Let $f(z)$ be a transcendental meromorphic function. The paper investigates, using the hyperbolic metric, the relation between the forward orbit $P(f)$ of singularities of $f^{-1}$ and limit functions of iterations of $f$ in its Fatou components. It is mainly proved, among other things, that for a wandering domain $U$ , all the limit functions of $\{f^n\vert_U\}$ lie in the derived set of $P(f)$ and that if $f^{np}\vert_V\rightarrow q(n\rightarrow +\infty)$ for a Fatou component $V$ , then either $q$ is in the derived set of $S_p(f)$ or $f^p(q) = q$ . As applications of main theorems, some sufficient conditions of the non-existence of wandering domains and Baker domains are given.

Let $\Omega \subset \R^2$ be a bounded Lipschitz domain and let \[ F: \Omega \times {\mathbb R}_{+}^{2 \times 2} \longrightarrow {\mathbb R} \] be a Carathèodory integrand such that $F\left(x, \cdot\right)$ is polyconvex for ${\mathcal L}^2$-a.e. $x \in \Omega$. Moreover assume that $F$ is bounded from below and satisfies the condition $F\left(x, \xi\right) \searrow \infty$ as $\det \xi \searrow 0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional \[ {\mathbb F} \left[u\right] := \int_{\Omega } F\left(x, \nabla u\left(x\right)\right) \, dx,\] where the map $u$ lies in the Sobolev space $W_{\rm {id}}^{1,p} (\Omega, {\mathbb R}^2)$ with $p \geqslant 2$ and satisfies the pointwise condition $\det \nabla u\left(x\right) >0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The question is settled by establishing that ${\mathbb F}\left[\cdot\right]$ admits a set of strong local minimizers on {W^{1,p}_{\rm{id}}(\Omega, {\mathbb R}^2)} that can be indexed by the group $\Pn \oplus \Z^n$, the direct sum of Artin's pure braid group on $n$ strings and $n$ copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes $n$ in $\Omega$ and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.

Necessary and sufficient conditions are given for a Banach space operator with the single-valued extension property to satisfy Weyl's theorem and $a$-Weyl's theorem. It is shown that if $T$ or $T^{\ast}$ has the single-valued extension property and $T$ is transaloid, then Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma (T))$. When $T^{\ast}$ has the single-valued extension property, $T$ is transaloid and $T$ is $a$-isoloid, then <formula form="inline" disc="math" id="ffm012"><formtex notation="AMSTeX">$a$-Weyl's theorem holds for $f(T)$ for every $f\in H(\sigma(T))$. It is also proved that if $T$ or $T^{\ast}$ has the single-valued extension property, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

The paper proves an almost-orthogonality principle for maximal operators over arbitrary sets of directions in ${\bb R}^2$. Namely, the Lp-bounds for an operator of this type are obtained from the corresponding Lp-bounds of the maximal functions associated to a certain partition of the set of directions, and from the particular structure of this partition. Applications to several types of maximal operators are provided.

It is shown that every special Lagrangian cone in $\C^3$ determines, and is determined by, a primitive harmonic surface in the 6-symmetric space ${\rm SU}_3/{\rm SO}_2$. For cones over tori, this allows the classification theory of harmonic tori to be used to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.

For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.

A connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension are constructed. The construction is based on certain structures on the bundle, that is, connections and bundle splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures.

The class of linearly recurrent Cantor systems contains the substitution subshifts and some odometers. For substitutionsubshifts, measure-theoretical and continuous eigenvalues are the same. It is natural to ask whether this rigidity property remains true for the class of linearly recurrent Cantor systems. Partial answers are given to this question.

The paper describes some qualitative properties of minimizers on a manifold $\mathcal M$ endowed with a discontinuous metric. The discontinuity occurs on a hypersurface $\Sigma$ disconnecting $\mathcal M$. Denote by $\Omega_1$ and $\Omega_2$ the open subsets of $\mathcal M$ such that $\mathcal M\setminus\Sigma=\Omega_1\cup\Omega_2$. Assume that $\overline\Omega_1$ and $\overline\Omega_2$ are endowed with metrics $\left\langle\cdot,\cdot\right\rangle_{\left(1\right)}$ and $\left\langle\cdot,\cdot\right\rangle_{\left(2\right)}$, respectively, such that $\overline\Omega_i (i=1, 2)$ is convex or concave. The existence of a minimizer of the length functional on curves joining two given points of $\mathcal M$ is proved. The qualitative properties obtained allows the refraction law in a very general situation to be described.

Let $\Omega X$ be the space of Moore loops on a finite, $q$-connected, $n$-dimensional CW complex $X$, and let $R\subset\Q$ be a subring containing 1/2. Let $\rho\left(R\right)$ be the least non-invertible prime in $R$. For a graded $R$-module $M$ of finite type, let $FM = M / {\rm Torsion}\,M$. We show that the inclusion $P \subset FH_{*}\left(\Omega X;R\right)$ of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras $$UP \overset{\cong}{\longrightarrow} FH_{*}\left(\Omega X;R\right)},$$ provided that $\rho\left(R\right) \geqslant n/q$. Furthermore, the Hurewicz homomorphism induces an embedding of $F(\pi_{*}\left(\Omega X\right)\otimes R)$ in $P$, with $P/F(\pi_{*}\left(\Omega X\right)\otimes R)$ torsion. As a corollary, if $X$ is elliptic, then $FH_{*}\left(\Omega X;R\right)$ is a finitely generated $R$-algebra.

Let $1 < p <\infty, 0 < v < p^\prime$ , let $\Omega$ be a bounded domain in ${\bb R}^n$ , and denote by ${\rm id}_{\omega}$ the limiting compact embedding of the Besov space $B^{n/p}_{pp}({\bb R}^n)$ into the exponential Orlicz space $L_{\exp (t^v)}(\Omega)$ , mapping a function $f$ onto its restriction $f\vert_{\Omega}$ . In 1993 Triebel established, among others, two-sided estimates for the entropy numbers of ${\rm id}_{\omega}$ , which are even asymptotically optimal for ‘small’ $\nu$ . The aim of the paper is to improve the upper bounds in the case of ‘large’ $\nu$ , where Triebel's estimates are not yet sharp, thus making a further step towards the conjectured correct asymptotic behaviour.

Sufficient conditions are found for stochastic convolution integrals driven by a Wiener process in a Hilbert space to belong to the Orlicz space exp $L^2$ ; standard exponential tail estimates follow from these results. Proofs are based on the extrapolation theory and are rather simple.

A random walk that is certain to visit $(0, \infty)$ has associated with it, via a suitable $h$ -transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let $\phi(x) = \log\log x$ . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below $x$ ultimately lies between $Lx^2/\phi(x)$ and $Ux^2\phi(x)$ , for suitable (non-random) positive $L$ and finite $U$ , as $x$ goes to infinity. For Bessel-3, the best $L$ and $U$ are identified.