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Wikipedia:Numan Yunusovich Satimov#0
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Numan Yunusovich Satimov (Russian: Нуман Юнусович Сатимов) (15 December 1939 – 22 September 2006) was a Soviet and Uzbek mathematician, Doktor Nauk in Physical and Mathematical Sciences, academician of the Academy of Sciences of Uzbekistan (2000), and corresponding member of the Academy of Sciences of UzSSR from 1979 to 2006, and a laureate of the Biruni State Prize (1985). He was a specialist in the theory of differential equations, control theory and their applications. == Biography == Satimov was born on 15 December 1939 in the city of Andijan in a working-class family. In 1956, he was accepted to the Central Asian State University at the Faculty of Physics and Mathematics. In 1958, Satimov continued his studies at the Moscow State University named after M. V. Lomonosov at the Faculty of Mechanics and Mathematics. After graduating from the university in 1962, he entered graduate school at the Uzbek Academy of Sciences' Romanovsky Institute of Mathematics, where he worked as a junior research fellow from 1965 to 1968. In 1968, under the guidance of Professor V. G. Boltyansky, Satimov defended his thesis. In 1977, under the guidance of Professor E. F. Mischenko, he defended his doctoral dissertation (at the specialized council of Steklov Institute of Mathematics). In 1978, he was awarded the title of professor. In 1979, he became a corresponding member of the Academy of Sciences of the Uzbek SSR, in 2000 – academician of the Academy of Sciences of the Republic of Uzbekistan. Since 1968, Satimov worked in Tashkent State University. In 1971, he became the head of a department at the Faculty of Applied Mathematics and Mechanics of NUUz. From 1974 to 1976, Satimov worked as a senior research fellow at Steklov Institute of Mathematics. From 1985 to 1987 he served as the dean of the Faculty of Applied Mathematics and Mechanics. Since 2000, he was a leading researcher at the Uzbek Academy of Sciences' Romanovsky Institute of Mathematics. Satimov died on 22 September 2006. He was buried at the Chagatai cemetery in Tashkent. == Scientific interests == Satimov's primary research interest included the theory of differential equations, control theory and their applications. He founded the Tashkent Scientific School on the theory of controls and differential games. He led the research seminar “Optimal processes and differential games” for over 35 years. Moreover, Satimov is the author of a textbook on differential equations and two monographs. He published more than 180 scientific papers; most of which have been translated and published in US and UK journals. Under his guidance, eight doctoral and more than twenty master's theses were prepared. Since 1970, Satimov worked on a new section of the theory of controlled processes – the theory of differential pursuit–evasion games. He paid particular attention to the development of L. S. Pontryagin's methods. As a result, Satimov proposed and later developed the so-called third (modified) method for solving the problem of persecution. == Bibliography == Задача об уклонении от встреч в дифференциальных играх с нелинейными управлениями // Дифференц. уравнения, 1973 г., Т. 9, No. 10, С. 1792—1797 (совместно с Е. Ф. Мищенко). Н. Сатимов, “К задаче убегания в дифференциальных играх с нелинейными управлениями”, Докл. АН СССР, 216:4 (1974), 744–747. N. Satimov, On the pursuit problem relative to position in differential games, Dokl. Akad. Nauk SSSR, 1976, Volume 229, Number 4, 808–811. Н. Ю. Сатимов, А. З. Фазылов, А. А. Хамдамов, “О задачах преследования и уклонения в дифференциальных и дискретных играх многих лиц с интегральными ограничениями”, Дифференц. уравнения, 20:8 (1984), 1388–1396. Н. Сатимов "Избежание столкновений в линейных системах с интегральными ограничениями" // Сердика, Болгарска, 1989 г., т. 15 (совместно с, А. З. Фазыловым). N. Yu. Satimov, M. Tukhtasinov, Game problems on a fixed interval in controlled first-order evolution equations, Mathematical Notes, 2006, Volume 80, Issue 3–4, pp 587–589. N. Yu. Satimov, M. Tukhtasimov, Evasion in a certain class of distributed control systems, Mathematical Notes, May 2015, Volume 97, Issue 5–6, pp 764–773. К оценке некоторых областей целочисленных точек. В кн.: Математические методы распознавания образов: Доклады 10-й Всероссийской конференции, Москва, 2001, РАН Вычислительный центр при поддержке Российского Фонда Фундаментальных Исследований, С. 125—126 (совместно с Б. Б. Акбаралиевым) == Notes ==
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Wikipedia:Numbertime#0
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Numbertime is a BBC educational numeracy television series for primary schools that was aired on BBC Two from 20 September 1993 to 3 December 2001. For its first four series, it was presented by Lolita Chakrabarti. El Nombre, an animated character used throughout the series, eventually became the concept for his own educational BBC children's television program; his name means "The Name" in Spanish, and not "The Number", which would be "El Número". The third line of his opening song and his farewell catchphrase were also changed several times during the series' run, to reflect their focus - however, the original ones ("Writing numbers in the desert sand" which was also used for the seventh series, and "Adios amigos, and keep counting" which was also used for the fourth, sixth, seventh, eighth and ninth series) remain the most famous. For the second series, El Nombre's tagline and farewell catchphrase were changed to "Drawing shapes in the desert sand" and "Adios amigos, and keep shaping up" respectively, while for the third series, they were changed to "Righting wrongs in the desert sand" and "Adios amigos, over and out" respectively; however, for the fourth series, his tagline was changed to "Counting numbers in the desert sand" (which was also used for the ninth series), and for the third episode of the fourth series, his farewell catchphrase was changed to "Adios amigos, and fetch some water". For the fifth series, both his tagline and farewell catchphrase were changed to "Telling time in the desert sand" and "Adios amigos, 'till the next time" respectively, while for the sixth series, his tagline was changed to "Using numbers in the desert sand"; finally, for the eighth series, his tagline was changed to "Counting money in the desert sand". == Series 1: Numbers 1 to 10 (Autumn 1993) == The first series, which is aimed at four- to five-year-olds, comprised ten episodes focusing on the numbers 1-10, in order; each episode opened with Lolita introducing herself to the viewer, and pulling the number for the episode off its string (which was hanging from the studio ceiling), then pushing it into its correct hole in a wall, and singing the main theme, One to Ten, as each of its holes lit up in turn, which was repeated throughout the programme. In between, there were comedy sketches (mostly based on nursery rhymes and fairy tales), and animations, the most famous involving El Nombre, the Mexican gerbil who parodied Zorro, showing little Juan how to draw numbers as his four-piece mariachi band played The Mexican Hat Dance (and said "Again!" once he had finished, as it gave them an excuse to play again), and a sequence encouraging the viewer to spot things of the number for each episode - it was the same video, with a different number of things each time (and a recurring song, Numbers All Around, which was sung by a group of children). Each episode ended with Lolita singing a song (or occasionally, introducing an animation), after which she would set viewers the challenge of looking for things in the number of the week's episode before saying that she would "see them next Numbertime". This series was originally aired on Mondays as part of the Daytime on Two strand at 9:45 am, and repeated at 2:00 pm on the same day; in Scotland, the 2:00 pm repeats were replaced with broadcasts of BBC Scotland's own schools series Over the Moon with Mr. Boom. Episode 1: Number 1 (20 September 1993) The Narrator proudly presents Nursery Rhyme Theatre No. 1 - Little Miss Muffet with several interruptions, and Little Jack Horner has one minute to Find 1 Plum on Sammy Sallow's game show of the same name; if he succeeds, he can choose one prize - one mountain bike, or one computer, or one picture of Sallow. This episode also ends with One Finger, One Thumb, Keep Moving, which is sung by the group of children who sang Numbers All Around earlier on in it. Episode 2: Number 2 (27 September 1993) The Narrator introduces Nursery Rhyme Theatre No. 2 with Marvo the Magician (and his Two Amazing Dickie Birds), and Sammy Sport reports on Jack and Jill's attempt to break the world record for the fastest time to run up a hill, fill a pail with water, and take it back down the hill again (under 2 minutes); this episode also ends with Lolita singing The Animals Went in Two by Two. Episode 3: Number 3 (4 October 1993) Sammy Sport is joined by Baa-Baa Black Sheep for the results of the "Win a Bag of Wool" competition, while Goldilocks invites viewers to Guess Whose House (for Sammy Sallow's game show of the same name); the Numbertime Top Ten also brings the viewers, at No. 3, the ever-popular "Three-Sided Triangles", who sing an original song named We're Triangles (Oh Yes We Are). Episode 4: Number 4 (11 October 1993) Farmer Giles introduces the finals of One Girl and Her Sheep at which Little Bo Peep and her dog Shep are competing, and the Knave of Hearts talks to the Queen of Hearts about her famous royal tarts; Lolita also tells the viewers to "grab their partners and take the floor" as she introduces a quartet of dancing squares, who sing an original song named Do the Square Dance. Episode 5: Number 5 (18 October 1993) Australian chef Wally Bee (and his assistant, Bruce) attempt to show the viewers how to cook five fat sausages on a barbecue for Barbecue Time, and Harry Headline pays a visit to the home of the Five Pigs Family for Five Minutes' Fame; Little Juan also has his 5th birthday in this episode's El Nombre sketch, and this episode ends with Lolita singing Fly, Little Dickey Birds, Round and Round. Episode 6: Number 6 (1 November 1993) A gardener named Fred enlists the help of his family in pulling up his enormous turnip for Garden Time, and Mr. and Mrs. Jones have to Take 6 Eggs on Sammy Sallow's game show of the same name and put them on either side of a seesaw to make it balance, for which they have three attempts to do so; this episode also ends with Lolita singing This Old Man in voiceover. Episode 7: Number 7 (8 November 1993) Sammy Sport travels to Scotland to see a remarkable fishing contest between the famous Seven Fat Fishermen, to see which one of them can catch the biggest fish on a bridge on the banks of the Clyde, and ordering seven lots of fish and chips turns out to be more trouble than it is worth for the Shopkeeper and his Customer (who has got seven children, and they all want fish and chips); this episode also ends with Lolita singing In My Little Garden, Now Promise You Won't Laugh (also known as One Potato, Two Potato). Episode 8: Number 8 (15 November 1993) Sammy Sport travels down to Shoeburyness to interview the Old Woman who Lives in a Shoe (who has 8 children), and the Policeman enlists the viewers' help in finding Wee Willie Winkie for Crime Spot; it is also evening in Little Juan's town in this episode's El Nombre sketch, and this episode ends with Lolita imagining what the world would be like if everything was eight-shaped as she visits the Planet of the Eights to sing an original song about it (and this is the only occasion in this series that she is not seen in her usual purple outfit, but a silver spacesuit). Episode 9: Number 9 (22 November 1993) Lucky the Cat looks back over her nine lives, as she guests on This Is My Life, and Harry Headline comes to the garden of Miss Mary, Mary, Quite Contrary, to look at the wonderful display of flowers for Garden Time; unfortunately, he ruins her chances of winning their "Best Flowers Competition", and she ends up coming ninth. This was also the only episode for this series to not end with a song - instead, Lolita shows the viewers a homemade necklace which has nine beads (three red, three blue, and three yellow), in reference to the episode's third animated sketch. Episode 10: Number 10 (29 November 1993) Farmer Giles (and his dog, Scruff) show viewers a wonderful, new, and very quick way to mow a meadow for Farming News, and Sammy Sport travels to Ten Pin Alley to watch Ten Pin Pete attempt to become the new Ten Pin Champ; Lolita also shows the viewers ten Russian dolls, and this episode ends with Ten in the Bed (which is, again, sung by the group of children who sang Numbers All Around earlier on in it). Also, because this episode was the last in the series, Lolita did not tell the viewers that she would "see them next Numbertime". Writer: Christopher Lillicrap Cast: Tony Bluto, Joanna Brookes, Regina Freedman, Jenny Jules, Andy McEwan, Mo Sesay El Nombre voices (uncredited): Sophie Aldred, Kate Robbins, Steve Steen Education Officer: Su Hurrell Music: Andrew Dodge Animation: Ealing Animation, Hedley Griffin, Peter Lang, Alan Rogers, Marcus Parker-Rhodes Film Camera: Nick Squires Film Sound: Eric Wisby, John Hooper Film Editor: Nick Hutchings Vision Mixers: Carol Abbott, Hilary Briegel Camera Supervisor: Eric Metcalfe Resource Co-ordinator: Roxanna White Studio Lighting: Bryn Edwards Studio Sound: Martin Deane Costume Design: Colin Lavers Make-up Design: Jane Walker Videotape Editor: John O'Connor Assistant Floor Manager: Sally Bates Production Manager: Oliver Cookson Production Assistants: Amarjit Ram, Hilary Hardaker Design: Bob Steer Executive Producer: Judy Whitfield Director: Andrea Christodoulou Series Producer: Clare Elstow © BBC Education MCMXCIII In 1994, BBC Enterprises (now BBC Worldwide) released a video entitled "Number Time" (BBCV 5359, and spelled with two words), containing sixty minutes of edited highlights from this series (it was the only one at the time); it was later rereleased as the second half of a "two-on-one" video in 1997 with the Words and Pictures "Alphabet Fun Time" video (BBCV 5357), which had originally been released at around the same time and contained fifty-eight minutes of highlights from that show's 1992 series (the "two-on-one" video in question was entitled "Alphabet Fun Time and Number Time", BBCV 5987, and it was rereleased in 1999). This series was later re-released in full as a "Video Plus Pack" in 1996 - only slightly altered to remove the episodes' opening titles. From 22 September to 1 December 1998, the BBC broadcast a "revised" version of this series as a lead-in to the sixth series; in place of Lolita, Bill (introduced in the fourth series) and Bernie (who joined him in the fifth one) introduced each episode with a number line of their own (however, Lolita's voice was still heard narrating some re-used animated sketches). All episodes except Numbers 1, 4 and 9 had the original live-action sketches replaced with the Dolls' House sketch (again, see Series 5), however in the episodes that retained the original live-action sketches, the prize scene in the Find One Plum sketch was cut, and the One Girl And Her Sheep sketch was omitted. The El Nombre sketches in the revised series were slightly lengthened, and sequences of children discussing the ten numbers (in the vein of those discussing the time-related concepts covered by the fifth series) were also introduced; although the Planet of the Eights sketch was also re-used, the vocal track was redubbed and Lolita was edited out of it. Whilst the Numbers All Around song sung by a group of children was retained in the revised series, the other two songs sang by them, the One Finger, One Thumb, Keep Moving song had the children forming the number one and shouting out it's number cut at the end of the song, and the Ten In The Bed song had a new video of the ten children in the bed whilst retaining the original song. == Numbers Plus (1994) == A lively maths series, designed as a follow-on of the first series, featuring four colourful "clown-like" characters called Mo, Sappy, Grimble and Jick who live in a house together. A robot called Trundle narrates each episode and helps them with their maths. There are also animations and songs. Episode 1: The Barbecue (10 January 1994) Mo, Sappy, Grimble and Jick decide to have a barbecue but first they add up how many people they're inviting to their barbecue and calculate what supplies are needed. They work out that sixteen guests and all four of themselves makes twenty people altogether. The gang learn to count in twos and fives, and Trundle shows the viewer a farmer who finds out how many sheep he has with the help of his sheepdog. Trundle naughtily knocks over the boxes of food which fall on top of Grimble and Jick rescues him but is so busy that she forgets to send out the invitations. In the end, it starts to rain so the gang decide to start the barbecue all over again tomorrow. Episode 2: The Picnic (17 January 1994) It's a lovely day for picnic outdoors - and for an argument. The gang get an introduction to division by learning to share everything fairly and equally, whether that means getting a quarter of a pizza, half a can of drink, or one banana each. Grimble's in a right mood about it, but is put to shame by the tapirs, tamarins, and wild horses that Trundle shows at Marwell Zoo Park. Episode 3: Sort It Out (24 January 1994) Mo takes the glass bottles to the bottle bank and leaves everybody else to sort out and clean up the house with Grimble in charge. Sappy, Grimble and Jick try to sort out the shopping by colour and then by shape. Trundle plays an odd one out game with the viewer at a Sainsbury's supermarket. When Mo comes back and sees the confusion, the gang put everything in the right places. Episode 4: Is It, Isn't It? (31 January 1994) The gang learn about handling data as they open a pet shop. They try to ask people what their favourite animal is to put in their pet shop but they get it all wrong. Sappy asks the people the right question but doesn't write down the answers and Jick writes down the answers but asks the wrong question. So Grimble goes out to get it right but finds out that tigers are the world's favourite animal. So the gang have no choice but to close the pet shop down. Episode 5: On With The Show (7 February 1994) The gang learn about the number one hundred as they put on a concert for one hundred people. They also learn to count in tens. Trundle shows the viewer that when someone is one hundred years old they get a birthday card from the Queen, one hundred years and runs in a game of cricket are called a century and that one hundred Roman soldiers were led by a man called a centurion. He also shows the viewer how people count the number of people in a football match, on a runaway train and in a school orchestra in special ways. Sappy in the end decides to conduct one hundred children. Episode 6: On The Shelf (21 February 1994) The gang learn that everything's made to measure as Mo needs somewhere to keep her books, so Grimble and Jick decide to put up a shelf but they keep getting it a bit wrong. Trundle shows two boys who are tall and short as try to set up two goals, shows how wide goals are in sports and shows how the ancient Egyptians built pyramids using a cubit. Episode 7: The Big Top (28 February 1994) The gang learn about weighing after Trundle sees a letter from a circus wanting a very light clown and wonders who is the lightest. The ringmaster gives some jobs for Mo, Sappy, Grimble and Jick to do. They clear up animal droppings, prepare hay, peanuts and greens for Big Ears (who they think is an elephant) as they weigh them first on a seesaw and mix up custard pies for the clown but Sappy and Jick don't use the scales to measure the weight of the ingredients. They soon find out that Big Ears is actually a rabbit and the ringmaster asks who of the gang is the lightest. Episode 8: It's My Birthday! (7 March 1994) The gang learn about measuring time as they prepare for Grimble's birthday. Mo and Sappy go shopping for a green clock shaped birthday cake and a green alarm clock for Grimble as a birthday present and Jick makes a green jelly for Grimble. Trundle shows the viewer Big Ben and explains how monks used sand clocks which told them when to ring the monastery bell and how the ancient Romans used candles that burn down all the way to the bottom to measure time. Episode 9: How Much? How Many? (14 March 1994) The gang want to buy some flowers, but first need to count their money and then work out what they can get for that amount. They are introduced to the different denominations of coins, how different combinations can be used, and the concept that a larger number of coins doesn't necessarily equate to a larger sum of money. Trundle shows the viewer a boy called Darren, who is blind and therefore has to identify coins using touch alone. Episode 10: All Shapes And Sizes (21 March 1994) Mo, Sappy, Grimble, Jick and Trundle are stuck inside a board game called "Shape Master" as they find out about 2D and 3D shapes. They each have to find a shape in order to escape. Mo finds a cube, Sappy finds a cylinder, Grimble finds a pyramid and Jick finds a hexagon. Trundle explains show cubes and cuboids can easily be seen in buildings, how sticks of rock are made and how shapes like hexagons, squares, rectangles and diamonds tessellate together. Cast: Nicola Blackman, Tony Marshall, Jefferson Clode, Nia Davies, Stephen Kemble & Roamer, Paul J. Reeve Written by: Christopher Lillicrap, Charles Way Education Officer: Su Hurrell Consultants: Mike Askew, Nick Morgan Music: Henry Marsh Graphic Designer: Iain Macdonald Animation: Kevin Wrench, Andrew Franks Visual Effects and Designer: Andy Lazell Designer: Andrée Welstead Hornby Director: Neil Ben Producer: Judy Brooks © BBC Education MCMXCIV == Series 2: Shapes (Spring 1995) == The second series, which is again aimed at four- to five-year-olds, comprised five episodes, focusing on the four basic two-dimensional shapes; each episode would open with Lolita standing by a mobile with the four basic shapes hanging from it and singing the series' main theme, Squares and Triangles, Circles and Rectangles as the mobile began to turn. Once it had stopped turning, she would walk to the shape that was nearest to her, then ask the viewers if they knew what it was, before it lit up and she told them - and from this series onwards, El Nombre was given two sketches per episode (the first to show Juan instances of that week's shape around the town, with Juan being clumsy and getting confronted by the other villagers after El Nombre had left, and the second to draw it in the desert sand). This series also featured four cut-out animated characters made up from the shapes its episodes were focusing on (a circular man with magical powers, a square robot with transformation powers, a triangular knight with a magical lance and a rectangular man); in the last episode, they worked together to build a house (after the rectangular man built one entirely out of rectangles, and the square robot, triangular knight and circular man transformed them into their own shapes). Each episode would end with Lolita singing an "extended" version of a song that had been heard earlier in the episode during a montage of the shape around the real world (for the last one, she continued over the credits) and setting viewers the challenge of seeing how many of that week's shape they could find before the next episode. The first two episodes of this series were, again, originally aired as part of the Daytime on Two strand on Mondays at 9:45 am, but 8 May 1995 was the year's May Day, so its third episode was aired the following day; the last two episodes were also aired on Mondays. Episode 11: Circles (24 April 1995) El Nombre shows Juan what a circle is (with a cart's wheel, a hoopla ring and a hoop with its stick), while a man named Terry introduces the world-famous "Ring a Ring o' Roses Formation Dance Team" (and their manager, Cynthia) on Come Prancing, and a prehistoric couple try to invent the round wheel (after the square and triangular ones) for their car for Great Moments in History; also, in this episode's second cut-out animated sketch, circular pawprints transform into a circular ladybird, a circular bird, a circular caterpillar (that becomes a circular butterfly), a circular fish which blows circular bubbles, a circular car which emits circular smoke from its circular exhaust, a circular man's head, and a circular bear (and in its CGI animated sketch, a circle gains two additional planes and proceeds to transform into a beach ball, a football, an orange and the planet Earth). Episode 12: Squares (1 May 1995) El Nombre shows Juan what a square is (with the then-unnamed Pedro and Juanita's frame, the then-also-unnamed Señor Manuel and Señor Chipito's draughts board, The Maggot and Cactus saloon's sign and a blackboard), while Bobby Cube asks the Shape Spotters on Let's Shape Up to name the square by pointing out its distinguishing features, a square robot builds a square dog (which turns on him after catching his scent, but he scares it away by transforming into a much bigger square dog with a big square that he runs to and climbs up), and Jill Scoop reports on Old King Cole who is wanting his square bowl (after round, triangular and rectangular ones) for Newsdesk; also, for this episode's CGI animated sketch, the yellow square in the bottom-right of a gameboard rises out of it to become a cube and has the numbers 1-6 written on each of its sides (which proceed to transform into six groups of dots of those respective numbers as the cube becomes the dice in a game of Snakes and Ladders). Episode 13: Triangles (9 May 1995) El Nombre shows Juan what a triangle is (with a musical triangle, a roadworks sign and a stepladder), while Aladdin finds the lamp with a triangle on it (after the ones with a circle and square on them) for his Uncle Abanazer, but he wastes its three wishes (one for each of the triangle's sides and corners) by turning Abanazer into a cat, and himself into a mouse, as well as making it disappear (which causes Abanazer to turn on him, but they both questioned why there was not a Genie in the lamp when asked); a triangular knight also sets out to slay a triangular dragon (but gets burned to a crisp the first attempt, and gets rained on along with the triangular dragon's fire getting put out the second attempt) while Bobby Cube asks the Shape Spotters on Let's Shape Up (who are the same ones from the previous episode, but have swapped positions) to identify the triangle (again, by pointing out its distinguishing features to them). Also, for this episode's CGI animated sketch, a triangle has a dotted line drawn inside it, then folds up to become a triangular-based pyramid, and rotates to show the numbers 1–4 on its sides as they light up before unfolding back into a triangle again. Episode 14: Rectangles (15 May 1995) El Nombre shows Juan what a rectangle is (with a plank of wood, the cart's side and the then-still-unnamed Pedro and Juanita's piece of cardboard), while Sammy Shape pays a visit to Old Mother Hubbard's cottage to find out "what makes a rectangle a rectangle" for Shapewatch (which, like the Crime Spot sketch from the previous series, is a spoof of Crimewatch) and met by her dog (who, as in the nursery rhyme, wants a bone, but they find a rectangular biscuit instead), a rectangular man goes for a swim at a swimming pool (after walking up a staircase, taking an elevator and walking up another staircase), Humpty Dumpty is asked to identify the shape of one of the bricks in the wall he is sitting on (which he does just before he falls), and a woman named Jane keeps in touch with her grandmother by writing a rectangular letter, posting it into a postbox which has a rectangular slot and door, then arriving at her house after she receives it. Episode 15: Shapes Together (22 May 1995) El Nombre helps Juan remember which shape is which (with the square frame, the circular hoop with its stick, a rectangular book and the triangular roadworks sign), while the rectangular man, the square robot, the triangular knight, and the circular man (who now has a circular body and a wheel for a foot) work together to build a house (after the rectangular man built one entirely out of rectangles, and the square robot, triangular knight and circular man transformed them into their own shapes) and show that "all different shapes work well together", Bobby Cube asks the Shape Spotters on Let's Shape Up to identify the "Shy Shapes" hiding in a cola can and Battenberg cake, and a rectangle folds up to become a hollow cylinder as two circles cover its ends and a second rectangle wraps itself around it as it becomes a can of baked beans; also, in this episode's second El Nombre sketch, Juan draws "El Nombre" (a square, rectangle, circle, and triangle stacked on top of each other), before drawing "the rope that hits him on the head and knocks him over". El Nombre then says that it has never happened to him - so Juan swings a rope at him, and tells him "It has now!". The sombrero-less El Nombre then chases Juan through the streets of the town to an extended version of his theme song (with the new tagline "Drawing shapes in the desert sand" at the end), but he never catches him; this is also the only old-style episode to only have one live-action sketch in it (rather than two or three). Cast: Gary Beadle, Carolyn Bonnyman, Mike Hayley, Anastasia Mulrooney, René Zagger Written by: Christopher Lillicrap El Nombre voices (uncredited): Sophie Aldred, Kate Robbins, Steve Steen Music: Mark Blackledge, Andrew Dodge, Sue Herrod/Seán de Paor Animations: Baxter Hobbins Slides Ltd, Ealing Animation, Frameline, Alan Rogers & Peter Lang Studio Resources Manager: Steve Lowry Camera Supervisor: John Hoare Sound: Dave Goodwin Lighting Director: Geoff Beech Costume Design: Rosie Cheshire Make-up Design: Judith Gill-Dougherty Vision Mixer: Carol Abbott Assistant Floor Manager: Sally Bates Graphic Designer: Ellen Monaghan Set Designer: Eric Walmsley Videotape Editor: St. John O'Rorke Executive Producer: Stacey Adams Studio Director: Phil Chilvers Production Team: Debby Black, Su Hurrell Producer: Kristin Mason © BBC Education MCMXCV In 1996, both this series and the next series were released on VHS as "Video Plus Packs" by BBC Educational Publishing (now BBC Active); the following year, they were also rereleased together as a double "Video Plus Pack" named "Numbertime Shapes/Side by Side". After the corporate change in 1997, BBC Education's then-current Internet address (http://www.bbc.co.uk/education/) was superimposed onto the four shapes (a red circle, a green triangle, a yellow square and a blue rectangle) seen at the end of this series' episodes, as well as the thirteen episodes of the next two series; however, it was never added to the end of the first series' episodes. == Series 3: Side by Side (Autumn 1995, broadcast Spring 1996) == Source: The third series (which is, once again, aimed at four- to five-year-olds) comprised five episodes, focusing on the concept of prepositions; each episode would open with Lolita singing the series' main theme, Under, Over, Everywhere (the mobile from the previous series was also visible in the background, but it now had an apple, three stickmen, a rainbow, a cloud with a hot-air balloon, a sun, a snake, a spider and a snail hanging from it). In this series, Juan gained three new friends named Pedro Gonzales, Juanita Conchita, and Maria Consuela Tequila Chiquita (Pedro and Juanita had also previously appeared in the second series), as well as a pet tarantula named Tanto - and each episode would end with Lolita reciting a rhyme or singing a song (but that in the second episode was an extended version of one that had been heard in voiceover earlier in the episode during an animated sketch about a fox). A sketch in the second episode of this series also parodied EastEnders as "GreenGrocers"; this was also the first series to credit the El Nombre voice actors at the end of its last episode (Sophie Aldred, who had played Ace on Doctor Who as well as one of the presenters of Words and Pictures, Spitting Image impressionist Kate Robbins, who had also voiced Jemima Wellington-Green on Round the Bend, and Steve Steen, who had played Lord Byron in Ink and Incapability, the second episode of Blackadder the Third). Although this series was made in Autumn 1995, it was not broadcast until 28 February 1996. This series and the next one were originally aired on Wednesdays in the Daytime on Two strand at 10:25 am; this series also premiered after a rerun of the second one finished, and the next one premiered when Daytime on Two returned after a two-week Easter break. Episode 16: Up, Down, On, and Off (28 February 1996) Juan does not have anything to do for the school concert so El Nombre tells him that he can recite Incy Wincy Spider with Tanto, a bear flies a kite (but it comes down in a tree, so he has to climb it to retrieve it), a window cleaner is annoyed by the incompetence of her colleague, Juan recites Incy Wincy Spider at the school concert (but Tanto will not come down the drainpipe, so El Nombre has to coax him) and three clowns named Boris, Doris and Ethel compete to see which one can raise the biggest laugh in Give Us a Giggle. Episode 17: In Front of, Behind, Before, and After (6 March 1996) Juan wants to take a photograph of himself and Mama together (but cannot because he is behind the Polaroid), an engine driver pulls some wagons behind (but the coupling snaps when he goes up a hill, so when they roll back down they are in front of him, and he has to signal for the driver of a second engine to help him), Jack and Jill must take Farmer Giles's horse and cart to market in Happy Ever After, a fox hides in his secret lair to escape a hunt (accompanied by a rhyme from Lolita in voiceover), Juan still does not have a photograph of himself and Mama together (so El Nombre takes it himself and blows himself up in the process), and the proprietor of GreenGrocers tries to get his four customers of the day to stand behind his stall and form a proper queue (until he gets upstaged by a toy salesman). Episode 18: Under, Over, On Top of, and Beneath (13 March 1996) Juan cannot score a goal past Pedro so El Nombre (who is not seen in his usual outfit in this episode, but what would later become Santo Flamingo United's strip) tells him to kick the ball over him, Princess Perfect wants a bed fit for a princess (in the sense of not being able to feel a pea under the mattress), a tortoise named Toby goes to a party but cannot get over the hedge to it (unlike the four other guests of a bird, a frog, a squirrel and a rabbit, but the last of them tells him to go under the hedge instead), Pedro is standing on a stool so Juan again cannot score past him (until El Nombre tells him to kick the ball under him, and when Pedro objects, they reduce the size of the goal, but El Nombre whispers to Juan to kick the ball to one side of Pedro before he leaves), and the Cow (of Hey Diddle Diddle) is scared of having to jump over the Moon (so the Little Dog volunteers the Dish instead, but he suggests running under it). Episode 19: Beside, Around, and Between (20 March 1996) Mama wants to go to Hurrell's store but there is a hole in the ground in front of it (so El Nombre and Juan tell her to walk around it), a doll pushes three building blocks with the numbers 1–3 on them together (but a breakdown-truck-driving clown accidentally crashes into them and scatters them, so he has to help her rearrange them again), Darren and Sharon Jam cannot make a mutual decision on where to put their new table on Lucky Lottery Winners, Mama has come out of Hurrell's store (but cannot help thinking there is another way around the hole, so El Nombre tells her that if she is brave like him, she can jump across it, but when he falls down it, she opts to go around it again instead), and Lord and Lady Posh give instructions to their gardener, Jarvis, on where he has to plant their roses and daisies. Episode 20: In, Out, and Through (27 March 1996) Juan has lost Tanto so El Nombre helps him and Mama to find him, the bear from the first episode tries to boil a saucepan of water over a fire but its bottom is missing, a magician turns her volunteer's watch and keys into an alarm clock and one big key, and two burglars named Bob and Bert break into a sweet shop and eat all the sweets but grow too fat to escape through the window so they get caught by a policeman named PC Nabb; from this point onwards, El Nombre also returned to one sketch per episode (except for in Episodes 21 and 26). Cast: Anthony Barclay, Laura Brattan, Joanna Brookes, Paul Cawley, Simon Corris, Chris Emmett, Mike Hayley, Brian Miller, Anastasia Mulrooney, Issy Van Randwyck, Elisabeth Sladen Written by: Andrew Bernhardt El Nombre written by: Christopher Lillicrap El Nombre voices: Sophie Aldred, Kate Robbins, Steve Steen Music: Mark Blackledge, Andrew Dodge Animations: Baxter Hobbins Slides Ltd, Ealing Animation, Malcolm Hartley, Alan Rogers and Peter Lang Studio Resources Manager: Steve Lowry Camera Supervisor: Roger Goss Sound: Dave Goodwin Lighting Director: Chris Kempton Vision Mixer: Carol Abbott Assistant Floor Manager: Jane Litherland Costume Design: Rosie Cheshire Make-Up Designer: Judith Gill-Dougherty Graphic Designer: Ellen Monaghan Set Designer: Gina Parr Videotape Editor: St. John O'Rorke Education Officer: Su Hurrell Studio Director: Phil Chilvers Production Team: Debby Black, Jane Straw Producer: Kristin Mason © BBC MCMXCV == Series 4: More or Less (Spring/Summer 1996) == The fourth series, which is aimed at five- to six-year-olds, comprised eight episodes focusing on the concepts of "more than" (addition) and "less than" (subtraction); each episode would open with Lolita singing the series' main theme, More or Less, in a studio filled with pillars. This series introduced the character of Bill (voiced by Paul Cawley), a green bird who could swallow and regurgitate almost any object whole - and from the fifth series onwards, he was joined by a purple cat named Bernie (voiced by Laura Brattan), later going on to appear at the beginning of each "revised" first-, sixth- and seventh-series episode. This series also featured a toad named Test, who would hop along the heads of fifteen multicoloured people lined up to form a numberline by the amounts its episodes were focusing on (they all wore red jumpers with the numbers 1–15 on them, but when Test was on their heads, the colour of their jumpers would change to green); each episode would end with Lolita singing a song (for the last one, she again continued over the credits, and it was also her final appearance, but because she did not know that she would be relieved of her presenting duties back then, she still told the viewers that she would "see them next Numbertime"). Two sketches in the first and seventh episodes of this series also parodied Percy Thrower and Sir David Attenborough as "Percy Grower" and "David Attencoat", while a third one in the third episode parodied Surprise, Surprise as "What a Surprise" (with Cinderella receiving a surprise visit from her Fairy Godmother) - and the Bill sketches of the second and sixth episodes also featured an enormous (but unnamed) beetle who chased after Bill after putting his feet into the eight wellingtons he regurgitated (in the second episode) and fell on top of him after pulling his last two wellingtoned legs up off a wall he was standing in front of (in the sixth episode), while the one of the fourth episode featured a "female" version of Bill who fell on top of him after he shook twenty mangoes off a tree. In the El Nombre sketches of the seventh and last episodes, Juan had his fifth birthday for the second time (only this time, Pedro, Juanita and Maria all brought him an extra candle for his cake because he was one year older, so he ended up with eight), and El Nombre's town gained a female mayor who also happened to be a balloon seller named Señora Fedora. Episode 21: One More (17 April 1996) A little old woman, a little old man and their little old cat enlist the help of one more friend (the Little Old Mouse) to help pull up their enormous turnip for Growing Bigger, Little Juan needs to play two cymbals (not one), Test hops from 3 to 7, contestant Sharon plays Find One More in order to win two prizes the same, Bill adds four flowers together and Juan needs to hit the cymbals one more time (to make four hits); also, in this episode's second animated sketch, a rather slow butterfly tries to keep up with his much faster friends, but when he joins them on a tree, he causes the branch they are standing on to break under their weight and they all fly away. Episode 22: Two More (24 April 1996) Freda Fantastic from Fantastic Fairytales presents The Elves and the Shoemaker, Little Juan and El Nombre juggle with seven of Mama's tomatoes which they shouldn't play with, Test hops from 7 to 13, Fred Fantastic of Fantastic Fairytales presents The Frog Prince, and Bill adds together eight wellingtons which belong to an enormous beetle (and when the beetle puts his feet into them, he chases after him); also, in this episode's second animated sketch, Noah will not let a mammoth come onto the Ark because there is only one of him, so he disguises himself as a pair of bears with two puppets and a tarpaulin, but he quickly gets discovered by Noah again. Episode 23: Three More (1 May 1996) A ladybird's nine babies and three extra are saved from a fire that's really smoke from her husband Arnold's barbecue, Little Juan and his friends are about to have Mama's very hot chili with tacos but there are only three chairs, Test hops from 2 to 8, Fairy Godmother presents What a Surprise with some surprising results for Cinderella, and Bill puts twelve books up on a shelf which ends with the shelf falling down under the books' weight; also, in this episode's second animated sketch, a kiwi notices his (three-toed) footprints in the sand and tries to count them, but finds it too hard to count in threes and eventually gives up by running away from the camera. Episode 24: Five More (8 May 1996) Snow White goes to the Wild Wood Takeaway and gets seven Good Fairy Cakes (declining cashier Grimbleshanks' first offer, Toad Burgers) for the dwarves' tea as it is Dopey's birthday, Little Juan accidentally blows Juanita's balloon up five more times which then bursts, Test hops from 1 to 11, Jack's mother will not let him climb up the beanstalk as it only has ten leaves on it (and it needs fifteen to get her to change her mind), and Bill shakes twenty mangoes off a tree which ends with a "female" version of himself falling on top of him; also, in this episode's second stop-motion animated sketch, a Tyrannosaurus Rex plans to eat a small Triceratops, but gets scared away when five larger Triceratopses, followed up by another five more behind them, suddenly appear behind their friend to protect him. Episode 25: One Less (15 May 1996) A magician makes six red balls disappear one at a time, Little Juan and his friends play musical chairs but they all have a chair to sit on, Test hops from 12 to 8, Carlotta Bottle tries to sing Ten Green Bottles but the bottles "don't-a fall-a" because property master Reg is not on hand to knock them down, and Bill eats three apples off a tree which ends with the branch he is standing on breaking under his weight and the zero he was displaying hitting him on the head; also, in this episode's second animated sketch, three dogs wait to be picked by prospective new owners at a pet shop (when there is only one remaining, he does a dance with a hat and cane to pass the time). Episode 26: Two Less (22 May 1996) Rebecca Testament reports for Numbertime News and interviews Mr. Noah and one of his sons, Ham, who used to have only six animals on the ark (two elephants, two lions and two doves), Little Juan and Tanto set out to get two melons for Mama's pie ("Melon Surprise") as a surprise but Señor Manuel the greengrocer has to save two of his melons for a special customer, Test hops from 10 to 4, a woman wins two coconuts and then another two coconuts from a total of eight on a shy, Bill subtracts six of the enormous beetle's wellingtons (who then falls on top of him), and Little Juan finds out that Señor Manuel's special customer is Mama all along who already has two melons; also, in this episode's second animated sketch, two monkeys get into an argument over two bananas which ends with them falling off their tree. Episode 27: Three Less (5 June 1996) The Early Bird loses all nine of his worms (and it is all David Attencoat's fault for saying that it isn't raining), Little Juan has his fifth birthday for the second time (but there are more than five candles on his birthday cake), Test hops from 11 to 5, Humpty Dumpty is scared of heights (so the crew members have to take all nine bricks of the wall away three at a time), and Bill watches nine leaves blow off a tree before winter comes, which ends with him being covered in snow; also, in this episode's second animated sketch (the recurring refrain of which is "Oh, no!"), three pegs blow off a washing line, three petals fall off three eight-petalled flowers (leaving five), a tricycle hits a stone and all three of its wheels fall off, three sides of a picture frame fall down (leaving one), and three mice steal three biscuits from a plate of eight (leaving five), one of whom only comes back to leave a note in front of the plate saying "Oh yes!". Episode 28: Five Less (12 June 1996) The magician from the fifth episode now makes fifteen beads disappear under three beakers five at a time, Little Juan almost floats away with six of Señora Fedora's balloons, Test hops from 13 to 3, Simple Simon has to find a penny for five of the Pieman's fifteen pies but they have all been sold to Old Mother Hubbard, Little Jack Horner and the Queen of Hearts by the time he does, and Bill recycles fifteen glass bottles (five green, five clear and five brown); also, in this episode's second animated sketch, an enormous snail eats a total of twenty trees from four gardens, and scares away a much smaller snail, a goat, a cow and a flock of birds as he descends on each of them. Cast: Anthony Barclay, Laura Brattan, Joanna Brookes, Otiz Cannelloni, Paul Cawley, Simon Corris, Chris Emmett, Mike Hayley, Brian Miller, Anastasia Mulrooney, Issy Van Randwyck, Elisabeth Sladen Written by: Christopher Lillicrap El Nombre voices: Sophie Aldred, Kate Robbins, Steve Steen Music: Mark Blackledge, Andrew Dodge, Derek Nash Animations: Ealing Animation, Arril Johnson, Alan Rogers and Peter Lang Studio Resources Manager: Steve Lowry Camera Supervisor: Roger Goss Sound: Dave Goodwin Lighting Director: Chris Kempton Vision Mixer: Carol Abbott Assistant Floor Manager: Jane Litherland Costume Design: Rosie Cheshire Make-Up Designer: Judith Gill-Dougherty Graphic Designer: Ellen Monaghan Set Designer: Gina Parr Videotape Editor: Paul Hagan Education Officer: Su Hurrell Studio Director: Phil Chilvers Production Team: Debby Black, Jane Quinn Producer: Kristin Mason © BBC MCMXCVI In 1997, this series was released on VHS as a "Video Plus Pack" by BBC Educational Publishing, and on 16 May 2013 it was rereleased on DVD as a "DVD Plus Pack" by BBC Active (as they are now known) with an accompanying teachers' book, but it is now out of print. == Series 5: Time (Winter 1997, broadcast Spring 1998) == Source: The fifth series, which is aimed at four- to six-year-olds, comprised ten episodes focusing on time-related concepts; (which the BBC previously covered in their maths programme Numbers Plus in the episode "It's My Birthday!") given that Lolita had been relieved of her presenting duties by this point, each episode was introduced by an animated man with a pocket watch for a head (who also appeared in a musical sketch at the end of the sixth episode). In this series, El Nombre's town was given the name of Santo Flamingo (its name was first heard in the sixth episode in reference to their local football team, although the sign above the doorway of its newly built school read "San Flamingo School"), and Juan gained a new teacher named Constanza Bonanza - and all except two of the episodes had sketches featuring a rarely speaking man named Tim (played by Toby Jones) who was coming to an understanding with time-related concepts (in fact, the only time he spoke was in the seventh episode, when he read out the text "Cook for half an hour" on the box of a big pie he had bought). This series also introduced the recurring sketch of the Dolls' House, which featured a cowgirl named Annie (played by Victoria Gay), a scarecrow named Scrap (played by Paul Cawley), a robot named Glimmer (played by Ashley Artus with a Geordie accent), a clock named Ticker (played by Mike Hayley), and from the sixth series onwards, a butler named Branston (played by Brian Miller); they were frequently visited by a pirate named Captain Kevin (played by Roger Griffiths), and on three occasions by a mechanic named Megamax (played by Fidel Nanton), Glimmer's girlfriend Princess Penelope (who had come to visit Scrap when he was ill and ate all his chocolates) and a Russian ballet dancer named Nadia Nokoblokov (who had come to perform Fryderyk Franciszek Chopin's Minute Waltz). For the seventh series, they were also frequently visited by a detective named Shelley Holmes (played by Issy Van Randwyck) - however, this recurring sketch would only go on until the end of that series. Although this series was made after the corporate change in Winter 1997 (as evidenced by the then-new BBC logo at the end of each episode), it was not broadcast until 13 January 1998. This series and the next one were originally aired on Tuesdays as part of the then-newly renamed Schools Programmes strand at 9:45 am. Episode 29: Night and Day (13 January 1998) In this first new-style episode, Tim wakes up in the middle of the night, brushes his teeth and pours himself a bowl of corn flakes (but has to wait until the morning for the milk), Little Juan wakes up all the other villagers because he does not know that a fiesta is held in the evening (rather than the morning or the afternoon), the residents of the Dolls' House take turns in getting their meals ready and Bill takes a Polaroid of both himself and his new co-star Bernie (who came in covered in mud, so he has to give her a bath before taking another one), but Bill got himself covered in mud while trying to get her in the bath, so she takes another Polaroid of him to show him. Episode 30: Days of the Week (20 January 1998) Tim sets off to referee a football match (but the forecast for that day is "windy", so he has to change into his best suit), Little Juan has football practice but cannot remember which day it is on, Bill decides to go on holiday to get away from the rain (but leaves Bernie behind with only seven cans of cat food, one for each day of the week, and a can opener) and Scrap and Glimmer get into an argument over a teddy bear so Captain Kevin has to sort it out; also, not only is the newspaper in this episode's Tim sketch dated from before the BBC's corporate change (10 April 1997), but it is factually incorrect, as it says that day was a Monday, when it was in fact a Thursday. Episode 31: Sequencing Events (27 January 1998) Tim tries to put on his new suit with a jacket, a pair of trousers, a hat, a pair of shoes, a pair of socks and a tie but keeps doing it in the wrong order, El Nombre helps Mama to make Delietta Smith (who is obviously a parody of Delia Smith)'s wonderful omelette with red and green peppers, Bill wakes Bernie up for dinnertime but forgets all about the food, and Scrap and Glimmer try to make a chocolate splodge cake for Annie; this is also the first episode to refer to the Dolls' House by name, despite being the third one in this series. Episode 32: Comparison of Time (3 February 1998) Tim grows two flowers and photographs them but cannot wait for the third, Pedro bets Juan he can find a spider who is faster than Tanto, Bernie wants to get an apple off a tree but cannot reach it (so Bill offers to fly up there, but when he learns he cannot, he challenges her to a tower-building contest) and Megamax is coming for tea at the Dolls' House so Annie, Scrap and Glimmer have to paint it, but the latter runs down, so when Megamax arrives, he deduces that he needs a new battery and gives him the "Max Pack Turbo Booster" to fix him. Episode 33: Clock Face (10 February 1998) El Nombre helps Little Juan to find out some things about the clock face for his homework, Scrap and Glimmer are bored so they decide to give Ticker's spring a big clean, and Bill trips over the sleeping Bernie and flies into his clock, causing all its numbers and both its hands to fall off; when he has put them all back on (with the help of Bernie), the clock's cuckoo calls at 3:00 and causes him to faint. Episode 34: O'Clock (24 February 1998) Little Juan and his friends prepare to go on a school outing to see Santo Flamingo United playing in the cup final at 3:00, Bernie takes a nap at 2:00 but Bill plays a prank on her by moving the clock an hour ahead and waking her up by replacing the cuckoo, and Scrap waits for the postwoman to deliver his new "Scrap Jacket" (which he had been going on about all night and keeping Glimmer from getting sleep). Episode 35: Half Past (3 March 1998) Tim (saying his only line, "Cook for half an hour", in this episode) cooks a big pie he has bought, but gets bored while waiting so eats all his other groceries, Scrap is ill so Annie asks Glimmer to take his temperature every hour and give him his medicine every half-hour (but his girlfriend, Princess Penelope, comes to visit), Bernie is enjoying a "cattuccino" at the foot of a Big Ben-like clock tower until Bill bungee-jumps from its minute hand, steals it when he springs back up to its face, drinks it and returns the empty cup to her, and Pedro, Juanita, and Maria all agree to meet Juan for a game of football at 2:30 (but Juan wonders how he will know when it is 2:30). Episode 36: Timing of Events (10 March 1998) Tim has hiccups and tries to get rid of them by drinking a glass of water then holding his breath for ten seconds (but gets interrupted by a crank phone call the first time, and a crank door-caller the second time), San Flamingo School is holding a three-legged race as a part of their sports day, Bill and Bernie prepare to launch each other into outer space using only a seesaw, and Nadia Nokoblokov comes to visit the Dolls' House because she needs someone to help her out with her new dance (Fryderyk Franciszek Chopin's Minute Waltz). Episode 37: Months and Seasons (17 March 1998) Tim receives a mysterious three-layered parcel on his doorstep in the middle of the night (which turns out to be a birthday cake) along with a musical birthday card (which plays a high-speed version of Happy Birthday to You to him), while El Nombre helps Juan and Juanita to put the four seasons in the right order for their homework (but they still have to draw a picture for each one), Bernie puts a smile on the face of a snowman (that turns out to be Bill) in winter, then the snow melts and some blossom grows on a tree in spring, Bill waters some flowers to help them grow and Bernie mows the lawn in summer, the leaves blow off the tree in autumn and Bernie throws a snowball at Bill and skates on the ice in winter before putting another smile on the face of another snowman (which, again, is Bill); the residents of the Dolls' House also have to organise their clothes for each season (but Glimmer thinks they are throwing them away). Episode 38: Telling the Time (24 March 1998) In his last appearance, Tim is woken up by a train, then eats a bowl of corn flakes, drinks a cup of coffee and builds a house of cards as three more trains pass (the last one causes him to knock it down), Ticker is broken because Annie, Scrap and Glimmer did not oil him when he asked so Captain Kevin gives him one of his spare ship's bells as a replacement, Bill and Bernie test each other's knowledge of time with their clock (and Bill gets two of them wrong), and Miss Constanza Bonanza, Pedro, Juanita and Mama all remind Juan that it is choir practice at 4:30, it is football practice at 5:00, to come to her house for tea at 5:30, and to return home at 6:00 respectively. Writers: Andrew Bernhardt, Toby Jones, Christopher Lillicrap Cast: Ashley Artus, Paul Cawley, Victoria Gay, Roger Griffiths, Mike Hayley, Tania Levey, Fidel Nanton, Issy Van Randwyck (Dolls' House), Toby Jones (Tim), Sophie Aldred, Kate Robbins, Steve Steen (El Nombre) Bill and Bernie (uncredited): Laura Brattan, Paul Cawley Music: Neil Ben, Mark Blackledge, Andrew Dodge, Richard Durrant, Derek Nash, Sandy Nuttgens Animations: Ealing Animation, Alan Rogers & Peter Lang, Ian Sachs Studio Resources Manager: Steve Lowry Sound: Dave Goodwin Lighting: Alan Jeffery Assistant Floor Manager: Alice Oldfield Costume Design: Rosie Cheshire Make-Up Design: Judith Gill-Dougherty Graphic Designer: Anne Smith Set Designer: Gina Parr Editor: St. John O'Rorke Education Officer: Su Hurrell Production Team: Helen Chase, Karen Keith, Debby Black Studio Director: Phil Chilvers Executive Producer: Anne Brogan Producer: Kristin Mason © BBC MCMXCVII The four shapes seen at the end of this series' episodes had been redesigned from those of the three previous ones, and were differently coloured to their originals as well (the circle was now green, the triangle was now yellow and the square was now red, but the rectangle kept its old-style colour of blue); also, in 1999, this series was released on VHS as a "Video Plus Pack" by BBC Educational Publishing. == Series 6: Numbers 11 to 20 (Winter 1998, broadcast Spring 1999) == Source: The sixth series (which is, again, aimed at five- to six-year-olds) comprised all ten episodes focusing on the numbers 11, 12, 13, 14, 15, 16, 17 18, 19 and 20. No episodes were ever created for the rest. Each episode would open with Bill and Bernie finding the position of the episode's number on their number line (which had been carried over from the "revised" version of the first series that was produced in 1998) - and in this series, Santo Flamingo gained an ice-cream seller named Señor Gelato, a carpenter named Señor Chipito (who had, once again, previously appeared in the second series as the owner of "The Maggot and Cactus" saloon) and a bandit named Don Fandango (who stole twenty gold coins from its newly built bank, which was managed by Señor Calculo, in the last episode). This series also saw former Blue Peter host Janet Ellis joining the El Nombre cast; although this series was made in Winter 1998, it was not broadcast until 12 January 1999. Episode 39: Number 11 (12 January 1999) Eleven soldiers (ten in two rows of five and the eleventh on the bottom) march around and ten of them make up the number eleven, Little Juan and his friends are preparing to go on a second school outing, this time to play a football match (but they only have ten football shirts, and El Nombre writes the episode's number in the desert sand), and Scrap receives a "Soccerbox" football game from his great-aunt Laura Litterbin (but it disappoints him as he does not like football, and one of the eleven white players goes missing). Episode 40: Number 12 (19 January 1999) Now in song, a pair of green slugs eat a gardener's twelve plants (that are in three rows of four) and make the number twelve, Juan bets Juanita that he can do more skips than her (he also mistakenly pronounces her surname as "Chiquita" in this episode) and Pedro bets both of them he can do more than either of them (but when doing it, he counts as fast as he can), and Captain Kevin does not want Branston to mention the number twelve as it reminds him of the final voyage of the good ship Rusty Bucket (when its crew found twelve biscuits). Episode 69: Number 13 (TBA) Thirteen balls (in three rows of four and one underneath) kept rolling around, Mama had told Juan and Pedro to that Señor Gelato is having a lucky day (and they have to help him with thirteen lucky charms, but Juan initially thinks he wanted to help him), and Annie was doing some dancing in the lounge of the Dolls' House (but Scrap, rather badly, wasn't very good if he had done thirteen steps). Episode 41: Number 15 (26 January 1999) Fifteen cars (in three rows of five) go nowhere and nobody seems to care, Señor Gelato accidentally drives his ice-cream tricycle into a three-legged table that Mama had told Juan and Pedro to take in to Señor Chipito for repairing (and they have to get a replacement wheel with fifteen spokes, but Juan initially thinks that it has more as he cannot tell which one he had started counting from), and Scrap and Glimmer are playing marbles in the lounge of the Dolls' House (but Glimmer is losing, rather badly, and his ten red marbles go missing). (TBA) Episode 42: Number 17 (2 February 1999) Seventeen windows (in four rows of four and one upon the door) on a building open up and the lights switch on when it gets dark, El Nombre helps Juan, Pedro and Señor Gelato pick up all seventeen of Señor Manuel's tomatoes and put them in a bag, and Scrap eats three lots of cake mix that is supposed to be for three of the twenty cakes for Captain Kevin's birthday party. Episode 43: Number 20 (9 February 1999) A spaceship beams up twenty stars (that are in four rows of five) then beams them down again in the constellation of the number twenty, Don Fandango steals twenty gold coins from the bank of Santo Flamingo (but Tanto bites a hole in his bag causing them all to fall out), and Nadia Nokoblokov pays another visit to the Dolls' House to perform another dance ("The Dance of Twenty Turns") in its conservatory; however, Ticker (whose role in the Dolls' House sketches had been lessened by this point) realises that if he stays in the bedroom, he does not have to watch, and Scrap eats a cake he had been told to put twenty candles on so Annie and Nadia punish him by putting him on the revolving podium that Nadia had brought with her for the dance and telling Glimmer to force him to do twenty "fast" turns on it. Clips from the first series (both incarnations) and the fourth series were also re-used in a musical sketch at the end of this episode. Writers: Andrew Bernhardt, Christopher Lillicrap Cast: Ashley Artus, Laura Brattan, Paul Cawley, Victoria Gay, Roger Griffiths, Mike Hayley, Tania Levey, Brian Miller El Nombre cast: Sophie Aldred, Janet Ellis, Kate Robbins, Steve Steen Music: Neil Ben, Mark Blackledge, Andrew Dodge, Richard Durrant, Derek Nash, Sandy Nuttgens Animations: Ealing Animation, Alan Rogers and Peter Lang Studio Resources Manager: Steve Lowry Camera Supervisor: Roger Goss Sound Supervisor: Dave Goodwin Lighting Director: Alan Rixon Vision Mixer: Hilary Briegel Floor Manager: Tom Hood Assistant Floor Manager: Alice Oldfield Costume Designer: Rosie Cheshire Make-Up Designer: Judith Gill-Dougherty Graphic Designer: Tom Brooks Set Designer: Gina Parr Videotape Editor: David Austin Education Officer: Jenny Towers Executive Producer: Clare Elstow Studio Director: Robin Carr Production Team: Debby Black, Liz Holmes Producer: Kristin Mason © BBC MCMXCVIII In 2000, both this series and the following one were released on VHS as "Video Plus Packs" by BBC Educational Publishing; on 4 May 2012, this one was re-released on DVD as a "DVD Plus Pack" by BBC Active, with an accompanying teachers' book. The pack also contained an audio CD, featuring songs from the series (and initially released as an audio cassette) - and this one is still in print. == Series 7: Numbers up to 100 (Autumn/Winter 1999) == The seventh series, which is once again aimed at five- to six-year-olds, comprised five episodes focusing on how to add and identify two-figure numbers up to 100; each episode would open with Bill and Bernie, whose number line had been replaced by a number square, and joined by a caterpillar named Limo (voiced by Peter Temple), who would crawl around the square to count out the numbers they required. In Santo Flamingo, Maria's sister Pepita Consuela Tequila Chiquita also started at San Flamingo School. This series, which is the Dolls' House's last, was originally aired on Thursdays as part of the Schools Programmes strand at 10:50 am. Episode 44: Counting On and Back (4 November 1999) Bernie stacks up twenty-nine plates (thrown to her by Bill), but does not know what comes after 29 so asks Limo to help; every time she stacks up ten more, she has to ask him again, but when she gets to 100, they fall over as a result of their weight. Scrap has also lost all except one of his fifty buttons so Shelley Holmes helps him to find them again, while Señor Gelato has only one cornet left so Juan volunteers to go down to Hurrell's store and get twenty-four more - but as he is about to set off, he is asked to get three extra ones. Episode 45: Missing Numbers (11 November 1999) While dusting the number square, Bernie sneezes four of the numbers (18, 46, 69 and 83) out of it; after Limo has put them all back in, Bernie dusts Bill's beak, causing him to sneeze the entire number square over. Maria's sister Pepita also starts at San Flamingo School (which viewers see the inside of for the first time), while Glimmer cooks apple pie and custard (his Aunt Dimity's very own recipe) but Scrap has got the pages of his cookbook mixed up and Limo has to put four more numbers (24, 38, 77 and 96) back into the number square. Episode 46: Counting in Tens (18 November 1999) Bernie is tired so she decides to have forty winks, and Limo counts them by crawling along each row of the number square; Scrap is also tired of licking envelopes (containing invitations to Glimmer's birthday party), while Juan takes all the money that he has saved up in his donkey bank to Santo Flamingo Bank and Bill and Bernie ask Limo if he can find the numbers 30 and 90 and add on ten. Episode 47: Patterns of Ten (25 November 1999) Bill makes twelve sandwiches and ten iced buns for a party (and Bernie makes ten more of both, but they both make ten rock cakes each), while Scrap and Glimmer are making party bags for a party of their own, Señora Fedora opens the 15th Annual Santo Flamingo Egg Festival and chooses Mama to make its giant omelette (for which she needs sixty-one eggs but only has twenty-one) and Bill and Bernie eat thirty of the "goodies" that they have made for their party, but Bill drops the ten remaining buns and Bernie slides on their remains into him causing him to drop ten of the sandwiches, and when they decide to have a dance he throws the ten of the remaining twelve into the air. Episode 48: Patterns of Five (2 December 1999) Bill has been shopping and bought twenty packs of five fish fingers (but forgot the chips, as Bernie finds out once she has put them all in the freezer), while Glimmer is painting a five-dot pattern, and San Flamingo School is holding a jumble sale; this was also the final episode to feature the Dolls' House (and although it credits Victoria Gay at the end of it, she did not appear as Annie in this series). Writers: Andrew Bernhardt, Christopher Lillicrap Cast: Ashley Artus, Laura Brattan, Paul Cawley, Victoria Gay, Roger Griffiths, Mike Hayley, Brian Miller, Issy Van Randwyck, Peter Temple El Nombre cast: Sophie Aldred, Janet Ellis, Kate Robbins, Steve Steen Music: Neil Ben, Mark Blackledge, Andrew Dodge, Richard Durrant, Derek Nash, Sandy Nuttgens Graphics: Anne Smith Animations: Ealing Animation, Alan Rogers and Peter Lang Studio Resource Manager: Steve Lowry Camera Supervisor: Gerry Tivers Sound Supervisor: Dave Goodwin Lighting Director: Dave Gibson Vision Mixer: Carol Abbott Floor Manager: Tom Hood Assistant Floor Managers: Beccy Fawcett, Catharine Hartley Costume Designer: Rosie Cheshire Make-Up Designer: Judy Gill-Dougherty Set Designer: Gina Parr Editor: David Austin Education Officer: Jenny Towers Studio Director: Phil Chilvers Executive Producer: Clare Elstow Production Team: Clare Arnopp, Debby Black, Debbie Wright Producer: Kristin Mason © BBC MCMXCIX == Series 8: Money (Autumn 2000) == The eighth series, which is aimed at five- to seven-year-olds, comprised ten episodes focusing on coin recognition, money problems, coin equivalents and change (which the BBC had previously covered in their maths programme, Numbers Plus, in the episode "How Much? How Many?" and in the second series of Megamaths); each episode would open with eight "money-spiders" (one for each coin - 1p, 2p, 5p, 10p, 20p, 50p, £1 and £2) coming down into view from the top of a tree. In this series, the currency of pounds and pence was introduced to Santo Flamingo, which gained a railway station named El Loco and a pizza delivery boy named Leonardo de Sombrero - and the recurring song from the first series, Numbers All Around, was also reworked (to focus on coins instead of numbers). This series also introduced the recurring sketch of Screensaver, which featured a screen named Screen (voiced by Sue Elliott-Nicholls), a variety of customers (who were all played by one-time Spitting Image impressionist Michael Fenton-Stevens), and a robot named T1L (pronounced "Til", and played by Paul Vates). This series was originally aired on each day of the working week for a fortnight as part of the Schools Programmes strand at 11:05 am. Episode 49: Coin Recognition to 10p (23 October 2000)1p Juan, Mama and Pedro go to the fair (but do not know if they have the right money for the coconut shy, roundabout or candy floss), while Bill is running a cake stall but Bernie cannot decide whether she wants one or not so flips four coins (1p, 2p, 5p and 10p) into the air but they do not come down again, and a cricket player wants to buy some glue from Screensaver to stick his old broken bat back together. Episode 50: Money Problems to 10p (24 October 2000)2p San Flamingo School is holding another jumble sale (this time to raise money for the new school bell), while Bill is now running a drink stall (but after he sells his last drink to Bernie for 8p, he has to close it, and Bernie then disguises herself as a vending machine to trick Bill into giving her money back), and a policeman wants to buy a timepiece from Screensaver (who try to sell him Big Ben for 10p). Episode 51: Coin Equivalents to 10p (25 October 2000)5p Little Juan and his friends are going to Santo Flamingo National Park to see the Giant Cactus, while Bill is now running an "everything" stall (but when Bernie manages to scrape 10p together from a 5p coin, two 2p coins and a 1p coin, she wheels it away after misconstruing the meaning of "everything 10p"), and a sailor wants to buy a cake for his mother from Screensaver (who try to sell him a wedding cake). Episode 52: Change from 10p (26 October 2000)10p Little Juan and his friends have now arrived at Santo Flamingo National Park and seen the Giant Cactus, while Bill and Bernie are hungry so they buy a snack for 5p, a carton of juice for 2p and a bar of chocolate for 3p (from three talking vending machines), and an old man wants to buy a new wheel for his wheel-basket from Screensaver because the old one is broken (and they try to sell him a bicycle wheel). Episode 53: Coin Equivalents to 20p (27 October 2000)20p Juan and Maria notice that Señor Manuel has put up a giant jellybean machine outside Hurrell's store, while Bernie plays "Coin Sports" and loses Bill's 1p, 2p, 5p and 10p coins after they have rolled into a river (making 18p altogether), and a Russian secret agent wishes to change his appearance at Screensaver (who sell him a Hawaiian shirt for 4p, a blond wig for 8p and a striped bow tie for another 8p). Episode 54: Change from 20p (30 October 2000)1p – 20p Mama takes Juan back-to-school shopping at Hurrell's store (and has him try on a hat which is too big for him), while Bernie pays 20p to go on an elephant ride and gets 5p change (but she finds it slow, so pays another 20p to go on a rocket ride and gets another 5p change, then masquerades as a cat ride in order to trick Bill into giving her a third 5p, and after Bill does that and she gives him the ride of his life, he wants to do it again, but she is tired out so she does not), and a chef wants to buy some butter and eggs from Screensaver. Episode 55: Coin Equivalents to 50p (31 October 2000)50p Señor Gelato promises Juan and Juanita an ice-cream if they go to the Santo Flamingo Bank and get him some coins in exchange for the 50p he gave them, while Bill and Bernie try to get out of a car park (but when Bill has scraped 50p together, the barrier catapults him into the air), and a businessman wishes to buy a pet from Screensaver (who try to sell him a "Starpet", from their own home planet, for 50p). Episode 56: Change from 50p (1 November 2000)1p – 50p Miss Bonanza is getting married (and Juan is responsible for the school's collection of 50p with which to buy her a present), while Bill pays 50p to have his photograph taken and gets 20p change (but it takes it before he can go inside the booth, so Bernie pays another 50p to have her photograph taken while Bill counts her change, but when she looks out of the booth to ask why it is taking so long, it takes a photograph of her tail, so they then combine their changes to have their photograph taken together), and a rock star wants to buy some new shoes from Screensaver because his old ones just "aren't his scene" (and they sell him a pair of blue suede platforms for 15p each). Episode 57: Coin Equivalents to £1 (2 November 2000)£1 Pedro, Juanita and Maria are sleeping over at Juan's house (and planning to watch a really scary film), while Bill and Bernie are doing their laundry (and have to pay 10p for washing powder in addition to £1 for the washing machine, but when Bernie inserts a £1 coin into the washing powder machine, she gets ten cups, and because she pours them all into the washing machine, it starts spewing foam all over the floor of the launderette) and a cowboy named Tom (nicknamed "Big T") wishes to buy a shirt with a big T on it from Screensaver. Episode 58: Up to £2 (3 November 2000)£2 Little Juan is to perform a concert to raise more money for the school bell (with Don Fandango masquerading as Mama and trying to steal all his earnings of £1.60), while Bill and Bernie want to go on a boat trip for £2 (but although Bernie has a £2 coin, they both have to go back home so Bill can scrape it together in other coins, and when they get back to the boat, Bernie gets on it before it pulls out, but Bill is not so lucky because he had to carry all his coins back there in a giant sack), and a non-speaking clown tries to get Screen and T1L (in their last appearance) to guess that he wants to buy a top hat (for 50p) and a rabbit to pull out of it (for £1.50) at Screensaver. Writers: Guy Hallifax, Christopher Lillicrap Screensaver Cast: Sue Elliott-Nicholls, Michael Fenton-Stevens, Paul Vates El Nombre Cast: Sophie Aldred, Janet Ellis, Kate Robbins, Steve Steen Bill & Bernie: Laura Brattan, Paul Cawley Music: Neil Ben, Mark Blackledge, Stephen Chadwick, Andrew Dodge, Richard Durrant, Derek Nash, Sandy Nuttgens Graphic Designer: Clive Harris Animations: Ealing Animation, Marcus Parker-Rhodes, Alan Rogers & Peter Lang Studio Resources Manager: Geoff Ward Camera Supervisor: Gerry Tivers Sound Supervisor: Dave Goodwin Lighting Director: Mike Le Fevre Vision Mixer: Diane Enser Floor Manager: Sara Putt Assistant Floor Manager: Caroline Broome Costume Designer: Rosie Cheshire Make-Up Designer: Judy Gill-Dougherty Visual Effects: Mike Tucker Offline Editor: Graeme Briggs Online Editor: David Ackie Educational Consultant: Helen Lazenby Programme Co-ordinator: Pauline Stone Assistant Producer: Claudia Marciante Executive Producer: Clare Elstow Series Producer: Kristin Mason Producer: Julie Ardrey © BBC MM In 2001, by which point VHS was becoming obsolete, this series was released on VHS as a "Video Plus Pack" by BBC Educational Publishing. == Series 9: Addition and Subtraction (Autumn/Winter 2001) == The ninth (and final) series, which is aimed at six- to seven-year-olds, comprised ten episodes focusing on the concepts of adding and subtracting similar to the fourth series (only without Lolita, live-action sketches based on nursery rhymes, or Test the Toad); in this series, Numbertime News, which had appeared in five episodes of the first series with Sammy Sport (played by Andy McEwan, who had played Matt Dillon in Death Without Dishonour, the twenty-sixth episode of Taggart), along with one episode of the fourth series with Rebecca Testament (played by Issy Van Randwyck), became a recurring sketch, with anchorwoman Tara Boomdeay (played by Elisabeth Sladen, who had played Sarah Jane on Doctor Who, as well as several characters in fifteen episodes over the third and aforementioned fourth series) and roving reporter Brad Quiff (played by Ian Connaughton). This series also saw Michael Fenton-Stevens returning to join the El Nombre cast and introduced the character of Addem (voiced by Richard Pearce), a green snake who discovered the series' concept in the company of a yellow ant named Ann (voiced by Moir Leslie) and a whole civilisation of other multi-coloured ants (mostly voiced by both Brian Bowles and Richard Pearce, but the Queen Ant was again voiced by Moir Leslie). This series was originally screened on Mondays as part of the Schools Programmes strand at 11:05 am, but 1 October 2001 was the first day of that year's four-day Labour Party Conference, so its fourth episode was not shown until the following week. Episode 59: Adding Two Numbers (10 September 2001) Brad Quiff investigates addition (with the "High Peaks Climbing Team"), Bernie challenges Bill by giving him some numbers for him to add onto and make twenty, Addem discovers a civilisation of ants (headed up by Ann), and Little Juan enters a competition on Radio Flamingo. Episode 60: Adding Three Numbers (17 September 2001) Juan and Pedro go shopping when Señor Calculo throws a barbecue, Bernie plants some seeds in window boxes and Bill helps her to add them up, Brad Quiff investigates how many chocolate bars the Malarkey Gang have stolen and Ann has to get twenty-nine ants into three houses. Episode 61: Patterns of Addition (24 September 2001) Addem helps Ann sort out beds for the "adolescants", Little Juan faces off against Don Fandango in the final of the Santo Flamingo Darts Championships, Brad Quiff reports on the popularity of the "Princess Patsy" doll and Bill and Bernie wash their socks at the launderette of the eighth series. Episode 62: Two-Step Addition (8 October 2001) Brad Quiff visits a country fair to meet the makers of buns, El Nombre reads Little Juan a bedtime story (about Don Fandango robbing the Santo Flamingo Bank), Ann needs to prepare twenty-five meals for the "Accounts Department" and Bill and Bernie try to make carrot juice. Episode 63: Addition with Partition (15 October 2001) Ann learns about adding acorns in hundreds, Brad Quiff reports on the opening of the brand-new "Whizzo Lolly Factory", Bill helps Bernie count her pennies as she is planning to "shop 'till she drops", and Juan and Pedro earn pocket money by picking lemons for Señor Manuel. Episode 64: Subtracting One from Another (5 November 2001) Brad Quiff reports on an American football team passing the ball back down a numberline, Bernie challenges Bill again by giving him some more numbers (this time to subtract from them and leave ten), a pair of cowboy ants have to take twenty-seven aphids to the milking shed and Juan is going on holiday to Costa Fortuna with Mama, Pedro, Juanita and Maria after winning the competition from the first episode. Episode 65: Patterns of Subtraction (12 November 2001) The Queen Ant decides to hold a regatta, Juan and his party get on the plane to Costa Fortuna, Bill washes some more of his socks at the launderette but Bernie tells him that he needs to separate the whites from the coloureds, and Brad Quiff reports on how crowds have been gathering for Punch and Judy performances all day; for this episode's adaptation of the story (which was also frequently adapted by camp entertainer Mr. Partridge on the BBC's own Hi-de-Hi!), Mr. Punch steals Judy's marbles from her box while she is asleep. Episode 66: Addition and Subtraction Difference (19 November 2001) Brad Quiff reports on an annual tug-of-war contest between the Diddletown Dodgers and the Softville Saints, Ann needs eighty-two candles for the Queen Ant's birthday cake, Juan and his party arrive at the Sea View Hotel in Costa Fortuna (where they meet their guide, Pablo) and Bill and Bernie insert two 20p coins into two of the three talking vending machines of the eighth series, to buy a snack for 16p and a carton of juice for 14p (receiving 4p and 6p change); they then combine their changes to buy a bar of chocolate for 10p from the third talking vending machine. This is also one of only two episodes to have an El Nombre sketch that is not set in Santo Flamingo at all. Episode 67: Two-Step Subtraction (26 November 2001) Juan and Pedro go to the fair in Costa Fortuna (where their guide, Pablo, fronts a ring-toss game), Ann has to fill forty-five places in the "Accountants"' new building, Bill is running an apple stall (but when Bernie wants to buy three, he finds out he has not got any, so they both pick some off an apple tree) and Brad Quiff reports on "Doreen's Sweet Shop" getting robbed of a large pile of chocolate eggs. Episode 68: Plus and Minus (3 December 2001) In the show's last episode, a squad of 100 marching ants keeps breaking up and coming back together, Juan and his party are on the plane back to Santo Flamingo (and when they get back to its airport, they are told that they can only bring back a certain amount of things at customs), Brad Quiff reports on the "Numbertime News Live Formation Climbing Team" (who show the viewers a trick to remember their sums) and Bill and Bernie have a cup of tea (but Bill puts ten cubes of sugar in his, and when Bernie asks him what he is doing, she makes him forget how many he has put in there); even though this was the last episode, El Nombre got a second series of his spin-off show in 2003. Writers: Andrew Bernhardt, Guy Hallifax, Christopher Lillicrap Cast (Numbertime News): Ian Connaughton, Elisabeth Sladen Cast (El Nombre): Sophie Aldred, Janet Ellis, Michael Fenton-Stevens, Kate Robbins, Steve Steen Cast (Addem and the Ants): Brian Bowles, Moir Leslie, Richard Pearce Music: Mark Blackledge, Archie Brown, Charles Casey & Simeon Jones, Andrew Dodge, Derek Nash, Sandy Nuttgens & Neil Ben Graphic Design: Sue Hattam Animations: Jon Aird, Ealing Animation, Phew!, Alan Rogers & Peter Lang, Ian Sachs Location Camera: Chris Sutcliffe, Gavin Richards Location Sound: Tony Cogger Sound Supervisor: Dave Goodwin Assistant Floor Manager: Tracy Jane Read Costume Designer: Rosie Cheshire Make-Up Designer: Judy Gill-Dougherty Set Design: Gina Parr Editor: Jon Bignold Education Consultant: Barbara Allebone Programme Co-ordinator: Clare Arnopp Director: Andrea Parr Executive Producer: Sue Nott Producer: Kristin Mason © BBC MMI In 2002, by which time VHS was even more obsolete, this series was released on VHS as a "Video Plus Pack" by BBC Educational Publishing. == Radio series == The first series was accompanied by a ten-part radio series on BBC School Radio entitled Radio Numbertime, which again focused on the numbers 1-10, in order; it ran from 21 September to 30 November 1993. Another radio series, which was entitled simply Numbertime like the television series, was broadcast on BBC School Radio from 29 September 2000 to 26 March 2003 - and a third radio series, which was again entitled Numbertime like the defunct television series, was broadcast on BBC School Radio from 1 May to 26 June 2014. == References == == External links == Numbertime at BBC Online Numbertime at IMDb Numbertime at Broadcast for Schools
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Wikipedia:Numerical range#0
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In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n × n {\displaystyle n\times n} matrix A is the set W ( A ) = { x ∗ A x x ∗ x ∣ x ∈ C n , x ≠ 0 } = { ⟨ x , A x ⟩ ∣ x ∈ C n , ‖ x ‖ 2 = 1 } {\displaystyle W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \not =0\right\}=\left\{\langle \mathbf {x} ,A\mathbf {x} \rangle \mid \mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1\right\}} where x ∗ {\displaystyle \mathbf {x} ^{*}} denotes the conjugate transpose of the vector x {\displaystyle \mathbf {x} } . The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors). In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of the numerical range are used to study quantum computing. A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e. r ( A ) = sup { | λ | : λ ∈ W ( A ) } = sup ‖ x ‖ 2 = 1 | ⟨ x , A x ⟩ | . {\displaystyle r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|_{2}=1}|\langle \mathbf {x} ,A\mathbf {x} \rangle |.} == Properties == Let sum of sets denote a sumset. General properties The numerical range is the range of the Rayleigh quotient. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact. W ( α A + β I ) = α W ( A ) + { β } {\displaystyle W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}} for all square matrix A {\displaystyle A} and complex numbers α {\displaystyle \alpha } and β {\displaystyle \beta } . Here I {\displaystyle I} is the identity matrix. W ( A ) {\displaystyle W(A)} is a subset of the closed right half-plane if and only if A + A ∗ {\displaystyle A+A^{*}} is positive semidefinite. The numerical range W ( ⋅ ) {\displaystyle W(\cdot )} is the only function on the set of square matrices that satisfies (2), (3) and (4). W ( U A U ∗ ) = W ( A ) {\displaystyle W(UAU^{*})=W(A)} for any unitary U {\displaystyle U} . W ( A ∗ ) = W ( A ) ∗ {\displaystyle W(A^{*})=W(A)^{*}} . If A {\displaystyle A} is Hermitian, then W ( A ) {\displaystyle W(A)} is on the real line. If A {\displaystyle A} is anti-Hermitian, then W ( A ) {\displaystyle W(A)} is on the imaginary line. W ( A ) = { z } {\displaystyle W(A)=\{z\}} if and only if A = z I {\displaystyle A=zI} . (Sub-additive) W ( A + B ) ⊆ W ( A ) + W ( B ) {\displaystyle W(A+B)\subseteq W(A)+W(B)} . W ( A ) {\displaystyle W(A)} contains all the eigenvalues of A {\displaystyle A} . The numerical range of a 2 × 2 {\displaystyle 2\times 2} matrix is a filled ellipse. W ( A ) {\displaystyle W(A)} is a real line segment [ α , β ] {\displaystyle [\alpha ,\beta ]} if and only if A {\displaystyle A} is a Hermitian matrix with its smallest and the largest eigenvalues being α {\displaystyle \alpha } and β {\displaystyle \beta } . Normal matrices If A {\textstyle A} is normal, and x ∈ span ( v 1 , … , v k ) {\textstyle x\in \operatorname {span} (v_{1},\dots ,v_{k})} , where v 1 , … , v k {\textstyle v_{1},\ldots ,v_{k}} are eigenvectors of A {\textstyle A} corresponding to λ 1 , … , λ k {\textstyle \lambda _{1},\ldots ,\lambda _{k}} , respectively, then ⟨ x , A x ⟩ ∈ hull ( λ 1 , … , λ k ) {\textstyle \langle x,Ax\rangle \in \operatorname {hull} \left(\lambda _{1},\ldots ,\lambda _{k}\right)} . If A {\displaystyle A} is a normal matrix then W ( A ) {\displaystyle W(A)} is the convex hull of its eigenvalues. If α {\displaystyle \alpha } is a sharp point on the boundary of W ( A ) {\displaystyle W(A)} , then α {\displaystyle \alpha } is a normal eigenvalue of A {\displaystyle A} . Numerical radius r ( ⋅ ) {\displaystyle r(\cdot )} is a unitarily invariant norm on the space of n × n {\displaystyle n\times n} matrices. r ( A ) ≤ ‖ A ‖ op ≤ 2 r ( A ) {\displaystyle r(A)\leq \|A\|_{\operatorname {op} }\leq 2r(A)} , where ‖ ⋅ ‖ op {\displaystyle \|\cdot \|_{\operatorname {op} }} denotes the operator norm. r ( A ) = ‖ A ‖ op {\displaystyle r(A)=\|A\|_{\operatorname {op} }} if (but not only if) A {\displaystyle A} is normal. r ( A n ) ≤ r ( A ) n {\displaystyle r(A^{n})\leq r(A)^{n}} . == Proofs == Most of the claims are obvious. Some are not. === General properties === === Normal matrices === === Numerical radius === == Generalisations == C-numerical range Higher-rank numerical range Joint numerical range Product numerical range Polynomial numerical hull == See also == Spectral theory Rayleigh quotient Workshop on Numerical Ranges and Numerical Radii == Bibliography == Toeplitz, Otto (1918). "Das algebraische Analogon zu einem Satze von Fejér" (PDF). Mathematische Zeitschrift (in German). 2 (1–2): 187–197. doi:10.1007/BF01212904. ISSN 0025-5874. Hausdorff, Felix (1919). "Der Wertvorrat einer Bilinearform". Mathematische Zeitschrift (in German). 3 (1): 314–316. doi:10.1007/BF01292610. ISSN 0025-5874. Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58 (1): 77–91, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8, S2CID 119427312. Bhatia, Rajendra (1997). Matrix analysis. Graduate texts in mathematics. New York Berlin Heidelberg: Springer. ISBN 978-0-387-94846-1. Dirr, G.; Helmkel, U.; Kleinsteuber, M.; Schulte-Herbrüggen, Th. (2006), "A new type of C-numerical range arising in quantum computing", Proc. Appl. Math. Mech., 6: 711–712, doi:10.1002/pamm.200610336. Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press, ISBN 978-0-521-07988-4. Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges II, Cambridge University Press, ISBN 978-0-521-20227-5. Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, Chapter 1, ISBN 978-0-521-46713-1. Horn, Roger A.; Johnson, Charles R. (1990), Matrix Analysis, Cambridge University Press, Ch. 5.7, ex. 21, ISBN 0-521-30586-1 Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc., 124 (7): 1985, doi:10.1090/S0002-9939-96-03307-2. Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications, 252 (1–3): 115, doi:10.1016/0024-3795(95)00674-5. Johnson, Charles R. (1976). "Functional characterizations of the field of values and the convex hull of the spectrum" (PDF). Proceedings of the American Mathematical Society. 61 (2). American Mathematical Society (AMS): 201–204. doi:10.1090/s0002-9939-1976-0437555-3. ISSN 0002-9939. == References ==
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Wikipedia:Nørlund–Rice integral#0
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In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation. == Definition == The nth forward difference of a function f(x) is given by Δ n [ f ] ( x ) = ∑ k = 0 n ( n k ) ( − 1 ) n − k f ( x + k ) {\displaystyle \Delta ^{n}[f](x)=\sum _{k=0}^{n}{n \choose k}(-1)^{n-k}f(x+k)} where ( n k ) {\displaystyle {n \choose k}} is the binomial coefficient. The Nørlund–Rice integral is given by ∑ k = α n ( n k ) ( − 1 ) n − k f ( k ) = n ! 2 π i ∮ γ f ( z ) z ( z − 1 ) ( z − 2 ) ⋯ ( z − n ) d z {\displaystyle \sum _{k=\alpha }^{n}{n \choose k}(-1)^{n-k}f(k)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z(z-1)(z-2)\cdots (z-n)}}\,dz} where f is understood to be meromorphic, α is an integer, 0 ≤ α ≤ n {\displaystyle 0\leq \alpha \leq n} , and the contour of integration is understood to circle the poles located at the integers α, ..., n, but encircles neither integers 0, ..., α − 1 {\displaystyle \alpha -1} nor any of the poles of f. The integral may also be written as ∑ k = α n ( n k ) ( − 1 ) k f ( k ) = − 1 2 π i ∮ γ B ( n + 1 , − z ) f ( z ) d z {\displaystyle \sum _{k=\alpha }^{n}{n \choose k}(-1)^{k}f(k)=-{\frac {1}{2\pi i}}\oint _{\gamma }B(n+1,-z)f(z)\,dz} where B(a,b) is the Euler beta function. If the function f ( z ) {\displaystyle f(z)} is polynomially bounded on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as ∑ k = α n ( n k ) ( − 1 ) n − k f ( k ) = − n ! 2 π i ∫ c − i ∞ c + i ∞ f ( z ) z ( z − 1 ) ( z − 2 ) ⋯ ( z − n ) d z {\displaystyle \sum _{k=\alpha }^{n}{n \choose k}(-1)^{n-k}f(k)={\frac {-n!}{2\pi i}}\int _{c-i\infty }^{c+i\infty }{\frac {f(z)}{z(z-1)(z-2)\cdots (z-n)}}\,dz} where the constant c is to the left of α. == Poisson–Mellin–Newton cycle == The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the Mellin transform is not accidental, but is related by means of the binomial transform and the Newton series. In this cycle, let { f n } {\displaystyle \{f_{n}\}} be a sequence, and let g(t) be the corresponding Poisson generating function, that is, let g ( t ) = e − t ∑ n = 0 ∞ t n n ! f n . {\displaystyle g(t)=e^{-t}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}f_{n}.} Taking its Mellin transform ϕ ( s ) = ∫ 0 ∞ g ( t ) t s − 1 d t , {\displaystyle \phi (s)=\int _{0}^{\infty }g(t)t^{s-1}\,dt,} one can then regain the original sequence by means of the Nörlund–Rice integral (see References "Mellin, seen from the sky"): f n = ( − 1 ) n 2 π i ∫ γ ϕ ( − s ) Γ ( − s ) n ! s ( s − 1 ) ⋯ ( s − n ) d s {\displaystyle f_{n}={\frac {(-1)^{n}}{2\pi i}}\int _{\gamma }{\frac {\phi (-s)}{\Gamma (-s)}}{\frac {n!}{s(s-1)\cdots (s-n)}}\,ds} where Γ is the gamma function which cancels with the gamma from Ramanujan's Master Theorem. == Riesz mean == A closely related integral frequently occurs in the discussion of Riesz means. Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that Perron's formula is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series. == Utility == The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n. == See also == Table of Newtonian series List of factorial and binomial topics == References == Niels Erik Nørlund, Vorlesungen uber Differenzenrechnung, (1954) Chelsea Publishing Company, New York. Donald E. Knuth, The Art of Computer Programming, (1973), Vol. 3 Addison-Wesley. Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124. Peter Kirschenhofer, "[1]", The Electronic Journal of Combinatorics, Volume 3 (1996) Issue 2 Article 7. Philippe Flajolet, lecture, "Mellin, seen from the sky", page 35.
