Direction vector - Revision history

en>Dicklyon: rm contradiction and redundancy from lead; ce heading case

2014-09-28T04:11:43Z

rm contradiction and redundancy from lead; ce heading case

← Older revision		Revision as of 06:11, 28 September 2014
Line 1:		Line 1:
	~~{{Redirect6\|Solovay\|alleged internet "Spam King"\|Robert Soloway\|people with similar names\|Soloway (disambiguation)}}~~		Hi there. Allow me begin by introducing the writer, her title is Sophia Boon but she never really liked that title. What I adore doing is football but I don't have the time lately. Ohio is exactly where her house is. Credit authorising is how she tends to make a residing.<br><br>Feel free to visit my web site - best psychic ([http://isaworld.pe.kr/?document_srl=392088 isaworld.pe.kr])
	~~{{no footnotes\|date=February 2013}}~~
	~~{{Infobox scientist~~
	~~\| name = Robert M. Solovay~~
	~~\| image = Robert Solovay~~.~~jpeg~~
	~~\| image_size = 200px~~
	~~\| caption = Robert Solovay in 1972 (photo~~ by ~~George Bergman)~~
	~~\| birth_date = {{Birth date and age\|1938\|12\|15\|mf=y}}~~
	~~\| birth_place = [[Brooklyn~~, ~~New York]], U.S.~~
	~~\| death_date =~~
	~~\| death_place =~~
	~~\| nationality = [[United States\|American]]~~
	~~\| fields = [[Mathematics]]~~
	~~\| workplaces = [[University of California, Berkeley]]~~
	~~\| alma_mater = [[University of Chicago]]~~
	~~\| doctoral_advisor = [[Saunders Mac Lane]]~~
	~~\| doctoral_students = [[Matthew Foreman]]<br>[[Kenneth McAloon]]<br>[[Judith Roitman]]<br>[[W~~. ~~Hugh Woodin]]~~
	~~\| known_for =~~
	~~\| awards =~~
	}}
	~~'''Robert Martin Solovay''' (born December 15, 1938)~~ is ~~an [[United States\|American]] [[mathematician]] specializing in [[set theory]].~~

	~~Solovay earned his [[Ph.D.]] from the [[University of Chicago]] in 1964 under the direction of [[Saunders Mac Lane]], with a dissertation on '~~'~~A Functorial Form of~~ the ~~Differentiable [[Riemann–Roch theorem]]''~~. ~~Solovay has spent his career at the [[University of California]] at Berkeley,~~ where ~~his notable Ph.D. students include [[W. Hugh Woodin]] and [[Matthew Foreman]]~~.

	~~Solovay's noted accomplishments include:~~
	* [[Solovay's theorem]] showing that, if one assumes the existence of an [[inaccessible cardinal]], then the statement "every [[Set (mathematics)\|set]] of [[real number]]s is ~~[[Lebesgue measurable]]" is consistent with [[Zermelo–Fraenkel set theory\|ZF]] without the [[axiom of choice]];~~
	* Isolating the notion of [[zero sharp\|0<sup>#</sup>]];
	* Proving that the existence of a ~~[[measurable cardinal#Real-valued measurable\|real valued measurable cardinal]] is [[equiconsistency\|equiconsistent]] with the existence of a measurable cardinal;~~
	* Proving that if <~~math~~>~~\lambda~~<~~/math~~> ~~is a strong limit [[singular cardinal]], greater than a [[strongly compact cardinal]] then <math>2^\lambda=\lambda^+</math> holds;~~
	* Proving that if <math>\kappa</math> is an uncountable regular cardinal, and <math>S\subseteq\kappa</math> is a [[stationary set]], then <math>S</math> can be decomposed into the union of <math>\kappa</math> disjoint stationary sets;
	* Outside of set theory, developing (with [[Volker Strassen]]) the [[Solovay–Strassen primality test]], used to ~~identify large [[natural number]]s that are [[prime number\|prime]] with high [[probability]]. This method has had important ramifications for [[cryptography]].~~
	* Proving that GL (~~the~~ [~~[normal modal logic]] which has the instances of the schema <math>\Box(\Box A\to A)\to\Box A<~~/~~math> as additional axioms) completely axiomatizes the logic of the provability predicate of [[Peano Arithmetic]].~~

