User:Drorbn/06-1350-HW4 - Revision history

Drorbn: /* A Mathematica Verification */

2006-11-20T23:41:48Z

A Mathematica Verification

← Older revision		Revision as of 19:41, 20 November 2006
Line 87:		Line 87:

	{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]		{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
	+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3~~, x4, x5] + rho3[x1 + x3, x2~~, x4, x5]		+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5]
	- rho3[x1 + x4, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4]~~</nowiki>}}~~		+ rho3[x1 + x3, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5]
			+ rho3[x1 + x5, x2, x3, x4]</nowiki>}}

	{{InOut\|n=4\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>\|out=<nowiki>0</nowiki>}}		{{InOut\|n=4\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>\|out=<nowiki>0</nowiki>}}

Drorbn: /* The "B around B" Syzygy */

2006-11-20T23:40:00Z

The "B around B" Syzygy

← Older revision		Revision as of 19:40, 20 November 2006
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	\|[[Image:06-1350-BAroundB.svg\|center]]		\|[[Image:06-1350-BAroundB.svg\|center]]
	\|-		\|-
	\|align=right\|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality ~~picture~~ are also acceptable)		\|align=right\|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality pictures are also acceptable)
	\|}		\|}

Drorbn: /* A Mathematica Verification */

2006-11-20T23:36:32Z

A Mathematica Verification

← Older revision		Revision as of 19:36, 20 November 2006
Line 75:		Line 75:
	{{In\|n=1\|in=<nowiki>d1 = {		{{In\|n=1\|in=<nowiki>d1 = {
	rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +		rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
	bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - ~~bp[x1 + x4, x2, x3]~~		bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] -
			bp[x1 + x4, x2, x3]
	};		};
	d2 = {		d2 = {

Drorbn: /* A Mathematica Verification */

2006-11-20T23:35:54Z

A Mathematica Verification

← Older revision		Revision as of 19:35, 20 November 2006
Line 74:		Line 74:

	{{In\|n=1\|in=<nowiki>d1 = {		{{In\|n=1\|in=<nowiki>d1 = {
	rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +		rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
	bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]		bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
	};		};
	d2 = {		d2 = {
	BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +		BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
	rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -		rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
	rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -		rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
	rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +		rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
	rho3[x1 + x3, x2, x4, x5]		rho3[x1 + x3, x2, x4, x5]
	};</nowiki>}}		};</nowiki>}}

	{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]		{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]

Drorbn: /* A Mathematica Verification */

2006-11-20T23:34:27Z

A Mathematica Verification

← Older revision		Revision as of 19:34, 20 November 2006
Line 74:		Line 74:

	{{In\|n=1\|in=<nowiki>d1 = {		{{In\|n=1\|in=<nowiki>d1 = {
	rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + ~~bp[x1, x3, x4] -~~		rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] +
	bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]		bp[x1, x3, x4] - bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
	};		};
	d2 = {		d2 = {
	BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + ~~rho3[x1 + x5, x2, x3, x4] -~~		BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] +
	rho3[x1 + ~~x2, x3, x4,~~ x5~~] - rho3[x1~~, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -		rho3[x1 + x5, x2, x3, x4] - rho3[x1 + x2, x3, x4, x5] -
	rho3[x1, x2~~, x3, x4] + rho3[x1, x3~~, x4, x5] + rho3[x1 + x3, x2, x4, x5]		rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
			rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] +
⚫	};</nowiki>}}
			rho3[x1 + x3, x2, x4, x5]
		⚫	};</nowiki>}}

	{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>-rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]		{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>- rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
	+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]		+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
	- rho3[x1 + x4, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4]</nowiki>}}		- rho3[x1 + x4, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4]</nowiki>}}

	{{InOut\|n=4\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>\|out=<nowiki>0</nowiki>}}		{{InOut\|n=4\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>\|out=<nowiki>0</nowiki>}}

Drorbn at 23:29, 20 November 2006

2006-11-20T23:29:02Z

← Older revision		Revision as of 19:29, 20 November 2006
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	====The "B around B" Syzygy====		====The "B around B" Syzygy====

			The picture, with all shielding removed, is
	{\| align=center		{\| align=center
	\|- align=center		\|- align=center
	\|[[Image:06-1350-BAroundB.svg\|center]]		\|[[Image:06-1350-BAroundB.svg\|center]]
	\|-		\|-
	\|align=right\|(Drawn with [http://www.inkscape.org/ Inkscape])		\|align=right\|(Drawn with [http://www.inkscape.org/ Inkscape])<br>(note that lower quality picture are also acceptable)
	\|}		\|}

