In January 2004 the world mathematical community lost Professor Olga Alexandrovna Ladyzhenskaya who was a great mathematician and a member of several Academies of Science. She made significant and important contributions to the area of partial differential equations, particularly the Navier--Stokes equations, nonlinear elliptic and parabolic equations.

Topological properties of the eigenfields dependence on the eigenvalue position is discussed for the cases, where the variety of the eigenfield vanishing does not divide the oscillating domain into pieces.

We prove, given a closed differential $k$-form $\omega$ in an arbitrary open set $D \subset {\mathbb R}^n$, and a parameter dependent smooth map $F(\cdot,\lambda)$ from an arbitrary open set $G \subset {\mathbb R}^m$ into $D$, that the derivative with respect to $\lambda$ of the pull-back $F(\cdot,\lambda)^{*}\omega$ is exact in $G$. We give applications to various theorems in topology, dynamics and hydrodynamics.

We extend the notion of a categorial Conley-Morse index, as defined in [K. P. Rybakowski, The Morse index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces, J. Differential Equations 47 (1987), 66–98], to the case based on a more general concept of an index pair introduced in [R. D. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988), 270–287]. We also establish a naturality result of the long exact sequence of attractor-repeller pairs with respect to the choice of index triples. In particular, these results immediately give a complete and rigorous existence result for homology index braids in infinite dimensional Conley index theory.

Finally, we describe some general regular and singular continuation results for homology index braids obtained in our recent papers [M. C. Carbinatto and K. P. Rybakowski, Nested sequences of index filtrations and continuation of the connection matrix, J. Differential Equations 207 (2004), 458–488] and [M. C. Carbinatto and K. P. Rybakowski, Continuation of the connection matrix in singular perturbation problems].

In this paper we discuss some geometric aspects concerning almost flat bundles, notion introduced by Connes, Gromov and Moscovici [Conjecture de Novikov et fibrés presque plats, C. R. Acad. Sci. Paris Sér. I 310 (1990), 273–277]. Using a natural construction of [B. Hanke and T. Schick, Enlargeability and index theory, preprint, 2004], we present here a simple description of such bundles. For this we modify the notion of almost flat structure on bundles over smooth manifolds and extend this notion to bundles over arbitrary CW-spaces using quasi-connections [N. Teleman, Distance function, Linear quasi-connections and Chern character, IHES/M/04/27].

Connes, Gromov and Moscovici [Conjecture de Novikov et fibrés presque plats, C. R. Acad. Sci. Paris Sér. I 310 (1990), 273–277] showed that for any almost flat bundle $\alpha$ over the manifold $M$, the index of the signature operator with values in $\alpha$ is a homotopy equivalence invariant of $M$. From here it follows that a certain integer multiple $n$ of the bundle $\alpha$ comes from the classifying space $B\pi_{1}(M)$. The geometric arguments discussed in this paper allow us to show that the bundle $\alpha$ itself, and not necessarily a certain multiple of it, comes from an arbitrarily large compact subspace $Y\subset B\pi_{1}(M)$ trough the classifying mapping.

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.

We prove a new multiplicity result for nodal solutions of the Dirichlet problem for the singularly perturbed equation $-\varepsilon^2 \Delta u+u =f(u)$ for $\varepsilon> 0$ small on a bounded domain $\Omega\subset{\mathbb R}^N$. The nonlinearity $f$ grows superlinearly and subcritically. We relate the topology of the configuration space $C\Omega=\{(x,y)\in\Omega\times\Omega:x\not=y\}$ of ordered pairs in the domain to the number of solutions with exactly two nodal domains. More precisely, we show that there exist at least ${\rm cupl}(C\Omega)+2$ nodal solutions, where ${\rm cupl}$ denotes the cuplength of a topological space. We furthermore show that ${\rm cupl}(C\Omega)+1$ of these solutions have precisely two nodal domains, and the last one has at most three nodal domains.

In this paper we prove existence results for first and second order semilinear impulsive differential inclusions in Banach spaces.

We state and prove a version of Sharkovskiǐ's theorem for forced interval maps in which the forcing flow is minimal (Birkhoff recurrent). This setup includes quasiperiodically forced interval maps as a special case. We find that it is natural to substitute the concept of "fixed point" with that of "core strip." Core strips are frequently of almost automorphic type.

We consider nonlinear elliptic systems with Dirichlet boundary condition on a bounded domain in $\mathbb R^{N}$ which is invariant with respect to the action of some group $G$ of orthogonal transformations. For every subgroup $K$ of $G$ we give a simple criterion for the existence of infinitely many solutions which are $K$-invariant but not $G$-invariant. We include a detailed discussion of the case $N=3$.