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It can be argued, based on their similar shape and stance, that Calder's earliest object mobiles have very little to do with kinetic art or moving art. By the 1960s, most art critics believed that Calder had perfected the style of object mobiles in such creations as the Cat Mobile (1966). In this piece, Calder allows the cat's head and its tail to be subject to random motion, but its body is stationary. Calder did not start the trend in suspended mobiles, but he was the artist that became recognized for his apparent originality in mobile construction. One of his earliest suspended mobiles, McCausland Mobile (1933), is different from many other contemporary mobiles simply because of the shapes of the two objects. Most mobile artists such as Rodchenko and Tatlin would never have thought to use such shapes because they didn't seem malleable or even remotely aerodynamic.
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Despite the fact that Calder did not divulge most of the methods he used when creating his work, he admitted that he used mathematical relationships to make them. He only said that he created a balanced mobile by using direct variation proportions of weight and distance. Calder's formulas changed with every new mobile he made, so other artists could never precisely imitate the work. Virtual movement By the 1940s, new styles of mobiles, as well as many types of sculpture and paintings, incorporated the control of the spectator. Artists such as Calder, Tatlin, and Rodchenko produced more art through the 1960s, but they were also competing against other artists who appealed to different audiences. When artists such as Victor Vasarely developed a number of the first features of virtual movement in their art, kinetic art faced heavy criticism. This criticism lingered for years until the 1960s, when kinetic art was in a dormant period.
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Materials and electricity Vasarely created many works that were considered to be interactive in the 1940s. One of his works Gordes/Cristal (1946) is a series of cubic figures that are also electrically powered. When he first showed these figures at fairs and art exhibitions, he invited people up to the cubic shapes to press the switch and start the color and light show. Virtual movement is a style of kinetic art that can be associated with mobiles, but from this style of movement there are two more specific distinctions of kinetic art.
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Apparent movement and op art Apparent movement is a term ascribed to kinetic art that evolved only in the 1950s. Art historians believed that any type of kinetic art that was mobile independent of the viewer has apparent movement. This style includes works that range from Pollock's drip technique all the way to Tatlin's first mobile. By the 1960s, other art historians developed the phrase "op art" to refer to optical illusions and all optically stimulating art that was on canvas or stationary. This phrase often clashes with certain aspects of kinetic art that include mobiles that are generally stationary.
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In 1955, for the exhibition Mouvements at the Denise René gallery in Paris, Victor Vasarely and Pontus Hulten promoted in their "Yellow manifesto" some new kinetic expressions based on optical and luminous phenomenon as well as painting illusionism. The expression "kinetic art" in this modern form first appeared at the Museum für Gestaltung of Zürich in 1960, and found its major developments in the 1960s. In most European countries, it generally included the form of optical art that mainly makes use of optical illusions, such as op art, represented by Bridget Riley, as well as art based on movement represented by Yacov Agam, Carlos Cruz-Diez, Jesús Rafael Soto, Gregorio Vardanega, Martha Boto or Nicolas Schöffer. From 1961 to 1968, GRAV (Groupe de Recherche d’Art Visuel) founded by François Morellet, Julio Le Parc, Francisco Sobrino, Horacio Garcia Rossi, Yvaral, Joël Stein and Vera Molnár was a collective group of opto-kinetic artists. According to its 1963 manifesto, GRAV appealed
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to the direct participation of the public with an influence on its behavior, notably through the use of interactive labyrinths.
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Contemporary work In November 2013, the MIT Museum opened 5000 Moving Parts, an exhibition of kinetic art, featuring the work of Arthur Ganson, Anne Lilly, Rafael Lozano-Hemmer, John Douglas Powers, and Takis. The exhibition inaugurates a "year of kinetic art" at the Museum, featuring special programming related to the artform. Neo-kinetic art has been popular in China where you can find interactive kinetic sculptures in many public places, including Wuhu International Sculpture Park and in Beijing. Changi Airport, Singapore has a curated collection of artworks including large-scale kinetic installations by international artists ART+COM and Christian Moeller. Selected works Selected kinetic sculptors
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Yaacov Agam Uli Aschenborn David Ascalon Fletcher Benton Mark Bischof Daniel Buren Alexander Calder Gregorio Vardanega Martha Boto U-Ram Choe Angela Conner Carlos Cruz-Diez Marcel Duchamp Lin Emery Rowland Emett Arthur Ganson Nemo Gould Gerhard von Graevenitz Bruce Gray Ralfonso Gschwend Rafael Lozano-Hemmer Chuck Hoberman Anthony Howe Irma Hünerfauth Tim Hunkin Theo Jansen Ned Kahn Roger Katan Starr Kempf Frederick Kiesler Viacheslav Koleichuk Gyula Kosice Gilles Larrain Julio Le Parc Liliane Lijn Len Lye Sal Maccarone Heinz Mack Phyllis Mark László Moholy-Nagy Alejandro Otero Robert Perless Otto Piene George Rickey Ken Rinaldo Barton Rubenstein Nicolas Schöffer Eusebio Sempere Jesús Rafael Soto Mark di Suvero Takis Jean Tinguely Wen-Ying Tsai Marc van den Broek Panayiotis Vassilakis Lyman Whitaker Ludwig Wilding Selected kinetic op artists
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Nadir Afonso Getulio Alviani Marina Apollonio Carlos Cruz-Díez Ronald Mallory Youri Messen-Jaschin Vera Molnár Abraham Palatnik Bridget Riley Eusebio Sempere Grazia Varisco Victor Vasarely Jean-Pierre Yvaral See also Gas sculpture Lumino kinetic art Robotic art Sound art Sound installation References Further reading External links Kinetic Art Organization (KAO) - KAO - Largest International Kinetic Art Organisation (Kinetic Art film and book library, KAO Museum planned) Modern art Types of sculpture Motion (physics) Contemporary art Visual arts genres
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The Mountain Between Us is a 2017 American drama film directed by Hany Abu-Assad and written by Chris Weitz and J. Mills Goodloe, based on the 2011 novel of the same name by Charles Martin. It stars Idris Elba and Kate Winslet as a surgeon and a journalist, respectively, who survive a plane crash, with a dog, and are stranded in the High Uintas Wilderness with injuries and harsh weather conditions. The film premiered on September 9, 2017, at the 2017 Toronto International Film Festival, and was theatrically released in the United States on October 6, 2017, by 20th Century Fox.
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Plot After their flight is canceled due to stormy weather, neurosurgeon Dr. Ben Bass (Idris Elba) and photojournalist Alex Martin (Kate Winslet) hire private pilot Walter (Beau Bridges) to get them from Idaho to Denver for connecting flights to Alex's wedding in New York and Ben's emergency surgery appointment in Baltimore. Walter, who has not filed a flight plan, suffers a fatal stroke mid-flight, and the plane crashes on a mountaintop in the High Uintas Wilderness. Ben, Alex, and Walter's Labrador Retriever survive the crash but Alex has injured her leg quite badly. Ben attends to her cuts and injured knee, and buries the pilot while she is unconscious.
