mori cones and intersection rings of arbitrary dimension
and can also take specific triangulations of reflexive polytopes as input.
Furthermore, the program nef.x is enhanced by an option that allows the user to
enter reflexive Gorenstein cones as input. The present documentation is
complemented by a Wiki which is available online.Comment: 71 pages, to appear in "Strings, Gauge Fields, and the Geometry
Behind - The Legacy of Maximilian Kreuzer". PALP Wiki available at
http://palp.itp.tuwien.ac.at/wiki/index.php/Main_Pag
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
A theory of spectral partitions of metric graphs
We introduce an abstract framework for the study of clustering in metric
graphs: after suitably metrising the space of graph partitions, we restrict
Laplacians to the clusters thus arising and use their spectral gaps to define
several notions of partition energies; this is the graph counterpart of the
well-known theory of spectral minimal partitions on planar domains and includes
the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012),
815--838] as a special case. We focus on the existence of optimisers for a
large class of functionals defined on such partitions, but also study their
qualitative properties, including stability, regularity, and parameter
dependence. We also discuss in detail their interplay with the theory of nodal
partitions. Unlike in the case of domains, the one-dimensional setting of
metric graphs allows for explicit computation and analytic -- rather than
numerical -- results. Not only do we recover the main assertions in the theory
of spectral minimal partitions on domains, as studied in [Conti \textit{et al},
Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\
Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we
can also generalise some of them and answer (the graph counterparts of) a few
open questions
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