The RKrGLm method is a numerical method for solving initial value problems in ordinary differential equations of the form y@u'=f(x,y) and is based on a combination of a Runge-Kutta method of order r ...and m-point Gauss-Legendre quadrature. In this paper we describe the propagation of local errors in this method, and we give an inductive proof of the form of the global error in RKrGLm. We show that, for a suitable choice of r and m, the global order of RKrGLm is expected to be r+1, one better than the underlying Runge-Kutta method. We show that this gain in order is due to a reduction or ''quenching'' of the accumulated local error at every (m+1)th node. We also show how a Hermite interpolating polynomial of degree 2m+1 may be employed to estimate f(x,y) if the nodes to be used for the Gauss-Legendre quadrature component are not suitably placed.
Context . Super-luminous supernovae (SLSNe) are rare events defined as being significantly more luminous than normal terminal stellar explosions. The source of the additional power needed to achieve ...such luminosities is still unclear. Discoveries in the local Universe (i.e. z < 0.1) are scarce, but afford dense multi-wavelength observations. Additional low-redshift objects are therefore extremely valuable.
Aims . We present early-time observations of the type I SLSN ASASSN-18km/SN 2018bsz. These data are used to characterise the event and compare to literature SLSNe and spectral models. Host galaxy properties are also analysed.
Methods . Optical and near-IR photometry and spectroscopy were analysed. Early-time ATLAS photometry was used to constrain the rising light curve. We identified a number of spectral features in optical-wavelength spectra and track their time evolution. Finally, we used archival host galaxy photometry together with H( II )region spectra to constrain the host environment.
Results . ASASSN-18km/SN 2018bsz is found to be a type I SLSN in a galaxy at a redshift of 0.0267 (111 Mpc), making it the lowest-redshift event discovered to date. Strong C- II lines are identified in the spectra. Spectral models produced by exploding a Wolf-Rayet progenitor and injecting a magnetar power source are shown to be qualitatively similar to ASASSN-18km/SN 2018bsz, contrary to most SLSNe-I that display weak or non-existent C (II) lines. ASASSN-18km/SN 2018bsz displays along, slowly rising, red plateau of >26 days, before a steeper, faster rise to maximum. The host has an absolute magnitude of -19.8 mag (r), a mass of M-* = 1.5(-0.33)(+0.08) x 10(9) M-circle dot, and a star formation rate of =0.50(-0.19)(+2.22) M-circle dot yr(-1). A nearby H (II) region has an oxygen abundance (O3N2) of 8.31 +/- 0.01 dex.
An algorithm for error control (absolute and relative) in the five-point finite-difference method applied to Poisson's equation is described. The algorithm is based on discretization of the domain of ...the problem by means of three rectilinear grids, each of different resolution. We discuss some hardware limitations associated with the algorithm, which are mainly due to its second-order nature. A generalization of the algorithm for finite-difference methods of arbitrary order is presented. We believe that the algorithm is a valuable addition to typical textbook discussions of the five-point finite-difference method for Poisson's equation.
ABSTRACT
We present observations and analysis of 18 stripped-envelope supernovae observed during 2013–2018. This sample consists of five H/He-rich SNe, six H-poor/He-rich SNe, three narrow lined SNe ...Ic, and four broad lined SNe Ic. The peak luminosity and characteristic time-scales of the bolometric light curves are calculated, and the light curves modelled to derive 56Ni and ejecta masses (MNi and Mej). Additionally, the temperature evolution and spectral line velocity curves of each SN are examined. Analysis of the O i line in the nebular phase of eight SNe suggests their progenitors had initial masses <20 M⊙. The bolometric light curve properties are examined in combination with those of other SE events from the literature. The resulting data set gives the Mej distribution for 80 SE–SNe, the largest such sample in the literature to date, and shows that SNe Ib have the lowest median Mej, followed by narrow-lined SNe Ic, H/He-rich SNe, broad-lined SNe Ic, and finally gamma-ray burst SNe. SNe Ic-6/7 show the largest spread of Mej ranging from ∼1.2–11 M⊙, considerably greater than any other subtype. For all SE–SNe = 2.8 ± 1.5 M⊙ which further strengthens the evidence that SE–SNe arise from low-mass progenitors which are typically <5 M⊙ at the time of explosion, again suggesting MZAMS <25 M⊙. The low and lack of clear bimodality in the distribution implies <30 M⊙ progenitors and that envelope stripping via binary interaction is the dominant evolutionary pathway of these SNe.
Calcified plaque in the coronary arteries is a marker for atheromatous-plaque burden and is predictive of future risk of cardiovascular events. We examined the relationship between estrogen therapy ...and coronary-artery calcium in the context of a randomized clinical trial.
