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Wave drag

In aeronautics, wave drag is a component of the aerodynamic drag on aircraft wings and fuselage, propeller blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects, and tends to present itself as a sudden and dramatic increase in drag as the vehicle increases speed to the critical Mach number. It is the sudden and dramatic rise of wave drag that leads to the concept of a sound barrier.

Overview
Wave drag is a component of pressure drag due to compressibility effects. It is caused by the formation of shock waves around a body. Shock waves create a considerable amount of drag, which can result in extreme drag on the body. Although shock waves are typically associated with supersonic flow, they can form at subsonic aircraft speeds on areas of the body where local airflow accelerates to supersonic speed. The effect is typically seen on aircraft at transonic speeds (about Mach 0.8), but it is possible to notice the problem at any speed over that of the critical Mach of that aircraft. It is so pronounced that, prior to 1947, it was thought that aircraft engines would not be powerful enough to overcome the enhanced drag, or that the forces would be so great that aircraft would be at risk of breaking up in midflight. It led to the concept of a sound barrier.

Research
In 1947, studies into wave drag led to the development of perfect shapes to reduce wave drag as much as theoretically possible. For a fuselage the resulting shape was the Sears–Haack body, which suggested a perfect cross-sectional shape for any given internal volume. The von Kármán ogive was a similar shape for bodies with a blunt end, like a missile. Both were based on long narrow shapes with pointed ends, the main difference being that the ogive was pointed on only one end.

Reduction of drag
A number of new techniques developed during and just after World War II were able to dramatically reduce the magnitude of wave drag, and by the early 1950s the latest fighter aircraft could reach supersonic speeds.

These techniques were quickly put to use by aircraft designers. One common solution to the problem of wave drag was to use a swept wing, which had actually been developed before World War II and used on some German wartime designs. Sweeping the wing makes it appear thinner and longer in the direction of the airflow, making a conventional teardrop wing shape closer to that of the von Kármán ogive, while still remaining useful at lower speeds where curvature and thickness are important.

The wing need not be swept when it is possible to build a wing that is extremely thin. This solution was used on a number of designs, beginning with the Bell X-1, the first manned aircraft to fly at the speed of sound. The downside to this approach is that the wing is so thin it is no longer possible to use it for storage of fuel or landing gear. Such wings are very common on missiles, although, in that field, they are often referred to as "fins".

Fuselage shaping was similarly changed with the introduction of the Whitcomb area rule. Whitcomb had been working on testing various airframe shapes for transonic drag when, after watching a presentation by Adolf Busemann in 1952, he realized that the Sears-Haack body had to apply to the entire aircraft, not just the fuselage. This meant that the fuselage needed to be made narrower where it joined the wings, so that the cross-section of the entire aircraft matched the Sears-Haack body.

Application of the area rule can also be seen in the use of anti-shock bodies on transonic aircraft, including some jet airliners. Anti-shock bodies, which are pods along the trailing edges of the wings, serve the same role as the narrow waist fuselage design of other transonic aircraft.

Other drag reduction methods
Several other attempts to reduce wave drag have been introduced over the years. The supercritical airfoil is a type that results in reasonable low speed lift like a normal airfoil, but has a profile considerably closer to that of the von Kármán ogive. All modern civil airliners use forms of supercritical aerofoil and have substantial supersonic flow over the wing upper surface.

For flat plate aerofoil
$$cd_w = 4*\frac{\alpha^2}{\sqrt{(M^2 - 1)}}$$

For double-wedge aerofoil
$$cd_w = 4*\frac{\alpha^2 + (t/c)^2}{\sqrt{(M^2 - 1)}}$$

Where:

cd_w - Coefficient of drag from wave drag

α - Angle of attack

t/c - Thickness to Chord ratio

M - Freestream Mach number

These equations are applicable at low angles of attack (α < 5°)

title: Wave drag


text:
User:Acid burnd

Derrick acid_burnd@excite.com

Articles That I Initiated

 * Emma Peel
 * Fox Mulder
 * Arthur Hastings
 * Chief Inspector Japp

Articles To Which I Made More Than Minor Edits

 * Hercule Poirot
 * added description of the character and edited discussion of recurring characters
 * Barry White
 * added Ally McBeal reference and mentioned Barry's wanting to be remembered for being a good person
 * Xander Harris
 * added description of character's comic relief role
 * Ren and Stimpy
 * added description of Ren & Stimpy and a discussion of the show's gross-out humor

Some Favorite Quotes
"What about home school? You know, it's not just for scary religious people anymore!" -- Buffy Summers

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

title: User:Acid burnd


text:
OCD (disambiguation)

OCD is an abbreviation that may stand for:

Medical uses

 * Obsessive–compulsive disorder (OCD), a mental disorder
 * Obsessive–compulsive personality disorder (OCPD), a mental disorder that is often mistakenly called OCD
 * On-call duty, a long shift that may be assigned to medical residents
 * Ortho-Clinical Diagnostics, an American company
 * Osteochondritis dissecans, a painful joint condition in humans and other animals

Computing

 * OCD, the file suffix used by OCAD mapping software
 * Object collision detection
 * On-chip debugging
 * OpenOCD (Open On-Chip Debugger), a debugger mostly used in embedded software development.

Offices and institutions

 * Office of Civil Defense, the National Emergency Management Agency of the Philippines
 * Office of Civilian Defense, a World War II–era U.S. government agency
 * Office of Community Development, a part of the U.S. Department of Agriculture's rural development activities
 * Office of Cultural Development, a part of the Louisiana Department of Culture, Recreation and Tourism

Christianity

 * OCD, the acronym for the Order of Discalced Carmelites (from the Latin, Ordo Carmelitarum Discalceatorum)

Other

 * "O.C.D.", a song by Suicide Silence from their third album The Black Crown
 * Obsessive Completion Distinction, an achievement demanding extraordinary attention to detail in the video game World of Goo
 * Octavian County Day School, a school attended by characters of Lisi Harrison's The Clique books
 * OCD overdrive pedal by Fulltone
 * "OCD", a song by Fitz and the Tantrums from All the Feels, 2019
 * "OCD", a song by Logic featuring Dwn2Earth
 * "OCD", a song by the Godfathers from Alpha Beta Gamma Delta, 2022
 * A 1999 cassette album by the noise musician Prurient
 * Offshore Development Center
 * Okanogan Conservation District
 * Oxford Classical Dictionary

title: OCD (disambiguation)


text:
Jørgen Nash

Jørgen Nash (March 16, 1920 – May 17, 2004) was a Danish artist, writer and central proponent of Situationism.

Life
He was born in Vejrum, Jutland, Denmark, baptized Jørgen Axel Jørgensen, the brother of Asger Jorn. He later changed his family name from Jørgensen to Nash. He was married three times and had six children. His last marriage was with the painter Lis Zwick and would last until his death in 2004. The last forty years of his life he lived in the artist collective Drakabygget outside Örkelljunga, Sweden.

From 1948 to 1951 he was involved with CoBrA. He went on to join the Situationist International.

In 1959 he produced Stavrim, Sonetter a book of poetry illustrated by Jorn.

In 1960 he founded the Situationist Bauhaus based at the farm Drakabygget with Jorn.

He was the editor of Drakabygget, a journal about art and social philosophy.

In 1963 became an honorary artist of the Royal Danish Academy of Art.

Exhibitions
He has exhibited at:

 * Galerias del Palacio de Belles Artes, Mexico City
 * Kobe Art Museum, Kobe, Japan
 * Museum of Modern Art New York
 * Moderna Museet
 * National Museum of Modern Art, Tokyo
 * Museum de Art Moderno, Mexico City
 * Ueno Royal Museum, Tokyo
 * Rooseum, Sweden
 * The Modern Art Gallery Silkeborg, Denmark
 * Museo Carillo Gil, Mexico City
 * Klingspor-Museum, Offenbach am Main, Germany
 * Yokohama City Art Hall, Yokohama, Japan
 * Tessin Institute, Paris
 * Museo National de Belles Artes and Centro Wifredo Lam, Havana, Cuba
 * Skissernas Museum, Lund, Sweden
 * Kongelige Kobberstiksamling, Copenhagen, Denmark
 * Centre Pompidou, Paris
 * Royal Academi of Art, Stockholm
 * Museu Dárt Contemporani, Barcelona, Spain

Exclusion from the SI
On 10 February 1962, Nash voted with the majority of the Central Council of the Situationist International, for the exclusion of the Spur group. The majority was 5 to 1, with Dieter Kunzelmann be the only opposer. Nash rebuked those responsible for Spur, was initially in favor of them publishing a retraction, but stopped short of demanding an exclusion. After a debate on this subject, however, Nash decided on the option of exclusion. Kunzelmann too, the only opposing vote, and himself part of Spur, expressed his approval of all the Central Council's critiques towards the other Spur members.

In Sweden, in March 1962, Jörgen Nash suddenly attacked the SI in the name of the Scandinavian section, with the tract ''Danger! Do Not Lean Out'', and planned to convert the section in another Swedish "Bauhaus". Reckoning on the considerable geographical dispersion of the Scandinavian situationists, Nash had not even consulted with all of them before his putsch. Surprised at not being unanimously followed and at finding himself countered on the spot by the partisans of the SI majority — Jeppesen Victor Martin, who immediately circulated a definitive repudiation of his imposture — Nash at first feigned astonishment that things had gone to the point of a complete break with the situationists; as if the fact of launching a public surprise attack full of lies was compatible with carrying on a dialogue, on the basis of some sort of Nashist Scandinavian autonomy.

According to the SI, Nash's main goal was to use the seal of "situationism" to attract a few highly profitable art dealers. The SI stated that this was confirmed by the fact that Nash's new Swedish "Bauhaus", which consisted of two or three Scandinavian ex-situationists plus "a mass of unknowns flocking to the feast", immediately plunged into "the most shopworn forms of artistic production". According to the SI, the development of this conspiracy was no doubt precipitated by the recent elimination of the SI's right wing, the Spur group, on whose support the Nashists had relied.