Drorbn: /* A Mathematica Verification */

2006-11-19T18:15:39Z

A Mathematica Verification

← Older revision		Revision as of 14:15, 19 November 2006
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	===A Mathematica Verification===		===A Mathematica Verification===

	The following simulated Mathematica session proves that for our ~~signle~~ relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!		The following simulated Mathematica session proves that for our single relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

	{{In\|n=1\|in=<nowiki>d1 = {		{{In\|n=1\|in=<nowiki>d1 = {

Drorbn at 17:38, 19 November 2006

2006-11-19T17:38:27Z

← Older revision		Revision as of 13:38, 19 November 2006
Line 47:		Line 47:
	====The "B around B" Syzygy====		====The "B around B" Syzygy====

			{\| align=center
⚫	[[Image:06-1350-BAroundB.svg\|center]]
			\|- align=center
		⚫	\|[[Image:06-1350-BAroundB.svg\|center]]
			\|-
			\|align=right\|(Drawn with [http://www.inkscape.org/ Inkscape])
			\|}

			The functional form of this syzygy is

			{\| align=center
			\|-
			\|<math>BB(x_1,x_2,x_3,x_4,x_5) = </math>
			\|<math>\rho_3(x_1, x_2, x_3, x_5) + \rho_3(x_1 + x_5, x_2, x_3, x_4) - \rho_3(x_1 + x_2, x_3, x_4, x_5)</math>
			\|-
			\|
			\|<math>- \rho_3(x_1, x_2, x_4, x_5) - \rho_3(x_1 + x_4, x_2, x_3, x_5) - \rho_3(x_1, x_2, x_3, x_4)</math>
			\|-
			\|
			\|<math>+ \rho_3(x_1, x_3, x_4, x_5) + \rho_3(x_1 + x_3, x_2, x_4, x_5).</math>
			\|}

	===A Mathematica Verification===		===A Mathematica Verification===

			The following simulated Mathematica session proves that for our signle relation and single syzygy, <math>d^2=0</math>. Copy paste it into a live Mathematica session to see that it's right!

			{{In\|n=1\|in=<nowiki>d1 = {
			rho3[x1_, x2_, x3_, x4_] :> bp[x1, x2, x3] + bp[x1 + x3, x2, x4] + bp[x1, x3, x4] -
			bp[x1 + x2, x3, x4] - bp[x1, x2, x4] - bp[x1 + x4, x2, x3]
			};
			d2 = {
			BAroundB[x1_, x2_, x3_, x4_, x5_] :> rho3[x1, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4] -
			rho3[x1 + x2, x3, x4, x5] - rho3[x1, x2, x4, x5] - rho3[x1 + x4, x2, x3, x5] -
			rho3[x1, x2, x3, x4] + rho3[x1, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
			};</nowiki>}}

			{{InOut\|n=3\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2</nowiki>\|out=<nowiki>-rho3[x1, x2, x3, x4] + rho3[x1, x2, x3, x5] - rho3[x1, x2, x4, x5]
			+ rho3[x1, x3, x4, x5] - rho3[x1 + x2, x3, x4, x5] + rho3[x1 + x3, x2, x4, x5]
			- rho3[x1 + x4, x2, x3, x5] + rho3[x1 + x5, x2, x3, x4]</nowiki>}}

			{{InOut\|n=4\|in=<nowiki>BAroundB[x1, x2, x3, x4, x5] /. d2 /. d1</nowiki>\|out=<nowiki>0</nowiki>}}

Drorbn: /* The "B around B" Syzygy */

2006-11-19T16:06:18Z

The "B around B" Syzygy

← Older revision		Revision as of 12:06, 19 November 2006
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	====The "B around B" Syzygy====		====The "B around B" Syzygy====

	[[Image:~~BonBSyzygy~~.svg\|center~~\|600px~~]]		[[Image:06-1350-BAroundB.svg\|center]]

	===A Mathematica Verification===		===A Mathematica Verification===

Drorbn at 23:44, 18 November 2006

2006-11-18T23:44:11Z

← Older revision		Revision as of 19:44, 18 November 2006
Line 44:		Line 44:

	===The Syzygies===		===The Syzygies===

			====The "B around B" Syzygy====

			[[Image:BonBSyzygy.svg\|center\|600px]]

	===A Mathematica Verification===		===A Mathematica Verification===