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Belle's Good Cide

Belle's Good Cide (March 17, 1993 – March 8, 2003) was an Oklahoma-bred Thoroughbred racing mare best known as the dam of Funny Cide.

Background
Belle's Good Cide was a chestnut mare bred by John Smicklas. She was sired by Slewacide out of Belle of Killarney (who also produced the stakes winning Belle of Cozzene). As a yearling she was sold for $2,800 at the 1994 Heritage Place Fall sale.

Racing career
Belle's Good Cide raced for three years, starting 26 times, winning twice, and earning $26,696.

Breeding record
As a broodmare Belle's Good Side's best offspring was the gelding Funny Cide, by Distorted Humor, who won the Kentucky Derby in 2003. Her other foals included Coincide (a grey colt by Cozzene) and Rockside (a bay filly by Personal Flag

In December 2001 she was consigned to the Fasig-Tipton Midlantic mixed sale and was bought for $3,500 by the Boniface family's Bonita Farm agentShe finished her life at McMahon of Saratoga Thoroughbreds in Saratoga Springs, New York where Funny Cide was born.

Belle's Good Cide was euthanized due to complications from colic in 2003 just after giving birth to her fourth foal, a chestnut colt by Mojave Moon and bred by Bonita Farm of Maryland. This happened shortly before Funny Cide's Kentucky Derby and Preakness Stakes victories. The foal was called Homicide and began racing in the year 2006. In 2008 he proved to be a successful point to point horse in Maryland.

title: Belle's Good Cide


text: Category:Guitarists

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Category:Guitarists


text:
Dillie Keane

Louise Miriam "Dillie" Keane (born 23 May 1952 ) is an actress, singer and comedian. She has been a member of the comedy cabaret trio Fascinating Aïda since its 1983 inception, and has also pursued a solo career. In 1995, with Fascinating Aïda, she was nominated for the Laurence Olivier Award for Best Entertainment.

Early life
Born in Portsmouth in 1952, Keane is the daughter of Frank Keane, a doctor from County Mayo, and Miriam Slattery, originally from Tralee, County Kerry, and was brought up in Portsmouth as a Roman Catholic. "My mother was a dragon," Keane said in 2008.

Keane was educated at the strict Roman Catholic Woldingham School (or Sacred Heart), where she sang in the school choir and played the guitar on the Folk Mass album recorded by some of the girls at Abbey Road Studios in 1967. She has described the school as disorganised. At the age of eighteen, she was expelled for going to see Fellini's Satyricon in London with boys from Worth School.

Keane then crammed for A-levels and studied music at Trinity College, Dublin, but left the four-year course after three years and went on to study acting for three years at the London Academy of Music and Dramatic Art.

Theatre and Fascinating Aïda
Keane was nominated for a Perrier Award at the Edinburgh Festival in 1990 for her one-woman theatre show Single Again (Sean Hughes would go on to win the award), returning the following year with Citizen Keane. In 1995, with Fascinating Aïda, she was nominated for the Laurence Olivier Award for Best Entertainment. Between 1999 and 2001 she toured a one-woman show, Back With You, taking in the UK and Germany, and winning the Best Comedy Award at the Moers Comedy Festival in 2001.

In 2002, Keane wrote the songs for Sandi Toksvig's musical comedy Big Night Out at the Little Palace Theatre (starring in the show itself at the Watford Palace Theatre with Toksvig and Bonnie Langford). Other works include the plays A Slice of Life (1981) and Boat People (1983). She has written songs for two pantomimes with Adèle Anderson.

Keane continued her acting career, including touring versions of Dancing at Lughnasa and Charley's Aunt, Juno and the Paycock at the Leicester Haymarket; The Plough and the Stars at the West Yorkshire Playhouse in Leeds; Accommodating Eva at the King's Head Theatre in Islington and Present Laughter at Birmingham Repertory Theatre. She was included in the premiere production of The Vagina Monologues in Dublin in 2002. She appeared with Kit and The Widow in Tomfoolery during 2005 and in the premiere one night only staging of a new musical version of Little Women at the Theatre Royal, Drury Lane.

In 2006, Keane appeared with Jenny Eclair and Linda Robson, in Grumpy Old Women Live, and as the Duchess in Me and My Girl (2006) on a new national tour. She performed the role of Dolly in Frank and Dolly a new play by Lizzie Hopley at the Edinburgh Festival 2007, for which she was nominated for the Stage Best Actress Award, before getting back on tour with Grumpy Old Women Live. She then toured round Ireland performing the role of Lady Bracknell in The Importance of Being Earnest, before returning to England, and beginning to write new songs for the 25th anniversary of Fascinating Aïda, which performed for a few weeks at the Jermyn Street Theatre.

On 29 June 2008, she appeared in (and produced with Barb Jungr) The Lovely Russell Concert, which celebrated the life of her friend and colleague Russell Churney. In 2009, she toured England with Fascinating Aïda, and completed a new album.

Television
As well as appearing in documentaries about Fascinating Aïda, Keane has appeared with Richard Griffiths and Samantha Janus in Pie in the Sky, and with Phil Cool on a number of his series.

Radio
Keane has written and presented on radio for shows such as Stop the Week, 4th Column, Booked! and Call Me When You're in Something (which won the Prix Monte Carlo).

Writing
Keane was a columnist for the Mail on Sunday (1993–95), and published two books connected with Fascinating Aïda: The Joy of Sequins—The Fascinating Aïda Songbook (1995) and Fascinating Who? (1985).

Personal life
Keane was in a relationship with John O'Neill for over 22 years until his death from a stroke on 13 August 2022. She lives in Oxfordshire. She travels for her work, and particularly enjoys working in Ireland, the homeland of her parents.

In 2003, she was interviewed about driving a Ford Transit: "I'm colossally uninterested in cars... but vans are different. They're incredible fun to drive. You are practically always the bigger dog. People always back up in small country lanes to let you pass. And even lorry drivers are much nicer; they'll flash you to let you in."

In the local elections of 2017, she stood as a candidate for the Liberal Democrats in the Ploughley division of Oxfordshire County Council. She came fourth, and was not elected.

Honours
In 2009, Keane was made an honorary Doctor of Letters by the University of Portsmouth, in recognition of her musical career and her links with the city.

title: Dillie Keane


text:
Awake (disambiguation)

To be awake is to experience wakefulness, the state of being conscious.

Awake may also refer to:

Film and television

 * Awake (2007 film), an American conspiracy thriller film by Joby Harold
 * Awake: The Life of Yogananda, a 2014 American documentary film
 * Awake (2019 film), an American crime action thriller film directed by Aleksandr Chernyaev
 * Awake (2021 film), an American apocalyptic science fiction thriller film directed by Mark Raso
 * Awake (TV series), a 2012 American police procedural fantasy drama series starring Jason Isaacs
 * Awake: The Million Dollar Game, a 2019 streaming game show
 * "Awake" (Once Upon a Time), a television episode
 * "Awake", a season 4 episode of Servant (TV series)

Albums

 * Awake (Alison Wonderland album) or the title song, 2018
 * Awake (Dream Theater album), 1994
 * Awake (Godsmack album), or the title song (see below), 2000
 * Awake (Hillsong Worship album), 2019
 * Awake (Illenium album), 2017
 * Awake (Josh Groban album), 2006
 * Awake (John Wesley Harding album), 1998
 * Awake (Julian Marley album) or the title song, 2009
 * Awake (L'Arc-en-Ciel album), 2005
 * Awake (Secondhand Serenade album) or the title song, 2007
 * Awake (Skillet album), 2009
 * Awake (Tycho album) or the title song, 2014
 * Awake (Wands album) or the title song, 1999
 * Awake? (Zao album) or the title song, 2009
 * Awake: The Best of Live, by Live, 2004
 * Awake, by Crematory album), 1997
 * Awake, by DeLon, 2011
 * Awake, by Klinik, 1996

EPs

 * Awake (KNK EP), 2016
 * Awake (Tamta EP), 2020
 * Awake, by Bleed the Dream, 2003
 * Awake, by Trash Talk, 2011

Songs

 * "Awake" (Crash Karma song), 2009
 * "Awake" (Donkeyboy song), 2009
 * "Awake" (Godsmack song), 2000
 * "Awake" (Mutiny Within song), 2009
 * "Awake" (Snoop Dogg song), 2015
 * "Awake", by Asia from Aura, 2001
 * "Awake", by Black Rebel Motorcycle Club from B.R.M.C., 2001
 * "Awake", by BTS from Wings, 2016
 * "Awake", by the Clay People from The Clay People, 1998
 * "Awake", by Finch from What It Is to Burn, 2002
 * "Awake", by Holy Knights from Between Daylight and Pain, 2012
 * "Awake", by Pillar from The Reckoning, 2006
 * "Awake", by Seventh Day Slumber from Finally Awake, 2007

Other uses

 * Awake!, a magazine published by Jehovah's Witnesses
 * AWAKE (Advanced WAKEfield Experiment), a plasma acceleration facility at CERN
 * "Awake", the first episode of the video game Life Is Strange: Before the Storm
 * Auake people, or Awake, an ethnic group of South America
 * Awaké language, a language of South America

title: Awake (disambiguation)


text:
Talk:Names of large numbers

Maximunssextillion
Say the number of maximunssextillion Luccasantana (talk) 22:57, 2 February 2024 (UTC)
 * That's not a real number. Black Yoshi (Yoshi! | Yoshi's Eggs) 22:58, 2 February 2024 (UTC)

