Independence numbers of some double vertex graphs and pair graphs

The combinatorial properties of double vertex graphs has been widely studied since the 90's. However only very few results are know about the independence number of such graphs. In this paper we obtain the independence numbers of the double vertex graphs of fan graphs and wheel graphs. Also we obtain the independence numbers of the pair graphs, that is a generalization of the double vertex graphs, of some families of graphs.Comment: 17 pages. Minor changes in the proof

On the triviality of the shocked map

The (non-spanning) tree-decorated quadrangulation is a random pair formed by a quadrangulation and a subtree chosen uniformly over the set of pairs with prescribed size. In this paper we study the tree-decorated quadrangulation in the critical regime: when the number of faces of the map, $f$, is proportional to the square of the size of the tree. We show that with high probability in this regime, the diameter of the tree is between $o(f^{1/4})$ and $f^{1/4}/\log^\alpha(f)$, for $\alpha >1$. Thus after scaling the distances by $f^{-1/4}$, the critical tree-decorated quadrangulation converges to a Brownian disk where the boundary has been identified to a point. These results imply the triviality of the shocked map: the metric space generated by gluing a Brownian disk with a continuous random tree.Comment: 29 pages. 10 beautiful figure

Infinite dimensional holomorphic non-extendability and algebraic genericity

In this note, the linear structure of the family He(G) of holomorphic functions in a domain G of a complex Banach space that are not holomorphically continuable beyond the boundary of G is analyzed. More particularly, we prove that He(G) contains, except for zero, a closed (and a dense) vector space having maximal dimension, as well as a maximally generated free algebra. The results obtained complete a number of previous ones by several authors.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Multiplicative structures of hypercyclic functions for convolution operators

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.Comment: 12 page

Semiclassical propagator of the Wigner function

Propagation of the Wigner function is studied on two levels of semiclassical propagation, one based on the van-Vleck propagator, the other on phase-space path integration. Leading quantum corrections to the classical Liouville propagator take the form of a time-dependent quantum spot. Its oscillatory structure depends on whether the underlying classical flow is elliptic or hyperbolic. It can be interpreted as the result of interference of a \emph{pair} of classical trajectories, indicating how quantum coherences are to be propagated semiclassically in phase space. The phase-space path-integral approach allows for a finer resolution of the quantum spot in terms of Airy functions.Comment: 4 pages, 3 figure

Non-Lipschitz differentiable functions on slit domains

It is proved the existence of large algebraic structures –including large vector subspaces or infinitely generated free algebras– inside the family of non-Lipschitz differentiable real functions with bounded gradient defined on special non-convex plane domains. In particular, this yields that there are many differentiable functions on plane domains that do not satisfy the Mean Value Theorem.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