HoriAsso - Documentation - Revision history

Karenechu: /* 1. Scalar multiplication and Addition */

2006-08-17T19:53:45Z

1. Scalar multiplication and Addition

← Older revision		Revision as of 15:53, 17 August 2006
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	At the moment the coefficients for the FLn elements need to be Integers. This will be replaced with rational numbers soon.		At the moment the coefficients for the FLn elements need to be Integers. This will be replaced with rational numbers soon.

	~~Sum of~~ Linear combination of ~~Word~~:		Linear combination of Words:
	{{InOut\|n=13\|in =		{{InOut\|n=13\|in =
	LinCombOfWords = <nowiki> 2 RComb[a, b, a, b] + LW[a, b, b, a, b, b, b]		LinCombOfWords = <nowiki> 2 RComb[a, b, a, b] + LW[a, b, b, a, b, b, b]
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	Sum of Free Lie Algebra elements (a linear combination of words ~~wraps~~ ~~around~~		Sum of Free Lie Algebra elements (linear combination of words wrapped with the FLn head){{InOut\|n=14\|in =
			SumOfFL3Elements = <nowiki>FL3[ -4 LW[a, b, a, b, a, b, b]]
	{{InOut\|n=14\|in =
	~~SumOfFL3Elements~~ = ~~<nowiki>FL3[~~ -4 ~~LW[a,~~ b, a, b, a, b, ~~b]]~~ + FL3[2LW[a, b, b, a, b, b, b]		+ FL3[2LW[a, b, b, a, b, b, b] +RComb[a, b, a, b]]</nowiki>\|
	+RComb[a, b, a, b]]</nowiki>\|
	out=FL3[-LW[a, a, b, b] - 4 LW[a, b, a, b, a, b, b] + 2 LW[a, b, b, a, b, b, b]]}}		out=FL3[-LW[a, a, b, b] - 4 LW[a, b, a, b, a, b, b] + 2 LW[a, b, b, a, b, b, b]]}}


	Note that the FLn elements itself is not automatically ~~reduce~~ to the Lyndon basis form, but a sum of the FLn elements is.		Note that the FLn elements itself is not automatically reduced to the Lyndon basis form, but a sum of the FLn elements is.


	=====2. Reduce to linear combination of Lyndon basis elements=====		=====2. Reduce to linear combination of Lyndon basis elements=====

Karenechu: /* Notation */

2006-08-17T19:52:05Z

Notation

← Older revision		Revision as of 15:52, 17 August 2006
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	:You can check to see if a word satisfies the definition of a Lyndon Word:		:You can check to see if a word satisfies the definition of a Lyndon Word:
	{{InOut\|n=11\|in = <nowiki>IsLyndon[LW[a,b,a]</nowiki>\|out=False}}		{{InOut\|n=11\|in = <nowiki>IsLyndon[LW[a,b,a]]</nowiki>\|out=False}}

	:'''''Important note''''': In ''Java'', you could replace the left or right subtree of a Lie tree by any tree, resulting in tree structures that are neither right-combs or Lyndon. However, this Mathematica interface assumes that if a Lie tree is not a Lyndon Word, it would be a right-comb; using those Java methods (namely ''javatree@setLeft(or right)Subtree(new_left(or_right)_subtree))'' directly in ''Mathematica'' will cause errors in calculations.		:'''''Important note''''': In ''Java'', you could replace the left or right subtree of a Lie tree by any tree, resulting in tree structures that are neither right-combs or Lyndon. However, this Mathematica interface assumes that if a Lie tree is not a Lyndon Word, it would be a right-comb; using those Java methods (namely ''javatree@setLeft(or right)Subtree(new_left(or_right)_subtree))'' directly in ''Mathematica'' will cause errors in calculations.

128.100.68.131: /* 2. Reduce to linear combination of Lyndon basis elements */

2006-08-17T16:51:04Z

2. Reduce to linear combination of Lyndon basis elements

← Older revision		Revision as of 12:51, 17 August 2006
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	For reducing a word to Lyndon form, the word length is only limited by performance. For a good size example:		For reducing a word to Lyndon form, the word length is only limited by performance. For a good size example:
	{{InOut\|n=15\|in = <nowiki> ReduceToLyndon[RComb[a, b, b, a, a, b, a, b, a, b, b, a, b]]</nowiki>\|		{{InOut\|n=15\|in = <nowiki> ReduceToLyndon[RComb[a, b, b, a, a, b, a, b, a, b, b, a, b]]</nowiki>\|
	out = <nowiki> LW[a, a, a, a, a, a, b, b, b, b, b, b, b] + 4 LW[a, a, a, a, a, b, a, b, b, b, b, b, b]		out = <nowiki> LW[a, a, a, a, a, a, b, b, b, b, b, b, b] + 4 LW[a, a, a, a, a, b, a, b, b, b, b, b, b]
	+ 6 LW[a, a, a, a, a, b, b, a, b, b, b, b, b] + 3 LW[a, a, a, a, a, b, b, b, a, b, b, b, b]		+ 6 LW[a, a, a, a, a, b, b, a, b, b, b, b, b] + 3 LW[a, a, a, a, a, b, b, b, a, b, b, b, b]
	- LW[a, a, a, a, a, b, b, b, b, a, b, b, b] - 5 LW[a, a, a, a, a, b, b, b, b, b, a, b, b]		- LW[a, a, a, a, a, b, b, b, b, a, b, b, b] - 5 LW[a, a, a, a, a, b, b, b, b, b, a, b, b]