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Once conscious, Alex thinks Ben has a better chance of finding help if he leaves her behind, but Ben refuses. Stranded for days with dwindling supplies, Alex grows skeptical that they will be rescued, although Ben wants to wait for help with the plane's wreckage. He agrees to climb a ridge to see if there is any sign of a road, but sees nothing but mountains and narrowly avoids falling down the side. Alex goes through Ben's things and listens to a message from his wife saying, 'I'm glad to have had this time with you'. Alex is found by a cougar who attacks the dog. She shoots a flare at the big cat, killing it. The dog returns and later, when Ben comes back he tends to the dog’s wounds. They cook the cougar giving them, Ben thinks, ten days of food.
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The two argue over waiting for rescue or descending the mountain to find help/a phone signal. Alex starts a lone descent down the mountain. Ben catches up, having located the tail end of the plane he finds a beacon - but it is smashed. The two hike down to the tree-line and spend the night in a cave. Using her telephoto lens, Alex thinks she sees a cabin. At the same time that Alex falls into freezing water, Ben comes across the empty cabin. He pulls her out, but she remains unconscious and severely dehydrated. Ben again saves her life by fashioning an IV. They stay there for several days while Alex recovers; Ben reveals that his wife died two years prior from a brain tumor. Eventually they have sex. As he sleeps, Alex takes his picture. Later, she again tells Ben to leave her behind to find help. Ben initially agrees but soon returns; they press forward again.
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The dog alerts them to a nearby timber yard. On their way toward it, Ben's leg gets caught in a bear trap. Alex cannot free him, but she reaches the yard and collapses in front of an approaching truck. Ben awakens in a hospital and goes to Alex's room, where he finds her with Mark (Dermot Mulroney), her fiancé. After a brief conversation, Ben leaves, heartbroken.
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Ben and Alex go their separate ways after the hospital, with Ben keeping the dog. Alex tries calling Ben, but he ignores her calls until she sends him photos she had taken on the mountain, writing that only he can understand them. This encourages Ben to call Alex. They meet at a restaurant in New York, where it is revealed that Alex is now a part-time teacher, and Ben is a consultant at trauma clinics in London because his frostbitten hands will not recover sufficiently for him to perform surgery again. Ben says he did not call Alex because he thought she had married; Alex says she could not go through with it because she fell in love with Ben. Outside the restaurant, Ben admits to Alex that they survived because they fell in love. Alex dismisses her feelings and reminds Ben of something he said on the mountain: "the heart is just a muscle." She tells him she doesn’t know how they could be together in the real world. They hug goodbye, and begin to depart in opposite directions. While
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walking away both become distraught, and finally turn and begin running back to each other. A split second before the two embrace, the screen cuts to black and the credits roll.
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Cast Idris Elba as Dr. Ben Bass, a neurosurgeon Kate Winslet as Alex Martin, a photojournalist Dermot Mulroney as Mark Robertson, Alex’s fiancé Beau Bridges as Walter, the pilot of the charter plane Raleigh and Austin as Walter's dog Production The project was first developed in January 2012, with Mexican director Gerardo Naranjo set to direct a script by J. Mills Goodloe. In August 2012, Scott Frank was hired to re-write the script. In November 2014, Hany Abu-Assad replaced Naranjo, and Chris Weitz was later hired to re-write the script.
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Abu-Assad has said of the film, "I really didn’t see an epic love story against the background of survival, I think optimism and hope is crucial to survive. And to go on with your life even if you’ve had a lot of bad luck. So if you give (in) to the bad luck, you will die. (But) if you fight the bad luck, you have a better chance to survive and make your life better. This is very simple wisdom, yes? But still very crucial especially in these kind of days, when everybody feels entitled to their good luck." Casting The film went through several lead casting changes. In March 2012, it was announced that Michael Fassbender would star as Bass, but by September 2014, Fassbender dropped out due to a scheduling conflict, and Charlie Hunnam replaced him. Margot Robbie also came on board to star as Alex. In November 2014, Robbie dropped out of the project, and Rosamund Pike entered negotiations for the lead role. In December 2015, both Hunnam and Pike dropped out.
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In February 2016, Idris Elba came on board, followed by Kate Winslet in June 2016. Dermot Mulroney joined the cast as Winslet's character's fiancé in early February 2017. Filming Principal photography started on December 5, 2016, in Vancouver, and continued until February 24, 2017. Elba and Winslet filmed scenes at the Vancouver International Airport and Abbotsford International Airport on December 7, 2016. Filming stopped for Christmas holidays, from December 20, 2016 to January 3, 2017. Filming resumed around Invermere and Panorama Mountain Village on January 4, 2017.
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Most of the filming took place in Canada, on the border of Alberta and British Columbia. Abu-Assad has described the locations as having very cold temperatures, and tough and harsh filming conditions. Many scenes were shot on a mountaintop, and he and the crew had to drive 40 minutes before reaching the film's base camp. When the weather was okay, they could board the helicopter to reach their destination along with their supplies.
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Music German-Iranian composer Ramin Djawadi composed & conducted the music for the film. The official trailer for the movie was released with "Dusk Till Dawn" by Zayn Malik and Sia Furler as the official soundtrack. The score from the film is now released at Lakeshore Records. Soundtrackdreams reviewed, 'The main theme from “The mountain between us” is the best advertisement for this score; a sweeping piano and violin theme, both grandiose and intimate at the same time, a proper dramatic opening that gets emotional from the first minutes without an adjustment period. The first cue is the kind of piece that could have very well sat at the end as the dramatic climax of the movie. The horn buildup towards the end joins the rolling piano in making sure this theme will end up as one of the most memorable he has ever written.
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Releases The film premiered at the 2017 Toronto International Film Festival, on September 9, 2017. The film was initially set for release in the United States on October 20, 2017 but was later moved up to October 6, 2017. Box office , The Mountain Between Us has grossed $30.3 million in the United States and Canada, and $30.7 million in other territories, for a worldwide total of $62.3 million, against a production budget of $35 million. In the United States and Canada, the film was released alongside Blade Runner 2049 and My Little Pony: The Movie, and was expected to gross $11–12 million from 3,088 theaters in its opening weekend. It ended up debuting to $10.1 million, finishing second at the box office, behind Blade Runner 2049 ($32.5 million). The film dropped 47% in its second weekend, making $5.7 million and falling to 5th.
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Critical response On Rotten Tomatoes, the film has an approval rating of 38%, based on 173 reviews, with an average rating of 5.10/10. The site's critical consensus reads, "The Mountain Between Us may be too far-fetched for some viewers to appreciate, but it's elevated by reliably engaging performances from Idris Elba and Kate Winslet." On Metacritic, the film has a weighted average score of 48 out of 100, based on 37 critics, indicating "mixed or average reviews". Audiences polled by CinemaScore gave the film an average grade of "A−" on an A+ to F scale.
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Jordan Mintzer of The Hollywood Reporter described the film as "an easily digestible love story-cum-survival tale that tosses two excellent actors in the snow and lets them do their thing," before concluding that "what really helps Mountain overcome its far-fetched scenario is the pairing of Winslet and Elba, who know how to turn up the charm tenfold yet make Alex and Ben seem (mostly) like real people." Tim Grierson of ScreenDaily noted that the film "struggles to balance its life-or-death stakes with its far more florid love story," but added that "the considerable chemistry between Kate Winslet and Idris Elba certainly helps sell this tearjerker."
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In his review for Variety, Peter Debruge described it as "a movie in which neither the subzero temperature nor the romantic heat penetrates more than skin deep." Also criticizing the film, Steve Pond of TheWrap found it unnecessarily lengthy and said that "a love story cheapens the grand survival story." Ignatiy Vishnevetsky of The A.V. Club criticized the central characters as "a couple of one-note personality-test types" with "zero romantic chemistry," and wrote that the script "actually tones down the howling outrageousness of Martin’s novel, which seems to miss the point. But, structurally, it’s the same junk. Problems pop out of nowhere and resolve themselves, while torturous motivations attempt to explain why characters would withhold basic information from one another for weeks..." References External links
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2010s adventure drama films 2010s disaster films 2017 romantic drama films 2010s survival films 2017 films 20th Century Fox films American adventure drama films American disaster films American films American romantic drama films American survival films 2010s English-language films Films about aviation accidents or incidents Films based on American novels Films based on romance novels Films directed by Hany Abu-Assad Films scored by Ramin Djawadi Films set in Utah Films shot in Alberta Films shot in London Films shot in Vancouver Films about interracial romance Mountaineering films Films with screenplays by Chris Weitz Chernin Entertainment films
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City, University of London, is a public research university in London, United Kingdom, and a member institution of the federal University of London. It was founded in 1894 as the Northampton Institute, and became a university when The City University was created by royal charter in 1966. The Inns of Court School of Law, which merged with City in 2001, was established in 1852, making it the university's oldest constituent part. City joined the federal University of London on 1 September 2016, becoming part of the eighteen colleges and ten research institutes that then made up that university.
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City has strong links with the City of London, and the Lord Mayor of London serves as the university's rector. The university has its main campus in Central London in the London Borough of Islington, with additional campuses in Islington, the city, the West End and East End. The annual income of the institution for 2019–20 was £245.0 million, of which £11.1 million was from research grants and contracts, with an expenditure of £218.4 million. It is organised into five schools, within which there are around forty academic departments and centres, including the Department of Journalism, the Business School, and City Law School which incorporates the Inns of Court School of Law.
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City is a founding member of the WC2 University Network which developed for collaboration between leading universities of the heart of major world cities particularly to address cultural, environmental and political issues of common interest to world cities and their universities. The university is a member of the Association of MBAs, EQUIS and Universities UK. Alumni of City include a Founding Father, members of Parliament of the United Kingdom, Prime Ministers of the United Kingdom, governors, politicians and CEOs. History Origins
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City traces its origin to the Northampton Institute, established in 1852, which was named after the Marquess of Northampton who donated the land on which the institute was built, between Northampton Square and St John Street in Islington. The institute was established to provide for the education and welfare of the local population. It was constituted under the City of London Parochial Charities Act (1883), with the objective of "the promotion of the industrial skill, general knowledge, health and well-being of young men and women belonging to the poorer classes". Northampton Polytechnic Institute was an institute of technology in Clerkenwell, London, founded in 1894. Its first Principal was Robert Mullineux Walmsley.
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Alumni include Colin Cherry, Stuart Davies and Anthony Hunt. Arthur George Cocksedge, a British gymnast who competed in the 1920 Summer Olympics, was a member of the Northampton Polytechnic Institute's Gymnastics Club and was Champion of the United Kingdom in 1920. In 1937 Maurice Dennis of the (Northampton Polytechnic ABC) was the 1937 ABA Middleweight Champion. Frederick Handley Page was a lecturer in aeronautics at the institute. The Handley Page Type A, the first powered aircraft designed and built by him, ended up as an instructional airframe at the school. The novelist Eric Ambler studied engineering at the institute. The six original departments at the institute were Applied Physics and Electrical Engineering; Artistic Crafts; Domestic Economy and Women's Trades; Electro-Chemistry; Horology (the science of time and art of clock-making); and Mechanical Engineering and Metal Trades.
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20th century A separate technical optics department was established in 1903–04. In 1909, the first students qualified for University of London BSc degrees in engineering as internal students. The Institute had been involved in aeronautics education since that year, and the School of Engineering and Mathematical Sciences celebrated the centenary of aeronautics at City in 2009. The institute was used for the 1908 Olympic Games; boxing took place there. In 1957, the institute was designated a "College of Advanced Technology". The institute's involvement in information science began in 1961, with the introduction of a course on "Collecting and Communicating Scientific Knowledge". City received its royal charter in 1966, becoming "The City University" to reflect the institution's close links with the City of London. The Apollo 15 astronauts visited City in 1971, and presented the Vice-Chancellor, Tait, with a piece of heat shield from the Apollo 15 rocket.
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In October 1995, it was announced that City University would merge with both the St Bartholomew School of Nursing & Midwifery and the Charterhouse College of Radiography, doubling the number of students in City's Institute of Health Sciences to around 2,500. 21st century The university formed a strategic alliance with Queen Mary, University of London, in April 2001. In May 2001, a fire in the college building gutted the fourth-floor offices and roof. In August 2001 City and the Inns of Court School of Law agreed to merge. Following a donation from Sir John Cass's Foundation, a multimillion-pound building was built at 106 Bunhill Row for the Business School. A new £23 million building to house the School of Social Sciences and the Department of Language and Communication Science was opened in 2004. The reconstruction and redevelopment of the university's Grade II listed college building (following the fire in 2001) was completed in July 2006.
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In 2007 the School of Arts received a £10m building refurbishment. A new students' union venue opened in October 2008 called "TEN squared", which provides a hub for students to socialise in during the day and hosts a wide range of evening entertainment including club nights, society events and quiz nights. In January 2010, premises were shared with the University of East Anglia (UEA) London, following City's partnership with INTO University Partnerships. Since then City has resumed its own International Foundation Programme to prepare students for their pre-university year. City was ranked among the top 30 higher education institutions in the UK by the Times Higher Education Table of Tables. In April 2011, it was announced that the current halls of residence and Saddler's Sports Centre will be closed and demolished for rebuilding in June 2011. The new student halls and sports facility, now known as CitySport, opened in 2015.
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In September 2016 The City University became a member institution of the federal University of London and changed its name to City, University of London. Campus City has sites throughout London, with the main campus located at Northampton Square in the Finsbury area of Islington. The Rhind Building which houses the School of Arts and Social Sciences is directly west of Northampton Square. A few buildings of the main campus are located in nearby Goswell Road in Clerkenwell. Other academic sites are: The City Law School (incorporating the former Inns of Court School of Law) in Holborn, Camden Bayes Business School in St Luke's, Islington, and at 200 Aldersgate in Smithfield, City of London INTO City in Spitalfields, Tower Hamlets Organisation and administration
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The rector of City, University of London, is ex officio the Lord Mayor of the City of London. The day-to-day running of the university is the responsibility of the president. The current president is Sir Anthony Finkelstein. Schools City, University of London, is organised into five schools: The City Law School, incorporating The Centre for Legal Studies and the Inns of Court School of Law School of Health Sciences, incorporating St Bartholomew School of Nursing & Midwifery School of Arts and Social Science, including the Department of Journalism School of Mathematics, Computer Science and Engineering Bayes Business School (Formerly Cass Business School)
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Finances In the financial year ended 31 July 2011, City had a total income (including share of joint ventures) of £178.6 million (2008/09 – £174.4 million) and total expenditure of £183.62 million (2008/09 – £178.82 million). Key sources of income included £39.58 million from Funding Council grants (2008/09 – £39.52 million), £116.91 million from tuition fees and education contracts (2008/09 – £104.39 million), £7.86 million from research grants and contracts (2008/09 – £9.29 million), £1.04 from endowment and investment income (2008/09 – £1.83 million) and £15.05 million from other income (2008/09 – £19.37 million). During the 2010/11 financial year, City had a capital expenditure of £9.77 million (2008/09 – £16.13 million). At year end, City had reserves and endowments of £112.89 million (2009/10 – £110.05 million) and total net assets of £147.64 million (2008/09 – £147.27 million). Academic profile Courses and rankings
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City, University of London, offers Bachelor's, Master's, and Doctoral degrees as well as certificates and diplomas at both undergraduate and postgraduate level. More than two-thirds of City's programmes are recognised by the appropriate professional bodies such as the BCS, BPS, CILIP, ICE, RICS, HPC etc. in recognition of the high standards of relevance to the professions. The university also has an online careers network where over 2,000 former students offer practical help to current students. The City Law School offers courses for undergraduates, postgraduates, master graduates and professional courses leading to qualification as a solicitor or barrister, as well as continuing professional development. Its Legal Practice Course has the highest quality rating from the Solicitors Regulation Authority.
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The Department of Radiography (part of the School of Community and Health Sciences) offers two radiography degrees, the BSc (Hons) Radiography (Diagnostic Imaging) and BSc (Hons) Radiography (Radiotherapy and Oncology), both of which are recognised by the Health Professions Council (HPC). Partnerships and collaborations CETL Queen Mary, University of London, and City, University of London, were jointly awarded Centre for Excellence in Teaching and Learning (CETL) status by the Higher Education Funding Council for England (HEFCE) in recognition of their work in skills training for 3,000 students across six healthcare professions. City of London City, University of London, has links with businesses in the City of London. City has also joined forces with other universities such as Queen Mary and the Institute of Education (both part of the University of London) with which it jointly delivers several leading degree programmes.
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LCACE London Centre for Arts and Cultural Exchange is a consortium of nine universities. It was established in 2004 to foster collaboration and to promote and support the exchange of knowledge between the consortium's partners and London's arts and cultural sectors. The nine institutions involved are: University of the Arts London; Birkbeck, University of London; City, University of London; The Courtauld Institute of Art; Goldsmiths, University of London; Guildhall School of Music & Drama; King's College London; Queen Mary, University of London, and Royal Holloway, University of London.
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WC2 University Network City is a founding member of the WC2 University Network, a network of universities developed with the goal of bringing together leading universities located in the heart of major world cities in order to address cultural, environmental and political issues of common interest to world cities and their universities. In addition to City, University of London, the founding members of WC2 members are: City University of New York, Technische Universität Berlin, Universidade de São Paulo, Hong Kong Polytechnic University, Universidad Autonoma Metropolitana, Saint Petersburg State Polytechnical University, Politecnico di Milano, University of Delhi, Northeastern University Boston and Tongji University.
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Erasmus Mundus MULTI City was selected as the sole British university to take part in the selective Erasmus Mundus MULTI programme, funded by the European Commission to promote scientific exchange between Europe and the industrialised countries of South-East Asia. It is the first Erasmus program to involve universities outside of Europe. In addition to City, the partner universities are: Aix-Marseille University (France), Univerzita Karlova v Praze (Czech Republic), Freie Universität Berlin (Germany), Universität des Saarlandes (Germany), Università di Pisa (Italy), Universidad de Sevilla (Spain), The Hong Kong Polytechnic University (Hong Kong, SAR China), Universiti Brunei Darussalam (Brunei), University of Macau (Macau, SAR China), Nanyang Technological University (Singapore), and National Taiwan University (Taiwan).
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UCL Partners City has joined the executive group of UCL Partners, one of five accredited academic health science groups in the UK. City was invited to join the partnership in recognition of its expertise in nursing, allied health, health services research and evaluation and health management. Student life Students' Union The City Students' Union is run primarily by students through three elected sabbatical officers, an executive committee and a union council, with oversight by a trustee board. The Students' Union provides support, representation, facilities, services, entertainment and activities for its members. It is run for students, by students.
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Student media City currently has two student-run media outlets, including Carrot Radio, which was co-founded by journalism postgraduates Jordan Gass-Poore' and Winston Lo in the autumn of 2018. Carrot Radio currently records weekday podcasts. The second is the student-led online magazine, Carrot Magazine. They recently released their first print magazine in December 2017. Other For a number of years, City students have taken part in the annual Lord Mayor's Show, representing the university in one of the country's largest and liveliest parades. Sustainability ranking City ranked joint 5th out of the 168 universities surveyed in the 2019 People & Planet league table of the most sustainable UK universities having climbed from 7th place in the 2016 league. In both the 2016 and 2019 rankings, it was the highest ranking University of London institution, and one of only four London institutions in the top twenty.
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The league table's Fossil Free Scorecard report, drawn from Freedom of Information requests, found that £800,000 (6.4%) of City's £12.5m endowment was invested in fossil fuels, and that the institution had not made a public commitment to fossil fuel divestment. It also noted nearly £1m of research funding into renewables since 2001 with just £64k of total funding from fossil fuel companies; and no honorary degrees or board positions held by fossil fuel executives. Notable people Notable alumni Government, politics and society Mahatma Gandhi – Leader of the Indian Independence Movement, graduated in 1891 from the Inns of Court School of Law (now part of The City Law School)
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Muhammad Ali Jinnah – founder of Pakistan, first Governor-General of Pakistan graduated from the Inns of Court school of Law (now part of The City Law School) Margaret Thatcher – Conservative Party Prime Minister of the United Kingdom from 1979 to 1990, graduated from the Inns of Court School of Law (now part of The City Law School) Clement Attlee – Labour Prime Minister of the United Kingdom from 1945 to 1951 H. H. Asquith – Liberal Prime Minister of the United Kingdom from 1908 to 1916 Tony Blair – Labour Party Prime Minister of the United Kingdom from 1997 to 2007, graduated from the Inns of Court School of Law (now part of The City Law School) Christos Staikouras – Finance Minister of Greece from 2019 to present Roderic Bowen – Welsh Liberal Party politician Robert Chote – chief of the Office for Budget Responsibility; former director of Institute for Fiscal Studies Ali Dizaei – former police commander
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Jody Dunn – Liberal Democrat politician, and a barrister specialising in family law Sir James Dutton – Royal Marine general and former deputy commander of the International Security Assistance Force Chloë Fox – Australian politician, former Labor MP for the South Australian electoral district of Bright Syed Sharifuddin Pirzada – Noted Pakistani lawyer & Politician. Also served as 5th secretary general of Organisation of Islamic Cooperation. James Hart – Commissioner of the City of London Police David Heath – Politician and Liberal Democrat Member of Parliament for Somerton and Frome Syed Kamall – Conservative Party politician and Member of the European Parliament for the London European Parliament constituency David Lammy – Labour MP for Tottenham Barbara Mensah, judge Liu Mingkang – Chinese Politician and Businessman, current Chairman of the China Banking Regulatory Commission, former Vice-Governor of the China Development Bank
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Jawaharlal Nehru – First Prime Minister of the Republic of India Houda Nonoo – Bahraini Ambassador to the United States Patrick O'Flynn – UK Independence Party MEP Stav Shaffir – Youngest member of the Israeli Knesset, leader of the social justice movement Aris Spiliotopoulos – Minister of Greek Tourism Ivy Williams – First woman to be called to the English bar
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Arts, science and academia
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L. Bruce Archer – British mechanical engineer and Professor of Design Research at the Royal College of Art Susan Bickley – Mezzo-soprano in opera and classical music George Daniels – Horologist, regarded as the greatest watchmaker of modern times and inventor of the coaxial escapement Jerry Fishenden – Technologist, former Microsoft National Technology Officer for the UK Julia Gomelskaya – Ukrainian contemporary music composer, professor of Odessa State Music Academy in Ukraine Norman Gowar – Professor of Mathematics at the Open University and Principal of Royal Holloway College, University of London Michel Guillon - British optometrist and researcher Clare Hammond – Concert pianist David Hirsh – Academic and sociologist Muhammad Iqbal – Muslim poet, philosopher and politician, born in present-day Pakistan, graduated from the Inns of Court School of Law and University of Cambridge
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John Loder – Sound engineer, record producer and founder of Southern Studios, as well as a former member of EXIT Sharon Maguire – Director of Bridget Jones's Diary Rhodri Marsden – Journalist, musician and blogger; columnist for The Independent Robin Milner – Computer scientist and recipient of the 1991 ACM Turing Award Bernard Miles - Actor and founder of the Mermaid Theatre. John Palmer – Instrumental and electroacoustic music composer Sebastian Payne – Journalist Ziauddin Sardar – Academic and scholar of Islamic issues, Commissioner of the Equality and Human Rights Commission Theresa Wallach – Pioneer female engineer, motorcycle adventurer, author, educator and entrepreneur, holder of Brooklands Gold Star. John Hodge – Aeronautical Engineer who played a key role in NASA and America's space race.
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Business and finance
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Winston Set Aung – Politician, Economist and Management Consultant, incumbent Deputy Governor of the Central Bank of Myanmar Brendan Barber – General Secretary of the Trades Union Congress Jonathan Breeze – Founder and CEO of Jet Republic, private jet airline company in Europe Michael Boulos – associate director of Callian Capital Group, and partner of Tiffany Trump William Castell – former Chairman of the Wellcome Trust and a Director of General Electric and BP, former CEO of Amersham plc Peter Cullum – British entrepreneur James J. Greco – former CEO and President of Sbarro Sir Stelios Haji-Ioannou – Founder of easyGroup Tom Ilube CBE, British entrepreneur and Chair of the RFU Bob Kelly – former CEO of Bank of New York Mellon and CFO of Mellon Financial Corporation and Wachovia Corporation Muhtar Kent – former CEO and Chairman of The Coca-Cola Company William Lewis – former CEO Dow Jones Publisher, The Wall Street Journal
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Ian Livingstone – chairman and co-owner, London & Regional Properties Liu Mingkang – former Chairman of the China Banking Regulatory Commission Dick Olver – former Chairman of BAE Systems, member of the board of directors at Reuters Syed Ali Raza – former president and Chairman of the National Bank of Pakistan Martin Wheatley – former CEO of the Financial Conduct Authority Brian Wynter – Governor of the Bank of Jamaica Durmuş Yılmaz – Governor of the Central Bank of Turkey
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Media and entertainment
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Samira Ahmed – Channel 4 News presenter, BBC News presenter, writer and journalist Decca Aitkenhead – journalist Joanna Blythman – Non-fiction writer, Britain's leading investigative food journalist Emily Buchanan – BBC World Affairs correspondent Sally Bundock – BBC Presenter Ellie Crisell – BBC Presenter Imogen Edwards-Jones – Novelist Gamal Fahnbulleh – Sky News Presenter and journalist Mimi Fawaz, BBC presenter and journalist Michael Fish – BBC weatherman Adam Fleming – CBBC Reporter Lourdes Garcia-Navarro – Journalist, Jerusalem foreign correspondent for National Public Radio (NPR) Alex Graham – Chairman of PACT and the Scott Trust Michael Grothaus – Novelist and journalist; author of Epiphany Jones Rachel Horne – BBC and Virgin Radio presenter and journalist Faisal Islam – BBC News Economics Editor Gillian Joseph – Sky News Presenter Kirsty Lang – BBC Presenter and journalist Ellie Levenson – Freelance Journalist and Author
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William Lewis – Journalist and editor of The Daily Telegraph Donal MacIntyre – Investigative journalist Sharon Maguire – Writer and Director, directed Bridget Jones's Diary Rhodri Marsden – Journalist, musician and blogger; columnist for The Independent Sharon Mascall – Journalist, broadcaster and writer; lecturer at the University of South Australia Lucrezia Millarini – Freelance Journalist and ITV Newsreader Dermot Murnaghan – Presenter on Sky News Tiff Needell – Grand Prix driver, Presenter of Fifth Gear on Five Maryam Nemazee – Presenter for Al Jazeera London Linda Papadopoulos – Psychologist, appearing occasionally on TV Catherine Pepinster – journalist, religion writer Raj Persaud – British consultant psychiatrist, broadcaster, and author on psychiatry Richard Preston – Novelist Gavin Ramjaun – Television presenter and journalist Sophie Raworth – Newsreader, presenter on BBC One O'Clock News Apsara Reddy – journalist
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Joel Rubin – World-renowned klezmer clarinetist Ian Saville – British magician Barbara Serra – Presenter for Al Jazeera London Sarah Walker – BBC Radio 3 presenter Josh Widdicombe - Comedian and presenter Mark Worthington – BBC Correspondent
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Notable faculty and staff Rosemary Crompton – Professor of Sociology Roy Greenslade – Journalist Steven Haberman – Professor of Actuarial Science at City, University of London Corinna Hawkes – Professor of Food Policy Rosemary Hollis – Professor of International Politics at City, University of London Jamal Nazrul Islam – Physicist, Mathematician, Cosmologist, Astronomer Ernest Krausz (1931-2018) - Israeli professor of sociology and President at Bar Ilan University David Leigh – Journalist David Marks – Psychologist Penny Marshall – Journalist Stewart Purvis – Broadcaster Denis Smalley – Composer Bill Thompson – Journalist David Willets – Conservative Member of Parliament for Havant; Shadow Secretary of State for Education and Skills
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Vice-Chancellors (Pre-2016)/Presidents (Post-2016) 1966–1974: Sir James Sharp Tait 1974–1978: Sir Edward W. Parkes 1978–1998: Raoul Franklin 1998–2007: David William Rhind 2007–2009: Malcolm Gillies 2009–2010: Julius Weinberg (acting) 2010–2021: Sir Paul Curran 2021–Present: Sir Anthony Finkelstein In popular culture City University's Bastwick Street Halls of Residence in Islington was the first home of MasterChef following its 2005 revival. References External links City, University of London City, University of London, Students Union Lists of Northampton Polytechnic Institute students List of Northampton Polytechnic Institute military personnel, 1914–1918 Optometry schools Schools of informatics Educational institutions established in 1894 1894 establishments in England Venues of the 1908 Summer Olympics Olympic boxing venues Universities UK
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In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order which are all isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties
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A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size. A finite field of order exists if and only if is a prime power (where is a prime number and is a positive integer). In a field of order , adding copies of any element always results in zero; that is, the characteristic of the field is . If , all fields of order are isomorphic (see below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, and they are unambiguously denoted , or , where the letters GF stand for "Galois field".
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In a finite field of order , the polynomial has all elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.) The simplest examples of finite fields are the fields of prime order: for each prime number , the prime field of order , , may be constructed as the integers modulo , . The elements of the prime field of order may be represented by integers in the range . The sum, the difference and the product are the remainder of the division by of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see ).
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Let be a finite field. For any element in and any integer , denote by the sum of copies of . The least positive such that is the characteristic of the field. This allows defining a multiplication of an element of by an element of by choosing an integer representative for . This multiplication makes into a -vector space. It follows that the number of elements of is for some integer . The identity (sometimes called the freshman's dream) is true in a field of characteristic . This follows from the binomial theorem, as each binomial coefficient of the expansion of , except the first and the last, is a multiple of . By Fermat's little theorem, if is a prime number and is in the field then . This implies the equality for polynomials over . More generally, every element in satisfies the polynomial equation .
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Any finite field extension of a finite field is separable and simple. That is, if is a finite field and is a subfield of , then is obtained from by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect. A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence is a finite field. Existence and uniqueness Let be a prime power, and be the splitting field of the polynomial
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over the prime field . This means that is a finite field of lowest order, in which has distinct roots (the formal derivative of is , implying that , which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of are roots of , as well as the multiplicative inverse of a root of . In other words, the roots of form a field of order , which is equal to by the minimality of the splitting field. The uniqueness up to isomorphism of splitting fields implies thus that all fields of order are isomorphic. Also, if a field has a field of order as a subfield, its elements are the roots of , and cannot contain another subfield of order . In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:
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The order of a finite field is a prime power. For every prime power there are fields of order , and they are all isomorphic. In these fields, every element satisfies and the polynomial factors as It follows that contains a subfield isomorphic to if and only if is a divisor of ; in that case, this subfield is unique. In fact, the polynomial divides if and only if is a divisor of . Explicit construction Non-prime fields Given a prime power with prime and , the field may be explicitly constructed in the following way. One first chooses an irreducible polynomial in of degree (such an irreducible polynomial always exists). Then the quotient ring of the polynomial ring by the ideal generated by is a field of order .
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More explicitly, the elements of are the polynomials over whose degree is strictly less than . The addition and the subtraction are those of polynomials over . The product of two elements is the remainder of the Euclidean division by of the product in . The multiplicative inverse of a non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions. Except in the construction of , there are several possible choices for , which produce isomorphic results. To simplify the Euclidean division, one commonly chooses for a polynomial of the form
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which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic , irreducible polynomials of the form may not exist. In characteristic , if the polynomial is reducible, it is recommended to choose with the lowest possible that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" , as polynomials of degree greater than , with an even number of terms, are never irreducible in characteristic , having as a root. A possible choice for such a polynomial is given by Conway polynomials. They ensure a certain compatibility between the representation of a field and the representations of its subfields. In the next sections, we will show how the general construction method outlined above works for small finite fields.
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Field with four elements The smallest non-prime field is the field with four elements, which is commonly denoted or It consists of the four elements such that and for every the other operation results being easily deduced from the distributive law. See below for the complete operation tables. This may be deduced as follows from the results of the preceding section. Over , there is only one irreducible polynomial of degree : Therefore, for the construction of the preceding section must involve this polynomial, and Let denote a root of this polynomial in . This implies that and that and are the elements of that are not in . The tables of the operations in result from this, and are as follows:
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A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of by , the values of must be read in the left column, and the values of in the top row. (Because for every in every ring the division by 0 has to remain undefined.) The map is the non-trivial field automorphism, called Frobenius automorphism, which sends into the second root of the above mentioned irreducible polynomial GF(p2) for an odd prime p For applying the above general construction of finite fields in the case of , one has to find an irreducible polynomial of degree 2. For , this has been done in the preceding section. If is an odd prime, there are always irreducible polynomials of the form , with in .
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More precisely, the polynomial is irreducible over if and only if is a quadratic non-residue modulo (this is almost the definition of a quadratic non-residue). There are quadratic non-residues modulo . For example, is a quadratic non-residue for , and is a quadratic non-residue for . If , that is , one may choose as a quadratic non-residue, which allows us to have a very simple irreducible polynomial . Having chosen a quadratic non-residue , let be a symbolic square root of , that is a symbol which has the property , in the same way as the complex number is a symbolic square root of . Then, the elements of are all the linear expressions with and in . The operations on are defined as follows (the operations between elements of represented by Latin letters are the operations in ): GF(8) and GF(27) The polynomial
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is irreducible over and , that is, it is irreducible modulo and (to show this, it suffices to show that it has no root in nor in ). It follows that the elements of and may be represented by expressions where are elements of or (respectively), and is a symbol such that The addition, additive inverse and multiplication on and may thus be defined as follows; in following formulas, the operations between elements of or , represented by Latin letters, are the operations in or , respectively: GF(16) The polynomial is irreducible over , that is, it is irreducible modulo . It follows that the elements of may be represented by expressions where are either or (elements of ), and is a symbol such that
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(that is, is defined as a root of the given irreducible polynomial). As the characteristic of is , each element is its additive inverse in . The addition and multiplication on may be defined as follows; in following formulas, the operations between elements of , represented by Latin letters are the operations in . The field has eight primitive elements (the elements that have all nonzero elements of as integer powers). These elements are the four roots of and their multiplicative inverses. In particular, is a primitive element, and the primitive elements are with less than and coprime with 15 (that is, 1, 2, 4, 7, 8, 11, 13, 14). Multiplicative structure
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The set of non-zero elements in is an abelian group under the multiplication, of order . By Lagrange's theorem, there exists a divisor of such that for every non-zero in . As the equation has at most solutions in any field, is the lowest possible value for . The structure theorem of finite abelian groups implies that this multiplicative group is cyclic, that is, all non-zero elements are powers of a single element. In summary: The multiplicative group of the non-zero elements in is cyclic, and there exists an element , such that the non-zero elements of are . Such an element is called a primitive element. Unless , the primitive element is not unique. The number of primitive elements is where is Euler's totient function. The result above implies that for every in . The particular case where is prime is Fermat's little theorem. Discrete logarithm
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If is a primitive element in , then for any non-zero element in , there is a unique integer with such that . This integer is called the discrete logarithm of to the base . While can be computed very quickly, for example using exponentiation by squaring, there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols, see Discrete logarithm for details. When the nonzero elements of are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo . However, addition amounts to computing the discrete logarithm of . The identity allows one to solve this problem by constructing the table of the discrete logarithms of , called Zech's logarithms, for (it is convenient to define the discrete logarithm of zero as being ).
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Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table of the same size as the order of the field. Roots of unity Every nonzero element of a finite field is a root of unity, as for every nonzero element of . If is a positive integer, an th primitive root of unity is a solution of the equation that is not a solution of the equation for any positive integer . If is a th primitive root of unity in a field , then contains all the roots of unity, which are . The field contains a th primitive root of unity if and only if is a divisor of ; if is a divisor of , then the number of primitive th roots of unity in is (Euler's totient function). The number of th roots of unity in is .
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In a field of characteristic , every th root of unity is also a th root of unity. It follows that primitive th roots of unity never exist in a field of characteristic . On the other hand, if is coprime to , the roots of the th cyclotomic polynomial are distinct in every field of characteristic , as this polynomial is a divisor of , whose discriminant is nonzero modulo . It follows that the th cyclotomic polynomial factors over into distinct irreducible polynomials that have all the same degree, say , and that is the smallest field of characteristic that contains the th primitive roots of unity. Example: GF(64) The field has several interesting properties that smaller fields do not share: it has two subfields such that neither is contained in the other; not all generators (elements with minimal polynomial of degree over ) are primitive elements; and the primitive elements are not all conjugate under the Galois group.
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The order of this field being , and the divisors of being , the subfields of are , , , and itself. As and are coprime, the intersection of and in is the prime field . The union of and has thus elements. The remaining elements of generate in the sense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree over . This implies that, over , there are exactly irreducible monic polynomials of degree . This may be verified by factoring over . The elements of are primitive th roots of unity for some dividing . As the 3rd and the 7th roots of unity belong to and , respectively, the generators are primitive th roots of unity for some in . Euler's totient function shows that there are primitive th roots of unity, primitive st roots of unity, and primitive rd roots of unity. Summing these numbers, one finds again elements.
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By factoring the cyclotomic polynomials over , one finds that: The six primitive th roots of unity are roots of and are all conjugate under the action of the Galois group. The twelve primitive st roots of unity are roots of They form two orbits under the action of the Galois group. As the two factors are reciprocal to each other, a root and its (multiplicative) inverse do not belong to the same orbit. The primitive elements of are the roots of They split into six orbits of six elements each under the action of the Galois group. This shows that the best choice to construct is to define it as . In fact, this generator is a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division. Frobenius automorphism and Galois theory In this section, is a prime number, and is a power of . In , the identity implies that the map
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is a -linear endomorphism and a field automorphism of , which fixes every element of the subfield . It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. Denoting by the composition of with itself times, we have It has been shown in the preceding section that is the identity. For , the automorphism is not the identity, as, otherwise, the polynomial would have more than roots. There are no other -automorphisms of . In other words, has exactly -automorphisms, which are In terms of Galois theory, this means that is a Galois extension of , which has a cyclic Galois group. The fact that the Frobenius map is surjective implies that every finite field is perfect. Polynomial factorization If is a finite field, a non-constant monic polynomial with coefficients in is irreducible over , if it is not the product of two non-constant monic polynomials, with coefficients in .
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As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field may be factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. They are a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, every computer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields. Irreducible polynomials of a given degree The polynomial factors into linear factors over a field of order . More precisely, this polynomial is the product of all monic polynomials of degree one over a field of order .
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This implies that, if then is the product of all monic irreducible polynomials over , whose degree divides . In fact, if is an irreducible factor over of , its degree divides , as its splitting field is contained in . Conversely, if is an irreducible monic polynomial over of degree dividing , it defines a field extension of degree , which is contained in , and all roots of belong to , and are roots of ; thus divides . As does not have any multiple factor, it is thus the product of all the irreducible monic polynomials that divide it. This property is used to compute the product of the irreducible factors of each degree of polynomials over ; see Distinct degree factorization. Number of monic irreducible polynomials of a given degree over a finite field The number of monic irreducible polynomials of degree over is given by where is the Möbius function. This formula is almost a direct consequence of above property of .
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By the above formula, the number of irreducible (not necessarily monic) polynomials of degree over is . A (slightly simpler) lower bound for is One may easily deduce that, for every and every , there is at least one irreducible polynomial of degree over . This lower bound is sharp for . Applications In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis of several widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, a secure internet connection to Wikipedia involved the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field. In coding theory, many codes are constructed as subspaces of vector spaces over finite fields.
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Finite fields are used by many error correction codes, such as Reed–Solomon error correction code or BCH code. The finite field almost always has characteristic of 2, since computer data is stored in binary. For example, a byte of data can be interpreted as an element of . One exception is PDF417 bar code, which is . Some CPUs have special instructions that can be useful for finite fields of characteristic 2, generally variations of carry-less product. Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing them modulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization and linear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then reconstruction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.
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Similarly many theoretical problems in number theory can be solved by considering their reductions modulo some or all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry were motivated by the need to enlarge the power of these modular methods. Wiles' proof of Fermat's Last Theorem is an example of a deep result involving many mathematical tools, including finite fields. The Weil conjectures concern the number of points on algebraic varieties over finite fields and the theory has many applications including exponential and character sum estimates. Finite fields have widespread application in combinatorics, two well known examples being the definition of Paley Graphs and the related construction for Hadamard Matrices. In arithmetic combinatorics finite fields and finite field models are used extensively, such as in Szemerédi's theorem on arithmetic progressions. Extensions
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Algebraic closure A finite field is not algebraically closed: the polynomial has no roots in , since for all in . Fix an algebraic closure of . The map sending each to is called the th power Frobenius automorphism. The subfield of fixed by the th iterate of is the set of zeros of the polynomial , which has distinct roots since its derivative in is , which is never zero. Therefore that subfield has elements, so it is the unique copy of in . Every finite extension of in is this for some , so The absolute Galois group of is the profinite group
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Like any infinite Galois group, may be equipped with the Krull topology, and then the isomorphisms just given are isomorphisms of topological groups. The image of in the group is the generator , so corresponds to . It follows that has infinite order and generates a dense subgroup of , not the whole group, because the element has infinite order and generates the dense subgroup One says that is a topological generator of . Quasi-algebraic closure Although finite fields are not algebraically closed, they are quasi-algebraically closed, which means that every homogeneous polynomial over a finite field has a non-trivial zero whose components are in the field if the number of its variables is more than its degree. This was a conjecture of Artin and Dickson proved by Chevalley (see Chevalley–Warning theorem).
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Wedderburn's little theorem A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn's little theorem states that all finite division rings are commutative, and hence are finite fields. This result holds even if we relax the associativity axiom to alternativity, that is, all finite alternative division rings are finite fields, by the Artin–Zorn theorem. See also Quasi-finite field Field with one element Finite field arithmetic Finite ring Finite group Elementary abelian group Hamming space Notes References W. H. Bussey (1905) "Galois field tables for pn ≤ 169", Bulletin of the American Mathematical Society 12(1): 22–38, W. H. Bussey (1910) "Tables of Galois fields of order < 1000", Bulletin of the American Mathematical Society 16(4): 188–206, External links Finite Fields at Wolfram research.
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Jack Spigot (4 May 1818 –June 1843) was a British Thoroughbred racehorse that won the 1821 St. Leger Stakes and was a sire in the early 19th century. His paternity is attributed to either Ardrossan or Marmion. His mother was a blind mare with a difficult temperament, whose unpredictable behavior necessitated that he be raised by a foster mare. He was named after one of his owner's tenant farmers, Jack Faucet. He won four of his six career starts before being retired from racing in early 1823. He is not considered to be a good sire. Jack Spigot died in June 1843 and was buried at Bolton Hall. Background Jack Spigot was foaled on 4 May 1818 on Middleham Moor near Bolton Hall in Leyburn, the family seat of his breeder Thomas Orde-Powlett, who was a younger brother of the Baron Bolton. Thomas Orde-Powlett was also a cousin of William Orde, Jr. the owner of Beeswing.
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Jack Spigot's dam, a sister to the stallion Bourbon, was bred by two stallions in the year preceding Jack Spigot's birth. She was first bred to Marmion on 14 April and then by Ardrossan in June 1817. Given the timing of Jack Spigot's birth, Ardrossan is likely his sire. Ardrossan was an unbeaten racehorse in three starts and stood at Rushyford near Durham. Marmion was only defeated once in eight career starts and is an ancestor of the influential broodmare Pocahontas.
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Jack Spigot's dam had a fractious temperament and had been blind and barren for four years before Jack Spigot's birth. Her first foal died at a young age in 1814 and she was owned by H. Peirse until she was sold to Powlett for a small sum. While owned by Peirse, the mare had killed other horses by kicking, notably Reveller's dam Rosette in 1816. She kicked about so much in the foaling paddock after Jack Spigot's birth that in order to prevent the death of her second foal, he had to be raised by a surrogate. A dapple grey foster mare similar in colouring to his own dam was procured from one of Powlett's tenants, Jack Faucet, to raise the foal. The sister to Bourbon mare also produced the grey colt Isaac in 1831 (sired by Figaro), who was considered to be the "best Cup horse of his day", running in flat and steeplechase races until he was 15 years old winning 53 races out of 172 starts.
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Before the horse's racing career, Powlett suggested that the colt be named "Jack Faucet" in honour of John Faucet. Faucet objected to the choice on the grounds that the horse was a good candidate for winning that year's St. Leger. Powlett allegedly quipped, "Well John, a Faucet's nothing without a Spigot" and the colt was subsequently named "Jack Spigot." Jack Spigot was the first horse that John F. Herring painted from life and the artist painted several portraits of the racehorse during and after his racing career. Racing career Jack Spigot was trained at Middleham by Isaac Blades ("J. Blades") who worked exclusively for Powlett. The colt was ridden in his early engagements by Bill Scott. After his St. Leger win, Jack Spigot developed an intense dislike of the jockey and would react violently if Scott attempted to approach the horse or even spoke in his presence. Jack Spigot was not raced extensively, starting only six times and winning four races including the St Leger.
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1820: two-year-old season In his only start as a two-year-old, Jack Spigot, ridden by Bill Scott, won a 320 guinea sweepstakes race at Doncaster Racecourse beating Mr. Riddell's colt Colwell. 1821: three-year-old season Jack Spigot did not run until the autumn Doncaster meeting and on 17 September started in the St. Leger Stakes against 12 other horses including the 1821 Derby winner Gustavus. Coronation was the front runner until three quarters of a mile when Lunatic took the lead for another half-mile. Jack Spigot and the mare Fortuna overtook Coronation with Jack Spigot edging out Fortuna at the finish by a margin of half a length. Two days later he won the Foal Stakes over a distance of one and a half miles beating his only competitor, the filly My Lady, in a "very excellent race."
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1822–1823: four and five-year-old seasons Returning as a four-year-old and refusing to be ridden by Bill Scott, Jack Spigot won the 450-guinea Newcastle Convivial Stakes ridden by Robert Johnson. In his only other start of 1822, he was third and last in the 3.25-mile Preston Cup, losing to Reveller and the 11-year-old Dr. Syntax. Jack Spigot only ran once more, on 14 April 1823 he was unplaced in the Craven Stakes won by the Duke of Rutland's colt Scarborough. Thomas Orde-Powlett retired from racing in September 1823 and put his horses up for sale at the Doncaster meeting. Jack Spigot was retained as a breeding stallion at Bolton Hall.
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Stud career Jack Spigot was retired to stud duty at Bolton Hall in 1824. He is not considered to be a good sire. Jack Spigot died in June 1843 and was buried in Yew Tree Court at Bolton Hall, his grave at one time surrounded by eight yew trees. Some of his bones were exhumed when a water pipe was laid across his grave years after his death. A cannon bone was recovered, set in silver and used to make a letter weight for Lord Bolton. Pedigree References 1818 racehorse births 1843 racehorse deaths Racehorses bred in the United Kingdom Racehorses trained in the United Kingdom Thoroughbred family 5-a St Leger winners
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"Jesus of Suburbia" is a song by American rock band Green Day. It was released as the fifth and final single from the group's seventh studio album, American Idiot, and the second song on the album. With the song running for 9 minutes and 8 seconds, it is Green Day's second longest song (with the band's longest song being fellow American Idiot song "Homecoming", which runs for 9 minutes and 18 seconds) and the group's longest song to be released as a single. The studio version of the song was considered to be unfriendly for radio, so it was cut down to 6½ minutes for the radio edit. The long version was still played on many album rock and alternative rock radio stations. At most live shows on the first leg of the group's 21st Century Breakdown World Tour, the band would pick a member from the audience to play guitar for the song. The single has sold 205,000 copies as of July 2010. Despite its commercial success, the song is the only hit single from the American Idiot album not to be
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included on the band's greatest hits album God's Favorite Band.
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Background American Idiot is a concept album that describes the story of a central character named Jesus of Suburbia, an anti-hero created by Billie Joe Armstrong. It is written from the perspective of a lower-middle-class suburban American teen, raised on a diet of "soda pop and Ritalin." Jesus hates his town and those close to him, so he leaves for The City. "Jesus of Suburbia" was the second multi-part song the group formed. Armstrong said it took "a long time" to write the song. Dirnt said that it came about from natural rehearsing between the trio. The song was an extension of Armstrong's desire to write the "Bohemian Rhapsody" of the future.
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As the song changes into different sections, Armstrong’s guitars were recorded differently. The musicians would "split the signal from the guitar and send it into an amp while simultaneously going direct with it," to achieve a sound reminiscent of "Revolution" by the Beatles or the style of David Bowie guitarist Mick Ronson. In addition, an overdrive pedal was employed to accentuate gain from the instrument, producing a "punchy" sound to each chord. For the first two sections of the song, Cool emulated Ginger Baker and Charlie Watts, two English drummers from the 1960s. For the final three, he drums in his style: "I'm tipping my hat to all these great drummers that I love, and then I kick the door down and do it … my style." In addition to Watts, Cool pulled inspiration from Keith Moon and Alex Van Halen. The song was composed by Green Day (with Billie Joe Armstrong writing the lyrics), and was co-produced by Rob Cavallo. "Jesus of Suburbia" has five movements: