Study plan for 2011 is shaping up

The Study Plan for 2011 is shaping up.

With M336 Group Theory forthcoming in 2012 I will keep my Abstract Algebra 'warm' by reading:
- Goodman; Algebra Abstract and Concrete. ( Free e-book );
- Rose, Harvey E; A Course on Finite Groups;
- Cox, David; Galois Theory;

I will probably have some difficulty ( 'uneasyness', if you like ) studying MST209 because the materials will not be presented in mathematical but in scientific format. Since I am not very good at long reading sessions ( I get distracted too easily ) I prefer the Theorem / Proof / Example presentation because it leaves a lot to the reader, i.e.: read a bit, then DO a lot. Mathematica experiments for example. For that purpose I selected the books
- Kelley, Peterson; The Theory of Differential Equations - Classical and Qualitative. From the preface: "...Previous experience with differential equations is helpful but not required. Consequently, this book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students. ...)", and
- Differential Equations DeMystified as an antidote to the previous book and as a guide to the HOW-TO's of solving DE's.

Top priority will of course have MST209 because it adds 60 points to my balance. I'll have to decide if I want and / or can add either M337 Complex Analysis or M381 Number Theory to the workload.

The Mobius strip

I blogged many times about the Mobius strip, a fascinating object and superb to introduce mathematics to the uninitiated. Algebraic Geometry is the field of mathematics where they try ( more often do ) to catch these objects in polynomial equations. A quick browse ( or read ) of this PDF may give you an idea of the field.

http://data.imaginary-exhibition.com/IMAGINARY-Moebiusband-Stephan-Klaus.pdf

Found while surfing IMAGINARY.

Analytical function representing the Fibonacci series

The MST209 exam has a different format than MS221 and M208 have. In 2006, for example, the format was as follows:
Part A. 15 multiple-choice questions, 2 marks each = 30 points. ( 1 hour )
Part B. 8 questions, 5 marks each = 40 points. ( 1 hour 15 min )
Part C. 3 out of 7, 15 marks each = 45 points. ( 45 min )
Yes. Maximum score is 115. Scores above 100 are set to 100.
The exam looks doable. And again questions on eigenvalues and eigenvectors. That would be three in a row: MS221, M208 and MST209. Considering the fact that one can prove Binet's formula for the Fibonacci numbers with them it's worthwhile having it firm under your math-belt.

There is also an analytical function for the Fibonacci numbers which rounded, gives an exact Fibonacci number if the input variable is an integer. Here is the related math.

Let $F_n = F_{n-1} + F_{n-2}, F_0=0, F_1=1$, show that $Fa_n=\frac{1}{\sqrt{5}}e^{n \cdot \log{\phi}}$, where $\phi$ is the Golden Ratio or $\frac{1+\sqrt{5}}{2}$. ( Round $Fa_n$ to get $F_n$. )

If we define the elements $F_{n}$ and $F_{n+1}$ as the vector $s_n= \left(
\begin{array}{c}
F_{n+1}\\
F_{n}
\end{array}
\right)$
then $F_n$ simply becomes
$F_n= \left(
\begin{array}{cc}
1 & 1\\
1 & 0
\end{array}
\right)^n
\cdot s_{0}$.

We can calculate the power of a matrix by diagonalizing the matrix. And this is where eigenvalues and vectors come in. If $\lambda_1, \lambda_2$ are eigenvectors with respective eigenvectors $E= \left( e_1, e_2 \right)$ we get $F_n= E^{-1}
\cdot
\left(
\begin{array}{cc}
\lambda_1^n & 0\\
0 & \lambda_2^n
\end{array}
\right)
\cdot
E
\cdot s_{0}$

The eigenvectors are the roots of the characteristic equation $\left|
\begin{array}{cc}
1-\lambda & 1\\
1 & -\lambda
\end{array} \right| = 0$ and are thus $\frac{1}{2} + \frac{1+\sqrt{5}}{2}$ and $\frac{1}{2} - \frac{1+\sqrt{5}}{2}$.

