Barrelledness in Topological and Ordered Vector Spaces by T. Husain, S.M. Khaleelulla

By T. Husain, S.M. Khaleelulla

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What is nice in this reformulation, is that one can immediately notice something. Let us remember that we are interested in solutions of PVI, and even if we believe that most of them are nonclassical, it may happen that our solution is classical. And, generally, let us look whether there are classical solutions. One of them is seen immediately; if all αj are zero, then we get the equation d2 z/dτ 2 = 0, whose all solutions are linear functions. By the way, in the initial form of the equation this solution is not seen.

Technically, the problem I am speaking about concerns the attempt to understand the structure of the potential of quantum cohomology of the projective plane. I am just going to write down explicitly what this means. The projective plane, say over the field of complex numbers, yields its three-dimensional cohomology space, which is spanned by the cohomology classes of the whole plane ∆0 , of a line ∆1 , and of a point ∆2 . This three-dimensional space has coordinates which I denote by x, y, z: H ∗ (P 2 , C) = {x∆0 + y∆1 + z∆2 }.

This being not very easy to do, however, a little of algebraic geometry enables one to expose this argument correctly. It is striking that these numbers N (d) satisfy absolutely nontrivial bilinear relations allowing us to express them consecutively, one through another. Removing from Φ(x, y, z) the classical contribution 12 (xy 2 +x2 z) whose sense I explained yesterday in my lecture at the Moscow Mathematical Society, one gets the function ∞ z 3d−1 dy φ(y, z) = e N (d) (3d − 1)! d=1 that already does not depend on x.

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