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Wednesday, December 28, 2005

A 6****** star math book

I don't rate something 5***** stars easily. I mean what's beyond 5? Today I found a book for which a 5 star rating is simply too low. The book is extremely:
- different
- relevant
- interactive ( even more so than some math 'websites' )
- challenging
- fun.
If there is a book which I ever rated as a 5***** it must be Concrete Mathematics from Graham, Knuth and Patashnik. Polynomials, that's how it's called, is like that but much more coherent. It is a * * * * * * rating, to me anyway. Must have, must read, can't wait, buy now.

Oh, eh, it is out of print, sorry. Not for sale. ;-)

Friday, December 16, 2005

Group Explorer 2.0

"... Group Explorer is mathematical visualization software for the abstract algebra classroom. Users can explore dozens of finite groups (and morphisms among them) visually and interactively. ..."
GE 2.0 developer Nathan Carter.

Where GAP is targeted to professional algebra researchers and Mathematica Abstract Algebra to students who already have completed one or two abstract algebra courses and know the ins- and outs of Mathematica, Group Explorer is meant for beginners. A beautiful tool to demonstrate properties of a limited number of small groups. Version 2.0 is primarily a "look and feel" release. The program is free and downloadable as C++ source or executable.

Thursday, December 15, 2005

Lie Algebras

e^ A, e to the power matrix A, e to the power {{2,0},{0,1}}.
Matrices as power coefficients.
Is that cool, or what?

Friday, November 11, 2005

TeXnicCenter

As far as mathematics is concerned I am more or less illiterate. I can read a bit but when it comes to writing I freeze entirely. What to write? I just can't formulate my thoughts in a mathematical correct manner. What is worse? Well, I can't even reproduce a text from some math book. Look at me, on my age, learning to type. It is depressing. Look at the bright side, I have TeXnicCenter. That's a start. You can't write a letter without a pen either.

Sunday, October 23, 2005

Groups of order 18

There are 5 groups of order 18:

gap> grp(18);
Group 1 has structure: D18 and is Non-Abelian
Group 2 has structure: C18 and is Abelian
Group 3 has structure: C3 x S3 and is Non-Abelian
Group 4 has structure: (C3 x C3) : C2 and is Non-Abelian
Group 5 has structure: C6 x C3 and is Abelian

The workhorse here is the function ConstructAllGroups and is part of the GAP package GrpConst.


#
# Prints all groups of order n
# IN: Group
#
grp:=function(n)
local listGroups;
listGroups:=ConstructAllGroups(n);
i:=1;
while(i<(Length(listGroups)+1)) do
Print("Group ");
Print(i);
Print(" has structure: ");
Print(StructureDescription(listGroups[i]));
Print(" and is ");
if (IsAbelian(listGroups[i])) then
Print("Abelian");
else
Print("Non-Abelian");
fi;
Print("\n");
i:=i+1;
od;
end;

Sunday, October 2, 2005

Permutations

Whatever you have to do with a structure-endowed entity S try to determine its group of automorphisms . . . You can expect to gain a deep insight into the constitution of S in this way.
Herman Weyl, Symmetry ( Quote found in this PDF )

Is it possible to translate the above sentence to "Jip en Janneke"-language? Before anything else I should give it a try myself.

Sunday, September 25, 2005

Self-study trick

Linear Algebra, extremely important stuff for 3D programming. This book
must be one of the best. It is from Springer, written by the well known Serge Lang and it is a 2nd edition. Must be good. Not? Thing is that I rather spend my money on other things like... bills, other bills and more bills. Solution? Use the table of contents from this book as a sort of plan and get the knowledge from free sources elsewhere. It really works. My Rubik program is the proof of that. Examples? This one to get started: Elements of Abstract and Linear Algebra.

Tuesday, September 6, 2005

Learning Tex and other preparations

If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities. Serious: I learned some new stuff today. Semi-Direct Groups ( at last ) and some first glimpses of Polya Burnside Enumeration.

