title: Young Americans

text: During this time, former Beatle John Lennon was working at the Record Plant on his 1975 covers album "Rock 'n' Roll". Lennon, who was in his famous "" period, had previously met Bowie in Los Angeles at a party hosted by actress Elizabeth Taylor in September 1974. The two connected and decided to record together. With Alomar, the two convened at Electric Lady Studios in New York in January 1975 and recorded "Fame" and a cover of Lennon's Beatles song "Across the Universe". In Visconti's absence, the session was co-produced by Bowie and Maslin. Alongside Alomar, Bowie invited guitarist Earl Slick and drummer Dennis Davis, making their debuts on a Bowie record, as well as bassist Emir Ksasan from the Soul tour band. Newcomers were percussionist Ralph MacDonald and backing vocalists Jean Fineberg and Jean Millington.


title: Young Americans

text: Mixing for "Young Americans" was completed at the Record Plant on 12January 1975. Bowie contacted Visconti about the collaborations with Lennon two weeks later. According to Pegg, Bowie was apologetic and asked if two tracks could be replaced by "Across the Universe" and "Fame"; the tracks replaced were "Who Can I Be Now?" and "It's Gonna Be Me". Commenting about the replacement, Visconti said: "Beautiful songs and it made me sick when he decided not to use them. I think it was the personal content of the songs which he was a bit reluctant to release, although it was so obscure I don't think even I knew what he was on about in them!"


title: Young Americans

text: "Young Americans" presented a new musical direction for Bowie. Although songs on "Diamond Dogs", including "Rock 'n' Roll with Me" and "1984", exhibited a funk and soul direction, Stephen Thomas Erlewine of AllMusic wrote that the blue-eyed soul showcased on "Young Americans" "came as a shock". Along with blue-eyed soul, the music has been described as R&B and Philadelphia soul, while Bowie himself labeled the sound of the album as "plastic soul", describing it as "the squashed remains of ethnic music as it survives in the age of Muzak rock, written and sung by a white limey". Ashley Naftule of "Consequence of Sound" described the album as "a blue-eyed soul album that plays matchmaker between Bowie's artsy rocker tendencies and the warm earnestness of soul and R&B." Biographer Christopher Sandford writes that the album is "a record of high spirits and lively, colliding ideas". Biographer Marc Spitz felt that the album doesn't showcase "Bowie does black music", but rather "Bowie and black music do each other".


title: Young Americans

text: The opening track is the title track, which Bowie said was "just [about] young Americans", more specifically "a newly-wed couple who don't know if they really like each other. Well, they do, but they don't know if they do or don't." The song also presents new lyrical directions for the artist: instead of "shady" characters living in worlds fraught with darkness, "Young Americans" shows typical American teenagers. References are made to the Watergate scandal and McCarthyism, while the line "I heard the news today, oh boy" is from the Beatles' song "A Day in the Life", acknowledging Lennon's influence on Bowie and their imminent collaboration later on in the album. Author Peter Doggett writes that the song introduced the world to an entirely new Bowie, catching everyone by surprise. Bowie wrote "Win" about people who "don't work very hard". According to author James Perone, the lyrics are more abstract than the previous track and thus are open to interpretation. Saxophones and strings feature throughout, while the backing vocalists are more relaxed and in line with Bowie's lead. O'Leary cites "Win" as the track on "Young Americans" that mostly foreshadows Bowie's direction on his next album "Station to Station" (1976). Buckley calls the track the album's standout and "one of the most gorgeous melodies Bowie has ever written."


title: Young Americans

text: "Fascination" evolved out of a Vandross track titled "Funky Music (Is a Part of Me)", which Bowie added new lyrics to. Bowie kept most of Vandross' structure but changed the interplay of the backing vocalists. Doggett cites elements in the novels "City of Night" (1963) and "The Occult Reich" (1974) as inspirations for the title, while Buckley writes that it reaffirms the 'strange fascination' motif of Bowie's 1971 track "Changes". "Right" is the only track on "Young Americans" to feature Bowie's old friend Geoff MacCormack. The call and response between Bowie and the backing singers "lends an air of immaculate sophistication to the lyric's paean to positive thinking", according to Pegg. In 1975, Bowie called the song a mantra: "People forget what the sound of Man's instinct is—it's a drone, a mantra. And people say, 'Why are so many things popular that just drone on and on?' But that's the point really. It reaches a particular vibration, not necessarily a musical level." Perone states that the track is a "classic example" of a "groove piece". Bowie, Vandross, Clark and Cherry are seen rehearsing the song in the BBC documentary "Cracked Actor".


title: Young Americans

text: Originally developed from "I Am Divine", the title of "Somebody Up There Likes Me" was taken from the 1956 film of the same name starring Paul Newman. Similar to "Right", it uses a call-and-response structure and is embellished in strings, saxophone and synthesisers which hide its rather dark lyrics. Pegg states that the lyrics discuss the idea of celebrity and the "hollowness of fame and adulation". Bowie himself described the song as a "Watch out mate, Hitler's on your back" warning. Buckley finds the lyrics inauthentic as a soul song, comparing them to his 1969 track "Cygnet Committee". Bowie's rendition of "Across the Universe" is a blue-eyed soul reworking that features Lennon on guitar and backing vocals. Bowie had previously called the Beatles' original version "very watery" and wanted to "hammer the hell out of it". His cover has been maligned by critics and biographers, and is often considered as one of Bowie's low points in what Pegg calls his "golden years". Nevertheless, Perone finds that as a "groove piece", the cover succeeds.


title: Young Americans

text: After Lulu's version of "Take It in Right" failed to come to fruition, Bowie rerecorded it under the title "Can You Hear Me?". O'Leary describes it as contemporary R&B, while Doggett believes its style is more reminiscent of southern music rather than Philly soul. Pegg praises Visconti's string arrangement as one of the album's highlights, further noting that it foreshadows the "majestic ballad style" of "Station to Station". Perone likens Bowie's vocal performance to singer Al Green. In 1975, Bowie stated that the song was "written for somebody" but declined to disclose who; his biographers agree that it was most likely for Cherry.


title: Young Americans

text: "Fame" was co-written by Bowie, Alomar and Lennon. The lyrics represent Bowie and Lennon's dissatisfaction with the troubles of fame and stardom. Alomar originally developed the guitar riff for Bowie's cover of "Footstompin'" by the Flairs, which Bowie then used to create "Fame". Lennon's voice is heard interjecting the falsetto "Fame" throughout the song. Sources differ as to the extent of Lennon's contributions. Although Doggett argues that Lennon only made the "briefest lyrical contributions", Bowie would later say that Lennon was the "energy" and the "inspiration" for "Fame", and that's why he received a co-writing credit. Lennon would later contradict this story in a 1980 interview where he said: "We took some Stevie Wonder middle eight and did it backwards, you know, and we made a record out of it!"


title: Young Americans

text: "Who Can I Be Now?" reflects the theme of self-identity. Over its runtime, it builds to a what Pegg calls a "gospel-choir climax". O'Leary writes that Bowie would revisit a similar theme of Gnostic teachings on "Station to Station". Doggett writes that its title summarises Bowie's career up to this point, sharing a similar theme as "Changes". "It's Gonna Be Me" is a ballad similar in style to Aretha Franklin. Originally titled "Come Back My Baby", it is lyrically similar to "Can You Hear Me?", in that it follows a casual seducer who realises the error of his ways and works to redeem himself. Biographers have generally praised the track as one of the most overlooked gems of Bowie's entire career. "John, I'm Only Dancing (Again)" is a radical seven-minute funk and disco reworking of Bowie's 1972 glam rock single "John, I'm Only Dancing". Although it mostly retains the original song's chorus, Bowie wrote new verses and an entirely different melody. Biographers have generally praised Bowie's vocal performance. It was released as a single in 1979.


title: Young Americans

text: For the album cover artwork, Bowie initially wanted to commission Norman Rockwell to create a painting but retracted the offer when he heard that Rockwell would need at least six months to do the job. According to Pegg, another rejected idea was a full-length portrait of Bowie in a "flying suit" and white scarf, standing in front of an American flag and raising a glass. The final cover photo, a back-lit and airbrushed photo of Bowie, was taken in Los Angeles on 30August 1974 by photographer Eric Stephen Jacobs. Using that photo, Craig DeCamps designed the final cover at RCA Records' New York City office. Sandford calls it one of the "classic" album covers.


title: Young Americans

text: After recording much of the album's material in August 1974, Bowie was eager to perform his new work live. Embarking on the second half of the Diamond Dogs tour, lasting 2September to 1December 1974, this portion has been given the nickname the Soul Tour, due to the influence of the new material. Because of this, the shows were heavily altered, no longer featuring elaborate set-pieces, partly due to Bowie's exhaustion with the design and wanting to explore the new sound he was creating. Songs from the previous leg were dropped, while new ones, including some from the new album, were added. During this time, a documentary was filmed that depicts Bowie on the Diamond Dogs tour in Los Angeles, using a mixture of sequences filmed in limousines, hotels and concert footage, most of which was taken from a show at the Los Angeles Universal Amphitheatre on 2September. Directed by Alan Yentob and broadcast on BBC1 in the UK on 26January 1975, "Cracked Actor" is notable for being a primary source of footage of the Diamond Dogs tour, while also showing Bowie's decaying mental state during this period due to his increasing cocaine addiction. Although "Cracked Actor" has never received an official release, Pegg calls it "arguably the finest documentary made about David Bowie". After seeing an advanced screening of the film, director Nicolas Roeg immediately contacted Bowie to discuss a role in "The Man Who Fell to Earth" (1976).


