By P. J. Higgins

Different types and Groupoids (Mathematics experiences)

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PROOF. '* '* The next result generalizes one of the forms of the Schauder fixed point theorem; it follows that the principle of KKM-maps is in fact equivalent to the Brouwer fixed point theorem. 5) Theorem (Ky Fan [10)). Let C be a compact convex set in a normed space E and let /: C - E be continuous. Assume further that, for each x E C with x j(x) the line segment [x,j(x)] contains at least two points of C. Then / has a fixed point. '* PROOF. Define G : C - 2E by G(x) = {y E C Illy - /(y) I :s Ilx - /(y)II}.

There was even a serious journal, Acta Arithmetica (which continues to this day), devoted to Number Theory, but this beautiful and important area was far from the forefront of mathematical preoccupation in Poland before World War II. I cannot remember at all how I came to think about number theoretic problems in connection with Probability Theory, but I do remember making what appeared to me then to be a great discovery (it wasn't). If cp(n) is the familiar Euler function one has q,~) II (1 - = i) which can be written in the form q,(n) -n II (1 _~(n) ) p A PERSONAL HISTORY OF THE SCOTTISH BOOK 25 where Qp (n) ={ 0,1, pfn.

PROOF. EI be the family of all continuous seminorms in E. For each i E I set AI = {y E C I PI(y - f(y» = a}. A point Yo E C is a fixed point for J, if and only if Yo E n lElA I' By compactness of C we need show only that njEJAj f) for each finite subset J C I. Define G : C - 2E by '* G(x) = {y E C I E PJ{Yo - f(y» :5 jEJ E PAX - f(y»}. jEJ It is easy to verify that G is a KKM-map; consequently there is a point Yo E C such that E PJ{Yo - f(yo» :5 jEJ E PJ{x - f(yo» jEJ for all x E C. This clearly implies that PJ{Yo - f(yo» = 0 for j E J and thus Yo E njEJAj.

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Categories and groupoids by P. J. Higgins

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