24 research outputs found
The algebra of set functions I: The product theorem and duality
AbstractWe give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions
On the number of fully packed loop configurations with a fixed associated matching
We show that the number of fully packed loop configurations corresponding to
a matching with nested arches is polynomial in if is large enough,
thus essentially proving two conjectures by Zuber [Electronic J. Combin. 11
(2004), Article #R13].Comment: AnS-LaTeX, 43 pages; Journal versio
The Algebra of Set Functions II : An Enumerative Analogue of Hall's Theorem for Bipartite Graphs
Abstract Triesch This formula follows from a general duality theory which we develop for counting matchings. Moreover, we make use of generating functions for set functions as introduced i
The potential of discs from a "mean Green function"
By using various properties of the complete elliptic integrals, we have
derived an alternative expression for the gravitational potential of axially
symmetric bodies, which is free of singular kernel in contrast with the
classical form. This is mainly a radial integral of the local surface density
weighted by a regular "mean Green function" which depends explicitly on the
body's vertical thickness. Rigorously, this result stands for a wide variety of
configurations, as soon as the density structure is vertically homogeneous.
Nevertheless, the sensitivity to vertical stratification | the Gaussian profile
has been considered | appears weak provided that the surface density is
conserved. For bodies with small aspect ratio (i.e. geometrically thin discs),
a first-order Taylor expansion furnishes an excellent approximation for this
mean Green function, the absolute error being of the fourth order in the aspect
ratio. This formula is therefore well suited to studying the structure of
self-gravitating discs and rings in the spirit of the "standard model of thin
discs" where the vertical structure is often ignored, but it remains accurate
for discs and tori of finite thickness. This approximation which perfectly
saves the properties of Newton's law everywhere (in particular at large
separations), is also very useful for dynamical studies where the body is just
a source of gravity acting on external test particles.Comment: Accepted for publication in MNRAS, 11 page
Séries génératrices de matroïde
Inspired by set functions and the greedy algorithm, we introduce matroid power series to provide short formulations and proofs for many classical and new results in the theory of graphs, matroids and oriented matroids, emphasizing the interest to study a matroid together with its minors in a single object