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Friday, September 24, 2010

Group Theory - Exercise ( 24/9-'10 )

Which group is represented by the following representation :
$(a,b|a^5=1,b^2=1,(a \circ b)^3=1)$.

Although I don't expect a question like this on the M208 or MS221 exams on Group Theory due to the ugly '5-min-to-think' constraint, candidates for M208 ( and possibly MS221 ) are well prepared to solve it.

Thursday, September 23, 2010

Competition leads to innovation!

This site does not load particular fast in Internet Explorer 9beta or Opera 10, which are my preferred browsers. Thank God, we have a choice in browsers. Different browsers for different people. Functionality wise I still love Opera, although I don't like its default look and feel I have found a sexy skin for it. - It takes IE and Opera a few seconds to completely load this blog, i.e. all widgets and processing the MathJax. I tried the latest Google Chrome today and it looked as though the site was loaded in about a second. Further inspection showed it was an optical illusion because left down I noticed the MathJax was still processing. Still, the Google Chrome has everything of a fast browser. Lean and mean like a Porsche car, sort of. I am sure my friends in Norway, the Opera whiz-kids, will leap and frog Chrome once more.

Competition leads to innovation. It is the reason why browsers keep getting better and Microsoft Word and Excel are getting more lame by every new "release".

Tuesday, September 14, 2010

English mathematics

If you aren't a native but still intend to study with the UK Open University you are in for some real surprises. Take Group Theory for example. ( MS221, M208 and M336 ).

According to GAP we have:

gap> (1,2,7,5)(3,8,4)*(1,3,6,7,5);
(1,2,5,3,8,4,6,7)

the Open University however, says otherwise:

gap> (1,2,7,5)(3,8,4)*(1,3,6,7,5);
(1,8,4,3,6,2,5,7).

GAP is THE leading, -state-of-the-art- Abstract Algebra package used by research mathematicians everywhere on the planet. ( If I am correct parts are or have been developed in Scotland, so they can adjust if required. )

The general consensus is that we evaluate from left to right. Not so in England, however. I even lost marks in MST121 because I was using commas and decimalpoints in the incorrect manner.

I am reading Representations and characters of groups, by Gordon James and Martin Liebeck published by Cambridge University Press. They consistently use xf, instead of f(x). The book is readable IF you consistenly translate their vgf to f(g*v) and so forth.

Saturday, September 11, 2010

9/11-'10

What can I possibly say about 9/11, besides that I don't believe the official story, AT ALL ?

See: http://www.911truth.org/links.php 

Euler Diagram or Venn Diagram: what's the difference ?

Got an interesting reply to my post on the UK Venn Diagram. This is actually an Euler Diagram. Again Euler!

Saturday, September 4, 2010

Goedel's surprise

Goedel (l), Einstein (r) pic from LIFE mag

Mathematical surprises, a blog article by Dave Richeson. Very interesting list. I don't know how it is with you ( fellow math students ) but I often have the feeling that I know absolutely nothing about mathematics. Like today for example, even though I understood about 15 or so topics on the list of 23. Strange. - Godels incompleteness theorems ( suprise #1 and I agree ) are a major topic of the M381 Number Theory / Logic course. I am definitely looking forward to that course! - I found this article because I subscribed to some twitter accounts of mathematicians. I already was but I am now even more convinced that Twitter does have a function.

Thursday, September 2, 2010

3x+1 look alike

Number theory fascinates me. I studied elementary number theory, the next step,  analytical number theory requires knowledge of group and representatuon theory ( characters ) and complex analysis. I am working hard on that. In the meanwhile I like to study the famous but accessible problems of number theory. An interesting problem is the following.

It has been conjectured that the function $x_n = f(x_{n-1})$ with $x_0 \in \mathbf{N}$ always ends in $1,4,2,1,4,2,1 \cdots$ where

$$f: x \mapsto \begin{cases}
\frac{x}{2} \text{ if } x \equiv 0 \mod{2} \\
3x+1 \text{ if } x \equiv 1 \mod{2}
\end{cases}$$

Example:
27
82
41
124
62
31
94
47
142
71
214
107
322
161
484
242
121
364
182
91
274
137
412
206
103
310
155
466
233
700
350
175
526
263
790
395
1186
593
1780
890
445
1336
668
334
167
502
251
754
377
1132
566
283
850
425
1276
638
319
958
479
1438
719
2158
1079
3238
1619
4858
2429
7288
3644
1822
911
2734
1367
4102
2051
6154
3077
9232
4616
2308
1154
577
1732
866
433
1300
650
325
976
488
244
122
61
184
92
46
23
70
35
106
53
160
80
40
20
10
5
16
8
4
2
1

A fornal proof that this sequence ends like this is unknown. According to Conway it is even undecidable. Many papers have been written on the problem and it has several dedicated websites.


Very similar, but nevertheless generally generating smaller sequences is the function
$$f: x \mapsto \begin{cases}

1 \text{ if } x = 2 \\
\text{ else } x \mapsto \begin{cases}
\frac{x}{4} \text{ if } x \equiv 0 \mod{4} \\
6x+2 \text{ if } x \equiv 1 \text{ or } 3 \mod{4} \\
\frac{x-2}{4} \text{ if } x \equiv 2 \mod{4}
\end{cases}
\end{cases}$$

 27
164
41
248
62
15
92
23
140
35
212
53
320
80
20
5
32
8
2
1

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Mathematics: is it the fabric of MEST?
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To create new theorems and proofs
To boldly go where no man has gone before




(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)