On the Sr I 4607 A Hanle depolarization signals in the quiet Sun

The Hanle depolarization signals of Sr 4607 A have been used to estimate the unsigned magnetic flux and magnetic energy existing in the quiet Sun photosphere. However, the Sr 4607 A Hanle signals are not sensitive to the unsigned flux and energy. They only bear information on the fraction of photosphere occupied by magnetic field strengths smaller than the Hanle saturation, which do not contribute to the unsigned flux and energy. We deduce an approximate expression for the relationship between magnetic fill factor and Hanle signal. When applied to existing Hanle depolarization measurements, it indicates that only 40% of the quiet Sun is filled by magnetic fields with a strength smaller than 60 G. The remaining 60% of the surface has field strengths above this limit. Such constraint will be needed to determine the distribution of magnetic field strengths existing in the quiet Sun.Comment: To appear in A&

On the insertion of n-powers

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other natural pseudovarieties of ordered monoids and of monoids associated with the Burnside pseudovariety of groups defined by the identity $x^n=1$. In particular, we are interested in determining the pseudovariety of monoids that it generates, which can be viewed as the problem of determining the Boolean closure of the class of regular languages closed under $n$-power insertions. We exhibit a simple upper bound and show that it satisfies all pseudoidentities which are provable from $1\le x^n$ in which both sides are regular elements with respect to the upper bound

Generalized Euler-Lagrange equations for variational problems with scale derivatives

We obtain several Euler-Lagrange equations for variational functionals defined on a set of H\"older curves. The cases when the Lagrangian contains multiple scale derivatives, depends on a parameter, or contains higher-order scale derivatives are considered.Comment: Submitted on 03-Aug-2009; accepted for publication 16-March-2010; in "Letters in Mathematical Physics"