128.100.68.131: /* 2. Reduce to linear combination of Lyndon basis elements */

2006-08-17T16:50:44Z

2. Reduce to linear combination of Lyndon basis elements

← Older revision		Revision as of 12:50, 17 August 2006
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	For reducing a word to Lyndon form, the word length is only limited by performance. For a good size example:		For reducing a word to Lyndon form, the word length is only limited by performance. For a good size example:
	{{InOut\|n=15\|in = <nowiki> ReduceToLyndon[RComb[a, b, b, a, a, b, a, b, a, b, b, a, b]]</nowiki>\|		{{InOut\|n=15\|in = <nowiki> ReduceToLyndon[RComb[a, b, b, a, a, b, a, b, a, b, b, a, b]]</nowiki>\|
	out = <nowiki> LW[a, a, a, a, a, a, b, b, b, b, b, b, b] + 4 LW[a, a, a, a, a, b, a, b, b, b, b, b, b]		out = <nowiki> LW[a, a, a, a, a, a, b, b, b, b, b, b, b] + 4 LW[a, a, a, a, a, b, a, b, b, b, b, b, b]
	+ 6 LW[a, a, a, a, a, b, b, a, b, b, b, b, b] + 3 LW[a, a, a, a, a, b, b, b, a, b, b, b, b]		+ 6 LW[a, a, a, a, a, b, b, a, b, b, b, b, b] + 3 LW[a, a, a, a, a, b, b, b, a, b, b, b, b]
	- LW[a, a, a, a, a, b, b, b, b, a, b, b, b] - 5 LW[a, a, a, a, a, b, b, b, b, b, a, b, b]		- LW[a, a, a, a, a, b, b, b, b, a, b, b, b] - 5 LW[a, a, a, a, a, b, b, b, b, b, a, b, b]
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	''Note'': the two notations will probably merge.		''Note'': the two notations will probably merge.

	=====3. Lie Brackets=====		=====3. Lie Brackets=====

128.100.68.131: /* Java Class Documentation (not ready) */

2006-08-17T16:44:15Z

Java Class Documentation (not ready)

← Older revision		Revision as of 12:44, 17 August 2006
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	==[[Installation and Setup for using Java in Mathematica]]==		==[[Installation and Setup for using Java in Mathematica]]==

	==[[Java Class Documentation]] (not ~~ready~~)==		==[[Java Class Documentation]] (not formatted yet)==

	==How to use the Free Lie Algebra (first part in the HoriAsso) package ==		==How to use the Free Lie Algebra (first part in the HoriAsso) package ==

128.100.68.131: /* Java Class Documentation */

2006-08-17T16:43:53Z

Java Class Documentation

← Older revision		Revision as of 12:43, 17 August 2006
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	==[[Installation and Setup for using Java in Mathematica]]==		==[[Installation and Setup for using Java in Mathematica]]==

	==[[Java Class Documentation]]==		==[[Java Class Documentation]] (not ready)==

	==How to use the Free Lie Algebra (first part in the HoriAsso) package ==		==How to use the Free Lie Algebra (first part in the HoriAsso) package ==

Karenechu: /* 3. Lie Brackets */

2006-08-17T16:43:02Z

3. Lie Brackets

← Older revision		Revision as of 12:43, 17 August 2006
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	{{InOut\|n=18\| in=<nowiki> Bracket[FL4[LW[a, b, c, a, c] + 2RComb[a, b, c, b]],		{{InOut\|n=18\| in=<nowiki> Bracket[ FL4[LW[a, b, c, a, c] + 2RComb[a, b, c, b]],
	FL4[-2RComb[a, b, b, c, a, b] + 4RComb[b, a, c]]]</nowiki>\|		FL4[-2RComb[a, b, b, c, a, b] + 4RComb[b, a, c]]]</nowiki>\|
	out=<nowiki> FL4[8 LW[a, b, b, c, a, c, b] - 4 LW[a, b, c, a, c, a, c, b]		out=<nowiki> FL4[ 8 LW[a, b, b, c, a, c, b] - 4 LW[a, b, c, a, c, a, c, b]
	- 4 LW[a, b, c, a, c, b, a, c] + 4 LW[a, a, b, c, b, b, a, b, b, c]		- 4 LW[a, b, c, a, c, b, a, c] + 4 LW[a, a, b, c, b, b, a, b, b, c]
	+ 4 LW[a, a, c, b, b, b, a, b, b, c] - 2 LW[a, a, b, c, b, b, a, b, c, a, c]		+ 4 LW[a, a, c, b, b, b, a, b, b, c] - 2 LW[a, a, b, c, b, b, a, b, c, a, c]

Karenechu: /* Notation */

2006-08-17T16:41:21Z

Notation

← Older revision		Revision as of 12:41, 17 August 2006
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	2. Only two tree structures are used and allowed, the right Comb, and the Lyndon Word.		2. Only two tree structures are used and allowed, the right Comb, and the Lyndon Word.

