Education in the US

MIT has it's OpenCourseWare program. Except for a few courses on video ( Linear Algebra and Differential Equations ) the entire ' openness ' is nothing more than publishing the syllabi and some handwritten lecture notes of students. - What a hoax. What is the big deal?

Today, I watched the first videolecture of Harvard's E-222 course on Abstract Algebra. The lectures are about the book Algebra by M. Artin. ( Well known to me, I self-studied it in 2007 ). Anyway Benedict Gross, the lecturer, was talking about other Harvard courses in numbers. I tried to look up the topics of these courses and then I found out that a login is required for that.

That's it! There is secrecy about the actual content of the mathematics programmes of the various universities. That is called capitalism. Competition among universities. It is also a way to hoax-up the so-called 'quality' of the educational material. They are asking ludicrous fees for reading out loud Artin's books. It's a system to introduce classes in society.

Everyone who read Ramanujan's life story probably agrees with me that Mathematics should be free and accessible to everyone on the planet wether studying in Cambridge, through the Open University or at home self-studying a copy of Artin's book downloaded from internet.

What I like about the US is, that it is home of Mathematica -AND- Sage, of Bush/Obama -AND- Ron Paul, and a zillion other contradictions.

When closure is sufficient for a subset to be a subgroup.

While browsing Group Theory I ( Suzuki ) I noticed the following proposition.

If G is a finite group and S is a subset of G then closure in S suffices for S to be a subgroup.

Proof:
S is a subgroup if for all a,b,c in S
( i ) ab in S - closure
( ii ) a(bc)= (ab)c - associativity
( iii ) e in S - has identity
( iv ) a^(-1) in S - has inverse
let's prove them one by one:
( i) is proposed to be true ;
( ii ) is true for G and thus true for S ;
( iii ) since G is finite there is an integer n such that a^n = e thus e in S
( iv) since a a^(n-1) = e all a have an inverse.
[]

When in exercises the word 'finite' is added to group like 'G is a finite group ' we know that for G the group axioms are true like (i) to (iv) above AND that there is an integer n such that for all g in G g^n = e.

Michio Suzuki

Michio Suzuki (1926-1998) was one of the 20th century pioneers of modern Group Theory. When I found his books Group Theory I and Group Theory II my first thought was that I had to wait a while before any books of him become accessible to me. I was pleasantly surprised that, unlike say Lang, his writings about Group Theory are written to explain Group Theory to the uninitiated, instead of documenting Group Theory for the experts. At least, that's my expression. So these are my preferred books on Group Theory for now.

Links:
- In memoriam M. Suzuki
- Suzuki, Group Theory I 

Skiena's CSE 547

Discrete Math Lectures on Video

10 years old - 1999 was clearly a small bandwidth area.

Nice to have anyway.

Video lectures Abstract Algebra

I found a set of video lectures on Abstract Algebra.
MATH E-222 Abstract Algebra - http://www.extension.harvard.edu/openlearning/math222/Enjoy!

( Update 1-feb/'11: )
You might also like:
- Video lectures number theory
- [News] - Video Lectures Algebraic Topology ( for Undergraduates )
- video lectures complex analysis

M336 - Groups and Geometry

M336 starts in feb 2010. 4TMA's and 1 examination, like MS221. The M336 course covers two related topics: groups and geometry and is delivered in 16 well known OU type of books including exercises, solutions, summaries and so on. The group theory-stream consists of the following:

Axioms and examples
Subgroups
Generating subgroups
Cyclic groups
Group actions

Group axioms
Subgroups and cosets
Normal subgroups and quotient groups
Isomorphisms and homomorphisms
Generators and relations

Equivalent colourings
Group actions
The counting lemma
The cycle index
Polya's enumeration formula.

Direct products
Abelian groups and groups of small orders
Cyclic groups
Subgroups and quotient groups of cyclic groups
Direct products of cyclic groups

Finitely presented abelian groups
The reduction algorithm
Existence and uniqueness of torsion coefficients and rank
Finitely generated abelian groups

Finite abelian groups
Subgroups of abelian groups
Permutation groups
Conjugacy - p-groups

Sylow p-subgroups
Sylow's first and second theorems
Sylow's third theorem
Applications of the Sylow theorems
Subgroups of prime power order

Review
Groups of order 2p
Groups of order 12
Where now?

Plan for 2010.

OK people,
Just decided to do MST 209 first instead of M208. As far as possible from the Open University website I compared both courses. My main conclusion is that the pay off in "skills" is much higher from MST 209. Why start on the theoretic Real Analysis if your can't solve the basic differential equations? And MST 209 has to be done anyway. I can still add Groups and Symmetry to my schedule for next year.
Yep. As far as MST 209 is concerned I made up my mind.

About MS221 exam (3)

A commenter asked if the TMA results count in any way.

Well, yes and no.

Your final result is the lowest of the two!

So a 100 for TMA's and a 40 for exam = 40.

so a 40 for TMA's and a 100 for exam = 40.

At the end of the day only the exam counts, that is if your TMA's were 85+.

