The Taylor Center

in 3-body free fall

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
1759-1803
Introduced the main formulas for n-order derivatives including what was later named the Faa di-Bruno formula

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 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 Labworks, called Exploratorium, which enables experimentation and visualization for 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 one hundred of other annotated simulations (without the textbook) are supplied or accessible under Demo menu item. The Taylor Center therefore now took 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.

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^*) 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.

(Updated first - 7/15/2014, last - 8/8/2024)