499 research outputs found
Intrinsic circle domains
Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain
Ω
\Omega
in a compact Riemann surface
S
S
. This means that each connected component
B
B
of
S
∖
Ω
S\setminus \Omega
is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface
(
Ω
∪
B
)
(\Omega \cup B)
. Moreover, the pair
(
Ω
,
S
)
(\Omega , S)
is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally, we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.</p
Conical limit sets and continued fractions
Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, S2. By identifying S2 with the boundary of three-dimensional hyperbolic space, H3, we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H3. Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets; for example, it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions
Functional Large Deviations for Cox Processes and Queues, with a Biological Application
We consider an infinite-server queue into which customers arrive according to
a Cox process and have independent service times with a general distribution.
We prove a functional large deviations principle for the equilibrium queue
length process. The model is motivated by a linear feed-forward gene regulatory
network, in which the rate of protein synthesis is modulated by the number of
RNA molecules present in a cell. The system can be modelled as a tandem of
infinite-server queues, in which the number of customers present in a queue
modulates the arrival rate into the next queue in the tandem. We establish
large deviation principles for this queueing system in the asymptotic regime in
which the arrival process is sped up, while the service process is not scaled.Comment: 36 pages, 2 figures, to appear in Annals of Applied Probabilit
Cluster growth in the dynamical Erd\H{o}s-R\'{e}nyi process with forest fires
We investigate the growth of clusters within the forest fire model of
R\'{a}th and T\'{o}th [22]. The model is a continuous-time Markov process,
similar to the dynamical Erd\H{o}s-R\'{e}nyi random graph but with the addition
of so-called fires. A vertex may catch fire at any moment and, when it does so,
causes all edges within its connected cluster to burn, meaning that they
instantaneously disappear. Each burned edge may later reappear.
We give a precise description of the process of the size of the cluster
of a tagged vertex, in the limit as the number of vertices in the model tends
to infinity. We show that is an explosive branching process with a
time-inhomogeneous offspring distribution and instantaneous return to on
each explosion. Additionally, we show that the characteristic curves used to
analyse the Smoluchowski-type coagulation equations associated to the model
have a probabilistic interpretation in terms of the process .Comment: 31 page
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