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The material below was presented in the "Semantic Representation of Mathematical Knowledge Workshop", February 3-5, 2016 at the Fields Institute, University of Toronto. It shows the approach I used for “digitizing mathematical knowledge" regarding Differential Equations, Special Functions and Solutions to Einstein's equations. While for these areas using databases of information helps (for example textbooks frequently contain these sort of databases), these are areas that, at the same time, are very suitable for using algorithmic mathematical approaches, that result in much richer mathematics than what can be hard-coded into a database. The material also focuses on an interesting cherry-picked collection of Maple functionality, that I think is beautiful, not well know, and seldom focused inter-related as here.

 

 

Digitizing of special functions,

differential equations,

and solutions to Einstein’s equations

within a computer algebra system

 

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Editor, Computer Physics Communications

 

 

Digitizing (old paradigm)

 

• 

Big amounts of knowledge available to everybody in local machines or through the internet
• 

Take advantage of basic computer functionality, like searching and editing

 

 

Digitizing (new paradigm)

• 

By digitizing mathematical knowledge inside appropriate computational contexts that understand about the topics, one can use the digitized knowledge to automatically generate more and higher level knowledge

 

 

Challenges


1) how to identify, test and organize the key blocks of information,

 

2) how to access it: the interface,

 

3) how to mathematically process it to automatically obtain more information on demand

 

 

 

 

                                           Three examples

Mathematical Functions
 

"Mathematical functions, are defined by algebraic expressions. So consider algebraic expressions in general ..."

The FunctionAdvisor (basic)
 

"Supporting information on definitions, identities, possible simplifications, integral forms, different types of series expansions, and mathematical properties in general"

Examples
   

General description
   

References
   

 

Differential equation representation for generic nonlinear algebraic expressions - their use
 

"Compute differential polynomial forms for arbitrary systems of non-polynomial equations ..."

The Differential Equations representing arbitrary algebraic expresssions
   

Deriving knowledge: ODE solving methods
   

Extending the mathematical language to include the inverse functions
   

Solving non-polynomial algebraic equations by solving polynomial differential equations
   

References
   

 

Branch Cuts of algebraic expressions
 

"Algebraically compute, and visualize, the branch cuts of arbitrary mathematical expressions"

Examples
   

References
   

 

Algebraic expresssions in terms of specified functions
 

"A conversion network for arbitrary mathematical expressions, to rewrite them in terms of different functions in flexible ways"

Examples
   

General description
   

References
   

 

Symbolic differentiation of algebraic expressions
 

"Perform symbolic differentiation by combining different algebraic techniques, including functions of symbolic sequences and Faà di Bruno's formula"

Examples
   

References
   

 

Ordinary Differential Equations
 

"Beyond the concept of a database, classify an arbitrary ODE and suggest solution methods for it"

General description
   

Examples
   

References
   

 

Exact Solutions to Einstein's equations
 

 

Lambda*g[mu, nu]+G[mu, nu] = 8*Pi*T[mu, nu]

 

"The authors of "Exact solutions toEinstein's equations" reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, organized the whole material into chapters according to the physical properties of these solutions. These solutions are key in the area of general relativity, are now all digitized and become alive in a worksheet"


The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords (old paradigm) changes the game.

More important, within a computer algebra system this knowledge becomes alive (new paradigm).

• 

The solutions are turned active by a simple call to one commend, called the g_  spacetime metric.
• 

Everything else gets automatically derived and set on the fly ( Christoffel symbols  , Ricci  and Riemann  tensors orthonormal and null tetrads , etc.)
• 

Almost all of the mathematical operations one can perform on these solutions are implemented as commands in the Physics  and DifferentialGeometry  packages.
• 

All the mathematics within the Maple library are instantly ready to work with these solutions and derived mathematical objects.

 

Finally, in the Maple PDEtools package , we have all the mathematical tools to tackle the equivalence problem around these solutions.

Examples
   

References
   

 

Download:  Digitizing_Mathematical_Information.mw,    Digitizing_Mathematical_Information.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft
physics special-function differential-equations

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