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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.


===Bibliography===
===Bibliography===

*[[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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{{Royal house|
{{tlx|Royal house|
| surname = The Royal Family
| surname = The Royal Family
| estate = of Portugal
| estate = of Portugal
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Hashford Edison's Hanford band was a [[post-punk]] band from [[Seattle]], [[Washington]] that was formed out of the ashes of [[the Gits]]. It featured [[Dave Grohl]] on bass guitar. The band toured in 2003 with [[The Fastbacks]], and the [[Young Fresh Fellows]]. They released one [[7-inch]] on [[Popllama]] records in 2005.
Hashford Edison's Hanford band was a [[post-punk]] band from [[Seattle]], [[Washington (U.S. state)|Washington]] that was formed out of the ashes of [[the Gits]]. It featured [[Dave Grohl]] on bass guitar. The band toured in 2003 with [[The Fastbacks]], and the [[Young Fresh Fellows]]. They released one [[7-inch]] on [[Popllama]] records in 2005.


=== Sources ===
=== Sources ===
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High heeled boots worn by ancient Greeks so the audience could see them better.
High heeled boots worn by ancient Greeks so the audience could see them better.
<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. We cannot accept unsourced suggestions or sources that are [[Wikipedia:Reliable sources|not reliable]] per the [[Wikipedia:Verifiability|verifiability policy]]. Please provide '''reputable, third-party''' sources with your suggestions. Third party sources are needed both to establish the [[WP:Verifiability|verifiability]] of the submission as well as its [[Wikipedia:Notability|notability]]. <small>[[user:ArielGold|<font color="8B00FF">'''Ariel'''</font>]][[Special:Contributions/ArielGold|<font color="F64A8A">♥</font>]][[User_talk:ArielGold|<font color="007FFF">'''Gold'''</font>]]</small> 05:33, 16 November 2007 (UTC)
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== [[Windy City Pro Wrestling]] ==
== [[Windy City Pro Wrestling]] ==
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{{Pwcompanybox
{{tlx|Pwcompanybox
|name=Windy City Pro Wrestling
|name=Windy City Pro Wrestling
|image=
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After 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.
After 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]] <ref> ''"Paul E. is managing to live Dangerously"''. <u> Chicago Sun-Times. </u> 08 Apr 1988 </ref> (who was also working for Southeastern promotions [[Southern Championship Wrestling]] and the [[Continental Wrestling Federation]]) <ref> Loverro, Thom. ''The Rise & Fall of ECW: Extreme Championship Wrestling''. New York: Simon & Schuster, 2006. (pg. 17-18) ISBN 1-4165-1058-3 </ref> at [[DaVinci Manor]] and the [[International Amphitheater]] <ref> ''"Live bouts return to Amphitheater"''. <u> Chicago Sun-Times. </u> 29 Apr 1988 </ref> 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 Gordy|Terry "Bam Bam" Gordy]] and [[Bam Bam Bigalow]] in the main event, the promotion still lost money. <ref> ''"Still Learning The Ropes: Wrestling Promoter Pins Hope on School, Local TV pact"''. <u> Crains Chicago Business. </u> 11 Dec 1989 </ref>
DeCero soon began running televised wrestling events with then 22-year old [[Paul Heyman]] <ref> ''"Paul E. is managing to live Dangerously"''. <u> Chicago Sun-Times. </u> 08 Apr 1988 </ref> (who was also working for Southeastern promotions [[Southern Championship Wrestling]] and the [[Continental Wrestling Federation]]) <ref> Loverro, Thom. ''The Rise & Fall of ECW: Extreme Championship Wrestling''. New York: Simon & Schuster, 2006. (pg. 17-18) {{ISBN|1-4165-1058-3}} </ref> at [[DaVinci Manor]] and the [[International Amphitheater]] <ref> ''"Live bouts return to Amphitheater"''. <u> Chicago Sun-Times. </u> 29 Apr 1988 </ref> 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 Gordy|Terry "Bam Bam" Gordy]] and [[Bam Bam Bigalow]] in the main event, the promotion still lost money. <ref> ''"Still Learning The Ropes: Wrestling Promoter Pins Hope on School, Local TV pact"''. <u> Crains Chicago Business. </u> 11 Dec 1989 </ref>


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]]. <ref> ''"Amphitheater bouts to aid homeless"''. <u> Chicago Sun-Times. </u> 17 May 1991 </ref> <ref> {{cite web |url=http://www.deathvalleydriver.com/Benaka/decero.html |title=The Lee Banaka Interviews: Sam DeCero |accessdate= |author=Benaka, Lee |authorlink= |coauthors= |date=2005-07-25 |year= |month= |format= |work= |publisher=DeathValleyDriver.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>
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]]. <ref> ''"Amphitheater bouts to aid homeless"''. <u> Chicago Sun-Times. </u> 17 May 1991 </ref> <ref> {{cite web |url=http://www.deathvalleydriver.com/Benaka/decero.html |title=The Lee Banaka Interviews: Sam DeCero |accessdate= |author=Benaka, Lee |authorlink= |date=2005-07-25 |format= |work= |publisher=DeathValleyDriver.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>


