1. Center of Complex Systems and Visualization
  2. Lehre
  3. Oberseminar

In dem regelmäßig stattfindenden Oberseminar tragen Gäste aus aller Welt über Forschungsarbeiten zu Themen vor, die mit der Arbeit von CeVis und MeVis in Verbindung stehen, und Mitarbeiter von CeVis und MeVis präsentieren ihre neusten Ergebnisse.

C(M)eVis-Oberseminar am 4.7.2001

Datum: Mittwoch, der 4.7.2001
Zeit: 11.00 Uhr s.t.
Ort: Seminarraum Mandelbrot
Referent: Prof. Dr. Semeon Bogatyi (Faculty of Mechanics and Mathematics, Moscow State University)

k-regular mappings: interpolations, Chebyshev approximations, transversal Tverberg theorem

A mapping f: X -> R^m is called k-regular, if the images f(x_0),...,f(x_k) of no (k+1) points x_0,...,x_k from X are in a (k-1)-dimensional plane of R^m. 1-regular maps are exactly the embeddings. Any (k+1)-regular map is k-regular. Haar-Kolmogorov-Rubinshtein theorem connects k-regular maps with possibility to interpolate and with the dimension of the Chebyshev's best approximators. Boltyanski proved that for a n-dimensional polyhedron P the conditions (1) and (2) are equivalent:

(1) m >= kn+n+k;

(2) the set of all k-regular maps of P into R^m is dense in the space of all continuous maps of P into R^m.

We show that for an odd number k analog of Van Kampen-Flores theorem is true. For any continuous map of n-dimensional skeleton of kn+n+k+(k+1)/2-dimensional simplex into R^m, where m=kn+n+k-1, there exists k+1 pare-wise disjoint simplexes, whose images have a common (k-1)-dimensional transversal. Relation with transversal Tverberg theorem is discussed. Any set in R^m of cardinality (q-1-d)(m-d)+q can be partitioned onto q parts M_1,...M_q, whose convex hulls have a common d-dimensional transversal, that is for some d-dimensional plane P the sets conv M_i\cap \Pi, i=1,...q, are non-empty. The case d=0 is the Tverberg theorem, generalizing the famous Radon's theorem (q=2).