LIBOR market model - Revision history

199.172.169.97 at 18:02, 30 October 2014

2014-10-30T18:02:49Z

← Older revision		Revision as of 20:02, 30 October 2014
Line 1:		Line 1:
	~~{{about\|figure-eight shaped curves in algebraic geometry\|other uses\|Lemniscate (disambiguation)}}~~		Alyson Meagher is the name her parents gave her but she doesn't like when people use her full title. North Carolina is exactly where we've been residing for years and will by no means transfer. He is an info officer. To play domino is some thing I really appreciate performing.<br><br>Take a look at my web page; [http://cspl.postech.ac.kr/zboard/Membersonly/144571 free psychic]
	In [[algebraic geometry]], a '''lemniscate''' may refer to any of several figure-eight or <big>[[∞]]</big>-shaped [[curve]]s.<ref name="lemniscatomy"/><ref name="erickson"/> The word comes from the [[Latin language\|Latin]] "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of [[Lemnos]] where ribbons were worn as decorations,<ref name="erickson">{{citation\|title=Beautiful Mathematics\|series=MAA Spectrum\|publisher=[[Mathematical Association of America]]\|first=Martin J.\|last=Erickson\|year=2011\|isbn=9780883855768\|contribution=1.1 Lemniscate\|pages=1–3\|url=http://books.google.com/books?id=LgeP62-ZxikC&pg=PA1}}.</ref> or alternatively may refer to the [[wool]] from which the ribbons were made.<ref name="lemniscatomy"/>

	Although the name "lemniscate" dates to the late 17th century, the consideration of curves with a figure eight shape can be traced back to [[Proclus]], a Greek [[Neoplatonist]] philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a [[torus]] by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is [[tangent]] to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a [[horse]] [[Legcuffs\|fetter]] (a device for holding two feet of a horse together). The Greek phrase for a horse fetter became the word [[hippopede]], the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the [[quartic polynomial]] {{math\|(''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>2</sup> − ''cx''<sup>2</sup> − ''dy''~~<sup>2</sup>}}~~ when ~~the parameter ''d''~~ is ~~negative. For positive values of~~ '~~'d'' one instead obtains an oval-shaped curve, the [[oval of Booth]].<ref name="lemniscatomy">{{citation~~
	~~\| last = Schappacher \| first = Norbert~~
	~~\| contribution = Some milestones of lemniscatomy~~
	~~\| location = New York~~
	~~\| mr = 1483331~~
	~~\| pages = 257–290~~
	~~\| publisher = Dekker~~
	~~\| series = Lecture Notes in Pure~~ and ~~Applied Mathematics~~
	~~\| title = Algebraic Geometry (Ankara, 1995)~~
	~~\| volume = 193~~
	~~\| year = 1997}}.</ref>~~

	~~In 1694, [[Johann Bernoulli]] studied the zero set of the polynomial equation <math>(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)</math> in connection with a problem of "isochrones" that had been posed earlier~~ by ~~[[Leibniz]]~~. ~~Bernoulli's brother [[Jacob Bernoulli]] also studied the same curve in the same year, and gave it its name, the lemniscate.<ref name="bos">{{citation~~
	~~\| last = Bos \| first = H. J. M.~~
	~~\| contribution = The lemniscate of Bernoulli~~
	~~\| location = Dordrecht~~
	~~\| mr = 774250~~
	~~\| pages = 3–14~~
	~~\| publisher = Reidel~~
	~~\| series = Boston Stud. Philos. Sci~~.~~, XV~~
	~~\| title = For Dirk Struik~~
	~~\| url = http://books.google.com/books?id=OvK9orJNezwC&pg=PA3~~
	~~\| year = 1974}}.</ref> It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.<ref>{{citation~~
	~~\| last1 = Langer \| first1 = Joel C.~~
	~~\| last2 = Singer \| first2 = David A.~~
	~~\| doi = 10.1007/s00032-010-0124-5~~
	~~\| issue = 2~~
	~~\| journal = Milan Journal of Mathematics~~
	~~\| mr = 2781856~~
	~~\| pages = 643–682~~
	~~\| title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem~~
	~~\| volume = 78~~
	~~\| year = 2010}}.</ref> It~~ is ~~a special case of the hippopede, with <math>d=-c</math>, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other~~.<~~ref name="lemniscatomy"/~~> The [[lemniscatic elliptic function]]s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the [[Gauss's constant\|lemniscate constants]] arise in evaluating the [[arc length]] of this lemniscate.

