1,239 research outputs found
Chebyshev polynomials and the Frohman-Gelca formula
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the
Kauffman bracket skein module of the torus. This basis is especially useful
because the Jones-Kauffman product can be described via a very simple
Product-to-Sum formula. Presented in this work is a diagrammatic proof of this
formula, which emphasizes and demystifies the role played by Chebyshev
polynomials.Comment: 13 page
Information filtering via Iterative Refinement
With the explosive growth of accessible information, expecially on the
Internet, evaluation-based filtering has become a crucial task. Various systems
have been devised aiming to sort through large volumes of information and
select what is likely to be more relevant. In this letter we analyse a new
ranking method, where the reputation of information providers is determined
self-consistently.Comment: 10 pages, 3 figures. Accepted for publication on Europhysics Letter
Time-averaged MSD of Brownian motion
We study the statistical properties of the time-averaged mean-square
displacements (TAMSD). This is a standard non-local quadratic functional for
inferring the diffusion coefficient from an individual random trajectory of a
diffusing tracer in single-particle tracking experiments. For Brownian motion,
we derive an exact formula for the Laplace transform of the probability density
of the TAMSD by mapping the original problem onto chains of coupled harmonic
oscillators. From this formula, we deduce the first four cumulant moments of
the TAMSD, the asymptotic behavior of the probability density and its accurate
approximation by a generalized Gamma distribution
The Ehrenfest urn revisited: Playing the game on a realistic fluid model
The Ehrenfest urn process, also known as the dogs and fleas model, is
realistically simulated by molecular dynamics of the Lennard-Jones fluid. The
key variable is Delta z, i.e. the absolute value of the difference between the
number of particles in one half of the simulation box and in the other half.
This is a pure-jump stochastic process induced, under coarse graining, by the
deterministic time evolution of the atomic coordinates. We discuss the Markov
hypothesis by analyzing the statistical properties of the jumps and of the
waiting times between jumps. In the limit of a vanishing integration time-step,
the distribution of waiting times becomes closer to an exponential and,
therefore, the continuous-time jump stochastic process is Markovian. The random
variable Delta z behaves as a Markov chain and, in the gas phase, the observed
transition probabilities follow the predictions of the Ehrenfest theory.Comment: Accepted by Physical Review E on 4 May 200
Generalized Master Equations for Non-Poisson Dynamics on Networks
The traditional way of studying temporal networks is to aggregate the
dynamics of the edges to create a static weighted network. This implicitly
assumes that the edges are governed by Poisson processes, which is not
typically the case in empirical temporal networks. Consequently, we examine the
effects of non-Poisson inter-event statistics on the dynamics of edges, and we
apply the concept of a generalized master equation to the study of
continuous-time random walks on networks. We show that the equation reduces to
the standard rate equations when the underlying process is Poisson and that the
stationary solution is determined by an effective transition matrix whose
leading eigenvector is easy to calculate. We discuss the implications of our
work for dynamical processes on temporal networks and for the construction of
network diagnostics that take into account their nontrivial stochastic nature
The effect of discrete vs. continuous-valued ratings on reputation and ranking systems
When users rate objects, a sophisticated algorithm that takes into account
ability or reputation may produce a fairer or more accurate aggregation of
ratings than the straightforward arithmetic average. Recently a number of
authors have proposed different co-determination algorithms where estimates of
user and object reputation are refined iteratively together, permitting
accurate measures of both to be derived directly from the rating data. However,
simulations demonstrating these methods' efficacy assumed a continuum of rating
values, consistent with typical physical modelling practice, whereas in most
actual rating systems only a limited range of discrete values (such as a 5-star
system) is employed. We perform a comparative test of several co-determination
algorithms with different scales of discrete ratings and show that this
seemingly minor modification in fact has a significant impact on algorithms'
performance. Paradoxically, where rating resolution is low, increased noise in
users' ratings may even improve the overall performance of the system.Comment: 6 pages, 2 figure
OPEN EDUCATIONAL RESOURCES IN THE CONTEXT OF SCHOOL EDUCATION: BARRIERS AND POSSIBLE SOLUTIONS
Due to the increasing professional mobility of their parents, pupils often find themselves in new and unfamiliar learning scenarios in foreign contexts and countries. Besides having to leave their familiar environments, these pupils additionally may face language barriers, different curricula, and have to cope with foreign cultures. Printed textbooks, which are the most commonly used educational resources in schools, provide little support for these pupils to manage the new challenges. Teachers are the professionals designated to provide the necessary support. However, they often may not fully appreciate the pupils’ individual challenges. Possible solutions could be the provision of alternative learning contents in the pupils’ native languages and an international open exchange of knowledge and experiences amongst schoolteachers. These issues are addressed by the Open Discovery Space platform. In order to empower this platform to provide the best possible support to teachers, we explored barriers to adoption of Open Educational Practices in the context of school education and asked for manageable solutions. The investigation took place in an action research scenario. After an introduction of the ODS project, we will present the identified barriers and recommendations for solutions to overcome these, and the mechanisms which we are going to implement in the ODS platform in order to provide the best possible support to the community
Dephasing by a Continuous-Time Random Walk Process
Stochastic treatments of magnetic resonance spectroscopy and optical
spectroscopy require evaluations of functions like ,
where t is time, Q(s) is the value of a stochastic process at time s, and the
angular brackets denote ensemble averaging. This paper gives an exact
evaluation of these functions for the case where Q is a continuous-time random
walk process. The continuous time random walk describes an environment that
undergoes slow, step-like changes in time. It also has a well-defined Gaussian
limit, and so allows for non-Gaussian and Gaussian stochastic dynamics to be
studied within a single framework. We apply the results to extract
qubit-lattice interaction parameters from dephasing data of P-doped Si
semiconductors (data collected elsewhere), and to calculate the two-dimensional
spectrum of a three level harmonic oscillator undergoing random frequency
modulations.Comment: 25 pages, 4 figure
Formulating genome-scale kinetic models in the post-genome era
The biological community is now awash in high-throughput data sets and is grappling with the challenge of integrating disparate data sets. Such integration has taken the form of statistical analysis of large data sets, or through the bottom–up reconstruction of reaction networks. While progress has been made with statistical and structural methods, large-scale systems have remained refractory to dynamic model building by traditional approaches. The availability of annotated genomes enabled the reconstruction of genome-scale networks, and now the availability of high-throughput metabolomic and fluxomic data along with thermodynamic information opens the possibility to build genome-scale kinetic models. We describe here a framework for building and analyzing such models. The mathematical analysis challenges are reflected in four foundational properties, (i) the decomposition of the Jacobian matrix into chemical, kinetic and thermodynamic information, (ii) the structural similarity between the stoichiometric matrix and the transpose of the gradient matrix, (iii) the duality transformations enabling either fluxes or concentrations to serve as the independent variables and (iv) the timescale hierarchy in biological networks. Recognition and appreciation of these properties highlight notable and challenging new in silico analysis issues
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