36,907 research outputs found
Mean Rational Approximation for Some Compact Planar Subsets
In 1991, J. Thomson obtained celebrated structural results for
Later, J. Brennan (2008) generalized Thomson's theorem to when the
diameters of the components of are bounded below. The
results indicate that if is pure, then is the "same as" the algebra of bounded analytic functions on
\mbox{abpe}(R^t(K, \mu)), the set of analytic bounded point evaluations. We
show that if the diameters of the components of are
allowed to tend to zero, then even though \text{int}(K) = \mbox{abpe}(R^t(K,
\mu)) and the algebra may "be equal to" a proper sub-algebra of bounded analytic
functions on where functions in the sub-algebra are
"continuous" on certain portions of the inner boundary of Comment: arXiv admin note: text overlap with arXiv:1904.0644
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
Mean Rational Approximation for Compact Subsets with Thin Boundaries
In 1991, J. Thomson obtained a celebrated decomposition theorem for
the closed subspace of spanned by the analytic
polynomials, when 1 \le t < \i. In 2008, J. Brennan \cite{b08} generalized
Thomson's theorem to the closed subspace of spanned
by the rational functions with poles off a compact subset containing the
support of when the diameters of the components of are bounded below. We obtain a necessary and sufficient condition for
to ensure such a decomposition theorem holdsComment: arXiv admin note: text overlap with arXiv:1904.06446,
arXiv:2212.0539
The Commutant of Multiplication by z on the Closure of Rational Functions in
For a compact set a finite positive Borel measure
on and 1 \le t < \i, let be the set of rational
functions with poles off and let be the closure of
in For a bounded Borel subset let \area_{\mathcal D} denote the area (Lebesgue) measure
restricted to and let H^\i (\mathcal D) be the weak-star closed
sub-algebra of L^\i(\area_{\mathcal D}) spanned by bounded and analytic
on for some compact subset We show that if contains no non-trivial
direct summands, then there exists a Borel subset
whose closure contains the support of and there exists an isometric
isomorphism and a weak-star homeomorphism from onto such that for all
Consequently, we obtain some structural decomposition
theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081
Precise tail asymptotics of fixed points of the smoothing transform with general weights
We consider solutions of the stochastic equation ,
where is a fixed constant, are independent, identically distributed
random variables and are independent copies of , which are independent
both from 's and . The hypotheses ensuring existence of solutions are
well known. Moreover under a number of assumptions the main being
and , the
limit exists. In the present
paper, we prove positivity of .Comment: Published at http://dx.doi.org/10.3150/13-BEJ576 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The short-time behavior of kinetic spherical model with long-ranged interactions
The kinetic spherical model with long-ranged interactions and an arbitrary
initial order m_{0} quenched from a very high temperature to T < T_{c} is
solved. In the short-time regime, the bulk order increases with a power law in
both the critical and phase-ordering dynamics. To the latter dynamics, a power
law for the relative order m_{r} ~ -t^{-k} is found in the intermediate
time-regime. The short-time scaling relation of small m_{0} are generalized to
an arbitrary m_{0} and all the time larger than t_{mic}. The characteristic
functions for the scaling of m_{0} and for
T'=T/T_{c} are obtained. The crossover between scaling regimes is discussed in
detail.Comment: 22 pages, 3 figure
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