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Properties of a hypothetical exotic complex structure on ℂ P 3

J. R. Brown — 2007

Mathematica Bohemica

We consider almost-complex structures on ℂ P 3 whose total Chern classes differ from that of the standard (integrable) almost-complex structure. E. Thomas established the existence of many such structures. We show that if there exists an “exotic” integrable almost-complex structures, then the resulting complex manifold would have specific Hodge numbers which do not vanish. We also give a necessary condition for the nondegeneration of the Frölicher spectral sequence at the second level.

Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle

Nachman Aronszajn;  R. D. Brown;  R. S. Butcher — 1973

Annales de l'institut Fourier

A technique is developed for constructing the solution of Δ 2 u = F in R = { ( x , y ) : | x | < a , | y | < b } , subject to boundary conditions u = φ , ∂ u ∂ n = ψ on ∂ R . The problem is reduced to that of finding the orthogonal projection P w of w in L 2 ( R ) onto the subspace H of square integrable functions harmonic in R . This problem is solved by decomposition H into the closed direct (not orthogonal) sum of two subspaces H ( 1 ) , H ( 2 ) for which complete orthogonal bases are known. P is expressed in terms of the projections P ( 1 ) , P ( 2 ) of L 2 ( R ) onto H ( 1 ) , H ( 2 ) respectively. The resulting construction...

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