5,692 research outputs found
Generalization of Uniqueness Theorems for Entire and Meromorphic Functions
In this paper, we deal with the uniqueness problems on entire and meromorphic functions concerning
differential polynomials that share fixed-points. Moreover, we generalise and improve
some results of Weichuan Lin, Hongxun Yi, Meng Chao, C. Y. Fang, M. L. Fang and Junfeng xu
Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem
We establish an extension of Liouville's classical representation theorem for
solutions of the partial differential equation and combine
this result with methods from nonlinear elliptic PDE to construct holomorphic
maps with prescribed critical points and specified boundary behaviour. For
instance, we show that for every Blaschke sequence in the unit disk
there is always a Blaschke product with as its set of critical
points. Our work is closely related to the Berger-Nirenberg problem in
differential geometry.Comment: 21 page
Value-sharing of meromorphic functions on a Riemann surface
We present some results on two meromorphic functions from S to the Riemann
sphere sharing a number of values where S is a Riemann surface of one of the
following types: compact, compact minus finitely many points, the unit disk, a
torus, the complex plane.Comment: 15 page
On the Relationship between the Uniqueness of the Moonshine Module and Monstrous Moonshine
We consider the relationship between the conjectured uniqueness of the
Moonshine Module, , and Monstrous Moonshine, the genus zero
property of the modular invariance group for each Monster group Thompson
series. We first discuss a family of possible meromorphic orbifold
constructions of based on automorphisms of the Leech
lattice compactified bosonic string. We reproduce the Thompson series for all
51 non-Fricke classes of the Monster group together with a new relationship
between the centralisers of these classes and 51 corresponding Conway group
centralisers (generalising a well-known relationship for 5 such classes).
Assuming that is unique, we then consider meromorphic
orbifoldings of and show that Monstrous Moonshine holds if
and only if the only meromorphic orbifoldings of give
itself or the Leech theory. This constraint on the
meromorphic orbifoldings of therefore relates Monstrous
Moonshine to the uniqueness of in a new way.Comment: 53 pages, PlainTex, DIAS-STP-93-0
On the uniqueness theorem of Holmgren
We rereview the classical Cauchy-Kovalevskaya theorem and the related
uniqueness theorem of Holmgren, in the simple setting of powers of the
Laplacian and a smooth curve segment in the plane. As a local problem, the
Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the
existence and uniqueness issues. Here, we consider a global uniqueness problem
of Holmgren's type. Perhaps surprisingly, we obtain a connection with the
theory of quadrature identities, which demonstrates that rather subtle
algebraic properties of the curve come into play. For instance, if is
the interior domain of an ellipse, and is a proper arc of the ellipse
, then there exists a nontrivial biharmonic function in
which vanishes to degree three on (i.e., all partial derivatives
of of order vanish on ) if and only if the ellipse is a circle.
Finally, we consider a three-dimensional case, and analyze it partially using
analogues of the square of the 2X2 Cauchy-Riemann operator.Comment: 14 page
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