My Operahello world...
Sunday, December 18, 2005 5:05:18 PM
Just looking around what kind of functionality MyOpera offers. Looks good. I would move if Blogger import worked, but unfortunately it doesn't.
Intermediate Goal
Sunday, December 18, 2005 9:35:35 AM
To understand all of the following...:
StructureDescription(<G>) ::=
1 ; trivial group
| C<size> ; cyclic group
| A<degree> ; alternating group
| S<degree> ; symmetric group
| D<size> ; dihedral group
| Q<size> ; quaternion group
| QD<size> ; quasidihedral group
| PSL(<n>,<q>) ; projective special linear group
| SL(<n>,<q>) ; special linear group
| GL(<n>,<q>) ; general linear group
| PSU(<n>,<q>) ; proj. special unitary group
| O(2<n>+1,<q>) ; orthogonal group, type B
| O+(2<n>,<q>) ; orthogonal group, type D
| O-(2<n>,<q>) ; orthogonal group, type 2D
| PSp(2<n>,<q>) ; proj. special symplectic group
| Sz(<q>) ; Suzuki group
| Ree(<q>) ; Ree group (type 2F or 2G)
| E(6,<q>) | E(7,<q>) | E(8,<q>) ; Lie group of exceptional type
| 2E(6,<q>) | F(4,<q>) | G(2,<q>)
| 3D(4,<q>) ; Steinberg triality group
| M11 | M12 | M22 | M23 | M24
| J1 | J2 | J3 | J4 | Co1 | Co2
| Co3 | Fi22 | Fi23 | Fi24' | Suz
| HS | McL | He | HN | Th | B
| M | ON | Ly | Ru ; sporadic simple group
| 2F(4,2)' ; Tits group
| PerfectGroup(<size>,<id>) ; the indicated group from the
; library of perfect groups
| A x B ; direct product
| N : H ; semidirect product
| C(G) . G/C(G) = G' . G/G' ; non-split extension
; (equal alternatives and
; trivial extensions omitted)
| Phi(G) . G/Phi(G) ; non-split extension:
; Frattini subgroup and
; Frattini factor group
Progress Saturday
Saturday, December 17, 2005 10:48:09 PM
Worked on the character table of C3 x S3. Did not complete it due to time constraints but I felt much more comfortable working on character tables than a while back. I should, must, want to complete a few tables real soon.
A start, at last...! Sym^2 V = The subspace of V (x) V fixed under s. I found a definition. Pfff. What is s? Will see later. I really thought I would never understand that stuff. I blame it under 'poorly documented' stuff.
Associative Algebras . Learned about a new algebraic structure today. Not really new because I have worked a lot with instances of this structure like the n x n matrices and the quaternions.
*** ALERT! *** Today I stumbled over the following group while I was doing some simple calculations:
gap> StructureDescription(AllGroups(32)[8]);
"C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)"
I knew of the Group Direct Product and Semi-direct Product, how I can use them in GAP, and how I can construct them manually. It is however the first time that I see a group description in this sort of format. I checked the GAP Reference and found that it had to with 'splitting extensions'. To make a long story short: I found ( a lot of ?? ) uncovered ground in Group Theory. I didn't expect such a surprise with a group of only order 32. To Do? Group Extension , Wreath Product , Hall Subgroup , , Nilpotent Group , Frattini Subgroup , Fitting Subgroup to start with. I have to plan this stuff.
A start, at last...! Sym^2 V = The subspace of V (x) V fixed under s. I found a definition. Pfff. What is s? Will see later. I really thought I would never understand that stuff. I blame it under 'poorly documented' stuff.
Associative Algebras . Learned about a new algebraic structure today. Not really new because I have worked a lot with instances of this structure like the n x n matrices and the quaternions.
*** ALERT! *** Today I stumbled over the following group while I was doing some simple calculations:
gap> StructureDescription(AllGroups(32)[8]);
"C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)"
I knew of the Group Direct Product and Semi-direct Product, how I can use them in GAP, and how I can construct them manually. It is however the first time that I see a group description in this sort of format. I checked the GAP Reference and found that it had to with 'splitting extensions'. To make a long story short: I found ( a lot of ?? ) uncovered ground in Group Theory. I didn't expect such a surprise with a group of only order 32. To Do? Group Extension , Wreath Product , Hall Subgroup , , Nilpotent Group , Frattini Subgroup , Fitting Subgroup to start with. I have to plan this stuff.
Now present on Sourceforge.
Saturday, December 17, 2005 8:54:56 PM
Since the xmSQL XML JDBC Driver has been terminated infrastructure came available for a new project. In some cases we have to work with the name xmsql. No problem, at all. At not before long anyway. We are glad to be on Sourceforge. See the Project
Summary Page
.
GAP programming
Saturday, December 17, 2005 12:19:36 AM
GAP programming is an essential skill for an algebraist. The best material I have found sofar on GAP programming ( besides the official Reference ) are the handouts of
Alice C. Niemeyer
's ( what a
beauty...!
)
Group Theory Learning Workshop
.
Weekend (2)
Friday, December 16, 2005 8:27:36 PM
If I study I should do any of the following activities
- study exterior Product
- study induced representation
- study Young diagrams (once more)
- create character table of C3 x S3
- exercise set
- work on summary ( in TeX ), ( virtual lecture )
if I want to stay on course, that is.
- study exterior Product
- study induced representation
- study Young diagrams (once more)
- create character table of C3 x S3
- exercise set
- work on summary ( in TeX ), ( virtual lecture )
if I want to stay on course, that is.
(^^) Goal (2)
Friday, December 16, 2005 7:06:43 PM
With Polyhedra Inspector (P.I.) I want to
- create the computer version of the BrainTwist game
- analyze and program the Transformation Group of the BrainTwist
- solve the BrainTwist by computer
- find and prove the optimal solution
- and more...
- create the computer version of the BrainTwist game
- analyze and program the Transformation Group of the BrainTwist
- solve the BrainTwist by computer
- find and prove the optimal solution
- and more...
Weekend
Friday, December 16, 2005 12:24:21 AM
Could, would like to do exercises ( full analysis C3 x S3, RT Set 2 and solve the Serganova question ), summarize, or even write, one article using GAP ), reading in "Lie Groups". And brainstorming about P.I. Last weekend was not OK.
The matrix exponential
Thursday, December 15, 2005 7:44:33 PM
e
^ A
e to the power matrix A.
| 2 0 |
| 0 1 |
e
Matrices as power coefficients.
Is that cool, or what?
Lie Algebras.
e to the power matrix A.
| 2 0 |
| 0 1 |
e
Matrices as power coefficients.
Is that cool, or what?
Lie Algebras.
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