What is the name of the 1089 digit emirp?
There is a reversed prime number (emirp) defined by Jens Kruse Andersen from Denmark (http://primerecords.dk) that has 1089 digits. I don't seem to find a name for it in Wikipedia. 115.64.52.116 (talk) 07:04, 16 March 2024 (UTC)

HM and UM
The table for googol and googolplex has UM as an authority, but it hasn't been mentioned before and has no link. I believe it's supposed to represent the HM reference, but it doesn't say that. 130.226.139.26 (talk) 07:24, 22 April 2024 (UTC)
 * Fixed. Black Yoshi (Yoshi! | Yoshi's Eggs) 12:20, 22 April 2024 (UTC)

Base -illion orders of magnitude
Does this make 1,000 the zeroth base -illion, and 1 the negative first base -illion? 2601:1C0:847C:50C0:A54D:85AC:91BC:7681 (talk) 15:29, 5 May 2024 (UTC)

 * In a sense yes, since a n-illion is 103n+3. However, this observation is original research and we are right to leave it out of the article. Certes (talk) 15:57, 5 May 2024 (UTC)

1,000,000,000,000,000,000
Quintillion has 18 zeros 77.100.228.242 (talk) 19:32, 7 May 2024 (UTC)

 * Yes, as stated in the article. It can also have 30 zeros in long scale.  Does something need to be amended? Certes (talk) 21:42, 7 May 2024 (UTC)

10^100 (Ten Duotrigintillion)
Ten Duotrigintillion Is Also call Googol 10^100 77.100.228.242 (talk) 19:34, 7 May 2024 (UTC)

 * Yes, as stated in the article. Certes (talk) 21:46, 7 May 2024 (UTC)

One Hundred Quinsexagintillion is also call gargoogol (10^200
Gargoogol has 200 Zeros 77.100.228.242 (talk) 19:35, 7 May 2024 (UTC)

 * This number is documented on several wikis such as Fandom, but as user-contributed content it may not be a sufficiently reliable source to support inclusion in the article. Certes (talk) 21:46, 7 May 2024 (UTC)

Ten Billion is also call diosge
Disoge has 10 zeros 77.100.228.242 (talk) 19:36, 7 May 2024 (UTC)

 * Does any reliable source use this term? Certes (talk) 21:38, 7 May 2024 (UTC)

Googolcrime has 1,000 zeros It's 10× Bigger than Googol
1,000 zeros Is so many 77.100.228.242 (talk) 19:37, 7 May 2024 (UTC)

 * One YouTuber seems to think that "googolcrime" is a number, but they don't appear to be a reliable source. (Also, that ratio is 10900 rather than 10.) Certes (talk) 21:40, 7 May 2024 (UTC)

title: Talk:Names of large numbers


text: Category:Bass guitarists

title: Category:Bass guitarists


text: Category:Musicians

title: Category:Musicians


text:
Category:Drummers

Musicians who play drums or drum set (including drums played with hands as well as those played with sticks). For musicians who play drums as well as other percussion instruments that are not drums, use the category Category:Percussionists.

title: Category:Drummers


text: Category:Afrika Bambaataa albums

title: Category:Afrika Bambaataa albums


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Talk:X-Factor (comics)

Definition
what does the general term x-factor mean?
 * An unknown factor affecting the outcome of a process. -Sean Curtin 05:59, 31 May 2004 (UTC)

X Factor
I fail to understand why 'X Factor' point here, not to the disambig page. I'm going to move this off accordingly--XmarkX 08:14, 10 Jan 2005 (UTC)

My bad, I didn't realise that this page was 'X-factor' with a hyphen--XmarkX 08:15, 10 Jan 2005 (UTC)

X-Factor (1986-1991)
I just wanted to take issue with the last paragraph of this section: I find the assertion that X-Factor existed "awkwardly" within the broader mythos to be a little too subjective. This might be the case when looking backward from where the franchise was during the early- to mid-nineties, but I would argue that this series should be viewed as the very vehicle that brought those solid original characters back into the fold. It also lent a consistency to the mythos, because it served as both a means to move the 'mutant story' forward while still containing a direct link to its rich past. This was, I feel, especially important during the late 80s when Claremont's 'Uncanny' stories (though I love them) left the parent team in a state of near-constant flux. I think that this period of X-Factor made the movement into what one might call the 'modern age' of X-Men(the early '90s, based on sales, anyway) a lot more fluid. RobM22 11:15, 2 February 2007 (UTC)RobM22

 * I agree that the "awkwardly" sentence is problematic. For one thing, I'm not sure what it's supposed to mean. Does it mean that it was less closely connected with the other X-titles. Because I think, arguably, sometime between Fall of the Mutants and Inferno, X-Factor became more tightly integrated with Uncanny X-Men than New Mutants was. And Excalibur was arguably the most loosely connected of all. Does it mean that events in X-Factor were significant to the overall X-Men stories? If so, I strongly disagree. Lots of events in X-Factor had repercussions in other titles. Does it mean that X-Factor was awkwardly written. I also disagree with that; I think that statement would need support. Some readers don't like the original X-Factor because it brought Jean Grey back. However, I don't think the article should give that particular the opinion the imprimatur of broad fan consensus. --JamesAM 21:57, 3 April 2007 (UTC)

Jean Grey and the Pheonix
My understanding is that it was John Byrne, not Kurt Busiek, who devised the idea of having Jean Grey be a separate entity from Phoenix. In fact, I recall him saying at the time that he'd been asked how Phoenix could be brought back, and he said, "Well, this is how I'd do it...", outlining the approach eventually used. And, of course, it was Byrne's story in Fantastic Four which offically revealed what "happened". I've never heard of Busiek being involved at all. -mhr 23:20, 5 Aug 2004 (UTC)

 * Busiek suggested it to Roger Stern, who suggested it to Byrne. From the rec.arts.comics.marvel.universe FAQ:
 * 2-4. Why is Kurt Busiek credited in FANTASTIC FOUR vol. 1 #286?
 * When Marvel launched X-FACTOR, a title (originally) starring the five original X-Men, it had a teensy problem: Founding X-Woman Jean Grey had died in X-MEN vol. 1 #137. So a prologue to X-FACTOR was done in AVENGERS vol. 1 #263 and FANTASTIC FOUR vol. 1 #286 that brought Jean back, allowing her to join the team.
 * Without getting into the details of Jean's "death" and return (which would be on-topic elsewhere), the idea that was used to put her back into play was conceived by Kurt Busiek. Kurt had been a fan when he'd come up with this, but by the time Marvel needed Jean back several pros had gotten wind of it and he was working at Marvel. So Busiek was credited and paid for his concept, but he didn't actually contribute anything else to FF #286. This was _not_ Kurt's first published comics work or how he broke into comics; he'd written for Marvel and other publishers prior to having his idea used here.
 * One reason for the confusion is that FF writer/artist John Byrne was not credited. This was because he was dissatisfied with unspecified changes made to the issue. He _did_ write the story, and _did_ support the retcon of his own work in X-MEN #137.
 * There have been many Usenet posts from Busiek confirming that this is how things went, if you feel like searching Google Groups for them. -Sean Curtin 06:22, 6 Aug 2004 (UTC)

Check out the Pheonix Rising trade paperback which features and introduction by Busiek himself where he explains the whole story. Evidently it was an idea he came up with in college and mentioned it to Roger Stern early in Busiek's career and, well, you all know what happened next.

Requested move

 * X-Factor (superhero) → X-Factor (comics) … Rationale: "X-Factor" isn't the name of an individual superhero; it's the name of a series and the team that it's about. … Please share your opinion at Talk:X-Factor (superhero). --Phoenix-forgotten 18:52, 5 June 2006 (UTC)

Survey

 * Add *Support or *Oppose followed by an optional one-sentence explanation, then sign your opinion with  ~

face="Arial" color="#FFFFFF">Ardenn ]] 20:10, 5 June 2006 (UTC)
 * Support See above. Phoenix-forgotten 18:51, 5 June 2006 (UTC)
 * Support per nom. [[User talk:Ardenn|<font style="background: green"
 * Support: not all incarnations would classify as superhero (ie volume 2 and 3) Dizzy D 22:11, 5 June 2006 (UTC)
 * Support (comics) is the preferred naming. --waffle iron talk 22:21, 5 June 2006 (UTC)

Check out the ''Pheonix Rising' trade paperback which features and introduction by Busiek himself where he explains the whole story. Evidently it was an idea he came up with in college and mentioned it to Roger Stern early in Busiek's career and, well, you all know what happened next.

Discussion

 * Add any additional comments

Phoenix-forgotten, I actually put in for a requested move in the June 1 section of the Requested moves page. I suggested the same move. You might want to strike or remove yours from June 5. --waffle iron talk 22:21, 5 June 2006 (UTC)
 * Done. — Nightst a  llion  (?) 10:31, 6 June 2006 (UTC)

Disambiguation link fixing
I've gone through and fixed quite a few of the disambig links from the move. There are no more articles that link to X-Factor (superhero), but there are a lot of links to X-Factor (which redirects to X Factor) that are comics related.