title: Young Americans

text: On 29October 1974, Bowie appeared on "The Dick Cavett Show" and performed "1984", "Young Americans" and a version of "Footstompin'". During his interview, he was visibly drugged, barely being able to talk and nose sniffing constantly. RCA later released the title track as the lead single on 21February 1975, backed by the "Ziggy Stardust" track "Suffragette City". In the US, it was released in edited form, with a length of 3:11, omitting two verses and a chorus. It managed to reach number 18 on the UK Singles Chart while it charted at number 28 on the "Billboard" Hot 100, his second top 40 entry and second-highest chart peak in the US up to that point.


title: Young Americans

text: RCA rush-released "Young Americans" on 7March 1975, with the catalogue number RS 1006. It reached number nine on the US "Billboard" Top LPs & Tape chart and remained on the chart for 51 weeks. It stayed on the UK Albums Chart for 17 weeks, peaking at number two, being kept off the top spot by Tom Jones's "20 Greatest Hits". However, sales were overall lower than "Diamond Dogs" according to Buckley. The second single "Fame" was released on 25July, with album track "Right" as the B-side. Although it only reached number 17 in the UK, "Fame" topped the charts in the US. Its chart success was a surprise to Bowie, who recalled in 1990: "Even though [Lennon] had contributed to it and everything, and I had no idea, as with 'Let's Dance', that that was what a commercial single is. I haven't got a clue when it comes to singles. I just don't know about them, I don't get it, and 'Fame' was really out of left-field for me." He appeared on ABC TV's "Soul Train" in early November 1975, where he gave a mimed performance of "Fame" and his most recent single "Golden Years"; he was one of the first white artists to appear on the programme. He then sang "Fame" and "Can You Hear Me?" live on CBS's "The Cher Show" on 23November.


title: Young Americans

text: "Young Americans" was released to a generally favorable reception, particularly in America. "Billboard" wrote that the album "should not only endear Bowie even more to his current fans but should open up an entirely new avenue of fans for him", and selected it among their "Top Album Picks" for the week of 15 March 1975. It was further described by "Record World" as his "most compelling album to date", while "Cashbox" called the artist "the brightest star in the pop music constellation with this latest RCA release". Amongst mixed reviews, some enjoyed certain tracks but disregarded the collection as a whole. "Rolling Stone"s Jon Landau praised the title-track and thought that "the rest of the album works best when Bowie combines his renewed interest in soul with his knowledge of English pop, rather than opting entirely for one or the other." "Record Mirror" described the sound as "spasmodic, awkward, frustrating" and having "a joyless energy".


title: Young Americans

text: Several critics were negative. Reviewing for "The Village Voice", Robert Christgau described the record "an almost total failure" and said "although the amalgam of rock and Philly soul is so thin it's interesting, it overwhelms David's voice, which is even thinner." He nonetheless appreciated Bowie's renewed "generosity of spirit to risk failure" following "Diamond Dogs" and "David Live", which Christgau had found disappointing. Writing for "Phonograph Record", John Mendelsohn felt the album was overall very weak, finding the melodies "as good as non-existent", and further criticising the lyrics and Bowie's vocal performance. In the "NME", Ian MacDonald felt the record was more of a transitional one, created out of a confused state of mind not knowing where to take his career next. Nevertheless, he greatly enjoyed it despite its flaws. In "Melody Maker", Michael Watts praised the backing band, the title track and "Fame", but found the record too "pastiche" to be credible and Bowie's worst release up to that point.


title: Young Americans

text: Retrospectively, "Young Americans" continues to receive mixed reviews from critics and fans. Erlewine felt the album was affected by a lack of strong songwriting. Although he praised the title track and "Fame", he concludes ""Young Americans" is more enjoyable as a stylistic adventure than as a substantive record." Douglas Wolk regarded it as "distinctly a transitional record" in "Pitchfork", stating: "It doesn't have the mad theatrical scope of "Diamond Dogs" or the formal audacity of "Station to Station"; at times, it comes off as an artist trying very hard to demonstrate how unpredictable he is." Nevertheless, Wolk also praised the fact that "while there had already been a handful of disco hits on the pop charts, no other established rock musician had yet tried to do anything similar." "Ultimate Classic Rock"s Jeff Giles gave the album a positive review, saying "it remains a beloved bright spot in a discography with more than its share".


title: Young Americans

text: As a whole, Perone finds that "Young Americans" lacks the musical cohesiveness of its predecessors and "Station to Station", while some of the tracks, including "Across the Universe", "Can You Hear Me" and "Fame", act better as standalone tracks. Nevertheless, he commends the record's musical accessibility and concludes: "[The album] would better balance [Bowie's] fascination of more exotic characters with his observations about more widely universal characters better than any of his pre-"Young Americans" albums." Buckley considers "Young Americans" to be one of Bowie's most influential records and writes that it brought fans of both glam rock and soul together in the wake of the disco era. Pegg similarly states that "By jumping on the Stax/George McCrae bandwagon" with the album, "Bowie had undertaken the first significant excursion into Black Soul by a mainstream white artist," and paved the way for other artists to engage in similar styles. Sandford adds that while many British rockers have tried and failed to experiment with black musical styles, Bowie was one of the first to succeed. In the UK, artists who would follow in Bowie's footsteps were Elton John (with his single "Philadelphia Freedom"), Roxy Music (with their single "Love Is the Drug") and Rod Stewart (with his album "Atlantic Crossing"). Pegg writes that Bowie's foray into soul and funk would influence numerous bands in ensuing years, including Talking Heads, Spandau Ballet, Japan and ABC.


title: Young Americans

text: Spitz writes that "Young Americans" was also Bowie's first album in three years to not feature Ziggy Stardust, but Bowie himself. By not featuring Ziggy, Bowie showcased maturity, which Sandford believes was his ticket into the US market. Indeed, the album turned Bowie from "a mildly unsavoury cult artist to a chat-show friendly showbiz personality" in the US. Biographer Paul Trynka states that although the album as a whole is inconsistent, it nevertheless at the time restored Bowie's momentum after "David Live", and the "impressionist working methods" used in its making would "underpin Bowie's career through the rest of the decade". In 2016, Joe Lynch of "Billboard" argued that "Fame" and "Young Americans" as a whole served as an influence not only on other funk artists (naming George Clinton and Parliament-Funkadelic's song "Give Up the Funk (Tear the Roof off the Sucker)"), but also early hip hop artists and the West Coast G-funk genre of the early 1990s.


title: Young Americans

text: Bowie himself expressed mixed statements about "Young Americans" throughout his lifetime. In late 1975 he described it as "the phoniest R&B I've ever heard. If I ever would have got my hands on that record when I was growing up I would have cracked it over my knee." He would further voice his dislike for the record and describe it as "a phase" in a 1976 interview with "Melody Maker". Bowie would later reverse his stance in the 1990s, speaking to "Q" magazine in 1990: "I shouldn't have been quite so hard on myself, because looking back it was pretty good white, blue-eyed soul."


title: Young Americans

text: Despite the overall mixed reception, "Young Americans" was voted Bowie's ninth best album in a 2013 readers' poll for "Rolling Stone". The magazine argued that its style shift helped introduce Bowie to a wider audience. That same year, "NME" ranked the album at number 175 in its list of the 500 Greatest Albums of All Time. The album was also included in the 2018 edition of Robert Dimery's book "1001 Albums You Must Hear Before You Die". The aggregate website Acclaimed Music lists "Young Americans" as the 19th most acclaimed album of 1975, the 234th most acclaimed album of the 1970s and the 920th most acclaimed album in history.


title: Young Americans

text: The album was originally released on CD by RCA in 1984, and then by Rykodisc/EMI in 1991, with three bonus tracks. This reissue charted at number 54 on the UK Albums Chart for one week in April 1991. A 1999 rerelease by EMI featured 24-bit digitally remastered sound and no extra tracks. The 2007 reissue, marketed as a "Special Edition," included an accompanying DVD, containing 5.1 surround sound mixes of the album and video footage from "The Dick Cavett Show". In 2016, the album was remastered for the "Who Can I Be Now? (1974–1976)" box set, which also includes an earlier, rawer-sounding draft of the album, titled "The Gouster". It was released in CD, vinyl, and digital formats, both as part of this compilation and separately.


title: Young Americans

text: The 1991 and 2007 reissues featured, as bonus tracks, "Who Can I Be Now?", "John, I'm Only Dancing (Again)", and "It's Gonna Be Me"; the latter was released in an alternate version with strings on the 2007 edition. The 1991 reissue replaced the original versions of "Win", "Fascination" and "Right" with alternate mixes, but later reissues restored the original mixes. Another outtake, "After Today", appeared on the 1989 box set "Sound + Vision", as did the alternate mix of "Fascination".


title: Charlbi Dean

text: Charlbi Dean Kriek ( ; 5 February 1990 – 29 August 2022) was a South African actress and model. She was best known for her roles in the "Spud" films (2010–2013), the superhero drama series "Black Lightning" (2018) and the Palme d'Or winner "Triangle of Sadness" (2022).


title: Charlbi Dean

text: Dean was born on 5 February 1990 to Joanne Muller and Johan Kriek. She had a brother. She began modelling at the age of 6, appearing in commercials and catalogues. She signed with Alfa Model Management when she was 12 and was homeschooled from the age of 14. She attended Waterfront Theatre School in her hometown of Cape Town, traveling widely for her career.


title: Charlbi Dean

text: In October 2008, Dean and fellow model Ashton Schnehage survived a car crash. Her injuries included a broken wrist, four broken ribs, and a collapsed lung. She was hospitalised at Milnerton Medi-Clinic and received life-saving surgery. She took a break from her career after the accident.