	~~== Selected publications ==~~
	* {{cite journal\|doi=10.2307/~~1970696\|author=Solovay, Robert M~~.~~\|title=A model of set-theory in which every set of reals is Lebesgue measurable\|journal=Annals of Mathematics. Second Series\|volume=92\|issue=1\|year=1970\|pages=1–56}}~~
	* {{cite journal\|author=Solovay, Robert M.~~\|title=A nonconstructible ''Δ<sup>1</sup><sub>3<~~/~~sub>'' set of integers\|journal=Transactions of the American Mathematical Society\|volume~~=~~127\|year=1967\|pages=50–75\|doi=10~~.~~2307/1994631\|issue=1\|publisher=American Mathematical Society\|jstor=1994631}}~~
	* {{cite journal\|author=Solovay, Robert M. and Volker Strassen\|journal=SIAM Journal on Computing\|title=A fast Monte-Carlo test for primality\|volume=6\|year=1977\|issue=1\|pages=84–85\|doi=10.~~1137/0206006}}~~

	~~== See also ==~~
	* [[Provability logic]]

	~~== External links ==~~
	* {{MathGenealogy\|id=6522}}
	* {{DBLP\|id=Solovay:Robert}}

	~~{{Kanellakis Award laureates}}~~

	~~{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->~~
	~~\| NAME = Solovay, Robert M.~~
	~~\| ALTERNATIVE NAMES =~~
	~~\| SHORT DESCRIPTION = American mathematician~~
	~~\| DATE OF BIRTH = December 15, 1938~~
	~~\| PLACE OF BIRTH = [[Brooklyn, New York]], U.S.~~
	~~\| DATE OF DEATH =~~
	~~\| PLACE OF DEATH =~~
	}}
	~~{{DEFAULTSORT:Solovay, Robert M.}}~~
	~~[[Category:American logicians]]~~
	~~[[Category:Members of the United States National Academy of Sciences]]~~
	~~[[Category:20th-century mathematicians]]~~
	~~[[Category:Set theorists]]~~
	~~[[Category:1938 births]]~~
	~~[[Category:Living people]]~~

	~~{{US-mathematician-stub}}~~

en>Pbierre: word parametrically is not appropriate here, since the representation does not depend on a parameter t

2013-10-07T05:07:52Z

word parametrically is not appropriate here, since the representation does not depend on a parameter t

← Older revision		Revision as of 07:07, 7 October 2013
Line 1:		Line 1:
	~~Hi there~~. ~~Allow me start~~ by ~~introducing the author~~, ~~her name~~ is ~~Sophia~~. ~~One~~ of the ~~very best things in~~ the ~~globe for him is performing ballet and he~~'~~ll be beginning something else along with it~~. ~~Office supervising is exactly~~ where ~~my primary income arrives from but I~~'~~ve always wanted my personal business. For years she~~'s ~~are psychics~~ real ([~~http:~~//~~netwk.hannam.ac.kr~~/xe/~~data_2~~/~~85669 http:~~//~~netwk~~.~~hannam~~.ac.kr/~~xe/data_2/85669]) been residing~~ in ~~Kentucky but her husband desires them to move~~.<br><br>~~Here is my [http:~~//~~www~~.~~youronlinepublishers~~.~~com~~/~~authWiki/AdolphvhBladenqq tarot readings~~] ~~page~~ - ~~live psychic reading (~~[~~http~~:~~//gcjcteam~~.~~org/index~~.~~php?mid~~=~~etc_video&document_srl~~=~~696611&sort_index~~=~~regdate&order_type~~=~~desc read this article~~])		{{Redirect6\|Solovay\|alleged internet "Spam King"\|Robert Soloway\|people with similar names\|Soloway (disambiguation)}}
			{{no footnotes\|date=February 2013}}
			{{Infobox scientist
			\| name = Robert M. Solovay
			\| image = Robert Solovay.jpeg
			\| image_size = 200px
			\| caption = Robert Solovay in 1972 (photo by George Bergman)
			\| birth_date = {{Birth date and age\|1938\|12\|15\|mf=y}}
			\| birth_place = [[Brooklyn, New York]], U.S.
			\| death_date =
			\| death_place =
			\| nationality = [[United States\|American]]
			\| fields = [[Mathematics]]
			\| workplaces = [[University of California, Berkeley]]
			\| alma_mater = [[University of Chicago]]
			\| doctoral_advisor = [[Saunders Mac Lane]]
			\| doctoral_students = [[Matthew Foreman]]<br>[[Kenneth McAloon]]<br>[[Judith Roitman]]<br>[[W. Hugh Woodin]]
			\| known_for =
			\| awards =
			}}
			'''Robert Martin Solovay''' (born December 15, 1938) is an [[United States\|American]] [[mathematician]] specializing in [[set theory]].