( TO BE CONTINUED ... )

Mathematical Deceptions - ( Coast to Coast AM 21-oct-2010 )

C2CAM had a 3 hour during interview with Prof. C. Seife on mathematics

Prof. Charles Seife discussed the art of using pure mathematics for impure ends. He used the term "proofiness," (an analogue of Stephen Colbert's term "truthiness") to describe how people use numbers to prove things that they believe in their hearts are true, even if they're not factually true. People often lie in polls to make themselves feel better, he reported. For instance, in surveys men will typically overestimate the number of sexual partners they have had, while women will underestimate the number.

Advertising often uses numbers fabricated out of whole cloth, like saying a moisturizer delivers 70% more moisture in every drop. Ads are just "larded with...completely meaningless numbers that are supposedly backed by some sort of research which is either dubious or non-existent," he declared. Politicians sometimes manipulate the scale of graphs to make it appear like a dramatic effect or change is occurring, such as with tax cuts or budgets. Al Gore, in his film An Inconvenient Truth, exaggerated climate projections, as a way to further his agenda, Seife added.

Numbers are often inflated to make a person or organization seem more important, such as the number of people in attendance at a rally, or the circulation of readers for newspapers and magazines, he noted. Seife also discussed various developments in science, such as the Large Hadron Collider experiments at CERN. If "a certain pattern of missing energy turns up, it could be a very good sign that we have found another dimension beyond our own," he reflected.

For subscribers: http://www.coasttocoastam.com/show/2010/10/21

Otherwise: http://www.demonoid.com/files/details/2427492/2340388/ ( Opera Browser has a built-in BitTorrent Client )

Bernouilli equations

I watched approximately half of video lecture 4 of 18.03 today. Topic of lecture 4 is solving differential equations by direct or indirect substitution.

Direct and indirect substitution originates from Calculus Integration, I suppose. Take for example, the integral:
$$\int{x \sin(x^2)\ dx}$$
requires the direct substitutions $y=x^2, dy=2x \ dx$ to solve. However, the integral
$$\int{\frac{1}{\sqrt{1-x^2}}\ dx}$$
requires the indirect substitution $x=sin(u), dx=cos(u) \ du$ to solve.

A Bernoulli equation is a DE of type:
$$y' = p(x) \cdot y + q(x) \cdot y^{n}$$
Rearrange as follows:
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{y}{y^{n}} + q(x) \cdot \frac{y^{n}}{y^{n}}$$
$$\frac{y'}{y^{n}} = p(x) \cdot \frac{1}{y^{n-1}} + q(x) $$
Now substitute $v=\frac{1}{y^{n-1}}$, and thus $v' = (1-n) \frac{y'}{y^{n}}$:
$$\frac{v'}{1-n} = p(x) \cdot v + q(x)$$
$$v' + (n-1)p(x) \cdot v = (1-n)q(x) $$
The last equation is a linear ODE in standard form.

Recipe for solving linear first order ODE

As I blogged before I am going to make more notes, summaries and stuff and store everything in a plex in a Personal Brain database. The following note basically summarizes 18.03 lecture 3.

$$a(x) \cdot y' + b(x) \cdot y = c(x)$$
$$y' + p(x) \cdot y = q(x)$$
$$e^{\int{p(x) \ dx}} \cdot y' + p(x) e^{\int{p(x) \ dx}} \cdot y = e^{\int{p(x) \ dx}} q(x) $$
$$( e^{\int{p(x) \ dx}} \cdot y )'= e^{\int{p(x) \ dx}} q(x) $$
$$e^{\int{p(x) \ dx}} \cdot y= \int{e^{\int{p(x) \ dx}} q(x) \ dx} $$
$$y= \frac{\int{e^{\int{p(x) \ dx}} q(x) \ dx}}{e^{\int{p(x) \ dx}}} + \frac{C}{e^{\int{p(x) \ dx}}}$$

( Would need some explanations but as a personal note it is clear to me. )

Watched MIT 18.03 videos 2 and 3.

