Moyal bracket - Revision history

en>Cuzkatzimhut: /* top */

2014-06-27T18:12:59Z

top

← Older revision		Revision as of 20:12, 27 June 2014
Line 1:		Line 1:
	~~{{no footnotes\|date=May 2012}}~~		The author is called Irwin Wunder but it's not the most masucline title out there. To gather coins is one of the issues I love most. Minnesota is where he's been residing for many years. Hiring is my profession.<br><br>Look at my web page [http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/ http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/]
	~~In [[mathematics]], '''Cutler's bar notation''' is a [[Mathematical notation\|notation system]] for [[large number]]s, introduced by Mark Cutler in 2004.~~ The ~~idea~~ is ~~based on [[iterated exponentiation]] in much the same way that [[exponentiation]] is [[iteration\|iterated]] [[multiplication]].~~

	~~==Introduction==~~
	~~A regular [[exponentiation\|exponential]] can be expressed as such:~~

	~~:<math>~~
	~~\begin{matrix}~~
	~~a^b & = & \underbrace{a_{} \times a \times\dots \times a} \\~~
	~~& & b\mbox{ copies of }a~~
	~~\end{matrix}~~
	~~</math>~~

	~~However, these expressions become arbitrarily large when dealing with systems such as [[Knuth~~'s ~~up-arrow notation]]. Take~~ the ~~following:~~

	~~:<math>~~
	~~\begin{matrix}~~
	~~& \underbrace{a_{}^{a^{{}^{.\,^{.\,^{~~.~~\,^a}}}}}} &~~
	\\
	~~& b\mbox{ copies of }a~~
	~~\end{matrix}~~
	~~</math>~~

	~~Cutler's bar notation shifts these exponentials counterclockwise, forming <math>{^b} \bar a</math>. A bar~~ is ~~placed above the variable to denote this change. As such:~~

	~~:<math>~~
	~~\begin{matrix}~~
	~~{^b} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &~~
	\\
	~~& b\mbox{ copies of }a~~
	~~\end{matrix}~~
	~~</math>~~

	~~This system becomes effective with multiple exponent, when regular denotation becomes too cumbersome.~~

	~~:<math>~~
	~~\begin{matrix}~~
	~~^{^b{b}} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &~~
	\\
	~~& {{^b} \bar a}\mbox{ copies~~ of }a
	~~\end{matrix}~~
	~~</math>~~

	~~At any time, this can be further shortened by rotating~~ the ~~exponential counter-clockwise once more~~.

	~~:<math>~~
	~~\begin{matrix}~~
	~~\underbrace{b_{}^{b^{{}^{.\,^{.\,^{.\,^b}}}}}} \bar a = {_c} \bar a~~
	\\
	~~c \mbox{ copies of } b~~
	~~\end{matrix}~~
	~~</math>~~

	~~The same pattern could be iterated a fourth time, becoming <math>\bar a_{d}</math>. For this reason, it~~ is ~~sometimes referred to as '''Cutler~~'s ~~circular notation'''.~~

	~~==Advantages and drawbacks==~~

	~~The Cutler Bar Notation can be used to easily express other notation systems in exponent form. It also allows~~ for ~~a flexible summarisation of multiple copies of the same exponents, where any number of stacked exponents can be shfted counter-clockwise and shortened to a single variable~~. ~~The Bar Notation also allows for farily rapid composure of very large numbers~~. ~~For instance, the number~~ <~~math~~>~~\bar {10}_{10}~~<~~/math~~> ~~would contain more than a~~ [~~[googolplex]] digits, whilst remaining fairly simple to write with and remember~~.

	~~However, the system reaches a problem when dealing with different exponents in a single expression~~. ~~For instance, the expression <math> ^{a^{b^{b^{c}}}}<~~/~~math> could not be summarised in Bar notation. Additionally, the exponent can only be shifted thrice before it returns~~ to ~~its original position, making a five degree shift indistinguishable from a one degree shift. Some have suggested using a double and triple bar in subsequent rotations, though this presents problems~~ when ~~dealing~~ with ~~ten and twenty degree shifts.~~

	~~==See also==~~
	*[[Mathematical notation]]

	~~==References==~~

	*Mark Cutler, ''Physical Infinity'', 2004
	*[[Daniel Geisler]], ''[http://www.~~tetration~~.~~org~~/ ~~tetration.org]''~~
	*R. Knobel. "Exponentials Reiterated." ''[[American Mathematical Monthly]]'' '''88''', (1981)

	~~[[Category:Mathematical notation]]~~
	~~[[Category:Large numbers]~~]

en>Felix116: Moved link for Dirac higher up the page.