			Solovay earned his [[Ph.D.]] from the [[University of Chicago]] in 1964 under the direction of [[Saunders Mac Lane]], with a dissertation on ''A Functorial Form of the Differentiable [[Riemann–Roch theorem]]''. Solovay has spent his career at the [[University of California]] at Berkeley, where his notable Ph.D. students include [[W. Hugh Woodin]] and [[Matthew Foreman]].

			Solovay's noted accomplishments include:
			* [[Solovay's theorem]] showing that, if one assumes the existence of an [[inaccessible cardinal]], then the statement "every [[Set (mathematics)\|set]] of [[real number]]s is [[Lebesgue measurable]]" is consistent with [[Zermelo–Fraenkel set theory\|ZF]] without the [[axiom of choice]];
			* Isolating the notion of [[zero sharp\|0<sup>#</sup>]];
			* Proving that the existence of a [[measurable cardinal#Real-valued measurable\|real valued measurable cardinal]] is [[equiconsistency\|equiconsistent]] with the existence of a measurable cardinal;
			* Proving that if <math>\lambda</math> is a strong limit [[singular cardinal]], greater than a [[strongly compact cardinal]] then <math>2^\lambda=\lambda^+</math> holds;
			* Proving that if <math>\kappa</math> is an uncountable regular cardinal, and <math>S\subseteq\kappa</math> is a [[stationary set]], then <math>S</math> can be decomposed into the union of <math>\kappa</math> disjoint stationary sets;
			* Outside of set theory, developing (with [[Volker Strassen]]) the [[Solovay–Strassen primality test]], used to identify large [[natural number]]s that are [[prime number\|prime]] with high [[probability]]. This method has had important ramifications for [[cryptography]].
			* Proving that GL (the [[normal modal logic]] which has the instances of the schema <math>\Box(\Box A\to A)\to\Box A</math> as additional axioms) completely axiomatizes the logic of the provability predicate of [[Peano Arithmetic]].

			== Selected publications ==
			* {{cite journal\|doi=10.2307/1970696\|author=Solovay, Robert M.\|title=A model of set-theory in which every set of reals is Lebesgue measurable\|journal=Annals of Mathematics. Second Series\|volume=92\|issue=1\|year=1970\|pages=1–56}}
			* {{cite journal\|author=Solovay, Robert M.\|title=A nonconstructible ''Δ<sup>1</sup><sub>3</sub>'' set of integers\|journal=Transactions of the American Mathematical Society\|volume=127\|year=1967\|pages=50–75\|doi=10.2307/1994631\|issue=1\|publisher=American Mathematical Society\|jstor=1994631}}
			* {{cite journal\|author=Solovay, Robert M. and Volker Strassen\|journal=SIAM Journal on Computing\|title=A fast Monte-Carlo test for primality\|volume=6\|year=1977\|issue=1\|pages=84–85\|doi=10.1137/0206006}}

			== See also ==
			* [[Provability logic]]

			== External links ==
			* {{MathGenealogy\|id=6522}}
			* {{DBLP\|id=Solovay:Robert}}

