19,518 research outputs found
Extending Whitney's extension theorem: nonlinear function spaces
We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains , with non-smooth
boundary, in possibly non-compact manifolds. Assuming is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition
A.3, which makes no impact on proofs or results. Version submitted for
publication. v3 small changes in response to referee comments, title
extended. v4 crucial gap filled, results not affected. v5 final version to
appear in Annales de l'Institut Fourie
PyDEC: Software and Algorithms for Discretization of Exterior Calculus
This paper describes the algorithms, features and implementation of PyDEC, a
Python library for computations related to the discretization of exterior
calculus. PyDEC facilitates inquiry into both physical problems on manifolds as
well as purely topological problems on abstract complexes. We describe
efficient algorithms for constructing the operators and objects that arise in
discrete exterior calculus, lowest order finite element exterior calculus and
in related topological problems. Our algorithms are formulated in terms of
high-level matrix operations which extend to arbitrary dimension. As a result,
our implementations map well to the facilities of numerical libraries such as
NumPy and SciPy. The availability of such libraries makes Python suitable for
prototyping numerical methods. We demonstrate how PyDEC is used to solve
physical and topological problems through several concise examples.Comment: Revised as per referee reports. Added information on scalability,
removed redundant text, emphasized the role of matrix based algorithms,
shortened length of pape
Polyfolds: A First and Second Look
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding
commonalities in the analytic framework for a variety of geometric elliptic
PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to
systematically address the common difficulties of compactification and
transversality with a new notion of smoothness on Banach spaces, new local
models for differential geometry, and a nonlinear Fredholm theory in the new
context. We shine meta-mathematical light on the bigger picture and core ideas
of this theory. In addition, we compiled and condensed the core definitions and
theorems of polyfold theory into a streamlined exposition, and outline their
application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version,
to appear in the EMS Surv. Math. Sc
Aspects of p-adic non-linear functional analysis
The article provides an introduction to infinite-dimensional differential
calculus over topological fields and surveys some of its applications, notably
in the areas of infinite-dimensional Lie groups and dynamical systems.Comment: 19 pages; LaTe
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