In the tract ''Danger! Do Not Lean Out'', Nash, with Ansgar Elde and Jacqueline de Jong, stated that although there may be grounds to criticise the Spur group (Ervin Eisch, Lothar Fischer, Dieter Kunzelmann, Renee Nele, Heimrad Prem, Gretel Stadler, Helmut Sturm and Hand-Peter Zimmer), the way they had been expelled, and the SI itself, were totalitarian. However, the decision followed a debate and was voted by the majority 5 to 1, with Nash also voting in favor of it. Nash also implied that Ansgar Elde was present at the Council and that Jacqueline de Jong was part of if, which was false. The SI, at the behest of Jeppesen Victor Martin, described those they had expelled as nashists.

On 24 April 1964 Nash with other members of Bauhaus Situationniste decapitated the statue of The Little Mermaid in Copenhagen harbour. This manifestation was part of the movements campaign against consumerist society.

Following his exclusion he became involved in the Second Situationist International and the creation in 1974 of the Situationist Antinational.

title: Jørgen Nash


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Bernie Taupin

Bernard John Taupin (born 22 May 1950) is an English lyricist and visual artist. He is best known for his songwriting partnership with Elton John, recognised as one of the most successful partnerships of its kind in history. Taupin is behind the majority of John's songs, dating back to the 1960s.

In 1967, Taupin answered an advertisement in the music paper New Musical Express placed by Liberty Records, a company that was seeking new songwriters. John responded to the same advertisement and they were brought together, collaborating on many albums since. Taupin and John were inducted into the Songwriters Hall of Fame in 1992.

In 2020, Taupin and John received the Oscar for Best Original Song for "(I'm Gonna) Love Me Again" from the film Rocketman. Taupin was inducted into the Rock and Roll Hall of Fame in the Musical Excellence Award category in 2023. In 2024, Taupin and John were awarded the Gershwin Prize for Popular Song. Also in 2024, Taupin won the Ivor Novello Award for Outstanding Contribution to British Music.

Birth and childhood
Taupin was born at Flatters House, a farmhouse located between the village of Anwick and the town of Sleaford in Lincolnshire, the son of Robert Taupin and Daphne (Cort). His paternal grandparents were French, the Taupin family having come to London at the turn of the 20th century to set up a wine-importing business.

Taupin's father was educated in Dijon and was employed as a stockman by a large farm estate near the town of Market Rasen, Lincolnshire. Taupin's mother spent most of the Second World War living in Switzerland. On her return to London, she worked as a governess for the Taupin family, in which connection she met Robert Taupin, whom she married in 1947. The family later moved to Rowston Manor, where they lived rent-free because of Robert's promotion to farm manager. This was a significant step up from Flatters farmhouse, which had no electricity.

In 1959, Taupin's father decided to try independent farming, and the family moved to the north Lincolnshire village of Owmby-by-Spital, where they lived at the run-down 10-acre Maltkiln Farm, the income from which derived from battery-farming chickens for eggs. The house lacked heating and up-to-date plumbing. Taupin's 11-year-younger brother, Kit, was born there.

Unlike his older brother, Tony, who attended a grammar school and later went to university, Taupin was not a diligent student, although he showed an early flair for writing. At 15, he left school and started work as a trainee in the print room of the local newspaper, the Lincolnshire Standard, with aspirations of becoming a journalist. Taupin soon left that job, and spent the rest of his teenage years hanging out with friends, hitchhiking the country roads to attend youth club dances in the surrounding villages, playing snooker in the Aston Arms pub in Market Rasen, and drinking. Taupin had worked at several part-time jobs when, at 17, he answered the advertisement that eventually led to his collaboration with Elton John.

Early influences
Taupin's mother had studied French literature, and his maternal grandfather John Leonard Patchett "Poppy" Cort, a classics teacher and graduate of the University of Cambridge, instilled in him an appreciation for nature and literature and narrative poetry, both of which influenced his early lyrics.

Collaboration with Elton John (1967–present)
In 1967, Taupin answered an advertisement for talent that was placed in the New Musical Express by Liberty Records A&R man Ray Williams. Elton John answered the same advert. Neither Taupin nor John passed the audition for Liberty Records. Elton told the man behind the desk that he could not write lyrics, so the man handed Elton a sealed envelope from the pile of people submitting lyrics, which he opened on the London Underground ride home. The envelope contained poems by Taupin.

The duo have collaborated on more than thirty albums since. The team took some time off from each other for a while between 1977 and 1979, while Taupin worked with other songwriters, including Alice Cooper, and John worked with other lyricists, including Gary Osborne and Tom Robinson. (The 1978 single-only A side "Ego" was their only collaboration of note during the period, although John/Taupin B-sides such as "Lovesick" and "I Cry at Night" were issued with the respective singles "Song for Guy" and "Part-time Love" from the album A Single Man.)

John and Taupin resumed writing together on an occasional basis in 1980, with Taupin contributing lyrics to several songs on albums such as 21 at 33, The Fox, and Jump Up!. However, by 1983's Too Low for Zero, the two renewed their partnership on a full-time basis and from that point forward Taupin was again John's primary lyricist for his solo releases. John often works with other lyricists on specific theatrical or film projects such as 1994's The Lion King and 2000's Aida, both of which featured lyrics by Tim Rice, and 2005's Billy Elliot, which has script and lyrics by the original film's screenwriter, Lee Hall. Hall also wrote the screenplay for the Elton John biopic Rocketman, in which Taupin features as a major character.

Taupin wrote the lyrics for "Rocket Man", "Levon", "Crocodile Rock", "Honky Cat", "Tiny Dancer", "Candle in the Wind", "Saturday Night's Alright for Fighting", "Bennie and the Jets", "Goodbye Yellow Brick Road", "Mona Lisas and Mad Hatters", "Don't Let the Sun Go Down on Me", "The Bitch is Back", "Daniel", and 1970's "Your Song", their first hit. Hits in the 1980s include "I'm Still Standing", "I Guess That's Why They Call It The Blues", "Sad Songs", and "Nikita." In the 1990s, Taupin and John had more hits, including "The One", "Simple Life", "The Last Song", "Club at the End of the Street" and "Believe". In September 1997, Taupin rewrote the lyrics of "Candle in the Wind" for "Candle in the Wind 1997", a tribute to the late Diana, Princess of Wales.

The 1991 film documentary Two Rooms described the John/Taupin writing style, which involves Taupin writing the lyrics on his own and John then putting them to music, with no further interaction between the two. The process is still fundamentally the same, with John composing to Taupin's words, but the two interact on songs far more today, with Taupin joining John in the studio as the songs are written and occasionally during recording sessions.

Taupin and John had their first Broadway musical open in March 2006 with Lestat: The Musical. Taupin wrote lyrics for 10 songs (and an 11th completed non-album track "Across the River Thames") for John's 2006 album The Captain & The Kid (sequel to Captain Fantastic and the Brown Dirt Cowboy) and appeared on the cover with him for the first time marking their 40th anniversary of working together. ("Across the River Thames" was issued as an Internet-only download as a bonus with certain editions of The Captain & the Kid.)

On 25 March 2007, Taupin made a surprise appearance at John's 60th birthday celebration at Madison Square Garden, briefly discussing their 40-year songwriting partnership. Of Taupin's importance to their careers, as recorded on the Elton 60 – Live at Madison Square Garden DVD, John told the audience that without Taupin there probably would not be an 'Elton John' as the public has come to know him. Taupin and John also composed several songs for The Union, a collaboration album between Elton and his longtime hero Leon Russell released in October 2010. They also collaborated on five original songs for the 2011 Miramax movie Gnomeo and Juliet, including the Golden Globe-nominated "Hello Hello". Taupin and John's most-recent album collaboration is Wonderful Crazy Night, released in 2016.

In 2018, Taupin and John collaborated on two original songs for the animated film Sherlock Gnomes. In 2019, the movie Rocketman was released, containing an original song written by Taupin and John for the film, "(I'm Gonna) Love Me Again", which plays over the closing credits. The film depicts the personal friendship of Taupin (played by Jamie Bell) and John (played by Taron Egerton) during the 1970s and 1980s. They won the Academy Award for Best Original Song for "(I'm Gonna) Love Me Again".

Collaboration with other artists
In addition to writing for Elton John, Taupin has also written lyrics for use by other composers, with notable successes including "We Built This City", which was recorded by Starship, and "These Dreams," recorded by Heart (both of which were collaborations with English composer/musician Martin Page). In 1978, he co-wrote the album From the Inside with Alice Cooper.

Taupin also produced American Gothic for singer-songwriter David Ackles. Released in 1972, it did not enjoy big sales, but the album was highly acclaimed by music critics in the US and UK. The influential British music critic Derek Jewell of the UK Sunday Times described the album as being "the Sgt. Pepper of folk." Of Ackles's four albums, it was the only one recorded in England rather than in the United States. Taupin and Ackles had become acquainted when Ackles was selected to be the co-headlining act for Elton John's 1970 American debut at the Troubadour in Los Angeles. Taupin was mentioned specifically as being one of the reasons American Gothic was selected by the writers and editors for inclusion in the book, 1001 Albums You Must Hear Before You Die. He also collaborated on the book Burning Cold with photographer Gary Bernstein. In the late 1980s and early 1990s Taupin also collaborated with French American musician, Josquin Des Pres on at least 13 songs in his collection that have been performed and recorded by artists worldwide.

In 2002, Willie Nelson and Kid Rock recorded "Last Stand in Open Country" for Nelson's album The Great Divide. The song was the title track of the first album from Taupin's band Farm Dogs (see below). Nelson's album included two other Taupin songs, "This Face" and "Mendocino County Line". The latter, a duet between Nelson and Lee Ann Womack, was made into a video and released as the album's first single. The song won the 2003 Grammy for best vocal collaboration in country music. In 2004, he co-wrote Courtney Love's song "Uncool", from her 2004 debut solo album America's Sweetheart. In 2005, he co-wrote the title track to What I Really Want For Christmas with Brian Wilson for his first seasonal album. In 2006, he won a Golden Globe Award for his lyrics to the song "A Love That Will Never Grow Old" from the film Brokeback Mountain. The music of the song was composed by Argentine producer and songwriter Gustavo Santaolalla.

Works as a performer
In 1971, Taupin recorded a spoken-word album titled Taupin, in which he recites some of his early poems against a background of impromptu, sitar-heavy music created by some members of Elton's band, including Davey Johnstone and Caleb Quaye. Side one, "Child", contains poems about his early childhood in southern Lincolnshire. The first poem, "The Greatest Discovery", which looks at his birth from the perspective of his older brother Tony, was also set to music by Elton John and included on John's second album, Elton John. There are poems about Taupin's first two childhood homes, Flatters and Rowston Manor, and others about his relationship with his brother and grandfather. Side Two includes a variety of poems of varying obscurity, from a marionette telling her own story to a rat catcher who falls victim to his prey. Taupin stated in interviews that he was not pleased with the album.

