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## Thursday, December 31, 2009

### Lecture 23 of Abstract Algebra E-222

Another lecture by Peter.
Summary.
- Sylow Theorems
- Groups of order p*q
- Groups of order p^2 * q
- S5
Topic of the lecture: A5.
- Prove the following proposition: " If G is a simple group of order 60 then G is isomorphic to A5."

And more A5 stuff.

## Wednesday, December 30, 2009

### Jokes about math and mathematicians

The stereotype mathematician in jokes is a male writing some incomprehensible formula on a blackboard and not being able to communicate the meaning of it to his audience. There are however jokes where the mathematician is viewed more favourable. I found two of them on the site from Simon Singh. ( Simon Singh is an English author, journalist and TV producer, specialising in science and mathematics. He wrote a book on Fermat's Last Theorem. As I mentioned Singh collects jokes, you'll find his jokes on his website. )

An assemblage of the most gifted minds in the world were all posed  the following question:"What is 2 + 2 ?"

The engineer whips out his calculator, taps away at it for a while and finally announces "3.99".

The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02".

The mathematician cogitates for a while, oblivious to the rest of the world, then announces: "I don't know what the answer is, but I can prove an answer exists!".

The philosopher strokes his chin for several days, finally asking:  "But what do you mean by 2 + 2?"

Finally the accountant closes all the doors and windows,  looks around carefully then asks "What do you want the answer to be?"

( By Helen Arney )

An astronomer, a physicist and a mathematician were holidaying in Scotland. Glancing from a train window, they observed a black sheep in the middle of a field.

"How interesting," observed the astronomer, "all Scottish sheep are black!"

To which the physicist responded, "No, no! Some Scottish sheep are black!"

The mathematician gazed heavenward in supplication, and then intoned,  "In Scotland there exists at least one field, containing at least one sheep,  at least one side of which is black."

( By Stephen Oman )

## Tuesday, December 29, 2009

### Math cartoon

Cartoons about math are usually not funny.

Don't you agree?

## Monday, December 28, 2009

### Symmetry is everywhere

Another pic from the beautiful book, I wrote earlier about, Visual Symmetry.

( 23rd Street, New York City )

## Sunday, December 27, 2009

### Mathematica now supports handwriting recognition

In this blog-entry I wrote that I couldn't wait for handwriting recognition in Mathematica. Well, it's here. That is it works on my laptop. I have been very careful with installing / upgrading Windows 7. I have been burned too many times by ( Microsoft ) software. I first did a fresh install on a PC I have ready and available for test purposes. ( Although I have addressed it as the 'Database Server', have to finish the apps first. But that's off-topic here. ) Anyway. There were problems, of course. A driver issue of the built-in wireless network card. Some more, I can't remember. I fixed all problems eventually. Then it was time for one of my 'production' PC's: the laptop. Considering what Windows 7 adds to Vista, which is not much ( really, except handwriting recognition in Mathematica, it is nothing really ) I decided I should go for an upgrade. The idea of re-installing all my apps was too much.

I had a copy of Windows 7 Ultimate. UK version. That did not work as an upgrade of a Dutch version of Vista. Yakh. That delayed the upgrade several weeks. When I finally got hold of a Dutch 7 Ultimate I continued. MAJOR PROBLEMS, this time. The Wacom Driver ( I have a tablet PC ) did not work anymore, screen rotation portrait / landscape did not work, the HP specific buttons did not work anymore. I had to download Windows 7 drivers for most of the hardware. There is however no Wacom tablet driver for Windows 7. Bye, bye tablet. I thought for a day anyway. While browsing some forum I read a story of someone who experienced the same problems and was so kind to document what he did. He just -reinstalled the Wacom Vista drivers-. And that worked. There were several other issues but I managed to solve all of them. The entire upgrade took me about 12 ( Twelve ) hours. That includes the waiting and searching for problem solutions.

The good thing is that I can now use handwriting to enter mathematics into Mathematica. And that is extremely cool. It takes practice though, as with everything. Still discovering the how-to's. The tablet features of Windows 7 did got a major upgrade compared to Vista. ( And Vista was already much better than XP in that respect. ) - For now I decided to leave my desktop AS IS: Vista. An upgrade from Vista to Windows 7 is nice but it is not a must. Unless there is this specific feature you are after, like me with math handwriting recognition.

One more thing: initially handwriting did not work as described in the Mathematica documentation. After re-installing Mathematica it did however work.

## Thursday, December 24, 2009

### Oxford numbers ( math mystery movie )

Definitely NOT a family Christmas movie: too much sex and nudity. Don't know who wrote the script, probably written by a dreamer, a wannabee of some kind. Young Ph.D. student from Arizona moves to Cambridge as an overseas student. He managed to get a room at the house of one of the people ( an older woman living together with her young attractive cello playing daughter, hint. ) who together with Alan Turing (-himself-) cracked the enigma. In the first scene he walks into the room. And says "that's an enigma", "No, merely a copy, the real one is at the museum", she says. More math stuff. The story plays in 1993 when a certain Wilkins announces that he cracked the 300-year old Bormat Theorem. It turned out that Wilkins stole the idea from a student. ( Maybe the entire reason for the movie was to leak the fact that Andrew Wiles stole the idea for cracking Fermat's theorem from a Ph.D. student. It wouldn't be the first time. Highly speculative of course. ) Anyway, all the girls the Ph.D. student meet fall instaneously for his charmes and seduce him. How surreal. Nerds and math students are instant turn-offs for women, like nude men wearing white socks, sort-of. The Oxford Murders is a paradox in many ways. It's a murder mystery. It's custom not to give away the murderer so I won't. The paradox is that it's a very bad movie, but still so much better than a Beautiful Mind ( except for Jennifer Connolly of course ) which was an award winning movie. There is -some- math in the movie. Fibonacci sequence, just as in PI, which is still the best math movie by far, if you ask me.

## Wednesday, December 23, 2009

### Mathematics in Movies ( 2 )

I got two math movies today. Fermat's Room and the Oxford Numbers. Both from 2008, maybe that explains why neither are on the page I mentioned. Haven't seen them yet though. Something for Christmas, perhaps.
I am not sure, but I think a movie about Ramanujan will be released in 2010. A must see. Ramanujan and Hardy in Cambridge. - Haven't watched Numb3rs in a while. I saved two seasons, for 'someday'. Well, maybe that day should be soon. Although Charlie is completely surreal, I mean he is an expert in -every- subject and has time to lecture, write books and be a full-time consultant to the FBI as well. Secret: I adopted one of his habits: I often use a noise-reduction headphone just to create silence around me, not to listen to music. It's great, you should try it while studying.

## Tuesday, December 22, 2009

### Lecture 22 of Abstract Algebra E-222

Richard Taylor ( once Ph.D. student of Andrew Wiles, the Fermat Theorem guy ) gave a lecture about the Symmetric group.
He demonstrated a short way of calculating the conjugate of a permutation and he talked a bit about S5, including the number of conjugacy classes it has. Each partition of n in Sn represents a conjugacy class. So for 5 it is:
-5 : 24
-41 : 30
-32 : 20
-311 : 20
-221 : 15
-211 : 10
-11111 : 1
Total 120=5!
The Stirling numbers of the first kind are used to calculate these numbers, which incidentally produce a Pascal triangle type-of matrix.

## Sunday, December 20, 2009

### Lecture 21 of Abstract Algebra E-222

Again the Sylow theorems and a summary of the proofs.
Then finite groups of certain orders were classified.
Order p, p prime. Cyclic groups of order p.
Order p * q, p > 2, p < q. Cyclic of order p*q and if p / (q-1): Cq : Cp.
Order 2p, cyclic of order 2p and the Dihedral groups Dp.
Order p^2*q. Started with order 12:
- C12
- C6 X C2
- A4
- D6
- C4 : C3.
- A4 = ( C2 X C2 ) : C3.

