12,866 research outputs found
Infinite products of matrices and the Gibbs properties of Bernoulli convolutions
We consider the infinite sequences (A\_n)\_{n\in\NN} of matrices
with nonnegative entries, where the are taken in a finite set of
matrices. Given a vector V=\pmatrix{v\_1\cr v\_2} with , we give
a necessary and sufficient condition for to converge uniformly. In application we prove that the
Bernoulli convolutions related to the numeration in Pisot quadratic bases are
weak Gibbs
Unified bijections for planar hypermaps with general cycle-length constraints
We present a general bijective approach to planar hypermaps with two main
results. First we obtain unified bijections for all classes of maps or
hypermaps defined by face-degree constraints and girth constraints. To any such
class we associate bijectively a class of plane trees characterized by local
constraints. This unifies and greatly generalizes several bijections for maps
and hypermaps. Second, we present yet another level of generalization of the
bijective approach by considering classes of maps with non-uniform girth
constraints. More precisely, we consider "well-charged maps", which are maps
with an assignment of "charges" (real numbers) on vertices and faces, with the
constraints that the length of any cycle of the map is at least equal to the
sum of the charges of the vertices and faces enclosed by the cycle. We obtain a
bijection between charged hypermaps and a class of plane trees characterized by
local constraints
Unified bijections for maps with prescribed degrees and girth
This article presents unified bijective constructions for planar maps, with
control on the face degrees and on the girth. Recall that the girth is the
length of the smallest cycle, so that maps of girth at least are
respectively the general, loopless, and simple maps. For each positive integer
, we obtain a bijection for the class of plane maps (maps with one
distinguished root-face) of girth having a root-face of degree . We then
obtain more general bijective constructions for annular maps (maps with two
distinguished root-faces) of girth at least . Our bijections associate to
each map a decorated plane tree, and non-root faces of degree of the map
correspond to vertices of degree of the tree. As special cases we recover
several known bijections for bipartite maps, loopless triangulations, simple
triangulations, simple quadrangulations, etc. Our work unifies and greatly
extends these bijective constructions. In terms of counting, we obtain for each
integer an expression for the generating function
of plane maps of girth with root-face of
degree , where the variable counts the non-root faces of degree .
The expression for was already obtained bijectively by Bouttier, Di
Francesco and Guitter, but for the expression of is new. We
also obtain an expression for the generating function
\G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees
and , such that cycles separating the two root-faces have length at
least while other cycles have length at least . Our strategy is to
obtain all the bijections as specializations of a single "master bijection"
introduced by the authors in a previous article. In order to use this approach,
we exhibit certain "canonical orientations" characterizing maps with prescribed
girth constraints
Weak Gibbs property and system of numeration
We study the selfsimilarity and the Gibbs properties of several measures
defined on the product space \Omega\_r:=\{0,1,...,\break r-1\}^{\mathbb N}.
This space can be identified with the interval by means of the
numeration in base . The last section is devoted to the Bernoulli
convolution in base , called the Erd\H os measure, and
its analogue in base , that we study by means of a
suitable system of numeration
Connection between the Burgers equation with an elastic forcing term and a stochastic process
We present a complete analytical resolution of the one dimensional Burgers
equation with the elastic forcing term ,
. Two methods existing for the case are adapted
and generalized using variable and function transformations, valid for all
values of space an time. The emergence of a Fokker-Planck equation in the
method allows to connect a fluid model, depicted by the Burgers equation, with
an Ornstein-Uhlenbeck process
The glass transition in a nutshell: a source of inspiration to describe the subcritical transition to turbulence
The starting point of the present work is the observation of possible
analogies, both at the phenomenological and at the methodological level,
between the subcritical transition to turbulence and the glass transition.
Having recalled the phenomenology of the subcritical transition to turbulence,
we review the theories of the glass transition at a very basic level, focusing
on the history of their development as well as on the concepts they have
elaborated. Doing so, we aim at attracting the attention on the above mentioned
analogies, which we believe could inspire new developments in the theory of the
subcritical transition to turbulence. We then briefly describe a model inspired
by one of the simplest and most inspiring model of the glass transition, the
so-called Random Energy Model, as a first step in that direction.Comment: 9 pages, 1 figure; to appear in a topical issue of Eur. Phys. J. E
dedicated to Paul Mannevill
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