Integer-valued function - Revision history

Boleyn: has ref

2024-10-08T21:50:49Z

has ref

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	{{~~unreferenced~~\|date=June 2013}}		{{onesource\|date=June 2013}}
	[[Image:Floor function.svg\|thumb\|right\|288px\|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.]]		[[Image:Floor function.svg\|thumb\|right\|288px\|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.]]
	{{Functions}}		{{Functions}}

Manoomin: Added a standard textbook as a reference for one of the mathematical results, since previously this article had no references.

2024-10-07T21:49:06Z

Added a standard textbook as a reference for one of the mathematical results, since previously this article had no references.

← Previous revision		Revision as of 21:49, 7 October 2024
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	== Algebraic properties ==		== Algebraic properties ==
	On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of addition and multiplication, ~~and also an [[algebra over a ring\|algebra]] over the ring~~ {{~~math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered~~ ~~ring]]:~~		On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of addition and multiplication,<ref>{{cite book
			\| last1 = Dummit
			\| first1 = David S.
			\| last2 = Foote
			\| first2 = Richard M.
			\| date = July 2003
			\| title = Abstract Algebra
			\| edition = 3rd
			\| publisher = John Wiley and Sons, Inc.
			\| page = 225
			\| isbn = 978-0-471-43334-7
			}}</ref> and also an [[algebra over a ring\|algebra]] over the ring {{math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:
	:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>		:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>

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	== References ==		== References ==
			{{Reflist}}

			==Further reading==
	* https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf		* https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf

Bera678: Added a reference

2024-01-02T04:49:20Z

Added a reference

← Previous revision		Revision as of 04:49, 2 January 2024
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	* [[Rank (disambiguation)#Mathematics]]		* [[Rank (disambiguation)#Mathematics]]
	* [[Grade (disambiguation)#In mathematics]]		* [[Grade (disambiguation)#In mathematics]]

			== References ==

			* https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyIntegerValuedEntireFunctions.pdf

	[[Category:Types of functions]]		[[Category:Types of functions]]

Joel Brennan: added wikilinks

2023-01-17T21:49:09Z

added wikilinks

← Previous revision		Revision as of 21:49, 17 January 2023
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	[[Image:Floor function.svg\|thumb\|right\|288px\|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.]]		[[Image:Floor function.svg\|thumb\|right\|288px\|The floor function on real numbers. Its discontinuities are pictured with white discs outlines with blue circles.]]
	{{Functions}}		{{Functions}}
	In mathematics, an '''integer-valued function''' is a [[function (mathematics)\|function]] whose ~~[[codomain\|~~values]] are [[integer]]s. In other words, it is a function that assigns an integer to each member of its [[domain of a function\|domain]].		In [[mathematics]], an '''integer-valued function''' is a [[function (mathematics)\|function]] whose values are [[integer]]s. In other words, it is a function that assigns an integer to each member of its [[domain of a function\|domain]].

	[[~~Floor~~ and ceiling functions]] are examples of an integer-valued [[function of a real variable]], but on [[real number]]s and generally, on (non-disconnected) [[topological space]]s integer-valued functions are not especially useful. Any such function on a [[connected space]] either has [[discontinuity (mathematics)\|discontinuities]] or is [[constant function\|constant]]. On the other hand, on [[discrete space\|discrete]] and other [[totally disconnected space]]s integer-valued functions have roughly the same importance as [[real-valued function]]s have on non-discrete spaces.		The [[floor and ceiling functions]] are examples of integer-valued [[function of a real variable\|functions of a real variable]], but on [[real number]]s and, generally, on (non-disconnected) [[topological space]]s integer-valued functions are not especially useful. Any such function on a [[connected space]] either has [[discontinuity (mathematics)\|discontinuities]] or is [[constant function\|constant]]. On the other hand, on [[discrete space\|discrete]] and other [[totally disconnected space]]s integer-valued functions have roughly the same importance as [[real-valued function]]s have on non-discrete spaces.

	Any function with [[natural number\|natural]], or [[non-negative]] integer values is a partial case of integer-valued function.		Any function with [[natural number\|natural]], or [[non-negative]] integer values is a partial case of an integer-valued function.

	== Examples ==		== Examples ==
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	== Algebraic properties ==		== Algebraic properties ==
	On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of [[addition]] and [[multiplication]], and also an [[algebra ~~(ring~~ ~~theory)~~\|algebra]] over the ring {{math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:		On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of addition and multiplication, and also an [[algebra over a ring\|algebra]] over the ring {{math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:
	:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>		:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>

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	=== Mathematical logic and computability theory ===		=== Mathematical logic and computability theory ===
	In [[mathematical logic]] such concepts as a [[primitive recursive function]] and a [[μ-recursive function]] represent integer-valued functions of several natural variables or, in other words, functions on {{math\|'''N'''~~[[coordinate space\|~~<sup>''n''</sup>]]}}. [[Gödel numbering]], defined on [[well-formed formula]]e of some [[formal language]], is a natural-valued function.		In [[mathematical logic]], such concepts as [[primitive recursive function]]s and [[μ-recursive function]]s represent integer-valued functions of several natural variables or, in other words, functions on {{math\|'''N'''<sup>''n''</sup>}}. [[Gödel numbering]], defined on [[well-formed formula]]e of some [[formal language]], is a natural-valued function.