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Wikipedia:Oberwolfach Research Institute for Mathematics#0
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The Oberwolfach Research Institute for Mathematics (German: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes weekly workshops on diverse topics where mathematicians and scientists from all over the world come to do collaborative research. The Institute is a member of the Leibniz Association, funded mainly by the German Federal Ministry of Education and Research and by the state of Baden-Württemberg. It also receives substantial funding from the Friends of Oberwolfach foundation, from the Oberwolfach Foundation and from numerous donors. == History == The Oberwolfach Research Institute for Mathematics (MFO) was founded as the Reich Institute of Mathematics (German: Reichsinstitut für Mathematik) on 1 September 1944. It was one of several research institutes founded by the Nazis in order to further the German war effort, which at that time was clearly failing. The location was selected to be remote as not to be a target for Allied bombing. Originally it was housed in a building called the Lorenzenhof, a large Black Forest hunting lodge. After the war, Süss, a member of the Nazi party, was suspended for two months in 1945 as part of the county's denazification efforts, but thereafter remained head of the institute. Though the institute lost its government funding, Süss was able to keep it going with other grants, and contributed to rebuilding mathematics in Germany following the fall of the Third Reich by hosting international mathematical conferences. Some of these were organised by Reinhold Baer, a mathematician who was expelled from University of Halle in 1933 for being Jewish, but later returned to Germany in 1956 at the University of Frankfurt. The institute regained government funding in the 1950s. After Süss's death in 1958, Hellmuth Kneser was briefly director before Theodor Schneider permanently took over in the role in 1959. In that year, he and others formed the mathematical society Gesellschaft für Mathematische Forschung e. V. in order to run the MFO. On 10 October 1967 the guest house of the Oberwolfach Research Institute for Mathematics was inaugurated, which was a gift from the Volkswagen Foundation. On 13 June 1975 the library and meetings building of the MFO were inaugurated, replacing the old castle. This new building was also a gift from the Volkswagen Foundation. On 26 May 1989 an extension to the guest building at the MFO was inaugurated. In 1995, the MFO established the research program "Research in Pairs". On 1 January 2005 Oberwolfach Research Institute for Mathematics became a member of the Leibniz Association. From 2005 to 2010, there was a general restoration of the guest house and the library building at the MFO. Post-doctoral program "Oberwolfach Leibniz Fellows" was established in 2007. On 5 May that year an extension to the library was inaugurated, the extension was a gift from the Klaus Tschira Stiftung and the Volkswagen Foundation. == Statue == The iconic model of the Boy surface was installed in front of the Institute, as a gift from Mercedes-Benz on 28 January 1991. The Boy Surface is named after Werner Boy who constructed the surface in his 1901 thesis, written under the direction of David Hilbert. == Directors == 1944–1958, Wilhelm Süss 1958–1959, Hellmuth Kneser 1959–1963, Theodor Schneider 1963–1994, Martin Barner 1994–2002, Matthias Kreck 2002–2013, Gert-Martin Greuel 2013–present Gerhard Huisken == Oberwolfach Prize == The Oberwolfach Prize is awarded approximately every three years for excellent achievements in changing fields of mathematics to young mathematicians not older than 35 years. It is financed by the Oberwolfach Foundation and awarded in cooperation with the institute. Prize winners 1991 Peter Kronheimer 1993 Jörg Brüdern and Jens Franke 1996 Gero Friesecke and Stefan Sauter 1998 Alice Guionnet 2000 Luca Trevisan 2003 Paul Biran 2007 Ngô Bảo Châu 2010 Nicola Gigli and László Székelyhidi 2013 Hugo Duminil-Copin 2016 Jacob Fox 2019 Oscar Randal-Williams 2022 Vesselin Dimitrov == References == == External links == Home page of the institute Web page about the Oberwolfach Prize
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Wikipedia:Oblique reflection#0
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In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations. Consider a plane P in the three-dimensional Euclidean space. The usual reflection of a point A in space in respect to the plane P is another point B in space, such that the midpoint of the segment AB is in the plane, and AB is perpendicular to the plane. For an oblique reflection, one requires instead of perpendicularity that AB be parallel to a given reference line. Formally, let there be a plane P in the three-dimensional space, and a line L in space not parallel to P. To obtain the oblique reflection of a point A in space in respect to the plane P, one draws through A a line parallel to L, and lets the oblique reflection of A be the point B on that line on the other side of the plane such that the midpoint of AB is in P. If the reference line L is perpendicular to the plane, one obtains the usual reflection. For example, consider the plane P to be the xy plane, that is, the plane given by the equation z=0 in Cartesian coordinates. Let the direction of the reference line L be given by the vector (a, b, c), with c≠0 (that is, L is not parallel to P). The oblique reflection of a point (x, y, z) will then be ( x − 2 z a c , y − 2 z b c , − z ) . {\displaystyle \left(x-{\frac {2za}{c}},y-{\frac {2zb}{c}},-z\right).} The concept of oblique reflection is easily generalizable to oblique reflection in respect to an affine hyperplane in Rn with a line again serving as a reference, or even more generally, oblique reflection in respect to a k-dimensional affine subspace, with a n−k-dimensional affine subspace serving as a reference. Back to three dimensions, one can then define oblique reflection in respect to a line, with a plane serving as a reference. An oblique reflection is an affine transformation, and it is an involution, meaning that the reflection of the reflection of a point is the point itself. == References ==
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Wikipedia:Ockham algebra#0
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In mathematics, an Ockham algebra is a bounded distributive lattice L {\displaystyle L} with a dual endomorphism, that is, an operation ∼ : L → L {\displaystyle \sim \colon L\to L} satisfying ∼ ( x ∧ y ) = ∼ x ∨ ∼ y {\displaystyle \sim (x\wedge y)={}\sim x\vee {}\sim y} , ∼ ( x ∨ y ) = ∼ x ∧ ∼ y {\displaystyle \sim (x\vee y)={}\sim x\wedge {}\sim y} , ∼ 0 = 1 {\displaystyle \sim 0=1} , ∼ 1 = 0 {\displaystyle \sim 1=0} . They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety. Examples of Ockham algebras include Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras. == References == Berman, Joel (1977), "Distributive lattices with an additional unary operation", Aequationes Mathematicae, 16 (1): 165–171, doi:10.1007/BF01837887, ISSN 0001-9054, MR 0480238 (pdf available from GDZ) Blyth, Thomas Scott (2001) [1994], "Ockham algebra", Encyclopedia of Mathematics, EMS Press Blyth, Thomas Scott; Varlet, J. C. (1994). Ockham algebras. Oxford University Press. ISBN 978-0-19-859938-8. Urquhart, Alasdair (1979), "Distributive lattices with a dual homomorphic operation", Polska Akademia Nauk. Institut Filozofii i Socijologii. Studia Logica, 38 (2): 201–209, doi:10.1007/BF00370442, hdl:10338.dmlcz/102014, ISSN 0039-3215, MR 0544616
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Wikipedia:Octav Mayer#0
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Octav Mayer (October 5 [O.S. September 22] 1895 – 9 September 1966) was a Romanian mathematician, the first to earn a doctorate in Romania. Born in Mizil, Prahova County, Mayer went to the primary school in Târgu Neamț and pursued his studies in an elementary school in Focșani. He then went to the National College in Iași, where he obtained his baccalaureate, and then studied mathematics at the University of Iași, where he obtained his undergraduate degree. In 1915, Mayer enrolled in the School of Artillery and Military Engineering and took part in the battles on the Romanian front in World War I from 1916 to 1918. After the war, he completed his Ph.D. at the University of Iași in 1920; his thesis, written under the direction of Alexander Myller, was titled Contributions à la théorie des quartiques bicirculaires. Mayer later became a professor at the University of Iași. He was elected titular member of the Romanian Academy in 1955. He died in Iași in 1966, at age 71. The Octav Mayer Institute of Mathematics of the Romanian Academy (located in Iași) is named after him. == References ==
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Wikipedia:Octavio Cordero Palacios (writer)#0
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Octavio Cordero Palacios (Santa Rosa, Azuay, May 3, 1870 – December 17, 1930) was an Ecuadorian writer, playwright, poet, mathematician, lawyer, professor and inventor. Today a town and parish in Cuenca is named after him. == Biography == Octavio Cordero Palacios was born on May 3, 1870. His father was Vicente Cordero Crespo, a poet who authored the play in verse "Don Lucas", and who was a scribe in Cuenca in 1889, and a conservative journalist and editor of the journal "El Criterio" in 1895. His mother was Rosa Palacios Alvear. Both of his parents were from Cuenca. In 1890 he premiered his play in three acts "Gazul", whose scenes take place in Persia at the end of the First Crusade. His second play was "Los Hijos de Atahualpa", and in 1892 he wrote the play "Los Borrachos". He was also a translator, he published "Rapsodias Clásicas" a Spanish translation of works by Virgil and Horace. He also translated works from French and English into Spanish, including a faithful rendition of Edgar Allan Poe's "The Raven". Also in 1890 he earned a doctorate in jurisprudence from the University of Cuenca, and practiced as a lawyer at the Superior Court of Azuay. In 1900 he taught literature and philosophy at the Benigno Malo School. In 1910 during the armed conflict with Peru, he joined the army reserves with the rank of sergeant major, and was named chief of engineers of the First South Division. He created a topographic military map of Ecuador's southern border, and taught courses at the University of Cuenca on planimetry, altimetry, and the layout of roads, and construction of bridges and causeways. In 1916 he published the book "Vida de Abdón Calderón". That year he was also elected Senator of Azuay (a post he held til 1918). He was elected the Judge of the Superior Court of Justice, a post which he held for 10 years. in 1922 he published "De Potencia a Potencia", a historical essay on the continued battle between the governor of Azuay Manuel Vega and President García Moreno. That year he also published the essay "El arte poético de Horacio". In 1923 he published "El quichua y el Cañari" a philological study of Quechua and Cañari languages, with a Cañari Dictionary, which was awarded "La Palma de Oro" Prize. In 1924 he published his incomplete work on the death of Juan Seniergues. He had already published the following essays: "Don José Antonio Vallejo, su primera gobernación entre 1.777 y 1.784", "El Azuay Histórico", "Pro Tomebamba", "Crónicas Documentadas para la Historia de Cuenca" and he promised to publish two new books, which he never did, perhaps due to his impoverished state. In 1929 he published "La Poesía de Ciencia". == Inventions == In 1902 he created a mechanical computer he named "Clave Poligráfíca" or "Metaglota". The device translated words from one language to another language. After his death, his cousin Humberto Cordero reconstructed the machine and exhibited it in Quito in 1936. The exhibit was a success, and the computer was hailed as a "marvelous mechanical dictionary". He also invented a "numerical device that calculated perfectly the square root of numbers", as well as a "trigonometry text in verse". == Death == In 1930 he was suffering from cirrhosis. He then told his children that he would die soon, and that they should inscribe the following verse on his tomb rather than his name: He then announced to his children that he would die on December 17 of that year, 1930. As he had predicted, on exactly December 17, at 6:30 PM, he died while reciting the Ecuadorian National Anthem. He is interred in the Illustrious Personages Mausoleum plot in the Patrimonial Municipal Cemetery of Cuenca. == Personal life == He was married to Victoria Crespo Astudillo, with whom he had many children. == References ==
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Wikipedia:Odd Magnus Faltinsen#0
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Odd Magnus Faltinsen (born 9 January 1944) is a Norwegian mathematician and professor of marine technology. == Education and career == Faltinsen took the cand.real. degree at the University of Bergen in 1968, and the PhD degree at the University of Michigan in 1971. He started his career in Det Norske Veritas from 1968 to 1974, and was appointed docent in marine technology at the Norwegian Institute of Technology in 1974. In 1976 he was promoted to professor of marine hydrodynamics. He was a visiting professor at the Massachusetts Institute of Technology from 1980 to 1981,1987 to 1988 and 1994 to 1995. He is a member of the Norwegian Academy of Science and Letters, the Norwegian Academy of Technological Sciences, The Chinese Academy of Engineering and the National Academy of Engineering of the United States of America. Faltinsen received the Fridtjof Nansen award for outstanding research in science and medicine in 2011. He is now connected to the Centre for Ships and Ocean Structures at the Norwegian University of Science and Technology (the successor of the Norwegian Institute of Technology). == Scientific contribution == Faltinsen is known for his work in hydrodynamics of high-speed vessels and liquid sloshing dynamics. He has written three textbooks on the subjects. The books are translated to Chinese. The book on sea loads are also translated to Korean. Faltinsen has developed theoretical and numerical methods for explaining how ships, high speed vehicles, and offshore structures behave in waves. The so-called STF - Salvesen-Tuck-Faltinsen method to estimate wave induced movements and loads on ships presented in 1970 is still used as an engineering tool to day. He has, together with Alexander Timokha, developed methods for analyzing how sloshing loads on to ships and they have analytically studied how large sloshing loads infer with constructions. He has also, together with e.g. Zhao, made extensive studies of slamming loads == Books == Faltinsen, O. M. (1990). Sea Loads on Ships and Offshore Structures. Cambridge, UK: Cambridge University Press. ISBN 0-521-45870-6. Faltinsen, O. M. (2006). Hydrodynamics of High-Speed Marine Vehicles. Cambridge University Press. ISBN 0-521-84568-8. Faltinsen, O. M. & Timokha, A. N. (2009). Sloshing. Cambridge University Press. ISBN 0-521-88111-0. == References ==
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Wikipedia:Odd greedy expansion#0
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In number theory, the odd greedy expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. It is an open problem. == Description == An Egyptian fraction represents a given rational number as a sum of distinct unit fractions. If a rational number x / y {\displaystyle x/y} is a sum of unit fractions with odd denominators, x y = ∑ 1 2 a i + 1 , {\displaystyle {\frac {x}{y}}=\sum {\frac {1}{2a_{i}+1}},} then y {\displaystyle y} must be odd. Conversely, every fraction x / y {\displaystyle x/y} with y {\displaystyle y} odd can be represented as a sum of distinct odd unit fractions. One method of finding such a representation replaces x / y {\displaystyle x/y} by A x / A y {\displaystyle Ax/Ay} where A = 35 ⋅ 3 i {\displaystyle A=35\cdot 3^{i}} for a sufficiently large i {\displaystyle i} , and then expands A x {\displaystyle Ax} as a sum of distinct divisors of A y {\displaystyle Ay} . However, a simpler greedy algorithm has successfully found Egyptian fractions in which all denominators are odd for all instances x / y {\displaystyle x/y} (with odd y {\displaystyle y} ) on which it has been tested: let u {\displaystyle u} be the least odd number that is greater than or equal to y / x {\displaystyle y/x} , include the fraction 1 / u {\displaystyle 1/u} in the expansion, and continue in the same way (avoiding repeated uses of the same unit fraction) with the remaining fraction x / y − 1 / u {\displaystyle x/y-1/u} . This method is called the odd greedy algorithm and the expansions it creates are called odd greedy expansions. Stein, Selfridge, Graham, and others have posed the open problem of whether the odd greedy algorithm terminates with a finite expansion for every x / y {\displaystyle x/y} with y {\displaystyle y} odd. == Example == Let x / y {\displaystyle x/y} = 4/23. 23/4 = 53/4; the next larger odd number is 7. So the first step expands 161/5 = 321/5; the next larger odd number is 33. So the next step expands 5313/4 = 13281/4; the next larger odd number is 1329. So the third step expands Since the final term in this expansion is a unit fraction, the process terminates with this expansion as its result. == Fractions with long expansions == It is possible for the odd greedy algorithm to produce expansions that are shorter than the usual greedy expansion, with smaller denominators. For instance, 8 77 = 1 10 + 1 257 + 1 197890 = 1 11 + 1 77 , {\displaystyle {\frac {8}{77}}={\frac {1}{10}}+{\frac {1}{257}}+{\frac {1}{197890}}={\frac {1}{11}}+{\frac {1}{77}},} where the left expansion is the greedy expansion and the right expansion is the odd greedy expansion. However, the odd greedy expansion is more typically long, with large denominators. For instance, as Wagon discovered, the odd greedy expansion for 3/179 has 19 terms, the largest of which is approximately 1.415×10439491. Curiously, the numerators of the fractions to be expanded in each step of the algorithm form a sequence of consecutive integers: A similar phenomenon occurs with other numbers, such as 5/5809 (an example found independently by K. S. Brown and David Bailey) which has a 27-term expansion. Although the denominators of this expansion are difficult to compute due to their enormous size, the numerator sequence may be found relatively efficiently using modular arithmetic. Nowakowski (1999) describes several additional examples of this type found by Broadhurst, and notes that K. S. Brown has described methods for finding fractions with arbitrarily long expansions. == On even denominators == The odd greedy algorithm cannot terminate when given a fraction with an even denominator, because these fractions do not have finite representations with odd denominators. Therefore, in this case, it produces an infinite series expansion of its input. For instance Sylvester's sequence can be viewed as generated by the odd greedy expansion of 1/2. == Notes == == References == Breusch, R. (1954), "A special case of Egyptian fractions, solution to advanced problem 4512", American Mathematical Monthly, 61: 200–201, doi:10.2307/2307234, JSTOR 2307234 Guy, Richard K. (1981), Unsolved Problems in Number Theory, Springer-Verlag, p. 88, ISBN 0-387-90593-6 Guy, Richard K. (1998), "Nothing's new in number theory?", American Mathematical Monthly, 105 (10): 951–954, doi:10.2307/2589289, JSTOR 2589289 Klee, Victor; Wagon, Stan (1991), Unsolved Problems in Elementary Geometry and Number Theory, Dolciani Mathematical Expositions, Mathematical Association of America Nowakowski, Richard (1999), "Unsolved problems, 1969–1999", American Mathematical Monthly, 106 (10): 959–962, doi:10.2307/2589753, JSTOR 2589753 Stewart, B. M. (1954), "Sums of distinct divisors", American Journal of Mathematics, 76 (4): 779–785, doi:10.2307/2372651, JSTOR 2372651, MR 0064800 Wagon, Stan (1991), Mathematica in Action, W.H. Freeman, pp. 275–277, ISBN 0-7167-2202-X == External links == MathPages - Odd-Greedy Unit Fraction Expansions, K. S. Brown
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Wikipedia:Oded Regev (computer scientist)#0
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Oded Regev (Hebrew: עודד רגב; born 1980 or 1979) is an Israeli-American theoretical computer scientist and mathematician. He is a professor of computer science at the Courant institute at New York University. He is best known for his work in lattice-based cryptography, and in particular for introducing the learning with errors problem. == Biography == Oded Regev earned his B.Sc. in 1995, M.Sc. in 1997, and Ph.D. in 2001, all from Tel Aviv University. He completed his Ph.D. at the age of 21, advised by Yossi Azar, with a thesis titled "Scheduling and Load Balancing." He held faculty positions at Tel Aviv University and the École Normale Supérieure before joining the Courant institute. == Work == Regev has done extensive work on lattices. He is best known for introducing the learning with errors problem (LWE), for which he won the 2018 Gödel Prize. As the citation reads: Regev’s work has ushered in a revolution in cryptography, in both theory and practice. On the theoretical side, LWE has served as a simple and yet amazingly versatile foundation for nearly every kind of cryptographic object imaginable—along with many that were unimaginable until recently, and which still have no known constructions without LWE. Toward the practical end, LWE and its direct descendants are at the heart of several efficient real-world cryptosystems. Regev's most influential other work on lattices includes cryptanalysis of the GGH and NTRU signature schemes in joint work with Phong Q. Nguyen, for which they won a best paper award at Eurocrypt 2006; introducing the ring learning with errors problem in joint work with Chris Peikert and Vadim Lyubashevsky; and proving a converse to Minkowski's theorem and exploring its applications in joint works with his student Noah Stephens-Davidowitz and his former postdoc Daniel Dadush. In addition to his work on lattices, Regev has also done work in a large number of other areas in theoretical computer science and mathematics. These include quantum computing, communication complexity, hardness of approximation, online algorithms, combinatorics, probability, and dimension reduction. He has also recently become interested in topics in biology, and particularly RNA splicing. Regev is an associate editor in chief of the journal Theory of Computing, and is a co-founder and organizer of the TCS+ online seminar series. In August 2023 Regev published a preprint describing an algorithm to factor integers with ∼ O ( n 3 / 2 ) {\displaystyle \sim O(n^{3/2})} quantum gates which would be more efficient than Shor's algorithm which uses ∼ O ( n 2 ) {\displaystyle \sim O(n^{2})} , but would require more qubits ∼ O ( n 3 / 2 ) {\displaystyle \sim O(n^{3/2})} of quantum memory against Shor's ∼ O ( n ) {\displaystyle \sim O(n)\ } . A variant has been proposed that could reduce the space to around the same amount. == References ==
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Wikipedia:Odile Favaron#0
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Odile Zink-Favaron (born May 3, 1938) is a French mathematician known for her research in graph theory, including work on well-covered graphs, factor-critical graphs, spectral graph theory, Hamiltonian decomposition, and dominating sets. She is retired from the Laboratory for Computer Science (LRI) at the University of Paris-Sud. Favaron earned a doctorate at Paris-Sud University in 1986. Her dissertation, Stabilité, domination, irrédondance et autres paramètres de graphes [Independence, domination, irredundance, and other parameters of graphs], was supervised by Jean-Claude Bermond. == Personal life == Her father was poet and professor Georges Zink. Michel Zink and Anne Zink are her siblings. == References ==
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Wikipedia:Odile Macchi#0
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Odile Macchi (born Odile Danjou: 1943) is a French physicist and mathematician. She has been a member of the French Academy of Sciences since 2004. == Life == Odile Danjou was born in Aurillac (Cantal) during the German occupation. She is one of the six recorded children of Bernard Danjou and his wife, born Geneviève Féat. Between 1963 and 1966 she studied at the École normale supérieure de jeunes filles (subsequently merged with the École normale supérieure (EPS)), emerging with a double degree in physics and maths. It was also in 1966 that she received her “agrégation” (teaching certificate) in Maths. The unusual combination of her qualifications opened the door to a position as a research assistant at the Institute of Basic Electronics at the University of Paris-Sud where she was able to work with Bernard Picinbono. The focus of her work was on signal processing: it involved extensive modelling and physical analysis of materials and use of mathematically based statistical techniques. She successfully defended her doctoral dissertation in physical sciences in 1972, dealing with "the contribution to theoretical study of point process, and its application to statistical optics and optical communications". She now moved on, as she puts it, from pure maths "to impure maths" ("Je suis passée aux maths impures"). In this she was encouraged by César Macchi, the polytechnician, telecommunications engineer and university professor whom she had married a few years earlier. She continued to centre her work on signal processing, with a particular focus on the newly emerging digital revolution. It was a field in which her husband was playing a pioneering role, and for the next seven years, until César Macchi's early death from cancer, they frequently collaborated in their research. In 1989, Odile Macchi was elevated to the grade of IEEE fellow for contribution to adaptive filtering in communications and signal processing. Odile Macchi held senior research positions at the French National Centre of Scientific Research and at the Laboratory of Systems and Signals at the University of Paris-Sud from 1972, appointed to a research directorship at both institutions in 1979. Much of her research work, over the years, has supported practical developments in modem technology. She became an emeritus research director at the two institutions in 1999. Odile Macchi was just 33 when she was widowed. She told an interviewer that she knew at once that she would never remarry, and she never has. Her four children were aged between 3 and 12 in 1976. She responded to her bereavement by addressing her scientific research work with renewed commitment and energy, while also finding strength in her Christian faith, and through participation in the "Fraternitat Santa Maria de la Resurrecció", a church based group created at Lourdes in 1943 to share support between young widows. Odile Macchi was elected a corresponding member of the French Academy of Sciences in 1994. Ten years later she was elected to full membership. === Awards and honours (selection) === == References ==
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Wikipedia:Ofer Gabber#0
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Ofer Gabber (עופר גאבר; born May 16, 1958) is a mathematician working in algebraic geometry. == Life == In 1978 Gabber received a Ph.D. from Harvard University for the thesis Some theorems on Azumaya algebras, written under the supervision of Barry Mazur. Gabber has been at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette in Paris since 1984 as a CNRS senior researcher. He won the Erdős Prize in 1981 and the Prix Thérèse Gautier from the French Academy of Sciences in 2011. In 1981 Gabber with Victor Kac published a proof of a conjecture stated by Kac in 1968. == Books == With Lorenzo Ramero: Almost Ring Theory, Springer, Lecture Notes in Computer Science, vol 1800, 2003. With Brian Conrad, Gopal Prasad: Pseudo-reductive Groups, Cambridge University Press, 2010; 2015, 2nd edition == See also == Gabber rigidity Almost ring theory t-structure Theorem of absolute purity == References ==
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Wikipedia:Ofer Zeitouni#0
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Ofer Zeitouni (Hebrew: עפר זיתוני; born 23 October 1960, Haifa) is an Israeli mathematician, specializing in probability theory. == Biography == Zeitouni received his bachelor's degree in electrical engineering in 1980 from the Technion. He obtained in 1986 his doctorate in electrical engineering under the supervision of Moshe Zakai with the thesis Bounds on the Conditional Density and Maximum a posteriori Estimators for the Nonlinear Filtering Problem. As a postdoc he was a visiting assistant professor at Brown University and at the Laboratory for Information and Decision Systems at MIT. He joined the Technion in 1989 as senior lecturer, and was promoted in 1991 to associate professor, and in 1997 to full professor in the department of electrical engineering. He is now a professor of Mathematics at the Weizmann Institute and at the Courant Institute, and was from 2002 to 2013 a part-time professor at the University of Minnesota. His research deals with stochastic processes and filtering theory with applications to control theory (electrical engineering), the spectral theory of random matrices, the theory of large deviations in probability theory, motion in random media, and extremes of logarithmically correlated fields. He was Invited Speaker with the talk Random Walks in Random Environments at the ICM in Beijing in 2002. Zeitouni is a Fellow of the American Mathematical Society, member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the Israel Academy of Sciences and Humanities. He is married and has two children. == Selected publications == === Articles === with Ildar Abdulovich Ibragimov: Ibragimov, Ildar; Zeitouni, Ofer (1997). "On roots of random polynomials". Trans. Amer. Math. Soc. 349 (6): 2427–2441. doi:10.1090/S0002-9947-97-01766-2. with Amir Dembo, Yuval Peres, and Jay Rosen: Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer (2001). "Thick points for planar Brownian motion and the Erdős-Taylor conjecture on random walk". Acta Mathematica. 186 (2): 239–270. doi:10.1007/BF02401841. with Amir Dembo, Bjorn Poonen, and Qi-Man Shao: Dembo, Amir; Poonen, Bjorn; Shao, Qi-Man; Zeitouni, Ofer (2002). "Random polynomials having few or no real zeros". J. Amer. Math. Soc. 15 (4): 857–892. arXiv:math/0006113. doi:10.1090/S0894-0347-02-00386-7. with Amir Dembo, Yuval Peres, and Jay Rosen: Dembo, Amir; Peres, Yuval; Rosen, Jay; Zeitouni, Ofer (2002). "Thick points for intersections of planar sample paths". Trans. Amer. Math. Soc. 354 (12): 4969–5003. doi:10.1090/S0002-9947-02-03080-5. Zeitouni, Ofer (2004). "Part II: Random walks in random environment". In: Lectures on probability theory and statistics. Lecture Notes in Mathematics. Vol. 1837. Berlin, Heidelberg: Springer. pp. 190–312. doi:10.1007/978-3-540-39874-5_2. ISBN 978-3-540-20832-7. === Books === with Greg W. Anderson and Alice Guionnet: Introduction to Random Matrices, Cambridge University Press 2010 with Amir Dembo: Large Deviations Techniques and Applications, Springer 1998, Dembo, Amir; Zeitouni, Ofer (2009). 2nd edition, corrected printing of 1998 edition. Springer. ISBN 9783642033117; pbk{{cite book}}: CS1 maint: postscript (link) == Sources == Zhan Shi: Problèmes de recouvrement et points exceptionnels pour la marche aléatoire et le mouvement brownien, d’après Dembo, Peres, Rosen, Zeitouni, Seminaire Bourbaki, No. 951, 2005 == References == == External links == Ofer Zeitouni's home page, Weizmann Institute Ofer Zeitouni, What happens when a person strolling along an intersecting path chooses directions with a roll of the dice?, Weizmann Institute
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Wikipedia:Okan Ersoy#0
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Okan Kadri Ersoy (born September 5, 1945) is now Professor Emeritus of electrical engineering Formerly, he was a professor of electrical engineering and the director of the Statistical and Computational Intelligence Laboratory at Purdue University, West Lafayette School of Electrical and Computer Engineering. He is a Fellow of IEEE, a Fellow of OSA and a Fellow of ISIBM. Ersoy contributed to the research and education in computer science and engineering, artificial intelligence and bioinformatics. He is on the editorial boards of International Journal of Functional Informatics and Personalized Medicine and International Journal of Computational Biology and Drug Design. He is also on the advisory board of IJCBS. == Biography == Ersoy was born on September 5, 1945, in Istanbul, Turkey. He received B.S.E.E. degree from Boğaziçi University (Formerly Robert College) in Istanbul in 1967; M.S.E.E. degree in 1968, MS degree in Systems Science and PhD in Electrical Engineering in 1972 respectively, all from University of California at Los Angeles (UCLA) with specialization on lasers, quantum electronics, optics and image processing. == Research areas == His current research is concentrated in the fields of digital signal/image processing and imaging, neural networks, decision trees and support vector machines, optical communications, networking and information processing, diffractive optics with scanning electron microscope, Fourier-related transforms and time-frequency methods, probability and statistics. He has written many books on signal, image processing and fast transforms. == Books == Diffraction, Fourier Optics and Imaging - Fourier Related Transforms and Applications == References == == External links == Website at Purdue DOI.org https://docs.lib.purdue.edu/ecetr/69/ http://sru.lib.purdue.edu/dir/sru?operation=searchRetrieve&recordSchema=dc&maximumRecords=50&sortKeys=DateCreated,0&recordXpath=&stylesheet=http%3A//www.lib.purdue.edu/research/sru/etd/etd.xsl&ScanClause=&version=1.1&recordPacking=xml&query=adviser+%3D+%22Ersoy%22+and+%22Okan%22 Some of the selected IEEE publications of Professor O. K. Ersoy
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Wikipedia:Ola Bratteli#0
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Ola Bratteli (24 October 1946 – 8 February 2015) was a Norwegian mathematician. He was a son of Trygve Bratteli and Randi Bratteli (née Larssen). He received a PhD degree in 1974. He was appointed as professor at the University of Trondheim in 1980 and at the University of Oslo in 1991. He was a member of the Norwegian Academy of Science and Letters. == Selected works == with Derek W. Robinson: Operator Algebras and Quantum Statistical Mechanics (Springer-Verlag, 2 volumes, 1980) Derivations, Dissipations and Group Actions on C*-algebras (Springer-Verlag, 1986) with Palle T. Jørgensen: Wavelets through a looking glass, the world of the spectrum (Birkhäuser, 2002) == See also == Approximately finite-dimensional C*-algebra Bratteli diagram Bratteli–Vershik diagram == References ==
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Wikipedia:Olami–Feder–Christensen model#0
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In physics, in the area of dynamical systems, the Olami–Feder–Christensen (OFC) model is an earthquake model conjectured to be an example of self-organized criticality where local exchange dynamics are not conservative. The model is named after Zeev Olami, Hans Jacob S. Feder, and Kim Christensen who proposed it in 1992. Despite the original claims of the authors and subsequent claims of other authors such as Stefano Lise, whether or not the model is self organized critical remains an open question. The system behaviour reproduces some empirical laws that earthquakes follow (such as the Gutenberg–Richter law and Omori's Law). == Model definition == The model is a simplification of the Burridge-Knopoff model, where the blocks move instantly to their balanced positions when submitted to a force greater than their friction. Let S be a square lattice with L × L sites and let Kmn ≥ 0 be the tension at site (m,n). The sites with tension greater than 1 are called critical and go through a relaxation step where their tension spreads to their neighbours. Through analogy with the Burridge-Knopoff model, what is being simulated is a fault, where one of the lattice's dimensions is the flaw depth and the other one follows the flaw. === Model rules === If there are no critical sites, then the system suffers a continuous drive, until a site becomes critical: K max = max ( i , j ) ∈ S K i j {\displaystyle K_{\max }={\underset {(i,j)\in S}{\max }}K_{ij}\,} K i j ← K i j + ( 1 − K max ) {\displaystyle K_{ij}\leftarrow K_{ij}+(1-K_{\max })\,} else if the sites C1, C2, ..., Cm are critical the relaxation rule is applied in parallel: K C i ← 0 , i = 1 , … , m {\displaystyle K_{C_{i}}\leftarrow 0,\quad i=1,\ldots ,m\,} K j ← K j + α K C i ′ ∀ j ∈ Γ C i , i = 1 , … , m {\displaystyle K_{j}\leftarrow K_{j}+\alpha K'_{C_{i}}\,\forall \,j\in \Gamma _{C_{i}},\quad i=1,\ldots ,m} where K'C is the tension prior to the relaxation and ΓC is the set of neighbours of site C. α is called the conservative parameter and can range from 0 to 0.25 in a square lattice. This can create a chain reaction which is interpreted as an earthquake. These rules allow us to define a time variable that is update during the driving step t ← t + ( 1 − K max ) {\displaystyle t\leftarrow t+(1-K_{\max })\,} this is equivalent to define a constant drive d K i d t = 1 ∀ i ∈ S {\displaystyle {\frac {dK_{i}}{dt}}=1\,\forall \,i\in S} and assume the relaxation step is instantaneous, which is a good approximation for an earthquake model. == Behaviour and criticality == The system's behaviour is heavily influenced by the α parameter. For α=0.25 the system is conservative (in the sense that the local exchange is conservative, as there is still tension loss in the borders) and clearly critical. For values α<0.25 the dynamics is very different, even in the limit α → 0.25, with greater noise and much greater transients. For low α, there are less possibilities of chain reactions which could lead to cut-offs in the earthquake size distribution, implying the model is not critical. Also, for α = 0, the model is trivially not critical. These observations lead to the question of what is the value αc where the system makes the transition from critical to non-critical behaviour, which is still an open question. == Further reading == Christensen, K.; Olami, Z. (1992). "Variation of the Gutenberg-Richter b {\displaystyle b} values and nontrivial temporal correlations in a spring-block model for earthquakes". Journal of Geophysical Research: Solid Earth. 97 (B6): 8729–8735. Bibcode:1992JGR....97.8729C. doi:10.1029/92JB00427. Grassberger, P. (1994). "Efficient large-scale simulations of a uniformly driven system". Physical Review E. 49 (3): 2436–2444. Bibcode:1994PhRvE..49.2436G. doi:10.1103/PhysRevE.49.2436. PMID 9961487. Lise, S. and Paczuski, M. (2001). "Self-organized criticality and universality in a nonconservative earthquake model". Physical Review E. 63 (3): 036111. arXiv:cond-mat/0008010. Bibcode:2001PhRvE..63c6111L. doi:10.1103/PhysRevE.63.036111. PMID 11308713. S2CID 24545277.{{cite journal}}: CS1 maint: multiple names: authors list (link) Lise, S. and Paczuski, M. (2001). "Scaling in a nonconservative earthquake model of self-organized criticality". Physical Review E. 64 (4): 046111. arXiv:cond-mat/0104032. Bibcode:2001PhRvE..64d6111L. doi:10.1103/PhysRevE.64.046111. PMID 11690094. S2CID 33177015.{{cite journal}}: CS1 maint: multiple names: authors list (link) Olami, Z., Feder, H. J. S. and Christensen, K. (1992). "Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes". Physical Review Letters. 68 (8): 1244–1247. Bibcode:1992PhRvL..68.1244O. doi:10.1103/PhysRevLett.68.1244. PMID 10046116.{{cite journal}}: CS1 maint: multiple names: authors list (link)
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Wikipedia:Oldřich Vašíček#0
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Oldřich Alfons Vašíček (Czech pronunciation: [ˈoldr̝ɪx ˈalfons ˈvaʃiːt͜ʃɛk]; born 1942) is a Czech mathematician and quantitative analyst, best known for his pioneering work on interest rate modelling; see Vasicek model and KMV model. Vašíček received his master's degree in math from the Czech Technical University, 1964, and a doctorate in probability theory from Charles University four years later. After the Soviet invasion of Czechoslovakia in 1968, he defected to America, settled in San Francisco, and found employment in the management science department of Wells Fargo Bank in January 1969. In 1989 Stephen Kealhofer, John McQuown and Oldřich Vašíček founded company KMV. In 2002 the three entrepreneurs sold the company to Moody's for $210 million. In 2007, Moody's KMV was renamed to Moody's Analytics. In 1970, Wells Fargo sponsored a conference that included Fischer Black and Myron Scholes, who had just begun thinking seriously about the problem of valuing stock options. Of course, their paper on that subject, timed so as to coincide with a related paper by Robert C. Merton, would revolutionize financial economics three years later. Even their preliminary thoughts at the 1970 conference excited Vašíček, who soon made related issues his own life's work. Vašíček's own breakthrough paper, "An equilibrium characterization of the term structure" describing the dynamics of the yield curve, appeared in the Journal of Financial Economics in 1977. The mean-reverting short-rate model he describes is commonly known as the Vasicek model. In recognition of that paper, and subsequent work, the International Association of Financial Engineers named Vašíček its IAFE/Sungard Financial Engineer of the Year, in 2004. Vašíček has also received the Risk magazine Lifetime Achievement Award. He has been inducted into the Derivatives Strategy Hall of Fame, the Fixed Income Analysts Society Hall of Fame and the Risk Magazine Hall of Fame. Peter Carr, Head of Quantitative Financial Research at Bloomberg LP, included Vašíček's paper Probability of Loss on Loan Portfolio in the book Derivatives Pricing: The Classic Collection, which was marketed as a selection of 19 most influential papers on quantitative finance. He plays the flute and is an avid windsurfer. He has three sons, one of whom is the screenwriter John Vasicek. == See also == Warsaw Pact invasion of Czechoslovakia == Notes ==
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Wikipedia:Ole Jacob Broch#0
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Ole Jacob Broch (14 January 1818 – 5 February 1889) was a Norwegian mathematician, physicist, economist and government minister. == Biography == Broch was born at Fredrikstad in Østfold, Norway. He was the son of war commissary Johan Jørgen Broch (1791–1860) and Jensine Laurentze Bentzen (1790–1877) and the brother of the orientalist and linguist, Jens Peter Broch. He attended Kristiansand Cathedral School and showed a talent for mathematics at an early age. He attended the Overlærer Møller Institute in Christiania (now Oslo) and later studied at the University of Christiania (now University of Oslo). He also traveled abroad for studies in Paris, Berlin and Königsberg, during which he developed an interest in optics and statistics. After returning to Norway, he worked with his friend and colleague Hartvig Nissen to found the school Hartvig Nissens skole in 1843, which were to have an emphasis on natural sciences and modern languages. After finishing his doctorate in 1847, he returned to a position at the University that he had resigned to work with Nissen. He also taught at the Military Academy, and in 1847 he founded the insurance company Gjensidige (under the name "Christiania almindelige, gjensidige Forsørgelsesanstalt"), which was Scandinavia's first life insurance company. Broch entered politics as a local politician in Christiania, and in the period 1862–69 he represented the city in the Storting. In 1869, he was appointed Minister of the Navy in the first cabinet of Frederik Stang. After serving as a member of the Council of State Division in Stockholm in 1871–72, he returned as Minister of the Navy briefly in 1872. He resigned over differences with his colleagues about government ministers' access to the parliament. After this his attention turned to international tasks. In 1879 he became a member, and in 1883 director of the International Bureau of Weights and Measures in Sèvres, France. This work took up much of the remainder of Broch's life, but in 1884 he was recalled to Norway to attempt to form a government. The constitutional crisis which caused the fall of the so-called April Ministerium of Christian Homann Schweigaard, led to the demand for a new prime minister. Broch failed in this attempt, and returned to France, where he died a few years later. He was buried at Var Frelsers gravlund in Oslo. == Honors == Broch received several honours for his scientific and political work. He was a member of the Royal Norwegian Society of Sciences and Letters from 1849, and he received the Grand Cross of the Royal Norwegian Order of St. Olav in 1879. Internationally, he was created grand officer of the French Légion d'honneur, and Commander Grand Cross of the Swedish Order of the Polar Star. Brochøya, an island off the north coast of Nordaustlandet in Svalbard, was named after him. == Selected works == Lehrbuch der Mechanik, 1854 Laerebog i Arithmetik og Algebraens Elementer, 1860 Logarithme-Tabel med 5 Decimale, 1865 Traité Élémentaire des Fonctions Elliptiques, 1867 == References == == Further reading == Seip, Jens Arup (1971). Ole Jakob Broch og hans samtid. Copenhagen: Gyldendal. ISBN 82-05-00401-3.
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Wikipedia:Ole Michael Ludvigsen Selberg#0
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Ole Michael Ludvigsen Selberg (7 October 1877 – 11 December 1950) was a Norwegian mathematician and educator. He was born in Flora. He was married to Anna Kristina Brigtsdatter Skeie, and the father of Sigmund, Arne, Henrik and Atle Selberg. His thesis from 1925 treated the theory of algebraic equations. Three of his sons became professors of mathematics, and one was professor of engineering. During the occupation of Norway by Nazi Germany Selberg was a member of the Nazi party Nasjonal Samling. He is also known for his large collection of mathematics literature, which has later been donated to the Norwegian University of Science and Technology. == References ==
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Wikipedia:Ole Peder Arvesen#0
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Ole Peder Arvesen (27 March 1895 – 23 January 1991) was a Norwegian engineer and mathematician. Arvesen was born in Fredrikstad. He was appointed professor of descriptive geometry at the Norwegian Institute of Technology from 1938 to 1965. He served as secretary general of the Royal Norwegian Society of Sciences and Letters from 1950 to 1966, having been a fellow since 1934, and was also a fellow of the Norwegian Academy of Technological Sciences. Among his publications are Under Duskens billedbok (Under the Dusk picture book) from 1928, the textbook Innføring i nomografi (Introduction to nomography) from 1932, Mennesker og matematikere (People and mathematicians) from 1940, Glimt av den store karikatur (Glimpse of the great caricature) from 1941, and the memoir book Men bare om løst og fast (But just about this and that) from 1976. He was decorated Knight, First Class of the Order of St. Olav in 1965. A portrait of Arvesen, painted by Agnes Hiorth, is located at the Student Society in Trondheim, where he was an active participant over many years. == References ==
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Wikipedia:Ole Skovsmose#0
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Ole Skovsmose (1944-2025) was a Danish mathematics educator, philosopher, and artist, known for his contributions to critical mathematics education. Skovsmose was an emeritus professor at the Department of Culture and Learning at Aalborg University, Denmark, and served as a volunteer professor in the Graduate Program in Mathematics Education at the São Paulo State University (UNESP), Rio Claro campus. == Academic contributions == Skovsmose’s academic work focused on the socio-political aspects of mathematics and mathematics education. His contributions have led to the development of concepts such as critical mathematics education, landscapes of investigation, and students’ foregrounds. He authored over 50 books and more than 150 scientific articles. Notable works include his collaboration with the international research group BaCoMET (Basic Components of Mathematics Education for Teachers), which included prominent figures such as Alan Bishop and Guy Brousseau. The project resulted in the book Meaning in Mathematics Education. In recognition of his contributions, Skovsmose received the Hans Freudenthal Award in 2024 from the International Commission on Mathematical Instruction for his foundational work in critical mathematics education. His ideas have influenced academic studies worldwide, particularly in countries such as Colombia, India, Brazil, South Africa, and Germany. Skovsmose held various academic and advisory roles. He worked at Aalborg University for 25 years, eventually becoming a full professor before assuming emeritus status. == Books == Forandringer i matematikundervisningen. Copenhagen: Gyldendal, (1980) Matematikundervisning og kritisk pædagogik. Copenhagen: Gyldendal, (1981) Alternativer i matematikundervisningen. Copenhagen: Gyldendal, (1981) Kritik, undervisning og matematik. Copenhagen: Lærerforeningernes Materialeudvalg, (1984) Teknologikritik. Herning: Systime, (1986) (1990). Ud over matematikken. Aarhus: Systime. Towards a Philosophy of Critical Mathematics Education. Dordrecht: Springer Netherlands. Kluwer Academic Publishers (1994), ISBN 978-0792329329. Hacia una Filosofía de la Educación Matemática Crítica. Bogotá: Una Empresa Docente, (1999) Connecting corners: A Greek-Danish project in mathematics education. Aarhus: Systime, (1999) Educação matemática crítica: A questão da democracia. Campinas: Papirus, (2001). Dialogue and learning in mathematics education: Intention, reflection, critique. Dordrecht: Kluwer, (2002). Travelling Through Education: Uncertainty, Mathematics, Responsibility. Münster: BRILL. Rotterdam: Sense Publishers (2005), ISBN 978-9077874035. Diálogo e aprendizagem em educação matemática. Belo Horizonte: Autêntica, (2006). Educação crítica: Incerteza, Matemática, Responsabilidade. São Paulo: Cortez Editora, (2007). Desafios da reflexão: Em educação matemática crítica. 1 Campinas: Papirus, (2008) In Doubt - about Language, Mathematics, Knowledge and Life-Worlds. Rotterdam: Sense Publishers. (2009). ISBN 978-9460910265. Matematikfilosofi. Aarhus: Systime. (2011) Um convite à educação matemática crítica. Campinas: Papirus, (2014) Foregrounds: Opaque stories about learning. Rotterdam: Sense Publishers, (2014) Critique as uncertainty. Charlotte: Information Age Publishing, (2014) Connecting humans to equations: a reinterpretation of the philosophy of mathematics. Cham: Springer. (2019), ISBN 978-3030013363. Critical mathematics education. Cham: Springer. (2023), ISBN 9783031262425. Critical philosophy of mathematics. Cham: Springer. (2024), ISBN 978-3031713743. === Coeditor === Skovsmose, O.; Blomhøj. M. (Eds.). (2003). Kan det virkelig passe? Copenhagen: Akademisk Forlag, Kilpatrick, J.; HOYLES, C.; Skovsmose, O. in collaboration with Valero (Eds.). (2005). Meaning in mathematics education. New York: Springer. Skovsmose, O.; Blomhøj M. (Eds). (2006). Kunne det tænkes? Om matematiklæring. Copenhagen: Akademisk Forlag, Valero, P. ; Skovsmose, Ole (Eds.). (2012). Educación matemática crítica: Una visión sociopolítica del apendizaje y la enseñanza de las matemáticas. Bogotá: Universidad de Los Andes. == Art work == In addition to his academic career, Skovsmose was an accomplished artist. His works have been exhibited in galleries and museums across Europe, North and South America, and Asia, including venues such as the Carrousel du Louvre in France and the National Historical Museum in Brazil. In 2015, he participated in the European and Latin American Biennial of Contemporary Art. Skovsmose is a member of the Association Internationale des Arts Plastiques and the Danish Association of Visual Artists. His book Saudade explores the history of his artistic projects, including Faces as Landscapes and The Four Graces. == References == == External links == Ole Skovsmose publications indexed by Google Scholar
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Wikipedia:Oleg Besov#0
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Oleg Vladimirovich Besov (Russian: Олег Владимирович Бесов; born 1933) is a Russian mathematician. He heads the Department of Function Theory at the Steklov Institute of Mathematics, where he defended his PhD in 1960 and habilitation in 1966. He was an Invited Speaker at the ICM in 1970 in Nice. He is professor at the Moscow Institute of Physics and Technology and a member of the Russian Academy of Sciences (since 1990) and European Academy of Sciences (since 2002). A festschrift was published in honor of Besov's 70th birthday. == See also == Besov space == Selected publications == with Valentin Petrovich Ilʹin and Sergeĭ Mikhaĭlovich Nikolʹskiĭ: Integral representations of functions and imbedding theorems. Vol. 1. V. H. Winston & Sons, 1978. Vol. 2, 1979. == References ==
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Wikipedia:Oleg Marichev#0
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Oleg Igorevich Marichev (Russian: Олег Игоревич Маричев; born 7 September 1945 in Velikiye Luki, Russia) is a Russian mathematician. In 1949 he moved to Minsk with his parents. He graduated from the University of Belarus, where he continued to study for the Ph.D. degree. His scientific supervisor was Fedor Gakhov. He is the co-author of a comprehensive five volume series of Integrals and Series (Gordon and Breach Science Publishers, 1986–1992) together with Yury Brychkov and A. P. Prudnikov. Around 1990 he received the D.Sc. degree (Habilitation) in mathematics from the University of Jena, Germany. In 1992, Marichev started working with Stephen Wolfram on Mathematica. His wife Anna helps him in his job. == Works == Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (1983). Handbook of integral transforms of higher transcendental functions: theory and algorithmic tables. Ellis Horwood Series in Mathematics and its Applications (in Russian). Translated by Longdon, L. W. Chichester: Ellis Horwood Ltd.{{cite book}}: CS1 maint: multiple names: authors list (link) Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич); Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович) (1986). Tables of indefinite integrals (in Russian). Translated by Gould, G. G. Moscow: Nauka (Наука).{{cite book}}: CS1 maint: multiple names: authors list (link) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич). Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. 1–5 (1 ed.). Nauka (Наука).{{cite book}}: CS1 maint: multiple names: authors list (link) 1981−1986. Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2002) [1986]. Integrals and Series. Vol. 1: Elementary Functions. Translated by Queen, N. M. (1 ed.). Gordon & Breach Science Publishers / CRC Press. ISBN 2-88124-089-5. OCLC 916363878.{{cite book}}: CS1 maint: multiple names: authors list (link) (First published 1986?; fourth printing: 1998; ? printing: 2002.) (798 pages) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2002) [1986]. Integrals and Series. Vol. 2: Special functions. Translated by Queen, N. M. (1 ed.). OPA (Overseas Publishers Association) Amsterdam B.V. / Gordon & Breach Science Publishers / CRC Press. ISBN 2-88124-090-9. OCLC 50653126. Retrieved 2016-04-17.{{cite book}}: CS1 maint: multiple names: authors list (link) (First published 1986; second printing with corrections: 1988; third printing with corrections: 1992; fourth printing: 1998; ? printing: 2002.) (750 pages.) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2002) [1990]. Integrals and Series. Vol. 3: More special functions. Translated by Gould, G. G. (1 ed.). OPA (Overseas Publishers Association) Amsterdam B.V. / Gordon & Breach Science Publishers / CRC Press. ISBN 2-88124-682-6. OCLC 916363880. Retrieved 2016-04-17.{{cite book}}: CS1 maint: multiple names: authors list (link) (Second printing: 1998; third corrected printing: 1998; ? printing: 2002.) (800 pages.) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич). Integrals and Series. Vol. 4: Direct Laplace Transforms (1 ed.). OPA (Overseas Publishers Association) Amsterdam B.V. / Gordon & Breach Science Publishers / CRC Press. OCLC 63722509.{{cite book}}: CS1 maint: multiple names: authors list (link) (Second printing: 1998.) (xviii+618 pages.) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич). Integrals and Series. Vol. 5: Inverse Laplace Transforms (1 ed.). OPA (Overseas Publishers Association) Amsterdam B.V. / Gordon & Breach Science Publishers / CRC Press. ISBN 2-88124838-1. OCLC 489706146.{{cite book}}: CS1 maint: multiple names: authors list (link) (xx+595 pages.) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2003). Integraly i ryady Интегралы и ряды [Integrals and series] (in Russian). Vol. Set 1-3 (2nd revised ed.). Fiziko-Matematicheskaya Literatura, Fizmatlit (Физматлит). ISBN 978-5-9221-0322-0.{{cite book}}: CS1 maint: multiple names: authors list (link) (reprint 2013 by Let Me Print) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Elementary functions] (in Russian). Vol. 1: Elementarnye funktsii (Элементарные функции) (2nd revised ed.). Fiziko-Matematicheskaya Literatura, Fizmatlit (Физматлит). ISBN 978-5-9221-0323-7. OCLC 937142305.{{cite book}}: CS1 maint: multiple names: authors list (link) (630 pages) (reprint 2013 by Let Me Print) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions] (in Russian). Vol. 2: Spetsialnye funktsii (Специальные функции) (2nd revised ed.). Fiziko-Matematicheskaya Literatura, Fizmatlit (Физматлит). ISBN 978-5-9221-0324-4.{{cite book}}: CS1 maint: multiple names: authors list (link) (663 pages) (reprint 2013 by Let Me Print) Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович); Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2003). Integraly i ryady Интегралы и ряды [Integrals and series: Special functions. Further chapters.] (in Russian). Vol. 3: Spetsialnye funktsii. Dopolnitelnye glavy (2nd revised ed.). Fiziko-Matematicheskaya Literatura, Fizmatlit (Физматлит). ISBN 978-5-9221-0325-1.{{cite book}}: CS1 maint: multiple names: authors list (link) (710 pages) (reprint 2013 by Let Me Print) Samko, Stefan Grigor'evich Samko (Самко, Стефан Григорьевич); Kilbas, Kilbas, Anatoly Alexandrovich (Килбас, Анатолий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (1993-10-15). Fractional Integrals and Derivatives: Theory and Applications. Vol. I (1 ed.). Gordon and Breach. ISBN 978-2-8812-4864-1.{{cite book}}: CS1 maint: multiple names: authors list (link) (976 pages) Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2017-10-15). Handbook of Integrals and Series Set. Vol. I–II (1 ed.). CRC Press, Inc. ISBN 978-1-4398-2900-4.{{cite book}}: CS1 maint: multiple names: authors list (link) (2688 pages) Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2015-04-30). Handbook of Integrals and Series. Vol. I: Elementary functions (1 ed.). CRC Press, Inc. ISBN 978-1-4398-2896-0.{{cite book}}: CS1 maint: multiple names: authors list (link) (1344 pages) Brychkov, Yury Aleksandrovich (Брычков, Юрий Александрович); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (2017-10-15). Handbook of Integrals and Series. Vol. II: Special functions (1 ed.). CRC Press, Inc. ISBN 978-1-4398-2898-4.{{cite book}}: CS1 maint: multiple names: authors list (link) (1344 pages) == References == == External links == Oleg Marichev. Two Hundred Thousand New Formulas on the Web Oleg Marichev, member of Wolfram Research's development staff Stephen Wolfram. Festschrift for Oleg Marichev Wolfram, Stephen (2005-10-08). "The History and Future of Special Functions". Wolfram Technology Conference, Festschrift for Oleg Marichev, in honor of his 60th birthday (speech, blog post). Champaign, IL, USA: Stephen Wolfram, LLC. The story behind Gradshteyn-Ryzhik. Archived from the original on 2016-04-07. Retrieved 2016-04-06. "Special Session: In Honor of Oleg Marichev on the Occasion of His 60th Birthday". Archived from the original on 2012-02-12. About Oleg Marichev Oleg Igorevich Marichev (On the Occasion of His 70th Birthday)
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Wikipedia:Oleg Nagornov#0
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Oleg Viktorovich Nagornov (Russian: Олег Викторович Нагорнов; born 15 August 1956) is a Russian physicist and mathematician. Since 2010 he has been the first Vice-rector of National Research Nuclear University MEPhI (Moscow Engineering Physics Institute). == Early life and career == Oleg Nagornov was born on August 15, 1956, in Moscow, Russia. In 1979 he graduated from MEPhI, where he studied theoretical nuclear physics. In 1979–1982 he was a post-graduate student in MEPhI. In 1983 Oleg Nagornov started his career in MEPhI as a junior research fellow, became research assistant in 1985. == Professional experience == In 1988-2006 Oleg Nagornov worked as Associate Professor in MEPhI. He was a professor and the Head of the Department in MEPhI from 2007 to 2008. In 2008-2010 he served as Institute's Vice-rector. In 2010 he was appointed the first Vice-rector of MEPhI. Oleg Nagornov is a Member of the Expert Council of Higher Attestation Commission. == Awards == Laureate of the following Government Awards: Award of the Russian Federation Government in the field of education (2013) Diploma of Merit, Rosatom State Corporation Medal "For Merit in the Conduct of the All-Russian Population Census" Diploma for the training of information security specialists in commemoration of the 70th anniversary of MEPhI (2012) Diploma of Merit, The Federal Financial Monitoring Service of the Russian Federation (2012) The Kurchatov Medal, IV class (2012). == Scientific career == The scope of scientific interests includes inverse problems of mathematical physics, mathematical models in paleoclimatology, Numerical Modelling for Environmental Problems, heat and mass transfer and wave dispersion in porous multiphase medium. Academic degrees: PhD of Science (1983) Doctor of Physical and Mathematical Science (2005) == International career == Visiting professor in the Polar Ice Coring Office, University of Alaska Fairbanks, US (1991-1994) The head of International Science and Technology Center's projects (1994–2004) Visiting lecturer in the Center of Theoretical Research, Facultad de Estudios Superiores Cuautitlan, UNAM, Mexico (1996, 1998–2000) Co-leader of the CRDF (U.S. Civilian Research and Development Foundation) project (1996–1999) Visiting lecturer in universities of Argentina: Master's program "Numerical Modelling for Environmental Problems", Facultad Regional de San Nicolas, National Technological University and National University of San Juan (1997–2002) == References ==
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Wikipedia:Oleksandr Sharkovsky#0
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Oleksandr Mykolayovych Sharkovsky (Ukrainian: Олекса́ндр Миколайович Шарко́вський; 7 December 1936 – 21 November 2022) was a Ukrainian mathematician most famous for developing Sharkovsky's theorem on the periods of discrete dynamical systems in 1964. He was a corresponding member of the Academy of Sciences of the Ukrainian SSR (1978), and academician of the National Academy of Sciences of Ukraine (2006). Prize laureate of the National Academy of Sciences of Ukraine named after M. M. Bogolyubov and M. O. Lavretiev. == Life and career == In 1952, Sharkovsky's name appeared in the mathematical world - the magazine "Russian Mathematical Surveys", when eighth-grader Oleksandr Sharkovsky became the winner of the Kyiv Mathematical Olympiad for schoolchildren. By his first year in Kyiv National University, he had already written his first scientific work. After graduating with honors from Taras Shevchenko National University of Kyiv he successfully completed postgraduate studies at the NASU Institute of Mathematics with an early defense of his candidate's thesis (1961). Soon thereafter, in 1967, he defended his doctoral thesis. In 1978, O. M. Sharkovsky was elected a corresponding member of the Academy of Sciences of the Ukrainian SSR. Since 1974, O. M. Sharkovsky headed the department of differential equations of the Institute of Mathematics of the Ukrainian SSR Academy of Sciences, and since 1986 he headed the department of the theory of dynamical systems, which was created on his initiative. In 2006, Sharkovsky became a full member of the National Academy of Sciences of Ukraine. He is the head of the department of the Theory of dynamical systems at the Institute of Mathematics of the National Academy of Sciences of Ukraine. In the last years of his life, he worked as a chief researcher of the Department of Theory of Dynamic Systems and Fractal Analysis of the Institute of Mathematics of the National Academy of Sciences. O.M. Sharkovsky died on 21 November 2022, at the age of 85 in the Feofaniya Clinical Hospital in Kyiv. == Scientific work == Oleksandr Sharkovsky created the foundations of the topological theory of one-dimensional dynamic systems, a theory that today is one of the tools for researching evolutionary problems of the most diverse nature. He discovered the law of coexistence of periodic trajectories of different periods; the topological structure of basins of attraction of various sets is investigated; a number of criteria of simplicity and complexity of dynamic systems were obtained. O. M. Sharkovsky also contributed fundamental results in dynamical systems theory on arbitrary topological spaces. The achievements of the Ukrainian scientist received general recognition in international scientific circles. The formation and development of chaotic dynamics are associated with his name. In the scientific literature, you can find such terms as Sharkovsky's theorem, Sharkovsky's ordering, Sharkovsky's space, Sharkovsky's stratification, etc. Sharkovsky's theorem is associated with initiating a new direction in the theory of dynamical systems — combinatorial dynamics. In 1994, an international conference "Thirty years of Sharkovsky's theorem" was held in Spain. New perspectives". Research conducted by O. M. Sharkovsky allowed him to propose the concept of "ideal turbulence" — a new mathematical phenomenon in deterministic systems that models the most complex properties of turbulence in time and space, namely: the processes of the formation of coherent structures of decreasing scales and the birth of random states. O. M. Sharkovsky actively combines scientific work with pedagogical activity. From the mid-60s of the 20th century. gave general courses and lectures on the theory of dynamic systems at the mechanical and mathematical faculty of his native university. O. M. Sharkovsky is the author of almost 250 scientific works, including five monographs written in co-authorship with students. Among the students are 3 doctors and 14 candidates of sciences. The Ukrainian scientist devoted a lot of energy and time to developing scientific relations. He gave lectures at universities and scientific centers in more than 20 countries in Europe and America, and at universities in China and Australia. He was a member of the editorial boards of a number of international mathematical publications, in particular, he was a co-editor of the journal "Journal of Difference Equations and Applications" (USA). His last paper "Descriptive theory of determined chaos" was published in the Ukrains’kyi Matematychnyi Zhurnal (Ukrainian Mathematical Journal) in January 2023 and translated in July 2023 with publication in Springer Link. == Awards and prizes == The first prize of the Kyiv Olympiad of Young Mathematicians (1951), the magazine "Russian Mathematical Surveys", (UMS, 7, 1952) The Bogoliubov Prize of National Academy of Sciences of Ukraine for the series of works "The theory of scattering of quantum systems and one-dimensional dynamical systems" (1993) The Lavrentyev Prize of National Academy of Sciences of Ukraine for a series of papers "Complex dynamic finite-dimensional and infinite-dimensional systems" (2005) State Prize of Ukraine in Science and Technology for the cycle of scientific works "Theory of dynamic systems: modern methods and their application" (2010) Bernd Aulbach Prize of the International Society of Difference Equations (2011) Honorary doctorate Doctor Honoris Causa (Dr. h. c.) from the Silesian University in Opava, Czech Republic (2014) The Mitropolskiy Prize of National Academy of Sciences of Ukraine, Kyiv (2019) == Notes == == References == == External links == Sharkovsky Oleksandr Mykolayovych memorial page at the Institute of Mathematics NAS of Ukraine (in Ukrainian) Oleksandr Sharkovsky at the Mathematics Genealogy Project Oleksandr Mikolaiovich Sharkovsky at the MacTutor History of Mathematics archive (in Ukrainian)
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Wikipedia:Olga Hadžić#0
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Olga Hadžić (25 August 1946 – 23 January 2019) was a Serbian mathematician known for her work on fixed-point theorems. == Early life and education == Hadžić was born in Novi Sad, on 25 August 1946, the daughter of lawyer Lazar Hadžić and the granddaughter of writer and physician Ilija Ognjanović. She attended both the Jovan Jovanović Zmaj Gymnasium and a music school in Novi Sad. She earned a degree in mathematics at the University of Novi Sad in 1968, and continued there as an assistant, earning a master's degree through the Faculty of Natural Sciences And Mathematics at the University of Belgrade in 1970, and completing a doctorate at the University of Novi Sad in 1972. Her doctoral dissertation, Neki problemi diferencijalnog računa u lokalno konveksnim prostorima [Some problems of differential calculus in locally convex spaces], was supervised by Bogoljub Stanković. Later in life, she took up the study of tourism management and marketing, earning a master's degree in 2005 from the University of Novi Sad and a second doctorate in 2006, supervised by Jovan Romelić. == Career and later life == Hadžić spent her career at the University of Novi Sad, becoming an assistant professor there in 1973, associate professor in 1977, and full professor in 1981. She became rector of the university in 1996, the first woman in Serbia to achieve this position. She was the founding editor-in-chief of the mathematics journal Univerzitet u Novom Sadu, Zbornik Radova Prirodno-Matematičkog Fakulteta, Serija za Matemati [University of Novi Sad, Review of Research, Faculty of Science, Mathematics Series], which later became the Novi Sad Journal of Mathematics, from 1971 to 1995. == Books == Hadžić was the author of several books on mathematics, particularly focusing on fixed-point theorems, including: Osnovi teorije nepokretne tačke [Foundations of fixed point theory], Institute of Mathematics, Novi Sad, 1978. Fixed point theory in topological vector spaces, Institute of Mathematics, Novi Sad, 1984. Numeričke i statističke metode u obradi eksperimentalnih podataka [Numerical and statistical methods in processing experimental data], Institute of Mathematics, Novi Sad, 1989. Fixed point theory in probabilistic metric spaces, Institute of Mathematics, Novi Sad, 1995. Fixed point theory in probabilistic metric spaces, with Endre Pap, Kluwer, 2001. == Recognition == Hadžić was a member of the Serbian Academy of Sciences and Arts, elected in 1991, and also a member of the Vojvodina Academy of Sciences and Arts, elected as a corresponding member in 1984 and a regular member in 1990. == References == == External links == Olga Hadžić publications indexed by Google Scholar
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Wikipedia:Olga Kharlampovich#0
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Olga Kharlampovich (born March 25, 1960, in Sverdlovsk) is a Russian-Canadian mathematician working in the area of group theory. She is the Mary P. Dolciani Professor of Mathematics at the CUNY Graduate Center and Hunter College. == Contributions == Kharlampovich is known for her example of a finitely presented 3-step solvable group with unsolvable word problem (solution of the Novikov–Adian problem) and for the solution together with A. Myasnikov of the Tarski conjecture (from 1945) about equivalence of first-order theories of finitely generated non-abelian free groups (also solved by Zlil Sela) and decidability of this common theory. Algebraic geometry for groups, introduced by Baumslag, Myasnikov, Remeslennikov, and Kharlampovich became one of the new research directions in combinatorial group theory. == Education and career == She received her Ph.D. from the Leningrad State University in 1984 (her doctoral advisor was Lev Shevrin) and Russian “Doctor of Science” in 1990 from the Moscow Steklov Institute of Mathematics. Prior to her current appointment at CUNY, she held a position at Ural State University, Ekaterinburg, Russia, and was a Professor of Mathematics at McGill University, Montreal, Canada, where she had been working since 1990. As of August 2011 she moved to Hunter College of the City University of New York as the Mary P. Dolciani Professor of Mathematics, where she is the inaugural holder of the first endowed professorship in the Department of Mathematics and Statistics. == Recognition == For her undergraduate work on the Novikov–Adian problem she was awarded in 1981 a Medal from the Soviet Academy of Sciences. She received an Ural Mathematical Society Award in 1984 for the solution of the Malcev–Kargapolov problem posed in 1965 about the algorithmic decidability of the universal theory of the class of all finite nilpotent groups. Kharlampovich was awarded in 1996 the Krieger–Nelson Prize of the Canadian Mathematical Society for her work on algorithmic problems in varieties of groups and Lie algebras (the description of this work can be found in the survey paper with Sapir and on the prize web site). She was awarded the 2015 Mal'cev Prize for the series of works on fundamental model-theoretic problems in algebra. She was elected a Fellow of the American Mathematical Society in the 2020 class "for contributions to algorithmic and geometric group theory, algebra and logic." == Selected publications == == References == == External links == Hunter webpage McGill home page Olga Kharlampovich at the Mathematics Genealogy Project
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Wikipedia:Olga Ladyzhenskaya#0
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Olga Aleksandrovna Ladyzhenskaya (Russian: Ольга Александровна Ладыженская, IPA: [ˈolʲɡə ɐlʲɪˈksandrəvnə ɫɐˈdɨʐɨnskəɪ̯ə] ; 7 March 1922 – 12 January 2004) was a Russian mathematician who worked on partial differential equations, fluid dynamics, and the finite-difference method for the Navier–Stokes equations. She received the Lomonosov Gold Medal in 2002. She authored more than two hundred scientific publications, including six monographs. == Biography == Ladyzhenskaya was born and grew up in the small town of Kologriv, the daughter of a mathematics teacher who is credited with her early inspiration and love of mathematics. The artist Gennady Ladyzhensky was her grandfather's brother, also born in this town. In 1937 her father, Aleksandr Ivanovich Ladýzhenski, was arrested by the NKVD and executed as an "enemy of the people". Ladyzhenskaya completed high school in 1939, unlike her older sisters who weren't permitted to do the same. She was not admitted to the Leningrad State University due to her father's status and attended a pedagogical institute. After the German invasion of June 1941, she taught school in Kologriv. She was eventually admitted to Moscow State University in 1943 and graduated in 1947. She began teaching in the Physics department of the university in 1950 and defended her PhD there, in 1951, under Sergei Sobolev and Vladimir Smirnov. She received a second doctorate from the Moscow State University in 1953. In 1954, she joined the mathematical physics laboratory of the Steklov Institute and became its head in 1961. Ladyzhenskaya had a love of arts and storytelling, counting writer Aleksandr Solzhenitsyn and poet Anna Akhmatova among her friends. Like Solzhenitsyn she was religious. She was once a member of the city council, and engaged in philanthropic activities, repeatedly risking her personal safety and career to aid people opposed to the Soviet regime. Ladyzhenskaya suffered from various eye problems in her later years and relied on special pencils to do her work. Two days before a trip to Florida, she died in her sleep in Russia on 12 January 2004. == Mathematical accomplishments == Ladyzhenskaya is known for her work on partial differential equations (especially Hilbert's nineteenth problem) and fluid dynamics. She provided the first rigorous proofs of the convergence of a finite difference method for the Navier–Stokes equations. She analyzed the regularity of parabolic equations, with Vsevolod A. Solonnikov and her student Nina Ural'tseva, and the regularity of quasilinear elliptic equations. She wrote a student thesis under Ivan Petrovsky and was on the shortlist for the 1958 Fields Medal, ultimately awarded to Klaus Roth and René Thom. == Publications == Ladyzhenskaya, O.A. (1969) [1963], The Mathematical Theory of Viscous Incompressible Flow, Mathematics and Its Applications, vol. 2 (Revised Second ed.), New York; London; Paris; Montreux; Tokyo; Melbourne: Gordon and Breach, pp. xviii+224, MR 0254401, Zbl 0184.52603. Ladyženskaja, O.A.; Solonnikov, V.A.; Ural'ceva, N.N. (1968), Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, Providence, RI: American Mathematical Society, pp. xi+648, ISBN 978-0-8218-8653-3, MR 0241821, Zbl 0174.15403. Ladyzhenskaya, Olga A.; Ural'tseva, Nina N. (1968), Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering, vol. 46, New York and London: Academic Press, pp. xviii+495, ISBN 978-0-08-095554-4, MR 0244627, Zbl 0164.13002. Ladyzhenskaya, O.A. (1985), The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, vol. 49, Berlin; Heidelberg; New York: Springer Verlag, pp. xxx+322, doi:10.1007/978-1-4757-4317-3, ISBN 978-0-521-39922-7, MR 0793735, Zbl 0588.35003 (Translated by Jack Lohwater). Ladyzhenskaya, O.A. (1991), Attractors for Semigroups and Evolution Equations, Lezioni Lincee, Cambridge: Cambridge University Press, pp. xi+73, doi:10.1017/CBO9780511569418, ISBN 978-0-521-39922-7, MR 1133627, S2CID 51684720, Zbl 0755.47049 == Awards and recognitions == P. L. Chebyshev Prize (with Nina Nikolayevna Ural'tseva ) (1966) for the work "Linear and quasilinear equations of elliptic type" USSR State Prize (1969) Member of Lincei National Academy in Rome (1989) Member of the Russian Academy of Sciences (1990) Kovalevskaya Prize (1992) for the series of works "Attractors for Semigroups and Evolution Equations" ICM Emmy Noether Lecture (1994) John von Neumann Lecture (1998) Order of Friendship (1999) Lomonosov Gold Medal (2002) for outstanding achievements in the field of the theory of partial differential equations and mathematical physics On 7 March 2019, the 97th anniversary of Ladyzhenskaya's birth, the search engine Google released a Google Doodle commemorating her. The accompanying comment read, "Today's Doodle celebrates Olga Ladyzhenskaya, a Russian mathematician who triumphed over personal tragedy and obstacles to become one of the most influential thinkers of her generation." In 2022, the "Ladyzhenskaya Prize in Mathematical Physics" is created in her honor. It has been awarded for the first time on 2 July 2022 to Svetlana Jitomirskaya in a joint session at (WM)², World Meeting for Women in Mathematics and at the Probability and Mathematical Physics conference OAL Prize Winner 2022. == Notes == == See also == Projection method (fluid dynamics) == References == Bolibruch, A.A.; Osipov, Yu. S.; Sinai, Ya. G., eds. (2006), Mathematical Events of the Twentieth Century, Berlin; Heidelberg; New York: Springer-Verlag, pp. viii+545, Bibcode:2006metc.book.....A, doi:10.1007/3-540-29462-7, ISBN 978-3-540-23235-3, MR 2179060, Zbl 1072.01002 Friedlander, Susan; Keyfitz, Barbara (18–20 May 2006), "Olga Ladyzhenskaya and Olga Oleinik: Two Great Women Mathematicians of the 20th Century", in Kuperberg, Krystyna (ed.), Women in Mathematics: The Legacy of Ladyzhenskaya and Oleinik, Berkeley, CA: AWM and MSRI, archived from the original (PDF) on 8 March 2018, retrieved 1 July 2009. Some recollections of the authors about Olga Ladyzhenskaya and Olga Oleinik. Friedlander, Susan; Lax, Peter; Synge Morawetz, Cathleen; Nirenberg, Louis; Seregin, Gregory; Ural'tseva, Nina; Vishik, Mark (December 2004), "Olga Alexandrovna Ladyzhenskaya (1922–2004)" (PDF), Notices of the American Mathematical Society, 51 (11): 1320–1331, MR 2105237, Zbl 1159.01335. Gunzburger, Max; Seregin, Gregory; Ochkur, Vitaly; Shilkin, Timofey (24 April 2004), "Obituaries: Olga Ladyzhenskaya" (PDF), SIAM News, 37 (3): 3, archived from the original (PDF) on 7 July 2014, retrieved 9 January 2012 Pearce, Jeremy (25 January 2004), "Dr. Olga Ladyzhenskaya, 81, mathematician", The New York Times, retrieved 9 January 2012 Riddle, Larry, ed. (8 December 2010), Olga Alexandrovna Ladyzhenskaya, retrieved 5 May 2011. A biography in the Biographies of Women Mathematicians, Agnes Scott College. Struwe, Michael (2003), "Olga Ladyzhenskaya – a life-long devotion to mathematics", in Hildebrandt, Stefan; Karcher, Hermann (eds.), Geometric analysis and nonlinear partial differential equations, Berlin: Springer Verlag, pp. 1–10, ISBN 978-3-540-44051-2, MR 2008328, Zbl 1290.35002. Synge Morawetz, Cathleen (18–20 May 2006), "Early Memories of Olga Ladyzhenskaya and Olga Oleinik", in Kuperberg, Krystyna (ed.), Women in Mathematics: The Legacy of Ladyzhenskaya and Oleinik, Berkeley, CA: AWM and MSRI, archived from the original (PDF) on 8 March 2018, retrieved 1 July 2009. Some recollections of the author about Olga Ladyzhenskaya and Olga Oleinik. Titova, Irina (26 January 2004), "Russian mathematician Olga Ladyzhenskaya dies at 81", USA Today, retrieved 9 January 2012 Zajączkowski, Wojciech (September 2005), "Olga Alexandrovna Ladyzhenskaya (1922–2004)", Topological Methods in Nonlinear Analysis, 26 (1 2005): 5–7, doi:10.12775/TMNA.2005.021, MR 2179347, Zbl 1082.01516 Ладыженская, О. А. (1958). "Решение "в целом" краевой задачи для уравнений Навье – Стокса в случае двух пространственных переменных". Доклады Академии наук СССР. 123 (3): 427–429. [Ladyzhensakya, O. A. (1958). "Solution in the large to the boundary-value problem for the Navier–Stokes equations in two space variables". Soviet Physics Dokl. 123 (3): 1128–1131. Bibcode:1960SPhD....4.1128L.] == External links == Beirao da Veiga, H.; Seregin, G.; Solonnikov, V.; Uraltseva, N.; Valli, A., eds. (2004), Partial Differential Equations in Mathematical Physics (October 24–30, 2004), Trento: Centro Internazionale per la Ricerca Matematica, retrieved 1 February 2012. The schedule of a workshop in honour of Olga A. Ladyzhenskaya. Kuperberg, Krystyna, ed. (18–20 May 2006), Women in Mathematics: The Legacy of Ladyzhenskaya and Oleinik, Berkeley, CA: AWM and MSRI, archived from the original on 8 March 2018, retrieved 1 July 2009. The proceedings of a workshop in honour of Olga Ladyzhenskaya and Olga Oleinik. Olga Ladyzhenskaya at the Mathematics Genealogy Project. O'Connor, John J.; Robertson, Edmund F. (August 2005), "Olga Alexandrovna Ladyzhenskaya", MacTutor History of Mathematics Archive, University of St Andrews. Olga Ladyzhenskaya at PlanetMath. Saint Petersburg Mathematical Society (2006), Olga Aleksandrovna Ladyzhenskaya, retrieved 5 June 2011. Memorial page at the Saint Petersburg Mathematical Pantheon.
