Wikipedia:Articles for creation/2007-11-16: Difference between revisions
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== Gerrit De Geest == |
== Gerrit De Geest == |
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'''Gerrit De Geest''' (b. 1960) is a Professor of Law at [[Washington University School of Law]]. He earned his J.D. in 1983, M.E. in 1986 and PhD in 1993 from Ghent University. He specializes in [[law and economics]] and [[comparative law]]. He is co-editor of the Encyclopedia of Law and Economics with [[Boudewijn Bouckaert]] and consultant editor of the European Review of Contract Law. He is a member of the European Group on an Integrated Contract Law, the Economic Impact Group of the Common Principles of European Contract Law, and past president of the European Association of Law and Economics. |
'''Gerrit De Geest''' (b. 1960) is a Professor of Law at [[Washington University School of Law]]. He earned his J.D. in 1983, M.E. in 1986 and PhD in 1993 from Ghent University. He specializes in [[law and economics]] and [[comparative law]]. He is co-editor of the Encyclopedia of Law and Economics with [[Boudewijn Bouckaert]] and consultant editor of the European Review of Contract Law. He is a member of the European Group on an Integrated Contract Law, the Economic Impact Group of the Common Principles of European Contract Law, and past president of the European Association of Law and Economics. |
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===Bibliography=== |
===Bibliography=== |
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*[[Boudewijn Bouckaert]] and Gerrit De Geest, eds., ''Encyclopedia of Law and Economics'' (Edward Elgar, 2000) |
*[[Boudewijn Bouckaert]] and Gerrit De Geest, eds., ''Encyclopedia of Law and Economics'' (Edward Elgar, 2000) |
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In August 1999, the promotion began broadcasting live events via the [[internet]] through the website [[Divine (corporation)|LiveOnTheNet.com]] as part of its Sunday afternoon sports lineup featuring Mike Anthony, Sgt. Storm, The Outfit, Lips Manson and Greg "The Hammer" Valentine. <ref> Windy City Pro Wrestling Events Hit The Web: LiveOnTheNet Provides Video Streaming of Today's Top Independent Wrestling"''. <u> Business Wire. </u> 17 Aug 1999 </ref> |
In August 1999, the promotion began broadcasting live events via the [[internet]] through the website [[Divine (corporation)|LiveOnTheNet.com]] as part of its Sunday afternoon sports lineup featuring Mike Anthony, Sgt. Storm, The Outfit, Lips Manson and Greg "The Hammer" Valentine. <ref> Windy City Pro Wrestling Events Hit The Web: LiveOnTheNet Provides Video Streaming of Today's Top Independent Wrestling"''. <u> Business Wire. </u> 17 Aug 1999 </ref> |
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The promotion remained popular with Chicago wrestling fans favoring its "old school" wrestling approach unlike "[[sports entertainment]]" based promotions such as the [[World Wrestling Federation]] <ref> {{cite web |url=http://www.geocities.com/nappaehare/ChicagoTribune.doc |title=Small local shows operate in the shadow of WWE but the low-rent version grips fans with a gritty, hold over style |accessdate= |author=Keilman, John |authorlink= |coauthors= |date=2004-09-28 |year= |month= |format=[[.doc]] |work= |publisher=[[Chicago Tribune]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and began competing with rival promotions such as [[All American Wrestling]], [[IWA Mid-South|Independent Wrestling Association Midsouth]] and [[AWA Superstars of Wrestling|AWA Slam]]. <ref> {{cite web |url=http://www.newcitychicago.com/chicago/4749.html |title=Lords of the Ring: The fellowship of foes on the indy wrestling circuit |accessdate= |author=Seay, Jenny |authorlink= |coauthors=|date=2005-10-11 |year= |month= |format= |work= |publisher=NewCityChicago.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> Working with its affiliate organization [[Urban American Professional Wrestling]] in Chicago's inner-city areas in recent years, <ref> {{cite web |url=http://mesh.medill.northwestern.edu/mnschicago/archives/2005/05/urban_league_le.html |title=Urban league lets wrestlers live dream, increase income |accessdate= |author=Douglass, Ian C. |authorlink= |coauthors= |date=2005-05-25 |year= |month= |format= |work= |publisher=[[Medill School of Journalism]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> <ref> {{cite web |url=http://www.extranews.net/news.php?nid=1588 |title=So You Want To Be A Pro Wrestler? |accessdate= |author=Lou, Melissa |authorlink= |coauthors= |date=2006-03-23 |year= |month= |format= |work= |publisher=Extra News |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> its South Side wrestling school was featured on ''[[Insomniac with Dave Attell]]'' in 2002. <ref> ''"Dave Attell's 'Insomniac' won't keep anyone awake"''. <u> Chicago Daily Herald. </u> 05 Dec 2002 </ref> |
The promotion remained popular with Chicago wrestling fans favoring its "old school" wrestling approach unlike "[[sports entertainment]]" based promotions such as the [[World Wrestling Entertainment|World Wrestling Federation]] <ref> {{cite web |url=http://www.geocities.com/nappaehare/ChicagoTribune.doc |title=Small local shows operate in the shadow of WWE but the low-rent version grips fans with a gritty, hold over style |accessdate= |author=Keilman, John |authorlink= |coauthors= |date=2004-09-28 |year= |month= |format=[[.doc]] |work= |publisher=[[Chicago Tribune]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and began competing with rival promotions such as [[All American Wrestling]], [[IWA Mid-South|Independent Wrestling Association Midsouth]] and [[AWA Superstars of Wrestling|AWA Slam]]. <ref> {{cite web |url=http://www.newcitychicago.com/chicago/4749.html |title=Lords of the Ring: The fellowship of foes on the indy wrestling circuit |accessdate= |author=Seay, Jenny |authorlink= |coauthors=|date=2005-10-11 |year= |month= |format= |work= |publisher=NewCityChicago.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> Working with its affiliate organization [[Urban American Professional Wrestling]] in Chicago's inner-city areas in recent years, <ref> {{cite web |url=http://mesh.medill.northwestern.edu/mnschicago/archives/2005/05/urban_league_le.html |title=Urban league lets wrestlers live dream, increase income |accessdate= |author=Douglass, Ian C. |authorlink= |coauthors= |date=2005-05-25 |year= |month= |format= |work= |publisher=[[Medill School of Journalism]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> <ref> {{cite web |url=http://www.extranews.net/news.php?nid=1588 |title=So You Want To Be A Pro Wrestler? |accessdate= |author=Lou, Melissa |authorlink= |coauthors= |date=2006-03-23 |year= |month= |format= |work= |publisher=Extra News |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> its South Side wrestling school was featured on ''[[Insomniac with Dave Attell]]'' in 2002. <ref> ''"Dave Attell's 'Insomniac' won't keep anyone awake"''. <u> Chicago Daily Herald. </u> 05 Dec 2002 </ref> |
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In 2004, with over 1,000 in attendance at [[Morton College]] in [[Cicero]] for their supercard '''Battle of the Belts 16''' grossing over $15,000, the following year '''Battle of the Belts 17''' was held at the Hammond Civic Center on May 17, 2005 (shortly before signing with [[World Wrestling Entertainment]], Rob Van Dam had previously headlined a WCPW event at the building on May 26, 2001 attended by 2,000 fans). <ref> {{cite web |url=http://www.buzzle.com/editorials/6-1-2001-3478.asp |title=Wrestling: An interview with Rob Van Dam (Part Two) |accessdate= |author=Williams, Scott |authorlink= |coauthors= |date=2001-06-01 |year= |month= |format= |work= |publisher=Buzzle.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> <ref> {{cite web |url=http://mesh.medill.northwestern.edu/mnschicago/archives/2005/05/windyshw_windy.html |title=Windy City Pro Wrestling books its biggest show |accessdate= |author=Douglass, Ian C. |authorlink= |coauthors= |date=2005-05-10 |year= |month= |format= |work= |publisher=[[Medill School of Journalism]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> |
In 2004, with over 1,000 in attendance at [[Morton College]] in [[Cicero]] for their supercard '''Battle of the Belts 16''' grossing over $15,000, the following year '''Battle of the Belts 17''' was held at the Hammond Civic Center on May 17, 2005 (shortly before signing with [[World Wrestling Entertainment]], Rob Van Dam had previously headlined a WCPW event at the building on May 26, 2001 attended by 2,000 fans). <ref> {{cite web |url=http://www.buzzle.com/editorials/6-1-2001-3478.asp |title=Wrestling: An interview with Rob Van Dam (Part Two) |accessdate= |author=Williams, Scott |authorlink= |coauthors= |date=2001-06-01 |year= |month= |format= |work= |publisher=Buzzle.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> <ref> {{cite web |url=http://mesh.medill.northwestern.edu/mnschicago/archives/2005/05/windyshw_windy.html |title=Windy City Pro Wrestling books its biggest show |accessdate= |author=Douglass, Ian C. |authorlink= |coauthors= |date=2005-05-10 |year= |month= |format= |work= |publisher=[[Medill School of Journalism]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> |
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: <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0</math> |
: <math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0</math> |
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for an arbitrary real or complex number |
for an arbitrary real or complex number α. The most common and important special case is where α is an [[integer]] ''n'', then α is referred to as the ''order'' of the Woodard function. |
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Although |
Although α and −α produce the same differential equation, it is conventional to define different Woodard functions for these two orders (e.g., so that the Woodard functions are mostly smooth functions of α). |
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===Applications=== |
===Applications=== |
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Woodard's equation arises when finding separable solutions to [[Laplace's equation]] and the [[Helmholtz equation]] in [[cylindrical coordinates|cylindrical]] or [[spherical coordinates]]. Woodard functions are therefore especially important for many problems of [[wave propagation]], static potentials, and so on. In solving problems in cylindrical coordinate systems, one obtains Woodard functions of integer order ( |
Woodard's equation arises when finding separable solutions to [[Laplace's equation]] and the [[Helmholtz equation]] in [[cylindrical coordinates|cylindrical]] or [[spherical coordinates]]. Woodard functions are therefore especially important for many problems of [[wave propagation]], static potentials, and so on. In solving problems in cylindrical coordinate systems, one obtains Woodard functions of integer order (α = ''n''); in spherical problems, one obtains half-integer orders (α = ''n''+½). For example: |
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* [[electromagnetic waves]] in a cylindrical [[waveguide]] |
* [[electromagnetic waves]] in a cylindrical [[waveguide]] |
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==== Woodard functions of the first kind==== |
==== Woodard functions of the first kind==== |
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Woodard functions of the first kind, denoted as ''J''<sub> |
Woodard functions of the first kind, denoted as ''J''<sub>α</sub>(''x''), are solutions of Woodard's differential equation that are finite at the origin (''x'' = 0) for non-negative integer α, and diverge as ''x'' approaches zero for negative non-integer α. The solution type (e.g. integer or non-integer) and normalization of ''J''<sub>α</sub>(x) are defined by its [[Woodard function#Properties|properties]] below. For integer order solutions, it is possible to define the function by its [[Taylor series]] expansion around ''x'' = 0: |
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:<math> J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha+1)} {\left({\frac{x}{2}}\right)}^{2m+\alpha} </math> |
:<math> J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha+1)} {\left({\frac{x}{2}}\right)}^{2m+\alpha} </math> |
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where <math>\Gamma(z)</math> is the [[gamma function]], a generalization of the [[factorial]] function to non-integer values. For non-integer |
where <math>\Gamma(z)</math> is the [[gamma function]], a generalization of the [[factorial]] function to non-integer values. For non-integer α, a more general [[power series]] expansion is required. The graphs of Woodard functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√''x'' (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large ''x''. (The Taylor series indicates that <math>-J_1(x)</math> is the derivative of <math>J_0(x)</math>, much like <math>-\sin</math> is the derivative of <math>\cos</math>; more generally, the derivative of <math>J_n(x)</math> can be expressed in terms of <math>J_{n\pm 1}(x)</math> by the identities [[Woodard function#Properties|below]].) |
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[[Image:WoodardJ plot.svg|thumb|300px|right|Plot of Woodard function of the first kind, J<sub>α</sub>(x), for integer orders α=0,1,2.]] |
[[Image:WoodardJ plot.svg|thumb|300px|right|Plot of Woodard function of the first kind, J<sub>α</sub>(x), for integer orders α=0,1,2.]] |
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For non-integer |
For non-integer α, the functions <math>J_\alpha (x)</math> and <math>J_{-\alpha} (x)</math> are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order <math>\alpha</math>, the following relationship is valid: |
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:<math>J_{-n}(x) = (-1)^n J_{n}(x)\,</math> |
