- From: Eoin Lane <eoin.lane@capeclear.com>
- Date: Mon, 13 May 2002 20:22:54 +0100
- To: <www-math@w3.org>
I have a bit of confusion here. I have followed the instruction on the
http://www.w3.org/Math/XSL/ page. I have a xhtml document with mathml
and xhtml which I am trying to view locally first and I have all the
relevant *.xsl in the current directory, so there should be no security
issues. I have included the document that I am trying to render below.
When I include the dtd:
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
"http://www.w3.org/TR/MathML2/dtd/xhtml-math11-f.dtd" [
<!ENTITY mathml "http://www.w3.org/1998/Math/MathML">
<!ATTLIST maction id ID #IMPLIED>
]>
I get "Parameter entity must be declared before it is used".
2. And when I leave it out, as I know that IE has a problem with this
dtd I get "Reference: to undefined entity Sum"
Has anyone any idea on how I might be able to get this to render in IE
5.5 or 6.0
Thanks in advance.
Eoin.
Here is the xhtml dtd
<?xml version="1.0"?>
<?xml-stylesheet type="text/xsl" href="mathml.xsl"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
"http://www.w3.org/TR/MathML2/dtd/xhtml-math11-f.dtd" [
<!ENTITY mathml "http://www.w3.org/1998/Math/MathML">
<!ATTLIST maction id ID #IMPLIED>
]>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<!--<link href="http://inconn.dyndns.org/maths/maths.css"
rel="stylesheet" type="text/css"/>-->
<title>CapScience MathML Tutorial</title>
</head>
<body>
<h2>A bit of Maths</h2>
<p>For this tutorial we will be considering three types of
number prime numbers, mersenne prime and perfect numbers. </p>
<p>A <a
href="http://www.utm.edu/research/primes/index.html"><b>Prime</b></a>
number is any number that is only divisible by itself and one. So
examples of prime number are
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mo stretchy="true">{</mo>
<mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo>
<mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>1</mn> <mn>1</mn>
<mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow>
<mo stretchy="true">}</mo>
</mrow>
</math>
</p>
<p>
A <a
href="http://www.utm.edu/research/primes/mersenne/index.html"><b>Mersenn
e</b></a> prime is a special type of prime number in that it is a prime
number that can be written in the form of
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
</math>. Example of the first few mersenne number are
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mo stretchy="true">{</mo>
<mrow>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
<mo>,</mo>
<mn>5</mn>
<mo>,</mo>
<mn>7</mn>
<mo>,</mo>
<mn>1</mn>
<mn>3</mn>
<mo>,</mo>
<mn>1</mn>
<mn>7</mn>
<mo>,</mo>
<mn>1</mn>
<mn>9</mn>
<mo>,</mo>
<mn>3</mn>
<mn>1</mn>
<mo>,</mo>
<mn>6</mn>
<mn>7</mn>
<mo>,</mo>
<mn>1</mn>
<mn>2</mn>
<mn>7</mn>
<mo>,</mo>
<mn>2</mn>
<mn>5</mn>
<mn>7</mn>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</mrow>
<mo stretchy="true">}</mo>
</mrow>
</math>
</p>
<p>
A <a
href="http://primes.utm.edu/glossary/page.php/PerfectNumber.html"><b>
Perfect</b></a> numbers, on the other hand is, that which is equal to
the sum of it parts. Put another way, a whole number is <b>perfect</b>
if it is equal to the sum of its proper divisors. So for example the
number 6 is perfect because it proper divisors are 1, 2, 3 and 1 + 2 + 3
= 6. So also are 28 (1 + 2 + 4 + 7 +14); 496 and 8128. There numbers
constitute the first four perfect numbers and as is evident for the
series there are <i>not</i> that many of them.
</p>
<p>So what is the relationship between these numbers? From its
definition it is clear that a mersenne prime is also a prime numbers. A
consequence of this is, that k in the
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
</math>
equation must also be prime.
Also, a relationship between prime numbers and perfect numbers had been
known since antiquity and formulated by <a
href="http://members.fortunecity.com/kokhuitan/euclid.html">Euclid</a>
in the following Theorem. </p>
<p> <b>Theorem. </b><i>If <math
xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
</math> is prime and if
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>N</mi>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="true">(</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mo stretchy="true">)</mo>
</mrow>
</math>, then N is perfect.</i></p>
<p>
But it wasn't until almost 2000 year later that the exact relationship
between these types of numbers was resolved by <a
href="http://members.fortunecity.com/kokhuitan/euler.html">Euler</a> in
the following theorem.</p> <p><b>Theorem. </b><i>If N is an even perfect
number, then
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>N</mi>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="true">(</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mo stretchy="true">)</mo>
</mrow>
</math>
, where
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
</math>
is prime</i>
</p><p>In plain language an even perfect number has a mesenne
prime associated with it.</p>
<h2>A bit of UML</h2>
<p>A mersenne prime then is a special type of prime number. In UML this
is represented as an <i>'is a'</i> relationship <i>i.e.</i> A mersenne
prime <i>is a</i> special type of prime number. Also since k in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
</math>
is also prime, then there is also an <i>'has a'</i>
relationship <i>i.e.</i> A Mersenne number <i>has a</i> Prime number.
</p> <p><img alt="" src="./images/mersennePrime.gif"/></p>
<p>From Euler theorem on perfect numbers above it is clear that there
are two types Odd Perfect numbers and Even Perfect numbers.
Again this can be represented in UML as follows:</p><p><img
src="./images/perfectNumber.gif" alt=""/>
</p><p>It is interesting to note that an odd perfect number has never
been found even though some extraordinary properties about them are
known, among which are:</p><ul>
<li>An odd perfect number cannot be divided by 105</li>
<li>An odd perfect number must contain at least 8 different prime
factors</li>
<li>The smallest odd perfect number must exceed
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>300</mn>
</mrow>
</msup>
</mrow>
</math>
</li>
<li>The second largest prime factor of an odd number exceeds
1000</li>
<li>The sum of the reciprocials of all odd perfect number id
finite. Symbolically
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<munder>
<mrow>
<mo>∑</mo>
</mrow>
<mrow>
<mi>odd perfect</mi>
</mrow>
</munder>
<mfrac>
<mrow>
<mn>1</mn>
</mrow>
<mrow>
<mi>n</mi>
</mrow>
</mfrac>
<mo><</mo>
</mrow>
<mrow>
<mi>∞</mi>
</mrow>
</math>
</li>
</ul>
<p>Look at item three on the list; imagine trying to do some long
division sums in your head with that number</p>
<p>Finally, again from Eular theorem, perfect numbers are related to
prime numbers in that an all even perfect number can be expressed in the
form
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>N</mi>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="true">(</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
</mrow>
</msup>
<mo>-</mo>
<mn>1</mn>
</mrow>
<mo stretchy="true">)</mo>
</mrow>
</math>
<i>i.e.</i>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>N</mi>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math>
(a mersenne number). Therefore every even perfect number <i>has a</i>
mersenne number. Once more in UML this can be diagrammatically depicted
as follows:</p>
<p><img src="./images/perfectPrimeNumber.gif" alt=""/>
</p>
</body>
</html>
----------------------------------
Eoin Lane (PhD)
Technical Analyst
Tel: ++44 (0) 20 8899 6565
Fax: ++44 (0) 20 8899 6156
Mob: ++44 (0) 7813 928412
<http://www.capeclear.com/>
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Received on Monday, 13 May 2002 15:25:34 UTC