During the late 1980s, the promotion featured many popular wrestlers of the era including [[Dick Murdoch]], [[Shigeri Akabane]], [[Dennis Condrey]], <ref> ''"`Lover Boy' Condrey revs up for slugfest at Amphitheatre"''. <u> Chicago Sun-Times. </u> 12 Aug 1988 </ref> [[Steve Regal]] <ref> ''"Windy City champion Regal is ready to `electrify' Cicero"''. <u> Chicago Sun-Times. </u> 10 Feb 1989 </ref> and [[Bob Duane Sabre|George Ringo]], who acted as honorary commissioner. Prior to his death in [[Puerto Rico]], [[Bruiser Brody]] had been scheduled to face [[John Nord|Nord the Barbarian]] at the International Amphitheater in Chicago, Illinois on August 12, 1988. <ref> {{cite web |url=http://members.shaw.ca/Bruiser_Brody/brody_article_8_txt.html |title=Frank "King Kong" Brody dies, outlaw wrestler |accessdate= |author=Grahnke, Lon |authorlink= |coauthors= |date= |year=1988 |month=July |format= |work= |publisher=The Unofficial Bruiser Brody Memorial Page |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> Other mainstays included [[Mike Anthony (wrestler)|"Mean" Mike Anthony]], <ref> {{cite web |url=http://www.wrestlingobserver.com/wo/news/headlines/default.asp?aID=17676 |title=Wrestling Observer Headlines, 10/19/06 |accessdate= |author=Meltzer, Dave |authorlink=Dave Meltzer |coauthors= |date=2006-10-19 |year= |month= |format= |work= |publisher=[[Wrestling Observer]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Trevor Blanchard]], <ref> {{cite web |url=http://www.deathvalleydriver.com/Benaka/blanchard.html |title=The Lee Benaka Interviews: Trevor Blanchard |accessdate= |author=Banaka, Lee |authorlink= |coauthors= |date=2005-07-25 |year= |month= |format= |work= |publisher=DeathValleyDriver.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Rockin' Randy]] <ref> {{cite web |url=http://wrestlerevue.tripod.com/Dec01/interview.htm |title=Rockin' Randy: "From Ethiopia to Edgerton, I'll beat any wrestler out there!" |accessdate= |author= |authorlink= |coauthors= |date= |year=2001 |month=December |format= |work= |publisher=WrestleRevue.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>, [[Tony Montana (wrestler)|Tony Montana]] <ref> {{cite web |url=http://findarticles.com/p/articles/mi_qn4155/is_20041121/ai_n12569614 |title=Former pro wrestler prefers Hollywood ring |accessdate= |author=Pearlman, Cindy |authorlink= |coauthors= |date=2004-11-21 |year= |month= |format= |work= |publisher=[[Chicago Sun-Times]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and '''The Power Twins''' (Larry & David Sontag).
During the late 1980s, the promotion featured many popular wrestlers of the era including [[Dick Murdoch]], [[Shigeri Akabane]], [[Dennis Condrey]], <ref> ''"`Lover Boy' Condrey revs up for slugfest at Amphitheatre"''. <u> Chicago Sun-Times. </u> 12 Aug 1988 </ref> [[Steve Regal]] <ref> ''"Windy City champion Regal is ready to `electrify' Cicero"''. <u> Chicago Sun-Times. </u> 10 Feb 1989 </ref> and [[Bob Duane Sabre|George Ringo]], who acted as honorary commissioner. Prior to his death in [[Puerto Rico]], [[Bruiser Brody]] had been scheduled to face [[John Nord|Nord the Barbarian]] at the International Amphitheater in Chicago, Illinois on August 12, 1988. <ref> {{cite web |url=http://members.shaw.ca/Bruiser_Brody/brody_article_8_txt.html |title=Frank "King Kong" Brody dies, outlaw wrestler |accessdate= |author=Grahnke, Lon |authorlink= | date=July 1988 |format= |work= |publisher=The Unofficial Bruiser Brody Memorial Page |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> Other mainstays included [[Mike Anthony (wrestler)|"Mean" Mike Anthony]], <ref> {{cite web |url=http://www.wrestlingobserver.com/wo/news/headlines/default.asp?aID=17676 |title=Wrestling Observer Headlines, 10/19/06 |accessdate= |author=Meltzer, Dave |authorlink=Dave Meltzer |date=2006-10-19 |format= |work= |publisher=[[Wrestling Observer]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Trevor Blanchard]], <ref> {{cite web |url=http://www.deathvalleydriver.com/Benaka/blanchard.html |title=The Lee Benaka Interviews: Trevor Blanchard |accessdate= |author=Banaka, Lee |authorlink= |date=2005-07-25 |format= |work= |publisher=DeathValleyDriver.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Rockin' Randy]] <ref> {{cite web |url=http://wrestlerevue.tripod.com/Dec01/interview.htm |title=Rockin' Randy: "From Ethiopia to Edgerton, I'll beat any wrestler out there!" |accessdate= |author= |authorlink= | date=December 2001 |format= |work= |publisher=WrestleRevue.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>, [[Tony Montana (wrestler)|Tony Montana]] <ref> {{cite web |url=http://findarticles.com/p/articles/mi_qn4155/is_20041121/ai_n12569614 |title=Former pro wrestler prefers Hollywood ring |accessdate= |author=Pearlman, Cindy |authorlink= |date=2004-11-21 |format= |work= |publisher=[[Chicago Sun-Times]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> 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 <ref> ''"Raquel Copies Liz in Krantz's 'Torch' Affair"''. <u> Chicago Sun-Times. </u> 22 May 1993 </ref> and, two years later, presented [[Sailor Art Thomas]] a ''Lifetime Service to Sport Award'' on May 16, 1995. <ref> ''"'The Sailor' To Be Honored"''. <u> Wisconsin State Journal. </u> 20 Apr 1995 </ref> 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. <ref> ''"Razzle dazzle"''. <u> Telegraph Herald. </u> 25 Feb 1996 </ref> 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'''''. <ref> {{cite web |url=http://www.sfifer.com/images/2185859_9.doc |title=Media/Intellectual Property Case Summeries |accessdate= |author=Fifer, Samuel |authorlink= |coauthors= |date= |year= |month= |format=[[.doc]] |work= |publisher=Sfifer.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>
Although going into a slump during the early 1990s, its annual '''Battle of the Belts''' supercard was aired on [[SportsChannel]] on May 22, 1993 <ref> ''"Raquel Copies Liz in Krantz's 'Torch' Affair"''. <u> Chicago Sun-Times. </u> 22 May 1993 </ref> and, two years later, presented [[Sailor Art Thomas]] a ''Lifetime Service to Sport Award'' on May 16, 1995. <ref> ''"'The Sailor' To Be Honored"''. <u> Wisconsin State Journal. </u> 20 Apr 1995 </ref> 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. <ref> ''"Razzle dazzle"''. <u> Telegraph Herald. </u> 25 Feb 1996 </ref> 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'''''. <ref> {{cite web |url=http://www.sfifer.com/images/2185859_9.doc |title=Media/Intellectual Property Case Summeries |accessdate= |author=Fifer, Samuel |authorlink= |format=[[.doc]] |work= |publisher=Sfifer.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>