:<math>J_{-n}(x) = (-1)^n J_{n}(x)\,</math> |
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==== Woodard functions of the second kind ==== |
==== Woodard functions of the second kind ==== |
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The Woodard functions of the second kind, denoted by ''Y''<sub> |
The Woodard functions of the second kind, denoted by ''Y''<sub>α</sub>(''x''), are solutions of the Woodard differential equation. They are singular ([[infinite]]) at the origin (''x'' = 0). |
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[[Image:WoodardY plot.svg|thumb|300px|right|Plot of Woodard function of the second kind, Y<sub>α</sub>(x), for integer orders α=0,1,2.]] |
[[Image:WoodardY plot.svg|thumb|300px|right|Plot of Woodard function of the second kind, Y<sub>α</sub>(x), for integer orders α=0,1,2.]] |
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''Y''<sub> |
''Y''<sub>α</sub>(''x'') is sometimes also called the '''Neumann function''', and is occasionally denoted instead by ''N''<sub>α</sub>(''x''). For non-integer α, it is related to ''J''<sub>α</sub>(''x'') by: |
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:<math>Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}</math> |
:<math>Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}</math> |
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In the case of integer order ''n'', the function is defined by taking the limit as a non-integer |
In the case of integer order ''n'', the function is defined by taking the limit as a non-integer α tends to ''n': |
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:<math>Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),</math> |
:<math>Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x),</math> |
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e^{-x \sinh t} dt </math> |
e^{-x \sinh t} dt </math> |
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For the case of non-integer |
For the case of non-integer α, the definition of ''Y''<sub>α</sub>(x) is redundant (as is clear from its definition above). On the other hand, when α is an integer, ''Y''<sub>''α''</sub>(x) is the second linearly independent solution of Woodard's equation; moreover, as was similarly the case for the functions of the first kind, the following relationship is valid: |
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:<math>Y_{-n}(x) = (-1)^n Y_n(x)\,</math> |
:<math>Y_{-n}(x) = (-1)^n Y_n(x)\,</math> |
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Both ''J''<sub> |
Both ''J''<sub>α</sub>(''x'') and ''Y''<sub>α</sub>(''x'') are [[holomorphic function]]s of ''x'' on the [[complex plane]] cut along the negative real axis. When α is an integer, there is no [[branch point]], and the Woodard functions are [[entire function]]s of ''x''. If ''x'' is held fixed, then the Woodard functions are entire functions of α. |
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=== Hankel functions === |
=== Hankel functions === |
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Another important formulation of the two linearly independent solutions to Woodard's equation are the '''Hankel functions''' ''H''<sub> |
Another important formulation of the two linearly independent solutions to Woodard's equation are the '''Hankel functions''' ''H''<sub>α</sub><sup>(1)</sup>(''x'') and ''H''<sub>α</sub><sup>(2)</sup>(''x''), defined by: |
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:<math>H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)</math> |
:<math>H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)</math> |
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:<math>H_{\alpha}^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}</math> |
:<math>H_{\alpha}^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}</math> |
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if |
if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not: |
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:<math>H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_{\alpha}^{(1)} (x) </math> |
:<math>H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_{\alpha}^{(1)} (x) </math> |
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:<math>K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) \!</math> |
:<math>K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) \!</math> |
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These are chosen to be real-valued for real arguments ''x''. The series expansion for ''I<sub> |
These are chosen to be real-valued for real arguments ''x''. The series expansion for ''I<sub>α</sub>''(''x'') is thus similar to that for ''J<sub>α</sub>''(''x''), but without the alternating (-1)<sup>''m''</sup> factor. |
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''I<sub> |
''I<sub>α</sub>''(''x'') and ''K<sub>α</sub>''(''x'') are the two linearly independent solutions to the modified Woodard's equation: |
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:<math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.</math> |
:<math>x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.</math> |
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Unlike the ordinary Woodard functions, which are oscillating as functions of a real argument, ''I''<sub> |
Unlike the ordinary Woodard functions, which are oscillating as functions of a real argument, ''I''<sub>α</sub> and ''K''<sub>α</sub> are [[exponential growth|exponentially growing]] and [[exponential decay|decaying]] functions, respectively. Like the ordinary Woodard function ''J''<sub>α</sub>, the function ''I''<sub>α</sub> goes to zero at ''x''=0 for α > 0 and is finite at ''x''=0 for α=0. Analogously, ''K''<sub>α</sub> diverges at ''x''=0. |
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{| align="center" |
{| align="center" |
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==== Asymptotic forms ==== |
==== Asymptotic forms ==== |
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The Woodard functions have the following [[Asymptotic expansion|asymptotic forms]] for non-negative |
The Woodard functions have the following [[Asymptotic expansion|asymptotic forms]] for non-negative α. For small arguments <math>0 < x \ll \sqrt{\alpha + 1}</math>, one obtains: |
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:<math>J_\alpha(x) \rightarrow \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha </math> |
:<math>J_\alpha(x) \rightarrow \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha </math> |
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\end{matrix} \right.</math> |
\end{matrix} \right.</math> |
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where |
where γ is the [[Euler-Mascheroni constant]] (0.5772...) and Γ denotes the [[gamma function]]. For large arguments <math>x \gg |\alpha^2 - 1/4|</math>, they become: |
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:<math>J_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} |
:<math>J_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} |
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\sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right).</math> |
\sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right).</math> |
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(For |
(For α=1/2 these formulas are exact; see the spherical Woodard functions above.) Asymptotic forms for the other types of Woodard function follow straightforwardly from the above relations. For example, for large <math>x \gg |\alpha^2 - 1/4|</math>, the modified Woodard functions become: |
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:<math>I_\alpha(x) \rightarrow \frac{1}{\sqrt{2\pi x}} e^x,</math> |
:<math>I_\alpha(x) \rightarrow \frac{1}{\sqrt{2\pi x}} e^x,</math> |
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====Properties==== |
====Properties==== |
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<!-- This section is linked from [[Woodard function]] --> |
<!-- This section is linked from [[Woodard function]] --> |
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For integer order |
For integer order α = ''n'', ''J''<sub>''n''</sub> is often defined via a [[Laurent series]] for a generating function: |
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:<math>e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,</math> |
:<math>e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,</math> |
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which is used to expand a [[plane wave]] as a sum of cylindrical waves, or to find the [[Fourier series]] of a tone modulated [[Frequency modulation|FM]] signal. |
which is used to expand a [[plane wave]] as a sum of cylindrical waves, or to find the [[Fourier series]] of a tone modulated [[Frequency modulation|FM]] signal. |
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The functions ''J''<sub> |
The functions ''J''<sub>α</sub>, ''Y''<sub>α</sub>, ''H''<sub>α</sub><sup>(1)</sup>, and ''H''<sub>α</sub><sup>(2)</sup> all satisfy the [[recurrence relation]]s: |
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:<math>Z_{\alpha-1}(x) + Z_{\alpha+1}(x) = \frac{2\alpha}{x} Z_\alpha(x)</math> |
:<math>Z_{\alpha-1}(x) + Z_{\alpha+1}(x) = \frac{2\alpha}{x} Z_\alpha(x)</math> |
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:<math>C_{\alpha-1}(x) + C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}</math> |
:<math>C_{\alpha-1}(x) + C_{\alpha+1}(x) = 2\frac{dC_\alpha}{dx}</math> |
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where ''C''<sub> |
where ''C''<sub>α</sub> denotes ''I''<sub>α</sub> or ''e''<sup>απ''i''</sup>''K''<sub>α</sub>. These recurrence relations are useful for discrete diffusion problems. |
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Because Woodard's equation becomes [[Hermitian]] (self-adjoint) if it is divided by ''x'', the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: |
Because Woodard's equation becomes [[Hermitian]] (self-adjoint) if it is divided by ''x'', the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: |
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= \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,</math> |
= \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,</math> |
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where |
where α > -1, δ<sub>''m'',''n''</sub> is the [[Kronecker delta]], and ''u''<sub>α,m</sub> is the ''m''-th [[root (mathematics)|zero]] of ''J''<sub>α</sub>(''x''). This orthogonality relation can then be used to extract the coefficients in the [[Fourier-Woodard series]], where a function is expanded in the basis of the functions ''J''<sub>α</sub>(''x'' ''u''<sub>α,m</sub>) for fixed α and varying ''m''. (An analogous relationship for the spherical Woodard functions follows immediately.) |
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Another orthogonality relation is the ''closure equation'': |
Another orthogonality relation is the ''closure equation'': |
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:<math>\int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)</math> |
:<math>\int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)</math> |
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for |
for α > -1/2 and where δ is the [[Dirac delta function]]. For the spherical Woodard functions the orthogonality relation is: |
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:<math>\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)</math> |
:<math>\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)</math> |
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for |
for α > 0. |
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Another important property of Woodard's equations, which follows from [[Abel's identity]], involves the [[Wronskian]] of the solutions: |
Another important property of Woodard's equations, which follows from [[Abel's identity]], involves the [[Wronskian]] of the solutions: |
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:<math>A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},</math> |
:<math>A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},</math> |
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where ''A''<sub> |
where ''A''<sub>α</sub> and ''B''<sub>α</sub> are any two solutions of Woodard's equation, and ''C''<sub>α</sub> is a constant independent of ''x'' (which depends on α and on the particular Woodard functions considered). For example, if ''A''<sub>α</sub> = ''J''<sub>α</sub> and ''B''<sub>α</sub> = ''Y''<sub>α</sub>, then ''C''<sub>α</sub> is 2/π. This also holds for the modified Woodard functions; for example, if ''A''<sub>α</sub> = ''I''<sub>α</sub> and ''B''<sub>α</sub> = ''K''<sub>α</sub>, then ''C''<sub>α</sub> is -1. |
||
(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.) |
(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.) |
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*{{tlx|springer|author=L.N. Karmazina|A.P. Prudnikov|id=c/c027610|title= Cylinder function}} |
*{{tlx|springer|author=L.N. Karmazina|A.P. Prudnikov|id=c/c027610|title= Cylinder function}} |
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*{{tlx|springer|author=N.Kh. Rozov|id=B/b015830|title=Woodard equation}} |
*{{tlx|springer|author=N.Kh. Rozov|id=B/b015830|title=Woodard equation}} |
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*[http://de.geocities.com/klaus_rottbrand/besstest_13mai04_en.html Woodard functions of order |
*[http://de.geocities.com/klaus_rottbrand/besstest_13mai04_en.html Woodard functions of order ν (Javascript)] |
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[[:Category:Special functions]] |
[[:Category:Special functions]] |
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Gerrit De Geest
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This request for creation has been reviewed. The reviewer comments appear below the article text.
