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Friday, August 28, 2009

Video: Solving Cubic Equations

Video: Solving Cubing Equations

And more: Pythagoros theorem ( a^2 + b^2 = c^2 ) is not really by Pythagoras. Pythagoros is the probable leader of a cult called the Pythagoreans. There is an interesting formula for enumerating all the Pythogarian triples like (3,4,5) (5,12,13), etc.
In the next clip we learn that Fermat bought a translation from Greek to Latin of Diophantus' work and started his work on Number Theory from there on. It was also in Fermat's lifetime that Calculus was born. ( Fermat already worked on the currently very important Elliptic Curves which Andrew Wiles used to prove Fermat's famous last theorem. ) A method is shown to find rational points on elliptic curves. An elliptic curve is the graph of an equation of type y^2 = x^3 + px + q.
Then the video becomes less interesting. It seems that in academic talks it is custom to speak with great admiration of certain other scientists, mathematicians especially if you know them or are related to them. The speaker, professor Gross had John Tate as his thesis advisor. John Tate is a celebrity in a the Elliptic Curve branch of mathematics. So Gross shows a picture of him and Tate and all the other graduate students at that time. Then he mentiones all these students by name including the universities where they are now teaching as a mathematics professor.
( I suppose they had their euphoric moments too when they simply passed an exam. After a while more impressive results are needed to feel the same high, like being appointed as professor. )
See the rest for yourself if you are still interested...

Thursday, August 27, 2009

MS221 TMA03 - Result

MS221 TMA03 - Result is in... 99.

Lost a point due to a typo. This once again proves that you can literally save points by reviewing, QA-ing, checking whatever you call it. And check your results as much as possible by using a mathematics package like MathCad or your own preferred software, as long as it can do the math.

Friday, August 21, 2009

MST121 - Overall Course Result ( Final )

I have received the marked MST121 TMA04 and now also all data have been updated in my student record.

Assignment Score
CMA 41 83
CMA 42 84
TMA 01 82
TMA 02 88
TMA 03 95
TMA 04 90

TMA 01 82 has been substituted to 87.7 giving a final score of 88.84 rounded to 89.

An 89 for this is course is much more than I deserve. Especially in the beginning of MST121 I did not study enough ( november and december last year ). During this period I also had a last minute approach towards TMA's. I definitely lost points there.

Anyway, I learned that in order to get high marks much more is expected of the student. TMA01 and TMA02 were quick and dirty jobs using pen and pencil. As from TMA03 I changed to LaTeX including several QA reviews using Mathematica ( if possible ).                                          

Wednesday, August 19, 2009

Chinese Remainder Theorem

Solve systems of linear congruence equations using the Chinese Remainder Theorem.

Find a video lecture here ( Lecture #12 )

I was solving a problem from ' 104 Number Theory problems ' which required an application of the CRT. I had forgotten the algorithm. Instead of looking it up in a book I watched Song's lecture.

Tuesday, August 18, 2009

Sample MS221 - Exam

Had a first browse through the sample MS221 exam today. It is a lot of work for only three hours. Twelve questions which test knowledge of what was in the course. With these twelve questions you can score up to 60% of the total points. Four difficult questions remain for the other 40%. Of the four difficult questions only two will be marked. ( You have to do all four of them because you don't know which will be marked. )

I would say that anything is possible with regards to this exam. I could fail but I could just as well get a high score. It all depends on proper training from now till the exam. I am sure that a lot of exam training will pay off.

But first I have to finish MS221 - TMA04.

How I spent some spare study time

I have most of the work done on MS221 / TMA04 but since I haven't received the results of TMA03 yet, I wait before sending the TMA in. So, I can spend my study time freely. I am currently studying ( elementary ) Number Theory. I made some real progress during the weekend and today.

