483 research outputs found
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
The descriptive theory of represented spaces
This is a survey on the ongoing development of a descriptive theory of
represented spaces, which is intended as an extension of both classical and
effective descriptive set theory to deal with both sets and functions between
represented spaces. Most material is from work-in-progress, and thus there may
be a stronger focus on projects involving the author than an objective survey
would merit.Comment: survey of work-in-progres
Topological Wiener-Wintner theorems for amenable operator semigroups
Inspired by topological Wiener-Wintner theorems we study the mean ergodicity
of amenable semigroups of Markov operators on and show the connection to
the convergence of strong and weak ergodic nets. The results are then used to
characterize mean ergodicity of Koopman semigroups corresponding to skew
product actions on compact group extensions.Comment: 26 pages, referee's suggestions incorporated, to appear in "Ergodic
Theory and Dynamical Systems
Integration on the Surreals
Conway's real closed field No of surreal numbers is a sweeping generalization
of the real numbers and the ordinals to which a number of elementary functions
such as log and exponentiation have been shown to extend. The problems of
identifying significant classes of functions that can be so extended and of
defining integration for them have proven to be formidable. In this paper, we
address this and related unresolved issues by showing that extensions to No,
and thereby integrals, exist for most functions arising in practical
applications. In particular, we show they exist for a large subclass of the
resurgent functions, a subclass that contains the functions that at infinity
are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as
well as generic solutions to linear and nonlinear systems of ODEs possibly
having irregular singularities. We further establish a sufficient condition for
the theory to carry over to ordered exponential subfields of No more generally
and illustrate the result with structures familiar from the surreal literature.
We work in NBG less the Axiom of Choice (for both sets and proper classes),
with the result that the extensions of functions and integrals that concern us
here have a "constructive" nature in this sense. In the Appendix it is shown
that the existence of such constructive extensions and integrals of
substantially more general types of functions (e.g. smooth functions) is
obstructed by considerations from the foundations of mathematics.Comment: This paper supersedes the positive portion of O. Costin, P. Ehrlich
and H. Friedman, "Integration on the surreals: a conjecture of Conway,
Kruskal and Norton", arXiv:1505.02478v3, 24 Aug 2015. A separate paper
superseding the negative portion of the earlier arXiv preprint is in
preparation by H. Friedman and O. Costi
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