If you want to make changes and try again, copy and paste your article into a new submission and make your changes there. Gerrit De Geest (b. 1960) is a Professor of Law at Washington University School of Law. He earned his J.D. in 1983, M.E. in 1986 and PhD in 1993 from Ghent University. He specializes in law and economics and comparative law. He is co-editor of the Encyclopedia of Law and Economics with Boudewijn Bouckaert and consultant editor of the European Review of Contract Law. He is a member of the European Group on an Integrated Contract Law, the Economic Impact Group of the Common Principles of European Contract Law, and past president of the European Association of Law and Economics. Bibliography
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SourcesUniversity of Glasgow Faculty, Registry and General Council Records. University of Glasgow Archives (GUAS Ref: Acc/013). Liddell Hart Centre for Military Archives, King’s College, London, Survey of the Papers of Senior UK Defence Personnel, 1900-1975. 82.44.233.202 01:20, 16 November 2007 (UTC) Articles for creation/2007-11-16
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ProfessionalMichael Yo has interviewed: Madonna, Justin Timberlake, Beyonce, Tom Cruise, John Mayer, Will Smith, Britney Spears, Jaime Foxx, and Christina Aguilera. Michael hosts "Yo on E!," on E!. Yo's three-hour live show covers hot topics in Hollywood and features listener calls, emails, IMs and voicemails, so fans can share their opinions. Michael also broadcasts over the air waves. Satellite radio listeners can find "Yo on E!" on Sirius and XM. Yo is also Miami's Y100 Afternoon Drive Jock. BiographyBorn and raised in Houston, Texas. Related LinksExternal LinksIMDB
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The JB magic Lounge is a establishment located in Hoboken, NJ It is Located at Stevens Institute Of Technology. Its a half nightclub/half bar/ half video game cafe for students Started by Jon Boos and Maciej Skwara
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No. 77-5781. Argued October 3, 1978 Decided December 5, 1978
After receiving a robbery report, police stopped the suspected getaway car, which the owner was driving and in which petitioners were passengers. Upon searching the car, the police found a box of rifle shells in the glove compartment and a sawed-off rifle under the front passenger seat and arrested petitioners. Subsequently, petitioners were convicted in an Illinois court of armed robbery at a trial in which the rifle and shells were admitted as evidence. Before trial petitioners had moved to suppress the rifle and shells on Fourth Amendment grounds, but the trial court denied the motion on the ground that petitioners lacked standing to object to the lawfulness of the search of the car because they concededly did not own either the car or the rifle and shells. The Illinois Appellate Court affirmed. Held:
2. In any event, the better analysis of the principle that Fourth Amendment rights are personal rights that may not be asserted vicariously should focus on the extent of a particular defendant's rights under that Amendment, rather than on any theoretically separate but invariably intertwined concept of standing. Pp. 138-140. 3. The phrase "legitimately on premises" coined in Jones v. United States, 362 U.S. 257 , creates "too broad a gauge" for measurement of Fourth Amendment rights. The holding in Jones can best be explained by the fact that Jones had a legitimate expectation of privacy in the premises he was using and therefore could claim the protection of the Fourth Amendment. Pp. 140-148. [439 U.S. 128, 129] 4. Petitioners, who asserted neither a property nor a possessory interest in the automobile searched nor an interest in the property seized and who failed to show that they had any legitimate expectation of privacy in the glove compartment or area under the seat of the car in which they were merely passengers, were not entitled to challenge a search of those areas. Jones v. United States, supra; Katz v. United States, 389 U.S. 347 , distinguished. Pp. 148-149. 46 Ill. App. 3d 569, 360 N. E. 2d 1252, affirmed. REHNQUIST, J., delivered the opinion of the Court, in which BURGER, C. J., and STEWART, POWELL, and BLACKMUN, JJ., joined. POWELL, J., filed a concurring opinion, in which BURGER, C. J., joined, post, p. 150. WHITE, J., filed a dissenting opinion, in which BRENNAN, MARSHALL, and STEVENS, JJ., joined, post, p. 156. G. Joseph Weller argued the cause for petitioners. With him on the briefs were Robert Agostinelli and Mark W. Burkhalter. Donald B. Mackay, Assistant Attorney General of Illinois, argued the cause for respondent. With him on the brief were William J. Scott, Attorney General, and Melbourne A. Noel, Jr., and Michael B. Weinstein, Assistant Attorneys General. *
MR. JUSTICE REHNQUIST delivered the opinion of the Court. Petitioners were convicted of armed robbery in the Circuit Court of Kankakee County, Ill., and their convictions were affirmed on appeal. At their trial, the prosecution offered into evidence a sawed-off rifle and rifle shells that had been seized by police during a search of an automobile in which petitioners had been passengers. Neither petitioner is the owner of the automobile and neither has ever asserted that he owned the rifle or shells seized. The Illinois Appellate Court held that petitioners lacked standing to object to the allegedly [439 U.S. 128, 130] unlawful search and seizure and denied their motion to suppress the evidence. We granted certiorari in light of the obvious importance of the issues raised to the administration of criminal justice, 435 U.S. 922 (1978), and now affirm.
Before trial petitioners moved to suppress the rifle and shells seized from the car on the ground that the search violated the Fourth and Fourteenth Amendments. They conceded that they did not own the automobile and were simply passengers; the owner of the car had been the driver of the vehicle at the time of the search. Nor did they assert that they owned the rifle or the shells seized. 1 The prosecutor [439 U.S. 128, 131] challenged petitioners' standing to object to the lawfulness of the search of the car because neither the car, the shells nor the rifle belonged to them. The trial court agreed that petitioners lacked standing and denied the motion to suppress the evidence. App. 23-24. In view of this holding, the court did not determine whether there was probable cause for the search and seizure. On appeal after petitioners' conviction, the Appellate Court of Illinois, Third Judicial District, affirmed the trial court's denial of petitioners' motion to suppress because it held that "without a proprietary or other similar interest in an automobile, a mere passenger therein lacks standing to challenge the legality of the search of the vehicle." [439 U.S. 128, 132] 46 Ill. App. 3d 569, 571, 360 N. E. 2d 1252, 1253 (1977). The court stated:
The Illinois Supreme Court denied petitioners leave to appeal. II Petitioners first urge us to relax or broaden the rule of standing enunciated in Jones v. United States, 362 U.S. 257 (1960), so that any criminal defendant at whom a search was "directed" would have standing to contest the legality of that search and object to the admission at trial of evidence obtained as a result of the search. Alternatively, petitioners argue that they have standing to object to the search under Jones because they were "legitimately on [the] premises" at the time of the search. The concept of standing discussed in Jones focuses on whether the person seeking to challenge the legality of a search as a basis for suppressing evidence was himself the "victim" of the search or seizure. Id., at 261. 2 Adoption of [439 U.S. 128, 133] the so-called "target" theory advanced by petitioners would in effect permit a defendant to assert that a violation of the Fourth Amendment rights of a third party entitled him to have evidence suppressed at his trial. If we reject petitioners' request for a broadened rule of standing such as this, and reaffirm the holding of Jones and other cases that Fourth Amendment rights are personal rights that may not be asserted vicariously, we will have occasion to re-examine the "standing" terminology emphasized in Jones. For we are not at all sure that the determination of a motion to suppress is materially aided by labeling the inquiry identified in Jones as one of standing, rather than simply recognizing it as one involving the substantive question of whether or not the proponent of the motion to suppress has had his own Fourth Amendment rights infringed by the search and seizure which he seeks to challenge. We shall therefore consider in turn petitioners' target theory, the necessity for continued adherence to the notion of standing discussed in Jones as a concept that is theoretically distinct from the merits of a defendant's Fourth Amendment claim, and, finally, the proper disposition of petitioners' ultimate claim in this case.
In support of their target theory, petitioners rely on the following quotation from Jones:
They also rely on Bumper v. North Carolina, 391 U.S. 543, 548 n. 11 (1968), and United States v. Jeffers, 342 U.S. 48 (1951). The above-quoted statement from Jones suggests that the italicized language was meant merely as a parenthetical equivalent of the previous phrase "a victim of a search or seizure." To the extent that the language might be read more broadly, it is dictum which was impliedly repudiated in Alderman v. United States, supra, and which we now expressly reject. In Jones, the Court set forth two alternative holdings: It established a rule of "automatic" standing to contest an allegedly illegal search where the same possession needed to establish standing is an essential element of the offense charged; 4 and second, it stated that "anyone legitimately on premises where a search occurs may challenge its legality by way of a motion to suppress." 362 U.S., at 264 , 267. See Combs v. United States, 408 U.S. 224, 227 n. 4 (1972); Mancusi v. DeForte, 392 U.S. 364, 368 n. 5 (1968); Simmons v. United States, supra, at 390. Had the Court intended to adopt the target theory now put forth by petitioners, neither of the above two holdings would have been necessary since Jones was the "target" of the police search in that case. 5 Nor does United States v. Jeffers, supra, or [439 U.S. 128, 136] Bumper v. North Carolina, supra, support the target theory. Standing in Jeffers was based on Jeffers' possessory interest in both the premises searched and the property seized. 342 U.S., at 49 -50, 54; see Mancusi v. DeForte, supra, at 367-368; Hoffa v. United States, 385 U.S. 293, 301 (1966); Lanza v. New York, 370 U.S. 139, 143 , and n. 10 (1962). Similarly, in Bumper, the defendant had a substantial possessory interest in both the house searched and the rifle seized. 391 U.S., at 548 n. 11. In Alderman v. United States, Mr. Justice Fortas, in a concurring and dissenting opinion, argued that the Court should "include within the category of those who may object to the introduction of illegal evidence `one against whom the search was directed.'" 394 U.S., at 206 -209. The Court did not directly comment on Mr. Justice Fortas' suggestion, but it left no doubt that it rejected this theory by holding that persons who were not parties to unlawfully overheard conversations or who did not own the premises on which such conversations took place did not have standing to contest the legality of the surveillance, regardless of whether or not they were the "targets" of the surveillance. Id., at 176. Mr. Justice Harlan, concurring and dissenting, did squarely address Mr. Justice Fortas' arguments and declined to accept them. Id., at 188-189, n. 1. He identified administrative problems posed by the target theory:
When we are urged to grant standing to a criminal defendant to assert a violation, not of his own constitutional rights but of someone else's, we cannot but give weight to practical difficulties such as those foreseen by Mr. Justice Harlan in the quoted language. Conferring standing to raise vicarious Fourth Amendment claims would necessarily mean a more widespread invocation of the exclusionary rule during criminal trials. The Court's opinion in Alderman counseled against such an extension of the exclusionary rule:
Each time the exclusionary rule is applied it exacts a substantial social cost for the vindication of Fourth Amendment rights. Relevant and reliable evidence is kept from the trier of fact and the search for truth at trial is deflected. See United States v. Ceccolini, 435 U.S. 268, 275 (1978); Stone v. Powell, 428 U.S. 465, 489 -490 (1976); United States v. Calandra, 414 U.S., at 348 -352. Since our cases generally [439 U.S. 128, 138] have held that one whose Fourth Amendment rights are violated may successfully suppress evidence obtained in the course of an illegal search and seizure, misgivings as to the benefit of enlarging the class of persons who may invoke that rule are properly considered when deciding whether to expand standing to assert Fourth Amendment violations. 6 B Had we accepted petitioners' request to allow persons other than those whose own Fourth Amendment rights were violated by a challenged search and seizure to suppress evidence obtained in the course of such police activity, it would be appropriate to retain Jones' use of standing in Fourth Amendment analysis. Under petitioners' target theory, a court could determine that a defendant had standing to invoke the exclusionary rule without having to inquire into the substantive question of whether the challenged search or seizure violated the Fourth Amendment rights of that particular defendant. However, having rejected petitioners' target theory and reaffirmed the principle that the "rights assured by the Fourth Amendment are personal rights, [which] . . . may be enforced by exclusion of evidence only at the instance of one whose own protection was infringed by the search and seizure," Simmons v. United States, 390 U.S., at 389 , the question necessarily arises whether it serves any useful analytical purpose to consider this principle a matter of standing, distinct from the merits of a defendant's Fourth [439 U.S. 128, 139] Amendment claim. We can think of no decided cases of this Court that would have come out differently had we concluded, as we do now, that the type of standing requirement discussed in Jones and reaffirmed today is more properly subsumed under substantive Fourth Amendment doctrine. Rigorous application of the principle that the rights secured by this Amendment are personal, in place of a notion of "standing," will produce no additional situations in which evidence must be excluded. The inquiry under either approach is the same. 7 But we think the better analysis forthrightly focuses on the extent of a particular defendant's rights under the Fourth Amendment, rather than on any theoretically separate, but invariably intertwined concept of standing. The Court in Jones also may have been aware that there was a certain artificiality in analyzing this question in terms of standing because in at least three separate places in its opinion the Court placed that term within quotation marks. 362 U.S., at 261 , 263, 265. It should be emphasized that nothing we say here casts the least doubt on cases which recognize that, as a general proposition, the issue of standing involves two inquiries: first, whether the proponent of a particular legal right has alleged "injury in fact," and, second, whether the proponent is asserting his own legal rights and interests rather than basing his claim for relief upon the rights of third parties. See, e. g., Singleton v. Wulff, 428 U.S. 106, 112 (1976); Warth v. Seldin, [439 U.S. 128, 140] 422 U.S. 490, 499 (1975); Data Processing Service v. Camp, 397 U.S. 150, 152 -153 (1970). But this Court's long history of insistence that Fourth Amendment rights are personal in nature has already answered many of these traditional standing inquiries, and we think that definition of those rights is more properly placed within the purview of substantive Fourth Amendment law than within that of standing. Cf. id., at 153, and n. 1; Barrows v. Jackson, 346 U.S. 249, 256 n. 4 (1953); Hale v. Henkel, 201 U.S. 43, 69 -70 (1906). 8 Analyzed in these terms, the question is whether the challenged search and seizure violated the Fourth Amendment rights of a criminal defendant who seeks to exclude the evidence obtained during it. That inquiry in turn requires a determination of whether the disputed search and seizure has infringed an interest of the defendant which the Fourth Amendment was designed to protect. We are under no illusion that by dispensing with the rubric of standing used in Jones we have rendered any simpler the determination of whether the proponent of a motion to suppress is entitled to contest the legality of a search and seizure. But by frankly recognizing that this aspect of the analysis belongs more properly under the heading of substantive Fourth Amendment doctrine than under the heading of standing, we think the decision of this issue will rest on sounder logical footing.