Video 3 is about a straightforward recipe for solving a linear ODE of the first order:


Video 2 is on numerically solving DE's with Euler's method:

Getting started on MST209 Differential Equations

In order to get a feeling for MST209 I will be watching the video's of MIT 18.03 Differential Equations by Prof. Arthur Mattuck. I watched lecture 1 today. Since there are 33 video lectures in total I doubt if I'll get through all of them by february though. The purpose is getting back into differential equation stuff. OU OpenLearn has several booklets from the MST209 course available which give a reasonable idea on what to expect of MST209. Both MIT 18.03 and MST209 are less formal than M208, they are mainstream courses teaching mainly how-to's, i.e. how-to solve a particular type of differential equation. - My choice for MST209 next year is more or less certain by now. Three options remain. 1) do just MST209 and concentrate and spend time on self-study of topics that really interest me (i.e. Abstract Algebra stuff ). 2) Add M381 Number Theory + Logic. 3) Add M337 Complex Analysis. Well, I haven't decided yet. MST209 involves 7 TMA's, 2CMA's and probably a MS221, M208 'type-of-exam', i.e. 12 questions for 70, plus 2 out of 5 for 30.

Announcing two new websites.

I received this message earlier today.

Hi Nilo Just to let you know I've gone live the link is http://chrisfmathsphysicsmusic.blogspot.com/ Hope you find it interesting Best wishes Chris

Go here: http://chrisfmathsphysicsmusic.blogspot.com/.

This might be the moment to 'open-up' a website in its very early beginnings. The site will change often and is going to contain my ( mathematical ) notes. Anything I ever might publish will go through there. At least, that's the idea. I don't have a domain yet, let alone a webserver so I used Opera Unite's out-of-the-box Webserver. The address is http://home.ndroock1.operaunite.com/webserver/

See also the Bookmarks page

Nicolas Bourbaki

From Preface to the Third Edition of Galois Theory by Ian Stewart

Nicolas Bourbaki is the pseudonym of a group of mathematicians — mostly French, mostly young — who tidied up the mathematics of the mid-20th century in a lengthy series of books. Their guiding principle was never to prove a theorem if it could be deduced as a special case of a more general theorem. To study planar geometry, work in n dimensions and then let n = 2.

Note-taking

I found a way to enter LaTeX in Personal Brain thoughts. Exporting a plex to a website is simple. I hope to demo it soon. I publish my work via Opera Unite which means that I don't have to upload files to a server because the web server runs as an Opera process / thread on my computer. I don't leave it on 24/7 though. The exciting part is that you can use Personal Brain to:
- manage your mathematical notes in LaTeX ( plexes of 300,000 thoughts exist );
- query thoughts;
- reorganize database structure instantly;
- publish thoughts to a website;
- export thoughts to other programs;
- import thoughts ( Mathematica, for example can export notebooks to LaTex ).

Personal Brain is a sort of Mind Manager without the constraint that everything has to fit in a tree structure. They have a Microsoft sort of version scheme: home, business, enterprise. The home version is free, it's not an expiring trial.

Personal Brain website

Books on Galois Theory

Starting a self-study project involves finding the right books. I found the following books on Galois Theory which are aimed at beginners in the topic. For my purpose the following books are the most useful.

Galois Theory by David Cox, Wiley 2004
This is a very beautiful book of close to 600 pages, every page shows that the author loves the subject and really tries to explain the subject. It has sections on Galois Theory in Mathematica. ( In study-tech terminology: it effectively handles the first barrier to study, i.e. lack of mass, by making the subject tangible in the form of Mathematica functions. The student can explore the subject in a concrete fashion. ) Essential for self-study it has hints to selected exercises.

Exploratory Galois Theory by John Swallow, Cambridge University Press 2004
The author wrote software of his own in the Mathematica language which is available for download.