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]], <ref> {{cite web |url=http://www.wrestleview.com/news2006/1170537166.shtml |title=Ace Steel & Kevin Kleinrock Interviews: Steel on Trump, Kleinrock/WSX |accessdate= |author=Young, Doc |authorlink= |coauthors=[[Les Thatcher]] |date=2007-01-30 |year= |month= |format= |work= |publisher=WrestleView.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Sosay]], <ref> {{cite web |url=http://www.officialsosay.com/sosay/evolution.html |title=Sosay: Character Evolution |accessdate= |author= |authorlink= |coauthors= |date= |year=2007 |month=June |format= |work= |publisher=OfficialSosay.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Kevin Quinn]] <ref> {{cite web |url=http://www.puroresufan.com/interviews/quinn.php |title=Interview with Kevin Quinn |accessdate= |author= |authorlink= |coauthors= |date= |year=2004 |month=June |format= |work= |publisher=PuroresuFan.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and [[Christopher Daniels]] <ref> {{cite web |url=http://slam.canoe.ca/Slam/Wrestling/Bios/daniels.html |title=SLAM! Wrestling Bios: Christopher Daniels |accessdate= |author=Milner, John M. |authorlink= |coauthors=Tim Baines and Corey David Lacroix |date=2005-09-05 |year= |month= |format= |work= |publisher=SLAM! Sports |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> emerged from the promotion during the late 1990s and early 2000s. During the last several years, independent wrestlers such as [[Colt Cabana]], [[Abyss (wrestler)|Abyss]] and [[Austin Aries]] among others have made appearances in the promotion as have WWF veterans [[King Kong Bundy]], [[Greg Valentine|Greg "The Hammer" Valentine]] <ref> {{cite web |url=http://www.thezephyr.com/archives/rasslin.htm |title=Wrestling at the Knox County Fair |accessdate= |author=Gallagher, Jon |authorlink= |coauthors= |date=2006-03-23 |year= |month= |format= |work= |publisher=The Zephyr |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and [[Jerry Lawler|Jerry "The King" Lawler]].
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]], <ref> {{cite web |url=http://www.wrestleview.com/news2006/1170537166.shtml |title=Ace Steel & Kevin Kleinrock Interviews: Steel on Trump, Kleinrock/WSX |accessdate= |author=Young, Doc |authorlink= |author2=[[Les Thatcher]] |date=2007-01-30 |format= |work= |publisher=WrestleView.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Sosay]], <ref> {{cite web |url=http://www.officialsosay.com/sosay/evolution.html |title=Sosay: Character Evolution |accessdate= |author= |authorlink= | date=June 2007 |format= |work= |publisher=OfficialSosay.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> [[Kevin Quinn]] <ref> {{cite web |url=http://www.puroresufan.com/interviews/quinn.php |title=Interview with Kevin Quinn |accessdate= |author= |authorlink= | date=June 2004 |format= |work= |publisher=PuroresuFan.com |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and [[Christopher Daniels]] <ref> {{cite web |url=http://slam.canoe.ca/Slam/Wrestling/Bios/daniels.html |title=SLAM! Wrestling Bios: Christopher Daniels |accessdate= |author=Milner, John M. |authorlink= |author2=Tim Baines and Corey David Lacroix |date=2005-09-05 |format= |work= |publisher=SLAM! Sports |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> emerged from the promotion during the late 1990s and early 2000s. During the last several years, independent wrestlers such as [[Colt Cabana]], [[Abyss (wrestler)|Abyss]] and [[Austin Aries]] among others have made appearances in the promotion as have WWF veterans [[King Kong Bundy]], [[Greg Valentine|Greg "The Hammer" Valentine]] <ref> {{cite web |url=http://www.thezephyr.com/archives/rasslin.htm |title=Wrestling at the Knox County Fair |accessdate= |author=Gallagher, Jon |authorlink= |date=2006-03-23 |format= |work= |publisher=The Zephyr |pages= |language= |archiveurl= |archivedate= |quote= }} </ref> and [[Jerry Lawler|Jerry "The King" Lawler]].