title: Charlbi Dean

text: In 2010, Dean made her acting debut in the film adaptation of "Spud" as Amanda, a role she reprised in the sequel "". She went on to star in the films "Don't Sleep" in 2017 and "An Interview with God" in 2018. Also in 2018, she landed the role of Syonide, a recurring character she played for two seasons of the Arrowverse series "Black Lightning". In February 2020, she was cast in a leading role in Ruben Östlund's satirical film "Triangle of Sadness", which premiered at the 2022 Cannes Film Festival and won the Palme d'Or.


title: Charlbi Dean

text: At the time of her death, Dean was engaged to fellow South African actor and model Luke Volker, who she had been with since 2018.


title: Charlbi Dean

text: On 29 August 2022, Dean was admitted to a hospital in New York City after feeling unwell. While her initial symptoms were mild, her condition deteriorated rapidly and she died several hours later; she was 32. Pending autopsy results, a cause of death was not announced, although it was found that she had a viral infection in her lungs. Her spleen had been removed after her 2008 car crash, which increases the risk of serious infections.


title: Charlbi Dean

text: Dean's death occurred shortly before the international release of "Triangle of Sadness". Peter Bradshaw of "The Guardian" wrote that she "was a true star-in-the-making. Her loss is a huge one... [she] had a singular style and enormous promise."


title: Weather forecasting

text: Weather forecasting is the application of science and technology to predict the conditions of the atmosphere for a given location and time. People have attempted to predict the weather informally for millennia and formally since the 19th century. Weather forecasts are made by collecting quantitative data about the current state of the atmosphere, land, and ocean and using meteorology to project how the atmosphere will change at a given place.


title: Weather forecasting

text: Once calculated manually based mainly upon changes in barometric pressure, current weather conditions, and sky condition or cloud cover, weather forecasting now relies on computer-based models that take many atmospheric factors into account. Human input is still required to pick the best possible forecast model to base the forecast upon, which involves pattern recognition skills, teleconnections, knowledge of model performance, and knowledge of model biases. The inaccuracy of forecasting is due to the chaotic nature of the atmosphere, the massive computational power required to solve the equations that describe the atmosphere, the land, and the ocean, the error involved in measuring the initial conditions, and an incomplete understanding of atmospheric and related processes. Hence, forecasts become less accurate as the difference between current time and the time for which the forecast is being made (the "range" of the forecast) increases. The use of ensembles and model consensus help narrow the error and provide confidence level in the forecast.


title: Weather forecasting

text: There is a vast variety of end uses to weather forecasts. Weather warnings are important forecasts because they are used to protect life and property. Forecasts based on temperature and precipitation are important to agriculture, and therefore to traders within commodity markets. Temperature forecasts are used by utility companies to estimate demand over coming days. On an everyday basis, many use weather forecasts to determine what to wear on a given day. Since outdoor activities are severely curtailed by heavy rain, snow and wind chill, forecasts can be used to plan activities around these events, and to plan ahead and survive them.


title: Weather forecasting

text: Weather forecasting is a part of the economy, for example, in 2009, the US spent approximately $5.1 billion on weather forecasting, producing benefits estimated at six times as much.


title: Weather forecasting

text: For millennia, people have tried to forecast the weather. In 650 BC, the Babylonians predicted the weather from cloud patterns as well as astrology. In about 350 BC, Aristotle described weather patterns in "Meteorologica". Later, Theophrastus compiled a book on weather forecasting, called the "Book of Signs". Chinese weather prediction lore extends at least as far back as 300 BC, which was also around the same time ancient Indian astronomers developed weather-prediction methods. In New Testament times, Jesus himself referred to deciphering and understanding local weather patterns, by saying, "When evening comes, you say, 'It will be fair weather, for the sky is red', and in the morning, 'Today it will be stormy, for the sky is red and overcast.' You know how to interpret the appearance of the sky, but you cannot interpret the signs of the times."


title: Weather forecasting

text: In 904 AD, Ibn Wahshiyya's "Nabatean Agriculture", translated into Arabic from an earlier Aramaic work, discussed the weather forecasting of atmospheric changes and signs from the planetary astral alterations; signs of rain based on observation of the lunar phases; and weather forecasts based on the movement of winds.


title: Weather forecasting

text: Ancient weather forecasting methods usually relied on observed patterns of events, also termed pattern recognition. For example, it was observed that if the sunset was particularly red, the following day often brought fair weather. This experience accumulated over the generations to produce weather lore. However, not all of these predictions prove reliable, and many of them have since been found not to stand up to rigorous statistical testing.


title: Weather forecasting

text: It was not until the invention of the electric telegraph in 1835 that the modern age of weather forecasting began. Before that, the fastest that distant weather reports could travel was around 160 kilometres per day (100 mi/d), but was more typically 60–120 kilometres per day (40–75 mi/day) (whether by land or by sea). By the late 1840s, the telegraph allowed reports of weather conditions from a wide area to be received almost instantaneously, allowing forecasts to be made from knowledge of weather conditions further upwind.


title: Weather forecasting

text: The two men credited with the birth of forecasting as a science were an officer of the Royal Navy Francis Beaufort and his protégé Robert FitzRoy. Both were influential men in British naval and governmental circles, and though ridiculed in the press at the time, their work gained scientific credence, was accepted by the Royal Navy, and formed the basis for all of today's weather forecasting knowledge.


title: Weather forecasting

text: Beaufort developed the Wind Force Scale and Weather Notation coding, which he was to use in his journals for the remainder of his life. He also promoted the development of reliable tide tables around British shores, and with his friend William Whewell, expanded weather record-keeping at 200 British coast guard stations.


title: Weather forecasting

text: Robert FitzRoy was appointed in 1854 as chief of a new department within the Board of Trade to deal with the collection of weather data at sea as a service to mariners. This was the forerunner of the modern Meteorological Office. All ship captains were tasked with collating data on the weather and computing it, with the use of tested instruments that were loaned for this purpose.


title: Weather forecasting

text: A storm in October 1859 that caused the loss of the "Royal Charter" inspired FitzRoy to develop charts to allow predictions to be made, which he called ""forecasting the weather"", thus coining the term "weather forecast". Fifteen land stations were established to use the telegraph to transmit to him daily reports of weather at set times leading to the first gale warning service. His warning service for shipping was initiated in February 1861, with the use of telegraph communications. The first daily weather forecasts were published in "The Times" in 1861. In the following year a system was introduced of hoisting storm warning cones at the principal ports when a gale was expected. The ""Weather Book"" which FitzRoy published in 1863 was far in advance of the scientific opinion of the time.


title: Weather forecasting

text: As the electric telegraph network expanded, allowing for the more rapid dissemination of warnings, a national observational network was developed, which could then be used to provide synoptic analyses. Instruments to continuously record variations in meteorological parameters using photography were supplied to the observing stations from Kew Observatory – these cameras had been invented by Francis Ronalds in 1845 and his barograph had earlier been used by FitzRoy.


title: Weather forecasting

text: To convey accurate information, it soon became necessary to have a standard vocabulary describing clouds; this was achieved by means of a series of classifications first achieved by Luke Howard in 1802, and standardized in the "International Cloud Atlas" of 1896.


title: Weather forecasting

text: It was not until the 20th century that advances in the understanding of atmospheric physics led to the foundation of modern numerical weather prediction. In 1922, English scientist Lewis Fry Richardson published "Weather Prediction By Numerical Process", after finding notes and derivations he worked on as an ambulance driver in World War I. He described therein how small terms in the prognostic fluid dynamics equations governing atmospheric flow could be neglected, and a finite differencing scheme in time and space could be devised, to allow numerical prediction solutions to be found.


title: Weather forecasting

text: Richardson envisioned a large auditorium of thousands of people performing the calculations and passing them to others. However, the sheer number of calculations required was too large to be completed without the use of computers, and the size of the grid and time steps led to unrealistic results in deepening systems. It was later found, through numerical analysis, that this was due to numerical instability. The first computerised weather forecast was performed by a team composed of American meteorologists Jule Charney, Philip Thompson, Larry Gates, and Norwegian meteorologist Ragnar Fjørtoft, applied mathematician John von Neumann, and ENIAC programmer Klara Dan von Neumann. Practical use of numerical weather prediction began in 1955, spurred by the development of programmable electronic computers.


title: Weather forecasting

text: The first ever daily weather forecasts were published in "The Times" on August 1, 1861, and the first weather maps were produced later in the same year. In 1911, the Met Office began issuing the first marine weather forecasts via radio transmission. These included gale and storm warnings for areas around Great Britain. In the United States, the first public radio forecasts were made in 1925 by Edward B. "E.B." Rideout, on WEEI, the Edison Electric Illuminating station in Boston. Rideout came from the U.S. Weather Bureau, as did WBZ weather forecaster G. Harold Noyes in 1931.


title: Weather forecasting

text: The world's first televised weather forecasts, including the use of weather maps, were experimentally broadcast by the BBC in November 1936. This was brought into practice in 1949, after World War II. George Cowling gave the first weather forecast while being televised in front of the map in 1954. In America, experimental television forecasts were made by James C. Fidler in Cincinnati in either 1940 or 1947 on the DuMont Television Network. In the late 1970s and early 80s, John Coleman, the first weatherman on ABC-TV's "Good Morning America", pioneered the use of on-screen weather satellite information and computer graphics for television forecasts. Coleman was a co-founder of The Weather Channel (TWC) in 1982. TWC is now a 24-hour cable network. Some weather channels have started broadcasting on live broadcasting programs such as YouTube and Periscope to reach more viewers.


