Macaulay 2

Today I installed Macaulay 2, 'software system devoted to supporting research in algebraic geometry and commutative algebra'.

Example 1: calculate (x+y)^2 in the polynomial ring Z[x,y].

i107 : R=ZZ[x,y]
o107 = R
o107 : PolynomialRing
i108 : (x+y)^2
        2           2
o108 = x  + 2x*y + y
o108 : R

Example 2: calculate (x+y)^2 in the polynomial ring Z/2[x,y].

i111 : R=ZZ/2[x,y]
o111 = R
o111 : PolynomialRing
i112 : (x+y)^2
        2    2
o112 = x  + y
o112 : R

Helpful people

Yesterday evening I ran out of ideas on how to implement something in GAP. Intuitively I knew it was possible. Within my understanding of mathematics, anything must bepossible with GAP, I mean GAP is used by professional researchers who speak GAP. Today, when I took a short break at work, I posted my question in the GAP forum. I immediately received two replies and that happened before. There is always someone who helps you out. Somehow that means a lot to me, it's why I believe in the future despite the tremendous problems we have in our world.

Dear Nilo,
You have a ring R, and you want to treat it as a group. This group is
written additively in GAP; so you must convert it to a multiplicative
group first.

Here is a simple example of what you (probably) want to do. Hope it helps:

gap> r := GroupRing(GF(2),SymmetricGroup(3));

gap> rplus := Group(List(Basis(r),AdditiveElementAsMultiplicativeElement));

gap> StructureDescription(rplus);
"C2 x C2 x C2 x C2 x C2 x C2"
--
Laurent Bartholdi \ laurent.bartholdi*****com
EPFL SB SMA IMB MAD \ Téléphone: +41 21-*******
Station 8 \ Secrétaire: +41 21-*******
CH-1015 Lausanne, Switzerland \ Fax: +41 21-*******

Thanks again Mr. Bartholdi.

GroupRing(ZmodnZ(9),CyclicGroup(5))

The SAGE Notebook interface has been updated. It is now possible to select the computeralgebra dialect from a dropdownlist including gap, mathematica, maple and singular, well, basically everything out there except Cocoa perhaps. Besides that, it is possible to use lisp and python as programming languages within SAGE.

To the point, as I was playing a bit in SAGE to get a feeling for the interface I managed to create the following beast.

RG:=GroupRing(ZmodnZ(9),CyclicGroup(5))
free left module over (Integers mod 9), 
and ring-with-one, with 1 generators
Size(RG)
59049

Definitely a beast to my liking.

Implementing change (2)

Daily ( evening / weekend ) study is a process I have to manage carefully, it's my life, basically. Earlier this month I wrote that I have been keeping track of various metrics regarding my study. Well, I tried. Up until now staying motivated and making a sufficient number of hours has been my prime area of concern. I did manage to record my study hours reasonably accurate so I do have some actual statistics at the moment. As far as the other four metrics are concerned I might even change their definition.
* motivation: study hrs/wk
* focus: SuperMemo statistics ( TBD )
* support: Pipeline statistics ( TBD )
* success: ( not defined yet )
* efficiency. ( not defined yet )
Motivation dropped to 63% last week.

Topology

Let X be any set and let T be a family of subsets of X. Then T is a topology on X if
- both the empty set and X are elements of T,
- any union of elements of T is an element of T, and
- any intersection of finitely many elements of T is an element of T.
If T is a topology on X, then X together with T is called a topological space.

Example:
X = {a,b,c} is a set.
T = {0, {a,b,c}, {a,b}, {a,c}, {b,c}. {a}, {b}, {c} } is a family of subsets.
[X, T] is a topological space.

From "Topology Without Tears" by

Sidney A. Morris.

Maximal ideal

Let R be a ring with identity. A proper ideal I ⊆ R is a maximal ideal if I is not a proper subset of any other ideal of R.

All maximal ideals are prime ideals. If R is commutative, an ideal I ⊂ R is maximal if and only if the quotient ring R is a field.

Example:
Let m, n be ideals of Z.
I = (6) = {..., -12, -6, 0, 6, 12, 18, 24, ...}
J = (3) = {..., -6, -3, 0, 3, 6, 9, 12, 15, ...}
The ideal m is not maximal because I ⊂ J, while J is maximal because there is no ideal K such that J ⊂ K.

Ideals in Z

Let I,J be ideals in Z.

If I=(m), J=(n) then I+J=(GCD(m,n)), and I∩J=(LCM(m,n)).

Elliptic Curves

It has been a while but I had another of those euphoric a-ha moments. I have seen a glimpse of what's all the fuzz on Elliptic Curves about. Yes, they are spectaculair.

( More later. )

LaTeX IDE's: shortlist

I have been looking at various LaTeX editors. The cycle of producing mathematics documents is very similar to producing (Java) software so LaTeX editors are in fact complete IDE's. That's why both NetBeans and Eclipse, popular Java IDE's have a LaTeX module ( NetBeans) or plug-in(s) ( Eclipse ).
I have looked at
* Lyx
* TeXnicCenter
* LaTeX for NetBeans
* TeXlipse (Eclipse plug-in)
* LEd

Lyx tries to be wysiwyg, and therefore does not work for high level math. TeXnicCenter is the tool I have been using from the beginning. Excellent, really, but it lacks features like code-folding. LaTeX for NetBeans. It would be extremely cool to be able to produce math in NetBeans. It can be done but the feature set does not even come close to TeXnicCenter, although it has code-folding. TeXlipse is worse than LaTeX for NetBeans. The document structure viewer does not even support multiple file documents. There is something like the Full LaTeX view but it does not work in many cases. LEd. Excellent stuff but it is not very friendly to projects created with other tools. I was not able to import an existing set of tex files.

My choice is TeXnicCenter.

Implementing change

Since the beginning of March I keep track of various metrics regarding my study. It's a way of staying focused. The set of metrics I used was just a try-out. I changed the metrics a bit today. As long as it serves my purpose.
* motivation: study hrs/wk
* focus: repeat-sessions;
* support: new repeat-items
* success: #exercises done
* efficiency. (not defined yet)
A sort of personal CMM.
* 'Motivated'
* 'Focused'
* 'Supported'
* 'Successful'
* 'Efficient'
I am motivated and working on the other levels to achieve. I need to focus more on actually 'doing' math, solving problems, doing exercises and proofs and writing papers. Not to publish them but to get experienced in the mechanics of it. Writing beautiful TeX to start with. I have more than enough ideas, things I want to research. More on that later. - Exercises: I found a few exercises on Rings and Fields, with answers, and here is another set. It is my goal to do most of them in the forthcoming month.

Fermat's Last Theorem

In about 1630 Fermat was reading a recently published translation of Arithmetica by Diophantus of Alexandria. He was making notes in the margin and at one point he entered:

To divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable proof of this, but the margin is too narrow to contain it.

Today I have seen the documentary about the Andrew Wiles' proof of Fermat's theorem. ( Link to video: [ here ] )

Cornucopia

As seen on MathWorld ( Click and drag )

name="input_archive" value="zip/cornucop.zip" />

alt="Cornucopia" />

SAGE

I downloaded and watched a colloquium on SAGE by William Stein. Stein talks about the past, present and future of SAGE. He is very much in favor of open source software but couldn't find the perfect open source math program so he decided to write one himself. His plan was to reverse-engineer existing open source code and port it to a new Python program. He soon realized it would take several lifetimes to implement so he used existing software as a sort of library behind GAPE. ( ... )

He briefly mentioned none of the open source programs are in any way multi-threaded. I suppose that's both a technical and mathematical challenge. Interesting...