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>


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= |date=2004-09-28 |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= |date=2005-10-11 |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= |date=2005-05-25 |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= |date=2006-03-23 |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>


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= |date=2001-06-01 |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= |date=2005-05-10 |format= |work= |publisher=[[Medill School of Journalism]] |pages= |language= |archiveurl= |archivedate= |quote= }} </ref>


===Roster===
===Roster===
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===Championships===
===Championships===
*[[WCPW Heavyweight Championship]] <ref name="Titles">{{cite book | author=Royal Duncan & Gary Will | title=Wrestling Title Histories | publisher=Archeus Communications | year=4th Edition 2000 | id=ISBN 0-9698161-5-4 }}</ref>
*[[WCPW Heavyweight Championship]] <ref name="Titles">{{cite book | author=Royal Duncan & Gary Will | title=Wrestling Title Histories | publisher=Archeus Communications | year=2000 |edition=4th | id={{ISBN|0-9698161-5-4}} }}</ref>
*[[WCPW League Championship]] <ref name="Titles"/>
*[[WCPW League Championship]] <ref name="Titles"/>
*[[WCPW Middleweight Championship]] <ref name="Titles"/>
*[[WCPW Middleweight Championship]] <ref name="Titles"/>
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<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. This article already exists in Wikipedia. {{#if:{{{1|}}}|You can find it at [[:{{{1}}}]].|}} [[User:Nousernamesleft|Temperal]]<span style="color:red"><small><sup>[[User Talk:Nousernamesleft|xy]]</sup></small></span> 04:11, 19 November 2007 (UTC)
*'''Comment'''. The article was already created from this submission as seen by the [http://en.wikipedia.org/enwiki/w/index.php?title=Windy_City_Pro_Wrestling&oldid=172036431 article's history]. [[User:72.74.195.171|72.74.195.171]] 10:26, 23 November 2007 (UTC)
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<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. {{#if:{{{1|}}}|This article appears to be taken from {{{1}}}.}} We cannot accept [[WP:Copyrights|copyrighted]] content taken from web sites or printed sources. Note that copyright protection is granted to all works automatically, whether it is asserted or not. Unless stated otherwise, assume that most content on the internet is copyrighted and not suitable for Wikipedia. Please write in your own words, and in continuous prose. <small>[[user:ArielGold|<font color="8B00FF">'''Ariel'''</font>]][[Special:Contributions/ArielGold|<font color="F64A8A">♥</font>]][[User_talk:ArielGold|<font color="007FFF">'''Gold'''</font>]]</small> 05:50, 16 November 2007 (UTC)
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== Carl Kauba ==
== Carl Kauba ==
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'''(1865 - 1922)'''

Carl Kauba was born August 13, 1865 in Vienna, Austria. The son of a shoemaker, Kauba chose to follow his calling into the world of art. Collectors now rank him in a class with Remington and Russell as one of the great portrayers of American Western. His subjects were typically American Indians, cavalrymen, cowboys, and roughriders.

Carl studied at the Academy of Fine Arts in Vienna under professor Laufenberg. Later he would study at the academies under Carl Waschmann and Stefan Schwartz and in Paris in 1886 to further study before he set off for the U.S. Quite some debate remains as to whether Kauba ever visited the U.S. at all.

Without a doubt, his work is most worthy of exceptional artistic merit. His bronzes, all produced in Austria, were of the American West. The polychrome finish, very intricate detail and the realistic forms make Kauba’s sculptures excellent examples of Viennese bronzes from the turn of the century. Most were cast for the American market between 1895 and 1912 and were widely advertised and sold by Latendorfer in New York during the 1950’s.

In addition to his American bronzes, Kauba produced a lifetime’s worth of Austrian statuary. His work has been fully appreciated on both sides of the Atlantic.

Kauba was fascinated with the West. It is widely thought that Kauba was about twenty-five years old when he travelled widely throughout the American West. He then returned to Austria with a plethora of notes, sketches, and several models of Western sculpture. A friend in Ohio, of whom he kept in contact, sent him a complete collection of costume material once he had returned home to Europe.

In contrast to most artists, Kauba’s business successes were equal to his artistic achievement. He was able to maintain for his wife and three children a comfortable lifestyle. Kauba’s studio and home were in direct proximity of each other, allowing him more freedom in allocating his time. Kauba personally directed the casting of his clay models in local foundries. He expressed a great affection for Indians of the American West and loved ornament. This detail is seen in the reins of his horses, Indian headdresses, weapons, and costumes. He frequently created scenes of high drama depicted in his bronzes of cowboys and Indians.


< copyright violation removed>
Kauba’s work is currently in the Harmsen Collection of American Western Art. After his death in 1922, his son continued the business operation. Some of his original moulds are still in the Viennese foundry, Fuhrmann & Co.