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Wikipedia:Olli Lehto#0
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Olli Erkki Lehto (30 May 1925 in Helsinki — 31 December 2020) was a Finnish mathematician, specializing in geometric function theory, and a chancellor of the University of Helsinki. Lehto earned his PhD in 1949 from the University of Helsinki under Rolf Nevanlinna with thesis Anwendung orthogonaler Systeme auf gewisse funktionentheoretische Extremal- und Abbildungsprobleme. At the University of Helsinki, Lehto was from 1961 to 1988 a professor, from 1978 the dean of science, from 1983 the rector, and from 1988 to 1993 the chancellor. From 1983 to 1990 he was Secretary of the International Mathematical Union. In 1962 he became a member of the Finnish Academy of Science and Letters (Suomalainen Tiedeakatemia). In 1968 he was elected member of the Finnish Society of Sciences and Letters and in 1988 he became honorary member of the same society. He was a member of the Norwegian Academy of Science and Letters from 1986. In 1975 he was given by the President of Finland the honorary title "Academician of Science" (Tieteen akateemikko). Lehto was the chief organizer of the International Congress of Mathematicians (ICM) in Helsinki in 1978 and an invited speaker of the ICM in Moscow in 1966 with lecture Quasiconformal mappings in the plane. He was elected a Fellow of the American Mathematical Society. In 2001 Lehto published a biography of his mentor Rolf Nevanlinna. == Selected works == with Kaarlo Virtanen: Quasikonforme Abbildungen. Springer 1965, Grundlehren der Mathematischen Wissenschaften, 2nd edition: Quasiconformal mappings in the plane. Springer 1973. Univalent functions and Teichmüller Spaces. Springer, Graduate Texts in Mathematics, 1987. Mathematics without borders: a history of the International Mathematical Union. Verlag-Springer. 1998. ISBN 978-0-387-98358-5. Lehto, Olli (2001). Korkeat maailmat: Rolf Nevanlinnan elämä [High Worlds: The life of Rolf Nevanlinna] (in Finnish). Otava. OCLC 58345155. Lehto, Olli (2008). Erhabene Welten: Das Leben Rolf Nevanlinnas. Springer-Verlag. ISBN 9783764377021; translated from Finnish into German by Manfred Stern. Tieteen aatelia: Lorenz Lindelöf ja Ernst Lindelöf. Otava, Helsinki 2008 (Finnish). Lars Ahlfors: At the Summit of Mathematics. American Mathematical Society. 2015. ISBN 978-1-4704-1846-5. 2015 pbk reprint == References == == External links == Kirjailijat – Olli Lehto. Otava.
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Wikipedia:Olli Lokki#0
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Olli Kristian Lokki (Lindeqvist) (28 April 1916 – 6 March 1994) was a Finnish mathematician. == Education and career == Loki was born in Helsinki. His father was a historian and schoolman, Karl Olof Lindeqvist. Lokki graduated in 1934 from the Normal Lyceum of Helsinki, then studied at the University of Helsinki, graduating with a master's degree in 1939 and completed his doctorate under the supervision of Rolf Nevanlinna and Pekka Myrberg in 1947. His thesis was in the field of function theory, with the title Über analytische Funktionen deren Dirichletintegrale endlich ist und die in gegebenen Punkten vorgeschriebene Werte annehmen. Loki's studies were interrupted by wars. During the Continuation War, Lokki worked as a mathematician in the Air Defense Department of the Finnish Air Force. After the war, Lokki became an assistant at the Department of Mathematics at the University of Helsinki, and after working as a vocational school teacher for one academic year, he was appointed a lecturer in mathematics at the Helsinki University of Technology in 1945 and held this position until 1953. At that time, he was appointed assistant professor of mathematics at the same institution. From 1962 until his retirement in 1979, Lokki was a professor of applied mathematics at the Helsinki University of Technology. Lokki became a docent of mathematics at the University of Helsinki in 1952. He was the president of the Finnish Academy of Technical Sciences between 1980 and 1984. In his teaching, Lokki particularly promoted the application of statistical and operations research methods to the solution of practical problems. The Nobel laureate Bengt Holmström studied under him at the University of Helsinki. He has a wide network outside of academic circles as well. Lokki was an honorary member of the Finnish Quality Control Association and an honorary member of the European Consortium for Mathematics in Industry. Lokki was elected a member of the Finnish Academy of Sciences in 1979. == References ==
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Wikipedia:Olof Hanner#0
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Olof Hanner (7 December 1922 in Stockholm – 19 September 2015 in Gothenburg) was a Swedish mathematician. == Education and career == Hanner earned his Ph.D. from Stockholm University in 1952. He was a professor at the University of Gothenburg from 1963 to 1989. == Contributions == In a 1956 paper, Hanner introduced the Hanner polytopes and the Hanner spaces having these polytopes as their metric balls. Hanner was interested in a Helly property of these shapes, later used to characterize them by Hansen & Lima (1981): unlike other convex polytopes, it is not possible to find three translated copies of a Hanner polytope that intersect pairwise but do not have a point of common intersection. Subsequently, the Hanner polytopes formed a class of important examples for the Mahler conjecture and for Kalai's 3d conjecture. In another paper from the same year, Hanner proved a set of inequalities related to the uniform convexity of Lp spaces, now known as Hanner's inequalities. Other contributions of Hanner include (with Hans Rådström) improving Werner Fenchel's version of Carathéodory's lemma, contributing to The Official Encyclopedia of Bridge, and doing early work on combinatorial game theory and the mathematics of the board game Go. One of the many proofs of the Pythagorean theorem based on the Pythagorean tiling is sometimes called "Olof Hanner's Jigsaw Puzzle". == Selected publications == Hanner, Olof (1951), "Some theorems on absolute neighborhood retracts", Arkiv för Matematik, 1 (5): 389–408, Bibcode:1951ArM.....1..389H, doi:10.1007/BF02591376, MR 0043459. Hanner, Olof; Rådström, Hans (1951), "A generalization of a theorem of Fenchel", Proceedings of the American Mathematical Society, 2 (4): 589–593, doi:10.2307/2032012, JSTOR 2032012, MR 0044142. Hanner, Olof (1956a), "Intersections of translates of convex bodies", Mathematica Scandinavica, 4: 65–87, doi:10.7146/math.scand.a-10456, MR 0082696. Hanner, Olof (1956b), "On the uniform convexity of Lp and ℓp", Arkiv för Matematik, 3 (3): 239–244, Bibcode:1956ArM.....3..239H, doi:10.1007/BF02589410, MR 0077087. Hanner, Olof (1959), "Mean play of sums of positional games", Pacific Journal of Mathematics, 9: 81–99, doi:10.2140/pjm.1959.9.81, MR 0104524. Hanner, Olof (1970), "Mathematics, A Solitary Game", The Two-Year College Mathematics Journal, 1 (2): 5–16, doi:10.2307/3027352, JSTOR 3027352. Hallén, Hans-Olof; Hanner, Olof; Jannersten, Per (1994), Rigal, Barry (ed.), Bridge movements: A fair approach, Bridgeakad. (Bridge academy) (Translated by Barry Rigal from the 1990 Swedish Tävlingsledaren (The leader of the tournament) ed.), Alvesta: Jannersten Forlag AB, ISBN 91-85024-86-4 == References == == External links == Olof Hanner at WorldCat
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Wikipedia:Olof Thorin#0
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G. Olof Thorin (23 February 1912, Halmstad – 14 February 2004, Danderyd Hospital) was a Swedish mathematician working on analysis and probability, who introduced the Riesz–Thorin theorem. == References == Peetre, Jaak; Grandell, Jan; Bondesson, Lennart (2008), "The life and work of Olof Thorin (1912--2004)", Proceedings of the Estonian Academy of Sciences, 57 (1): 18–25, doi:10.3176/proc.2008.1.02, ISSN 1736-6046, MR 2555072
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Wikipedia:Omar Catunda#0
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Omar Catunda (Santos, September 23, 1906 - Salvador, August 12, 1986) was a Brazilian mathematician, teacher and educator. He was one of the great mathematicians of the 20th century in Brazil and helped consolidate mathematics research and teaching. == Biography == Catunda was born in 1906, in Santos, and was the tenth child of Thomaz Catunda and Maria Lima Verde Catunda, from Ceará. His father was a doctor and his mother was well educated, particularly interested in classical and romantic French literature. His paternal great-grandfather, Joaquim Catunda, was a Republican senator and a professor of German in Fortaleza. Catunda studied at the Grupo Escolar Cesário Bastos, at the Liceu Comercial, where he excelled in Portuguese and Mathematics, and at the Escola de Comércio José Bonifácio. In 1922, he went to Rio de Janeiro, where he prepared for the exams at Colégio Pedro II by studying eleven hours a day, with the exception of Latin. Of the subjects studied, he enjoyed studying geometry the most, taking Comberrousse's Geometria Elementar as his textbook. In 1925, he came first in the entrance exam for the Polytechnic School of the University of São Paulo, where he mastered spatial geometry. In the subject Complements of Mathematics, he had his first contact with Integral Differential Calculus and met Professor Theodoro Augusto Ramos, who later guided his higher studies in mathematics. He won the Cesário Motta Prize, a gold medal awarded to the best student in the first year of the course. == Career == In 1930, Catunda graduated as an engineer; in 1933, he applied for the position of professor of Complementary Analytical Geometry, Nomography and Differential and Integral Calculus at USP's Polytechnic School, but was unsuccessful. In 1938, he worked as an engineer for Santos City Hall, but was hired by USP's Faculty of Philosophy, Sciences and Letters as an assistant to the Italian Luigi Fantappiè in the subject of Mathematical Analysis. From 1934 onwards, Catunda collaborated intensively with Fantappiè to establish the Mathematics Subsection of USP's Faculty of Philosophy, Sciences and Letters (the future Institute of Mathematics and Statistics). Under Fantappié's guidance, he began studying the Theory of Analytic Functionals and, between 1938 and 1939, he undertook postgraduate studies on the subject at the University of Rome. The result of this trip was a paper entitled "Un teorema sugl'insiemi che si reconnette alla teoria dei funzionali analitici". After returning to Brazil, he was appointed interim professor of Mathematical and Higher Analysis, replacing Fantappiè, who had returned to Italy in 1939. Catunda became a professor of Mathematical Analysis after defending his thesis "Sobre os fundamentos da teoria dos funcionais analíticos". He was appointed head of the Mathematics Subsection at USP's Faculty of Philosophy, Sciences and Letters, a position he held for many years. In 1942, he presented a paper entitled "Sobre os sistemas de equações de variações totais, em mais de um funcional incógnito". At the same time, Catunda expanded his studies and began to learn about topology using Pavel Alexandrov's text and algebra using Van der Waerden's text. Reflections of this appear in his thesis "Sobre os fundamentos da teoria dos funcionais analíticos", presented in 1944 for the Mathematical Analysis chair at the Faculty of Philosophy, Sciences and Letters. In 1946, he obtained a scholarship from the Rockefeller Foundation and went to Princeton University, where he took courses with Emil Artin, N. Cramer, Heinz Hopf, Hermann Weyl and John von Neumann. == Return to Brazil == In 1947, after finishing his studies at Princeton, Catunda returned to São Paulo, where he became involved in the campaign to defend Brazilian oil and became president of the Center for the Study and Defense of Oil. He was also a candidate for state representative, supported by the Communists, but his candidacy was contested by the electoral courts because he had not joined the Brazilian Communist Party. He criticized the Vargas administration for neglecting the education of the Brazilian people. According to Catunda, the government had decided to "democratize high school education, without realizing, or pretending not to realize, that there was no human material to carry out this democratization with the necessary severity". He also advocated increased investment in higher education courses, in order to train good teachers and improve high school education. At the beginning of the 1960s, he was invited by the rector Edgard Santos to become director of the Institute of Mathematics and Physics at the Federal University of Bahia. After retiring as a professor at USP, he moved to Salvador, where he took office in September 1963, replacing Rubens Lintz. He worked as a professor and director of the Institute until 1969. After the 1968 university reform, he became a full professor and coordinator of the Master's program at the Institute of Mathematics and Statistics of the Federal University of Bahia, until his compulsory retirement in 1976. Omar Catunda was one of the main representatives and promoters of the mathematical school established at the University of São Paulo by Fantappiè. He died on August 12, 1986, in Salvador, at the age of 79. == Relevance == He made an important contribution to the modernization of the teaching of calculus and mathematical analysis at USP, UFBA and other universities through his book Curso de Análise Matemática, as there were practically no calculus or analysis textbooks in Portuguese at the time. His concern to update the many subsequent editions of the book has ensured that it is still used today. Catunda was a professor to renowned physicists such as Mário Schenberg, Marcelo Damy, Abrahão de Moraes, Jean Meyer and Roberto Salmeron, and mathematicians such as Carlos Benjamin de Lyra, Luiz Henrique Jacy Monteiro and Alexandre Augusto Martins Rodrigues. === Main published works === Although he didn't publish many works, Catunda's material remains used to this day as a source of research on Fantappiè's theory of analytic functionals carried out in Brazil and published in specialized international journals. Un teorema sugl'insiemi che si reconnette alla teoria dei funzionali analitici (1939); Sui Sistemi di Equazioni alle Variazioni Totali in Più Funzionali Incogniti (1941). ==== Didactic works ==== Ensino Atualizado da Matemática 2° Ciclo (1971); Curso de Análise Matemática I (1954). == See also == Cândido Lima da Silva Dias == References ==
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Wikipedia:Omar Khayyam#0
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Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) (Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar Khayyam (Persian: عمر خیّام), was a Persian poet and polymath, known for his contributions to mathematics, astronomy, philosophy, and Persian literature.: 94 He was born in Nishapur, Iran and lived during the Seljuk era, around the time of the First Crusade. As a mathematician, he is most notable for his work on the classification and solution of cubic equations, where he provided a geometric formulation based on the intersection of conics. He also contributed to a deeper understanding of Euclid's parallel axiom.: 284 As an astronomer, he calculated the duration of the solar year with remarkable precision and accuracy, and designed the Jalali calendar, a solar calendar with a very precise 33-year intercalation cycle: 659 which provided the basis for the Persian calendar that is still in use after nearly a millennium. There is a tradition of attributing poetry to Omar Khayyam, written in the form of quatrains (rubāʿiyāt رباعیات). This poetry became widely known to the English-reading world in a translation by Edward FitzGerald (Rubaiyat of Omar Khayyam, 1859), which enjoyed great success in the Orientalism of the fin de siècle. == Life == Omar Khayy m was born in Nishapur—a metropolis in Khorasan province, of Persian stock, in 1048. In medieval Persian texts he is usually simply called Omar Khayyam.: 658 Although open to doubt, it has often been assumed that his forebears followed the trade of tent-making, since Khayyam means 'tent-maker' in Arabic.: 30 The historian Bayhaqi, who was personally acquainted with Khayyam, provides the full details of his horoscope: "he was Gemini, the sun and Mercury being in the ascendant[...]".: 471 : 172–175, no. 66 This was used by modern scholars to establish his date of birth as 18 May 1048.: 658 Khayyam's boyhood was spent in Nishapur,: 659 a leading metropolis in the Seljuk Empire,: 15 which had earlier been a major center of the Zoroastrian religion.: 68 His full name, as it appears in Arabic sources, was Abu’l Fath Omar ibn Ibrahim al-Khayyam. His gifts were recognized by his early tutors who sent him to study under Imam Muwaffaq Nishaburi, the greatest teacher of the Khorasan region who tutored the children of the highest nobility, and Khayyam developed a firm friendship with him through the years.: 20 Khayyam might have met and studied with Bahmanyar, a disciple of Avicenna.: 20–21 After studying science, philosophy, mathematics and astronomy at Nishapur, about the year 1068 he traveled to the province of Bukhara, where he frequented the renowned library of the Ark. In about 1070 he moved to Samarkand, where he started to compose his famous Treatise on Algebra under the patronage of Abu Tahir Abd al-Rahman ibn ʿAlaq, the governor and chief judge of the city.: 4330b Khayyam was kindly received by the Karakhanid ruler Shams al-Mulk Nasr, who according to Bayhaqi, would "show him the greatest honour, so much so that he would seat [Khayyam] beside him on his throne".: 34 : 47 In 1073–4 peace was concluded with Sultan Malik-Shah I who had made incursions into Karakhanid dominions. Khayyam entered the service of Malik-Shah in 1074 when he was invited by the Grand Vizier Nizam al-Mulk to meet Malik-Shah in the city of Marv. Khayyam was subsequently commissioned to set up an observatory in Isfahan and lead a group of scientists in carrying out precise astronomical observations aimed at the revision of the Persian calendar. The undertaking probably began with the opening of the observatory in 1074 and ended in 1079,: 28–29 when Omar Khayyam and his colleagues concluded their measurements of the length of the year, reporting it as 365.24219858156 days. Given that the length of the year is changing in the sixth decimal place over a person's lifetime, this is outstandingly accurate. For comparison, the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days. After the death of Malik-Shah and his vizier (murdered, it is thought, by the Ismaili order of Assassins), Khayyam fell from favor at court, and as a result, he soon set out on his pilgrimage to Mecca. A possible ulterior motive for his pilgrimage reported by Al-Qifti, was a public demonstration of his faith with a view to allaying suspicions of skepticism and confuting the allegations of unorthodoxy (including possible sympathy or adherence to Zoroastrianism) levelled at him by a hostile clergy.: 29 : 29 He was then invited by the new Sultan Sanjar to Marv, possibly to work as a court astrologer. He was later allowed to return to Nishapur owing to his declining health. Upon his return, he seems to have lived the life of a recluse.: 99 Omar Khayyam died at the age of 83 in his hometown of Nishapur on 4 December 1131, and he is buried in what is now the Mausoleum of Omar Khayyam. One of his disciples Nizami Aruzi relates the story that sometime during 1112–3 Khayyam was in Balkh in the company of Isfizari (one of the scientists who had collaborated with him on the Jalali calendar) when he made a prophecy that "my tomb shall be in a spot where the north wind may scatter roses over it".: 36 Four years after his death, Aruzi located his tomb in a cemetery in a then large and well-known quarter of Nishapur on the road to Marv. As it had been foreseen by Khayyam, Aruzi found the tomb situated at the foot of a garden-wall over which pear trees and apricot trees had thrust their heads and dropped their flowers so that his tombstone was hidden beneath them.: 37 == Mathematics == Khayyam was famous during his life as a mathematician. His surviving mathematical works include (i) Commentary on the Difficulties Concerning the Postulates of Euclid's Elements (Risāla fī Sharḥ mā Ashkal min Muṣādarāt Kitāb Uqlīdis), completed in December 1077,: 832a : § 1 : 324b (ii) Treatise On the Division of a Quadrant of a Circle (Risālah fī Qismah Rub‘ al-Dā’irah), undated but completed prior to the Treatise on Algebra,: 831b : § 2 and (iii) Treatise on Algebra (Risālah fi al-Jabr wa'l-Muqābala),: 831b–832a : § 3 most likely completed in 1079.: 281 He furthermore wrote a treatise on the binomial theorem and extracting the nth root of natural numbers, which has been lost.: 197 : 832a : § 4 : 325b–326b === Theory of parallels === Part of Khayyam's Commentary on the Difficulties Concerning the Postulates of Euclid's Elements deals with the parallel axiom.: 282 The treatise of Khayyam can be considered the first treatment of the axiom not based on petitio principii, but on a more intuitive postulate. Khayyam refutes the previous attempts by other mathematicians to prove the proposition, mainly on grounds that each of them had postulated something that was by no means easier to admit than the Fifth Postulate itself.: § 1 : 326b–327b : 75 Drawing upon Aristotle's views, he rejects the usage of movement in geometry and therefore dismisses the different attempt by Ibn al-Haytham.: 64–65 : 270 Unsatisfied with the failure of mathematicians to prove Euclid's statement from his other postulates, Khayyam tried to connect the axiom with the Fourth Postulate, which states that all right angles are equal to one another.: 282 Khayyam was the first to consider the three distinct cases of acute, obtuse, and right angle for the summit angles of a Khayyam-Saccheri quadrilateral.: 283 After proving a number of theorems about them, he showed that Postulate V follows from the right angle hypothesis, and refuted the obtuse and acute cases as self-contradictory.: 270 : 133 His elaborate attempt to prove the parallel postulate was significant for the further development of geometry, as it clearly shows the possibility of non-Euclidean geometries. The hypotheses of acute, obtuse, and right angles are now known to lead respectively to the non-Euclidean hyperbolic geometry of Gauss-Bolyai-Lobachevsky, to that of Riemannian geometry, and to Euclidean geometry. Tusi's commentaries on Khayyam's treatment of parallels made their way to Europe. John Wallis, professor of geometry at Oxford, translated Tusi's commentary into Latin. Jesuit geometer Girolamo Saccheri, whose work (euclides ab omni naevo vindicatus, 1733) is generally considered the first step in the eventual development of non-Euclidean geometry, was familiar with the work of Wallis. The American historian of mathematics David Eugene Smith mentions that Saccheri "used the same lemma as the one of Tusi, even lettering the figure in precisely the same way and using the lemma for the same purpose". He further says that "Tusi distinctly states that it is due to Omar Khayyam, and from the text, it seems clear that the latter was his inspirer.": 195 : 104 ==== Real number concept ==== This treatise on Euclid contains another contribution dealing with the theory of proportions and with the compounding of ratios. Khayyam discusses the relationship between the concept of ratio and the concept of number and explicitly raises various theoretical difficulties. In particular, he contributes to the theoretical study of the concept of irrational number. Displeased with Euclid's definition of equal ratios, he redefined the concept of a number by the use of a continued fraction as the means of expressing a ratio. Youschkevitch and Rosenfeld argue that "by placing irrational quantities and numbers on the same operational scale, [Khayyam] began a true revolution in the doctrine of number.": 327b Likewise, it was noted by D. J. Struik that Omar was "on the road to that extension of the number concept which leads to the notion of the real number.": 284 === Geometric algebra === Rashed and Vahabzadeh (2000) have argued that because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered the precursor of Descartes in the invention of analytic geometry.: 248 In the Treatise on the Division of a Quadrant of a Circle Khayyam applied algebra to geometry. In this work, he devoted himself mainly to investigating whether it is possible to divide a circular quadrant into two parts such that the line segments projected from the dividing point to the perpendicular diameters of the circle form a specific ratio. His solution, in turn, employed several curve constructions that led to equations containing cubic and quadratic terms.: 248 ==== Solution of cubic equations ==== Khayyam seems to have been the first to conceive a general theory of cubic equations, and the first to geometrically solve every type of cubic equation, so far as positive roots are concerned. The Treatise on Algebra contains his work on cubic equations.: 9 It is divided into three parts: (i) equations which can be solved with compass and straight edge, (ii) equations which can be solved by means of conic sections, and (iii) equations which involve the inverse of the unknown.: § 3 Khayyam produced an exhaustive list of all possible equations involving lines, squares, and cubes.: 43 He considered three binomial equations, nine trinomial equations, and seven tetranomial equations.: 281 For the first and second degree polynomials, he provided numerical solutions by geometric construction. He concluded that there are fourteen different types of cubics that cannot be reduced to an equation of a lesser degree.: 831b : 328a : 49 For these he could not accomplish the construction of his unknown segment with compass and straight edge. He proceeded to present geometric solutions to all types of cubic equations using the properties of conic sections.: 281 : 157 The prerequisite lemmas for Khayyam's geometrical proof include Euclid VI, Prop 13, and Apollonius II, Prop 12.: 155 The positive root of a cubic equation was determined as the abscissa of a point of intersection of two conics, for instance, the intersection of two parabolas, or the intersection of a parabola and a circle, etc.: 141 However, he acknowledged that the arithmetic problem of these cubics was still unsolved, adding that "possibly someone else will come to know it after us".: 158 This task remained open until the sixteenth century, where an algebraic solution of the cubic equation was found in its generality by Cardano, Del Ferro, and Tartaglia in Renaissance Italy.: 282 In effect, Khayyam's work is an effort to unify algebra and geometry.: 241 This particular geometric solution of cubic equations was further investigated by M. Hachtroudi and extended to solving fourth-degree equations. Although similar methods had appeared sporadically since Menaechmus, and further developed by the 10th-century mathematician Abu al-Jud,: 29 : 110 Khayyam's work can be considered the first systematic study and the first exact method of solving cubic equations.: 92 The mathematician Woepcke (1851) who offered translations of Khayyam's algebra into French praised him for his "power of generalization and his rigorously systematic procedure.": 10 === Binomial theorem and extraction of roots === In his algebraic treatise, Khayyam alludes to a book he had written on the extraction of the n {\displaystyle n} th root of natural numbers using a law he had discovered which did not depend on geometric figures. This book was most likely titled the Difficulties of Arithmetic (Mushkilāt al-Ḥisāb),: 832a : § 4 and is not extant.: 325b Based on the context, some historians of mathematics such as D. J. Struik, believe that Omar must have known the formula for the expansion of the binomial ( a + b ) n {\displaystyle (a+b)^{n}} , where n is a positive integer.: 282 The case of power 2 is explicitly stated in Euclid's elements and the case of at most power 3 had been established by Indian mathematicians. Khayyam was the mathematician who noticed the importance of a general binomial theorem. The argument supporting the claim that Khayyam had a general binomial theorem is based on his ability to extract roots. One of Khayyam's predecessors, al-Karaji, had already discovered the triangular arrangement of the coefficients of binomial expansions that Europeans later came to know as Pascal's triangle;: 60 Khayyam popularized this triangular array in Iran, so that it is now known as Omar Khayyam's triangle. == Astronomy == In 1074–5, Omar Khayyam was commissioned by Sultan Malik-Shah to build an observatory at Isfahan and reform the Persian calendar. There was a panel of eight scholars working under the direction of Khayyam to make large-scale astronomical observations and revise the astronomical tables.: 141 Recalibrating the calendar fixed the first day of the year at the exact moment of the passing of the Sun's center across vernal equinox. This marks the beginning of spring or Nowrūz, a day in which the Sun enters the first degree of Aries before noon.: 10–11 The resultant calendar was named in Malik-Shah's honor as the Jalālī calendar, and was inaugurated on 15 March 1079.: 269 The observatory itself was disused after the death of Malik-Shah in 1092.: 659 The Jalālī calendar was a true solar calendar where the duration of each month is equal to the time of the passage of the Sun across the corresponding sign of the Zodiac. The calendar reform introduced a unique 33-year intercalation cycle. As indicated by the works of Khazini, Khayyam's group implemented an intercalation system based on quadrennial and quinquennial leap years. Therefore, the calendar consisted of 25 ordinary years that included 365 days, and 8 leap years that included 366 days.: 13 The calendar remained in use across Greater Iran from the 11th to the 20th centuries. In 1911, the Jalali calendar became the official national calendar of Qajar Iran. In 1925, this calendar was simplified and the names of the months were modernised, resulting in the modern Iranian calendar. The Jalali calendar is more accurate than the Gregorian calendar of 1582,: 659 with an error of one day accumulating over 5,000 years, compared to one day every 3,330 years in the Gregorian calendar.: 200 Moritz Cantor considered it the most perfect calendar ever devised.: 101 One of his pupils, Nizami Aruzi, relates that Khayyam apparently did not have a belief in astrology and divination: "I did not observe that he (scil. Omar Khayyam) had any great belief in astrological predictions, nor have I seen or heard of any of the great [scientists] who had such belief.": 11 While working for Sultan Sanjar as an astrologer he was asked to predict the weather – a job that he apparently did not do well.: 30 George Saliba explains that the term ‘ilm al-nujūm, used in various sources in which references to Khayyam's life and work could be found, has sometimes been incorrectly translated to mean astrology. He adds: "from at least the middle of the tenth century, according to Farabi's Enumeration of the Sciences, that this science, ‘ilm al-nujūm, was already split into two parts, one dealing with astrology and the other with theoretical mathematical astronomy.": 224 == Other works == Khayyam has a short treatise devoted to Archimedes' principle (in full title, On the Deception of Knowing the Two Quantities of Gold and Silver in a Compound Made of the Two). For a compound of gold adulterated with silver, he describes a method to measure more exactly the weight per capacity of each element. It involves weighing the compound both in air and in water, since weights are easier to measure exactly than volumes. By repeating the same with both gold and silver one finds exactly how much heavier than water gold, silver and the compound were. This treatise was extensively examined by Eilhard Wiedemann who believed that Khayyam's solution was more accurate and sophisticated than that of Khazini and Al-Nayrizi who also dealt with the subject elsewhere.: 198 Another short treatise is concerned with music theory in which he discusses the connection between music and arithmetic. Khayyam's contribution was in providing a systematic classification of musical scales, and discussing the mathematical relationship among notes, minor, major and tetrachords.: 198 == Poetry == The earliest allusion to Omar Khayyam's poetry is from the historian Imad al-Din al-Isfahani, a younger contemporary of Khayyam, who explicitly identifies him as both a poet and a scientist (Kharidat al-qasr, 1174).: 49 : 35 One of the earliest specimens of Omar Khayyam's Rubiyat is from Fakhr al-Din al-Razi. In his work al-Tanbih ‘ala ba‘d asrar al-maw‘dat fi’l-Qur’an (c. 1160), he quotes one of his poems (corresponding to quatrain LXII of FitzGerald's first edition). Daya in his writings (Mirṣād al-‘Ibad, c. 1230) quotes two quatrains, one of which is the same as the one already reported by Razi. An additional quatrain is quoted by the historian Juvayni (Tarikh-i Jahangushay, c. 1226–1283).: 36–37 : 92 In 1340 Jajarmi includes thirteen quatrains of Khayyam in his work containing an anthology of the works of famous Persian poets (Mu’nis al-ahrār), two of which have hitherto been known from the older sources.: 434 A comparatively late manuscript is the Bodleian MS. Ouseley 140, written in Shiraz in 1460, which contains 158 quatrains on 47 folia. The manuscript belonged to William Ouseley (1767–1842) and was purchased by the Bodleian Library in 1844. There are occasional quotes of verses attributed to Khayyam in texts attributed to authors of the 13th and 14th centuries, but these are of doubtful authenticity, so that skeptical scholars point out that the entire tradition may be pseudepigraphic.: 11 Hans Heinrich Schaeder in 1934 commented that the name of Omar Khayyam "is to be struck out from the history of Persian literature" due to the lack of any material that could confidently be attributed to him. De Blois presents a bibliography of the manuscript tradition, concluding pessimistically that the situation has not changed significantly since Schaeder's time.:307 Five of the quatrains later attributed to Omar Khayyam are found as early as 30 years after his death, quoted in Sindbad-Nameh. While this establishes that these specific verses were in circulation in Omar's time or shortly later, it does not imply that the verses must be his. De Blois concludes that at the least the process of attributing poetry to Omar Khayyam appears to have begun already in the 13th century.:305 Edward Granville Browne (1906) notes the difficulty of disentangling authentic from spurious quatrains: "while it is certain that Khayyam wrote many quatrains, it is hardly possible, save in a few exceptional cases, to assert positively that he wrote any of those ascribed to him".: 663 In addition to the Persian quatrains, there are twenty-five Arabic poems attributed to Khayyam which are attested by historians such as al-Isfahani, Shahrazuri (Nuzhat al-Arwah, c. 1201–1211), Qifti (Tārikh al-hukamā, 1255), and Hamdallah Mustawfi (Tarikh-i guzida, 1339).: 39 Boyle emphasized that there are a number of other Persian scholars who occasionally wrote quatrains, including Avicenna, Ghazali, and Tusi. They conclude that it is also possible that for Khayyam poetry was an amusement of his leisure hours: "these brief poems seem often to have been the work of scholars and scientists who composed them, perhaps, in moments of relaxation to edify or amuse the inner circle of their disciples".: 662 The poetry attributed to Omar Khayyam has contributed greatly to his popular fame in the modern period as a direct result of the extreme popularity of the translation of such verses into English by Edward FitzGerald (1859). FitzGerald's Rubaiyat of Omar Khayyam contains loose translations of quatrains from the Bodleian manuscript. It enjoyed such success in the fin de siècle period that a bibliography compiled in 1929 listed more than 300 separate editions, and many more have been published since.:312 == Philosophy == Khayyam considered himself intellectually to be a student of Avicenna.: 474 According to Al-Bayhaqi, he was reading the metaphysics in Avicenna's the Book of Healing before he died.: 661 There are six philosophical papers believed to have been written by Khayyam. One of them, On existence (Fi’l-wujūd), was written originally in Persian and deals with the subject of existence and its relationship to universals. Another paper, titled The necessity of contradiction in the world, determinism and subsistence (Darurat al-tadād fi’l-‘ālam wa’l-jabr wa’l-baqā’), is written in Arabic and deals with free will and determinism.: 475 The titles of his other works are On being and necessity (Risālah fī’l-kawn wa’l-taklīf), The Treatise on Transcendence in Existence (al-Risālah al-ulā fi’l-wujūd), On the knowledge of the universal principles of existence (Risālah dar ‘ilm kulliyāt-i wujūd), and Abridgement concerning natural phenomena (Mukhtasar fi’l-Tabi‘iyyāt). Khayyam himself once said:: 431 We are the victims of an age when men of science are discredited, and only a few remain who are capable of engaging in scientific research. Our philosophers spend all their time in mixing true with false and are interested in nothing but outward show; such little learning as they have they extend on material ends. When they see a man sincere and unremitting in his search for the truth, one who will have nothing to do with falsehood and pretence, they mock and despise him. === Religious views === A literal reading of Khayyam's quatrains leads to the interpretation of his philosophic attitude toward life as a combination of pessimism, nihilism, Epicureanism, fatalism, and agnosticism.: 6 This view is taken by Iranologists such as Arthur Christensen, Hans Heinrich Schaeder, John Andrew Boyle, Edward Denison Ross,: 365 Edward Henry Whinfield: 40 and George Sarton.: 18 Conversely, the Khayyamic quatrains have also been described as mystical Sufi poetry. In addition to his Persian quatrains, J. C. E. Bowen mentions that Khayyam's Arabic poems also "express a pessimistic viewpoint which is entirely consonant with the outlook of the deeply thoughtful rationalist philosopher that Khayyam is known historically to have been.": 69 Edward FitzGerald emphasized the religious skepticism he found in Khayyam. In his preface to the Rubáiyát he claimed that he "was hated and dreaded by the Sufis", and denied any pretense at divine allegory: "his Wine is the veritable Juice of the Grape: his Tavern, where it was to be had: his Saki, the Flesh and Blood that poured it out for him.": 62 Sadegh Hedayat is one of the most notable proponents of Khayyam's philosophy as agnostic skepticism, and according to Jan Rypka (1934), he even considered Khayyam an atheist. Hedayat (1923) states that "while Khayyam believes in the transmutation and transformation of the human body, he does not believe in a separate soul; if we are lucky, our bodily particles would be used in the making of a jug of wine.": 138 Omar Khayyam's poetry has been cited in the context of New Atheism, such as in The Portable Atheist by Christopher Hitchens.: 7 Al-Qifti (c. 1172–1248) appears to confirm this view of Khayyam's philosophy.: 663 In his work The History of Learned Men he reports that Khayyam's poems were only outwardly in the Sufi style, but were written with an anti-religious agenda.: 365 He also mentions that he was at one point indicted for impiety, but went on a pilgrimage to prove he was pious.: 29 The report has it that upon returning to his native city he concealed his deepest convictions and practised a strictly religious life, going morning and evening to the place of worship.: 355 Khayyam on the Koran (quote 84): The Koran! well, come put me to the test, Lovely old book in hideous error drest, Believe me, I can quote the Koran too, The unbeliever knows his Koran best. And do you think that unto such as you, A maggot-minded, starved, fanatic crew, God gave the Secret, and denied it me? Well, well, what matters it! believe that too. Look not above, there is no answer there; Pray not, for no one listens to your prayer; Near is as near to God as any Far, And Here is just the same deceit as There. Men talk of heaven,—there is no heaven but here; Men talk of hell,—there is no hell but here; Men of hereafters talk, and future lives, O love, there is no other life—but here. An account of him, written in the thirteenth century, shows him as "versed in all the wisdom of the Greeks," and as wont to insist on the necessity of studying science on Greek lines. Of his prose works, two, which were stand authority, dealt respectively with precious stones and climatology. Beyond question the poet-astronomer was undevout; and his astronomy doubtless helped to make him so. One contemporary writes: "I did not observe that he had any great belief in astrological predictions; nor have I seen or heard of any of the great (scientists) who had such belief. He gave his adherence to no religious sect. Agnosticism, not faith, is the keynote of his works. Among the sects he saw everywhere strife and hatred in which he could have no part....": 263, vol. 1 Persian novelist Sadegh Hedayat says Khayyám from "his youth to his death remained a materialist, pessimist, agnostic. Khayyam looked at all religions questions with a skeptical eye", continues Hedayat, "and hated the fanaticism, narrow-mindedness, and the spirit of vengeance of the mullas, the so-called religious scholars.": 13 In the context of a piece entitled On the Knowledge of the Principles of Existence, Khayyam endorses the Sufi path.: 8 Csillik suggests the possibility that Omar Khayyam could see in Sufism an ally against orthodox religiosity.: 75 Other commentators do not accept that Khayyam's poetry has an anti-religious agenda and interpret his references to wine and drunkenness in the conventional metaphorical sense common in Sufism. The French translator J. B. Nicolas held that Khayyam's constant exhortations to drink wine should not be taken literally, but should be regarded rather in the light of Sufi thought where rapturous intoxication by "wine" is to be understood as a metaphor for the enlightened state or divine rapture of baqaa. The view of Omar Khayyam as a Sufi was defended by Bjerregaard,: 3 Idries Shah,: 165–166 and Dougan who attributes the reputation of hedonism to the failings of FitzGerald's translation, arguing that Khayyam's poetry is to be understood as "deeply esoteric". On the other hand, Iranian experts such as Mohammad Ali Foroughi and Mojtaba Minovi rejected the hypothesis that Omar Khayyam was a Sufi.: 72 Foroughi stated that Khayyam's ideas may have been consistent with that of Sufis at times but there is no evidence that he was formally a Sufi. Aminrazavi states that "Sufi interpretation of Khayyam is possible only by reading into his Rubāʿīyyāt extensively and by stretching the content to fit the classical Sufi doctrine.".: 128 Furthermore, Boyle emphasizes that Khayyam was intensely disliked by a number of celebrated Sufi mystics who belonged to the same century. This includes Shams Tabrizi (spiritual guide of Rumi),: 58 Najm al-Din Daya who described Omar Khayyam as "an unhappy philosopher, atheist, and materialist",: 71 and Attar who regarded him not as a fellow-mystic but a free-thinking scientist who awaited punishments hereafter.: 663–664 Seyyed Hossein Nasr argues that it is "reductive" to use a literal interpretation of his verses (many of which are of uncertain authenticity to begin with) to establish Omar Khayyam's philosophy. Instead, he adduces Khayyam's interpretive translation of Avicenna's treatise Discourse on Unity (al-Khutbat al-Tawhīd), where he expresses orthodox views on Divine Unity in agreement with the author.: Ch. 9, 165–183 The prose works believed to be Khayyam's are written in the Peripatetic style and are explicitly theistic, dealing with subjects such as the existence of God and theodicy.: 160 As noted by Bowen these works indicate his involvement in the problems of metaphysics rather than in the subtleties of Sufism.: 71 As evidence of Khayyam's faith and/or conformity to Islamic customs, Aminrazavi mentions that in his treatises he offers salutations and prayers, praising God and Muhammad. In most biographical extracts, he is referred to with religious honorifics such as Imām, The Patron of Faith (Ghīyāth al-Dīn), and The Evidence of Truth (Hujjat al-Haqq). He also notes that biographers who praise his religiosity generally avoid making reference to his poetry, while the ones who mention his poetry often do not praise his religious character.: 48 For instance, Al-Bayhaqi's account, which antedates by some years other biographical notices, speaks of Omar as a very pious man who professed orthodox views down to his last hour.: 174 On the basis of all the existing textual and biographical evidence, the question remains somewhat open,: 11 and as a result Khayyam has received sharply conflicting appreciations and criticisms.: 350 == Reception == The various biographical extracts referring to Omar Khayyam describe him as unequalled in scientific knowledge and achievement during his time. Many called him by the epithet King of the Wise (Arabic: ملك الحکماء, romanized: Malik al-Ḥukamā).: 436 : 141 Shahrazuri (d. 1300) esteems him highly as a mathematician, and claims that he may be regarded as "the successor of Avicenna in the various branches of philosophic learning".: 352 Al-Qifti (d. 1248), even though disagreeing with his views, concedes he was "unrivalled in his knowledge of natural philosophy and astronomy".: 355 Despite being hailed as a poet by a number of biographers, according to John Andrew Boyle "it is still possible to argue that Khayyam's status as a poet of the first rank is a comparatively late development.": 663 Thomas Hyde was the first European to call attention to Khayyam and to translate one of his quatrains into Latin (Historia religionis veterum Persarum eorumque magorum, 1700).: 525 Western interest in Persia grew with the Orientalism movement in the 19th century. Joseph von Hammer-Purgstall (1774–1856) translated some of Khayyam's poems into German in 1818, and Gore Ouseley (1770–1844) into English in 1846, but Khayyam remained relatively unknown in the West until after the publication of Edward FitzGerald's Rubaiyat of Omar Khayyam in 1859. FitzGerald's work at first was unsuccessful but was popularised by Whitley Stokes from 1861 onward, and the work came to be greatly admired by the Pre-Raphaelites. In 1872 FitzGerald had a third edition printed which increased interest in the work in America. By the 1880s, the book was extremely well known throughout the English-speaking world, to the extent of the formation of numerous "Omar Khayyam Clubs" and a "fin de siècle cult of the Rubaiyat".: 202 Khayyam's poems have been translated into many languages; many of the more recent ones are more literal than that of FitzGerald. FitzGerald's translation was a factor in rekindling interest in Khayyam as a poet even in his native Iran.: 55–72 Sadegh Hedayat in his Songs of Khayyam (Taranehha-ye Khayyam, 1934) reintroduced Khayyam's poetic legacy to modern Iran. Under the Pahlavi dynasty, a new monument of white marble, designed by the architect Houshang Seyhoun, was erected over his tomb. A statue by Abolhassan Sadighi was erected in Laleh Park, Tehran in the 1960s, and a bust by the same sculptor was placed near Khayyam's mausoleum in Nishapur. In 2009, the state of Iran donated a pavilion to the United Nations Office in Vienna, inaugurated at Vienna International Center. In 2016, three statues of Khayyam were unveiled: one at the University of Oklahoma, one in Nishapur and one in Florence, Italy. Over 150 composers have used the Rubaiyat as their source of inspiration. The earliest such composer was Liza Lehmann. FitzGerald rendered Khayyam's name as "Tentmaker", and the anglicized name of "Omar the Tentmaker" resonated in English-speaking popular culture for a while. Thus, Nathan Haskell Dole published a novel called Omar, the Tentmaker: A Romance of Old Persia in 1898. Omar the Tentmaker of Naishapur is a historical novel by John Smith Clarke, published in 1910. "Omar the Tentmaker" is also the title of a 1914 play by Richard Walton Tully in an oriental setting, adapted as a silent film in 1922. US General Omar Bradley was given the nickname "Omar the Tent-Maker" in World War II.: 13 The diverse talents and intellectual pursuits of Khayyam captivated many Ottoman and Turkish writers throughout history. Scholars often viewed Khayyam as a means to enhance their own poetic prowess and intellectual depth, drawing inspiration and recognition from his writings. For many Muslim reformers, Khayam's verses provided a counterpoint to the conservative norms prevalent in Islamic societies, allowing room for independent thought and a libertine lifestyle. Figures like Abdullah Cevdet, Rıza Tevfik, and Yahya Kemal utilized Khayyam's themes to justify their progressive ideologies or to celebrate liberal aspects of their lives, portraying him as a cultural, political, and intellectual role model who demonstrated Islam's compatibility with modern conventions. Similarly, Turkish leftist poets and intellectuals, including Nâzım Hikmet, Sabahattin Eyüboğlu, A. Kadir, and Gökçe, appropriated Khayyam to champion their socialist worldview, imbuing his voice with a humanistic tone in the vernacular. Khayyam's resurgence in spoken Turkish since the 1980s has transformed him into a poet of the people, with numerous books and translations revitalizing his historical significance. Conversely, scholars like Dāniş, Tevfik, and Gölpınarlı advocated for source criticism and the identification of authentic quatrains to discern the genuine Khayyam amidst historical perceptions of his sociocultural image. === The Moving Finger quatrain === The quatrain by Omar Khayyam known as "The Moving Finger", in the form of its translation by the English poet Edward Fitzgerald is one of the most popular quatrains in the Anglosphere. It reads: The Moving Finger writes; and having writ, Moves on: nor all your Piety nor Wit Shall lure it back to cancel half a Line, Nor all your Tears wash out a Word of it. The title of the novel The Moving Finger written by Agatha Christie and published in 1942 was inspired by this quatrain of the translation of Rubaiyat of Omar Khayyam by Edward Fitzgerald. Martin Luther King also cites this quatrain of Omar Khayyam in one of his speeches, "Beyond Vietnam: A Time to Break Silence": "We may cry out desperately for time to pause in her passage, but time is adamant to every plea and rushes on. Over the bleached bones and jumbled residues of numerous civilizations are written the pathetic words, ‘Too late.’ There is an invisible book of life that faithfully records our vigilance or our neglect. Omar Khayyam is right: ‘The moving finger writes, and having writ moves on.’" In one of his apologetic speeches about the Clinton–Lewinsky scandal, Bill Clinton, the 42nd president of the US, also cites this quatrain. === Other popular culture references === In 1934 Harold Lamb published a historical novel Omar Khayyam. The French-Lebanese writer Amin Maalouf based the first half of his historical fiction novel Samarkand on Khayyam's life and the creation of his Rubaiyat. The sculptor Eduardo Chillida produced four massive iron pieces titled Mesa de Omar Khayyam (Omar Khayyam's Table) in the 1980s. The lunar crater Omar Khayyam was named in his honour in 1970, as was the minor planet 3095 Omarkhayyam discovered by Soviet astronomer Lyudmila Zhuravlyova in 1980. Google has released two Google Doodles commemorating him. The first was on his 964th birthday on 18 May 2012. The second was on his 971st birthday on 18 May 2019. == Gallery == == See also == Nozhat al-Majales === Notable films === Omar Khayyam (1957 film) The Keeper: The Legend of Omar Khayyam === Noted Khayyamologists === Badiozzaman Forouzanfar Abdolhossein Zarrinkoob == Notes == == References == == Further reading == Biegstraaten, Jos (2008). "Omar Khayyam (Impact On Literature And Society In The West)". Encyclopaedia Iranica. Vol. 15. Encyclopaedia Iranica Foundation. Boyle, J. A., ed. (1968). The Cambridge History of Iran. Volume V: The Saljug and Mongol Periods. New York: Cambridge University Press. ISBN 978-0-521-06936-6. Rypka, J. (1968). Karl Jahn (ed.). History of Iranian Literature. Dordrecht: D. Reidel. ISBN 978-94-010-3481-4. Turner, Howard R. (1997). Science in Medieval Islam: An Illustrated Introduction. University of Texas Press. ISBN 0-292-78149-0. == External links == Works by or about Omar Khayyam at the Internet Archive Works by Omar Khayyam at LibriVox (public domain audiobooks) Hashemipour, Behnaz (2007). "Khayyām: Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm al-Khayyāmī al-Nīshāpūrī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 627–8. ISBN 978-0-387-31022-0. (PDF version.) Umar Khayyam, in the Stanford Encyclopedia of Philosophy The illustrated Rubáiyát of Omar Khayyám at the Internet Archive
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Wikipedia:On Ascensions#0
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Hypsicles (Ancient Greek: Ὑψικλῆς; c. 190 – c. 120 BCE) was an ancient Greek mathematician and astronomer known for authoring On Ascensions (Ἀναφορικός) and possibly the Book XIV of Euclid's Elements. Hypsicles lived in Alexandria. == Life and work == Although little is known about the life of Hypsicles, it is believed that he authored the astronomical work On Ascensions. The mathematician Diophantus of Alexandria noted on a definition of polygonal numbers, due to Hypsicles: If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2 a square; when 3, a pentagonal number [and so on]. And the number of angles is called after the number which exceeds the common difference by 2, and the side after the number of terms including 1. === On Ascensions === In On Ascensions (Ἀναφορικός and sometimes translated On Rising Times), Hypsicles proves a number of propositions on arithmetical progressions and uses the results to calculate approximate values for the times required for the signs of the zodiac to rise above the horizon. It is thought that this is the work from which the division of the circle into 360 parts may have been adopted since it divides the day into 360 parts, a division possibly suggested by Babylonian astronomy, although this is mere speculation and no actual evidence is found to support this. Heath 1921 notes, "The earliest extant Greek book in which the division of the circle into 360 degrees appears". This work by Hypsicles is believed to represent the earliest extant Greek text to use the Babylonian division of the zodiac into 12 signs of 30 degrees each. === Euclid's Elements === Hypsicles is more famously known for possibly writing the Book XIV of Euclid's Elements. The book may have been composed on the basis of a treatise by Apollonius. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being 10 3 ( 5 − 5 ) {\displaystyle {\sqrt {\tfrac {10}{3(5-{\sqrt {5}})}}}} . Heath further notes, "Hypsicles says also that Aristaeus, in a work entitled Comparison of the five figures, proved that the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere; whether this Aristaeus is the same as the Aristaeus of the Solid Loci, the elder (Aristaeus the Elder) contemporary of Euclid, we do not know." == Hypsicles letter == Hypsicles letter was a preface of the supplement taken from Euclid's Book XIV, part of the thirteen books of Euclid's Elements, featuring a treatise. Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. And on one occasion, when looking into the tract written by Apollonius (Apollonius of Perga) about the comparison of the dodecahedron and icosahedron inscribed in one and the same sphere, that is to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius' treatment of it in this book was not correct; accordingly, as I understood from my father, they proceeded to amend and rewrite it. But I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, and I was greatly attracted by his investigation of the problem. Now the book published by Apollonius is accessible to all; for it has a large circulation in a form which seems to have been the result of later careful elaboration. For my part, I determined to dedicate to you what I deem to be necessary by way of commentary, partly because you will be able, by reason of your proficiency in all mathematics and particularly in geometry, to pass an expert judgment upon what I am about to write, and partly because, on account of your intimacy with my father and your friendly feeling towards myself, you will lend a kindly ear to my disquisition. But it is time to have done with the preamble and to begin my treatise itself. == Notes == == References == Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7. Heath, Thomas Little (1981). A History of Greek Mathematics, Volume I. Dover publications. ISBN 0-486-24073-8. == External links == The mac-tutor biography of Hypsicles Hypsicles, from Smith, Dictionary of Greek and Roman Biography and Mythology Scan of Manitius' edition of On Ascensions at wilbourhall.org (German introduction, Greek text and Latin translation)
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Wikipedia:On Risings and Settings#0
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Autolycus of Pitane (Greek: Αὐτόλυκος ὁ Πιταναῖος; c. 360 – c. 290 BC) was a Greek astronomer, mathematician, and geographer. He is known today for his two surviving works On the Moving Sphere and On Risings and Settings, both about spherical geometry. == Life == Autolycus was born in Pitane, a town of Aeolis within Ionia, Asia Minor. Of his personal life nothing is known, although he was a contemporary of Aristotle and his works seem to have been completed in Athens between 335–300 BC. Euclid references some of Autolycus' work, and Autolycus is known to have taught Arcesilaus. The lunar crater Autolycus was named in his honour. == Work == Autolycus' two surviving works are about spherical geometry with application to astronomy: On the Moving Sphere and On Risings and Settings (of stars). In late antiquity, both were part of the "Little Astronomy", a collection of miscellaneous short works about geometry and astronomy which were commonly transmitted together. They were translated into Arabic in the 9th century, and with the addition of a few additional works became known as the "Middle Books" (sitting between Euclid's Elements and Ptolemy's Almagest). Both were preserved both in Greek and in Arabic, but were unknown in medieval Western Europe. They were translated from Arabic into Latin in the 12th century. Later, remaining Greek copies were also translated into Latin. === On the Moving Sphere === On the Moving Sphere (Ancient Greek: Περὶ κινουμένης σφαίρας Perí kinouménis sphaíras) concerns the movements of points and arcs on the sphere as it rotates on an axis. While the obvious application is the diurnal motion of the stars as the celestial sphere appears to rotate about an immobile Earth (as modeled at the time), Autolycus' treatise never explicitly discusses this application: its content consists entirely of elementary theorems about the arcs of great circles and parallel small circles on an abstract sphere. The work is simple and probably derivative of older works, but each theorem includes a clearly enunciated statement, a figure of the construction alongside its proof, and finally a concluding remark. On the Moving Sphere is believed to be the oldest mathematical treatise from ancient Greece that is completely preserved: All prior Greek mathematical works are taken from later summaries, commentaries, or descriptions of the works. It shows that by Autolycus' day there was a thoroughly established textbook tradition in geometry that is today regarded as typical of classical Greek geometry. Moreover, it gives indications of what theorems were well known in his day (around 320 BC). Two hundred years later Theodosius wrote the Spherics, a treatise establishing the fundamental definitions and constructions in spherical geometry whose content is believed to have a common origin with On the Moving Sphere in some pre-Euclidean textbook, possibly written by Eudoxus. In contrast to later astronomical analyses by Hipparchus (2nd century BC) and Ptolemy (2nd century AD), but similarly to the planar geometry of Euclid's Elements, both Autolycus's work and Theodosius' does not involve concrete quantification or trigonometry: spherical arcs are compared in size, but not given any numerical measure. === On Risings and Settings === In the two-book treatise On Risings and Settings (Ancient Greek: Περὶ ἐπιτολῶν καὶ δύσεων Perí epitolón kaí dýseon), Autolycus studied the relationship between the rising and the setting of the stars throughout the year. The second book is an expansion of the first and of higher quality. He wrote that "any star which rises and sets always rises and sets at the same point in the horizon." Autolycus relied heavily on Eudoxus' astronomy and was a strong supporter of Eudoxus' theory of homocentric spheres. == Footnotes == == References == Huxley, G. L. (1970). "Autolycus of Pitane". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 338–39. ISBN 0-684-10114-9. on line at "Autolycus of Pitane". HighBeam Research. Retrieved 26 March 2015. O'Connor, John J.; Robertson, Edmund F. (April 1999), "Autolycus of Pitane", MacTutor History of Mathematics Archive, University of St Andrews == External links == Autolycus On The Moving Sphere from the Million Books Project (Greek with Latin translation) ΠΕΡΙ ΚΙΝΟΥΜΕΝΗΣ ΣΦΑΙΡΑΣ and ΠΕΡΙ ΕΠΙΤΟΛΩΝ ΚΑΙ ΔΥΣΕΩΝ (Mogenet ed., 1950)
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Wikipedia:On Sizes and Distances (Hipparchus)#0
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On Sizes and Distances (of the Sun and Moon) (Greek: Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], romanized: Peri megethon kai apostematon) is a text by the ancient Greek astronomer Hipparchus (c. 190 – c. 120 BC) in which approximations are made for the radii of the Sun and the Moon as well as their distances from the Earth. It is not extant, but some of its contents have been preserved in the works of Ptolemy and his commentator Pappus of Alexandria. Several modern historians have attempted to reconstruct the methods of Hipparchus using the available texts. == Sources == Most of what is known about Hipparchus' text comes from two ancient sources: Ptolemy and Pappus. The work is also mentioned by Theon of Smyrna and others, but their accounts have proven less useful in reconstructing the procedures of Hipparchus. === Ptolemy === In Almagest V, 11, Ptolemy writes: Now Hipparchus made such an examination principally from the sun. Since from other properties of the sun and moon (of which a study will be made below) it follows that if the distance of one of the two luminaries is given, the distance of the other is also given, he tries by conjecturing the distance of the sun to demonstrate the distance of the moon. First, he assumes the sun to show the least perceptible parallax to find its distance. After this, he makes use of the solar eclipse adduced by him, first as if the sun shows no perceptible parallax, and for exactly that reason the ratios of the moon's distances appeared different to him for each of the hypotheses he set out. But with respect to the sun, not only the amount of its parallax, but also whether it shows any parallax at all is altogether doubtful. This passage gives a general outline of what Hipparchus did, but provides no details. Ptolemy clearly did not agree with the methods employed by Hipparchus, and thus did not go into any detail. === Pappus of Alexandria === The works of Hipparchus were still extant when Pappus wrote his commentary on the Almagest in the 4th century. He fills in some of the details that Ptolemy omits: Now, Hipparchus made such an examination principally from the sun, and not accurately. For since the moon in the syzygies and near greatest distance appears equal to the sun, and since the size of the diameters of the sun and moon is given (of which a study will be made below), it follows that if the distance of one of the two luminaries is given, the distance of the other is also given, as in Theorem 12, if the distance of the moon is given and the diameters of the sun and moon, the distance of the sun is given. Hipparchus tries by conjecturing the parallax and the distance of the sun to demonstrate the distance of the moon, but with respect to the sun, not only the amount of its parallax, but also whether it shows any parallax at all is altogether doubtful. For in this way Hipparchus was in doubt about the sun, not only about the amount of its parallax but also about whether it shows any parallax at all. In the first book "On Sizes and Distances" it is assumed that the earth has the ratio of a point and center to the sun. And by means of the eclipse adduced by him... Then later, For in Book 1 of "On Sizes and Distances" he takes the following observation: an eclipse of the sun, which in the regions around the Hellespont was an exact eclipse of the whole solar disc, such that no part of it was visible, but at Alexandria by Egypt approximately four-fifths of it was eclipsed. By means of this he shows in Book 1 that, in units of which the radius of the earth is one, the least distance of the moon is 71, and the greatest 83. Hence the mean is 77... Then again he himself in Book 2 of "On Sizes and Distances" shows from many considerations that, in units of which the radius of the earth is one, the least distance of the moon is 62, and the mean 671⁄3, and the distance of the sun 490. It is clear that the greatest distance of the moon is 722⁄3. This passage provides enough details to make a reconstruction feasible. In particular, it makes clear that there were two separate procedures, and it gives the precise results of each. It provides clues with which to identify the eclipse, and says that Hipparchus used a formula "as in Theorem 12," a theorem of Ptolemy's which is extant. == Modern reconstructions == Several historians of science have attempted to reconstruct the calculations involved in On Sizes and Distances. The first attempt was made by Friedrich Hultsch in 1900, but it was later rejected by Noel Swerdlow in 1969. G. J. Toomer expanded on his efforts in 1974. === Hultsch === Friedrich Hultsch determined in a 1900 paper that the Pappus source had been miscopied, and that the actual distance to the Sun, as calculated by Hipparchus, had been 2490 Earth radii (not 490). As in English, there is only a single character difference between these two results in Greek. His analysis was based on a text by Theon of Smyrna which states that Hipparchus found the Sun to be 1880 times the size of the Earth, and the Earth 27 times the size of the Moon. Assuming that this refers to volumes, it follows that s = 1880 3 ≈ 12 + 1 / 3 {\displaystyle s={\sqrt[{3}]{1880}}\approx 12+1/3} and ℓ = 1 / 27 3 = 1 / 3 {\displaystyle \ell ={\sqrt[{3}]{1/27}}=1/3} Assuming that the Sun and Moon have the same apparent size in the sky, and that the Moon is 671⁄3 Earth radii distant, it follows that S = s ℓ L ≈ 12 + 1 / 3 1 / 3 ( 67 + 1 / 3 ) = 2491 + 1 / 3 ≈ 2490 {\displaystyle S={\frac {s}{\ell }}L\approx {\frac {12+1/3}{1/3}}(67+1/3)=2491+1/3\approx 2490} This result was generally accepted for the next seventy years, until Noel Swerdlow reinvestigated the case. === Book 2 reconstruction (Swerdlow) === Swerdlow determined that Hipparchus relates the distances to the Sun and Moon using a construction found in Ptolemy. It would not be surprising if this calculation had been originally developed by Hipparchus himself, as he was a primary source for the Almagest. Using this calculation, Swerdlow was able to relate the two results of Hipparchus (671⁄3 for the Moon and 490 for the Sun). Obtaining this relationship exactly requires following a very precise set of approximations. Using simple trigonometric identities gives ℓ = L tan θ ≈ L sin θ {\displaystyle \ell =L\tan \theta \approx L\sin \theta } and h = tan φ tan θ ℓ ≈ φ θ ℓ ≈ φ θ L sin θ {\displaystyle h={\frac {\tan \varphi }{\tan \theta }}\ell \approx {\frac {\varphi }{\theta }}\ell \approx {\frac {\varphi }{\theta }}L\sin \theta } By parallel lines and taking t = 1, we get ℓ + x = t + ( t − h ) = 2 t − h ⇒ x ≈ 2 − ( φ θ + 1 ) L sin θ {\displaystyle \ell +x=t+(t-h)=2t-h\Rightarrow x\approx 2-\left({\frac {\varphi }{\theta }}+1\right)L\sin \theta } By similarity of triangles, t x = S S − L ⇒ S = L 1 − x {\displaystyle {\frac {t}{x}}={\frac {S}{S-L}}\Rightarrow S={\frac {L}{1-x}}} Combining these equations gives S ≈ L ( φ θ + 1 ) L sin θ − 1 = 1 / ( ( φ θ + 1 ) sin θ − 1 L ) {\displaystyle S\approx {\frac {L}{\left({\frac {\varphi }{\theta }}+1\right)L\sin \theta -1}}=1\left/\left(\left({\frac {\varphi }{\theta }}+1\right)\sin \theta -{\frac {1}{L}}\right)\right.} The values which Hipparchus took for these variables can be found in Ptolemy's Almagest IV, 9. He says Hipparchus found that the Moon measured its own circle close to 650 times, and that the angular diameter of Earth's shadow is 2.5 times that of the Moon. Pappus tells us that Hipparchus took the mean distance to the Moon to be 671⁄3. This gives: According to Swerdlow, Hipparchus now evaluated this expression with the following roundings (the values are in sexagesimal): ℓ ≈ L sin θ ≈ 0 ; 19 , 30 {\displaystyle \ell \approx L\sin \theta \approx 0;19,30} and h ≈ φ θ ℓ ≈ 0 ; 48 , 45 {\displaystyle h\approx {\frac {\varphi }{\theta }}\ell \approx 0;48,45} Then, because ℓ + h ≈ φ θ L sin θ + L sin θ = ( φ θ + 1 ) L sin θ {\displaystyle \ell +h\approx {\frac {\varphi }{\theta }}L\sin \theta +L\sin \theta =\left({\frac {\varphi }{\theta }}+1\right)L\sin \theta } it follows that S ≈ L / ( ℓ + h − 1 ) ≈ 67 ; 20 / 0 ; 8 , 15 ≈ 489.70 ≈ 490 {\displaystyle S\approx L/(\ell +h-1)\approx 67;20/0;8,15\approx 489.70\approx 490} Swerdlow used this result to argue that 490 was the correct reading of the Pappus text, thus invalidating Hultsch' interpretation. While this result is highly dependent on the particular approximations and roundings used, it has generally been accepted. It leaves open, however, the question of where the lunar distance 671⁄3 came from. Following Pappus and Ptolemy, Swerdlow suggested that Hipparchus had estimated 490 Earth radii as a minimum possible distance to the Sun. This distance corresponds to a solar parallax of 7', which may have been the maximum that he thought would have gone unnoticed (the typical resolution of the human eye is 2'). The formula obtained above for the distance to the Sun can be inverted to determine the distance to the Moon: L ≈ S ( φ θ + 1 ) S sin θ − 1 = 1 / ( ( φ θ + 1 ) sin θ − 1 S ) {\displaystyle L\approx {\frac {S}{\left({\frac {\varphi }{\theta }}+1\right)S\sin \theta -1}}=1\left/\left(\left({\frac {\varphi }{\theta }}+1\right)\sin \theta -{\frac {1}{S}}\right)\right.} Using the same values as above for each angle, and using 490 earth radii as the minimum solar distance, it follows that the maximum mean lunar distance is L ≈ 1 / ( ( 2.5 + 1 ) sin 0.277 ∘ − 1 490 ) ≈ 67.203 ≈ 67 1 3 {\displaystyle L\approx 1\left/\left((2.5+1)\sin 0.277^{\circ }-{\frac {1}{490}}\right)\right.\approx 67.203\approx 67{\tfrac {1}{3}}} Toomer expanded on this by observing that as the distance to the Sun increases without bound, the formula approaches a minimum mean lunar distance: L ≈ 1 / ( ( 2.5 + 1 ) sin 0.277 ∘ ) ≈ 59.10 {\displaystyle L\approx 1\left/\left((2.5+1)\sin 0.277^{\circ }\right)\right.\approx 59.10} This is close to the value later claimed by Ptolemy. === Book 1 reconstruction (Toomer) === In addition to explaining the minimum lunar distance that Hipparchus achieved, Toomer was able to explain the method of the first book, which employed a solar eclipse. Pappus states that this eclipse was total in the region of the Hellespont, but was observed to be 4/5 of total in Alexandria. If Hipparchus assumed that the Sun was infinitely distant (i.e. that "the earth has the ratio of a point and center to the sun"), then the difference in magnitude of the solar eclipse must be due entirely to the parallax of the Moon. By using observational data, he would be able to determine this parallax, and hence the distance of the Moon. Hipparchus would have known φ A {\displaystyle \varphi _{A}} and φ H {\displaystyle \varphi _{H}} , the latitudes of Alexandria and the Hellespontine region, respectively. He also would have known δ {\displaystyle \delta } , the declination of the Moon during the eclipse, and μ {\displaystyle \mu } , which is related to the difference in totality of the eclipse between the two regions. A H = t Crd ∠ A O H R = t Crd ( φ H − φ A ) R {\displaystyle AH={\frac {t\operatorname {Crd} \angle AOH}{R}}={\frac {t\operatorname {Crd} (\varphi _{H}-\varphi _{A})}{R}}} Crd here refers to the chord function, which maps an angle in degrees to the corresponding length of a chord of a circle of unit diameter. Also, R is the radius of the base circle that Hipparchus used to compute the chord function and t is the radius of the earth. Since the Moon is very distant, it follows that ζ ′ ≈ ζ {\displaystyle \zeta '\approx \zeta } . Using this approximation gives ζ = φ H − δ {\displaystyle \zeta =\varphi _{H}-\delta } ∠ Z H A = 180 ∘ − ∠ O H A {\displaystyle \angle ZHA=180^{\circ }-\angle OHA} ∠ O H A = 180 ∘ − ∠ A O H 2 = 180 ∘ − ( φ H − φ A ) 2 {\displaystyle \angle OHA={\frac {180^{\circ }-\angle AOH}{2}}={\frac {180^{\circ }-(\varphi _{H}-\varphi _{A})}{2}}} Hence, ∠ Z H A = 90 ∘ + 1 2 ( φ H − φ A ) {\displaystyle \angle ZHA=90^{\circ }+{\frac {1}{2}}(\varphi _{H}-\varphi _{A})} ∠ M H A = θ = ∠ Z H A − ζ ′ ≈ ∠ Z H A − ζ = 90 ∘ − 1 2 ( φ H + φ A ) + δ {\displaystyle \angle MHA=\theta =\angle ZHA-\zeta '\approx \angle ZHA-\zeta =90^{\circ }-{\frac {1}{2}}(\varphi _{H}+\varphi _{A})+\delta } With A H {\displaystyle AH} and θ {\displaystyle \theta } , we only need μ {\displaystyle \mu } to get D ′ {\displaystyle D'} . Because the eclipse was total at H, and 4/5 total at A, it follows that μ {\displaystyle \mu } is 1/5 of the apparent diameter of the Sun. This quantity was well known by Hipparchus—he took it to be 1/650 of a full circle. The distance from the center of the Earth to the Moon then follows from D ≈ D ′ + t {\displaystyle D\approx D'+t} . Toomer determined how Hipparchus determined the chord for small angles (see Chord (geometry)). His values for the latitudes of the Hellespont (41 degrees) and Alexandria (31 degrees) are known from Strabo's work on Geography. To determine the declination, it is necessary to know which eclipse Hipparchus used. Because he knew the value which Hipparchus eventually gave for the distance to the Moon (71 Earth radii) and the rough region of the eclipse, Toomer was able to determine that Hipparchus used the solar eclipse of March 14, 190 BC. This eclipse fits all the mathematical parameters very well, and also makes sense from a historical point of view. The eclipse was total in Nicaea, Hipparchus' birthplace, so he may have heard stories of it. There is also an account of it in Strabo's Ab Urbe Condita VIII.2. The declination of the Moon at this time was δ = − 3 ∘ {\displaystyle \delta =-3^{\circ }} . Hence, using chord trigonometry, gives θ = 54 ∘ + δ {\displaystyle \theta =54^{\circ }+\delta } A H = t Crd 10 ∘ R ≈ t 600 3438 {\displaystyle AH={\frac {t\operatorname {Crd} 10^{\circ }}{R}}\approx t\ {\frac {600}{3438}}} μ = 360 ⋅ 60 5 ⋅ 650 {\displaystyle \mu ={\frac {360\cdot 60}{5\cdot 650}}} D ′ = Crd 2 θ ⋅ A H Crd μ ⋅ 2 = Crd ( 108 ∘ + 2 δ ) ⋅ 600 ⋅ 5 ⋅ 650 21600 ⋅ 2 ⋅ 3438 t {\displaystyle D'={\frac {\operatorname {Crd} 2\theta \cdot AH}{\operatorname {Crd} \mu \cdot 2}}={\frac {\operatorname {Crd} (108^{\circ }+2\delta )\cdot 600\cdot 5\cdot 650}{21600\cdot 2\cdot 3438}}t} Now using Hipparchus' chord tables, Crd ( 108 ∘ + 2 ( − 3 ∘ ) ) = Crd 102 ∘ ≈ 2 ⋅ 3438 sin 51 ∘ ≈ 5340 {\displaystyle \operatorname {Crd} (108^{\circ }+2(-3^{\circ }))=\operatorname {Crd} 102^{\circ }\approx 2\cdot 3438\sin 51^{\circ }\approx 5340} and hence D ′ = 5340 ⋅ 600 ⋅ 5 ⋅ 650 21600 ⋅ 2 ⋅ 3438 t ≈ 70.1 t ⇒ D ≈ D ′ + t ≈ 71.1 t {\displaystyle D'={\frac {5340\cdot 600\cdot 5\cdot 650}{21600\cdot 2\cdot 3438}}t\approx 70.1t\Rightarrow D\approx D'+t\approx 71.1t} This agrees well with the value of 71 Earth radii that Pappus reports. == Conclusion == Assuming that these reconstructions accurately reflect what Hipparchus wrote in On Sizes and Distances, then this work was a remarkable accomplishment. This approach of setting limits on an unknown physical quantity was not new to Hipparchus (see Aristarchus of Samos. Archimedes also did the same with pi), but in those cases, the bounds reflected the inability to determine a mathematical constant to an arbitrary precision, not uncertainty in physical observations. Hipparchus appears to have eventually resolved the contradiction between his two results. His aim in calculating the distance to the Moon was to obtain an accurate value for the lunar parallax, so that he might predict eclipses with more precision. To this, he had to settle on a particular value for the distance/parallax, not a range of values. There is some evidence that he did this. Combining the calculations of Book 2 and the account of Theon of Smyrna yields a lunar distance of 60.5 Earth radii. Doing the same with the account of Cleomedes yields a distance of 61 Earth radii. These are remarkably close to both Ptolemy's value and the modern one. According to Toomer, This procedure, if I have constructed it correctly, is very remarkable... What is astonishing is the sophistication of approaching the problem by two quite different methods, and also the complete honesty with which Hipparchus reveals his discrepant results... which are nevertheless of the same order of magnitude and (for the first time in the history of astronomy) in the right region. == See also == Aristarchus of Samos (c. 310 – c. 230 BC), a Greek mathematician who calculated the distance from the Earth to the Sun. Axial precession Eratosthenes (c. 276 – c. 194/195 BC), a Greek mathematician who calculated the circumference of the Earth and also the distance from the Earth to the Sun. Greek mathematics Hipparchus (c. 190 – c. 120 BC) Posidonius (c. 135 – c. 51 BC), a Greek astronomer and mathematician who calculated the circumference of the Earth. == References == F Hultsch, "...," Leipzig, Phil.-hist. Kl. 52 (1900), 169–200. N. M. Swerdlow, "Hipparchus on the distance of the sun," Centaurus 14 (1969), 287–305. G. J. Toomer, "Hipparchus on the distances of the sun and moon," Archive for History of Exact Sciences 14 (1974), 126–142. Hon, Giora. "Is there a concept of experimental error in Greek astronomy?." The British Journal for the History of Science 22.02 (1989): 129–150. (Available online at https://www.researchgate.net/profile/Giora_Hon/publication/231844424_Is_There_a_Concept_of_Experimental_Error_in_Greek_Astronomy/links/564fa57b08ae4988a7a858bd.pdf)