Galois Theory (3rd ed) by Ian Stewart, Chapman & Hall, 2004
Ian Stewart (*) is well know in England ( and beyond ) and this was his first book. I have added this book to my list because there are four different proofs of the Fundamental Theorem of Galois Theory in it.

Most good introductory books on Abstract Algebra have a few chapters on Fields and Galois Theory but it seems they are merely included to create an appetite for more.

Choosing books is a critical phase in self-study. I started with the book of Weintraub which he probably started very enthusiastically but it got denser and denser almost by page. ( Good book but needs extensive lecture notes by a teacher. ) Via Google Books -and other sources- you can find and browse any book on the subject. The reviews on Amazon can be helpful too.

The books are decided upon. On to phase 2: planning.

Short note about writing math exams

( I'm doing a bit of random surfing. )

Some quotes.
"... I'm starting to feel the pressure of fitting an entire linear algebra course in just 5 weeks. ( ... ) I am covering 2 sections of text every day, which amounts to covering 2 chapters a week. ..."

What I have learned is ( and you might, or might not, agree ) that mathematics needs a lot of time TO SINK IN. And doing mathematics is like doing physical exercise or playing a musical instrument: if you don't practice every day you lose the skill and/or knowledge. I think that's what the Open University is doing. Topics are introduced in MST121, come back in MS221, M208 and they keep coming back with each repeat covering some new ground.

About designing exam questions:
"...It's just very hard to come up with clever ways to present problems that cover knowledge of the material in a way such that those that know the material will be able to do the problems without much difficulty but those that did not put the work will have a fair amount of difficulty. ..."

It explains why some students find an exam easy while it was hard for another: it had been deliberately designed this way!

Bookmarks

In case you haven't noticed yet, at the top of this blog there are two links: Home and Bookmarks. On the bookmark page I publish some of my bookmarks on mathematics.

When I surf and come across an interesting site, I store the url at my Delicious page. When I save a bookmark on Delicious, an automatic Tweet is generated, which is then sent to my Facebook profile.

Today I came across a nice site called MathOverflow. On this site we can see what is on the mind of some research mathematicians...

Galois Theory (3)

Equations of degree 5 and higher can't be solved. That's what I thought. I knew that certain equations of degree 5 could be solved however like $x^5 - a = 0$. Abel's impossibility theorem ( aka Abel-Ruffini theorem ) says that the quintic can't be solved in general by radicals. In general. What they in fact proved is that some equations can't be solved by radicals. Galois developed a technique for determining if an equation can be solved by radicals or not. This technique was the start of Group- and Galois Theory. A polynomial has a corresponding group, it's so called Galois Group. If this group is solvable, i.e. is a solvable group then the equation can be solved by radicals.

What next ?

I have written earlier that I would probably do MST209 and 2 30p Level 3s in 2011. No way! Even if it would be possible: it is no fun. I think I will like MST209, so it will be MST209 + ( maybe ) another course. Number Theory would be ideal, but, and this makes me sooo angry, they merged Logic/Computability into M381.

Mandelbrot

( *** Changed YT video to a better one. *** )

Have you ever been asked "If you study mathematics then you must know a lot about fractals, right?" Wrong. Unfortunately. I have learned that understanding fractals requires -at least- knowledge of Complex Analysis ( M337 ). A student of mathematics must be patient.

Last week's mathematical news was of course the death of Benoit Mandelbrot. The magazine Math Factor published a video-lecture ( 20 min ) of him recorded February, 2010. On that page I also found this video of the Mandelbox.


Enjoy!

Galois Theory (2)

The polynomial F(x) = x^8 - 40x^6 + 352x^4 +960x^2 - 576 has 8 ( real ) roots : +/- Sqrt(2) +/- Sqrt(3) +/- Sqrt(5). ( See image above. ) The symmetry group of the roots is abelian and has order 8, all elements have order 2, so the group is isomorphic to C2 X C2 X C2.