title: Weather forecasting

text: The basic idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics and thermodynamics to estimate the state of the fluid at some time in the future. The main inputs from country-based weather services are surface observations from automated weather stations at ground level over land and from weather buoys at sea. The World Meteorological Organization acts to standardize the instrumentation, observing practices and timing of these observations worldwide. Stations either report hourly in METAR reports, or every six hours in SYNOP reports. Sites launch radiosondes, which rise through the depth of the troposphere and well into the stratosphere. Data from weather satellites are used in areas where traditional data sources are not available. Compared with similar data from radiosondes, the satellite data has the advantage of global coverage, however at a lower accuracy and resolution. Meteorological radar provide information on precipitation location and intensity, which can be used to estimate precipitation accumulations over time. Additionally, if a pulse Doppler weather radar is used then wind speed and direction can be determined. These methods, however, leave an in situ observational gap in the lower atmosphere (from 100 m to 6 km above ground level). To reduce this gap, in the late 1990s weather drones started to be considered for obtaining data from those altitudes. Research has been growing significantly since the 2010s, and weather drones data may in future be added to numerical weather models.


title: Weather forecasting

text: Commerce provides pilot reports along aircraft routes, and ship reports along shipping routes. Research flights using reconnaissance aircraft fly in and around weather systems of interest such as tropical cyclones. Reconnaissance aircraft are also flown over the open oceans during the cold season into systems that cause significant uncertainty in forecast guidance, or are expected to be of high impact 3–7 days into the future over the downstream continent.


title: Weather forecasting

text: Models are "initialized" using this observed data. The irregularly spaced observations are processed by data assimilation and objective analysis methods, which perform quality control and obtain values at locations usable by the model's mathematical algorithms (usually an evenly spaced grid). The data are then used in the model as the starting point for a forecast. Commonly, the set of equations used to predict the physics and dynamics of the atmosphere are called primitive equations. These equations are initialized from the analysis data and rates of change are determined. The rates of change predict the state of the atmosphere a short time into the future. The equations are then applied to this new atmospheric state to find new rates of change, and these new rates of change predict the atmosphere at a yet further time into the future. This "time stepping" procedure is continually repeated until the solution reaches the desired forecast time.


title: Weather forecasting

text: The length of the time step chosen within the model is related to the distance between the points on the computational grid, and is chosen to maintain numerical stability. Time steps for global models are on the order of tens of minutes, while time steps for regional models are between one and four minutes. The global models are run at varying times into the future. The Met Office's Unified Model is run six days into the future, the European Centre for Medium-Range Weather Forecasts model is run out to 10 days into the future, while the Global Forecast System model run by the Environmental Modeling Center is run 16 days into the future. The visual output produced by a model solution is known as a prognostic chart, or "prog". The raw output is often modified before being presented as the forecast. This can be in the form of statistical techniques to remove known biases in the model, or of adjustment to take into account consensus among other numerical weather forecasts. MOS or model output statistics is a technique used to interpret numerical model output and produce site-specific guidance. This guidance is presented in coded numerical form, and can be obtained for nearly all National Weather Service reporting stations in the United States. As proposed by Edward Lorenz in 1963, long range forecasts, those made at a range of two weeks or more, are impossible to definitively predict the state of the atmosphere, owing to the chaotic nature of the fluid dynamics equations involved. In numerical models, extremely small errors in initial values double roughly every five days for variables such as temperature and wind velocity.


title: Weather forecasting

text: Essentially, a model is a computer program that produces meteorological information for future times at given locations and altitudes. Within any modern model is a set of equations, known as the primitive equations, used to predict the future state of the atmosphere. These equations—along with the ideal gas law—are used to evolve the density, pressure, and potential temperature scalar fields and the velocity vector field of the atmosphere through time. Additional transport equations for pollutants and other aerosols are included in some primitive-equation mesoscale models as well. The equations used are nonlinear partial differential equations, which are impossible to solve exactly through analytical methods, with the exception of a few idealized cases. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods: some global models use spectral methods for the horizontal dimensions and finite difference methods for the vertical dimension, while regional models and other global models usually use finite-difference methods in all three dimensions.


title: Weather forecasting

text: The simplest method of forecasting the weather, persistence, relies upon today's conditions to forecast the conditions tomorrow. This can be a valid way of forecasting the weather when it is in a steady state, such as during the summer season in the tropics. This method of forecasting strongly depends upon the presence of a stagnant weather pattern. Therefore, when in a fluctuating weather pattern, this method of forecasting becomes inaccurate. It can be useful in both short range forecasts and long range forecasts.


title: Weather forecasting

text: Measurements of barometric pressure and the pressure tendency (the change of pressure over time) have been used in forecasting since the late 19th century. The larger the change in pressure, especially if more than , the larger the change in weather can be expected. If the pressure drop is rapid, a low pressure system is approaching, and there is a greater chance of rain. Rapid pressure rises are associated with improving weather conditions, such as clearing skies.


title: Weather forecasting

text: Along with pressure tendency, the condition of the sky is one of the more important parameters used to forecast weather in mountainous areas. Thickening of cloud cover or the invasion of a higher cloud deck is indicative of rain in the near future. High thin cirrostratus clouds can create halos around the sun or moon, which indicates an approach of a warm front and its associated rain. Morning fog portends fair conditions, as rainy conditions are preceded by wind or clouds that prevent fog formation. The approach of a line of thunderstorms could indicate the approach of a cold front. Cloud-free skies are indicative of fair weather for the near future. A bar can indicate a coming tropical cyclone. The use of sky cover in weather prediction has led to various weather lore over the centuries.


title: Weather forecasting

text: The forecasting of the weather within the next six hours is often referred to as nowcasting. In this time range it is possible to forecast smaller features such as individual showers and thunderstorms with reasonable accuracy, as well as other features too small to be resolved by a computer model. A human given the latest radar, satellite and observational data will be able to make a better analysis of the small scale features present and so will be able to make a more accurate forecast for the following few hours. However, there are now expert systems using those data and mesoscale numerical model to make better extrapolation, including evolution of those features in time. Accuweather is known for a Minute-Cast, which is a minute-by-minute precipitation forecast for the next two hours.


title: Weather forecasting

text: In the past, the human forecaster was responsible for generating the entire weather forecast based upon available observations. Today, human input is generally confined to choosing a model based on various parameters, such as model biases and performance. Using a consensus of forecast models, as well as ensemble members of the various models, can help reduce forecast error. However, regardless how small the average error becomes with any individual system, large errors within any particular piece of guidance are still possible on any given model run. Humans are required to interpret the model data into weather forecasts that are understandable to the end user. Humans can use knowledge of local effects that may be too small in size to be resolved by the model to add information to the forecast. While increasing accuracy of forecast models implies that humans may no longer be needed in the forecast process at some point in the future, there is currently still a need for human intervention.


title: Weather forecasting

text: The analog technique is a complex way of making a forecast, requiring the forecaster to remember a previous weather event that is expected to be mimicked by an upcoming event. What makes it a difficult technique to use is that there is rarely a perfect analog for an event in the future. Some call this type of forecasting pattern recognition. It remains a useful method of observing rainfall over data voids such as oceans, as well as the forecasting of precipitation amounts and distribution in the future. A similar technique is used in medium range forecasting, which is known as teleconnections, when systems in other locations are used to help pin down the location of another system within the surrounding regime. An example of teleconnections are by using El Niño-Southern Oscillation (ENSO) related phenomena.


title: Weather forecasting

text: Most end users of forecasts are members of the general public. Thunderstorms can create strong winds and dangerous lightning strikes that can lead to deaths, power outages, and widespread hail damage. Heavy snow or rain can bring transportation and commerce to a stand-still, as well as cause flooding in low-lying areas. Excessive heat or cold waves can sicken or kill those with inadequate utilities, and droughts can impact water usage and destroy vegetation.


title: Weather forecasting

text: Several countries employ government agencies to provide forecasts and watches/warnings/advisories to the public to protect life and property and maintain commercial interests. Knowledge of what the end user needs from a weather forecast must be taken into account to present the information in a useful and understandable way. Examples include the National Oceanic and Atmospheric Administration's National Weather Service (NWS) and Environment Canada's Meteorological Service (MSC). Traditionally, newspaper, television, and radio have been the primary outlets for presenting weather forecast information to the public. In addition, some cities had weather beacons. Increasingly, the internet is being used due to the vast amount of specific information that can be found. In all cases, these outlets update their forecasts on a regular basis.


title: Weather forecasting

text: A major part of modern weather forecasting is the severe weather alerts and advisories that the national weather services issue in the case that severe or hazardous weather is expected. This is done to protect life and property. Some of the most commonly known of severe weather advisories are the severe thunderstorm and tornado warning, as well as the severe thunderstorm and tornado watch. Other forms of these advisories include winter weather, high wind, flood, tropical cyclone, and fog. Severe weather advisories and alerts are broadcast through the media, including radio, using emergency systems as the Emergency Alert System, which break into regular programming.


title: Weather forecasting

text: The low temperature forecast for the current day is calculated using the lowest temperature found between 7pm that evening through 7am the following morning. So, in short, today's forecasted low is most likely tomorrow's low temperature.