In 1980, Taupin recorded his first album as a singer, He Who Rides the Tiger. The album failed to make a dent in the charts. Taupin later suggested in interviews that he did not have the creative control he would have liked over the album. In 1987, he recorded another album, Tribe. The songs were co-written with Martin Page. "Citizen Jane" and "Friend of the Flag" were released as singles. Videos of both singles featured Rene Russo, the sister of Taupin's then wife Toni.

In 1996, Taupin formed a band called Farm Dogs, whose two albums were conscious (and successful) throwbacks to the grittier, earthier sound of Tumbleweed Connection. While Taupin wrote the lyrics, the music was a collaborative effort among the band members. Their first album, 1996's Last Stand in Open Country, received critical praise but little airplay. The title track was later recorded by Willie Nelson and Kid Rock for Nelson's 2002 album The Great Divide. In 1998, Farm Dogs released its second and final album, Immigrant Sons. The album was unsuccessful despite a tour of small clubs across America.

Non-musical projects
In 1973, Taupin collected all his lyrics up through the Goodbye Yellow Brick Road album into a book, Bernie Taupin: The One Who Writes the Words for Elton John. In addition to the lyrics from the albums, this book contained the lyrics to all the single B-sides, various rarities, and Taupin's 1970 spoken-word album. The songs are illustrated by various artists, friends, and celebrity guests such as John Lennon and Joni Mitchell. The book is in black & white except for the cover.

In 1977, Taupin collaborated with rock photographer David Nutter on It's A Little Bit Funny, adding text and helping chronicle Elton John's year-long Louder Than Concorde Tour. The now-collectible book was published in hard and soft cover editions by Penguin Books. It collects the better part of one year's worth of personal adventures and memories of Elton and the band, aboard his private plane, on the beaches of Barbados, at backstage gatherings and in some quieter off-stage moments with friends (including some famous faces that Elton and Bernie met and palled around with in their travels).

In 1978, Taupin also appeared in an episode of The Hardy Boys/Nancy Drew Mysteries, "The Hardy Boys & Nancy Drew Meet Dracula", singing backup to Shaun Cassidy. In 1988, Taupin published an autobiography of his childhood, A Cradle of Haloes: Sketches of a Childhood. The book was released only in the UK. It tells the tale of a childhood fuelled by fantasy in rural Lincolnshire in the 1950s and 1960s, ending in 1969 as Taupin gets on the train to seek his fortune in London.

In 1991, Taupin self-published a book of poems called The Devil at High Noon. In 1994, Taupin's lyrics up through the Made in England album were collected into a hardcover book, Elton John & Bernie Taupin: The Complete Lyrics, published by Hyperion. However, it does not appear that Taupin was intimately involved in this project, as it contains multiple misspellings and outright misrenderings of the lyrics. It is also missing some of the rarities and B-sides found in the earlier collection. As with the 1973 collection, the songs are illustrated by various artists, this time in full colour throughout.

In 1992, Taupin was asked to produce a benefit for AIDS Project Los Angeles. The event featured no songs written by the writer, instead opening with an acoustic set of performances of material chosen by the performers followed by selections from the musical West Side Story, chosen for its "timeless message of tolerance that is relevant to every decade."

Visual art career
In addition to his music, much of his time is spent creating his visual art. Attributing his passion for art to his mother, Daphne, Taupin began displaying and selling his original artwork in 2010. Consisting of large, mixed media, contemporary assemblages, the art has been shown and collected across the United States and Canada.

Taupin's artwork has been seen at Art Miami, Art Southampton, the LA Art Fair, and in various galleries. His artwork often incorporates the American flag.

Personal life
Taupin has been married four times. His first three marriages ended in divorce. His wives were: Maxine Feibelman (1971–77); Toni Lynn Russo (1979–91), sister of actress Rene Russo; Stephanie Haymes Roven (1993–98), daughter of entertainers Dick Haymes and Fran Jeffries; and Heather Kidd (March 2004–present), with whom he has two daughters.

Taupin moved to the United States from England in the mid-1970s. In 1990 he became a US citizen and lives in Santa Barbara County, California.

Taupin was appointed Commander of the Order of the British Empire (CBE) in the 2022 New Year Honours for services to music.

Solo albums

 * 1971 – Taupin
 * 1980 – He Who Rides the Tiger
 * 1987 – Tribe

With Farm Dogs

 * 1996 – Last Stand in Open Country
 * 1998 – Immigrant Sons

title: Bernie Taupin


text:
Thomas Tusser

Thomas Tusser (c. 1524 – 3 May 1580) was an English poet and farmer, best known for his instructional poem Five Hundred Points of Good Husbandry, an expanded version of his original title, A Hundreth Good Pointes of Husbandrie, first published in 1557. For Tusser the garden was the domain of the housewife, and the 1562 text expands on this theme. Scholars also consider it a text of interest for its defence of enclosures. It was among the best selling poetry books of the Elizabethan age.

Early life
Tusser was born in Rivenhall, Essex, about 1524, the son of William and Isabella Tusser. At a very early age he became a chorister in St Nicholas' Collegiate Chapel in Wallingford Castle, Wallingford, Oxfordshire. He appears to have been pressed for service in the King's Chapel, the choristers of which were usually afterwards placed by the King in one of the Royal Foundations at Oxford or Cambridge but Tusser entered the choir of St. Paul's Cathedral, and from there went to Eton College. He has left a quaint account of his privations at Wallingford, and of the severities of Nicholas Udall at Eton.

He was elected to King's College, Cambridge in 1543, a date which sets the earliest limit of his birth-year, as he would have been ineligible at nineteen. From King's College he moved to Trinity Hall, Cambridge. On leaving Cambridge, he went to court in the service of William Paget, 1st Baron Paget of Beaudesart, as a musician. After ten years of life at court, he married and settled as a farmer at Cattawade, Suffolk, near the River Stour.

Literary career
There he wrote A Hundreth Good Pointes of Husbandrie, a long poem in rhyming couplets recording the country year. This work was first printed in London in 1557 by the publisher Richard Tottel, and was frequently reprinted. Tottel published an enlarged edition Five Hundreth Pointes of Good Husbandrie in 1573. Tusser includes a homely mix of instructions and observations about farming and country customs which offer insight into life in Tudor England, and his work records many terms and proverbs in print for the first time. In this work, he also famously presents ten characteristics the perfect cheese must have: Not like Gehazi, i.e., dead white, like a leper

Not like Lot's wife, all salt

Not like Argus, full of eyes

Not like Tom Piper, "hoven and puffed"

Not like Crispin, leathery

Not like Lazarus, poor

Not like Esau, hairy

Not like Mary Magdalene, full of whey or maudlin

Not like the Gentiles, full of maggots

Not like a Bishop, made of burnt milk

He never remained long in one place. For the sake of his wife's health he removed to Ipswich. After her death he married again and farmed for some time at West Dereham in Norfolk. He then became a singing man in Norwich Cathedral, where he found a good patron in the Dean, John Salisbury.

Samples
Five Hundred Points contains these rhyming couplets: Swéete April showers,

Doo spring Maie flowers.

as well as

At Christmas play and make good cheere,

for Christmas comes but once a yeere.

and

A foole and his monie be soone at debate,

which after with sorrow repents him too late.

The latter is an early version of the proverb A fool and his money are soon parted.

Later life
After another experiment in farming at Fairstead, Essex, he moved once again to London, whence he was driven by the plague of 1572–1573 to find refuge at Trinity Hall, being matriculated as a servant of the college in 1573. At the time of his death he was in possession of a small estate at Chesterton, near Cambridge, and his will proves that he was not, as has sometimes been stated, in poverty of any kind, but had in some measure the thrift he preached. Thomas Fuller says he "traded at large in oxen, sheep, dairies, grain of all kinds, to no profit"; that he "spread his bread with all sorts of butter, yet none would stick thereon."

Death
Tusser died on 3 May 1580 at the age of about 55. An erroneous inscription at Manningtree, Essex, asserts that he was 65 years old.

According to John Stow's Survey of London, Cheape Ward, Thomas Tusser was buried in the now lost church of St Mildred in the Poultry. The inscription on his tomb there was as follows: "Here Thomas Tusser, clad in earth, doth lie,

That sometime made the pointes of Husbandrie;

By him then learne thou maiest; here learne we must,

When all is done, we sleepe, and turne to dust:

And yet, through Christ, to Heaven we hope to goe;

Who reades his bookes, shall find his faith was so." Stow's editor adds the following epigram on Tusser from a volume called The More the Merrier (1608), by 'H. P.': Ad Tusserum

"Tusser, they tell me, when thou wert alive,

Thou, teaching thrift, thyselfe couldst never thrive.

So, like the whetstone, many men are wont

To sharpen others, when themselves are blunt."

title: Thomas Tusser


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Symmetric difference

In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets $$\{1,2,3\}$$ and $$\{3,4\}$$ is $$\{1,2,4\}$$.

The symmetric difference of the sets A and B is commonly denoted by $$A \operatorname\Delta B$$ (alternatively, $$A \operatorname\vartriangle B$$), $$A \oplus B$$, or $$A \ominus B$$. It can be viewed as a form of addition modulo 2.

The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.

Properties
[[File:Venn 0110 1001.svg|thumb| Venn diagram of $$~(A \Delta B)  \Delta C$$

Venn 0110 0110.svg $~ \Delta~$ Venn 0000 1111.svg $$~=~$$ ]]

The symmetric difference is equivalent to the union of both relative complements, that is:

 * $$A\, \Delta\,B = \left(A \setminus B\right) \cup \left(B \setminus A\right),$$

The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation:

 * $$A\mathbin{ \Delta}B = \{x : (x \in A) \oplus (x \in B)\}.$$

The same fact can be stated as the indicator function (denoted here by $$\chi$$) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: $$\chi_{(A\, \Delta\,B)} = \chi_A \oplus \chi_B$$ or using the Iverson bracket notation $$[x \in A\, \Delta\,B] = [x \in A] \oplus [x \in B]$$.