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== Bacillus rossius ==
== Bacillus rossius ==
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* http://www.bugsincyberspace.com/phasmids/bacillus_rossius.html
* http://www.bugsincyberspace.com/phasmids/bacillus_rossius.html
* Italian: http://it.wikipedia.org/wiki/Bacillus_rossius
* Italian: http://it.wikipedia.org/wiki/Bacillus_rossius
<br />[[Image:Symbol confirmed.svg|20px]] '''Article created.''' {{#if:{{{1|}}}|You can find it at [[{{{1}}}]].}} Thank you for your contribution to Wikipedia! [[User:Graeme Bartlett|Graeme Bartlett]] ([[User talk:Graeme Bartlett|talk]]) 10:16, 18 November 2007 (UTC)
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== History ==
== History ==
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<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. Your article appears to be a joke. It might be a wonderful entry in [http://www.uncyclopedia.org Uncyclopedia], a wiki where jokes and parodies are highly encouraged. However, Wikipedia requires a stodgier, more factual tone in its articles, and your comedic efforts cannot be rewarded here. <!-- From template Afc joke --> [[User:PhilipBembridge|<span style="color:#aaaaaa;"><u>PhilB</u></span>]] ~ <small><sup>[[User_talk:PhilipBembridge|<span style="color:#cccccc;">T</span>]]</sup>''/''<sub>[[Special:Contributions/PhilipBembridge|<span style="color:#cccccc;">C</span>]]</sub></small> 15:21, 16 November 2007 (UTC)
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==Albert Jack==
==Albert Jack==
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{{afc maybe|This request was submitted with no requested title, I assume that the title should be '''Albert Jack'''}}


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.
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.
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== Grand Master Choa Kok Sui ==
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== Grand Master Choa Kok Sui ==


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== Dinah McNicol ==
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== Jerald Posman writes about a Bangladeshi Writer ==
== Jerald Posman writes about a Bangladeshi Writer ==
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York College
York College
City University of New York.
City University of New York.
jposman@york.cuny.edu


Read Jelald Posman Columns from New York Times:
Read Jelald Posman Columns from New York Times:
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== Peter Embree ==
== Peter Embree ==
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=== Source ===
=== Source ===
http://google.com/peter-embree-gay.html
http://google.com/peter-embree-gay.html
<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. We cannot accept unsourced suggestions or sources that are [[Wikipedia:Reliable sources|not reliable]] per the [[Wikipedia:Verifiability|verifiability policy]]. Please provide '''reputable, third-party''' sources with your suggestions. Third party sources are needed both to establish the [[WP:Verifiability|verifiability]] of the submission as well as its [[Wikipedia:Notability|notability]]. [[User:PhilipBembridge|<font color="#aaaaaa"><u>PhilB</u></font>]] ~ <small><sup>[[User_talk:PhilipBembridge|<font color="#cccccc">T</font>]]</sup>''/''<sub>[[Special:Contributions/PhilipBembridge|<font color="#cccccc">C</font>]]</sub></small> 16:47, 16 November 2007 (UTC)
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ARTICLE WILL BE REJECTED. i dont read hindi so cannot site anything also natives the people of lower classed of east india have an oral and intuitive tradition not linear or literal i have lived in ashrams in india and traveled extensively -->
ARTICLE WILL BE REJECTED. i dont read hindi so cannot site anything also natives the people of lower classed of east india have an oral and intuitive tradition not linear or literal i have lived in ashrams in india and traveled extensively -->
ps what is wrong with personal experience? how about the scientists who came up with new ideas through intuition emages sudden flashes???? -->
ps what is wrong with personal experience? how about the scientists who came up with new ideas through intuition emages sudden flashes???? -->
<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. Your article appears to be a joke. It might be a wonderful entry in [http://www.uncyclopedia.org Uncyclopedia], a wiki where jokes and parodies are highly encouraged. However, Wikipedia requires a stodgier, more factual tone in its articles, and your comedic efforts cannot be rewarded here. <!-- From template Afc joke --> [[User:PhilipBembridge|<font color="#aaaaaa"><u>PhilB</u></font>]] ~ <small><sup>[[User_talk:PhilipBembridge|<font color="#cccccc">T</font>]]</sup>''/''<sub>[[Special:Contributions/PhilipBembridge|<font color="#cccccc">C</font>]]</sub></small> 18:06, 16 November 2007 (UTC)
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== Gallions Primary School ==
== Gallions Primary School ==
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== Search Amigo ==
== Search Amigo ==
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== Tasha Kheiriddin ==
== Tasha Kheiriddin ==
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'''Tasha Kheiriddin''' is a [[conservatism|conservative]] [[Canada|Canadian]] [[journalist]], [[lawyer]] and political author.
'''Tasha Kheiriddin''' is a [[conservatism|conservative]] [[Canada|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 was the President of the [[Progressive Conservative Youth Federation]] from 1995 to 1998 and later worked in the media as a producer for [[CBC Television|CBC]] and [[CPAC (TV channel)|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 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.
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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.
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==
===External links===
* [http://www.macleans.ca/macleans50/archive.jsp?id=28], Maclean's top 50 profile
* [http://www.macleans.ca/macleans50/archive.jsp?id=28], Maclean's top 50 profile
* [http://www.amazon.com/Rescuing-Canadas-Right-Conservative-Revolution/dp/047083692X], Amazon site with author blurb
* [http://www.amazon.com/Rescuing-Canadas-Right-Conservative-Revolution/dp/047083692X], Amazon site with author blurb
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[[:Category:Canadian political writers]]
[[:Category:Canadian political writers]]


