Substitution tiling - Revision history

188.182.238.181: peacock word

2015-01-10T02:51:35Z

peacock word

← Older revision		Revision as of 04:51, 10 January 2015
Line 1:		Line 1:
	~~In [[mathematics]], the simplest~~ '~~''real analytic Eisenstein series''' is a special function of two variables~~. It is ~~used in the [[representation theory]] of [[SL2(R)\|SL(2,'''R''')]] and in [[analytic number theory]]~~. It is ~~closely related to~~ the ~~Epstein zeta function.~~		Alyson is what my spouse loves to contact me but I don't like when people use my complete title. Invoicing is my profession. North Carolina is the location he enjoys most but now he is considering other options. To perform lacross is the factor I adore most of all.<br><br>My website - real psychic - [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ mouse click for source] -

	~~There are many generalizations associated to more complicated groups.~~

	~~==Definition==~~
	~~The Eisenstein series ''E''(''z'', ''s'') for ''z'' = ''x'' + ''iy'' in the [[upper half-plane]]~~ is ~~defined by~~

	~~:<math>E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over\|mz+n\|^{2s}}</math>~~

	~~for Re(''s'') > 1, and by analytic continuation for~~ other ~~values of the complex number ''s''~~. ~~The sum~~ is ~~over all pairs of coprime integers.~~

	~~'''Warning''': there are several other slightly different definitions. Some authors omit~~ the factor of ~~½, and some sum over~~ all ~~pairs of integers that are not both zero; which changes the function by a factor of ζ(2''s'').~~

	~~==Properties==~~

	~~===As a function on ''z''===~~
	Viewed as a function of ''z'', ''E''(''z'',''s'') is a real-analytic [[eigenfunction]] of the [[Laplace operator]] on '''H''' with the eigenvalue ''s''(''s''-1). In other words, it satisfies the [[elliptic partial differential equation]]

	~~: <math> y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)E(z,s) = s(s-1)E(z,s), </math> &ensp;&ensp; where <math>z=x+yi.</math>~~

	The function ''E''(''z'', ''s'') is invariant under the action of SL(2,'''Z''') on ''z'' in the upper half plane by [[fractional linear transformation]]s. Together with the previous property, this means that the Eisenstein series is a [[Maass form]], a real-analytic analogue of a classical elliptic [[modular function]].

	~~'''Warning''': ''E''(''z'', ''s'') is not a square-integrable function of ''z'' with respect to the invariant Riemannian metric on '''H'''.~~

	~~===As a function on ''s''===~~
	The Eisenstein series converges for Re(''s'')>1, but can be [[analytic continuation\|analytically continued]] to a meromorphic function of ''s'' on the entire complex plane, with a unique pole of residue π at ''s'' = 1 (for all ''z'' in '''H'''). The constant term of the pole at ''s'' = 1 is described by the [[Kronecker limit formula]].

	~~The modified function~~

	:<~~math~~> ~~E^*(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s)\~~ <~~/math~~>

	~~satisfies the functional equation~~

	~~:<math>E^(z,s) = E^(z,1-s)\ </math>~~

	~~analogous to the functional equation for the [[Riemann zeta function]] ζ(''s'').~~

	~~Scalar product of two different Eisenstein series ''E''(''z'', ''s'') and ''E''(''z'', ''t'') is given by the [[Maass~~-~~Selberg relation]]s.~~

	~~==Epstein zeta function==~~
	The '''Epstein zeta function''' ζ<sub>''Q''</sub>(''s'') {{harv\|Epstein\|1903}} for a positive definite integral quadratic form ''Q''(''m'', ''n'') = ''cm''<sup>2</sup> + ''bmn'' +''an''<sup>2</sup> is defined by

	~~:<math> \zeta_Q(s) = \sum_{(m,n)\ne (0,0)} {1\over Q(m,n)^s}.\ </math>~~

	~~It is essentially a special case of the~~ real ~~analytic Eisenstein series for a special value of ''z'', since~~

	~~:<math> Q(m,n) = a\|mz+n\|^2\ </math>~~

	~~for~~

	~~:<math> z = {~~-~~b\over 2a} + i {\sqrt{4ac-b^2}\over 2a}.\ </math>~~

	~~This zeta function was named after [[Paul Epstein]].~~

	~~==Generalizations==~~
	The real analytic Eisenstein series ''E''(''z'', ''s'') is really the Eisenstein series associated to the discrete subgroup [[modular group\|SL(2,'''Z''')]] of [[SL2(R)\|SL(2,'''R''')]]. [[Atle Selberg\|Selberg]] described generalizations to other discrete subgroups Γ of SL(2,'''R'''), and used these to study the representation of SL(2,'''R''') on L<sup>2</sup>(SL(2,'''R''')/Γ). [[Robert Langlands\|Langlands]] extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by [[Joseph Bernstein]].

	~~==See also==~~
	*[[Eisenstein series]]
	*[[Kronecker limit formula]]

	~~==References==~~
	*J. Bernstein, [http://~~www~~.~~math.uchicago.edu/~arinkin~~/~~langlands~~/~~Bernstein~~/ ~~''Meromorphic continuation of Eisenstein series'']~~
	*{{citation\|last=Epstein\|first=P.\|authorlink=Paul Epstein\|title=Zur Theorie allgemeiner Zetafunktionen I\|journal=Math. Ann.\|volume=56\|pages=614–644\|year=1903\|doi=10.1007/~~BF01444309\|issue=4}}.~~
	*{{springer\|id=E/~~e120130\|title=Epstein zeta~~-~~function\|author=A. Krieg}}~~
	*{{citation\|first=T.\|last=Kubota\|authorlink=Tomio Kubota\|title=Elementary theory of Eisenstein series\|isbn=0-~~470~~-~~50920~~-~~1\|year=1973\|publisher=Kodansha\|location=Tokyo}}.~~
	*{{citation\|first=Robert P.\|last=Langlands\|url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html\|title=On the functional equations satisfied by Eisenstein series\|isbn=0-~~387~~-~~07872-X\|year=1976\|publisher=Springer-Verlag\|location=Berlin}}.~~
	*A. Selberg, ''Discontinuous groups and harmonic analysis'', Proc. Int. Congr. Math., 1962.
	*[[D. Zagier]~~], ''Eisenstein series and the Riemann zeta~~-~~function''.~~
	~~[[Category:Modular forms]]~~
	~~[[Category:Special functions]]~~
	~~[[Category:Representation theory of Lie groups]]~~
	~~[[Category:Analytic number theory]]~~

en>Brirush: Added subdivision rules

2013-09-17T12:23:36Z

Added subdivision rules

← Older revision		Revision as of 14:23, 17 September 2013
Line 1:		Line 1:
	~~Golda~~ [~~http~~:/~~/gcjcteam~~.~~org/index~~.~~php?mid~~=~~etc_video&document_srl~~=~~696611~~&~~sort_index=regdate~~&~~order_type~~=~~desc free psychic~~] is ~~what~~'s ~~created on my beginning certificate although it~~ is not the ~~name~~ on ~~my birth certificate~~. ~~Distributing production is where my primary income comes from and it~~'s ~~some thing I really appreciate. I am really fond of handwriting~~ but I can'~~t make it my occupation really~~. ~~Kentucky~~ is ~~exactly where I've always been residing~~.<br><br>~~Also visit my weblog~~ :~~: psychics~~ [~~http~~://~~kpupf~~.~~com~~/xe/~~talk~~/~~735373 free online tarot card readings~~], [http://~~Galab~~-~~work~~.cs.~~pusan~~.ac.kr/~~Sol09B~~/~~?document_srl~~=~~1489804 just click~~ the ~~up coming web site~~],		In [[mathematics]], the simplest '''real analytic Eisenstein series''' is a special function of two variables. It is used in the [[representation theory]] of [[SL2(R)\|SL(2,'''R''')]] and in [[analytic number theory]]. It is closely related to the Epstein zeta function.

			There are many generalizations associated to more complicated groups.

			==Definition==
			The Eisenstein series ''E''(''z'', ''s'') for ''z'' = ''x'' + ''iy'' in the [[upper half-plane]] is defined by

			:<math>E(z,s) ={1\over 2}\sum_{(m,n)=1}{y^s\over\|mz+n\|^{2s}}</math>

			for Re(''s'') > 1, and by analytic continuation for other values of the complex number ''s''. The sum is over all pairs of coprime integers.

			'''Warning''': there are several other slightly different definitions. Some authors omit the factor of ½, and some sum over all pairs of integers that are not both zero; which changes the function by a factor of ζ(2''s'').

			==Properties==

			===As a function on ''z''===
			Viewed as a function of ''z'', ''E''(''z'',''s'') is a real-analytic [[eigenfunction]] of the [[Laplace operator]] on '''H''' with the eigenvalue ''s''(''s''-1). In other words, it satisfies the [[elliptic partial differential equation]]

			: <math> y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)E(z,s) = s(s-1)E(z,s), </math> &ensp;&ensp; where <math>z=x+yi.</math>

			The function ''E''(''z'', ''s'') is invariant under the action of SL(2,'''Z''') on ''z'' in the upper half plane by [[fractional linear transformation]]s. Together with the previous property, this means that the Eisenstein series is a [[Maass form]], a real-analytic analogue of a classical elliptic [[modular function]].

			'''Warning''': ''E''(''z'', ''s'') is not a square-integrable function of ''z'' with respect to the invariant Riemannian metric on '''H'''.

			===As a function on ''s''===
			The Eisenstein series converges for Re(''s'')>1, but can be [[analytic continuation\|analytically continued]] to a meromorphic function of ''s'' on the entire complex plane, with a unique pole of residue π at ''s'' = 1 (for all ''z'' in '''H'''). The constant term of the pole at ''s'' = 1 is described by the [[Kronecker limit formula]].

			The modified function

			:<math> E^*(z,s) = \pi^{-s}\Gamma(s)\zeta(2s)E(z,s)\ </math>

			satisfies the functional equation

			:<math>E^(z,s) = E^(z,1-s)\ </math>

			analogous to the functional equation for the [[Riemann zeta function]] ζ(''s'').

			Scalar product of two different Eisenstein series ''E''(''z'', ''s'') and ''E''(''z'', ''t'') is given by the [[Maass-Selberg relation]]s.

			==Epstein zeta function==
			The '''Epstein zeta function''' ζ<sub>''Q''</sub>(''s'') {{harv\|Epstein\|1903}} for a positive definite integral quadratic form ''Q''(''m'', ''n'') = ''cm''<sup>2</sup> + ''bmn'' +''an''<sup>2</sup> is defined by

			:<math> \zeta_Q(s) = \sum_{(m,n)\ne (0,0)} {1\over Q(m,n)^s}.\ </math>

			It is essentially a special case of the real analytic Eisenstein series for a special value of ''z'', since

			:<math> Q(m,n) = a\|mz+n\|^2\ </math>

			for

			:<math> z = {-b\over 2a} + i {\sqrt{4ac-b^2}\over 2a}.\ </math>

			This zeta function was named after [[Paul Epstein]].

			==Generalizations==
			The real analytic Eisenstein series ''E''(''z'', ''s'') is really the Eisenstein series associated to the discrete subgroup [[modular group\|SL(2,'''Z''')]] of [[SL2(R)\|SL(2,'''R''')]]. [[Atle Selberg\|Selberg]] described generalizations to other discrete subgroups Γ of SL(2,'''R'''), and used these to study the representation of SL(2,'''R''') on L<sup>2</sup>(SL(2,'''R''')/Γ). [[Robert Langlands\|Langlands]] extended Selberg's work to higher dimensional groups; his notoriously difficult proofs were later simplified by [[Joseph Bernstein]].

			==See also==
			*[[Eisenstein series]]
			*[[Kronecker limit formula]]

			==References==
			*J. Bernstein, [http://www.math.uchicago.edu/~arinkin/langlands/Bernstein/ ''Meromorphic continuation of Eisenstein series'']
			*{{citation\|last=Epstein\|first=P.\|authorlink=Paul Epstein\|title=Zur Theorie allgemeiner Zetafunktionen I\|journal=Math. Ann.\|volume=56\|pages=614–644\|year=1903\|doi=10.1007/BF01444309\|issue=4}}.
			*{{springer\|id=E/e120130\|title=Epstein zeta-function\|author=A. Krieg}}
			*{{citation\|first=T.\|last=Kubota\|authorlink=Tomio Kubota\|title=Elementary theory of Eisenstein series\|isbn=0-470-50920-1\|year=1973\|publisher=Kodansha\|location=Tokyo}}.
			*{{citation\|first=Robert P.\|last=Langlands\|url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/automorphic.html\|title=On the functional equations satisfied by Eisenstein series\|isbn=0-387-07872-X\|year=1976\|publisher=Springer-Verlag\|location=Berlin}}.
			*A. Selberg, ''Discontinuous groups and harmonic analysis'', Proc. Int. Congr. Math., 1962.
			*[[D. Zagier]], ''Eisenstein series and the Riemann zeta-function''.
			[[Category:Modular forms]]
			[[Category:Special functions]]
			[[Category:Representation theory of Lie groups]]
			[[Category:Analytic number theory]]

en>Cydebot: Robot - Speedily moving category Tiling to :Category:Tessellation per CFDS.

2012-02-16T15:52:43Z

Robot - Speedily moving category Tiling to Category:Tessellation per CFDS.

New page

Golda [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc free psychic] is what's created on my beginning certificate although it is not the name on my birth certificate. Distributing production is where my primary income comes from and it's some thing I really appreciate. I am really fond of handwriting but I can't make it my occupation really. Kentucky is exactly where I've always been residing.<br><br>Also visit my weblog :: psychics [http://kpupf.com/xe/talk/735373 free online tarot card readings], [http://Galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 just click the up coming web site],