W. Kulpa

THE POINCARE-MIRANDA THEOREM

equivalent result was established earlier by Bohl [2] in 1904. It was Hadamard [4] who in 1910 gave (using the Kroneckex index) the first proof fox arbitrary n. In 1912 Brouwer gave another proof using the simplicial approximation technique, and notions of degree. A short and simple proof of the BohlwBrouwer theorem was given in 1929 by Knaster-Kuratowski-Mazurkiewicz [7]; the proof was based on the lemma of Sperner [12]. In this paper, we also apply combinatorial methods of proof taken from the Sperner lemma, but our proof seems to be simpler because it does not require the notion of barycentric coordinates and barycentric subdivision. Let us begin by discussing the Poincare-Miranda Theorem Let k > 1 be a given natural number and let Zk := {i/k:i c Z}, whexe Z denotes the set of integers. Let Zkn denote the Cartesian product of n copies of the set Zk Zkn {z {l, * * *, n} Zk 0 z is a map}

Some remarks on Park's abstract convex spaces

We discuss S. Parks abstract convex spaces and their relevance to classical convexieties and

L1 cheapest paths in “Fjord Scenery”

L^{\ast }

On extensions of the domain invariance theorem

-operators. We construct an example of a space satisfying the partial KKM principle that is not a KKM space. The existence of such a space solves a problem by S. Park.

Intersection properties of Helly families

Abstract We show a simple proof of the existence of a path on the “border of water and rocks” based on combinatorial induction procedure and we present an algorithm for computing L 1 shortest path in “Fjord Scenery”. The proposed algorithm is a version of the Dijkstra technique adapted to a rectangle map with a square network. A few pre-processing modifications of the algorithm following from the combinatorial procedure are included. The validity of this approach is shown by numerical calculations for an example.

On the size of a map

Abstract If f maps continuously a compact subset X of R n into R n and x is a point whose distance from the boundary ∂X is greater than double diameter of the fibres of the points in f ( ∂X ) then f ( x ) is in the interior of f ( X ). This theorem extends some results due to Borsuk and Sitnikov.

Equivalent forms of the Brouwer fixed point theorem

Abstract The Helly convex-set theorem is extended onto topological spaces. From our results it follows that if there are given m +2 convex subsets of an m -dimensional contractible Hausdorff space and the intersection of each collection of m +1 the subsets is a nonempty contractible set, then the intersection of the whole collection of m +2 subsets is a nonempty set. Our results are stated in terms of Helly families, the definition of which involves k -connectedness of intersections of m − k sets for k =−1,0,…, m −1.

The Sorgenfrey line has no connected compactification

Some properties depending on an upper bound of the diameter of fibers of a continuous map f from the n-dimensional unit cube In to the Euclidean space are investigated. In particular, we consider the problem when the image f(In) has the nonempty interior. Obtained results are consequences of the Poincaré theorem and some theorems on extensions of maps. Generalizations of the De Marco theorem and the Borsuk theorem are presented.

Poincaré and domain invariance theorem

In this paper we survey a set of Brouwer fixed point theorem equivalents. These equivalents are divided into four loops related to (1) the Borsuk retraction theorem, (2) the Himmelberg fixed point theorem, (3) the Gale lemma and (4) the Nash equilibrium theorem.