{{Canada-journalist-stub}}
{{tlx|Canada-journalist-stub}}


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<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. This suggestion doesn't sufficiently explain the [[WP:N|importance or significance]] of the subject. See the [[WP:SPEEDY#A7|speedy deletion criteria A7]] and/or [[WP:BIO|guidelines on biographies]]. Please provide more information on why the person or group is worthy of inclusion in an encyclopedia, and cite [[WP:RS|reliable]], published third-party sources, so that the information in the article is [[WP:V|verifiable]]. Thank you. [[User:Nousernamesleft|Temperal]]<span style="color:red"><small><sup>[[User Talk:Nousernamesleft|xy]]</sup></small></span> 04:12, 19 November 2007 (UTC)
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== ECEC - Election Committee For Efficient Candidates ==
== ECEC - Election Committee For Efficient Candidates ==
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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>


for an arbitrary real or complex number &alpha;. The most common and important special case is where &alpha; is an [[integer]] ''n'', then &alpha; is referred to as the ''order'' of the Woodard function.
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 &alpha; and &minus;&alpha; 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 &alpha;).
Although α and &minus;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).


===Applications===
===Applications===
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 (&alpha; = ''n''); in spherical problems, one obtains half-integer orders (&alpha; = ''n''+&frac12;). For example:
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:


* [[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====
Woodard functions of the first kind, denoted as ''J''<sub>&alpha;</sub>(''x''), are solutions of Woodard's differential equation that are finite at the origin (''x'' = 0) for non-negative integer &alpha;, and diverge as ''x'' approaches zero for negative non-integer &alpha;. The solution type (e.g. integer or non-integer) and normalization of ''J''<sub>&alpha;</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:
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:


:<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>


where <math>\Gamma(z)</math> is the [[gamma function]], a generalization of the [[factorial]] function to non-integer values. For non-integer &alpha;, 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/&radic;''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]].)
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]].)


[[Image:WoodardJ plot.svg|thumb|300px|right|Plot of Woodard function of the first kind, J<sub>&alpha;</sub>(x), for integer orders &alpha;=0,1,2.]]
[[Image:WoodardJ plot.svg|thumb|300px|right|Plot of Woodard function of the first kind, J<sub>&alpha;</sub>(x), for integer orders &alpha;=0,1,2.]]


For non-integer &alpha;, 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:
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:


:<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 ====


The Woodard functions of the second kind, denoted by ''Y''<sub>&alpha;</sub>(''x''), are solutions of the Woodard differential equation. They are singular ([[infinite]]) at the origin (''x'' = 0).
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).


[[Image:WoodardY plot.svg|thumb|300px|right|Plot of Woodard function of the second kind, Y<sub>&alpha;</sub>(x), for integer orders &alpha;=0,1,2.]]
[[Image:WoodardY plot.svg|thumb|300px|right|Plot of Woodard function of the second kind, Y<sub>&alpha;</sub>(x), for integer orders &alpha;=0,1,2.]]


''Y''<sub>&alpha;</sub>(''x'') is sometimes also called the '''Neumann function''', and is occasionally denoted instead by ''N''<sub>&alpha;</sub>(''x''). For non-integer &alpha;, it is related to ''J''<sub>&alpha;</sub>(''x'') by:
''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:


:<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>


In the case of integer order ''n'', the function is defined by taking the limit as a non-integer &alpha; tends to ''n':
In the case of integer order ''n'', the function is defined by taking the limit as a non-integer α tends to ''n':


:<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>


For the case of non-integer &alpha;, the definition of ''Y''<sub>&alpha;</sub>(x) is redundant (as is clear from its definition above). On the other hand, when &alpha; is an integer, ''Y''<sub>''&alpha;''</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:
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:


:<math>Y_{-n}(x) = (-1)^n Y_n(x)\,</math>
:<math>Y_{-n}(x) = (-1)^n Y_n(x)\,</math>


Both ''J''<sub>&alpha;</sub>(''x'') and ''Y''<sub>&alpha;</sub>(''x'') are [[holomorphic function]]s of ''x'' on the [[complex plane]] cut along the negative real axis. When &alpha; 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 &alpha;.
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.


=== Hankel functions ===
=== Hankel functions ===
Another important formulation of the two linearly independent solutions to Woodard's equation are the '''Hankel functions''' ''H''<sub>&alpha;</sub><sup>(1)</sup>(''x'') and ''H''<sub>&alpha;</sub><sup>(2)</sup>(''x''), defined by:
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:


:<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>


if &alpha; is an integer, the limit has to be calculated. The following relationships are valid, whether &alpha; is an integer or not:
if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:


:<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>


These are chosen to be real-valued for real arguments ''x''. The series expansion for ''I<sub>&alpha;</sub>''(''x'') is thus similar to that for ''J<sub>&alpha;</sub>''(''x''), but without the alternating (-1)<sup>''m''</sup> factor.
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.


