Download PDF by A. Barlotti, etc.: Combinatorics ’81, in honour of Beniamino Segre: proceedings

By A. Barlotti, etc.

ISBN-10: 0444865462

ISBN-13: 9780444865465

Curiosity in combinatorial suggestions has been enormously stronger by means of the purposes they could provide in reference to computing device know-how. The 38 papers during this quantity survey the cutting-edge and file on contemporary leads to Combinatorial Geometries and their applications.

Contributors: V. Abatangelo, L. Beneteau, W. Benz, A. Beutelspacher, A. Bichara, M. Biliotti, P. Biondi, F. Bonetti, R. Capodaglio di Cocco, P.V. Ceccherini, L. Cerlienco, N. Civolani, M. de Soete, M. Deza, F. Eugeni, G. Faina, P. Filip, S. Fiorini, J.C. Fisher, M. Gionfriddo, W. Heise, A. Herzer, M. Hille, J.W.P. Hirschfield, T. Ihringer, G. Korchmaros, F. Kramer, H. Kramer, P. Lancellotti, B. Larato, D. Lenzi, A. Lizzio, G. Lo Faro, N.A. Malara, M.C. Marino, N. Melone, G. Menichetti, ok. Metsch, S. Milici, G. Nicoletti, C. Pellegrino, G. Pica, F. Piras, T. Pisanski, G.-C. Rota, A. Sappa, D. Senato, G. Tallini, J.A. Thas, N. Venanzangeli, A.M. Venezia, A.C.S. Ventre, H. Wefelscheid, B.J. Wilson, N. Zagaglia Salvi, H. Zeitler.

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Additional info for Combinatorics ’81, in honour of Beniamino Segre: proceedings of the International Conference on Combinatorial Geometrics and their Applications, Rome, June 7-12, 1981

Example text

B) Show that the statement in (a) is also implied by the Borsuk–Ulam theorem. 4. , onto). 1 for n implies the validity of all the statements for n−1. 6. (Generalized Lyusternik–Shnirel’man theorem [Gre02]) Derive the following common generalization of (LS-c) and (LS-o): Whenever S n is covered by n+1 sets A1 , A2 , . . , An+1 , each Ai open or closed, there is an i such that Ai ∩ (−Ai ) = ∅. 7. Does the Lyusternik–Shnirel’man theorem remain valid for coverings of S n by n+1 sets, each of which can be obtained from open sets by finitely many set-theoretic operations (union, intersection, difference)?

In the forthcoming proof we also need an additional condition on the triangulation T of B n in Tucker’s lemma. For k = 0, 1, 2, . . , n−1, we define Hk+ = {x ∈ S n−1 : xk+1 ≥ 0, xk+2 = xk+3 = · · · = xn = 0}, Hk− = {x ∈ S n−1 : xk+1 ≤ 0, xk+2 = xk+3 = · · · = xn = 0}. , and finally, H0+ and H0− are a pair of antipodal points. We assume that T respects this structure: For each i = 0, 1, . . , n−1, there are subcomplexes that triangulate Hi+ and Hi− (such triangulations can be constructed, for instance, as refinements of +n ).

We define g(x) as the point in which the ray originating in f (x) and going through x intersects S n−1 . This g contradicts (BU2b). Notes. The earliest reference for what is now commonly called the Borsuk–Ulam theorem is probably Lyusternik and Shnirel’man [LS30] from 1930 (the covering version (LS-c)). Borsuk’s paper [Bor33] is from 1933. The only written reference concerning Ulam’s role in the matter seems to be Borsuk’s footnote quoted above. Since then, hundreds of papers with various new proofs, variations of old proofs, generalizations, and applications have appeared; the most comprehensive survey known to me, Steinlein [Ste85] from 1985, lists nearly 500 items in the bibliography.

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Combinatorics ’81, in honour of Beniamino Segre: proceedings of the International Conference on Combinatorial Geometrics and their Applications, Rome, June 7-12, 1981 by A. Barlotti, etc.

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