title: Weather forecasting

text: There are a number of sectors with their own specific needs for weather forecasts and specialist services are provided to these users as given below:


title: Weather forecasting

text: Because the aviation industry is especially sensitive to the weather, accurate weather forecasting is essential. Fog or exceptionally low ceilings can prevent many aircraft from landing and taking off. Turbulence and icing are also significant in-flight hazards. Thunderstorms are a problem for all aircraft because of severe turbulence due to their updrafts and outflow boundaries, icing due to the heavy precipitation, as well as large hail, strong winds, and lightning, all of which can cause severe damage to an aircraft in flight. Volcanic ash is also a significant problem for aviation, as aircraft can lose engine power within ash clouds. On a day-to-day basis airliners are routed to take advantage of the jet stream tailwind to improve fuel efficiency. Aircrews are briefed prior to takeoff on the conditions to expect en route and at their destination. Additionally, airports often change which runway is being used to take advantage of a headwind. This reduces the distance required for takeoff, and eliminates potential crosswinds.


title: Weather forecasting

text: Commercial and recreational use of waterways can be limited significantly by wind direction and speed, wave periodicity and heights, tides, and precipitation. These factors can each influence the safety of marine transit. Consequently, a variety of codes have been established to efficiently transmit detailed marine weather forecasts to vessel pilots via radio, for example the MAFOR (marine forecast). Typical weather forecasts can be received at sea through the use of RTTY, Navtex and Radiofax.


title: Weather forecasting

text: Farmers rely on weather forecasts to decide what work to do on any particular day. For example, drying hay is only feasible in dry weather. Prolonged periods of dryness can ruin cotton, wheat, and corn crops. While corn crops can be ruined by drought, their dried remains can be used as a cattle feed substitute in the form of silage. Frosts and freezes play havoc with crops both during the spring and fall. For example, peach trees in full bloom can have their potential peach crop decimated by a spring freeze. Orange groves can suffer significant damage during frosts and freezes, regardless of their timing.


title: Weather forecasting

text: Weather forecasting of wind, precipitations and humidity is essential for preventing and controlling wildfires. Different indices, like the "Forest fire weather index" and the "Haines Index", have been developed to predict the areas more at risk to experience fire from natural or human causes. Conditions for the development of harmful insects can be predicted by forecasting the evolution of weather, too.


title: Weather forecasting

text: Electricity and gas companies rely on weather forecasts to anticipate demand, which can be strongly affected by the weather. They use the quantity termed the degree day to determine how strong of a use there will be for heating (heating degree day) or cooling (cooling degree day). These quantities are based on a daily average temperature of . Cooler temperatures force heating degree days (one per degree Fahrenheit), while warmer temperatures force cooling degree days. In winter, severe cold weather can cause a surge in demand as people turn up their heating. Similarly, in summer a surge in demand can be linked with the increased use of air conditioning systems in hot weather. By anticipating a surge in demand, utility companies can purchase additional supplies of power or natural gas before the price increases, or in some circumstances, supplies are restricted through the use of brownouts and blackouts.


title: Weather forecasting

text: Increasingly, private companies pay for weather forecasts tailored to their needs so that they can increase their profits or avoid large losses. For example, supermarket chains may change the stocks on their shelves in anticipation of different consumer spending habits in different weather conditions. Weather forecasts can be used to invest in the commodity market, such as futures in oranges, corn, soybeans, and oil.


title: Weather forecasting

text: The UK Royal Navy, working with the UK Met Office, has its own specialist branch of weather observers and forecasters, as part of the Hydrographic and Meteorological (HM) specialisation, who monitor and forecast operational conditions across the globe, to provide accurate and timely weather and oceanographic information to submarines, ships and Fleet Air Arm aircraft.


title: Weather forecasting

text: A mobile unit in the RAF, working with the UK Met Office, forecasts the weather for regions in which British, allied servicemen and women are deployed. A group based at Camp Bastion provides forecasts for the British armed forces in Afghanistan.


title: Weather forecasting

text: Similar to the private sector, military weather forecasters present weather conditions to the war fighter community. Military weather forecasters provide pre-flight and in-flight weather briefs to pilots and provide real time resource protection services for military installations. Naval forecasters cover the waters and ship weather forecasts. The United States Navy provides a special service to both themselves and the rest of the federal government by issuing forecasts for tropical cyclones across the Pacific and Indian Oceans through their Joint Typhoon Warning Center.


title: Weather forecasting

text: Within the United States, Air Force Weather provides weather forecasting for the Air Force and the Army. Air Force forecasters cover air operations in both wartime and peacetime operations and provide Army support; United States Coast Guard marine science technicians provide ship forecasts for ice breakers and other various operations within their realm; and Marine forecasters provide support for ground- and air-based United States Marine Corps operations. All four military branches take their initial enlisted meteorology technical training at Keesler Air Force Base. Military and civilian forecasters actively cooperate in analyzing, creating and critiquing weather forecast products.


title: Weather forecasting

text: These are academic or governmental meteorology organizations. Most provide at least a limited forecast for their area of interest on their website.


title: Frederick W. Smith

text: Frederick Wallace Smith (born August 11, 1944) is an American business magnate and investor. He is the founder and chairman of FedEx Corporation, the world's largest transportation company. On June 1, 2022, Smith stepped down as CEO to become executive chairman, and was replaced by Raj Subramaniam. He is considered one of the most successful transportation entrepreneurs in the world.


title: Frederick W. Smith

text: Frederick Smith was born in Marks, Mississippi, the son of James Frederick "Fred" Smith, the founder of the Toddle House restaurant chain and the Smith Motor Coach Company (renamed the Dixie Greyhound Lines after The Greyhound Corporation bought a controlling interest in 1931). The elder Smith died when his son was only 4, and the boy was raised by his mother and uncles.


title: Frederick W. Smith

text: He attended elementary school at Presbyterian Day School in Memphis and high school at Memphis University School, and became an amateur pilot as a teen.


title: Frederick W. Smith

text: In 1962, Smith entered Yale University. While attending Yale, he wrote a paper for an economics class, outlining overnight delivery service, on which he received a “C”.


title: Frederick W. Smith

text: Additionally, Smith became a member and eventually the president of the Delta Kappa Epsilon (DKE) fraternity and the Skull and Bones secret society. He received his bachelor's degree in economics in 1966. In his college years, he was a friend and DKE fraternity brother of future U.S. president George W. Bush. Smith was also friends with future U.S. Senator and Secretary of State John Kerry; the two shared an enthusiasm for aviation and were flying partners.


title: Frederick W. Smith

text: After graduation, Smith was commissioned in the U.S. Marine Corps, serving for three years (from 1966 to 1969) as a platoon leader and a forward air controller (FAC), flying in the back seat of the OV-10.


title: Frederick W. Smith

text: He served two tours of duty in Vietnam and was honorably discharged in 1969 with the rank of captain, having received the Silver Star, the Bronze Star, and two Purple Hearts .


title: Frederick W. Smith

text: The President of the United States of America takes pleasure in presenting the Silver Star to First Lieutenant Frederick Wallace Smith, United States Marine Corps, for conspicuous gallantry and intrepidity in action while serving as Commanding Officer of Company K, 3rd Battalion, 5th Marines, 1st Marine Division in connection with operations against the enemy in the Republic of Vietnam. On the morning of 27 May 1968, while conducting a search and destroy operation, Company K became heavily engaged with a North Vietnamese Army battalion occupying well-entrenched emplacements on Goi Noi Island in Quang Nam Province. As Lieutenant Smith led his men in an aggressive assault upon the enemy positions, the North Vietnamese force launched a determined counterattack, supported by mortars, on the Marines' left flank. Unhesitatingly rushing through the intense hostile fire to the position of heaviest contact, Lieutenant Smith fearlessly removed several casualties from the hazardous area and, shouting words of encouragement to his men, directed their fire upon the advancing enemy soldiers, successfully repulsing the hostile attack. Moving boldly across the fire-swept terrain to an elevated area, he calmly disregarded repeated North Vietnamese attempts to direct upon him as he skillfully adjusted artillery fire and air strikes upon the hostile positions to within fifty meters of his own location and continued to direct the movement of his unit. Accurately assessing the confusion that supporting arms was causing among the enemy soldiers, he raced across the fire-swept terrain to the right flank of his company and led an enveloping attack on the hostile unit's weakest point, routing the North Vietnamese unit and inflicting numerous casualties. His aggressive tactics and calm presence of min [sic] under fire inspired all who observed him and were instrumental in his unit accounting for the capture of two hostile soldiers as well as numerous documents and valuable items of equipment. By his courage, aggressive leadership and unfaltering devotion to duty at great personal risk, Lieutenant Smith upheld the highest traditions of the Marine Corps and of the United States Naval Service."


title: Frederick W. Smith

text: In 1970, Smith purchased the controlling interest in an aircraft maintenance company, Ark Aviation Sales, and by 1971 turned its focus to trading used jets. On June 18, 1971, Smith founded Federal Express with his $4 million inheritance (approximately $23 million in 2013 dollars) and raised $91 million (approximately $525 million in 2013 dollars) in venture capital. In 1973, the company began offering service to 25 cities, and it began with small packages and documents and a fleet of 14 Falcon 20 (DA-20) jets. His focus was on developing an integrated air-ground system. Smith developed FedEx on the business idea of a shipment version of a bank clearing house where one bank clearing house was located in the middle of the representative banks and all their representatives would be sent to the central location to exchange materials.


title: Frederick W. Smith

text: In the early days of FedEx, Smith had to go to great lengths to keep the company afloat. In one instance, after a crucial business loan was denied, he took the company's last $5,000 to Las Vegas and won $27,000 gambling on blackjack to cover the company's $24,000 fuel bill. It kept FedEx alive for one more week.


title: Frederick W. Smith

text: In 2003 Smith, along with Robert Rothman and Dwight Schar, purchased a minority share of the Washington Commanders, an American football franchise belonging to the National Football League. The three owned a total of 40% of the team until 2021, when they sold their stake to majority owner Dan Snyder following discontentment with Snyder. Smith also owns or co-owns several other entertainment companies, such as Alcon Entertainment.


