Nowhere-zero flow - Revision history

en>(don't talk secrets): /* Definition */

2015-01-07T14:17:10Z

Definition

← Older revision		Revision as of 16:17, 7 January 2015
Line 1:		Line 1:
	~~In mathematics~~, ~~the word~~ '''''undefined''''' has several different meanings, depending on the context. In [[geometry]], simple words such as "point" and "line" are taken as undefined terms. In [[arithmetic]], some arithmetic operations are called "undefined". The most famous example is that [[division by zero]] is undefined. In [[algebra]], a [[function (mathematics)\|function]] is said to be "undefined" at ~~points not in its domain~~. ~~For example, in the [[real number]] system, <math> f(x)=\sqrt{x} </math>~~ is ~~undefined~~ for ~~negative <math>x</math>.~~		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is what she does for a residing. I am truly fond of handwriting but I can't make it my profession really. Alaska is exactly where he's usually been residing.<br><br>Also visit my webpage ... psychic phone ([http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ try what she says])

	~~==Undefined terms in geometry==~~

	~~In ancient times, geometers attempted to define every term. For example, [[Euclid of Alexandria\|Euclid]] defined~~ a ~~point as "that which has no part"~~. In modern times, mathematicians recognized that attempting to define every word inevitably led to [[circular definition]]s, and in geometry left some words, "point" for example, as undefined. See [[primitive notion]].

	~~==Undefined operations in arithmetic==~~
	~~The reasoning behind leaving [[division by zero]] undefined is as follows. Division is the inverse~~ of multiplication. If <math> a\div b=c</math>, then <math>b\times c=a</math>. But if <math> b=0 </math>, then any multiple of <math> b </math> is also <math> 0 </math>, and so if <math> a\ne 0 </math>, no such <math> c </math> exists. On the other hand, if <math> a </math> and <math> b </math> are both zero, then every real number <math> c </math> satisfies <math> b\times c=a </math>. Either way, it ~~is impossible to assign a particular real number to the quotient when the divisor is zero~~.

	~~In calculus, <math> 0/0 </math>~~ is ~~sometimes used as a symbol, and is called an [[indeterminate form]], but the symbol does not represent division in the sense the word is used in ordinary arithmetic~~.

	~~Another common operation that is undefined is that of raising zero to the zero power. On the one hand, if <math> x\ne 0 </math>, then~~ <~~math~~> ~~x^{0}=1~~<~~/math~~>. ~~On the other hand, if <math> y </math> is any positive number, <math> 0^{y}=0 </math>, while if <math> y </math> is negative, <math> 0^y </math> leads to division by zero, which is undefined~~. ~~Thus, to make the [[laws of exponents]] work in every case where exponents are defined, <math>0^0</math> is left undefined~~. ~~That said, there are branches of higher mathematics where various definitions of zero to the zero power are given~~ (~~see~~: ~~[[Exponentiation]]).~~

	~~==Values for which functions are undefined==~~
	The set of numbers for which a [[function (mathematics)\|function]] is defined is called the ''domain'' of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are <math> f(x)=\frac{1}{x}</math> which is undefined for <math>x=0</math>, and <math> f(x)=\sqrt{x}</~~math>, which is undefined (in the real number system) for negative <math> x <~~/~~math>~~.

	~~==Notation using ↓ and ↑==~~
	In [[computability theory (computer science)\|computability theory]], if ''f'' is a [[partial function]] on ''S'' and ''a'' is an element of ''S'', then this is written as ''f''(''a'')↓ and is read "''f''(''a'') is ''defined''."

	~~If ''a'' is not in the domain of ''f'', then ''f''(''a'')↑ is written and is read as "''f''(''a'') is ''undefined''"~~.

