163 research outputs found
Topologies of nodal sets of random band limited functions
It is shown that the topologies and nestings of the zero and nodal sets of
random (Gaussian) band limited functions have universal laws of distribution.
Qualitative features of the supports of these distributions are determined. In
particular the results apply to random monochromatic waves and to random real
algebraic hyper-surfaces in projective space.Comment: An announcement of recent results. Includes an announcement of the
resolution of some open questions from the older version. 11 pages, 6 figure
Integral points on Markoff type cubic surfaces
For integers , we consider the affine cubic surface given by
. We show that for
almost all the Hasse Principle holds, namely that is
non-empty if is non-empty for all primes , and that
there are infinitely many 's for which it fails. The Markoff morphisms act
on with finitely many orbits and a numerical study points
to some basic conjectures about these "class numbers" and Hasse failures. Some
of the analysis may be extended to less special affine cubic surfaces.Comment: 57 pages, many figures, revised Introduction, Sec. 5 and Appendi
Real zeros of holomorphic Hecke cusp forms
This note is concerned with the zeros of holomorphic Hecke cusp forms of
large weight on the modular surface. The zeros of such forms are symmetric
about three geodesic segments and we call those zeros that lie on these
segments, real. Our main results give estimates for the number of real zeros as
the weight goes to infinity
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