title: Frederick W. Smith

text: In 2000, Smith made an appearance as himself in the Tom Hanks movie "Cast Away", when Tom's character is welcomed back, which was filmed on location at FedEx's home facilities in Memphis, Tennessee. A DKE Fraternity Brother of George W. Bush while at Yale, after Bush's 2000 election, there was some speculation that Smith might be appointed to the Bush Cabinet as Defense Secretary.


title: Frederick W. Smith

text: While Smith was Bush's first choice for the position, he declined for medical reasons — Donald Rumsfeld was named instead. Although Smith was friends with both 2004 major candidates, John Kerry and George W. Bush, Smith chose to endorse Bush's re-election in 2004. When Bush decided to replace Rumsfeld, Smith was offered the position again, but he declined in order to spend time with his terminally ill daughter.


title: Frederick W. Smith

text: Smith was a supporter of Senator John McCain's 2008 Presidential bid, and had been named McCain's National Co-Chairman of his campaign committee.


title: Frederick W. Smith

text: Smith was inducted into the Junior Achievement U.S. Business Hall of Fame and also awarded the Golden Plate Award of the American Academy of Achievement in 1998. He was inducted into the SMEI Sales & Marketing Hall of Fame in 2000. His other awards include "CEO of the Year 2004" by Chief Executive Magazine and the 2008 Kellogg Award for Distinguished Leadership, presented by the Kellogg School of Management on May 29, 2008. He was also awarded the 2008 Bower Award for Business Leadership from The Franklin Institute in Philadelphia, Pennsylvania. He is the 2011 recipient of the Tony Jannus Award for distinguished contributions to commercial aviation.


title: Frederick W. Smith

text: While CEO of FedEx in 2008, Frederick W. Smith earned a total compensation of $10,434,589, which included a base salary of $1,430,466, a cash bonus of $2,705,000, stocks granted of $0, and options granted of $5,461,575.


title: Frederick W. Smith

text: In June 2009, Smith expressed interest in purchasing the controlling share (60%) of the St. Louis Rams from Chip Rosenbloom and Lucia Rodriguez. In 2009, Frederick W. Smith earned a total compensation of $7,740,658, which included a base salary of $1,355,028, a cash bonus of $0, stocks granted of $0, options granted of $5,079,191, and other compensation totaling $1,306,439.


title: Frederick W. Smith

text: In March 2022, Smith announced that he will step down as CEO and become executive chairman. He named long-time FedEx executive Raj Subramaniam as his successor.


title: Frederick W. Smith

text: On January 31, 1975, Fred Smith was indicted for forgery by a federal grand jury. The suit was filed by Smith's two half-sisters. The lawsuit alleged Smith had forged documents to obtain a $2 million bank loan and he and executives of his family's trust fund had sold stock from the fund for a loss of $14 million. A warrant for Smith's arrest was issued for which Smith posted bond with federal authorities in Memphis. Smith was later found not guilty on the forgery charge.


title: Frederick W. Smith

text: The same evening of his forgery indictment Smith was involved in a fatal hit and run in which he killed a 54-year-old handyman named George C. Sturghill. Smith was arrested and charged with leaving the scene of a crash and driving with an expired license. He was released on a $250 bond. All charges were later dismissed.


title: Frederick W. Smith

text: This was not the first time Smith was involved in a fatal car crash. During his first summer break from Yale, Smith was back in Memphis driving out to a lake with friends when he lost control of the car he was driving, causing the vehicle to flip and killing the passenger in the front seat. The cause of the crash was never determined.


title: Frederick W. Smith

text: Smith has ten children, including photographer Windland Smith Rice, film producer Molly Smith and Atlanta Falcons head coach Arthur Smith.


title: Analytics

text: Analytics is the systematic computational analysis of data or statistics. It is used for the discovery, interpretation, and communication of meaningful patterns in data. It also entails applying data patterns toward effective decision-making. It can be valuable in areas rich with recorded information; analytics relies on the simultaneous application of statistics, computer programming, and operations research to quantify performance.


title: Analytics

text: Organizations may apply analytics to business data to describe, predict, and improve business performance. Specifically, areas within analytics include descriptive analytics, diagnostic analytics, predictive analytics, prescriptive analytics, and cognitive analytics. Analytics may apply to a variety of fields such as marketing, management, finance, online systems, information security, and software services. Since analytics can require extensive computation (see big data), the algorithms and software used for analytics harness the most current methods in computer science, statistics, and mathematics. According to IDC, global spending on big data and business analytics (BDA) solutions is estimated to reach $215.7 billion in 2021. As per Gartner, the overall analytic platforms software market grew by $25.5 billion in 2020.


title: Analytics

text: Data analysis focuses on the process of examining past data through business understanding, data understanding, data preparation, modeling and evaluation, and deployment. It is a subset of data analytics, which takes multiple data analysis processes to focus on why an event happened and what may happen in the future based on the previous data. Data analytics is used to formulate larger organization decisions.


title: Analytics

text: Data analytics is a multidisciplinary field. There is extensive use of computer skills, mathematics, statistics, the use of descriptive techniques and predictive models to gain valuable knowledge from data through analytics. There is increasing use of the term "advanced analytics", typically used to describe the technical aspects of analytics, especially in the emerging fields such as the use of machine learning techniques like neural networks, decision trees, logistic regression, linear to multiple regression analysis, and classification to do predictive modeling. It also includes unsupervised machine learning techniques like cluster analysis, Principal Component Analysis, segmentation profile analysis and association analysis.


title: Analytics

text: Marketing organizations use analytics to determine the outcomes of campaigns or efforts, and to guide decisions for investment and consumer targeting. Demographic studies, customer segmentation, conjoint analysis and other techniques allow marketers to use large amounts of consumer purchase, survey and panel data to understand and communicate marketing strategy.


title: Analytics

text: Marketing analytics consists of both qualitative and quantitative, structured and unstructured data used to drive strategic decisions about brand and revenue outcomes. The process involves predictive modelling, marketing experimentation, automation and real-time sales communications. The data enables companies to make predictions and alter strategic execution to maximize performance results.


title: Analytics

text: Web analytics allows marketers to collect session-level information about interactions on a website using an operation called sessionization. Google Analytics is an example of a popular free analytics tool that marketers use for this purpose. Those interactions provide web analytics information systems with the information necessary to track the referrer, search keywords, identify the IP address, and track activities of the visitor. With this information, a marketer can improve marketing campaigns, website creative content, and information architecture.


title: Analytics

text: Analysis techniques frequently used in marketing include marketing mix modeling, pricing and promotion analyses, sales force optimization and customer analytics e.g.: segmentation. Web analytics and optimization of websites and online campaigns now frequently work hand in hand with the more traditional marketing analysis techniques. A focus on digital media has slightly changed the vocabulary so that "marketing mix modeling" is commonly referred to as "attribution modeling" in the digital or marketing mix modeling context.


title: Analytics

text: These tools and techniques support both strategic marketing decisions (such as how much overall to spend on marketing, how to allocate budgets across a portfolio of brands and the marketing mix) and more tactical campaign support, in terms of targeting the best potential customer with the optimal message in the most cost-effective medium at the ideal time.


title: Analytics

text: People analytics uses behavioral data to understand how people work and change how companies are managed.


title: Analytics

text: People analytics is also known as workforce analytics, HR analytics, talent analytics, people insights, talent insights, colleague insights, human capital analytics, and HRIS analytics. HR analytics is the application of analytics to help companies manage human resources. Additionally, HR analytics has become a strategic tool in analyzing and forecasting Human related trends in the changing labor markets, using Career Analytics tools. The aim is to discern which employees to hire, which to reward or promote, what responsibilities to assign, and similar human resource problems.


title: Analytics

text: For example, inspection of the strategic phenomenon of employee turnover utilizing People Analytics Tools may serve as an important analysis at times of disruption.


title: Analytics

text: It has been suggested that People Analytics is a separate discipline to HR analytics, representing a greater focus on business issues rather than administrative processes, and that People Analytics may not really belong within Human Resources in organizations. However, experts disagree on this, with many arguing that Human Resources will need to develop People Analytics as a key part of a more capable and strategic business function in the changing world of work brought on by automation. Instead of moving People Analytics outside HR, some experts argue that it belongs in HR, albeit enabled by a new breed of HR professional who is more data-driven and business savvy.


title: Analytics

text: A common application of business analytics is portfolio analysis. In this, a bank or lending agency has a collection of accounts of varying value and risk. The accounts may differ by the social status (wealthy, middle-class, poor, etc.) of the holder, the geographical location, its net value, and many other factors. The lender must balance the return on the loan with the risk of default for each loan. The question is then how to evaluate the portfolio as a whole.


title: Analytics

text: The least risk loan may be to the very wealthy, but there are a very limited number of wealthy people. On the other hand, there are many poor that can be lent to, but at greater risk. Some balance must be struck that maximizes return and minimizes risk. The analytics solution may combine time series analysis with many other issues in order to make decisions on when to lend money to these different borrower segments, or decisions on the interest rate charged to members of a portfolio segment to cover any losses among members in that segment.


title: Analytics

text: Predictive models in the banking industry are developed to bring certainty across the risk scores for individual customers. Credit scores are built to predict an individual's delinquency behavior and are widely used to evaluate the credit worthiness of each applicant. Furthermore, risk analyses are carried out in the scientific world and the insurance industry. It is also extensively used in financial institutions like online payment gateway companies to analyse if a transaction was genuine or fraud. For this purpose, they use the transaction history of the customer. This is more commonly used in Credit Card purchases, when there is a sudden spike in the customer transaction volume the customer gets a call of confirmation if the transaction was initiated by him/her. This helps in reducing loss due to such circumstances.


