678 research outputs found
Tilings of quadriculated annuli
Tilings of a quadriculated annulus A are counted according to volume (in the
formal variable q) and flux (in p). We consider algebraic properties of the
resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p
must be roots of unity and for q > 0, real negative.Comment: 33 pages, 12 figures; Minor changes were made to make some passages
cleare
The geometry of the critical set of nonlinear periodic Sturm-Liouville operators
We study the critical set C of the nonlinear differential operator F(u) = -u"
+ f(u) defined on a Sobolev space of periodic functions H^p(S^1), p >= 1. Let
R^2_{xy} \subset R^3 be the plane z = 0 and, for n > 0, let cone_n be the cone
x^2 + y^2 = tan^2 z, |z - 2 pi n| 0}
cone_n. For a generic smooth nonlinearity f: R -> R with surjective derivative,
we show that there is a diffeomorphism between the pairs (H^p(S^1), C) and
(R^3, Sigma) x H where H is a real separable infinite dimensional Hilbert
space.Comment: Added references, fixed typos; 24 pages, 4 figure
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