320 research outputs found
Uniform asymptotics of the coefficients of unitary moment polynomials
Keating and Snaith showed that the absolute moment of the
characteristic polynomial of a random unitary matrix evaluated on the unit
circle is given by a polynomial of degree . In this article, uniform
asymptotics for the coefficients of that polynomial are derived, and a maximal
coefficient is located. Some of the asymptotics are given in explicit form.
Numerical data to support these calculations are presented. Some apparent
connections between random matrix theory and the Riemann zeta function are
discussed.Comment: 31 pages, 1 figure, 2 tables. A few minor misprints fixe
Optimization in task--completion networks
We discuss the collective behavior of a network of individuals that receive,
process and forward to each other tasks. Given costs they store those tasks in
buffers, choosing optimally the frequency at which to check and process the
buffer. The individual optimizing strategy of each node determines the
aggregate behavior of the network. We find that, under general assumptions, the
whole system exhibits coexistence of equilibria and hysteresis.Comment: 18 pages, 3 figures, submitted to JSTA
Predicting the size and probability of epidemics in a population with heterogeneous infectiousness and susceptibility
We analytically address disease outbreaks in large, random networks with
heterogeneous infectivity and susceptibility. The transmissibility
(the probability that infection of causes infection of ) depends on the
infectivity of and the susceptibility of . Initially a single node is
infected, following which a large-scale epidemic may or may not occur. We use a
generating function approach to study how heterogeneity affects the probability
that an epidemic occurs and, if one occurs, its attack rate (the fraction
infected). For fixed average transmissibility, we find upper and lower bounds
on these. An epidemic is most likely if infectivity is homogeneous and least
likely if the variance of infectivity is maximized. Similarly, the attack rate
is largest if susceptibility is homogeneous and smallest if the variance is
maximized. We further show that heterogeneity in infectious period is
important, contrary to assumptions of previous studies. We confirm our
theoretical predictions by simulation. Our results have implications for
control strategy design and identification of populations at higher risk from
an epidemic.Comment: 5 pages, 3 figures. Submitted to Physical Review Letter
Propagation on networks: an exact alternative perspective
By generating the specifics of a network structure only when needed
(on-the-fly), we derive a simple stochastic process that exactly models the
time evolution of susceptible-infectious dynamics on finite-size networks. The
small number of dynamical variables of this birth-death Markov process greatly
simplifies analytical calculations. We show how a dual analytical description,
treating large scale epidemics with a Gaussian approximations and small
outbreaks with a branching process, provides an accurate approximation of the
distribution even for rather small networks. The approach also offers important
computational advantages and generalizes to a vast class of systems.Comment: 8 pages, 4 figure
Combinatorics and Boson normal ordering: A gentle introduction
We discuss a general combinatorial framework for operator ordering problems
by applying it to the normal ordering of the powers and exponential of the
boson number operator. The solution of the problem is given in terms of Bell
and Stirling numbers enumerating partitions of a set. This framework reveals
several inherent relations between ordering problems and combinatorial objects,
and displays the analytical background to Wick's theorem. The methodology can
be straightforwardly generalized from the simple example given herein to a wide
class of operators.Comment: 8 pages, 1 figur
From Quantum Mechanics to Quantum Field Theory: The Hopf route
We show that the combinatorial numbers known as {\em Bell numbers} are
generic in quantum physics. This is because they arise in the procedure known
as {\em Normal ordering} of bosons, a procedure which is involved in the
evaluation of quantum functions such as the canonical partition function of
quantum statistical physics, {\it inter alia}. In fact, we shall show that an
evaluation of the non-interacting partition function for a single boson system
is identical to integrating the {\em exponential generating function} of the
Bell numbers, which is a device for encapsulating a combinatorial sequence in a
single function. We then introduce a remarkable equality, the Dobinski
relation, and use it to indicate why renormalisation is necessary in even the
simplest of perturbation expansions for a partition function. Finally we
introduce a global algebraic description of this simple model, giving a Hopf
algebra, which provides a starting point for extensions to more complex
physical systems
Counting, generating and sampling tree alignments
Pairwise ordered tree alignment are combinatorial objects that appear in RNA
secondary structure comparison. However, the usual representation of tree
alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce
identical sets of matches between identical pairs of trees. This ambiguity is
uninformative, and detrimental to any probabilistic analysis.In this work, we
consider tree alignments up to equivalence. Our first result is a precise
asymptotic enumeration of tree alignments, obtained from a context-free grammar
by mean of basic analytic combinatorics. Our second result focuses on
alignments between two given ordered trees and . By refining our grammar
to align specific trees, we obtain a decomposition scheme for the space of
alignments, and use it to design an efficient dynamic programming algorithm for
sampling alignments under the Gibbs-Boltzmann probability distribution. This
generalizes existing tree alignment algorithms, and opens the door for a
probabilistic analysis of the space of suboptimal RNA secondary structures
alignments.Comment: ALCOB - 3rd International Conference on Algorithms for Computational
Biology - 2016, Jun 2016, Trujillo, Spain. 201
Reactive Control Improvisation
Reactive synthesis is a paradigm for automatically building
correct-by-construction systems that interact with an unknown or adversarial
environment. We study how to do reactive synthesis when part of the
specification of the system is that its behavior should be random. Randomness
can be useful, for example, in a network protocol fuzz tester whose output
should be varied, or a planner for a surveillance robot whose route should be
unpredictable. However, existing reactive synthesis techniques do not provide a
way to ensure random behavior while maintaining functional correctness. Towards
this end, we generalize the recently-proposed framework of control
improvisation (CI) to add reactivity. The resulting framework of reactive
control improvisation provides a natural way to integrate a randomness
requirement with the usual functional specifications of reactive synthesis over
a finite window. We theoretically characterize when such problems are
realizable, and give a general method for solving them. For specifications
given by reachability or safety games or by deterministic finite automata, our
method yields a polynomial-time synthesis algorithm. For various other types of
specifications including temporal logic formulas, we obtain a polynomial-space
algorithm and prove matching PSPACE-hardness results. We show that all of these
randomized variants of reactive synthesis are no harder in a
complexity-theoretic sense than their non-randomized counterparts.Comment: 25 pages. Full version of a CAV 2018 pape
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