12,331 research outputs found
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Robustness and epistasis in mutation-selection models
We investigate the fitness advantage associated with the robustness of a
phenotype against deleterious mutations using deterministic mutation-selection
models of quasispecies type equipped with a mesa shaped fitness landscape. We
obtain analytic results for the robustness effect which become exact in the
limit of infinite sequence length. Thereby, we are able to clarify a seeming
contradiction between recent rigorous work and an earlier heuristic treatment
based on a mapping to a Schr\"odinger equation. We exploit the quantum
mechanical analogy to calculate a correction term for finite sequence lengths
and verify our analytic results by numerical studies. In addition, we
investigate the occurrence of an error threshold for a general class of
epistatic landscape and show that diminishing epistasis is a necessary but not
sufficient condition for error threshold behavior.Comment: 20 pages, 14 figure
Evolutionary game dynamics in phenotype space
Evolutionary dynamics can be studied in well-mixed or structured populations.
Population structure typically arises from the heterogeneous distribution of
individuals in physical space or on social networks. Here we introduce a new
type of space to evolutionary game dynamics: phenotype space. The population is
well-mixed in the sense that everyone is equally likely to interact with
everyone else, but the behavioral strategies depend on distance in phenotype
space. Individuals might behave differently towards those who look similar or
dissimilar. Individuals mutate to nearby phenotypes. We study the `phenotypic
space walk' of populations. We present analytic calculations that bring
together ideas from coalescence theory and evolutionary game dynamics. As a
particular example, we investigate the evolution of cooperation in phenotype
space. We obtain a precise condition for natural selection to favor cooperators
over defectors: for a one-dimensional phenotype space and large population size
the critical benefit-to-cost ratio is given by b/c=1+2/sqrt{3}. We derive the
fundamental condition for any evolutionary game and explore higher dimensional
phenotype spaces.Comment: version 2: minor changes; equivalent to final published versio
The variance of identity-by-descent sharing in the Wright-Fisher model
Widespread sharing of long, identical-by-descent (IBD) genetic segments is a
hallmark of populations that have experienced recent genetic drift. Detection
of these IBD segments has recently become feasible, enabling a wide range of
applications from phasing and imputation to demographic inference. Here, we
study the distribution of IBD sharing in the Wright-Fisher model. Specifically,
using coalescent theory, we calculate the variance of the total sharing between
random pairs of individuals. We then investigate the cohort-averaged sharing:
the average total sharing between one individual and the rest of the cohort. We
find that for large cohorts, the cohort-averaged sharing is distributed
approximately normally. Surprisingly, the variance of this distribution does
not vanish even for large cohorts, implying the existence of "hyper-sharing"
individuals. The presence of such individuals has consequences for the design
of sequencing studies, since, if they are selected for whole-genome sequencing,
a larger fraction of the cohort can be subsequently imputed. We calculate the
expected gain in power of imputation by IBD, and subsequently, in power to
detect an association, when individuals are either randomly selected or
specifically chosen to be the hyper-sharing individuals. Using our framework,
we also compute the variance of an estimator of the population size that is
based on the mean IBD sharing and the variance in the sharing between inbred
siblings. Finally, we study IBD sharing in an admixture pulse model, and show
that in the Ashkenazi Jewish population the admixture fraction is correlated
with the cohort-averaged sharing.Comment: Includes Supplementary Materia
Definition by Induction in Frege's Grundgesetze der Arithmetik
This paper discusses Frege's account of definition by induction in Grundgesetze and the two key theorems Frege proves using it
The Inflation Technique for Causal Inference with Latent Variables
The problem of causal inference is to determine if a given probability
distribution on observed variables is compatible with some causal structure.
The difficult case is when the causal structure includes latent variables. We
here introduce the for tackling this problem. An
inflation of a causal structure is a new causal structure that can contain
multiple copies of each of the original variables, but where the ancestry of
each copy mirrors that of the original. To every distribution of the observed
variables that is compatible with the original causal structure, we assign a
family of marginal distributions on certain subsets of the copies that are
compatible with the inflated causal structure. It follows that compatibility
constraints for the inflation can be translated into compatibility constraints
for the original causal structure. Even if the constraints at the level of
inflation are weak, such as observable statistical independences implied by
disjoint causal ancestry, the translated constraints can be strong. We apply
this method to derive new inequalities whose violation by a distribution
witnesses that distribution's incompatibility with the causal structure (of
which Bell inequalities and Pearl's instrumental inequality are prominent
examples). We describe an algorithm for deriving all such inequalities for the
original causal structure that follow from ancestral independences in the
inflation. For three observed binary variables with pairwise common causes, it
yields inequalities that are stronger in at least some aspects than those
obtainable by existing methods. We also describe an algorithm that derives a
weaker set of inequalities but is more efficient. Finally, we discuss which
inflations are such that the inequalities one obtains from them remain valid
even for quantum (and post-quantum) generalizations of the notion of a causal
model.Comment: Minor final corrections, updated to match the published version as
closely as possibl
Inference of historical population-size changes with allele-frequency data
With up to millions of nearly neutral polymorphisms now being routinely sampled in population-genomic surveys, it is possible to estimate the site-frequency spectrum of such sites with high precision. Each frequency class reflects a mixture of potentially unique demographic histories, which can be revealed using theory for the probability distributions of the starting and ending points of branch segments over all possible coalescence trees. Such distributions are completely independent of past population history, which only influences the segment lengths, providing the basis for estimating average population sizes separating tree-wide coalescence events. The history of population-size change experienced by a sample of polymorphisms can then be dissected in a model-flexible fashion, and extension of this theory allows estimation of the mean and full distribution of long-term effective population sizes and ages of alleles of specific frequencies. Here, we outline the basic theory underlying the conceptual approach, develop and test an efficient statistical procedure for parameter estimation, and apply this to multiple population-genomic datasets for the microcrustacean Daphnia pulex
Maximum principle and mutation thresholds for four-letter sequence evolution
A four-state mutation-selection model for the evolution of populations of
DNA-sequences is investigated with particular interest in the phenomenon of
error thresholds. The mutation model considered is the Kimura 3ST mutation
scheme, fitness functions, which determine the selection process, come from the
permutation-invariant class. Error thresholds can be found for various fitness
functions, the phase diagrams are more interesting than for equivalent
two-state models. Results for (small) finite sequence lengths are compared with
those for infinite sequence length, obtained via a maximum principle that is
equivalent to the principle of minimal free energy in physics.Comment: 25 pages, 16 figure
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