617 research outputs found
A generalisation of a partition theorem of Andrews to overpartitions
In 1969, Andrews proved a theorem on partitions with difference conditions
which generalises Schur's celebrated partition identity. In this paper, we
generalise Andrews' theorem to overpartitions. The proof uses q-differential
equations and recurrences
On Dyson's crank conjecture and the uniform asymptotic behavior of certain inverse theta functions
In this paper we prove a longstanding conjecture by Freeman Dyson concerning
the limiting shape of the crank generating function. We fit this function in a
more general family of inverse theta functions which play a key role in
physics.Comment: Some error bounds have been fixe
An overpartition analogue of the -binomial coefficients
We define an overpartition analogue of Gaussian polynomials (also known as
-binomial coefficients) as a generating function for the number of
overpartitions fitting inside the rectangle. We call these new
polynomials over Gaussian polynomials or over -binomial coefficients. We
investigate basic properties and applications of over -binomial
coefficients. In particular, via the recurrences and combinatorial
interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan
type partition theorem.Comment: v2: new section added about another way of proving our theorems using
q-series identitie
Asymptotic formulae for partition ranks
Using an extension of Wright's version of the circle method, we obtain
asymptotic formulae for partition ranks similar to formulae for partition
cranks which where conjectured by F. Dyson and recently proved by the first
author and K. Bringmann
A generalisation of two partition theorems of Andrews
International audienceIn 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from -difference equations on generating functions to recurrence equations on their coefficients.En 1968 et 1969, Andrews a prouvé deux identités de partitions du type Rogers-Ramanujan qui généralisent le célèbre théorème de Schur (1926). Ces deux généralisations sont devenues deux des théorèmes les plus importants de la théorie des partitions, avec des applications en combinatoire, en théorie des représentations et en algèbre quantique. Dans ce papier, nous généralisons les deux théorèmes de Andrews aux surpartitions. Les preuves utilisent une nouvelle technique qui consiste à faire des allers-retours entre équations aux -différences sur les séries génératrices et équations de récurrence sur leurs coefficients
VANET Connectivity Analysis
Vehicular Ad Hoc Networks (VANETs) are a peculiar subclass of mobile ad hoc
networks that raise a number of technical challenges, notably from the point of
view of their mobility models. In this paper, we provide a thorough analysis of
the connectivity of such networks by leveraging on well-known results of
percolation theory. By means of simulations, we study the influence of a number
of parameters, including vehicle density, proportion of equipped vehicles, and
radio communication range. We also study the influence of traffic lights and
roadside units. Our results provide insights on the behavior of connectivity.
We believe this paper to be a valuable framework to assess the feasibility and
performance of future applications relying on vehicular connectivity in urban
scenarios
Connectivity vs Capacity in Dense Ad Hoc Networks
We study the connectivity and capacity of finite area ad hoc wireless networks, with an increasing number of nodes (dense networks). We find that the properties of the network strongly depend on the shape of the attenuation function. For power law attenuation functions, connectivity scales, and the available rate per node is known to decrease like 1/sqrt(n). On the contrary, if the attenuation function does not have a singularity at the origin and is uniformly bounded, we obtain bounds on the percolation domain for large node densities, which show that either the network becomes disconnected, or the available rate per node decreases like 1/n
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