82 research outputs found
Maps conjugating holomorphic maps in C^n
If f is a bijection from C^n onto a complex manifold M, which conjugates
every holomorphic map in C^n to an endomorphism in M, then we prove that f is
necessarily biholomorphic or antibiholomorphic. This extends a result of A.
Hinkkanen to higher dimensions. As a corollary, we prove that if there is an
epimorphism from the semigroup of all holomorphic endomorphisms of C^n to the
semigroup of holomorphic endomorphisms in M, or an epimorphism in the opposite
direction for a doubly-transitive M, then it is given by conjugation by some
biholomorphic or antibiholomorphic map. We show also that there are two
unbounded domains in C^n with isomorphic endomorphism semigroups but which are
neither biholomorphically nor antibiholomorphically equivalent.Comment: 10 page
Algebraic surfaces holomorphically dominable by C^2
Using the Kodaira dimension and the fundamental group of X, we succeed in
classifying algebraic surfaces which are dominable by C^2 except for certain
cases in which X is an algebraic surface of Kodaira dimension zero and the case
when X is rational without any logarithmic 1-form. More specifically, in the
case when X is compact (namely projective), we need to exclude only the case
when X is birationally equivalent to a K3 surface (a simply connected compact
complex surface which admits a globally non-vanishing holomorphic 2-form) that
is neither elliptic nor Kummer.
With the exceptions noted above, we show that for any algebraic surface of
Kodaira dimension less than 2, dominability by C^2 is equivalent to the
apparently weaker requirement of the existence of a holomorphic image of C
which is Zariski dense in the surface. With the same exceptions, we will also
show the very interesting and revealing fact that dominability by C^2 is
preserved even if a sufficiently small neighborhood of any finite set of points
is removed from the surface. In fact, we will provide a complete classification
in the more general category of (not necessarily algebraic) compact complex
surfaces before tackling the problem in the case of non-compact algebraic
surfaces
Hyperbolic automorphisms and holomorphic motions in C<sup>2</sup>
This article does not have an abstract
Generative Plug and Play: Posterior Sampling for Inverse Problems
Over the past decade, Plug-and-Play (PnP) has become a popular method for
reconstructing images using a modular framework consisting of a forward and
prior model. The great strength of PnP is that an image denoiser can be used as
a prior model while the forward model can be implemented using more traditional
physics-based approaches. However, a limitation of PnP is that it reconstructs
only a single deterministic image.
In this paper, we introduce Generative Plug-and-Play (GPnP), a generalization
of PnP to sample from the posterior distribution. As with PnP, GPnP has a
modular framework using a physics-based forward model and an image denoising
prior model. However, in GPnP these models are extended to become proximal
generators, which sample from associated distributions. GPnP applies these
proximal generators in alternation to produce samples from the posterior. We
present experimental simulations using the well-known BM3D denoiser. Our
results demonstrate that the GPnP method is robust, easy to implement, and
produces intuitively reasonable samples from the posterior for sparse
interpolation and tomographic reconstruction. Code to accompany this paper is
available at https://github.com/gbuzzard/generative-pnp-allerton .Comment: 8 pages, submitted to 2023 IEEE Allerton Conferenc
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