Intermezzo

No courses for two / three months. How shall I spend the free study time, if any? I am currently self-studying the book Introduction to Analytic Number Theory by Tom M. Apostol, this book is used in Analytic Number Theory I ( M823 ) and Analytic Number Theory II ( M829 ). Lots of new subjects. I also have a problem book with exercises about the subject so that keeps me going.

About the MS221 exam ( 2 )

Ok, let's compare my earlier prediction with how I see things now

I
2nd order recurrence system (6)
conics (6)
matrix calculation (6)
composite isometry (4)
linear transformation rotation / reflection (4)
differentiation (3)
integration(3)
taylor series (1)
function iteration (1)
logic (1)
complex numbers (4)
number theory (4)
group theory (6)
49

IIconic with xy component ( rotated )
iteration( 4 )
logic ( 8 )
12
So that is 61 or a grade 3 pass.

In a TMA @home situation I would have a 99/100 no doubt about it whatsover.

About the MS221 exam

The exam was held in The Hague, there werea about 20 people taking the MS221 exam. There were 8 exams going on in total. Can't remember which ones. What was the crowd like? I suppose +/- 25% were like me... after-career type of people finally doing / studying what they really like / love. The rest were late 20-ers, early 30-ers going for their first degree, I suppose. MS221 is not a real math-exam, the S says it all: Math Sciences, still there was only one ( 1 ) woman in the group. There was more balance in the other exams though.

The exam itself. Well, it was hard, hard, HARD! Everybody was still writing after three hours. I just finished the last question at 17.30. Wow, did time fly! I didn't know time could go -that- fast. The questions were as expected just as I predicted in an earlier post. 12 + 2 questions in three hours is a lot. No time to check, double-check, evaluate and play around with the exercises as I was used to when doing TMA's.

A distinction is impossible. I underestimated the time-pressure component. There is no time to think all you can do is robotically do the exercise as if on auto-pilot. But you can train for this so I think distinctions will be possible in future exams.

I am confident I'll pass but on a lower grade I had in mind. I am 200% motivated to continue my mathematics study. Either Pure Mathematics (60) + Groups and Symmetry (30) or maybe just Pure Mathematics (60) depending on other commitments.

The question on group theory was really easy, I saw all answers immediately in a flash, but writing all the answers down still took me 10 minutes ( or more ).

Anyway, my WIN of today is that I:
- am confident I'll pass
- am full of motivation to continue ( will register very soon )
- have learned from the experience in order to do better in future exams.

40m before the exam

Getting crowded @ the ORG.
Preparing to leave.
Sort of excited.

2h30 before MS221 exam

2h30 and counting.
Staying calm is definitely a quality.
Preparing at Uni.
My goal was ( is ) a distinction.
Had another look at the specimen exam. In an at home TMA like situation with Mathematica a 90+ score is very well possible.
But with only three hours at the worst possible moment of the day (14.30 - 17.30) and with only a calculator, I am not sure at all.
Thank G_d, we are allowed to use the MST121/MS221 handbook.

Speculating...
To get a grade 4 pass...

33 + 7.
11 / 12 questions @ 3/6 + 1/2 @ 7/14.
That is still a lot.
OR
5/12 questions @ 6/6 + 2/2 @ 5/14.

To get a grade 2 pass...
12 /12 questions @ 4/6 2/2 @ 11/14

More realistically, questions are expected on the following subjects ( expected result )
I
2nd order recurrence system (6)
conics (3)
composite isometry (3)
function iteration (3)
linear transformation rotation / reflection (3)
matrix calculation (3)
differentiation (6)
integrationb (6)
taylor series (0)
complex numbers (3)
number theory (6)
group theory (6)
logic (6)
II
conic with xy component ( rotated )
eigenvalue problem ( 11 )
volume of 3D body
proof by induction ( 17 )

54 + 17 = 71 grade 2 pass

More tomorrow or after the exam.

Result MS221 - TMA04 is in

All TMA's are done now. - Due to the substitution rule which is used at the OU the 87 for TMA02 has been replaced by 91,25.

What's it worth? Well, it means I am placed for distinction but that has to be proved at the exam next tuesday. I haven't got a clue how I'll perform during the exam. It's in the middle of my afternoon dip: 14.30 - 17.30. There isn't a single topic I don't understand that's why I was able to score high TMA's. But an exam has a time limit. 

I am not afraid of the exam because even at a score of 15 I am entitled to a resit and 40 is a grade 4 pass. But, but, but: I need 85 for a distinction. Still three days to go to the exam. 

The story of mathematics (4)

Yesterday I have seen Story of Mathematics, part 4. ( See previous posts on 1,2 and 3 ). Among other topics it was about Cantor's math on infinity. Cantor introduced an entire array of infinities. The 'smallest' infinity is the cardinality ( number of elements of a set ) of N, the set of natural numbers. The paradox that the sets {1,2,3, ... } and {10,20,30,...} have the same number of elements was shown. A nice graphic followed about how Cantor reasoned that Q, the set of all fractions, has the same cardinality as Z. It went more or less like this.

Make an infinite square of all fractions such that the first row contains all fractions with numerator 1, the second row with numerator 2, etc. Do the same for the denominators but by colomn. The square should look like:

1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 ...
2/1 2/2 2/3 2/4 2/5 2/6 2/7 ...
3/1 3/2 3/3 3/4 3/5 3/6 ...
4/1 4/2 4/3 4/4 4/5 ...
5/1 5/2 5/3 5/4 ...
...

It is then possible to traverse this square like
1/1
2/1 1/2
3/1 2/2 1/3
4/1 3/2 2/3 1/4
The sum of the numerator and denominator is constant by row. Because this traversal includes every fraction it is possible to map the fractions in a 1-to-1 relation with the natural numbers and thus the set of fractions has the same number of elements as N. ( It is easy to include the negative fractions in the traversal, the result is the same. ) It turned out that a similar reasoning is -not- possible for the reals R and thus R has an infinity which is more infinite than the natural numbers. ( It wasn't in the documentary but I know that there is a third level of infinity which is the same as the total number of possible curves in a 2D-space ( plane ). Theoretically there is an infinity of possible infinities ( I think ).

The mathematician Hilbert and his famous problems for the 20th century was introduced. The first problem was called the Continuum Hypothesis which asked if there was a set in between Q and R. Around that time Kurt Godel proved that there are statements in every possible system of mathematics which cannot be proved. Until the 30's of the last century Europe had the world centers of mathematics in cities like Gottingen and Paris. Then the Nazis came but many scientists and artists left. In the US Princeton was established to become the next world center of science and mathematics. Among others Godel and Einstein lectured there. Needless to say both were Jewish.

Many famous mathematicians had some sort of psychological issue. Godel suffered paranoia and Cantor had a bipolar disorder. In the 30's there were no anti-depressants or ADHD type of drugs so they must have had a dark life. Yet, I don't think having a disorder is a prerequisite for being able to produce excellent math because there are many more mathematicians without a disorder. I am not even sure if the mathematical community has more psychological issues than other groups of scientists.

At Princeton there was a young mathematician Cohen who took on Hilbert's first problem. His answer wasn't what the community expected but because Godel approved his paper the community accepted that there are two types of mathematics: one where the Hypothesis is false and one where the hypothesis is false.

Base conversion

Example base 10 to base 3 conversion:

What is 777 in base 3 ?

We repeatedly divide by 3 and store the remainder until 0 is left:
777 = 3 * 259 + 0
259 = 3 * 86 + 1
86 = 3 * 28 + 2
28 = 3 * 9 + 1
9 = 3 * 3 + 0
3 = 3 * 1 + 0
1 = 0 * 3 + 1

Now the number is equal to the stored remainders in reverse order:
1001210

Let's check if the value in base 10 is indeed 777:
1 * 729 = 729
0 * 243
0 * 81
1 * 27 = 27
2 * 9 = 18
1 * 3 = 3
0 * 1
Finally we add 729 + 27 + 18 + 3 = 777.

Don works for basically any coniguration,

An identity

Were you aware of

3^3 + 4^3 + 5^3 = 6^3?

Cool.

Pythagorian triples were known in 1900BC

Just read In A primer of analytic number theory by Jeffrey Stopple, 2003 that there is a Babylonian cuneiform tablet (designated Plimpton 322 in the archives of Columbia University) from the nineteenth century b.c. that lists fifteen very large Pythagorean triples; for example, 127092^2 + 135002^2 = 185412^2.

This means that they must have known how to generate these numbers with
x = 2*s*t, y = s^2 − t^2, z = s^2 + t^2

For s=2,.t=1 we get the well known x=4, y=3, z=5 and
for s=3, t=1 we get 6, 8, 10 and
for s=4, t=2 we get 16, 12, 20.

1900 BC.

So 1900BC is much closer than I thought. The current blazing speed of scientic development has only been reacned since the last few hundred years or so.

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Blog or diary?

I think the words 'blog' and 'diary' contradict each other. This should be either a blog or a diary now it is an attempt to be both while the endresult is neither. My blog/diary contains a lot of introvert babble like 'TMA done'. Who cares? So I completed MST121 TMA02 early this year. The course is completed. Done. I am going to write more about mathematics.

With the MS221 in less than two weeks I'll be free to study whatever I like until at least february 2010 when I will be doing 'Pure Mathematics' and 'Groups and Geometry'. I think I am going to self-study Introduction to Analytic Number Theory by Apostol. That will prepare me for three OU courses: Number Theory (M381) and Analytic Number Theory I ( M823 ) and II ( M829 ).

I would very much like to be able to prove, or at least understand the proof of, The Prime Number Theorem.

Thinking of the future

Received the Open University Prospectus Mathematics and Statistics 2009 / 2010 today. Although still in the beginning of my studies I already more or less planned my undergraduate studies so I looked at the MSc. programme. It is possible to study for a M.Sc. degree after you have finished the 360 points B.Sc. course. The M.Sc. course requires 180 points from five 30 point modules and a 30 point dissertation. With my interests I probably do Analytic number theory I and II ( 2 * 30 ), Coding Theory ( 30 ) so still 60 to choose and the Dissertation ( 30 ) of course.

We'll see.