Mathematics in the internet

Mika Seppälä

University of Helsinki and

Florida State University

We need to be able to

• Cut and paste mathematical objects

• Search information using the context of a formula as a key word

• Exchange scientific data so that the meaning gets understood automatically

• Produce scientific web material with live formulae

Proper standard encoding helps to avoid costly mistakes

• Ariane V explosion

• Mars orbiter

Standard encoding of mathematics

Needed for proper educational web material used for a virtual university

OpenMath and MathML

• Languages to define the meaning of mathematical formulae

• MathML covers K12 mathematics

• OpenMath extends MathML to all areas of mathematics (via extensible content dictionaries)

OpenMath solution

OpenMath language as defined by CD’s

Maple

Mathem.

Editor

Database

Browser

Translation between OM and proprietary presentations by phrasebooks

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1

 

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1Canonical URL: http://www.openmath.org/cd/arith1.ocd

 

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1Canonical URL: http://www.openmath.org/cd/arith1.ocd CD File: arith1.ocd CD as XML Encoded OpenMath: arith1.omcd

 

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1Canonical URL: http://www.openmath.org/cd/arith1.ocd CD File: arith1.ocd CD as XML Encoded OpenMath: arith1.omcd Defines: abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times, unary_minus

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1Canonical URL: http://www.openmath.org/cd/arith1.ocd CD File: arith1.ocd CD as XML Encoded OpenMath: arith1.omcd Defines: abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times, unary_minus Date: 2001-03-12 Version: 2 Review Date: 2003-04-01

Preamble of an OpenMath CD

OpenMath Content Dictionary: arith1Canonical URL: http://www.openmath.org/cd/arith1.ocd CD File: arith1.ocd CD as XML Encoded OpenMath: arith1.omcd Defines: abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times, unary_minus Date: 2001-03-12 Version: 2 Review Date: 2003-04-01 Status: official Uses CD: alg1, fns1, integer1, interval1, linalg2, logic1, quant1, relation1, set1, setname1, transc1

 

OpenMath CD definitions

lcm The symbol to represent the n-ary function to return the least common multiple of its arguments. Commented Mathematical property (CMP): lcm(a,b) = a*b/gcd(a,b)

OpenMath CD definitions

Formal Mathematical property (FMP): <OMOBJ> <OMA> <OMS cd="relation1" name="eq"/> <OMA> <OMS cd="arith1" name="lcm"/> <OMV name="a"/> <OMV name="b"/> </OMA> <OMA> <OMS cd="arith1" name="divide"/> <OMA> <OMS cd="arith1" name="times"/>

<OMV name="a"/><OMV name="b"/>

</OMA> <OMA> <OMS cd="arith1" name="gcd"/> <OMV name="a"/> <OMV name="b"/> </OMA> </OMA> </OMA></OMOBJ>eq (lcm ( a, b) , divide (times ( a, b) , gcd ( a, b) ) )

 

OM definition of sin(x2+1)

x2

<OMOBJ>………. <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMI>2</OMI>……….

</OMOBJ>

OM definition of sin(x2+1)

x2+1

<OMOBJ>………. <OMA> <OMS cd = "arith1" name="plus"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMI>2</OMI> </OMA> <OMI>1</OMI> </OMA>……….

</OMOBJ>

OM definition of sin(x2+1)

(x2+1)

<OMOBJ>…… <OMATP> <OMS cd="presentation" name="left"/> <OMSTR>(</OMSTR> <OMS cd="presentation" name="right"/> <OMSTR>)</OMSTR> </OMATP> <OMA> <OMS cd = "arith1" name="plus"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMI>2</OMI> </OMA> <OMI>1</OMI> </OMA> </OMATTR> </OMA></OMOBJ>

OM definition of sin(x2+1)

sin(x2+1)

<OMOBJ><OMA> <OMS cd="transc1" name="sin"/> <OMATTR> <OMATP> <OMS cd="presentation" name="left"/> <OMSTR>(</OMSTR> <OMS cd="presentation" name="right"/> <OMSTR>)</OMSTR> </OMATP> <OMA> <OMS cd = "arith1" name="plus"/> <OMA> <OMS cd="arith1" name="power"/> <OMV name="x"/> <OMI>2</OMI> </OMA> <OMI>1</OMI> </OMA> </OMATTR> </OMA></OMOBJ>

OpenMath and MathML Organizations

• W3 Math working group (chaired by A. Diaz and P. Ion)

• NAOMI

• OpenMath Society

• OpenMath Esprit project

NAOMI

• North American OpenMath Initiative

• Started by S. Braham (SFU) and R. Sutor (IBM)

• Partner organizations include FSU, NCS, Drexel, UC Berkeley

OpenMath Society

• Registered international society in Helsinki

• President: A. Cohen (Eindhoven, The Netherlands)

• Executive committee: A. Cohen, G. Gonnet, M. Seppälä, S. Watt, S. Braham, M. Dewar, M. Gaetano

• openmath.org hosted by FSU

OpenMath Esprit project

• Coordinated by M. Dewar (NAG, Oxford)

• Funding over 3 M$ for 1997-2000

• Included several academic and industrial partners in Europe

• Produced the OM standard, phrasebooks and tools to create and edit OM objects

Industrial partners of the OpenMath projects

• IBM (techexplorer group)

• Waterloo Maple

• Springer Verlag

• Other European partners

• North American partners

Example of a next generation OM application

Cellular telephone as a scientific calculator(following slides provided by mofile.com, inc.)