	[[Computability theory]] is essentially based on natural numbers and natural (or integer) functions on them.		[[Computability theory]] is essentially based on natural numbers and natural (or integer) functions on them.
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	=== Computer science ===		=== Computer science ===
	In [[computer programming]] many [[function (programming)\|functions]] return values of [[Integer (computer science)\|integer type]] due to simplicity of implementation.		In [[computer programming]], many [[function (programming)\|functions]] return values of [[Integer (computer science)\|integer type]] due to simplicity of implementation.

	== See also ==		== See also ==

Fehufanga: Reverted edits by 179.209.57.112 (talk) (HG) (3.4.10)

2022-02-18T00:28:06Z

Reverted edits by 179.209.57.112 (talk) (HG) (3.4.10)

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	* [[Grade (disambiguation)#In mathematics]]		* [[Grade (disambiguation)#In mathematics]]

	[[Category:Types of functions]]~~12171~~		[[Category:Types of functions]]

179.209.57.112 at 00:27, 18 February 2022

2022-02-18T00:27:55Z

← Previous revision		Revision as of 00:27, 18 February 2022
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	* [[Grade (disambiguation)#In mathematics]]		* [[Grade (disambiguation)#In mathematics]]

	[[Category:Types of functions]]		[[Category:Types of functions]]12171

Mohdaamir825: /* Number theory */ Fixed grammar

2015-05-06T14:25:42Z

Number theory: Fixed grammar

← Previous revision		Revision as of 14:25, 6 May 2015
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	=== Number theory ===		=== Number theory ===
	In [[number theory]], many [[arithmetic function]]s are integer-valued. ~~dust ffgtrr~~		In [[number theory]], many [[arithmetic function]]s are integer-valued.

	=== Computer science ===		=== Computer science ===

Mohdaamir825: /* Number theory */ Fixed typo

2015-05-06T14:25:04Z

Number theory: Fixed typo

← Previous revision		Revision as of 14:25, 6 May 2015
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	=== Number theory ===		=== Number theory ===
	In [[number theory]], many [[arithmetic function]]s are integer-valued.		In [[number theory]], many [[arithmetic function]]s are integer-valued. dust ffgtrr

	=== Computer science ===		=== Computer science ===

Magioladitis: Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (3) using AWB (10331)

2014-07-31T07:41:35Z

Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (3) using AWB (10331)

← Previous revision		Revision as of 07:41, 31 July 2014
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	== Algebraic properties ==		== Algebraic properties ==
	On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of [[addition]] and [[multiplication]], and also an [[algebra (ring theory)\|algebra]] over the ring {{math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:		On an arbitrary [[set (mathematics)\|set]] {{mvar\|X}}, integer-valued functions form a [[ring (mathematics)\|ring]] with [[pointwise]] operations of [[addition]] and [[multiplication]], and also an [[algebra (ring theory)\|algebra]] over the ring {{math\|'''Z'''}} of integers. Since the latter is an [[ordered ring]], the functions form a [[partially ordered ring]]:
	:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>		:<math>f \le g \quad\iff\quad \forall x: f(x) \le g(x).</math>

	== Uses ==		== Uses ==

	=== Graph theory and algebra ===		=== Graph theory and algebra ===
	Integer-valued functions are ubiquitous in [[graph theory]]. They also have similar uses in [[geometric group theory]], where ''[[length function]]'' represents the concept of [[norm (mathematics)\|norm]], and ''[[word metric]]'' represents the concept of [[metric (mathematics)\|metric]].		Integer-valued functions are ubiquitous in [[graph theory]]. They also have similar uses in [[geometric group theory]], where ''[[length function]]'' represents the concept of [[norm (mathematics)\|norm]], and ''[[word metric]]'' represents the concept of [[metric (mathematics)\|metric]].
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	=== Mathematical logic and computability theory ===		=== Mathematical logic and computability theory ===
	In [[mathematical logic]] such concepts as a [[primitive recursive function]] and a [[μ-recursive function]] represent integer-valued functions of several natural variables or, in other words, functions on {{math\|'''N'''[[coordinate space\|<sup>''n''</sup>]]}}. [[Gödel numbering]], defined on [[well-formed formula]]e of some [[formal language]], is a natural-valued function.		In [[mathematical logic]] such concepts as a [[primitive recursive function]] and a [[μ-recursive function]] represent integer-valued functions of several natural variables or, in other words, functions on {{math\|'''N'''[[coordinate space\|<sup>''n''</sup>]]}}. [[Gödel numbering]], defined on [[well-formed formula]]e of some [[formal language]], is a natural-valued function.

	[[Computability theory]] is essentially based on natural numbers and natural (or integer) functions on them.		[[Computability theory]] is essentially based on natural numbers and natural (or integer) functions on them.
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	* [[Integer-valued polynomial]]		* [[Integer-valued polynomial]]
	* [[Semi-continuity]]		* [[Semi-continuity]]
	* [[Rank (disambiguation) #Mathematics]]		* [[Rank (disambiguation)#Mathematics]]
	* [[Grade (disambiguation) #In mathematics]]		* [[Grade (disambiguation)#In mathematics]]


	[[Category:Types of functions]]		[[Category:Types of functions]]

David Eppstein: remove ridiculous list of bad categories

2014-07-28T03:46:12Z

remove ridiculous list of bad categories

← Previous revision		Revision as of 03:46, 28 July 2014
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	[[Category:Types of functions]]		[[Category:Types of functions]]
	[[Category:Ring theory]]
	[[Category:Geometric group theory]]
	[[Category:Graph theory]]
	[[Category:Mathematical logic]]
	[[Category:Discrete mathematics]]