1,325 research outputs found
Discrete Nodal Domain Theorems
We give a detailed proof for two discrete analogues of Courant's Nodal Domain
Theorem
A Simple Data-Adaptive Probabilistic Variant Calling Model
Background: Several sources of noise obfuscate the identification of single
nucleotide variation (SNV) in next generation sequencing data. For instance,
errors may be introduced during library construction and sequencing steps. In
addition, the reference genome and the algorithms used for the alignment of the
reads are further critical factors determining the efficacy of variant calling
methods. It is crucial to account for these factors in individual sequencing
experiments.
Results: We introduce a simple data-adaptive model for variant calling. This
model automatically adjusts to specific factors such as alignment errors. To
achieve this, several characteristics are sampled from sites with low mismatch
rates, and these are used to estimate empirical log-likelihoods. These
likelihoods are then combined to a score that typically gives rise to a mixture
distribution. From these we determine a decision threshold to separate
potentially variant sites from the noisy background.
Conclusions: In simulations we show that our simple proposed model is
competitive with frequently used much more complex SNV calling algorithms in
terms of sensitivity and specificity. It performs specifically well in cases
with low allele frequencies. The application to next-generation sequencing data
reveals stark differences of the score distributions indicating a strong
influence of data specific sources of noise. The proposed model is specifically
designed to adjust to these differences.Comment: 19 pages, 6 figure
Cover-Encodings of Fitness Landscapes
The traditional way of tackling discrete optimization problems is by using
local search on suitably defined cost or fitness landscapes. Such approaches
are however limited by the slowing down that occurs when the local minima that
are a feature of the typically rugged landscapes encountered arrest the
progress of the search process. Another way of tackling optimization problems
is by the use of heuristic approximations to estimate a global cost minimum.
Here we present a combination of these two approaches by using cover-encoding
maps which map processes from a larger search space to subsets of the original
search space. The key idea is to construct cover-encoding maps with the help of
suitable heuristics that single out near-optimal solutions and result in
landscapes on the larger search space that no longer exhibit trapping local
minima. We present cover-encoding maps for the problems of the traveling
salesman, number partitioning, maximum matching and maximum clique; the
practical feasibility of our method is demonstrated by simulations of adaptive
walks on the corresponding encoded landscapes which find the global minima for
these problems.Comment: 15 pages, 4 figure
Convex Cycle Bases
Convex cycles play a role e.g. in the context of product graphs. We introduce convex cycle bases and describe a polynomial-time algorithm that recognizes whether a given graph has a convex cycle basis and provides an explicit construction in the positive case. Relations between convex cycles bases and other types of cycles bases are discussed. In particular we show that if G has a unique minimal cycle bases, this basis is convex. Furthermore, we characterize a class of graphs with convex cycles bases that includes partial cubes and hence median graphs. (authors' abstract)Series: Research Report Series / Department of Statistics and Mathematic
Square Property, Equitable Partitions, and Product-like Graphs
Equivalence relations on the edge set of a graph that satisfy restrictive
conditions on chordless squares play a crucial role in the theory of Cartesian
graph products and graph bundles. We show here that such relations in a natural
way induce equitable partitions on the vertex set of , which in turn give
rise to quotient graphs that can have a rich product structure even if
itself is prime.Comment: 20 pages, 6 figure
Replicator Dynamics in Protocells
Replicator equations have been studied for three decades as a generic dynamical
system modelling replication processes. Here we show how they arise naturally in
models of self-replicating polymers and discuss some of their basic properties. We
then concentrate on a minimal dynamic model of a protocell by coupling replicating
polymers with a growing membrane
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