715 research outputs found
Matrix Factorizations and Homological Mirror Symmetry on the Torus
We consider matrix factorizations and homological mirror symmetry on the
torus T^2 using a Landau-Ginzburg description. We identify the basic matrix
factorizations of the Landau-Ginzburg superpotential and compute the full
spectrum, taking into account the explicit dependence on bulk and boundary
moduli. We verify homological mirror symmetry by comparing three-point
functions in the A-model and the B-model.Comment: 41 pages, 9 figures, v2: reference added, minor corrections and
clarifications, version published in JHE
Efficient data augmentation for fitting stochastic epidemic models to prevalence data
Stochastic epidemic models describe the dynamics of an epidemic as a disease
spreads through a population. Typically, only a fraction of cases are observed
at a set of discrete times. The absence of complete information about the time
evolution of an epidemic gives rise to a complicated latent variable problem in
which the state space size of the epidemic grows large as the population size
increases. This makes analytically integrating over the missing data infeasible
for populations of even moderate size. We present a data augmentation Markov
chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic
epidemic model parameters, in which measurements are augmented with
subject-level disease histories. In our MCMC algorithm, we propose each new
subject-level path, conditional on the data, using a time-inhomogeneous
continuous-time Markov process with rates determined by the infection histories
of other individuals. The method is general, and may be applied, with minimal
modifications, to a broad class of stochastic epidemic models. We present our
algorithm in the context of multiple stochastic epidemic models in which the
data are binomially sampled prevalence counts, and apply our method to data
from an outbreak of influenza in a British boarding school
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
Implicit ODE solvers with good local error control for the transient analysis of Markov models
Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.Peer ReviewedPostprint (published version
Topological Signals of Singularities in Ricci Flow
We implement methods from computational homology to obtain a topological
signal of singularity formation in a selection of geometries evolved
numerically by Ricci flow. Our approach, based on persistent homology, produces
precise, quantitative measures describing the behavior of an entire collection
of data across a discrete sample of times. We analyze the topological signals
of geometric criticality obtained numerically from the application of
persistent homology to models manifesting singularities under Ricci flow. The
results we obtain for these numerical models suggest that the topological
signals distinguish global singularity formation (collapse to a round point)
from local singularity formation (neckpinch). Finally, we discuss the
interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure
A generalized method for the transient analysis of Markov models of fault-tolerant systems with deferred repair
Randomization is an attractive alternative for the transient analysis of continuous
time Markov models. The main advantages of the method are numerical stability,
well-controlled computation error, and ability to specify the computation error
in advance. However, the fact that the method can be computationally expensive
limits its applicability. Recently, a variant of the (standard) randomization method, called split regenerative randomization has been proposed for the efficient analysis of reliability-like models of fault-tolerant systems with deferred repair. In this article, we generalize that method so that it covers more general reward measures: the expected transient reward rate and the expected averaged reward rate. The generalized method has the same good properties as the standard randomization method and, for large models and large values of the time t at which the
measure has to be computed, can be significantly less expensive. The method
requires the selection of a subset of states and a regenerative state satisfying some
conditions. For a class of continuous time Markov models, class C'_2, including
typical failure/repair reliability models with exponential failure and repair time
distributions and deferred repair, natural selections for the subset of states and
the regenerative state exist and results are available assessing approximately the
computational cost of the method in terms of “visible” model characteristics. Using
a large model class C'_2 example, we illustrate the performance of the method and show that it can be significantly faster than previously proposed randomizationbased methods.Postprint (published version
On effective sigma-boundedness and sigma-compactness
We prove several theorems on sigma-bounded and sigma-compact pointsets. We
start with a known theorem by Kechris, saying that any lightface \Sigma^1_1 set
of the Baire space either is effectively sigma-bounded (that is, covered by a
countable union of compact lightface \Delta^1_1 sets), or contains a
superperfect subset (and then the set is not sigma-bounded, of course). We add
different generalizations of this result, in particular, 1) such that the
boundedness property involved includes covering by compact sets and equivalence
classes of a given finite collection of lightface \Delta^1_1 equivalence
relations, 2) generalizations to lightface \Sigma^1_2 sets, 3) generalizations
true in the Solovay model.
As for effective sigma-compactness, we start with a theorem by Louveau,
saying that any lightface \Delta^1_1 set of the Baire space either is
effectively sigma-compact (that is, is equal to a countable union of compact
lightface \Delta^1_1 sets), or it contains a relatively closed superperfect
subset. Then we prove a generalization of this result to lightface \Sigma^1_1
sets.Comment: arXiv admin note: substantial text overlap with arXiv:1103.106
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