You can find the full list at Special:Whatlinkshere/X-Factor. Any help would be appreciated. --waffle iron talk 23:28, 7 June 2006 (UTC)

 * All of the disambiguation pages for X-Factor should redirect to X Factor. Any pages specifically about the comic book that link there, should be fixed to link to X-Factor (comics). Fortdj33 (talk) 19:50, 2 June 2009 (UTC)

Merge
Should X-Factor Investigations be merged with this article?

Survey

 * Support: This article says itself that "X-Factor" is the name of several different teams and contains roughly the same information as is in X-Factor Investigations. Moreover, the book is still entitled X-Factor. --Newt ΨΦ 00:15, 8 July 2006 (UTC)
 * Oppose X-Factor has had so many incarnations that lumping them all together makes the article too large. The members of X-Factor Investigations aren't the most continuous with previous incarnations. The way I see it, the articles are currently written so that the X-Factor article is about the team that used that name, not the publication history of the comic books named X-Factor. --waffle iron talk 00:26, 8 July 2006 (UTC)
 * Support --InShaneee 00:36, 8 July 2006 (UTC)
 * Support --Chris Griswold 06:45, 8 July 2006 (UTC)
 * Oppose. This is essentially conflating several different series that have little or no overlap except for a common word in their titles. Cribcage 23:06, 12 July 2006 (UTC)
 * Comment I agree. That's the idea I was grasping for but couldn't put into the right words. --waffle iron talk 23:46, 12 July 2006 (UTC)
 * Comment A number of characters in the new series were also featured characters in the first series. --Chris Griswold 08:41, 13 July 2006 (UTC)
 * Comment The three teams that have used a variation of the name have been quite dissimilar: a new younger X-Men team, a set of more international heros under the control of X-Corp and a dectective agency. To argue absolute continuity just because of a name and one or two members seems silly. --waffle iron talk 15:51, 13 July 2006 (UTC)
 * Comment Perhaps the focus of the article should be changed to the publication history of the X-Factor books rather than about the fictional teams that are featured in them. This would be more in line with the Manual of Style (writing about fiction), and would limit the in-universe treatment of the fictional history contained in the book. It does not seem it would be too difficult to change. --Newt ΨΦ 16:09, 13 July 2006 (UTC)
 * Comment I went ahead and modified the lead to show it's possible to change the focus. I admit it will take further editing of the rest of the article, however, the headings could be seen to already treat the material under the new focus. The focus seemed scattered before anyway, switching between talking about the book and the team without much segue or coherence.  --Newt ΨΦ 16:22, 13 July 2006 (UTC)

 * Oppose - I agree with Cribcage -- Ipstenu ( talk | contribs ) 21:07, 13 July 2006 (UTC)
 * Support --Aznsilver67 15:51, 14 July 2006 (UTC)
 * Merge --Jamdav86 12:53, 16 July 2006 (UTC)
 * Oppose Keeps size of the main article down. --Basique 16:40, 22 July 2006 (UTC)
 * Comment Having X-Factor Investigations cover the latest incarnation on a separate page hurts nothing, and there's already a "see also" on this page in case anyone is interested in the current version. That X-Factor Investigations page does need a little bit of love. It needs an "ongoing storyline" tag. A lot has happened in the last nine issues, and someone needs to chronicle that, I don't even see a Civil War update, External links, or a history section. Give it some love and hopefully the merger threat will go away. --Basique 16:40, 22 July 2006 (UTC)

 * Oppose - The two articles only similarity is the name. Keep it seperate, but add an if you were looking for disambig tage at the top.-Giant89 17:54, 25 July 2006 (UTC)
 * Support Bradtcordeiro 23:22, 25 July 2006 (UTC)
 * Oppose Substatic 09:56, 26 July 2006 (UTC)

Decision closed with no decision to merge. --Chris Griswold 08:41, 31 July 2006 (UTC)

Discussion

 * I did my best to edit X-Factor Investigations to remove extraneous and/or detailed plot information, add a publication history, and cite it. There's not a lot there.  It could definitely be merged, and as I found out trying to write the pub history, would probably make more sense if thought of as a continuation of Marvel's usage of the X-Factor title.  --Newt ΨΦ 19:09, 27 July 2006 (UTC)

Award nomination
--- Another Believer ( Talk ) 00:46, 29 January 2021 (UTC)
 * 32nd GLAAD Media Awards

title: Talk:X-Factor (comics)


text: Category:Against Me! albums

title: Category:Against Me! albums


text:
Goose Goslin

Leon Allen "Goose" Goslin (October 16, 1900 – May 15, 1971) was an American professional baseball left fielder. He played in Major League Baseball for the Washington Senators, St. Louis Browns, and Detroit Tigers, from 1921 until 1938.

Goslin led the American League (AL) in triples two times and finished the season with a batting average of over .300 eleven times. He won the AL batting title in 1928 with a .379 batting average which set a Washington Senators record. He led the AL in assists five times, putouts four times and his 4,141 putouts and 181 assists as a left fielder are both 5th all time. His 173 triples are 22nd all time and his .316 batting average is 7th all time among left fielders with over 2,000 games played. A two time World Series winner, he was elected to the National Baseball Hall of Fame in 1968 via the Veterans Committee.

Early years
Born in Salem, New Jersey, Goslin was 16 when he left home to play on a touring semipro circuit of the Eastern seaboard, and by 19 had moved into the minor leagues in South Carolina as a pitcher. Goslin was discovered by famed scout Joe Engel. After hearing from Engel, Senators owner Clark Griffith personally scouted Goslin and attended a Sally League game in which Goslin was playing for Columbia, South Carolina. A fly ball hit Goslin on the head, and another barely missed him. Goslin hit three home runs in the game, and Griffith decided to take a chance on him.

Goslin's difficulty in judging fly balls contributed to his nickname "Goose." Opposing players said Goslin resembled a bird flapping its wings when he ran after a ball with his arms waving. While not a great fielder, Goslin did have a good throwing arm, leading the American League in assists by an outfielder in 1924 and 1925. However, one year during spring training, Goslin wandered to an adjacent field where a track and field team was working out. Goslin tried the shot put, and his throwing arm was never the same afterward.

The 20-year-old Goslin was called up to the major leagues to play for the Washington Senators for the last two weeks of the 1921 season. He had a promising .351 on-base percentage in 14 games in 1921 and became a starter for the Senators in 1922. Goslin played 93 games in 1922 and became a fixture for the Senators in left field until 1930. Goslin hit .324 in his first full season in 1922, followed by a .300 season in 1923 with 99 RBIs. Showing speed on the base paths, Goslin led the American League with 18 triples in 1923.

1924 and 1925 World Series
In 1924, Goslin established himself as one of the league's top run producers, as he led the American League with 129 RBIs and finished seventh in batting average (.344). At age 23, Goslin also hit for the cycle and was among the league leaders with 17 triples (second best), 299 total bases (fourth best) and 199 hits (fifth best). After the Senators had losing records in 1922 and 1923, Goslin helped to spark the team to a 92-win season and their first World Series championship in 1924. With a 36-year-old Walter Johnson contributing 23 wins and the young Goslin knocking in 129 runs (50 more RBI than any other player on the team), the Senators finished two games ahead of the Yankees and defeated the New York Giants in the 1924 World Series. Goslin hit .344 with three home runs, seven RBI and a .656 slugging percentage in that World Series. Goslin also set a World Series record in 1924 with six consecutive hits, spread across three games (3-5). That record was tied in 1976 by Thurman Munson and later broken in 1990 by Billy Hatcher, who had seven consecutive hits in that World Series.

Goslin contributed another strong performance to the 1925 Senators, batting .334, with 72 extra base hits and 113 RBI. His 20 triples led the American League. Once again, he batted in far more runs than any other Senators hitter – 26 more than Sam Rice. The Senators easily won their second consecutive pennant, finishing $8 1/2$ games ahead of the Philadelphia Athletics. Despite Goslin's three home runs, six RBI and a .692 slugging percentage in the 1925 World Series, the Senators were defeated in seven games by a Pittsburgh Pirates team led by Pie Traynor, Kiki Cuyler and Max Carey after having taken a 3–1 lead earlier in the series.

The American League MVP award for 1924 and 1925 went to Goslin's teammates Walter Johnson and Roger Peckinpaugh.

1928 batting title
Goslin continued as one of the American League's best batters with averages of .354 and .334 in 1926 and 1927, but his best season came in 1928. That year, he won the American League batting crown with a career-high .379 batting average. He also finished among the league leaders with a .442 on-base percentage (third best in the league), a .614 slugging percentage (third best), 17 home runs (third best) and 63 extra base hits (fourth best).

The 1928 batting title was not decided until the last day of the season. Goslin and Heinie Manush of the St. Louis Browns were tied going into the final game, and the Senators and Browns played each other in the final game. Goslin was leading Manush when his turn came to bat in the ninth inning. If Goslin made an out, he would lose the batting crown. In Lawrence Ritter's 1966 oral history, "The Glory of Their Times", Goslin described the events that followed. Manager Bucky Harris left the decision to Goslin on whether to bat or sit. Goslin decided to sit and take the batting crown, but his teammates (particularly Joe Judge) goaded him that he would appear yellow if he didn't bat. Goslin was persuaded to bat and promptly took two strikes. At that point, Goslin recalled that he unsuccessfully tried to get ejected from the game, as the at bat would then disappear. Goslin began berating the home plate umpire about the strike calls, only to have the umpire tell him that he was not going to get ejected, and wasn't going to get a walk, so he better step back up and swing. Goslin ended up with what he called a "lucky hit" to beat Manush by a fraction of a point.

Goslin's years with the Browns and return to the Senators
In 1929, Goslin's batting average dropped to .288. Two months into the 1930 season, with Goslin struggling with a .271 batting average, the Senators traded him to the St. Louis Browns for Heinie Manush and Alvin Crowder. Goslin batted .326 with a career-high .652 slugging percentage for the Browns in 1930. In 101 games for the Browns, Goslin had 30 home runs, 100 RBI, and 62 extra base hits. Goslin had another solid year for the Browns in 1931, batting .328 with a career-high 42 doubles, 76 extra base hits and 105 RBI.

While Goslin's average slipped to .299 in 1932, he still drove in 104 runs for the Browns. On Opening Day, April 12, Goslin came up to the plate against the Chicago White Sox with a bat that featured 12 longitudinal green stripes. The bat was thrown out of the game, and on the following day, American League President William Harridge declared the "zebra bat" illegal, as it caused a distraction to the fielding team. At the end of the 1932 season, Goslin was traded back to the Senators. Goslin led the team back to the 1933 World Series. The Senators played in only three World Series in their history, and Goslin played for the Senators in every game of those Series. However, Goslin did not have his usual power in 1933. His 64 RBI and .452 slugging percentage were Goslin's lowest numbers since his rookie season.

The "G-Men" lead Detroit to the 1934 and 1935 World Series
After the Senators lost the 1933 World Series, the Senators traded Goslin to the Detroit Tigers for John Stone. Goslin later recounted that owner Clark Griffith told him that he simply couldn't afford to pay him. Even though the Senators had made it to the World Series, the team was not making money. With stars Charlie Gehringer and Hank Greenberg already in place, the Tigers added future Hall of Famers Goslin and Mickey Cochrane in the off-season. The 1934 Tigers became known as the "G-Men", with the team's top stars being Gehringer, Greenberg and Goslin. The 1934 Tigers sailed to the pennant with a 101–53 record and were matched up in a classic World Series between the "G-Men" and St. Louis's "Gashouse Gang." The Cardinals won a hard-fought seven-game series, which was filled with controversial calls and an infamous play in Game 7 which resulted in the Detroit crowd pelting Joe Medwick with fruit in left field.

Goslin and the Tigers returned to the World Series in 1935. The Tigers won the 1935 World Series on Goslin's game-winning single in the bottom of the ninth inning of Game 6. With the game tied 3–3 in the top of the ninth inning, the Cubs' Stan Hack hit a lead-off triple, but Detroit ace Tommy Bridges struck out the next batter, followed by a ground-out and a fly-out caught by Goslin in left field. In the bottom of the ninth, Goslin came to bat with two outs and Mickey Cochrane on second base. Goslin singled to right, driving in Cochrane for the winning run. Detroit had its first championship, and the "G-Men" were the toast of Detroit.

Goslin is one of only three players to be the last hitter of two World Series, having struck out to end the 1925 World Series and won the 1935 World Series with his walk-off RBI single. Édgar Rentería won the 1997 Series with a walk-off RBI single and ended the 2004 World Series with a ground out to the pitcher. Boss Schmidt of the Detroit Tigers ended the 1907 World Series by popping up and the 1908 World Series by grounding out. Goslin and Mickey Cochrane were the only two non-Yankees in the American League to play in five World Series during the original Yankees dynasty from 1921 to 1964.

Final years in baseball
Goslin played two more seasons with the Tigers in 1936 and 1937, batting .315 in 1936 but dropping to .238 in 1937. On July 28, 1936, Goslin hit one of the most unusual home runs in baseball history. Goslin drove the ball into the gap. The Yankees' right fielder (Joe DiMaggio) and center fielder (Myril Hoag) collided and were knocked unconscious while sprinting for the ball. Goslin rounded the bases with an inside-the-park home run. The Tigers ended up releasing Goslin after his .238 season, and Goslin later recounted (in "The Glory of Their Times") that he received a call from his old boss, Clark Griffith, asking him if he'd be interested in ending his career back where it began in Washington. Goslin jumped at the opportunity and batted .158 in 38 games for the Senators in the 1938 season. In 1939, Goslin became a player-manager for the Trenton Senators of the Interstate League before retiring as a player.

Goslin finished his career after 18 major league seasons with a .316 batting average and a .500 slugging percentage. His 4,325 total bases, 2,735 hits, 921 extra base hits, 1,612 RBI, 500 doubles and 173 triples all rank among the top 50 in each category in major league history. Goslin had 11 seasons with at least 100 RBI, and his league leadership in RBI in 1924 deprived Babe Ruth of the triple crown. He hit .300 or better in 11 seasons in his career. He also hit 3 home runs in a game three times and had 5 hits in a game five times, along with 41 four-hit games in his career.

Goslin holds the record for career home runs at Yankee Stadium (1923–2008) by a visiting player, with 32.

Life after baseball
Goslin was elected to the Baseball Hall of Fame in 1968, along with Kiki Cuyler, by the Veterans Committee. When he was inducted into the Hall, Goslin broke down and cried. "I have been lucky", he said, "I want to thank God, who gave me the health and strength to compete with these great players. I will never forget this. I will take this to my grave." In 1999, he ranked number 89 on The Sporting News list of Baseball's Greatest Players, and was nominated as a finalist for the Major League Baseball All-Century Team. Goslin ranked #6 on the Sports Illustrated list of The 50 Greatest New Jersey Sports Figures.

After retiring from baseball, Goslin operated a boat rental company on Delaware Bay for many years, until he retired in 1969. He died in Bridgeton, New Jersey, aged 70, and is buried in the Baptist Cemetery, Salem, New Jersey.

title: Goose Goslin


text: Category:Vocalists

title: Category:Vocalists


text:
Category:Matrix theory

Matrix theory is a branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra, but soon grew to include subjects related to graph theory, algebra, combinatorics and statistics.

title: Category:Matrix theory


text: Category:Human rights

title: Category:Human rights


text:
W12 engine

A W12 engine is a twelve-cylinder piston engine where either three banks of four cylinders, or four banks of three cylinders are arranged in a W configuration around a common crankshaft.

W12 engines with three banks of four cylinders were used by several aircraft engines from 1917 until the 1930s. A three-bank design was also used for an unsuccessful W12 engine which was intended to compete in Formula One in 1990.

W12 engines are less common than V12 engines as only a handful of automobile manufacturers use them. The WR12 engine has been produced by the Volkswagen Group since 2001. This four-bank engine – based on two VR6 engines with a common crankshaft – has only been used in flagship high performance car models produced by the Volkswagen Group and by Spyker.

Aircraft engines
The Napier Lion was a three-bank design (also called "broad arrow" design) W12 engine produced in the United Kingdom from 1917 to the late 1930s. It had a capacity of 24 L and produced 450 - 900 hp. As well as use in various military and racing airplanes, the Lion was also used in land speed record cars (such as the Napier-Railton and the Napier-Campbell Blue Bird) and racing boats (such as the Miss Britain III).

During the 1920s, the Farman 12We aircraft engine was produced in France. The 12We produced 500 hp and was one of the company's best-selling engines in the 1920s.

The Lorraine 12E Courlis is another W12 aircraft engine that was produced in France during the 1920s and 1930s.

Motor racing engines

For the 1990 Formula One season, the Italian team Life Racing Engines built a three-bank W12 engine with a displacement of 3.5 L. The Life Racing Engine F35 used a central master connecting rod, with a slave rod locating onto each side of the master rod, rather than directly onto the crank pin. This meant that there was no offset between the cylinders, reducing the length of the crankpins. The engine was used in rounds 1 to 12 of the 1990 season, however it was unreliable and lacking in power, and the car failed to pre-qualify for any races. The W12 engine was replaced by a third-party V8 engine after round 12.

Road car engines

The only mass-production W12 engine is the Volkswagen 6.0 WR12 48v, a four-bank design that was first released in 2001. The engine is constructed by mating two narrow-angle 15° VR6 engines at an inclined angle of 72°. The narrow angle of each set of cylinders allows just two overhead camshafts to drive each pair of banks, so the WR12 engine has both a crankshaft design, and the same number of camshafts as the V-12 engine. Where the VR6 is a staggered-bank straight 6, the WR12 in turn is a staggered-bank V-12 in terms of engineering design. As a consequence, it cannot be considered a true three-block W-engine, in the manner of the Life F-1 engine, or the Napier-Railton.

Within Volkswagen, the engine is designated a WR engine, terminology in derivation of its use of the VR6 geometry for each bank. Early Volkswagen and Audi cars using this engine were badged with a W with inset V, to emphasise the engine design. Later badging (after 2001 approximately) of cars with this engine did not make use of that particular badging, leading to confusion of the staggered-bank V-12 with a true W engine in the public mind.

The naturally aspirated variant of the WR12 engine has been used in the brands of Audi and Volkswagen. In 2003, a twin-turbocharged version of the WR12 engine was produced to be used exclusively in the Bentley Continental GT and Bentley Flying Spur.

The WR12 engine has a very compact design for a 12-cylinder engine, with the overall size of the 6.0 L engine being smaller than Volkswagen's contemporary 4.2 L V8 engine.

The first application of the Volkswagen WR12 was the 2001 Volkswagen W12, a mid-engined concept car which set the 24‑hour world endurance record in 2001 with a distance of 7085.7 km and an average speed of 295 km/h.

The first production car to use the WR12 engine was the 2001 Audi A8 (D2). The succeeding Audi A8 (D3) and Audi A8 (D4) also had the WR12 as an upgrade engine only for the long-wheelbase variant, the Audi A8L W12, which was last available in the 2018 model year. The Audi A8L W12 with its all-wheel drive was unique among its German contemporaries, as 12-cylinder rivals from BMW and Mercedes-Benz used V12 engines with rear-wheel drive.

Other cars to use the WR12 engine are the 2003–present Bentley Continental GT, 2005–present Bentley Flying Spur, 2015–present Bentley Bentayga, 2004–2011 Volkswagen Phaeton W12 and the 2005–2010 Volkswagen Touareg W12. The engine was also used in the 2006 Spyker C12 La Turbie and 2008 Spyker C12 Zagato low-volume sports cars.

title: W12 engine


text: Category:Departments of Colombia

title: Category:Departments of Colombia


text:
Transformation matrix

In linear algebra, linear transformations can be represented by matrices. If $$T$$ is a linear transformation mapping $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$ and $$\mathbf x$$ is a column vector with $$n$$ entries, then $$T( \mathbf x ) = A \mathbf x$$ for some $$m \times n$$ matrix $$A$$, called the transformation matrix of $$T$$. Note that $$A$$ has $$m$$ rows and $$n$$ columns, whereas the transformation $$T$$ is from $$\mathbb{R}^n$$ to $$\mathbb{R}^m$$. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.

Uses
Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation. This also allows transformations to be composed easily (by multiplying their matrices).

Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space Rn can be represented as linear transformations on the n+1-dimensional space Rn+1. These include both affine transformations (such as translation) and projective transformations. For this reason, 4×4 transformation matrices are widely used in 3D computer graphics. These n+1-dimensional transformation matrices are called, depending on their application, affine transformation matrices, projective transformation matrices, or more generally non-linear transformation matrices. With respect to an n-dimensional matrix, an n+1-dimensional matrix can be described as an augmented matrix.

In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system (change of basis). The distinction between active and passive transformations is important. By default, by transformation, mathematicians usually mean active transformations, while physicists could mean either.

Put differently, a passive transformation refers to description of the same object as viewed from two different coordinate frames.

Finding the matrix of a transformation
If one has a linear transformation $$T(x)$$ in functional form, it is easy to determine the transformation matrix A by transforming each of the vectors of the standard basis by T, then inserting the result into the columns of a matrix. In other words, $$A = \begin{bmatrix} T( \mathbf e_1 ) & T( \mathbf e_2 ) & \cdots & T( \mathbf e_n ) \end{bmatrix}$$

For example, the function $$T(x) = 5x$$ is a linear transformation. Applying the above process (suppose that n = 2 in this case) reveals that $$T( \mathbf{x} ) = 5 \mathbf{x} = 5I\mathbf{x} = \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \mathbf{x}$$

The matrix representation of vectors and operators depends on the chosen basis; a similar matrix will result from an alternate basis. Nevertheless, the method to find the components remains the same.

To elaborate, vector $$\mathbf v$$ can be represented in basis vectors, $$E = \begin{bmatrix}\mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n\end{bmatrix}$$ with coordinates $$ [\mathbf v]_E = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}^\mathrm{T}$$: $$\mathbf v = v_1 \mathbf e_1 + v_2 \mathbf e_2 + \cdots + v_n \mathbf e_n = \sum_i v_i \mathbf e_i = E [\mathbf v]_E$$

Now, express the result of the transformation matrix A upon $$\mathbf v$$, in the given basis: $$\begin{align} A(\mathbf v)   &= A \left(\sum_i v_i \mathbf e_i \right) = \sum_i {v_i A(\mathbf e_i)} \\ &= \begin{bmatrix}A(\mathbf e_1) & A(\mathbf e_2) & \cdots & A(\mathbf e_n)\end{bmatrix} [\mathbf v]_E = A \cdot [\mathbf v]_E \\[3pt] &= \begin{bmatrix}\mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n \end{bmatrix} \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \\ \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n\end{bmatrix} \end{align}$$

The $$a_{i,j}$$ elements of matrix A are determined for a given basis E by applying A to every $$\mathbf e_j = \begin{bmatrix} 0 & 0 & \cdots & (v_j=1) & \cdots & 0 \end{bmatrix}^\mathrm{T}$$, and observing the response vector $$A \mathbf e_j = a_{1,j} \mathbf e_1 + a_{2,j} \mathbf e_2 + \cdots + a_{n,j} \mathbf e_n = \sum_i a_{i,j} \mathbf e_i.$$

This equation defines the wanted elements, $$a_{i,j}$$, of j-th column of the matrix A.

Eigenbasis and diagonal matrix
Yet, there is a special basis for an operator in which the components form a diagonal matrix and, thus, multiplication complexity reduces to $n$. Being diagonal means that all coefficients $$a_{i,j} $$ except $$a_{i,i}$$ are zeros leaving only one term in the sum $\sum a_{i,j} \mathbf e_i$ above. The surviving diagonal elements, $$a_{i,i}$$, are known as eigenvalues and designated with $$\lambda_i$$ in the defining equation, which reduces to $$A \mathbf e_i = \lambda_i \mathbf e_i$$. The resulting equation is known as eigenvalue equation. The eigenvectors and eigenvalues are derived from it via the characteristic polynomial.

With diagonalization, it is often possible to translate to and from eigenbases.

Examples in 2 dimensions
Most common geometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix.

Stretching
A stretch in the xy-plane is a linear transformation which enlarges all distances in a particular direction by a constant factor but does not affect distances in the perpendicular direction. We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the form $x' = kx$; $y' = y$ for some positive constant $k$. (Note that if $k > 1$, then this really is a "stretch"; if $k < 1$, it is technically a "compression", but we still call it a stretch. Also, if $k = 1$, then the transformation is an identity, i.e. it has no effect.)

The matrix associated with a stretch by a factor $k$ along the x-axis is given by: $$\begin{bmatrix} k & 0 \\ 0 & 1 \end{bmatrix} $$

Similarly, a stretch by a factor k along the y-axis has the form $x' = x$; $y' = ky$, so the matrix associated with this transformation is $$\begin{bmatrix} 1 & 0 \\ 0 & k \end{bmatrix} $$

Squeezing
If the two stretches above are combined with reciprocal values, then the transformation matrix represents a squeeze mapping: $$\begin{bmatrix} k & 0 \\ 0 & 1/k \end{bmatrix} .$$ A square with sides parallel to the axes is transformed to a rectangle that has the same area as the square. The reciprocal stretch and compression leave the area invariant.

Rotation
For rotation by an angle θ counterclockwise (positive direction) about the origin the functional form is $$x' = x \cos \theta - y \sin \theta$$ and $$y' = x \sin \theta + y \cos \theta$$. Written in matrix form, this becomes: $$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin\theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

Similarly, for a rotation clockwise (negative direction) about the origin, the functional form is $$x' = x \cos \theta + y \sin \theta$$ and $$y' = -x \sin \theta + y \cos \theta$$ the matrix form is: $$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta &  \sin\theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

These formulae assume that the x axis points right and the y axis points up.

Shearing
For shear mapping (visually similar to slanting), there are two possibilities.

A shear parallel to the x axis has $$x' = x + ky$$ and $$y' = y$$. Written in matrix form, this becomes: $$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

A shear parallel to the y axis has $$x' = x$$ and $$y' = y + kx$$, which has matrix form: $$ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ k & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Reflection
For reflection about a line that goes through the origin, let $$\mathbf{l} = (l_x, l_y)$$ be a vector in the direction of the line. Then use the transformation matrix: $$\mathbf{A} = \frac{1}{\lVert\mathbf{l}\rVert^2} \begin{bmatrix} l_x^2 - l_y^2 & 2 l_x l_y \\ 2 l_x l_y & l_y^2 - l_x^2 \end{bmatrix}$$

Orthogonal projection
To project a vector orthogonally onto a line that goes through the origin, let $$\mathbf{u} = (u_x, u_y)$$ be a vector in the direction of the line. Then use the transformation matrix: $$\mathbf{A} = \frac{1}{\lVert\mathbf{u}\rVert^2} \begin{bmatrix} u_x^2 & u_x u_y \\ u_x u_y & u_y^2 \end{bmatrix}$$

As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation.

Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix, homogeneous coordinates can be used.

Rotation
The matrix to rotate an angle θ about any axis defined by unit vector (x,y,z) is $$\begin{bmatrix} xx(1-\cos \theta)+\cos\theta & yx(1-\cos\theta)-z\sin\theta & zx(1-\cos\theta)+y\sin\theta\\ xy(1-\cos\theta)+z\sin\theta & yy(1-\cos\theta)+\cos\theta & zy(1-\cos\theta)-x\sin\theta \\ xz(1-\cos\theta)-y\sin\theta & yz(1-\cos\theta)+x\sin\theta & zz(1-\cos\theta)+\cos\theta \end{bmatrix}.$$

Reflection
To reflect a point through a plane $$ax + by + cz = 0$$ (which goes through the origin), one can use $$\mathbf{A} = \mathbf{I} - 2\mathbf{NN}^\mathrm{T} $$, where $$\mathbf{I}$$ is the 3×3 identity matrix and $$\mathbf{N}$$ is the three-dimensional unit vector for the vector normal of the plane. If the L2 norm of $$a$$, $$b$$, and $$c$$ is unity, the transformation matrix can be expressed as: $$\mathbf{A} = \begin{bmatrix} 1 - 2 a^2 & - 2 a b & - 2 a c \\ - 2 a b  & 1 - 2 b^2 & - 2 b c  \\ - 2 a c & - 2 b c & 1 - 2c^2 \end{bmatrix}$$

Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): $$\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 - 2 a^2 & - 2 a b & - 2 a c & - 2 a d \\ - 2 a b  & 1 - 2 b^2 & - 2 b c & - 2 b d \\ - 2 a c & - 2 b c & 1 - 2c^2 & - 2 c d \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} $$ where $$d = -\mathbf{p} \cdot \mathbf{N}$$ for some point $$\mathbf{p}$$ on the plane, or equivalently, $$ax + by + cz + d = 0$$.

If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix. See homogeneous coordinates and affine transformations below for further explanation.

Composing and inverting transformations
One of the main motivations for using matrices to represent linear transformations is that transformations can then be easily composed and inverted.

Composition is accomplished by matrix multiplication. Row and column vectors are operated upon by matrices, rows on the left and columns on the right. Since text reads from left to right, column vectors are preferred when transformation matrices are composed:

If A and B are the matrices of two linear transformations, then the effect of first applying A and then B to a column vector $$\mathbf{x}$$ is given by: $$\mathbf{B}(\mathbf{A} \mathbf x) = (\mathbf{BA}) \mathbf x.$$

In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices.

When A is an invertible matrix there is a matrix A−1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating in opposite direction) and then composing them in reverse order. Reflection matrices are a special case because they are their own inverses and don't need to be separately calculated.

Affine transformations

To represent affine transformations with matrices, we can use homogeneous coordinates. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. Using this system, translation can be expressed with matrix multiplication. The functional form $$x' = x + t_x; y' = y + t_y$$ becomes: $$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}.$$

All ordinary linear transformations are included in the set of affine transformations, and can be described as a simplified form of affine transformations. Therefore, any linear transformation can also be represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example, the counter-clockwise rotation matrix from above becomes: $$\begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Using transformation matrices containing homogeneous coordinates, translations become linear, and thus can be seamlessly intermixed with all other types of transformations. The reason is that the real plane is mapped to the $w = 1$ plane in real projective space, and so translation in real Euclidean space can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear).

More affine transformations can be obtained by composition of two or more affine transformations. For example, given a translation T' with vector $$(t'_x, t'_y),$$ a rotation R by an angle θ counter-clockwise, a scaling S with factors $$(s_x, s_y)$$ and a translation T of vector $$(t_x, t_y),$$ the result M of T'RST is: $$\begin{bmatrix} s_x \cos \theta & - s_y \sin \theta & t_x s_x \cos \theta - t_y s_y \sin \theta + t'_x \\ s_x \sin \theta & s_y \cos \theta & t_x s_x \sin \theta + t_y s_y \cos \theta + t'_y \\ 0     & 0 & 1 \end{bmatrix}$$

When using affine transformations, the homogeneous component of a coordinate vector (normally called w) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections.

Perspective projection

Another type of transformation, of importance in 3D computer graphics, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer (see also reciprocal function).

The simplest perspective projection uses the origin as the center of projection, and the plane at $$z = 1$$ as the image plane. The functional form of this transformation is then $$x' = x / z$$; $$y' = y / z$$. We can express this in homogeneous coordinates as: $$\begin{bmatrix} x_c \\ y_c \\ z_c \\ w_c \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}=\begin{bmatrix} x \\ y \\ z \\ z \end{bmatrix} $$

After carrying out the matrix multiplication, the homogeneous component $$w_c$$ will be equal to the value of $$z$$ and the other three will not change. Therefore, to map back into the real plane we must perform the homogeneous divide or perspective divide by dividing each component by $$w_c$$: $$\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \frac{1}{w_c} \begin{bmatrix} x_c \\ y_c \\ z_c \\ w_c \end{bmatrix}=\begin{bmatrix} x / z \\ y / z \\ 1 \\ 1 \end{bmatrix}$$

More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.

title: Transformation matrix


text: Category:A-ha albums

title: Category:A-ha albums


text:
Charlie Ruggles

Charles Sherman Ruggles (February 8, 1886 – December 23, 1970) was an American comic character actor. In a career spanning six decades, Ruggles appeared in close to 100 feature films, often in mild-mannered and comic roles. He was also the elder brother of director, producer, and silent film actor Wesley Ruggles (1889–1972).

Career
Ruggles was born in Los Angeles, California, in 1886. Despite training to be a doctor, Ruggles soon found himself on the stage, appearing in a stock production of Nathan Hale in 1905. In 1912, he worked in the stock company at Alcazar Theatre in San Francisco in a stage production of The Dawn of a Tomorrow. At Los Angeles's Majestic Theatre, he played Private Jo Files in L. Frank Baum and Louis F. Gottschalk's musical The Tik-Tok Man of Oz in 1913.

He moved to Broadway to appear in Help Wanted in 1914. His first screen role came in the silent Peer Gynt the following year. Throughout the 1910s and 1920s, Ruggles continued to appear in silent movies, though his passion remained the stage, appearing in long-running productions such as The Passing Show of 1918, The Demi-Virgin, Ladies' Night in a Turkish Bath, and Battling Buttler. One of his most famous stage hits was Queen High, one of his last before a nearly 30-year hiatus, produced in 1926. He also played Peter Braley in Spring Is Here, which ran for 104 performances in 1929.

From 1929, Ruggles appeared in talking pictures. His first was Gentleman of the Press in which he played a comic, alcoholic newspaper reporter. Throughout the 1930s, he was teamed with comic actress Mary Boland in a string of domestic farces, notably If I Had a Million, Six of a Kind, Ruggles of Red Gap, and People Will Talk. Ruggles is best remembered today as the big-game hunter in Bringing Up Baby and billionaire Michael J. "Mike" O'Connor in It Happened on Fifth Avenue.

In 1944, he had a summer radio series, The Charlie Ruggles Show on CBS. In 1949, Ruggles halted his film career to return to the stage and to move into television. He was the headline character in the TV series The Ruggles (1949–52), a family comedy in which he played a character also called Charlie Ruggles and was again the headline character in the daily sitcom The World of Mr. Sweeney, which ran for 345 episodes in 1954–55.

Ruggles returned to the big screen in 1961, playing Charles McKendrick in The Parent Trap and Mackenzie Savage in The Pleasure of His Company. In the latter film, he reprised the role for which he had won a Tony Award in 1959. In 1963 he memorably played the grandfather of the silent film star Corinne Griffith in Papa's Delicate Condition. Griffith had written the book of her early life on which the film is based.

Ruggles made guest appearances in episodes of various television series through the 1950s and 1960s, such as a time-traveling librarian in "Man from 1997," a 1956 science fiction episode of the television series Conflict, and a 1961 appearance as a wealthy neighbor who offers to finance a European trip for Hassie McCoy on The Real McCoys, in the season 5 episode "Hassie's European Tour". Ruggles had a recurring guest role on The Beverly Hillbillies in the mid-1960s as Lowell Redlings Farquhar, father-in-law of Milburn Drysdale (Raymond Bailey). Ruggles also played Aunt Clara's (Marion Lorne) old flame, the warlock Hedley Partridge, as well as a Mr. Caldwell, whose company marketed soup, in the television series Bewitched. In Wagon Train, he played, Jameson Hershey, the owner of an elderly horse, Herman, that joins up with the wagon train. He played Congressman John Canfield on an episode of The Andy Griffith Show called "Aunt Bee, The Swinger", and appeared as a driving instructor on The Munsters. Ruggles also lent his voice to the Aesop and Son features in Jay Ward's The Rocky and Bullwinkle Show.

One of Ruggles' last television appearances before his death was a starring role in the syndicated television special The Wonder Circus, where he played Charlie Wonder, a retired ringmaster who talked about his life leading a circus.

Personal life
His marriage to Adele Rowland (1914–1916) ended in divorce after two years. He then married Barbara Guillan and they remained married until her death in 1941. He married Marion LaBarba in 1942; the couple remained wed until his death in 1970.

Death
Ruggles died of cancer at Saint John's Health Center in Santa Monica, California, on December 23, 1970, at the age of 84.

He is interred at Forest Lawn Memorial Park in Glendale, California in the Garden of Memory near his brother Wesley Ruggles.

Legacy
Charlie Ruggles has three stars on the Hollywood Walk of Fame: one for his contributions to motion pictures on 6200 Hollywood Boulevard, one for his radio work on 6300 Hollywood Boulevard, and one for television on 1600 Vine Street.

Complete filmography

 * The Patchwork Girl of Oz (1914) – Minor Role (unconfirmed)
 * The Majesty of the Law (1915) – Lawrence Evans
 * Peer Gynt (1915) – The Button Molder
 * The Reform Candidate (1915) – Loony Jim
 * The Heart Raider (1923) – Gaspard McMahon (an insurance clerk)
 * Gentlemen of the Press (1929) – Charlie Haven
 * The Lady Lies (1929) – Charlie Tayler
 * The Battle of Paris (1929) – Zizi
 * Roadhouse Nights (1930) – Willie Bindbugel
 * Young Man of Manhattan (1930) – Shorty Ross
 * Queen High (1930) – T. Boggs Johns
 * Her Wedding Night (1930) – Bertie Bird
 * Charley's Aunt (1930) – Lord Fancourt Babberley
 * Honor Among Lovers (1931) – Monty Dunn
 * The Smiling Lieutenant (1931) – Max
 * The Girl Habit (1931) – Charlie Floyd
 * The Beloved Bachelor (1931) – Jerry Wells
 * Husband's Holiday (1931) – Clyde Saunders
 * This Reckless Age (1932) – Goliath Whitney
 * One Hour with You (1932) – Adolph
 * This Is the Night (1932) – Bunny West
 * Make Me a Star (1932) – Charles Ruggles (uncredited)
 * Love Me Tonight (1932) – Viscount Gilbert de Varèze
 * 70,000 Witnesses (1932) – Johnny Moran
 * The Night of June 13 (1932) – Philo Strawn
 * Trouble in Paradise (1932) – The Major
 * Evenings for Sale (1932) – Bimpfl
 * If I Had a Million (1932) – Henry Peabody
 * Madame Butterfly (1932) – Lt. Barton
 * Murders in the Zoo (1933) – Peter Yates
 * Terror Aboard (1933) – Blackie Witherspoon
 * Melody Cruise (1933) – Pete Wells
 * Mama Loves Papa (1933) – Wilbur Todd
 * Goodbye Love (1933) – Oswald Groggs
 * Girl Without a Room (1933) – Vergil Crock
 * Alice in Wonderland (1933) – March Hare
 * Six of a Kind (1933) – J. Pinkham Whinney
 * Melody in Spring (1934) – Warren Blodgett
 * Murder in the Private Car (1934) – Godfrey D. Scott
 * Friends of Mr. Sweeney (1934) – Asaph 'Ace' Holliday
 * The Pursuit of Happiness (1934) – Aaron Kirkland
 * Ruggles of Red Gap (1935) – Egbert Floud
 * People Will Talk (1935) – Henry Wilton
 * No More Ladies (1935) – Edgar
 * The Big Broadcast of 1936 (1935) – Wilbur Sealingsworth
 * Anything Goes (1936) – Moonface Martin
 * The Preview Murder Mystery (1936) – Charles Ruggles (uncredited)
 * Early to Bed (1936) – Chester Beatty
 * Hearts Divided (1936) – Henry
 * Yours for the Asking (1936) – Sunbather (uncredited)
 * Hollywood Boulevard (1936) – Himself – Actor – Cameo Appearance (uncredited)
 * Wives Never Know (1936) – Homer Bigelow
 * Mind Your Own Business (1936) – Orville Shanks
 * Turn Off the Moon (1937) – J. Elliott Dinwiddy
 * Exclusive (1937) – Tod Swain
 * Bringing Up Baby (1938) – Major Applegate
 * Hollywood Handicap (1938, Short) – Himself
 * Screen Snapshots Series 17, No. 12 (1938, Documentary short) – Himself
 * Breaking the Ice (1938) – Samuel Terwilliger
 * Service de Luxe (1938) – Scott Robinson
 * His Exciting Night (1938) – Adam Tripp
 * Boy Trouble (1939) – Homer C. Fitch
 * Sudden Money (1939) – Sweeney J. Patterson
 * Invitation to Happiness (1939) – Henry 'Pop' Hardy
 * Night Work (1939) – Homer C. Fitch
 * Balalaika (1939) – Nicki Popoff
 * The Farmer's Daughter (1940) – Nickie North
 * Opened by Mistake (1940) – Buzz Nelson
 * Maryland (1940) – Dick Piper
 * Public Deb No. 1 (1940) – Milburn
 * No Time for Comedy (1940) – Philo Swift
 * The Invisible Woman (1940) – George
 * Honeymoon for Three (1941) – Harvey Wilson
 * Model Wife (1941) – Milo Everett
 * The Parson of Panamint (1941) – Chuckawalla Bill Redfield
 * Go West, Young Lady (1941) – Jim Pendergast
 * The Perfect Snob (1941) – Dr. Edgar Mason
 * Friendly Enemies (1942) – Heinrich Block
 * Dixie Dugan (1943) – Pa Dugan
 * The Shining Future (1944, Short) – Mr. Ames
 * The Road to Victory (1944, Short) – Mr. Ames (uncredited)
 * The Doughgirls (1944) – Stanley Slade
 * Our Hearts Were Young and Gay (1944) – Otis Skinner
 * Three Is a Family (1944) – Sam Whitaker
 * Bedside Manner (1945) – Dr. J.H. 'Doc' Fredericks
 * Incendiary Blonde (1945) – Cherokee Jim
 * A Stolen Life (1946) – Freddie Linley
 * Gallant Journey (1946) – Jim Montgomery
 * The Perfect Marriage (1947) – Dale Williams, Sr.
 * My Brother Talks to Horses (1947) – Richard Pennington Roeder
 * Ramrod (1947) – Ben Dickason
 * It Happened on 5th Avenue (1947) – Michael J. O'Connor
 * Give My Regards to Broadway (1948) – Toby Helper
 * The Lovable Cheat (1949) – Claude Mercadet
 * Look for the Silver Lining (1949) – Caro 'Pop' Miller
 * Ben and Me (1953, Short) – Ben Franklin (voice, uncredited)
 * The Bells of St. Mary's (1959, TV movie) – Horace Bogardus
 * Once Upon a Christmas Time (1959, TV movie) – Mayor
 * All in a Night's Work (1961) – Dr. Warren Kingsley Sr.
 * The Parent Trap (1961) – Charles McKendrick
 * The Pleasure of His Company (1961) – Mackenzie Savage
 * The Ginger Rogers Show (1961, TV movie) – Eli Harcourt
 * Ernestine (1962, TV movie)
 * Son of Flubber (1963) – Judge Murdock
 * Papa's Delicate Condition (1963) – Anthony Ghio
 * I'd Rather Be Rich (1964) – Dr. Charles Crandall
 * The Ugly Dachshund (1966) – Dr. J.L. Pruitt
 * Follow Me, Boys! (1966) – John Everett Hughes
 * Carousel (1967, TV movie) – The Starkeeper / Dr. Selden

Television credits

 * Father Knows Best (1960) (1 episode) as Jim's co-worker
 * Rocky and His Friends (1960–1961) as Voice of Aesop (uncredited)
 * The Bullwinkle Show (1961) as Voice of Aesop (uncredited)
 * The Real McCoys (1961) as Mr. Deveraux
 * Burke's Law (1963–1964) as O.B. Danberry / Charles Wingfield / I.A. Bugg / Mr. Gregory
 * My Living Doll (1964, "I'll Leave It to You") as Jonas Clay
 * Beverly Hillbillies (1964) as Mrs. Drysdale's father, Lowell Redlings Farquhar in the episodes "Mrs. Drysdale's Father" and "Mr. Farquhar Stays On"
 * Bewitched as Mr. Caldwell (1964, episode: "Help Help, Don't Save Me") and Hedley Partridge (1965, episode: "Aunt Clara's Old Flame")
 * Wagon Train as Mr. Caldwell (1965, episode:"Herman")
 * The Man from U.N.C.L.E. (1965) (1 episode) as Governor Callahan
 * The Andy Griffith Show (1965) (episode "Aunt Bee, The Swinger") as John Canfield
 * The Munsters (1965) (1 episode, "Herman's Driving Test") as Charlie Wiggens
 * Bonanza (1966) (1 episode, "Horse of a Different Hue") as Col. Robert Fairchild
 * The Wonder Circus (1966) (television special) as Charlie Wonder
 * The Danny Thomas Hour (1968) (1 episode, "One for My Baby") as Stimson

title: Charlie Ruggles


text:
Rotation operator (quantum mechanics)

This article concerns the rotation operator, as it appears in quantum mechanics.

Quantum mechanical rotations
With every physical rotation $$R$$, we postulate a quantum mechanical rotation operator $$D(R)$$ which rotates quantum mechanical states. $$| \alpha \rangle_R = D(R) |\alpha \rangle$$

In terms of the generators of rotation, $$D (\mathbf{\hat n},\phi) = \exp \left( -i \phi \frac{\mathbf{\hat n} \cdot \mathbf J }{ \hbar} \right),$$ where $$\mathbf{\hat n}$$ is rotation axis, $$ \mathbf{J} $$ is angular momentum, and $$\hbar$$ is the reduced Planck constant.

The translation operator
The rotation operator $$\operatorname{R}(z, \theta)$$, with the first argument $$z$$ indicating the rotation axis and the second $$\theta$$ the rotation angle, can operate through the translation operator $$\operatorname{T}(a)$$ for infinitesimal rotations as explained below. This is why, it is first shown how the translation operator is acting on a particle at position x (the particle is then in the state $$|x\rangle$$ according to Quantum Mechanics).