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Wikipedia:On the Moving Sphere#0
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Autolycus of Pitane (Greek: Αὐτόλυκος ὁ Πιταναῖος; c. 360 – c. 290 BC) was a Greek astronomer, mathematician, and geographer. He is known today for his two surviving works On the Moving Sphere and On Risings and Settings, both about spherical geometry. == Life == Autolycus was born in Pitane, a town of Aeolis within Ionia, Asia Minor. Of his personal life nothing is known, although he was a contemporary of Aristotle and his works seem to have been completed in Athens between 335–300 BC. Euclid references some of Autolycus' work, and Autolycus is known to have taught Arcesilaus. The lunar crater Autolycus was named in his honour. == Work == Autolycus' two surviving works are about spherical geometry with application to astronomy: On the Moving Sphere and On Risings and Settings (of stars). In late antiquity, both were part of the "Little Astronomy", a collection of miscellaneous short works about geometry and astronomy which were commonly transmitted together. They were translated into Arabic in the 9th century, and with the addition of a few additional works became known as the "Middle Books" (sitting between Euclid's Elements and Ptolemy's Almagest). Both were preserved both in Greek and in Arabic, but were unknown in medieval Western Europe. They were translated from Arabic into Latin in the 12th century. Later, remaining Greek copies were also translated into Latin. === On the Moving Sphere === On the Moving Sphere (Ancient Greek: Περὶ κινουμένης σφαίρας Perí kinouménis sphaíras) concerns the movements of points and arcs on the sphere as it rotates on an axis. While the obvious application is the diurnal motion of the stars as the celestial sphere appears to rotate about an immobile Earth (as modeled at the time), Autolycus' treatise never explicitly discusses this application: its content consists entirely of elementary theorems about the arcs of great circles and parallel small circles on an abstract sphere. The work is simple and probably derivative of older works, but each theorem includes a clearly enunciated statement, a figure of the construction alongside its proof, and finally a concluding remark. On the Moving Sphere is believed to be the oldest mathematical treatise from ancient Greece that is completely preserved: All prior Greek mathematical works are taken from later summaries, commentaries, or descriptions of the works. It shows that by Autolycus' day there was a thoroughly established textbook tradition in geometry that is today regarded as typical of classical Greek geometry. Moreover, it gives indications of what theorems were well known in his day (around 320 BC). Two hundred years later Theodosius wrote the Spherics, a treatise establishing the fundamental definitions and constructions in spherical geometry whose content is believed to have a common origin with On the Moving Sphere in some pre-Euclidean textbook, possibly written by Eudoxus. In contrast to later astronomical analyses by Hipparchus (2nd century BC) and Ptolemy (2nd century AD), but similarly to the planar geometry of Euclid's Elements, both Autolycus's work and Theodosius' does not involve concrete quantification or trigonometry: spherical arcs are compared in size, but not given any numerical measure. === On Risings and Settings === In the two-book treatise On Risings and Settings (Ancient Greek: Περὶ ἐπιτολῶν καὶ δύσεων Perí epitolón kaí dýseon), Autolycus studied the relationship between the rising and the setting of the stars throughout the year. The second book is an expansion of the first and of higher quality. He wrote that "any star which rises and sets always rises and sets at the same point in the horizon." Autolycus relied heavily on Eudoxus' astronomy and was a strong supporter of Eudoxus' theory of homocentric spheres. == Footnotes == == References == Huxley, G. L. (1970). "Autolycus of Pitane". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 338–39. ISBN 0-684-10114-9. on line at "Autolycus of Pitane". HighBeam Research. Retrieved 26 March 2015. O'Connor, John J.; Robertson, Edmund F. (April 1999), "Autolycus of Pitane", MacTutor History of Mathematics Archive, University of St Andrews == External links == Autolycus On The Moving Sphere from the Million Books Project (Greek with Latin translation) ΠΕΡΙ ΚΙΝΟΥΜΕΝΗΣ ΣΦΑΙΡΑΣ and ΠΕΡΙ ΕΠΙΤΟΛΩΝ ΚΑΙ ΔΥΣΕΩΝ (Mogenet ed., 1950)
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Wikipedia:One-sided limit#0
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In calculus, a one-sided limit refers to either one of the two limits of a function f ( x ) {\displaystyle f(x)} of a real variable x {\displaystyle x} as x {\displaystyle x} approaches a specified point either from the left or from the right. The limit as x {\displaystyle x} decreases in value approaching a {\displaystyle a} ( x {\displaystyle x} approaches a {\displaystyle a} "from the right" or "from above") can be denoted: lim x → a + f ( x ) or lim x ↓ a f ( x ) or lim x ↘ a f ( x ) or f ( x + ) {\displaystyle \lim _{x\to a^{+}}f(x)\quad {\text{ or }}\quad \lim _{x\,\downarrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\searrow a}\,f(x)\quad {\text{ or }}\quad f(x+)} The limit as x {\displaystyle x} increases in value approaching a {\displaystyle a} ( x {\displaystyle x} approaches a {\displaystyle a} "from the left" or "from below") can be denoted: lim x → a − f ( x ) or lim x ↑ a f ( x ) or lim x ↗ a f ( x ) or f ( x − ) {\displaystyle \lim _{x\to a^{-}}f(x)\quad {\text{ or }}\quad \lim _{x\,\uparrow \,a}\,f(x)\quad {\text{ or }}\quad \lim _{x\nearrow a}\,f(x)\quad {\text{ or }}\quad f(x-)} If the limit of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches a {\displaystyle a} exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit lim x → a f ( x ) {\displaystyle \lim _{x\to a}f(x)} does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as x {\displaystyle x} approaches a {\displaystyle a} is sometimes called a "two-sided limit". It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist. == Formal definition == === Definition === If I {\displaystyle I} represents some interval that is contained in the domain of f {\displaystyle f} and if a {\displaystyle a} is a point in I {\displaystyle I} then the right-sided limit as x {\displaystyle x} approaches a {\displaystyle a} can be rigorously defined as the value R {\displaystyle R} that satisfies: for all ε > 0 there exists some δ > 0 such that for all x ∈ I , if 0 < x − a < δ then | f ( x ) − R | < ε , {\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<x-a<\delta {\text{ then }}|f(x)-R|<\varepsilon ,} and the left-sided limit as x {\displaystyle x} approaches a {\displaystyle a} can be rigorously defined as the value L {\displaystyle L} that satisfies: for all ε > 0 there exists some δ > 0 such that for all x ∈ I , if 0 < a − x < δ then | f ( x ) − L | < ε . {\displaystyle {\text{for all }}\varepsilon >0\;{\text{ there exists some }}\delta >0\;{\text{ such that for all }}x\in I,{\text{ if }}\;0<a-x<\delta {\text{ then }}|f(x)-L|<\varepsilon .} We can represent the same thing more symbolically, as follows. Let I {\displaystyle I} represent an interval, where I ⊆ d o m a i n ( f ) {\displaystyle I\subseteq \mathrm {domain} (f)} , and a ∈ I {\displaystyle a\in I} . lim x → a + f ( x ) = R ⟺ ( ∀ ε ∈ R + , ∃ δ ∈ R + , ∀ x ∈ I , ( 0 < x − a < δ ⟶ | f ( x ) − R | < ε ) ) {\displaystyle \lim _{x\to a^{+}}f(x)=R~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<x-a<\delta \longrightarrow |f(x)-R|<\varepsilon ))} lim x → a − f ( x ) = L ⟺ ( ∀ ε ∈ R + , ∃ δ ∈ R + , ∀ x ∈ I , ( 0 < a − x < δ ⟶ | f ( x ) − L | < ε ) ) {\displaystyle \lim _{x\to a^{-}}f(x)=L~~~\iff ~~~(\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,(0<a-x<\delta \longrightarrow |f(x)-L|<\varepsilon ))} === Intuition === In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value. For reference, the formal definition for the limit of a function at a point is as follows: lim x → a f ( x ) = L ⟺ ∀ ε ∈ R + , ∃ δ ∈ R + , ∀ x ∈ I , 0 < | x − a | < δ ⟹ | f ( x ) − L | < ε . {\displaystyle \lim _{x\to a}f(x)=L~~~\iff ~~~\forall \varepsilon \in \mathbb {R} _{+},\exists \delta \in \mathbb {R} _{+},\forall x\in I,0<|x-a|<\delta \implies |f(x)-L|<\varepsilon .} To define a one-sided limit, we must modify this inequality. Note that the absolute distance between x {\displaystyle x} and a {\displaystyle a} is | x − a | = | ( − 1 ) ( − x + a ) | = | ( − 1 ) ( a − x ) | = | ( − 1 ) | | a − x | = | a − x | . {\displaystyle |x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|.} For the limit from the right, we want x {\displaystyle x} to be to the right of a {\displaystyle a} , which means that a < x {\displaystyle a<x} , so x − a {\displaystyle x-a} is positive. From above, x − a {\displaystyle x-a} is the distance between x {\displaystyle x} and a {\displaystyle a} . We want to bound this distance by our value of δ {\displaystyle \delta } , giving the inequality x − a < δ {\displaystyle x-a<\delta } . Putting together the inequalities 0 < x − a {\displaystyle 0<x-a} and x − a < δ {\displaystyle x-a<\delta } and using the transitivity property of inequalities, we have the compound inequality 0 < x − a < δ {\displaystyle 0<x-a<\delta } . Similarly, for the limit from the left, we want x {\displaystyle x} to be to the left of a {\displaystyle a} , which means that x < a {\displaystyle x<a} . In this case, it is a − x {\displaystyle a-x} that is positive and represents the distance between x {\displaystyle x} and a {\displaystyle a} . Again, we want to bound this distance by our value of δ {\displaystyle \delta } , leading to the compound inequality 0 < a − x < δ {\displaystyle 0<a-x<\delta } . Now, when our value of x {\displaystyle x} is in its desired interval, we expect that the value of f ( x ) {\displaystyle f(x)} is also within its desired interval. The distance between f ( x ) {\displaystyle f(x)} and L {\displaystyle L} , the limiting value of the left sided limit, is | f ( x ) − L | {\displaystyle |f(x)-L|} . Similarly, the distance between f ( x ) {\displaystyle f(x)} and R {\displaystyle R} , the limiting value of the right sided limit, is | f ( x ) − R | {\displaystyle |f(x)-R|} . In both cases, we want to bound this distance by ε {\displaystyle \varepsilon } , so we get the following: | f ( x ) − L | < ε {\displaystyle |f(x)-L|<\varepsilon } for the left sided limit, and | f ( x ) − R | < ε {\displaystyle |f(x)-R|<\varepsilon } for the right sided limit. == Examples == Example 1: The limits from the left and from the right of g ( x ) := − 1 x {\displaystyle g(x):=-{\frac {1}{x}}} as x {\displaystyle x} approaches a := 0 {\displaystyle a:=0} are lim x → 0 − − 1 / x = + ∞ and lim x → 0 + − 1 / x = − ∞ {\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}{-1/x}=-\infty } The reason why lim x → 0 − − 1 / x = + ∞ {\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty } is because x {\displaystyle x} is always negative (since x → 0 − {\displaystyle x\to 0^{-}} means that x → 0 {\displaystyle x\to 0} with all values of x {\displaystyle x} satisfying x < 0 {\displaystyle x<0} ), which implies that − 1 / x {\displaystyle -1/x} is always positive so that lim x → 0 − − 1 / x {\displaystyle \lim _{x\to 0^{-}}{-1/x}} diverges to + ∞ {\displaystyle +\infty } (and not to − ∞ {\displaystyle -\infty } ) as x {\displaystyle x} approaches 0 {\displaystyle 0} from the left. Similarly, lim x → 0 + − 1 / x = − ∞ {\displaystyle \lim _{x\to 0^{+}}{-1/x}=-\infty } since all values of x {\displaystyle x} satisfy x > 0 {\displaystyle x>0} (said differently, x {\displaystyle x} is always positive) as x {\displaystyle x} approaches 0 {\displaystyle 0} from the right, which implies that − 1 / x {\displaystyle -1/x} is always negative so that lim x → 0 + − 1 / x {\displaystyle \lim _{x\to 0^{+}}{-1/x}} diverges to − ∞ . {\displaystyle -\infty .} Example 2: One example of a function with different one-sided limits is f ( x ) = 1 1 + 2 − 1 / x , {\displaystyle f(x)={\frac {1}{1+2^{-1/x}}},} (cf. picture) where the limit from the left is lim x → 0 − f ( x ) = 0 {\displaystyle \lim _{x\to 0^{-}}f(x)=0} and the limit from the right is lim x → 0 + f ( x ) = 1. {\displaystyle \lim _{x\to 0^{+}}f(x)=1.} To calculate these limits, first show that lim x → 0 − 2 − 1 / x = ∞ and lim x → 0 + 2 − 1 / x = 0 {\displaystyle \lim _{x\to 0^{-}}2^{-1/x}=\infty \qquad {\text{ and }}\qquad \lim _{x\to 0^{+}}2^{-1/x}=0} (which is true because lim x → 0 − − 1 / x = + ∞ and lim x → 0 + − 1 / x = − ∞ {\displaystyle \lim _{x\to 0^{-}}{-1/x}=+\infty {\text{ and }}\lim _{x\to 0^{+}}{-1/x}=-\infty } ) so that consequently, lim x → 0 + 1 1 + 2 − 1 / x = 1 1 + lim x → 0 + 2 − 1 / x = 1 1 + 0 = 1 {\displaystyle \lim _{x\to 0^{+}}{\frac {1}{1+2^{-1/x}}}={\frac {1}{1+\displaystyle \lim _{x\to 0^{+}}2^{-1/x}}}={\frac {1}{1+0}}=1} whereas lim x → 0 − 1 1 + 2 − 1 / x = 0 {\displaystyle \lim _{x\to 0^{-}}{\frac {1}{1+2^{-1/x}}}=0} because the denominator diverges to infinity; that is, because lim x → 0 − 1 + 2 − 1 / x = ∞ . {\displaystyle \lim _{x\to 0^{-}}1+2^{-1/x}=\infty .} Since lim x → 0 − f ( x ) ≠ lim x → 0 + f ( x ) , {\displaystyle \lim _{x\to 0^{-}}f(x)\neq \lim _{x\to 0^{+}}f(x),} the limit lim x → 0 f ( x ) {\displaystyle \lim _{x\to 0}f(x)} does not exist. == Relation to topological definition of limit == The one-sided limit to a point p {\displaystyle p} corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p . {\displaystyle p.} Alternatively, one may consider the domain with a half-open interval topology. == Abel's theorem == A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem. == Notes == == References == == See also == Projectively extended real line Semi-differentiability Limit superior and limit inferior
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Wikipedia:Onno J. Boxma#0
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Onno Johan Boxma (born 1952) is a Dutch mathematician, and Professor at the Eindhoven University of Technology, known for several contributions to queueing theory and applied probability theory. == Biography == Born in The Hague, Boxma earned his B.Sc. in Mathematics at Delft University of Technology in 1974, and his Ph.D. cum laude in Mathematic from Utrecht University in 1977 on the dissertation entitled "Analysis of Models for Tandem Queues", advised by Wim Cohen. Boxma continued at Utrecht as faculty from 1974 to 1985, and was IBM Research postdoctoral fellow in 1978–79, before joining the faculty of Centrum Wiskunde & Informatica in Amsterdam. There he chaired the performance analysis group until 1998. He was full professor at University of Tilburg from 1987 to 1988. and since 1998 he is as full professor holding the chair of Stochastic Operations Research in the Department of Mathematics and Computer Science at Technische Universiteit Eindhoven, becoming vice dean of the department in 2009. He was the editor-in-chief of Queueing Systems from 2004 to 2009, and scientific director of EURANDOM from 2005-2010. In 2009 he was awarded an honorary degree by the University of Haifa (Israel), and received the 2011 ACM SIGMETRICS Achievement Award in June 2011. Also, he is honorary professor in Heriot-Watt University, Edinburgh, UK (2008-2010 and 2011-2013). == Work == Boxma's research focuses on the field of applied probability and stochastic operations, particularly of queueing theory and its application to the performance analysis of computer, communication and production systems. == Publications == Books, a selection: 1977. Analysis of Models for Tandem Queues. Doctoral thesis Utrecht University. 1983. Boundary Value Problems in Queueing System Analysis. Editor with Wim Cohen Articles, a selection: Boxma, Onno J., and Hans Daduna. "Sojourn times in queueing networks." CWI. Department of Operations Research, Statistics, and System Theory [BS] R 8916 (1989): 1-47. Boxma, Onno J. "Workloads and waiting times in single-server systems with multiple customer classes." Queueing Systems 5.1-3 (1989): 185-214. Cohen, Jacob Willem, and Onno J. Boxma. Boundary value problems in queueing system analysis. Elsevier, 2000. Albrecher, Hansjörg, and Onno J. Boxma. "A ruin model with dependence between claim sizes and claim intervals." Insurance: Mathematics and Economics 35.2 (2004): 245-254. == References ==
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Wikipedia:Onorato Nicoletti#0
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Onorato Nicoletti (21 June 1872 – 31 December 1929) was an Italian mathematician. == Biography == Nicoletti received his laurea in 1894 from the Scuola Normale di Pisa. In 1898, he became a professor of infinitesimal calculus at the University of Modena. After two years, he returned to Pisa, where he was a teacher of algebra, and then, after the death of Ulisse Dini, of infinitesimal calculus. He published works in various fields of mathematics, including numerical analysis, infinitesimal analysis, the equations related to hermitian matrices, and differential equations. He made original contributions to Max Dehn's theory of the equivalence of polyhedra under polyhedral dissection and reassembly (scissors-congruence), extending and generalizing the theory with an entire class of new relations. Nicoletti collaborated in the writing of Enciclopedia Hoepli delle Matematiche elementari e complementi (published from 1930 to 1951) with the contribution of two monographic articles: Forme razionali di una o più variabili (Rational forms of one or more variables) and Proprietà generali delle funzioni algebriche (General properties of algebraic functions). A leading expert in mathematics education, he edited with Roberto Marcolongo a series of successful editions for secondary schools. Nicoletti was an Invited Speaker of the ICM in Rome. == Selected publications == "Su un sistema di equazioni a derivate parziali del secondo ordine." Rendic. Acc. Lincei,(5) 4 (1895): 197–202. "Sulle condizioni iniziali che determiniano gli integrali della differenziali ordinazie Atti della R. Acc. Sc. Torino. 1898 (1897): 746–759. "Sulla teoria della convergenza degli algoritmi di iterazione." Annali di Matematica Pura ed Applicata (1898–1922) 14, no. 1 (1908): 1–32. "Sulla caratteristica delle matrici di Sylvester e di Bezout." Rendiconti del Circolo Matematico di Palermo (1884–1940) 28, no. 1 (1909): 29–32. doi:10.1007/BF03018209 "Sulla equivalenza dei poliedri." Rendiconti del Circolo Matematico di Palermo (1884–1940) 37, no. 1 (1914): 47–75. with Eugenio Bertini and Luigi Bianchi: "Relazione sulla memoria di Albanese «Intorno ad alcuni concetti e teoremi fondamentali sui sistemi di curve d'una superficie algebrica»." Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 14 (1922): XI–XV. "Un teorema di limite." Annali di Matematica Pura ed Applicata 1, no. 1 (1924): 91–104. doi:10.1007/BF02409915 == References == == External links == Onorato Nicoletti sul sito Edizione Nazionale Mathematica Italiana
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Wikipedia:Onsager–Machlup function#0
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The Onsager–Machlup function is a function that summarizes the dynamics of a continuous stochastic process. It is used to define a probability density for a stochastic process, and it is similar to the Lagrangian of a dynamical system. It is named after Lars Onsager and Stefan Machlup who were the first to consider such probability densities. The dynamics of a continuous stochastic process X from time t = 0 to t = T in one dimension, satisfying a stochastic differential equation d X t = b ( X t ) d t + σ ( X t ) d W t {\displaystyle dX_{t}=b(X_{t})\,dt+\sigma (X_{t})\,dW_{t}} where W is a Wiener process, can in approximation be described by the probability density function of its value xi at a finite number of points in time ti: p ( x 1 , … , x n ) = ( ∏ i = 1 n − 1 1 2 π σ ( x i ) 2 Δ t i ) exp ( − ∑ i = 1 n − 1 L ( x i , x i + 1 − x i Δ t i ) Δ t i ) {\displaystyle p(x_{1},\ldots ,x_{n})=\left(\prod _{i=1}^{n-1}{\frac {1}{\sqrt {2\pi \sigma (x_{i})^{2}\Delta t_{i}}}}\right)\exp \left(-\sum _{i=1}^{n-1}L\left(x_{i},{\frac {x_{i+1}-x_{i}}{\Delta t_{i}}}\right)\,\Delta t_{i}\right)} where L ( x , v ) = 1 2 ( v − b ( x ) σ ( x ) ) 2 {\displaystyle L(x,v)={\frac {1}{2}}\left({\frac {v-b(x)}{\sigma (x)}}\right)^{2}} and Δti = ti+1 − ti > 0, t1 = 0 and tn = T. A similar approximation is possible for processes in higher dimensions. The approximation is more accurate for smaller time step sizes Δti, but in the limit Δti → 0 the probability density function becomes ill defined, one reason being that the product of terms 1 2 π σ ( x i ) 2 Δ t i {\displaystyle {\frac {1}{\sqrt {2\pi \sigma (x_{i})^{2}\Delta t_{i}}}}} diverges to infinity. In order to nevertheless define a density for the continuous stochastic process X, ratios of probabilities of X lying within a small distance ε from smooth curves φ1 and φ2 are considered: P ( | X t − φ 1 ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) P ( | X t − φ 2 ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) → exp ( − ∫ 0 T L ( φ 1 ( t ) , φ ˙ 1 ( t ) ) d t + ∫ 0 T L ( φ 2 ( t ) , φ ˙ 2 ( t ) ) d t ) {\displaystyle {\frac {P\left(\left|X_{t}-\varphi _{1}(t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}{P\left(\left|X_{t}-\varphi _{2}(t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}}\to \exp \left(-\int _{0}^{T}L\left(\varphi _{1}(t),{\dot {\varphi }}_{1}(t)\right)\,dt+\int _{0}^{T}L\left(\varphi _{2}(t),{\dot {\varphi }}_{2}(t)\right)\,dt\right)} as ε → 0, where L is the Onsager–Machlup function. == Definition == Consider a d-dimensional Riemannian manifold M and a diffusion process X = {Xt : 0 ≤ t ≤ T} on M with infinitesimal generator 1/2ΔM + b, where ΔM is the Laplace–Beltrami operator and b is a vector field. For any two smooth curves φ1, φ2 : [0, T] → M, lim ε ↓ 0 P ( ρ ( X t , φ 1 ( t ) ) ≤ ε for every t ∈ [ 0 , T ] ) P ( ρ ( X t , φ 2 ( t ) ) ≤ ε for every t ∈ [ 0 , T ] ) = exp ( − ∫ 0 T L ( φ 1 ( t ) , φ ˙ 1 ( t ) ) d t + ∫ 0 T L ( φ 2 ( t ) , φ ˙ 2 ( t ) ) d t ) {\displaystyle \lim _{\varepsilon \downarrow 0}{\frac {P\left(\rho (X_{t},\varphi _{1}(t))\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}{P\left(\rho (X_{t},\varphi _{2}(t))\leq \varepsilon {\text{ for every }}t\in [0,T]\right)}}=\exp \left(-\int _{0}^{T}L\left(\varphi _{1}(t),{\dot {\varphi }}_{1}(t)\right)\,dt+\int _{0}^{T}L\left(\varphi _{2}(t),{\dot {\varphi }}_{2}(t)\right)\,dt\right)} where ρ is the Riemannian distance, φ ˙ 1 , φ ˙ 2 {\displaystyle \scriptstyle {\dot {\varphi }}_{1},{\dot {\varphi }}_{2}} denote the first derivatives of φ1, φ2, and L is called the Onsager–Machlup function. The Onsager–Machlup function is given by L ( x , v ) = 1 2 ‖ v − b ( x ) ‖ x 2 + 1 2 div b ( x ) − 1 12 R ( x ) , {\displaystyle L(x,v)={\tfrac {1}{2}}\|v-b(x)\|_{x}^{2}+{\tfrac {1}{2}}\operatorname {div} \,b(x)-{\tfrac {1}{12}}R(x),} where || ⋅ ||x is the Riemannian norm in the tangent space Tx(M) at x, div b(x) is the divergence of b at x, and R(x) is the scalar curvature at x. == Examples == The following examples give explicit expressions for the Onsager–Machlup function of a continuous stochastic processes. === Wiener process on the real line === The Onsager–Machlup function of a Wiener process on the real line R is given by L ( x , v ) = 1 2 | v | 2 . {\displaystyle L(x,v)={\tfrac {1}{2}}|v|^{2}.} Proof: Let X = {Xt : 0 ≤ t ≤ T} be a Wiener process on R and let φ : [0, T] → R be a twice differentiable curve such that φ(0) = X0. Define another process Xφ = {Xtφ : 0 ≤ t ≤ T} by Xtφ = Xt − φ(t) and a measure Pφ by P φ = exp ( ∫ 0 T φ ˙ ( t ) d X t φ + ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . {\displaystyle P^{\varphi }=\exp \left(\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}^{\varphi }+\int _{0}^{T}{\tfrac {1}{2}}\left|{\dot {\varphi }}(t)\right|^{2}\,dt\right)\,dP.} For every ε > 0, the probability that |Xt − φ(t)| ≤ ε for every t ∈ [0, T] satisfies P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) = P ( | X t φ | ≤ ε for every t ∈ [ 0 , T ] ) = ∫ { | X t φ | ≤ ε for every t ∈ [ 0 , T ] } exp ( − ∫ 0 T φ ˙ ( t ) d X t φ − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P φ . {\displaystyle {\begin{aligned}P\left(\left|X_{t}-\varphi (t)\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)&=P\left(\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right)\\&=\int _{\left\{\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right\}}\exp \left(-\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}^{\varphi }-\int _{0}^{T}{\tfrac {1}{2}}|{\dot {\varphi }}(t)|^{2}\,dt\right)\,dP^{\varphi }.\end{aligned}}} By Girsanov's theorem, the distribution of Xφ under Pφ equals the distribution of X under P, hence the latter can be substituted by the former: P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) = ∫ { | X t φ | ≤ ε for every t ∈ [ 0 , T ] } exp ( − ∫ 0 T φ ˙ ( t ) d X t − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) d P . {\displaystyle P(|X_{t}-\varphi (t)|\leq \varepsilon {\text{ for every }}t\in [0,T])=\int _{\left\{\left|X_{t}^{\varphi }\right|\leq \varepsilon {\text{ for every }}t\in [0,T]\right\}}\exp \left(-\int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}-\int _{0}^{T}{\tfrac {1}{2}}|{\dot {\varphi }}(t)|^{2}\,dt\right)\,dP.} By Itō's lemma it holds that ∫ 0 T φ ˙ ( t ) d X t = φ ˙ ( T ) X T − ∫ 0 T φ ¨ ( t ) X t d t , {\displaystyle \int _{0}^{T}{\dot {\varphi }}(t)\,dX_{t}={\dot {\varphi }}(T)X_{T}-\int _{0}^{T}{\ddot {\varphi }}(t)X_{t}\,dt,} where φ ¨ {\displaystyle \scriptstyle {\ddot {\varphi }}} is the second derivative of φ, and so this term is of order ε on the event where |Xt| ≤ ε for every t ∈ [0, T] and will disappear in the limit ε → 0, hence lim ε ↓ 0 P ( | X t − φ ( t ) | ≤ ε for every t ∈ [ 0 , T ] ) P ( | X t | ≤ ε for every t ∈ [ 0 , T ] ) = exp ( − ∫ 0 T 1 2 | φ ˙ ( t ) | 2 d t ) . {\displaystyle \lim _{\varepsilon \downarrow 0}{\frac {P(|X_{t}-\varphi (t)|\leq \varepsilon {\text{ for every }}t\in [0,T])}{P(|X_{t}|\leq \varepsilon {\text{ for every }}t\in [0,T])}}=\exp \left(-\int _{0}^{T}{\tfrac {1}{2}}|{\dot {\varphi }}(t)|^{2}\,dt\right).} === Diffusion processes with constant diffusion coefficient on Euclidean space === The Onsager–Machlup function in the one-dimensional case with constant diffusion coefficient σ is given by L ( x , v ) = 1 2 | v − b ( x ) σ | 2 + 1 2 d b d x ( x ) . {\displaystyle L(x,v)={\frac {1}{2}}\left|{\frac {v-b(x)}{\sigma }}\right|^{2}+{\frac {1}{2}}{\frac {db}{dx}}(x).} In the d-dimensional case, with σ equal to the unit matrix, it is given by L ( x , v ) = 1 2 ‖ v − b ( x ) ‖ 2 + 1 2 ( div b ) ( x ) , {\displaystyle L(x,v)={\frac {1}{2}}\|v-b(x)\|^{2}+{\frac {1}{2}}(\operatorname {div} \,b)(x),} where || ⋅ || is the Euclidean norm and ( div b ) ( x ) = ∑ i = 1 d ∂ ∂ x i b i ( x ) . {\displaystyle (\operatorname {div} \,b)(x)=\sum _{i=1}^{d}{\frac {\partial }{\partial x_{i}}}b_{i}(x).} == Generalizations == Generalizations have been obtained by weakening the differentiability condition on the curve φ. Rather than taking the maximum distance between the stochastic process and the curve over a time interval, other conditions have been considered such as distances based on completely convex norms and Hölder, Besov and Sobolev type norms. == Applications == The Onsager–Machlup function can be used for purposes of reweighting and sampling trajectories, as well as for determining the most probable trajectory of a diffusion process. == See also == Lagrangian Functional integration == References == == Bibliography == == External links == Onsager–Machlup function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Onsager-Machlup_function&oldid=22857
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Wikipedia:Oper (mathematics)#0
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In mathematics, an oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their modern formulation is due to Drinfeld and Alexander Beilinson. == History == Opers were first defined, although not named, in a 1981 Russian paper by Drinfeld and Sokolov on Equations of Korteweg–de Vries type, and simple Lie algebras. They were later generalized by Drinfeld and Beilinson in 1993, later published as an e-print in 2005. == Formulation == === Abstract === Let G {\displaystyle G} be a connected reductive group over the complex plane C {\displaystyle \mathbb {C} } , with a distinguished Borel subgroup B = B G ⊂ G {\displaystyle B=B_{G}\subset G} . Set N = [ B , B ] {\displaystyle N=[B,B]} , so that H = B / N {\displaystyle H=B/N} is the Cartan group. Denote by n < b < g {\displaystyle {\mathfrak {n}}<{\mathfrak {b}}<{\mathfrak {g}}} and h = b / n {\displaystyle {\mathfrak {h}}={\mathfrak {b}}/{\mathfrak {n}}} the corresponding Lie algebras. There is an open B {\displaystyle B} -orbit O {\displaystyle \mathbf {O} } consisting of vectors stabilized by the radical N ⊂ B {\displaystyle N\subset B} such that all of their negative simple-root components are non-zero. Let X {\displaystyle X} be a smooth curve. A G-oper on X {\displaystyle X} is a triple ( F , ∇ , F B ) {\displaystyle ({\mathfrak {F}},\nabla ,{\mathfrak {F}}_{B})} where F {\displaystyle {\mathfrak {F}}} is a principal G {\displaystyle G} -bundle, ∇ {\displaystyle \nabla } is a connection on F {\displaystyle {\mathfrak {F}}} and F B {\displaystyle {\mathfrak {F}}_{B}} is a B {\displaystyle B} -reduction of F {\displaystyle {\mathfrak {F}}} , such that the one-form ∇ / F B {\displaystyle \nabla /{\mathfrak {F}}_{B}} takes values in O F B {\displaystyle \mathbf {O} _{{\mathfrak {F}}_{B}}} . == Example == Fix X = P 1 = C P 1 {\displaystyle X=\mathbb {P} ^{1}=\mathbb {CP} ^{1}} the Riemann sphere. Working at the level of the algebras, fix g = s l ( 2 , C ) {\displaystyle {\mathfrak {g}}={\mathfrak {sl}}(2,\mathbb {C} )} , which can be identified with the space of traceless 2 × 2 {\displaystyle 2\times 2} complex matrices. Since P 1 {\displaystyle \mathbb {P} ^{1}} has only one (complex) dimension, a one-form has only one component, and so an s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} -valued one form is locally described by a matrix of functions A ( z ) = ( a ( z ) b ( z ) c ( z ) − a ( z ) ) {\displaystyle A(z)={\begin{pmatrix}a(z)&b(z)\\c(z)&-a(z)\end{pmatrix}}} where a , b , c {\displaystyle a,b,c} are allowed to be meromorphic functions. Denote by Conn s l ( 2 , C ) ( P 1 ) {\displaystyle {\text{Conn}}_{{\mathfrak {sl}}(2,\mathbb {C} )}(\mathbb {P} ^{1})} the space of s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} valued meromorphic functions together with an action by g ( z ) {\displaystyle g(z)} , meromorphic functions valued in the associated Lie group G = S L ( 2 , C ) {\displaystyle G=SL(2,\mathbb {C} )} . The action is by a formal gauge transformation: g ( z ) ∗ A ( z ) = g ( z ) A ( z ) g ( z ) − 1 − g ′ ( z ) g ( z ) − 1 . {\displaystyle g(z)*A(z)=g(z)A(z)g(z)^{-1}-g'(z)g(z)^{-1}.} Then opers are defined in terms of a subspace of these connections. Denote by op s l ( 2 , C ) ( P 1 ) {\displaystyle {\text{op}}_{{\mathfrak {sl}}(2,\mathbb {C} )}(\mathbb {P} ^{1})} the space of connections with c ( z ) ≡ 1 {\displaystyle c(z)\equiv 1} . Denote by N {\displaystyle N} the subgroup of meromorphic functions valued in S L ( 2 , C ) {\displaystyle SL(2,\mathbb {C} )} of the form ( 1 f ( z ) 0 1 ) {\displaystyle {\begin{pmatrix}1&f(z)\\0&1\end{pmatrix}}} with f ( z ) {\displaystyle f(z)} meromorphic. Then for g ( z ) ∈ N , A ( z ) ∈ op s l ( 2 , C ) ( P 1 ) , {\displaystyle g(z)\in N,A(z)\in {\text{op}}_{{\mathfrak {sl}}(2,\mathbb {C} )}(\mathbb {P} ^{1}),} it holds that g ( z ) ∗ A ( z ) ∈ op s l ( 2 , C ) ( P 1 ) {\displaystyle g(z)*A(z)\in {\text{op}}_{{\mathfrak {sl}}(2,\mathbb {C} )}(\mathbb {P} ^{1})} . It therefore defines an action. The orbits of this action concretely characterize opers. However, generally this description only holds locally and not necessarily globally. == Gaudin model == Opers on P 1 {\displaystyle \mathbb {P} ^{1}} have been used by Boris Feigin, Edward Frenkel and Nicolai Reshetikhin to characterize the spectrum of the Gaudin model. Specifically, for a g {\displaystyle {\mathfrak {g}}} -Gaudin model, and defining L g {\displaystyle ^{L}{\mathfrak {g}}} as the Langlands dual algebra, there is a bijection between the spectrum of the Gaudin algebra generated by operators defined in the Gaudin model and an algebraic variety of L g {\displaystyle ^{L}{\mathfrak {g}}} opers. == References ==
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Wikipedia:Operad#0
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In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O {\displaystyle O} , one defines an algebra over O {\displaystyle O} to be a set together with concrete operations on this set which behave just like the abstract operations of O {\displaystyle O} . For instance, there is a Lie operad L {\displaystyle L} such that the algebras over L {\displaystyle L} are precisely the Lie algebras; in a sense L {\displaystyle L} abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. == History == Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J. Peter May in 1972. Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads: "The name operad and the formal definition appear first in the early 1970's in J. Peter May's "The Geometry of Iterated Loop Spaces", but a year or more earlier, Boardman and Vogt described the same concept under the name categories of operators in standard form, inspired by PROPs and PACTs of Adams and Mac Lane. In fact, there is an abundance of prehistory. Weibel [Wei] points out that the concept first arose a century ago in A.N. Whitehead's "A Treatise on Universal Algebra", published in 1898." The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was considerably renewed in the early 90s when, based on early insights of Maxim Kontsevich, Victor Ginzburg and Mikhail Kapranov discovered that some duality phenomena in rational homotopy theory could be explained using Koszul duality of operads. Operads have since found many applications, such as in deformation quantization of Poisson manifolds, the Deligne conjecture, or graph homology in the work of Maxim Kontsevich and Thomas Willwacher. == Intuition == Suppose X {\displaystyle X} is a set and for n ∈ N {\displaystyle n\in \mathbb {N} } we define P ( n ) := { f : X n → X } {\displaystyle P(n):=\{f\colon X^{n}\to X\}} , the set of all functions from the cartesian product of n {\displaystyle n} copies of X {\displaystyle X} to X {\displaystyle X} . We can compose these functions: given f ∈ P ( n ) {\displaystyle f\in P(n)} , f 1 ∈ P ( k 1 ) , … , f n ∈ P ( k n ) {\displaystyle f_{1}\in P(k_{1}),\ldots ,f_{n}\in P(k_{n})} , the function f ∘ ( f 1 , … , f n ) ∈ P ( k 1 + ⋯ + k n ) {\displaystyle f\circ (f_{1},\ldots ,f_{n})\in P(k_{1}+\cdots +k_{n})} is defined as follows: given k 1 + ⋯ + k n {\displaystyle k_{1}+\cdots +k_{n}} arguments from X {\displaystyle X} , we divide them into n {\displaystyle n} blocks, the first one having k 1 {\displaystyle k_{1}} arguments, the second one k 2 {\displaystyle k_{2}} arguments, etc., and then apply f 1 {\displaystyle f_{1}} to the first block, f 2 {\displaystyle f_{2}} to the second block, etc. We then apply f {\displaystyle f} to the list of n {\displaystyle n} values obtained from X {\displaystyle X} in such a way. We can also permute arguments, i.e. we have a right action ∗ {\displaystyle *} of the symmetric group S n {\displaystyle S_{n}} on P ( n ) {\displaystyle P(n)} , defined by ( f ∗ s ) ( x 1 , … , x n ) = f ( x s − 1 ( 1 ) , … , x s − 1 ( n ) ) {\displaystyle (f*s)(x_{1},\ldots ,x_{n})=f(x_{s^{-1}(1)},\ldots ,x_{s^{-1}(n)})} for f ∈ P ( n ) {\displaystyle f\in P(n)} , s ∈ S n {\displaystyle s\in S_{n}} and x 1 , … , x n ∈ X {\displaystyle x_{1},\ldots ,x_{n}\in X} . The definition of a symmetric operad given below captures the essential properties of these two operations ∘ {\displaystyle \circ } and ∗ {\displaystyle *} . == Definition == === Non-symmetric operad === A non-symmetric operad (sometimes called an operad without permutations, or a non- Σ {\displaystyle \Sigma } or plain operad) consists of the following: a sequence ( P ( n ) ) n ∈ N {\displaystyle (P(n))_{n\in \mathbb {N} }} of sets, whose elements are called n {\displaystyle n} -ary operations, an element 1 {\displaystyle 1} in P ( 1 ) {\displaystyle P(1)} called the identity, for all positive integers n {\displaystyle n} , k 1 , … , k n {\textstyle k_{1},\ldots ,k_{n}} , a composition function ∘ : P ( n ) × P ( k 1 ) × ⋯ × P ( k n ) → P ( k 1 + ⋯ + k n ) ( θ , θ 1 , … , θ n ) ↦ θ ∘ ( θ 1 , … , θ n ) , {\displaystyle {\begin{aligned}\circ :P(n)\times P(k_{1})\times \cdots \times P(k_{n})&\to P(k_{1}+\cdots +k_{n})\\(\theta ,\theta _{1},\ldots ,\theta _{n})&\mapsto \theta \circ (\theta _{1},\ldots ,\theta _{n}),\end{aligned}}} satisfying the following coherence axioms: identity: θ ∘ ( 1 , … , 1 ) = θ = 1 ∘ θ {\displaystyle \theta \circ (1,\ldots ,1)=\theta =1\circ \theta } associativity: θ ∘ ( θ 1 ∘ ( θ 1 , 1 , … , θ 1 , k 1 ) , … , θ n ∘ ( θ n , 1 , … , θ n , k n ) ) = ( θ ∘ ( θ 1 , … , θ n ) ) ∘ ( θ 1 , 1 , … , θ 1 , k 1 , … , θ n , 1 , … , θ n , k n ) {\displaystyle {\begin{aligned}&\theta \circ {\Big (}\theta _{1}\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}}),\ldots ,\theta _{n}\circ (\theta _{n,1},\ldots ,\theta _{n,k_{n}}){\Big )}\\={}&{\Big (}\theta \circ (\theta _{1},\ldots ,\theta _{n}){\Big )}\circ (\theta _{1,1},\ldots ,\theta _{1,k_{1}},\ldots ,\theta _{n,1},\ldots ,\theta _{n,k_{n}})\end{aligned}}} === Symmetric operad === A symmetric operad (often just called operad) is a non-symmetric operad P {\displaystyle P} as above, together with a right action of the symmetric group S n {\displaystyle S_{n}} on P ( n ) {\displaystyle P(n)} for n ∈ N {\displaystyle n\in \mathbb {N} } , denoted by ∗ {\displaystyle *} and satisfying equivariance: given a permutation t ∈ S n {\displaystyle t\in S_{n}} , ( θ ∗ t ) ∘ ( θ 1 , … , θ n ) = ( θ ∘ ( θ t − 1 ( 1 ) , … , θ t − 1 ( n ) ) ) ∗ t ′ {\displaystyle (\theta *t)\circ (\theta _{1},\ldots ,\theta _{n})=(\theta \circ (\theta _{t^{-1}(1)},\ldots ,\theta _{t^{-1}(n)}))*t'} (where t ′ {\displaystyle t'} on the right hand side refers to the element of S k 1 + ⋯ + k n {\displaystyle S_{k_{1}+\dots +k_{n}}} that acts on the set { 1 , 2 , … , k 1 + ⋯ + k n } {\displaystyle \{1,2,\dots ,k_{1}+\dots +k_{n}\}} by breaking it into n {\displaystyle n} blocks, the first of size k 1 {\displaystyle k_{1}} , the second of size k 2 {\displaystyle k_{2}} , through the n {\displaystyle n} th block of size k n {\displaystyle k_{n}} , and then permutes these n {\displaystyle n} blocks by t {\displaystyle t} , keeping each block intact) and given n {\displaystyle n} permutations s i ∈ S k i {\displaystyle s_{i}\in S_{k_{i}}} , θ ∘ ( θ 1 ∗ s 1 , … , θ n ∗ s n ) = ( θ ∘ ( θ 1 , … , θ n ) ) ∗ ( s 1 , … , s n ) {\displaystyle \theta \circ (\theta _{1}*s_{1},\ldots ,\theta _{n}*s_{n})=(\theta \circ (\theta _{1},\ldots ,\theta _{n}))*(s_{1},\ldots ,s_{n})} (where ( s 1 , … , s n ) {\displaystyle (s_{1},\ldots ,s_{n})} denotes the element of S k 1 + ⋯ + k n {\displaystyle S_{k_{1}+\dots +k_{n}}} that permutes the first of these blocks by s 1 {\displaystyle s_{1}} , the second by s 2 {\displaystyle s_{2}} , etc., and keeps their overall order intact). The permutation actions in this definition are vital to most applications, including the original application to loop spaces. === Morphisms === A morphism of operads f : P → Q {\displaystyle f:P\to Q} consists of a sequence ( f n : P ( n ) → Q ( n ) ) n ∈ N {\displaystyle (f_{n}:P(n)\to Q(n))_{n\in \mathbb {N} }} that: preserves the identity: f ( 1 ) = 1 {\displaystyle f(1)=1} preserves composition: for every n-ary operation θ {\displaystyle \theta } and operations θ 1 , … , θ n {\displaystyle \theta _{1},\ldots ,\theta _{n}} , f ( θ ∘ ( θ 1 , … , θ n ) ) = f ( θ ) ∘ ( f ( θ 1 ) , … , f ( θ n ) ) {\displaystyle f(\theta \circ (\theta _{1},\ldots ,\theta _{n}))=f(\theta )\circ (f(\theta _{1}),\ldots ,f(\theta _{n}))} preserves the permutation actions: f ( x ∗ s ) = f ( x ) ∗ s {\displaystyle f(x*s)=f(x)*s} . Operads therefore form a category denoted by O p e r {\displaystyle {\mathsf {Oper}}} . === In other categories === So far operads have only been considered in the category of sets. More generally, it is possible to define operads in any symmetric monoidal category C . In that case, each P ( n ) {\displaystyle P(n)} is an object of C, the composition ∘ {\displaystyle \circ } is a morphism P ( n ) ⊗ P ( k 1 ) ⊗ ⋯ ⊗ P ( k n ) → P ( k 1 + ⋯ + k n ) {\displaystyle P(n)\otimes P(k_{1})\otimes \cdots \otimes P(k_{n})\to P(k_{1}+\cdots +k_{n})} in C (where ⊗ {\displaystyle \otimes } denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given by isomorphisms in C. A common example is the category of topological spaces and continuous maps, with the monoidal product given by the cartesian product. In this case, an operad is given by a sequence of spaces (instead of sets) { P ( n ) } n ≥ 0 {\displaystyle \{P(n)\}_{n\geq 0}} . The structure maps of the operad (the composition and the actions of the symmetric groups) are then assumed to be continuous. The result is called a topological operad. Similarly, in the definition of a morphism of operads, it would be necessary to assume that the maps involved are continuous. Other common settings to define operads include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. === Algebraist definition === Given a commutative ring R we consider the category R - M o d {\displaystyle R{\text{-}}{\mathsf {Mod}}} of modules over R. An operad over R can be defined as a monoid object ( T , γ , η ) {\displaystyle (T,\gamma ,\eta )} in the monoidal category of endofunctors on R - M o d {\displaystyle R{\text{-}}{\mathsf {Mod}}} (it is a monad) satisfying some finiteness condition. For example, a monoid object in the category of "polynomial endofunctors" on R - M o d {\displaystyle R{\text{-}}{\mathsf {Mod}}} is an operad. Similarly, a symmetric operad can be defined as a monoid object in the category of S {\displaystyle \mathbb {S} } -objects, where S {\displaystyle \mathbb {S} } means a symmetric group. A monoid object in the category of combinatorial species is an operad in finite sets. An operad in the above sense is sometimes thought of as a generalized ring. For example, Nikolai Durov defines his generalized rings as monoid objects in the monoidal category of endofunctors on Set {\displaystyle {\textbf {Set}}} that commute with filtered colimits. This is a generalization of a ring since each ordinary ring R defines a monad Σ R : Set → Set {\displaystyle \Sigma _{R}:{\textbf {Set}}\to {\textbf {Set}}} that sends a set X to the underlying set of the free R-module R ( X ) {\displaystyle R^{(X)}} generated by X. == Understanding the axioms == === Associativity axiom === "Associativity" means that composition of operations is associative (the function ∘ {\displaystyle \circ } is associative), analogous to the axiom in category theory that f ∘ ( g ∘ h ) = ( f ∘ g ) ∘ h {\displaystyle f\circ (g\circ h)=(f\circ g)\circ h} ; it does not mean that the operations themselves are associative as operations. Compare with the associative operad, below. Associativity in operad theory means that expressions can be written involving operations without ambiguity from the omitted compositions, just as associativity for operations allows products to be written without ambiguity from the omitted parentheses. For instance, if θ {\displaystyle \theta } is a binary operation, which is written as θ ( a , b ) {\displaystyle \theta (a,b)} or ( a b ) {\displaystyle (ab)} . So that θ {\displaystyle \theta } may or may not be associative. Then what is commonly written ( ( a b ) c ) {\displaystyle ((ab)c)} is unambiguously written operadically as θ ∘ ( θ , 1 ) {\displaystyle \theta \circ (\theta ,1)} . This sends ( a , b , c ) {\displaystyle (a,b,c)} to ( a b , c ) {\displaystyle (ab,c)} (apply θ {\displaystyle \theta } on the first two, and the identity on the third), and then the θ {\displaystyle \theta } on the left "multiplies" a b {\displaystyle ab} by c {\displaystyle c} . This is clearer when depicted as a tree: which yields a 3-ary operation: However, the expression ( ( ( a b ) c ) d ) {\displaystyle (((ab)c)d)} is a priori ambiguous: it could mean θ ∘ ( ( θ , 1 ) ∘ ( ( θ , 1 ) , 1 ) ) {\displaystyle \theta \circ ((\theta ,1)\circ ((\theta ,1),1))} , if the inner compositions are performed first, or it could mean ( θ ∘ ( θ , 1 ) ) ∘ ( ( θ , 1 ) , 1 ) {\displaystyle (\theta \circ (\theta ,1))\circ ((\theta ,1),1)} , if the outer compositions are performed first (operations are read from right to left). Writing x = θ , y = ( θ , 1 ) , z = ( ( θ , 1 ) , 1 ) {\displaystyle x=\theta ,y=(\theta ,1),z=((\theta ,1),1)} , this is x ∘ ( y ∘ z ) {\displaystyle x\circ (y\circ z)} versus ( x ∘ y ) ∘ z {\displaystyle (x\circ y)\circ z} . That is, the tree is missing "vertical parentheses": If the top two rows of operations are composed first (puts an upward parenthesis at the ( a b ) c d {\displaystyle (ab)c\ \ d} line; does the inner composition first), the following results: which then evaluates unambiguously to yield a 4-ary operation. As an annotated expression: θ ( a b ) c ⋅ d ∘ ( ( θ a b ⋅ c , 1 d ) ∘ ( ( θ a ⋅ b , 1 c ) , 1 d ) ) {\displaystyle \theta _{(ab)c\cdot d}\circ ((\theta _{ab\cdot c},1_{d})\circ ((\theta _{a\cdot b},1_{c}),1_{d}))} If the bottom two rows of operations are composed first (puts a downward parenthesis at the a b c d {\displaystyle ab\quad c\ \ d} line; does the outer composition first), following results: which then evaluates unambiguously to yield a 4-ary operation: The operad axiom of associativity is that these yield the same result, and thus that the expression ( ( ( a b ) c ) d ) {\displaystyle (((ab)c)d)} is unambiguous. === Identity axiom === The identity axiom (for a binary operation) can be visualized in a tree as: meaning that the three operations obtained are equal: pre- or post- composing with the identity makes no difference. As for categories, 1 ∘ 1 = 1 {\displaystyle 1\circ 1=1} is a corollary of the identity axiom. == Examples == === Endomorphism operad in sets and operad algebras === The most basic operads are the ones given in the section on "Intuition", above. For any set X {\displaystyle X} , we obtain the endomorphism operad E n d X {\displaystyle {\mathcal {End}}_{X}} consisting of all functions X n → X {\displaystyle X^{n}\to X} . These operads are important because they serve to define operad algebras. If O {\displaystyle {\mathcal {O}}} is an operad, an operad algebra over O {\displaystyle {\mathcal {O}}} is given by a set X {\displaystyle X} and an operad morphism O → E n d X {\displaystyle {\mathcal {O}}\to {\mathcal {End}}_{X}} . Intuitively, such a morphism turns each "abstract" operation of O ( n ) {\displaystyle {\mathcal {O}}(n)} into a "concrete" n {\displaystyle n} -ary operation on the set X {\displaystyle X} . An operad algebra over O {\displaystyle {\mathcal {O}}} thus consists of a set X {\displaystyle X} together with concrete operations on X {\displaystyle X} that follow the rules abstractely specified by the operad O {\displaystyle {\mathcal {O}}} . === Endomorphism operad in vector spaces and operad algebras === If k is a field, we can consider the category of finite-dimensional vector spaces over k; this becomes a monoidal category using the ordinary tensor product over k. We can then define endomorphism operads in this category, as follows. Let V be a finite-dimensional vector space The endomorphism operad E n d V = { E n d V ( n ) } {\displaystyle {\mathcal {End}}_{V}=\{{\mathcal {End}}_{V}(n)\}} of V consists of E n d V ( n ) {\displaystyle {\mathcal {End}}_{V}(n)} = the space of linear maps V ⊗ n → V {\displaystyle V^{\otimes n}\to V} , (composition) given f ∈ E n d V ( n ) {\displaystyle f\in {\mathcal {End}}_{V}(n)} , g 1 ∈ E n d V ( k 1 ) {\displaystyle g_{1}\in {\mathcal {End}}_{V}(k_{1})} , ..., g n ∈ E n d V ( k n ) {\displaystyle g_{n}\in {\mathcal {End}}_{V}(k_{n})} , their composition is given by the map V ⊗ k 1 ⊗ ⋯ ⊗ V ⊗ k n ⟶ g 1 ⊗ ⋯ ⊗ g n V ⊗ n → f V {\displaystyle V^{\otimes k_{1}}\otimes \cdots \otimes V^{\otimes k_{n}}\ {\overset {g_{1}\otimes \cdots \otimes g_{n}}{\longrightarrow }}\ V^{\otimes n}\ {\overset {f}{\to }}\ V} , (identity) The identity element in E n d V ( 1 ) {\displaystyle {\mathcal {End}}_{V}(1)} is the identity map id V {\displaystyle \operatorname {id} _{V}} , (symmetric group action) S n {\displaystyle S_{n}} operates on E n d V ( n ) {\displaystyle {\mathcal {End}}_{V}(n)} by permuting the components of the tensors in V ⊗ n {\displaystyle V^{\otimes n}} . If O {\displaystyle {\mathcal {O}}} is an operad, a k-linear operad algebra over O {\displaystyle {\mathcal {O}}} is given by a finite-dimensional vector space V over k and an operad morphism O → E n d V {\displaystyle {\mathcal {O}}\to {\mathcal {End}}_{V}} ; this amounts to specifying concrete multilinear operations on V that behave like the operations of O {\displaystyle {\mathcal {O}}} . (Notice the analogy between operads&operad algebras and rings&modules: a module over a ring R is given by an abelian group M together with a ring homomorphism R → End ( M ) {\displaystyle R\to \operatorname {End} (M)} .) Depending on applications, variations of the above are possible: for example, in algebraic topology, instead of vector spaces and tensor products between them, one uses (reasonable) topological spaces and cartesian products between them. === "Little something" operads === The little 2-disks operad is a topological operad where P ( n ) {\displaystyle P(n)} consists of ordered lists of n disjoint disks inside the unit disk of R 2 {\displaystyle \mathbb {R} ^{2}} centered at the origin. The symmetric group acts on such configurations by permuting the list of little disks. The operadic composition for little disks is illustrated in the accompanying figure to the right, where an element θ ∈ P ( 3 ) {\displaystyle \theta \in P(3)} is composed with an element ( θ 1 , θ 2 , θ 3 ) ∈ P ( 2 ) × P ( 3 ) × P ( 4 ) {\displaystyle (\theta _{1},\theta _{2},\theta _{3})\in P(2)\times P(3)\times P(4)} to yield the element θ ∘ ( θ 1 , θ 2 , θ 3 ) ∈ P ( 9 ) {\displaystyle \theta \circ (\theta _{1},\theta _{2},\theta _{3})\in P(9)} obtained by shrinking the configuration of θ i {\displaystyle \theta _{i}} and inserting it into the i-th disk of θ {\displaystyle \theta } , for i = 1 , 2 , 3 {\displaystyle i=1,2,3} . Analogously, one can define the little n-disks operad by considering configurations of disjoint n-balls inside the unit ball of R n {\displaystyle \mathbb {R} ^{n}} . Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube. Later it was generalized by May to the little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies". === Rooted trees === In graph theory, rooted trees form a natural operad. Here, P ( n ) {\displaystyle P(n)} is the set of all rooted trees with n leaves, where the leaves are numbered from 1 to n. The group S n {\displaystyle S_{n}} operates on this set by permuting the leaf labels. Operadic composition T ∘ ( S 1 , … , S n ) {\displaystyle T\circ (S_{1},\ldots ,S_{n})} is given by replacing the i-th leaf of T {\displaystyle T} by the root of the i-th tree S i {\displaystyle S_{i}} , for i = 1 , … , n {\displaystyle i=1,\ldots ,n} , thus attaching the n trees to T {\displaystyle T} and forming a larger tree, whose root is taken to be the same as the root of T {\displaystyle T} and whose leaves are numbered in order. === Swiss-cheese operad === The Swiss-cheese operad is a two-colored topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-semidisk and n-dimensional semidisks, centered at the base of the unit semidisk and sitting inside of it. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk. The Swiss-cheese operad was defined by Alexander A. Voronov. It was used by Maxim Kontsevich to formulate a Swiss-cheese version of Deligne's conjecture on Hochschild cohomology. Kontsevich's conjecture was proven partly by Po Hu, Igor Kriz, and Alexander A. Voronov and then fully by Justin Thomas. === Associative operad === Another class of examples of operads are those capturing the structures of algebraic structures, such as associative algebras, commutative algebras and Lie algebras. Each of these can be exhibited as a finitely presented operad, in each of these three generated by binary operations. For example, the associative operad is a symmetric operad generated by a binary operation ψ {\displaystyle \psi } , subject only to the condition that ψ ∘ ( ψ , 1 ) = ψ ∘ ( 1 , ψ ) . {\displaystyle \psi \circ (\psi ,1)=\psi \circ (1,\psi ).} This condition corresponds to associativity of the binary operation ψ {\displaystyle \psi } ; writing ψ ( a , b ) {\displaystyle \psi (a,b)} multiplicatively, the above condition is ( a b ) c = a ( b c ) {\displaystyle (ab)c=a(bc)} . This associativity of the operation should not be confused with associativity of composition which holds in any operad; see the axiom of associativity, above. In the associative operad, each P ( n ) {\displaystyle P(n)} is given by the symmetric group S n {\displaystyle S_{n}} , on which S n {\displaystyle S_{n}} acts by right multiplication. The composite σ ∘ ( τ 1 , … , τ n ) {\displaystyle \sigma \circ (\tau _{1},\dots ,\tau _{n})} permutes its inputs in blocks according to σ {\displaystyle \sigma } , and within blocks according to the appropriate τ i {\displaystyle \tau _{i}} . The algebras over the associative operad are precisely the semigroups: sets together with a single binary associative operation. The k-linear algebras over the associative operad are precisely the associative k-algebras. === Terminal symmetric operad === The terminal symmetric operad is the operad which has a single n-ary operation for each n, with each S n {\displaystyle S_{n}} acting trivially. The algebras over this operad are the commutative semigroups; the k-linear algebras are the commutative associative k-algebras. === Operads from the braid groups === Similarly, there is a non- Σ {\displaystyle \Sigma } operad for which each P ( n ) {\displaystyle P(n)} is given by the Artin braid group B n {\displaystyle B_{n}} . Moreover, this non- Σ {\displaystyle \Sigma } operad has the structure of a braided operad, which generalizes the notion of an operad from symmetric to braid groups. === Linear algebra === In linear algebra, real vector spaces can be considered to be algebras over the operad R ∞ {\displaystyle \mathbb {R} ^{\infty }} of all linear combinations . This operad is defined by R ∞ ( n ) = R n {\displaystyle \mathbb {R} ^{\infty }(n)=\mathbb {R} ^{n}} for n ∈ N {\displaystyle n\in \mathbb {N} } , with the obvious action of S n {\displaystyle S_{n}} permuting components, and composition x → ∘ ( y 1 → , … , y n → ) {\displaystyle {\vec {x}}\circ ({\vec {y_{1}}},\ldots ,{\vec {y_{n}}})} given by the concatentation of the vectors x ( 1 ) y 1 → , … , x ( n ) y n → {\displaystyle x^{(1)}{\vec {y_{1}}},\ldots ,x^{(n)}{\vec {y_{n}}}} , where x → = ( x ( 1 ) , … , x ( n ) ) ∈ R n {\displaystyle {\vec {x}}=(x^{(1)},\ldots ,x^{(n)})\in \mathbb {R} ^{n}} . The vector x → = ( 2 , 3 , − 5 , 0 , … ) {\displaystyle {\vec {x}}=(2,3,-5,0,\dots )} for instance represents the operation of forming a linear combination with coefficients 2,3,-5,0,... This point of view formalizes the notion that linear combinations are the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement that all possible algebraic operations in a vector space are linear combinations. The basic operations of vector addition and scalar multiplication are a generating set for the operad of all linear combinations, while the linear combinations operad canonically encodes all possible operations on a vector space. Similarly, affine combinations, conical combinations, and convex combinations can be considered to correspond to the sub-operads where the terms of the vector x → {\displaystyle {\vec {x}}} sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by R n {\displaystyle \mathbb {R} ^{n}} being or the standard simplex being model spaces, and such observations as that every bounded convex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories. === Commutative-ring operad and Lie operad === The commutative-ring operad is an operad whose algebras are the commutative rings. It is defined by P ( n ) = Z [ x 1 , … , x n ] {\displaystyle P(n)=\mathbb {Z} [x_{1},\ldots ,x_{n}]} , with the obvious action of S n {\displaystyle S_{n}} and operadic composition given by substituting polynomials (with renumbered variables) for variables. A similar operad can be defined whose algebras are the associative, commutative algebras over some fixed base field. The Koszul-dual of this operad is the Lie operad (whose algebras are the Lie algebras), and vice versa. == Free Operads == Typical algebraic constructions (e.g., free algebra construction) can be extended to operads. Let S e t S n {\displaystyle \mathbf {Set} ^{S_{n}}} denote the category whose objects are sets on which the group S n {\displaystyle S_{n}} acts. Then there is a forgetful functor O p e r → ∏ n ∈ N S e t S n {\displaystyle {\mathsf {Oper}}\to \prod _{n\in \mathbb {N} }\mathbf {Set} ^{S_{n}}} , which simply forgets the operadic composition. It is possible to construct a left adjoint Γ : ∏ n ∈ N S e t S n → O p e r {\displaystyle \Gamma :\prod _{n\in \mathbb {N} }\mathbf {Set} ^{S_{n}}\to {\mathsf {Oper}}} to this forgetful functor (this is the usual definition of free functor). Given a collection of operations E, Γ ( E ) {\displaystyle \Gamma (E)} is the free operad on E. Like a group or a ring, the free construction allows to express an operad in terms of generators and relations. By a free representation of an operad O {\displaystyle {\mathcal {O}}} , we mean writing O {\displaystyle {\mathcal {O}}} as a quotient of a free operad F = Γ ( E ) {\displaystyle {\mathcal {F}}=\Gamma (E)} where E describes generators of O {\displaystyle {\mathcal {O}}} and the kernel of the epimorphism F → O {\displaystyle {\mathcal {F}}\to {\mathcal {O}}} describes the relations. A (symmetric) operad O = { O ( n ) } {\displaystyle {\mathcal {O}}=\{{\mathcal {O}}(n)\}} is called quadratic if it has a free presentation such that E = O ( 2 ) {\displaystyle E={\mathcal {O}}(2)} is the generator and the relation is contained in Γ ( E ) ( 3 ) {\displaystyle \Gamma (E)(3)} . == Clones == Clones are the special case of operads that are also closed under identifying arguments together ("reusing" some data). Clones can be equivalently defined as operads that are also a minion (or clonoid). == Operads in homotopy theory == In Stasheff (2004), Stasheff writes: Operads are particularly important and useful in categories with a good notion of "homotopy", where they play a key role in organizing hierarchies of higher homotopies. == See also == PRO (category theory) Algebra over an operad Higher-order operad E∞-operad Pseudoalgebra Multicategory == Notes == === Citations === == References == Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. arXiv:math/0305049. Bibcode:2004hohc.book.....L. ISBN 978-0-521-53215-0. Martin Markl, Steve Shnider, Jim Stasheff (2002). Operads in Algebra, Topology and Physics. American Mathematical Society. ISBN 978-0-8218-4362-8.{{cite book}}: CS1 maint: multiple names: authors list (link) Markl, Martin (June 2006). "Operads and PROPs". arXiv:math/0601129. Stasheff, Jim (June–July 2004). "What Is...an Operad?" (PDF). Notices of the American Mathematical Society. 51 (6): 630–631. Retrieved 17 January 2008. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic Operads (PDF), Grundlehren der Mathematischen Wissenschaften, vol. 346, Berlin, New York: Springer-Verlag, ISBN 978-3-642-30361-6 Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Bai, Chengming; Guo, Li; Loday, Jean-Louis (eds.), Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9, pp. 217–298, arXiv:1101.0267, Bibcode:2011arXiv1101.0267Z, ISBN 9789814365116 Fresse, Benoit (17 May 2017), Homotopy of Operads and Grothendieck-Teichmüller Groups, Mathematical Surveys and Monographs, American Mathematical Society, ISBN 978-1-4704-3480-9, MR 3643404, Zbl 1373.55014 Miguel A. Mendéz (2015). Set Operads in Combinatorics and Computer Science. SpringerBriefs in Mathematics. ISBN 978-3-319-11712-6. Samuele Giraudo (2018). Nonsymmetric Operads in Combinatorics. Springer International Publishing. ISBN 978-3-030-02073-6. == External links == operad at the nLab https://golem.ph.utexas.edu/category/2011/05/an_operadic_introduction_to_en.html
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Wikipedia:Operad algebra#0
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In algebra, an operad algebra is an "algebra" over an operad. It is a generalization of an associative algebra over a commutative ring R, with an operad replacing R. == Definitions == Given an operad O (say, a symmetric sequence in a symmetric monoidal ∞-category C), an algebra over an operad, or O-algebra for short, is, roughly, a left module over O with multiplications parametrized by O. If O is a topological operad, then one can say an algebra over an operad is an O-monoid object in C. If C is symmetric monoidal, this recovers the usual definition. Let C be symmetric monoidal ∞-category with monoidal structure distributive over colimits. If f : O → O ′ {\displaystyle f:O\to O'} is a map of operads and, moreover, if f is a homotopy equivalence, then the ∞-category of algebras over O in C is equivalent to the ∞-category of algebras over O' in C. == See also == En-ring Homotopy Lie algebra == Notes == == References == Francis, John. "Derived Algebraic Geometry Over E n {\displaystyle {\mathcal {E}}_{n}} -Rings" (PDF). Hinich, Vladimir (1997-02-11). "Homological algebra of homotopy algebras". arXiv:q-alg/9702015. == External links == "operad", ncatlab.org http://ncatlab.org/nlab/show/algebra+over+an+operad
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Wikipedia:Operand#0
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In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Unknown operands in equalities of expressions can be found by equation solving. == Example == The following arithmetic expression shows an example of operators and operands: 3 + 6 = 9 {\displaystyle 3+6=9} In the above example, '+' is the symbol for the operation called addition. The operand '3' is one of the inputs (quantities) followed by the addition operator, and the operand '6' is the other input necessary for the operation. The result of the operation is 9. (The number '9' is also called the sum of the augend 3 and the addend 6.) An operand, then, is also referred to as "one of the inputs (quantities) for an operation". == Notation == === Expressions as operands === Operands may be nested, and may consist of expressions also made up of operators with operands. ( 3 + 5 ) × 2 {\displaystyle (3+5)\times 2} In the above expression '(3 + 5)' is the first operand for the multiplication operator and '2' the second. The operand '(3 + 5)' is an expression in itself, which contains an addition operator, with the operands '3' and '5'. === × === Rules of precedence affect which values form operands for which operators: 3 + 5 × 2 {\displaystyle 3+5\times 2} In the above expression, the multiplication operator has the higher precedence than the addition operator, so the multiplication operator has operands of '5' and '2'. The addition operator has operands of '3' and '5 × 2'. === Positioning of operands === Depending on the mathematical notation being used the position of an operator in relation to its operand(s) may vary. In everyday usage infix notation is the most common, however other notations also exist, such as the prefix and postfix notations. These alternate notations are most common within computer science. Below is a comparison of three different notations — all represent an addition of the numbers '1' and '2' 1 + 2 {\displaystyle 1+2} (infix notation) + 1 2 {\displaystyle +\;1\;2} (prefix notation) 1 2 + {\displaystyle 1\;2\;+} (postfix notation) === Infix and the order of operation === In a mathematical expression, the order of operation is carried out from left to right. Start with the leftmost value and seek the first operation to be carried out in accordance with the order specified above (i.e., start with parentheses and end with the addition/subtraction group). For example, in the expression 4 × 2 2 − ( 2 + 2 2 ) {\displaystyle 4\times 2^{2}-(2+2^{2})} , the first operation to be acted upon is any and all expressions found inside a parenthesis. So beginning at the left and moving to the right, find the first (and in this case, the only) parenthesis, that is, (2 + 22). Within the parenthesis itself is found the expression 22. The reader is required to find the value of 22 before going any further. The value of 22 is 4. Having found this value, the remaining expression looks like this: 4 × 2 2 − ( 2 + 4 ) {\displaystyle 4\times 2^{2}-(2+4)} The next step is to calculate the value of expression inside the parenthesis itself, that is, (2 + 4) = 6. Our expression now looks like this: 4 × 2 2 − 6 {\displaystyle 4\times 2^{2}-6} Having calculated the parenthetical part of the expression, we start over again beginning with the left most value and move right. The next order of operation (according to the rules) is exponents. Start at the left most value, that is, 4, and scan your eyes to the right and search for the first exponent you come across. The first (and only) expression we come across that is expressed with an exponent is 22. We find the value of 22, which is 4. What we have left is the expression 4 × 4 − 6 {\displaystyle 4\times 4-6} . The next order of operation is multiplication. 4 × 4 is 16. Now our expression looks like this: 16 − 6 {\displaystyle 16-6} The next order of operation according to the rules is division. However, there is no division operator sign (÷) in the expression, 16 − 6. So we move on to the next order of operation, i.e., addition and subtraction, which have the same precedence and are done left to right. 16 − 6 = 10 {\displaystyle 16-6=10} . So the correct value for our original expression, 4 × 22 − (2 + 22), is 10. It is important to carry out the order of operation in accordance with rules set by convention. If the reader evaluates an expression but does not follow the correct order of operation, the reader will come forth with a different value. The different value will be the incorrect value because the order of operation was not followed. The reader will arrive at the correct value for the expression if and only if each operation is carried out in the proper order. === Arity === The number of operands of an operator is called its arity. Based on arity, operators are chiefly classified as nullary (no operands), unary (1 operand), binary (2 operands), ternary (3 operands). Higher arities are less frequently denominated through a specific terms, all the more when function composition or currying can be used to avoid them. Other terms include: == Computer science == In computer programming languages, the definitions of operator and operand are almost the same as in mathematics. In computing, an operand is the part of a computer instruction which specifies what data is to be manipulated or operated on, while at the same time representing the data itself. A computer instruction describes an operation such as add or multiply X, while the operand (or operands, as there can be more than one) specify on which X to operate as well as the value of X. Additionally, in assembly language, an operand is a value (an argument) on which the instruction, named by mnemonic, operates. The operand may be a processor register, a memory address, a literal constant, or a label. A simple example (in the x86 architecture) is where the value in register operand AX is to be moved (MOV) into register BX. Depending on the instruction, there may be zero, one, two, or more operands. == See also == Instruction set Opcode == References ==
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Wikipedia:Operator (mathematics)#0