			{{Kanellakis Award laureates}}

			{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
			\| NAME = Solovay, Robert M.
			\| ALTERNATIVE NAMES =
			\| SHORT DESCRIPTION = American mathematician
			\| DATE OF BIRTH = December 15, 1938
			\| PLACE OF BIRTH = [[Brooklyn, New York]], U.S.
			\| DATE OF DEATH =
			\| PLACE OF DEATH =
			}}
			{{DEFAULTSORT:Solovay, Robert M.}}
			[[Category:American logicians]]
			[[Category:Members of the United States National Academy of Sciences]]
			[[Category:20th-century mathematicians]]
			[[Category:Set theorists]]
			[[Category:1938 births]]
			[[Category:Living people]]

			{{US-mathematician-stub}}

en>Bruno321: /* Direction vector for a line in R2 / it was plain wrong. The vector (a,b) is a normal* vector, which is precisely orthogonal to a direction vector.

2012-03-13T21:15:52Z

Direction vector for a line in R2: it was plain wrong. The vector (a,b) is a *normal* vector, which is precisely orthogonal to a *direction* vector.

New page

Hi there. Allow me start by introducing the author, her name is Sophia. One of the very best things in the globe for him is performing ballet and he'll be beginning something else along with it. Office supervising is exactly where my primary income arrives from but I've always wanted my personal business. For years she's are psychics real ([http://netwk.hannam.ac.kr/xe/data_2/85669 http://netwk.hannam.ac.kr/xe/data_2/85669]) been residing in Kentucky but her husband desires them to move.<br><br>Here is my [http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq tarot readings] page - live psychic reading ([http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc read this article])

← Older revision		Revision as of 06:11, 28 September 2014
Line 1:		Line 1:
	~~{{Redirect6\|Solovay\|alleged internet "Spam King"\|Robert Soloway\|people with similar names\|Soloway (disambiguation)}}~~		Hi there. Allow me begin by introducing the writer, her title is Sophia Boon but she never really liked that title. What I adore doing is football but I don't have the time lately. Ohio is exactly where her house is. Credit authorising is how she tends to make a residing.<br><br>Feel free to visit my web site - best psychic ([http://isaworld.pe.kr/?document_srl=392088 isaworld.pe.kr])
	~~{{no footnotes\|date=February 2013}}~~
	~~{{Infobox scientist~~
	~~\| name = Robert M. Solovay~~
	~~\| image = Robert Solovay~~.~~jpeg~~
	~~\| image_size = 200px~~
	~~\| caption = Robert Solovay in 1972 (photo~~ by ~~George Bergman)~~
	~~\| birth_date = {{Birth date and age\|1938\|12\|15\|mf=y}}~~
	~~\| birth_place = [[Brooklyn~~, ~~New York]], U.S.~~
	~~\| death_date =~~
	~~\| death_place =~~
	~~\| nationality = [[United States\|American]]~~
	~~\| fields = [[Mathematics]]~~
	~~\| workplaces = [[University of California, Berkeley]]~~
	~~\| alma_mater = [[University of Chicago]]~~
	~~\| doctoral_advisor = [[Saunders Mac Lane]]~~
	~~\| doctoral_students = [[Matthew Foreman]]<br>[[Kenneth McAloon]]<br>[[Judith Roitman]]<br>[[W~~. ~~Hugh Woodin]]~~
	~~\| known_for =~~
	~~\| awards =~~
	}}
	~~'''Robert Martin Solovay''' (born December 15, 1938)~~ is ~~an [[United States\|American]] [[mathematician]] specializing in [[set theory]].~~

	~~Solovay earned his [[Ph.D.]] from the [[University of Chicago]] in 1964 under the direction of [[Saunders Mac Lane]], with a dissertation on '~~'~~A Functorial Form of~~ the ~~Differentiable [[Riemann–Roch theorem]]''~~. ~~Solovay has spent his career at the [[University of California]] at Berkeley,~~ where ~~his notable Ph.D. students include [[W. Hugh Woodin]] and [[Matthew Foreman]]~~.

