Lanczos resampling - Revision history

en>Senator2029: using {{Lang}} to mark foreign language text

2014-10-14T16:22:33Z

using {{Lang}} to mark foreign language text

← Older revision		Revision as of 18:22, 14 October 2014
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	~~Alyson is the title individuals use~~ to ~~call me~~ and ~~I think~~ it ~~sounds fairly great when you say~~ it. ~~What I adore doing~~ is ~~soccer but I don't have the time lately~~. I've ~~always~~ cherished ~~living~~ in ~~Alaska. Since I was eighteen~~ I'~~ve been working as a bookkeeper but quickly my spouse and I will start our own business~~.<br><br>~~Also visit~~ my ~~weblog~~ - ~~psychic readers (~~[http://~~www~~.~~chk~~.~~woobi~~.~~co.kr~~/xe/~~?document_srl=346069 mouse click the next site~~])		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. One of the issues she loves most is canoeing and she's been performing it for quite a whilst. Office supervising is my occupation. I've usually cherished residing in Kentucky but now I'm contemplating other options.<br><br>Feel free to surf to my web-site [http://165.132.39.93/xe/visitors/372912 love psychic readings]

en>Monkbot: Fix CS1 deprecated date parameter errors (test)

2014-02-25T14:35:59Z

Fix CS1 deprecated date parameter errors (test)

← Older revision		Revision as of 16:35, 25 February 2014
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	~~{{refimprove\|date=August 2011}}~~		Alyson is the title individuals use to call me and I think it sounds fairly great when you say it. What I adore doing is soccer but I don't have the time lately. I've always cherished living in Alaska. Since I was eighteen I've been working as a bookkeeper but quickly my spouse and I will start our own business.<br><br>Also visit my weblog - psychic readers ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 mouse click the next site])
	~~In [[mathematical logic]], an '''arithmetical set''' (or '''arithmetic set''')~~ is ~~a [[set (mathematics)\|set]] of [[natural number]]s that can be defined by a formula of first-order [[Peano arithmetic]]. The arithmetical sets are classified by~~ the ~~[[arithmetical hierarchy]].~~

	The definition can be extended to an arbitrary [[countable set]] ''A'' (e.g. the set of n-[[tuple]]s of [[integers]], the set of [[rational numbers]], the set of formulas in some [[formal language]], etc.) by using [[Gödel number]]s to ~~represent elements of the set~~ and ~~declaring a subset of ''A'' to be arithmetical if the set of corresponding Gödel numbers is arithmetical~~.

	~~A function <math>f:\subseteq \mathbb{N}^k \to \mathbb{N}</math>~~ is ~~called '''arithmetically definable''~~' if the ~~[[graph of a function\|graph]] of <math>f</math> is an arithmetical set~~.

	~~A [[real number]] is called~~ '~~''arithmetical''' if the set of all smaller rational numbers is arithmetical. A [[complex number]] is called arithmetical if its [[real and imaginary parts]] are both arithmetical~~.

	~~== Formal definition ==~~

	~~A set~~ '~~'X'' of natural numbers is '''arithmetical''' or '''arithmetically definable''' if there is~~ a ~~formula φ(''n'') in the language of Peano arithmetic such that each number ''n'' is in ''X'' if~~ and ~~only if φ(''n'') holds in the standard model of arithmetic~~. ~~Similarly, a ''k''-ary relation~~
	<~~math~~>~~R(n_1,\ldots,n_k)~~<~~/math~~> ~~is arithmetical if there is a formula~~
	~~<math>\psi~~(~~n_1,\ldots,n_k)</math> such that <math>R(n_1,\ldots,n_k) \Leftrightarrow \psi(n_1,\ldots,n_k)<~~/~~math> holds for all ''k''-tuples <math>(n_1,\ldots,n_k)<~~/~~math> of natural numbers~~.

	~~A [[finitary]] function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation~~.