Gross sofar never talked about GAP, Mathematica, Magma or Maple. Group theory can get very complicated without the aid of a mathematics package. The groups were classified solely on the basis of the Sylow theorems in this lecture. The semi-direct product has not been lectured yet. For those new I would say these lectures were real hard. Unnecessarily hard in my opinion. The next lecture is about the Symmetry group by a famous number theory guy ( forgot his name ).

Anyway, I discovered an interesting fact about S6 today, the symmetry group on 6 letters. It is the only symmetric group whose automorphism group is not eqaul to the group itself. - Now, why would that be? Seems like some deep fact to me. - This is were mathematics gets a grip on you. You must know why.

## Saturday, December 19, 2009

### Linear Algebra ( Books for 2010 )

Professor Gross ( Harvard ) said in one of the abstract algebra ( e-222 ) lectures that " you can't learn too much linear algebra ". I always liked linear algebra but I thought I knew most of it. I could not have been more ignorant. I discovered a book today 'Advanced Linear Algebra' by Steven Roman, a book from the Springer Graduate Texts in Mathematics series. It contains two parts, part 1: basic linear algebra contains ten chapters called:
- 1. Vector spaces
- 2. Linear transformations
- 3. Isomorphism theorems
- 4. Modules I
- 5. Modules II
- 6. Modules over a PID
- 7. The structure of a linear operator
- 8. Eigenvalues and eigenvectors
- 9. Real and complex inner product spaces
- 10. Structure theory for normal operators
Part 2: topics, contains another 9 chapters.
I think I am ready for part 1, since M208 contains linear algebra as well, this book definitely comes on my 2010 list. I just decided I am going to make a list for the math books I want to study in 2010 besides M208, MT365.

I was actually studying a book on Group Representation Theory. That's a topic which relies heavily on linear algebra. When I was studying Maschke's Theorem I realized I had to review my linear algebra, especially inner product spaces. And then I found Roman's book. My understanding of vector spaces (1) is ok, linear transformations (2) as well, and if not, M208 has lots of stuff on that. I studied the isomorphism theorems (3) in group theory, I think they are more or less the same in linear algebra. I have some notion about modules (4,5,6): like vector spaces but with scalars from a ring instead of from a field. Have to study them deeper, I suppose. Maybe the CG Modules ( vector spaces where the vectors can be multiplied with group elements as well as scalars ) I studied are a sort of modules, have to check it out. Looking forward to learn more about structure of linear operators (7) and (10) as well. The stuff in Eigenvalues and eigenvectors
(8) and Real and complex inner product spaces (9) is familiar but most likely goes much deeper here.

From a first browse through the book I can say that I like the style. Clearly written and enough examples. I hate books that don't have examples. I think that authors who don't include examples in their books are - A) too lazy, or B) not really willing to communicate their knowledge, or C) sadistic. In all cases bad people. - The book is packed with exercises but alas for the self-study student: no answers. I haven't made my mind up about authors who do provide exercises but keep the answers to themselves. Fortunately I found some problem books on advanced linear algebra as well. More on that another time, perhaps.

## Thursday, December 17, 2009

### Assessment strategy M208, MT365.

For MT365: one exam, and the OCAS consists of 4 TMA's and 4 CMA's. I wasn't expecting the CMA's. The TMA's count for 15% a piece and the CMA's 10% a piece.

For M208: one exam, and the OCAS consists of 7 (not 4) TMA's.

Had a brief look at some past exams from both courses. Both exams seem doable, passable. Most of the MT365 is entirely new to me, so I must be careful not to underestimate this one. Rating the course at level 3 must have had a reason.

For the moment I'll stop preparing, pre-reading for M208, MT365, will do other math stuff.

## Wednesday, December 16, 2009

### Registered for MT365

Just committed myself to MT365. - I had doubts if I should go for 90 or not. Had to think about it for a while. I still think 60 is the best to do but the group theory part in M208 can't be difficult for me. I can spend that time entirely on MT365.

What's in MT365? ( Although it's a level 3 course there aren't much hard pre-requisites, just a certain mathematical maturity whatever that is. )

The course is divided into three related areas: graphs, networks and design. The Introduction introduces two themes of the course, combinatorics and mathematical modelling, and illustrates them with examples from the three areas.

Graphs 1: Graphs and digraphs discusses graphs and digraphs in general, and describes the use of graph theory in genetics, ecology and music, and of digraphs in the social sciences. We discuss Eulerian and Hamiltonian graphs and related problems; one of these is the well-known KĂ¶nigsberg bridges problem.

Networks 1: Network flows is concerned with the problem of finding the maximum amount of a commodity (gas, water, passengers) that can pass between two points of a network in a given time. We give an algorithm for solving this problem, and discuss important variations that frequently arise in practice.

Design 1: Geometric design, concerned with geometric configurations, discusses two-dimensional patterns such as tiling patterns, and the construction and properties of regular and semi-regular tilings, and of polyominoes and polyhedra.

Graphs 2: Trees Trees are graphs occurring in areas such as branching processes, decision procedures and the representation of molecules. After discussing their mathematical properties we look at their applications, such as the minimum connector problem and the travelling salesman problem.

Networks 2: Optimal paths How does an engineering manager plan a complex project encompassing many activities? This application of graph theory is called ‘critical path planning’. It is one of the class of problems in which the shortest or longest paths in a graph or digraph must be found.

Design 2: Kinematic design The mechanical design of table lamps, robot manipulators, car suspension systems, space-frame structures and other artefacts depends on the interconnection of systems of rigid bodies. This unit discusses the contribution of combinatorial ideas to this area of engineering design.

Graphs 3: Planarity and colouring When can a graph be drawn in the plane without crossings? Is it possible to colour the countries of any map with just four colours so that neighbouring countries have different colours? These are two of several apparently unrelated problems considered in this unit.

Networks 3: Assignment and transportation If there are ten applicants for ten jobs and each is suitable for only a few jobs, is it possible to fill all the jobs? If a manufacturer supplies several warehouses with a product made in several factories, how can the warehouses be supplied at the least cost? These problems of the system-management type are answered in this unit.

Design 3: Design of codes Redundant information in a communication system can be used to overcome problems of imperfect reception. This section discusses the properties of certain codes that arise in practice, in particular cyclic codes and Hamming codes, and some codes used in space probes.

Graphs 4: Graphs and computing describes some important uses of graphs in computer science, such as depth-first and breadth-first search, quad trees, and the knapsack and travelling salesman problems.

Networks 4: Physical networks Graph theory provides a unifying method for studying the current through an electrical network or water flow through pipes. This unit describes the graphical representation of such networks.

Design 4: Block designs If an agricultural research station wants to test different varieties of a crop, how can a field be designed to minimise bias due to variations in the soil? The answer lies in block designs. This unit explains the concepts of balanced and resolvable designs and gives methods for constructing block designs.

Conclusion In this unit, many of the ideas and problems encountered in the course are reviewed, showing how they can be generalised and extended, and the progress made in finding solutions is discussed.

### Registered for M208

Just committed myself to M208.

What's in M208?

Introduction Real Functions and Graphs is a reminder of the principles underlying the sketching of graphs of functions and other curves. Mathematical Language covers the writing of pure mathematics and some of the methods used to construct proofs. Number Systems looks at the systems of numbers most widely used in mathematics: the integers, rational numbers, real numbers, complex numbers and modular or ‘clock’ arithmetics.