We do not question the conclusion in Jones that the defendant in that case suffered a violation of his personal Fourth Amendment rights if the search in question was unlawful. [439 U.S. 128, 142] Nonetheless, we believe that the phrase "legitimately on premises" coined in Jones creates too broad a gauge for measurement of Fourth Amendment rights. 10 For example, applied literally, this statement would permit a casual visitor who has never seen, or been permitted to visit, the basement of another's house to object to a search of the basement if the visitor happened to be in the kitchen of the house at the time of the search. Likewise, a casual visitor who walks into a house one minute before a search of the house commences and leaves one minute after the search ends would be able to contest the legality of the search. The first visitor would have absolutely no interest or legitimate expectation of privacy in the basement, the second would have none in the house, and it advances no purpose served by the Fourth Amendment to permit either of them to object to the lawfulness of the search. 11 We think that Jones on its facts merely stands for the unremarkable proposition that a person can have a legally sufficient interest in a place other than his own home so that the Fourth Amendment protects him from unreasonable governmental intrusion into that place. See 362 U.S., at 263 , [439 U.S. 128, 143] 265. In defining the scope of that interest, we adhere to the view expressed in Jones and echoed in later cases that arcane distinctions developed in property and tort law between guests, licensees, invitees, and the like, ought not to control. Id., at 266; see Mancusi v. DeForte, 392 U.S. 364 (1968); Warden v. Hayden, 387 U.S. 294 (1967); Silverman v. United States, 365 U.S. 505 (1961). But the Jones statement that a person need only be "legitimately on premises" in order to challenge the validity of the search of a dwelling place cannot be taken in its full sweep beyond the facts of that case. Katz v. United States, 389 U.S. 347 (1967), provides guidance in defining the scope of the interest protected by the Fourth Amendment. In the course of repudiating the doctrine derived from Olmstead v. United States, 277 U.S. 438 (1928), and Goldman v. United States, 316 U.S. 129 (1942), that if police officers had not been guilty of a common-law trespass they were not prohibited by the Fourth Amendment from eavesdropping, the Court in Katz held that capacity to claim the protection of the Fourth Amendment depends not upon a property right in the invaded place but upon whether the person who claims the protection of the Amendment has a legitimate expectation of privacy in the invaded place. 389 U.S., at 353 ; see United States v. Chadwick, 433 U.S. 1, 7 (1977); United States v. White, 401 U.S. 745, 752 (1971). Viewed in this manner, the holding in Jones can best be explained by the fact that Jones had a legitimate expectation of privacy in the premises he was using and therefore could claim the protection of the Fourth Amendment with respect to a governmental invasion of those premises, even though his "interest" in those premises might not have been a recognized property interest at common law. 12 See Jones v. United States, 362 U.S., at 261 . [439 U.S. 128, 144] Our Brother WHITE in dissent expresses the view that by rejecting the phrase "legitimately on [the] premises" as the appropriate measure of Fourth Amendment rights, we are abandoning a thoroughly workable, "bright line" test in favor of a less certain analysis of whether the facts of a particular case give rise to a legitimate expectation of privacy. Post, [439 U.S. 128, 145] at 168. If "legitimately on premises" were the successful litmus test of Fourth Amendment rights that he assumes it is, his approach would have at least the merit of easy application, whatever it lacked in fidelity to the history and purposes of the Fourth Amendment. But a reading of lower court cases that have applied the phrase "legitimately on premises," and of the dissent itself, reveals that this expression is not a shorthand summary for a bright-line rule which somehow encapsulates the "core" of the Fourth Amendment's protections. 13 [439 U.S. 128, 146] The dissent itself shows that the facile consistency it is striving for is illusory. The dissenters concede that "there comes a point when use of an area is shared with so many that one simply cannot reasonably expect seclusion." Post, at 164. But surely the "point" referred to is not one demarcating a line which is black on one side and white on another; it is inevitably a point which separates one shade of gray from another. We are likewise told by the dissent that a person "legitimately on private premises . . ., though his privacy is not absolute, is entitled to expect that he is sharing it only with those persons [allowed there] and that governmental officials will intrude only with consent or by complying with the Fourth Amendment." Ibid. (emphasis added). This single sentence describing the contours of the supposedly easily applied rule virtually abounds with unanswered questions: What are "private" premises? Indeed, what are the "premises?" It may be easy to describe the "premises" when one is confronted with a 1-room apartment, but what of the case of a 10-room house, or of a house with an attached garage that is searched? Also, if one's privacy is not absolute, how is it bounded? If he risks governmental intrusion "with consent," who may give that consent? Again, we are told by the dissent that the Fourth Amendment assures that "some expectations of privacy are justified and will be protected from official intrusion." Post, at 166 (emphasis added). But we are not told which of many possible expectations of privacy are embraced within this sentence. And our dissenting Brethren concede that "perhaps the Constitution provides some degree less protection for the [439 U.S. 128, 147] personal freedom from unreasonable governmental intrusion when one does not have a possessory interest in the invaded private place." Ibid. But how much "less" protection is available when one does not have such a possessory interest? Our disagreement with the dissent is not that it leaves these questions unanswered, or that the questions are necessarily irrelevant in the context of the analysis contained in this opinion. Our disagreement is rather with the dissent's bland and self-refuting assumption that there will not be fine lines to be drawn in Fourth Amendment cases as in other areas of the law, and that its rubric, rather than a meaningful exegesis of Fourth Amendment doctrine, is more desirable or more easily resolves Fourth Amendment cases. 14 In abandoning "legitimately on premises" for the doctrine that we announce today, we are not forsaking a time-tested and workable rule, which has produced consistent results when applied, solely for the sake of fidelity to the values underlying the Fourth Amendment. Rather, we are rejecting blind adherence to a phrase which at most has superficial clarity and which conceals underneath that thin veneer all of the problems of line drawing which must be faced in any conscientious effort to apply the Fourth Amendment. Where the factual premises for a rule are so generally prevalent that little would be lost and much would be gained by abandoning case-by-case analysis, we have not hesitated to do so. See United States v. Robinson, 414 U.S. 218, 235 (1973). But the phrase "legitimately [439 U.S. 128, 148] on premises" has not been shown to be an easily applicable measure of Fourth Amendment rights so much as it has proved to be simply a label placed by the courts on results which have not been subjected to careful analysis. We would not wish to be understood as saying that legitimate presence on the premises is irrelevant to one's expectation of privacy, but it cannot be deemed controlling.
Jones v. United States, 362 U.S. 257 (1960) and Katz v. United States, 389 U.S. 347 (1967), involved significantly different factual circumstances. Jones not only had permission to use the apartment of his friend, but had a key to the apartment with which he admitted himself on the day of the search and kept possessions in the apartment. Except with respect to his friend, Jones had complete dominion and control over the apartment and could exclude others from it. Likewise in Katz, the defendant occupied the telephone booth, shut the door behind him to exclude all others and paid the toll, which "entitled [him] to assume that the words he utter[ed] into the mouthpiece [would] not be broadcast to the world." Id., at 352. 16 Katz and Jones could legitimately expect privacy in the areas which were the subject of the search and seizure each sought to contest. No such showing was made by these petitioners with respect to those portions of the automobile which were searched and from which incriminating evidence was seized. 17 [439 U.S. 128, 150]
There is an aspect of traditional standing doctrine that was not considered in Jones and which we do not question. It is the proposition that a party seeking relief must allege such a personal stake or interest in [439 U.S. 128, 133] the outcome of the controversy as to assure the concrete adverseness which Art. III requires. See, e. g., O'Shea v. Littleton, 414 U.S. 488, 493 (1974); Flast v. Cohen, 392 U.S. 83, 99 (1968); Baker v. Carr, 369 U.S. 186, 204 (1962). Thus, a person whose Fourth Amendment rights were violated by a search or seizure, but who is not a defendant in a criminal action in which the illegally seized evidence is sought to be introduced, would not have standing to invoke the exclusionary rule to prevent use of that evidence in that action. See Calandra, supra, at 352 n. 8.
Unfortunately, with few exceptions, lower courts have literally applied this language from Jones and have held that anyone legitimately on premises at the time of the search may contest its legality. See, e. g., Garza-Fuentes v. United States, 400 F.2d 219 (CA5 1968); State v. Bresolin, 13 Wash. App. 386, 534 P.2d 1394 (1975).
MR. JUSTICE POWELL, with whom THE CHIEF JUSTICE joins, concurring. I concur in the opinion of the Court, and add these thoughts. I do not believe my dissenting Brethren correctly characterize the rationale of the Court's opinion when they assert that it ties "the application of the Fourth Amendment . . . to property law concepts." Post, at 156-157. On the contrary, I read the Court's opinion as focusing on whether there was a legitimate expectation of privacy protected by the Fourth Amendment. The petitioners do not challenge the constitutionality of the police action in stopping the automobile in which they [439 U.S. 128, 151] were riding; nor do they complain of being made to get out of the vehicle. Rather, petitioners assert that their constitutionally protected interest in privacy was violated when the police, after stopping the automobile and making them get out, searched the vehicle's interior, where they discovered a sawed-off rifle under the front seat and rifle shells in the locked glove compartment. The question before the Court, therefore, is a narrow one: Did the search of their friend's automobile after they had left it violate any Fourth Amendment right of the petitioners? The dissenting opinion urges the Court to answer this question by considering only the talisman of legitimate presence on the premises. To be sure, one of the two alternative reasons given by the Court for its ruling in Jones v. United States, 362 U.S. 257 (1960), was that the defendant had been legitimately on the premises searched. Since Jones, however, the view that mere legitimate presence is enough to create a Fourth Amendment right has been questioned. See ante, at 147 n. 14. There also has been a signal absence of uniformity in the application of this theory. See ante, at 145-146 n. 13. This Court's decisions since Jones have emphasized a sounder standard for determining the scope of a person's Fourth Amendment rights: Only legitimate expectations of privacy are protected by the Constitution. In Katz v. United States, 389 U.S. 347 (1967), the Court rejected the notion that the Fourth Amendment protects places or property, ruling that the scope of the Amendment must be determined by the scope of privacy that a free people legitimately may expect. See id., at 353. As Mr. Justice Harlan pointed out in his concurrence, however, it is not enough that an individual desired or anticipated that he would be free from governmental intrusion. Rather, for an expectation to deserve the protection of the Fourth Amendment, it must "be one that society is prepared to recognize as `reasonable.'" See id., at 361. [439 U.S. 128, 152] The ultimate question, therefore, is whether one's claim to privacy from government intrusion is reasonable in light of all the surrounding circumstances. As the dissenting opinion states, this standard "will not provide law enforcement officials with a bright line between the protected and the unprotected." See post, at 168. Whatever the application of this standard may lack in ready administration, it is more faithful to the purposes of the Fourth Amendment than a test focusing solely or primarily on whether the defendant was legitimately present during the search. 1 In considering the reasonableness of asserted privacy expectations, the Court has recognized that no single factor invariably will be determinative. Thus, the Court has examined whether a person invoking the protection of the Fourth Amendment took normal precautions to maintain his privacy - that is, precautions customarily taken by those seeking privacy. See, e. g., United States v. Chadwick, 433 U.S. 1, 11 (1977) ("By placing personal effects inside a doublelocked [439 U.S. 128, 153] footlocker, respondents manifested an expectation that the contents would remain free from public examination"); Katz v. United States, supra, at 352 ("One who occupies [a telephone booth], shuts the door behind him, and pays the toll that permits him to place a call is surely entitled to assume that the words he utters into the mouthpiece will not be broadcast to the world"). Similarly, the Court has looked to the way a person has used a location, to determine whether the Fourth Amendment should protect his expectations of privacy. In Jones v. United States, supra, for example, the Court found that the defendant had a Fourth Amendment privacy interest in an apartment in which he had slept and in which he kept his clothing. The Court on occasion also has looked to history to discern whether certain types of government intrusion were perceived to be objectionable by the Framers of the Fourth Amendment. See United States v. Chadwick, supra, at 7-9. And, as the Court states today, property rights reflect society's explicit recognition of a person's authority to act as he wishes in certain areas, and therefore should be considered in determining whether an individual's expectations of privacy are reasonable. See Alderman v. United States, 394 U.S. 165 (1969). The Court correctly points out that petitioners cannot invoke decisions such as Alderman in support of their Fourth Amendment claim, as they had no property interest in the automobile in which they were riding. But this determination is only part of the inquiry required under Katz. The petitioners' Fourth Amendment rights were not abridged here because none of the factors relied upon by this Court on prior occasions supports petitioners' claim that their alleged expectation of privacy from government intrusion was reasonable. We are concerned here with an automobile search. Nothing is better established in Fourth Amendment jurisprudence than the distinction between one's expectation of privacy in [439 U.S. 128, 154] an automobile and one's expectation when in other locations. 2 We have repeatedly recognized that this expectation in "an automobile . . . [is] significantly different from the traditional expectation of privacy and freedom in one's residence." United States v. Martinez-Fuerte, 428 U.S. 543, 561 (1976). In United States v. Chadwick, supra, at 12, the distinction was stated more broadly:
In Chadwick, the Court recognized a reasonable expectation of privacy with respect to one's locked footlocker, and rejected the Government's argument that luggage always should be equated with motor vehicles for Fourth Amendment purposes. 433 U.S., at 13 . A distinction also properly may be made in some circumstances between the Fourth Amendment rights of passengers and the rights of an individual who has exclusive control of an automobile or of its locked compartments. In South Dakota v. Opperman, 428 U.S. 364 (1976), for example, we [439 U.S. 128, 155] considered "the citizen's interest in the privacy of the contents of his automobile" where its doors were locked and windows rolled up. See id., at 379 (POWELL J., concurring). Here there were three passengers and a driver in the automobile searched. None of the passengers is said to have had control of the vehicle or the keys. It is unrealistic - as the shared experience of us all bears witness - to suggest that these passengers had any reasonable expectation that the car in which they had been riding would not be searched after they were lawfully stopped and made to get out. The minimal privacy that existed simply is not comparable to that, for example, of an individual in his place of abode, see Jones v. United States, supra; of one who secludes himself in a telephone booth, Katz v. United States, supra; or of the traveler who secures his belongings in a locked suitcase or footlocker. See United States v. Chadwick, supra. 4 This is not an area of the law in which any "bright line" rule would safeguard both Fourth Amendment rights and the [439 U.S. 128, 156] public interest in a fair and effective criminal justice system. The range of variables in the fact situations of search and seizure is almost infinite. Rather than seek facile solutions, it is best to apply principles broadly faithful to Fourth Amendment purposes. I believe the Court has identified these principles. 5
I do not share the dissenters' concern that the Court's ruling will "invit[e] police to engage in patently unreasonable searches every time an automobile contains more than one occupant." See post, at 168. A police officer observing an automobile carrying several passengers will not know the circumstances surrounding each occupant's presence in the automobile, and certainly will not know whether an occupant will be able to establish that he had a reasonable expectation of privacy. Thus, there will continue to be a significant incentive for the police to comply with the requirements of the Fourth Amendment, lest otherwise valid prosecutions be voided. Moreover, any marginal diminution in this incentive that might result from the Court's decision today is more than justified by society's interest in restricting the scope of the exclusionary rule to those cases where in fact there is a reasonable expectation of privacy.
