Birmingham

We prove that if $\frac{\log^{117} n}{n} \leq p \leq 1 -

n^{-1/8}$, then asymptotically almost surely the edges of $G(n,p)$ can

be covered by $\lceil \Delta(G(n,p))/2 \rceil$ Hamilton cycles. This

is clearly best possible and improves an approximate result of Glebov,

Krivelevich and Szab\'o, which holds for $p \geq n^{-1 + \varepsilon}$.

Based on joint work with Daniela Kuhn, John Lapinskas and Deryk Osthus.

An embedding of a graph H in a graph G is an injective mapping of the vertices of H to the vertices of G such that edges of H are mapped to edges of G. Embedding problems have been extensively studied. A very powerful tool in this area is Szemeredi's Regularity Temma. It approximates the host graph G by a quasirandom graph which inherits many of the properties of G. Unfortunately the direct use of Szemeredi's Regularity Lemma is useless if the host graph G is sparse.

During the talk I shall expose a technique to deal with embedding trees in sparse graphs. This technique has been developed by Ajtai, Komlos,Simonovits and Szemeredi to solve the Erdos-Sos conjecture. Presently the author together with Hladky, Komlos, Simonovits, Stein and Szemeredi apply this method to solve the related conjecture of Loebl, Komlos and Sos (approximate version).

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characterstics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.

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