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Title and Author(s) |
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Olga Alexandrovna Ladyzhenskaya
Wojciech M. Zajaczkowski
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5
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On the topology of eigenfields
Vladimir I. Arnold
| ABSTRACT.
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.
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9
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Parameter dependent pull-back of closed differential forms and invariant integrals
Jean Mawhin
| ABSTRACT.
We prove, given a closed differential k-form \omega in an
arbitrary open set D \subset {\Bbb R}^n, and a parameter dependent
smooth map F(\,\cdot\,,\lambda) from an arbitrary open set G \subset {\Bbb 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.
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17
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Homology index braids in infinite-dimensional Conley index theory
Maria C. Carbinatto and Krzysztof P. Rybakowski
| ABSTRACT.
We extend the notion of a categorial Conley-Morse index, as defined
in [R1], to the case based on a more general
concept of an index pair introduced in [FM]. 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 [CR6] and [CR7].
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35
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Almost flat bundles and almost flat structures
Alexandr Mishchenko and Nicolae Teleman
| ABSTRACT.
In this paper we discuss some geometric aspects concerning almost
flat bundles, notion introduced by Connes, Gromov and Moscovici
[2]. Using a natural construction of [1], 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 [3].
Connes, Gromov and Moscovici [2] 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.
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75
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On multiple solutions of the exterior Neumann problem involving critical Sobolev exponent
Jan Chabrowski and Michael Willem
| ABSTRACT.
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.
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89
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The effect of the domain's configuration space on the number of nodal solutions of singularly perturbed elliptic equations
Thomas Bartsch and Tobias Weth
| ABSTRACT.
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\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 cupl(C\Omega)+2
nodal solutions, where cupl denotes the cuplength of a topological
space. We furthermore show that cupl(C\Omega)+1 of these solutions
have precisely two nodal domains, and the last one has at most three nodal
domains.
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109
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Existence results for first and second order semilinear impulsive differential inclusions
Lech Górniewicz, Sotiris K. Ntouyas and Donal O'Regan
| ABSTRACT.
In this paper we prove existence results for first and second order semilinear impulsive differential inclusions in Banach spaces.
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135
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A Sharkovskii-type theorem for minimally forced interval maps
Roberta Fabbri, Tobias Jaeger, Russel Johnson and Gerhard Keller
| ABSTRACT.
We state and prove a version of Sharkovskii'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.
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135
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Symmetry breaking solutions of nonlinear elliptic systems
Javier Bracho, Monica Clapp and Waclaw Marzantowicz
| ABSTRACT.
We consider nonlinear elliptic systems with Dirichlet boundary condition on
a bounded domain in RN 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.
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189
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