In our ancillary substudy of the Women's Health Initiative trial of conjugated equine estrogens (0.625 mg per day) as compared with placebo in women who had undergone hysterectomy, we performed computed tomography of the heart in 1064 women aged 50 to 59 years at randomization. Imaging was conducted at 28 of 40 centers after a mean of 7.4 years of treatment and 1.3 years after the trial was completed (8.7 years after randomization). Coronary-artery calcium (or Agatston) scores were measured at a central reading center without knowledge of randomization status.
The mean coronary-artery calcium score after trial completion was lower among women receiving estrogen (83.1) than among those receiving placebo (123.1) (P=0.02 by rank test). After adjustment for coronary risk factors, the multivariate odds ratios for coronary-artery calcium scores of more than 0, 10 or more, and 100 or more in the group receiving estrogen as compared with placebo were 0.78 (95% confidence interval, 0.58 to 1.04), 0.74 (0.55 to 0.99), and 0.69 (0.48 to 0.98), respectively. The corresponding odds ratios among women with at least 80% adherence to the study estrogen or placebo were 0.64 (P=0.01), 0.55 (P<0.001), and 0.46 (P=0.001). For coronary-artery calcium scores of more than 300 (vs. <10), the multivariate odds ratio was 0.58 (P=0.03) in an intention-to-treat analysis and 0.39 (P=0.004) among women with at least 80% adherence.
Among women 50 to 59 years old at enrollment, the calcified-plaque burden in the coronary arteries after trial completion was lower in women assigned to estrogen than in those assigned to placebo. However, estrogen has complex biologic effects and may influence the risk of cardiovascular events and other outcomes through multiple pathways. (ClinicalTrials.gov number, NCT00000611.)
To assess the suitability of the
13C-octanoic acid breath test for measuring gastric emptying in circumstances other than the post-absorptive state, a preliminary study was performed where 6 hourly ...spaced isoenergetic meals preceded the determination of gastric emptying of a subsequent 2
MJ meal. Emptying was measured in three individuals on four separate occasions, with a reproducibility of 8%.
A crossover study was then conducted to test the hypothesis that meal frequency can modulate the gastric emptying of a subsequent meal, with the potential to influence appetite regulation. Sixteen subjects were fed to energy balance, receiving food either as 2 isoenergetic meals 3
h apart or 6 isoenergetic meals fed hourly. Gastric emptying of a subsequent 2
MJ meal was investigated. Visual analogue scales were used throughout to assess appetite.
The maximum rate of gastric emptying was unchanged but the onset of emptying was delayed by the more frequent feeding pattern. There was no significant difference in subjective appetite before or after the test meal.
In conclusion, short-term increases in feeding frequency delayed the gastric emptying of a subsequent meal, but significant effects on post-meal appetite could not be demonstrated.
We describe composite Gauss-Legendre quadrature for determining definite integrals, including a means of controlling the approximation error. We compare the form and performance of the algorithm with ...standard Newton-Cotes quadrature.
A simple nonstiff linear initial-value problem is used to demonstrate the amplification of round-off error in the course of using a second-order Runge-Kutta method. This amplification is understood ...in terms of an appropriate expression for the global error. An implicit method is then used to show how the roundoff error may actually be suppressed. This article describes a phenomenon not usually considered in general purpose textbooks on numerical analysis, and is therefore potentially interesting and useful to students of the subject.
We introduce the RKGL method for the numerical solution of initial-value problems of the form
y
′
=
f
(
x
,
y
)
,
y
(
a
)
=
α
. The method is a straightforward modification of a classical explicit ...Runge–Kutta (RK) method, into which Gauss–Legendre (GL) quadrature has been incorporated. The idea is to enhance the efficiency of the method by reducing the number of times the derivative
f
(
x
,
y
)
needs to be computed. The incorporation of GL quadrature serves to enhance the global order of the method by, relative to the underlying RK method. Indeed, the RKGL method has a global error of the form
Ah
r
+
1
+
Bh
2
m
, where
r is the order of the RK method and
m is the number of nodes used in the GL component. In this paper we derive this error expression and show that RKGL is consistent, convergent and strongly stable.
We describe an algorithm, based on Euler's method, for solving Volterra integro-differential equations. The algorithm approximates the relevant integral by means of the composite Trapezium Rule, ...using the discrete nodes of the independent variable as the required nodes for the integration variable. We have developed an error control device, using Richardson extrapolation, and we have achieved accuracy better than 1e-12 for all numerical examples considered.
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