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

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{{06-1350/Navigation}}
 
{{06-1350/Navigation}}
 
{{In Preparation}}
 
  
 
==Today's Agenda==
 
==Today's Agenda==
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 +
===Formulas are a Chore (Bore?)===
  
 
* Sweeping clean a tree and <math>{\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)})</math>.
 
* Sweeping clean a tree and <math>{\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)})</math>.
 
[[Image:06-1350-TRPhiB.png|center|500px]]
 
[[Image:06-1350-TRPhiB.png|center|500px]]
 +
 
* <math>{\mathcal A}(\uparrow_n)</math> is a VS-algebra (see more at [[VS, TS and TG Algebras]]).
 
* <math>{\mathcal A}(\uparrow_n)</math> is a VS-algebra (see more at [[VS, TS and TG Algebras]]).
 +
 
* In the coordinates above, write the <math>TR\Phi B</math> relations in various algebraic notations.
 
* In the coordinates above, write the <math>TR\Phi B</math> relations in various algebraic notations.
** R4: <math>(1230)^\star B^\pm\cdot(1213)^\star B^\pm\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B^\pm</math>.
+
** 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>.
** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</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>.
** R3, R2, R1
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** 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>.
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** R1: <math>(B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2})</math>.
 +
* But for now, skip the writing of the following relations:
 
** Symmetry of <math>\Phi</math> and of <math>B^{\pm}</math>.
 
** Symmetry of <math>\Phi</math> and of <math>B^{\pm}</math>.
 
** <math>u</math>, <math>d</math> and <math>\#</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>.
 
** Idempotence for <math>T</math>, <math>R</math>, <math>\Phi</math> and <math>B^{\pm}</math>.
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** <math>B^{\pm}</math> in terms of <math>\Phi</math> and <math>R</math> and <math>R</math> in terms of <math>T</math>.
 +
 +
===Exponentiation is a Miracle===
 +
 +
* Description of the problem.
 +
* Beads and strands.
 +
* The perturbative approach, linearization.
 +
* The syzygies: relations between the errors.
 +
* The Hochschild complex and homology.

Latest revision as of 10:29, 16 November 2006

Today's Agenda

Formulas are a Chore (Bore?)

  • Sweeping clean a tree and {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}).
06-1350-TRPhiB.png
  • In the coordinates above, write the 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^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1).
    • R1: (B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2}).
  • But for now, skip the writing of the following relations:
    • 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.

Exponentiation is a Miracle

  • Description of the problem.
  • Beads and strands.
  • The perturbative approach, linearization.
  • The syzygies: relations between the errors.
  • The Hochschild complex and homology.