Translation of the particle at position $$x$$ to position $$x + a$$: $$\operatorname{T}(a)|x\rangle = |x + a\rangle$$

Because a translation of 0 does not change the position of the particle, we have (with 1 meaning the identity operator, which does nothing): $$\operatorname{T}(0) = 1$$ $$\operatorname{T}(a) \operatorname{T}(da)|x\rangle = \operatorname{T}(a)|x + da\rangle = |x + a + da\rangle = \operatorname{T}(a + da)|x\rangle \Rightarrow \operatorname{T}(a) \operatorname{T}(da) = \operatorname{T}(a + da)$$

Taylor development gives: $$\operatorname{T}(da) = \operatorname{T}(0) + \frac{d\operatorname{T}(0)}{da} da + \cdots = 1 - \frac{i}{\hbar} p_x da$$ with $$p_x = i \hbar \frac{d\operatorname{T}(0)}{da}$$

From that follows: $$\operatorname{T}(a + da) = \operatorname{T}(a) \operatorname{T}(da) = \operatorname{T}(a)\left(1 - \frac{i}{\hbar} p_x da\right) \Rightarrow \frac{\operatorname{T}(a + da) - \operatorname{T}(a)}{da} = \frac{d\operatorname{T}}{da} = - \frac{i}{\hbar} p_x \operatorname{T}(a)$$

This is a differential equation with the solution

$$\operatorname{T}(a) = \exp\left(- \frac{i}{\hbar} p_x a\right).$$

Additionally, suppose a Hamiltonian $$H$$ is independent of the $$x$$ position. Because the translation operator can be written in terms of $$p_x$$, and $$[p_x,H] = 0$$, we know that $$[H, \operatorname{T}(a)]=0.$$ This result means that linear momentum for the system is conserved.

In relation to the orbital angular momentum
Classically we have for the angular momentum $$\mathbf L = \mathbf r \times \mathbf p.$$ This is the same in quantum mechanics considering $$\mathbf r$$ and $$\mathbf p$$ as operators. Classically, an infinitesimal rotation $$dt$$ of the vector $$\mathbf r = (x,y,z)$$ about the $$z$$-axis to $$\mathbf r' = (x',y',z)$$ leaving $$z$$ unchanged can be expressed by the following infinitesimal translations (using Taylor approximation): $$\begin{align} x' &= r \cos(t + dt) = x - y \, dt + \cdots \\ y' &= r \sin(t + dt) = y + x \, dt + \cdots \end{align}$$

From that follows for states: $$\operatorname{R}(z, dt)|r\rangle = \operatorname{R}(z, dt)|x, y, z\rangle = |x - y \, dt, y + x \, dt, z\rangle = \operatorname{T}_x(-y \, dt) \operatorname{T}_y(x \, dt)|x, y, z\rangle = \operatorname{T}_x(-y \, dt) \operatorname{T}_y(x \, dt) |r\rangle$$

And consequently: $$\operatorname{R}(z, dt) = \operatorname{T}_x (-y \, dt) \operatorname{T}_y(x \, dt)$$

Using $$T_k(a) = \exp\left(- \frac{i}{\hbar} p_k a\right)$$ from above with $$k = x,y$$ and Taylor expansion we get: $$\operatorname{R}(z,dt)=\exp\left[-\frac{i}{\hbar} \left(x p_y - y p_x\right) dt\right] = \exp\left(-\frac{i}{\hbar} L_z dt\right) = 1-\frac{i}{\hbar}L_z dt + \cdots$$ with $$L_z = x p_y - y p_x$$ the $$z$$-component of the angular momentum according to the classical cross product.

To get a rotation for the angle $$t$$, we construct the following differential equation using the condition $$\operatorname{R}(z, 0) = 1 $$:

$$\begin{align} &\operatorname{R}(z, t + dt) = \operatorname{R}(z, t) \operatorname{R}(z, dt) \\[1.1ex] \Rightarrow {} & \frac{d\operatorname{R}}{dt} = \frac{\operatorname{R}(z, t + dt) - \operatorname{R}(z, t)}{dt} = \operatorname{R}(z, t) \frac{\operatorname{R}(z, dt) - 1}{dt} = - \frac{i}{\hbar} L_z \operatorname{R}(z, t) \\[1.1ex] \Rightarrow {}& \operatorname{R}(z, t) = \exp\left(- \frac{i}{\hbar}\, t \, L_z\right) \end{align}$$

Similar to the translation operator, if we are given a Hamiltonian $$H$$ which rotationally symmetric about the $$z$$-axis, $$[L_z,H]=0$$ implies $$[\operatorname{R}(z,t),H]=0$$. This result means that angular momentum is conserved.

For the spin angular momentum about for example the $$y$$-axis we just replace $$L_z$$ with $S_y = \frac{\hbar}{2} \sigma_y$ (where $$\sigma_y$$ is the Pauli Y matrix) and we get the spin rotation operator $$\operatorname{D}(y, t) = \exp\left(- i \frac{t}{2} \sigma_y\right).$$

Effect on the spin operator and quantum states
Operators can be represented by matrices. From linear algebra one knows that a certain matrix $$A$$ can be represented in another basis through the transformation $$A' = P A P^{-1}$$ where $$P$$ is the basis transformation matrix. If the vectors $$b$$ respectively $$c$$ are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle $$t$$ between them. The spin operator $$S_b$$ in the first basis can then be transformed into the spin operator $$S_c$$ of the other basis through the following transformation: $$S_c = \operatorname{D}(y, t) S_b \operatorname{D}^{-1}(y, t)$$

From standard quantum mechanics we have the known results $S_b |b+\rangle = \frac{\hbar}{2} |b+\rangle$ and $S_c |c+\rangle = \frac{\hbar}{2} |c+\rangle$  where  $$|b+\rangle$$ and $$|c+\rangle$$ are the top spins in their corresponding bases. So we have: $$\frac{\hbar}{2} |c+\rangle = S_c |c+\rangle = \operatorname{D}(y, t) S_b \operatorname{D}^{-1}(y, t) |c+\rangle \Rightarrow$$ $$S_b \operatorname{D}^{-1}(y, t) |c+\rangle = \frac{\hbar}{2} \operatorname{D}^{-1}(y, t) |c+\rangle$$

Comparison with $S_b |b+\rangle = \frac{\hbar}{2} |b+\rangle$ yields $$|b+\rangle = D^{-1}(y, t) |c+\rangle$$.

This means that if the state $$|c+\rangle$$ is rotated about the $$y$$-axis by an angle $$t$$, it becomes the state $$|b+\rangle$$, a result that can be generalized to arbitrary axes.

title: Rotation operator (quantum mechanics)


text:
Russell Davies

Robert Russell Davies (born 5 April 1946) is a British journalist and broadcaster.

Davies was born in Barmouth, North Wales. He attended Manchester Grammar School, according to his own statement on a November 2010 Brain of Britain programme. Also according to the programme, his grandfather was a mole-catcher. During his time at MGS (1957–64) he acted in dramatic society productions and was appointed school vice-captain.

He gained a scholarship to St John's College, Cambridge, and was awarded a first class degree in Modern and Mediaeval Languages in 1967, but soon abandoned his post-graduate studies in German literature when the opportunity arose to tour with the Cambridge Footlights revue. During his time in Cambridge, he contributed topical cartoons to the news pages of Varsity, the undergraduate newspaper, under the pseudonym Dai.

In 1970, Davies co-starred with fellow Footlights alumni Pete Atkin, Clive James and Julie Covington across three London Weekend Television series. The Party's Moving On and What Are You Doing After the Show? were revue-style shows, featuring original music written by Atkin and James along with sketches and commentary. Think Twice focused on discussion of the arts between Davies and co-host Clive James.

As a journalist, Davies worked as a film and television critic for The Observer and The Sunday Times, features writer and sports columnist for The Daily Telegraph and The Sunday Telegraph, a caricaturist for The Times Literary Supplement and was a deputy editor of Punch. He edited Kenneth Williams's diaries and letters for publication, despite being the target of Williams's acid pen in those same diaries. He regularly appeared on television in the 1980s, presenting Saturday Review for BBC2.

In 1988, Davies was the presenter of a 12-part BBC radio series Radio Fun chronicling the history of comedy on the radio from the thirties to the present day. The series was repeated from 2007 to 2010 on BBC Radio 7 (now BBC Radio 4 Extra).

Davies is a jazz trombonist and has presented television and radio documentaries on jazz, including in 1999 a year-long history for BBC Radio 3, Jazz Century. He has written and presented other radio programmes including BBC Radio 4's series Word of Mouth (winner of the 1996 European Radio, ONDAS prize), and a series on Radio 2 about songwriters Rodgers and Hart, Legends of Light Music. In 2003, Russell Davies wrote and presented Quest for Perfection, a film about jazz clarinetist and bandleader Artie Shaw, for BBC Four and produced by John Warburton (shortlisted for the 2004 Grierson Award). By July 2005, his film had been broadcast nine times.

His documentary on the life of the cartoonist Ronald Searle was shown on the BBC in 2006.

He presents Brain of Britain on Radio 4, first temporarily to cover the illness of presenter Robert Robinson, then permanently from 2010 when Robinson retired. From 1998 (when he took over from Benny Green) until 29 September 2013, he presented a Sunday radio programme on BBC Radio 2 called 'The Russell Davies Song Show' which showcased 'The Art, Craft & Inspiration of the Popular Song'. The show was produced by Roy Oakshott and predominantly featured music from 'The Great American Songbook' and the pre-rock 'n' roll jazz era. He has also presented the BBC Radio 4 music quiz Counterpoint.

title: Russell Davies


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