	~~A set ''A'' is said to be '''arithmetical in''' a set ''B'' if ''A'' is definable by an arithmetical formula which has ''B'' as a set parameter.~~

	~~== Examples ==~~

	* The set of all [[prime number]]s is arithmetical.
	* Every [[recursively enumerable set]] is arithmetical.
	* Every [[computable function]] is arithmetically definable.
	* The set encoding the [[Halting problem]] is arithmetical.
	* [[Chaitin's constant Ω]] is an arithmetical real number.
	* [[Tarski's indefinability theorem]] shows that the set of true formulas of first order arithmetic is not arithmetically definable.

	~~== Properties ==~~

	* The [[complement (set theory)\|complement]] of an arithmetical set is an arithmetical set.
	* The [[Turing jump]] of an arithmetical set is an arithmetical set.
	* The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumerates all arithmetical sets.
	* The set of real arithmetical numbers is [[denumerable]], [[Dense order\|dense]] and [[order-isomorphic]] to the set of rational numbers.

	~~== Implicitly arithmetical sets ==~~

	~~Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set~~. ~~An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property~~.

	A set ''Y'' of natural numbers is '''implicitly arithmetical''' or '''implicitly arithmetically definable''' if it is definable with an arithmetical formula that is able to use ''Y'' as a parameter. That is, if there is a formula <math>\theta(Z)</~~math> in the language of Peano arithmetic with no free number variables and a new set parameter ''Z'' and set membership relation <math>\in<~~/~~math> such that ''Y'' is the unique set ''Z'' such that <math>\theta(Z)</math> holds.~~

	~~Every arithmetical set is implicitly arithmetical; if ''X'' is arithmetically defined by φ(''n'') then it is implicitly defined by~~ the ~~formula~~
	~~:<math>\forall n [n \in Z \Leftrightarrow \phi(n)]</math>.~~
	~~Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.~~

	~~== See also ==~~

	* [[Arithmetical hierarchy]]
	* [[Computable set]]
	* [[Computable number]]

	~~== Further reading ==~~
	*Rogers, H. (1967)~~. ''Theory of recursive functions and effective computability.'' McGraw-Hill. {{ISBN missing\|date=September 2013}}~~

	~~{{Number systems}}~~
	~~[[Category:Effective descriptive set theory]]~~
	~~[[Category:Mathematical logic hierarchies]]~~
	~~[[Category:Computability theory]]~~

en>Dicklyon: /* Lanczos kernel */ comma

2014-01-25T05:11:31Z

Lanczos kernel: comma

← Older revision		Revision as of 07:11, 25 January 2014
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	~~Golda~~ is ~~what~~'s ~~written on my birth certificate although it~~ is ~~not~~ the ~~name on my beginning certification~~. I'~~ve always loved residing in Alaska~~. ~~She~~ is ~~truly fond~~ of ~~caving but she doesn~~'~~t have~~ the ~~time recently. Office supervising~~ is ~~where her main earnings arrives from but she~~'~~s already~~ ~~psychics online~~, ~~[http:~~//~~brazil~~.~~amor-amore~~.~~com/irboothe brazil~~.~~amor-amore~~.~~com~~], ~~utilized~~ for ~~another one~~.<br><br>~~[http:~~//~~myfusionprofits~~.~~com~~/~~groups/find-out-about-self-improvement-tips-here/ free psychic] My site~~ ... ~~online psychic~~ reading (~~[http://formalarmour~~.~~com/index~~.~~php?do~~=~~/profile-26947/info/ mouse click the next webpage~~])		{{refimprove\|date=August 2011}}
			In [[mathematical logic]], an '''arithmetical set''' (or '''arithmetic set''') is a [[set (mathematics)\|set]] of [[natural number]]s that can be defined by a formula of first-order [[Peano arithmetic]]. The arithmetical sets are classified by the [[arithmetical hierarchy]].

			The definition can be extended to an arbitrary [[countable set]] ''A'' (e.g. the set of n-[[tuple]]s of [[integers]], the set of [[rational numbers]], the set of formulas in some [[formal language]], etc.) by using [[Gödel number]]s to represent elements of the set and declaring a subset of ''A'' to be arithmetical if the set of corresponding Gödel numbers is arithmetical.

