Stanisław Spież

Moving compacta in Rm apart

Abstract We prove that if X and Y are compacta such that dim( X × Y ) m and 2 dim X + dim Y ⩽ 2 m - 2, then any two mappings X → R m and Y → R m admit arbitrarily close approximations with disjoint images. This is a consequence of a more general result which is stated below as Theorem 3 and gives a sufficient condition under which two mappings into R m admit e-approximation with disjoint images.

Embeddings of polyhedra in Rm and the deleted product obstruction

Abstract Weber has proved that if 2 m ⩾ 3( n + 1) then an n -dimensional polyhedron K embeds in R m if and only if there exists an equivariant map from the deleted product K ∗ into the sphere S m − 1 . As a consequence he has obtained that in the same range of dimensions an n -dimensional polyhedron embeds in R m if and only if it quasi embeds in R m . We show that for m ⩾ max(4, n ) the dimension restrictions in Webers results are necessary in all cases. This leaves only two open cases remaining (namely m = 3 and n = 2 or 3) in related questions about embeddings.

Generalized manifolds in products of curves

The intent of this article is to distinguish and study some n-dimensional compacta (such as weak n-manifolds) with respect to embeddability into products of n curves. We show that if X is a locally connected weak n-manifold lying in a product of n curves, then rank H 1 (X) > n. If rank H 1 (X) = n, then X is an n-torus. Moreover, if rank H 1 (X) < 2n, then X can be presented as a product of an m-torus and a weak (n ― m)-manifold, where m ≥ 2n ― rank H 1 (X). If rank H 1 (X) < ∞, then X is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak 2-manifold X lying in a product of two graphs such that H 2 (X) = 0.

Strong shape of uniform spaces

Abstract A strong shape category for finitistic uniform spaces is constructed and it is shown, that certain nice properties known from strong shape theory of compact Hausdorff spaces carry over to this setting. These properties include a characterization of the new category as localization of the homotopy category, the product property and obstruction theory.

Equilibrium existence and topology in some repeated games with incomplete information

This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals. This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping F on a compact convex set P C R n , we give further conditions which imply that every point p 0 ∈ P belongs to the convex hull of a finite subset P 0 of the domain of F satisfying ∩ x ∈ P0 F(x) ¬= O.

Remarks on the Classical Threshold Secret Sharing Schemes

We survey some results related to classical secret sharing schemes defined in Shamir [10] and Blakley [1], and developed in Brickell [2] and Lai and Ding [4]. Using elementary symmetric polynomials, we describe in a unified way which allocations of identities to participants define Shamirs threshold scheme, or its generalization by Lai and Ding, with a secret placed as a fixed coefficient of the scheme polynomial. This characterization enabled proving in Schinzel et al. [8], [9] and Spiez et al. [13] some new and non-trivial properties of such schemes. Also a characterization of matrices corresponding to the threshold secret sharing schemes of Blakley and Brickells type is given. Using Gaussian elimination we provide an algorithm to construct all such matrices which is efficient in the case of relatively small matrices. The algorithm may be useful in constructing systems where dynamics is important (one may generate new identities using it). It can also be used to construct all possible MDS codes. MSC: primary 94A62; secondary 11T71; 11C20

Equilibria in a class of games and topological results implying their existence.

We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.ResumenExponemos resultados relacionados con el problema de la existencia de equilibrios en algunas clases de juegos bipersonales infinitamente repetidos con información incompleta por una de las partes, considerados por primera vez por Aumann, Maschler y Stearns. Generalizamos este marco a uno más amplio de problemas de agentes principales. También discutimos los resultados topológicos necesarios, presentándolos dualmente (usando cohomología en lugar de homología) y de modo más sistemático que en nuestros artículos anteriores.

On questions of strict contractibility

Abstract A space X is strictly contractible to a point x 0 ∈ X if there exists a homotopy H :X×[0,1]→X starting at the identity such that H ( x , t )= x 0 if and only if either x = x 0 or t =1. Michael proved that if E is a locally compact and non-compact space then E ×0 is a perfect retract of the product E ×[0,1) if and only if the one-point compactification E ∗ =E∪x ∗ of E is strictly contractible to x ∗ . We answer some questions posed by Michael [Closed retracts and perfect retracts, Topology Appl., to appear] and we characterize strictly contractible ANRs in shape-theoretic terms.

A parametrized version of the Borsuk Ulam theorem

We show that for a ‘continuous’ family of Borsuk–Ulam situations, parametrized by points of a compact manifold W, its solution set also depends ‘continuously’ on the parameter space W. By such a family we understand a compact set Z⊂W×Sm×ℝm, the solution set consists of points (w, x, v)∈Z such that also (w,−x, v)∈Z. Here, ‘continuity’ means that the solution set supports a homology class that maps onto the fundamental class of W. We also show how to construct such a family starting from a ‘continuous’ family Y⊂∂ W×ℝm when W is a compact top-dimensional subset in ℝm+1. This solves a problem related to a conjecture that is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with ℤ/2-coefficients.

On constructing privileged coalitions in Shamirʼs type scheme

Abstract We consider Shamirʼs type secret sharing scheme with the secret placed as a coefficient a i of the scheme polynomial f ( x ) = ∑ i = 0 k − 1 a i x i over F q . A coalition of shareholders equipped with pairwise different public identities t j belonging to F q is called ( k , i ) -authorized if the holders using their secret shares y j = f ( t j ) ( 1 ⩽ j ⩽ n ) are able to reconstruct the secret by themselves. A ( k , i ) -authorized coalition of k − 1 or less shareholders is called a ( k , i ) -privileged coalition. We construct all ( k , i ) -privileged coalitions of k − 1 shareholders if i ≠ 0 , k − 1 . This construction allows us to deduce a new estimate for the number of such coalitions, which implies that they exist if q > 2 k − 1 . We also give a method of extending a given minimal ( k , i ) -privileged coalition of r shareholders to a coalition of n shareholders ( n ⩾ r ) each of whose ( k , i ) -privileged subcoalitions contains the given one provided q ⩾ n + r ( n − 2 k − 2 ) .