I have found another introductory book on Galois Theory for undergraduates ( to support my study of Weintraub's Galois Theory I think, for the moment anyway. )


It comes with AlgFields ( download here ), a system of Maple and Mathematica routines for calculating with low-dimensional number fields and finite fields.

Galois Theory

While I was procrastinating on revising M208 stuff I explored new mathematical territory: Galois Theory. What have I discovered sofar?

- In between the fields Q and R there is another ( perhaps hypothetical field ) called A, the field of algebraic numbers. It contains of all quotients plus all numbers that are solutions to polynomial equations with coefficients in Q. For example Sqrt(2) is not a quotient but can be expressed as the solution of the equation x^2-2=0. So A is equal to Q plus all numbers like Sqrt(2).

- Something very interesting happens if we add ( adjoin ) Sqrt(2) to Q: Q remains a field! ( The field Q is an abelian group for + and *, the operations + and * are related via the distributive laws. Identities are 0 for + and 1 for * ). It can be proved trivially that { x | x = a + b*Sqrt(2) , a,b in Q } is a field. This field is written like Q(Srt(2)), or Q/(X^2-2) and is called an extension field.

- If we put on our Linear Algebra glasses we could say that a + b*Sqrt(2) is in fact a vector (a,b) over the basis {1, Sqrt(2)}.

- The roots of the equation X^2-2 have a C2 symmetry, the roots of X^3-2 have a Dihedral Group 3 symmetry. Investigating the symmetry of the roots of equations is a task in Galois Theory. The symmetry group is called the Galois Group, Gal(E/F). In our example E=Q(Sqrt(2)) and F=Q.

- Now the Fundamental Theory of Galois Theory ( FTGT ) says that there is a 1-to-1 correspondence between subgroups of Gal(E/F) and fields intermediate E and F.

Fascinating stuff. Unfortunately Galois Theory is not part of any Open University course I know of.

The book I am reading on Galois Theory is:

M208 is done.

A rather dark side of my personality manifested itself the last two months or so. Don't worry I killed the monster. I sort of digressed to the period when I was in high school, i.e. I did not send my work on TMAs 6 and 7 to my tutor since I more or less had it with that reptilian. Goodbye wished for, hoped for distinction. More about this in another post. In this post I'll report about the exam itself.

The exam took place in The Hague. I counted four people coming in doing M208 but there could have been more. In the examination room with about 16 or so people I have seen at least three different sets of papers of which S320 was one. It was mentioned that two people did not show up. When we entered the room we got our papers and could choose a place to work for ourselves.

At 2.30 PM ( local time ) we were allowed to open the paper. Same format as MS221, i.e. two parts A and B. Part A had 12 questions with a maximum score of 70. And Part B was a 2 out of 5 set. We could choose two questions of 15 points each. From what I recall the questions in part A were about ( not in order ):
1. Graph of (2x+3) / (3-x), incl. asymptotes and axes-intercepts.
2. Solving an inequality.
3. Diagonalizing a matrix using eigenvectors.
4. Question about [{1,2,4,9,10,12}, mult mod 13]
5. Question about Homomorphism z->|z|^2.
6. Question involving R2 geometry using vectors and inner product
7. Question on series
8. Finding an integral
9. Question on a Taylor series
10. Question about permutations
11. Question about the symmetry group of the pentagon.
12. ( Forgotten )

At 4.30 I had completed 9 out of 12 questions. Last year with MS221 I continued working on part A with the result that I only completed half of a question of part B. I think it was a correct decision. I had three questions left which would take me at least 40 minutes, leaving 20 minutes for part B. I could score 18 max. Continuing with part B at this point gave me an opportunity to collect 30 points although part B questions are somewhat more difficult.
I was getting tired and somewhat stresssed. Again, I did not complete all 12 questions in 2 hours. I did not take time to read all five questions. I started to work on question 13 which was on group theory.