	~~==The symbols of infinity==~~

	~~In [[analysis]], [[measure theory]], and other mathematical disciplines, the symbol <math>\infty<~~/~~math> is frequently used to denote an infinite pseudo~~-~~number, in real analysis alongside with its negative <math>~~ -~~\infty</math>. The symbol has no well~~-~~defined meaning by itself, but an expression like <math>\left\{a_n\right\}\rightarrow\infty</math> is a shorthand~~ for ~~a [[divergent sequence]] which is eventually larger than any given real number.~~

	~~Arithmetic with the symbols <math>\pm\infty</math> is undefined. The following conventions of addition and multiplication are in common use:~~
	* <math>x+\infty=\infty</math>   <math>\forall x\in\mathbb{R}\cup\{\infty\};-~~\infty+x=-\infty</math>   <math>\forall x\in\mathbb{R}\cup\{~~-~~\infty\}</math>.~~
	* <math>x\cdot\infty=\infty</math>   <math>\forall x\in\mathbb{R}^{+}</math>.

	~~No sensible extension of addition and multiplication with <math>\infty</math> exist in the following cases:~~
	* <math>\infty-~~\infty<~~/~~math>~~
	* <math>0\cdot\infty</math> (although in [[measure theory]~~], this is often defined as <math>0</math>~~)
	* <math>\frac{\infty}{\infty}</math>

	~~See [[extended real number line]] for more information.~~

	~~==Singularities in complex analysis==~~

	In [[complex analysis]], a point <math>z\in\mathbb{C}</math> where a [[holomorphic function]] is undefined is called a [[Mathematical singularity\|singularity]]. One distinguishes between [[removable singularity\|removable singularities]] (the function can be extended holomorphically to <math>z</math>, [[Pole (complex analysis)\|poles]] (the function can be extended [[meromorphic function\|meromorphically]] to <math>z</math>), and [[essential singularity\|essential singularities]], where no meromorphic extension to <math>z</math> exists.

	~~==References==~~

	* James R. Smart, ''Modern Geometries'' Third Edition, Brooks/Cole, 1988, ISBN 0-534-08310-2

	~~[[Category:Mathematical terminology]]~~
	~~[[Category:Calculus]]~~

en>Bender2k14: /* Definition */ added a related definition

2013-06-18T15:55:56Z

Definition: added a related definition

← Older revision		Revision as of 17:55, 18 June 2013
Line 1:		Line 1:
	~~Hello~~ and ~~welcome~~. ~~My title~~ is ~~Ling~~. ~~Bookkeeping~~ is ~~what she does~~. The ~~preferred pastime for him~~ and ~~his children~~ is to ~~play badminton~~ but he is ~~having difficulties~~ to ~~find time~~ for it. ~~I presently live~~ in ~~Alabama~~.<br><br>~~my web blog~~ - [~~http~~://~~Phoenixbiofuel~~.~~com~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/60/~~userId~~/~~184800~~/~~Default~~.~~aspx Phoenixbiofuel~~.~~com~~]		In mathematics, the word '''''undefined''''' has several different meanings, depending on the context. In [[geometry]], simple words such as "point" and "line" are taken as undefined terms. In [[arithmetic]], some arithmetic operations are called "undefined". The most famous example is that [[division by zero]] is undefined. In [[algebra]], a [[function (mathematics)\|function]] is said to be "undefined" at points not in its domain. For example, in the [[real number]] system, <math> f(x)=\sqrt{x} </math> is undefined for negative <math>x</math>.

			==Undefined terms in geometry==

			In ancient times, geometers attempted to define every term. For example, [[Euclid of Alexandria\|Euclid]] defined a point as "that which has no part". In modern times, mathematicians recognized that attempting to define every word inevitably led to [[circular definition]]s, and in geometry left some words, "point" for example, as undefined. See [[primitive notion]].

			==Undefined operations in arithmetic==
			The reasoning behind leaving [[division by zero]] undefined is as follows. Division is the inverse of multiplication. If <math> a\div b=c</math>, then <math>b\times c=a</math>. But if <math> b=0 </math>, then any multiple of <math> b </math> is also <math> 0 </math>, and so if <math> a\ne 0 </math>, no such <math> c </math> exists. On the other hand, if <math> a </math> and <math> b </math> are both zero, then every real number <math> c </math> satisfies <math> b\times c=a </math>. Either way, it is impossible to assign a particular real number to the quotient when the divisor is zero.

			In calculus, <math> 0/0 </math> is sometimes used as a symbol, and is called an [[indeterminate form]], but the symbol does not represent division in the sense the word is used in ordinary arithmetic.