	~~Another lemniscate, the [[lemniscate of Gerono]] or lemniscate of Huygens, is the zero set of the quartic polynomial equation~~ <~~math~~>~~y^2=x^2(~~a^2-x^2)</math>.<ref>{{citation\|title=An elementary treatise on cubic and quartic curves\|first=Alfred Barnard\|last=Basset\|publisher=Deighton, Bell\|year=1901\|pages=171–172\|url=http://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171\|contribution=The Lemniscate of Gerono}}.</ref><ref>{{citation\|title=Newton's Principia for the common reader\|first=S\|last=Chandrasekhar\|publisher=Oxford University Press\|year=2003\|isbn=9780198526759\|page~~=133\|url=http://books.google.com/books?id=qomP58txKQwC&pg=PA133}}.</ref> [~~[Viviani's curve]], a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.<ref>{{citation\|first1=Luisa Rossi\|last1=Costa\|first2=Elena\|last2=Marchetti\|contribution=Mathematical and Historical Investigation on Domes and Vaults\|pages=73–80\|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004\|year=2005\|location=Mammendorf\|publisher=Pro Literatur\|editor1-last=Weber\|editor1-first=Ralf\|editor2-last=Amann\|editor2-first=Matthias Albrecht}}.</ref>

	~~Other figure-eight shaped algebraic curves include~~
	* The [[Devil's curve]], a curve defined by the quartic equation <math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math> in which one connected component has a figure-eight shape,<ref>{{citation\|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes\|first=David\|last=Darling\|publisher=John Wiley & Sons\|year=2004\|isbn=9780471667001\|contribution=devil's curve\|pages=91–92\|url=http://~~books~~.~~google~~.~~com/books?id=HrOxRdtYYaMC&pg=PA91}}~~.</~~ref>~~
	* [[Watt's curve]], a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation <math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</~~math> and has the lemniscate of Bernoulli as a special case.~~

	~~==See also==~~
	* [[Analemma]], the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
	* [[Lorenz attractor]], a three-dimensional dynamic system exhibiting a lemniscate shape
	* [[Polynomial lemniscate]], a level set of the absolute value of a complex polynomial

	~~==References==~~
	~~{{Reflist\|30em}}~~

	~~==External links==~~
	* {{springer\|title=Lemniscates\|id=p/~~l058130}}~~

	~~[[Category:Mathematical terminology]]~~
	~~[[Category:Curves]~~]

en>Cfries: /* External links */ Added link to Java source code of the model.

2013-04-28T21:29:44Z

External links: Added link to Java source code of the model.

← Older revision		Revision as of 23:29, 28 April 2013
Line 1:		Line 1:
	~~Andrew Simcox~~ is the title ~~his mothers~~ and ~~fathers~~ gave ~~him and he totally loves this~~ name. ~~Credit authorising is exactly where my primary earnings comes~~ from. ~~To climb~~ is ~~some thing she would by no means give up~~. I'~~ve usually loved residing~~ in ~~Mississippi~~.<br><br>~~Also visit my homepage~~ ... ~~psychic love readings~~ - [http://~~165~~.~~132~~.39.93/xe/~~visitors/372912 click~~ the ~~next webpage~~],		{{about\|figure-eight shaped curves in algebraic geometry\|other uses\|Lemniscate (disambiguation)}}
			In [[algebraic geometry]], a '''lemniscate''' may refer to any of several figure-eight or <big>[[∞]]</big>-shaped [[curve]]s.<ref name="lemniscatomy"/><ref name="erickson"/> The word comes from the [[Latin language\|Latin]] "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of [[Lemnos]] where ribbons were worn as decorations,<ref name="erickson">{{citation\|title=Beautiful Mathematics\|series=MAA Spectrum\|publisher=[[Mathematical Association of America]]\|first=Martin J.\|last=Erickson\|year=2011\|isbn=9780883855768\|contribution=1.1 Lemniscate\|pages=1–3\|url=http://books.google.com/books?id=LgeP62-ZxikC&pg=PA1}}.</ref> or alternatively may refer to the [[wool]] from which the ribbons were made.<ref name="lemniscatomy"/>

			Although the name "lemniscate" dates to the late 17th century, the consideration of curves with a figure eight shape can be traced back to [[Proclus]], a Greek [[Neoplatonist]] philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a [[torus]] by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is [[tangent]] to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a [[horse]] [[Legcuffs\|fetter]] (a device for holding two feet of a horse together). The Greek phrase for a horse fetter became the word [[hippopede]], the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the [[quartic polynomial]] {{math\|(''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>2</sup> − ''cx''<sup>2</sup> − ''dy''<sup>2</sup>}} when the parameter ''d'' is negative. For positive values of ''d'' one instead obtains an oval-shaped curve, the [[oval of Booth]].<ref name="lemniscatomy">{{citation
			\| last = Schappacher \| first = Norbert
			\| contribution = Some milestones of lemniscatomy
			\| location = New York
			\| mr = 1483331
			\| pages = 257–290
			\| publisher = Dekker
			\| series = Lecture Notes in Pure and Applied Mathematics
			\| title = Algebraic Geometry (Ankara, 1995)
			\| volume = 193
			\| year = 1997}}.</ref>

			In 1694, [[Johann Bernoulli]] studied the zero set of the polynomial equation <math>(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)</math> in connection with a problem of "isochrones" that had been posed earlier by [[Leibniz]]. Bernoulli's brother [[Jacob Bernoulli]] also studied the same curve in the same year, and gave it its name, the lemniscate.<ref name="bos">{{citation
			\| last = Bos \| first = H. J. M.
			\| contribution = The lemniscate of Bernoulli
			\| location = Dordrecht
			\| mr = 774250
			\| pages = 3–14
			\| publisher = Reidel
			\| series = Boston Stud. Philos. Sci., XV
			\| title = For Dirk Struik
			\| url = http://books.google.com/books?id=OvK9orJNezwC&pg=PA3
			\| year = 1974}}.</ref> It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.<ref>{{citation
			\| last1 = Langer \| first1 = Joel C.
			\| last2 = Singer \| first2 = David A.
			\| doi = 10.1007/s00032-010-0124-5
			\| issue = 2
			\| journal = Milan Journal of Mathematics
			\| mr = 2781856
			\| pages = 643–682
			\| title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
			\| volume = 78
			\| year = 2010}}.</ref> It is a special case of the hippopede, with <math>d=-c</math>, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other.<ref name="lemniscatomy"/> The [[lemniscatic elliptic function]]s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the [[Gauss's constant\|lemniscate constants]] arise in evaluating the [[arc length]] of this lemniscate.

			Another lemniscate, the [[lemniscate of Gerono]] or lemniscate of Huygens, is the zero set of the quartic polynomial equation <math>y^2=x^2(a^2-x^2)</math>.<ref>{{citation\|title=An elementary treatise on cubic and quartic curves\|first=Alfred Barnard\|last=Basset\|publisher=Deighton, Bell\|year=1901\|pages=171–172\|url=http://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171\|contribution=The Lemniscate of Gerono}}.</ref><ref>{{citation\|title=Newton's Principia for the common reader\|first=S\|last=Chandrasekhar\|publisher=Oxford University Press\|year=2003\|isbn=9780198526759\|page=133\|url=http://books.google.com/books?id=qomP58txKQwC&pg=PA133}}.</ref> [[Viviani's curve]], a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.<ref>{{citation\|first1=Luisa Rossi\|last1=Costa\|first2=Elena\|last2=Marchetti\|contribution=Mathematical and Historical Investigation on Domes and Vaults\|pages=73–80\|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004\|year=2005\|location=Mammendorf\|publisher=Pro Literatur\|editor1-last=Weber\|editor1-first=Ralf\|editor2-last=Amann\|editor2-first=Matthias Albrecht}}.</ref>

			Other figure-eight shaped algebraic curves include
			* The [[Devil's curve]], a curve defined by the quartic equation <math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math> in which one connected component has a figure-eight shape,<ref>{{citation\|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes\|first=David\|last=Darling\|publisher=John Wiley & Sons\|year=2004\|isbn=9780471667001\|contribution=devil's curve\|pages=91–92\|url=http://books.google.com/books?id=HrOxRdtYYaMC&pg=PA91}}.</ref>
			* [[Watt's curve]], a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation <math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</math> and has the lemniscate of Bernoulli as a special case.

			==See also==
			* [[Analemma]], the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
			* [[Lorenz attractor]], a three-dimensional dynamic system exhibiting a lemniscate shape
			* [[Polynomial lemniscate]], a level set of the absolute value of a complex polynomial

			==References==
			{{Reflist\|30em}}

			==External links==
			* {{springer\|title=Lemniscates\|id=p/l058130}}

			[[Category:Mathematical terminology]]
			[[Category:Curves]]

en>Btyner: /* References */ +cat

2012-07-11T02:59:13Z

References: +cat

New page

Andrew Simcox is the title his mothers and fathers gave him and he totally loves this name. Credit authorising is exactly where my primary earnings comes from. To climb is some thing she would by no means give up. I've usually loved residing in Mississippi.<br><br>Also visit my homepage ... psychic love readings - [http://165.132.39.93/xe/visitors/372912 click the next webpage],

← Older revision		Revision as of 20:02, 30 October 2014
Line 1:		Line 1:
	~~{{about\|figure-eight shaped curves in algebraic geometry\|other uses\|Lemniscate (disambiguation)}}~~		Alyson Meagher is the name her parents gave her but she doesn't like when people use her full title. North Carolina is exactly where we've been residing for years and will by no means transfer. He is an info officer. To play domino is some thing I really appreciate performing.<br><br>Take a look at my web page; [http://cspl.postech.ac.kr/zboard/Membersonly/144571 free psychic]
	In [[algebraic geometry]], a '''lemniscate''' may refer to any of several figure-eight or <big>[[∞]]</big>-shaped [[curve]]s.<ref name="lemniscatomy"/><ref name="erickson"/> The word comes from the [[Latin language\|Latin]] "lēmniscātus" meaning "decorated with ribbons", which in turn may come from the ancient Greek island of [[Lemnos]] where ribbons were worn as decorations,<ref name="erickson">{{citation\|title=Beautiful Mathematics\|series=MAA Spectrum\|publisher=[[Mathematical Association of America]]\|first=Martin J.\|last=Erickson\|year=2011\|isbn=9780883855768\|contribution=1.1 Lemniscate\|pages=1–3\|url=http://books.google.com/books?id=LgeP62-ZxikC&pg=PA1}}.</ref> or alternatively may refer to the [[wool]] from which the ribbons were made.<ref name="lemniscatomy"/>

	Although the name "lemniscate" dates to the late 17th century, the consideration of curves with a figure eight shape can be traced back to [[Proclus]], a Greek [[Neoplatonist]] philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a [[torus]] by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is [[tangent]] to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a [[horse]] [[Legcuffs\|fetter]] (a device for holding two feet of a horse together). The Greek phrase for a horse fetter became the word [[hippopede]], the name for this figure-eight shaped curve, which is also called the lemniscate of Booth. It may be defined algebraically as the zero set of the [[quartic polynomial]] {{math\|(''x''<sup>2</sup> + ''y''<sup>2</sup>)<sup>2</sup> − ''cx''<sup>2</sup> − ''dy''~~<sup>2</sup>}}~~ when ~~the parameter ''d''~~ is ~~negative. For positive values of~~ '~~'d'' one instead obtains an oval-shaped curve, the [[oval of Booth]].<ref name="lemniscatomy">{{citation~~
	~~\| last = Schappacher \| first = Norbert~~
	~~\| contribution = Some milestones of lemniscatomy~~
	~~\| location = New York~~
	~~\| mr = 1483331~~
	~~\| pages = 257–290~~
	~~\| publisher = Dekker~~
	~~\| series = Lecture Notes in Pure~~ and ~~Applied Mathematics~~
	~~\| title = Algebraic Geometry (Ankara, 1995)~~
	~~\| volume = 193~~
	~~\| year = 1997}}.</ref>~~

	~~In 1694, [[Johann Bernoulli]] studied the zero set of the polynomial equation <math>(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)</math> in connection with a problem of "isochrones" that had been posed earlier~~ by ~~[[Leibniz]]~~. ~~Bernoulli's brother [[Jacob Bernoulli]] also studied the same curve in the same year, and gave it its name, the lemniscate.<ref name="bos">{{citation~~
	~~\| last = Bos \| first = H. J. M.~~
	~~\| contribution = The lemniscate of Bernoulli~~
	~~\| location = Dordrecht~~
	~~\| mr = 774250~~
	~~\| pages = 3–14~~
	~~\| publisher = Reidel~~
	~~\| series = Boston Stud. Philos. Sci~~.~~, XV~~
	~~\| title = For Dirk Struik~~
	~~\| url = http://books.google.com/books?id=OvK9orJNezwC&pg=PA3~~
	~~\| year = 1974}}.</ref> It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.<ref>{{citation~~
	~~\| last1 = Langer \| first1 = Joel C.~~
	~~\| last2 = Singer \| first2 = David A.~~
	~~\| doi = 10.1007/s00032-010-0124-5~~
	~~\| issue = 2~~
	~~\| journal = Milan Journal of Mathematics~~
	~~\| mr = 2781856~~
	~~\| pages = 643–682~~
	~~\| title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem~~
	~~\| volume = 78~~
	~~\| year = 2010}}.</ref> It~~ is ~~a special case of the hippopede, with <math>d=-c</math>, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other~~.<~~ref name="lemniscatomy"/~~> The [[lemniscatic elliptic function]]s are analogues of trigonometric functions for the lemniscate of Bernoulli, and the [[Gauss's constant\|lemniscate constants]] arise in evaluating the [[arc length]] of this lemniscate.

	~~Another lemniscate, the [[lemniscate of Gerono]] or lemniscate of Huygens, is the zero set of the quartic polynomial equation~~ <~~math~~>~~y^2=x^2(~~a^2-x^2)</math>.<ref>{{citation\|title=An elementary treatise on cubic and quartic curves\|first=Alfred Barnard\|last=Basset\|publisher=Deighton, Bell\|year=1901\|pages=171–172\|url=http://books.google.com/books?id=T40LAAAAYAAJ&pg=PA171\|contribution=The Lemniscate of Gerono}}.</ref><ref>{{citation\|title=Newton's Principia for the common reader\|first=S\|last=Chandrasekhar\|publisher=Oxford University Press\|year=2003\|isbn=9780198526759\|page~~=133\|url=http://books.google.com/books?id=qomP58txKQwC&pg=PA133}}.</ref> [~~[Viviani's curve]], a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection.<ref>{{citation\|first1=Luisa Rossi\|last1=Costa\|first2=Elena\|last2=Marchetti\|contribution=Mathematical and Historical Investigation on Domes and Vaults\|pages=73–80\|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004\|year=2005\|location=Mammendorf\|publisher=Pro Literatur\|editor1-last=Weber\|editor1-first=Ralf\|editor2-last=Amann\|editor2-first=Matthias Albrecht}}.</ref>

	~~Other figure-eight shaped algebraic curves include~~
	* The [[Devil's curve]], a curve defined by the quartic equation <math>y^2 (y^2 - a^2) = x^2 (x^2 - b^2)</math> in which one connected component has a figure-eight shape,<ref>{{citation\|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes\|first=David\|last=Darling\|publisher=John Wiley & Sons\|year=2004\|isbn=9780471667001\|contribution=devil's curve\|pages=91–92\|url=http://~~books~~.~~google~~.~~com/books?id=HrOxRdtYYaMC&pg=PA91}}~~.</~~ref>~~
	* [[Watt's curve]], a figure-eight shaped curve formed by a mechanical linkage. Watt's curve is the zero set of the degree-six polynomial equation <math>(x^2+y^2)(x^2+y^2-d^2)^2+4a^2y^2(x^2+y^2-b^2)=0</~~math> and has the lemniscate of Bernoulli as a special case.~~

	~~==See also==~~
	* [[Analemma]], the figure-eight shaped curve traced by the noontime positions of the sun in the sky over the course of a year
	* [[Lorenz attractor]], a three-dimensional dynamic system exhibiting a lemniscate shape
	* [[Polynomial lemniscate]], a level set of the absolute value of a complex polynomial

	~~==References==~~
	~~{{Reflist\|30em}}~~

	~~==External links==~~
	* {{springer\|title=Lemniscates\|id=p/~~l058130}}~~

	~~[[Category:Mathematical terminology]]~~
	~~[[Category:Curves]~~]