the set of squares $n^2$, $n<2^k$, is considered and the sum of binary digits $s(n^2)$ is split up into two parts $s_{[<k]}(n^2)+s_{[\ge k]}(n^2)$, where $s_{[<k]}(n^2) = s(n^2{\rm mod}2^k)$ collects the first $k$ digits and $s_{[\ge k]}(n^2) = s(\lfloor n^2/2^k\rfloor)$ collects the remaining digits. very precise results on the distribution of $s_{[<k]}(n^2)$ and $s_{[\ge k]}(n^2)$ are presented. for example, asymptotic formulae are provided for the numbers $\#\{n< 2^k{:} s_{[<k]}(n^2) = m\}$ and $\#\{n< 2^k {:} s_{[\ge k]}(n^2) = m\}$ and it is shown that these partial sum-of-digits functions are asymptotically equidistributed in residue classes. these results are prompted by a conjecture by gelfond saying that the (total) sum-of-digits function $s(n^2)$ is asymptotically equidistributed in residue classes.

the convergence case of a khintchine-type theorem for a large class of hyperplanes is obtained. the approach to the problem is from a dynamical viewpoint, and a method due to kleinbock and margulis is modified to prove the result.

it is conjectured that a rational map whose coefficients are algebraic over $\mathbb q_p$ has no wandering components of the fatou set. benedetto has shown that any counterexample to this conjecture must have a wild recurrent critical point. we provide the first examples of rational maps whose coefficients are algebraic over $\mathbb q_p$ and that have a (wild) recurrent critical point. in fact, it is shown that there is such a rational map in every one-parameter family of rational maps that is defined over a finite extension of $\mathbb q_p$ and that has a misiurewicz bifurcation.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.

a new viewpoint is offered on some of the generic structures constructed using hrushovski's predimensions and it is shown that they are natural reducts of quite straightforward trivial, one-based stable structures.

let $l$ denote a right-invariant sub-laplacian on an exponential, hence solvable lie group $g$, endowed with a left-invariant haar measure. depending on the structure of $g$, and possibly also that of $l, l$ may admit differentiable $l^p$-functional calculi, or may be of holomorphic $l^p$-type for a given $p{\ne} 2$. ‘holomorphic $l^p$-type’ means that every $l^p$-spectral multiplier for $l$ is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the $l^2$-spectrum of $l$. this can in fact only arise if the group algebra $l^1(g)$ is non-symmetric.

assume that $p{\ne} 2$. for a point $\ell$ in the dual $\frak{g}^*$ of the lie algebra $\frak{g}$ of $g$, denote by $\omega(\ell){=}\ad^*(g)\ell$ the corresponding coadjoint orbit. it is proved that every sub-laplacian on $g$ is of holomorphic $l^p$-type, provided that there exists a point $\ell{\in} \frak{g}^*$ satisfying boidol's condition (which is equivalent to the non-symmetry of $l^1(g)$), such that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed. this work improves on results in previous work by christ and müller and ludwig and müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group $g$, and on the other hand, for the case $p{>}1$, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits.

it seems likely that the condition that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed could be replaced by the weaker condition that the orbit $\omega(\ell)$ itself is closed. this would then prove one implication of a conjecture by ludwig and müller, according to which there exists a sub-laplacian of holomorphic $l^1$ (or, more generally, $l^p$) type on $g$ if and only if there exists a point $\ell{\in} \frak{g}^*$ whose orbit is closed and which satisfies boidol's condition.

a cauchy problem for a nonlinear convection-diffusion equation with periodic rapidly oscillating coefficients is studied. under the assumption that the convection term is large, it is proved that the limit (homogenized) equation is a nonlinear diffusion equation which shows dispersion effects. the convergence of the homogenization procedure is justified by using a new version of a two-scale convergence technique adapted to rapidly moving coordinates.

an approximation theorem in inhomogeneous orlicz–sobolev spaces is proved which allows a second-order parabolic equation in orlicz spaces to be solved. a trace result is also given which shows that the solutions are continuous with respect to time.