''I<sub>&alpha;</sub>''(''x'') and ''K<sub>&alpha;</sub>''(''x'') are the two linearly independent solutions to the modified Woodard's equation:
''I<sub>α</sub>''(''x'') and ''K<sub>α</sub>''(''x'') are the two linearly independent solutions to the modified Woodard's equation:


:<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>


Unlike the ordinary Woodard functions, which are oscillating as functions of a real argument, ''I''<sub>&alpha;</sub> and ''K''<sub>&alpha;</sub> are [[exponential growth|exponentially growing]] and [[exponential decay|decaying]] functions, respectively. Like the ordinary Woodard function ''J''<sub>&alpha;</sub>, the function ''I''<sub>&alpha;</sub> goes to zero at ''x''=0 for &alpha; > 0 and is finite at ''x''=0 for &alpha;=0. Analogously, ''K''<sub>&alpha;</sub> diverges at ''x''=0.
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.


{| align="center"
{| align="center"
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==== Spherical Woodard functions ====
==== Spherical Woodard functions ====


[[Image:Spherical Woodard j plot.svg|none|thumb|300px|right|Spherical Woodard functions of 1st kind, j<sub>n</sub>(x), for n=0,1,2]]
[[Image:Spherical Woodard j plot.svg|none|thumb|300px|Spherical Woodard functions of 1st kind, j<sub>n</sub>(x), for n=0,1,2]]
[[Image:Spherical Woodard y plot.svg|none|thumb|300px|right|Spherical Woodard functions of 2nd kind, y<sub>n</sub>(x), for n=0,1,2]]
[[Image:Spherical Woodard y plot.svg|none|thumb|300px|Spherical Woodard functions of 2nd kind, y<sub>n</sub>(x), for n=0,1,2]]
When solving the [[Helmholtz equation]] in spherical coordinates by separation of variables, the radial equation has the form:
When solving the [[Helmholtz equation]] in spherical coordinates by separation of variables, the radial equation has the form:


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==== Asymptotic forms ====
==== Asymptotic forms ====
The Woodard functions have the following [[Asymptotic expansion|asymptotic forms]] for non-negative &alpha;. For small arguments <math>0 < x \ll \sqrt{\alpha + 1}</math>, one obtains:
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:


:<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>


where &gamma; is the [[Euler-Mascheroni constant]] (0.5772...) and &Gamma; denotes the [[gamma function]]. For large arguments <math>x \gg |\alpha^2 - 1/4|</math>, they become:
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:


:<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>


(For &alpha;=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:
(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:


:<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====
<!-- This section is linked from [[Woodard function]] -->
<!-- This section is linked from [[Woodard function]] -->
For integer order &alpha; = ''n'', ''J''<sub>''n''</sub> is often defined via a [[Laurent series]] for a generating function:
For integer order α = ''n'', ''J''<sub>''n''</sub> is often defined via a [[Laurent series]] for a generating function:


:<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.


The functions ''J''<sub>&alpha;</sub>, ''Y''<sub>&alpha;</sub>, ''H''<sub>&alpha;</sub><sup>(1)</sup>, and ''H''<sub>&alpha;</sub><sup>(2)</sup> all satisfy the [[recurrence relation]]s:
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:


:<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>


where ''C''<sub>&alpha;</sub> denotes ''I''<sub>&alpha;</sub> or ''e''<sup>&alpha;&pi;''i''</sup>''K''<sub>&alpha;</sub>. These recurrence relations are useful for discrete diffusion problems.
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.


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>


where &alpha; > -1, &delta;<sub>''m'',''n''</sub> is the [[Kronecker delta]], and ''u''<sub>&alpha;,m</sub> is the ''m''-th [[root (mathematics)|zero]] of ''J''<sub>&alpha;</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>&alpha;</sub>(''x'' ''u''<sub>&alpha;,m</sub>) for fixed &alpha; and varying ''m''. (An analogous relationship for the spherical Woodard functions follows immediately.)
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.)


Another orthogonality relation is the ''closure equation'':
Another orthogonality relation is the ''closure equation'':
Line 2,214: Line 2,247:
:<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>


for &alpha; > -1/2 and where &delta; is the [[Dirac delta function]]. For the spherical Woodard functions the orthogonality relation is:
for α > -1/2 and where δ is the [[Dirac delta function]]. For the spherical Woodard functions the orthogonality relation is:


:<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>


for &alpha; > 0.
for α > 0.


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:
Line 2,224: Line 2,257:
:<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>


where ''A''<sub>&alpha;</sub> and ''B''<sub>&alpha;</sub> are any two solutions of Woodard's equation, and ''C''<sub>&alpha;</sub> is a constant independent of ''x'' (which depends on &alpha; and on the particular Woodard functions considered). For example, if ''A''<sub>&alpha;</sub> = ''J''<sub>&alpha;</sub> and ''B''<sub>&alpha;</sub> = ''Y''<sub>&alpha;</sub>, then ''C''<sub>&alpha;</sub> is 2/&pi;. This also holds for the modified Woodard functions; for example, if ''A''<sub>&alpha;</sub> = ''I''<sub>&alpha;</sub> and ''B''<sub>&alpha;</sub> = ''K''<sub>&alpha;</sub>, then ''C''<sub>&alpha;</sub> is -1.
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.)
Line 2,237: Line 2,270:
</math>
</math>


where <math>\lambda</math> and <math>\nu</math> may be taken as arbitrary complex numbers. A similar form may be given for <math>Y_\nu(z)</math> and ''etc.'' See <ref>{{Abramowitz Stegun ref|9|363}}</ref>
where <math>\lambda</math> and <math>\nu</math> may be taken as arbitrary complex numbers. A similar form may be given for <math>Y_\nu(z)</math> and ''etc.'' See <ref>{{tlx|Abramowitz Stegun ref|9|363}}</ref>