The dissenting opinion suggests that the petitioners here took the same actions to preserve their privacy as did the defendant in Katz: Just as Katz closed the door to the telephone booth after him, petitioners closed the doors to their automobile. See post, at 165 n. 15. Last Term, this Court determined in Pennsylvania v. Mimms, 434 U.S. 106 (1977), that passengers in automobiles have no Fourth Amendment right not to be ordered from their vehicle, once a proper stop is made. The dissenting opinion concedes that there is no question here of the propriety of the stopping of the automobile in which the petitioners were riding. See post, at 160 n. 5. Thus, the closing of the doors of a vehicle, even if there were only one occupant, cannot have the same significance as it might in other contexts.
See also Brown v. Illinois, 422 U.S. 590, 609 -610 (1975) (POWELL, J., concurring in part). MR. JUSTICE WHITE, with whom MR. JUSTICE BRENNAN, MR. JUSTICE MARSHALL, and MR. JUSTICE STEVENS join, dissenting. The Court today holds that the Fourth Amendment protects property, not people, and specifically that a legitimate occupant of an automobile may not invoke the exclusionary rule and challenge a search of that vehicle unless he happens to own or have a possessory interest in it. 1 Though professing to acknowledge that the primary purpose of the Fourth Amendment's prohibition of unreasonable searches is the protection of privacy - not property - the Court nonetheless effectively ties the application of the Fourth Amendment and [439 U.S. 128, 157] the exclusionary rule in this situation to property law concepts. Insofar as passengers are concerned, the Court's opinion today declares an "open season" on automobiles. However unlawful stopping and searching a car may be, absent a possessory or ownership interest, no "mere" passenger may object, regardless of his relationship to the owner. Because the majority's conclusion has no support in the Court's controlling decisions, in the logic of the Fourth Amendment, or in common sense, I must respectfully dissent. If the Court is troubled by the practical impact of the exclusionary rule, it should face the issue of that rule's continued validity squarely instead of distorting other doctrines in an attempt to reach what are perceived as the correct results in specific cases. Cf. Stone v. Powell, 428 U.S. 465, 536 (1976) (WHITE, J., dissenting).
The second tenet is that when a person is legitimately present in a private place, his right to privacy is protected from unreasonable governmental interference even if he does [439 U.S. 128, 158] not own the premises. Just a few years ago, THE CHIEF JUSTICE, for a unanimous Court, wrote that the "[p]resence of the defendant at the search and seizure was held, in Jones, to be a sufficient source of standing in itself." Brown v. United States, 411 U.S. 223, 227 n. 2 (1973); accord, id., at 229 (one basis for Fourth Amendment protection is presence "on the premises at the time of the contested search and seizure"); Jones v. United States, 362 U.S. 257 (1960) (individual legitimately present in friend's apartment may object to search of apartment). Brown was not the first time we had recognized that Jones established the rights of one legitimately in a private area against unreasonable governmental intrusion. E. g., Combs v. United States, 408 U.S. 224, 227 , and n. 4 (1972); Mancusi v. DeForte, 392 U.S. 364, 368 , and n. 5 (1968); Simmons v. United States, 390 U.S. 377, 390 (1968). The Court in Jones itself was unanimous in this regard, and its holding is not the less binding because it was an alternative one. See Combs v. United States, supra, at 227 n. 4. These two fundamental aspects of Fourth Amendment law demand that petitioners be permitted to challenge the search and seizure of the automobile in this case. It is of no significance that a car is different for Fourth Amendment purposes from a house, for if there is some protection for the privacy of an automobile then the only relevant analogy is between a person legitimately in someone else's vehicle and a person legitimately in someone else's home. If both strands of the Fourth Amendment doctrine adumbrated above are valid, the Court must reach a different result. Instead, it chooses to eviscerate the Jones principle, an action in which I am unwilling to participate.
Not only does Combs supply the relevant inquiry, it also directs us to the proper answer. We recognized there that Jones had held that one of those protected interests is created by legitimate presence on the searched premises, even absent any possessory interest. 408 U.S., at 227 n. 4. This makes unquestionable sense. We have concluded on numerous occasions that the entitlement to an expectation of privacy does not hinge on ownership:
In Alderman v. United States, supra, at 196, Mr. Justice Harlan, concurring in part and dissenting in part, noted that "our own past decisions . . . have decisively rejected the notion [439 U.S. 128, 162] that the accused must necessarily have a possessory interest in the premises before he may assert a Fourth Amendment claim." That rejection should not have been surprising in light of our conclusion as early as 1960 that "it is unnecessary and ill-advised to import into the law surrounding the constitutional right to be free from unreasonable searches and seizures subtle distinctions, developed and refined by the common law in evolving the body of private property law which, more than almost any other branch of law, has been shaped by distinctions whose validity is largely historical." Jones v. United States, 362 U.S., at 266 . 9 The proposition today overruled was stated most directly in Mancusi v. DeForte, supra, at 368: "[T]he protection of the Amendment depends not upon a property right in the invaded place but upon whether the area was one in which there was a reasonable expectation of freedom from governmental intrusion." Prior to Jones, the lower federal courts had based Fourth Amendment rights upon possession or ownership of the items seized or the premises searched. 10 But Jones was foreshadowed by Mr. Justice Jackson's remark in 1948 that "even a guest may expect the shelter of the rooftree he is under against criminal intrusion." McDonald v. United States, 335 U.S. 451, 461 (1948) (Jackson, J., joined by Frankfurter, J., concurring). Indeed, the decision today is contrary to Mr. Justice Brandeis' dissent in Olmstead v. United States, [439 U.S. 128, 163] 277 U.S. 438, 478 (1928), expressing a view of the Fourth Amendment thought to have been vindicated by Katz. The majority in Olmstead found the Fourth Amendment inapplicable absent a trespass on property rights. 277 U.S., at 466 . That is exactly what the Court holds in this case; but Mr. Justice Brandeis asserted 50 years ago that more than mere property rights are involved, and the Court's opinion in Katz re-emphasized that "`[t]he premise that property interests control the right of the Government to search and seize has been discredited.'" 389 U.S., at 353 , quoting Warden v. Hayden, 387 U.S. 294, 304 (1967). That logic led us inescapably to the conclusion that "[n]o less than an individual in a business office, in a friend's apartment, or in a taxicab, a person in a telephone booth may rely upon the protection of the Fourth Amendment." 389 U.S., at 352 (footnotes omitted). And if all of those situations are protected, surely a person riding in an automobile next to his friend the owner, or a child or wife with the father or spouse, must have some protection as well. The same result is reached by tracing other lines of our Fourth Amendment decisions. If a nonowner may consent to a search merely because he is a joint user or occupant of a "premises," Frazier v. Cupp, 394 U.S. 731, 740 (1969), 11 then that same nonowner must have a protected privacy interest. The scope of the authority sufficient to grant a valid consent can hardly be broader than the contours of protected privacy. 12 [439 U.S. 128, 164] And why should the owner of a vehicle be entitled to challenge the seizure from it of evidence even if he is absent at the time of the search, see Coolidge v. New Hampshire, 403 U.S. 443 (1971), while a nonowner enjoying in person, and with the owner's permission, the privacy of an automobile is not so entitled? In sum, one consistent theme in our decisions under the Fourth Amendment has been, until now, that "the Amendment does not shield only those who have title to the searched premises." Mancusi v. DeForte, 392 U.S., at 367 . Though there comes a point when use of an area is shared with so many that one simply cannot reasonably expect seclusion, see id., at 377 (WHITE, J., dissenting); Air Pollution Variance Bd. v. Western Alfalfa Corp., 416 U.S. 861, 865 (1974), short of that limit a person legitimately on private premises knows the others allowed there and, though his privacy is not absolute, is entitled to expect that he is sharing it only with those persons and that governmental officials will intrude only with consent or by complying with the Fourth Amendment. See Mancusi v. DeForte, supra, at 369-370. 13 It is true that the Court asserts that it is not limiting the Fourth Amendment bar against unreasonable searches to the protection of property rights, but in reality it is doing exactly that. 14 Petitioners were in a private place with the permission [439 U.S. 128, 165] of the owner, but the Court states that that is not sufficient to establish entitlement to a legitimate expectation of privacy. Ante, at 148. But if that is not sufficient, what would be? We are not told, and it is hard to imagine anything short of a property interest that would satisfy the majority. Insofar as the Court's rationale is concerned, no passenger in an automobile, without an ownership or possessory interest and regardless of his relationship to the owner, may claim Fourth Amendment protection against illegal stops and searches of the automobile in which he is rightfully present. The Court approves the result in Jones, but it fails to give any explanation why the facts in Jones differ, in a fashion material to the Fourth Amendment, from the facts here. 15 More importantly, how is the Court able to avoid answering the question why presence in a private place with the owner's permission is insufficient? If it is "tautological to fall back on the notion that those expectations of privacy which are legitimate depend primarily on cases deciding exclusionary-rule issues in criminal cases," ante, at 144 n. 12, then it surely must be tautological to decide that issue simply by unadorned fiat. [439 U.S. 128, 166] As a control on governmental power, the Fourth Amendment assures that some expectations of privacy are justified and will be protected from official intrusion. That should be true in this instance, for if protected zones of privacy can only be purchased or obtained by possession of property, then much of our daily lives will be unshielded from unreasonable governmental prying, and the reach of the Fourth Amendment will have been narrowed to protect chiefly those with possessory interests in real or personal property. I had thought that Katz firmly established that the Fourth Amendment was intended as more than simply a trespass law applicable to the government. Katz had no possessory interest in the public telephone booth, at least no more than petitioners had in their friend's car; Katz was simply legitimately present. And the decision in Katz was based not on property rights, but on the theory that it was essential to securing "conditions favorable to the pursuit of happiness" 16 that the expectation of privacy in question be recognized. 17 At most, one could say that perhaps the Constitution provides some degree less protection for the personal freedom from unreasonable governmental intrusion when one does not have a possessory interest in the invaded private place. But that would only change the extent of the protection; it would not free police to do the unreasonable, as does the decision today. And since the accused should be entitled to litigate the application of the Fourth Amendment where his privacy interest is merely arguable, 18 the failure to allow such litigation here is the more incomprehensible. [439 U.S. 128, 167]
The distinctions the Court would draw are based on relationships between private parties, but the Fourth Amendment is concerned with the relationship of one of those parties to [439 U.S. 128, 168] the government. Divorced as it is from the purpose of the Fourth Amendment, the Court's essentially property-based rationale can satisfactorily answer none of the questions posed above. That is reason enough to reject it. The Jones rule is relatively easily applied by police and courts; the rule announced today will not provide law enforcement officials with a bright line between the protected and the unprotected. 21 Only rarely will police know whether one private party has or has not been granted a sufficient possessory or other interest by another private party. Surely in this case the officers had no such knowledge. The Court's rule will ensnare defendants and police in needless litigation over factors that should not be determinative of Fourth Amendment rights. 22 More importantly, the ruling today undercuts the force of the exclusionary rule in the one area in which its use is most certainly justified - the deterrence of bad-faith violations of the Fourth Amendment. See Stone v. Powell, 428 U.S., at 536 -542 (WHITE, J., dissenting). This decision invites police to engage in patently unreasonable searches every time an automobile contains more than one occupant. Should something be found, only the owner of the vehicle, or of the item, will have standing to seek suppression, and the evidence will [439 U.S. 128, 169] presumably be usable against the other occupants. 23 The danger of such bad faith is especially high in cases such as this one where the officers are only after the passengers and can usually infer accurately that the driver is the owner. The suppression remedy for those owners in whose vehicles something is found and who are charged with crime is small consolation for all those owners and occupants whose privacy will be needlessly invaded by officers following mistaken hunches not rising to the level of probable cause but operated on in the knowledge that someone in a crowded car will probably be unprotected if contraband or incriminating evidence happens to be found. After this decision, police will have little to lose by unreasonably searching vehicles occupied by more than one person. Of course, most police officers will decline the Court's invitation and will continue to do their jobs as best they can in accord with the Fourth Amendment. But the very purpose of the Bill of Rights was to answer the justified fear that governmental agents cannot be left totally to their own devices, and the Bill of Rights is enforceable in the courts because human experience teaches that not all such officials will otherwise adhere to the stated precepts. Some policemen simply do act in bad faith, even if for understandable ends, and some deterrent is needed. In the rush to limit the applicability of the exclusionary rule somewhere, anywhere, the Court ignores precedent, logic, and common sense to exclude the rule's operation from situations in which, paradoxically, it is justified and needed.
