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A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

Robert Skiba (2014)

Open Mathematics

We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.

A construction of noncontractible simply connected cell-like two-dimensional Peano continua

Katsuya Eda, Umed H. Karimov, Dušan Repovš (2007)

Fundamenta Mathematicae

Using the topologist sine curve we present a new functorial construction of cone-like spaces, starting in the category of all path-connected topological spaces with a base point and continuous maps, and ending in the subcategory of all simply connected spaces. If one starts from a noncontractible n-dimensional Peano continuum for any n > 0, then our construction yields a simply connected noncontractible (n + 1)-dimensional cell-like Peano continuum. In particular, starting from the circle 𝕊¹,...

A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

Stephen Montgomery-Smith, Milan Pokorný (2002)

Commentationes Mathematicae Universitatis Carolinae

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...

A dimensional property of Cartesian product

Michael Levin (2013)

Fundamenta Mathematicae

We show that the Cartesian product of three hereditarily infinite-dimensional compact metric spaces is never hereditarily infinite-dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.

A fixed point theorem in o-minimal structures

Mário J. Edmundo (2007)

Annales de l’institut Fourier

Here we prove an o-minimal fixed point theorem for definable continuous maps on definably compact definable sets, generalizing Brumfiel’s version of the Hopf fixed point theorem for semi-algebraic maps.

A formula for the rational LS-category of certain spaces

Luis Lechuga, Aniceto Murillo (2002)

Annales de l’institut Fourier

In this paper we find a formula for the rational LS-category of certain elliptic spaces which generalizes or complements previous work of the subject. This formula is given in terms of the minimal model of the space.

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