13. A question about some finite group with 16 elements. The Cayley Table was given, nothing else.
- Find a cyclic subgroup, call it H.
- Prove that H is normal and that group K ( given ) is not normal.
- List elements of quotient qroup G/H
- Determine the structure of the group.
- ( One more question, forgotten )

It was now 17.00, I chose the next question on Linear Algebra
14. About a linear transformation in R3.
- Find the dimension of the kernel
- What is the geometry of the kernel
- Find a basis of the image
- What is the geoemtry of the image and find an equation
- Given two sets of 3equations with 3 unknowns determine the number of solutions

I feel about the same as last year with MS221. I am fairly sure ( 99% ) of a pass.

All in all, I have learned a lot this year. About mathematics of course, but also about myself and about the effect a study like this can have on a person. Not everything has been said about this study year. More next time.

M208 Exam

I am off to the exam.

From AbstractAlgebra for Mathematica to GAP.

A few notes about the Mathematica video's I created on YouTube.

AbstractAlgebra is a Mathematica add-on package. It is open source, has been written entirely in the Mathematica programming language by two mathematics professors Hibbard and Levasseur. The download url is http://www.central.edu/EAAM/Downloads/AAPackage.asp.

The functions I used like FormGroupoid, GenerateGroupoidByRelations and FormMorphoid are defined in a Mathematica package called Master in the AbstractAlgebra directory. The group I created as G1=FormGroupoid[Range[0,11],Mod[#1+#2,12]&] is by default available as Z[12] ( when SwitchStructureTo[Group] has been used ), in fact all groups I used are available as defaults. I wanted to demonstrate that you can define any group you want, in any case much more than the familiar 'textbook example groups'.

The AbstractAlgebra package is meant as a tool to visualize the often abstract concepts in Group Theory and other topics in Abstract Algebra. There is a book available from which you can learn Abstract Algebra with Labs and Exercises in Mathematica with the Abstract Algebra package. Other algebraic structures which can be created are Rings, Polynomials, Polynomials over Rings, Galois Fields, Permutations, Permutation Cycles and more.

The package AbstractAlgebra is NOT a tool for Computational Group Theory. Group Theory is alive as it is being actively researched. A state-of-the-art tool for Computational Group Theory is GAP. ( I might give some GAP demo's soon. )

Group Theory and Mathematica - 2

I have uploaded another Mathematica demo video.

For this and other math videos go to my YouTube channel.

Or watch here.



In "Group Theory and Mathematica-2" I show how to create groups by using generators and relations and how to create and visualize homomorphisms and kernels of homomorphisms.

Group Theory and Mathematica

My first video with audio...

About blogging math

My post in progress about AutGrps proved that it is quite hard to write math in the Blogger format. Although it is possible to write about math in this format, writing actual math like an expose on automorphism groups proves to be hard even today with tools like MathJax. I could continue and publish in PDF but in that case I would violate the Blogger format. I am not a math writer anyway. I hope to become one though, maybe one day after many years of intense study. - I want to communicate about math so I am going to try to create mini-lectures on YouTube. I'll finish the automorphism article in some form but it won't be the first in a series.

Automorphism Groups #3

Introduction

In this post I will explain the concept of an Automorphism Group. We will make a list of the automorphism groups of all 24 groups of order less than or equal to 12 and (to our surprise) we will see that one of these groups has as much as 168 elements and that different groups can share the same automorphism group. Finally, we will make a strategy that can be of help in finding automorphism groups in general. So far we have come across direct products of groups when we studied groups of type C2 X C2 or C2 X C2 X C2. The study of automorphism groups prepares us for the study of another type of group product, the semi-direct product. ( Which I might discuss in detail in a future post. ) In this post I consider all groups to be finite. I'll try to use as much examples as I can at first and formalize later in final wrap up. Let's begin!

Generating sets

" In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. " As so very often is the case, simple things can be hard to catch in words. Let's simplify this a bit to "If [the only subgroup of G containing X = G] then [X generates G]." So it is all about -the only subgroup of G containing X-. But we don't know X! Time for examples.

