Related papers: Systematic approximations for the period of a fini…
A simple approximation formula is derived here for the dependence of the period of a simple pendulum on amplitude that only requires a pocket calculator and furnishes an error of less than 0.25% with respect to the exact period. It is shown…
A heuristic but pedagogical derivation is given of an explicit formula which accurately reproduces the period of a simple pendulum even for large amplitudes. The formula is compared with others in the literature.
In this paper we deal with the care one must have in adopting approximations in regard with terms he chooses to leave behind in the particular case of the expression valid for the maximum period of a long pendulum oscillating near Earth's…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…
An effective radius of convergence is defined and computed for any truncated Taylor series. Applications to well known series are performed and is shown that a range of good coincidence for actual and approximative plot can always be found.…
Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational…
Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude…
Truncated sum rules have been used to calculate the fundamental limits of the nonlinear susceptibilities; and, the results have been consistent with all measured molecules. However, given that finite-state models result in inconsistencies…
We find convergent double series expansions for Legendre's third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighbourhood of the logarithmic…
An exact series representation of the even frequency moments of the dynamic structure factor is derived. Truncations are proposed that allow to evaluate the explicitly unknown second, fourth and fifth frequency moments for the finite…
For random maps, the expected value of the order (i.e. the period of the sequence of compositional iterates) is approximated asymptotically. It is much smaller than the expected value for the product of the cycle lengths.
We present new tight bounds for averaging differential inclusions, which we apply to multi-frequency inclusions consisting of a sum of time periodic set-valued mappings. For this family of inclusions we establish an a tight estimate of…
The harmonizable Piranashvili-type stochastic processes are approximated by finite time shifted average sampling sums. Explicit truncation error upper bounds are established. Various corollaries and special cases are discussed.
The Carleman linearization is one of the mainstream approaches to lift a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system with the promise of providing accurate approximations of the original…
We present an analysis of the adiabatic approximation to understand when it applies, in view of the recent criticisms and studies for the validity of the adiabatic theorem. We point out that this approximation is just the leading order of a…
In truncated partial-wave analysis, one fits observables that are bilinear in the amplitudes rather than the amplitudes themselves. Truncation is therefore not merely a restriction of the amplitude basis, but of the bilinear interference…
For a sample of Exponentially distributed durations we aim at point estimation and a confidence interval for its parameter. A duration is only observed if it has ended within a certain time interval, determined by a Uniform distribution.…
One aspect of Poisson approximation is that the support of the random variable of interest is often finite while the support of the Poisson distribution is not. In this paper we will remedy this by examining truncated negative binomial (of…
We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en,…
Context: Several approaches to estimate frequency, phase and amplitude errors in time series analyses were reported in the literature, but they are either time consuming to compute, grossly overestimating the error, or are based on…