The symmetric difference can also be expressed as the union of the two sets, minus their intersection:

 * $$A\, \Delta\,B = (A \cup B) \setminus (A \cap B),$$

In particular, $$A \mathbin{ \Delta} B\subseteq A\cup B$$; the equality in this non-strict inclusion occurs if and only if $$A$$ and $$B$$ are disjoint sets. Furthermore, denoting $$D = A \mathbin{ \Delta} B$$ and $$I = A \cap B$$, then $$D$$ and $$I$$ are always disjoint, so $$D$$ and $$I$$ partition $$A \cup B$$. Consequently, assuming intersection and symmetric difference as primitive operations, the union of two sets can be well defined in terms of symmetric difference by the right-hand side of the equality

 * $$A\,\cup\,B = (A\, \Delta\,B)\, \Delta\,(A \cap B)$$.

The symmetric difference is commutative and associative:

 * $$\begin{align}

A\, \Delta\,B &= B\, \Delta\,A, \\ (A\, \Delta\,B)\, \Delta\,C &= A\, \Delta\,(B\, \Delta\,C). \end{align}$$

The empty set is neutral, and every set is its own inverse:
 * $$\begin{align}

A\, \Delta\,\varnothing &= A, \\ A\, \Delta\,A &= \varnothing. \end{align}$$

Thus, the power set of any set X becomes an abelian group under the symmetric difference operation. (More generally, any field of sets forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has order 2) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set. In the case where X has only two elements, the group thus obtained is the Klein four-group.

Equivalently, a Boolean group is an elementary abelian 2-group. Consequently, the group induced by the symmetric difference is in fact a vector space over the field with 2 elements Z2. If X is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of X. This construction is used in graph theory, to define the cycle space of a graph.

From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed. In particular:

 * $$(A\, \Delta\,B)\, \Delta\,(B\, \Delta\,C) = A\, \Delta\,C.$$

This implies triangle inequality: the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C.

Intersection distributes over symmetric difference:
 * $$A \cap (B\, \Delta\,C) = (A \cap B)\, \Delta\,(A \cap C),$$

and this shows that the power set of X becomes a ring, with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a Boolean ring.

Further properties of the symmetric difference include:

 * $$A \mathbin{ \Delta} B = \emptyset$$ if and only if $$A = B$$.
 * $$A \mathbin{ \Delta} B = A^c \mathbin{ \Delta} B^c$$, where $$A^c$$, $$B^c$$ is $$A$$'s complement, $$B$$'s complement, respectively, relative to any (fixed) set that contains both.
 * $$\left(\bigcup_{\alpha\in\mathcal{I}}A_\alpha\right) \Delta\left(\bigcup_{\alpha\in\mathcal{I}}B_\alpha\right)\subseteq\bigcup_{\alpha\in\mathcal{I}}\left(A_\alpha \mathbin{ \Delta} B_\alpha\right)$$, where $$\mathcal{I}$$ is an arbitrary non-empty index set.
 * If $$f : S \rightarrow T$$ is any function and $$A, B \subseteq T$$ are any sets in $$f$$'s codomain, then $$f^{-1}\left(A \mathbin{ \Delta} B\right) = f^{-1}\left(A\right) \mathbin{ \Delta} f^{-1}\left(B\right).$$

The symmetric difference can be defined in any Boolean algebra, by writing
 * $$ x\, \Delta\,y = (x \lor y) \land \lnot(x \land y) = (x \land \lnot y) \lor (y \land \lnot x) = x \oplus y.$$

This operation has the same properties as the symmetric difference of sets.

n-ary symmetric difference
Repeated symmetric difference is in a sense equivalent to an operation on a multitude of sets (possibly with multiple appearances of the same set) giving the set of elements which are in an odd number of sets.

The symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection: $$ \Delta M = \left\{ a \in \bigcup M: \left|\{A \in M:a \in A\}\right| \text{ is odd}\right\}.$$

Evidently, this is well-defined only when each element of the union $\bigcup M$ is contributed by a finite number of elements of $$M$$.

Suppose $$M = \left\{M_1, M_2, \ldots, M_n\right\}$$ is a multiset and $$n \ge 2$$. Then there is a formula for $$| \Delta M|$$, the number of elements in $$ \Delta M$$, given solely in terms of intersections of elements of $$M$$: $$| \Delta M| = \sum_{l=1}^n (-2)^{l-1} \sum_{1 \leq i_1 < i_2 < \ldots < i_l \leq n} \left|M_{i_1} \cap M_{i_2} \cap \ldots \cap M_{i_l}\right|.$$

Symmetric difference on measure spaces
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are.

First consider a finite set S and the counting measure on subsets given by their size. Now consider two subsets of S and set their distance apart as the size of their symmetric difference. This distance is in fact a metric, which makes the power set on S a metric space. If S has n elements, then the distance from the empty set to S is n, and this is the maximum distance for any pair of subsets.

Using the ideas of measure theory, the separation of measurable sets can be defined to be the measure of their symmetric difference. If μ is a σ-finite measure defined on a σ-algebra Σ, the function
 * $$d_\mu(X, Y) = \mu(X\, \Delta\,Y)$$

is a pseudometric on Σ. dμ becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if $$\mu(X\, \Delta\,Y) = 0$$. It is sometimes called Fréchet-Nikodym metric. The resulting metric space is separable if and only if L2(μ) is separable.

If $$\mu(X), \mu(Y) < \infty$$, we have: $$|\mu(X) - \mu(Y)| \leq \mu(X\, \Delta\,Y)$$. Indeed,
 * $$\begin{align}

|\mu(X) - \mu(Y)| &=   \left|\left(\mu\left(X \setminus Y\right) + \mu\left(X \cap Y\right)\right) - \left(\mu\left(X \cap Y\right) + \mu\left(Y \setminus X\right)\right)\right| \\ &=   \left|\mu\left(X \setminus Y\right) - \mu\left(Y \setminus X\right)\right| \\ &\leq \left|\mu\left(X \setminus Y\right)\right| + \left|\mu\left(Y \setminus X\right)\right| \\ &=   \mu\left(X \setminus Y\right) + \mu\left(Y \setminus X\right) \\ &=   \mu\left(\left(X \setminus Y\right) \cup \left(Y \setminus X\right)\right) \\ &=   \mu\left(X\,  \Delta \, Y\right) \end{align}$$

If $$S = \left(\Omega, \mathcal{A},\mu\right)$$ is a measure space and $$F, G \in \mathcal{A}$$ are measurable sets, then their symmetric difference is also measurable: $$F \Delta G \in \mathcal{A}$$. One may define an equivalence relation on measurable sets by letting $$F$$ and $$G$$ be related if $$\mu\left(F \Delta G\right) = 0$$. This relation is denoted $$F = G\left[\mathcal{A}, \mu\right]$$.

Given $$\mathcal{D}, \mathcal{E} \subseteq \mathcal{A}$$, one writes $$\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A}, \mu\right]$$ if to each $$D\in\mathcal{D}$$ there's some $$E \in \mathcal{E}$$ such that $$D = E\left[\mathcal{A}, \mu\right]$$. The relation "$$\subseteq\left[\mathcal{A}, \mu\right]$$" is a partial order on the family of subsets of $$\mathcal{A}$$.

We write $$\mathcal{D} = \mathcal{E}\left[\mathcal{A}, \mu\right]$$ if $$\mathcal{D}\subseteq\mathcal{E}\left[\mathcal{A}, \mu\right]$$ and $$\mathcal{E} \subseteq \mathcal{D}\left[\mathcal{A}, \mu\right]$$. The relation "$$= \left[\mathcal{A}, \mu\right]$$" is an equivalence relationship between the subsets of $$\mathcal{A}$$.

The symmetric closure of $$\mathcal{D}$$ is the collection of all $$\mathcal{A}$$-measurable sets that are $$= \left[\mathcal{A}, \mu\right]$$ to some $$D \in \mathcal{D}$$. The symmetric closure of $$\mathcal{D}$$ contains $$\mathcal{D}$$. If $$\mathcal{D}$$ is a sub-$$\sigma$$-algebra of $$\mathcal{A}$$, so is the symmetric closure of $$\mathcal{D}$$.

$$F = G\left[\mathcal{A}, \mu\right]$$ iff $$\left|\mathbf{1}_F - \mathbf{1}_G\right| = 0$$ $$\left[\mathcal{A}, \mu\right]$$ almost everywhere.

Hausdorff distance vs. symmetric difference
The Hausdorff distance and the (area of the) symmetric difference are both pseudo-metrics on the set of measurable geometric shapes. However, they behave quite differently. The figure at the right shows two sequences of shapes, "Red" and "Red ∪ Green". When the Hausdorff distance between them becomes smaller, the area of the symmetric difference between them becomes larger, and vice versa. By continuing these sequences in both directions, it is possible to get two sequences such that the Hausdorff distance between them converges to 0 and the symmetric distance between them diverges, or vice versa.

title: Symmetric difference


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Talk:Toledo War

Casualties and losses
Is it zero casualties or 22,000? I can find no other source that says 22,000 people were injured. Other than a Michigan sheriff who was stabbed in a tavern scuffle. SiXiam (talk) 01:41, 23 May 2020 (UTC)
 * The current revision of the article is correct at 1 wounded and no deaths. Any other number is vandalism.  Imzadi 1979  →   02:13, 23 May 2020 (UTC)

FA concerns
In order to meet WP:FACR #1c, this older FA should really incorporated Faber's 2008 work on the Toledo War, especially given the degree that this article relies on 19th-century sources. Mary Karl George's 1971 The Rise and Fall of Toledo, Michigan... The Toledo War! should probably be consulted as well. Way is a major source on this topic, but more modern materials should be used when possible. The reliability of captain-johns.com is questionable. An article titled "A Long-Overdue Evaluation of Benjamin Hough's Observation for Latitude of the North Cape of Maumee Bay, Ohio's Northern Border." has been published in Michigan Historical Review; while it isn't directly relevant on much of the article content, it would be useful for fleshing out some of the background surveying that led to the dispute. "Collecting the Toledo War" from the same journal is mainly a bibliography and is not all that useful, but does contain about four or so pages of good secondary material. There's also a need to spot-check some of the sourcing; for instance I'm not finding the April 3 date from "On April 3, 1835, Jackson sent two representatives from Washington, D.C., " in the cited source although most of my spot checks had no issues. Hog Farm Talk 22:38, 1 April 2022 (UTC)

 * No response, listing at WP:FARGIVEN. Sandy Georgia (Talk)  03:42, 5 July 2022 (UTC)

Name, Result, and Bloodshed in 1835 section
The name, "Toledo War," seems to be a bit misleading. No casualties occurred, and one person was non-fatally wounded. "Toledo Conflict" seems to be more of an appropriate name due to the nature of the conflict.

