06-1350/Class Notes for Thursday November 16 - Revision history

Drorbn at 14:29, 16 November 2006

2006-11-16T14:29:35Z

← Older revision		Revision as of 10:29, 16 November 2006
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	* Description of the problem.		* Description of the problem.
			* Beads and strands.
	* The perturbative approach, linearization.		* The perturbative approach, linearization.
	* The syzygies: relations between the errors.		* The syzygies: relations between the errors.

Drorbn at 14:28, 16 November 2006

2006-11-16T14:28:13Z

← Older revision		Revision as of 10:28, 16 November 2006
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	{{06-1350/Navigation}}		{{06-1350/Navigation}}

	{{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.

Drorbn at 02:56, 16 November 2006

2006-11-16T02:56:29Z

← Older revision		Revision as of 22:56, 15 November 2006
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	** 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>.
	** 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: <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>.		** 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>.
		⚫	** <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.

Drorbn at 02:36, 16 November 2006

2006-11-16T02:36:51Z

← Older revision		Revision as of 22:36, 15 November 2006
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	* <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> or <math>(~~B^\pm_~~{1a}~~B^\pm_~~{2a}\Phi_{1a}; ~~B^\pm_~~{1b}~~B^\pm_~~{2b}; ~~B^\pm_~~{1c}~~B^\pm_~~{2b}\Phi_{1b}; ~~B^\pm_~~{2c}\Phi_{1c}) = (\Phi_{2a}~~B^\pm_~~{3a}; \Phi_{2a}~~B^\pm_~~{3b}; \Phi_{2b}~~B^\pm_~~{3c}; \Phi_{2c}~~B^\pm_~~{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>.
			** 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, R1
	** <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>.
⚫	** ~~R3,~~ R2, R1
	** 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>

Drorbn at 02:02, 16 November 2006

2006-11-16T02:02:00Z

← Older revision		Revision as of 22:02, 15 November 2006
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	* <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^\pm\cdot(1213)^\star B^\pm\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B^\pm</math> or <math>(B^\pm_{1a}B^\pm_{2a}\Phi_{1a}; B^\pm_{1b}B^\pm_{2b}; B^\pm_{1c}B^\pm_{2b}\Phi_{1b}; B^\pm_{2c}\Phi_{1c}) = (\Phi_{2a}B^\pm_{3a}; \Phi_{2a}B^\pm_{3b}; \Phi_{2b}B^\pm_{3c}; \Phi_{2c}B^\pm_{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>.		** <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, R2, R1		** R3, R2, R1

Drorbn at 14:19, 15 November 2006

2006-11-15T14:19:06Z

New page

{{06-1350/Navigation}}

{{In Preparation}}

==Today's Agenda==

* Sweeping clean a tree and <math>{\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)})</math>.
[[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]]).
* 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>.
** <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, R2, R1
** 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>.