2,430 research outputs found
Multiperiodicity, modulations and flip-flops in variable star light curves I. Carrier fit method
The light curves of variable stars are commonly described using simple
trigonometric models, that make use of the assumption that the model parameters
are constant in time. This assumption, however, is often violated, and
consequently, time series models with components that vary slowly in time are
of great interest. In this paper we introduce a class of data analysis and
visualization methods which can be applied in many different contexts of
variable star research, for example spotted stars, variables showing the
Blazhko effect, and the spin-down of rapid rotators. The methods proposed are
of explorative type, and can be of significant aid when performing a more
thorough data analysis and interpretation with a more conventional method.Our
methods are based on a straightforward decomposition of the input time series
into a fast "clocking" periodicity and smooth modulating curves. The fast
frequency, referred to as the carrier frequency, can be obtained from earlier
observations (for instance in the case of photometric data the period can be
obtained from independently measured radial velocities), postulated using some
simple physical principles (Keplerian rotation laws in accretion disks), or
estimated from the data as a certain mean frequency. The smooth modulating
curves are described by trigonometric polynomials or splines. The data
approximation procedures are based on standard computational packages
implementing simple or constrained least-squares fit-type algorithms.Comment: 14 pages, 23 figures, submitted to Astronomy and Astrophysic
How many zeros of a random polynomial are real?
We provide an elementary geometric derivation of the Kac integral formula for
the expected number of real zeros of a random polynomial with independent
standard normally distributed coefficients. We show that the expected number of
real zeros is simply the length of the moment curve
projected onto the surface of the unit sphere, divided by . The
probability density of the real zeros is proportional to how fast this curve is
traced out. We then relax Kac's assumptions by considering a variety of random
sums, series, and distributions, and we also illustrate such ideas as integral
geometry and the Fubini-Study metric.Comment: 37 page
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