"Result" in the table on the right side should probably say something along the lines of, "Ohio gained the Toledo strip permanently" instead of "inconclusive." The conflict ended with the result of Ohio taking Toledo as its own territory, and I think adding this would help prevent misconstrued information.

Lastly, the "Bloodshed in 1835" section seems to be a bit too beefy. Summarizing the most important and relevant information about the conflict would be less confusing. All of this is just coming from the perspective of a new audience member for this article. Flamingosmoothie (talk) Flamingosmoothie (talk) 21:59, 26 February 2023 (UTC)

 * @Flamingosmoothie: regarding the name, since it is the most common name in sources, and since we follow those sources, that's the name we use. It is not up to us to invent new names when they've already been named.  Imzadi 1979  →   23:59, 26 February 2023 (UTC)

title: Talk:Toledo War


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Harmonic progression

Harmonic progression may refer to:

 * Chord progression in music
 * Harmonic progression (mathematics)
 * Sequence (music)

title: Harmonic progression


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Faro, Yukon

Faro is a town in central Yukon, Canada, the home of the now abandoned Faro Mine. It was the largest open-pit lead–zinc mine in the world as well as a significant producer of silver and other natural resources. The mine was built by the Ralph M. Parsons Construction Company of the United States with General Enterprises Ltd. of Whitehorse being the main subcontractor. As of 2021, the population is 440, down from its peak population of 1,652 in 1981. Faro was named after the card game of the same name.

As these industries have declined over the past decade, Faro is attempting to attract ecotourism to the region to view such animals as Dall and Stone sheep. Both species of sheep are almost unique to the surrounding area. Several viewing platforms have been constructed in and around the town.

One unusual feature of Faro is that it has a golf course running through the main part of town.

History
The area was prospected in the 1950s and 1960s by Al Kulan, credited with discovering several significant deposits of lead and zinc ore and playing a major role in the discovery of the Faro Mine, which became Canada's largest lead-zinc mine. The Cyprus Anvil Mining Corporation established the first operations to mine the deposits, and established the town of Faro. A new highway was built between Carmacks and Ross River to serve the Faro area – initially numbered Highway 9, it is today part of the Robert Campbell Highway, Yukon Highway 4. A forest fire in 1969 destroyed the newly built homes, which had to be re-built.

On 26 October 2021, two people were killed and another critically injured in an active shooter situation in Faro. A suspect, identified as a local resident, was taken into custody by police and charged with two counts each of first-degree murder, attempted murder, and aggravated assault. Flags were lowered to half-mast in Whitehorse and Dawson City in response to the killings.

Demographics
In the 2021 Census of Population conducted by Statistics Canada, Faro had a population of $440$ living in $210$ of its $423$ total private dwellings, a change of NaN% from its 2016 population of $348$. With a land area of 199.89 km2, it had a population density of in 2021.

Faro Mine
"Faro Mine was once the largest open-pit lead-zinc mine in the world. Today, it is the site of one of the most complex abandoned mine remediation projects in Canada. The mine spans 25 km2, an area roughly the size of the city of Victoria, British Columbia."

The mine was opened by the Cyprus Anvil Mining Corporation in 1969, which was formed to mine the deposit. From the late 1960s until 1982 the mine was one of the largest lead-zinc mines in the world. At one time it was the largest open-pit mine in the world. During its history, 320 million tonnes of waste rock was removed to reach the ore. The mine remained in more-or-less constant production until 1982. Trucks carried the ore concentrate from the mill by highway to Whitehorse, where the buckets were lifted from the trucks and lowered onto cars of the White Pass and Yukon Route railway. The trains took the buckets another 170 km to Skagway, Alaska, where the contents were poured out into the holds of ships.

During those years, Cyprus Anvil was purchased by Dome Petroleum. World prices for metals fell in 1982, and the mine owners announced in May a two-month halt to production starting in June, 1982. In July, the mine owners extended the shutdown to four months. In September, the owners announced that the shutdown would be indefinite. The shock to the Yukon economy, where the mine produced approximately 40% of territorial GDP, was enormous. With government assistance an 18-month waste-rock stripping program was operated in 1983 and 1984. Curragh Resources purchased the property in 1985 and production resumed in 1986. This time, ore was trucked in ore pots from Faro directly to Skagway, bypassing the railway. This operation ended in 1993, not long after Curragh Resources suffered a coal mining disaster at the Westray Mine in Plymouth, Pictou County, Nova Scotia. A third operation, by the Anvil Range Mining Corporation, opened in 1995 and ceased production in January 1998, followed by the bankruptcy of Anvil Range. Much of the heavy mining and milling equipment was sold and removed from the Yukon.

Any prospect for further mining of the lead-zinc resource would now require significant investment to bring in mining equipment, and it would need to come entirely by road unless a British Columbia-Alaska railway is built or the White Pass route is reopened to freight traffic to Whitehorse. Cost of clean-up is estimated at close to a billion dollars.

When the mine closed it left 70 million tonnes of tailings and 320 million tonnes of waste rock that require remediation to protect the health of humans, land, water, and wildlife. With the bankruptcy of Anvil Range the clean up bill for the mine and tailings pond fell to the federal government. Early estimates put the cost at $450–590 million over 40 years. In 2016 the federal treasury reported that between $250 million and $350 million had been spent maintaining the mine site, without starting clean-up. The $40 million annual costs are to run pumps to prevent the tailings from breaching the dams and the construction and operations of three water treatment plants. The clean-up itself could cost a further $1-billion.

In a report by the Canadian Broadcasting Corporation in 2017, the Canadian government stated that they were prepared to embark on the remediation process of the Faro Mine. The concerns listed by federal officials include the leaching of toxic metals into the Pelly and Yukon rivers as the acidic rock drainage worsens over time. The remediation work will include rerouting regional creeks to avoid tailings and waste rock areas, treating contaminated water, and avoiding clean water from becoming contaminated.

Infrastructure
The town is served by the Faro Airport and the Robert Campbell Highway passes through just south of Faro.

Climate
Faro has a subarctic climate (Köppen Dfc) with mild summers and severely cold winters. Extreme seasonal swings are common, although less severe than farther east in Canada due to the warm spells in summers and occasional mild spells in winter.

People connected with Faro

 * Lorne Greene, famous for his work in Bonanza and Lorne Greene's New Wilderness, once narrated a film about Faro called A New World in the Yukon.
 * The digital artist Astra Zero grew up in Faro

title: Faro, Yukon


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User talk:Composer333

Hi there. welcome to the wikipedia. If you have any questions, stop by the Village Pump. Note that we don't put writer credits on articles -- the page history takes care of that for us. cheers -- Tarquin 18:42 6 Jul 2003 (UTC)

title: User talk:Composer333


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Public company

A public company is a company whose ownership is organized via shares of stock which are intended to be freely traded on a stock exchange or in over-the-counter markets. A public (publicly traded) company can be listed on a stock exchange (listed company), which facilitates the trade of shares, or not (unlisted public company). In some jurisdictions, public companies over a certain size must be listed on an exchange. In most cases, public companies are private enterprises in the private sector, and "public" emphasizes their reporting and trading on the public markets.

Public companies are formed within the legal systems of particular states, and therefore have associations and formal designations which are distinct and separate in the polity in which they reside. In the United States, for example, a public company is usually a type of corporation (though a corporation need not be a public company), in the United Kingdom it is usually a public limited company (plc), in France a "société anonyme" (SA), and in Germany an Aktiengesellschaft (AG). While the general idea of a public company may be similar, differences are meaningful, and are at the core of international law disputes with regard to industry and trade.

Securities
Usually, the securities of a publicly traded company are owned by many investors while the shares of a privately held company are owned by relatively few shareholders. A company with many shareholders is not necessarily a publicly traded company. Conversely, a publicly traded company typically (but not necessarily) has many shareholders. In the United States, in some instances, companies with over 500 shareholders may be required to report under the Securities Exchange Act of 1934; companies that report under the 1934 Act are generally deemed public companies.

Advantages
Public companies possess some advantages over privately held businesses.

 * Publicly traded companies are able to raise funds and capital through the sale (in the primary or secondary market) of shares of stock. This is the reason publicly traded corporations are important; prior to their existence, it was very difficult to obtain large amounts of capital for private enterprises, as significant capital could only come from a smaller set of wealthy investors or banks willing to risk typically large investments. The profit on stock is gained in form of dividend or capital gain to the holders.
 * The financial media, analysts, and the public are able to access additional information about the business, since the business is commonly legally bound, and naturally motivated (so as to secure further capital), to publicly disseminate information regarding the financial status and future of the company to its many shareholders and the government.
 * Because many people have a vested interest in the company's success, the company may be more popular or recognizable than a private company.
 * The initial shareholders of the company are able to share risk by selling shares to the public. If one were to hold a 100% share of the company, they would have to pay all of the business's debt; however, if an individual were to hold a 50% share, they would only need to pay 50% of the debt. This increases asset liquidity and the company does not need to depend on funding from a bank. For example, in 2013 Facebook founder Mark Zuckerberg owned 29.3% of the company's class A shares, which gave him enough voting power to control the business, while allowing Facebook to raise capital from, and distribute risk to, the remaining shareholders. Facebook was a privately held company prior to its initial public offering in 2012.
 * If some shares are given to managers or other employees, potential conflicts of interest between employees and shareholders (an instance of principal–agent problem) will be remitted. As an example, in many tech companies, entry-level software engineers are given stock in the company upon being hired (thus they become shareholders). Therefore, the engineers have a vested interest in the company succeeding financially, and are incentivized to work harder and more diligently to ensure that success.
 * Public companies have fiduciary duty to their shareholders, in addition to the directors' fiduciary duty to the company itself.

