Differential operator - Revision history

en>Enyokoyama: /* Definition */ resolve ambiguity, distribution->distribution (mathematics)

2015-01-08T10:12:13Z

Definition: resolve ambiguity, distribution->distribution (mathematics)

← Older revision		Revision as of 12:12, 8 January 2015
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en>Wavelength: inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

2014-02-27T17:35:22Z

inserting 1 hyphen: —> "three-dimensional"—wikt:three-dimensional

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	~~In [[abstract algebra]],~~ a branch of [[mathematics]], an '''Archimedean group''' is an [[algebraic structure]] consisting of a [[Set (mathematics)\|set]] together with a [[binary operation]] and [[binary relation]] satisfying certain axioms detailed below. We can ~~also say that an Archimedean group is a [[linearly ordered group]]~~ for ~~which the [[Archimedean property]] holds~~. ~~For example~~, the ~~set '''R'''~~ of ~~[[real number]]s together with~~ the ~~operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after [[Archimedes]]~~.

	~~==Definition==~~
	~~In the subsequent, we use the notation~~ <~~math~~>na</~~math> (where <math>n<~~/~~math> is in the set '''N'''~~ of ~~[[natural number~~]~~]s) for the sum of ''~~a'' with ~~itself ''n'' times~~.

	~~An '''Archimedean group''' (''G''~~, ~~+, ≤) is a [[linearly ordered group]] subject~~ to ~~the following condition:~~

	~~for any ''a'' and ''b''~~ in ~~''G''~~ which are ~~greater than ''0'',~~ the ~~inequality ''na'' ≤ ''b'' holding for every ''n'' in '''N''' implies ''a'' = 0~~.

	~~==Examples of Archimedean groups==~~
	The ~~sets of the [[integer]]s~~, the [~~[rational number]]~~s~~, the [[real number~~]~~]s, together with~~ the ~~operation~~ of ~~addition and the usual ordering (≤), are Archimedean groups~~.

	~~==Examples~~ of ~~non-Archimedean groups==~~
	~~An ordered group (''G''~~, +, ~~≤) defined~~ as ~~follows~~ is not ~~Archimedean:~~
	* ''G'' = '''R''' × '''R'''.
	* Let ''a~~'' = (''u''~~, ~~''v'') and ''b'' = (''x''~~, ~~''y'') then ''~~a~~'' + ''b'' = (''u'' + ''x'', ''v'' + ''y'')~~
	* ''a'' ≤ ''b'' [[iff]] ''v'' < ''y'' or (''v'' = ''y'' and ''u'' ≤ ''x'') ([[lexicographical order]] with the least-significant ~~number on~~ the ~~left)~~.

	~~'''Proof:''' Consider~~ the ~~elements (1~~, 0) and (0, ~~1). For all ''n''~~ in ~~'''N''' one evidently has ''n'' (1~~, ~~0) < (0, 1).~~

	~~For another example, see [[p~~-~~adic number]]~~.

	~~==Theorems==~~
	~~For each ''~~a'', '~~'b'' in ''G'' there exist ''m''~~, ~~''n''~~ in ~~'''N''' such~~ that ~~''ma'' ≤ ''b''~~ and ~~''a'' ≤ ''nb''~~.

	~~{{DEFAULTSORT:Archimedean Group}}~~
	~~[[Category:Ordered groups]]~~

27.252.64.22: /* Formal adjoint in one variable */

2013-12-12T02:00:08Z

Formal adjoint in one variable

← Older revision		Revision as of 04:00, 12 December 2013
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			In [[abstract algebra]], a branch of [[mathematics]], an '''Archimedean group''' is an [[algebraic structure]] consisting of a [[Set (mathematics)\|set]] together with a [[binary operation]] and [[binary relation]] satisfying certain axioms detailed below. We can also say that an Archimedean group is a [[linearly ordered group]] for which the [[Archimedean property]] holds. For example, the set '''R''' of [[real number]]s together with the operation of addition and usual ordering relation (≤) is an Archimedean group. The concept is named after [[Archimedes]].

			==Definition==
			In the subsequent, we use the notation <math>na</math> (where <math>n</math> is in the set '''N''' of [[natural number]]s) for the sum of ''a'' with itself ''n'' times.

			An '''Archimedean group''' (''G'', +, ≤) is a [[linearly ordered group]] subject to the following condition:

			for any ''a'' and ''b'' in ''G'' which are greater than ''0'', the inequality ''na'' ≤ ''b'' holding for every ''n'' in '''N''' implies ''a'' = 0.

			==Examples of Archimedean groups==
			The sets of the [[integer]]s, the [[rational number]]s, the [[real number]]s, together with the operation of addition and the usual ordering (≤), are Archimedean groups.

			==Examples of non-Archimedean groups==
			An ordered group (''G'', +, ≤) defined as follows is not Archimedean:
			* ''G'' = '''R''' × '''R'''.
			* Let ''a'' = (''u'', ''v'') and ''b'' = (''x'', ''y'') then ''a'' + ''b'' = (''u'' + ''x'', ''v'' + ''y'')
			* ''a'' ≤ ''b'' [[iff]] ''v'' < ''y'' or (''v'' = ''y'' and ''u'' ≤ ''x'') ([[lexicographical order]] with the least-significant number on the left).

			'''Proof:''' Consider the elements (1, 0) and (0, 1). For all ''n'' in '''N''' one evidently has ''n'' (1, 0) < (0, 1).

			For another example, see [[p-adic number]].

			==Theorems==
			For each ''a'', ''b'' in ''G'' there exist ''m'', ''n'' in '''N''' such that ''ma'' ≤ ''b'' and ''a'' ≤ ''nb''.

			{{DEFAULTSORT:Archimedean Group}}
			[[Category:Ordered groups]]

en>Maschen: /* See also */ add links for examples

2012-08-18T01:46:33Z