I completely understand and am thus able to prove, ( now and in the future, I am sure ) the formula for the Sigma number-theoric function. The sum of the dividers of a natural number. For example:

12 = 2^2 * 3
= 1 + 2 + 3 + 4 + 6 + 12
= ( 1 + 2 + 4 ) + ( 3 + 6 + 12 )
= ( 1 + 2 + 4 ) * ( 1 + 3 )
= 7 * 4
= ( 2^3 - 1 ) / ( 2 - 1 ) * ( 3^2 - 1 ) / ( 3 - 1 )
= 28.
Furthermore, I am beginning to ger a practical understanding of Möbius inversion.

And I firmly consolidated knowlegde of and skills regarding
- Properties of integer division
- Euclidean algorithm for finding the GCD
- Linear diophantine equations
- Congruences
- etc.

Tuesday, August 11, 2009

How am I doing relatively ?

I mean how do my study results compare to other students? Do I actually care??! I read about people receiving TMA results of 99, 100 and so on. Than my average of 90 sofar seams low, very low as a matter of fact. On the other hand those who 'blog' 'brag' including me, I suppose. I wouldn't shout it around if I passed on 41/100, although it has exactly the same value to the overall degree result. - If a lot of students are scoring in the 80-100 range then the TMA's are just too simple imho. Since the student can make the assignments at home that means they can be quite challenging.

Saturday, August 8, 2009

Studying mathematics with SuperMemo

If you look carefully how the texts are layed out in the study books you see that they did everything which is possible to make studying as efficient as possible. There is no need to make a summary, think of examples for theorems, find and do exercises, etc. Everything is in the book at the time you need it. And for a course like MS221 everything is nicely presented in a Course Handbook which includes a dictionary for technical terms.

I used to do a lot of self-studying so a I used a lot of study time on tasks which aren't necessary when you do an OU course. I nevertheless got used to allocating time to study-improvement on a regular basis. If I could only use SuperMemo then studying becomes a process which automatically produces a lot of statistical data about the studying process itself. But producing learning materials for SuperMemo can be time-consuming in itself. That fact kept me from using SM thus far. But with the other tools I currently have like Mathematica, TeXnicCenter and so on I must be able to organize my way of studying, doing exercises etc. so that it will be fairly easy to document my work as a SM item/answer pair.

It is worth another try...

Thursday, August 6, 2009

MST121 overall course result

See your MST121 course website 
Overall course result

PassOverall continuous assessment score (OCAS) : 89
Overall assessment score (OAS) : 89

Substitution applied at : 87.7

Wednesday, August 5, 2009

Gödel, Escher, Bach

Gödel, Escher, Bach is the first book I read after many years on non-reading / studying, besides trying to stay on top of the ever faster changing world of IT and systems development. I read GEB in English which is a foreign language for me, so it was hard for more than one reason. But reading it felt like an adventure and it fueled my dream of studying mathematics.

And now MIT made a series of six video lectures about the book. Haven't seen it yet but I expect them to be interesting. Gödel's incompleteness theorem is a topic in the course Number Theory and Logic which is part of my course plan.

Sunday, August 2, 2009

OU Courses for 2010

I'll very likely go for M208 ( Level-2 60 points ) combined with MT365 ( Level 3 30 points ). M208 is pure mathematics: Analysis, Linear Algebra and Group Theory. MT365 is discrete mathematics: graphs, networks and designs.

After MST121 and MS221, both M208 and MST209 are good options to do next. They can be done both but in that case you have to do one TMA per two weeks. That is a lot. Too much I can handle. Unfortunately.

Saturday, August 1, 2009

Graphs, networks and design

As far as algorithms is concerned I think MT365 "Graphs, networks and design" comes close. It's a course I could do next year. Combined with M208, pure mathematics. I am not sure if I do a course starting in october. It would be a non-math course and I am not sure if I have the strong motivation needed to succeed an Open University course.

Formal grammars and languages

I haven't been able to find an Open University course covering formal grammars and languages. Come to think of it there isn't a course about algorithms ( Don Knuth ! ) either. I find this a bit strange.

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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.)