485 research outputs found
Topological and algebraic structures on the ring of Fermat reals
The ring of Fermat reals is an extension of the real field containing
nilpotent infinitesimals, and represents an alternative to Synthetic
Differential Geometry in classical logic. In the present paper, our first aim
is to study this ring from using standard topological and algebraic structures.
We present the Fermat topology, generated by a complete pseudo-metric, and the
omega topology, generated by a complete metric. The first one is closely
related to the differentiation of (non standard) smooth functions defined on
open sets of Fermat reals. The second one is connected to the differentiation
of smooth functions defined on infinitesimal sets. Subsequently, we prove that
every (proper) ideal is a set of infinitesimals whose order is less than or
equal to some real number. Finally, we define and study roots of
infinitesimals. A computer implementation as well as an application to
infinitesimal Taylor formulas with fractional derivatives are presented.Comment: 43 page
A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint
Some data processing requirements for precision Nap-Of-the-Earth (NOE) guidance and control of rotorcraft
Nap-Of-the-Earth (NOE) flight in a conventional helicopter is extremely taxing for two pilots under visual conditions. Developing a single pilot all-weather NOE capability will require a fully automatic NOE navigation and flight control capability for which innovative guidance and control concepts were examined. Constrained time-optimality provides a validated criterion for automatically controlled NOE maneuvers if the pilot is to have confidence in the automated maneuvering technique. A second focus was to organize the storage and real-time updating of NOE terrain profiles and obstacles in course-oriented coordinates indexed to the mission flight plan. A method is presented for using pre-flight geodetic parameter identification to establish guidance commands for planned flight profiles and alternates. A method is then suggested for interpolating this guidance command information with the aid of forward and side looking sensors within the resolution of the stored data base, enriching the data content with real-time display, guidance, and control purposes. A third focus defined a class of automatic anticipative guidance algorithms and necessary data preview requirements to follow the vertical, lateral, and longitudinal guidance commands dictated by the updated flight profiles and to address the effects of processing delays in digital guidance and control system candidates. The results of this three-fold research effort offer promising alternatives designed to gain pilot acceptance for automatic guidance and control of rotorcraft in NOE operations
Automatic modular abstractions for template numerical constraints
We propose a method for automatically generating abstract transformers for
static analysis by abstract interpretation. The method focuses on linear
constraints on programs operating on rational, real or floating-point variables
and containing linear assignments and tests. In addition to loop-free code, the
same method also applies for obtaining least fixed points as functions of the
precondition, which permits the analysis of loops and recursive functions. Our
algorithms are based on new quantifier elimination and symbolic manipulation
techniques. Given the specification of an abstract domain, and a program block,
our method automatically outputs an implementation of the corresponding
abstract transformer. It is thus a form of program transformation. The
motivation of our work is data-flow synchronous programming languages, used for
building control-command embedded systems, but it also applies to imperative
and functional programming
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