3,285 research outputs found
Exploring the tree of numerical semigroups
In this paper we describe an algorithm visiting all numerical semigroups up
to a given genus using a well suited representation. The interest of this
algorithm is that it fits particularly well the architecture of modern
computers allowing very large optimizations: we obtain the number of numerical
semigroups of genus g 67 and we confirm the Wilf conjecture for g 60.Comment: 14 page
Differential equation approximations of stochastic network processes: an operator semigroup approach
The rigorous linking of exact stochastic models to mean-field approximations
is studied. Starting from the differential equation point of view the
stochastic model is identified by its Kolmogorov equations, which is a system
of linear ODEs that depends on the state space size () and can be written as
. Our results rely on the convergence of the transition
matrices to an operator . This convergence also implies that the
solutions converge to the solution of . The limiting ODE
can be easily used to derive simpler mean-field-type models such that the
moments of the stochastic process will converge uniformly to the solution of
appropriately chosen mean-field equations. A bi-product of this method is the
proof that the rate of convergence is . In addition, it turns
out that the proof holds for cases that are slightly more general than the
usual density dependent one. Moreover, for Markov chains where the transition
rates satisfy some sign conditions, a new approach for proving convergence to
the mean-field limit is proposed. The starting point in this case is the
derivation of a countable system of ordinary differential equations for all the
moments. This is followed by the proof of a perturbation theorem for this
infinite system, which in turn leads to an estimate for the difference between
the moments and the corresponding quantities derived from the solution of the
mean-field ODE
Semigroup approach to diffusion and transport problems on networks
Models describing transport and diffusion processes occurring along the edges
of a graph and interlinked by its vertices have been recently receiving a
considerable attention. In this paper we generalize such models and consider a
network of transport or diffusion operators defined on one dimensional domains
and connected through boundary conditions linking the end-points of these
domains in an arbitrary way (not necessarily as the edges of a graph are
connected). We prove the existence of -semigroups solving such problems
and provide conditions fully characterizing when they are positive
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