A central limit theorem for a generalized Harper operator on a crystal lattice is obtained. As the limit, the continuous semigroup of a uniform magnetic Schrödinger operator is captured on a vector space equipped with a special Euclidean structure. The standard realization of the crystal lattice is a key to the Euclidean structure and a linear vector potential on the Euclidean space from combinatorial data of the generalized Harper operator.

Time-variant filters based on Calderón and Gabor reproducing formulas are important tools in time-frequency analysis. The paper studies the behavior of the eigenvalues of these filters. Optimal two-sided estimates of the number of eigenvalues contained in the interval $(\delta_1,\delta_2)$ , where $0<\delta_1<\delta_2<1$ , are obtained. The estimates cover large classes of localization domains and generating functions.

Suppose that A is a pointed CW-complex. The paper looks at how difficult it is to construct an A-cellular space B from copies of A by repeatedly taking homotopy colimits; this is determined by an ordinal number called the complexity of B. Studying the complexity leads to an iterative technique, based on resolutions, for constructing the A-cellular approximation CWA(X) of an arbitrary space X.

A partial differential operator associated with natural oscillations of an incompressible fluid in the neighbourhood of an elliptical flow is considered. The differentiation is only taken with respect to the angular variable, and thus the operator becomes a family of ordinary differential operators parametrized by the radial variable. It is shown that the spectra of these ordinary differential operators completely determine the spectrum of the given operator which turns out to have a kind of skeleton structure.

The interaction between harmonic analysis and operator theory has been fruitful. In [1] Arveson related results in the theory of operator algebras to spectral synthesis. He defined synthesis for subspace lattices and proved that certain classes of lattices are synthetic. The main result in this paper is a generalization of a result of Arveson for the case of subspace maps. In order to describe in more detail the content of the present work, we need to introduce some definitions and facts from [2] and [9].

Let ${\cal H}_1$ and ${\cal H}_2$ be separable complex Hilbert spaces, and ${\cal P}_i$ be the lattice of all (orthogonal) projections on ${\cal H}_i, i = 1, 2$ . Following Erdos [1], we let ${\cal M}({\cal P}_1, {\cal P}_2)$ denote the set of all maps $\varphi : {\cal P}_1 \longrightarrow {\cal P}_2$ which are 0-preserving and $\vee$ -continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each $\varphi \in {\cal M}({\cal P}_1, {\cal P}_2)$ uniquely defines semi-lattices ${\cal S}_{1\varphi} \subseteq {\cal P}_1$ and ${\cal S}_{2\varphi} = \varphi({\cal P}_1) \subseteq {\cal P}_{2}$ such that $\varphi$ is a bijection between ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ and is uniquely determined by its restriction to ${\cal S}_{1\varphi}$ . Moreover, ${\cal S}_{1\varphi}$ is meet-complete and contains the identity projection while ${\cal S}_{2\varphi}$ is join-complete and contains the zero projection. If ${\cal S}_{1\varphi}$ and ${\cal S}_{2\varphi}$ are commutative, we say that $\varphi$ is a commutative subspace map.

It is shown that a weakly closed subspace ${\cal S}$ of a nest algebra ${\cal A}$ is closed under conjugation by invertible elements in ${\cal A}$ , that is, $a^{-1}{\cal S}\,a={\cal S}$ if and only if ${\cal S}$ is a Lie ideal. A similar result holds for not-necessarily-closed subspaces of algebras of infinite multiplicity. An explicit characterisation of weakly closed Lie ideals in a nest algebra is given.

It is shown that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case (that is, the double local case). The normal form of transversely joined two families of plane curves with second-order contact at the envelope is also given.

A new invariant of (integrable) homotopy type for foliations is introduced: the tangential category of a foliated manifold.

The classical Lusternik–Schnirelmann theory is generalized to foliations and the relations of the tangential category with other known invariants such as the fibrewise and the equivariant category are studied. Cohomological lower bounds are provided in terms of foliated cohomology.

If the foliation is a product, the tangential category coincides with the ordinary category of the leaves. In general it is just bounded below. Estimates are given of the tangential category for compact-Hausdorff foliations and suspensions. Examples show that the lower and upper bounds are realized.

The following theorem is proved: if $(M^{4n+1},g)$ is a complete Riemannian manifold and $\Sigma\subset M$ is an oriented hypersurface partitioning $M$ and with non-zero signature, then the spectrum of the Hodge–deRham Laplacian is $[0,\infty[$ . This result is obtained by a new Callias-type index. This new formula links half-bound harmonic forms (that is, nearly $L^2$ but not in $L^2$ ) with the signature of $\Sigma$ .

The paper gives a necessary and sufficient criterion on the Lévy measure that determines whether a Lévy process creeps. (The author says that a Lévy process creeps if it can continuously pass a fixed level.)