Disadvantages
Many stock exchanges require that publicly traded companies have their accounts regularly audited by outside auditors, and then publish the accounts to their shareholders. Besides the cost, this may make useful information available to competitors. Various other annual and quarterly reports are also required by law. In the United States, the Sarbanes–Oxley Act imposes additional requirements. The requirement for audited books is not imposed by the exchange known as OTC Pink. The shares may be maliciously held by outside shareholders and the original founders or owners may lose benefits and control. The principal–agent problem, or the agency problem is a key weakness of public companies. The separation of a company's ownership and control is especially prevalent in such countries as the UK and the US.

Stockholders
In the United States, the Securities and Exchange Commission requires that firms whose stock is traded publicly report their major shareholders each year. The reports identify all institutional shareholders (primarily, firms owning stock in other companies), all company officials who own shares in their firm, and any individual or institution owning more than 5% of the firm's stock.

General trend
For many years, newly created companies were privately held but held initial public offering to become publicly traded company or to be acquired by another company if they became larger and more profitable or had promising prospects. More infrequently, some companies—such as investment banking firm Goldman Sachs and logistics services provider United Parcel Service (UPS)—chose to remain privately held for a long period of time after maturity into a profitable company.

However, from 1997 to 2012, the number of corporations publicly traded on American stock exchanges dropped 45%. According to one observer (Gerald F. Davis), "public corporations have become less concentrated, less integrated, less interconnected at the top, shorter lived, less remunerative for average investors, and less prevalent since the turn of the 21st century". Davis argues that technological changes such as the decline in price and increasing power, quality and flexibility of computer numerical control machines and newer digitally enabled tools such as 3D printing will lead to smaller and more local organization of production.

Privatization
In corporate privatization, more often called "going private", a group of private investors or another company that is privately held can buy out the shareholders of a public company, taking the company off the public markets. This is typically done through a leveraged buyout and occurs when the buyers believe the securities have been undervalued by investors. In some cases, public companies that are in severe financial distress may also approach a private company or companies to take over ownership and management of the company. One way of doing this would be to make a rights issue designed to enable the new investor to acquire a supermajority. With a super-majority, the company could then be relisted, i.e. privatized.

Alternatively, a publicly traded company may be purchased by one or more other publicly traded companies, with the target company becoming either a subsidiary or joint venture of the purchaser(s), or ceasing to exist as a separate entity, its former shareholders receiving compensation in the form of either cash, shares in the purchasing company or a combination of both. When the compensation is primarily shares then the deal is often considered a merger. Subsidiaries and joint ventures can also be created de novo—this often happens in the financial sector. Subsidiaries and joint ventures of publicly traded companies are not generally considered to be privately held companies (even though they themselves are not publicly traded) and are generally subject to the same reporting requirements as publicly traded companies. Finally, shares in subsidiaries and joint ventures can be (re)-offered to the public at any time—firms that are sold in this manner are called spin-outs.

Most industrialized jurisdictions have enacted laws and regulations that detail the steps that prospective owners (public or private) must undertake if they wish to take over a publicly traded corporation. This often entails the would-be buyer(s) making a formal offer for each share of the company to shareholders.

Trading and valuation
The shares of a publicly traded company are often traded on a stock exchange. The value or "size" of a company is called its market capitalization, a term which is often shortened to "market cap". This is calculated as the number of shares outstanding (as opposed to authorized but not necessarily issued) times the price per share. For example, a company with two million shares outstanding and a price per share of US$40 has a market capitalization of US$80 million. However, a company's market capitalization should not be confused with the fair market value of the company as a whole since the price per share are influenced by other factors such as the volume of shares traded. Low trading volume can cause artificially low prices for securities, due to investors being apprehensive of investing in a company they perceive as possibly lacking liquidity.

For example, if all shareholders were to simultaneously try to sell their shares in the open market, this would immediately create downward pressure on the price for which the share is traded unless there were an equal number of buyers willing to purchase the security at the price the sellers demand. So, sellers would have to either reduce their price or choose not to sell. Thus, the number of trades in a given period of time, commonly referred to as the "volume" is important when determining how well a company's market capitalization reflects true fair market value of the company as a whole. The higher the volume, the more the fair market value of the company is likely to be reflected by its market capitalization.

Another example of the impact of volume on the accuracy of market capitalization is when a company has little or no trading activity and the market price is simply the price at which the most recent trade took place, which could be days or weeks ago. This occurs when there are no buyers willing to purchase the securities at the price being offered by the sellers and there are no sellers willing to sell at the price the buyers are willing to pay. While this is rare when the company is traded on a major stock exchange, it is not uncommon when shares are traded over-the-counter (OTC). Since individual buyers and sellers need to incorporate news about the company into their purchasing decisions, a security with an imbalance of buyers or sellers may not feel the full effect of recent news.

title: Public company


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Secondary chord

A secondary chord is an analytical label for a specific harmonic device that is prevalent in the tonal idiom of Western music beginning in the common practice period: the use of diatonic functions for tonicization.

Secondary chords are a type of altered or borrowed chord, chords that are not part of the music piece's key. They are the most common sort of altered chord in tonal music. Secondary chords are referred to by the function they have and the key or chord in which they function. Conventionally, they are written with the notation "function/key". Thus, the most common secondary chord, the dominant of the dominant, is written "V/V" and read as "five of five" or "the dominant of the dominant". The major or minor triad on any diatonic scale degree may have any secondary function applied to it; secondary functions may even be applied to diminished triads in some special circumstances.

Secondary chords were not used until the Baroque period and are found more frequently and freely in the Classical period, even more so in the Romantic period. Composers began to use them less frequently with the breakdown of conventional harmony in modern classical music—but secondary dominants are a cornerstone of popular music and jazz in the 20th century.

Secondary dominant
A secondary dominant (also applied dominant, artificial dominant, or borrowed dominant) is a major triad or dominant seventh chord built and set to resolve to a scale degree other than the tonic, with the dominant of the dominant (written as V/V or V of V) being the most frequently encountered. The chord that the secondary dominant is the dominant of is said to be a temporarily tonicized chord. The secondary dominant is normally, though not always, followed by the tonicized chord. Tonicizations that last longer than a phrase are generally regarded as modulations to a new key (or new tonic).

According to music theorists David Beach and Ryan C. McClelland, "[t]he purpose of the secondary dominant is to place emphasis on a chord within the diatonic progression." The secondary-dominant terminology is still usually applied even if the chord resolution is nonfunctional. For example, the V/ii label is still used even if the V/ii chord is not followed by ii.

Definition
The major scale contains seven basic chords, which are named with Roman numeral analysis in ascending order. Because tonic triads are either major or minor, one would not expect to find diminished chords (either the vii in major or the ii in minor) tonicized by a secondary dominant. It would also not make sense for the tonic of the key itself to be tonicized.

In the key of C major, the five remaining chords are:

Of these chords, the V chord (G major) is said to be the dominant of C major. However, each of the chords from ii to vi also has its own dominant. For example, V (G major) has a D major triad as its dominant. These extra dominant chords are not part of the key of C major as such because they include notes that are not part of the C major scale. Instead, they are secondary dominants.

The notation below shows the secondary-dominant chords for C major. Each chord is accompanied by its standard number in harmonic notation. In this notation, a secondary dominant is usually labeled with the formula "V of ..." (dominant chord of); thus "V of ii" stands for the dominant of the ii chord, "V of iii" for the dominant of iii, and so on. A shorter notation, used below, is "V/ii", "V/iii", etc.

Like most chords, secondary dominants may be seventh chords or chords with other upper extensions. Dominant seventh chords are commonly used as secondary dominants. The notation below shows the same secondary dominants as above but with dominant seventh chords.

Note that the triad V/IV is the same as the I triad. When a seventh is added (V7/IV), it becomes an altered chord because the seventh is not a diatonic pitch. Beethoven's Symphony No. 1 begins with a V7/IV chord:

According to the principles exposed above, in fact, V7/IV, which means the C7 chord, i.e. the dominant seventh chord on the F major scale (C–E–G–B♭), does not represent the tonic because it contains a B♭, which isn't included in the main key, as Beethoven's Symphony No. 1 is written in the key of C major. The chord then resolves on the natural IV (F major) and in the following bar the V7, i.e. G7 (dominant seventh chord on the C major key), is presented.

Chromatic mediants, for example VI is also a secondary dominant of ii (V/ii) and III is V/vi, are distinguished from secondary dominants with context and analysis revealing the distinction.

History
Before the 20th century, in the music of J.S. Bach, Mozart, Beethoven, and Brahms, a secondary dominant, along with its chord of resolution, was considered a modulation. Since this was a rather self-contradictory description, theorists in the early 1900s, such as Hugo Riemann (who used the term "Zwischendominante"—"intermediary dominant", still the usual German term for a secondary dominant), searched for a better description of the phenomenon.

Walter Piston first used the analysis "V7 of IV" in a monograph entitled Principles of Harmonic Analysis. (Notably, Piston's analytical symbol always used the word "of"—e.g. "V7 of IV" rather than the virgule "V7/IV.) In his 1941 book Harmony, Piston used the term "secondary dominant". At around the same time (1946–48), Arnold Schoenberg created the expression "artificial dominant" to describe the same phenomenon, in his posthumously published book Structural Functions of Harmony.

In the fifth edition of Walter Piston's Harmony, a passage from the last movement of Mozart's Piano Sonata K. 283 in G major serves as one illustration of secondary dominants. This passage has three secondary dominants. The final four chords form a circle of fifths progression, ending in a standard dominant-tonic cadence, which concludes the phrase.

In jazz and popular music

In jazz harmony, a secondary dominant is any dominant seventh chord on a weak beat and resolves downward by a perfect fifth. Thus, a chord is a secondary dominant when it functions as the dominant of some harmonic element other than the key's tonic and resolves to that element. This is slightly different from the traditional use of the term, where a secondary dominant does not have to be a seventh chord, occur on a weak beat, or resolve downward. If a non-diatonic dominant chord is used on a strong beat, it is considered an extended dominant. If it doesn't resolve downward, it may be a borrowed chord.

Secondary dominants are used in jazz harmony in the bebop blues and other blues progression variations, as are substitute dominants and turnarounds. In some jazz tunes, all or almost all of the chords that are used are dominant chords. For example, in the standard jazz chord progression ii–V–I, which would normally be Dm–G7–C in the key of C major, some tunes will use D7–G7–C7. Since jazz tunes are often based on the circle of fifths, this creates long sequences of secondary dominants.