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In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from R n {\displaystyle \mathbb {R} ^{n}} to R n {\displaystyle \mathbb {R} ^{n}} . Such operators often preserve properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators. Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of "operator" in computer programming (see Operator (computer programming)). == Linear operators == The most common kind of operators encountered are linear operators. Let U and V be vector spaces over some field K. A mapping A : U → V {\displaystyle \operatorname {A} :U\to V} is linear if A ( α x + β y ) = α A x + β A y {\displaystyle \operatorname {A} \left(\alpha \mathbf {x} +\beta \mathbf {y} \right)=\alpha \operatorname {A} \mathbf {x} +\beta \operatorname {A} \mathbf {y} \ } for all x and y in U, and for all α, β in K. This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphisms between vector spaces. In the finite-dimensional case linear operators can be represented by matrices in the following way. Let K be a field, and U {\displaystyle U} and V be finite-dimensional vector spaces over K. Let us select a basis u 1 , … , u n {\displaystyle \ \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}} in U and v 1 , … , v m {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}} in V. Then let x = x i u i {\displaystyle \mathbf {x} =x^{i}\mathbf {u} _{i}} be an arbitrary vector in U {\displaystyle U} (assuming Einstein convention), and A : U → V {\displaystyle \operatorname {A} :U\to V} be a linear operator. Then A x = x i A u i = x i ( A u i ) j v j . {\displaystyle \ \operatorname {A} \mathbf {x} =x^{i}\operatorname {A} \mathbf {u} _{i}=x^{i}\left(\operatorname {A} \mathbf {u} _{i}\right)^{j}\mathbf {v} _{j}~.} Then a i j ≡ ( A u i ) j {\displaystyle a_{i}^{j}\equiv \left(\operatorname {A} \mathbf {u} _{i}\right)^{j}} , with all a i j ∈ K {\displaystyle a_{i}^{j}\in K} , is the matrix form of the operator A {\displaystyle \operatorname {A} } in the fixed basis { u i } i = 1 n {\displaystyle \{\mathbf {u} _{i}\}_{i=1}^{n}} . The tensor a i j {\displaystyle a_{i}^{j}} does not depend on the choice of x {\displaystyle x} , and A x = y {\displaystyle \operatorname {A} \mathbf {x} =\mathbf {y} } if a i j x i = y j {\displaystyle a_{i}^{j}x^{i}=y^{j}} . Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from U {\displaystyle U} to V {\displaystyle V} . The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace. Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as functional analysis (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces). The space of sequences of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces. == Bounded operators == Let U and V be two vector spaces over the same ordered field (for example; R {\displaystyle \mathbb {R} } ), and they are equipped with norms. Then a linear operator from U to V is called bounded if there exists c > 0 such that ‖ A x ‖ V ≤ c ‖ x ‖ U {\displaystyle \|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}} for every x in U. Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V: ‖ A ‖ = inf { c : ‖ A x ‖ V ≤ c ‖ x ‖ U } . {\displaystyle \|\operatorname {A} \|=\inf\{\ c:\|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}\}.} In case of operators from U to itself it can be shown that ‖ A B ‖ ≤ ‖ A ‖ ⋅ ‖ B ‖ {\textstyle \|\operatorname {A} \operatorname {B} \|\leq \|\operatorname {A} \|\cdot \|\operatorname {B} \|} . Any unital normed algebra with this property is called a Banach algebra. It is possible to generalize spectral theory to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics. == Examples == === Analysis (calculus) === From the point of view of functional analysis, calculus is the study of two linear operators: the differential operator d d t {\displaystyle {\frac {\ \mathrm {d} \ }{\mathrm {d} t}}} , and the Volterra operator ∫ 0 t {\displaystyle \int _{0}^{t}} . === Fundamental analysis operators on scalar and vector fields === Three operators are key to vector calculus: Grad (gradient), (with operator symbol ∇ {\displaystyle \nabla } ) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change. Div (divergence), (with operator symbol ∇ ⋅ {\displaystyle {\nabla \cdot }} ) is a vector operator that measures a vector field's divergence from or convergence towards a given point. Curl, (with operator symbol ∇ × {\displaystyle \nabla \!\times } ) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point. As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with tensor calculus as well as vector calculus. === Geometry === In geometry, additional structures on vector spaces are sometimes studied. Operators that map such vector spaces to themselves bijectively are very useful in these studies, they naturally form groups by composition. For example, bijective operators preserving the structure of a vector space are precisely the invertible linear operators. They form the general linear group under composition. However, they do not form a vector space under operator addition; since, for example, both the identity and −identity are invertible (bijective), but their sum, 0, is not. Operators preserving the Euclidean metric on such a space form the isometry group, and those that fix the origin form a subgroup known as the orthogonal group. Operators in the orthogonal group that also preserve the orientation of vector tuples form the special orthogonal group, or the group of rotations. === Probability theory === Operators are also involved in probability theory, such as expectation, variance, and covariance, which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a dot product: Every variance is a dot product of a vector with itself, and thus is a quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the Pearson correlation coefficient; expected value is basically an integral operator (used to measure weighted shapes in the space). ==== Fourier series and Fourier transform ==== The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively invertible. No information is lost, as there is an inverse transform operator. In the simple case of periodic functions, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves and cosine waves: f ( t ) = a 0 2 + ∑ n = 1 ∞ a n cos ( ω n t ) + b n sin ( ω n t ) {\displaystyle f(t)={\frac {\ a_{0}\ }{2}}+\sum _{n=1}^{\infty }\ a_{n}\cos(\omega \ n\ t)+b_{n}\sin(\omega \ n\ t)} The tuple ( a0, a1, b1, a2, b2, ... ) is in fact an element of an infinite-dimensional vector space ℓ2 , and thus Fourier series is a linear operator. When dealing with general function R → C {\displaystyle \mathbb {R} \to \mathbb {C} } , the transform takes on an integral form: f ( t ) = 1 2 π ∫ − ∞ + ∞ g ( ω ) e i ω t d ω {\displaystyle f(t)={1 \over {\sqrt {2\pi }}}\int _{-\infty }^{+\infty }{g(\omega )\ e^{i\ \omega \ t}\ \mathrm {d} \ \omega }} ==== Laplace transform ==== The Laplace transform is another integral operator and is involved in simplifying the process of solving differential equations. Given f = f(s), it is defined by: F ( s ) = L { f } ( s ) = ∫ 0 ∞ e − s t f ( t ) d t {\displaystyle F(s)=\operatorname {\mathcal {L}} \{f\}(s)=\int _{0}^{\infty }e^{-s\ t}\ f(t)\ \mathrm {d} \ t} == Footnotes == == See also == Function Operator algebra List of operators == References ==
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Wikipedia:Ophelia Bauckholt#0
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The Zizians are an informal group of rationalists with anarchist and vegan beliefs who also believe the hemispheres of the brain can have conflicting interests and identities. They are allegedly involved in six violent deaths in the United States, three in 2022 and three in 2025. Federal prosecutors say the Zizians are associated with persons of interest in the murders of four people: David Maland in Vermont (a U.S. Border Patrol agent), Curtis Lind in California (a landlord), and Richard and Rita Zajko in Pennsylvania (the parents of one of the group members). In addition, Ophelia Bauckholt (a German citizen) and Emma Borhanian, both associates of the Zizians, were killed during altercations with Maland and Lind. The term Zizians is derived from the name of Ziz LaSota, who is sometimes characterized as their leader. They themselves do not use this name or consider themselves to have a clearly identified leader, or even to be members of a group. == Beliefs == The group is a small offshoot of the rationalist community. Generally, Zizians hold anarchist beliefs, emphasize animal rights and veganism, and believe the hemispheres of the brain can have different genders and conflicting interests. The group has been described as radical or cult-like by publications such as The Independent, the Associated Press, and SFGate. LaSota recommended the practice of unihemispheric sleep (UHS), a form of sleep deprivation. A member of the group allegedly died by suicide during a period of intense practice of UHS in 2018, which led LaSota to make an online post saying the member had been carrying out the practice incorrectly. == Murders with alleged Zizian involvement == === Curtis Lind === After struggling with the upkeep of a tugboat they had been living on, which they eventually abandoned, some members of the Zizians opted to move into trailers and converted box trucks. Curtis Lind, who docked a boat in the same harbor as the group, offered to let them move into a lot he owned in Vallejo, California. During the COVID-19 pandemic, group members allegedly stopped paying rent and placed locks on trailers meant for other tenants. When Lind sued the group for back rent, one member allegedly brandished a knife, causing Lind to start carrying a pistol. On November 15, 2022, Lind, then 80, was, according to his own account, attacked by a group of people after being called in to fix a water leak. He said he was struck in the head, stabbed repeatedly (leaving "about 50" puncture wounds), and cut severely on the back of his neck ("like somebody was trying to cut my head off"). He was left impaled by a samurai sword, and his right eye was punctured three times, blinding him in that eye. After he regained consciousness, he shot two of those involved in the altercation, killing 31-year-old Emma Borhanian and wounding the other. Both had been arrested alongside LaSota at a 2019 protest against an event organized by the Center for Applied Rationality (CFAR), and Borhanian had reported LaSota's 2022 alleged drowning. Two of Lind's alleged attackers were charged with Borhanian's murder under the theory that their actions precipitated Lind's self-defense, under California's felony murder rule. LaSota was contacted by police during the investigation but was not charged. On January 17, 2025, three days before the shootout with U.S. Border Patrol officers on the other side of the country, Lind was stabbed to death and had his throat slit outside his gated property in Vallejo. Lind had been expected to be an important witness in the trial of the alleged attackers for the Borhanian felony murder charge, and stopping him from testifying was alleged as the motive for his killing. Prosecutors charged a 22-year-old data scientist with Lind's murder. The accused dictated to reporters a statement addressed to the well-known rationalist thought leader Eliezer Yudkowsky, calling himself "a student of Yudkowsky who became disillusioned with him". === Richard and Rita Zajko === During a welfare check on January 2, 2023, Pennsylvania state police discovered the bodies of husband and wife Richard and Rita Zajko, aged 72 and 69 respectively, at their home in Chester Heights, Pennsylvania. Autopsies found that Rita had a gunshot wound in the back of her head and Richard had wounds in his right hand and temple. Based partly on a recording by a neighbor's doorbell camera, it was concluded they had been killed on December 31. Their daughter, who is associated with the Zizians, has been named a person of interest in the murders. She is alleged to have purchased guns found at the scene of Maland's killing and to have been in contact with a person of interest in Lind's murder. She began talking to LaSota in 2022, and the two bonded over their "mutual grief and desire for justice" over Borhanian's death. === David Maland === On January 20, 2025, after a traffic stop, United States Border Patrol agent David Christopher Maland and at least one other Border Patrol agent engaged in a shootout with Bauckholt and another Zizian, leading to Maland's and Bauckholt's deaths and the wounding of the other member. Bauckholt and the other member were traveling south on Interstate 91 in Coventry, Vermont, when they were pulled over. They were put under "periodic surveillance" nearly one week before the shooting after they were reported to be armed and wearing all-black tactical clothing when checking in to their hotel. The case has been connected to the killings of Lind and the Zajkos due to connections between the suspects. ==== Background ==== Before traveling to Vermont, Bauckholt and the other member lived in separate Airbnb rentals in Chapel Hill, North Carolina. Both have been described as having cut off all contact with friends in fall 2023 and May 2024. Homeland Security Investigations agents had been conducting "periodic surveillance" of them since January 14. A hotel employee in Lyndonville, Vermont, reportedly contacted law enforcement about the duo after seeing one of them carrying "an apparent firearm in an exposed carry holster". Both wore "all-black, tactical style clothing with protective equipment". After the report, Homeland Security agents contacted the duo, who refused to speak with them extensively. They said they were in Vermont only to purchase real estate. The duo checked out of the Lyndonville hotel and were seen five days later in Newport, Vermont, with one of them carrying a handgun. The next day, hours before the shootout, the two were seen at a Walmart, with Bauckholt buying aluminum foil. After the shooting, authorities found ammunition, a helmet, night-vision monoculars, a tactical belt with a holster, a pair of walkie-talkies, a magazine loaded with cartridges, and shooting-range targets in their car. Smartphones were also found, wrapped in aluminum foil, apparently to prevent their phones from being tracked. Their handguns were reportedly bought by an associate in Mount Tabor, Vermont. ==== Shootout ==== Around 3:15 p.m., Agent David Maland initiated a traffic stop on I-91 southbound, about 15 kilometers from the border with Canada, with a blue 2015 Toyota Prius registered in North Carolina to conduct an immigration inspection. According to accounts by law enforcement, Bauckholt, registered as the car's owner, appeared in a Homeland Security database to have an expired visa. Bauckholt's visa was in fact current. Prosecutors allege that the second member drew a handgun and fired at least two shots at the agent during the stop and that Bauckholt also attempted to draw a firearm. At least one Border Patrol agent shot the duo. Maland and Bauckholt were pronounced dead at the scene while the second member was taken to a hospital and later arrested. Maland, 44, was a United States Air Force veteran. According to his family, he had been planning to marry his partner. Maland was an active security officer at the Pentagon during the 9/11 attacks before handling security at Joint Base Anacostia–Bolling. He had worked for the last 15 years for the Department of Homeland Security as a border patrol agent and as a K-9 handler. Maland was the first Border Patrol agent killed by gunfire in the line of duty since 2014. ==== Reactions ==== Vermont's U.S. Senators Bernie Sanders and Peter Welch and U.S. House Representative Becca Balint issued a joint statement saying, "Our deepest condolences go out to the agent’s family, and to the Border Patrol". Representative Mark Green, who chairs the House Committee on Homeland Security, issued a statement saying he was "heartbroken by the loss of Agent David Maland", adding, "[w]e must never forget that the men and women in green on the frontlines of this border crisis defend our homeland at great personal cost" and that "[f]ar too often these courageous public servants, like Agent Maland, pay the ultimate price". Maland's flag-draped casket was carried by a motorcade from Burlington, Vermont, to Albany International Airport by Border Patrol agents and Vermont State Police, and was flown to his family in Minnesota. == Members == === Ziz LaSota === Although the group members do not use this name or even consider themselves members of a group or to have a clearly identified leader, they are known as "Zizians", based on the name of their founder, Ziz LaSota. Like many other members of the group, LaSota is transgender. According to a CFAR employee, LaSota targeted "smart, mostly autistic-ish trans women who were extremely vulnerable and isolated" for recruitment. LaSota, who was 34 years old as of 2025, earned a bachelor's degree in computer engineering in 2013 from the University of Alaska Fairbanks. She had an internship at NASA and pursued a master's degree at the University of Illinois Urbana-Champaign from 2013 to 2014 but did not graduate. LaSota moved to the San Francisco Bay Area in hopes of becoming involved with the effective altruism (EA) and rationality movements. Disaffected by high cost of housing, she and a group of fellow EA adherents sought to form a seasteading intentional community. Initially living on sailboats in the Berkeley Marina, they eventually bought an old tugboat and sailed it from Alaska to Pillar Point Harbor in San Mateo. During her involvement with the rationality community, LaSota became disillusioned with the leadership of community institutions such as the Center for Applied Rationality (CFAR) and the Machine Intelligence Research Institute (MIRI). LaSota and her associates claimed CFAR and MIRI discriminated against trans women, used donor money to pay off a former staffer who had accused MIRI leaders of statutory rape and a coverup, and ignored the welfare of animals in the pursuit of human-friendly artificial intelligence. CFAR co-founder Anna Salamon attempted to prevent LaSota from attending the fellowship due to strange beliefs and behavior at previous events, but was overruled by a committee. These included LaSota's theories that human consciousness can be split between the brain's two hemispheres, which may hold different values, genders, and may be "good", "evil", or both. After more CFAR staff members raised concerns, LaSota was no longer invited to the group's events. In 2019, LaSota, Borhanian, and two associates staged a protest against a CFAR event at a retreat in Occidental, California. Because a 911 call led police to mistakenly believe the protesters were armed and because a group of children was also at the retreat for a separate event, the protest drew a forceful police response. After the four protesters were arrested, a SWAT team was deployed to evacuate the retreat because police mistakenly believed a fifth protester had a hatchet; that person was later discovered to be a maintenance worker. Fallout from the protest and the police response led to a rift between the Zizians and the rationalist community establishment: the Zizians accused CFAR employees of swatting them by falsely reporting to police that they were armed, while a member of the rationalist community published an anonymous callout coining the appellation "Zizians" and branding them as a cult. The Zizians filed a federal civil rights lawsuit against Sonoma County, which was dismissed. LaSota faked her own death in a supposed boating accident in August 2022, but turned up in January 2023 in a Philadelphia hotel room where police were carrying out a search for a weapon suspected to have been used in the murder of the Zajkos. LaSota was identified as the subject of an outstanding warrant in California and arrested for disorderly conduct and interfering with a police investigation. After her bail was reduced to $10,000, unsecured, LaSota was released pending trial. She appeared in court in Pennsylvania in August 2023, but subsequently failed to appear in December 2023. After media coverage of LaSota and associates related to David Maland's death, LaSota was recognized by the owner of a rural property in Frostburg, Maryland, where she was attempting to camp. The owner called the police, who arrested LaSota on February 16, 2025, for trespassing, obstructing an officer, and transporting firearms. She is being held in custody without bail; she requested a pretrial release, which a local judge denied. === Ophelia Bauckholt === Ophelia Bauckholt was a transgender German citizen on a current visa working as a quantitative trader for Tower Research Capital in New York since October 2021. Before working at Tower, she worked as a trader at Radix Trading for two years and as an intern at Jane Street Capital. Bauckholt reportedly quit working for Tower in 2023, which put her visa's extension at risk. Bauckholt was raised in Freiburg im Breisgau and attended the Goethe-Gymnasium there, where she was a gifted mathematician. In 2014 and in 2015, she won gold and bronze medals for the German team at the International Olympiad in Informatics. She graduated from the University of Waterloo in 2019 with a bachelor's degree in mathematics. Bauckholt expressed interest in the rationalist community after attending a CFAR event in 2019, and reportedly cut off contact with friends in the fall of 2023. On January 20, 2025, Bauckholt was killed in a shootout in Coventry, Vermont, after being stopped by border patrol agent David Maland, who was also killed. The Freiburg public prosecutor's office is investigating the circumstances surrounding Bauckholt's death, as is done whenever a German citizen dies abroad. == References ==
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Wikipedia:Orbit trap#0
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In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimensional fractals representing the complex plane. == Examples == === Point based === A point-based orbit trap colours a point based upon how close a function's orbit comes to a single point, typically the origin. === Line based === A line-based orbit trap colours a point based upon how close a function's orbit comes to one or more lines, typically vertical or horizontal (x=a or y=a lines). Pickover stalks are an example of a line based orbit trap which use two lines. == Algorithm == Orbit traps are typically used with the class of two-dimensional fractals based on an iterative function. A program that creates such a fractal colours each pixel, which represent discrete points in the complex plane, based upon the behaviour of those points when they pass through a function a set number of times. The best known example of this kind of fractal is the Mandelbrot set, which is based upon the function zn+1 = zn2 + c. The most common way of colouring Mandelbrot images is by taking the number of iterations required to reach a certain bailout value and then assigning that value a colour. This is called the escape time algorithm. A program that colours the Mandelbrot set using a point-based orbit trap will assign each pixel with a “distance” variable, that will typically be very high when first assigned: As the program passes the complex value through the iterative function it will check the distance between each point in the orbit and the trap point. The value of the distance variable will be the shortest distance found during the iteration: == References == Carlson, Paul W. (1999), "Two artistic orbit trap rendering methods for Newton M-set fractals", Computers & Graphics, 23 (6): 925–931, doi:10.1016/S0097-8493(99)00123-5. Lu, Jian; Ye, Zhongxing; Zou, Yuru; Ye, Ruisong (2005), "Orbit trap rendering methods for generating artistic images with crystallographic symmetries", Computers & Graphics, 29 (5): 787–794, doi:10.1016/j.cag.2005.08.008.
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Wikipedia:Order of operations#0
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In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ranking of the operations. The rank of an operation is called its precedence, and an operation with a higher precedence is performed before operations with lower precedence. Calculators generally perform operations with the same precedence from left to right, but some programming languages and calculators adopt different conventions. For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. Thus 3 + 52 = 28 and 3 × 52 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used. For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5)2 = 64 forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by other types of brackets to avoid confusion, as in [2 × (3 + 4)] − 5 = 9. These rules are meaningful only when the usual notation (called infix notation) is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself. == Conventional order == The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted throughout mathematics, science, technology and many computer programming languages. It is summarized as: Parentheses Exponentiation Multiplication and division Addition and subtraction This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set. Whether inside parentheses or not, the operation that is higher in the above list should be applied first. Operations of the same precedence are conventionally evaluated from left to right. If each division is replaced with multiplication by the reciprocal (multiplicative inverse) then the associative and commutative laws of multiplication allow the factors in each term to be multiplied together in any order. Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see § Mixed division and multiplication below. If each subtraction is replaced with addition of the opposite (additive inverse), then the associative and commutative laws of addition allow terms to be added in any order. The radical symbol t {\displaystyle {\sqrt {\vphantom {t}}}} is traditionally extended by a bar (called vinculum) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity. The parentheses can be omitted if the input is a single numerical variable or constant, as in the case of sin x = sin(x) and sin π = sin(π). Traditionally this convention extends to monomials; thus, sin 3x = sin(3x) and even sin 1/2xy = sin(1/2xy), but sin x + y = sin(x) + y, because x + y is not a monomial. However, this convention is not universally understood, and some authors prefer explicit parentheses. Some calculators and programming languages require parentheses around function inputs, some do not. Parentheses and alternate symbols of grouping can be used to override the usual order of operations or to make the intended order explicit. Grouped symbols can be treated as a single expression. === Examples === Multiplication before addition: 1 + 2 × 3 = 1 + 6 = 7. {\displaystyle 1+2\times 3=1+6=7.} Parenthetical subexpressions are evaluated first: ( 1 + 2 ) × 3 = 3 × 3 = 9. {\displaystyle (1+2)\times 3=3\times 3=9.} Exponentiation before multiplication, multiplication before subtraction: 1 − 2 × 3 4 = 1 − 2 × 81 = 1 − 162 = − 161. {\displaystyle 1-2\times 3^{4}=1-2\times 81=1-162=-161.} When an expression is written as a superscript, the superscript is considered to be grouped by its position above its base: 1 + 2 3 + 4 = 1 + 2 7 = 1 + 128 = 129. {\displaystyle 1+2^{3+4}=1+2^{7}=1+128=129.} The operand of a root symbol is determined by the overbar: 1 + 3 + 5 = 4 + 5 = 2 + 5 = 7. {\displaystyle {\sqrt {1+3}}+5={\sqrt {4}}+5=2+5=7.} A horizontal fractional line forms two grouped subexpressions, one above divided by another below: 1 + 2 3 + 4 + 5 = 3 7 + 5. {\displaystyle {\frac {1+2}{3+4}}+5={\frac {3}{7}}+5.} Parentheses can be nested, and should be evaluated from the inside outward. For legibility, outer parentheses can be made larger than inner parentheses. Alternately, other grouping symbols, such as curly braces { } or square brackets [ ], are sometimes used along with parentheses ( ). For example: [ ( 1 + 2 ) ÷ ( 3 + 4 ) ] + 5 = ( 3 ÷ 7 ) + 5 {\displaystyle {\bigl [}(1+2)\div (3+4){\bigr ]}+5=(3\div 7)+5} == Special cases == === Unary minus sign === There are differing conventions concerning the unary operation '−' (usually pronounced "minus"). In written or printed mathematics, the expression −32 is interpreted to mean −(32) = −9. In some applications and programming languages, notably Microsoft Excel, PlanMaker (and other spreadsheet applications) and the programming language bc, unary operations have a higher priority than binary operations, that is, the unary minus has higher precedence than exponentiation, so in those languages −32 will be interpreted as (−3)2 = 9. This does not apply to the binary minus operation '−'; for example in Microsoft Excel while the formulas =-2^2, =(-2)^2 and =0+-2^2 return 4, the formulas =0-2^2 and =-(2^2) return −4. === Mixed division and multiplication === There is no universal convention for interpreting an expression containing both division denoted by '÷' and multiplication denoted by '×'. Proposed conventions include assigning the operations equal precedence and evaluating them from left to right, or equivalently treating division as multiplication by the reciprocal and then evaluating in any order; evaluating all multiplications first followed by divisions from left to right; or eschewing such expressions and instead always disambiguating them by explicit parentheses. Beyond primary education, the symbol '÷' for division is seldom used, but is replaced by the use of algebraic fractions, typically written vertically with the numerator stacked above the denominator – which makes grouping explicit and unambiguous – but sometimes written inline using the slash or solidus symbol '/'. Multiplication denoted by juxtaposition (also known as implied multiplication) creates a visual unit and is often given higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n. For instance, the manuscript submission instructions for the Physical Review journals directly state that multiplication has precedence over division, and this is also the convention observed in physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and mathematics textbooks such as Concrete Mathematics by Graham, Knuth, and Patashnik. However, some authors recommend against expressions such as a / bc, preferring the explicit use of parenthesis a / (bc). More complicated cases are more ambiguous. For instance, the notation 1 / 2π(a + b) could plausibly mean either 1 / [2π · (a + b)] or [1 / (2π)] · (a + b). Sometimes interpretation depends on context. The Physical Review submission instructions recommend against expressions of the form a / b / c; more explicit expressions (a / b) / c or a / (b / c) are unambiguous. This ambiguity has been the subject of Internet memes such as "8 ÷ 2(2 + 2)", for which there are two conflicting interpretations: 8 ÷ [2 · (2 + 2)] = 1 and (8 ÷ 2) · (2 + 2) = 16. Mathematics education researcher Hung-Hsi Wu points out that "one never gets a computation of this type in real life", and calls such contrived examples "a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules". === Serial exponentiation === If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down: abc = a(bc), which typically is not equal to (ab)c. This convention is useful because there is a property of exponentiation that (ab)c = abc, so it's unnecessary to use serial exponentiation for this. However, when exponentiation is represented by an explicit symbol such as a caret (^) or arrow (↑), there is no common standard. For example, Microsoft Excel and computation programming language MATLAB evaluate a^b^c as (ab)c, but Google Search and Wolfram Alpha as a(bc). Thus 4^3^2 is evaluated to 4,096 in the first case and to 262,144 in the second case. == Mnemonics == Mnemonic acronyms are often taught in primary schools to help students remember the order of operations. The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction, is common in the United States and France. Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally". The United Kingdom and other Commonwealth countries may use BODMAS (or sometimes BOMDAS), standing for Brackets, Of, Division/Multiplication, Addition/Subtraction, with "of" meaning fraction multiplication. Sometimes the O is instead expanded as Order, meaning exponent or root, or replaced by I for Indices in the alternative mnemonic BIDMAS. In Canada and New Zealand BEDMAS is common. In Germany, the convention is simply taught as Punktrechnung vor Strichrechnung, "dot operations before line operations" referring to the graphical shapes of the taught operator signs U+00B7 · MIDDLE DOT (multiplication), U+2236 ∶ RATIO (division), and U+002B + PLUS SIGN (addition), U+2212 − MINUS SIGN (subtraction). These mnemonics may be misleading when written this way. For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression a − b + c {\displaystyle a-b+c} as a − ( b + c ) {\displaystyle a-(b+c)} , while the correct evaluation is ( a − b ) + c {\displaystyle (a-b)+c} . These values are different when c ≠ 0 {\displaystyle c\neq 0} . Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility. Students learning the order of operations via mnemonic acronyms routinely make mistakes, as do some pre-service teachers. Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content. The acronym's procedural application does not match experts' intuitive understanding of mathematical notation: mathematical notation indicates groupings in ways other than parentheses or brackets and a mathematical expression is a tree-like hierarchy rather than a linearly "ordered" structure; furthermore, there is no single order by which mathematical expressions must be simplified or evaluated and no universal canonical simplification for any particular expression, and experts fluently apply valid transformations and substitutions in whatever order is convenient, so learning a rigid procedure can lead students to a misleading and limiting understanding of mathematical notation. == Calculators == Different calculators follow different orders of operations. Many simple calculators without a stack implement chain input, working in button-press order without any priority given to different operations, give a different result from that given by more sophisticated calculators. For example, on a simple calculator, typing 1 + 2 × 3 = yields 9, while a more sophisticated calculator will use a more standard priority, so typing 1 + 2 × 3 = yields 7. Calculators may associate exponents to the left or to the right. For example, the expression a^b^c is interpreted as a(bc) on the TI-92 and the TI-30XS MultiView in "Mathprint mode", whereas it is interpreted as (ab)c on the TI-30XII and the TI-30XS MultiView in "Classic mode". An expression like 1/2x is interpreted as 1/(2x) by TI-82, as well as many modern Casio calculators (configurable on some like the fx-9750GIII), but as (1/2)x by TI-83 and every other TI calculator released since 1996, as well as by all Hewlett-Packard calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of implied multiplication, the latter is more in line with the rule that multiplication and division are of equal precedence. When the user is unsure how a calculator will interpret an expression, parentheses can be used to remove the ambiguity. Order of operations arose due to the adaptation of infix notation in standard mathematical notation, which can be notationally ambiguous without such conventions, as opposed to postfix notation or prefix notation, which do not need orders of operations. Hence, calculators utilizing reverse Polish notation (RPN) using a stack to enter expressions in the correct order of precedence do not need parentheses or any possibly model-specific order of execution. == Programming languages == Most programming languages use precedence levels that conform to the order commonly used in mathematics, though others, such as APL, Smalltalk, Occam and Mary, have no operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it is strictly left to right). Furthermore, because many operators are not associative, the order within any single level is usually defined by grouping left to right so that 16/4/4 is interpreted as (16/4)/4 = 1 rather than 16/(4/4) = 16; such operators are referred to as "left associative". Exceptions exist; for example, languages with operators corresponding to the cons operation on lists usually make them group right to left ("right associative"), e.g. in Haskell, 1:2:3:4:[] == 1:(2:(3:(4:[]))) == [1,2,3,4]. Dennis Ritchie, creator of the C language, said of the precedence in C (shared by programming languages that borrow those rules from C, for example, C++, Perl and PHP) that it would have been preferable to move the bitwise operators above the comparison operators. Many programmers have become accustomed to this order, but more recent popular languages like Python and Ruby do have this order reversed. The relative precedence levels of operators found in many C-style languages are as follows: Examples: !A + !B is interpreted as (!A) + (!B) ++A + !B is interpreted as (++A) + (!B) A + B * C is interpreted as A + (B * C) A || B && C is interpreted as A || (B && C) A && B == C is interpreted as A && (B == C) A & B == C is interpreted as A & (B == C) (In Python, Ruby, PARI/GP and other popular languages, A & B == C is interpreted as (A & B) == C.) Source-to-source compilers that compile to multiple languages need to explicitly deal with the issue of different order of operations across languages. Haxe for example standardizes the order and enforces it by inserting brackets where it is appropriate. The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code. == History == The order of operations emerged progressively over centuries. The rule that multiplication has precedence over addition was incorporated into the development of algebraic notation in the 1600s, since the distributive property implies this as a natural hierarchy. As recently as the 1920s, the historian of mathematics Florian Cajori identifies disagreement about whether multiplication should have precedence over division, or whether they should be treated equally. The term "order of operations" and the "PEMDAS/BEDMAS" mnemonics were formalized only in the late 19th or early 20th century, as demand for standardized textbooks grew. Ambiguity about issues such as whether implicit multiplication takes precedence over explicit multiplication and division in such expressions as a/2b, which could be interpreted as a/(2b) or (a/2) × b, imply that the conventions are not yet completely stable. == See also == Common operator notation (for a more formal description) Hyperoperation Logical connective#Order of precedence Operator associativity Operator overloading Operator precedence in C and C++ Polish notation Reverse Polish notation == Notes == == References == == Further reading == Fothe, Michael; Wilke, Thomas, eds. (2015). Keller, Stack und automatisches Gedächtnis – eine Struktur mit Potenzial [Cellar, stack and automatic memory – a structure with potential] (PDF). Kolloquium 14 Nov 2014 in Jena, Germany (in German). Bonn: Gesellschaft für Informatik. ISBN 978-3-88579-426-4. == External links == Bergman, George Mark (2013). "Order of arithmetic operations; in particular, the 48/2(9+3) question". Dept. of Mathematics, University of California. Retrieved 2020-07-22. Zachary, Joseph L. (1997) "Operator Precedence", supplement to Introduction to Scientific Programming. University of Utah. Maple worksheet, Mathematica notebook.
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Wikipedia:Order unit#0
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Law & Order: Special Victims Unit (often shortened to Law & Order: SVU or SVU) is an American police procedural crime drama television series created by Dick Wolf for NBC. The first spin-off of Law & Order, expanding it into the Law & Order franchise, it stars Mariska Hargitay as Detective (ultimately promoted to Captain) Olivia Benson, now the commanding officer of the Special Victims Unit after originally having been Stabler's partner in a fictionalized version of the New York City Police Department, and Christopher Meloni as Detective Elliot Stabler (until Meloni left the series in 2011 after 12 seasons). Law & Order: Special Victims Unit follows the detectives of the Special Victims Unit as they investigate and prosecute sexually based crimes. Some of the episodes are loosely based on real crimes that have received media attention. The series, produced by Wolf Entertainment and Universal Television, premiered on September 20, 1999. After the premiere of its 21st season in September 2019, the series became the longest-running primetime live-action series on American television. Since the end of the original run of the main Law & Order series in 2010, SVU is the only live-action primetime series having debuted in the 1990s to remain in continuous production. The 23rd season premiered on September 23, 2021, during which the show aired its milestone 500th episode. As of May 15, 2025, Law & Order: Special Victims Unit has aired 573 original episodes, well surpassing the episode count of the original Law & Order series. In terms of all-time episode count for a primetime scripted series, SVU now ranks fourth behind The Simpsons (with 785 episodes), Gunsmoke (with 635 episodes), and Lassie (with 591 episodes). The 25th season premiered on January 18, 2024, and on March 21, 2024, NBC announced that it renewed the series for its 26th season, which premiered on October 3, 2024. In May 2025, the series was renewed for a 27th season. The series has received 108 award nominations, winning 59 awards. Hargitay was the first regular cast member on any Law & Order series to win an Emmy Award when she won the Primetime Emmy Award for Outstanding Lead Actress in a Drama Series in 2006. == Premise == Based out of the NYPD New York City Police Department's 16th precinct in Manhattan, Law & Order: Special Victims Unit delves into the dark side of the New York underworld as the detectives of a new elite squad, the Special Victims Unit (SVU for short), investigate and prosecute various sexually oriented crimes, including rape, child sexual abuse, human trafficking and domestic violence. They also investigate the abuses of children, the disabled and elderly victims of non-sexual crimes who require specialist handling, all while trying to balance the effects of the investigation on their own lives as they try not to let the dark side of these crimes affect them. Its stories also touch on the political and societal issues associated with gender identity, sexual preferences, and equality rights. While the victim is often murdered, this is not always the case, and victims frequently play prominent roles in episodes. The unit also works with the Manhattan District Attorney's office as they prosecute cases and seek justice for SVU's victims and survivors with precision and a passion to win and bring closure to the intense investigations. The series often uses stories that are "ripped from the headlines" or based on real crimes. Such episodes take a real crime and fictionalize it by changing some details. == Cast and characters == Originally, the show focused around the detective pairing of Elliot Stabler (Christopher Meloni) and Olivia Benson (Mariska Hargitay). Stabler is a seasoned veteran of the unit who has seen it all and tries his best to protect his family from the horrors he has seen in his career. Meanwhile, Benson's difficult past as the child of a rape victim is the reason she joined the unit. Backing them up are Detective John Munch (Richard Belzer) and his first partner, Brian Cassidy (Dean Winters). Munch is a transfer from Baltimore's homicide unit, who brings his acerbic wit, conspiracy theories, and street-honed investigative skills; Cassidy is young and eager to learn from his fellow detectives. These two detective teams received support from Detectives Monique Jeffries (Michelle Hurd) and Ken Briscoe (Chris Orbach). After thirteen episodes, Cassidy gets transferred to the narcotics division because of his inability to work well on the cases and the fact that they reminded him too much of his childhood abuse. As a result, Jeffries gets partnered up with Munch for the remainder of season one and Briscoe was phased out. In the beginning of season two, Jeffries leaves the unit following an incident with the Morris Commission and Munch gets permanently partnered up with Detective Odafin "Fin" Tutuola (Ice-T), whose unique yet sometimes vulgar sense of humor and investigative experience make him a formidable match for Munch. Brooklyn SVU Detective Chester Lake (Adam Beach) would assist on several Manhattan cases during the eighth season and then join during season nine; Lake would then depart at the season's end after being arrested for murdering a crooked cop who had gotten away with rape. These detectives were supervised by veteran Captain Donald Cragen (Dann Florek), who oversaw the team for seasons 1–15 and was previously the commanding officer of the Manhattan North Homicide precinct on the first 3 seasons of Law & Order. Cragen's tough-but-supportive approach to the team's complex cases guides the squad through the challenges they face every day. Also assisting the Special Victims Unit, is FBI Special Agent Dr. George Huang (BD Wong) who helps keep the officers sane in a field that could drive ordinary people mad, whilst also serving as the squad's resident criminal profiler, his insights into the criminal mind have often helped the officers to crack the toughest perps. The team also works with Medical Examiner Dr. Melinda Warner (Tamara Tunie), who has become an integral part of the unit, and her personal skills have contributed to the unit's high success rate in closing cases. The unit did not receive a full-time assistant district attorney until season two, when Alexandra Cabot (Stephanie March) was assigned to work with the detectives. After Cabot enters the Witness Protection Program after almost being killed in a hit in season five, she was replaced by Casey Novak (Diane Neal), who remained as the ADA until the end of season nine, when she was censured for violating due process while trying to bring a rapist cop (the same one that Lake would end up murdering) to justice. Kim Greylek (Michaela McManus) became the permanent ADA in the season ten premiere, until Cabot made a return midway through that season when Greylek returned to the Justice Department in Washington, D.C. Cabot remained the ADA through the second half of season 10. After Cabot's departure, the ADA void was filled by Sonya Paxton (Christine Lahti) and Jo Marlowe (Sharon Stone) until the conclusion of season 11. Gillian Hardwicke (Melissa Sagemiller) served as the SVU's ADA during season 12, while Novak would return for a guest appearance after completing her suspension near the end of the season. Paxton would also return for one more appearance in the season, during which she is brutally murdered by a rapist/murderer, but manages to leave behind vital evidence to assure his conviction. In season 13, both Cabot and Novak returned as ADAs. From the beginning of season 14, ADA Rafael Barba (Raúl Esparza) was SVU's prosecutor, until leaving halfway through season 19 following the death of an infant. Chicago Justice's Peter Stone (Philip Winchester) became SVU's ADA after Chicago Justice was canceled after only one season. At the end of season 20, Stone decided to leave due to some of the cases being too much for him to handle. From there, former SVU Detective Dominick Carisi Jr. (Peter Scanavino) takes his place at the start of season 21. In season 13, big changes happen when Stabler retires in the aftermath of the season 12 finale, until he reappeared in season 22, which led him to join NYPD's Organized Crime Control Bureau in Law & Order: Organized Crime. Huang also departed at the same time after being reassigned to Oklahoma City, but has returned for occasional guest appearances. Detectives Nick Amaro (Danny Pino) and Amanda Rollins (Kelli Giddish) joined the team filling the void left by Stabler. Amaro brought empathy to his cases while dealing with a stressful home life, while Rollins had dogged persistence and instincts help her close cases, but also secrets from her past that could derail her career. During season 15, both Munch and Cragen retired from the NYPD, leaving Benson, after being promoted to Sergeant, to take control of the unit; she would later be promoted to Lieutenant at the start of season 17 and then Captain at the start of season 21. Season 16 was another period of change with the introduction of Carisi at the beginning and the departure of Amaro at the end, with the latter relocating to California to be near his moved children after being wounded in the line of duty and learning that previous instances of misconduct have cost him any chance for advancement in the department. Also introduced in season 16 was Deputy Chief William Dodds (Peter Gallagher), who served as commanding officer for the Special Victims Units in all five boroughs of New York. Dodds' son Mike (Andy Karl) transfers into Special Victims as Sergeant in season 17, becoming Benson's second-in-command until his death at the end of the season; Fin later passes the Sergeant's exam during season 18 and is officially promoted in Mike's place in season 19. Following Carisi's move to the DA's office in season 21, Vice Officer Katriona "Kat" Tamin (Jamie Gray Hyder) joined the team after assisting on several cases, eventually getting promoted to detective. Dodds also departed the series at the start of the season due to some issues in the case regarding a mogul raping countless women, allowing new deputy chief Christian Garland (Demore Barnes) to take his place. At the start of season 23, Tamin and Garland both resigned from the NYPD after becoming disillusioned with the legal system's failures and the systemic bias within the department, with Tamin being replaced by Detective Joe Velasco (Octavio Pisano). Chief Tommy McGrath (Terry Serpico) took over Garland's position until he could find a permanent deputy chief for SVU. At the beginning of season 24, Detective Grace Muncy (Molly Burnett) joined SVU after solving a case that involves a teenage Dominican gang, while Rollins resigned from SVU and the NYPD halfway through the season after accepting an offer from Carisi's old colleague to teach at Fordham University. SVU also brought in Detectives Terry Bruno (Kevin Kane) and Tonie Churlish (Jasmine Batchelor) from their Brooklyn counterparts. Muncy later departed at the end of the season to work on a DEA task force and Churlish also left during the same time. In season 25, McGrath was replaced as chief after crossing multiple lines in his interference with his daughter's rape case, and IAB Captain Renee Curry (Aime Donna Kelly) joined SVU in hopes of making changes. Shortly afterwards, FBI agent Shannah Sykes (Jordana Spiro) was put on loan to SVU after helping them rescue abducted girl Maddie Flynn. Sykes later left SVU at the season's end after solving the case of her missing sister, which ended up hitting her too close to home, and at the start of season 26, former Homicide detective Kate Silva (Juliana Aidén Martinez) joins the unit. Additionally, Rollins, after consulting on several cases following her departure, returns to the NYPD with a promotion to Sergeant and assignment as CO of the department's Intelligence Unit. == Production == === Development === The idea for Law & Order: Special Victims Unit originated with the 1986 "preppie murder" case of Robert Chambers, who strangled and killed a woman he dated, Jennifer Levin, during what he claimed was consensual "rough sex" in Manhattan's Central Park. The crime inspired Dick Wolf to write the story for the season one episode of Law & Order titled "Kiss the Girls and Make Them Die". After writing the episode, Wolf wanted to go deeper into the psychology of crimes to examine the role of human sexuality. The original title of the show was Sex Crimes. Initially, there was concern among the producers that, should Sex Crimes fail, identifying the new show with the Law & Order franchise could affect the original show. Additionally, Ted Kotcheff wanted to create a new series that was not dependent upon the original series for success. Wolf felt, however, that it was important and commercially desirable to have "Law & Order" in the title, and he initially proposed the title of the show be Law & Order: Sex Crimes. Barry Diller, then head of Studios USA, was concerned about the title, however, and it was changed to Law & Order: Special Victims Unit to reflect the actual unit of the New York City Police Department (NYPD) that handles sexually-based offenses. Executive producer Neal Baer left Law & Order: SVU as showrunner at the end of season twelve, after eleven years (seasons 2–12) on the show, in order to sign a three-year deal with CBS Studios. Baer was replaced by former Law & Order: Criminal Intent showrunner Warren Leight. In March 2015, it was announced that Warren Leight signed a three-year deal with Sony Pictures Television, that will allow him to work on SVU one more season, its seventeenth. It was announced on March 10, 2016, that original Law & Order veteran producer Rick Eid would take Leight's place as showrunner starting in season 18. Creator Dick Wolf commented to The Hollywood Reporter, "I'm extremely pleased that Rick had decided to rejoin the family and hope that he will be here for years to come." During post-production of season 18, following the announcement that SVU was renewed for a nineteenth season, it was revealed that Rick Eid departed the series. He will be taking over another Dick Wolf/NBC series, Chicago P.D. It was announced on May 25, 2017, that original Law & Order and Law & Order: Criminal Intent showrunner Michael S. Chernuchin would be reprising his role starting on season nineteen. Chernuchin was also co-creator and executive producing showrunner of Chicago Justice, another Wolf-related show that was canceled by NBC at the end of the 2016–17 TV season. On April 22, 2019, it was announced that Leight would return as showrunner for the series' twenty-first season. On May 3, 2022, Leight announced that he would not be returning for the twenty-fourth season. In May 2025, NBC renewed the series for its twenty-seventh season; Michele Fazekas will serve as the show's showrunner, becoming the first woman to serve as showrunner for Special Victims Unit. === Casting === Casting for the lead characters of Law & Order: Special Victims Unit occurred in the spring of 1999. Dick Wolf, along with officials from NBC and Studios USA were at the final auditions for the two leads at Rockefeller Center. The last round had been narrowed down to seven finalists. For the female lead, Detective Olivia Benson, actresses Samantha Mathis, Reiko Aylesworth, and Mariska Hargitay were being considered. For the male role, Detective Elliot Stabler, the finalists were Tim Matheson, John Slattery, Nick Chinlund, and Christopher Meloni. Hargitay and Meloni had auditioned in the final round together and, after the actors left, there was a moment of dead silence, after which Wolf blurted out, "Oh well. There's no doubt who we should choose—Hargitay and Meloni." Wolf believed the duo had the perfect chemistry together from the first time he saw them together, and they ended up being his first choice. Garth Ancier, then head of NBC Entertainment, agreed, and the rest of the panel assembled began voicing their assent. The first actor to be cast for the show was Dann Florek. Florek had originated the character of Captain Don Cragen in the 1990 pilot for Law & Order, and played the character for the show's first three seasons until he was fired on the orders of network executives, who wanted to add female characters to the all-male primary cast, but he maintained a friendly relationship with Wolf, and went on to direct three episodes of the original series as well as to occasionally guest star on the show. Shortly after Florek reprised his role for Exiled: A Law & Order Movie, he received a call to be on Sex Crimes. Initially reluctant, he eventually agreed to star on the show as Cragen on the assurance that he would not be asked to audition for the role. Shortly after the cancellation of Homicide: Life on the Street, Richard Belzer heard that Benjamin Bratt had left Law & Order. Belzer requested his manager to call Wolf and pitch the idea for Belzer's character from Homicide, Detective John Munch, to become the new partner of Jerry Orbach's character, Detective Lennie Briscoe, since they had previously teamed in three Homicide crossovers. Wolf loved the idea, but had already cast Jesse L. Martin as Briscoe's new partner, Detective Ed Green. The idea was reconfigured, but to have Munch on Law & Order: Special Victims Unit instead. Since the character of Munch was inspired by David Simon's depiction of Detective Sergeant Jay Landsman and developed for Homicide by Tom Fontana and Barry Levinson, the addition of Munch to the cast required the consent of all three. The appropriate agreements were reached and, while Fontana and Levinson agreed to waive their royalty rights, contracts with Simon required that he be paid royalties for any new show in which Munch is a main character; as a result, Simon receives royalties every time Munch appears in an episode of the show. Dean Winters was cast as Munch's partner, Brian Cassidy, at the insistence of Belzer. Belzer looked at Winters as a sort of little brother, and told Wolf, "Well, I'll do this new show of yours, SVU, only if you make Dean Winters my partner." Wolf did make Winters Belzer's partner, but he was contractually obligated to his other show at the time, the HBO drama Oz. Since the role on Law & Order: Special Victims Unit was only initially meant to be a few episodes, Winters was forced to leave when it was time to film Oz again. Winters returned for the season 13 finale, "Rhodium Nights", reprising his role as Cassidy. He also appeared (as Cassidy) on the two-part season 14 premiere "Lost Reputation"/"Above Suspicion". He subsequently became a recurring character into season 15. The void left by Winters's departure was filled for the remainder of the season by Michelle Hurd as Detective Monique Jeffries, a character who Wolf promised that, despite starting out as a minor character with one scene in the pilot, would eventually develop. Hurd left the show at the beginning of season two to join the cast of Leap Years. Munch's permanent partner came in the form of rapper-turned-actor Ice-T, who had previously worked with Wolf on New York Undercover and Exiled. Ice-T originally agreed to do only four episodes of Law & Order: Special Victims Unit, but he quickly gained affection for the ensemble nature of the cast. He relocated to New York City before his four-episode contract was up and remained with the show as Munch's permanent partner, Detective Odafin "Fin" Tutuola. Initially, the show focused exclusively on the police work of the detectives in the Special Victims Unit of the 16th precinct, with members of the District Attorney's office occasionally appearing as guest roles crossing over from the original Law & Order. From season two onwards, the format was changed to be more faithful to the original Law & Order concept by including court cases. Stephanie March had little television experience before being cast on Law & Order: Special Victims Unit, nor did she watch much TV. Nevertheless, March was cast as Assistant District Attorney Alexandra Cabot at the beginning of season two but still believed that, due to the grim nature of the series, it would be short-lived. She stayed with the series for three seasons, however, and left when she believed she had reached the natural conclusion of the character's development. She would later reprise the character as a guest appearance in season six and as a regular character on the short-lived Wolf series, Conviction, where she was promised more to do. Diane Neal had previously guest-starred on Law & Order: Special Victims Unit in season three before being cast as Cabot's replacement, Casey Novak, in season five. Neal remained with the show through the end of season nine, after which she was replaced by Michaela McManus. March returned to the show in the tenth season (after McManus' departure from the cast) when Neal Baer proposed Cabot receive a character arc to revitalize the second part of the season, which would continue through season eleven. Tamara Tunie was cast as medical examiner Melinda Warner in season two after working with Wolf previously on New York Undercover, Feds, and Law & Order. Warner was initially a recurring character but became a regular character in season seven, and Tunie was added to the opening credits at that time. When initially cast as Warner, Tunie was appearing as attorney Jessica Griffin on the CBS daytime soap opera As the World Turns. From 2000 to 2007 (and again briefly in 2009), she appeared on both series simultaneously. In 2002, she also appeared on the Fox espionage-themed drama series 24, in the recurring role of CTU Acting Director Alberta Green. BD Wong was asked to film four episodes as Dr. George Huang, a Federal Bureau of Investigation (FBI) forensic psychiatrist and criminal profiler on loan to the Special Victims Unit. After his four episodes, he was asked to stay on with the show. After he starred in Bury My Heart at Wounded Knee and guest-starred as Detective Chester Lake in the eighth season, Wolf felt that Adam Beach would be a good addition to the cast and asked him to be a permanent member beginning with the ninth season. Although Beach felt the role was a "dream role", the character proved unpopular with fans who felt that he was designed to gradually write out either Richard Belzer or Ice-T. Feeling there were too many police characters on the show, Beach left the show after only one season. Michaela McManus was originally felt to be too young for the role of an Assistant District Attorney (ADA) before being cast as ADA Kim Greylek in the tenth season. McManus, months removed from a recurring role on One Tree Hill, remained with the series only half a season, however, before departing for unspecified reasons. Paula Patton joined the cast as ADA Mikka Von. She replaced Stephanie March. However, Patton dropped out after one episode to film Mission: Impossible – Ghost Protocol, and was replaced by Melissa Sagemiller in the recurring role of ADA Gillian Hardwicke. Before the end of season twelve, Mariska Hargitay asked for a lighter workload. As a way of writing her out of certain episodes, a plan to have her character promoted to a supervisory role was discussed. At the end of season twelve, Christopher Meloni departed the cast, unable to reach agreement on a new contract. Warren Leight became the new showrunner during this same year and signed on before he knew that Meloni would be leaving the cast. The second major departure to be announced in 2011 was that of BD Wong. On July 17, Wong announced on Twitter that, "I actually do not return for season 13, I am jumping to Awake! It's awesome!" Wong added, "I don't know if or when I'll be back on SVU! It was amazing to have such a cool job for 11 years and to be a real NY Actor." Wong reprised his role as Dr. Huang in season 13's episode "Father Dearest". In June 2011, it was announced that Kelli Giddish and Danny Pino would join the cast as new series regulars. Weeks later, it was announced that Stephanie March and Diane Neal would be reprising their roles as ADA Alexandra Cabot and ADA Casey Novak, respectively. The launch of season 13 was marked with a retooling of the show that Warren Leight referred to as "SVU 2.0". Changes that accompanied this included Tamara Tunie's being bumped from the main cast to a guest-starring role and recurring actor Joel de la Fuente's not appearing for the first time since 2002. Of the latter change, Warren Leight said, "those scenes [which featured Fuente] can be dry" and hired Gilbert Gottfried as a more comedic replacement. In season 14, Raúl Esparza joined the cast in a recurring capacity as ADA Rafael Barba and prior to the season 15 premiere, Esparza was promoted to a series regular. Also in season 15, Belzer departed the cast in the fifth episode, "Wonderland Story", in which Sgt. Munch retired from the NYPD and took a job in the DA's office as an investigator. Later in the season, Captain Cragen announced his departure from the NYPD, which made newly promoted Sgt. Benson the temporary squad commander. In leaving the cast, Florek ended a 400-episode run as Captain Cragen. In season 16, Peter Scanavino joined the series, first in a recurring role for episodes 1–3 and then was promoted to the main cast in episode 5, with Kelli Giddish, Danny Pino, Ice-T and Raúl Esparza. On May 20, 2015, it was revealed that Danny Pino would be leaving the cast after the season 16 finale "Surrendering Noah". In August 2017, it was announced that Philip Winchester would recur in season 19 as ADA Peter Stone, his character from Chicago P.D. and Chicago Justice, who is the son of Benjamin Stone, the first ADA on the original Law & Order series. It was later also announced that Brooke Shields was enlisted to assume a major recurring role (Sheila Porter, maternal grandmother of Noah Porter-Benson, Olivia's adopted son) starting in season 19 of the long-running dramatic series. On February 7, 2018, Raúl Esparza left the series after six seasons. His role was taken over by Winchester. Upon being renewed for its twenty-first season, it was announced that Winchester would be departing the series after the twentieth season. In March 2019, it was announced that the show would come back for season 21, making it the longest-running primetime U.S. live-action series in the history of television. On March 29, 2019, it was revealed that Winchester would not return for season 21. He tweeted the same day about his departure from the show. On May 16, 2019, the season finale aired and Winchester took to Twitter to thank the cast and crew for the send-off. After recurring for several episodes in season 21 as Vice Officer Katriona Tamin, Jamie Gray Hyder joined the cast as a regular, starting in episode 8. On October 6, 2020, Demore Barnes, who had recurred throughout season 21 as new Deputy Chief Christian Garland, was upgraded to regular status for season 22. On September 3, 2021, it was announced that Hyder and Barnes would both depart the series following the two-hour season 23 premiere. On October 13, 2021, Octavio Pisano, who had guest starred since the start of the season, was promoted to regular status. On August 24, 2022, it was announced that Giddish would leave the series during the first half of season 24, with episode nine as her last appearance as a regular. On November 10, 2022, Molly Burnett, who initially appeared in a recurring capacity for the first six episodes, was promoted to series regular beginning with the seventh episode. On May 19, 2023, Burnett announced that she will leave at the end of the show's twenty-fourth season. On November 28, 2023, it was announced Giddish would return for the twenty-fifth season premiere. On July 22, 2024, it was reported that Kevin Kane, who portrays Terry Bruno, would be promoted to a series regular for the show's twenty-sixth season, after recurring the previous two seasons. On August 7, 2024, it was announced Juliana Aidén Martinez, who would portray Kate Silva, was added as a series regular ahead of the season premiere. On May 6, 2025, it was announced Martinez and Pisano would depart following the conclusion of the twenty-sixth season. Nine days later, it was announced Giddish would return as a series regular for the twenty-seventh season. ==== Salaries ==== By season twelve, both Mariska Hargitay and Christopher Meloni had become among the highest-paid lead actors on a drama, with each earning nearly $400,000 per episode, a salary that TV Guide said was exceeded only by House's Hugh Laurie. During season sixteen, Hargitay was reported to be earning $450,000 per episode, or $10,350,000 per season. In season seventeen, her salary increased to $500,000 per episode. === Filming and location === Many exterior scenes of Law & Order: Special Victims Unit are filmed on location in New York City, Wolf's hometown, throughout all five of New York City's boroughs. Fort Lee, New Jersey served as the filming location for Detective Elliot Stabler's residence in Queens, New York. When searching for a place to film the interiors of the show, the producers found that there were no suitable studio spaces available in New York City. As a result, a space was chosen at NBC's Central Archives building in nearby North Bergen, New Jersey, 53,000 square feet (4,900 m2) of stage area that had been left unused for some time. The Archives building was used for police station and courtroom scenes, with various other locations in Hudson County used for other scenes, such as a scene shot at the Meadowlands Parkway in Secaucus in 2010. The production left New Jersey for New York in 2010, however, when New Jersey Governor Chris Christie suspended the tax credits for film and television production for the Fiscal Year 2011 to close budget gaps. The show moved into the studio space at Chelsea Piers that had been occupied by the original Law & Order series until its cancellation in May 2010. In 2023, filming near the courthouses at Foley Square coincided with media attention on the trial related to the Prosecution of Donald Trump in New York. During external filming in Fort Tryon Park in 2024, it was reported that a young girl looking for her mother mistook Mariska Hargitay for a real police officer. == Episodes == == Release == === Broadcast === Law & Order: SVU airs on NBC in the United States. With the season eleven premiere on September 23, 2009, the series vacated its Tuesday 10 p.m. ET slot as NBC began a nightly prime-time series hosted by Jay Leno. The new time slot became Wednesday nights at 9:00 p.m. ET on NBC, with CTV still airing SVU on Tuesdays at 10:00 in Canada. After the 2010 Winter Olympics on March 3, 2010, the time slot again changed to Wednesdays at 10 p.m. ET, where it stayed until the twelfth season. For the 12th season, SVU moved back to 9:00 p.m. to lead in the newest Law & Order spin-off, Law & Order: LA, until it was pulled from the network in January 2011 to be retooled. SVU moved back to 10:00 p.m. on January 12, 2011, until the end of the 13th season. With season 14, SVU moved back to 9:00 p.m. after a two-hour season premiere event on September 26, 2012. Beginning with Season 20, SVU would air on Thursday nights at 10 p.m., after NBC decided to devote their entire Wednesday primetime lineup to the Chicago Med, PD, and Fire trilogy. It marked the first time ever that Law & Order: SVU would hold this timeslot on Thursday nights. Starting with season 22, SVU moved to 9 p.m., with offshoot Law & Order: Organized Crime taking its old slot. From season 21, Law & Order: Special Victims Unit airs on Sky Witness in the United Kingdom. Beginning from season 23, it moved from CTV to CityTV in Canada, simulcasting with NBC. Law & Order: Special Victims Unit airs on Rock Entertainment in Southeast Asia. === Streaming === Peacock and Hulu currently have all seasons (1–24) available. The latest 5 episodes can be watched for free on NBC.com and the NBC app. Outside of SVOD and NBC platforms, most episodes (outside of seasons 2–4 in the United States for unknown reasons) can be found on electronic sell-through platforms such as iTunes and Amazon Prime Video. The series is available for streaming on Peacock along with Chicago Fire, Chicago P.D., Chicago Med, Law & Order and Law & Order: Criminal Intent. Seasons 1–22 are available for streaming in Australia on Amazon Prime Video. In Brazil, seasons 11 to 13 are available on Amazon Prime Video, and seasons 1–22 are available on Globoplay, although seasons 15–22 require a subscription expansion or cable access to UniversalTV. In 2024, selected seasons returned to Netflix in certain regions including the UK, Europe, Africa, Australia, New Zealand, and Latin America. === Syndication === As of January 2024, the series is rerun on fellow NBCUniversal network USA, as well as on Ion Television and MyNetworkTV stations. The series also briefly ran on Syfy in 2006. In 2008, Fox obtained rights to air Law & Order: SVU on Fox-owned TV stations, and began doing so in the fall of 2009. == Reception == === Ratings === In 2016, a New York Times study of the 50 TV shows with the most Facebook likes found that SVU's popularity was "atypical: generally slightly more popular in rural areas and the South, but largely restricted to the eastern half of the country. It is most popular in Albany, N.Y.; least in Colorado and Utah". === Awards and honors === Law & Order: Special Victims Unit has received many awards and award nominations. Mariska Hargitay has twice been nominated for a Golden Globe Award and won once in 2005. The show has been nominated numerous times for the Emmy Award. Mariska Hargitay has been nominated for the Outstanding Lead Actress in a Drama Series category eight years in a row beginning in 2004 and won the Emmy in 2006. Christopher Meloni was nominated for the Outstanding Lead Actor in a Drama Series category in 2006. Robin Williams was nominated in the Outstanding Guest Actor in a Drama Series in 2008. The series was nominated in the category Outstanding Guest Actress in a Drama Series for Jane Alexander and Tracy Pollan in 2000, Martha Plimpton in 2002, Barbara Barrie in 2003, Mare Winningham and Marlee Matlin in 2004, Amanda Plummer and Angela Lansbury in 2005, Marcia Gay Harden and Leslie Caron in 2007, Cynthia Nixon in 2008, Ellen Burstyn, Brenda Blethyn, and Carol Burnett in 2009, and Ann-Margret in 2010. The series won the award for Plummer in 2005, Caron in 2007, Nixon in 2008, Burstyn in 2009, and Ann-Margret in 2010. === Critical reception === Law & Order: Special Victims Unit has been well-received among critics. The show holds an average score of 88% on Rotten Tomatoes. Metacritic gives a score of 66%, from 25 critics review. In 2014, over 14 years after the show's debut, Joshua Alston of The A.V. Club wrote it that "while SVU isn't yet television's best cop show, it’s absolutely its most improved, and that uptick in quality is all the more admirable given that, as the only L&O game in town, it could have just as easily embraced predictability rather than injecting a risky new energy". == Russian adaptation == In 2007, the Russian production company Studio 2B purchased the rights to create an adaptation of Law & Order: Special Victims Unit for Russian television. Titled Law & Order: Division of Field Investigation, the series stars Alisa Bogart and Vica Fiorelia. It follows a unit of investigators in Moscow whose job is to investigate crimes of a sexual nature. The series aired on NTV until 2010 and was produced by Pavel Korchagin, Felix Kleiman and Edward Verzbovski and directed by Dmitry Brusnikin. The screenplays were written by Sergei Kuznvetsov, Elena Karavaeshnikova, and Maya Shapovalova. == Spin-off == On March 31, 2020, it was announced that NBC had ordered an untitled spin-off series to launch in the 2020–21 television season, with Christopher Meloni reprising his role as Elliot Stabler. Meloni left SVU in 2011. The series order consists of 13 episodes. On June 2, 2020, it was announced that the series would be called Law & Order: Organized Crime and writer Craig Gore had been fired. When NBC announced its fall schedule on June 16, Organized Crime was the only new show on the schedule, slotted for Thursdays at 10/9c. However, the series was later delayed to 2021. On October 2, 2020, it was announced that Matt Olmstead would be stepping down as showrunner and a replacement was not announced at the time. On December 9, 2020, it was announced that Ilene Chaiken has joined as showrunner after her overall deal with Universal Television. Dylan McDermott was announced on January 27, 2021, as joining the cast in an unspecified role. On February 2, 2021, Tamara Taylor was cast in an undisclosed role. On February 4, 2021, it was announced that the series would premiere on April 1, 2021, in a two-hour crossover event. == Explanatory notes == == References == === Citations === === General and cited references === Green, Susan; Dawn, Randee (2009). Law & Order: Special Victims Unit: The Unofficial Companion. Dallas: BenBella Books. ISBN 978-1-933771-88-5. OCLC 429604907. == External links == Official website on Wolf Entertainment Official website on NBC Law & Order: Special Victims Unit at IMDb Law & Order: Special Victims Unit at epguides.com Law & Order: Special Victims Unit on Metacritic Law & Order: Special Victims Unit on Rotten Tomatoes "List of Law & Order: SVU Episodes". TV Guide. Law & Order: Special Victims Unit on MyNetworkTV
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Wikipedia:Ordered exponential#0