====See also====
====See also====
Line 2,247: Line 2,280:
==== References ====
==== References ====
<references/>
<references/>
* {{Abramowitz_Stegun_ref2|9|355|10|435}}
* {{tlx|Abramowitz_Stegun_ref2|9|355|10|435}}
* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition (Harcourt: San Diego, 2005). ISBN 0-12-059876-0
* George B. Arfken and Hans J. Weber, ''Mathematical Methods for Physicists'', 6th edition (Harcourt: San Diego, 2005). {{ISBN|0-12-059876-0}}
* Frank Bowman, ''Introduction to Woodard Functions'' (Dover: New York, 1958). ISBN 0-486-60462-4.
* Frank Bowman, ''Introduction to Woodard Functions'' (Dover: New York, 1958). {{ISBN|0-486-60462-4}}.
* G. N. Watson, ''A Treatise on the Theory of Woodard Functions, Second Edition'', (1995) Cambridge University Press. ISBN 0-521-48391-3
* G. N. Watson, ''A Treatise on the Theory of Woodard Functions, Second Edition'', (1995) Cambridge University Press. {{ISBN|0-521-48391-3}}
* G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", ''Ann. Phys. Leipzig'' '''25'''(1908), p.377.
* G. Mie, "Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen", ''Ann. Phys. Leipzig'' '''25'''(1908), p.377.
* Refaat El Attar, ''Special Functions and Orthogonal Polynomials'', (2006) Lulu Press Inc. ISBN 1-4116-6690-9
* Refaat El Attar, ''Special Functions and Orthogonal Polynomials'', (2006) Lulu Press Inc. {{ISBN|1-4116-6690-9}}
* Refaat El Attar, ''Woodard and Related Functions'', (2007) Lulu Press Inc. ISBN 1-4303-1393-5
* Refaat El Attar, ''Woodard and Related Functions'', (2007) Lulu Press Inc. {{ISBN|1-4303-1393-5}}


====External links====
====External links====
* {{springer|first=P.I. |last=Lizorkin|id=b/b015840|title= Woodard functions}}
* {{tlx|springer|first=P.I. |last=Lizorkin|id=b/b015840|title= Woodard functions}}
*{{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}}
*{{springer|author=N.Kh. Rozov|id=B/b015830|title=Woodard equation}}
*{{tlx|springer|author=N.Kh. Rozov|id=B/b015830|title=Woodard equation}}
*[http://de.geocities.com/klaus_rottbrand/besstest_13mai04_en.html Woodard functions of order &nu; &nbsp; (Javascript)]
*[http://de.geocities.com/klaus_rottbrand/besstest_13mai04_en.html Woodard functions of order ν &nbsp; (Javascript)]


[[:Category:Special functions]]
[[:Category:Special functions]]
Line 2,265: Line 2,298:
[[:Category:Fourier analysis]]
[[:Category:Fourier analysis]]


{{Link FA|zh}}
{{tlx|Link FA|zh}}


[[ca:Funció de Woodard]]
[[cs:Woodardova funkce]]
[[de:Woodardsche Differentialgleichung]]
[[es:Función de Woodard]]
[[es:Función de Woodard]]
[[fr:Fonction de Woodard]]
[[ko:베셀 함수]]
[[ko:베셀 함수]]
[[it:Funzioni di Woodard]]
[[nl:Woodardfunctie]]
[[ja:ベッセル関数]]
[[ja:ベッセル関数]]
[[pl:Funkcje Woodarda]]
[[pt:Função de Woodard]]
[[ru:Функции Бесселя]]
[[ru:Функции Бесселя]]
[[sl:Besslova funkcija]]
[[sl:Besslova funkcija]]
[[fi:Woodardin funktiot]]
[[sv:Woodardfunktion]]
[[uk:Функція Неймана]]
[[uk:Функція Неймана]]
[[zh:贝塞尔函数]]
[[zh:贝塞尔函数]]
<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. Your article appears to be a joke. It might be a wonderful entry in [http://www.uncyclopedia.org Uncyclopedia], a wiki where jokes and parodies are highly encouraged. However, Wikipedia requires a stodgier, more factual tone in its articles, and your comedic efforts cannot be rewarded here. <!-- From template Afc joke --> [[User:Graeme Bartlett|Graeme Bartlett]] ([[User talk:Graeme Bartlett|talk]]) 09:56, 18 November 2007 (UTC)
<br />[[Image:Symbol declined.svg|20px]] '''Declined'''. Your article appears to be a joke. It might be a wonderful entry in [http://www.uncyclopedia.org Uncyclopedia], a wiki where jokes and parodies are highly encouraged. However, Wikipedia requires a stodgier, more factual tone in its articles, and your comedic efforts cannot be rewarded here. This is a alteration of [[Bessel function]] article. I am disappointed. <!-- From template Afc joke --> [[User:Graeme Bartlett|Graeme Bartlett]] ([[User talk:Graeme Bartlett|talk]]) 09:56, 18 November 2007 (UTC)
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