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Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Fixed Points of Generalized Conjugations
, 802
"... Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these space by meas of the duality product. Generalized ..."
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Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these space by meas of the duality product. Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optima transportation problems, variational analysis and also optimization. If the coupled spaces are equal, generalized conjugations define order reversing maps of a family of functions into itself. In this case, is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugateds. Here we prove that any generalized symmetric conjugation has fixed points. The basic tool of the proof is a variational principle involving the order reversing feature of the conjugation. As an application of this abstract result, we will extend to real linear topological spaces a fixedpoint theorem for Fitzpatrickâ€™s functions, previously proved in Banach spaces.