7,263 research outputs found
Persistent and susceptible bacteria with individual deaths
The aim of this paper is to study two models for a bacterial population
subject to antibiotic treatments. It is known that some bacteria are sensitive
to antibiotics. These bacteria are in a state called persistence and each
bacterium can switch from this state to a non-persistent (or susceptible) state
and back. Our models extend those introduced in [6] by adding a (random)
natural life cycle for each bacterium and by allowing bacteria in the
susceptible state to escape the action of the antibiotics with a fixed
probability 1-p (while every bacterium in a persistent state survives with
probability 1). In the first model we "inject" the antibiotics in the system at
fixed, deterministic times while in the second one the time intervals are
random. We show that, in order to kill eventually the whole bacterial
population, these time intervals cannot be "too large". The maximum admissible
length is increasing with respect to p and it decreases rapidly when p<1.Comment: 14 pages, 5 figures, corrected some misprint
On some properties of transitions operators
We study a general transition operator, generated by a random walk on a graph
; in particular we give necessary and sufficient condition on the matrix
coefficient (1-step transition probablilities) to be a bounded operator from
into itself. Moreover we characterize compact operators and we
relate this property to the behaviour of the associated random walk. We give a
necessary and sufficient condition for the pre-adjoint of the discrete Laplace
operator to be an injective map.Comment: 9 page
A mathematical model for the atomic clock error in case of jumps
We extend the mathematical model based on stochastic differential equations
describing the error gained by an atomic clock to the cases of anomalous
behavior including jumps and an increase of instability. We prove an exact
iterative solution that can be useful for clock simulation, prediction, and
interpretation, as well as for the understanding of the impact of clock error
in the overall system in which clocks may be inserted as, for example, the
Global Satellite Navigation Systems
Strong local survival of branching random walks is not monotone
The aim of this paper is the study of the strong local survival property for
discrete-time and continuous-time branching random walks. We study this
property by means of an infinite dimensional generating function G and a
maximum principle which, we prove, is satisfied by every fixed point of G. We
give results about the existence of a strong local survival regime and we prove
that, unlike local and global survival, in continuous time, strong local
survival is not a monotone property in the general case (though it is monotone
if the branching random walk is quasi transitive). We provide an example of an
irreducible branching random walk where the strong local property depends on
the starting site of the process. By means of other counterexamples we show
that the existence of a pure global phase is not equivalent to nonamenability
of the process, and that even an irreducible branching random walk with the
same branching law at each site may exhibit non-strong local survival. Finally
we show that the generating function of a irreducible BRW can have more than
two fixed points; this disproves a previously known result.Comment: 19 pages. The paper has been deeply reorganized and two pictures have
been added. arXiv admin note: substantial text overlap with arXiv:1104.508
Branching random walks and multi-type contact-processes on the percolation cluster of
In this paper we prove that, under the assumption of quasi-transitivity, if a
branching random walk on survives locally (at arbitrarily
large times there are individuals alive at the origin), then so does the same
process when restricted to the infinite percolation cluster
of a supercritical Bernoulli percolation. When no
more than individuals per site are allowed, we obtain the -type contact
process, which can be derived from the branching random walk by killing all
particles that are born at a site where already individuals are present. We
prove that local survival of the branching random walk on
also implies that for sufficiently large the associated -type contact
process survives on . This implies that the strong
critical parameters of the branching random walk on and on
coincide and that their common value is the limit of
the sequence of strong critical parameters of the associated -type contact
processes. These results are extended to a family of restrained branching
random walks, that is, branching random walks where the success of the
reproduction trials decreases with the size of the population in the target
site.Comment: Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Rumor processes in random environment on N and on Galton-Watson trees
The aim of this paper is to study rumor processes in random environment. In a
rumor process a signal starts from the stations of a fixed vertex (the root)
and travels on a graph from vertex to vertex. We consider two rumor processes.
In the firework process each station, when reached by the signal, transmits it
up to a random distance. In the reverse firework process, on the other hand,
stations do not send any signal but they "listen" for it up to a random
distance. The first random environment that we consider is the deterministic
1-dimensional tree N with a random number of stations on each vertex; in this
case the root is the origin of N. We give conditions for the
survival/extinction on almost every realization of the sequence of stations.
Later on, we study the processes on Galton-Watson trees with random number of
stations on each vertex. We show that if the probability of survival is
positive, then there is survival on almost every realization of the infinite
tree such that there is at least one station at the root. We characterize the
survival of the process in some cases and we give sufficient conditions for
survival/extinction.Comment: 28 page
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