Let $f:M \to N$ be a smooth map of a closed $n$-dimensional manifold $M$ into a $p$-dimensional manifold $N (n \geq p)$. When $f$ has only definite fold singular points, we call it a special generic map. Porto and Furuya introduced the notion of regular equivalence for such maps. In this paper, we define another equivalence relation for special generic maps, called weak regular equivalence and we classify special generic maps up to these equivalences in the case where $M$ is a connected orientable closed $n$-dimensional manifold with $n \geq 3$ and $N$ is the plane. We also show the existence of a pair of special generic maps which are $C^0$ right-left equivalent but are not $C^\infty$ right-left equivalent.

The annihilator $J^{\perp}$ of a weak*-closed inner ideal $J$ in the JBW*-triple $A$ consists of elements $b$ of $A$ for which $\{J\,b\,A\}$ is equal to zero, and the kernel $\mathrm{Ker}(J)$ of $J$ consists of those elements $b$ in $A$ for which $\{J\,b\,J\}$ is equal to zero. The annihilator $J^{\perp}$ is also a weak*-closed inner ideal in $A$, and $A$ enjoys the Peirce decomposition \begin{eqnarray*} A &=& J \oplus_M J^{\perp} \oplus J_1\\ &=& J_2 \oplus_M J_0 \oplus J_1, \end{eqnarray*} where $J_1$ is the intersection of the kernels of $J$ and $J^{\perp}$. When all of the usual Peirce arithmetical relations hold, $J$ is said to be a Peirce inner ideal. A pair $(J,K)$ of weak*-closed inner ideals is said to be compatible when \[A = \bigoplus_{j,k = 0}^2 J_j \cap K_k.\] By analysing the biannihilator $J^{\perp\perp}$ of a Peirce inner ideal $J$ it is shown that, if $J$ is a Peirce inner ideal, then $J^{\perp}$ is a Peirce inner ideal. Furthermore, if $K$ is a weak*-closed inner ideal in $A$ such that $(J,K)$ is a compatible pair, then $(J^{\perp},K)$ is a compatible pair.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus in the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes.

Using Fuchsian techniques, a large family of Gowdy vacuum spacetimes have been constructed for which one has detailed control over asymptotic behaviour. In this paper we formulate a condition on initial data yielding the same form of asymptotics.

Consider $e_n = \inf \exp \int \log \parallel x - f(x) \parallel dP(x)$, where $p$ is a probability measure on $\real^d$ and the infimum is taken over all measurable maps $f{:}\ \real^d \rightarrow \real^d$ with $| f(\real^d)| \leq n$. We study solutions $f$ of this minimization problem. For absolutely continuous distributions and for self-similar distributions we derive the exact rates of convergence to zero of the $n$th quantization error $e_n$ as $n \rightarrow \infty$. We establish a relationship between the quantization dimension that rules the rates and the Hausdorff dimension of $P$.

Let ${\cal F}$ be an oriented conformal foliation of connected, totally geodesic and 1-dimensional leaves in $\mathbb{R}^{n+1}$. We prove that if $n\geq 3$ then the Gauss map $\phi{:}\,\,U\,{\to}\,S^n$ of ${\cal F}$ is a non-constant $n$-harmonic morphism if and only if it is a radial projection.

We consider singular block operator problems of the type arising in the study of stability of the Ekman boundary layer. The essential spectrum is located, and an analysis of the $L^2$ solutions of a related first order system of differential equations allows the development of a Titchmarsh–Weyl coefficient $M(\lambda)$. This, in turn, permits a rigorous analysis of the convergence of approximations to the spectrum arising from regular problems. Numerical results illustrate the theory.