132,556 research outputs found
The geometric mean of two matrices from a computational viewpoint
The geometric mean of two matrices is considered and analyzed from a
computational viewpoint. Some useful theoretical properties are derived and an
analysis of the conditioning is performed. Several numerical algorithms based
on different properties and representation of the geometric mean are discussed
and analyzed and it is shown that most of them can be classified in terms of
the rational approximations of the inverse square root functions. A review of
the relevant applications is given
Continued Fractions and Linear Fractional Transformations
Rational approximations to a square root can be produced by
iterating the transformation starting from for
any positive integer . We show that these approximations coincide infinitely
often with continued fraction convergents if and only if is an
integer, in which case the continued fraction has a rich structure. It consists
of the concatenation of the continued fractions of certain explicitly definable
rational numbers, and it belongs to one of infinitely many families of
continued fractions whose terms vary linearly in two parameters. We also give
conditions under which the orbit consists exclusively of
convergents or semiconvergents and prove that with few exceptions it includes
all solutions to the Pell equation .Comment: 18 page
Fisher waves in the strong noise limit
We investigate the effects of strong number fluctuations on traveling waves
in the Fisher-Kolmogorov reaction-diffusion system. Our findings are in stark
contrast to the commonly used deterministic and weak-noise approximations. We
compute the wave velocity in one and two spatial dimensions, for which we find
a linear and a square-root dependence of the speed on the particle density.
Instead of smooth sigmoidal wave profiles, we observe fronts composed of a few
rugged kinks that diffuse, annihilate, and rarely branch; this dynamics leads
to power-law tails in the distribution of the front sizes.Comment: 4 pages, 2 figures, updat
Boundary effect of a partition in a quantum well
The paper wishes to demonstrate that, in quantum systems with boundaries,
different boundary conditions can lead to remarkably different physical
behaviour. Our seemingly innocent setting is a one dimensional potential well
that is divided into two halves by a thin separating wall. The two half wells
are populated by the same type and number of particles and are kept at the same
temperature. The only difference is in the boundary condition imposed at the
two sides of the separating wall, which is the Dirichlet condition from the
left and the Neumann condition from the right. The resulting different energy
spectra cause a difference in the quantum statistically emerging pressure on
the two sides. The net force acting on the separating wall proves to be nonzero
at any temperature and, after a weak decrease in the low temperature domain, to
increase and diverge with a square-root-of-temperature asymptotics for high
temperatures. These observations hold for both bosonic and fermionic type
particles, but with quantitative differences. We work out several analytic
approximations to explain these differences and the various aspects of the
found unexpectedly complex picture.Comment: LaTeX (with iopart.cls, iopart10.clo and iopart12.clo), 28 pages, 17
figure
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