List of subgroups of C3 (cyclic group of 3 elements).
1
1, a, a^2.
The sets which are only contained in C3 are: {a}, {a^2} and {a, a^2}. Note that we can reduce this list to {a}, {a^2}. ( We will do so immediately in the next examples.) Verify that C3 has two different generating sets. C3=(a)=(a^2).

List of subgroups of D3 (dihedral group of an equilateral triangle or 3-gon).
1
1, a, a^2.
1, b
1, ba
1, ba^2
1, a, a^2, b, ba, ba^2
The sets which are only contained in D3 are: {a,b}, {a,ba}, {a,ba^2}, {a^2,b}, {a^2,ba} and {a^2, ba^2}. Verify that D3 has six different generating sets.

Automorphisms

An isomorphism from a group G to a group H is a map which is surjective, injective and preserves the group operation. An automorphism is basically an isomorphism from a group to itself. Let's ilustrate this with some examples.

Let's establish an isomorphism first.

This is a CayleyTable from a group of order 8, we call the group G.


| 0 1 2 3 4 5 6 7
- | - - - - - - - -
0 | 0 1 2 3 4 5 6 7
1 | 1 0 3 2 5 4 7 6
2 | 2 3 0 1 6 7 4 5
3 | 3 2 1 0 7 6 5 4
4 | 4 5 6 7 0 1 2 3
5 | 5 4 7 6 1 0 3 2
6 | 6 7 4 5 2 3 0 1
7 | 7 6 5 4 3 2 1 0


This is another CayleyTable from a group of order 8, we call the group H.


| a b c e b**a c**a c**b c**b**a
- | - - - - - - - -
a | e b**a c**a a b c c**b**a c**b
b | b**a e c**b b a c**b**a c c**a
c | c**a c**b e c c**b**a a b b**a
e | a b c e b**a c**a c**b c**b**a
b**a | b a c**b**a b**a e c**b c**a c
c**a | c c**b**a a c**a c**b e b**a b
c**b | c**b**a c b c**b c**a b**a e a
c**b**a | c**b c**a b**a c**b**a c b a e


$\begin{array}{cccccccccc}
& | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
- & | & - & - & - & - & - & - & - & - \\
a & | & e & b\text{**}a & c\text{**}a & a & b & c & c\text{**}b\text{**}a & c\text{**}b \\
b & | & b\text{**}a & e & c\text{**}b & b & a & c\text{**}b\text{**}a & c & c\text{**}a \\
c & | & c\text{**}a & c\text{**}b & e & c & c\text{**}b\text{**}a & a & b & b\text{**}a \\
e & | & a & b & c & e & b\text{**}a & c\text{**}a & c\text{**}b & c\text{**}b\text{**}a \\
b\text{**}a & | & b & a & c\text{**}b\text{**}a & b\text{**}a & e & c\text{**}b & c\text{**}a & c \\
c\text{**}a & | & c & c\text{**}b\text{**}a & a & c\text{**}a & c\text{**}b & e & b\text{**}a & b \\
c\text{**}b & | & c\text{**}b\text{**}a & c & b & c\text{**}b & c\text{**}a & b\text{**}a & e & a \\
c\text{**}b\text{**}a & | & c\text{**}b & c\text{**}a & b\text{**}a & c\text{**}b\text{**}a & c & b & a & e
\end{array}$

( Got the table in TeX but as you can see, blogger format is too small... )

We will investigate if G,H are isomorphic and ( if so ) then define an isomorphism f: G-> H.
( Post in progress, thus more later... )

Don Knuth announces new release of TeX

If Alan Turing invented programming then Don Knuth invented data structures and algorithms. Don Knuth created TeX in the early 80's, the standard for mathematical typesetting.

Tne new release is called iTex ( what else ), watch him announce it here. Unconfirmed rumours have been spread that iTeX only runs on Mac OS X with 4GB internal memory. iTeX is commercial and only available via the iTunes shop.