Group Theory (A) Symmetry studies the symmetries of plane figures and solids, including the five ‘Platonic solids’, and leads to the definition of a group. Groups and Subgroups introduces the idea of a cyclic group, using a geometric viewpoint, as well as isomorphisms between groups. Permutations introduces permutations, the cycle decomposition of permutations, odd and even permutations, and the notion of conjugacy. Cosets and Lagrange’s Theorem is about ‘blocking’ a group table, and leads to the notions of normal subgroup and quotient group.

Linear Algebra Vectors and Conics is an introduction to vectors and to the properties of conic sections. Linear Equations and Matrices explains why simultaneous equations may have different numbers of solutions, and also explains the use of matrices. Vector Spaces generalises the plane and three-dimensional space, providing a common structure for studying seemingly different problems. Linear Transformations is about mappings between vector spaces that preserve many geometric and algebraic properties. Eigenvectors leads to the diagonal representation of a linear transformation, and applications to conics and quadric surfaces.

Analysis (A) Numbers deals with real numbers as decimals, rational and irrational numbers, and goes on to show how to manipulate inequalities between real numbers. Sequences explains the ‘null sequence’ approach, used to make rigorous the idea of convergence of sequences, leading to the definitions of pi and e. Series covers the convergence of series of real numbers and the use of series to define the exponential function. Continuity describes the sequential definition of continuity, some key properties of continuous functions, and their applications.

Group Theory (B) Conjugacy looks at conjugate elements and conjugate subgroups, and returns to the idea of normal subgroups in this context. Homomorphisms is a generalisation of isomorphisms, which leads to a greater understanding of normal subgroups. Group Actions is a way of relating groups to geometry, which can be used to count the number of different ways a symmetric object can be coloured.

Analysis (B) Limits introduces the epsilon-delta approach to limits and continuity, and relates these to the sequential approach to limits of functions. Differentiation studies differentiable functions and gives l’HĂ´pital’s rule for evaluating limits. Integration explains the fundamental theorem of calculus, the Maclaurin integral test and Stirling’s formula. Power Series is about finding power series representations of functions, their properties and applications.

## Saturday, December 12, 2009

### Real matrix representations of quaternions

The quaternions were invented in 1843 by Hamilton. There are three quaternions: i, j and k where i is the well known imaginary number i. The rules for the multiplication of quaternions are as follows :

i * j = k, j * i = -k
j * k = i, k * j = -i
k * i = j, i * k = -j
i^2 = j^2 = k^2 = -1.

The quaternion group has eight members { 1, -1, i, j, k, -i, -j, -k } and is non-abelian.

The most common presentation of Q8 is < a,b | a^4 = 1, a^2 = b^2, b^-1*a*b = a^-1 >.

Using only real numbers the quaternions can be represented as 4-by-4 matrices as follows.

## Thursday, December 10, 2009

### MS221: Grade 2 pass ( Cont. )

All in all, it has been a few hours since I know the result. I am happy with it, of course I am. I am fully motivated for the courses next year, that's what counts now. I can't wait to make another TMA actually.

Should register asap. Registration for the 2010 courses closes the 16th.

Initial reaction: happiness
2nd reaction: relief, not the feared grade 4 pass
3rd reaction: disappointment only 3pnts below distinction
4th reaction: doesn't mean a thing because 90% passed at eirher d, 2, 3 or 4
5th reaction: jealousy, 20% or so did get a distinction
Just recording what I felt when I read the pass.

I MUST DO BETTER NEXT TIME.

( I had 82/100 on the examinable component, that's really bad. Yakh. )

## Wednesday, December 9, 2009

### Playing games with groups

Doing mathematics is a pleasurable activity. It can be very much like playing a game. With the help of GAP as a juror, teacher there are zillions of games you can do by yourself. Some ideas:
- finding all groups ( of a specific order );
- finding the presentation formulas;
- finding the automorphism group;
- finding the character table;
- drawing the Cayley graph;
Once you start on this you get a sort of natural need for the use of either the Mathematica Abstract Algebra package, or even GAP ( or similar ), that is a mindset which makes learning the language rather easy.

## Tuesday, December 8, 2009

### Aut( Q8 ) = S4

The automorphism group of the quaternion group is the symmetric group on four letters. Sounds fascinating to me. But once I figured it out it's rather easy.

The elements of Q8 are:
1:1, 2:-1, 3:i, 4:j, 5:k, 6:-i, 7:-j, 8:-k.
The generators are:
3:i, 4:j.

Then there are 24 isomorphisms, since there are 6 possible generators i, j, k, -i, -j, and -k but after 1 has been chosen, 4 remain ( choose i then -i can't be chosen anymore ).

So up to here I have shown that Aut(Q8)=24, not yet that it is isomorphic to S4. ( That's for later. )

## Monday, December 7, 2009

### Extensions of a group

I want to understand the groups of order 16, in order to construct them I have to understand the extension problem. So I am currently studying chapter 7 of Rotman.

The groups of order 16 are:
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
Q16
QD16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)

At this moment I can construct
C16
C8 x C2
C4 x C4
C4 x C2 x C2
C2 x C2 x C2 x C2
D16
D8 x C2
Q8 x C2
C8 : C2
C4 : C4

Work to do on:
Q16
QD16
(C4 x C2) : C2 (a)
(C4 x C2) : C2 (b)

## Saturday, December 5, 2009

### C13 symmetry.

The presence of the number 13, a Fibonacci number, is not a coincidence in this picture.

(Picture from the book Visual Symmetry.) - Btw. that book could be a terrific Christmas present. Although interest in mathematics is not required, appreciation of beauty is.

### Breaking news ( I missed ): TeXnicCenter 2.0 alpha released

Last October TeXnicCenter 2.0 ( alpha 1 ) was released. I just read about it and immediately had to make a note of it in my blog. How could I have missed it? October had a lot of things going on for me, including the MS221 exam ( results expected on the 18th ), suppose that must be the reason. Since I am not working actively on TMA's at the moment I don't use TXC a lot right now.

To the point: the new version. It is really great news that the developers took the time to upgrade TXC. Although near perfect already new technologies in IDEs became standard in recent years which aren't in TXC 1. Cold folding is one of them and it has been implemented in TXC 2 according to the website. Cold-folding is really a must-have nowadays. I recently blew TXC and started to use an Eclipse based TeX IDE. After only one TMA I returned to TXC which simply works best for me.

I will install and test the version of TeXniCCenter 2.0 alpha and report my findings here.

TexNicCenter

## Friday, December 4, 2009

### Watched lecture 20 of Abstract Algebra E-222

The entire lecture was about proving the Sylow Theorems. The Sylow Theorems are easy to remember ( because they are so useful ) the proofs aren't. More on the Sylow Theorems here ( MathWorld ).

## Thursday, December 3, 2009

### Change of plan ?

Since I have to register for the 2010 courses before the 14th this month I have to make up my mind fast. I thought to be sure about 2010 but watching the E-222 videos changed my mind. It was my plan to do MST209 and M336 next year, thus leaving M208 for 2011. ( Ideal would be M208 + MST209 but is just not advised to take on a load like that. If you slip a few weeks for whatever reason it is impossible to catch up. So M208 + MST209 is out of the question. )

Both M208 and M336 contain Group Theory. M208 introduces the basic concepts: groups, subgroups, cosets, Lagrange's theorem, normal subgroups, and quotient groups in one module. A second module covers conjugacy classes, homomorphisms and group actions. All stuff I understand fairly well. That's why I though M336 was an option although M208 is a prerequisite. M336 reviews the M208 stuff and then covers counting with the aid of group actions ( necklace problem, I suppose ). The theory of abelian groups is covered fairly deeply just as the Sylow Theorems. These topics are half of the course. The other half is about geometry using group theory. The solids in two and three dimensions ( what I have just seen in the E222 videos ), tilings, frieze patterns, lattices and the wallpaper patterns.

