44,435 research outputs found
Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function
This paper revisits a recently developed methodology based on the matrix
Lambert W function for the stability analysis of linear time invariant, time
delay systems. By studying a particular, yet common, second order system, we
show that in general there is no one to one correspondence between the branches
of the matrix Lambert W function and the characteristic roots of the system.
Furthermore, it is shown that under mild conditions only two branches suffice
to find the complete spectrum of the system, and that the principal branch can
be used to find several roots, and not the dominant root only, as stated in
previous works. The results are first presented analytically, and then verified
by numerical experiments
Analytical results for a Fokker-Planck equation in the small noise limit
We present analytical results for the lowest cumulants of a stochastic
process described by a Fokker-Planck equation with nonlinear drift. We show
that, in the limit of small fluctuations, the mean, the variance and the
covariance of the process can be expressed in compact form with the help of the
Lambert W function. As an application, we discuss the interplay of noise and
nonlinearity far from equilibrium.Comment: 5 pages, 4 figure
Electromagnetic surface wave propagation in a metallic wire and the Lambert function
We revisit the solution due to Sommerfeld of a problem in classical
electrodynamics, namely, that of the propagation of an electromagnetic axially
symmetric surface wave (a low-attenuation single TM mode) in a
cylindrical metallic wire, and his iterative method to solve the transcendental
equation that appears in the determination of the propagation wave number from
the boundary conditions. We present an elementary analysis of the convergence
of Sommerfeld's iterative solution of the approximate problem and compare it
with both the numerical solution of the exact transcendental equation and the
solution of the approximate problem by means of the Lambert function.Comment: REVTeX double column, 9 pages, 3 figures, minor differences between
v3 and published version; "Editor's Pick" for June 2019 edition of AJ
Quasi-normal frequencies: Key analytic results
The study of exact quasi-normal modes [QNMs], and their associated
quasi-normal frequencies [QNFs], has had a long and convoluted history -
replete with many rediscoveries of previously known results. In this article we
shall collect and survey a number of known analytic results, and develop
several new analytic results - specifically we shall provide several new QNF
results and estimates, in a form amenable for comparison with the extant
literature. Apart from their intrinsic interest, these exact and approximate
results serve as a backdrop and a consistency check on ongoing efforts to find
general model-independent estimates for QNFs, and general model-independent
bounds on transmission probabilities. Our calculations also provide yet another
physics application of the Lambert W function. These ideas have relevance to
fields as diverse as black hole physics, (where they are related to the damped
oscillations of astrophysical black holes, to greybody factors for the Hawking
radiation, and to more speculative state-counting models for the Bekenstein
entropy), to quantum field theory (where they are related to Casimir energies
in unbounded systems), through to condensed matter physics, (where one may
literally be interested in an electron tunelling through a physical barrier).Comment: V1: 29 pages; V2: Reformatted, 31 pages. Title changed to reflect
major additions and revisions. Now describes exact QNFs for the double-delta
potential in terms of the Lambert W function. V3: Minor edits for clarity.
Four references added. No physics changes. Still 31 page
Lambert W random variables - a new family of generalized skewed distributions with applications to risk estimation
Originating from a system theory and an input/output point of view, I
introduce a new class of generalized distributions. A parametric nonlinear
transformation converts a random variable into a so-called Lambert
random variable , which allows a very flexible approach to model skewed
data. Its shape depends on the shape of and a skewness parameter .
In particular, for symmetric and nonzero the output is skewed.
Its distribution and density function are particular variants of their input
counterparts. Maximum likelihood and method of moments estimators are
presented, and simulations show that in the symmetric case additional
estimation of does not affect the quality of other parameter
estimates. Applications in finance and biomedicine show the relevance of this
class of distributions, which is particularly useful for slightly skewed data.
A practical by-result of the Lambert framework: data can be "unskewed." The
package http://cran.r-project.org/web/packages/LambertWLambertW developed
by the author is publicly available (http://cran.r-project.orgCRAN).Comment: Published in at http://dx.doi.org/10.1214/11-AOAS457 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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