	~~Solovay's noted accomplishments include:~~
	* [[Solovay's theorem]] showing that, if one assumes the existence of an [[inaccessible cardinal]], then the statement "every [[Set (mathematics)\|set]] of [[real number]]s is ~~[[Lebesgue measurable]]" is consistent with [[Zermelo–Fraenkel set theory\|ZF]] without the [[axiom of choice]];~~
	* Isolating the notion of [[zero sharp\|0<sup>#</sup>]];
	* Proving that the existence of a ~~[[measurable cardinal#Real-valued measurable\|real valued measurable cardinal]] is [[equiconsistency\|equiconsistent]] with the existence of a measurable cardinal;~~
	* Proving that if <~~math~~>~~\lambda~~<~~/math~~> ~~is a strong limit [[singular cardinal]], greater than a [[strongly compact cardinal]] then <math>2^\lambda=\lambda^+</math> holds;~~
	* Proving that if <math>\kappa</math> is an uncountable regular cardinal, and <math>S\subseteq\kappa</math> is a [[stationary set]], then <math>S</math> can be decomposed into the union of <math>\kappa</math> disjoint stationary sets;
	* Outside of set theory, developing (with [[Volker Strassen]]) the [[Solovay–Strassen primality test]], used to ~~identify large [[natural number]]s that are [[prime number\|prime]] with high [[probability]]. This method has had important ramifications for [[cryptography]].~~
	* Proving that GL (~~the~~ [~~[normal modal logic]] which has the instances of the schema <math>\Box(\Box A\to A)\to\Box A<~~/~~math> as additional axioms) completely axiomatizes the logic of the provability predicate of [[Peano Arithmetic]].~~

	~~== Selected publications ==~~
	* {{cite journal\|doi=10.2307/~~1970696\|author=Solovay, Robert M~~.~~\|title=A model of set-theory in which every set of reals is Lebesgue measurable\|journal=Annals of Mathematics. Second Series\|volume=92\|issue=1\|year=1970\|pages=1–56}}~~
	* {{cite journal\|author=Solovay, Robert M.~~\|title=A nonconstructible ''Δ<sup>1</sup><sub>3<~~/~~sub>'' set of integers\|journal=Transactions of the American Mathematical Society\|volume~~=~~127\|year=1967\|pages=50–75\|doi=10~~.~~2307/1994631\|issue=1\|publisher=American Mathematical Society\|jstor=1994631}}~~
	* {{cite journal\|author=Solovay, Robert M. and Volker Strassen\|journal=SIAM Journal on Computing\|title=A fast Monte-Carlo test for primality\|volume=6\|year=1977\|issue=1\|pages=84–85\|doi=10.~~1137/0206006}}~~

	~~== See also ==~~
	* [[Provability logic]]

	~~== External links ==~~
	* {{MathGenealogy\|id=6522}}
	* {{DBLP\|id=Solovay:Robert}}

	~~{{Kanellakis Award laureates}}~~

	~~{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->~~
	~~\| NAME = Solovay, Robert M.~~
	~~\| ALTERNATIVE NAMES =~~
	~~\| SHORT DESCRIPTION = American mathematician~~
	~~\| DATE OF BIRTH = December 15, 1938~~
	~~\| PLACE OF BIRTH = [[Brooklyn, New York]], U.S.~~
	~~\| DATE OF DEATH =~~
	~~\| PLACE OF DEATH =~~
	}}
	~~{{DEFAULTSORT:Solovay, Robert M.}}~~
	~~[[Category:American logicians]]~~
	~~[[Category:Members of the United States National Academy of Sciences]]~~
	~~[[Category:20th-century mathematicians]]~~
	~~[[Category:Set theorists]]~~
	~~[[Category:1938 births]]~~
	~~[[Category:Living people]]~~

	~~{{US-mathematician-stub}}~~

Direction vector - Revision history

en>Dicklyon: rm contradiction and redundancy from lead; ce heading case

en>Pbierre: word parametrically is not appropriate here, since the representation does not depend on a parameter t

en>Bruno321: /* Direction vector for a line in R2 */ it was plain wrong. The vector (a,b) is a *normal* vector, which is precisely orthogonal to a *direction* vector.

en>Bruno321: /* Direction vector for a line in R2 / it was plain wrong. The vector (a,b) is a normal* vector, which is precisely orthogonal to a direction vector.