255,845 research outputs found

### Symmetries of Analytic Curves

Analytic curves are classified w.r.t. their symmetries under a regular Lie
group action on an analytic manifold. We show that an analytic curve is either
exponential or splits into countably many analytic immersive curves; each of
them decomposing naturally into symmetry free subcurves mutually and uniquely
related by the group action. We conclude that a connected analytic
1-dimensional submanifold is either analytically diffeomorphic to the unit
circle or some interval, or that each point (except for at most countably many)
admits a symmetry free chart.Comment: 49 page

### STARRY: Analytic Occultation Light Curves

We derive analytic, closed form, numerically stable solutions for the total
flux received from a spherical planet, moon or star during an occultation if
the specific intensity map of the body is expressed as a sum of spherical
harmonics. Our expressions are valid to arbitrary degree and may be computed
recursively for speed. The formalism we develop here applies to the computation
of stellar transit light curves, planetary secondary eclipse light curves, and
planet-planet/planet-moon occultation light curves, as well as thermal
(rotational) phase curves. In this paper we also introduce STARRY, an
open-source package written in C++ and wrapped in Python that computes these
light curves. The algorithm in STARRY is six orders of magnitude faster than
direct numerical integration and several orders of magnitude more precise.
STARRY also computes analytic derivatives of the light curves with respect to
all input parameters for use in gradient-based optimization and inference, such
as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit
observed light curves to infer properties of a celestial body's surface map.Comment: 55 pages, 20 figures. Accepted to the Astronomical Journal. Check out
the code at https://github.com/rodluger/starr

### Definable sets of Berkovich curves

In this article, we functorially associate definable sets to $k$-analytic
curves, and definable maps to analytic morphisms between them, for a large
class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association
allows us to have definable versions of several usual notions of Berkovich
analytic geometry such as the branch emanating from a point and the residue
curve at a point of type 2. We also characterize the definable subsets of the
definable counterpart of $X$ and show that they satisfy a bijective relation
with the radial subsets of $X$. As an application, we recover (and slightly
extend) results of Temkin concerning the radiality of the set of points with a
given prescribed multiplicity with respect to a morphism of $k$-analytic
curves.
In the case of the analytification of an algebraic curve, our construction
can also be seen as an explicit version of Hrushovski and Loeser's theorem on
iso-definability of curves. However, our approach can also be applied to
strictly $k$-affinoid curves and arbitrary morphisms between them, which are
currently not in the scope of their setting.Comment: 53 pages, 1 figure. v2: Section 7.2 on weakly stable fields added and
other minor changes. Final version. To appear in Journal of the Institute of
Mathematics of Jussie

### Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for
meromorphic functions. This means that the meromorphic function in question is
supposed to be extendable along a sequence of complex curves, which are
arbitrary, not necessarily straight lines. Moreover, these curves are not
supposed to belong to any finite dimensional analytic family. The conclusion of
our theorem is that nevertheless the function in question meromorphically
extends along an (infinite dimensional) analytic family of complex curves and
its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and
improvements. To appear in Arkiv f\"or matemati

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