188 research outputs found
Statistical learning with phylogenetic network invariants
Phylogenetic networks provide a means of describing the evolutionary history
of sets of species believed to have undergone hybridization or gene flow during
their evolution. The mutation process for a set of such species can be modeled
as a Markov process on a phylogenetic network. Previous work has shown that a
site-pattern probability distributions from a Jukes-Cantor phylogenetic network
model must satisfy certain algebraic invariants. As a corollary, aspects of the
phylogenetic network are theoretically identifiable from site-pattern
frequencies. In practice, because of the probabilistic nature of sequence
evolution, the phylogenetic network invariants will rarely be satisfied, even
for data generated under the model. Thus, using network invariants for
inferring phylogenetic networks requires some means of interpreting the
residuals, or deviations from zero, when observed site-pattern frequencies are
substituted into the invariants. In this work, we propose a method of utilizing
invariant residuals and support vector machines to infer 4-leaf level-one
phylogenetic networks, from which larger networks can be reconstructed. Given
data for a set of species, the support vector machine is first trained on model
data to learn the patterns of residuals corresponding to different network
structures to classify the network that produced the data. We demonstrate the
performance of our method on simulated data from the specified model and
primate data.Comment: 27 pages, 8 figure
Principal Landau Determinants
We reformulate the Landau analysis of Feynman integrals with the aim of
advancing the state of the art in modern particle-physics computations. We
contribute new algorithms for computing Landau singularities, using tools from
polyhedral geometry and symbolic/numerical elimination. Inspired by the work of
Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we
define the principal Landau determinant of a Feynman diagram. We illustrate
with a number of examples that this algebraic formalism allows to compute many
components of the Landau singular locus. We adapt the GKZ framework by
carefully specializing Euler integrals to Feynman integrals. For instance,
ultraviolet and infrared singularities are detected as irreducible components
of an incidence variety, which project dominantly to the kinematic space. We
compute principal Landau determinants for the infinite families of one-loop and
banana diagrams with different mass configurations, and for a range of
cutting-edge Standard Model processes. Our algorithms build on the Julia
package Landau.jl and are implemented in the new open-source package PLD.jl
available at https://mathrepo.mis.mpg.de/PLD/.Comment: 72 page
Recommended from our members
Geometric and Topological Combinatorics
The 2007 Oberwolfach meeting âGeometric and Topological Combinatoricsâ presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions
Learning Algebraic Varieties from Samples
We seek to determine a real algebraic variety from a fixed finite subset of
points. Existing methods are studied and new methods are developed. Our focus
lies on aspects of topology and algebraic geometry, such as dimension and
defining polynomials. All algorithms are tested on a range of datasets and made
available in a Julia package
On the Heegaard Floer homology of Dehn surgery and unknotting number
n this thesis we generalise three theorems from the literature on Heegaard Floer
homology and Dehn surgery: one by Ozsv Ìath and Szab Ìo on deficiency symmetries in
half-integral
L
-space surgeries, and two by Greene which use Donaldsonâs diagonali-
sation theorem as an obstruction to integral and half-integral
L
-space surgeries. Our
generalisation is two-fold: first, we eliminate the
L
-space conditions, opening these
techniques up for use with much more general 3-manifolds, and second, we unify the
integral and half-integral surgery results into a broader theorem applicable to non-
zero rational surgeries in
S
3
which bound sharp, simply connected, negative-definite
smooth 4-manifolds. Such 3-manifolds are quite common and include, for example, a
huge number of Seifert fibred spaces.
Over the course of the first three chapters, we begin by introducing background
material on knots in 3-manifolds, the intersection form of a simply connected 4-
manifold, Spin- and Spin
c
-structures on 3- and 4-manifolds, and Heegaard Floer ho-
mology (including knot Floer homology). While none of the results in these chapters
are original, all of them are necessary to make sense of what follows. In Chapter 4,
we introduce and prove our main theorems, using arguments that are predominantly
algebraic or combinatorial in nature. We then apply these new theorems to the study
of unknotting number in Chapter 5, making considerable headway into the extremely
difficult problem of classifying the 3-strand pretzel knots with unknotting number
one. Finally, in Chapter 6, we present further applications of the main theorems,
ranging from a plan of attack on the famous Seifert fibred space realisation problem
to more biologically motivated problems concerning rational tangle replacement. An
appendix on the implications of our theorems for DNA topology is provided at the
end.Open Acces
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