Secondary dominants are also used in popular music. Examples include II7 (V7/V) in Bob Dylan's "Don't Think Twice, It's All Right" and III7 (V7/vi) in Betty Everett's "The Shoop Shoop Song (It's in His Kiss)". "Five Foot Two, Eyes of Blue" features chains of secondary dominants. "Sweet Georgia Brown" opens with V/V/V–V/V–V–I.

Extended dominant
An extended dominant chord is a secondary dominant seventh chord that resolves down by a fifth to another dominant seventh chord. A series of extended dominant chords continues to resolve downwards by the circle of fifths until it reaches the tonic chord. The most common extended dominant chord is the tertiary dominant, which resolves to a secondary dominant. For example, V/V/V (in C major, A(7)) resolves to V/V (D(7)), which resolves to V (G(7)), which resolves to I. Note that V/V/V is the same chord as V/ii, but differs in its resolution to a major dominant rather than a minor chord.

Quaternary dominants are rarer, but an example is the bridge section of the rhythm changes, which starts from V/V/V/V (in C major, E(7)). The example below from Chopin's Polonaises, Op. 26, No. 1 (1835) has a quaternary dominant in the second beat (V/ii = V/V/V, V/vi = V/V/V/V).

Secondary leading-tone
In music theory, a secondary leading-tone chord is a secondary chord that is rooted on a tone that is a leading-tone of (in short, has a strong affinity to resolve to) a tone just 1 semitone from that root (typically 1 semitone above, though can be below). Like the secondary dominant it can be used as tonicization of only one subsequent chord (which will be rooted in the resolution tone), or the music can continue with other chords/notes in the key of that chord's root for a phrase, or even longer to be considered a modulation to that key. This one-semitone-apart resolution of the secondary leading-tone is in contrast to the secondary dominant which resolves through a wider distance of perfect fifth below or perfect fourth above the chord's root (as per the two distances between dominant and tonic). While the root of a secondary leading-tone chord needs to be the leading-tone, the other notes may vary and form with it one of: the triad or one of the diminished sevenths (as in seventh scale degree or leading-tone, not necessarily seventh chord) where the type of the diminished seventh is typically related to the type of tonicized triad:
 * 1) If the tonicized triad is minor, the leading-tone chord is fully diminished seventh chord.
 * 2) If it is major, the leading-tone chord may be either half-diminished or fully diminished, though fully diminished chords are used more often.

Because of their symmetry, secondary leading-tone diminished seventh chords are also useful for modulation; all four notes may be considered the root of any diminished seventh chord. They may resolve to these major or minor diatonic triads:

 * In major keys: ii, iii, IV, V, vi
 * In minor keys: III, iv, V, VI

Especially in four-part writing, the seventh should resolve downwards by step and if possible the lower tritone should resolve appropriately, inwards if a diminished fifth and outwards if an augmented fourth, as the example below shows.

Secondary leading-tone chords were not used until the Baroque period and are found more frequently and less conventionally in the Classical period. They are found even more frequently and freely in the Romantic period, but they began to be used less frequently with the breakdown of conventional harmony.

The chord progression vii$7$/V–V–I is quite common in ragtime music.

Secondary supertonic
The secondary supertonic chord, or secondary second, is a secondary chord that is on the supertonic scale degree. Rather than tonicizing a degree other than the tonic, as does a secondary dominant, it creates a temporary dominant. Examples include ii7/III (F♯min.7, in C major).

Secondary subdominant
The secondary subdominant is the subdominant (IV) of the tonicized chord. For example, in C major, the subdominant chord is F major and the IV of IV chord is B♭ major.

Others
The other secondary functions are the secondary mediant, the secondary submediant, and the secondary subtonic.

title: Secondary chord


text:
Rayleigh distribution

In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribution is named after Lord Rayleigh.

A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions. Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, which is infrequent, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.

Definition
The probability density function of the Rayleigh distribution is

 * $$f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/(2\sigma^2)}, \quad x \geq 0,$$

where $$\sigma$$ is the scale parameter of the distribution. The cumulative distribution function is

 * $$F(x;\sigma) = 1 - e^{-x^2/(2\sigma^2)}$$

for $$x \in [0,\infty).$$

Relation to random vector length
Consider the two-dimensional vector $$ Y = (U,V) $$ which has components that are bivariate normally distributed, centered at zero, with equal variances $$\sigma^2$$, and independent. Then $$U$$ and $$V$$ have density functions

 * $$f_U(x; \sigma) = f_V(x;\sigma) = \frac{e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi\sigma^2}}.$$

Let $$X$$ be the length of $$Y$$. That is, $$X = \sqrt{U^2 + V^2}.$$ Then $$X$$ has cumulative distribution function

 * $$F_X(x; \sigma) = \iint_{D_x} f_U(u;\sigma) f_V(v;\sigma) \,dA,$$

where $$D_x$$ is the disk

 * $$D_x = \left\{(u,v) : \sqrt{u^2 + v^2} \leq x\right\}.$$

Writing the double integral in polar coordinates, it becomes

 * $$F_X(x; \sigma) = \frac{1}{2\pi\sigma^2} \int_0^{2\pi} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr\,d\theta = \frac 1 {\sigma^2} \int_0^x r e^{-r^2/(2\sigma^2)} \,dr.

$$

Finally, the probability density function for $$X$$ is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is

 * $$f_X(x;\sigma) = \frac d {dx} F_X(x;\sigma) = \frac x {\sigma^2} e^{-x^2/(2\sigma^2)},$$

which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2. There are also generalizations when the components have unequal variance or correlations (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).

Suppose $$Y$$ is a random vector with components $$u,v$$ that follows a multivariate t-distribution. If the components both have mean zero, equal variance and are independent, the bivariate Student's-t distribution takes the form:

 * $$f(u,v) = {1\over{2\pi\sigma^{2}}}\left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}$$

Let $$R = \sqrt{U^{2}+V^{2}}$$ be the magnitude of $$Y$$. Then the cumulative distribution function (CDF) of the magnitude is:

 * $$ F(r) = {1\over{2\pi\sigma^{2}}}\iint_{D_{r}} \left( 1 + {u^{2}+v^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}du \; dv $$

where $$D_{r}$$ is the disk defined by:

 * $$ D_{r} = \left\{ (u,v) : \sqrt{u^{2}+v^{2}} \leq r \right\} $$

Converting to polar coordinates leads to the CDF becoming:

 * $$ \begin{aligned} F(r) &= {1\over{2\pi\sigma^{2}}}\int_{0}^{r}\int_{0}^{2\pi} \rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1}d\theta \; d\rho \\ &= {1\over{\sigma^{2}}}\int_{0}^{r}\rho\left( 1 + {\rho^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} d\rho \\ &= 1-\left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2} \end{aligned} $$

Finally, the probability density function (PDF) of the magnitude may be derived:

 * $$ f(r) = F'(r) = {r\over{\sigma^{2}}} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} $$

In the limit as $$ \nu \rightarrow \infty $$, the Rayleigh distribution is recovered because:

 * $$ \lim_{\nu\rightarrow \infty} \left( 1 + {r^{2}\over{\nu \sigma^{2}}} \right)^{-\nu/2-1} = e^{-r^{2}/2\sigma^{2}} $$

Properties
The raw moments are given by:
 * $$\mu_j = \sigma^j2^{j/2}\,\Gamma\left(1 + \frac j 2\right),$$

where $$\Gamma(z)$$ is the gamma function.

The mean of a Rayleigh  random variable  is thus :
 * $$\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253\ \sigma.$$

The standard deviation of a Rayleigh  random variable  is:

 * $$\operatorname{std}(X) = \sqrt{\left (2-\frac{\pi}{2}\right)} \sigma \approx 0.655\ \sigma$$

The variance  of a  Rayleigh  random variable  is :

 * $$\operatorname{var}(X) = \mu_2-\mu_1^2 = \left(2-\frac{\pi}{2}\right) \sigma^2 \approx 0.429\ \sigma^2$$

The mode is $$\sigma,$$ and the maximum pdf is

 * $$ f_{\max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-1/2} \approx \frac{0.606}{\sigma}.$$

The skewness is given by:

 * $$\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^{3/2}} \approx 0.631$$

The excess kurtosis is given by:

 * $$\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245$$

The characteristic function is given by:

 * $$\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\operatorname{erfi}\left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]$$

where $$\operatorname{erfi}(z)$$ is the imaginary error function. The moment generating function is given by

M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\operatorname{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]$$

where $$\operatorname{erf}(z)$$ is the error function.

Differential entropy
The differential entropy is given by

 * $$H = 1 + \ln\left(\frac \sigma {\sqrt{2}}\right) + \frac \gamma 2 $$

where $$\gamma$$ is the Euler–Mascheroni constant.

Parameter estimation
Given a sample of N independent and identically distributed Rayleigh random variables $$x_i$$ with parameter $$\sigma$$,

 * $$\widehat{\sigma^2} = \!\,\frac{1}{2N}\sum_{i=1}^N x_i^2$$ is the maximum likelihood estimate and also is unbiased.

 * $$\widehat{\sigma}\approx \sqrt{\frac 1 {2N} \sum_{i=1}^N x_i^2}$$ is a biased estimator that can be corrected via the formula

 * $$\sigma = \widehat{\sigma} \frac {\Gamma(N)\sqrt{N}} {\Gamma\left(N + \frac 1 2\right)} = \widehat{\sigma} \frac {4^N N!(N-1)!\sqrt{N}} {(2N)!\sqrt{\pi}}$$ $$= \frac{\hat{\sigma}}{c_4(2N+1)}$$, where c4 is the correction factor used to unbias estimates of standard deviation for normal random variables.

Confidence intervals
To find the (1 − α) confidence interval, first find the bounds $$[a,b]$$ where:
 * $$P\left(\chi_{2N}^2 \leq a\right) = \alpha/2, \quad P\left(\chi_{2N}^2 \leq b\right) = 1 - \alpha/2$$

then the scale parameter will fall within the bounds
 * $$\frac{{N}\overline{x^2}}{b} \leq {\widehat{\sigma^2}} \leq \frac{{N}\overline{x^2}}{a}$$

Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

 * $$X=\sigma\sqrt{-2 \ln U}\,$$

has a Rayleigh distribution with parameter $$\sigma$$. This is obtained by applying the inverse transform sampling-method.

