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

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{{In Preparation}}


==Today's Agenda==
==Today's Agenda==

===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\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>.
** 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>.
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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>.
** 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>.
** R1: <math>(B^\pm_a; B^\pm_bB^\pm_c)=(1;T^{\pm 2})</math>.
** 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>
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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>.
** <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===
'''General Principle.''' Believe not, until you see a running program.

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

Revision as of 09:28, 16 November 2006

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On to 07-1352

Today's Agenda

Formulas are a Chore (Bore?)

  • Sweeping clean a tree and [math]\displaystyle{ {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}) }[/math].
06-1350-TRPhiB.png
  • In the coordinates above, write the [math]\displaystyle{ TR\Phi B }[/math] relations in various algebraic notations.
    • R4: [math]\displaystyle{ (1230)^\star B\cdot(1213)^\star B\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B }[/math] or [math]\displaystyle{ (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]\displaystyle{ (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]\displaystyle{ (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]\displaystyle{ (123)^\star B^\pm\cdot(132)^\star B^\mp=1_3 }[/math] or [math]\displaystyle{ (B^\pm_{1a}B^\mp_{2a}; B^\pm_{2b}B^\mp_{3c}; B^\pm_{1c}B^\mp_{2b}) = (1;1;1) }[/math].
    • R1: [math]\displaystyle{ (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]\displaystyle{ \Phi }[/math] and of [math]\displaystyle{ B^{\pm} }[/math].
    • [math]\displaystyle{ u }[/math], [math]\displaystyle{ d }[/math] and [math]\displaystyle{ \# }[/math]
    • Idempotence for [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math].
    • [math]\displaystyle{ B^{\pm} }[/math] in terms of [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ R }[/math] and [math]\displaystyle{ R }[/math] in terms of [math]\displaystyle{ T }[/math].

Exponentiation is a Miracle

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