2013-10-08T21:15:38Z

Moved link for Dirac higher up the page.

← Older revision		Revision as of 23:15, 8 October 2013
Line 1:		Line 1:
	~~Wilber Berryhill~~ is the ~~name his parents gave him and he totally digs~~ that ~~name~~. ~~Ohio~~ is where ~~his home is~~ and ~~his family members loves it~~. The ~~preferred hobby~~ for ~~him and his kids is~~ to ~~perform lacross~~ and ~~he'll be starting something else alongside~~ with ~~it. Office supervising is what she does for~~ a ~~residing~~.<br><br>~~My website :: accurate psychic predictions (~~[http://~~jplusfn~~.~~gaplus~~.kr/~~xe/qna/78647 find out this here~~])		{{no footnotes\|date=May 2012}}
			In [[mathematics]], '''Cutler's bar notation''' is a [[Mathematical notation\|notation system]] for [[large number]]s, introduced by Mark Cutler in 2004. The idea is based on [[iterated exponentiation]] in much the same way that [[exponentiation]] is [[iteration\|iterated]] [[multiplication]].

			==Introduction==
			A regular [[exponentiation\|exponential]] can be expressed as such:

			:<math>
			\begin{matrix}
			a^b & = & \underbrace{a_{} \times a \times\dots \times a} \\
			& & b\mbox{ copies of }a
			\end{matrix}
			</math>

			However, these expressions become arbitrarily large when dealing with systems such as [[Knuth's up-arrow notation]]. Take the following:

			:<math>
			\begin{matrix}
			& \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
			\\
			& b\mbox{ copies of }a
			\end{matrix}
			</math>

			Cutler's bar notation shifts these exponentials counterclockwise, forming <math>{^b} \bar a</math>. A bar is placed above the variable to denote this change. As such:

			:<math>
			\begin{matrix}
			{^b} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
			\\
			& b\mbox{ copies of }a
			\end{matrix}
			</math>

			This system becomes effective with multiple exponent, when regular denotation becomes too cumbersome.

			:<math>
			\begin{matrix}
			^{^b{b}} \bar a = & \underbrace{a_{}^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
			\\
			& {{^b} \bar a}\mbox{ copies of }a
			\end{matrix}
			</math>

			At any time, this can be further shortened by rotating the exponential counter-clockwise once more.

			:<math>
			\begin{matrix}
			\underbrace{b_{}^{b^{{}^{.\,^{.\,^{.\,^b}}}}}} \bar a = {_c} \bar a
			\\
			c \mbox{ copies of } b
			\end{matrix}
			</math>

			The same pattern could be iterated a fourth time, becoming <math>\bar a_{d}</math>. For this reason, it is sometimes referred to as '''Cutler's circular notation'''.

			==Advantages and drawbacks==

			The Cutler Bar Notation can be used to easily express other notation systems in exponent form. It also allows for a flexible summarisation of multiple copies of the same exponents, where any number of stacked exponents can be shfted counter-clockwise and shortened to a single variable. The Bar Notation also allows for farily rapid composure of very large numbers. For instance, the number <math>\bar {10}_{10}</math> would contain more than a [[googolplex]] digits, whilst remaining fairly simple to write with and remember.

			However, the system reaches a problem when dealing with different exponents in a single expression. For instance, the expression <math> ^{a^{b^{b^{c}}}}</math> could not be summarised in Bar notation. Additionally, the exponent can only be shifted thrice before it returns to its original position, making a five degree shift indistinguishable from a one degree shift. Some have suggested using a double and triple bar in subsequent rotations, though this presents problems when dealing with ten and twenty degree shifts.

			==See also==
			*[[Mathematical notation]]

			==References==

			*Mark Cutler, ''Physical Infinity'', 2004
			*[[Daniel Geisler]], ''[http://www.tetration.org/ tetration.org]''
			*R. Knobel. "Exponentials Reiterated." ''[[American Mathematical Monthly]]'' '''88''', (1981)

			[[Category:Mathematical notation]]
			[[Category:Large numbers]]

en>Cuzkatzimhut at 14:19, 27 August 2012

2012-08-27T14:19:02Z

New page

Wilber Berryhill is the name his parents gave him and he totally digs that name. Ohio is where his home is and his family members loves it. The preferred hobby for him and his kids is to perform lacross and he'll be starting something else alongside with it. Office supervising is what she does for a residing.<br><br>My website :: accurate psychic predictions ([http://jplusfn.gaplus.kr/xe/qna/78647 find out this here])