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Reconstructing the global dynamics of attractors via the Conley index

100%
McCord C.
Banach Center Publications
|
1999
|
tom 47
|
nr 1
145-156
EN
Given an unknown attractor 𝓐 in a continuous dynamical system, how can we discover the topology and dynamics of 𝓐? As a practical matter, how can we do so from only a finite amount of information? One way of doing so is to produce a semi-conjugacy from 𝓐 onto a model system 𝓜 whose topology and dynamics are known. The complexity of 𝓜 then provides a lower bound for the complexity of 𝓐. The Conley index can be used to construct a simplicial model and a surjective semi-conjugacy for a large class of attractors. The essential features of this construction are that the model 𝓜 can be explicitly described; and that the finite amount of information needed to construct it is computable.
2
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Combinatorial lemmas for polyhedrons I

88%
Idzik A., Junosza-Szaniawski K.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 3
439-338
EN
We formulate general boundary conditions for a labelling of vertices of a triangulation of a polyhedron by vectors to assure the existence of a balanced simplex. The condition is not for each vertex separately, but for a set of vertices of each boundary simplex. This allows us to formulate a theorem, which is more general than the Sperner lemma and theorems of Shapley; Idzik and Junosza-Szaniawski; van der Laan, Talman and Yang. A generalization of the Poincaré-Miranda theorem is also derived.
3
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Combinatorial lemmas for polyhedrons

75%
Idzik A., Junosza-Szaniawski K.
Discussiones Mathematicae Graph Theory
|
2005
|
tom 25
|
nr 1-2
95-102
EN
We formulate general boundary conditions for a labelling to assure the existence of a balanced n-simplex in a triangulated polyhedron. Furthermore we prove a Knaster-Kuratowski-Mazurkiewicz type theorem for polyhedrons and generalize some theorems of Ichiishi and Idzik. We also formulate a necessary condition for a continuous function defined on a polyhedron to be an onto function.
4
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Structure of the set of all minimal total dominating functions of some classes of graphs

63%
Kumar K., MacGillivray G.
Discussiones Mathematicae Graph Theory
|
2010
|
tom 30
|
nr 3
407-423
EN
In this paper we study some of the structural properties of the set of all minimal total dominating functions ($𝔉_T$) of cycles and paths and introduce the idea of function reducible graphs and function separable graphs. It is proved that a function reducible graph is a function separable graph. We shall also see how the idea of function reducibility is used to study the structure of $𝔉_T(G)$ for some classes of graphs.
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