	:Right-Combs represent the tree structure [a,[a,[a,[a,[a,[a,[a,[a...[a,b]]]]]]]]] and is represented by:		:Right-Combs represent the tree structure [a,[a,[a,[a,[a,[a,[a,[a...[a,b]]]]]]]]] and is represented, for example, by:
	{{In\|n=9\|in=<nowiki>RCombExample = RComb[a,a,a,a,a,a,a,a,...a,b] </nowiki>}}		{{In\|n=9\|in=<nowiki>RCombExample = RComb[a,a,a,a,a,a,a,a,...a,b] </nowiki>}}

	:Lyndon Words are words that are smaller by alphabetical order than any of its right subwords. Since they are Hall Words, ~~and~~ each word implies a unique tree structure. They are represented, for example, as		:Lyndon Words are words that are smaller by alphabetical order than any of its right subwords. Since they are Hall Words, each word implies a unique tree structure. They are represented, for example, by
	{{In\|n=10\|in=<nowiki>LyndonWordExample = LW[a,a,b,a,b,a,b,a,b] </nowiki>}}		{{In\|n=10\|in=<nowiki>LyndonWordExample = LW[a,a,b,a,b,a,b,a,b] </nowiki>}}

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	{{InOut\|n=11\|in = <nowiki>IsLyndon[LW[a,b,a]</nowiki>\|out=False}}		{{InOut\|n=11\|in = <nowiki>IsLyndon[LW[a,b,a]</nowiki>\|out=False}}

	:''Important note'': In ''Java'', you could replace the left or right subtree of a Lie tree by any tree, resulting in tree structures that are neither right-combs or Lyndon. However, this Mathematica interface assumes that if a Lie tree is not a Lyndon Word, it would be a right-comb; using those Java methods (namely ''javatree@setLeft(or right)Subtree(new_left(or_right)_subtree))'' directly in ''Mathematica'' will cause errors in calculations.		:'''''Important note''''': In ''Java'', you could replace the left or right subtree of a Lie tree by any tree, resulting in tree structures that are neither right-combs or Lyndon. However, this Mathematica interface assumes that if a Lie tree is not a Lyndon Word, it would be a right-comb; using those Java methods (namely ''javatree@setLeft(or right)Subtree(new_left(or_right)_subtree))'' directly in ''Mathematica'' will cause errors in calculations.

Karenechu: /* Notation */

2006-08-17T16:39:26Z

Notation

← Older revision		Revision as of 12:39, 17 August 2006
Line 72:		Line 72:
	2. Only two tree structures are used and allowed, the right Comb, and the Lyndon Word.		2. Only two tree structures are used and allowed, the right Comb, and the Lyndon Word.

	:Right-Combs represent the tree structure [a,[a,[a,[a,[a,[a,[a,[a...[a,b]]]]]]]]]~~...~~ is represented ~~in our setup, for example, as~~		:Right-Combs represent the tree structure [a,[a,[a,[a,[a,[a,[a,[a...[a,b]]]]]]]]] and is represented by:
	{{In\|n=9\|in=<nowiki>RCombExample = RComb[a,a,a,a,a,a,a,a,...a,b] </nowiki>}}		{{In\|n=9\|in=<nowiki>RCombExample = RComb[a,a,a,a,a,a,a,a,...a,b] </nowiki>}}

Karenechu: /* Demo ''Mathematica'' NoteBook */

2006-08-17T16:37:30Z

Demo ''Mathematica'' NoteBook

← Older revision		Revision as of 12:37, 17 August 2006
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	===Demo ''Mathematica'' NoteBook===		===Demo ''Mathematica'' NoteBook===

	A ''Mathematica'' notebook file named '''''HoriAssoFLnSpaceDemo.nb''''' to demonstrate the usage of this Free Lie Algebra package is included in the '''''FreeLieAlgebra''''' folder that you just downloaded from the [http://katlas.math.toronto.edu/svn/HoriAsso/trunk/FreeLieAlgebra repository]. After installation (as above), ~~simple~~ open the notebook in ''Mathematica'', follow the simple initialization steps below, (i.e., modifying path names and parameters in the demo file), and evaluate the cells as you wish.		A ''Mathematica'' notebook file named '''''HoriAssoFLnSpaceDemo.nb''''' to demonstrate the usage of this Free Lie Algebra package is included in the '''''FreeLieAlgebra''''' folder that you just downloaded from the [http://katlas.math.toronto.edu/svn/HoriAsso/trunk/FreeLieAlgebra repository]. After installation (as above), simply open the notebook in ''Mathematica'', follow the simple initialization steps below, (i.e., modifying path names and parameters in the demo file), and evaluate the cells as you wish.

	===Usage===		===Usage===