06-1350/Class Notes for Thursday November 16: Difference between revisions

Revision as of 22:56, 15 November 2006

#	Week of...	Links
1	Sep 11	About, Tue, Thu
2	Sep 18	Tue, Kurlin(P), Thu
3	Sep 25	Tue, Photo, Thu
4	Oct 2	HW1, Tue, Thu
5	Oct 9	Tue(P), Thu(P)
6	Oct 16	HW2, Tue(P), Thu(P)
7	Oct 23	Tue(P), Thu
8	Oct 30	HW3, Tue, Thu
9	Nov 6	Tue ( $\nu$ ), Thu
10	Nov 13	Tue, Thu
11	Nov 20	HW4(P), Thu
12	Nov 27	Thu
13	Dec 4	Syzygies in Asymptote, Final
	Jan 8	Grades
Note. (P) means "contains a problem that Dror cares about".
Add your name / see who's in!
On to 07-1352

In Preparation

The information below is preliminary and cannot be trusted! (v)

Sweeping clean a tree and ${\mathcal {A}}(\Gamma )\equiv {\mathcal {A}}(\uparrow _{b_{1}(\Gamma )})$ .

${\mathcal {A}}(\uparrow _{n})$ is a VS-algebra (see more at VS, TS and TG Algebras).
In the coordinates above, write the $TR\Phi B$ $TR\Phi B$ relations in various algebraic notations.
- R4: $(1230)^{\star }B\cdot (1213)^{\star }B\cdot (1023)^{\star }\Phi =(1123)^{\star }\Phi \cdot (1233)^{\star }B$ or $(B_{1a}B_{2a}\Phi _{1a};B_{1b}B_{2b};B_{1c}B_{2a}\Phi _{1b};B_{2c}\Phi _{1c})=(\Phi _{2a}B_{3a};\Phi _{2a}B_{3b};\Phi _{2b}B_{3c};\Phi _{2c}B_{3c})$ .
- R3: $(1230)^{\star }B\cdot (1213)^{\star }B\cdot (1023)^{\star }B=(1123)^{\star }B\cdot (1203)^{\star }B\cdot (1231)^{\star }B$ or $(B_{1a}B_{2a}B_{3a};B_{1b}B_{2b};B_{1c}B_{2a}B_{3b};B_{2c}B_{3c})=(B_{4a}B_{5a}B_{6a};B_{4a}B_{5b}B_{6b};B_{4b}B_{6c};B_{4c}B_{5c}B_{6a})$ .
- R2: $(123)^{\star }B^{\pm }\cdot (132)^{\star }B^{\mp }=1_{3}$ or $(B_{1a}^{\pm }B_{2a}^{\mp };B_{2b}^{\pm }B_{3c}^{\mp };B_{1c}^{\pm }B_{2b}^{\mp })=(1;1;1)$ .
- R1: $(B_{a}^{\pm };B_{b}^{\pm }B_{c}^{\pm })=(1;T^{\pm 2})$ .
- Symmetry of $\Phi$ and of $B^{\pm }$ .
- $u$ , $d$ and $\#$
- Idempotence for $T$ , $R$ , $\Phi$ and $B^{\pm }$ .
- $B^{\pm }$ in terms of $\Phi$ and $R$ and $R$ in terms of $T$ .

General Principle. Believe not, until you see a running program.

@@ Line 11: / Line 11: @@
 ** R4: <math>(1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B</math> or <math>(B_{1a}B_{2a}\Phi_{1a}; B_{1b}B_{2b}; B_{1c}B_{2a}\Phi_{1b}; B_{2c}\Phi_{1c}) = (\Phi_{2a}B_{3a}; \Phi_{2a}B_{3b}; \Phi_{2b}B_{3c}; \Phi_{2c}B_{3c})</math>.
 ** R3: <math>(1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star B = (1123)^\star B\cdot(1203)^\star B\cdot(1231)^\star B</math> or <math>(B_{1a}B_{2a}B_{3a}; B_{1b}B_{2b}; B_{1c}B_{2a}B_{3b}; B_{2c}B_{3c}) = (B_{4a}B_{5a}B_{6a}; B_{4a}B_{5b}B_{6b}; B_{4b}B_{6c}; B_{4c}B_{5c}B_{6a})</math>.
+** R2: <math>(123)^\star B^\pm\cdot(132)^\star B^\mp=1_3</math> or <math>(B^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1)</math>.
-** R2, R1
+** R1: <math>(B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2})</math>.
-** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</math>.
 ** Symmetry of <math>\Phi</math> and of <math>B^{\pm}</math>.
 ** <math>u</math>, <math>d</math> and <math>\#</math>
 ** Idempotence for <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>.
+** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</math>.
+'''General Principle.''' Believe not, until you see a running program.