2,065 research outputs found
From coinductive proofs to exact real arithmetic: theory and applications
Based on a new coinductive characterization of continuous functions we
extract certified programs for exact real number computation from constructive
proofs. The extracted programs construct and combine exact real number
algorithms with respect to the binary signed digit representation of real
numbers. The data type corresponding to the coinductive definition of
continuous functions consists of finitely branching non-wellfounded trees
describing when the algorithm writes and reads digits. We discuss several
examples including the extraction of programs for polynomials up to degree two
and the definite integral of continuous maps
Which electric fields are realizable in conducting materials?
In this paper we study the realizability of a given smooth periodic gradient
field defined in \RR^d, in the sense of finding when one can
obtain a matrix conductivity \si such that \si\nabla u is a divergence free
current field. The construction is shown to be always possible locally in
\RR^d provided that is non-vanishing. This condition is also
necessary in dimension two but not in dimension three. In fact the
realizability may fail for non-regular gradient fields, and in general the
conductivity cannot be both periodic and isotropic. However, using a dynamical
systems approach the isotropic realizability is proved to hold in the whole
space (without periodicity) under the assumption that the gradient does not
vanish anywhere. Moreover, a sharp condition is obtained to ensure the
isotropic realizability in the torus. The realizability of a matrix field is
also investigated both in the periodic case and in the laminate case. In this
context the sign of the matrix field determinant plays an essential role
according to the space dimension.Comment: 19 page
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
Infinite dimensional moment problem: open questions and applications
Infinite dimensional moment problems have a long history in diverse applied
areas dealing with the analysis of complex systems but progress is hindered by
the lack of a general understanding of the mathematical structure behind them.
Therefore, such problems have recently got great attention in real algebraic
geometry also because of their deep connection to the finite dimensional case.
In particular, our most recent collaboration with Murray Marshall and Mehdi
Ghasemi about the infinite dimensional moment problem on symmetric algebras of
locally convex spaces revealed intriguing questions and relations between real
algebraic geometry, functional and harmonic analysis. Motivated by this
promising interaction, the principal goal of this paper is to identify the main
current challenges in the theory of the infinite dimensional moment problem and
to highlight their impact in applied areas. The last advances achieved in this
emerging field and briefly reviewed throughout this paper led us to several
open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated
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