Let$f$and$g$be nonlinear entire functions. The relations between the dynamics of$f⊗g$and$g⊗f$are discussed. Denote byℐ (·) and$F$(·) the Julia and Fatou sets. It is proved that if$z$∈$C$, then$z$∈ℐ8464 ($f⊗g$) if and only if$g(z)$∈ℐ8464 ($g⊗f$); if$U$is a component of$F$($f$○$g$) and$V$is the component of$F$($g$○$g$) that contains$g(U)$, then$U$is wandering if and only if$V$is wandering; if$U$is periodic, then so is$V$and moreover,$V$is of the same type according to the classification of periodic components as$U$. These results are used to show that certain new classes of entire functions do not have wandering domains.

We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.

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We prove that $$ \mathop{ \lim \inf}\limits_{n \rightarrow \infty} \frac{p_{n+1}-p_{n}}{\sqrt{\log p_{n}} \left(\log \log p_{n}\right)^{2}}< \infty, $$ where$p$_{$n$}denotes the$n$th prime. Since on average$p$_{$n$+1}−$p$_{$n$}is asymptotically log_{$n$}, this shows that we can always find pairs of primes much closer together than the average. We actually prove a more general result concerning the set of values taken on by the differences$p$−$p$′ between primes which includes the small gap result above.