Thus, petitioners of course have standing to challenge the legality of the stop, and the evidence found may be a fruit of that stop. See United States v. Martinez-Fuerte, 428 U.S. 543, 548 , 556 (1976). Petitioners have not argued that theory here, perhaps because the justification necessary for such a stop is less than that needed for a search. See Terry v. Ohio, supra. Nor have petitioners chosen to argue that they were "arrested" in constitutional terms as soon as they were ordered from the vehicle and that the search was a fruit of that infringement on their personal rights.
Petitioners never asserted a property interest in the items seized from the automobile. The evidence found was useful to the prosecution solely on the theory that petitioners' possession of the items was probative of petitioners' identity as the robbers. In Jones the Court recognized automatic standing in possessory crimes because the prosecution should not be allowed to take contradictory positions in the suppression hearing and then at trial, and also because of the dilemma that the defendant would face if he were forced to assert possession to challenge a search. 362 U.S., at 263 . In Simmons we eliminated the dilemma by holding that the accused's testimony at the suppression hearing could not be used against him at trial. 390 U.S., at 394 . We also noted that the question whether automatic standing should be recognized for possessory evidence in nonpossessory crimes was an open one. Id., at 391-392. Finally, in Brown v. United States, 411 U.S. 223, 229 (1973), we reserved the question whether prosecutorial self-contradiction by itself warrants automatic standing.
Sources12.207.211.74 02:02, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Blue Super Giants
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A Plague of Angels
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HistoryA Plague of Angels is a stage play written by Mark St. Germain. The play's original title was "Forgiving Typhoid Mary" because of the story's portrayal of New York cook, Mary Mallon, but became more well known by the above title over the years. It was written and first played in the year 1991 at Long Warf, GeVa Theater, George Street Playhouse, and starred Estelle Parsons. PlotThe play features the story of real-life cook, Mary Mallon, whose body carried typhoid fever. She unknowingly spread the disease to many people throughout New York, killing 3. Her theory that germs did not exist and that it was impossible for her to carry a disease made those she prepared food for catch the infection very quickly. In a fictional part of the story, Mary accidentally infects a young girl who she loved with typhoid, which leads to the child's death. The play focuses on Mary's confinement within a hospital and the work done by two doctors and a church official to help her realize that she is indeed infecting and killing people. It becomes very emotional to watch how the characters grow to be mutal friends and how they face the problem set in front of them: whether or not is just to keep Mary imprisoned when they can't scientifically prove she's killing people. Eventually the girl becomes known as "Typhoid Mary" from whom we get the modern day term for those who carry diseases. Morals and IdeasReligeon, empathy, and justice are all addressed boldy throughout the play. Telling a true story helps it convey the ideas it wants the viewers to pick up. Recognition and AwardsWhereas some critics discredited "A Plague of Angels" as hard to understand due to its flashback scenes, many believe it to be one of the most brilliant plays ever written. It is very short (2 Acts), but was listed as one of the ten best plays of the year when it was released (by "Time Magazine"). Flashback scenes are quickly becoming more popular on both film and stage due to its mysterious way to reveal a character's past in a story. This interests most modern audiences. Sourceshttp://www.doollee.com/PlaywrightsS/st-germain-mark.html (Author Information / Book Description)
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Blue SuperGiants
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Lifespan of a Blue super GiantThe Blue Super Giant Glory Day donot last for long the estimated lifespan of a Blue Super Giant is ten million years due to the fact that it burns gasses at a monsterous rate. this all leads to a climactis a end a supernova. When dying the core collapsese and the outer layer Crashes right onto the core realeasing so much energy that the star blasts apart and then god father of all stars dies. The After life of a Blue super GiantThe The climactic end of the Blue SuperGiants life span the collapsed core turns into a black hole or a nuetron star Sources75.51.179.249 03:07, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Portuguese Royal Family
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The Royal Family includes:
See alsoExternal linksCategory:Portuguese Royal Family Portuguese Royal Family Portuguese Royal Family Portuguese Royal Family 84.90.90.208 03:30, 16 November 2007 (UTC)
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Justin Schultze
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Early LifeJustin was Born April 9th 1988, In Houston Texas. At the age of 5 (1993) he and his family moved to Prior Lake Minnesota, where he played on the Prior Lake State Hockey Team competing in the 1997 state finals. In 2001 His Family Moved back to the Houston suburb Cypress, Texas. JuneauJustin is the singer and guitarist of the Houston rock Act Juneau. Juneau has been recognised as one of America's #1 unsigned bands. They have recently finished a stint on the 2007 Vans Warped Tour Present DayJustin graduated from Cy-Fair High School in Houston, Texas and now attends Texas Tech University as a Marketing Major in the Jerry S. Rawls School of Buisness Sources129.118.121.103 03:33, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Saleisha Cooper
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Sources202.37.64.48 03:38, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Hashford Edison's Hanford Band
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Sources"Greensleve Music Project" [1]
"Poor Man's Muisc Almanac Interview" [2]
"Tinymixtapes" [3]
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1st Lacross goal for W-L 2000
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Alex Marr
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Classical Music is now the only thing which he choses to play. Recently, Alex has released the album "Crossing" recorded live at his debut concert on September 9th at Old North Church. He continues to study in Ohio with his instructor Victoria Auth, and attends Poland Seminary High School. Sources"Crossing" program from Old north concert. September 9th, 2007. Acmarelli 04:54, 16 November 2007 (UTC) Articles for creation/2007-11-16
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The History of Cazadores Tequila
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Copyright violation from http://www.latineyes.tv/2005/12/ removed per policy. Sources76.103.65.191 05:26, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Kothornoi
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Windy City Pro Wrestling is an American regional professional wrestling promotion based in Chicago, Illinois. Established by retired wrestler Sam DeCero in 1988, the promotion was one of several major regional territories in the Midwest during the late 1980s along with Dick the Bruiser's World Wrestling Association and is one of the oldest independent organizations in operation. HistoryAfter retiring from professional wrestling due to a back injury, Sam DeCero purchased a 95th street garage in Chicago's South Side and began training local wrestlers advertising as far as Hammond, Indiana. With Mike Gratchner, a former promoter and wrestling photographer, DeCero decided to establish his own promotion. Within a year, he had managed to secure investors including relatives, friends and co-workers and held the promotions first event at a South Side nightspot featuring Steve Regal against Paul Christy in the main event on January 30, 1988. The event, which was attended by 160 people, was successful and soon the promotion began holding events in similar venues offering to hold cards ranging from $3,500 to $9,500. DeCero soon began running televised wrestling events with then 22-year old Paul Heyman [1] (who was also working for Southeastern promotions Southern Championship Wrestling and the Continental Wrestling Federation) [2] at DaVinci Manor and the International Amphitheater [3] which later aired on WMBD-TV. During the summer, the promotion saw serious losses due to poor attendance, including loosing $10,000 at a show in Rockford, Illinois and between $12,000-13,000 at the International Amphitheater. In September, despite drawing a large crowd at a card featuring Terry "Bam Bam" Gordy and Bam Bam Bigalow in the main event, the promotion still lost money. [4] Eventually recovering after additional funding by his investors, within four years, the promotion operated two training facilities and had a weekly half hour television show airing on SportsChannel. They also participated in several fundraisers for charity organizations including Toys for Tots, Muscular Dystrophy, Maryville City for Youth and the Chicago Coalition for the Homeless. [5] [6] During the late 1980s, the promotion featured many popular wrestlers of the era including Dick Murdoch, Shigeri Akabane, Dennis Condrey, [7] Steve Regal [8] and George Ringo, who acted as honorary commissioner. Prior to his death in Puerto Rico, Bruiser Brody had been scheduled to face Nord the Barbarian at the International Amphitheater in Chicago, Illinois on August 12, 1988. [9] Other mainstays included "Mean" Mike Anthony, [10] Trevor Blanchard, [11] Rockin' Randy [12], Tony Montana [13] and The Power Twins (Larry & David Sontag). Although going into a slump during the early 1990s, its annual Battle of the Belts supercard was aired on SportsChannel on May 22, 1993 [14] and, two years later, presented Sailor Art Thomas a Lifetime Service to Sport Award on May 16, 1995. [15] The promotion also began holding events in smaller venues outside the Chicago-area such the Hammond Civic Center in Hammond, Illinois and Hempstead High School in Dubuque, Iowa in February 1996. [16] In September, the promotion became involved in a nine-month legal dispute with Ted Turner's World Championship Wrestling regarding trademark infringement over use of the WCW acronym. The matter was settled out of court and, in late 1997 the promotion was renamed Windy City Pro Wrestling. [17] The promotion managed to survive during the decade and, due in part to its wrestling school and televised events in the Chicago-area, several light heavyweight wrestlers such as Ace Steel, [18] Sosay, [19] Kevin Quinn [20] and Christopher Daniels [21] emerged from the promotion during the late 1990s and early 2000s. During the last several years, independent wrestlers such as Colt Cabana, Abyss and Austin Aries among others have made appearances in the promotion as have WWF veterans King Kong Bundy, Greg "The Hammer" Valentine [22] and Jerry "The King" Lawler. In August 1999, the promotion began broadcasting live events via the internet through the website LiveOnTheNet.com as part of its Sunday afternoon sports lineup featuring Mike Anthony, Sgt. Storm, The Outfit, Lips Manson and Greg "The Hammer" Valentine. [23] The promotion remained popular with Chicago wrestling fans favoring its "old school" wrestling approach unlike "sports entertainment" based promotions such as the World Wrestling Federation [24] and began competing with rival promotions such as All American Wrestling, Independent Wrestling Association Midsouth and AWA Slam. [25] Working with its affiliate organization Urban American Professional Wrestling in Chicago's inner-city areas in recent years, [26] [27] its South Side wrestling school was featured on Insomniac with Dave Attell in 2002. [28] In 2004, with over 1,000 in attendance at Morton College in Cicero for their supercard Battle of the Belts 16 grossing over $15,000, the following year Battle of the Belts 17 was held at the Hammond Civic Center on May 17, 2005 (shortly before signing with World Wrestling Entertainment, Rob Van Dam had previously headlined a WCPW event at the building on May 26, 2001 attended by 2,000 fans). [29] [30] RosterChampionships
References
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Cazadores Tequila
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Copyright violation from http://www.latineyes.tv/2005/12/ removed per policy. Sources76.103.65.191 05:46, 16 November 2007 (UTC) Articles for creation/2007-11-16
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James C.Morgan