Related distributions

 * $$R \sim \mathrm{Rayleigh}(\sigma)$$ is Rayleigh distributed if $$R = \sqrt{X^2 + Y^2}$$, where $$X \sim N(0, \sigma^2)$$ and $$Y \sim N(0, \sigma^2)$$ are independent normal random variables. This gives motivation to the use of the symbol $$\sigma$$ in the above parametrization of the Rayleigh density.
 * The magnitude $$|z|$$ of a standard complex normally distributed variable z is Rayleigh distributed.
 * The chi distribution with v = 2 is equivalent to the Rayleigh Distribution with σ = 1: $$R(\sigma) \sim \sigma\chi_2^{\,}\ .$$
 * If $$R \sim \mathrm{Rayleigh} (1)$$, then $$R^2$$ has a chi-squared distribution with 2 degrees of freedom: $$[Q=R(\sigma)^2] \sim \sigma^2\chi_2^2\ .$$
 * If $$R \sim \mathrm{Rayleigh}(\sigma)$$, then $$\sum_{i=1}^N R_i^2$$ has a gamma distribution with parameters $$N$$ and $$\frac{1}{2\sigma^2}$$
 * $$\left[Y=\sum_{i=1}^N R_i^2\right] \sim \Gamma\left(N,\frac{1}{2\sigma^2}\right) .$$
 * The Rice distribution is a noncentral generalization of the Rayleigh distribution: $$ \mathrm{Rayleigh}(\sigma) = \mathrm{Rice}(0,\sigma) $$.
 * The Weibull distribution with the shape parameter k = 2 yields a Rayleigh distribution. Then the Rayleigh distribution parameter $$\sigma$$ is related to the Weibull scale parameter according to $$\lambda = \sigma \sqrt{2} .$$
 * If $$X$$ has an exponential distribution $$X \sim \mathrm{Exponential}(\lambda)$$, then $$Y=\sqrt{X} \sim \mathrm{Rayleigh}(1/\sqrt{2\lambda}) .$$
 * The half-normal distribution is the one-dimensional equivalent of the Rayleigh distribution.
 * The Maxwell–Boltzmann distribution is the three-dimensional equivalent of the Rayleigh distribution.

Applications
An application of the estimation of σ can be found in magnetic resonance imaging (MRI). As MRI images are recorded as complex images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.

The Rayleigh distribution was also employed in the field of nutrition for linking dietary nutrient levels and human and animal responses. In this way, the parameter σ may be used to calculate nutrient response relationship.

In the field of ballistics, the Rayleigh distribution is used for calculating the circular error probable—a measure of a gun's precision.

In physical oceanography, the distribution of significant wave height approximately follows a Rayleigh distribution.

title: Rayleigh distribution


text:
Nonchord tone

A nonchord tone (NCT), nonharmonic tone, or embellishing tone is a note in a piece of music or song that is not part of the implied or expressed chord set out by the harmonic framework. In contrast, a chord tone is a note that is a part of the functional chord. Non-chord tones are most often discussed in the context of the common practice period of classical music, but they can be used in the analysis of other types of tonal music as well, such as Western popular music.

Nonchord tones are often categorized as accented non-chord tones and unaccented non-chord tones depending on whether the dissonance occurs on an accented or unaccented beat (or part of a beat).

Over time, some musical styles assimilated chord types outside of the common-practice style. In these chords, tones that might normally be considered nonchord tones are viewed as chord tones, such as the seventh of a minor seventh chord. For example, in 1940s-era bebop jazz, an F♯ played with a C chord would be considered a chord tone if the chord were analyzed as C7(♯11). In European classical music, "[t]he greater use of dissonance from period to period as a result of the dialectic of linear/vertical forces led to gradual normalization of ninth, eleventh, and thirteenth chords [in analysis and theory]; each additional non-chord tone above the foundational triad became frozen into the chordal mass."

Theory
Chord and nonchord tones are defined by their membership (or lack of membership) in a chord: "The pitches which make up a chord are called chord-tones: any other pitches are called non-chord-tones." They are also defined by the time at which they sound: "Nonharmonic tones are pitches that sound along with a chord but are not chord pitches." For example, if an excerpt from a piece of music implies or uses a C-major chord, then the notes C, E and G are members of that chord, while any other note played at that time (e.g., notes such as F♯) is a nonchord tone. Such tones are most obvious in homophonic music but occur at least as frequently in contrapuntal music.

According to Music in Theory and Practice, "Most nonharmonic tones are dissonant and create intervals of a second, fourth or seventh", which are required to resolve to a chord tone in conventional ways. If the note fails to resolve until the next change of harmony, it may instead create a seventh chord or extended chord. While theoretically in a three-note chord, there are nine possible nonchord tones in equal temperament, in practice nonchord tones are usually in the prevailing key. Augmented and diminished intervals are also considered dissonant, and all nonharmonic tones are measured from the bass note, or lowest note sounding in the chord except in the case of nonharmonic bass tones.

Nonharmonic tones generally occur in a pattern of three pitches, of which the nonharmonic tone is the center:
 * {| style="text-align: center;"

 * Chord tone || – || Nonchord tone || – || Chord tone
 * Preparation|| – ||Dissonance|| – ||Resolution
 * }
 * Preparation|| – ||Dissonance|| – ||Resolution
 * }
 * }

Nonchord tones are categorized by how they are used. The most important distinction is whether they occur on a strong or weak beat and are thus either accented or unaccented nonchord tones. They are also distinguished by their direction of approach and departure and the voice or voices in which they occur and the number of notes they contain.

Anticipation
An anticipation (ANT) occurs when this note is approached by step and then remains the same. It is basically a note of the second chord played early. In the example below, the dissonant B in bar 1 is approached by step and resolves when that same pitch becomes a chord tone in bar 2.

A portamento is the late Renaissance precursor to the anticipation, though today it refers to a glissando.

Neighbor tone
A neighbor tone (NT) or auxiliary note (AUX) is a nonchord tone that passes stepwise from a chord tone directly above or below it (which frequently causes the NT to create dissonance with the chord) and resolves to the same chord tone:

In practice and analysis, neighboring tones are sometimes differentiated depending upon whether or not they are lower or higher than the chord tones surrounding them. A neighboring tone that is a step higher than the surrounding chord tones is called an upper neighboring tone or an upper auxiliary note while a neighboring tone that is a step lower than the surrounding chord tones is a lower neighboring tone or lower auxiliary note. However, following Heinrich Schenker's usage in Free Composition, some authors reserve the term "neighbor note" to the lower neighbor a half step below the main note.

The German term Nebennote is a somewhat broader category, including all nonchord tones approached from the main note by step.

Escape tone
An escape tone (ET) or echappée is a particular type of unaccented incomplete neighbor tone that is approached stepwise from a chord tone and resolved by a skip in the opposite direction back to the harmony.

Passing tone
A passing tone (PT) or passing note is a nonchord tone prepared by a chord tone a step above or below it and resolved by continuing in the same direction stepwise to the next chord tone (which is either part of the same chord or of the next chord in the harmonic progression).

Where two nonchord tones are before the resolution they are double passing tones or double passing notes.

Passing tone
A tone that sits between two chord tones and is between them.

Neighbor tone
A neighbour tone is where there is a step up or down from a note (or chord tone) and then move back to the original note.

Suspension and retardation
"Endeavor, moreover, to introduce suspensions now in this voice, now in that, for it is incredible how much grace the melody acquires by this means. And every note which has a special function is rendered audible thereby."

- Johann Joseph Fux (1725) A suspension (SUS) (sometimes referred to as a syncope ) occurs when the harmony shifts from one chord to another, but one or more notes of the first chord (the preparation) are either temporarily held over into or are played again against the second chord (against which they are nonchord tones called the suspension) before resolving downwards to a chord tone by step (the resolution). The whole process is called a suspension as well as the specific nonchord tone(s).

Suspensions may be further described with two numbers: (1) the interval between the suspended note and the bass note and (2) the interval between the resolution and the bass note. The most common suspensions are 4–3 suspension, 7–6 suspension, or 9–8 suspension. Note that except for the 9–8 suspensions, the numbers are typically referred to using the simple intervals, so for instance, if the intervals are actually an 11th and a 10th (the first example below), one would typically call it a 4–3 suspension. If the bass note is suspended, then the interval is calculated between the bass and the part that is most dissonant with it, often resulting in a 2–3 suspension.

Suspensions must resolve downwards. If a tied note is prepared like a suspension but resolves upwards, it is called a retardation. Common retardations include 2–3 and 7–8 retardations.

Decorated suspensions are common and consist of portamentos or double eighth notes, the second being a lower neighbor tone.

A chain of suspensions constitutes the fourth species of counterpoint; an example may be found in the second movement of Arcangelo Corelli's Christmas Concerto.

Appoggiatura
An appoggiatura (APP) is a type of accented incomplete neighbor tone approached skip-wise from one chord tone and resolved stepwise to another chord tone ("overshooting" the chord tone).

Nonharmonic bass
Nonharmonic bass notes are bass notes that are not a member of the chord below which they are written. Examples include the Elektra chord. An example of a nonharmonic bass from the third movement of Stravinsky's Symphony of Psalms.

Changing tones
Changing tones (CT) are two successive nonharmonic tones. A chord tone steps to a nonchord tone which skips to another nonchord tone which leads by step to a chord tone, often the same chord tone. They may imply neighboring tones with a missing or implied note in the middle. Also called double neighboring tones or neighbor group.

Pedal point
Another form of nonchord tone is a pedal point or pedal tone (PD) or note, almost always the tonic or dominant, which is held through a series of chord changes. The pedal point is almost always in the lowest voice (the term originates from organ playing), but it may be in an upper voice; then it may be called an inverted pedal. It may also be between the upper and lower voices, in which case it is called an internal pedal.

Chromatic nonharmonic tone
A chromatic nonharmonic tone is a nonharmonic tone that is chromatic, or outside of the key and creates half-step motion. The use of which, especially chromatic appoggiaturas and chromatic passing tones, increased in the Romantic Period. The example below shows chromatic nonharmonic tones (in red) in the first four measures of Frédéric Chopin's Prelude No. 21, op. 28.

title: Nonchord tone


text:
United States congressional delegations from Minnesota

These are tables of congressional delegations from Minnesota to the United States House of Representatives and the United States Senate.

The current dean of the Minnesota delegation is Representative Betty McCollum (MN-4), having served in the House since 2001.

Current members
The House delegation has 8 members, including 4 Democrats and 4 Republicans.