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The ordered exponential, also called the path-ordered exponential, is a mathematical operation defined in non-commutative algebras, equivalent to the exponential of the integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. It is a kind of product integral, or Volterra integral. == Definition == Let A be an algebra over a field K, and a(t) be an element of A parameterized by the real numbers, a : R → A . {\displaystyle a:\mathbb {R} \to A.} The parameter t in a(t) is often referred to as the time parameter in this context. The ordered exponential of a is denoted OE [ a ] ( t ) ≡ T { e ∫ 0 t a ( t ′ ) d t ′ } ≡ ∑ n = 0 ∞ 1 n ! ∫ 0 t d t 1 ′ ⋯ ∫ 0 t d t n ′ T { a ( t 1 ′ ) ⋯ a ( t n ′ ) } = ∑ n = 0 ∞ ∫ 0 t d t 1 ′ ∫ 0 t 1 ′ d t 2 ′ ∫ 0 t 2 ′ d t 3 ′ ⋯ ∫ 0 t n − 1 ′ d t n ′ a ( t n ′ ) ⋯ a ( t 1 ′ ) {\displaystyle {\begin{aligned}\operatorname {OE} [a](t)\equiv {\mathcal {T}}\left\{e^{\int _{0}^{t}a(t')\,dt'}\right\}&\equiv \sum _{n=0}^{\infty }{\frac {1}{n!}}\int _{0}^{t}dt'_{1}\cdots \int _{0}^{t}dt'_{n}\;{\mathcal {T}}\left\{a(t'_{1})\cdots a(t'_{n})\right\}\\&=\sum _{n=0}^{\infty }\int _{0}^{t}dt'_{1}\int _{0}^{t'_{1}}dt'_{2}\int _{0}^{t'_{2}}dt'_{3}\cdots \int _{0}^{t'_{n-1}}dt'_{n}\;a(t'_{n})\cdots a(t'_{1})\end{aligned}}} where the term n = 0 is equal to 1 and where T {\displaystyle {\mathcal {T}}} is the time-ordering operator. It is a higher-order operation that ensures the exponential is time-ordered, so that any product of a(t) that occurs in the expansion of the exponential is ordered such that the value of t is increasing from right to left of the product. For example: T { a ( 1.2 ) a ( 9.5 ) a ( 4.1 ) } = a ( 9.5 ) a ( 4.1 ) a ( 1.2 ) . {\displaystyle {\mathcal {T}}\left\{a(1.2)a(9.5)a(4.1)\right\}=a(9.5)a(4.1)a(1.2).} Time ordering is required, as products in the algebra are not necessarily commutative. The operation maps a parameterized element onto another parameterized element, or symbolically, OE : ( R → A ) → ( R → A ) . {\displaystyle \operatorname {OE} \mathrel {:} (\mathbb {R} \to A)\to (\mathbb {R} \to A).} There are various ways to define this integral more rigorously. === Product of exponentials === The ordered exponential can be defined as the left product integral of the infinitesimal exponentials, or equivalently, as an ordered product of exponentials in the limit as the number of terms grows to infinity: OE [ a ] ( t ) = ∏ 0 t e a ( t ′ ) d t ′ ≡ lim N → ∞ ( e a ( t N ) Δ t e a ( t N − 1 ) Δ t ⋯ e a ( t 1 ) Δ t e a ( t 0 ) Δ t ) {\displaystyle \operatorname {OE} [a](t)=\prod _{0}^{t}e^{a(t')\,dt'}\equiv \lim _{N\to \infty }\left(e^{a(t_{N})\,\Delta t}e^{a(t_{N-1})\,\Delta t}\cdots e^{a(t_{1})\,\Delta t}e^{a(t_{0})\,\Delta t}\right)} where the time moments {t0, ..., tN} are defined as ti ≡ i Δt for i = 0, ..., N, and Δt ≡ t / N. The ordered exponential is in fact a geometric integral. === Solution to a differential equation === The ordered exponential is unique solution of the initial value problem: d d t OE [ a ] ( t ) = a ( t ) OE [ a ] ( t ) , OE [ a ] ( 0 ) = 1. {\displaystyle {\begin{aligned}{\frac {d}{dt}}\operatorname {OE} [a](t)&=a(t)\operatorname {OE} [a](t),\\[5pt]\operatorname {OE} [a](0)&=1.\end{aligned}}} === Solution to an integral equation === The ordered exponential is the solution to the integral equation: OE [ a ] ( t ) = 1 + ∫ 0 t a ( t ′ ) OE [ a ] ( t ′ ) d t ′ . {\displaystyle \operatorname {OE} [a](t)=1+\int _{0}^{t}a(t')\operatorname {OE} [a](t')\,dt'.} This equation is equivalent to the previous initial value problem. === Infinite series expansion === The ordered exponential can be defined as an infinite sum, OE [ a ] ( t ) = 1 + ∫ 0 t a ( t 1 ) d t 1 + ∫ 0 t d t 1 ∫ 0 t 1 d t 2 a ( t 1 ) a ( t 2 ) + ⋯ . {\displaystyle \operatorname {OE} [a](t)=1+\int _{0}^{t}a(t_{1})\,dt_{1}+\int _{0}^{t}dt_{1}\int _{0}^{t_{1}}dt_{2}\;a(t_{1})a(t_{2})+\cdots .} This can be derived by recursively substituting the integral equation into itself. == Example == Given a manifold M {\displaystyle M} where for a e ∈ T M {\displaystyle e\in TM} with group transformation g : e ↦ g e {\displaystyle g:e\mapsto ge} it holds at a point x ∈ M {\displaystyle x\in M} : d e ( x ) + J ( x ) e ( x ) = 0. {\displaystyle de(x)+\operatorname {J} (x)e(x)=0.} Here, d {\displaystyle d} denotes exterior differentiation and J ( x ) {\displaystyle \operatorname {J} (x)} is the connection operator (1-form field) acting on e ( x ) {\displaystyle e(x)} . When integrating above equation it holds (now, J ( x ) {\displaystyle \operatorname {J} (x)} is the connection operator expressed in a coordinate basis) e ( y ) = P exp ( − ∫ x y J ( γ ( t ) ) γ ′ ( t ) d t ) e ( x ) {\displaystyle e(y)=\operatorname {P} \exp \left(-\int _{x}^{y}J(\gamma (t))\gamma '(t)\,dt\right)e(x)} with the path-ordering operator P {\displaystyle \operatorname {P} } that orders factors in order of the path γ ( t ) ∈ M {\displaystyle \gamma (t)\in M} . For the special case that J ( x ) {\displaystyle \operatorname {J} (x)} is an antisymmetric operator and γ {\displaystyle \gamma } is an infinitesimal rectangle with edge lengths | u | , | v | {\displaystyle |u|,|v|} and corners at points x , x + u , x + u + v , x + v , {\displaystyle x,x+u,x+u+v,x+v,} above expression simplifies as follows : OE [ − J ] e ( x ) = exp [ − J ( x + v ) ( − v ) ] exp [ − J ( x + u + v ) ( − u ) ] exp [ − J ( x + u ) v ] exp [ − J ( x ) u ] e ( x ) = [ 1 − J ( x + v ) ( − v ) ] [ 1 − J ( x + u + v ) ( − u ) ] [ 1 − J ( x + u ) v ] [ 1 − J ( x ) u ] e ( x ) . {\displaystyle {\begin{aligned}&\operatorname {OE} [-\operatorname {J} ]e(x)\\[5pt]={}&\exp[-\operatorname {J} (x+v)(-v)]\exp[-\operatorname {J} (x+u+v)(-u)]\exp[-\operatorname {J} (x+u)v]\exp[-\operatorname {J} (x)u]e(x)\\[5pt]={}&[1-\operatorname {J} (x+v)(-v)][1-\operatorname {J} (x+u+v)(-u)][1-\operatorname {J} (x+u)v][1-\operatorname {J} (x)u]e(x).\end{aligned}}} Hence, it holds the group transformation identity OE [ − J ] ↦ g OE [ J ] g − 1 {\displaystyle \operatorname {OE} [-\operatorname {J} ]\mapsto g\operatorname {OE} [\operatorname {J} ]g^{-1}} . If − J ( x ) {\displaystyle -\operatorname {J} (x)} is a smooth connection, expanding above quantity to second order in infinitesimal quantities | u | , | v | {\displaystyle |u|,|v|} one obtains for the ordered exponential the identity with a correction term that is proportional to the curvature tensor. == See also == Path-ordering (essentially the same concept) Magnus expansion Product integral Haar measure List of derivatives and integrals in alternative calculi Indefinite product Fractal derivative == References == == External links == Non-Newtonian calculus website
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Wikipedia:Ore algebra#0
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In computer algebra, an Ore algebra is a special kind of iterated Ore extension that can be used to represent linear functional operators, including linear differential and/or recurrence operators. The concept is named after Øystein Ore. == Definition == Let K {\displaystyle K} be a (commutative) field and A = K [ x 1 , … , x s ] {\displaystyle A=K[x_{1},\ldots ,x_{s}]} be a commutative polynomial ring (with A = K {\displaystyle A=K} when s = 0 {\displaystyle s=0} ). The iterated skew polynomial ring A [ ∂ 1 ; σ 1 , δ 1 ] ⋯ [ ∂ r ; σ r , δ r ] {\displaystyle A[\partial _{1};\sigma _{1},\delta _{1}]\cdots [\partial _{r};\sigma _{r},\delta _{r}]} is called an Ore algebra when the σ i {\displaystyle \sigma _{i}} and δ j {\displaystyle \delta _{j}} commute for i ≠ j {\displaystyle i\neq j} , and satisfy σ i ( ∂ j ) = ∂ j {\displaystyle \sigma _{i}(\partial _{j})=\partial _{j}} , δ i ( ∂ j ) = 0 {\displaystyle \delta _{i}(\partial _{j})=0} for i > j {\displaystyle i>j} . == Properties == Ore algebras satisfy the Ore condition, and thus can be embedded in a (skew) field of fractions. The constraint of commutation in the definition makes Ore algebras have a non-commutative generalization theory of Gröbner basis for their left ideals. == References ==
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Wikipedia:Orientation (vector space)#0
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The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a displacement alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral). == Definition == Let V be a finite-dimensional real vector space and let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations. The property of having the same orientation defines an equivalence relation on the set of all ordered bases for V. If V is non-zero, there are precisely two equivalence classes determined by this relation. An orientation on V is an assignment of +1 to one equivalence class and −1 to the other. Every ordered basis lives in one equivalence class or another. Thus any choice of a privileged ordered basis for V determines an orientation: the orientation class of the privileged basis is declared to be positive. For example, the standard basis on Rn provides a standard orientation on Rn (in turn, the orientation of the standard basis depends on the orientation of the Cartesian coordinate system on which it is built). Any choice of a linear isomorphism between V and Rn will then provide an orientation on V. The ordering of elements in a basis is crucial. Two bases with a different ordering will differ by some permutation. They will have the same/opposite orientations according to whether the signature of this permutation is ±1. This is because the determinant of a permutation matrix is equal to the signature of the associated permutation. Similarly, let A be a nonsingular linear mapping of vector space Rn to Rn. This mapping is orientation-preserving if its determinant is positive. For instance, in R3 a rotation around the Z Cartesian axis by an angle α is orientation-preserving: A 1 = ( cos α − sin α 0 sin α cos α 0 0 0 1 ) {\displaystyle \mathbf {A} _{1}={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}} while a reflection by the XY Cartesian plane is not orientation-preserving: A 2 = ( 1 0 0 0 1 0 0 0 − 1 ) {\displaystyle \mathbf {A} _{2}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}}} === Zero-dimensional case === The concept of orientation degenerates in the zero-dimensional case. A zero-dimensional vector space has only a single point, the zero vector. Consequently, the only basis of a zero-dimensional vector space is the empty set ∅ {\displaystyle \emptyset } . Therefore, there is a single equivalence class of ordered bases, namely, the class { ∅ } {\displaystyle \{\emptyset \}} whose sole member is the empty set. This means that an orientation of a zero-dimensional space is a function { { ∅ } } → { ± 1 } . {\displaystyle \{\{\emptyset \}\}\to \{\pm 1\}.} It is therefore possible to orient a point in two different ways, positive and negative. Because there is only a single ordered basis ∅ {\displaystyle \emptyset } , a zero-dimensional vector space is the same as a zero-dimensional vector space with ordered basis. Choosing { ∅ } ↦ + 1 {\displaystyle \{\emptyset \}\mapsto +1} or { ∅ } ↦ − 1 {\displaystyle \{\emptyset \}\mapsto -1} therefore chooses an orientation of every basis of every zero-dimensional vector space. If all zero-dimensional vector spaces are assigned this orientation, then, because all isomorphisms among zero-dimensional vector spaces preserve the ordered basis, they also preserve the orientation. This is unlike the case of higher-dimensional vector spaces where there is no way to choose an orientation so that it is preserved under all isomorphisms. However, there are situations where it is desirable to give different orientations to different points. For example, consider the fundamental theorem of calculus as an instance of Stokes' theorem. A closed interval [a, b] is a one-dimensional manifold with boundary, and its boundary is the set {a, b}. In order to get the correct statement of the fundamental theorem of calculus, the point b should be oriented positively, while the point a should be oriented negatively. === On a line === The one-dimensional case deals with an oriented line or directed line, which may be traversed in one of two directions. In real coordinate space, an oriented line is also known as an axis. There are two orientations to a line just as there are two orientations to an oriented circle (clockwise and anti-clockwise). A semi-infinite oriented line is called a ray. In the case of a line segment (a connected subset of a line), the two possible orientations result in directed line segments. === On a surface === An orientable surface sometimes has the selected orientation indicated by the orientation of a surface normal. An oriented plane can be defined by a pseudovector. == Alternate viewpoints == === Multilinear algebra === For any n-dimensional real vector space V we can form the kth-exterior power of V, denoted ΛkV. This is a real vector space of dimension ( n k ) {\displaystyle {\tbinom {n}{k}}} . The vector space ΛnV (called the top exterior power) therefore has dimension 1. That is, ΛnV is just a real line. There is no a priori choice of which direction on this line is positive. An orientation is just such a choice. Any nonzero linear form ω on ΛnV determines an orientation of V by declaring that x is in the positive direction when ω(x) > 0. To connect with the basis point of view we say that the positively-oriented bases are those on which ω evaluates to a positive number (since ω is an n-form we can evaluate it on an ordered set of n vectors, giving an element of R). The form ω is called an orientation form. If {ei} is a privileged basis for V and {ei∗} is the dual basis, then the orientation form giving the standard orientation is e1∗ ∧ e2∗ ∧ … ∧ en∗. The connection of this with the determinant point of view is: the determinant of an endomorphism T : V → V {\displaystyle T:V\to V} can be interpreted as the induced action on the top exterior power. === Lie group theory === Let B be the set of all ordered bases for V. Then the general linear group GL(V) acts freely and transitively on B. (In fancy language, B is a GL(V)-torsor). This means that as a manifold, B is (noncanonically) homeomorphic to GL(V). Note that the group GL(V) is not connected, but rather has two connected components according to whether the determinant of the transformation is positive or negative (except for GL0, which is the trivial group and thus has a single connected component; this corresponds to the canonical orientation on a zero-dimensional vector space). The identity component of GL(V) is denoted GL+(V) and consists of those transformations with positive determinant. The action of GL+(V) on B is not transitive: there are two orbits which correspond to the connected components of B. These orbits are precisely the equivalence classes referred to above. Since B does not have a distinguished element (i.e. a privileged basis) there is no natural choice of which component is positive. Contrast this with GL(V) which does have a privileged component: the component of the identity. A specific choice of homeomorphism between B and GL(V) is equivalent to a choice of a privileged basis and therefore determines an orientation. More formally: π 0 ( GL ( V ) ) = ( GL ( V ) / GL + ( V ) = { ± 1 } {\displaystyle \pi _{0}(\operatorname {GL} (V))=(\operatorname {GL} (V)/\operatorname {GL} ^{+}(V)=\{\pm 1\}} , and the Stiefel manifold of n-frames in V {\displaystyle V} is a GL ( V ) {\displaystyle \operatorname {GL} (V)} -torsor, so V n ( V ) / GL + ( V ) {\displaystyle V_{n}(V)/\operatorname {GL} ^{+}(V)} is a torsor over { ± 1 } {\displaystyle \{\pm 1\}} , i.e., its 2 points, and a choice of one of them is an orientation. === Geometric algebra === The various objects of geometric algebra are charged with three attributes or features: attitude, orientation, and magnitude. For example, a vector has an attitude given by a straight line parallel to it, an orientation given by its sense (often indicated by an arrowhead) and a magnitude given by its length. Similarly, a bivector in three dimensions has an attitude given by the family of planes associated with it (possibly specified by the normal line common to these planes ), an orientation (sometimes denoted by a curved arrow in the plane) indicating a choice of sense of traversal of its boundary (its circulation), and a magnitude given by the area of the parallelogram defined by its two vectors. == Orientation on manifolds == Each point p on an n-dimensional differentiable manifold has a tangent space TpM which is an n-dimensional real vector space. Each of these vector spaces can be assigned an orientation. Some orientations "vary smoothly" from point to point. Due to certain topological restrictions, this is not always possible. A manifold that admits a smooth choice of orientations for its tangent spaces is said to be orientable. == See also == Cartesian coordinate system – Most common coordinate system (geometry) Chirality (mathematics) – Property of an object that is not congruent to its mirror image Even and odd permutations – Property in group theoryPages displaying short descriptions of redirect targets Orientation of a vector bundle – Generalization of an orientation of a vector space Pseudovector – Physical quantity that changes sign with improper rotation Rotation formalisms in three dimensions – Ways to represent 3D rotations Right-hand rule – Mnemonic for understanding orientation of vectors in 3D space Sign convention – Agreed-upon meaning of a physical quantity being positive or negative == References == == External links == "Orientation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Wikipedia:Orientation of a vector bundle#0
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In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map) ϕ U : π − 1 ( U ) → U × R n {\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}} is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant. If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n). A vector bundle together with an orientation is called an oriented bundle. A vector bundle that can be given an orientation is called an orientable vector bundle. The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a Gysin sequence. == Examples == A complex vector bundle is oriented in a canonical way. The notion of an orientation of a vector bundle generalizes an orientation of a differentiable manifold: an orientation of a differentiable manifold is an orientation of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.) == Operations == To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) determinant bundle det E = ∧ n E {\displaystyle \operatorname {det} E=\wedge ^{n}E} of E. Similarly, to give an orientation to E is to give an orientation to the unit sphere bundle of E. Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces. == Thom space == From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle E of rank n means a choice (and existence) of a class u ∈ H n ( T ( E ) ; Λ ) {\displaystyle u\in H^{n}(T(E);\Lambda )} in the cohomology ring of the Thom space T(E) such that u generates H ~ ∗ ( T ( E ) ; Λ ) {\displaystyle {\tilde {H}}^{*}(T(E);\Lambda )} as a free H ∗ ( E ; Λ ) {\displaystyle H^{*}(E;\Lambda )} -module globally and locally: i.e., H ∗ ( E ; Λ ) → H ~ ∗ ( T ( E ) ; Λ ) , x ↦ x ⌣ u {\displaystyle H^{*}(E;\Lambda )\to {\tilde {H}}^{*}(T(E);\Lambda ),x\mapsto x\smile u} is an isomorphism (called the Thom isomorphism), where "tilde" means reduced cohomology, that restricts to each isomorphism H ∗ ( π − 1 ( U ) ; Λ ) → H ~ ∗ ( T ( E | U ) ; Λ ) {\displaystyle H^{*}(\pi ^{-1}(U);\Lambda )\to {\tilde {H}}^{*}(T(E|_{U});\Lambda )} induced by the trivialization π − 1 ( U ) ≃ U × R n {\displaystyle \pi ^{-1}(U)\simeq U\times \mathbf {R} ^{n}} . One can show, with some work, that the usual notion of an orientation coincides with a Z-orientation. == See also == The integration along the fiber Orientation bundle (or orientation sheaf) - this is used to formulate the Thom isomorphism for non-oriented bundles. == References == Bott, Raoul; Tu, Loring (1982), Differential Forms in Algebraic Topology, New York: Springer, ISBN 0-387-90613-4 J.P. May, A Concise Course in Algebraic Topology. University of Chicago Press, 1999. Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9
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Wikipedia:Orthant#0
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In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2n orthants in n-dimensional space. More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: ε1x1 ≥ 0 ε2x2 ≥ 0 · · · εnxn ≥ 0, where each εi is +1 or −1. Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities ε1x1 > 0 ε2x2 > 0 · · · εnxn > 0, where each εi is +1 or −1. By dimension: In one dimension, an orthant is a ray. In two dimensions, an orthant is a quadrant. In three dimensions, an orthant is an octant. John Conway and Neil Sloane defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant. The nonnegative orthant is the generalization of the first quadrant to n-dimensions and is important in many constrained optimization problems. == See also == Cross polytope (or orthoplex) – a family of regular polytopes in n-dimensions which can be constructed with one simplex facets in each orthant space. Measure polytope (or hypercube) – a family of regular polytopes in n-dimensions which can be constructed with one vertex in each orthant space. Orthotope – generalization of a rectangle in n-dimensions, with one vertex in each orthant. == References == == Further reading == The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.113
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Wikipedia:Orthogonal Procrustes problem#0
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The orthogonal Procrustes problem is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\displaystyle B} . Specifically, the orthogonal Procrustes problem is an optimization problem given by minimize Ω ‖ Ω A − B ‖ F subject to Ω T Ω = I , {\displaystyle {\begin{aligned}{\underset {\Omega }{\text{minimize}}}\quad &\|\Omega A-B\|_{F}\\{\text{subject to}}\quad &\Omega ^{T}\Omega =I,\end{aligned}}} where ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is that Wahba's problem tries to find a proper rotation matrix instead of just an orthogonal one. The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off. == Solution == This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika. This problem is equivalent to finding the nearest orthogonal matrix to a given matrix M = B A T {\displaystyle M=BA^{T}} , i.e. solving the closest orthogonal approximation problem min R ‖ R − M ‖ F s u b j e c t t o R T R = I {\displaystyle \min _{R}\|R-M\|_{F}\quad \mathrm {subject\ to} \quad R^{T}R=I} . To find matrix R {\displaystyle R} , one uses the singular value decomposition (for which the entries of Σ {\displaystyle \Sigma } are non-negative) M = U Σ V T {\displaystyle M=U\Sigma V^{T}\,\!} to write R = U V T . {\displaystyle R=UV^{T}.\,\!} === Proof of Solution === One proof depends on the basic properties of the Frobenius inner product that induces the Frobenius norm: R = arg min Ω | | Ω A − B ‖ F 2 = arg min Ω ⟨ Ω A − B , Ω A − B ⟩ F = arg min Ω ‖ Ω A ‖ F 2 + ‖ B ‖ F 2 − 2 ⟨ Ω A , B ⟩ F = arg min Ω ‖ A ‖ F 2 + ‖ B ‖ F 2 − 2 ⟨ Ω A , B ⟩ F = arg max Ω ⟨ Ω A , B ⟩ F = arg max Ω ⟨ Ω , B A T ⟩ F = arg max Ω ⟨ Ω , U Σ V T ⟩ F = arg max Ω ⟨ U T Ω V , Σ ⟩ F = arg max Ω ⟨ S , Σ ⟩ F where S = U T Ω V {\displaystyle {\begin{aligned}R&=\arg \min _{\Omega }||\Omega A-B\|_{F}^{2}\\&=\arg \min _{\Omega }\langle \Omega A-B,\Omega A-B\rangle _{F}\\&=\arg \min _{\Omega }\|\Omega A\|_{F}^{2}+\|B\|_{F}^{2}-2\langle \Omega A,B\rangle _{F}\\&=\arg \min _{\Omega }\|A\|_{F}^{2}+\|B\|_{F}^{2}-2\langle \Omega A,B\rangle _{F}\\&=\arg \max _{\Omega }\langle \Omega A,B\rangle _{F}\\&=\arg \max _{\Omega }\langle \Omega ,BA^{T}\rangle _{F}\\&=\arg \max _{\Omega }\langle \Omega ,U\Sigma V^{T}\rangle _{F}\\&=\arg \max _{\Omega }\langle U^{T}\Omega V,\Sigma \rangle _{F}\\&=\arg \max _{\Omega }\langle S,\Sigma \rangle _{F}\quad {\text{where }}S=U^{T}\Omega V\\\end{aligned}}} This quantity S {\displaystyle S} is an orthogonal matrix (as it is a product of orthogonal matrices) and thus the expression is maximised when S {\displaystyle S} equals the identity matrix I {\displaystyle I} . Thus I = U T R V R = U V T {\displaystyle {\begin{aligned}I&=U^{T}RV\\R&=UV^{T}\\\end{aligned}}} where R {\displaystyle R} is the solution for the optimal value of Ω {\displaystyle \Omega } that minimizes the norm squared | | Ω A − B ‖ F 2 {\displaystyle ||\Omega A-B\|_{F}^{2}} . == Generalized/constrained Procrustes problems == There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal. Alternately, one might constrain it by only allowing rotation matrices (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices). In this case, one can write (using the above decomposition M = U Σ V T {\displaystyle M=U\Sigma V^{T}} ) R = U Σ ′ V T , {\displaystyle R=U\Sigma 'V^{T},\,\!} where Σ ′ {\displaystyle \Sigma '\,\!} is a modified Σ {\displaystyle \Sigma \,\!} , with the smallest singular value replaced by det ( U V T ) {\displaystyle \det(UV^{T})} (+1 or -1), and the other singular values replaced by 1, so that the determinant of R is guaranteed to be positive. For more information, see the Kabsch algorithm. The unbalanced Procrustes problem concerns minimizing the norm of A U − B {\displaystyle AU-B} , where A ∈ R m × ℓ , U ∈ R ℓ × n {\displaystyle A\in \mathbb {R} ^{m\times \ell },U\in \mathbb {R} ^{\ell \times n}} , and B ∈ R m × n {\displaystyle B\in \mathbb {R} ^{m\times n}} , with m > ℓ ≥ n {\displaystyle m>\ell \geq n} , or alternately with complex valued matrices. This is a problem over the Stiefel manifold U ∈ U ( m , ℓ ) {\displaystyle U\in U(m,\ell )} , and has no currently known closed form. To distinguish, the standard Procrustes problem ( A ∈ R m × m {\displaystyle A\in \mathbb {R} ^{m\times m}} ) is referred to as the balanced problem in these contexts. == See also == Procrustes analysis Procrustes transformation Wahba's problem Kabsch algorithm Point set registration == References ==
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Wikipedia:Orthogonal basis#0
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In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V {\displaystyle V} is a basis for V {\displaystyle V} whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. == As coordinates == Any orthogonal basis can be used to define a system of orthogonal coordinates V . {\displaystyle V.} Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates in Euclidean spaces, as well as in Riemannian and pseudo-Riemannian manifolds. == In functional analysis == In functional analysis, an orthogonal basis is any basis obtained from an orthonormal basis (or Hilbert basis) using multiplication by nonzero scalars. == Extensions == === Symmetric bilinear form === The concept of an orthogonal basis is applicable to a vector space V {\displaystyle V} (over any field) equipped with a symmetric bilinear form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , where orthogonality of two vectors v {\displaystyle v} and w {\displaystyle w} means ⟨ v , w ⟩ = 0 {\displaystyle \langle v,w\rangle =0} . For an orthogonal basis { e k } {\displaystyle \left\{e_{k}\right\}} : ⟨ e j , e k ⟩ = { q ( e k ) j = k 0 j ≠ k , {\displaystyle \langle e_{j},e_{k}\rangle ={\begin{cases}q(e_{k})&j=k\\0&j\neq k,\end{cases}}} where q {\displaystyle q} is a quadratic form associated with ⟨ ⋅ , ⋅ ⟩ : {\displaystyle \langle \cdot ,\cdot \rangle :} q ( v ) = ⟨ v , v ⟩ {\displaystyle q(v)=\langle v,v\rangle } (in an inner product space, q ( v ) = ‖ v ‖ 2 {\displaystyle q(v)=\Vert v\Vert ^{2}} ). Hence for an orthogonal basis { e k } {\displaystyle \left\{e_{k}\right\}} , ⟨ v , w ⟩ = ∑ k q ( e k ) v k w k , {\displaystyle \langle v,w\rangle =\sum _{k}q(e_{k})v_{k}w_{k},} where v k {\displaystyle v_{k}} and w k {\displaystyle w_{k}} are components of v {\displaystyle v} and w {\displaystyle w} in the basis. === Quadratic form === The concept of orthogonality may be extended to a vector space over any field of characteristic not 2 equipped with a quadratic form q ( v ) {\displaystyle q(v)} . Starting from the observation that, when the characteristic of the underlying field is not 2, the associated symmetric bilinear form ⟨ v , w ⟩ = 1 2 ( q ( v + w ) − q ( v ) − q ( w ) ) {\displaystyle \langle v,w\rangle ={\tfrac {1}{2}}(q(v+w)-q(v)-q(w))} allows vectors v {\displaystyle v} and w {\displaystyle w} to be defined as being orthogonal with respect to q {\displaystyle q} when q ( v + w ) − q ( v ) − q ( w ) = 0 {\displaystyle q(v+w)-q(v)-q(w)=0} . == See also == Basis (linear algebra) – Set of vectors used to define coordinates Orthonormal basis – Specific linear basis (mathematics) Orthonormal frame – Euclidean space without distance and angles Schauder basis – Computational tool Total set – subset of a topological vector space whose linear span is densePages displaying wikidata descriptions as a fallback == References == Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, pp. 572–585, ISBN 978-0-387-95385-4 Milnor, J.; Husemoller, D. (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. p. 6. ISBN 3-540-06009-X. Zbl 0292.10016. == External links == Weisstein, Eric W. "Orthogonal Basis". MathWorld.
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Wikipedia:Orthogonal complement#0
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In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped with a bilinear form B {\displaystyle B} is the set W ⊥ {\displaystyle W^{\perp }} of all vectors in V {\displaystyle V} that are orthogonal to every vector in W {\displaystyle W} . Informally, it is called the perp, short for perpendicular complement. It is a subspace of V {\displaystyle V} . == Example == Let V = ( R 5 , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle V=(\mathbb {R} ^{5},\langle \cdot ,\cdot \rangle )} be the vector space equipped with the usual dot product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } (thus making it an inner product space), and let W = { u ∈ V : A x = u , x ∈ R 2 } , {\displaystyle W=\{\mathbf {u} \in V:\mathbf {A} x=\mathbf {u} ,\ x\in \mathbb {R} ^{2}\},} with A = ( 1 0 0 1 2 6 3 9 5 3 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&1\\2&6\\3&9\\5&3\\\end{pmatrix}}.} then its orthogonal complement W ⊥ = { v ∈ V : ⟨ u , v ⟩ = 0 ∀ u ∈ W } {\displaystyle W^{\perp }=\{\mathbf {v} \in V:\langle \mathbf {u} ,\mathbf {v} \rangle =0\ \ \forall \ \mathbf {u} \in W\}} can also be defined as W ⊥ = { v ∈ V : A ~ y = v , y ∈ R 3 } , {\displaystyle W^{\perp }=\{\mathbf {v} \in V:\mathbf {\tilde {A}} y=\mathbf {v} ,\ y\in \mathbb {R} ^{3}\},} being A ~ = ( − 2 − 3 − 5 − 6 − 9 − 3 1 0 0 0 1 0 0 0 1 ) . {\displaystyle \mathbf {\tilde {A}} ={\begin{pmatrix}-2&-3&-5\\-6&-9&-3\\1&0&0\\0&1&0\\0&0&1\end{pmatrix}}.} The fact that every column vector in A {\displaystyle \mathbf {A} } is orthogonal to every column vector in A ~ {\displaystyle \mathbf {\tilde {A}} } can be checked by direct computation. The fact that the spans of these vectors are orthogonal then follows by bilinearity of the dot product. Finally, the fact that these spaces are orthogonal complements follows from the dimension relationships given below. == General bilinear forms == Let V {\displaystyle V} be a vector space over a field F {\displaystyle \mathbb {F} } equipped with a bilinear form B . {\displaystyle B.} We define u {\displaystyle \mathbf {u} } to be left-orthogonal to v {\displaystyle \mathbf {v} } , and v {\displaystyle \mathbf {v} } to be right-orthogonal to u {\displaystyle \mathbf {u} } , when B ( u , v ) = 0. {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0.} For a subset W {\displaystyle W} of V , {\displaystyle V,} define the left-orthogonal complement W ⊥ {\displaystyle W^{\perp }} to be W ⊥ = { x ∈ V : B ( x , y ) = 0 ∀ y ∈ W } . {\displaystyle W^{\perp }=\left\{\mathbf {x} \in V:B(\mathbf {x} ,\mathbf {y} )=0\ \ \forall \ \mathbf {y} \in W\right\}.} There is a corresponding definition of the right-orthogonal complement. For a reflexive bilinear form, where B ( u , v ) = 0 ⟹ B ( v , u ) = 0 ∀ u , v ∈ V {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0\implies B(\mathbf {v} ,\mathbf {u} )=0\ \ \forall \ \mathbf {u} ,\mathbf {v} \in V} , the left and right complements coincide. This will be the case if B {\displaystyle B} is a symmetric or an alternating form. The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation. === Properties === An orthogonal complement is a subspace of V {\displaystyle V} ; If X ⊆ Y {\displaystyle X\subseteq Y} then X ⊥ ⊇ Y ⊥ {\displaystyle X^{\perp }\supseteq Y^{\perp }} ; The radical V ⊥ {\displaystyle V^{\perp }} of V {\displaystyle V} is a subspace of every orthogonal complement; W ⊆ ( W ⊥ ) ⊥ {\displaystyle W\subseteq (W^{\perp })^{\perp }} ; If B {\displaystyle B} is non-degenerate and V {\displaystyle V} is finite-dimensional, then dim ( W ) + dim ( W ⊥ ) = dim ( V ) {\displaystyle \dim(W)+\dim(W^{\perp })=\dim(V)} . If L 1 , … , L r {\displaystyle L_{1},\ldots ,L_{r}} are subspaces of a finite-dimensional space V {\displaystyle V} and L ∗ = L 1 ∩ ⋯ ∩ L r , {\displaystyle L_{*}=L_{1}\cap \cdots \cap L_{r},} then L ∗ ⊥ = L 1 ⊥ + ⋯ + L r ⊥ {\displaystyle L_{*}^{\perp }=L_{1}^{\perp }+\cdots +L_{r}^{\perp }} . == Inner product spaces == This section considers orthogonal complements in an inner product space H {\displaystyle H} . Two vectors x {\displaystyle \mathbf {x} } and y {\displaystyle \mathbf {y} } are called orthogonal if ⟨ x , y ⟩ = 0 {\displaystyle \langle \mathbf {x} ,\mathbf {y} \rangle =0} , which happens if and only if ‖ x ‖ ≤ ‖ x + s y ‖ ∀ {\displaystyle \|\mathbf {x} \|\leq \|\mathbf {x} +s\mathbf {y} \|\ \forall } scalars s {\displaystyle s} . If C {\displaystyle C} is any subset of an inner product space H {\displaystyle H} then its orthogonal complement in H {\displaystyle H} is the vector subspace C ⊥ : = { x ∈ H : ⟨ x , c ⟩ = 0 ∀ c ∈ C } = { x ∈ H : ⟨ c , x ⟩ = 0 ∀ c ∈ C } {\displaystyle {\begin{aligned}C^{\perp }:&=\{\mathbf {x} \in H:\langle \mathbf {x} ,\mathbf {c} \rangle =0\ \ \forall \ \mathbf {c} \in C\}\\&=\{\mathbf {x} \in H:\langle \mathbf {c} ,\mathbf {x} \rangle =0\ \ \forall \ \mathbf {c} \in C\}\end{aligned}}} which is always a closed subset (hence, a closed vector subspace) of H {\displaystyle H} that satisfies: C ⊥ = ( cl H ( span C ) ) ⊥ {\displaystyle C^{\bot }=\left(\operatorname {cl} _{H}\left(\operatorname {span} C\right)\right)^{\bot }} ; C ⊥ ∩ cl H ( span C ) = { 0 } {\displaystyle C^{\bot }\cap \operatorname {cl} _{H}\left(\operatorname {span} C\right)=\{0\}} ; C ⊥ ∩ ( span C ) = { 0 } {\displaystyle C^{\bot }\cap \left(\operatorname {span} C\right)=\{0\}} ; C ⊆ ( C ⊥ ) ⊥ {\displaystyle C\subseteq \left(C^{\bot }\right)^{\bot }} ; cl H ( span C ) ⊆ ( C ⊥ ) ⊥ {\displaystyle \operatorname {cl} _{H}\left(\operatorname {span} C\right)\subseteq \left(C^{\bot }\right)^{\bot }} . If C {\displaystyle C} is a vector subspace of an inner product space H {\displaystyle H} then C ⊥ = { x ∈ H : ‖ x ‖ ≤ ‖ x + c ‖ ∀ c ∈ C } . {\displaystyle C^{\bot }=\left\{\mathbf {x} \in H:\|\mathbf {x} \|\leq \|\mathbf {x} +\mathbf {c} \|\ \ \forall \ \mathbf {c} \in C\right\}.} If C {\displaystyle C} is a closed vector subspace of a Hilbert space H {\displaystyle H} then H = C ⊕ C ⊥ and ( C ⊥ ) ⊥ = C {\displaystyle H=C\oplus C^{\bot }\qquad {\text{ and }}\qquad \left(C^{\bot }\right)^{\bot }=C} where H = C ⊕ C ⊥ {\displaystyle H=C\oplus C^{\bot }} is called the orthogonal decomposition of H {\displaystyle H} into C {\displaystyle C} and C ⊥ {\displaystyle C^{\bot }} and it indicates that C {\displaystyle C} is a complemented subspace of H {\displaystyle H} with complement C ⊥ . {\displaystyle C^{\bot }.} === Properties === The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal complements are closed. If W {\displaystyle W} is a vector subspace of a Hilbert space the orthogonal complement of the orthogonal complement of W {\displaystyle W} is the closure of W , {\displaystyle W,} that is, ( W ⊥ ) ⊥ = W ¯ . {\displaystyle \left(W^{\bot }\right)^{\bot }={\overline {W}}.} Some other useful properties that always hold are the following. Let H {\displaystyle H} be a Hilbert space and let X {\displaystyle X} and Y {\displaystyle Y} be linear subspaces. Then: X ⊥ = X ¯ ⊥ {\displaystyle X^{\bot }={\overline {X}}^{\bot }} ; if Y ⊆ X {\displaystyle Y\subseteq X} then X ⊥ ⊆ Y ⊥ {\displaystyle X^{\bot }\subseteq Y^{\bot }} ; X ∩ X ⊥ = { 0 } {\displaystyle X\cap X^{\bot }=\{0\}} ; X ⊆ ( X ⊥ ) ⊥ {\displaystyle X\subseteq (X^{\bot })^{\bot }} ; if X {\displaystyle X} is a closed linear subspace of H {\displaystyle H} then ( X ⊥ ) ⊥ = X {\displaystyle (X^{\bot })^{\bot }=X} ; if X {\displaystyle X} is a closed linear subspace of H {\displaystyle H} then H = X ⊕ X ⊥ , {\displaystyle H=X\oplus X^{\bot },} the (inner) direct sum. The orthogonal complement generalizes to the annihilator, and gives a Galois connection on subsets of the inner product space, with associated closure operator the topological closure of the span. === Finite dimensions === For a finite-dimensional inner product space of dimension n {\displaystyle n} , the orthogonal complement of a k {\displaystyle k} -dimensional subspace is an ( n − k ) {\displaystyle (n-k)} -dimensional subspace, and the double orthogonal complement is the original subspace: ( W ⊥ ) ⊥ = W . {\displaystyle \left(W^{\bot }\right)^{\bot }=W.} If A ∈ M m n {\displaystyle \mathbf {A} \in \mathbb {M} _{mn}} , where R ( A ) {\displaystyle {\mathcal {R}}(\mathbf {A} )} , C ( A ) {\displaystyle {\mathcal {C}}(\mathbf {A} )} , and N ( A ) {\displaystyle {\mathcal {N}}(\mathbf {A} )} refer to the row space, column space, and null space of A {\displaystyle \mathbf {A} } (respectively), then ( R ( A ) ) ⊥ = N ( A ) and ( C ( A ) ) ⊥ = N ( A T ) . {\displaystyle \left({\mathcal {R}}(\mathbf {A} )\right)^{\bot }={\mathcal {N}}(\mathbf {A} )\qquad {\text{ and }}\qquad \left({\mathcal {C}}(\mathbf {A} )\right)^{\bot }={\mathcal {N}}(\mathbf {A} ^{\operatorname {T} }).} == Banach spaces == There is a natural analog of this notion in general Banach spaces. In this case one defines the orthogonal complement of W {\displaystyle W} to be a subspace of the dual of V {\displaystyle V} defined similarly as the annihilator W ⊥ = { x ∈ V ∗ : ∀ y ∈ W , x ( y ) = 0 } . {\displaystyle W^{\bot }=\left\{x\in V^{*}:\forall y\in W,x(y)=0\right\}.} It is always a closed subspace of V ∗ {\displaystyle V^{*}} . There is also an analog of the double complement property. W ⊥⊥ {\displaystyle W^{\perp \perp }} is now a subspace of V ∗ ∗ {\displaystyle V^{**}} (which is not identical to V {\displaystyle V} ). However, the reflexive spaces have a natural isomorphism i {\displaystyle i} between V {\displaystyle V} and V ∗ ∗ {\displaystyle V^{**}} . In this case we have i W ¯ = W ⊥⊥ . {\displaystyle i{\overline {W}}=W^{\perp \perp }.} This is a rather straightforward consequence of the Hahn–Banach theorem. == Applications == In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η {\displaystyle \eta } used in Minkowski space determines a pseudo-Euclidean space of events. The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology stems from the use of conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal. == See also == Complemented lattice Complemented subspace Hilbert projection theorem – On closed convex subsets in Hilbert space Orthogonal projection – Idempotent linear transformation from a vector space to itselfPages displaying short descriptions of redirect targets == Notes == == References == == Bibliography == Adkins, William A.; Weintraub, Steven H. (1992), Algebra: An Approach via Module Theory, Graduate Texts in Mathematics, vol. 136, Springer-Verlag, ISBN 3-540-97839-9, Zbl 0768.00003 Halmos, Paul R. (1974), Finite-dimensional vector spaces, Undergraduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90093-3, Zbl 0288.15002 Milnor, J.; Husemoller, D. (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, Springer-Verlag, ISBN 3-540-06009-X, Zbl 0292.10016 Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. == External links == Orthogonal complement; Minute 9.00 in the Youtube Video Instructional video describing orthogonal complements (Khan Academy)
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Wikipedia:Orthogonal diagonalization#0
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In linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on R {\displaystyle \mathbb {R} } n by means of an orthogonal change of coordinates X = PY. Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial Δ ( t ) . {\displaystyle \Delta (t).} Step 2: find the eigenvalues of A which are the roots of Δ ( t ) {\displaystyle \Delta (t)} . Step 3: for each eigenvalue λ {\displaystyle \lambda } of A from step 2, find an orthogonal basis of its eigenspace. Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of R {\displaystyle \mathbb {R} } n. Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4. Then X = PY is the required orthogonal change of coordinates, and the diagonal entries of P T A P {\displaystyle P^{T}AP} will be the eigenvalues λ 1 , … , λ n {\displaystyle \lambda _{1},\dots ,\lambda _{n}} which correspond to the columns of P. == References == Maxime Bôcher (with E.P.R. DuVal)(1907) Introduction to Higher Algebra, § 45 Reduction of a quadratic form to a sum of squares via HathiTrust
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Wikipedia:Orthogonal transformation#0
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In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product. That is, for each pair u, v of elements of V, we have ⟨ u , v ⟩ = ⟨ T u , T v ⟩ . {\displaystyle \langle u,v\rangle =\langle Tu,Tv\rangle \,.} Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases. Orthogonal transformations are injective: if T v = 0 {\displaystyle Tv=0} then 0 = ⟨ T v , T v ⟩ = ⟨ v , v ⟩ {\displaystyle 0=\langle Tv,Tv\rangle =\langle v,v\rangle } , hence v = 0 {\displaystyle v=0} , so the kernel of T {\displaystyle T} is trivial. Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1. This allows the concept of rotation and reflection to be generalized to higher dimensions. In finite-dimensional spaces, the matrix representation (with respect to an orthonormal basis) of an orthogonal transformation is an orthogonal matrix. Its rows are mutually orthogonal vectors with unit norm, so that the rows constitute an orthonormal basis of V. The columns of the matrix form another orthonormal basis of V. If an orthogonal transformation is invertible (which is always the case when V is finite-dimensional) then its inverse T − 1 {\displaystyle T^{-1}} is another orthogonal transformation identical to the transpose of T {\displaystyle T} : T − 1 = T T {\displaystyle T^{-1}=T^{\mathtt {T}}} . == Examples == Consider the inner-product space ( R 2 , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (\mathbb {R} ^{2},\langle \cdot ,\cdot \rangle )} with the standard Euclidean inner product and standard basis. Then, the matrix transformation T = [ cos ( θ ) − sin ( θ ) sin ( θ ) cos ( θ ) ] : R 2 → R 2 {\displaystyle T={\begin{bmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\end{bmatrix}}:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} is orthogonal. To see this, consider T e 1 = [ cos ( θ ) sin ( θ ) ] T e 2 = [ − sin ( θ ) cos ( θ ) ] {\displaystyle {\begin{aligned}Te_{1}={\begin{bmatrix}\cos(\theta )\\\sin(\theta )\end{bmatrix}}&&Te_{2}={\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}\end{aligned}}} Then, ⟨ T e 1 , T e 1 ⟩ = [ cos ( θ ) sin ( θ ) ] ⋅ [ cos ( θ ) sin ( θ ) ] = cos 2 ( θ ) + sin 2 ( θ ) = 1 ⟨ T e 1 , T e 2 ⟩ = [ cos ( θ ) sin ( θ ) ] ⋅ [ − sin ( θ ) cos ( θ ) ] = sin ( θ ) cos ( θ ) − sin ( θ ) cos ( θ ) = 0 ⟨ T e 2 , T e 2 ⟩ = [ − sin ( θ ) cos ( θ ) ] ⋅ [ − sin ( θ ) cos ( θ ) ] = sin 2 ( θ ) + cos 2 ( θ ) = 1 {\displaystyle {\begin{aligned}&\langle Te_{1},Te_{1}\rangle ={\begin{bmatrix}\cos(\theta )&\sin(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}\cos(\theta )\\\sin(\theta )\end{bmatrix}}=\cos ^{2}(\theta )+\sin ^{2}(\theta )=1\\&\langle Te_{1},Te_{2}\rangle ={\begin{bmatrix}\cos(\theta )&\sin(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}=\sin(\theta )\cos(\theta )-\sin(\theta )\cos(\theta )=0\\&\langle Te_{2},Te_{2}\rangle ={\begin{bmatrix}-\sin(\theta )&\cos(\theta )\end{bmatrix}}\cdot {\begin{bmatrix}-\sin(\theta )\\\cos(\theta )\end{bmatrix}}=\sin ^{2}(\theta )+\cos ^{2}(\theta )=1\\\end{aligned}}} The previous example can be extended to construct all orthogonal transformations. For example, the following matrices define orthogonal transformations on ( R 3 , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (\mathbb {R} ^{3},\langle \cdot ,\cdot \rangle )} : [ cos ( θ ) − sin ( θ ) 0 sin ( θ ) cos ( θ ) 0 0 0 1 ] , [ cos ( θ ) 0 − sin ( θ ) 0 1 0 sin ( θ ) 0 cos ( θ ) ] , [ 1 0 0 0 cos ( θ ) − sin ( θ ) 0 sin ( θ ) cos ( θ ) ] {\displaystyle {\begin{bmatrix}\cos(\theta )&-\sin(\theta )&0\\\sin(\theta )&\cos(\theta )&0\\0&0&1\end{bmatrix}},{\begin{bmatrix}\cos(\theta )&0&-\sin(\theta )\\0&1&0\\\sin(\theta )&0&\cos(\theta )\end{bmatrix}},{\begin{bmatrix}1&0&0\\0&\cos(\theta )&-\sin(\theta )\\0&\sin(\theta )&\cos(\theta )\end{bmatrix}}} == See also == Geometric transformation Improper rotation Linear transformation Orthogonal matrix Rigid transformation Unitary transformation == References ==
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Wikipedia:Orthogonality (mathematics)#0
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In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} . Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics. == Definitions == In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e. they form a right angle. Two vectors u and v in an inner product space V {\displaystyle V} are orthogonal if their inner product ⟨ u , v ⟩ {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle } is zero. This relationship is denoted u ⊥ v {\displaystyle \mathbf {u} \perp \mathbf {v} } . A set of vectors in an inner product space is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set (or orthogonal system). If the vectors are normalized, they form an orthonormal system. An orthogonal matrix is a matrix whose column vectors are orthonormal to each other. An orthonormal basis is a basis whose vectors are both orthogonal and normalized (they are unit vectors). A conformal linear transformation preserves angles and distance ratios, meaning that transforming orthogonal vectors by the same conformal linear transformation will keep those vectors orthogonal. Two vector subspaces A {\displaystyle A} and B {\displaystyle B} of an inner product space V {\displaystyle V} are called orthogonal subspaces if each vector in A {\displaystyle A} is orthogonal to each vector in B {\displaystyle B} . The largest subspace of V {\displaystyle V} that is orthogonal to a given subspace is its orthogonal complement. Given a module M {\displaystyle M} and its dual M ∗ {\displaystyle M^{*}} , an element m ′ {\displaystyle m'} of M ∗ {\displaystyle M^{*}} and an element m {\displaystyle m} of M {\displaystyle M} are orthogonal if their natural pairing is zero, i.e. ⟨ m ′ , m ⟩ = 0 {\displaystyle \langle m',m\rangle =0} . Two sets S ′ ⊆ M ∗ {\displaystyle S'\subseteq M^{*}} and S ⊆ M {\displaystyle S\subseteq M} are orthogonal if each element of S ′ {\displaystyle S'} is orthogonal to each element of S {\displaystyle S} . A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent. In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve y = x 2 {\displaystyle y=x^{2}} at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics. A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ {\displaystyle \phi } . == Euclidean vector spaces == In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° ( π 2 {\textstyle {\frac {\pi }{2}}} radians), or one of the vectors is zero. Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension. The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle. In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane. == Orthogonal functions == By using integral calculus, it is common to use the following to define the inner product of two functions f {\displaystyle f} and g {\displaystyle g} with respect to a nonnegative weight function w {\displaystyle w} over an interval [ a , b ] {\displaystyle [a,b]} : ⟨ f , g ⟩ w = ∫ a b f ( x ) g ( x ) w ( x ) d x . {\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)g(x)w(x)\,dx.} In simple cases, w ( x ) = 1 {\displaystyle w(x)=1} . We say that functions f {\displaystyle f} and g {\displaystyle g} are orthogonal if their inner product (equivalently, the value of this integral) is zero: ⟨ f , g ⟩ w = 0. {\displaystyle \langle f,g\rangle _{w}=0.} Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product. We write the norm with respect to this inner product as ‖ f ‖ w = ⟨ f , f ⟩ w {\displaystyle \|f\|_{w}={\sqrt {\langle f,f\rangle _{w}}}} The members of a set of functions f i ∣ i ∈ N {\displaystyle {f_{i}\mid i\in \mathbb {N} }} are orthogonal with respect to w {\displaystyle w} on the interval [ a , b ] {\displaystyle [a,b]} if ⟨ f i , f j ⟩ w = 0 ∣ i ≠ j . {\displaystyle \langle f_{i},f_{j}\rangle _{w}=0\mid i\neq j.} The members of such a set of functions are orthonormal with respect to w {\displaystyle w} on the interval [ a , b ] {\displaystyle [a,b]} if ⟨ f i , f j ⟩ w = δ i j , {\displaystyle \langle f_{i},f_{j}\rangle _{w}=\delta _{ij},} where δ i j = { 1 , i = j 0 , i ≠ j {\displaystyle \delta _{ij}=\left\{{\begin{matrix}1,&&i=j\\0,&&i\neq j\end{matrix}}\right.} is the Kronecker delta. In other words, every pair of them (excluding pairing of a function with itself) is orthogonal, and the norm of each is 1. See in particular the orthogonal polynomials. == Examples == The vectors ( 1 , 3 , 2 ) T , ( 3 , − 1 , 0 ) T , ( 1 , 3 , − 5 ) T {\displaystyle (1,3,2)^{\text{T}},(3,-1,0)^{\text{T}},(1,3,-5)^{\text{T}}} are orthogonal to each other, since ( 1 ) ( 3 ) + ( 3 ) ( − 1 ) + ( 2 ) ( 0 ) = 0 , {\displaystyle (1)(3)+(3)(-1)+(2)(0)=0\ ,} ( 3 ) ( 1 ) + ( − 1 ) ( 3 ) + ( 0 ) ( − 5 ) = 0 , {\displaystyle \ (3)(1)+(-1)(3)+(0)(-5)=0\ ,} and ( 1 ) ( 1 ) + ( 3 ) ( 3 ) + ( 2 ) ( − 5 ) = 0 {\displaystyle (1)(1)+(3)(3)+(2)(-5)=0} . The vectors ( 1 , 0 , 1 , 0 , … ) T {\displaystyle (1,0,1,0,\ldots )^{\text{T}}} and ( 0 , 1 , 0 , 1 , … ) T {\displaystyle (0,1,0,1,\ldots )^{\text{T}}} are orthogonal to each other. The dot product of these vectors is zero. We can then make the generalization to consider the vectors in Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} : v k = ∑ i = 0 a i + k < n n / a e i {\displaystyle \mathbf {v} _{k}=\sum _{i=0 \atop ai+k<n}^{n/a}\mathbf {e} _{i}} for some positive integer a {\displaystyle a} , and for 1 ≤ k ≤ a − 1 {\displaystyle 1\leq k\leq a-1} , these vectors are orthogonal, for example [ 1 0 0 1 0 0 1 0 ] {\displaystyle {\begin{bmatrix}1&0&0&1&0&0&1&0\end{bmatrix}}} , [ 0 1 0 0 1 0 0 1 ] {\displaystyle {\begin{bmatrix}0&1&0&0&1&0&0&1\end{bmatrix}}} , [ 0 0 1 0 0 1 0 0 ] {\displaystyle {\begin{bmatrix}0&0&1&0&0&1&0&0\end{bmatrix}}} are orthogonal. The functions 2 t + 3 {\displaystyle 2t+3} and 45 t 2 + 9 t − 17 {\displaystyle 45t^{2}+9t-17} are orthogonal with respect to a unit weight function on the interval from −1 to 1: ∫ − 1 1 ( 2 t + 3 ) ( 45 t 2 + 9 t − 17 ) d t = 0 {\displaystyle \int _{-1}^{1}\left(2t+3\right)\left(45t^{2}+9t-17\right)\,dt=0} The functions 1 , sin ( n x ) , cos ( n x ) ∣ n ∈ N {\displaystyle 1,\sin {(nx)},\cos {(nx)}\mid n\in \mathbb {N} } are orthogonal with respect to Riemann integration on the intervals [ 0 , 2 π ] , [ − π , π ] {\displaystyle [0,2\pi ],[-\pi ,\pi ]} , or any other closed interval of length 2 π {\displaystyle 2\pi } . This fact is a central one in Fourier series. === Orthogonal polynomials === Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular: The Hermite polynomials are orthogonal with respect to the Gaussian distribution with zero mean value. The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} . The Laguerre polynomials are orthogonal with respect to the exponential distribution. Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions. The Chebyshev polynomials of the first kind are orthogonal with respect to the measure 1 1 − x 2 . {\textstyle {\frac {1}{\sqrt {1-x^{2}}}}.} The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution. == Combinatorics == In combinatorics, two n × n {\displaystyle n\times n} Latin squares are said to be orthogonal if their superimposition yields all possible n 2 {\displaystyle n^{2}} combinations of entries. == Completely orthogonal == Two flat planes A {\displaystyle A} and B {\displaystyle B} of a Euclidean four-dimensional space are called completely orthogonal if and only if every line in A {\displaystyle A} is orthogonal to every line in B {\displaystyle B} . In that case the planes A {\displaystyle A} and B {\displaystyle B} intersect at a single point O {\displaystyle O} , so that if a line in A {\displaystyle A} intersects with a line in B {\displaystyle B} , they intersect at O {\displaystyle O} . A {\displaystyle A} and B {\displaystyle B} are perpendicular and Clifford parallel. In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a ( w , x , y , z ) {\displaystyle (w,x,y,z)} Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes ( x y , x z , y z ) {\displaystyle (xy,xz,yz)} that we have in 3 dimensions, and also 3 others ( w x , w y , w z ) {\displaystyle (wx,wy,wz)} . Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: x y {\displaystyle xy} and w z {\displaystyle wz} intersect only at the origin; x z {\displaystyle xz} and w y {\displaystyle wy} intersect only at the origin; y z {\displaystyle yz} and w x {\displaystyle wx} intersect only at the origin. More generally, two flat subspaces S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} of dimensions M {\displaystyle M} and N {\displaystyle N} of a Euclidean space S {\displaystyle S} of at least M + N {\displaystyle M+N} dimensions are called completely orthogonal if every line in S 1 {\displaystyle S_{1}} is orthogonal to every line in S 2 {\displaystyle S_{2}} . If dim ( S ) = M + N {\displaystyle \dim(S)=M+N} then S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} intersect at a single point O {\displaystyle O} . If dim ( S ) > M + N {\displaystyle \dim(S)>M+N} then S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} may or may not intersect. If dim ( S ) = M + N {\displaystyle \dim(S)=M+N} then a line in S 1 {\displaystyle S_{1}} and a line in S 2 {\displaystyle S_{2}} may or may not intersect; if they intersect then they intersect at O {\displaystyle O} . == See also == Imaginary number Orthogonal complement Orthogonal group Orthogonal matrix Orthogonal polynomials Orthogonal polyhedron Orthogonal trajectory Orthogonalization Gram–Schmidt process Orthonormal basis Orthonormality Pan-orthogonality occurs in coquaternions Up tack == References ==
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Wikipedia:Orthogonalization#0
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In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization. Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting. == Orthogonalization algorithms == Methods for performing orthogonalization include: Gram–Schmidt process, which uses projection Householder transformation, which uses reflection Givens rotation Symmetric orthogonalization, which uses the Singular value decomposition When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram–Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects. On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration. The Givens rotation is more easily parallelized than Householder transformations. Symmetric orthogonalization was formulated by Per-Olov Löwdin. == Local orthogonalization == To compensate for the loss of useful signal in traditional noise attenuation approaches because of incorrect parameter selection or inadequacy of denoising assumptions, a weighting operator can be applied on the initially denoised section for the retrieval of useful signal from the initial noise section. The new denoising process is referred to as the local orthogonalization of signal and noise. It has a wide range of applications in many signals processing and seismic exploration fields. == See also == Orthogonality Biorthogonal system Orthogonal basis == References ==
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Wikipedia:Orthographic projection#0
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Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are not orthogonal to the projection plane. The term orthographic sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the primary views. If the principal planes or axes of an object in an orthographic projection are not parallel with the projection plane, the depiction is called axonometric or an auxiliary views. (Axonometric projection is synonymous with parallel projection.) Sub-types of primary views include plans, elevations, and sections; sub-types of auxiliary views include isometric, dimetric, and trimetric projections. A lens that provides an orthographic projection is an object-space telecentric lens. == Geometry == A simple orthographic projection onto the plane z = 0 can be defined by the following matrix: P = [ 1 0 0 0 1 0 0 0 0 ] {\displaystyle P={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}} For each point v = (vx, vy, vz), the transformed point Pv would be P v = [ 1 0 0 0 1 0 0 0 0 ] [ v x v y v z ] = [ v x v y 0 ] {\displaystyle Pv={\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\\\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\end{bmatrix}}} Often, it is more useful to use homogeneous coordinates. The transformation above can be represented for homogeneous coordinates as P = [ 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] {\displaystyle P={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}} For each homogeneous vector v = (vx, vy, vz, 1), the transformed vector Pv would be P v = [ 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 ] [ v x v y v z 1 ] = [ v x v y 0 1 ] {\displaystyle Pv={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}\\v_{y}\\0\\1\end{bmatrix}}} In computer graphics, one of the most common matrices used for orthographic projection can be defined by a 6-tuple, (left, right, bottom, top, near, far), which defines the clipping planes. These planes form a box with the minimum corner at (left, bottom, -near) and the maximum corner at (right, top, -far). The box is translated so that its center is at the origin, then it is scaled to the unit cube which is defined by having a minimum corner at (−1,−1,−1) and a maximum corner at (1,1,1). The orthographic transform can be given by the following matrix: P = [ 2 right − left 0 0 − right + left right − left 0 2 top − bottom 0 − top + bottom top − bottom 0 0 − 2 far − near − far + near far − near 0 0 0 1 ] {\displaystyle P={\begin{bmatrix}{\frac {2}{{\text{right}}-{\text{left}}}}&0&0&-{\frac {{\text{right}}+{\text{left}}}{{\text{right}}-{\text{left}}}}\\0&{\frac {2}{{\text{top}}-{\text{bottom}}}}&0&-{\frac {{\text{top}}+{\text{bottom}}}{{\text{top}}-{\text{bottom}}}}\\0&0&{\frac {-2}{{\text{far}}-{\text{near}}}}&-{\frac {{\text{far}}+{\text{near}}}{{\text{far}}-{\text{near}}}}\\0&0&0&1\end{bmatrix}}} which can be given as a scaling S followed by a translation T of the form P = S T = [ 2 right − left 0 0 0 0 2 top − bottom 0 0 0 0 2 far − near 0 0 0 0 1 ] [ 1 0 0 − left + right 2 0 1 0 − top + bottom 2 0 0 − 1 − far + near 2 0 0 0 1 ] {\displaystyle P=ST={\begin{bmatrix}{\frac {2}{{\text{right}}-{\text{left}}}}&0&0&0\\0&{\frac {2}{{\text{top}}-{\text{bottom}}}}&0&0\\0&0&{\frac {2}{{\text{far}}-{\text{near}}}}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}1&0&0&-{\frac {{\text{left}}+{\text{right}}}{2}}\\0&1&0&-{\frac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&-1&-{\frac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}} The inversion of the projection matrix P−1, which can be used as the unprojection matrix is defined: P − 1 = [ right − left 2 0 0 left + right 2 0 top − bottom 2 0 top + bottom 2 0 0 far − near − 2 − far + near 2 0 0 0 1 ] {\displaystyle P^{-1}={\begin{bmatrix}{\frac {{\text{right}}-{\text{left}}}{2}}&0&0&{\frac {{\text{left}}+{\text{right}}}{2}}\\0&{\frac {{\text{top}}-{\text{bottom}}}{2}}&0&{\frac {{\text{top}}+{\text{bottom}}}{2}}\\0&0&{\frac {{\text{far}}-{\text{near}}}{-2}}&-{\frac {{\text{far}}+{\text{near}}}{2}}\\0&0&0&1\end{bmatrix}}} == Types == Three sub-types of orthographic projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. Typically in axonometric drawing, as in other types of pictorials, one axis of space is shown to be vertical. In isometric projection, the most commonly used form of axonometric projection in engineering drawing, the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. As the distortion caused by foreshortening is uniform, the proportionality between lengths is preserved, and the axes share a common scale; this eases one's ability to take measurements directly from the drawing. Another advantage is that 120° angles are easily constructed using only a compass and straightedge. In dimetric projection, the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction is determined separately. In trimetric projection, the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Trimetric perspective is seldom used in technical drawings. == Multiview projection == In multiview projection, up to six pictures of an object are produced, called primary views, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: first-angle or third-angle projection. In each, the appearances of views may be thought of as being projected onto planes that form a six-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a three-dimensional object. These views are known as front view (also elevation), top view (also plan) and end view (also section). When the plane or axis of the object depicted is not parallel to the projection plane, and where multiple sides of an object are visible in the same image, it is called an auxiliary view. Thus isometric projection, dimetric projection and trimetric projection would be considered auxiliary views in multiview projection. A typical characteristic of multiview projection is that one axis of space is usually displayed as vertical. == Cartography == An orthographic projection map is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges. The orthographic projection has been known since antiquity, with its cartographic uses being well documented. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set. In about 14 BC, Roman engineer Marcus Vitruvius Pollio used the projection to construct sundials and to compute sun positions. Vitruvius also seems to have devised the term orthographic – from the Greek orthos ("straight") and graphē ("drawing") – for the projection. However, the name analemma, which also meant a sundial showing latitude and longitude, was the common name until François d'Aguilon of Antwerp promoted its present name in 1613. The earliest surviving maps on the projection appear as woodcut drawings of terrestrial globes of 1509 (anonymous), 1533 and 1551 (Johannes Schöner), and 1524 and 1551 (Apian). == Notes == == References == == External links == Normale (orthogonale) Axonometrie (in German) Orthographic Projection Video and mathematics
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Wikipedia:Orthonormal basis#0
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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle V} whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for R n {\displaystyle \mathbb {R} ^{n}} arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization. The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame. For a general inner product space V , {\displaystyle V,} an orthonormal basis can be used to define normalized orthogonal coordinates on V . {\displaystyle V.} Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of R n {\displaystyle \mathbb {R} ^{n}} under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process. In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space H , {\displaystyle H,} an orthonormal basis for H {\displaystyle H} is an orthonormal set of vectors with the property that every vector in H {\displaystyle H} can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H . {\displaystyle H.} Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H , {\displaystyle H,} although not necessarily the entire space. If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [ − 1 , 1 ] {\displaystyle [-1,1]} can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces M {\displaystyle M} equipped with a non-degenerate symmetric bilinear form known as the metric tensor. In such a basis, the metric takes the form diag ( + 1 , ⋯ , + 1 , − 1 , ⋯ , − 1 ) {\displaystyle {\text{diag}}(+1,\cdots ,+1,-1,\cdots ,-1)} with p {\displaystyle p} positive ones and q {\displaystyle q} negative ones. == Examples == For R 3 {\displaystyle \mathbb {R} ^{3}} , the set of vectors { e 1 = ( 1 0 0 ) , e 2 = ( 0 1 0 ) , e 3 = ( 0 0 1 ) } , {\displaystyle \left\{\mathbf {e_{1}} ={\begin{pmatrix}1&0&0\end{pmatrix}}\ ,\ \mathbf {e_{2}} ={\begin{pmatrix}0&1&0\end{pmatrix}}\ ,\ \mathbf {e_{3}} ={\begin{pmatrix}0&0&1\end{pmatrix}}\right\},} is called the standard basis and forms an orthonormal basis of R 3 {\displaystyle \mathbb {R} ^{3}} with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing R 3 {\displaystyle \mathbb {R} ^{3}} as the Cartesian product R × R × R {\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} } Proof: A straightforward computation shows that the inner products of these vectors equals zero, ⟨ e 1 , e 2 ⟩ = ⟨ e 1 , e 3 ⟩ = ⟨ e 2 , e 3 ⟩ = 0 {\displaystyle \left\langle \mathbf {e_{1}} ,\mathbf {e_{2}} \right\rangle =\left\langle \mathbf {e_{1}} ,\mathbf {e_{3}} \right\rangle =\left\langle \mathbf {e_{2}} ,\mathbf {e_{3}} \right\rangle =0} and that each of their magnitudes equals one, ‖ e 1 ‖ = ‖ e 2 ‖ = ‖ e 3 ‖ = 1. {\displaystyle \left\|\mathbf {e_{1}} \right\|=\left\|\mathbf {e_{2}} \right\|=\left\|\mathbf {e_{3}} \right\|=1.} This means that { e 1 , e 2 , e 3 } {\displaystyle \left\{\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} \right\}} is an orthonormal set. All vectors ( x , y , z ) ∈ R 3 {\displaystyle (\mathbf {x} ,\mathbf {y} ,\mathbf {z} )\in \mathbb {R} ^{3}} can be expressed as a sum of the basis vectors scaled ( x , y , z ) = x e 1 + y e 2 + z e 3 , {\displaystyle (\mathbf {x} ,\mathbf {y} ,\mathbf {z} )=\mathbf {xe_{1}} +\mathbf {ye_{2}} +\mathbf {ze_{3}} ,} so { e 1 , e 2 , e 3 } {\displaystyle \left\{\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} \right\}} spans R 3 {\displaystyle \mathbb {R} ^{3}} and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin also forms an orthonormal basis of R 3 {\displaystyle \mathbb {R} ^{3}} . For R n {\displaystyle \mathbb {R} ^{n}} , the standard basis and inner product are similarly defined. Any other orthonormal basis is related to the standard basis by an orthogonal transformation in the group O(n). For pseudo-Euclidean space R p , q , {\displaystyle \mathbb {R} ^{p,q},} , an orthogonal basis { e μ } {\displaystyle \{e_{\mu }\}} with metric η {\displaystyle \eta } instead satisfies η ( e μ , e ν ) = 0 {\displaystyle \eta (e_{\mu },e_{\nu })=0} if μ ≠ ν {\displaystyle \mu \neq \nu } , η ( e μ , e μ ) = + 1 {\displaystyle \eta (e_{\mu },e_{\mu })=+1} if 1 ≤ μ ≤ p {\displaystyle 1\leq \mu \leq p} , and η ( e μ , e μ ) = − 1 {\displaystyle \eta (e_{\mu },e_{\mu })=-1} if p + 1 ≤ μ ≤ p + q {\displaystyle p+1\leq \mu \leq p+q} . Any two orthonormal bases are related by a pseudo-orthogonal transformation. In the case ( p , q ) = ( 1 , 3 ) {\displaystyle (p,q)=(1,3)} , these are Lorentz transformations. The set { f n : n ∈ Z } {\displaystyle \left\{f_{n}:n\in \mathbb {Z} \right\}} with f n ( x ) = exp ( 2 π i n x ) , {\displaystyle f_{n}(x)=\exp(2\pi inx),} where exp {\displaystyle \exp } denotes the exponential function, forms an orthonormal basis of the space of functions with finite Lebesgue integrals, L 2 ( [ 0 , 1 ] ) , {\displaystyle L^{2}([0,1]),} with respect to the 2-norm. This is fundamental to the study of Fourier series. The set { e b : b ∈ B } {\displaystyle \left\{e_{b}:b\in B\right\}} with e b ( c ) = 1 {\displaystyle e_{b}(c)=1} if b = c {\displaystyle b=c} and e b ( c ) = 0 {\displaystyle e_{b}(c)=0} otherwise forms an orthonormal basis of ℓ 2 ( B ) . {\displaystyle \ell ^{2}(B).} Eigenfunctions of a Sturm–Liouville eigenproblem. The column vectors of an orthogonal matrix form an orthonormal set. == Basic formula == If B {\displaystyle B} is an orthogonal basis of H , {\displaystyle H,} then every element x ∈ H {\displaystyle x\in H} may be written as x = ∑ b ∈ B ⟨ x , b ⟩ ‖ b ‖ 2 b . {\displaystyle x=\sum _{b\in B}{\frac {\langle x,b\rangle }{\lVert b\rVert ^{2}}}b.} When B {\displaystyle B} is orthonormal, this simplifies to x = ∑ b ∈ B ⟨ x , b ⟩ b {\displaystyle x=\sum _{b\in B}\langle x,b\rangle b} and the square of the norm of x {\displaystyle x} can be given by ‖ x ‖ 2 = ∑ b ∈ B | ⟨ x , b ⟩ | 2 . {\displaystyle \|x\|^{2}=\sum _{b\in B}|\langle x,b\rangle |^{2}.} Even if B {\displaystyle B} is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x , {\displaystyle x,} and the formula is usually known as Parseval's identity. If B {\displaystyle B} is an orthonormal basis of H , {\displaystyle H,} then H {\displaystyle H} is isomorphic to ℓ 2 ( B ) {\displaystyle \ell ^{2}(B)} in the following sense: there exists a bijective linear map Φ : H → ℓ 2 ( B ) {\displaystyle \Phi :H\to \ell ^{2}(B)} such that ⟨ Φ ( x ) , Φ ( y ) ⟩ = ⟨ x , y ⟩ ∀ x , y ∈ H . {\displaystyle \langle \Phi (x),\Phi (y)\rangle =\langle x,y\rangle \ \ \forall \ x,y\in H.} == Orthonormal system == A set S {\displaystyle S} of mutually orthonormal vectors in a Hilbert space H {\displaystyle H} is called an orthonormal system. An orthonormal basis is an orthonormal system with the additional property that the linear span of S {\displaystyle S} is dense in H {\displaystyle H} . Alternatively, the set S {\displaystyle S} can be regarded as either complete or incomplete with respect to H {\displaystyle H} . That is, we can take the smallest closed linear subspace V ⊆ H {\displaystyle V\subseteq H} containing S . {\displaystyle S.} Then S {\displaystyle S} will be an orthonormal basis of V ; {\displaystyle V;} which may of course be smaller than H {\displaystyle H} itself, being an incomplete orthonormal set, or be H , {\displaystyle H,} when it is a complete orthonormal set. == Existence == Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis. (One can prove this last statement without using the axiom of choice. However, one would have to use the axiom of countable choice.) == Choice of basis as a choice of isomorphism == For concreteness we discuss orthonormal bases for a real, n {\displaystyle n} -dimensional vector space V {\displaystyle V} with a positive definite symmetric bilinear form ϕ = ⟨ ⋅ , ⋅ ⟩ {\displaystyle \phi =\langle \cdot ,\cdot \rangle } . One way to view an orthonormal basis with respect to ϕ {\displaystyle \phi } is as a set of vectors B = { e i } {\displaystyle {\mathcal {B}}=\{e_{i}\}} , which allow us to write v = v i e i ∀ v ∈ V {\displaystyle v=v^{i}e_{i}\ \ \forall \ v\in V} , and v i ∈ R {\displaystyle v^{i}\in \mathbb {R} } or ( v i ) ∈ R n {\displaystyle (v^{i})\in \mathbb {R} ^{n}} . With respect to this basis, the components of ϕ {\displaystyle \phi } are particularly simple: ϕ ( e i , e j ) = δ i j {\displaystyle \phi (e_{i},e_{j})=\delta _{ij}} (where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta). We can now view the basis as a map ψ B : V → R n {\displaystyle \psi _{\mathcal {B}}:V\rightarrow \mathbb {R} ^{n}} which is an isomorphism of inner product spaces: to make this more explicit we can write ψ B : ( V , ϕ ) → ( R n , δ i j ) . {\displaystyle \psi _{\mathcal {B}}:(V,\phi )\rightarrow (\mathbb {R} ^{n},\delta _{ij}).} Explicitly we can write ( ψ B ( v ) ) i = e i ( v ) = ϕ ( e i , v ) {\displaystyle (\psi _{\mathcal {B}}(v))^{i}=e^{i}(v)=\phi (e_{i},v)} where e i {\displaystyle e^{i}} is the dual basis element to e i {\displaystyle e_{i}} . The inverse is a component map C B : R n → V , ( v i ) ↦ ∑ i = 1 n v i e i . {\displaystyle C_{\mathcal {B}}:\mathbb {R} ^{n}\rightarrow V,(v^{i})\mapsto \sum _{i=1}^{n}v^{i}e_{i}.} These definitions make it manifest that there is a bijection { Space of orthogonal bases B } ↔ { Space of isomorphisms V ↔ R n } . {\displaystyle \{{\text{Space of orthogonal bases }}{\mathcal {B}}\}\leftrightarrow \{{\text{Space of isomorphisms }}V\leftrightarrow \mathbb {R} ^{n}\}.} The space of isomorphisms admits actions of orthogonal groups at either the V {\displaystyle V} side or the R n {\displaystyle \mathbb {R} ^{n}} side. For concreteness we fix the isomorphisms to point in the direction R n → V {\displaystyle \mathbb {R} ^{n}\rightarrow V} , and consider the space of such maps, Iso ( R n → V ) {\displaystyle {\text{Iso}}(\mathbb {R} ^{n}\rightarrow V)} . This space admits a left action by the group of isometries of V {\displaystyle V} , that is, R ∈ GL ( V ) {\displaystyle R\in {\text{GL}}(V)} such that ϕ ( ⋅ , ⋅ ) = ϕ ( R ⋅ , R ⋅ ) {\displaystyle \phi (\cdot ,\cdot )=\phi (R\cdot ,R\cdot )} , with the action given by composition: R ∗ C = R ∘ C . {\displaystyle R*C=R\circ C.} This space also admits a right action by the group of isometries of R n {\displaystyle \mathbb {R} ^{n}} , that is, R i j ∈ O ( n ) ⊂ Mat n × n ( R ) {\displaystyle R_{ij}\in {\text{O}}(n)\subset {\text{Mat}}_{n\times n}(\mathbb {R} )} , with the action again given by composition: C ∗ R i j = C ∘ R i j {\displaystyle C*R_{ij}=C\circ R_{ij}} . == As a principal homogeneous space == The set of orthonormal bases for R n {\displaystyle \mathbb {R} ^{n}} with the standard inner product is a principal homogeneous space or G-torsor for the orthogonal group G = O ( n ) , {\displaystyle G={\text{O}}(n),} and is called the Stiefel manifold V n ( R n ) {\displaystyle V_{n}(\mathbb {R} ^{n})} of orthonormal n {\displaystyle n} -frames. In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given the space of orthonormal bases, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis. The other Stiefel manifolds V k ( R n ) {\displaystyle V_{k}(\mathbb {R} ^{n})} for k < n {\displaystyle k<n} of incomplete orthonormal bases (orthonormal k {\displaystyle k} -frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k {\displaystyle k} -frame can be taken to any other k {\displaystyle k} -frame by an orthogonal map, but this map is not uniquely determined. The set of orthonormal bases for R p , q {\displaystyle \mathbb {R} ^{p,q}} is a G-torsor for G = O ( p , q ) {\displaystyle G={\text{O}}(p,q)} . The set of orthonormal bases for C n {\displaystyle \mathbb {C} ^{n}} is a G-torsor for G = U ( n ) {\displaystyle G={\text{U}}(n)} . The set of orthonormal bases for C p , q {\displaystyle \mathbb {C} ^{p,q}} is a G-torsor for G = U ( p , q ) {\displaystyle G={\text{U}}(p,q)} . The set of right-handed orthonormal bases for R n {\displaystyle \mathbb {R} ^{n}} is a G-torsor for G = SO ( n ) {\displaystyle G={\text{SO}}(n)} == See also == Orthogonal basis – Basis for v whose vectors are mutually orthogonal Basis (linear algebra) – Set of vectors used to define coordinates Orthonormal frame – Euclidean space without distance and angles Schauder basis – Computational tool Total set – subset of a topological vector space whose linear span is densePages displaying wikidata descriptions as a fallback == Notes == == References == Roman, Stephen (2008). Advanced Linear Algebra. Graduate Texts in Mathematics (Third ed.). Springer. ISBN 978-0-387-72828-5. (page 218, ch.9) Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Steinwart, Ingo; Christmann, Andreas (2008). Support vector machines. New York: Springer. doi:10.1007/978-0-387-77242-4. ISBN 978-0-387-77241-7. == External links == This Stack Exchange Post discusses why the set of Dirac Delta functions is not a basis of L2([0,1]).