			Another common operation that is undefined is that of raising zero to the zero power. On the one hand, if <math> x\ne 0 </math>, then <math> x^{0}=1</math>. On the other hand, if <math> y </math> is any positive number, <math> 0^{y}=0 </math>, while if <math> y </math> is negative, <math> 0^y </math> leads to division by zero, which is undefined. Thus, to make the [[laws of exponents]] work in every case where exponents are defined, <math>0^0</math> is left undefined. That said, there are branches of higher mathematics where various definitions of zero to the zero power are given (see: [[Exponentiation]]).

			==Values for which functions are undefined==
			The set of numbers for which a [[function (mathematics)\|function]] is defined is called the ''domain'' of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are <math> f(x)=\frac{1}{x}</math> which is undefined for <math>x=0</math>, and <math> f(x)=\sqrt{x}</math>, which is undefined (in the real number system) for negative <math> x </math>.

			==Notation using ↓ and ↑==
			In [[computability theory (computer science)\|computability theory]], if ''f'' is a [[partial function]] on ''S'' and ''a'' is an element of ''S'', then this is written as ''f''(''a'')↓ and is read "''f''(''a'') is ''defined''."

			If ''a'' is not in the domain of ''f'', then ''f''(''a'')↑ is written and is read as "''f''(''a'') is ''undefined''".

			==The symbols of infinity==

			In [[analysis]], [[measure theory]], and other mathematical disciplines, the symbol <math>\infty</math> is frequently used to denote an infinite pseudo-number, in real analysis alongside with its negative <math> -\infty</math>. The symbol has no well-defined meaning by itself, but an expression like <math>\left\{a_n\right\}\rightarrow\infty</math> is a shorthand for a [[divergent sequence]] which is eventually larger than any given real number.

			Arithmetic with the symbols <math>\pm\infty</math> is undefined. The following conventions of addition and multiplication are in common use:
			* <math>x+\infty=\infty</math>   <math>\forall x\in\mathbb{R}\cup\{\infty\};-\infty+x=-\infty</math>   <math>\forall x\in\mathbb{R}\cup\{-\infty\}</math>.
			* <math>x\cdot\infty=\infty</math>   <math>\forall x\in\mathbb{R}^{+}</math>.

			No sensible extension of addition and multiplication with <math>\infty</math> exist in the following cases:
			* <math>\infty-\infty</math>
			* <math>0\cdot\infty</math> (although in [[measure theory]], this is often defined as <math>0</math>)
			* <math>\frac{\infty}{\infty}</math>

			See [[extended real number line]] for more information.

			==Singularities in complex analysis==

			In [[complex analysis]], a point <math>z\in\mathbb{C}</math> where a [[holomorphic function]] is undefined is called a [[Mathematical singularity\|singularity]]. One distinguishes between [[removable singularity\|removable singularities]] (the function can be extended holomorphically to <math>z</math>, [[Pole (complex analysis)\|poles]] (the function can be extended [[meromorphic function\|meromorphically]] to <math>z</math>), and [[essential singularity\|essential singularities]], where no meromorphic extension to <math>z</math> exists.

			==References==

			* James R. Smart, ''Modern Geometries'' Third Edition, Brooks/Cole, 1988, ISBN 0-534-08310-2

			[[Category:Mathematical terminology]]
			[[Category:Calculus]]

en>Scotthue: /* Flow/coloring duality / aesthetics of G

2012-05-02T02:44:08Z

Flow/coloring duality: aesthetics of G*

New page

Hello and welcome. My title is Ling. Bookkeeping is what she does. The preferred pastime for him and his children is to play badminton but he is having difficulties to find time for it. I presently live in Alabama.<br><br>my web blog - [http://Phoenixbiofuel.com/ActivityFeed/MyProfile/tabid/60/userId/184800/Default.aspx Phoenixbiofuel.com]

Nowhere-zero flow - Revision history

en>(don't talk secrets): /* Definition */

en>Bender2k14: /* Definition */ added a related definition

en>Scotthue: /* Flow/coloring duality */ aesthetics of G*

en>Scotthue: /* Flow/coloring duality / aesthetics of G