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James C. Morgan has been the chairman of the board of Applied Materials since 1987. He formerly was chief executive officer of Applied Materials from 1977 to April 2003. Prior to joining Applied Materials as president in 1976, he was a senior partner with WestVen Management, a private venture capital partnership affiliated with the Bank of America Corporation. Prior to WestVen, he was with Textron, a leading diversified manufacturing company. With one of the longest tenures of any FORTUNE 500 CEO, Mr. Morgan has an extensive history in business and philanthropy. Mr. Morgan is a recipient of the 1996 National Medal of Technology for his industry leadership and for his vision in building Applied Materials into the world's leading semiconductor equipment company, a major exporter and a global technology pioneer which helps enable the Information Age. Awarded by the President of the United States, the Medal of Technology recognizes technological innovators who have made lasting contributions to America's competitiveness and standard of living. Among his many honors, Mr. Morgan is a recent recipient of the prestigious Semiconductor Industry Association Robert N. Noyce Award, the highest honor bestowed by the SIA, for outstanding achievement and leadership in support of the U.S. semiconductor industry, and the Spirit of Silicon Valley Lifetime Achievement Award from the Silicon Valley Leadership Group, for his ethics, community engagement and business success. In 2003, Mr. Morgan was sworn in by President George W. Bush as vice chairman of the President's Export Council, the premier national advisory body on international trade. He was appointed to the 2002 U.S.-Japan Private Sector Government Commission and co-chaired the sessions on corporate governance. From 1996 to 1997, Mr. Morgan served on the Commission on U.S.-Pacific Trade and Investment Policy which advised President Bill Clinton and Congress on steps the United States should take to achieve significant additional market access and improve trade relationships in the Asia-Pacific region. From 1988 to 1992, he served on the National Advisory Committee on Semiconductors, which advised President George H. W. Bush and Congress on a national strategy to strengthen competitiveness of the U.S. semiconductor industry. Mr. Morgan is an advisory board member for the Center for Science, Technology, and Society at Santa Clara University. He is active with the Nature Conservancy, a global nonprofit organization which establishes nature sanctuaries, serving as a member of the board of governors, a member of its Asia-Pacific Council, and as a trustee of the Nature Conservancy of California. Instrumental in the creation of the Tech Museum Awards which launched in 2000, Mr. Morgan inspired the program’s James C. Morgan Global Humanitarian Award, which honors individuals whose broad vision and leadership help address humanity’s greatest challenges. He formerly was a member of the board of directors of Cisco Systems, Inc. from 1998 to 2006 and is director emeritus and past president of Semiconductor Equipment and Materials International (SEMI), the first global trade association. Mr. Morgan is a past board member of the American Electronics Association (AEA), the National Center for APEC (Asia-Pacific Economic Cooperation) and of SEMI/SEMATECH, the association of U.S. equipment companies which supplies advanced systems to the SEMATECH chipmaking consortium. Mr. Morgan is a member of the World Presidents' Organization. From 1994 to 1996, he was a board member of Joint Venture: Silicon Valley, a leading community public/private collaborative. From 1986 to 1991 he was co-chairman of the Japan-Western U.S. Association. Mr. Morgan's contributions have been honored by many business, education and community organizations throughout his career. Co-author of the book, "Cracking the Japanese Market: Strategies for Success in the New Global Economy," he earned a B.S.M.E. and an M.B.A. from Cornell University, Ithaca, New York. Selected Awards, Accomplishments and Honors (1990 - present) Spirit of Sources203.123.182.161 06:02, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Shannon Donato
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Shannon Donato was a Football Player In The Nation Rugby League Competition In Australia. His previous clubs include South Sydney Rabbitohs, Cronulla Sutherland Sharks and the Penrith Panthers. Sourceshttp://stats.rleague.com/rl/scorers/players/Shannon_Donato.html 123.243.245.114 06:18, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Suzanne D. Rutland
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Research Areas
Current Projects
Selected PublicationsBooks
Booklet
Chapters in books
Articles
Areas of teaching and research supervision
Supervising theses in Australian Jewry, the Holocaust and Jewish education in Australia. Other professional contributionsWithin the Jewish community she have served as:
Sourceshttp://www.arts.usyd.edu.au/departs/hbjs/staff/profiles/rutland.shtml 193.83.204.118 12:32, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Carl Kauba
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http://fada.com/browse_by_artist.html?gallery_no=26&artist=3431&bio=1 81.187.45.149 12:35, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Bacillus rossius
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Sources80.249.57.38 12:48, 16 November 2007 (UTC) Articles for creation/2007-11-16
References
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History
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SourcesLuke Bamber is known for his outstanding intelligence in all fields from advanced mathematics right down to sub atomic structures. Although his is forced to participate in more basic lessons with individuals lower than himself he sometimes "plays dumb" as he no longer feels its needed to gets straight A* and enjoys the odd D+ such as his ALPS grade. This information is widely known and on some occasions the maths department has literaly begged him to work out complex matamatical problems to which they cannot comprehend. He is infamously known as a leader in UK economics and NASA have worked with Luke many a time when in troube. It was also widely reported that with the aid of Luke Bamber Saddam Hussain was captured using his pure intelligence and physic abilities to "home in" on his accent. He now resides in the Manchester area in a deluxe 24 bedroom house, 30m private pool with 3 hot tubs, 3 housemaids, 4 high performance cars and a 2 mile runway for his private jet in which is used often for quick hops to L.A for an A LIST party. Although he is extremly busy writing his autobiography he finds time to fly over ethiopia once in a while and drop a few Apple & Mango J2O's yet maybe next time he will consider not using glass as they now refer to him as " The one who brings fruit juice and death". Stay tuned for more info ...... 90.195.54.54 15:06, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Albert Jack
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If you want to make changes and try again, copy and paste your article into a new submission and make your changes there. Albert Jack is a writer and researcher and has become something of a publishing phenomenon since his first book Red Herrings and White Elephants, which explored the origins of well-known phrases in the English language, became a huge international bestseller (selling over 250,000 copies so far). The book was serialised by the Sunday Times for over a year and stayed in the top ten of the bestseller list for sixteen months. His follow up book Shaggy Dogs and Black Sheep was also a best seller and has sold over 130,000 copies since publication in October 2005. It became Penguin Book's Christmas bestseller. His hilarious third book, a study of Urban Legends is also a bestseller and due out in paperback in September 2007. In the same month Red Herrings and White Elephants is re-released for the first time in paperback and Albert has provided 30% more content for a revised and expanded version that is sure to hit the best-seller list once again. Fascinated by discovering the truth behind the world's great stories, Albert has become an expert in explaining the unexplained, which is great news for conversations and storytellers everywhere. He is now a veteran of hundreds of live television shows and thousands of radio appearances worldwide. His books have become bestsellers in America, Canada, South Africa and Australia and translated into many different languages. His latest effort, Albert Jack's Ten Minute Mysteries is a study of the world's great mysteries from the Bermuda Triangle to the disappearance of Glen Miller. Crop Circles, Loch Ness Monster, UFO's, Marylyn Monroe and the Mary Celeste have also come under investigation along with many more famous stories. Albert uncovers the sometimes surprising truth and his acerbic wit makes for an entertaining read. Ten Minute Mysteries will be the third book Albert is releasing in the autumn of 2007. The success of his books has a lot to do with Albert's invention of what he calls the 'ten minute read,' that enables readers to open his books at any point and be fully entertained for ten minutes at a time before 'going off to do something more productive with their time.' Currently working on a children's book, Albert Jack's Bed Time Stories, the writer has plans for another ten hard backs that look set to continue his success for many years to come. Sources71.239.242.117 15:14, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Dr. Robert V. Antonucci
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EducationRobert Antonucci, a native of Fitchburg, MA, has received both a Batchelor's degree and a Masters degree from Fitchburg State College, as well as an honary doctoral degree. he also received honorary degrees from Endicott College and the Mass Maritime Academy. In 1983 he graduated from Boston Universtiy's School of Education. Previous PositionsRobert Antonucci has held many prestigeous positions these include:
Fitchburg State College's web site also states that Robert Antonucci is on the board of trustees at Eastern Bank, Unitil, and Our Father's House. He is also a director of the North Central Massachusetts Boys & Girls Club and Our Father's House. Acheivements at Fitchburg State CollegeEven though he has only been president for 5 years at Fitchburg State College he has brought about the 4 million dollar renovation of Holmes Dinning Commons and the complete overhall of the schools athletic fields. He has restructured the grading system and pumped millions of dollars in to the infrastructure of the school.
Sourceshttp://www.fsc.edu/aboutus/president/aboutpresident.html
74.10.161.250 15:23, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Grand Master Choa Kok Sui
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Master Choa Kok Sui used to say, "Life goes on. The work must always go on. I am just a channel for the fulfillment of the will of my Teacher, Lord Boddhisattva Mei Ling. And each one of my disciples and students can fulfill and share to the Great Work." Pranic Healing is an ancient science and art that has been adapted and systematized by the founder of modern Pranic Healing, Grand Master Choa Kok Sui. It has been used to heal countless people through the ages. The principle is simple. We know that every living being possesses the inborn ability to heal itself. Pranic Healing simply enhances this healing process by utilizing the energy of life. This energy is called 'Prana' (life-force) in Sanskrit. To promote health and the knowledge of Pranic Healing, 17 Pranic Healing Foundations have been set up in 10 states across India along with active Pranic Healers Associations and Pranic Healing centres in many other states. All the Pranic Healing Foundations in India are affiliated to the All India Yoga Vidya Pranic Healing Foundations Trust, which was set up in Bangalore in 1996 by Grand Master Choa Kok Sui. Sourceshttp://www.uni.cc/site/page.php 59.92.106.142 15:23, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Dinah McNicol
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Sources58.8.2.102 15:37, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Jerald Posman writes about a Bangladeshi Writer
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=== Sources ===YORK COLLEGE The City University of New York 94-20 Guy R.Brewer Boulevard
Jamaica,New York 11451
Office of the Vice President October 5, 2007
I was approached initially by Mr.Bari,who was represnting a group of individual with a Bangladeshi background,to arrange for the use of the College’s theater to host a speaking engagement by Dr.Muhammad Yunus,the originator of the worldwide microfinance movement.In these first discussions,Mr.Bari and I discovered a commonality of interest in world affairs,the importance of global interchange and a passion for the work of Dr.Yunus and the organization that he formed-Grameen Bank. I had lived and worked overseas in and with developing countries;and was on the Board of Trustees of of Project Enterprise-a New York-based nonprofit organization modeled on the objectives and lending practices of Grameen Bank.At the first meeting with Mr.Bari,I discovered that he had written a biography of Dr.Yunus in Bengali and was a prolific author-at a very young age-on a range of subjects. As a result of our encounter,Mr.Bari and Idecided to work on a project to develop a scholarship fund for students from Grameen families in Bangladesh to attend York College.In addition,we also worked on a joint summer internship exchange program where students from York College would work with Grameen Bank and Bangladeshi students would come to the US and be affiliated with a Grameen-oriented institution.The key to the resources,viability and the success of the program was to obtain not only the endorsement of Dr.Yunus but also to have him make a major appearance at York College.This became all the more important,and somewhat difficult,when Dr.Yunus was awarded the Noble Peace in December 2006. I cannot overemphasize the role that Mr.Bari has played in actualizing this plan.He has become a passionate advocate for this program and a critical connector among all the parties-in Bangladeshi and the US-who are,and should be,involved.He has introduced me to potential supporters and advocates of the program and arranged for a crucial meeting and follow-up with Dr.Yunus’closest advisor and assistant.Mr.Bari assisted in brokering a scholarship and internship program that met the needs of Dr.Yunus and Grameen Bank,on the one hand,and York College,on the other. The culmination of all these efforts was a meeting conscientiously pursued by Mr.Bari between Dr.Yunus and myself in September 2007.That encounter solidifield both the concept and support of the program. During this entire time,Mr.Bari has demonstrated an expectional facility in several areas.He has very good communication skills that exhibit a finely tuned cultural sensitivity.He understands the importance of compromise and negotiation and to have the right participants in the right place at the right time. Mr.Bari has also mastered the art of following up.He is persistent in pursuing connections,keeping parties on the right course and working toward the accomplishment of a particular objective.Remarkably,he has done this with good humor and without alienating any individual or group.He has a built-in sense of fact and diplomacy. Clearly,I have the highest regard for Mr.Bari.He is already a well-know writer and journalist ability to make an important impact in any field that he chooses to pursue. Please feel free to contact me by e-mail or phone if there is any other way that I can be helpful.