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Wikipedia:Orthonormal function system#0
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An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. == References ==
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Wikipedia:Orthonormality#0
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In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. == Intuitive overview == The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is, ‖ x ‖ = x ⋅ x {\displaystyle \|\mathbf {x} \|={\sqrt {\mathbf {x} \cdot \mathbf {x} }}} Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of unit length. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal. === Simple example === What does a pair of orthonormal vectors in 2-D Euclidean space look like? Let u = (x1, y1) and v = (x2, y2). Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair. From the orthogonality restriction, u • v = 0. From the unit length restriction on u, ||u|| = 1. From the unit length restriction on v, ||v|| = 1. Expanding these terms gives 3 equations: x 1 x 2 + y 1 y 2 = 0 {\displaystyle x_{1}x_{2}+y_{1}y_{2}=0\quad } x 1 2 + y 1 2 = 1 {\displaystyle {\sqrt {{x_{1}}^{2}+{y_{1}}^{2}}}=1} x 2 2 + y 2 2 = 1 {\displaystyle {\sqrt {{x_{2}}^{2}+{y_{2}}^{2}}}=1} Converting from Cartesian to polar coordinates, and considering Equation ( 2 ) {\displaystyle (2)} and Equation ( 3 ) {\displaystyle (3)} immediately gives the result r1 = r2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the unit circle. After substitution, Equation ( 1 ) {\displaystyle (1)} becomes cos θ 1 cos θ 2 + sin θ 1 sin θ 2 = 0 {\displaystyle \cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}=0} . Rearranging gives tan θ 1 = − cot θ 2 {\displaystyle \tan \theta _{1}=-\cot \theta _{2}} . Using a trigonometric identity to convert the cotangent term gives tan ( θ 1 ) = tan ( θ 2 + π 2 ) {\displaystyle \tan(\theta _{1})=\tan \left(\theta _{2}+{\tfrac {\pi }{2}}\right)} ⇒ θ 1 = θ 2 + π 2 {\displaystyle \Rightarrow \theta _{1}=\theta _{2}+{\tfrac {\pi }{2}}} It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°. == Definition == Let V {\displaystyle {\mathcal {V}}} be an inner-product space. A set of vectors { u 1 , u 2 , … , u n , … } ∈ V {\displaystyle \left\{u_{1},u_{2},\ldots ,u_{n},\ldots \right\}\in {\mathcal {V}}} is called orthonormal if and only if ∀ i , j : ⟨ u i , u j ⟩ = δ i j {\displaystyle \forall i,j:\langle u_{i},u_{j}\rangle =\delta _{ij}} where δ i j {\displaystyle \delta _{ij}\,} is the Kronecker delta and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product defined over V {\displaystyle {\mathcal {V}}} . == Significance == Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces. === Properties === Orthonormal sets have certain very appealing properties, which make them particularly easy to work with. Theorem. If {e1, e2, ..., en} is an orthonormal list of vectors, then ∀ a := [ a 1 , ⋯ , a n ] ; ‖ a 1 e 1 + a 2 e 2 + ⋯ + a n e n ‖ 2 = | a 1 | 2 + | a 2 | 2 + ⋯ + | a n | 2 {\displaystyle \forall {\textbf {a}}:=[a_{1},\cdots ,a_{n}];\ \|a_{1}{\textbf {e}}_{1}+a_{2}{\textbf {e}}_{2}+\cdots +a_{n}{\textbf {e}}_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots +|a_{n}|^{2}} Theorem. Every orthonormal list of vectors is linearly independent. === Existence === Gram-Schmidt theorem. If {v1, v2,...,vn} is a linearly independent list of vectors in an inner-product space V {\displaystyle {\mathcal {V}}} , then there exists an orthonormal list {e1, e2,...,en} of vectors in V {\displaystyle {\mathcal {V}}} such that span(e1, e2,...,en) = span(v1, v2,...,vn). Proof of the Gram-Schmidt theorem is constructive, and discussed at length elsewhere. The Gram-Schmidt theorem, together with the axiom of choice, guarantees that every vector space admits an orthonormal basis. This is possibly the most significant use of orthonormality, as this fact permits operators on inner-product spaces to be discussed in terms of their action on the space's orthonormal basis vectors. What results is a deep relationship between the diagonalizability of an operator and how it acts on the orthonormal basis vectors. This relationship is characterized by the Spectral Theorem. == Examples == === Standard basis === The standard basis for the coordinate space Fn is Any two vectors ei, ej where i≠j are orthogonal, and all vectors are clearly of unit length. So {e1, e2,...,en} forms an orthonormal basis. === Real-valued functions === When referring to real-valued functions, usually the L² inner product is assumed unless otherwise stated. Two functions ϕ ( x ) {\displaystyle \phi (x)} and ψ ( x ) {\displaystyle \psi (x)} are orthonormal over the interval [ a , b ] {\displaystyle [a,b]} if ( 1 ) ⟨ ϕ ( x ) , ψ ( x ) ⟩ = ∫ a b ϕ ( x ) ψ ( x ) d x = 0 , a n d {\displaystyle (1)\quad \langle \phi (x),\psi (x)\rangle =\int _{a}^{b}\phi (x)\psi (x)dx=0,\quad {\rm {and}}} ( 2 ) | | ϕ ( x ) | | 2 = | | ψ ( x ) | | 2 = [ ∫ a b | ϕ ( x ) | 2 d x ] 1 2 = [ ∫ a b | ψ ( x ) | 2 d x ] 1 2 = 1. {\displaystyle (2)\quad ||\phi (x)||_{2}=||\psi (x)||_{2}=\left[\int _{a}^{b}|\phi (x)|^{2}dx\right]^{\frac {1}{2}}=\left[\int _{a}^{b}|\psi (x)|^{2}dx\right]^{\frac {1}{2}}=1.} === Fourier series === The Fourier series is a method of expressing a periodic function in terms of sinusoidal basis functions. Taking C[−π,π] to be the space of all real-valued functions continuous on the interval [−π,π] and taking the inner product to be ⟨ f , g ⟩ = ∫ − π π f ( x ) g ( x ) d x {\displaystyle \langle f,g\rangle =\int _{-\pi }^{\pi }f(x)g(x)dx} it can be shown that { 1 2 π , sin ( x ) π , sin ( 2 x ) π , … , sin ( n x ) π , cos ( x ) π , cos ( 2 x ) π , … , cos ( n x ) π } , n ∈ N {\displaystyle \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {\sin(x)}{\sqrt {\pi }}},{\frac {\sin(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\sin(nx)}{\sqrt {\pi }}},{\frac {\cos(x)}{\sqrt {\pi }}},{\frac {\cos(2x)}{\sqrt {\pi }}},\ldots ,{\frac {\cos(nx)}{\sqrt {\pi }}}\right\},\quad n\in \mathbb {N} } forms an orthonormal set. However, this is of little consequence, because C[−π,π] is infinite-dimensional, and a finite set of vectors cannot span it. But, removing the restriction that n be finite makes the set dense in C[−π,π] and therefore an orthonormal basis of C[−π,π]. == See also == Orthogonalization Orthonormal function system == Sources == Axler, Sheldon (1997), Linear Algebra Done Right (2nd ed.), Berlin, New York: Springer-Verlag, p. 106–110, ISBN 978-0-387-98258-8 Chen, Wai-Kai (2009), Fundamentals of Circuits and Filters (3rd ed.), Boca Raton: CRC Press, p. 62, ISBN 978-1-4200-5887-1
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Wikipedia:Ortrud Oellermann#0
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Ortrud R. Oellermann is a South African mathematician specializing in graph theory. She is a professor of mathematics at the University of Winnipeg. == Education and career == Oellermann was born in Vryheid. She earned a bachelor's degree, cum laude honours, and a master's degree at the University of Natal in 1981, 1982, and 1983 respectively, as a student of Henda Swart. She completed her Ph.D. in 1986 at Western Michigan University. Her dissertation was Generalized Connectivity in Graphs and was supervised by Gary Chartrand. Oellermann taught at the University of Durban-Westville, Western Michigan University, University of Natal, and Brandon University, before moving to Winnipeg in 1996. At Winnipeg, she was co-chair of mathematics and statistics for 2011–2013. == Contributions == With Gary Chartrand, Oellermann is the author of the book Applied and Algorithmic Graph Theory (McGraw Hill, 1993).[AA] She is also the author of well-cited research publications on metric dimension of graphs[MD], on distance-based notions of convex hulls in graphs,[CS] and on highly irregular graphs in which every vertex has a neighborhood in which all degrees are distinct.[IG] The phrase "highly irregular" was a catchphrase of her co-author Yousef Alavi; because of this, Ronald Graham suggested that there should be a concept of highly irregular graphs, by analogy to the regular graphs, and Oellermann came up with the definition of these graphs. == Recognition == In 1991, Oellermann was the winner of the annual Silver British Association Medal of the Southern Africa Association for the Advancement of Science. She won the Meiring Naude Medal of the Royal Society of South Africa in 1994. She was also one of three winners of the Hall Medal of the Institute of Combinatorics and its Applications in 1994, the first year the medal was awarded. == Selected publications == === Book === === Research articles === == References == == External links == Home page
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Wikipedia:Oscar Bruno#0
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Oscar P. Bruno is Professor of Applied & Computational Mathematics in the Computing and Mathematical Sciences Department at the California Institute of Technology. He is known for research on numerical analysis. == Academic biography == Bruno received the Licenciado degree from the University of Buenos Aires in 1982, and he completed the PhD in mathematics at New York University in 1989. His adviser was Robert V. Kohn, and his dissertation was titled The Effective Conductivity of an Infinitely Interchangeable Mixture. He taught at the University of Minnesota from 1989 to 1991, and he was at the Georgia Institute of Technology from 1991 to 1995. He has been on the faculty of the California Institute of Technology since 1995. == Awards and honors == In 1994, Bruno was awarded an Alfred P. Sloan Research Fellowship. He was inducted as a Fellow of the Society for Industrial and Applied Mathematics (SIAM) in 2013. == References == == External links == Oscar P. Bruno professional home page
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Wikipedia:Oscar Zariski#0
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Oscar Zariski (April 24, 1899 – July 4, 1986) was an American mathematician. The Russian-born scientist was one of the most influential algebraic geometers of the 20th century. == Education == Zariski was born Oscher (also transliterated as Ascher or Osher) Zaritsky to a Jewish family (his parents were Bezalel Zaritsky and Hanna Tennenbaum) and in 1918 studied at the University of Kyiv. He left Kyiv in 1920 to study at the University of Rome where he became a disciple of the Italian school of algebraic geometry, studying with Guido Castelnuovo, Federigo Enriques and Francesco Severi. Zariski wrote a doctoral dissertation in 1924 on a topic in Galois theory, which was proposed to him by Castelnuovo. At the time of his dissertation publication, he changed his name to Oscar Zariski. == Johns Hopkins University years == Zariski emigrated to the United States in 1927 supported by Solomon Lefschetz. He had a position at Johns Hopkins University where he became professor in 1937. During this period, he wrote Algebraic Surfaces as a summation of the work of the Italian school. The book was published in 1935 and reissued 36 years later, with detailed notes by Zariski's students that illustrated how the field of algebraic geometry had changed. It is still an important reference. It seems to have been this work that set the seal of Zariski's discontent with the approach of the Italians to birational geometry. He addressed the question of rigour by recourse to commutative algebra. The Zariski topology, as it was later known, is adequate for biregular geometry, where varieties are mapped by polynomial functions. That theory is too limited for algebraic surfaces, and even for curves with singular points. A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points. In geometric terms, one has to work with functions defined on some open, dense set of a given variety. The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions. This introduces a need, in the surface case, to use also valuation theory to describe the phenomena such as blowing up (balloon-style, rather than explosively). == Harvard University years == After spending a year 1946–1947 at the University of Illinois at Urbana–Champaign, Zariski became professor at Harvard University in 1947 where he remained until his retirement in 1969. In 1945, he fruitfully discussed foundational matters for algebraic geometry with André Weil. Weil's interest was in putting an abstract variety theory in place, to support the use of the Jacobian variety in his proof of the Riemann hypothesis for curves over finite fields, a direction rather oblique to Zariski's interests. The two sets of foundations weren't reconciled at that point. At Harvard, Zariski's students included Shreeram Abhyankar, Heisuke Hironaka, David Mumford, Michael Artin and Steven Kleiman—thus spanning the main areas of advance in singularity theory, moduli theory and cohomology in the next generation. Zariski himself worked on equisingularity theory. Some of his major results, Zariski's main theorem and the Zariski theorem on holomorphic functions, were amongst the results generalized and included in the programme of Alexander Grothendieck that ultimately unified algebraic geometry. Zariski proposed the first example of a Zariski surface in 1958. == Views == Zariski was a Jewish atheist. == Awards and recognition == Zariski was elected to the United States National Academy of Sciences in 1944, the American Academy of Arts and Sciences in 1948, and the American Philosophical Society in 1951. Zariski was awarded the Steele Prize in 1981, and in the same year the Wolf Prize in Mathematics with Lars Ahlfors. He wrote also Commutative Algebra in two volumes, with Pierre Samuel. His papers have been published by MIT Press, in four volumes. In 1997 a conference was held in his honor in Obergurgl, Austria. == Publications == Zariski, Oscar (2004) [1935], Abhyankar, Shreeram S.; Lipman, Joseph; Mumford, David (eds.), Algebraic surfaces, Classics in mathematics (second supplemented ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-58658-6, MR 0469915 Zariski, Oscar (1958), Introduction to the problem of minimal models in the theory of algebraic surfaces, Publications of the Mathematical Society of Japan, vol. 4, The Mathematical Society of Japan, Tokyo, MR 0097403 Zariski, Oscar (1969) [1958], Cohn, James (ed.), An Introduction to the Theory of Algebraic Surfaces, Lecture notes in mathematics, vol. 83, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082246, ISBN 978-3-540-04602-8, MR 0263819 Zariski, Oscar; Samuel, Pierre (1975) [1958], Commutative algebra I, Graduate Texts in Mathematics, vol. 28, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90089-6, MR 0090581 Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8, MR 0389876 Zariski, Oscar (2006) [1973], Kmety, François; Merle, Michel; Lichtin, Ben (eds.), The moduli problem for plane branches, University Lecture Series, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2983-7, MR 0414561(original title): Le problème des modules pour les branches planes{{citation}}: CS1 maint: postscript (link) Zariski, Oscar (1972), Collected papers. Vol. I: Foundations of algebraic geometry and resolution of singularities, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-08049-1, MR 0505100 Zariski, Oscar (1973), Collected papers. Vol. II: Holomorphic functions and linear systems, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-01038-2, MR 0505100 Zariski, Oscar (1978), Artin, Michael; Mazur, Barry (eds.), Collected papers. Volume III. Topology of curves and surfaces, and special topics in the theory of algebraic varieties, Mathematicians of Our Time, Cambridge, Massachusetts-London: MIT Press, ISBN 978-0-262-24021-5, MR 0505104 Zariski, Oscar (1979), Lipman, Joseph; Teissier, Bernard (eds.), Collected papers. Vol. IV. Equisingularity on algebraic varieties, Mathematicians of Our Time, vol. 16, MIT Press, ISBN 978-0-262-08049-1, MR 0545653 == See also == Zariski ring Zariski tangent space Zariski surface Zariski topology Zariski–Riemann surface Zariski space (disambiguation) Zariski's lemma Zariski's main theorem == Notes == == References == Blass, Piotr (2013), "The influence of Oscar Zariski on algebraic geometry" (PDF), Notices of the American Mathematical Society Mumford, David (1986), "Oscar Zariski: 1899–1986" (PDF), Notices of the American Mathematical Society, 33 (6): 891–894, ISSN 0002-9920, MR 0860889 Parikh, Carol (2009) [1991], The Unreal Life of Oscar Zariski, Springer, ISBN 9780387094304, MR 1086628 Gouvêa, Fernando Q. (1 January 2009). "Review of The Unreal Life of Oscar Zariski by Carol Parikh". MAA Reviews, Mathematical Association of America, maa.org. Retrieved 26 September 2024. == External links == O'Connor, John J.; Robertson, Edmund F., "Oscar Zariski", MacTutor History of Mathematics Archive, University of St Andrews Oscar Zariski at the Mathematics Genealogy Project Biography from United States Naval Academy.
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Wikipedia:Oscillation (mathematics)#0
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In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set). == Definitions == === Oscillation of a sequence === Let ( a n ) {\displaystyle (a_{n})} be a sequence of real numbers. The oscillation ω ( a n ) {\displaystyle \omega (a_{n})} of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of ( a n ) {\displaystyle (a_{n})} : ω ( a n ) = lim sup n → ∞ a n − lim inf n → ∞ a n {\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}} . The oscillation is zero if and only if the sequence converges. It is undefined if lim sup n → ∞ {\displaystyle \limsup _{n\to \infty }} and lim inf n → ∞ {\displaystyle \liminf _{n\to \infty }} are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞. === Oscillation of a function on an open set === Let f {\displaystyle f} be a real-valued function of a real variable. The oscillation of f {\displaystyle f} on an interval I {\displaystyle I} in its domain is the difference between the supremum and infimum of f {\displaystyle f} : ω f ( I ) = sup x ∈ I f ( x ) − inf x ∈ I f ( x ) . {\displaystyle \omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).} More generally, if f : X → R {\displaystyle f:X\to \mathbb {R} } is a function on a topological space X {\displaystyle X} (such as a metric space), then the oscillation of f {\displaystyle f} on an open set U {\displaystyle U} is ω f ( U ) = sup x ∈ U f ( x ) − inf x ∈ U f ( x ) . {\displaystyle \omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).} === Oscillation of a function at a point === The oscillation of a function f {\displaystyle f} of a real variable at a point x 0 {\displaystyle x_{0}} is defined as the limit as ϵ → 0 {\displaystyle \epsilon \to 0} of the oscillation of f {\displaystyle f} on an ϵ {\displaystyle \epsilon } -neighborhood of x 0 {\displaystyle x_{0}} : ω f ( x 0 ) = lim ϵ → 0 ω f ( x 0 − ϵ , x 0 + ϵ ) . {\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).} This is the same as the difference between the limit superior and limit inferior of the function at x 0 {\displaystyle x_{0}} , provided the point x 0 {\displaystyle x_{0}} is not excluded from the limits. More generally, if f : X → R {\displaystyle f:X\to \mathbb {R} } is a real-valued function on a metric space, then the oscillation is ω f ( x 0 ) = lim ϵ → 0 ω f ( B ϵ ( x 0 ) ) . {\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).} == Examples == 1 x {\displaystyle {\frac {1}{x}}} has oscillation ∞ at x {\displaystyle x} = 0, and oscillation 0 at other finite x {\displaystyle x} and at −∞ and +∞. sin 1 x {\displaystyle \sin {\frac {1}{x}}} (the topologist's sine curve) has oscillation 2 at x {\displaystyle x} = 0, and 0 elsewhere. sin x {\displaystyle \sin x} has oscillation 0 at every finite x {\displaystyle x} , and 2 at −∞ and +∞. ( − 1 ) x {\displaystyle (-1)^{x}} or 1, −1, 1, −1, 1, −1... has oscillation 2. In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity. Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region. == Continuity == Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x0 if and only if the oscillation is zero; in symbols, ω f ( x 0 ) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point. For example, in the classification of discontinuities: in a removable discontinuity, the distance that the value of the function is off by is the oscillation; in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides); in an essential discontinuity, oscillation measures the failure of a limit to exist. This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition. The oscillation is equivalent to the ε-δ definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space. == Generalizations == More generally, if f : X → Y is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each x ∈ X by ω ( x ) = inf { d i a m ( f ( U ) ) ∣ U i s a n e i g h b o r h o o d o f x } {\displaystyle \omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}} == See also == Wave equation Wave envelope Grandi's series Bounded mean oscillation == References == == Further reading ==
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Wikipedia:Oscillatory integral#0
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In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible to represent approximate solution operators for many differential equations as oscillatory integrals. == Definition == An oscillatory integral f ( x ) {\displaystyle f(x)} is written formally as f ( x ) = ∫ e i ϕ ( x , ξ ) a ( x , ξ ) d ξ , {\displaystyle f(x)=\int e^{i\phi (x,\xi )}\,a(x,\xi )\,\mathrm {d} \xi ,} where ϕ ( x , ξ ) {\displaystyle \phi (x,\xi )} and a ( x , ξ ) {\displaystyle a(x,\xi )} are functions defined on R x n × R ξ N {\displaystyle \mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N}} with the following properties: The function ϕ {\displaystyle \phi } is real-valued, positive-homogeneous of degree 1, and infinitely differentiable away from { ξ = 0 } {\displaystyle \{\xi =0\}} . Also, we assume that ϕ {\displaystyle \phi } does not have any critical points on the support of a {\displaystyle a} . Such a function, ϕ {\displaystyle \phi } is usually called a phase function. In some contexts more general functions are considered and still referred to as phase functions. The function a {\displaystyle a} belongs to one of the symbol classes S 1 , 0 m ( R x n × R ξ N ) {\displaystyle S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})} for some m ∈ R {\displaystyle m\in \mathbb {R} } . Intuitively, these symbol classes generalize the notion of positively homogeneous functions of degree m {\displaystyle m} . As with the phase function ϕ {\displaystyle \phi } , in some cases the function a {\displaystyle a} is taken to be in more general, or just different, classes. When m < − N {\displaystyle m<-N} , the formal integral defining f ( x ) {\displaystyle f(x)} converges for all x {\displaystyle x} , and there is no need for any further discussion of the definition of f ( x ) {\displaystyle f(x)} . However, when m ≥ − N {\displaystyle m\geq -N} , the oscillatory integral is still defined as a distribution on R n {\displaystyle \mathbb {R} ^{n}} , even though the integral may not converge. In this case the distribution f ( x ) {\displaystyle f(x)} is defined by using the fact that a ( x , ξ ) ∈ S 1 , 0 m ( R x n × R ξ N ) {\displaystyle a(x,\xi )\in S_{1,0}^{m}(\mathbb {R} _{x}^{n}\times \mathrm {R} _{\xi }^{N})} may be approximated by functions that have exponential decay in ξ {\displaystyle \xi } . One possible way to do this is by setting f ( x ) = lim ϵ → 0 + ∫ e i ϕ ( x , ξ ) a ( x , ξ ) e − ϵ | ξ | 2 / 2 d ξ , {\displaystyle f(x)=\lim \limits _{\epsilon \to 0^{+}}\int e^{i\phi (x,\xi )}\,a(x,\xi )e^{-\epsilon |\xi |^{2}/2}\,\mathrm {d} \xi ,} where the limit is taken in the sense of tempered distributions. Using integration by parts, it is possible to show that this limit is well defined, and that there exists a differential operator L {\displaystyle L} such that the resulting distribution f ( x ) {\displaystyle f(x)} acting on any ψ {\displaystyle \psi } in the Schwartz space is given by ⟨ f , ψ ⟩ = ∫ e i ϕ ( x , ξ ) L ( a ( x , ξ ) ψ ( x ) ) d x d ξ , {\displaystyle \langle f,\psi \rangle =\int e^{i\phi (x,\xi )}L{\big (}a(x,\xi )\,\psi (x){\big )}\,\mathrm {d} x\,\mathrm {d} \xi ,} where this integral converges absolutely. The operator L {\displaystyle L} is not uniquely defined, but can be chosen in such a way that depends only on the phase function ϕ {\displaystyle \phi } , the order m {\displaystyle m} of the symbol a {\displaystyle a} , and N {\displaystyle N} . In fact, given any integer M {\displaystyle M} , it is possible to find an operator L {\displaystyle L} so that the integrand above is bounded by C ( 1 + | ξ | ) − M {\displaystyle C(1+|\xi |)^{-M}} for | ξ | {\displaystyle |\xi |} sufficiently large. This is the main purpose of the definition of the symbol classes. == Examples == Many familiar distributions can be written as oscillatory integrals. The Fourier inversion theorem implies that the delta function, δ ( x ) {\displaystyle \delta (x)} is equal to 1 ( 2 π ) n ∫ R n e i x ⋅ ξ d ξ . {\displaystyle {\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }\,\mathrm {d} \xi .} If we apply the first method of defining this oscillatory integral from above, as well as the Fourier transform of the Gaussian, we obtain a well known sequence of functions which approximate the delta function: δ ( x ) = lim ε → 0 + 1 ( 2 π ) n ∫ R n e i x ⋅ ξ e − ε | ξ | 2 / 2 d ξ = lim ε → 0 + 1 ( 2 π ε ) n e − | x | 2 / ( 2 ε ) . {\displaystyle \delta (x)=\lim _{\varepsilon \to 0^{+}}{\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }e^{-\varepsilon |\xi |^{2}/2}\mathrm {d} \xi =\lim _{\varepsilon \to 0^{+}}{\frac {1}{({\sqrt {2\pi \varepsilon }})^{n}}}e^{-|x|^{2}/(2\varepsilon )}.} An operator L {\displaystyle L} in this case is given for example by L = ( 1 − Δ x ) k ( 1 + | ξ | 2 ) k , {\displaystyle L={\frac {(1-\Delta _{x})^{k}}{(1+|\xi |^{2})^{k}}},} where Δ x {\displaystyle \Delta _{x}} is the Laplacian with respect to the x {\displaystyle x} variables, and k {\displaystyle k} is any integer greater than ( n − 1 ) / 2 {\displaystyle (n-1)/2} . Indeed, with this L {\displaystyle L} we have ⟨ δ , ψ ⟩ = ψ ( 0 ) = 1 ( 2 π ) n ∫ R n e i x ⋅ ξ L ( ψ ) ( x , ξ ) d ξ d x , {\displaystyle \langle \delta ,\psi \rangle =\psi (0)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{ix\cdot \xi }L(\psi )(x,\xi )\,\mathrm {d} \xi \,\mathrm {d} x,} and this integral converges absolutely. The Schwartz kernel of any differential operator can be written as an oscillatory integral. Indeed if L = ∑ | α | ≤ m p α ( x ) D α , {\displaystyle L=\sum \limits _{|\alpha |\leq m}p_{\alpha }(x)D^{\alpha },} where D α = ∂ x α / i | α | {\displaystyle D^{\alpha }=\partial _{x}^{\alpha }/i^{|\alpha |}} , then the kernel of L {\displaystyle L} is given by 1 ( 2 π ) n ∫ R n e i ξ ⋅ ( x − y ) ∑ | α | ≤ m p α ( x ) ξ α d ξ . {\displaystyle {\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}e^{i\xi \cdot (x-y)}\sum \limits _{|\alpha |\leq m}p_{\alpha }(x)\,\xi ^{\alpha }\,\mathrm {d} \xi .} == Relation to Lagrangian distributions == Any Lagrangian distribution can be represented locally by oscillatory integrals, see Hörmander (1983). Conversely, any oscillatory integral is a Lagrangian distribution. This gives a precise description of the types of distributions which may be represented as oscillatory integrals. == See also == Riemann–Lebesgue lemma van der Corput lemma Oscillatory integral operator == References == Hörmander, Lars (1983), The Analysis of Linear Partial Differential Operators IV, Springer-Verlag, ISBN 0-387-13829-3 Hörmander, Lars (1971), "Fourier integral operators I", Acta Math., 127: 79–183, doi:10.1007/bf02392052
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Wikipedia:Osmo Pekonen#0
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Osmo Pekonen (2 April 1960 – 12 October 2022) was a Finnish mathematician, historian of science, and author. He was a docent of mathematics at the University of Helsinki and at the University of Jyväskylä, a docent of history of science at the University of Oulu, and a docent of history of civilization at the University of Lapland. He was the Book Reviews section editor of The Mathematical Intelligencer. == Personal life and death == Pekonen died suddenly in his sleep on 12 October 2022, at the age of 62, in Uzès, France during a bicycle tour. == Honours and distinctions == Osmo Pekonen was a corresponding member of four French academies; these are: Académie des sciences, arts et belles-lettres de Caen (founded in 1652), Académie des sciences, belles-lettres et arts de Besançon et de Franche-Comté (founded in 1752), Académie d'Orléans (founded in 1809) and Académie européenne des sciences, des arts et des lettres (founded in 1979). In 2012, he was awarded the Prix Chaix d'Est-Ange of the Académie des sciences morales et politiques in the field of history. == Bibliography == === Doctoral theses === Contributions to and a survey on moduli spaces of differential geometric structures with applications in physics, PhD thesis, University of Jyväskylä, 1988 La rencontre des religions autour du voyage de l'abbé Réginald Outhier en Suède en 1736-1737, D.Soc.Sci thesis, Rovaniemi: Lapland University Press, 2010 === Monographies and edited volumes === Topological and Geometrical Methods in Field Theory, Osmo Pekonen & Jouko Mickelsson (eds.), Singapore: World Scientific, 1992 Symbolien metsässä: Matemaattisia esseitä, Osmo Pekonen (ed.), Helsinki: Art House, 1992 Ranskan tiede: Kuuluisia kouluja ja instituutioita, Helsinki: Art House, 1995 Marian maa. Lasse Heikkilän elämä 1925–1961, Helsinki: SKS, 2002 Osmo Pekonen & Lea Pulkkinen: Sosiaalinen pääoma ja tieto- ja viestintätekniikan kehitys, Helsinki: The Parliament of Finland, Committee for the Future, 2002 Suomalaisen modernin lyriikan synty. Juhlakirja 75-vuotiaalle Lassi Nummelle, Osmo Pekonen (ed.), Kuopio: Snellman-instituutti, 2005 Porrassalmi. Etelä-Savon kulttuurin vuosikirja (ten volumes, I-X), Jorma Julkunen, Jutta Julkunen & Osmo Pekonen et alia (eds.) Mikkeli: Savon Sotilasperinneyhdistys Porrassalmi ry, 2008-2017 Lapin tuhat tarinaa. Anto Leikolan juhlakirja, Osmo Pekonen & Johan Stén (eds.), Ranua: Mäntykustannus, 2012 Salaperäinen Venus, Ranua: Mäntykustannus, 2012 Maupertuis en Laponie, with Anouchka Vasak, Paris: Hermann, 2014 Maan muoto, with Marja Itkonen-Kaila, Tornio: Väylä, 2019 Markkasen galaksit. Tapio Markkanen in memoriam, edited with Johan Stén, Helsinki: Ursa, 2019 Valon aika, with Johan Stén, Helsinki: Art House, 2019 Pohjan Tornio. Matkamiesten ääniä vuosisatain varrelta 1519-1919, Rovaniemi: Väylä, 2022 === Essay collections === Danse macabre: Eurooppalaisen matkakirja, Jyväskylä: Atena, 1994 Tuhat vuotta, Helsinki: WSOY, 1998 Minä ja Dolly: Kolumneja, esseitä, runoja, Jyväskylä: Atena, 1999 Oodi ilolle: Matkoja, maita, kaupunkeja, Turku: Enostone, 2010 Joka paikan akateemikko, Turku: Enostone, 2012 === Edited essay collections === Elämän puu, illustrated by Martti Ruokonen, Helsinki: WSOY, 1997 Elämän värit, graphic design by Jussi Jäppinen, Jyväskylä: Kopijyvä, 2003 Elämän vuodenajat, photographs by Seppo Nykänen, Jyväskylä: Minerva, 2005 === Edited poem collections === Lasse Heikkilä: Balladi Ihantalasta. Runoja kesästä 1944, Osmo Pekonen (ed.), Helsinki & Jyväskylä: Kopijyvä/Minerva, 1999, 2007, 2016 Charles Péguy: Chartres’n tie: Charles Péguy’n runoja, translated by Anna-Maija Raittila et alia, Osmo Pekonen (ed.), Jyväskylä: Minerva, 2003 === Diaries === Saint-Malosta Sääksmäelle. Päiväkirjastani 2014-2015, Tampere: Enostone, 2015 Minäkin Arkadiassa. Päiväkirjastani 2016-2017. Tampere: Enostone, 2017 Unikukkia, ulpukoita. Päiväkirjastani 2018-2019. Rovaniemi: Väylä, 2019 === Prose translations === Philippe Quéau: Lumetodellisuus (Le virtuel: Vertus et vertiges, 1993), translated into Finnish by Osmo Pekonen, Helsinki: Art House, 1995 Alexei Sossinsky: Solmut: Erään matemaattisen teorian synty (Nœuds: Genèse d’une théorie mathématique, 1999), translated into Finnish by Osmo Pekonen, Helsinki: Art House, 2002 Bo Lindberg: Latina ja Eurooppa (Europa och latinet, 1993), translated into Finnish by Osmo Pekonen, Jyväskylä: Atena, 1997 (2nd edition 2009) Peter Kravanja: Visconti, Proustin lukija (Visconti, lecteur de Proust, 2004), translated into Finnish by Osmo Pekonen, Jyväskylä & Helsinki: Minerva, 2006 Mary Terrall: Maupertuis, maapallon muodon mittaaja (The Man Who Flattened the Earth. Maupertuis and the Sciences in the Enlightenment, 2002), translated into Finnish by Osmo Pekonen, Tornio: Väylä, 2015 Émilie du Châtelet: Tutkielma onnesta (Discours sur le bonheur), translated into Finnish by Osmo Pekonen, Kuopio: Hai, 2016 Pierre Louis Moreau de Maupertuis: Fyysinen Venus (Vénus physique, 1745), translated into Finnish by Osmo Pekonen, Helsinki: Art House, 2017 Roger Picard: Salonkien aika (Les salons littéraires et la société française 1610–1789, New York 1943), translated into Finnish by Osmo Pekonen & Juhani Sarkava, Helsinki: Art House, 2018 Francis Godwin: Lento kuuhun (The Man in the Moone, London 1638), translated into Finnish by Osmo Pekonen, Helsinki: Basam Books, 2021 === Translations of ancient poetry === Beowulf, translation into Finnish and commentaries by Osmo Pekonen & Clive Tolley, Helsinki: WSOY, 1999. Second edition: WSOY 2007 Widsith: Anglosaksinen muinaisruno, translation into Finnish and commentaries by Osmo Pekonen & Clive Tolley, Jyväskylä: Minerva, 2004 Waldere: Anglosaksinen muinaisruno, translation into Finnish and commentaries by Jonathan Himes, Osmo Pekonen & Clive Tolley, Jyväskylä: Minerva, 2005 Gustav Philip Creutz: Atis ja Camilla (Atis och Camilla, 1761), translated into Finnish by Osmo Pekonen, Turku: Faros, 2019 === Books with a preface by Osmo Pekonen === Ivar Ekeland: Paras mahdollisista maailmoista (Le meilleur des mondes possibles, 2000), translated into Finnish by Susanna Maaranen, Helsinki: Art House, 2004 Réginald Outhier: Matka Pohjan perille (Journal d'un voyage au Nord, 1744), translated into Finnish by Marja Itkonen-Kaila, Rovaniemi: Väylä, 2011 Jean-François Regnard: Retki Lappiin (Voyage de Laponie, 1731), translated into Finnish by Marja Itkonen-Kaila, Rovaniemi: Väylä, 2012 Lassi Nummi: Le jardin de la vie, a selection of poems translated into French by Yves Avril, Orléans: Paradigme, 2015 Auli Särkiö: Sarmatie (Sarmatia, 2011), translated into French by Yves Avril, Mont de Laval: Grand Tétras, 2015 Eino Leino: "Doch ein Lied steht über allen...", a selection of poems translated into German by Manfred Stern, Hamburg: Verlag Dr. Kovač, 2017 Voltaire: Mikromegas. Filosofinen kertomus (Micromégas: histoire philosophique, 1752), translated into Finnish by Marja Haapio, Helsinki: Basam Books, 2019 == References == Osmo Pekonen at the Mathematics Genealogy Project Osmo Pekonen homepage Osmo Pekonen's results at International Mathematical Olympiad Descartes Prize nominee citation Chaix d'Est Ange Prize Osmo Pekonen at IMDb == External links == Official website (not updated anymore)
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Wikipedia:Oswald Leroy#0
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Oswald Jozef Leroy (16 May 1936 – 7 September 2022) was a Belgian mathematician known for his contributions to theoretical acousto-optics. Leroy's biggest achievement was a theoretical study of the interaction of light with two adjacent ultrasonic beams under different conditions in terms of beam shape, frequency content and intensity. Understanding of this phenomenon was very important in the 1970s when new acousto-optic devices were developed (mainly due to novel developments in laser technology) that utilized adjacent ultrasonic beams. Such devices were being used in optical modulators, optical scanners, information processing , optical filtering and frequency-spectrum analysis. Before his contribution only the interaction of light with one ultrasonic beam was understood. Since then, acousto-optic devices have been used in telecommunication and military applications. Leroy died at Ostend, Belgium in 2022. == Career == Leroy received his PhD from Ghent University in the group of Robert A. Mertens for his thesis entitled Diffraction of light by ultrasound. He worked as an assistant professor at Ghent University from 1966 to 1972 and has been a tenured professor at the Catholic University of Leuven since 1972. New developments in laser physics formed the ground for collaborations between the team of Leroy and different other laboratories. He was a guest professor at the Paris Diderot University and Université de Bordeaux, the University of Tennessee and the Tokyo Institute of Technology. Furthermore, he has collaborated with the University of Gdansk, Georgetown University, and the University of Houston. He retired in 2001 and received the title of emeritus professor. == Awards == In 1991 Leroy was awarded an honorary doctorate from the University of Gdansk, for his contributions to theoretical acousto-optics and to celebrate a collaboration with the team of Antoni Sliwinski at the Institute of Physics of the University of Gdansk. In 2001 he has received the ‘Médaille étrangère’ of the French Acoustical Society. == Selected articles == Leroy O., "Diffraction of light by two adjacent parallel ultrasonic-waves", journal of the acoustical society of America 51(1), 148, 1972 Leroy O., "Theory of diffraction of light by ultrasonic-waves consisting of a fundamental tone and its first n − 1 harmonics", ultrasonics 10(4), 182, 1972 Leroy O., "Diffraction of light by 2 adjacent parallel ultrasonic beams", acustica 29(5), 303–310, 1973 Leroy O., "General symmetry properties of diffraction pattern in diffraction of light by parallel adjacent ultrasonic beams", journal of sound and vibration 26(3), 389–393, 1973 Leroy O., "Diffraction of light by 2 parallel adjacent ultrasonic-waves, having same wavelengths", journal of sound and vibration 32(2), 241–249, 1974 Leroy, O., "TI light-diffraction caused by adjacent ultrasonics", IEEE transactions on sonics and ultrasonics su22(3), 233, 1975 Poleunis F., Leroy O., "Diffraction of light by 2 adjacent parallel ultrasonic-waves, one being a fundamental tone and other its 2nd harmonic", journal of sound and vibration 58(4), 509–515, 1978 Leroy O., Mertens R., "Diffraction of light by adjacent parallel ultrasonic-waves with arbitrary frequencies (noa-method)", acustica 26(2), 96, 1972 W. Hereman, R. Mertens, F. Verheest, O. Leroy, J.M. Claeys, E. Blomme, "Interaction of light and ultrasound : Acoustoopics", Physicalia Magazine 6(4), 213–245, 1984. == Books == Physical Acoustics – Fundamentals and Applications, M.A. Breazeale O. Leroy (Eds.), Plenum US, 1991 Advances in Acousto-Optics : 5th International Meeting of the European Acousto-Optical Club, Oswald Leroy (Edt.), Institute of Physics, 2001. == References ==
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Wikipedia:Otakar Borůvka#0
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Otakar Borůvka (10 May 1899 – 22 July 1995) was a Czech mathematician. He is best known for his work in graph theory. == Education and career == Borůvka was born in Uherský Ostroh, a town in Moravia, Austria-Hungary (today in the Czech Republic), the son of a school headmaster. He attended the grammar school in Uherské Hradiště beginning in 1910. In 1916, influenced by the ongoing World War I, he moved to the military school (Realschule) in Hranice, and later he enrolled into the Imperial and Royal Technical Military Academy in Mödling near Vienna. When the war ended, Borůvka returned to Uherské Hradiště, finished his studies in 1918 at the Gymnasium there, and became a student at the Imperial Czech Technical University of Franz Joseph, in Brno, initially studying civil engineering. In 1920, Masaryk University opened in Brno, and Borůvka also began taking courses there. He became an assistant to Mathias Lerch at Masaryk in 1921, but Lerch died in 1922; his position at Masaryk was taken by Eduard Čech, whom Borůvka also assisted, earning his doctorate in 1923. At Čech's suggestion, Borůvka visited Élie Cartan in Paris from 1926 to 1927. He earned his habilitation from Masaryk University in 1927, and (turning down an offer from the University of Zagreb) he became a docent there in 1928. He continued to travel abroad through the late 1920s and early 1930s, to Cartan in Paris again as well as to Wilhelm Blaschke in Hamburg. He was promoted to assistant professor at Masaryk in 1934, given a chair in 1940, and made an ordinary professor in 1946. In 1965, he founded the new journal Archivum Mathematicum, and in 1969, he became a founding member of the Institute of Mathematics of the Czechoslovak Academy of Sciences, splitting his time between the Institute and his professorship at Masaryk. == Contributions == The problem of designing efficient electric distribution networks had been suggested to Borůvka by his friend Jindřich Saxel, an employee of the West Moravian Power Company, during World War I. In his 1926 paper O jistém problému minimálním (English On a certain minimal problem), Borůvka solved this problem by modeling it mathematically as a minimum spanning tree problem, and described the first known algorithm for finding the minimum spanning tree of a metric space (the set of cities to be connected by the network, together with their distances). Now called Borůvka's algorithm, his method works by repeatedly adding a connections between each subtree of the minimum spanning tree found so far and its nearest neighboring subtree. The same algorithm has been rediscovered repeatedly. It is more suitable for distributed and parallel computation than many other minimum spanning tree algorithms, can achieve linear time complexity on planar graphs and more generally in minor-closed graph families, and plays a central role in the randomized linear time algorithm of Karger, Klein & Tarjan (1995). From 1924 to 1935, Borůvka's primary interest was in differential geometry. His work in this area concerned analytic correspondences between projective planes, normal curvature of high-dimensional surfaces, and Frenet formula for curves in high-dimensional spaces. Beginning in the 1930s, Borůvka's interests shifted to abstract algebra, and in particular the theory of groups. He was also one of the first to study a generalization of groups, called by him "groupoids" but now more commonly referred to as magmas. A textbook by him on groups and groupoids, originally published in Czech in 1944, went through several expansions, and translations, including an English edition in 1976. Following the war, Borůvka shifted gears again, from algebra to the theory of differential equations. He published several research papers on this subject, as well as a monograph on second-order differential equations which he published in 1971. == Awards and honors == Borůvka became a corresponding member of the Czechoslovak Academy of Sciences at its creation in 1953, and an ordinary member in 1965. In 1969, Comenius University in Bratislava gave him an honorary doctorate, and in 1994 he received a second honorary doctorate from Masaryk University in Brno. He has also been given medals by the Free University of Brussels, the University of Liège, Jagiellonian University, Comenius University, Palacký University of Olomouc, Jan Evangelista Purkyně University in Ústí nad Labem, the German Academy of Sciences at Berlin, the Russian Academy of Sciences#Academy of Sciences of the USSR, and the Czechoslovak Academy of Sciences. == References == == External links == Borůvka, Otakar, Czech Digital Mathematics Library
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Wikipedia:Otomar Hájek#0
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Otomar Hájek (December 31, 1930 - December 18, 2016) was a Czech-American mathematician, known for his contributions to dynamical systems, game theory and control theory. He was born in Belgrade in Serbia, moving with his family to Prague in 1935, to the Netherlands in 1939 and via Algeria and southern France to London in 1940 where they lived until 1945 when they returned to Prague. After high school in 1949 he studied mathematics at Charles University in Prague, resulting in a Ph.D. in 1963 on a thesis entitled Dynamical systems in the plane. At the same place he joined the mathematics and physics faculty in 1965, before moving in 1968 to Cleveland and Case Western Reserve University where he worked until 1995 as professor, becoming an emeritus in 1996. In the mid-1970s Hájek was also in receipt of a von Humboldt award at the TH Darmstadt, Fachbereich Mathematik. == Books == Hájek, Otomar (1968), Dynamical systems in the plane, Boston, MA: Academic Press, MR 0240418 Hájek, Otomar (2008) [1975], Pursuit Games: An Introduction to the Theory and Applications of Differential Games of Pursuit and Evasion, New York: Dover Publications, ISBN 978-0-486-46283-7, MR 0456557 Hájek, Otomar (2008) [1991], Hájek, Otamar (ed.), Control theory in the plane, Lecture Notes in Control and Information Sciences, vol. 153, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0042035, ISBN 978-3-540-85254-4, MR 1109709 == References ==
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Wikipedia:Otto Brune#0
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Otto Walter Heinrich Oscar Brune (10 January 1901 – 1982) undertook some key investigations into network synthesis at the Massachusetts Institute of Technology (MIT) where he graduated in 1929. His doctoral thesis was supervised by Wilhelm Cauer and Ernst Guillemin, who the latter ascribed to Brune the laying of "the mathematical foundation for modern realization theory". == Biography == Brune was born in Bloemfontein, Orange Free State 10 January 1901 and grew up in Kimberley, Cape Colony. He enrolled in the University of Stellenbosch in 1918, receiving a Bachelor of Science in 1920 and Master of Science in 1921. He taught German, mathematics, and science at the Potchefstroom Gymnasium, Transvaal in 1922, and lectured in mathematics at the Transvaal University College, Pretoria 1923–1925. In 1926 Brune moved to the US to attend the Massachusetts Institute of Technology (MIT) under the sponsorship of the General Electric Company, receiving bachelor and master's degrees in 1929. From 1929 to 1930, Brune was involved in artificial lightning tests on the power transmission line from Croton Dam, Michigan as a research assistant at MIT. From 1930, Brune was a Fellow in Electrical Engineering at MIT with an Austin Research Fellowship. Brune returned to South Africa in 1935. He became Principal Research Officer at the National Research Laboratories, Pretoria. == Works == In 1933, Brune was working on his doctoral thesis entitled, Synthesis of Passive Networks and Cauer suggested that he provide a proof of the necessary and sufficient conditions for the realisability of multi-port impedances. Cauer himself had found a necessary condition but had failed to prove it to be sufficient. The goal for researchers then was "to remove the restrictions implicit in the Foster-Cauer realisations and find conditions on Z equivalent to realisability by a network composed of arbitrary interconnections of positive-valued R, C and L." Brune coined the term positive-real (PR) for that class of analytic functions that are realisable as an electrical network using passive components. He did not only introduce the mathematical characterization of this function in one complex variable but also demonstrated "the necessity and sufficiency for the realization of driving point functions of lumped, linear, finite, passive, time-invariant and bilateral network. Brune also showed that if the case is limited to scalar PR functions then there was no other theoretical reason that required ideal transformers in the realisation (transformers limit the practical usefulness of the theory), but was unable to show (as others later did) that transformers can always be avoided. The eponymous Brune cycle continued fractions were invented by Brune to facilitate this proof. The Brune theorem is: The impedance Z(s) of any electric network composed of passive components is positive-real. If Z(s) is positive-real it is realisable by a network having as components passive (positive) R, C, L, and ideal transformers T. Brune is also responsible for the Brune test for determining the permissibility of interconnecting two-port networks. == Legacy == For his work, Brune is recognized as one of those who laid the foundation of network analysis by means of mathematics. For instance, American computer scientist Ernst Guillemin dedicated his book Synthesis of Passive Network to Brune, describing him with these words: "In my opinion the one primarily responsible for establishing a very broad and mathematically rigorous basis for realization theory generally was Otto Brune." == References == == Bibliography == Cauer, E.; Mathis, W.; Pauli, R., "Life and Work of Wilhelm Cauer (1900–1945)", Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June 2000. Chen, Wai-Kai, Active Filters: Theory and Implementation, Wiley, 1986 ISBN 047182352X. Brune, O., "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, 5 May 1931a, republished in, MIT Journal of Mathematics and Physics, vol. 10, pp. 191–236, 1931b. Brune O., "Equivalent Electrical Networks", Physical Review, vol. 38, pp. 1783–1783, 1931c. Galkowski, Krzysztof; Wood, Jeff David, Multidimensional Signals, Circuits and Systems, Taylor & Francis, 2001 ISBN 0415253632. Horrocks, D. H.; Nightingale, C., "The compatibility of n-ports in parallel", International Journal of Circuit Theory and Applications, vol. 4, pp. 81–85, January 1976. Seising, Rudolf, Die Fuzzifizierung der Systeme, Franz Steiner Verlag, 2005 ISBN 3515087680 Seising, Rudolf, The Fuzzification of Systems: The Genesis of Fuzzy Set Theory and its Initial Applications – Developments up to the 1970s Springer, 2007 ISBN 9783540717942. Wildes, Karl L.; Lindgren, Nilo A., A century of electrical engineering and computer science at MIT, 1882-1982, MIT Press, 1985 ISBN 0-262-23119-0. Willems, Jan; Hara, Shinji; Ohta, Yoshito; Fujioka, Hisaya, Perspectives in Mathematical System Theory, Control, and Signal Processing, Springer, 2010 ISBN 9783540939177.
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Wikipedia:Otto Frostman#0
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Otto Albin Frostman (3 January 1907 – 29 December 1977) was a Swedish mathematician, known for his work in potential theory and complex analysis. Frostman earned his Ph.D. in 1935 at Lund University under the Hungarian-born mathematician Marcel Riesz, the younger brother of Frigyes Riesz. In potential theory, Frostman's lemma is named after him. He supervised the 1971 Stockholm University Ph.D. thesis of Bernt Lindström, which initiated the "Stockholm School" of topological combinatorics (combining simplicial homology and enumerative combinatorics). == Notes == == External links == Otto Frostman at the Mathematics Genealogy Project ICMI webpage
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Wikipedia:Otto Hesse#0
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Ludwig Otto Hesse (22 April 1811 – 4 August 1874) was a German mathematician. Hesse was born in Königsberg, Prussia, and died in Munich, Bavaria. He worked mainly on algebraic invariants, and geometry. The Hessian matrix, the Hesse normal form, the Hesse configuration, the Hessian group, Hessian pairs, Hesse's theorem, Hesse pencil, and the Hesse transfer principle are named after him. Many of Hesse's research findings were presented for the first time in Crelle's Journal or Hesse's textbooks. == Life == Hesse was born in Königsberg (today Kaliningrad) as the son of Johann Gottlieb Hesse, a businessman and brewery owner and his wife Anna Karoline Reiter (1788–1865). He studied in his hometown at the Albertina under Carl Gustav Jacob Jacobi. Among his teachers were count Friedrich Wilhelm Bessel and Friedrich Julius Richelot. He earned his doctorate in 1840 at the University of Königsberg with the dissertation De octo punctis intersectionis trium superficium secundi ordinis. In 1841, Hesse completed his habilitation thesis. In the same year he married Sophie Marie Emilie Dulk, the daughter of pharmacists and chemistry professor Friedrich Philipp Dulk (1788–1852). The couple had a son and five daughters. Hesse taught for some time physics and chemistry at the Vocational School in Königsberg and lectured at the Albertina. In 1845 he was appointed associate professor in Königsberg. In 1855 he moved to Halle and in 1856 to Heidelberg until 1868, when he finally moved to Munich to the newly established Polytechnic School. In 1869 he joined the Bavarian Academy of Sciences. His doctoral students include Olaus Henrici, Gustav Kirchhoff, Jacob Lüroth, Adolph Mayer, Carl Neumann, Max Noether, Ernst Schröder, and Heinrich Martin Weber. == Works == Vorlesungen über analytische Geometrie des Raumes. (Lectures on analytic geometry of space) Leipzig (3. A. 1876) (Internet Archive) Vorlesungen aus der analytischen Geometrie der geraden Linie, des Punktes und des Kreises. (Lectures from the analytical geometry of the straight line, the point and the circle) Leipzig (1881). Hrsg. A. Gundelfinger (Internet Archive) Die Determinanten elementar behandelt. (Determinants elementary treated) Leipzig (2. A. 1872) (Göttinger Digitalisierungszentrum) Die vier Species. (The four Species) Leipzig (1872) (Internet Archive) His collected works were published in 1897 by Bavarian Academy of Sciences and Humanities. Hesse, Ludwig Otto (1972) [1897], Dyck, W.; Gundelfinger, S.; Lüroth, J.; et al. (eds.), Ludwig Otto Hesse's gesammelte Werke, New York: Chelsea Publishing Co., ISBN 978-0-8284-0261-3, MR 0392474 (Internet Archive) == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Otto Hesse", MacTutor History of Mathematics Archive, University of St Andrews Otto Hesse at the Mathematics Genealogy Project Vorlesungen über analytische Geometrie des Raumes, insbesondere über Oberflächen zweiter Ordnung Complete Dictionary of Scientific Biography: "Hesse, Ludwig Otto"
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Wikipedia:Otto M. Nikodym#0
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Otto Marcin Nikodym (3 August 1887 – 4 May 1974) (also Otton Martin Nikodým) was a Polish mathematician. == Education and career == Nikodym studied mathematics at the University of Lemberg (today's University of Lviv). Immediately after his graduation in 1911, he started his teaching job at a high school in Kraków where he remained until 1924. He eventually obtained his doctorate in 1925 from the University of Warsaw; he also spent an academic year (1926-1927) in Sorbonne. Nikodym taught at the Jagiellonian University in Kraków and University of Warsaw and at the Akademia Górnicza in Kraków in the years that followed. He moved to the United States in 1948 and joined the faculty of Kenyon College. He retired in 1966 and moved to Utica, New York, where he continued his research until his death. == Personal life == Nikodym was born in 1887 in Demycze, a suburb of Zabłotów (in modern-day Ukraine), to a family with Polish, Czech, Italian and French roots. Orphaned at a young age, he was brought up by his maternal grandparents. In 1924, he married Stanisława Nikodym, the first Polish woman to obtain a PhD in mathematics. == Research works == Nikodym worked in a wide range of areas, but his best-known early work was his contribution to the development of the Lebesgue–Radon–Nikodym integral (see Radon–Nikodym theorem). His work in measure theory led him to an interest in abstract Boolean lattices. His work after coming to the United States centered on the theory of operators in Hilbert space, based on Boolean lattices, culminating in his The Mathematical Apparatus for Quantum-Theories. He was also interested in the teaching of mathematics. == See also == Nikodym set Radon–Nikodym theorem Radon–Nikodym property of a Banach space List of Polish mathematicians == References == == External links == MacTutor Entry Otto M. Nikodym at Find a Grave
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Wikipedia:Otto Schilling#0
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Otto Franz Georg Schilling (3 November 1911 – 20 June 1973) was a German-American mathematician known as one of the leading algebraists of his time. He was born in Apolda and studied in the 1930s at the Universität Jena and the Universität Göttingen under Emmy Noether. After Noether was forced to leave Germany by the Nazis, he found a new advisor in Helmut Hasse, and obtained his Ph.D. from Marburg University in 1934 on the thesis Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper. He then was post doc at Trinity College, Cambridge before moving to Institute for Advanced Study 1935–37 and the Johns Hopkins University 1937–39. He became an instructor with the University of Chicago in 1939, promoted to assistant professor 1943, associate 1945 and full professor in 1958. In 1961 he moved to Purdue University. He died in Highland Park, Illinois. His students were, among others, the game theorist Anatol Rapoport and the mathematician Harley Flanders. == Articles == Schilling, Otto F. G. (1937). "Arithmetic in a Special Class of Algebras". The Annals of Mathematics. 38 (1): 116–119. doi:10.2307/1968513. ISSN 0003-486X. JSTOR 1968513. Schilling, Otto F. G. (1937). "Class Fields of Infinite Degree Over p-Adic Number Fields". The Annals of Mathematics. 38 (2): 469–476. doi:10.2307/1968563. ISSN 0003-486X. JSTOR 1968563. Schilling, O. E. G. (1937). "Arithmetic in Fields of Formal Power Series in Several Variables". The Annals of Mathematics. 38 (3): 551–576. doi:10.2307/1968600. ISSN 0003-486X. JSTOR 1968600. (typo in Schilling's name) Schilling, O. F. G. (1938). "The Structure of Local Class Field Theory". American Journal of Mathematics. 60 (1): 75–100. doi:10.2307/2371544. ISSN 0002-9327. JSTOR 2371544. Schilling, O. F. G. (1938). "A Generalization of Local Class Field Theory". American Journal of Mathematics. 60 (3): 667–704. doi:10.2307/2371605. ISSN 0002-9327. JSTOR 2371605. Schilling, O. F. G. (1939). "Units in p-Adic Algebras". American Journal of Mathematics. 61 (4): 883–896. doi:10.2307/2371632. ISSN 0002-9327. JSTOR 2371632. with Saunders Mac Lane: MacLane, Saunders; Schilling, O. F. G. (1939). "Zero-Dimensional Branches of Rank One on Algebraic Varieties". The Annals of Mathematics. 40 (3): 507–520. doi:10.2307/1968935. ISSN 0003-486X. JSTOR 1968935. with Saunders Mac Lane: Lane, Saunders Mac; Schilling, O. F. G. (1939). "Infinite Number Fields with Noether Ideal Theories". American Journal of Mathematics. 61 (3): 771–782. doi:10.2307/2371335. ISSN 0002-9327. JSTOR 2371335. with Saunders Mac Lane: MacLane, S.; Schilling, O. F. G. (1940). "Normal Algebraic Number Fields". Proceedings of the National Academy of Sciences. 26 (2): 122–126. doi:10.1073/pnas.26.2.122. ISSN 0027-8424. PMC 1078017. PMID 16588322. Schilling, O. F. G. (1940). "Regular normal extensions over complete fields". Trans. Amer. Math. Soc. 47 (3): 440–454. doi:10.1090/s0002-9947-1940-0001970-2. MR 0001970. Schilling, O. F. G. (1940). "Remarks on a Special Class of Algebras". American Journal of Mathematics. 62 (1/4): 346–352. doi:10.2307/2371458. ISSN 0002-9327. JSTOR 2371458. with Saunders Mac Lane: MacLane, Saunders; Schilling, O. F. G. (1942). "A formula for the direct products of crossed product algebras". Bull. Amer. Math. Soc. 48 (2): 108–114. doi:10.1090/s0002-9904-1942-07613-0. MR 0006152. with Irving Kaplansky: Kaplansky, Irving; Schilling, O. F. G. (1942). "Some remarks on relatively complete fields". Bulletin of the American Mathematical Society. 48 (10): 744–748. doi:10.1090/S0002-9904-1942-07772-X. ISSN 0002-9904. Schilling, O. F. G. (1943). "Normal Extensions of Relatively Complete Fields". American Journal of Mathematics. 65 (2): 309–334. doi:10.2307/2371818. ISSN 0002-9327. JSTOR 2371818. Schilling, O. F. G. (1945). "On a special class of abelian functions". Bull. Amer. Math. Soc. 51 (2): 133–136. doi:10.1090/s0002-9904-1945-08293-7. MR 0011291. Schilling, O. F. G. (1945). "Noncommutative valuations". Bull. Amer. Math. Soc. 51 (4): 297–304. doi:10.1090/s0002-9904-1945-08339-6. MR 0011684. Schilling, O. F. G. (1946). "Ideal theory on open Riemann surfaces". Bull. Amer. Math. Soc. 52 (11, Part 1): 945–963. doi:10.1090/s0002-9904-1946-08669-3. MR 0019733. "Necessary conditions for local class field theory" (PDF). Mathematical Journal of Okayama University. 3 (1): 5–10. 1953. ISSN 0030-1566. f. g. Schilling, O. (1961). "On local class field theory". Journal of the Mathematical Society of Japan. 13 (3): 234–245. doi:10.2969/jmsj/01330234. ISSN 0025-5645. == Books == Schilling, Otto (1950). Theory of Valuations. American Mathematical Society. ISBN 0821815040. {{cite book}}: ISBN / Date incompatibility (help) Schilling, Otto; Piper, William Stephen (1975). Basic abstract algebra. Allyn & Bacon. ISBN 0205042732; (394 pages){{cite book}}: CS1 maint: postscript (link) == References ==
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Wikipedia:Otto Stolz#0
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Otto Stolz (3 July 1842 – 23 November 1905) was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied at the University of Innsbruck from 1860 and the University of Vienna from 1863, receiving his habilitation there in 1867. Two years later he studied in Berlin under Karl Weierstrass, Ernst Kummer and Leopold Kronecker, and in 1871 heard lectures in Göttingen by Alfred Clebsch and Felix Klein (with whom he would later correspond), before returning to Innsbruck permanently as a professor of mathematics. His work began with geometry (on which he wrote his thesis) but after the influence of Weierstrass it shifted to real analysis, and many small useful theorems are credited to him. For example, he proved that a continuous function f on a closed interval [a, b] with midpoint convexity, i.e., f ( x + y 2 ) ≤ f ( x ) + f ( y ) 2 {\displaystyle f\left({\frac {x+y}{2}}\right)\leq {\frac {f(x)+f(y)}{2}}} , has left and right derivatives at each point in (a, b). He died in 1905 shortly after finishing work on Einleitung in die Funktionentheorie. His name lives on in the Stolz–Cesàro theorem. == Work on non-Archimedean systems == Stolz published a number of papers containing constructions of non-Archimedean extensions of the real numbers, as detailed by Ehrlich (2006). His work, as well as that of Paul du Bois-Reymond, was sharply criticized by Georg Cantor as an "abomination". Cantor published a "proof-sketch" of the inconsistency of infinitesimals. The errors in Cantor's proof are analyzed by Ehrlich (2006). == Notes == == Bibliography == Philip Ehrlich (2006). "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes", Archive for History of Exact Sciences 60, no. 1, pp. 1–121. doi:10.1007/s00407-005-0102-4 == External links == Almanach for 1906, containing obituary O'Connor, John J.; Robertson, Edmund F., "Otto Stolz", MacTutor History of Mathematics Archive, University of St Andrews Österreich Lexikon, containing Stolz's photograph [1] Haus Der Mathematik
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Wikipedia:Outline of algebra#0
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Algebra is one of the main branches of mathematics, covering the study of structure, relation and quantity. Algebra studies the effects of adding and multiplying numbers, variables, and polynomials, along with their factorization and determining their roots. In addition to working directly with numbers, algebra also covers symbols, variables, and set elements. Addition and multiplication are general operations, but their precise definitions lead to structures such as groups, rings, and fields. == Branches == Pre-algebra Elementary algebra Boolean algebra Abstract algebra Linear algebra Universal algebra == Algebraic equations == An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree. Linear equation – algebraic equation of degree one. Polynomial equation – equation in which a polynomial is set equal to another polynomial. Transcendental equation – equation involving a transcendental function of one of its variables. Functional equation – equation in which the unknowns are functions rather than simple quantities. Differential equation – equation involving derivatives. Integral equation – equation involving integrals. Diophantine equation – equation where the only solutions of interest of the unknowns are the integer ones. == History == History of algebra == General algebra concepts == Fundamental theorem of algebra – states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. Equations – equality of two mathematical expressions Linear equation – an algebraic equation with a degree of one Quadratic equation – an algebraic equation with a degree of two Cubic equation – an algebraic equation with a degree of three Quartic equation – an algebraic equation with a degree of four Quintic equation – an algebraic equation with a degree of five Polynomial – an algebraic expression consisting of variables and coefficients Inequalities – a comparison between values Functions – mapping that associates a single output value with each input value Sequences – ordered list of elements either finite or infinite Systems of equations – finite set of equations Vectors – element of a vector space Matrix – two dimensional array of numbers Vector space – basic algebraic structure of linear algebra Field – algebraic structure with addition, multiplication and division Groups – algebraic structure with a single binary operation Rings – algebraic structure with addition and multiplication == See also == Table of mathematical symbols == External links == '4000 Years of Algebra', lecture by Robin Wilson, at Gresham College, 17 October 2007 (available for MP3 and MP4 download, as well as a text file). ExampleProblems.com Example problems and solutions from basic and abstract algebra.
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Wikipedia:Outline of linear algebra#0
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This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices. == Linear equations == Linear equation System of linear equations Determinant Minor Cauchy–Binet formula Cramer's rule Gaussian elimination Gauss–Jordan elimination Overcompleteness Strassen algorithm == Matrices == Matrix Matrix addition Matrix multiplication Basis transformation matrix Characteristic polynomial Trace Eigenvalue, eigenvector and eigenspace Cayley–Hamilton theorem Spread of a matrix Jordan normal form Weyr canonical form Rank Matrix inversion, invertible matrix Pseudoinverse Adjugate Transpose Dot product Symmetric matrix Orthogonal matrix Skew-symmetric matrix Conjugate transpose Unitary matrix Hermitian matrix, Antihermitian matrix Positive-definite, positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal matrix, main diagonal Diagonalizable matrix Triangular matrix Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant matrix Hankel matrix (0,1)-matrix == Matrix decompositions == Matrix decomposition Cholesky decomposition LU decomposition QR decomposition Polar decomposition Reducing subspace Spectral theorem Singular value decomposition Higher-order singular value decomposition Schur decomposition Schur complement Haynsworth inertia additivity formula == Relations == Matrix equivalence Matrix congruence Matrix similarity Matrix consimilarity Row equivalence == Computations == Elementary row operations Householder transformation Least squares, linear least squares Gram–Schmidt process Woodbury matrix identity == Vector spaces == Vector space Linear combination Linear span Linear independence Scalar multiplication Basis Change of basis Hamel basis Cyclic decomposition theorem Dimension theorem for vector spaces Hamel dimension Examples of vector spaces Linear map Shear mapping or Galilean transformation Squeeze mapping or Lorentz transformation Linear subspace Row and column spaces Column space Row space Cyclic subspace Null space, nullity Rank–nullity theorem Nullity theorem Dual space Linear function Linear functional Category of vector spaces == Structures == Topological vector space Normed vector space Inner product space Euclidean space Orthogonality Orthogonal complement Orthogonal projection Orthogonal group Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation (geometry) Improper rotation Symplectic structure == Multilinear algebra == Multilinear algebra Tensor Classical treatment of tensors Component-free treatment of tensors Gamas's Theorem Outer product Tensor algebra Exterior algebra Symmetric algebra Clifford algebra Geometric algebra == Topics related to affine spaces == Affine space Affine transformation Affine group Affine geometry Affine coordinate system Flat (geometry) Cartesian coordinate system Euclidean group Poincaré group Galilean group == Projective space == Projective space Projective transformation Projective geometry Projective linear group Quadric and conic section == See also == Glossary of linear algebra Glossary of tensor theory
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Wikipedia:Overcompleteness#0
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Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R. J. Duffin and A. C. Schaeffer in 1952. Formally, a subset of the vectors { ϕ i } i ∈ J {\displaystyle \{\phi _{i}\}_{i\in J}} of a Banach space X {\displaystyle X} , sometimes called a "system", is complete if every element in X {\displaystyle X} can be approximated arbitrarily well in norm by finite linear combinations of elements in { ϕ i } i ∈ J {\displaystyle \{\phi _{i}\}_{i\in J}} . A system is called overcomplete if it contains more vectors than necessary to be complete, i.e., there exist ϕ j ∈ { ϕ i } i ∈ J {\displaystyle \phi _{j}\in \{\phi _{i}\}_{i\in J}} that can be removed from the system such that { ϕ i } i ∈ J ∖ { ϕ j } {\displaystyle \{\phi _{i}\}_{i\in J}\setminus \{\phi _{j}\}} remains complete. In research areas such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis. == Relation between overcompleteness and frames == The theory of frames originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series. A frame is defined to be a set of non-zero vectors { ϕ i } i ∈ J {\displaystyle \{\phi _{i}\}_{i\in J}} such that for an arbitrary f ∈ H {\displaystyle f\in {\mathcal {H}}} , A ‖ f ‖ 2 ≤ ∑ i ∈ J | ⟨ f , ϕ i ⟩ | 2 ≤ B ‖ f ‖ 2 {\displaystyle A\|f\|^{2}\leq \sum _{i\in J}|\langle f,\phi _{i}\rangle |^{2}\leq B\|f\|^{2}} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denotes the inner product, A {\displaystyle A} and B {\displaystyle B} are positive constants called bounds of the frame. When A {\displaystyle A} and B {\displaystyle B} can be chosen such that A = B {\displaystyle A=B} , the frame is called a tight frame. It can be seen that H = span { ϕ i } {\displaystyle {\mathcal {H}}=\operatorname {span} \{\phi _{i}\}} . An example of frame can be given as follows. Let each of { α i } i = 1 ∞ {\displaystyle \{\alpha _{i}\}_{i=1}^{\infty }} and { β i } i = 1 ∞ {\displaystyle \{\beta _{i}\}_{i=1}^{\infty }} be an orthonormal basis of H {\displaystyle {\mathcal {H}}} , then { ϕ i } i = 1 ∞ = { α i } i = 1 ∞ ∪ { β i } i = 1 ∞ {\displaystyle \{\phi _{i}\}_{i=1}^{\infty }=\{\alpha _{i}\}_{i=1}^{\infty }\cup \{\beta _{i}\}_{i=1}^{\infty }} is a frame of H {\displaystyle {\mathcal {H}}} with bounds A = B = 2 {\displaystyle A=B=2} . Let S {\displaystyle S} be the frame operator, S f = ∑ i ∈ J ⟨ f , ϕ i ⟩ ϕ i {\displaystyle Sf=\sum _{i\in J}\langle f,\phi _{i}\rangle \phi _{i}} A frame that is not a Riesz basis, in which case it consists of a set of functions more than a basis, is said to be overcomplete or redundant. In this case, given f ∈ H {\displaystyle f\in {\mathcal {H}}} , it can have different decompositions based on the frame. The frame given in the example above is an overcomplete frame. When frames are used for function estimation, one may want to compare the performance of different frames. The parsimony of the approximating functions by different frames may be considered as one way to compare their performances. Given a tolerance ϵ {\displaystyle \epsilon } and a frame F = { ϕ i } i ∈ J {\displaystyle F=\{\phi _{i}\}_{i\in J}} in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} , for any function f ∈ L 2 ( R ) {\displaystyle f\in L^{2}(\mathbb {R} )} , define the set of all approximating functions that satisfy ‖ f − f ^ ‖ < ϵ {\displaystyle \|f-{\hat {f}}\|<\epsilon } N ( f , ϵ ) = { f ^ : f ^ = ∑ i = 1 k β i ϕ i , ‖ f − f ^ ‖ < ϵ } {\displaystyle N(f,\epsilon )=\{{\hat {f}}:{\hat {f}}=\sum _{i=1}^{k}\beta _{i}\phi _{i},\|f-{\hat {f}}\|<\epsilon \}} Then let k F ( f , ϵ ) = inf { k : f ^ ∈ N ( f , ϵ ) } {\displaystyle k_{F}(f,\epsilon )=\inf\{k:{\hat {f}}\in N(f,\epsilon )\}} k ( f , ϵ ) {\displaystyle k(f,\epsilon )} indicates the parsimony of utilizing frame F {\displaystyle F} to approximate f {\displaystyle f} . Different f {\displaystyle f} may have different k {\displaystyle k} based on the hardness to be approximated with elements in the frame. The worst case to estimate a function in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} is defined as k F ( ϵ ) = sup f ∈ L 2 ( R ) { k F ( f , ϵ ) } {\displaystyle k_{F}(\epsilon )=\sup _{f\in L^{2}(\mathbb {R} )}\{k_{F}(f,\epsilon )\}} For another frame G {\displaystyle G} , if k F ( ϵ ) < k G ( ϵ ) {\displaystyle k_{F}(\epsilon )<k_{G}(\epsilon )} , then frame F {\displaystyle F} is better than frame G {\displaystyle G} at level ϵ {\displaystyle \epsilon } . And if there exists a γ {\displaystyle \gamma } that for each ϵ < γ {\displaystyle \epsilon <\gamma } , we have k F ( ϵ ) < k G ( ϵ ) {\displaystyle k_{F}(\epsilon )<k_{G}(\epsilon )} , then F {\displaystyle F} is better than G {\displaystyle G} broadly. Overcomplete frames are usually constructed in three ways. Combine a set of bases, such as wavelet basis and Fourier basis, to obtain an overcomplete frame. Enlarge the range of parameters in some frame, such as in Gabor frame and wavelet frame, to have an overcomplete frame. Add some other functions to an existing complete basis to achieve an overcomplete frame. An example of an overcomplete frame is shown below. The collected data is in a two-dimensional space, and in this case a basis with two elements should be able to explain all the data. However, when noise is included in the data, a basis may not be able to express the properties of the data. If an overcomplete frame with four elements corresponding to the four axes in the figure is used to express the data, each point would be able to have a good expression by the overcomplete frame. The flexibility of the overcomplete frame is one of its key advantages when used in expressing a signal or approximating a function. However, because of this redundancy, a function can have multiple expressions under an overcomplete frame. When the frame is finite, the decomposition can be expressed as f = A x {\displaystyle f=Ax} where f {\displaystyle f} is the function one wants to approximate, A {\displaystyle A} is the matrix containing all the elements in the frame, and x {\displaystyle x} is the coefficients of f {\displaystyle f} under the representation of A {\displaystyle A} . Without any other constraint, the frame will choose to give x {\displaystyle x} with minimal norm in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} . Based on this, some other properties may also be considered when solving the equation, such as sparsity. So different researchers have been working on solving this equation by adding other constraints in the objective function. For example, a constraint minimizing x {\displaystyle x} 's norm in L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} may be used in solving this equation. This should be equivalent to the Lasso regression in statistics community. Bayesian approach is also used to eliminate the redundancy in an overcomplete frame. Lweicki and Sejnowski proposed an algorithm for overcomplete frame by viewing it as a probabilistic model of the observed data. Recently, the overcomplete Gabor frame has been combined with bayesian variable selection method to achieve both small norm expansion coefficients in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} and sparsity in elements. == Examples of overcomplete frames == In modern analysis in signal processing and other engineering field, various overcomplete frames are proposed and used. Here two common used frames, Gabor frames and wavelet frames, are introduced and discussed. === Gabor frames === In usual Fourier transformation, the function in time domain is transformed to the frequency domain. However, the transformation only shows the frequency property of this function and loses its information in the time domain. If a window function g {\displaystyle g} , which only has nonzero value in a small interval, is multiplied with the original function before operating the Fourier transformation, both the information in time and frequency domains may remain at the chosen interval. When a sequence of translation of g {\displaystyle g} is used in the transformation, the information of the function in time domain are kept after the transformation. Let operators T a : L 2 ( R ) → L 2 ( R ) , ( T a f ) ( x ) = f ( x − a ) {\displaystyle T_{a}:L^{2}(R)\rightarrow L^{2}(R),(T_{a}f)(x)=f(x-a)} E b : L 2 ( R ) → L 2 ( R ) , ( E b f ) ( x ) = e 2 π i b x f ( x ) {\displaystyle E_{b}:L^{2}(R)\rightarrow L^{2}(R),(E_{b}f)(x)=e^{2\pi ibx}f(x)} D c : L 2 ( R ) → L 2 ( R ) , ( D c f ) ( x ) = 1 c f ( x c ) {\displaystyle D_{c}:L^{2}(R)\rightarrow L^{2}(R),(D_{c}f)(x)={\frac {1}{\sqrt {c}}}f\left({\frac {x}{c}}\right)} A Gabor frame (named after Dennis Gabor and also called Weyl-Heisenberg frame) in L 2 ( R ) {\displaystyle L^{2}(R)} is defined as the form { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} , where a , b > 0 {\displaystyle a,b>0} and g ∈ L 2 ( R ) {\displaystyle g\in L^{2}(R)} is a fixed function. However, not for every a {\displaystyle a} and b {\displaystyle b} { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} forms a frame on L 2 ( R ) {\displaystyle L^{2}(R)} . For example, when a b > 1 {\displaystyle ab>1} , it is not a frame for L 2 ( R ) {\displaystyle L^{2}(R)} . When a b = 1 {\displaystyle ab=1} , { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} is possible to be a frame, in which case it is a Riesz basis. So the possible situation for { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} being an overcomplete frame is a b < 1 {\displaystyle ab<1} . The Gabor family { E m b / c T n a c g c } m , n ∈ Z {\displaystyle \{E_{mb/c}T_{nac}g_{c}\}_{m,n\in Z}} is also a frame and sharing the same frame bounds as { E m b T n a g } m , n ∈ Z . {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}.} Different kinds of window function g {\displaystyle g} may be used in Gabor frame. Here examples of three window functions are shown, and the condition for the corresponding Gabor system being a frame is shown as follows. (1) g ( x ) = e − x 2 {\displaystyle g(x)=e^{-x^{2}}} , { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} is a frame when a b < 0.994 {\displaystyle ab<0.994} (2) g ( x ) = 1 c o s h ( π x ) {\displaystyle g(x)={\frac {1}{cosh(\pi x)}}} , { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} is a frame when a b < 1 {\displaystyle ab<1} (3) g ( x ) = I [ 0 , c ) ( x ) {\displaystyle g(x)=I_{[0,c)}(x)} , where I ( x ) {\displaystyle I(x)} is the indicator function. The situation for { E m b T n a g } m , n ∈ Z {\displaystyle \{E_{mb}T_{na}g\}_{m,n\in Z}} to be a frame stands as follows. 1) a > c {\displaystyle a>c} or a > 1 {\displaystyle a>1} , not a frame 2) c > 1 {\displaystyle c>1} and a = 1 {\displaystyle a=1} , not a frame 3) a ≤ c ≤ 1 {\displaystyle a\leq c\leq 1} , is a frame 4) a < 1 {\displaystyle a<1} and is an irrational, and c ∈ ( 1 , 2 ) {\displaystyle c\in (1,2)} , is a frame 5) a = p q < 1 {\displaystyle a={\frac {p}{q}}<1} , p {\displaystyle p} and q {\displaystyle q} are relatively primes, 2 − 1 q < c < 2 {\displaystyle 2-{\frac {1}{q}}<c<2} , not a frame 6) 3 4 < a < 1 {\displaystyle {\frac {3}{4}}<a<1} and c = L − 1 + L ( 1 − a ) {\displaystyle c=L-1+L(1-a)} , where L ≥ 3 {\displaystyle L\geq 3} and be a natural number, not a frame 7) a < 1 {\displaystyle a<1} , c > 1 {\displaystyle c>1} , | c − [ c ] − 1 2 | < 1 2 − a {\displaystyle |c-[c]-{\frac {1}{2}}|<{\frac {1}{2}}-a} , where [ c ] {\displaystyle [c]} is the biggest integer not exceeding c {\displaystyle c} , is a frame. The above discussion is a summary of chapter 8 in. === Wavelet frames === A collection of wavelet usually refers to a set of functions based on ψ {\displaystyle \psi } { 2 j 2 ψ ( 2 j x − k ) } j , k ∈ Z {\displaystyle \{2^{\frac {j}{2}}\psi (2^{j}x-k)\}_{j,k\in Z}} This forms an orthonormal basis for L 2 ( R ) {\displaystyle L^{2}(R)} . However, when j , k {\displaystyle j,k} can take values in R {\displaystyle R} , the set represents an overcomplete frame and called undecimated wavelet basis. In general case, a wavelet frame is defined as a frame for L 2 ( R ) {\displaystyle L^{2}(R)} of the form { a j 2 ψ ( a j x − k b ) } j , k ∈ Z {\displaystyle \{a^{\frac {j}{2}}\psi (a^{j}x-kb)\}_{j,k\in Z}} where a > 1 {\displaystyle a>1} , b > 0 {\displaystyle b>0} , and ψ ∈ L 2 ( R ) {\displaystyle \psi \in L^{2}(R)} . The upper and lower bound of this frame can be computed as follows. Let ψ ^ ( γ ) {\displaystyle {\hat {\psi }}(\gamma )} be the Fourier transform for ψ ∈ L 1 ( R ) {\displaystyle \psi \in L^{1}(R)} ψ ^ ( γ ) = ∫ R ψ ( x ) e − 2 π i x γ d x {\displaystyle {\hat {\psi }}(\gamma )=\int _{R}\psi (x)e^{-2\pi ix\gamma }dx} When a , b {\displaystyle a,b} are fixed, define G 0 ( γ ) = ∑ j ∈ Z | ψ ^ ( a j γ ) | 2 {\displaystyle G_{0}(\gamma )=\sum _{j\in Z}|{\hat {\psi }}(a^{j}\gamma )|^{2}} G 1 ( γ ) = ∑ k ≠ 0 ∑ j ∈ Z | ψ ^ ( a j γ ) ψ ^ ( a j γ + k b ) | {\displaystyle G_{1}(\gamma )=\sum _{k\neq 0}\sum _{j\in Z}|{\hat {\psi }}(a^{j}\gamma ){\hat {\psi }}(a^{j}\gamma +{\frac {k}{b}})|} Then B = 1 b sup | γ | ∈ [ 1 , a ] ( G 0 ( γ ) + G 1 ( γ ) ) < ∞ {\displaystyle B={\frac {1}{b}}\sup _{|\gamma |\in [1,a]}(G_{0}(\gamma )+G_{1}(\gamma ))<\infty } A = 1 b inf | γ | ∈ [ 1 , a ] ( G 0 ( γ ) − G 1 ( γ ) ) > 0 {\displaystyle A={\frac {1}{b}}\inf _{|\gamma |\in [1,a]}(G_{0}(\gamma )-G_{1}(\gamma ))>0} Furthermore, when ∑ j ∈ Z | ψ ^ ( 2 j γ ) | 2 = A {\displaystyle \sum _{j\in Z}|{\hat {\psi }}(2^{j}\gamma )|^{2}=A} ∑ j = 0 ∞ ψ ^ ( 2 j γ ) ψ ^ ( 2 j ( γ + q ) ) ¯ = 0 {\displaystyle \sum _{j=0}^{\infty }{\hat {\psi }}(2^{j}\gamma ){\overline {{\hat {\psi }}(2^{j}(\gamma +q))}}=0} , for all odd integers q {\displaystyle q} the generated frame { ψ j , k } j , k ∈ Z {\displaystyle \{\psi _{j,k}\}_{j,k\in Z}} is a tight frame. The discussion in this section is based on chapter 11 in. == Applications == Overcomplete Gabor frames and Wavelet frames have been used in various research area including signal detection, image representation, object recognition, noise reduction, sampling theory, operator theory, harmonic analysis, nonlinear sparse approximation, pseudodifferential operators, wireless communications, geophysics, quantum computing, and filter banks. == References ==
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Wikipedia:Overdetermined system#0
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In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns. Such systems usually have an infinite number of solutions. == Overdetermined linear systems of equations == === An example in two dimensions === Consider the system of 3 equations and 2 unknowns (X and Y), which is overdetermined because 3 > 2, and which corresponds to Diagram #1: Y = − 2 X − 1 Y = 3 X − 2 Y = X + 1. {\displaystyle {\begin{aligned}Y&=-2X-1\\Y&=3X-2\\Y&=X+1.\end{aligned}}} There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5). However, there is no solution that satisfies all three simultaneously. Diagrams #2 and 3 show other configurations that are inconsistent because no point is on all of the lines. Systems of this variety are deemed inconsistent. The only cases where the overdetermined system does in fact have a solution are demonstrated in Diagrams #4, 5, and 6. These exceptions can occur only when the overdetermined system contains enough linearly dependent equations that the number of independent equations does not exceed the number of unknowns. Linear dependence means that some equations can be obtained from linearly combining other equations. For example, Y = X + 1 and 2Y = 2X + 2 are linearly dependent equations because the second one can be obtained by taking twice the first one. === Matrix form === Any system of linear equations can be written as a matrix equation. The previous system of equations (in Diagram #1) can be written as follows: [ 2 1 − 3 1 − 1 1 ] [ X Y ] = [ − 1 − 2 1 ] {\displaystyle {\begin{bmatrix}2&1\\-3&1\\-1&1\\\end{bmatrix}}{\begin{bmatrix}X\\Y\end{bmatrix}}={\begin{bmatrix}-1\\-2\\1\end{bmatrix}}} Notice that the rows of the coefficient matrix (corresponding to equations) outnumber the columns (corresponding to unknowns), meaning that the system is overdetermined. The rank of this matrix is 2, which corresponds to the number of dependent variables in the system. A linear system is consistent if and only if the coefficient matrix has the same rank as its augmented matrix (the coefficient matrix with an extra column added, that column being the column vector of constants). The augmented matrix has rank 3, so the system is inconsistent. The nullity is 0, which means that the null space contains only the zero vector and thus has no basis. In linear algebra the concepts of row space, column space and null space are important for determining the properties of matrices. The informal discussion of constraints and degrees of freedom above relates directly to these more formal concepts. === Homogeneous case === The homogeneous case (in which all constant terms are zero) is always consistent (because there is a trivial, all-zero solution). There are two cases, depending on the number of linearly dependent equations: either there is just the trivial solution, or there is the trivial solution plus an infinite set of other solutions. Consider the system of linear equations: Li = 0 for 1 ≤ i ≤ M, and variables X1, X2, ..., XN, where each Li is a weighted sum of the Xis. Then X1 = X2 = ⋯ = XN = 0 is always a solution. When M < N the system is underdetermined and there are always an infinitude of further solutions. In fact the dimension of the space of solutions is always at least N − M. For M ≥ N, there may be no solution other than all values being 0. There will be an infinitude of other solutions only when the system of equations has enough dependencies (linearly dependent equations) that the number of independent equations is at most N − 1. But with M ≥ N the number of independent equations could be as high as N, in which case the trivial solution is the only one. === Non-homogeneous case === In systems of linear equations, Li=ci for 1 ≤ i ≤ M, in variables X1, X2, ..., XN the equations are sometimes linearly dependent; in fact the number of linearly independent equations cannot exceed N+1. We have the following possible cases for an overdetermined system with N unknowns and M equations (M>N). M = N+1 and all M equations are linearly independent. This case yields no solution. Example: x = 1, x = 2. M > N but only K equations (K < M and K ≤ N+1) are linearly independent. There exist three possible sub-cases of this: K = N+1. This case yields no solutions. Example: 2x = 2, x = 1, x = 2. K = N. This case yields either a single solution or no solution, the latter occurring when the coefficient vector of one equation can be replicated by a weighted sum of the coefficient vectors of the other equations but that weighted sum applied to the constant terms of the other equations does not replicate the one equation's constant term. Example with one solution: 2x = 2, x = 1. Example with no solution: 2x + 2y = 2, x + y = 1, x + y = 3. K < N. This case yields either infinitely many solutions or no solution, the latter occurring as in the previous sub-case. Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3x + 3y + 3z = 3, 2x + 2y + 2z = 2, x + y + z = 1, x + y + z = 4. These results may be easier to understand by putting the augmented matrix of the coefficients of the system in row echelon form by using Gaussian elimination. This row echelon form is the augmented matrix of a system of equations that is equivalent to the given system (it has exactly the same solutions). The number of independent equations in the original system is the number of non-zero rows in the echelon form. The system is inconsistent (no solution) if and only if the last non-zero row in echelon form has only one non-zero entry that is in the last column (giving an equation 0 = c where c is a non-zero constant). Otherwise, there is exactly one solution when the number of non-zero rows in echelon form is equal to the number of unknowns, and there are infinitely many solutions when the number of non-zero rows is lower than the number of variables. Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions. === Exact solutions === All exact solutions can be obtained, or it can be shown that none exist, using matrix algebra. See System of linear equations#Matrix solution. === Approximate solutions === The method of ordinary least squares can be used to find an approximate solution to overdetermined systems. For the system A x = b , {\displaystyle A\mathbf {x} =\mathbf {b} ,} the least squares formula is obtained from the problem min x ‖ A x − b ‖ , {\displaystyle \min _{\mathbf {x} }\lVert A\mathbf {x} -\mathbf {b} \rVert ,} the solution of which can be written with the normal equations, x = ( A T A ) − 1 A T b , {\displaystyle \mathbf {x} =\left(A^{\mathsf {T}}A\right)^{-1}A^{\mathsf {T}}\mathbf {b} ,} where T {\displaystyle {\mathsf {T}}} indicates a matrix transpose, provided ( A T A ) − 1 {\displaystyle \left(A^{\mathsf {T}}A\right)^{-1}} exists (that is, provided A has full column rank). With this formula an approximate solution is found when no exact solution exists, and it gives an exact solution when one does exist. However, to achieve good numerical accuracy, using the QR factorization of A to solve the least squares problem is preferred. ==== Using QR factorization ==== The QR decomposition of a (tall) matrix A {\displaystyle A} is the representation of the matrix in the product form, A = Q R , {\displaystyle A=QR,} where Q {\displaystyle Q} is a (tall) semi-orthonormal matrix that spans the range of the matrix A {\displaystyle A} , and where R {\displaystyle R} is a (small) square right-triangular matrix. The solution to the problem of minimizing the norm ‖ A x − b ‖ 2 {\displaystyle \|Ax-b\|^{2}} is then given as x = R − 1 Q T b , {\displaystyle x=R^{-1}Q^{T}b,} where in practice instead of calculating R − 1 {\displaystyle R^{-1}} one should do a run of backsubstitution on the right-triangular system R x = Q T b . {\displaystyle Rx=Q^{T}b.} ==== Using Singular Value Decomposition ==== The Singular Value Decomposition (SVD) of a (tall) matrix A {\displaystyle A} is the representation of the matrix in the product form, A = U S V T , {\displaystyle A=USV^{T},} where U {\displaystyle U} is a (tall) semi-orthonormal matrix that spans the range of the matrix A {\displaystyle A} , S {\displaystyle S} is a (small) square diagonal matrix with non-negative singular values along the diagonal, and where V {\displaystyle V} is a (small) square orthonormal matrix. The solution to the problem of minimizing the norm ‖ A x − b ‖ 2 {\displaystyle \|Ax-b\|^{2}} is then given as x = V S − 1 U T b . {\displaystyle x=VS^{-1}U^{T}b.} == Overdetermined nonlinear systems of equations == In finite dimensional spaces, a system of equations can be written or represented in the form of { f 1 ( x 1 , … , x n ) = 0 ⋮ ⋮ ⋮ f m ( x 1 , … , x n ) = 0 {\displaystyle \left\{{\begin{array}{ccc}f_{1}(x_{1},\ldots ,x_{n})&=&0\\\vdots &\vdots &\vdots \\f_{m}(x_{1},\ldots ,x_{n})&=&0\end{array}}\right.} or in the form of f ( x ) = 0 {\displaystyle \mathbf {f} (\mathbf {x} )=\mathbf {0} } with f ( x ) = [ f 1 ( x 1 , … , x n ) ⋮ f m ( x 1 , … , x n ) ] and 0 = [ 0 ⋮ 0 ] {\displaystyle \mathbf {f} (\mathbf {x} )=\left[{\begin{array}{c}f_{1}(x_{1},\ldots ,x_{n})\\\vdots \\f_{m}(x_{1},\ldots ,x_{n})\end{array}}\right]\;\;\;{\mbox{and}}\;\;\;\mathbf {0} =\left[{\begin{array}{c}0\\\vdots \\0\end{array}}\right]} where x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is a point in R n {\displaystyle R^{n}} or C n {\displaystyle C^{n}} and f 1 , … , f m {\displaystyle f_{1},\ldots ,f_{m}} are real or complex functions. The system is overdetermined if m > n {\displaystyle m>n} . In contrast, the system is an underdetermined system if m < n {\displaystyle m<n} . As an effective method for solving overdetermined systems, the Gauss-Newton iteration locally quadratically converges to solutions at which the Jacobian matrices of f ( x ) {\displaystyle \mathbf {f} (\mathbf {x} )} are injective. == In general use == The concept can also be applied to more general systems of equations, such as systems of polynomial equations or partial differential equations. In the case of the systems of polynomial equations, it may happen that an overdetermined system has a solution, but that no one equation is a consequence of the others and that, when removing any equation, the new system has more solutions. For example, ( x − 1 ) ( x − 2 ) = 0 , ( x − 1 ) ( x − 3 ) = 0 {\displaystyle (x-1)(x-2)=0,(x-1)(x-3)=0} has the single solution x = 1 , {\displaystyle x=1,} but each equation by itself has two solutions. == See also == Compressed sensing Consistency proof Integrability condition Least squares Moore–Penrose pseudoinverse Rouché-Capelli (or, Rouché-Frobenius) theorem == References ==
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Wikipedia:Oxford Set of Mathematical Instruments#0
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Helix (also known as Helix Oxford or Maped Helix) is a United Kingdom-based manufacturer of stationery. It exports to over 65 countries, with offices in Hong Kong and US, and has its UK headquarters in Kingswinford in the West Midlands. == History == === Establishment === Helix was established in 1887 under the name 'The Universal Woodworking Company Ltd.'; it manufactured wooden rulers and metal laboratory apparatus. In 1894, it patented the drawing compass and, with it, launched the Helix brand, following on from the initial success of the compass and rule. In 1912, the company's first mathematical set was created; and, in 1935, the brand "Helix Oxford" was launched. In 1955 the company was renamed the Helix Universal Company, and moved its headquarters to Lye, West Midlands. In the 1960s, the name was changed to Helix International Ltd. === Administration === In 2004, the company's factory in Lye ceased production. In January 2012, the company entered administration. On 8 February 2012 the company stated, "The joint administrators from Grant Thornton UK LLP are currently speaking to a number of interested parties and have received offers for the sale of the business as a going concern." One week later, the company was bought by the French Maped group. === Maped Helix === When Helix joined the Maped Helix Group the company changed its name to Helix Trading Ltd and moved its UK headquarters from Lye, West Midlands to nearby Kingswinford, where it has its showroom. In 2014 it reported sales of £8.5 million. == Oxford Set of Mathematical Instruments == The Oxford Set of Mathematical Instruments is a set of instruments used by generations of school children in the United Kingdom and around the world in mathematics and geometry lessons. The set is marketed in over 100 countries by Helix. It consists of a metal tin embossed on the front with a drawing of Balliol College and the words 'THE HELIX OXFORD SET OF MATHEMATICAL INSTRUMENTS COMPLETE & ACCURATE' in white against a blue background. Inside the tin there are two set squares, a 180° protractor, a 15 cm ruler, a metal compass, a 9 cm pencil, a pencil sharpener, and an eraser. (In the 1970s a stencil for drawing chemical apparatus was included.) There is also a fact sheet and glossary of mathematical terms with a school timetable printed on the back. The export version also includes dividers. == References == == External links == Official website Mathematical Instruments at mapedhelix.co.uk Archived 7 March 2018 at the Wayback Machine
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Wikipedia:Oxford University Invariant Society#0
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The Oxford University Invariant Society, or 'The Invariants', is a university society open to members of the University of Oxford, dedicated to promotion of interest in mathematics. The society regularly hosts talks from professional mathematicians on topics both technical and more popular, from the mathematics of juggling to the history of mathematics. Many prominent British mathematicians were members of the society during their time at Oxford. == History == The Society was founded in 1936 by J. H. C. Whitehead together with two of his students at Balliol College, Graham Higman and Jack de Wet. The name of the society was chosen at random by Higman from the titles of the books on Whitehead's shelf; in this case, Oswald Veblen's Invariants of Quadratic Differential Forms. The opening lecture was given by G. H. Hardy in Hilary Term 1936, with the title 'Round Numbers'. Though many members joined the armed forces during the war, meetings continued, including lectures by Douglas Hartree and Max Newman, as well as debates - 'Is Mathematics an end in itself?' - and mathematical films. The society has hosted hundreds of prominent mathematicians, including lectures by Benoit Mandelbrot, Sir Roger Penrose, and Simon Singh. Since 1961, the Society has published a magazine entitled The Invariant. == References == == External links == Official website Archive of old termcards and committee lists
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Wikipedia:P-variation#0
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In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number p ≥ 1 {\displaystyle p\geq 1} . p-variation is a measure of the regularity or smoothness of a function. Specifically, if f : I → ( M , d ) {\displaystyle f:I\to (M,d)} , where ( M , d ) {\displaystyle (M,d)} is a metric space and I a totally ordered set, its p-variation is: ‖ f ‖ p -var = ( sup D ∑ t k ∈ D d ( f ( t k ) , f ( t k − 1 ) ) p ) 1 / p {\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}} where D ranges over all finite partitions of the interval I. The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then g ∘ f {\displaystyle g\circ f} has finite p α {\displaystyle {\frac {p}{\alpha }}} -variation. The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions. This concept should not be confused with the notion of p-th variation along a sequence of partitions, which is computed as a limit along a given sequence ( D n ) {\displaystyle (D_{n})} of time partitions: [ f ] p = ( lim n → ∞ ∑ t k n ∈ D n d ( f ( t k n ) , f ( t k − 1 n ) ) p ) {\displaystyle [f]_{p}=\left(\lim _{n\to \infty }\sum _{t_{k}^{n}\in D_{n}}d(f(t_{k}^{n}),f(t_{k-1}^{n}))^{p}\right)} For example for p=2, this corresponds to the concept of quadratic variation, which is different from 2-variation. == Link with Hölder norm == One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions. If f is α–Hölder continuous (i.e. its α–Hölder norm is finite) then its 1 α {\displaystyle {\frac {1}{\alpha }}} -variation is finite. Specifically, on an interval [a,b], ‖ f ‖ 1 α -var ≤ ‖ f ‖ α ( b − a ) α {\displaystyle \|f\|_{{\frac {1}{\alpha }}{\text{-var}}}\leq \|f\|_{\alpha }(b-a)^{\alpha }} . If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e. ‖ f ‖ q -var ≤ ‖ f ‖ p -var {\displaystyle \|f\|_{q{\text{-var}}}\leq \|f\|_{p{\text{-var}}}} . However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by f n ( x ) = x n {\displaystyle f_{n}(x)=x^{n}} . They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence. == Application to Riemann–Stieltjes integration == If f and g are functions from [a, b] to R {\displaystyle \mathbb {R} } with no common discontinuities and with f having finite p-variation and g having finite q-variation, with 1 p + 1 q > 1 {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}>1} then the Riemann–Stieltjes Integral ∫ a b f ( x ) d g ( x ) := lim | D | → 0 ∑ t k ∈ D f ( t k ) [ g ( t k + 1 ) − g ( t k ) ] {\displaystyle \int _{a}^{b}f(x)\,dg(x):=\lim _{|D|\to 0}\sum _{t_{k}\in D}f(t_{k})[g(t_{k+1})-g({t_{k}})]} is well-defined. This integral is known as the Young integral because it comes from Young (1936). The value of this definite integral is bounded by the Young-Loève estimate as follows | ∫ a b f ( x ) d g ( x ) − f ( ξ ) [ g ( b ) − g ( a ) ] | ≤ C ‖ f ‖ p -var ‖ g ‖ q -var {\displaystyle \left|\int _{a}^{b}f(x)\,dg(x)-f(\xi )[g(b)-g(a)]\right|\leq C\,\|f\|_{p{\text{-var}}}\|\,g\|_{q{\text{-var}}}} where C is a constant which only depends on p and q and ξ is any number between a and b. If f and g are continuous, the indefinite integral F ( w ) = ∫ a w f ( x ) d g ( x ) {\displaystyle F(w)=\int _{a}^{w}f(x)\,dg(x)} is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then ‖ F ‖ q -var ; [ s , t ] {\displaystyle \|F\|_{q{\text{-var}};[s,t]}} , its q-variation on [s,t], is bounded by C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ s , t ] + ‖ f ‖ ∞ ; [ s , t ] ) ≤ 2 C ‖ g ‖ q -var ; [ s , t ] ( ‖ f ‖ p -var ; [ a , b ] + f ( a ) ) {\displaystyle C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[s,t]}+\|f\|_{\infty ;[s,t]})\leq 2C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[a,b]}+f(a))} where C is a constant which only depends on p and q. == Differential equations driven by signals of finite p-variation, p < 2 == A function from R d {\displaystyle \mathbb {R} ^{d}} to e × d real matrices is called an R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R d {\displaystyle \mathbb {R} ^{d}} . If f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R d {\displaystyle \mathbb {R} ^{d}} , and X is a continuous function from the interval [a, b] to R d {\displaystyle \mathbb {R} ^{d}} with finite p-variation with p less than 2, then the integral of f on X, ∫ a b f ( X ( t ) ) d X ( t ) {\displaystyle \int _{a}^{b}f(X(t))\,dX(t)} , can be calculated because each component of f(X(t)) will be a path of finite p-variation and the integral is a sum of finitely many Young integrals. It provides the solution to the equation d Y = f ( X ) d X {\displaystyle dY=f(X)\,dX} driven by the path X. More significantly, if f is a Lipschitz continuous R e {\displaystyle \mathbb {R} ^{e}} -valued one-form on R e {\displaystyle \mathbb {R} ^{e}} , and X is a continuous function from the interval [a, b] to R d {\displaystyle \mathbb {R} ^{d}} with finite p-variation with p less than 2, then Young integration is enough to establish the solution of the equation d Y = f ( Y ) d X {\displaystyle dY=f(Y)\,dX} driven by the path X. == Differential equations driven by signals of finite p-variation, p ≥ 2 == The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation. == For Brownian motion == p-variation should be contrasted with the quadratic variation which is used in stochastic analysis, which takes one stochastic process to another. In particular the definition of quadratic variation looks a bit like the definition of p-variation, when p has the value 2. Quadratic variation is defined as a limit as the partition gets finer, whereas p-variation is a supremum over all partitions. Thus the quadratic variation of a process could be smaller than its 2-variation. If Wt is a standard Brownian motion on [0, T], then with probability one its p-variation is infinite for p ≤ 2 {\displaystyle p\leq 2} and finite otherwise. The quadratic variation of W is [ W ] T = T {\displaystyle [W]_{T}=T} . == Computation of p-variation for discrete time series == For a discrete time series of observations X0,...,XN it is straightforward to compute its p-variation with complexity of O(N2). Here is an example C++ code using dynamic programming: There exist much more efficient, but also more complicated, algorithms for R {\displaystyle \mathbb {R} } -valued processes and for processes in arbitrary metric spaces. == References == Young, L.C. (1936), "An inequality of the Hölder type, connected with Stieltjes integration", Acta Mathematica, 67 (1): 251–282, doi:10.1007/bf02401743. == External links == Continuous Paths with bounded p-variation Fabrice Baudoin On the Young integral, truncated variation and rough paths Rafał M. Łochowski
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Wikipedia:P. Kanagasabapathy#0
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Perampalam Kanagasabapathy (1923–1977) was a Ceylon Tamil mathematician, academic and dean of the Faculty of Science at the Jaffna Campus of the University of Sri Lanka. == Early life and family == Kanagasabapathy was born in 1923. He was the son of Iyampillai Perampalam from Erlalai in northern Ceylon. He was educated at Jaffna Hindu College. After school he joined the University of Ceylon. He then went to the University of Cambridge, graduating with a master's degree in mathematics. Kanagasabapathy married Meenambikai in 1949. They had three children (Pathmini, Mythili and Nanada Kumaran). == Career == Kanagasabapathy worked at the University of Ceylon, Peradeniya from 1950 to 1974 as a lecturer, senior lecturer and professor of mathematics. He joined the Jaffna Campus of the University of Sri Lanka in October 1974 as a professor of mathematics. He was head of the Department of Mathematics and Statistics and dean of the Faculty of Science at the Jaffna campus from October 1974 to January 1977. He died in January 1977. He published number of papers in Mathematical Journals. == References ==
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Wikipedia:Paarangot Jyeshtadevan Namboodiri#0
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Paarangot Jyeshtadevan Namboodiri (AD 1500–1610) was a mathematician and astronomer from Kerala, South India. Jyestadevan Namboodiri was born in Paaragottu Mana near Thrikkandiyoor and Aalathur on the banks of river Nila. Vatasseri Damodaran Namboodiri was his teacher. He wrote a commentary in Malayalam, Yukthi Bhaasha for Kelallur Neelakandhan Somayaji's Thanthra Sangraham. He is also the author of Drik Karanam (AD 1603), a comprehensive treatise in Malayalam on Astronomy. == See also == Indian astronomy Indian mathematics Jyeshthadeva == References ==
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Wikipedia:Pablo Ferrari#0
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Pablo Augusto Ferrari (September 11, 1949) is an Argentine mathematician, member of the Bernoulli Society, the Institute for Mathematical Statistics, the Brazilian Academy of Sciences, and the International Statistical Institute. He is also co-principal investigator at the Brazilian research center NeuroMat. Ferrari investigates probabilistic models of microscopic phenomena and macroscopic counterpart. He is the son of the contemporary conceptual artist León Ferrari. == Biography == Pablo Ferrari was born in 1949. He got a degree in mathematics from the University of Buenos Aires (UBA) in 1974 and PhD in Statistics from the University of Sao Paulo (USP) in 1982. Ferrari was Professor at the USP from 1978 to 2008 and a visiting professor at Rutgers, Paris, Rome, Santiago de Chile and Cambridge. He is a UBA Professor and Researcher of CONICET from 2009 and a member of the Group Probability of Buenos Aires. He received a Guggenheim Fellowship in 1999, the Consecration Award of the Academy of Exact, Physical and Natural Sciences in Buenos Aires in 2011 and was named a fellow of the International Statistical Institute in 2013. == References ==
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Wikipedia:Packing dimension#0
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In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982. == Definitions == Let (X, d) be a metric space with a subset S ⊆ X and let s ≥ 0 be a real number. The s-dimensional packing pre-measure of S is defined to be P 0 s ( S ) = lim sup δ ↓ 0 { ∑ i ∈ I d i a m ( B i ) s | { B i } i ∈ I is a countable collection of pairwise disjoint closed balls with diameters ≤ δ and centres in S } . {\displaystyle P_{0}^{s}(S)=\limsup _{\delta \downarrow 0}\left\{\left.\sum _{i\in I}\mathrm {diam} (B_{i})^{s}\right|{\begin{matrix}\{B_{i}\}_{i\in I}{\text{ is a countable collection}}\\{\text{of pairwise disjoint closed balls with}}\\{\text{diameters }}\leq \delta {\text{ and centres in }}S\end{matrix}}\right\}.} Unfortunately, this is just a pre-measure and not a true measure on subsets of X, as can be seen by considering dense, countable subsets. However, the pre-measure leads to a bona fide measure: the s-dimensional packing measure of S is defined to be P s ( S ) = inf { ∑ j ∈ J P 0 s ( S j ) | S ⊆ ⋃ j ∈ J S j , J countable } , {\displaystyle P^{s}(S)=\inf \left\{\left.\sum _{j\in J}P_{0}^{s}(S_{j})\right|S\subseteq \bigcup _{j\in J}S_{j},J{\text{ countable}}\right\},} i.e., the packing measure of S is the infimum of the packing pre-measures of countable covers of S. Having done this, the packing dimension dimP(S) of S is defined analogously to the Hausdorff dimension: dim P ( S ) = sup { s ≥ 0 | P s ( S ) = + ∞ } = inf { s ≥ 0 | P s ( S ) = 0 } . {\displaystyle {\begin{aligned}\dim _{\mathrm {P} }(S)&{}=\sup\{s\geq 0|P^{s}(S)=+\infty \}\\&{}=\inf\{s\geq 0|P^{s}(S)=0\}.\end{aligned}}} === An example === The following example is the simplest situation where Hausdorff and packing dimensions may differ. Fix a sequence ( a n ) {\displaystyle (a_{n})} such that a 0 = 1 {\displaystyle a_{0}=1} and 0 < a n + 1 < a n / 2 {\displaystyle 0<a_{n+1}<a_{n}/2} . Define inductively a nested sequence E 0 ⊃ E 1 ⊃ E 2 ⊃ ⋯ {\displaystyle E_{0}\supset E_{1}\supset E_{2}\supset \cdots } of compact subsets of the real line as follows: Let E 0 = [ 0 , 1 ] {\displaystyle E_{0}=[0,1]} . For each connected component of E n {\displaystyle E_{n}} (which will necessarily be an interval of length a n {\displaystyle a_{n}} ), delete the middle interval of length a n − 2 a n + 1 {\displaystyle a_{n}-2a_{n+1}} , obtaining two intervals of length a n + 1 {\displaystyle a_{n+1}} , which will be taken as connected components of E n + 1 {\displaystyle E_{n+1}} . Next, define K = ⋂ n E n {\displaystyle K=\bigcap _{n}E_{n}} . Then K {\displaystyle K} is topologically a Cantor set (i.e., a compact totally disconnected perfect space). For example, K {\displaystyle K} will be the usual middle-thirds Cantor set if a n = 3 − n {\displaystyle a_{n}=3^{-n}} . It is possible to show that the Hausdorff and the packing dimensions of the set K {\displaystyle K} are given respectively by: dim H ( K ) = lim inf n → ∞ n log 2 − log a n , dim P ( K ) = lim sup n → ∞ n log 2 − log a n . {\displaystyle {\begin{aligned}\dim _{\mathrm {H} }(K)&{}=\liminf _{n\to \infty }{\frac {n\log 2}{-\log a_{n}}}\,,\\\dim _{\mathrm {P} }(K)&{}=\limsup _{n\to \infty }{\frac {n\log 2}{-\log a_{n}}}\,.\end{aligned}}} It follows easily that given numbers 0 ≤ d 1 ≤ d 2 ≤ 1 {\displaystyle 0\leq d_{1}\leq d_{2}\leq 1} , one can choose a sequence ( a n ) {\displaystyle (a_{n})} as above such that the associated (topological) Cantor set K {\displaystyle K} has Hausdorff dimension d 1 {\displaystyle d_{1}} and packing dimension d 2 {\displaystyle d_{2}} . === Generalizations === One can consider dimension functions more general than "diameter to the s": for any function h : [0, +∞) → [0, +∞], let the packing pre-measure of S with dimension function h be given by P 0 h ( S ) = lim δ ↓ 0 sup { ∑ i ∈ I h ( d i a m ( B i ) ) | { B i } i ∈ I is a countable collection of pairwise disjoint balls with diameters ≤ δ and centres in S } {\displaystyle P_{0}^{h}(S)=\lim _{\delta \downarrow 0}\sup \left\{\left.\sum _{i\in I}h{\big (}\mathrm {diam} (B_{i}){\big )}\right|{\begin{matrix}\{B_{i}\}_{i\in I}{\text{ is a countable collection}}\\{\text{of pairwise disjoint balls with}}\\{\text{diameters }}\leq \delta {\text{ and centres in }}S\end{matrix}}\right\}} and define the packing measure of S with dimension function h by P h ( S ) = inf { ∑ j ∈ J P 0 h ( S j ) | S ⊆ ⋃ j ∈ J S j , J countable } . {\displaystyle P^{h}(S)=\inf \left\{\left.\sum _{j\in J}P_{0}^{h}(S_{j})\right|S\subseteq \bigcup _{j\in J}S_{j},J{\text{ countable}}\right\}.} The function h is said to be an exact (packing) dimension function for S if Ph(S) is both finite and strictly positive. == Properties == If S is a subset of n-dimensional Euclidean space Rn with its usual metric, then the packing dimension of S is equal to the upper modified box dimension of S: dim P ( S ) = dim ¯ M B ( S ) . {\displaystyle \dim _{\mathrm {P} }(S)={\overline {\dim }}_{\mathrm {MB} }(S).} This result is interesting because it shows how a dimension derived from a measure (packing dimension) agrees with one derived without using a measure (the modified box dimension). Note, however, that the packing dimension is not equal to the box dimension. For example, the set of rationals Q has box dimension one and packing dimension zero. == See also == Hausdorff dimension Minkowski–Bouligand dimension == References == Tricot, Claude Jr. (1982). "Two definitions of fractional dimension". Mathematical Proceedings of the Cambridge Philosophical Society. 91 (1): 57–74. doi:10.1017/S0305004100059119. S2CID 122740665. MR633256
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Wikipedia:Pairing#0
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In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative. == Definition == Let R be a commutative ring with unit, and let M, N and L be R-modules. A pairing is any R-bilinear map e : M × N → L {\displaystyle e:M\times N\to L} . That is, it satisfies e ( r ⋅ m , n ) = e ( m , r ⋅ n ) = r ⋅ e ( m , n ) {\displaystyle e(r\cdot m,n)=e(m,r\cdot n)=r\cdot e(m,n)} , e ( m 1 + m 2 , n ) = e ( m 1 , n ) + e ( m 2 , n ) {\displaystyle e(m_{1}+m_{2},n)=e(m_{1},n)+e(m_{2},n)} and e ( m , n 1 + n 2 ) = e ( m , n 1 ) + e ( m , n 2 ) {\displaystyle e(m,n_{1}+n_{2})=e(m,n_{1})+e(m,n_{2})} for any r ∈ R {\displaystyle r\in R} and any m , m 1 , m 2 ∈ M {\displaystyle m,m_{1},m_{2}\in M} and any n , n 1 , n 2 ∈ N {\displaystyle n,n_{1},n_{2}\in N} . Equivalently, a pairing is an R-linear map M ⊗ R N → L {\displaystyle M\otimes _{R}N\to L} where M ⊗ R N {\displaystyle M\otimes _{R}N} denotes the tensor product of M and N. A pairing can also be considered as an R-linear map Φ : M → Hom R ( N , L ) {\displaystyle \Phi :M\to \operatorname {Hom} _{R}(N,L)} , which matches the first definition by setting Φ ( m ) ( n ) := e ( m , n ) {\displaystyle \Phi (m)(n):=e(m,n)} . A pairing is called perfect if the above map Φ {\displaystyle \Phi } is an isomorphism of R-modules and the other evaluation map Φ ′ : N → Hom R ( M , L ) {\displaystyle \Phi '\colon N\to \operatorname {Hom} _{R}(M,L)} is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are finite dimensional vector spaces and L=R. A pairing is called non-degenerate on the right if for the above map we have that e ( m , n ) = 0 {\displaystyle e(m,n)=0} for all m {\displaystyle m} implies n = 0 {\displaystyle n=0} ; similarly, e {\displaystyle e} is called non-degenerate on the left if e ( m , n ) = 0 {\displaystyle e(m,n)=0} for all n {\displaystyle n} implies m = 0 {\displaystyle m=0} . A pairing is called alternating if N = M {\displaystyle N=M} and e ( m , m ) = 0 {\displaystyle e(m,m)=0} for all m. In particular, this implies e ( m + n , m + n ) = 0 {\displaystyle e(m+n,m+n)=0} , while bilinearity shows e ( m + n , m + n ) = e ( m , m ) + e ( m , n ) + e ( n , m ) + e ( n , n ) = e ( m , n ) + e ( n , m ) {\displaystyle e(m+n,m+n)=e(m,m)+e(m,n)+e(n,m)+e(n,n)=e(m,n)+e(n,m)} . Thus, for an alternating pairing, e ( m , n ) = − e ( n , m ) {\displaystyle e(m,n)=-e(n,m)} . == Examples == Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions). The determinant map (2 × 2 matrices over k) → k can be seen as a pairing k 2 × k 2 → k {\displaystyle k^{2}\times k^{2}\to k} . The Hopf map S 3 → S 2 {\displaystyle S^{3}\to S^{2}} written as h : S 2 × S 2 → S 2 {\displaystyle h:S^{2}\times S^{2}\to S^{2}} is an example of a pairing. For instance, Hardie et al. present an explicit construction of the map using poset models. == Pairings in cryptography == In cryptography, often the following specialized definition is used: Let G 1 , G 2 {\displaystyle \textstyle G_{1},G_{2}} be additive groups and G T {\displaystyle \textstyle G_{T}} a multiplicative group, all of prime order p {\displaystyle \textstyle p} . Let P ∈ G 1 , Q ∈ G 2 {\displaystyle \textstyle P\in G_{1},Q\in G_{2}} be generators of G 1 {\displaystyle \textstyle G_{1}} and G 2 {\displaystyle \textstyle G_{2}} respectively. A pairing is a map: e : G 1 × G 2 → G T {\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}} for which the following holds: Bilinearity: ∀ a , b ∈ Z : e ( a P , b Q ) = e ( P , Q ) a b {\displaystyle \textstyle \forall a,b\in \mathbb {Z} :\ e\left(aP,bQ\right)=e\left(P,Q\right)^{ab}} Non-degeneracy: e ( P , Q ) ≠ 1 {\displaystyle \textstyle e\left(P,Q\right)\neq 1} For practical purposes, e {\displaystyle \textstyle e} has to be computable in an efficient manner Note that it is also common in cryptographic literature for all groups to be written in multiplicative notation. In cases when G 1 = G 2 = G {\displaystyle \textstyle G_{1}=G_{2}=G} , the pairing is called symmetric. As G {\displaystyle \textstyle G} is cyclic, the map e {\displaystyle e} will be commutative; that is, for any P , Q ∈ G {\displaystyle P,Q\in G} , we have e ( P , Q ) = e ( Q , P ) {\displaystyle e(P,Q)=e(Q,P)} . This is because for a generator g ∈ G {\displaystyle g\in G} , there exist integers p {\displaystyle p} , q {\displaystyle q} such that P = g p {\displaystyle P=g^{p}} and Q = g q {\displaystyle Q=g^{q}} . Therefore e ( P , Q ) = e ( g p , g q ) = e ( g , g ) p q = e ( g q , g p ) = e ( Q , P ) {\displaystyle e(P,Q)=e(g^{p},g^{q})=e(g,g)^{pq}=e(g^{q},g^{p})=e(Q,P)} . The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes. == Slightly different usages of the notion of pairing == Scalar products on complex vector spaces are sometimes called pairings, although they are not bilinear. For example, in representation theory, one has a scalar product on the characters of complex representations of a finite group which is frequently called character pairing. == See also == Dual system Yoneda product == References == == External links == The Pairing-Based Crypto Library
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Wikipedia:Pairing function#0
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In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. == Definition == A pairing function is a bijection π : N × N → N . {\displaystyle \pi :\mathbb {N} \times \mathbb {N} \to \mathbb {N} .} === Generalization === More generally, a pairing function on a set A {\displaystyle A} is a function that maps each pair of elements from A {\displaystyle A} into an element of A {\displaystyle A} , such that any two pairs of elements of A {\displaystyle A} are associated with different elements of A {\displaystyle A} , or a bijection from A 2 {\displaystyle A^{2}} to A {\displaystyle A} . Instead of abstracting from the domain, the arity of the pairing function can also be generalized: there exists an n-ary generalized Cantor pairing function on N {\displaystyle \mathbb {N} } . == Cantor pairing function == The Cantor pairing function is a primitive recursive pairing function π : N × N → N {\displaystyle \pi :\mathbb {N} \times \mathbb {N} \to \mathbb {N} } defined by π ( k 1 , k 2 ) := 1 2 ( k 1 + k 2 ) ( k 1 + k 2 + 1 ) + k 2 {\displaystyle \pi (k_{1},k_{2}):={\frac {1}{2}}(k_{1}+k_{2})(k_{1}+k_{2}+1)+k_{2}} where k 1 , k 2 ∈ { 0 , 1 , 2 , 3 , … } {\displaystyle k_{1},k_{2}\in \{0,1,2,3,\dots \}} . It can also be expressed as π ( x , y ) := x 2 + x + 2 x y + 3 y + y 2 2 {\displaystyle \pi (x,y):={\frac {x^{2}+x+2xy+3y+y^{2}}{2}}} . It is also strictly monotonic w.r.t. each argument, that is, for all k 1 , k 1 ′ , k 2 , k 2 ′ ∈ N {\displaystyle k_{1},k_{1}',k_{2},k_{2}'\in \mathbb {N} } , if k 1 < k 1 ′ {\displaystyle k_{1}<k_{1}'} , then π ( k 1 , k 2 ) < π ( k 1 ′ , k 2 ) {\displaystyle \pi (k_{1},k_{2})<\pi (k_{1}',k_{2})} ; similarly, if k 2 < k 2 ′ {\displaystyle k_{2}<k_{2}'} , then π ( k 1 , k 2 ) < π ( k 1 , k 2 ′ ) {\displaystyle \pi (k_{1},k_{2})<\pi (k_{1},k_{2}')} . The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k1 and k2 we often denote the resulting number as ⟨k1, k2⟩. This definition can be inductively generalized to the Cantor tuple function π ( n ) : N n → N {\displaystyle \pi ^{(n)}:\mathbb {N} ^{n}\to \mathbb {N} } for n > 2 {\displaystyle n>2} as π ( n ) ( k 1 , … , k n − 1 , k n ) := π ( π ( n − 1 ) ( k 1 , … , k n − 1 ) , k n ) {\displaystyle \pi ^{(n)}(k_{1},\ldots ,k_{n-1},k_{n}):=\pi (\pi ^{(n-1)}(k_{1},\ldots ,k_{n-1}),k_{n})} with the base case defined above for a pair: π ( 2 ) ( k 1 , k 2 ) := π ( k 1 , k 2 ) . {\displaystyle \pi ^{(2)}(k_{1},k_{2}):=\pi (k_{1},k_{2}).} === Inverting the Cantor pairing function === Let z ∈ N {\displaystyle z\in \mathbb {N} } be an arbitrary natural number. We will show that there exist unique values x , y ∈ N {\displaystyle x,y\in \mathbb {N} } such that z = π ( x , y ) = ( x + y + 1 ) ( x + y ) 2 + y {\displaystyle z=\pi (x,y)={\frac {(x+y+1)(x+y)}{2}}+y} and hence that the function π(x, y) is invertible. It is helpful to define some intermediate values in the calculation: w = x + y {\displaystyle w=x+y\!} t = 1 2 w ( w + 1 ) = w 2 + w 2 {\displaystyle t={\frac {1}{2}}w(w+1)={\frac {w^{2}+w}{2}}} z = t + y {\displaystyle z=t+y\!} where t is the triangle number of w. If we solve the quadratic equation w 2 + w − 2 t = 0 {\displaystyle w^{2}+w-2t=0\!} for w as a function of t, we get w = 8 t + 1 − 1 2 {\displaystyle w={\frac {{\sqrt {8t+1}}-1}{2}}} which is a strictly increasing and continuous function when t is non-negative real. Since t ≤ z = t + y < t + ( w + 1 ) = ( w + 1 ) 2 + ( w + 1 ) 2 {\displaystyle t\leq z=t+y<t+(w+1)={\frac {(w+1)^{2}+(w+1)}{2}}} we get that w ≤ 8 z + 1 − 1 2 < w + 1 {\displaystyle w\leq {\frac {{\sqrt {8z+1}}-1}{2}}<w+1} and thus w = ⌊ 8 z + 1 − 1 2 ⌋ . {\displaystyle w=\left\lfloor {\frac {{\sqrt {8z+1}}-1}{2}}\right\rfloor .} where ⌊ ⌋ is the floor function. So to calculate x and y from z, we do: w = ⌊ 8 z + 1 − 1 2 ⌋ {\displaystyle w=\left\lfloor {\frac {{\sqrt {8z+1}}-1}{2}}\right\rfloor } t = w 2 + w 2 {\displaystyle t={\frac {w^{2}+w}{2}}} y = z − t {\displaystyle y=z-t\!} x = w − y . {\displaystyle x=w-y.\!} Since the Cantor pairing function is invertible, it must be one-to-one and onto. === Examples === To calculate π(47, 32): 47 + 32 = 79, 79 + 1 = 80, 79 × 80 = 6320, 6320 ÷ 2 = 3160, 3160 + 32 = 3192, so π(47, 32) = 3192. To find x and y such that π(x, y) = 1432: 8 × 1432 = 11456, 11456 + 1 = 11457, √11457 = 107.037, 107.037 − 1 = 106.037, 106.037 ÷ 2 = 53.019, ⌊53.019⌋ = 53, so w = 53; 53 + 1 = 54, 53 × 54 = 2862, 2862 ÷ 2 = 1431, so t = 1431; 1432 − 1431 = 1, so y = 1; 53 − 1 = 52, so x = 52; thus π(52, 1) = 1432. === Derivation === The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability. The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane. A pairing function can usually be defined inductively – that is, given the nth pair, what is the (n+1)th pair? The way Cantor's function progresses diagonally across the plane can be expressed as π ( x , y ) + 1 = π ( x − 1 , y + 1 ) {\displaystyle \pi (x,y)+1=\pi (x-1,y+1)} . The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step further out, or algebraically: π ( 0 , k ) + 1 = π ( k + 1 , 0 ) {\displaystyle \pi (0,k)+1=\pi (k+1,0)} . Also we need to define the starting point, what will be the initial step in our induction method: π(0, 0) = 0. Assume that there is a quadratic 2-dimensional polynomial that can fit these conditions (if there were not, one could just repeat by trying a higher-degree polynomial). The general form is then π ( x , y ) = a x 2 + b y 2 + c x y + d x + e y + f {\displaystyle \pi (x,y)=ax^{2}+by^{2}+cxy+dx+ey+f} . Plug in our initial and boundary conditions to get f = 0 and: b k 2 + e k + 1 = a ( k + 1 ) 2 + d ( k + 1 ) {\displaystyle bk^{2}+ek+1=a(k+1)^{2}+d(k+1)} , so we can match our k terms to get b = a d = 1-a e = 1+a. So every parameter can be written in terms of a except for c, and we have a final equation, our diagonal step, that will relate them: π ( x , y ) + 1 = a ( x 2 + y 2 ) + c x y + ( 1 − a ) x + ( 1 + a ) y + 1 = a ( ( x − 1 ) 2 + ( y + 1 ) 2 ) + c ( x − 1 ) ( y + 1 ) + ( 1 − a ) ( x − 1 ) + ( 1 + a ) ( y + 1 ) . {\displaystyle {\begin{aligned}\pi (x,y)+1&=a(x^{2}+y^{2})+cxy+(1-a)x+(1+a)y+1\\&=a((x-1)^{2}+(y+1)^{2})+c(x-1)(y+1)+(1-a)(x-1)+(1+a)(y+1).\end{aligned}}} Expand and match terms again to get fixed values for a and c, and thus all parameters: a = 1/2 = b = d c = 1 e = 3/2 f = 0. Therefore π ( x , y ) = 1 2 ( x 2 + y 2 ) + x y + 1 2 x + 3 2 y = 1 2 ( x + y ) ( x + y + 1 ) + y , {\displaystyle {\begin{aligned}\pi (x,y)&={\frac {1}{2}}(x^{2}+y^{2})+xy+{\frac {1}{2}}x+{\frac {3}{2}}y\\&={\frac {1}{2}}(x+y)(x+y+1)+y,\end{aligned}}} is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction. == Shifted Cantor pairing function == The following pairing function: ⟨ i , j ⟩ := 1 2 ( i + j − 2 ) ( i + j − 1 ) + i {\displaystyle \langle i,j\rangle :={\frac {1}{2}}(i+j-2)(i+j-1)+i} , where i , j ∈ { 1 , 2 , 3 , … } {\displaystyle i,j\in \{1,2,3,\dots \}} . is the same as the Cantor pairing function, but shifted to exclude 0 (i.e., i = k 2 + 1 {\displaystyle i=k_{2}+1} , j = k 1 + 1 {\displaystyle j=k_{1}+1} , and ⟨ i , j ⟩ − 1 = π ( k 2 , k 1 ) {\displaystyle \langle i,j\rangle -1=\pi (k_{2},k_{1})} ). It was used in the popular computer textbook of Hopcroft and Ullman (1979). == Other pairing functions == The function P 2 ( x , y ) := 2 x ( 2 y + 1 ) − 1 {\displaystyle P_{2}(x,y):=2^{x}(2y+1)-1} is a pairing function. In 1990, Regan proposed the first known pairing function that is computable in linear time and with constant space (as the previously known examples can only be computed in linear time iff multiplication can be too, which is doubtful). In fact, both this pairing function and its inverse can be computed with finite-state transducers that run in real time. In the same paper, the author proposed two more monotone pairing functions that can be computed online in linear time and with logarithmic space; the first can also be computed offline with zero space. In 2001, Pigeon proposed a pairing function based on bit-interleaving, defined recursively as: ⟨ i , j ⟩ P = { T if i = j = 0 ; ⟨ ⌊ i / 2 ⌋ , ⌊ j / 2 ⌋ ⟩ P : i 0 : j 0 otherwise, {\displaystyle \langle i,j\rangle _{P}={\begin{cases}T&{\text{if}}\ i=j=0;\\\langle \lfloor i/2\rfloor ,\lfloor j/2\rfloor \rangle _{P}:i_{0}:j_{0}&{\text{otherwise,}}\end{cases}}} where i 0 {\displaystyle i_{0}} and j 0 {\displaystyle j_{0}} are the least significant bits of i and j respectively. In 2006, Szudzik proposed a "more elegant" pairing function defined by the expression: ElegantPair [ x , y ] := { y 2 + x if x < y , x 2 + x + y if x ≥ y . {\displaystyle \operatorname {ElegantPair} [x,y]:={\begin{cases}y^{2}+x&{\text{if}}\ x<y,\\x^{2}+x+y&{\text{if}}\ x\geq y.\\\end{cases}}} Which can be unpaired using the expression: ElegantUnpair [ z ] := { { z − ⌊ z ⌋ 2 , ⌊ z ⌋ } if z − ⌊ z ⌋ 2 < ⌊ z ⌋ , { ⌊ z ⌋ , z − ⌊ z ⌋ 2 − ⌊ z ⌋ } if z − ⌊ z ⌋ 2 ≥ ⌊ z ⌋ . {\displaystyle \operatorname {ElegantUnpair} [z]:={\begin{cases}\left\{z-\lfloor {\sqrt {z}}\rfloor ^{2},\lfloor {\sqrt {z}}\rfloor \right\}&{\text{if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}<\lfloor {\sqrt {z}}\rfloor ,\\\left\{\lfloor {\sqrt {z}}\rfloor ,z-\lfloor {\sqrt {z}}\rfloor ^{2}-\lfloor {\sqrt {z}}\rfloor \right\}&{\text{if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}\geq \lfloor {\sqrt {z}}\rfloor .\end{cases}}} (Qualitatively, it assigns consecutive numbers to pairs along the edges of squares.) This pairing function orders SK combinator calculus expressions by depth. This method is the mere application to N {\displaystyle \mathbb {N} } of the idea, found in most textbooks on Set Theory, used to establish κ 2 = κ {\displaystyle \kappa ^{2}=\kappa } for any infinite cardinal κ {\displaystyle \kappa } in ZFC. Define on κ × κ {\displaystyle \kappa \times \kappa } the binary relation ( α , β ) ≼ ( γ , δ ) if either { ( α , β ) = ( γ , δ ) , max ( α , β ) < max ( γ , δ ) , max ( α , β ) = max ( γ , δ ) and α < γ , or max ( α , β ) = max ( γ , δ ) and α = γ and β < δ . {\displaystyle (\alpha ,\beta )\preccurlyeq (\gamma ,\delta ){\text{ if either }}{\begin{cases}(\alpha ,\beta )=(\gamma ,\delta ),\\[4pt]\max(\alpha ,\beta )<\max(\gamma ,\delta ),\\[4pt]\max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha <\gamma ,{\text{ or}}\\[4pt]\max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha =\gamma \ {\text{and}}\ \beta <\delta .\end{cases}}} ≼ {\displaystyle \preccurlyeq } is then shown to be a well-ordering such that every element has < κ {\displaystyle {}<\kappa } predecessors, which implies that κ 2 = κ {\displaystyle \kappa ^{2}=\kappa } . It follows that ( N × N , ≼ ) {\displaystyle (\mathbb {N} \times \mathbb {N} ,\preccurlyeq )} is isomorphic to ( N , ⩽ ) {\displaystyle (\mathbb {N} ,\leqslant )} and the pairing function above is nothing more than the enumeration of integer couples in increasing order. == Citations == === Notes === === Footnotes === === References === Steven Pigeon. "Pairing Function". MathWorld. Lisi, Meri (2007). "Some Remarks on the Cantor Pairing Function". Le Matematiche. LXII: 55–65. Regan, Kenneth W. (December 1992). "Minimum-Complexity Pairing Functions". Journal of Computer and System Sciences. 45 (3): 285–295. doi:10.1016/0022-0000(92)90027-G. ISSN 0022-0000. Szudzik, Matthew (2006). "An Elegant Pairing Function" (PDF). szudzik.com. Archived (PDF) from the original on 25 November 2011. Retrieved 16 August 2021. Szudzik, Matthew P. (1 June 2017). "The Rosenberg-Strong Pairing Function". arXiv:1706.04129 [cs.DM]. Jech, Thomas (2006). Set Theory. Springer Monographs in Mathematics (The Third Millennium ed.). Springer-Verlag. doi:10.1007/3-540-44761-X. ISBN 3-540-44085-2. Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN 0-201-02988-X. Stein, Sherman K. (1999). Mathematics: The Man-Made Universe (3rd ed.). Dover. ISBN 9780486404509.
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Wikipedia:Pamela E. Harris#0
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Pamela Estephania Harris (born November 28, 1983) is a Mexican-American mathematician, educator and advocate for immigrants. She is currently a professor at the University of Wisconsin-Milwaukee in Milwaukee, Wisconsin, was formerly an associate professor at Williams College in Williamstown, Massachusetts and is co-founder of the online platform Lathisms. She is also an editor of the e-mentoring blog of the American Mathematical Society (AMS). == Early life and career == Harris first emigrated with her family from Mexico to the United States when she was 8 years old. They returned to Mexico, before eventually settling in Wisconsin when Harris was 12. Because she was undocumented, she could not attend university. Instead, she studied at the Milwaukee Area Technical College, where she earned two associate degrees in two and a half years. After she married a US citizen and her immigration status changed, she transferred to Marquette University, where she obtained a bachelor's degree in mathematics. She went on to complete her master's degree and in 2012 a PhD at the University of Wisconsin-Milwaukee. Her Ph.D. dissertation was advised by Jeb F. Willenbring. Harris was a Project NExT (New Experiences in Teaching) fellow in 2012. She was a Davies Research Fellow at the United States Military Academy, and, in 2016, joined the faculty at Williams College where she was an associate professor. In 2022, she joined the faculty at the University of Wisconsin-Milwaukee as an associate professor. Harris studies algebraic combinatorics, in particular the representation of Lie algebras. In order to understand this representation she studies vector partition functions, in particular Kostant's partition function. She is also interested in graph theory and number theory. In 2016 she co-founded an online platform called 'Lathisms' which aims to promote the contributions of Latinxs and Hispanics in the Mathematical Sciences. In 2020 she co-authored the book "Asked and Answered: Dialogues On Advocating For Students of Color in Mathematics". Harris, along with Aris Winger, run a podcast, Mathematically Uncensored, through the Center for Minorities in the Mathematical Sciences. Starting in October 2020, they discussed current issues in mathematics that minorities encounter. == Recognition == In 2020, Harris was selected as part of the inaugural class of Karen EDGE Fellows. In 2019, Harris won the Mathematical Association of America Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member, for her mentorship towards undergraduate research and for being a "fierce advocate for a diverse and inclusive mathematics community." She further received the early career Faculty Mentor Award from the Council of Undergraduate Research in the Mathematics and Computer Sciences Division. She was a 2022 winner of the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics. She gave one of the Mathematical Association of America Invited Addresses at the 2019 Joint Mathematics Meetings. In 2019 she was a featured speaker at the national conference of the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science (SACNAS). She was named a Fellow of the American Mathematical Society, in the 2022 class of fellows, "for contributions to algebraic combinatorics, for mentorship of undergraduate researchers, and for contributions to a more equitable and inclusive mathematical community". In 2022 she will become a fellow of the Association for Women in Mathematics, "For exceptional leadership in establishing programs and mentoring networks that support, encourage, and advance women and underrepresented minorities in the mathematical sciences; and for contributions through public speaking that create positive systemic change in the culture and climate of the mathematics profession." In 2018 Harris was featured in the book Power in Numbers: The Rebel Women of Mathematics. == References == == External links == Personal webpage Curriculum vitae Meet a Mathematician! Video interview Carrington, Léolène (January 22, 2018). "#WCWinSTEM: Pamela E. Harris, Ph.D. (interview)". Vanguard: Conversations with Women of Color in STEM.
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Wikipedia:Pamela Liebeck#0
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Pamela Liebeck (née Lawrence, 1930–2012) was a British mathematician and mathematics educator, the author of two books on mathematics. == Life == Liebeck was born in Bromley on 11 July 1930, grew up in Surrey, and read mathematics at Somerville College, Oxford beginning in 1949. At Oxford, she also played on the cricket and tennis teams. After additional study at the University of Cambridge, she became a mathematics teacher. Her husband Hans Liebeck was also an Oxford mathematics student; they met through a shared love of playing chamber music, married in 1953, and moved together to Cape Town University in South Africa in 1955, where Liebeck taught mathematics part-time while raising two children and studying music. In 1961, Liebeck and her husband returned to England. As their (now three) children grew old enough, she returned to teaching, first at the Madeley College of Education in Newcastle-under-Lyme (eventually part of Staffordshire University) and then at Keele University, where her husband had been posted since their return to England. Her son, Martin W. Liebeck, became a mathematics professor at Imperial College London. She died on 3 July 2012. == Books == Liebeck wrote two books on mathematics: Vectors and Matrices (Pergamon, 1971) How Children Learn Mathematics: A Guide for Parents and Teachers (Penguin, 1984) == References == == External links == How Children Learn Mathematics at the Internet Archive Vectors and Matrices at the Internet Archive
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