title: Analytics

text: Digital analytics is a set of business and technical activities that define, create, collect, verify or transform digital data into reporting, research, analyses, recommendations, optimizations, predictions, and automation. This also includes the SEO (search engine optimization) where the keyword search is tracked and that data is used for marketing purposes. Even banner ads and clicks come under digital analytics. A growing number of brands and marketing firms rely on digital analytics for their digital marketing assignments, where MROI (Marketing Return on Investment) is an important key performance indicator (KPI).


title: Analytics

text: Security analytics refers to information technology (IT) to gather security events to understand and analyze events that pose the greatest risk. Products in this area include security information and event management and user behavior analytics.


title: Analytics

text: Software analytics is the process of collecting information about the way a piece of software is used and produced.


title: Analytics

text: In the industry of commercial analytics software, an emphasis has emerged on solving the challenges of analyzing massive, complex data sets, often when such data is in a constant state of change. Such data sets are commonly referred to as big data. Whereas once the problems posed by big data were only found in the scientific community, today big data is a problem for many businesses that operate transactional systems online and, as a result, amass large volumes of data quickly.


title: Analytics

text: The analysis of unstructured data types is another challenge getting attention in the industry. Unstructured data differs from structured data in that its format varies widely and cannot be stored in traditional relational databases without significant effort at data transformation. Sources of unstructured data, such as email, the contents of word processor documents, PDFs, geospatial data, etc., are rapidly becoming a relevant source of business intelligence for businesses, governments and universities. For example, in Britain the discovery that one company was illegally selling fraudulent doctor's notes in order to assist people in defrauding employers and insurance companies is an opportunity for insurance firms to increase the vigilance of their unstructured data analysis.


title: Analytics

text: These challenges are the current inspiration for much of the innovation in modern analytics information systems, giving birth to relatively new machine analysis concepts such as complex event processing, full text search and analysis, and even new ideas in presentation. One such innovation is the introduction of grid-like architecture in machine analysis, allowing increases in the speed of massively parallel processing by distributing the workload to many computers all with equal access to the complete data set.


title: Analytics

text: Analytics is increasingly used in education, particularly at the district and government office levels. However, the complexity of student performance measures presents challenges when educators try to understand and use analytics to discern patterns in student performance, predict graduation likelihood, improve chances of student success, etc. For example, in a study involving districts known for strong data use, 48% of teachers had difficulty posing questions prompted by data, 36% did not comprehend given data, and 52% incorrectly interpreted data. To combat this, some analytics tools for educators adhere to an over-the-counter data format (embedding labels, supplemental documentation, and a help system, and making key package/display and content decisions) to improve educators' understanding and use of the analytics being displayed.


title: Analytics

text: Risks for the general population include discrimination on the basis of characteristics such as gender, skin colour, ethnic origin or political opinions, through mechanisms such as price discrimination or statistical discrimination.


title: Elizabeth LeCompte

text: Elizabeth LeCompte (born April 28, 1944) is an American director of experimental theater, dance, and media. A founding member of The Wooster Group, she has directed that ensemble since its emergence in the late 1970s.


title: Elizabeth LeCompte

text: LeCompte was born and grew up in New Jersey. She earned a Bachelor of Science degree in Fine Arts from Skidmore College. She met director and actor Willem Dafoe at The Performance Group and began a professional and personal relationship. Their son, Jack, was born in 1982.


title: Elizabeth LeCompte

text: With The Wooster Group, she has composed, designed, and directed over forty works for theater, dance, film and video, starting with "Sakonnet Point" in 1975. These works characteristically interweave performance with multimedia technologies and are strongly influenced by historical and contemporary visual arts and architecture. She is known both for taking apart and reworking classics such as "Hamlet", "The Emperor Jones", and "The Hairy Ape" as well as constructing new works from scratch.


title: Elizabeth LeCompte

text: Prior to her work with The Wooster Group, she was a member of the experimental theater company The Performance Group from 1970 to 1975. Subsequently, LeCompte and Spalding Gray founded The Wooster Group, along with Jim Clayburgh, Willem Dafoe, Peyton Smith, Kate Valk, and Ron Vawter. For her work with these groups, LeCompte was included in Mitter and Shevtsova's 2004 volume discussing 50 influential theater directors around the world. Other writers consistently include her in the lineage of experimental theater artists that passes through Meyerhold and Grotowski to the present generation of "postdramatic" theater makers. As a "New Yorker" writer put it: "Luminaries of the theatrical avant-garde—Richard Foreman, Robert Wilson, and Peter Sellars among them—describe her as first among equals".


title: Elizabeth LeCompte

text: LeCompte has lectured and taught at American University, the Art Institute of Chicago, Columbia University, Connecticut College, the Lincoln Center Theatre Directors Lab, Massachusetts Institute of Technology, New York University, Northeastern University, the O’Neill Center, Smith College, the University of London, and the Yale School of Drama. In 2018, "The New York Times" critics ranked "House/Lights" the 16th greatest American play since "Angels in America".


title: Elizabeth LeCompte

text: Among her honors, LeCompte has received the National Endowment for the Arts Distinguished Artists Fellowship for Lifetime Achievement in the American Theater, the MacArthur Fellowship, the Chevalier des Arts et des Lettres from the French Cultural Ministry, a Guggenheim Fellowship, a Rockefeller Foundation Fellowship, a United States Artists Fellowship, an Anonymous Was A Woman Award, the Theater Practitioner Award from Theatre Communications Group, The Skowhegan Medal for Performance, a Doris Duke Charitable Foundation Performance Artist Award and honorary doctorates from the New School for Social Research and the California Institute of the Arts. She was included in the 1993 Whitney Biennial. She won the 2016 Dorothy and Lillian Gish Prize.


title: Elizabeth LeCompte

text: In 1977 LeCompte began a relationship with actor Willem Dafoe. They never married and ended their relationship in 2004 after 27 years. The couple have one son, Jack.


title: Lola Bunny

text: Lola Bunny is a "Looney Tunes" cartoon character portrayed as an anthropomorphic female rabbit created by Warner Bros. Pictures. She is generally depicted as Bugs Bunny's girlfriend. She first appeared in the 1996 film "Space Jam".


title: Lola Bunny

text: The first character with the name "Honey Bunny" first appeared in the "Bugs Bunny's Album" comic book in 1953. That character was depicted as Bugs' cousin who is an explorer. The name was reused for a separate character intended as Bugs' love interest, who debuted in "Bugs Bunny Comic Book" #108 on November 15, 1966. Robert McKimson designed the prototype version of the character with Phil DeLara redesigning Honey and using her as a semi-regular in the "Looney Tunes" Gold Key Comics in the 1960s.


title: Lola Bunny

text: Honey's physical appearance varied over time. She was originally drawn with lop ears with a bow and pale yellow-tan fur. A female rabbit resembling this design appears at the end of the 1979 television special "Bugs Bunny's Thanksgiving Diet". Later merchandise using the character depicted her as more closely resembling Bugs, with grey fur but a more visibly feminine model and clothing, which was used until the early 1990s. She also made several video game cameos such as "The Bugs Bunny Crazy Castle", "Bugs Bunny's Birthday Ball", and "The Bugs Bunny Crazy Castle 2". Honey was also mentioned in the book "Looney Tunes: The Official Visual Guide".


title: Lola Bunny

text: In the mid-1990s, Warner Bros. started working on "Space Jam" (1996). There were plans to feature Honey Bunny. Some artists commented that Honey looked too much like Bugs, and eventually created Lola Bunny as a replacement.


title: Lola Bunny

text: Lola first appeared in the 1996 film "Space Jam." She is shown with tan fur, blonde bangs, and wears a cropped yellow tank top, purple shorts and a matching rubber band on both ears like a ponytail. She has light blue eyes and a shapely figure. Lola is voiced by Kath E. Soucie in the film.


title: Lola Bunny

text: Lola was created to serve as a romantic interest for Bugs. As soon as she appears, Bugs is instantly smitten and several other male characters ogle her. Throughout the film, there is a sub-plot of Bugs attempting to win her affection. Lola reciprocates Bugs feelings when she is nearly injured by one of the opponents in the basketball game, and Bugs saves her.


title: Lola Bunny

text: According to author Kevin Sandler, Lola's personality is a combination of the Hawksian woman, tomboy, and femme fatale archetypes. She is a straight-talking, no-nonsense woman who is extremely independent and confident. She is both highly athletic and extremely seductive in her behavior. Her catchphrase is "Don't ever call me 'Doll'". As animation director Tony Cervone explained, Lola was originally intended to be more of a "tomboy", but the production team feared that she would appear "too masculine" and chose to emphasize her "feminine attributes" instead.


title: Lola Bunny

text: Following "Space Jam", Lola has regularly appeared in solo stories in the monthly "Looney Tunes" comic published by DC Comics.


title: Lola Bunny

text: Lola appears as a news reporter twice, both in the direct-to-video film "Tweety's High-Flying Adventure" and the game "". Kath Soucie reprises her voice in both. In the series "Baby Looney Tunes", she is like her older counterpart in "Space Jam", having tomboyish traits and an affinity for basketball. She is voiced by Britt McKillip. In the action-comedy "Loonatics Unleashed", her descendant is Lexi Bunny.