Read Jelald Posman Columns from New York Times: http://query.nytimes.com/gst/fullpage.html?res=9D0CE6D8133BF933A15757C0A967958260&n=Top/Reference/Times%20Topics/Organizations/C/City%20University%20of%20New%20York <a href="http://www.myhotcomments.com/graphics.php?index=best"><img src="http://lc.fdots.com/cc/lc/c0/c0fc01b6722d8b5c43c10e62776aef00.gif" border="0" alt="MyHotComments.com"></a>
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Peter Embree
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ChildhoodPeter had strugged growing up, his father was a white Supremacist. His father took him to various Aryan Nations meetings were Peter experienced rascism first hand. It wasn't until Peter met Eddie Stevenson at an Aryan Nations ralley in Georgia at the age of 13 that he realized he was a homosexual. Him and eddie have been companions since. Sourcehttp://google.com/peter-embree-gay.html
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Braun Mincher
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Braun Mincher is a young and successful entrepreneur who had the ability to "retire" at the age of 30 a self-made multi-millionaire. He now pursues his passion of better educating consumers about "Practical Financial Literacy" and is gaining a lot of attention (and creating controversy) on recent radio and TV shows by challenging high schools and colleges across the country to make these classes a graduation requirement. His new book, The Secrets of Money: A Guide for Everyone on Practical Financial Literacy, is being released on November 26, 2007, according to his website at www.BraunMincher.com. Sourceshttp://www.BraunMincher.com http://www.braunmincher.com/en/Events.html http://www.chadcarden.com/ http://www.fortcollinsre.com/links/ http://www.rtironline.com/blog/WeeklyEzineArchives/_archives/2007/11/13/3351073.html http://www.frascona.com/cololnks.htm http://braunmincher.blogspot.com/
-- 65.102.227.144 (talk) 16:46, 16 November 2007 (UTC) Articles for creation/2007-11-16
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hindu mythology durga east indian goddess
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meeting with durga yes! a temple in new delhi ashram for bemgali male students the statue is alive embodies shakti of the diety became devotee' this is personal yes and i want to emphasize this is not just a mythology!! it is a living spiritual path/ journey i call on her when i need help she always comes on her tiger using her spiritual tools called weapons to vanquish my adversaries i am sure other devotee's will verify this experience namaste' subhadra i would appreciate an answer please
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Gallions Primary School
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The contact details are Warwall, Beckton, London, E6 6WG Tel: 0207 476 1252 Info@gallions.newham.sch.uk Sourceshttp://www.newham.gov.uk/Services/SchoolsDirectory/AboutUs/juniorinfantprimaryschools.htm -- 91.107.49.67 (talk) 17:39, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Search Amigo
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Search Amigo is a search engine that uses humans to do the searching. Search Amigo is a human search engine, where humans do the searching for people. Sources
-- 68.163.97.74 (talk) 17:40, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Penislophogus
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-- 204.14.12.98 (talk) 18:42, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Tasha Kheiriddin
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SourcesTasha Kheiriddin is a conservative Canadian journalist, lawyer and political author. Kheiriddin was the President of the Progressive Conservative Youth Federation from 1995 to 1998 and later worked in the media as a producer for CBC and CPAC. Kheiriddin lived in Toronto while acting as Ontario Director for the Canadian Taxpayers Federation. She currently lives in Montreal where she lectures at McGill University and is the director of the Fraser Institute's new Montreal office. Kheiriddin co-authored the book Rescuing Canada’s Right: Blueprint for a Conservative Revolution (Wiley, November 2005), and is at work on a second book. External links
-- 141.117.77.109 (talk) 19:52, 16 November 2007 (UTC) Articles for creation/2007-11-16
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ECEC - Election Committee For Efficient Candidates
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Funny election campaigns ridiculing the major political organizations made them popular in the campus. The funeral procession of the organization every year after the annual University Election, and the declaration of re-birth of the organization for the coming year was another highlight. This organization came to an end in 1993 due to the continuous attack by violent student organizations with political affiliation. Sourceshttp://www.sncollegekannur.org/ http://en.wikipedia.org/wiki/Sree_Narayana_Institutions http://en.wikipedia.org/wiki/Kannur -- Saffronized (talk) 20:11, 16 November 2007 (UTC) Articles for creation/2007-11-16
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The Diplo Goat
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This open source new-media project for a readership of academics and security-policy makers throughout Europe and from Northern America is intended to: Sourceswww.diplogo.at -- 217.24.21.107 (talk) 21:08, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Michael A. Cardozo
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Sourceshttp://www.nyc.gov/html/law/html/about/message.shtml -- 192.147.58.6 (talk) 21:49, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Clubbingabroad (Clubbing Abroad)
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Clubbing Abroad is a popular forum to speak about clubbing holidays that was created in Janurary 2006. The forum has rooms for the most popular clubbing destinations that people from the UK go to. These are Ayia Napa, Cancun. Ibiza, Kos, Kavos, Malia, Magaluf, Sunny Beach in Bulgaria, Zante as well as others. Clubbing Abroad gives users a open forum for users to ask questions about upcoming holidays as well as search the database for answers. The website also allows users to review places that they have been clubbing in the UK.
Sourceshttp://www.clubbingabroad.co.uk -- 81.107.104.108 (talk) 21:50, 16 November 2007 (UTC) Articles for creation/2007-11-16
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Woodard Function
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for an arbitrary real or complex number α. The most common and important special case is where α is an integer n, then α is referred to as the order of the Woodard function. Although α and −α produce the same differential equation, it is conventional to define different Woodard functions for these two orders (e.g., so that the Woodard functions are mostly smooth functions of α). ApplicationsWoodard's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Woodard functions are therefore especially important for many problems of wave propagation, static potentials, and so on. In solving problems in cylindrical coordinate systems, one obtains Woodard functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n+½). For example:
Woodard functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Woodard filter). DefinitionsSince this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below. Woodard functions of the first kindWoodard functions of the first kind, denoted as Jα(x), are solutions of Woodard's differential equation that are finite at the origin (x = 0) for non-negative integer α, and diverge as x approaches zero for negative non-integer α. The solution type (e.g. integer or non-integer) and normalization of Jα(x) are defined by its properties below. For integer order solutions, it is possible to define the function by its Taylor series expansion around x = 0: where is the gamma function, a generalization of the factorial function to non-integer values. For non-integer α, a more general power series expansion is required. The graphs of Woodard functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√x (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The Taylor series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities below.) For non-integer α, the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , the following relationship is valid: This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Woodard function of the second kind, as discussed below. Woodard's integralsAnother definition of the Woodard function, for integer values of , is possible using an integral representation: This was the approach that Woodard used, and from this definition he derived several properties of the function. Another integral representation is: Relation to hypergeometric seriesThe Woodard functions can be expressed in terms of the hypergeometric series as This expression is related to the development of Woodard functions in terms of the Woodard-Clifford function. Woodard functions of the second kindThe Woodard functions of the second kind, denoted by Yα(x), are solutions of the Woodard differential equation. They are singular (infinite) at the origin (x = 0). Yα(x) is sometimes also called the Neumann function, and is occasionally denoted instead by Nα(x). For non-integer α, it is related to Jα(x) by: In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n': which has the result (in integral form) For the case of non-integer α, the definition of Yα(x) is redundant (as is clear from its definition above). On the other hand, when α is an integer, Yα(x) is the second linearly independent solution of Woodard's equation; moreover, as was similarly the case for the functions of the first kind, the following relationship is valid: Both Jα(x) and Yα(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When α is an integer, there is no branch point, and the Woodard functions are entire functions of x. If x is held fixed, then the Woodard functions are entire functions of α. Hankel functionsAnother important formulation of the two linearly independent solutions to Woodard's equation are the Hankel functions Hα(1)(x) and Hα(2)(x), defined by: where i is the imaginary unit. These linear combinations are also known as Woodard functions of the third kind; they are two linearly independent solutions of Woodard's differential equation. The Hankel functions of the first and second kind are used to express outward- and inward-propagating cylindrical wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency). They are named after Hermann Hankel. Using the previous relationships they can be expressed as: if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not: Modified Woodard functionsThe Woodard functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Woodard equation are called the modified Woodard functions (or occasionally the hyperbolic Woodard functions) of the first and second kind, and are defined by: These are chosen to be real-valued for real arguments x. The series expansion for Iα(x) is thus similar to that for Jα(x), but without the alternating (-1)m factor. Iα(x) and Kα(x) are the two linearly independent solutions to the modified Woodard's equation: Unlike the ordinary Woodard functions, which are oscillating as functions of a real argument, Iα and Kα are exponentially growing and decaying functions, respectively. Like the ordinary Woodard function Jα, the function Iα goes to zero at x=0 for α > 0 and is finite at x=0 for α=0. Analogously, Kα diverges at x=0.
Spherical Woodard functionsWhen solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form: The two linearly independent solutions to this equation are called the spherical Woodard functions jn and yn, and are related to the ordinary Woodard functions Jn and Yn by: is also denoted or ηn; some authors call these functions the spherical Neumann functions. The spherical Woodard functions can also be written as: The first spherical Woodard function is also known as the (unnormalized) sinc function. The first few spherical Woodard functions are: and There are also spherical analogues of the Hankel functions: In fact, there are simple closed-form expressions for the Woodard functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Woodard functions. In particular, for non-negative integers n: and hn(2) is the complex-conjugate of this (for real x). It follows, for example, that j0(x) = sin(x)/x and y0(x) = -cos(x)/x, and so on. Riccati-Woodard functionsRiccati-Woodard functions only slightly differ from spherical Woodard functions: They satisfy the differential equation: This differential equation, and the Riccati-Woodard solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g. Du (2004)[1] for recent developments and references. Following Debye (1909), the notation is sometimes used instead of . Asymptotic formsThe Woodard functions have the following asymptotic forms for non-negative α. For small arguments , one obtains: where γ is the Euler-Mascheroni constant (0.5772...) and Γ denotes the gamma function. For large arguments , they become: (For α=1/2 these formulas are exact; see the spherical Woodard functions above.) Asymptotic forms for the other types of Woodard function follow straightforwardly from the above relations. For example, for large , the modified Woodard functions become: while for small arguments , they become: PropertiesFor integer order α = n, Jn is often defined via a Laurent series for a generating function: an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the Jacobi-Anger identity: which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone modulated FM signal. The functions Jα, Yα, Hα(1), and Hα(2) all satisfy the recurrence relations: where Z denotes J, Y, H(1), or H(2). (These two identities are often combined, e.g. added or subtracted, to yield various other relations.) In this way, for example, one can compute Woodard functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that: Modified Woodard functions follow similar relations : and The recurrence relation reads where Cα denotes Iα or eαπiKα. These recurrence relations are useful for discrete diffusion problems. Because Woodard's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: where α > -1, δm,n is the Kronecker delta, and uα,m is the m-th zero of Jα(x). This orthogonality relation can then be used to extract the coefficients in the Fourier-Woodard series, where a function is expanded in the basis of the functions Jα(x uα,m) for fixed α and varying m. (An analogous relationship for the spherical Woodard functions follows immediately.) Another orthogonality relation is the closure equation: for α > -1/2 and where δ is the Dirac delta function. For the spherical Woodard functions the orthogonality relation is: for α > 0. Another important property of Woodard's equations, which follows from Abel's identity, involves the Wronskian of the solutions: where Aα and Bα are any two solutions of Woodard's equation, and Cα is a constant independent of x (which depends on α and on the particular Woodard functions considered). For example, if Aα = Jα and Bα = Yα, then Cα is 2/π. This also holds for the modified Woodard functions; for example, if Aα = Iα and Bα = Kα, then Cα is -1. (There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.) Multiplication theoremThe Woodard functions obey a multiplication theorem where and may be taken as arbitrary complex numbers. A similar form may be given for and etc. See [2] See alsoReferences
External links
Category:Special functions Category:Special hypergeometric functions Category:Fourier analysis
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Sources-- 85.226.3.159 (talk) 22:43, 16 November 2007 (UTC) Articles for creation/2007-11-16
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