1,209 research outputs found
Multiply connected wandering domains of entire functions
The dynamical behaviour of a transcendental entire function in any periodic
component of the Fatou set is well understood. Here we study the dynamical
behaviour of a transcendental entire function in any multiply connected
wandering domain of . By introducing a certain positive harmonic
function in , related to harmonic measure, we are able to give the first
detailed description of this dynamical behaviour. Using this new technique, we
show that, for sufficiently large , the image domains contain
large annuli, , and that the union of these annuli acts as an absorbing
set for the iterates of in . Moreover, behaves like a monomial
within each of these annuli and the orbits of points in settle in the long
term at particular `levels' within the annuli, determined by the function .
We also discuss the proximity of and for large
, and the connectivity properties of the components of . These properties are deduced from new results about the behaviour
of an entire function which omits certain values in an annulus
Dynamics of meromorphic functions with direct or logarithmic singularities
We show that if a meromorphic function has a direct singularity over
infinity, then the escaping set has an unbounded component and the intersection
of the escaping set with the Julia set contains continua. This intersection has
an unbounded component if and only if the function has no Baker wandering
domains. We also give estimates of the Hausdorff dimension and the upper box
dimension of the Julia set of a meromorphic function with a logarithmic
singularity over infinity. The above theorems are deduced from more general
results concerning functions which have "direct or logarithmic tracts", but
which need not be meromorphic in the plane. These results are obtained by using
a generalization of Wiman-Valiron theory. The method is also applied to complex
differential equations.Comment: 29 pages, 2 figures; v2: some overall revision, with comments and
references added; to appear in Proc. London Math. So
A statistical evaluation of the effects of gender differences in assessment of acute inhalation toxicity
Acute inhalation toxicity of chemicals has conventionally been assessed by the median lethal concentration
(LC50) test (organisation for economic co-operation and development (OECD) TG 403). Two new methods,
the recently adopted acute toxic class method (ATC; OECD TG 436) and a proposed fixed concentration procedure
(FCP), have recently been considered, but statistical evaluations of these methods did not investigate
the influence of differential sensitivity between male and female rats on the outcomes. This paper presents an
analysis of data from the assessment of acute inhalation toxicity for 56 substances. Statistically significant differences
between the LC50 for males and females were found for 16 substances, with greater than 10-fold differences
in the LC50 for two substances. The paper also reports a statistical evaluation of the three test
methods in the presence of unanticipated gender differences. With TG 403, a gender difference leads to a
slightly greater chance of under-classification. This is also the case for the ATC method, but more pronounced
than for TG 403, with misclassification of nearly all substances from Globally Harmonised System (GHS) class 3
into class 4. As the FCP uses females only, if females are more sensitive, the classification is unchanged. If males
are more sensitive, the procedure may lead to under-classification. Additional research on modification of the
FCP is thus proposed
Connectedness properties of the set where the iterates of an entire function are unbounded
We investigate the connectedness properties of the set I+(f) of points where the iterates of an entire function f are unbounded. In particular, we show that I+(f) is connected whenever iterates of the minimum modulus of f tend to ∞. For a general transcendental entire function f, we show that I+(f)∪ \{\infty\} is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded
Seasonal Variability In The Ionosphere Of Uranus
Infrared ground-based observations using IRTF, UKIRT, and Keck II of Uranus have been analyzed as to identify the long-term behavior of the H-3(+) ionosphere. Between 1992 and 2008 there are 11 individual observing runs, each recording emission from the H-3(+) Q branch emission around 4 mu m through the telluric L' atmospheric window. The column-averaged rotational H-3(+) temperature ranges between 715 K in 1992 and 534 K in 2008, with the linear fit to all the run-averaged temperatures decreasing by 8 K year(-1). The temperature follows the fractional illumination curve of the planet, declining from solstice (1985) to equinox (2007). Variations in H-3(+) column density do not appear to be correlated to either solar cycle phase or season. The radiative cooling by H-3(+) is similar to 10 times larger than the ultraviolet solar energy being injected to the atmosphere. Despite the fact that the solar flux alone is incapable of heating the atmosphere to the observed temperatures, the geometry with respect to the Sun remains an important driver in determining the thermospheric temperature. Therefore, the energy source that heats the thermosphere must be linked to solar mechanisms. We suggest that this may be in the form of conductivity created by solar ionization of atmospheric neutrals and/or seasonally dependent magnetospherically driven current systems.STFC PP/E/000983/1, ST/G0022223/1RCUKGemini ObservatoryNational Aeronautics and Space Administration (NASA) NXX08A043G, NNX08AE38AAstronom
On multiply connected wandering domains of meromorphic functions
We describe conditions under which a multiply connected wandering domain of a
transcendental meromorphic function with a finite number of poles must be a
Baker wandering domain, and we discuss the possible eventual connectivity of
Fatou components of transcendental meromorphic functions. We also show that if
is meromorphic, is a bounded component of and is the
component of such that , then maps each component of
onto a component of the boundary of in \hat{\C}. We give
examples which show that our results are sharp; for example, we prove that a
multiply connected wandering domain can map to a simply connected wandering
domain, and vice versa.Comment: 18 pages. To be published in the Journal of the London Mathematical
Societ
Functions of small growth with no unbounded Fatou components
We prove a form of the theorem which gives strong estimates
for the minimum modulus of a transcendental entire function of order zero. We
also prove a generalisation of a result of Hinkkanen that gives a sufficient
condition for a transcendental entire function to have no unbounded Fatou
components. These two results enable us to show that there is a large class of
entire functions of order zero which have no unbounded Fatou components. On the
other hand we give examples which show that there are in fact functions of
order zero which not only fail to satisfy Hinkkanen's condition but also fail
to satisfy our more general condition. We also give a new regularity condition
that is sufficient to ensure that a transcendental entire function of order
less than 1/2 has no unbounded Fatou components. Finally, we observe that all
the conditions given here which guarantee that a transcendental entire function
has no unbounded Fatou components, also guarantee that the escaping set is
connected, thus answering a question of Eremenko for such functions
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