2,609 research outputs found
A Lost Theorem: Definite Integrals in Asymptotic Setting
We present a simple yet rigorous theory of integration that is based on two
axioms rather than on a construction involving Riemann sums. With several
examples we demonstrate how to set up integrals in applications of calculus
without using Riemann sums. In our axiomatic approach even the proof of the
existence of the definite integral (which does use Riemann sums) becomes
slightly more elegant than the conventional one. We also discuss an interesting
connection between our approach and the history of calculus. The article is
written for readers who teach calculus and its applications. It might be
accessible to students under a teacher's supervision and suitable for senior
projects on calculus, real analysis, or history of mathematics
Subwavelength position sensing using nonlinear feedback and wave chaos
We demonstrate a position-sensing technique that relies on the inherent
sensitivity of chaos, where we illuminate a subwavelength object with a complex
structured radio-frequency field generated using wave chaos and a nonlinear
feedback loop. We operate the system in a quasi-periodic state and analyze
changes in the frequency content of the scalar voltage signal in the feedback
loop. This allows us to extract the object's position with a one-dimensional
resolution of ~\lambda/10,000 and a two-dimensional resolution of ~\lambda/300,
where \lambda\ is the shortest wavelength of the illuminating source.Comment: 4 pages, 4 figure
Logarithmic periodicities in the bifurcations of type-I intermittent chaos
The critical relations for statistical properties on saddle-node bifurcations
are shown to display undulating fine structure, in addition to their known
smooth dependence on the control parameter. A piecewise linear map with the
type-I intermittency is studied and a log-periodic dependence is numerically
obtained for the average time between laminar events, the Lyapunov exponent and
attractor moments. The origin of the oscillations is built in the natural
probabilistic measure of the map and can be traced back to the existence of
logarithmically distributed discrete values of the control parameter giving
Markov partition. Reinjection and noise effect dependences are discussed and
indications are given on how the oscillations are potentially applicable to
complement predictions made with the usual critical exponents, taken from data
in critical phenomena.Comment: 4 pages, 6 figures, accepted for publication in PRL (2004
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