the paper considers the eigenvalue problem \[ -\delta u-\alpha u+\lambda g(x)u=0\quad \mbox{with}u\in h^{1}(\mathbb{r}^{n}),\;u\neq0, \] where $\alpha,\lambda\in\mathbb{r}$ and \[g(x)\equiv0\mbox{on}\overline{\omega},\quad\ g(x)\in(0,1]\mbox{on}{\mathbb{r}^{n}}\setminus{\overline{\omega}}\quad\mbox{and}\quad\lim_{|x|\rightarrow+\infty}g(x)=1 \] for some bounded open set $\omega\in\mathbb{r}^{n}$.

given $\alpha>0$, does there exist a value of $\lambda>0$ for which the problem has a positive solution? it is shown that this occurs if and only if $\alpha$ lies in a certain interval $(\gamma,\xi_{1})$ and that in this case the value of $\lambda$ is unique, $\lambda=\lambda(\alpha)$. the properties of the function $\lambda(\alpha)$ are also discussed.

the analytic concepts of martingale type $p$ and cotype $q$ of a banach space have an intimate relation with the geometric concepts of $p$-concavity and $q$-convexity of the space under consideration, as shown by pisier. in particular, for a banach space $x$, having martingale type $p$ for some $p{>}1$ implies that $x$ has martingale cotype $q$ for some $q{<}\infty$.

the generalisation of these concepts to linear operators was studied by the author, and it turns out that the duality above only holds in a weaker form. an example is constructed showing that this duality result is best possible.

so-called random martingale unconditionality estimates, introduced by garling as a decoupling of the unconditional martingale differences (umd) inequality, are also examined.

it is shown that the random martingale unconditionality constant of $l_\infty^{2^n}$ for martingales of length $n$ asymptotically behaves like $n$. this improves previous estimates by geiss, who needed martingales of length $2^n$ to show this asymptotic. at the same time the order in the paper is the best that can be expected.

the interplay between the behaviour of approximately convex (and approximately affine) functions on the unit ball of a banach space and the geometry of banach $k$-spaces is studied.

global weierstrass representations are derived for complete minimal surfaces obtained by substituting the planar end of the costa surface by scherk ends. this gives rise to singly periodic examples, of which the fundamental piece is proved to be embedded by a simple geometric technique. this technique can be generalised for embeddedness proof of future examples of the same nature.

the total space of the tangent bundle of a kähler manifold admits a canonical kähler structure. parallel translation identifies the space ${\mathbb{t}}$ of oriented affine lines in ${\mathbb{r}}^3$ with the tangent bundle of $s^2$. thus the round metric on $s^2$ induces a kähler structure on ${\mathbb{t}}$ which turns out to have a metric of neutral signature. it is shown that the identity component of the isometry group of this metric is isomorphic to the identity component of the isometry group of the euclidean metric on ${\mathbb{r}}^3$.

the geodesics of this metric are either planes or helicoids in ${\mathbb{r}}^3$. the signature of the metric induced on a surface $\sigma$ in ${\mathbb{t}}$ is determined by the degree of twisting of the associated line congruence in ${\mathbb{r}}^3$, and it is shown that, for $\sigma$ lagrangian, the metric is either lorentz or totally null. for such surfaces it is proved that the keller–maslov index counts the number of isolated complex points of ${\mathbb{j}}$ inside a closed curve on $\sigma$.

it is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. the proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

given an $n$-dimensional compact closed oriented manifold $m$ and a field $\mathbb k$, f. cohen and l. taylor have constructed a spectral sequence, ${\mathcal e}(m,n,\mathbb k)$, converging to the cohomology of the space of ordered configurations of $n$ points in $m$. the symmetric group $\sigma_n$ acts on this spectral sequence giving a spectral sequence of $\sigma_n$-differential graded commutative algebras. here, an explicit description is provided of the invariants algebra $(e_1,d_1)^{\sigma_n}$ of the first term of $\mathcal e(m,n,\mathbb q)$. this determination is applied in two directions.

(a) in the case of a complex projective manifold or of an odd-dimensional manifold $m$, the cohomology algebra $h^*(c_n(m);\mathbb q)$ of the space of unordered configurations of $n$ points in $m$ is obtained (the concrete example of $p^2(\mathbb c)$ is detailed).

(b) the degeneration of the spectral sequence formed of the $\sigma_n$-invariants $\mathcal e(m,n,\mathbb q)^{\sigma_n}$ at level 2 is proved for any manifold $m$.

these results use a transfer map and are also true with coefficients in a finite field $\mathbb f_p$ with $p>n$.