			A function <math>f:\subseteq \mathbb{N}^k \to \mathbb{N}</math> is called '''arithmetically definable''' if the [[graph of a function\|graph]] of <math>f</math> is an arithmetical set.

			A [[real number]] is called '''arithmetical''' if the set of all smaller rational numbers is arithmetical. A [[complex number]] is called arithmetical if its [[real and imaginary parts]] are both arithmetical.

			== Formal definition ==

			A set ''X'' of natural numbers is '''arithmetical''' or '''arithmetically definable''' if there is a formula φ(''n'') in the language of Peano arithmetic such that each number ''n'' is in ''X'' if and only if φ(''n'') holds in the standard model of arithmetic. Similarly, a ''k''-ary relation
			<math>R(n_1,\ldots,n_k)</math> is arithmetical if there is a formula
			<math>\psi(n_1,\ldots,n_k)</math> such that <math>R(n_1,\ldots,n_k) \Leftrightarrow \psi(n_1,\ldots,n_k)</math> holds for all ''k''-tuples <math>(n_1,\ldots,n_k)</math> of natural numbers.

			A [[finitary]] function on the natural numbers is called arithmetical if its graph is an arithmetical binary relation.

			A set ''A'' is said to be '''arithmetical in''' a set ''B'' if ''A'' is definable by an arithmetical formula which has ''B'' as a set parameter.

			== Examples ==

			* The set of all [[prime number]]s is arithmetical.
			* Every [[recursively enumerable set]] is arithmetical.
			* Every [[computable function]] is arithmetically definable.
			* The set encoding the [[Halting problem]] is arithmetical.
			* [[Chaitin's constant Ω]] is an arithmetical real number.
			* [[Tarski's indefinability theorem]] shows that the set of true formulas of first order arithmetic is not arithmetically definable.

			== Properties ==

			* The [[complement (set theory)\|complement]] of an arithmetical set is an arithmetical set.
			* The [[Turing jump]] of an arithmetical set is an arithmetical set.
			* The collection of arithmetical sets is countable, but there is no arithmetically definable sequence that enumerates all arithmetical sets.
			* The set of real arithmetical numbers is [[denumerable]], [[Dense order\|dense]] and [[order-isomorphic]] to the set of rational numbers.

			== Implicitly arithmetical sets ==

			Each arithmetical set has an arithmetical formula which tells whether particular numbers are in the set. An alternative notion of definability allows for a formula that does not tell whether particular numbers are in the set but tells whether the set itself satisfies some arithmetical property.

			A set ''Y'' of natural numbers is '''implicitly arithmetical''' or '''implicitly arithmetically definable''' if it is definable with an arithmetical formula that is able to use ''Y'' as a parameter. That is, if there is a formula <math>\theta(Z)</math> in the language of Peano arithmetic with no free number variables and a new set parameter ''Z'' and set membership relation <math>\in</math> such that ''Y'' is the unique set ''Z'' such that <math>\theta(Z)</math> holds.

			Every arithmetical set is implicitly arithmetical; if ''X'' is arithmetically defined by φ(''n'') then it is implicitly defined by the formula
			:<math>\forall n [n \in Z \Leftrightarrow \phi(n)]</math>.
			Not every implicitly arithmetical set is arithmetical, however. In particular, the truth set of first order arithmetic is implicitly arithmetical but not arithmetical.

			== See also ==

			* [[Arithmetical hierarchy]]
			* [[Computable set]]
			* [[Computable number]]

			== Further reading ==
			*Rogers, H. (1967). ''Theory of recursive functions and effective computability.'' McGraw-Hill. {{ISBN missing\|date=September 2013}}

			{{Number systems}}
			[[Category:Effective descriptive set theory]]
			[[Category:Mathematical logic hierarchies]]
			[[Category:Computability theory]]

en>M.O.X: Filling in 3 references using Reflinks

2012-07-31T00:55:08Z

Filling in 3 references using Reflinks

New page

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