We determine the space of 1-point correlation functions associated with the Moonshine module: they are precisely those modular forms of non-negative integral weight which are holomorphic in the upper half plane, have a pole of order at most 1 at infinity, and whose Fourier expansion has constant 0. There are Monster-equivariant analogues in which one naturally associates to each element in the Monster a modular form of fixed weight k, the case k=0 corresponding to the original ``Moonshine'' of Conway and Norton.

En m~moire de ma m~re polynomials

We prove global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampere equations which yield corresponding existence and regularity results. Our conditions are essentially optimal and our techniques, while drawing on previous investigations, are substantially new.

Here ~9(x):p(x; wl,w3) denotes the elliptic Weierstrass function with fundamental periods 2wl and 2w3 (Im(w3/Wl)#0). In the special case where Wl is real and w3 is purely imaginary, the potential q(x) in (1.1) is real-valued and Ince's striking result [51], in modern spectral-theoretic terminology, yields that the spectrum of the unique self-adjoint operator associated with the differential expression L2=d2/dx 2 +q(x) in L2(R) exhibits finitely many bands (and gaps, respectively), that is,

O. Introduction 1. Notation 2. Some exterior algebra 3. Differential forms in Lnm (i2, A n) 4. Differential systems for quasiregular mappings 5. Liouville Theorem in even dimensions 6. Hodge theory in LP(R n) 7. The Beltrami equation in even dimensions 8. The Beurling-Ahlfors operator 9. Regularity theorems for quasiregular mappings 10. The Caccioppoli type estimate 11. Removability theorems for quasiregular mappings 12. Some examples Appendix: The 4-dimensional case