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Brook Taylor
1685-1731 Introduced the Taylor expansion - the main foundation of differential calculus - in words of J. L. Lagrange |
The Taylor Center |
Louis
François Antoine Arbogast
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The software, the underlying theory, and the community
The title of this site represents both a particular
software, the theory underlying
it, and the community of interested
researchers.
The software is an advanced ODE solver called
the Taylor Center. The solver is
an All-in-One GUI application for
Windows (32-bit addressing) implementing the modern Taylor integration
method (Automatic
Differentiation) in a multi-functional user friendly environment with
unique features such as graphing and real time animation of the
trajectories in 2D and 3D stereo, and with the highest number of
significant digits –
the Intel extended
10 byte format with 63-bit
mantissa (19 decimal digits).
Based on this unique software, the Taylor Center offers a virtual
laboratory, a sort of Labwork, called Exploratorium, which enables
experimentation and visualization for all problems in applied ODEs in
various fields. For example, such labworks with the accompanying textbooks are already
provided for the Rigid Body motion (including the Dzhanibekov effect),
and for a portion of Celestial mechanics. Over hundred of other
simulations (without the textbook) are under Demo menu item. The Taylor
Center therefore plays now a role of the no more existing Consortium for
ODE Experiments (CODEE)*),
1992–1997.
The theory in the foundation of the Taylor method unifies
...
- the classical concept of elementary functions,
- Automatic Differentiation,
- the Taylor Method for integration of ODEs,
- analytic continuation, and
- the new type of the special points in holomorphic functions: the so called unremovable "removable" singularities.
However there is an open question in this
theory, the Conjecture
remaining unsolved since 2004. Due to the nature of its claim, the
Conjecture affects the ways of definition of general elementary
functions and therefore the outline of the entire theory. Thus, as an
organization, the Taylor Center seeks attention of the mathematical
community in order to sooner close the gap in this otherwise
comprehensive theory. This site therefore will provide a forum for
discussing and advancing the solution of the Conjecture, as well as the
updates and upgrades of the software.
Here
is the most brief presentation
(and pdf) of the gap
(slide 4), the Conjecture
(slides
6-8) and its
context.
_____________________
*) The only tool for their ODE experiments then
was the ODE Architect (a
simple application for Windows with 16-bit addressing, implementing the
Runge-Kutta method). Later CODEE was transformed into a "Community of
Ordinary Differential
Equations Educators" publishing an electronic CODEE Journal .
CODEE neither associates nor collaborates with the Taylor Center.