51,031 research outputs found
Logarithmic divergences in the -inflationary power spectra computed through the uniform approximation
We investigate a calculation method for solving the Mukhanov-Sasaki equation
in slow-roll -inflation based on the uniform approximation (UA) in
conjunction with an expansion scheme for slow-roll parameters with respect to
the number of -folds about the so-called \textit{turning point}. Earlier
works on this method has so far gained some promising results derived from the
approximating expressions for the power spectra among others, up to second
order with respect to the Hubble and sound flow parameters, when compared to
other semi-analytical approaches (e.g., Green's function and WKB methods).
However, a closer inspection is suggestive that there is a problem when
higher-order parts of the power spectra are considered; residual logarithmic
divergences may come out that can render the prediction physically
inconsistent. Looking at this possibility, we map out up to what order with
respect to the mentioned parameters several physical quantities can be
calculated before hitting a logarithmically divergent result. It turns out that
the power spectra are limited up to second order, the tensor-to-scalar ratio up
to third order, and the spectral indices and running converge to all orders.
This indicates that the expansion scheme is incompatible with the working
equations derived from UA for the power spectra but compatible with that of the
spectral indices. For those quantities that involve logarithmically divergent
terms in the higher-order parts, existing results in the literature for the
convergent lower-order parts calculated in the equivalent fashion should be
viewed with some caution; they do not rest on solid mathematical ground.Comment: version 4 : extended Section 6 on remarks on logarithmic divergence
Reliability-Constrained Economic Dispatch with Analytical Formulation of Operational Risk Evaluation
Operational reliability and the decision-making process of economic dispatch (ED) are closely related and important for power system operation. Consideration of reliability indices and reliability constraints together in the operation problem is very challenging due to the problem size and tight reliability constraints. In this paper, a comprehensive reliability-constrained economic dispatch model with analytical formulation of operational risk evaluation (RCED-AF) is proposed to tackle the operational risk problem of power systems. An operational reliability evaluation model considering the ED decision is designed to accurately assess the system behavior. A computation scheme is also developed to achieve efficient update of risk indices for each ED decision by approximating the reliability evaluation procedure with an analytical polynomial function. The RCED-AF model can be constructed with decision-dependent reliability constraints expressed by the sparse polynomial chaos expansion. Case studies demonstrate that the proposed RCED-AF model is effective and accurate in the optimization of the reliability and the cost for day-ahead economic dispatch
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Asymptotic Proportion of Hard Instances of the Halting Problem
Although the halting problem is undecidable, imperfect testers that fail on
some instances are possible. Such instances are called hard for the tester. One
variant of imperfect testers replies "I don't know" on hard instances, another
variant fails to halt, and yet another replies incorrectly "yes" or "no". Also
the halting problem has three variants: does a given program halt on the empty
input, does a given program halt when given itself as its input, or does a
given program halt on a given input. The failure rate of a tester for some size
is the proportion of hard instances among all instances of that size. This
publication investigates the behaviour of the failure rate as the size grows
without limit. Earlier results are surveyed and new results are proven. Some of
them use C++ on Linux as the computational model. It turns out that the
behaviour is sensitive to the details of the programming language or
computational model, but in many cases it is possible to prove that the
proportion of hard instances does not vanish.Comment: 18 pages. The differences between this version and arXiv:1307.7066v1
are significant. They have been listed in the last paragraph of Section 1.
Excluding layout, this arXiv version is essentially identical to the Acta
Cybernetica versio
On sparse representations of linear operators and the approximation of matrix products
Thus far, sparse representations have been exploited largely in the context
of robustly estimating functions in a noisy environment from a few
measurements. In this context, the existence of a basis in which the signal
class under consideration is sparse is used to decrease the number of necessary
measurements while controlling the approximation error. In this paper, we
instead focus on applications in numerical analysis, by way of sparse
representations of linear operators with the objective of minimizing the number
of operations needed to perform basic operations (here, multiplication) on
these operators. We represent a linear operator by a sum of rank-one operators,
and show how a sparse representation that guarantees a low approximation error
for the product can be obtained from analyzing an induced quadratic form. This
construction in turn yields new algorithms for computing approximate matrix
products.Comment: 6 pages, 3 figures; presented at the 42nd Annual Conference on
Information Sciences and Systems (CISS 2008
Testing for Changes in Kendall's Tau
For a bivariate time series we want to detect
whether the correlation between and stays constant for all . We propose a nonparametric change-point test statistic based on
Kendall's tau and derive its asymptotic distribution under the null hypothesis
of no change by means a new U-statistic invariance principle for dependent
processes. The asymptotic distribution depends on the long run variance of
Kendall's tau, for which we propose an estimator and show its consistency.
Furthermore, assuming a single change-point, we show that the location of the
change-point is consistently estimated. Kendall's tau possesses a high
efficiency at the normal distribution, as compared to the normal maximum
likelihood estimator, Pearson's moment correlation coefficient. Contrary to
Pearson's correlation coefficient, it has excellent robustness properties and
shows no loss in efficiency at heavy-tailed distributions. We assume the data
to be stationary and P-near epoch dependent on an
absolutely regular process. The P-near epoch dependence condition constitutes a
generalization of the usually considered -near epoch dependence, , that does not require the existence of any moments. It is therefore very
well suited for our objective to efficiently detect changes in correlation for
arbitrarily heavy-tailed data
The Inflationary Perturbation Spectrum
Motivated by the prospect of testing inflation from precision cosmic
microwave background observations, we present analytic results for scalar and
tensor perturbations in single-field inflation models based on the application
of uniform approximations. This technique is systematically improvable,
possesses controlled error bounds, and does not rely on assuming the slow-roll
parameters to be constant. We provide closed-form expressions for the power
spectra and the corresponding scalar and tensor spectral indices.Comment: 4 pages, 1 figur
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