Beware of ignorance ( or: just discovered SAGE )

Ignorance is a lack of knowledge. Ignorance is also the state of being ignorant or uninformed.

Until today I never heard of SAGE. ( ... )

What's so special about SAGE? From the website:

Use SAGE for studying a huge range of mathematics, including algebra, calculus, elementary to very advanced number theory, cryptography, numerical computation, commutative algebra, group theory, combinatorics, graph theory, and exact linear algebra.

SAGE makes it easy for you to use most mathematics software together. SAGE includes interfaces to Magma, Maple, Mathematica, MATLAB, and MuPAD, and the free programs Axiom, GAP, GP/PARI, Macaulay2, Maxima, Octave, and Singular.

You work with SAGE using the highly regarded scripting language Python instead of an obscure language designed for a particular mathematics program. You can write programs that combine serious mathematics with anything else.

Use SAGE from your web browser, which connects either to a program running on your computer, or a program running elsewhere. With the SAGE notebook you can create embedded graphics, beautifully typeset mathematical expressions, add and delete input, and start up and interrupt multiple calculations.

Introduction to SAGE (pdf)

CoCoA

CoCoA, an algebra package. So there is more than GAP. Comparison of computer algebra systems

Computational Commutative Algebra I

Working on Rotman, Chapter 3: 'Commutative Rings I', I often scan ahead. Today I looked at Computational Commutative Algebra 1.

"... This is one of the most refreshing mathematical books I have ever held in my hands. The authors do not believe in teaching by spreading out the material, but they introduce it via questions and discussions, they explore it in an intuitive fashion, exercise it through well-chosen examples, and start the reader on his own expeditions through numerous "tutorials", i.e., guided projects. This is academic teaching at its best: if I had not seen it, I should not have believed that it can be done so well. ... In conclusion, this book gives students a stimulating introduction to commutative algebra very much geared to their need, and it provides numerous useful ideas to those who teach the subject." (H.Stetter, IMN - Internationale Mathematische Nachrichten 2003, Vol. 57, Issue 193)

Books

Some books are to be tasted, others to be swallowed,
and some few to be chewed and digested.
Francis Bacon (1597)

From Artin to Rotman

Every Coin Has Two Sides. I have this corrupt file, including several corrupt backups, which I used to store notes from, and manage Artin's Algebra. The good thing is that I decided to start working in another book.

Mathematica 6 is slow

Fom some expert forum ( non-Wolfram of course )

I also read things said in wolfram forums. The issue is totally different. There is no way to change or play with options. In our case the integral is straight forward with no singularities at all! Something like this:

F[t_]:=NIntegrate[e^t ((a+b)-3a^2 )^8, {a,0,b},{b,0,3}]
Plot[F[t],{t,0,4}]

That's it. No singularities.. no cut-off's nothing. Pure and simple double integral. When you try with v6.0 it does it 5 or sometimes 7 times slower than v5.2.

Let's assume there is something we miss in these calculations.. ok?

Let's go back to Wolfram's built in function called Benchmark; to test how fast it handles 15 various problems. On both versions this command exists. You give a try on both.. you still come out with the result that overall Mathematica 6.0 is slower than Mathematica 5.2.

This is the reality of it. Maybe v6.0 while calculating it also thinks how to decorate the result and show it nicely with colors and so on.. But who needs to wait 5 times more for "loox" as you said, right?

I hope Wolfram will take care of this issue. Meanwhile I will stick to v5.2.
Or even 4.1 which is btw faster than 5.2 :)

Try before you buy, or better wait until 6.1 or so.

Example of a finite vector space

The vector space of dimension 2 over the finite field GF(3) is isomorphic to the abelian group Z3 X Z3.

gap> b:=One(GF(3))*[[1,0],[0,1]];
[ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ]
gap> V:=VectorSpace(GF(3),b);

gap> Dimension(V);
2
gap> Size(V);
9
gap> Elements(V);
[ 
[ 0*Z(3), 0*Z(3) ], 
[ 0*Z(3), Z(3)^0 ], 
[ 0*Z(3), Z(3) ], 
[ Z(3)^0, 0*Z(3) ],
[ Z(3)^0, Z(3)^0 ], 
[ Z(3)^0, Z(3) ], 
[ Z(3), 0*Z(3) ], 
[ Z(3), Z(3)^0 ], 
[ Z(3), Z(3) ] 
]

Wallpaper groups

Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).

Discover the wallpaper patterns with the Kali Java applet.

How it's like to remember 22,500 digits of Pi

The Boy with the Incredible Brain ( Channel 5 rip on Google Video )

Remember Max Cohen in the movie Pi when the little child with the calculator asks him to do some difficult calculation? Daniel Tammet, beats Max and Daniel is real. But probably just as mad as Max. His disease is called autism.

I don't believe in his brilliance. If he was so brilliant why does he spend time learning the digits of Pi instead of coming up with some new identities for Pi which will deepen the understanding of Pi in the entire mathematical community? The researchers researching him aren't mathematicians but people who think math and arithmetic are one and the same.

Tammet is clever in the way he exploits his ability. There is nothing wrong with that. I don't expect any breakthroughs in science of him though. The real geniuses are working hard, day and night, far away from the publicity.

In the movie he meets the real Rain Man. He is certainly amazing. When you ask Rainman a questions like: "If Churchill would still live today, how old would he be and on what day of the week would be his birthday?" he answers immediately, and correct, of course.

People like Rainman and Tammet are called savants.

Abstract Algebra

Algebra is a branch of mathematics.

Algebra may also mean:
* elementary algebra
* abstract algebra
* linear algebra
* universal algebra
* computer algebra

In addition, many mathematical objects are known as algebras.

In logic:
* Boolean algebra
* Heyting algebra

In set theory:
* algebra over a set
* sigma algebra

In linear algebra, and the study of vector spaces:
* algebra over a field
* associative algebra
* commutative, anticommutative, and super- algebras
* Lie algebra

In ring theory:
* algebra over a commutative ring (or R-algebra)

In category theory:
* F-algebra
* F-coalgebra

In the relational model:
* Relational algebra

Hilbert's Hotel

The hotel manager David Hilbert had a very large hotel, in fact, it had infinitely many rooms numbered 1, 2, 3, ... The hotel was very popular and every room was occupied. One day a new guest arrived.
-Is there any free room?
-No, Mr. Hilbert said.
-Oh, what a pity, the guest said and started to walk away.
-But wait, you can still get a room.

The new guest was very confused by this and asked how that could be possible.
-I'll just ask the guest in room number 1 to move to room number 2, the guest in number 2 move to room 3, the guest in room 3 move to number 4, and so on, and then you can have room number 1.

The guest was very happy with this and called his friends to tell them about this fantastic hotel. Then one day they all arrived at the same time.
-Hello, we are countably many people, and we want a room each. Mr. Hilbert felt reluctant to ask each guest to move to a new room an infinite number of times. That would be very unpleasant, and they will never finish either. Luckily he got a brilliant idea.
-I'll let guest in number 1 move into number 2, guest in number 2 move into number 4, guest in number 3 move into number 6, number 4 to number 8 and so on. Then you can move into any room with an odd number; they will all be free, and there will be rooms for all of you.

Now all the new guests got a room on their own.

By Daume on PlanetMath

Mathematics Podcast

Check out Math Mutation Podcast for short podcasts for people of all ages, where we explore fun, interesting, or just plain wierd corners of mathematics that you probably didn't hear in school. ( ... You never know what might bring you the next idea you'll be working on. ...)