title: Lola Bunny

text: Lola also appears in "The Looney Tunes Show", where she was voiced by Kristen Wiig. As opposed to her personality in "Space Jam", she is portrayed as an eccentric, airheaded, endearing, and cheerful young rabbit who tends to obsess over Bugs, whom she refers to as "Bun-Bun". She is very dedicated to achieving goals, but oftentimes tends to forget what she was doing. Lola is mostly involved in bizarre situations, either created by herself or when accompanied by her friend Daffy.


title: Lola Bunny

text: Bugs nevertheless appears to enjoy having her around, even surprising himself when declaring himself her boyfriend in "" where she helped Daffy get the courage to ask Tina Russo out on a date. Later in the series, Bugs and Lola are seen in multiple episodes spending time with each other.


title: Lola Bunny

text: Lola's wealthy parents Walter (voiced by John O'Hurley) and Patricia (voiced by Grey DeLisle in season 1, Wendi McLendon-Covey in season 2) appear in the show as well.


title: Lola Bunny

text: Lola is one of the two lead characters in the straight-to-video spin-off film "", however here she is voiced by Rachel Ramras.


title: Lola Bunny

text: The "New Looney Tunes" portrays Lola as a happy and friendly character. She appears in the segments "Hare to the Throne", "Lola Rider" and "Rhoda Derby". Her appearance is similar to "The Looney Tunes Show", although she wears a different outfit. She always shows eccentricity and maintains her carefree attitude. Her intrepid and adventurous side appears within some episodes, where she performs various sports. Lola returned in "" with an updated character design, voiced by Zendaya although Soucie was initially announced to be reprising the role.


title: Lola Bunny

text: Lola appeared in the preschool series "Bugs Bunny Builders". This version gained positive reception, for her defying gender stereotypes by working as a leader for a construction team.


title: Lola Bunny

text: Since Lola Bunny's first official appearance in 1996, the cartoon character has been voiced by a variety of voice actresses.


title: Lola Bunny

text: For the majority of the "Looney Tunes" series, Lola's character was voiced by Kath Soucie, an American voice actress. Soucie has voiced Lola in "Space Jam" (1996), "Tweety's High-Flying Adventure" (2000), "Looney Tunes Racing" (2000), "" (2000), the "Looney Tunes" webtoons (2001–2005), "Looney Tunes Dance Off" (2010) and "New Looney Tunes" (2015–2020).


title: Lola Bunny

text: From 2011 to 2014, American actress, comedian, writer, and producer Kristen Wiig, voiced Lola in "The Looney Tunes Show". For her portrayal, Wiig received several nominations and won the People's Choice Voice Acting Award in 2011.


title: Lola Bunny

text: Since her first appearance, Lola quickly became a fan favorite and an iconic character in the "Looney Tunes" franchise. She has frequently been regarded as an animated sex symbol. In 2020, she was named the "most attractive cartoon character across the world" based on global search volume per month. Shannon Carlin of Bustle.com praised Lola from "Space Jam," calling her "confident" and "talented". Dan Kahan of PopDust.com wrote that Lola was meant to be "ogled, both by in-world characters and viewers".


title: Lola Bunny

text: Lola from "The Looney Tunes Show" was well received by critics. CBR.com ranked Lola and Bugs No. 2 in their 10 Best Romances From Childhood Cartoons, stating that she is more "lively and vapid" than in "Space Jam", and is "pretty cute and funny to watch". IGN.com praised the character, calling her a "crazy but charming character" with Kristen Wiig doing "a phenomenal job". WhatCulture.com calls Lola more interesting compared to her first appearance, stating that the "Lola of this show is scatter-brained, strange, and incredibly off-putting, making her leagues more interesting and funny as a result." Jonathan North of Rotoscopers.com complimented Lola from the same series, saying that it "brought out Lola's character far better than her debut in "Space Jam" did."


title: Lola Bunny

text: In 2019, after watching the original "Space Jam" for the first time, Malcolm D. Lee, the director for "", said that he "felt off-guard on how Lola was too sexualized" and decided to turn her into the typical "strong woman" character of modern films, stating: "The original Lola Bunny was not politically correct...It's important to reflect the authenticity of strong, capable female characters." The new personality and look gained controversy, specifically among Twitter, as fans complained about her "nerfed" appearance that made her intentionally less physically appealing.


title: Lola Bunny

text: Author Kevin Sandler has said that Lola Bunny was created as a female merchandising counterpart to Bugs Bunny. The character's original merchandise now sells for far more than its original price on resale markets. For example, original dolls now sell for hundreds of dollars on eBay. However, Lola Bunny is not the only character to see a rise in contemporary popularity, as original "Looney Tunes" merchandise in general has gained nostalgic value.


title: Lola Bunny

text: Kristen Wiig, who portrayed Lola Bunny in "The Looney Tunes Show", received several nominations for her work and won the 1 Behind the Voice Actors Award.


title: Isomorphism

text: In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος "isos" "equal", and μορφή "morphe" "form" or "shape".


title: Isomorphism

text: The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are .


title: Isomorphism

text: An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a universal property), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every prime number , all fields with elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.


title: Isomorphism

text: The term is mainly used for algebraic structures. In this case, mappings are called homomorphisms, and a homomorphism is an isomorphism if and only if it is bijective.


title: Isomorphism

text: In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:


title: Isomorphism

text: Category theory, which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.


title: Isomorphism

text: Let formula_1 be the multiplicative group of positive real numbers, and let formula_2 be the additive group of real numbers.


title: Isomorphism

text: The logarithm function formula_3 satisfies formula_4 for all formula_5 so it is a group homomorphism. The exponential function formula_6 satisfies formula_7 for all formula_8 so it too is a homomorphism.


title: Isomorphism

text: The identities formula_9 and formula_10 show that formula_11 and formula_12 are inverses of each other. Since formula_11 is a homomorphism that has an inverse that is also a homomorphism, formula_11 is an isomorphism of groups.


title: Isomorphism

text: The formula_11 function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.


title: Isomorphism

text: Consider the group formula_16 the integers from 0 to 5 with addition modulo 6. Also consider the group formula_17 the ordered pairs where the "x" coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the "x"-coordinate is modulo 2 and addition in the "y"-coordinate is modulo 3.


title: Isomorphism

text: Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same. More generally, the direct product of two cyclic groups formula_22 and formula_23 is isomorphic to formula_24 if and only if "m" and "n" are coprime, per the Chinese remainder theorem.


title: Isomorphism

text: If one object consists of a set "X" with a binary relation R and the other object consists of a set "Y" with a binary relation S then an isomorphism from "X" to "Y" is a bijective function formula_25 such that:


title: Isomorphism

text: S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, well-order, strict weak order, total preorder (weak order), an equivalence relation, or a relation with any other special properties, if and only if R is.


title: Isomorphism

text: For example, R is an ordering ≤ and S an ordering formula_27 then an isomorphism from "X" to "Y" is a bijective function formula_25 such that


title: Isomorphism

text: In algebra, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example:


title: Isomorphism

text: Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing a common structure form a heap. Letting a particular isomorphism identify the two structures turns this heap into a group.


title: Isomorphism

text: In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.


title: Isomorphism

text: In graph theory, an isomorphism between two graphs "G" and "H" is a bijective map "f" from the vertices of "G" to the vertices of "H" that preserves the "edge structure" in the sense that there is an edge from vertex "u" to vertex "v" in "G" if and only if there is an edge from formula_31 to formula_32 in "H". See graph isomorphism.


title: Isomorphism

text: In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.


title: Isomorphism

text: In early theories of logical atomism, the formal relationship between facts and true propositions was theorized by Bertrand Russell and Ludwig Wittgenstein to be isomorphic. An example of this line of thinking can be found in Russell's "Introduction to Mathematical Philosophy".


title: Isomorphism

text: In cybernetics, the good regulator or Conant–Ashby theorem is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system.


title: Isomorphism

text: In category theory, given a category "C", an isomorphism is a morphism formula_33 that has an inverse morphism formula_34 that is, formula_35 and formula_36 For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.


title: Isomorphism

text: Two categories and are isomorphic if there exist functors formula_37 and formula_38 which are mutually inverse to each other, that is, formula_39 (the identity functor on ) and formula_40 (the identity functor on ).


title: Isomorphism

text: In a concrete category (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the category of topological spaces or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules), an isomorphism must be bijective on the underlying sets. In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).


title: Isomorphism

text: In certain areas of mathematics, notably category theory, it is valuable to distinguish between on the one hand and on the other. Equality is when two objects are exactly the same, and everything that is true about one object is true about the other, while an isomorphism implies everything that is true about a designated part of one object's structure is true about the other's. For example, the sets


title: Isomorphism

text: are ; they are merely different representations—the first an intensional one (in set builder notation), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets formula_42 and formula_43 are not —the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their cardinality (number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is


title: Isomorphism

text: and no one isomorphism is intrinsically better than any other. On this view and in this sense, these two sets are not equal because one cannot consider them : one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism.


title: Isomorphism

text: Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a finite-dimensional vector space "V" and its dual space formula_46 of linear maps from "V" to its field of scalars formula_47


title: Isomorphism

text: These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism formula_48


title: Isomorphism

text: This corresponds to transforming a column vector (element of "V") to a row vector (element of "V"*) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis".


title: Isomorphism

text: More subtly, there a map from a vector space "V" to its double dual formula_51 that does not depend on the choice of basis: For all formula_52


title: Isomorphism

text: This leads to a third notion, that of a natural isomorphism: while formula_54 and formula_55 are different sets, there is a "natural" choice of isomorphism between them.


title: Isomorphism

text: This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a natural transformation; briefly, that one may identify, or more generally map from, a finite-dimensional vector space to its double dual, formula_56 for vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory.