Angle bisector theorem - Revision history

en>Kmhkmh: /* Proof 1 */

2015-01-13T07:46:34Z

Proof 1

← Older revision		Revision as of 09:46, 13 January 2015
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	~~The '''Lanczos tensor''' or '''Lanczos potential''' is a [[tensor rank\|rank 3 tensor]] in [[general relativity]] that generates the [[Weyl tensor]]~~.~~<ref name="Takeno">Hyôitirô Takeno, "On~~ the spintensor of Lanczos", ''Tensor'', '''15''' (1964) pp. 103–119.</ref> It was first introduced by [[Cornelius Lanczos]] in 1949.<ref name="Lanczos1949">Cornelius Lanczos, "Lagrangian Multiplier and Riemannian Spaces", ''Rev. Mod. Phys.'', ~~'''21''' (1949) pp. 497–502. {{doi\|10.1103/RevModPhys.21.497}}</ref> The theoretical importance of the Lanczos tensor~~ is that it serves as the [[gauge field]] for the [[gravitational field]] in the same way that, by analogy, the [[electromagnetic four-potential]] generates the [[electromagnetic field]].<ref name="O'DonnellPye">P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', '''7''' (2010) pp. 327–350. {{url\|http://www.ejtp.com/articles/ejtpv7i24p327.pdf}}</ref><ref name="NovelloVelloso">M. Novello and A. L. ~~Velloso, "The Connection Between General Observers and Lanczos Potential", ''General Relativity and Gravitation'', '''19''' (1987) pp. 1251-1265. {{doi\|10.1007/BF00759104}}</ref>~~		Hi there. Allow me start by introducing the writer, her title is Sophia. She is truly fond of caving but she doesn't have the time lately. Ohio is where my home is but my spouse desires us to move. Credit authorising is how she tends to make a living.<br><br>My web blog [http://kpupf.com/xe/talk/735373 spirit messages]

	~~==Definition==~~
	~~The Lanczos tensor can be defined in a few different ways. The most common modern definition~~ is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor.<ref name="NovelloVelloso" /> These equations, presented below, were given by Takeno in 1964.<ref name="Takeno" /> The way that Lanczos introduced the tensor originally was as a [[Lagrange multiplier]]<ref name="Lanczos1949" /><ref>Cornelius Lanczos, "The Splitting of the Riemann Tensor", ''Rev. Mod. Phys.'', '''34''' (1962) pp. 379–389. {{doi\|10.1103/RevModPhys.34.379}}</ref> on constraint terms studied in the [[variational methods in general relativity\|variational approach to general relativity]].<ref>Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", ''Annals of Mathematics'', '''39''' (1938) pp. 842-850. {{url\|http://www.jstor.org/stable/1968467}}</ref> Under any definition, the Lanczos tensor exhibits the following symmetries:
	~~:<math>~~
	~~H_{abc}+H_{bac}=0,\,</math>~~
	~~:<math>~~
	~~H_{abc}+H_{bca}+H_{cab}=0.~~
	~~</math>~~

	The Lanczos tensor always exists in four dimensions<ref name="BC1983">F. Bampi and G. Caviglia, "Third-order tensor potentials for the Riemann and Weyl tensors", ''General Relativity and Gravitation'', '''15''' (1983) pp. 375-386. {{doi\|10.1007/BF00759166}}</ref> but ~~does not generalize to higher dimensions.<ref>S. B. Edgar, "Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions",~~ '~~'General Relativity and Gravitation'', '''26''' (1994) pp. 329-332. {{doi\|10.1007/BF02108015}}</ref> This highlights~~ the ~~[[spacetime#Privileged character of 3+1 spacetime\|specialness of four dimensions]]~~.<ref name="O'DonnellPye" /> Note further that the full [[Riemann tensor]] cannot in general be derived from derivatives of the Lanczos potential alone.<ref name="BC1983" /><ref>E. Massa and E. Pagani, "Is the Riemann tensor derivable from a tensor potential?", ''General Relativity and Gravitation'', '''16''' (1984) pp. 805-816. {{doi\|10.1007/BF00762934}}</ref> The [[Einstein field equations]] must provide the [[Ricci tensor]] to complete the components of the [[Ricci decomposition]].

	~~===Weyl–Lanczos equations===~~

	The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:<ref name="O'Donnell">P. O’Donnell, "A Solution of the Weyl–Lanczos Equations for the Schwarzschild Space-Time", ''General Relativity and Gravitation'', '''36''' (2004) pp. 1415-1422. {{doi\|10.1023/B:GERG.0000022577.11259.e0}}</ref>

	~~:<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\~~
	~~&\, \, \, \, \, + (H^e{}_{(ac);e} + H_{(a\|e\|}{}^e{}_{;c)})g_{bd} + (H^e{}_{(bd);e} + H_{(b\|e\|}{}^e{}_{;d)})g_{ac} \\~~
	~~&\, \, \, \, \, - (H^e{}_{(ad);e} + H_{(a\|e\|}{}^e{}_{;d)})g_{bc} - (H^e{}_{(bc);e} + H_{(b\|e\|}{}^e{}_{;c)})g_{ad} \\~~
	~~&\, \, \, \, \, -\frac{2}{3} H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})\end{align}</math>~~

	where ~~<math>C_{abcd}</math>~~ is the Weyl tensor, the semicolon denotes the [[covariant derivative]], and the subscripted parentheses indicate [[symmetric tensor\|symmetrization]]. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has [[gauge freedom]] under an [[affine group]].<ref>K. S. Hammon and L. K. Norris "The Affine Geometry of the Lanczos H-tensor Formalism", ''General Relativity and Gravitation'','''25''' (1993) pp. 55-80. {{doi\|10.1007/BF00756929}}</ref> If <math>\Phi^a</math> is an arbitrary [[vector field]], then the Weyl–Lanczos equations are invariant under the gauge transformation

	~~:<math>H'_{abc} = H_{abc}+\Phi_{[a} g_{b]c}</math>~~

	where the subscripted brackets indicate [[antisymmetric tensor\|antisymmetrization]]. An often convenient choice is the Lanczos algebraic gauge, <math>\Phi_a=-\frac{2}{3}H_{ab}{}^b,</math> which sets <math>H'_{ab}{}^b=0.</math> The gauge can be further restricted through the Lanczos differential gauge <math>H_{ab}{}^c{}_{;c}=0</math>. These gauge choices reduce the Weyl–Lanczos equations to the simpler form

	~~:<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\~~
	~~&\, \, \, \, \, +H^e{}_{ac;e}g_{bd}+H^e{}_{bd;e}g_{ac}-H^e{}_{ad;e}g_{bc}-H^e{}_{bc;e}g_{ad}.\end{align}</math>~~

	~~==Wave equation==~~
	The Lanczos potential tensor satisfies a wave equation<ref>P. Dolan and C. W. Kim "The wave equation for the Lanczos potential", ''Proc. R. Soc. Lond. A'', '''447''' (1994) pp. 557-575. {{doi\|10.1098/rspa.1994.0155}}</ref>

	~~:<math>\begin{align}\Box H_{abc} = & J_{abc}\\~~
	~~& {}- 2{R_c}^d H_{abd}+{R_a}^d H_{bcd}+{R_b}^d H_{acd}\\~~
	~~& {}+ \left( H_{dbe}g_{ac}-H_{dae}g_{bc} \right)R^{de}+\frac{1}{2}RH_{abc},\end{align}</math>~~

	~~where <math>\Box</math> is the [[d'Alembert operator]] and~~
	~~:<math>J_{abc} = R_{ca;b}-R_{cb;a}-\frac{1}{6}\left( g_{ca}R_{;b}-g_{cb}R_{;a} \right)</math>~~
	is known as the [[Cotton tensor]]. Since the Cotton tensor depends only on [[covariant derivative]]s of the [[Ricci tensor]], it can perhaps be interpreted as a kind of matter current.<ref name="Roberts1996">Mark D. Roberts, "The Physical Interpretation of the Lanczos Tensor." ''Nuovo Cim.B'' '''110''' (1996) 1165-1176. {{doi\|10.1007/BF02724607}} {{arXiv\|gr-qc/9904006}}</ref> The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the [[vacuum solution (general relativity)\|vacuum solutions]], where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor. Thus in vacuum, the [[Einstein field equations]] are equivalent to the [[homogeneous differential equation\|homogeneous]] wave equation <math>\Box H_{abc} = 0,</math> in perfect analogy to the vacuum wave equation <math>\Box A_{a} = 0</math> of the electromagnetic four-potential. This shows a formal similarity between [[gravitational wave]]s and [[electromagnetic wave]]s, with the Lanczos tensor well-suited for studying gravitational waves.<ref>J. L. López-Bonilla, G. Ovando and J. J. Peña, "A Lanczos Potential for Plane Gravitational Waves." ''Foundations of Physics Letters'' '''12''' (1999) 401-405. {{doi\|10.~~1023/A:1021656622094}}</ref>~~

	~~In the weak field approximation where <math>g_{ab}=\eta_{ab}+h_{ab}</math>, a convenient form for the Lanczos tensor in the Lanczos gauge~~ is~~<ref name="Roberts1996" />~~

	~~:<math>4H_{abc} \approx h_{ac,b}-h_{bc,a}-\frac{1}{6}(\eta_{ac}{h^d}_{d,b}-\eta_{bc}{h^d}_{d,a}) .</math>~~

	~~==Example==~~
	The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the [[Schwarzschild metric]].<ref name="NovelloVelloso" /> The simplest, explicit component representation in [[natural units]] for the Lanczos tensor in this case is
	~~:<math>H_{trt}=\frac{GM}{r^2}</math>~~
	~~with all other components vanishing up to symmetries. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are~~
	~~:<math>H_{trt}=\frac{2GM}{3r^2}</math>~~
	~~:<math>H_{r\theta\theta}=\frac{-GM}{3(1-2GM/r)}</math>~~
	~~:<math>H_{r\phi\phi}=\frac{-GM\sin^2 \theta}{3(1-2GM/r)}</math>~~

	~~It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced~~ to a ~~linear combination of the spin coefficients of the [[Newman–Penrose formalism]], which attests to the Lanczos tensor's fundamental nature~~.<~~ref name="O'Donnell" /~~> ~~Similar calculations have been used to construct arbitrary [[Petrov type D]] solutions.~~<~~ref~~>Zafar Ahsan and Mohd Bilal, "A Solution of Weyl-Lanczos Equations for Arbitrary Petrov Type D Vacuum Spacetimes." ''Int J Theor Phys'' '''49''' (2010) 2713-2722. {{doi\|10.1007/s10773-010-0464-5}}</ref>

	~~==See also==~~
	*[[Bach tensor]]
	*[[Schouten tensor]]
	*[[Ricci calculus]]

	~~==References==~~
	~~<references />~~

	~~==External links==~~
	*Peter O'Donnell, ''Introduction To 2-Spinors In General Relativity''. [http://~~www.worldscibooks~~.com/~~physics~~/~~5222.html World Scientific], 2003.~~

	~~[[Category:Gauge theories]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Tensors]]~~
	~~[[Category:Tensors in general relativity]]~~
	~~[[Category:1949 introductions]~~]

en>Ascom99: /* External links */ Looks like spam - removed

2013-11-12T01:54:24Z

External links: Looks like spam - removed

← Older revision		Revision as of 03:54, 12 November 2013
Line 1:		Line 1:
	The ~~person who wrote~~ the ~~article~~ is ~~known~~ as ~~Jayson Hirano~~ and ~~he completely digs~~ that ~~title~~. ~~Ohio~~ is ~~where my home~~ is but ~~my husband wants us to transfer~~. ~~Distributing manufacturing~~ is ~~how he tends~~ to ~~make~~ a ~~living~~. ~~She~~ is ~~truly fond~~ of ~~caving but she doesn~~'t have the ~~time lately~~.<br><br>~~Check out my web~~-~~site~~: [~~http~~:/~~/www~~.~~edmposts~~.~~com~~/~~build~~-a-~~beautiful~~-~~organic~~-~~garden~~-~~using~~-~~these~~-~~ideas~~/ ~~free psychic~~]		The '''Lanczos tensor''' or '''Lanczos potential''' is a [[tensor rank\|rank 3 tensor]] in [[general relativity]] that generates the [[Weyl tensor]].<ref name="Takeno">Hyôitirô Takeno, "On the spintensor of Lanczos", ''Tensor'', '''15''' (1964) pp. 103–119.</ref> It was first introduced by [[Cornelius Lanczos]] in 1949.<ref name="Lanczos1949">Cornelius Lanczos, "Lagrangian Multiplier and Riemannian Spaces", ''Rev. Mod. Phys.'', '''21''' (1949) pp. 497–502. {{doi\|10.1103/RevModPhys.21.497}}</ref> The theoretical importance of the Lanczos tensor is that it serves as the [[gauge field]] for the [[gravitational field]] in the same way that, by analogy, the [[electromagnetic four-potential]] generates the [[electromagnetic field]].<ref name="O'DonnellPye">P. O’Donnell and H. Pye, "A Brief Historical Review of the Important Developments in Lanczos Potential Theory", ''EJTP'', '''7''' (2010) pp. 327–350. {{url\|http://www.ejtp.com/articles/ejtpv7i24p327.pdf}}</ref><ref name="NovelloVelloso">M. Novello and A. L. Velloso, "The Connection Between General Observers and Lanczos Potential", ''General Relativity and Gravitation'', '''19''' (1987) pp. 1251-1265. {{doi\|10.1007/BF00759104}}</ref>

			==Definition==
			The Lanczos tensor can be defined in a few different ways. The most common modern definition is through the Weyl–Lanczos equations, which demonstrate the generation of the Weyl tensor from the Lanczos tensor.<ref name="NovelloVelloso" /> These equations, presented below, were given by Takeno in 1964.<ref name="Takeno" /> The way that Lanczos introduced the tensor originally was as a [[Lagrange multiplier]]<ref name="Lanczos1949" /><ref>Cornelius Lanczos, "The Splitting of the Riemann Tensor", ''Rev. Mod. Phys.'', '''34''' (1962) pp. 379–389. {{doi\|10.1103/RevModPhys.34.379}}</ref> on constraint terms studied in the [[variational methods in general relativity\|variational approach to general relativity]].<ref>Cornelius Lanczos, "A Remarkable Property of the Riemann–Christoffel Tensor in Four Dimensions", ''Annals of Mathematics'', '''39''' (1938) pp. 842-850. {{url\|http://www.jstor.org/stable/1968467}}</ref> Under any definition, the Lanczos tensor exhibits the following symmetries:
			:<math>
			H_{abc}+H_{bac}=0,\,</math>
			:<math>
			H_{abc}+H_{bca}+H_{cab}=0.
			</math>

			The Lanczos tensor always exists in four dimensions<ref name="BC1983">F. Bampi and G. Caviglia, "Third-order tensor potentials for the Riemann and Weyl tensors", ''General Relativity and Gravitation'', '''15''' (1983) pp. 375-386. {{doi\|10.1007/BF00759166}}</ref> but does not generalize to higher dimensions.<ref>S. B. Edgar, "Nonexistence of the Lanczos potential for the Riemann tensor in higher dimensions", ''General Relativity and Gravitation'', '''26''' (1994) pp. 329-332. {{doi\|10.1007/BF02108015}}</ref> This highlights the [[spacetime#Privileged character of 3+1 spacetime\|specialness of four dimensions]].<ref name="O'DonnellPye" /> Note further that the full [[Riemann tensor]] cannot in general be derived from derivatives of the Lanczos potential alone.<ref name="BC1983" /><ref>E. Massa and E. Pagani, "Is the Riemann tensor derivable from a tensor potential?", ''General Relativity and Gravitation'', '''16''' (1984) pp. 805-816. {{doi\|10.1007/BF00762934}}</ref> The [[Einstein field equations]] must provide the [[Ricci tensor]] to complete the components of the [[Ricci decomposition]].

			===Weyl–Lanczos equations===

			The Weyl–Lanczos equations express the Weyl tensor entirely as derivatives of the Lanczos tensor:<ref name="O'Donnell">P. O’Donnell, "A Solution of the Weyl–Lanczos Equations for the Schwarzschild Space-Time", ''General Relativity and Gravitation'', '''36''' (2004) pp. 1415-1422. {{doi\|10.1023/B:GERG.0000022577.11259.e0}}</ref>

			:<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\
			&\, \, \, \, \, + (H^e{}_{(ac);e} + H_{(a\|e\|}{}^e{}_{;c)})g_{bd} + (H^e{}_{(bd);e} + H_{(b\|e\|}{}^e{}_{;d)})g_{ac} \\
			&\, \, \, \, \, - (H^e{}_{(ad);e} + H_{(a\|e\|}{}^e{}_{;d)})g_{bc} - (H^e{}_{(bc);e} + H_{(b\|e\|}{}^e{}_{;c)})g_{ad} \\
			&\, \, \, \, \, -\frac{2}{3} H^{ef}{}_{f;e}(g_{ac}g_{bd}-g_{ad}g_{bc})\end{align}</math>

			where <math>C_{abcd}</math> is the Weyl tensor, the semicolon denotes the [[covariant derivative]], and the subscripted parentheses indicate [[symmetric tensor\|symmetrization]]. Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has [[gauge freedom]] under an [[affine group]].<ref>K. S. Hammon and L. K. Norris "The Affine Geometry of the Lanczos H-tensor Formalism", ''General Relativity and Gravitation'','''25''' (1993) pp. 55-80. {{doi\|10.1007/BF00756929}}</ref> If <math>\Phi^a</math> is an arbitrary [[vector field]], then the Weyl–Lanczos equations are invariant under the gauge transformation

			:<math>H'_{abc} = H_{abc}+\Phi_{[a} g_{b]c}</math>

			where the subscripted brackets indicate [[antisymmetric tensor\|antisymmetrization]]. An often convenient choice is the Lanczos algebraic gauge, <math>\Phi_a=-\frac{2}{3}H_{ab}{}^b,</math> which sets <math>H'_{ab}{}^b=0.</math> The gauge can be further restricted through the Lanczos differential gauge <math>H_{ab}{}^c{}_{;c}=0</math>. These gauge choices reduce the Weyl–Lanczos equations to the simpler form

			:<math>\begin{align}C_{abcd}&=H_{abc;d}+H_{cda;b}+H_{bad;c}+H_{dcb;a} \\
			&\, \, \, \, \, +H^e{}_{ac;e}g_{bd}+H^e{}_{bd;e}g_{ac}-H^e{}_{ad;e}g_{bc}-H^e{}_{bc;e}g_{ad}.\end{align}</math>

			==Wave equation==
			The Lanczos potential tensor satisfies a wave equation<ref>P. Dolan and C. W. Kim "The wave equation for the Lanczos potential", ''Proc. R. Soc. Lond. A'', '''447''' (1994) pp. 557-575. {{doi\|10.1098/rspa.1994.0155}}</ref>

			:<math>\begin{align}\Box H_{abc} = & J_{abc}\\
			& {}- 2{R_c}^d H_{abd}+{R_a}^d H_{bcd}+{R_b}^d H_{acd}\\
			& {}+ \left( H_{dbe}g_{ac}-H_{dae}g_{bc} \right)R^{de}+\frac{1}{2}RH_{abc},\end{align}</math>

			where <math>\Box</math> is the [[d'Alembert operator]] and
			:<math>J_{abc} = R_{ca;b}-R_{cb;a}-\frac{1}{6}\left( g_{ca}R_{;b}-g_{cb}R_{;a} \right)</math>
			is known as the [[Cotton tensor]]. Since the Cotton tensor depends only on [[covariant derivative]]s of the [[Ricci tensor]], it can perhaps be interpreted as a kind of matter current.<ref name="Roberts1996">Mark D. Roberts, "The Physical Interpretation of the Lanczos Tensor." ''Nuovo Cim.B'' '''110''' (1996) 1165-1176. {{doi\|10.1007/BF02724607}} {{arXiv\|gr-qc/9904006}}</ref> The additional self-coupling terms have no direct electromagnetic equivalent. These self-coupling terms, however, do not affect the [[vacuum solution (general relativity)\|vacuum solutions]], where the Ricci tensor vanishes and the curvature is described entirely by the Weyl tensor. Thus in vacuum, the [[Einstein field equations]] are equivalent to the [[homogeneous differential equation\|homogeneous]] wave equation <math>\Box H_{abc} = 0,</math> in perfect analogy to the vacuum wave equation <math>\Box A_{a} = 0</math> of the electromagnetic four-potential. This shows a formal similarity between [[gravitational wave]]s and [[electromagnetic wave]]s, with the Lanczos tensor well-suited for studying gravitational waves.<ref>J. L. López-Bonilla, G. Ovando and J. J. Peña, "A Lanczos Potential for Plane Gravitational Waves." ''Foundations of Physics Letters'' '''12''' (1999) 401-405. {{doi\|10.1023/A:1021656622094}}</ref>

			In the weak field approximation where <math>g_{ab}=\eta_{ab}+h_{ab}</math>, a convenient form for the Lanczos tensor in the Lanczos gauge is<ref name="Roberts1996" />

			:<math>4H_{abc} \approx h_{ac,b}-h_{bc,a}-\frac{1}{6}(\eta_{ac}{h^d}_{d,b}-\eta_{bc}{h^d}_{d,a}) .</math>

			==Example==
			The most basic nontrivial case for expressing the Lanczos tensor is, of course, for the [[Schwarzschild metric]].<ref name="NovelloVelloso" /> The simplest, explicit component representation in [[natural units]] for the Lanczos tensor in this case is
			:<math>H_{trt}=\frac{GM}{r^2}</math>
			with all other components vanishing up to symmetries. This form, however, is not in the Lanczos gauge. The nonvanishing terms of the Lanczos tensor in the Lanczos gauge are
			:<math>H_{trt}=\frac{2GM}{3r^2}</math>
			:<math>H_{r\theta\theta}=\frac{-GM}{3(1-2GM/r)}</math>
			:<math>H_{r\phi\phi}=\frac{-GM\sin^2 \theta}{3(1-2GM/r)}</math>

			It is further possible to show, even in this simple case, that the Lanczos tensor cannot in general be reduced to a linear combination of the spin coefficients of the [[Newman–Penrose formalism]], which attests to the Lanczos tensor's fundamental nature.<ref name="O'Donnell" /> Similar calculations have been used to construct arbitrary [[Petrov type D]] solutions.<ref>Zafar Ahsan and Mohd Bilal, "A Solution of Weyl-Lanczos Equations for Arbitrary Petrov Type D Vacuum Spacetimes." ''Int J Theor Phys'' '''49''' (2010) 2713-2722. {{doi\|10.1007/s10773-010-0464-5}}</ref>

			==See also==
			*[[Bach tensor]]
			*[[Schouten tensor]]
			*[[Ricci calculus]]

			==References==
			<references />

			==External links==
			*Peter O'Donnell, ''Introduction To 2-Spinors In General Relativity''. [http://www.worldscibooks.com/physics/5222.html World Scientific], 2003.

			[[Category:Gauge theories]]
			[[Category:Differential geometry]]
			[[Category:Tensors]]
			[[Category:Tensors in general relativity]]
			[[Category:1949 introductions]]

en>WestwoodMatt: Undid revision 508911771 by 202.164.53.117 (talk) nonsense characters

2012-08-26T00:09:38Z

Undid revision 508911771 by 202.164.53.117 (talk) nonsense characters

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The person who wrote the article is known as Jayson Hirano and he completely digs that title. Ohio is where my home is but my husband wants us to transfer. Distributing manufacturing is how he tends to make a living. She is truly fond of caving but she doesn't have the time lately.<br><br>Check out my web-site: [http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ free psychic]