Thursday, July 21, 2005

Collatz Conjecture

I think i will look at it with a hopefully fresh view. I almost forgot the main results of the work of earlier this year. It was after this period that I started to document my work. The formulas are 'recovered'. Thank God. Who knows what is 'out there'? I suppose that's the fascinating part.

Galois



The math this boy created is fascinating until today. Because he was so young when he wrote his papers he used his own words and style of math, not at all like the papers written by the profesional mathematicians of his age. Because of this it took months to understand the essence of his work.

Sunday, July 17, 2005

Galois

What an interesting life for someone who died at 21...

"... So, why is Galois theory called Galois theory? The answer is that it is named after a French mathematician Evariste Galois (1811-1832) who did some very important work in this area. He had a very dramatic and difficult life, failing to get much of his work recognised due to his great difficulty in expressing himself clearly. For example, he wasn't admitted to the leading university in Paris, the Ecole Polytechnique, and had to make do with the Ecole Normale. He also met with difficulty because of his political sympathies, he was a republican. This led to him being expelled from the Ecole Normale when he wrote a letter to a newspaper criticising the director of the school. He joined a republican branch of the militia and was later imprisoned (twice) because of his membership. The second time whilst in prison he fell in love with the daughter of the prison physician, Stephanie-Felice du Motel and after being released died in a duel with Perscheux d'Herbinville. The reasons for the duel are not entirely clear, but it seems likely it had something to do with Stephanie. His death started republican riots and rallies which lasted for several days.

Although Galois is often credited with inventing group theory and Galois theory, it seems that an Italian mathematician Paolo Ruffini (1765-1822) may have come up with many of the ideas first. Unfortunately his ideas were not taken seriously by the rest of the mathematical community at the time. There are some links at the end of this document for anyone interested in finding out more about the history of group theory and Galois theory. ..."

Friday, May 20, 2005

Schnirelmann density

At last... a tool to measure the density of a sequence. Shnirelmann density. Will it help? Maybe, maybe. What caused my fascination for mathematics, in the first place? It must have been Spirograph.

Thursday, May 12, 2005

Quaternions...!

Mathematics of Computer Games

This book attempts to teach you what an average math student learns in three years ( or more ). I suppose the difference between the bookreader and the student is that the bookreader has a bag full of tricks applicable for game coding while the mathematician has a configurable toolbox applicable almost in any field. The average Linear Algebra course takes a year. In this book? A few chapters, I suppose. All in all an excellent introductory math book with focus. The Final Test...Is the code for these amazing grativity balls somewhere in that book? Yep!

Monday, April 18, 2005

Grothendieck

... never heard of the man before in my life. He received a Fields medal in 1966, that's roughly equivalent to a Nobel Prize for mathematics. Silly me.

"...The mere enumeration of Grothendieck 's best known contributions is overwhelming: topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann-Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), étale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, 'standard conjectures', motives and the 'yoga of weights', tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind. ..."

It looks as though I haven't got a clue to what mathematics ( algebra ) is about. It is however possible that the concepts are in my mind that I only have to match the words. Like with groups and rings. Once you understand the concept it becomes so trivial that you think you have always known what a group was, or a Galois group for that matter.

Erdos

"This one's from the Book!"

Saturday, April 9, 2005

Paradox of Life...

Maybe Matthew R. Watkins is a true genious, perhaps someone who lost direction in his life. I feel as if I never had any direction in my life or am I still searching? That's a paradox since I have found what I have been looking for.

Monday, March 28, 2005

Don Knuth

Another series of videos from Knuth recorded in 1987. Why is he becoming more famous every day? Maybe because we are all getting older. So much older. What heroes are there for old men? ( Older heroes wanted. )

Games!

Thursday, January 20, 2005

Collatz Conjecture

Once more...

"... Many mathematicians have attacked the problem with no result. Legend says scientists in Los Alamos spent a good deal of their time with it, instead of working on the atomic bomb! ..."

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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.)