Let G be a group generated by a finite subset S; define S to be the set Ž . < n < of all products of at most n elements of S, and let a S s S be the n n Ž . Ž . Ž . Ž . number of elements in S . As a S satisfies 1 F a S F a S ? a S , n nqm n m Ž .1r n Ž . Ž .1r n the limit lim a S exists, and a S s lim a S G 1. Although the n n Ž . exact value of a S depends on the generating set S, it is well ...

We develop a new technics to catch the non linear growth of interfaces. We apply this analytical approach to the one dimensional Cahn-Hilliard equation. The dynamics is captured through a solubility condition performed over a particular family of quasi-static solutions. The main result is that the dynamics along this particular class of solutions can be captured in simple ordinary differential ...

Let Γ be a group and rn(Γ) the number of its n-dimensional irreducible complex representations. We define and study the associated representation zeta function ZΓ(s) = ∞ ∑ n=1 rn(Γ)n . When Γ is an arithmetic group satisfying the congruence subgroup property then ZΓ(s) has an “Euler factorization”. The “factor at infinity” is sometimes called the “Witten zeta function” counting the rational rep...

In this note we give a simple proof that Dehn twists have linear growth with respect to the word metric on the mapping class group of an oriented surface of nite type (Theorem 1). This answers a question of Ivanov in 3]. A proof in the case of punctured surfaces was given by Mosher 6] using his techniques for constructing an automatic structure on the mapping class group. This result implies in...

The electrostatics of micellar growth is reviewed and extended for solutions containing excess salt. In dilute solution the expansion of a linear micelle with increasing salt concentration is explained for a wide range of ionic strength. When the micellar charge density is very high, counterions condense nonuniformly onto the micellar rod. In that case the micelle may contract upon the addition...