1,403 research outputs found
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints
Regularization of ill-posed linear inverse problems via penalization
has been proposed for cases where the solution is known to be (almost) sparse.
One way to obtain the minimizer of such an penalized functional is via
an iterative soft-thresholding algorithm. We propose an alternative
implementation to -constraints, using a gradient method, with
projection on -balls. The corresponding algorithm uses again iterative
soft-thresholding, now with a variable thresholding parameter. We also propose
accelerated versions of this iterative method, using ingredients of the
(linear) steepest descent method. We prove convergence in norm for one of these
projected gradient methods, without and with acceleration.Comment: 24 pages, 5 figures. v2: added reference, some amendments, 27 page
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
Dynamics of the Tippe Top via Routhian Reduction
We consider a tippe top modeled as an eccentric sphere, spinning on a
horizontal table and subject to a sliding friction. Ignoring translational
effects, we show that the system is reducible using a Routhian reduction
technique. The reduced system is a two dimensional system of second order
differential equations, that allows an elegant and compact way to retrieve the
classification of tippe tops in six groups as proposed in [1] according to the
existence and stability type of the steady states.Comment: 16 pages, 7 figures, added reference. Typos corrected and a forgotten
term in de linearized system is adde
The road to deterministic matrices with the restricted isometry property
The restricted isometry property (RIP) is a well-known matrix condition that
provides state-of-the-art reconstruction guarantees for compressed sensing.
While random matrices are known to satisfy this property with high probability,
deterministic constructions have found less success. In this paper, we consider
various techniques for demonstrating RIP deterministically, some popular and
some novel, and we evaluate their performance. In evaluating some techniques,
we apply random matrix theory and inadvertently find a simple alternative proof
that certain random matrices are RIP. Later, we propose a particular class of
matrices as candidates for being RIP, namely, equiangular tight frames (ETFs).
Using the known correspondence between real ETFs and strongly regular graphs,
we investigate certain combinatorial implications of a real ETF being RIP.
Specifically, we give probabilistic intuition for a new bound on the clique
number of Paley graphs of prime order, and we conjecture that the corresponding
ETFs are RIP in a manner similar to random matrices.Comment: 24 page
Kolmogorov-Sinai entropy from recurrence times
Observing how long a dynamical system takes to return to some state is one of
the most simple ways to model and quantify its dynamics from data series. This
work proposes two formulas to estimate the KS entropy and a lower bound of it,
a sort of Shannon's entropy per unit of time, from the recurrence times of
chaotic systems. One formula provides the KS entropy and is more theoretically
oriented since one has to measure also the low probable very long returns. The
other provides a lower bound for the KS entropy and is more experimentally
oriented since one has to measure only the high probable short returns. These
formulas are a consequence of the fact that the series of returns do contain
the same information of the trajectory that generated it. That suggests that
recurrence times might be valuable when making models of complex systems
The effect of dietary calcium inclusion on broiler gastrointestinal pH: quantification and method optimization
There is little consensus as to the most appropriate methodology for the measurement of gastrointestinal pH in chickens. An experiment was conducted to establish the optimum sampling method for the determination of broiler digesta pH in birds fed differing levels of dietary calcium. Ross 308 broilers (n = 60) were fed one of two experimental diets, one containing 0.8% monocalcium phosphate and 2% limestone and one containing 0.4% monocalcium phosphate and 1% limestone. Four factors were investigated to determine the most appropriate method of measuring broiler gastrointestinal digesta pH: removal from the tract, prolonged air exposure, altering the temperature of the assay, and controlling the water content of the digesta. The conditions were assessed at bird ages from 7 to 42 d post hatch. Dietary Ca content had no significant effect on in situ pH, but it contributed towards variance in ex situ pH of both gizzard and duodenum digesta
Adverse events associated with peanut oral immunotherapy in children – a systematic review and meta-analysis
While peanut oral immunotherapy (POIT) represents a promising treatment for peanut allergies in children, safety concerns remain a common barrier to widespread adoption. We aimed to systematically assess available evidence to determine the risk and frequency of adverse events occurring during POIT, and examine study-level characteristics associated with their occurrence and severity. A systematic search of MEDLINE, EMBASE, and Web of Science was conducted through April 2019. Controlled and non-controlled studies evaluating POIT were eligible. Twenty-seven studies, involving 1488 subjects, were included. Adverse events to POIT were common and led to treatment discontinuation in 6.6% of children (95% CI 4.4–9.0; 27 studies, I² = 48.7%). Adverse events requiring treatment with epinephrine occurred among 7.6% (4.5–11.4; 26 studies, I² = 75.5%) of participants, at a rate of 2.0 per 10,000 doses (0.8–3.7; 15 studies, I² = 64.4). Use of a rush treatment phase and targeting a higher maintenance dose were associated with a higher risk and frequency of epinephrine use, while using co-treatments in addition to POIT was associated with a lower risk of treatment discontinuation due to adverse events. While adverse events to POIT are common, this study provides promising explorative evidence that certain modifications to existing treatment protocols could significantly improve treatment outcomes.Luke E. Grzeskowiak, Billy Tao, Emma Knight, Sarah Cohen-Woods, Timothy Chatawa
How much time does a measurement take?
We consider the problem of measurement using the Lindblad equation, which
allows the introduction of time in the interaction between the measured system
and the measurement apparatus. We use analytic results, valid for weak
system-environment coupling, obtained for a two-level system in contact with a
measurer (Markovian interaction) and a thermal bath (non-Markovian
interaction), where the measured observable may or may not commute with the
system-environment interaction. Analysing the behavior of the coherence, which
tends to a value asymptotically close to zero, we obtain an expression for the
time of measurement which depends only on the system-measurer coupling, and
which does not depend on whether the observable commutes with the system-bath
interaction. The behavior of the coherences in the case of strong
system-environment coupling, found numerically, indicates that an increase in
this coupling decreases the measurement time, thus allowing our expression to
be considered the upper limit for the duration of the process.Comment: REVISED VERSION: 17 pages, 2 figure
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