My conclusion is that M336 in 2010 and M208 in 2011 is not an option. M208 + M336 is an option though. M208 in 2010 and M336 in 2011 is not an option because M336 doesn't run in 2011, so M208/2010, M336/2012 is the second option.

Recapping...
MST209 + M336 - NOT

1. M208 + M336 - Option
2. M208 ( only ) - Option
3. M208 + MT365 - Option
Both M208, MT365 have MST121, MS221 as prerequisite, so that fits. ( That is if I have a pass for MS221 ).

Just MST209 is an option as well, I suppose, although in that case I might jeopardize the overall plan.

## Wednesday, December 2, 2009

### Watched lecture 19 of Abstract Algebra E-222

A superb lecture about the proof of the proposition that A5 is a simple group. I got new insights from this lecture on conjugacy classes and constructing group theory proofs.

## Tuesday, December 1, 2009

### Watched lecture 18 of Abstract Algebra E-222

Topics.
- Group actions.
- Counting formula.
- Conjugation action.
- Conjugacy classes.
- Class equation.
- Symmetry group of the Tetrahedron (A4),Icosahedron (A5).
- Nice proof of the fact that a group of order p^n always has a non-trivial centre.
- At the time this video was recorded mathematicians conjectured that shape of the universe was SO(3)/A5.
( For me not a particularly simple lecture, but am still following. )

## Monday, November 30, 2009

### Watched lectures 15,16 and 17 of Abstract Algebra E-222

My first encounter with the topic of discrete groups of motions. Gross treats the subject very abstract. Have to read this in Artin first and then re-watch the videos or just wait for when the topic comes along during M336. I think that about 50% of M336 is spent on discrete groups and other entirely new stuff for me. M336 won't be as easy as I thought it was going to be. As far as E-222 is concerned, I can continue watching the other lectures since in the next lectures group actions are discussed. And soon after that the Sylow theorems. It will be interesting to see how Gross introduces both topics.

## Sunday, November 29, 2009

### Extending C2 x C2 to Q8

What I never understood is that in books on GroupTheory Q8 is shown as a concrete group, i.e. the group of quaternions {i, j, k | i^2 = j^2 = k^2 = -1, i*j=k, j*k=i, k*i=j } and not as an abstract group. Well, I just discovered that it is fairly easy to construct Q8 from C2 x C2 ( which is often shown in abstract form and concrete form: the Klein4 group ).The group C2 x C2 has the following presentation:<a,b | a^2 = b^2 = 1, a*b = b*a >.The group Q8-abstract has the following presentation:<a,b,c | a^2*c = b^2*c = 1, a*b*c = b*a >,members of this group are:{ 1, a, b, a*b, c, a*c, b*c, a*b*c }.The following isomorphism can be established between Q8-abstract and Q8:f: Q8-abstract -> Q8by{ 1 |-> 1,  a |-> i,   b |-> j,  ab |-> k,  c |-> -1  ac |-> -i,  bc |-> -j,  abc |-> -k }.

Q8 is not something like ( C2 X C2 ) : C2, where X stands for direct product and : stands for semi-direct product, but it is very likely something similar. I read briefly that there are ways to construct groups other than using the direct or semi-direct product. Will / must take some time to check this out.
The fact that Q8 can be constructed and has a fairly simple presentation predicts that there must be similar methods for -all- other finite groups.

## Tuesday, November 24, 2009

### Watched lecture 14 of Abstract Algebra E-222

Abstract babble on transformation groups in R3. We are now supposed to be ready to study some Euclidean geometry in the next lectures.

Gross: "The most important theorem in calculus is the Intermediate Value Theorem."

### What is the error?

Have a look at this...

There is a clearly an error in this example. Do you see what it is?

Example taken from
Abstract Algebra
Theory and Applications
Thomas W. Judson
Stephen F. Austin State University
February 14, 2009

( See comment for further explanation )

## Sunday, November 22, 2009

### Watched lecture 13 of Abstract Algebra E-222

Gross provided answers to the questions of the exam the students had a few days before this lecture was held.

One question was as follows.
Q. Show that if a group has a unique element of order 2 then it is part of the center.
A. Order of element is equal to order of conjugate and because there is only one element of order 2 the following is true.
a = g a g^-1 for all g in G, or
a g = g a for all g in G,
and thus a belongs to the centre of G.

The lecture introduced subgroups of GL(n,F):
- the orthonogal group O(n,F) and
- the special linear group SL(n,F).
Where GL(n,F) consists of invertible matrices in O(n,F) this is further reduced to matrices with the property that the transposed matrix is equal to the inverse matrix. These matrices turn out to have determinants 1 or -1. ( Not true that all matrices with determinant 1 are orthogonal ). Matrices with +/- 1's on the diagonal are orthogonal as well as permutation matrices.
The elements of the Special Linear group are further reduced to those with a determinant value of 1.

A concrete orthogonal group is O(2,R) as subgroup of GL(2,R). This group consists of 2 by 2 matrices with elements from the real numbers. These matrices are linear transformations of vectors in R2.

More groups and geometry to follow.

## Friday, November 20, 2009

### Group Explorer 2.2

Nathan Carter's Group Explorer is now in release 2.2. He takes very good care of his program. He wrote a book about Group Theory as well: Visual Group Theory. A sort of Group Theory explained illustrated by Group Explorer. Cool.

### Watched lecture 12 of Abstract Algebra E-222

The well known eigenvalues and eigenvectors stuff. Near the end Gross tells about his favourite theorem: the Cayley Hamilton theorem which says that if you plug in the matrix itself into its characteristic equation then you end up with a zero-matrix.

### What is six billion ?

Six billion is 6.000.000.000, six billion dollar is the amount that Obama on behalf of the US tax-payers is lending from foreign countres in order to keep the government going. Per day. Two trillion per year, 69444 per second. How long can this go on? The US must be bankrupt, but nobody wants to say it aloud.

### Personal Brain 5.5 released

TheBrain.com released version 5.5 of Personal Brain. The update is free for version 5 users. Among 150 or so ( minor, but ) new features they have added ESP(!) to your brain. That is the brain monitors what you are doing in another application via clip- and/or keyboard and moves within the Plex accordingly. Cool feature. It, indeed, 'senses' what is required of the Brain. - Btw. I am now using PB consistently for Abstract Algebra.

## Wednesday, November 18, 2009

### Rotman

How time flies. Two and a half year ago I discovered Rotman's book on Group Theory. From what I understood of it I found it a fascinating book but I knew -and hated the fact- that I simply wasn't ready for it at the time. Wasn't I? I doubt if I have been studying hard enough. I am going to give the book another try.

As Rotman put it...
Exercises in a text generally have two functions: to reinforce the reader's grasp of the material and to provide puzzles whose solutions give a certain pleasure. Here, the exercises have a third function: to enable the reader to discover important facts, examples, and counterexamples. The serious reader should attempt all the exercises (many are not difficult), for subsequent proofs may depend on them; the casual reader should regard the exercises as part of the text proper.

I think that's the trick to understanding his book.

As a matter of fact I already started. As Rotman wrote ( many ) are not difficult. The following exercise is an easy one if you studied Stirling numbers, if you didn't I wouldn't call it simple.

Exercise 1.5 ( in Rotman ). If 1 < r < n, then there are (1/r) [n(n — 1)... (n — r + 1)] r-cycles in Sn.
Why doesn't he use Stirling numbers ? I am going to need to review Stirling numbers.

Now I remember, I encountered this before and decided I had to study some Discrete Math first. Well, I did. So that won't hold me up studying this book. I think all the prerequisites are 'in' now for attacking this book.

## Tuesday, November 17, 2009

### Professors

If I would have seen only Gross in the Algebra videos I would have thought about him as a good lecturer, abstract type of guy but that is his profession. The fact that, as a contrast, we also see Peter learning the profession of lecturing, it becomes clear how difficult lecturing in fact must be. I mean Peter ( the 2003 version of him I bet he is huge by now ) is already impressive but no match to Gross, what a performance that was in lecture 11.

### Watched lecture 11 of Abstract Algebra E-222

Basically the same stuff as in lecture 10 with some cool examples. And finally the well known A = B A B^-1 matrix conjugation for base conversion.


GL(2,2) is the general linear group of dimension 2 over GF(2).
Gross asked which group is isomorphic to GL(2,2) ?
( I stopped the video and gave it a try. )

GF(2) has the following tables for addition and multiplication.
+    0 1    x    0 1
0    0 1    0    0 0
1    1 0    1    0 1

The possible maps from F2 -> F2 have the following matrices :
0 0    1 0    0 1    1 1
0 0    0 0    0 0     0 0

0 0    1 0    0 1    1 1
0 1    0 1    0 1    0 1

0 0    1 0    0 1    1 1
1 0    1 0    1 0    1 0

0 0    1 0    0 1    1 1
1 1    1 1    1 1     1 1

Elements of GL(2,2) are the matrices which have determinant 1.
1 0    1 1    0 1    1 1    1 0    0 1
0 1    0 1    1 0     1 0    1 1    1 1

Order    1    2    2    3    2    3

We now see that GL(2,2) is generated by
0 1    1 1
1 0     1 0
and is isomorphic to S3.

Gross however had a different ( smarter ) approach as follows.
F2 is the following set:
{ (0,0), (1,0), (0,1), (1,1) }
A linear transformation from F2 to F2 must fix (0,0)
so the elements of GL(2,2) are the permutations of
(1,0), (0,1) and (1,1) with group S3.



### Watched lecture 10 of Abstract Algebra E-222

A lecture by Peter on Linear Algebra. I scanned parts of the video, didn't really like it. Maybe its because I think I am done with this part of Linear Algebra,( have to go through it once more for M208, I suppose ) or its simply because I am interested mainly in groups at the moment.

Anyway, a very abstract and fast talk starting with the definition of a vector space all leading up to the change of basis problem. The change of basis problem is in itself not easy and takes a lot of practive to get comfortable with. A lot of material to go through in just one lecture. It took me months to fully understand these topics. Just to compare: Strang took about 10 lectures to cover this material.

## Sunday, November 15, 2009

### Improving the study process

Studying is a process which can be improved continuously. There are tons of books written with advice on how to do this. I intend to try to improve my own way of studying. I have found several books on the subject, will browse them and select one or two which I will read through. But before I do that I want to define some key statistics which I can monitor so that I can measure the effects of changes I might implement. - In order to facilitate the monitoring I started to use Personal Brain as a study tool. - ( More later. )

P.S.
I had my life to live over again, I would have made a rule to read some poetry and listen to some music at least once a week; for perhaps the parts of my brain now atrophied would have thus been kept active through use. The loss of these tastes is a loss of happiness, and may possibly be injurious to the intellect, and more probably to the moral character, by enfeebling the emotional part of our nature.
—Charles Darwin

## Thursday, November 12, 2009

### Watched lecture 9 of Abstract Algebra E-222

( Gross is Jewish. )

Nothing much really in this lecture.
Vector spaces.
Spans.
Linear combinations.
Dimensions.

Maybe a hint to a very important notion in mathematics: all vectorspaces of dimension n are isomorphic.

Peter on Monday.

## Tuesday, November 10, 2009

### Watched lecture 8 of Abstract Algebra E-222

The following topics are discussed in this lecture:

Isomorphism Theorem

Vector spaces over an arbitrary field
- Definition field
- Examples of finite fields

Proof that Z/pZ is a field
- Added to what we know from Z/nZ as an additive subgroup of Z we must prove that each a in Z/pZ has a multiplicative inverse, so we must show that if a is not a multiple of p then there is an integer b such that a*b congruent 1 mod p.
( Actual proof is worked out in the video )

What are the finite fields beyond Z/pZ ?
- the finite fields are of order p^n where p is a prime and n>=1, so there are finite fields of order 2, 3, 4, 5, 7, 8, 9, 11, etc. ( note 6 = 2*3, 10=2*5 not of type p^n )

Definition of a vector space ( V )
- With a map f: VxF -> V which is called the scalar multiplication
- ( All rules are written down on board. )

Examples of vector spaces
- V={0}
- V=F
- V=F2
- V=Fn
- V=F[X], vector space of all polynomials p(x) with coefficients in F

Vector subspace
- A subgroup 'stable' under scalar multiplication

Vector space homomorphisms
- Linear transformations ( as we knew it ) are explained as group homomorphisms stable under scalar multiplication
- So for T: V-> W we can define the Kernel of T as a subspace of V and the image of T as a subspace of W. We can also define the quotient space V/W analog to the quotient group

### Book of beauty: Visual Symmetry

Visual Symmetry by Magdolna and Istvan Margittai is a beautiful book with hundreds of pictures about symmetry. Aftrer reading this book you will look at the world through different eyes, forever. This book is also an excellent conpanion to M336 imo.

## Saturday, November 7, 2009

### Watched lecture 7 of Abstract Algebra E-222

Gross formulated the following question: Can we put a group structure on the set of cosets {aH} for a subgroup H in G? He subsequently based the entire lecture on answering this ( simple ) question. The answer is ( of course ) yes if H is normal in G.

At the end, briefly for students with an interest in Algebraic Topology, Gross mentioned sequences like 1 -> H -> G -> G' -> 1.  With examples 1 -> Z3 -> Z6 -> Z2 -> 1 and 1 -> A3 -> S3 -> {1,-1} -> 1 which show that by knowing  Z3 ~ A3 and Z2 ~ {1,-1} does not say anything about the resultgroup.

Basicly a long abstract theoretical discussion about factorgroups, with basicly zero examples. How will group theory develop in people exposed to such lectures? I am not sure if I want to think about that.

## Friday, November 6, 2009

### The prototypical mathematician ( is NOT ).

A picture of the prototypical Linux type of person. As one gains experience in Linux one tends to start looking like the Ultimate Nerd. I wish I could understand why. Thank G_d, there are all sorts of mathematicians, maybe it's because mathematics in itself is so rich in subjects that there is no such thing as the prototypical mathematician. - I read somewhere that it is rather not done to say " I am a mathematician", the proper thing is " I studied math ", or something like that. One becomes a mathematician not before others ( in the field ) are calling you one. - I agree there is a difference between simply having done some math courses ( even if they add up to a B.Sc. or M.Sc ) and practicing mathematics at the research level.

## Wednesday, November 4, 2009

### Galois Theory

Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of  a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.

I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.

Update: the field we work in is Q.

### Watched lecture 6 of Abstract Algebra E-222

( A lecture by Peter again ).
Arithmetic congruent mod n.
Multiplication
How a congruent b mod n is in fact an equivalence relation.
And thus induces a partition of the integers.
Cosets are nZ, 1+nZ, 2+nZ, ... (n-1)+nZ
Addition can be defined on these cosets and then they have a group structure.
The map Z -> nZ is then a homomorphism with 0 as kernel.

( Around min 35 or so I lost interest... I fast forwarded watching minutes here and there, just to make sure there was not introduced anything I did not know already. I hope I am not losing interest in the series all together. We'll see. )

## Tuesday, November 3, 2009

### Watched lecture 5 of Abstract Algebra E-222

Defines the equivalence relation on a set as a partition in disjoint subsets whose union is the set.
Properties of an equivalence relation:
- reflexive: a~a
- symmetric: a~b <=> b~a
- transitive: a~b and b~c => a~c.

A homomorphism f: G->H with kernel K which is a normal subgroup of G implies an equivalence relation on G where K is one of the equivalence classes. The other equivalence classes have the form aK = { ak; k in K, for some a in G}. aK is also called a left coset of K. ( Gross writes complete proof of this proposition on board. )
A bit of mathematical history about Lagrange ( born in Italy! ) who writes a letter to Euler at age 17 containing some very sophisticated mathematics. Euler immediately recognizes the genius of Lagrange and arranges further education for Lagrange who until that time learned his math through self-study.
(The famous) Theorem of Lagrange.
If G is a finite group and H is a subgroup of G then the order of H divides the order ( size ) of G.
More propositions are discussed.
- Groups of order p are simple.
- Groups of order p^2 are abelian.
- An is simple for n>=5.
- Any finite, non-abelian group has even order.

( Next lecture Peter. )

## Monday, November 2, 2009

### Watched lecture 4 of Abstract Algebra E-222

Definition of homomorphism.
Proof that e is mapped to e by any homomorphism.
Proof that inverses are mapped to inverses by any homomorphism.
Definition of Image.
Definition of Kernel.
Properties of the Kernel.
- subgroup;
- normal subgroup.
Any normal subgroup is the kernel of a homomorphism.
Example homomorphism.
f: GL(n,R) -> R_x
f(A) = det(A)
f has as kernel the matrices with det=1, also called SL(n,R). ( Special linear group )
Example homomorphism.
f: Sn->GL(n,R)
f(p)=Ap ( permutation matrix associated with p )
f( (1,2,3) ) = {{0,0,1}, {1,0,0}, {0,1,0} }
Definition center of G.
Example homomorphism G-> Aut(G) i.e. Klein4 -> S3

### Watched lecture 3 of Abstract Algebra E-222

( Off-topic: Since I am ' in between jobs ', which happens if you are a freelance IT professional and there in an economic crisis, I should be studying new Oracle features or something like that. Instead I watched another Algebra lecture, well the day is still young. )

Watched lecture 3 of Abstract Algebra E-222. ( A lecture by Peter, Gross's assistant, if he isn't a professor yet, he will be soon, I suppose. )

Review of lectures 1 and 2.
- Groups. And examples of groups GL(n,R), Sn, Z+.
- Subgroups. Cyclic subgroups.
- ( Hom(Rn, Rn) has the structure of a vectorspace. )
- All subgroups of Zn are of the form bZ. ( Emphasis on importance of proof of this proposition.)
- ( Studying the subgroup structure of a group is in general very difficult. )
- Example of a cyclic subgroup of GL(2,R). The group generated by {{1 1}, {0,1}} is {{1 n}, {0,1}} n in Z.

Example of an isomorphism.
G1 = {i, -1, -i, 1}
G2 = {(1,2,3,4}, (1,3),(2,4), (1,4,3,2), ()}
G1 and G2 are isomorphic by i -> (1,2,3,4)
( Permutations are here in cyclic notation which are not introduced in the course yet. )

Example of an isomorphism.
G1 = {R,+}
G2 = {R\{0},*}
G1 and G2 are isomorphic by f: G1->G2; x |-> e^x
Proof:
f(x+y)=e^(x+y)=e^x * e^y = f(x)*f(y).

Klein4 group.
V={() , (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)} as a subgroup of S4.
V={ {{1 0}, {0,1}}, {{-1 0}, {0,1}}, {{1 0}, {0,-1}}, {{-1 0}, {0,-1}} as a subgroup of GL(2,R).

Definitions.
-Automorphism.
-Homomorphism.
-Image ( of a homomorphism)

Next lecture Gross on images of homomorphisms ( and more ).

( Thank you, Peter. )

## Sunday, November 1, 2009

### MS221 course result ...

... will be available by the 18th december 2009. Why does that take so long. And even after the expiry dates of registering for new courses?

### Watched lecture 2 of Abstract Algebra E222

Topics

Example of group: GL(n,R), invertible nXn matrices with elements a_ij taken from R.

Definition of a group
- operation is closed
- operation is associative
- group has identity element
- elements have inverses

Definition of S(n), group of all bijective maps f: S->S, the symmetry group of n elements, with as operation the composition of maps

Definition of a subgroup
- closed
- group has identity element
- elements have inverses

Examples
- S1
- S2
- S3. This example was very messy. Instead of correctly naming the elements e, s, s2, t, st, s2t he named them e, t, t', s, s' and s'', not clearly emphasizing that s' was in fact st and so on. Here he could have nicely drawn a Cayley Table but he didn't.

Definition of transposition as the exchange of only two elements. ( Very important concept )

Example of all the subgroups of Z.
- bZ, all multiples of an integer including {0}, so {+/-2, +/-4, ...}, {+/-3, +/-6,... } are all subgroups.

Proof that all subgroups of Z are of form bZ.
- bZ is a subgroup
- any subgroup is of type bZ
This last part was really excellent, he used the Euclidian division algorithm to complete this part of the proof.

Definition of cyclic subgroup. The smallest subgroup containing an element g. This is the collection {g,g^2,g^3,...}. This subgroup can be either finite or infinite.

Definition of the order of an element g of a group. The smallest positive integer e such that g^m=e.

Next time his colleague / assistent will be lecturing and he will start with Lagranges theorem about that the order of a subgroup is a divisor of the order of a group. So I'll expect cosets will be introduced as well.

Note:

Two surprising, remarkable comments from Benedict Gross of which I am not sure I agree:
- " You cannot learn too much Linear Algebra "
( I agree Linear Algebra is important and fun but imho it will never be able to grasp deep theorems in say Number Theory. I am aware of the importance of Linear Algebra in Group Theory especially Representation Theory )

- " I do not recommend writing out multiplication tables "
( Playing with Cayley Tables gave me definitely more insight in the structure of many groups, if you have Mathematica or GAP producing a Cayley table is not difficult. I doubt if Gross has actual experience with either of the two, or he hides it carefully until later. )

### It is Artin...

... at least for as lang as I watch these Abstract Algebra video lectures.

( While browsing through some old blog entries and I stumbled upon this entry about Lang or Artin. )

## Saturday, October 31, 2009

### 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.

## Friday, October 30, 2009

### 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.

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

## Sunday, October 25, 2009

### 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

## Saturday, October 24, 2009

### 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?

## Thursday, October 22, 2009

### 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.

## Tuesday, October 20, 2009

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.

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.

## Saturday, October 17, 2009

### 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.

## Wednesday, October 14, 2009

### 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.

## Monday, October 12, 2009

### 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,

## Sunday, October 11, 2009

### 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.

## Thursday, October 8, 2009

### ( Test Blogger Buddy )

Test. This is a post from within the Vista Sidebar using Blogger Buddy.

### 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.

## Friday, October 2, 2009

### 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.

## Tuesday, September 22, 2009

### Ramanujan's magic square formula

Today I started to read the Ramanujan biography ( The e-book version, of course. ) The book looks promising. What was it like to communicate with someone gifted with such powerful mathematical insights? I am hoping ( a bit less after every failed try ) for a book that pulls me in that world of it's own. I have been there before on many occassions, while reading other books of course. Scientologists have a special word for it: exteriorizing.
At a very young age he designed the following formula for a 3 by 3 magic square:

C+Q | A+P | B+R
A+R | B+Q | C+P
B+P | C+R | A+Q

where A,B,C are integers in arithmetic progression and so are P,Q,R.

Try it, it works!

## Wednesday, September 16, 2009

### MS221 - TMA04: Done

I reviewed MS221 - TMA04, made some minor corrections and decided that the A is ready for shipment to T. I nevertheless wait with shipping the TMA ( Cut-off date is 30 sep anyway ) because I haven't received TMA03 back yet. If I get a high enough mark on that one I might even get away with a low score ion TMA04. We'll see.

## Tuesday, September 15, 2009

### MS221 exam close by

I copied this from a post by "Tma Machine" :

Here’s how a rough outline of what’s in the 2008 paper (hopefully this won’t constitute copyright infringement!):

Part 1
Question 1: Finding a closed form for a recurrence system
Question 2: Identifying and sketching a conic
Question 3: Stating the rule for some isometries, and for a composite isometry, and then using the double-angle and half-angle formulas to show a result
Question 4: Classifying fixed points of a curve, then sketching the graph of the curve and using graphic iteration construction
Question 5: Identifying basic linear transformations, applying them to a vector, and stating an invariant line for each one
Question 6: Finding eigenvalues, eigenlines and eigenvectors.
Question 7: Differentiation
Question 8: Integration
Question 9: Finding and manipulating Taylor series about 0, and classifying stationary points
Question 10: Finding the modulus and argument of a complex number, converting them from Cartesian to polar form and vice versa, and using the formula for powers of complex numbers.
Question 11: Using Euclid’s Algorithm and working with exponential ciphers.
Question 12: Combining variable propositions, finding a case for which a given proposition is false, and finding the converse of a proposition.

Part 2
Question 13: Looks like it’s about conics, but I haven’t done this one yet.
Question 14: Linear transformations
Question 15: I haven’t done this one yet either, but it looks like it involves differentiation, integration and stationary points.
Question 16: Groups

I sort of expected this, so I am not surprised. But I am not entirely ready for the exam yet. I still have several weaknesses but I am confident I can handle these in time. To be honest I wished the exam was over and done with. It is obviously my objective to score in the 90-100 because that is what I score for TMAs. But I check, doublecheck, re-check my TMAs at least two times. That makes a lot of difference.

## Sunday, September 13, 2009

### Logic puzzle

In lecture 2 Dr Kamala describes a murder mystery which can be solved entirely using logic in a Sherlock Holmes or Hercule Poirot like manner! Very nice.

## Saturday, September 12, 2009

### Discrete Mathematical Structures

Lectures on YouTube by Prof. Kamala Krithivasan, Department of Computer Science and Engineering, IIT Madras.

Beautiful lectures for the whole planet to watch.

## Sunday, September 6, 2009

### Excercise

I think this is a nice exercise at the MST121 / MS221 level.

Show that : Sin ( 18 degrees ) = 1/4 * ( SquareRoot(5) - 1 ).

Will use it to practice the trig formulas for the MS221 exam.

### Sage or Mathematica ?

Sage and Mathematica are both very powerful mathematics tools. For an absolute beginner both packages will do, I suppose.

Mathematica is closed source and for the price of a full edition Mathematica package you can also buy a state-of-the-art Sony VAIO or other sexy netbook or tablet you are thinking of buying. The Mathematica programming language is a so called propriety language. That means that it is only used in Mathematica and the owners of Mathematica can change the language whenever they want. Mathematica has a steep learning curve. Now that I master the basics of it I am very satisfied with what Mathematica can do for me.

Because Sage says it is possible to integrate Mathematica into Sage ( haven't figured yet out how, I suppose I need the Mathematica for Linux version ) I am having another look at Sage. Sage is Linux only but you can run it entirely from within a browser. So you can install Sage on a Linux server ( for example in a VMWare virtual machine ) and run it on Windows in your browser.

( More later about my Sage findings. )

http://www.sagemath.org/

## Saturday, September 5, 2009

### Characteristic of a ring

The characteristic of a ring is the number of times you must add the multiplicative identity element in order to get the additive identity element. If adding the multiplicative identity element to itself, no matter how many times, never gives us the additive identity element, we say the characteristic is 0.

## 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
sigma(12)
= 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 MĂ¶bius 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

Overall course result

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

Substitution applied at : 87.7

## Wednesday, August 5, 2009

### GĂ¶del, Escher, Bach

GĂ¶del, 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. GĂ¶del'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.

## Sunday, July 26, 2009

### Math writing recognition in Mathematica and Windows 7

Have been waiting for this a long time. Already have the tablet PC with Vista on which handwriting recognition works perfect. So all I need now is Windows 7. That's at least a valid and worthwile reason to upgrade to Windows 7.

## Tuesday, July 21, 2009

### MS221 - TMA03 Completed

MS221 - TMA03 completed and ready to be shipped to T. I My prediction for this one is 82 / 100. We'll see. It takes at least five weeks before a result may come back. The cut-off date is August 5.
TMA03 questions
1. Derivatives
2. Graph sketching
3. Indefinite integrals
4. Area and volume calculations
5. Taylor series
6. ( more ) Taylor series

## Monday, July 13, 2009

### PI is female.

A formula for PI is
PI / 4 = 1 - 1/3 + 1/5 - 1/7 +1/9 - 1/11 + ....

I played a bit with this formula and it took me approx. 4000 terms to get the first four decimals stable. Beautiful but not practical.

I would say that PI is female.

## Tuesday, June 30, 2009

### MS221 - TMA03

Started to work on MS221 / TMA03. I completed Q1a and Q2 partially ( steps 1-4 ). I suppose one of the shared experiences of all OU students is that at some point one cognites that starting asap with the TMA's is a crucial succes factor. Resultwise and otherwise.

## Sunday, June 28, 2009

### Handbook of Mathematics

It has been a while since I bought a math book. Books are expensive and most of my budget goes to the Open University anyway. Browsing some books at the local bookstore I could not resist Handbook of Mathematics from Springer publishers. A book packed with math, a must have. I open it on a random page and then just follow my thoughts.

## Saturday, June 27, 2009

### Mathematica TIP(2): ComplexForm; PolarForm

PolarForm[z_] := {Abs[z], Arg[z]}

ComplexForm[{R_, Theta_}] := R*Cos[Theta] + R*Sin[Theta]*I

Example:

PolarForm[1+i]={ SQRT[2] ,PI/4 }

### Open University Students

Added a bloglist with blogs from ( other ) Open University students. Started with one blog, will add more later. If you know of any, pls let me know.

## Thursday, June 25, 2009

### MS221 TMA02 - Result is in

MS221 TMA02 - Result is in... 87. Very disappointing at first. I hoped for 90+. Looking back at the effort I put in book B I can only be satisfied with the 87 points. Had to do a lot of MathCad work. ( A program I can't get used to. I suppose that I am spoiled by Mathematica. )

VERY OFF-TOPIC: MJ

( His biographer made the right prediction.

I don't think he has it in him to take his own life. I don't see him putting a gun to his head. It'll be an accidental overdose - something like that. ( Stacy Brown, July 2005 )
When MJ announced the London concerts I had the feeling it wasn't Michael but a double. )

## Tuesday, June 23, 2009

### The Story of Mathematics ( 3 )

Watched episode 3 of The Story of Mathematics. Prof. Marcus du Sautoy continues his talk about the origins of mathematics. Topics in this episode are the lives of  Descartes, Fermat, Newton, Leibnitz, Bernouilli(brothers), Euler, Fourier, Gauss, Bolyai and Riemann.

Marcus du Sautoy clearly loves mathematics and he is an excellent storyteller. When I was young I saw a similar series by Jacob Bronowski which laid the foundation for my love of maths. I hope du Sautoy's program had a similar effect on young people watching it.

Still one episode to go on 20th century math.

## Sunday, June 21, 2009

### Symmetry and the Monster by Mark Ronan

Just got this promising book. A history of Group Theory from the beginning until the discovery of the Monster Group. Will publish a review here when I have read it.

### The Story of Mathematics (2)

Watched episode 2 of The Story of Mathematics. Prof. Marcus du Sautoy continues his talk about the origins of mathematics. Topics in this episode are the mathematics from China, India and the Arab cultures. It turns out that 'discoveries' made in Europe were actually known centuries before in China and India. Anyway, some topics in this episode are: the Fibonacci Sequence, the invention of the number zero, the decimal number system, how the distance between the earth and the sun can be calculated in a simple fashion, a 3 by 3 magic square and more! And lots of beautiful scenery to watch. Two more parts to watch. :-) - Category: Seek and you will find.

## Saturday, June 20, 2009

### The Story of Mathematics

Watched episode 1 of The Story of Mathematics. Prof. Marcus du Sautoy tells about the origins of mathematics. Topics in this episode are Egyptian fractions, the geometry of the pyramids, the base-60 number system of the Babylons and the Greek mathematicians Pythagoras and Euclid. There is a spectaculair animation of du Sautoy agains a background of the pyramids how they actually were 4000 years ago. - Category: Seek and you will find.

## Thursday, June 18, 2009

### Contemplating a Project

I order to acquire synergy with other areas in my life I am contemplating about a project which requires algebra, combinatorics, java and mathematica.

Contemplating... ... ...

( More later )

### Feedback Delay on MST121 CMA42

Earlier today I checked if the results of CMA42 were available. There was a notice 'Feedback Delay'. Now why would that be??! There were 28 questions and all computer-marked, isn't it simply every correct question adds 3+4/7 = 3.57 marks to the total?

Maybe some questions were too simple / difficult and are cancelled, leaving 24 questions which count just as in CMA41.

I wonder when the results do come in.

## Wednesday, June 17, 2009

### MS221 - TMA02 - ( 2 )

Completed question 1 on Sunday. Was surprised by the work required for question 2. Worked yesterday from 8am till 23.30 pm. Mailed the pack to the tutor today. After e-mailing him the pdf version and announcing the pack was to be shipped today.

( Pffhhh. )

Time to study new things again! Book C is on calculus, while book D has introductions to number theory, group theory and logic.

## Saturday, June 13, 2009

### MS221 - TMA02

Done question 3 today. Warming up. I have till Tuesday. Only 5 points and simple at first sight. That is a huge Red Flag! Only 5 points also means only 5 to lose from this trick question.

## Tuesday, June 9, 2009

### MST121:DONE

Just submitted CMA42. All I have to do now is wait for the marks on TMA04 and CMA42. In the resultsplanner I estimated 35% for TMA04 and 60% for CMA42. The 35% for TMA04 is way below what I got for the previous TMA's but I don't feel confident at all about the results. I am only 'sure' about 35 or so. Anyway, I suppose I get a pass for MST121.

By doing MST121 I primarily learned what it is like to study at the OU. I haven't seen any entirely new mathematics but that wasn't to be expected from a level 1 course. I am already busy with MS221 and I suppose it will be M208 and / or MST209 after that. All in all I have enjoyed the course.

## Monday, June 8, 2009

Had to ask for an extension for MS221 TMA02. Not good. Not good at all. MST121 CMA42 is due wednesday, I am working on that one right now.

## Sunday, June 7, 2009

### Protocol for reading a mathematics book

In the introduction to " An introduction to Combinatorics by Alan Slomson " I found a good explanation of the protocol for reading a math book.

Once upon a time there was a programme on the radio for young children called Listen with Mother. (In those days it was assumed that it would be the mother who would be at home with the children.) In the first programme in 1950 the storyteller, Julia Lang, introduced the story she was about to tell by saying 'Are you sitting comfortably? Then we'll begin'. Apparently this introduction was not planned, but it caught on, and was used regularly until the programme ceased in 1982. When it comes to reading mathematics, however, this is not an appropriate beginning. A mathematics book cannot be read like a novel, sitting in a comfortable chair, with a glass by your side. Mathematics books need to be worked at. You need to be sitting at a table or a desk, with pencil and paper, both to work through the theory and to tackle the problems. A good guide is the amount of time it takes you to read the book. A novel can be read at a rate of about 60 pages an hour, whereas when it comes to many mathematics books you are doing well if you can read five pages an hour. (It follows that, even at 12 times the price, a mathematics book is good value for money!)

I would like to add that a good mathematics book can be read over and over. Some math books are companions for life.

## Tuesday, May 26, 2009

### MST121 - TMA04 ( 3 )

All exercises are in TeX, MathCad and OUStats work captured with SnagIT and saved to PDF. Most of the work is done, one or two QA sessions before I send the work the tutor. MST121 TMA04 is the most difficult TMA / CMA I had to do sofar. I am however confident that I will pass the required 40% score. - I did not particularly enjoy working on the TMA. I have postponed working on it several times. Instead I worked on other areas of math I am (self-)studying like combinatorics. Currently I am giving Concrete Mathematics by Graham, Knuth and Patashnik another try. When the TMA is out the door, I'll focus on CMA41 though. And I am extremely tight on my MS221 TMA02 schedule, but that -is- a fun topic!

## Tuesday, May 19, 2009

### MST121 - TMA04 ( 2 )

I completed questions 1 and 2 today. I am fairly sure now that I have virtually passed the exams of the course. - I got the results back from CMA41 for which I scored 83/100 ( 20/24, all 4 errors on book D2 stuff ) which means that I have more than 60% already for the entire course. The only requirement left is completion of TMA04 with a score of at least 40%. I think the three questions which I already did are enough.

## Tuesday, May 12, 2009

### MST121 - TMA04

TMA04 is the TMA covering the entire course. Question 3 ( of 4 ), which I completed today, is on calculus. All four questions are valued at 25 marks. Question 3 basicly covers all subjects of Book C, differentiation, integration and differential equations.

## Saturday, May 9, 2009

### Layed the foundation for new TMA's

Layed foundation for MS221-TMA02, MST121-TMA04. That is I prepared two latex docs in Texlipse and added all chapters, headings, toc stuff. All I have to do now is do the math. May is an busy month in all areas.

## Wednesday, May 6, 2009

### MS221 second shipment arrived

Compliments to the Open University!

I proposed a rather exceptional way of sending a shipment of books. The OU handled it as if it was a regular request. "... The quality of an organisation can be measured by their capacity of handling exceptions. ..." Isn't that true? Things usually go wrong when questions are asked.

Anyway, I received books C and D of course MS221 today. Book D is more or less an introduction to M208, I suppose. D has topics on Complex Numbers, Number Theory, Group Theory and Logic. Book C is calculus, again. I just to find calculus so booooring. I must say I find it less boring today and am actually looking forward to MST209 about model building which has LOTS of stuff on solving DE's.

MST209, M208 are both 60-ers and are both next on my left. I don't think I can handle 120 in a year. 90 is do-able for me, but 120? Don't think so. Havent ruled out the option of doing both MST209 and M208 in 2010. I havent planned courses for sep 09 / jan 10 yet so I could do the video based MIT self-study course on DE's in that period which would be an excellent technical preparation for MST209. But again, all options are still open.

As in football... concentrate on winning the next match instead of winning the competition.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
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.)