This paper gives a partial desingularisation construction for hyperkähler quotients and a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperkähler quotients. This criterion is applied to some linear actions on hyperkähler vector spaces.

The dynamical behavior of non-uniform cellular automata is compared with the one of classical cellular automata. Several differences and similarities are pointed out by a series of examples. Decidability of basic properties like surjectivity and injectivity is also established. The final part studies a strong form of equicontinuity property specially suited for non-uniform cellular automata.

We show that the equation a2 +b6 = cn has no nontrivial positive integer solutions with (a, b) = 1 via a combination of techniques based upon the modularity of Galois representations attached to certain Q -curves, corresponding surjectivity results of Ellenberg for these representations, and extensions of multi-Frey curve arguments of Siksek.

We calculate the Beilinson regulators of motives associated to Fermat curves and express them by special values of generalized hypergeometric functions. As a result, we obtain surjectivity results of the regulator, which support the Beilinson conjecture on special values of L-functions.

In this paper we extend to algebraic extensions of local and global fields their completions some classical results due Borel-Serre, Tits, Conrad, Douai, Kneser Sansuc concerning the finiteness, surjectivity maps between Galois cohomology groups obstruction weak approximation related connected linear groups.