279 research outputs found
Growth of Solutions of Complex Differential Equations in a Sector of the Unit Disc
In this paper, we deal with the growth of solutions of homogeneous linear
complex differential equation by using the concept of lower
[\textit{p,q}]-order and lower [\textit{p,q}]-type in a sector of the unit disc
instead of the whole unit disc, and we obtain similar results as in the case of
the unit disc.Comment: 21 page
All admissible meromorphic solutions of Hayman's equation
We find all non-rational meromorphic solutions of the equation
, where , and
are rational functions of . In so doing we answer a question of
Hayman by showing that all such solutions have finite order. Apart from special
choices of the coefficient functions, the general solution is not meromorphic
and contains movable branch points. For some choices for the coefficient
functions the equation admits a one-parameter family of non-rational
meromorphic solutions. Nevanlinna theory is used to show that all such
solutions have been found and allows us to avoid issues that can arise from the
fact that resonances can occur at arbitrarily high orders. We actually solve
the more general problem of finding all meromorphic solutions that are
admissible in the sense of Nevanlinna theory, where the coefficients ,
and are meromorphic functions.Comment: 11 page
Lower bounds on nodal sets of eigenfunctions via the heat flow
We study the size of nodal sets of Laplacian eigenfunctions on compact
Riemannian manifolds without boundary and recover the currently optimal lower
bound by comparing the heat flow of the eigenfunction with that of an
artifically constructed diffusion process. The same method should apply to a
number of other questions; for example, we prove a sharp result saying that a
nodal domain cannot be entirely contained in a small neighbourhood of a
'reasonably flat' surface. We expect the arising concepts to have more
connections to classical theory and pose some conjectures in that direction
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