547 research outputs found
Deciphering infant mortality. Part 1: empirical evidence
This paper is not (or at least not only) about human infant mortality. In
line with reliability theory, "infant" will refer here to the time interval
following birth during which the mortality (or failure) rate decreases. This
definition provides a systems science perspective in which birth constitutes a
sudden transition which falls within the field of application of the "Transient
Shock" (TS) conjecture put forward in Richmond et al. (2016c). This conjecture
provides predictions about the timing and shape of the death rate peak. (i) It
says that there will be a death rate spike whenever external conditions change
abruptly and drastically. (ii) It predicts that after a steep rising there will
be a much longer hyperbolic relaxation process. These predictions can be tested
by considering living organisms for which birth is a multi-step process. Thus,
for fish there are three states: egg, yolk-sac phase, young adult. The TS
conjecture predicts a mortality spike at the end of the yolk-sac phase, and
this timing is indeed confirmed by observation. Secondly, the hyperbolic nature
of the relaxation process can be tested using high accuracy Swiss statistics
which give postnatal death rates from one hour after birth up to the age of 10
years. It turns out that since the 19th century despite a great overall
reduction in infant mortality, the shape of the age-specific death rate has
remained basically unchanged. This hyperbolic pattern is not specific to
humans. It can also be found in small primates as recorded in the archives of
zoological gardens. Our ultimate objective is to set up a chain of cases which
starts from simple systems and then moves up step by step to more complex
organisms. The cases discussed here can be seen as initial landmarks.Comment: 46 pages, 14 figures, 4 table
First applications of a formula for the error of finite sinc interpolation
In former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval. In the present work we demonstrate the relevance of the formula through several applications: correction of the interpolant through the insertion of derivatives to increase its order of convergence, improvement of the barycentric formula, rational sinc interpolants (with and without replacement of the (usually unknown) derivatives with finite differences), convergence acceleration through extrapolation and improvement of one-sided interpolant
A formula for the error of finite sinc-interpolation over a finite interval
Sinc-interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. It, however, requires that the interpolated function decreases rapidly or is periodic. We give an error formula for the case where neither of these conditions is satisfie
A formula for the error of finite sinc interpolation with an even number of nodes
Sinc interpolation is a very efficient infinitely differentiable approximation scheme from equidistant data on the infinite line. We give a formula for the error committed when the function neither decreases rapidly nor is periodic, so that the sinc series must be truncated for practical purposes. To do so, we first complete a previous result for an odd number of points, before deriving a formula for the more involved case of an even number of point
A formula for optimal integration in H2
AbstractThe weights âj of the optimal integration formula Q̂ = Σjâjf(zj) in H2 for given integration points zj are the exact integrals of the cardinal functions in the corresponding formula for optimal evaluation. By writing these cardinal functions as sums of their principal values, we very easily obtain a closed formula for the weights. In the case of real zj's, this formula makes explicit a series formula of Wilf. We compare numerically the accuracy of the optimal formula with that of some well-known integration formulae. For points equidistant on a circle of radius r, the formula allows an alternate derivation of a formula obtained by Golomb. We give also the barycentric formula for optimal evaluation with these points, as well as an experimentally stable sequence of radii r for integrating with an increasing number of points
A circular interpretation of the Euler–Maclaurin formula
The present work makes the case for viewing the Euler–Maclaurin formula as an expression for the effect of a jump on the accuracy of Riemann sums on circles and draws some consequences thereof, e.g., when the integrand has several jumps. On the way we give a construction of the Bernoulli polynomials tailored to the proof of the formula and we show how extra jumps may lead to a smaller quadrature error
Linear rational finite differences from derivatives of barycentric rational interpolants
Derivatives of polynomial interpolants lead in a natural way to approximations of derivatives of the interpolated function, e.g., through finite differences. We extend a study of the approximation of derivatives of linear barycentric rational interpolants and present improved finite difference formulas arising from these interpolants. The formulas contain the classical finite differences as a special case and are more stable for calculating one-sided derivatives as well as derivatives close to boundaries
Linear barycentric rational quadrature
Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical test
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