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Wikipedia:André Haefliger#0
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André Haefliger (Swiss Standard German pronunciation: [ˈandreː ˈhɛːflɪɡər]; 22 May 1929 – 7 March 2023) was a Swiss mathematician who worked primarily on topology. == Education and career == Haefliger went to school in Nyon and then attended his final years at Collège de Genève in Geneva. He studied mathematics at the University of Lausanne from 1948 to 1952. He worked for two years as a teaching assistant at École Polytechnique de l'Université de Lausanne. He then moved to University of Strasbourg, then he followed Charles Ehresmann in Paris, where he received his Ph.D. degree in 1958. His thesis was entitled "Structures feuilletées et cohomologie à valeurs dans un faisceau de groupoïdes" and was written under the supervision of Charles Ehresmann. Haefliger got a research fellowship for one year at the University of Paris, where he participated in the seminar of Henri Cartan, and then from 1959 to 1961 he worked at the Institute for Advanced Study in Princeton, New Jersey. Since 1962 he has been a full professor at the University of Geneva until his retirement in 1996. In 1966 Haefliger was invited speaker at the International Congress of Mathematicians in Moscow. In 1974–75, he was president of the Swiss Mathematical Society. Haefliger obtained a Doctorate honoris causa from the ETH Zurich in 1992 and from the University of Dijon in 1997. In 2020 Haefliger and Martin Bridson were awarded the American Mathematical Society's Leroy P. Steele Prize for Mathematical Exposition, for their book Metric Spaces of Non-Positive Curvature (Springer Verlag, 1999). Haefliger died on 7 March 2023, at the age of 93. == Research == Haefliger's main research interests were differential topology and geometry. Haefliger found the topological obstruction to the existence of a spin structure on an orientable Riemannian manifold. In two papers in the Annals he studied various embedding of spheres in relations to knot theory. He has also made important contributions in the theory of foliations, introducing the notion of Haefliger structures. He wrote more than 80 papers in peer review journals and had 20 Ph.D. students, including Augustin Banyaga and the future Field Medalist Vaughan Jones. == Selected works == Haefliger, André (1988). "Un aperçu de l'oeuvre de Thom en topologie différentielle (jusqu'en 1957)" (PDF). Publications Mathématiques de l'IHÉS. 68: 13–18. doi:10.1007/BF02698538. S2CID 123652699. "Travaux de Novikov sur les feuilletages." Séminaire Bourbaki 10 (1966-1968): 433-444. "Sur les classes caractéristiques des feuilletages." Séminaire Bourbaki 14 (1971-1972): 239-260. "Sphères d'homotopie nouées." Séminaire Bourbaki 9 (1964-1966): 57-68. "Feuilletages riemanniens." Séminaire Bourbaki 31 (1988-1989): 183-197. "Plongements de variétés dans le domaine stable." Séminaire Bourbaki 8 (1962-1964): 63-77. Haefliger, André (1987). "Des espaces homogènes à la résolution de Koszul" (PDF). Annales de l'Institut Fourier. 37 (4): 5–13. doi:10.5802/aif.1107. "Variétés feuilletées" (PDF). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 16 (4): 367–397. 1962. Haefliger, André (1976). "Sur la cohomologie de l'algèbre de Lie des champs de vecteurs" (PDF). Annales Scientifiques de l'École Normale Supérieure. 9 (4): 503–532. doi:10.24033/asens.1316. Haefliger, André (1992). "Extension of complexes of groups" (PDF). Annales de l'Institut Fourier. 42 (1–2): 275–311. doi:10.5802/aif.1292. Haefliger, André (1960). "Quelques remarques sur les applications différentiables d'une surface dans le plan" (PDF). Annales de l'Institut Fourier. 10: 47–60. doi:10.5802/aif.97. "Plongements différentiables de variétés dans variétés." Commentarii Mathematici Helvetici. 36: 47–82. 1961. "Structures feuilletées et cohomologie à valeur dans un faisceau de groupoides". Commentarii Mathematici Helvetici. 32: 248–329. 1957. (Ph.D. Thesis) Haefliger, André; Shapiro, Arnold (1962). "Plongements différentiables dans le domaine stable". Commentarii Mathematici Helvetici. 37: 155–176. doi:10.1007/BF02566970. S2CID 120702104. Haefliger, André (1966). "Enlacements de sphères en codimension supérieure à 2". Commentarii Mathematici Helvetici. 41: 51–72. doi:10.1007/BF02566868. S2CID 119701637. Haefliger, André; Valentin, Poenaru (1964). "La classification des immersions combinatoires" (PDF). Publications Mathématiques de l'IHÉS. 23: 75–91. doi:10.1007/BF02684311. MR 0172296. S2CID 122155839. Armand, Borel; Haefliger, André (1961). "La classe d'homologie fondamentale d'un espace analytique" (PDF). Bulletin de la Société Mathématique de France. 89: 461–513. doi:10.24033/bsmf.1571. MR 0149503. Bridson, Martin R.; Haefliger, André (1999). Metric Spaces of Non-Positive Curvature. Grundlehren der mathematischen Wissenschaften. Vol. 319. Berlin, Heidelberg: Springer. doi:10.1007/978-3-662-12494-9. ISBN 978-3-642-08399-0. MR 1744486. == References == == External links == André Haefliger in German, French and Italian in the online Historical Dictionary of Switzerland. "André Haefliger". Department of Mathematics, University of Geneva. Allyn Jackson (2019). "Interview with André Haefliger". Celebratio Mathematica.
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Wikipedia:André Sainte-Laguë#0
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The Webster method, also called the Sainte-Laguë method (French pronunciation: [sɛ̃t.la.ɡy]), is a highest averages apportionment method for allocating seats in a parliament among federal states, or among parties in a party-list proportional representation system. The Sainte-Laguë method shows a more equal seats-to-votes ratio for different sized parties among apportionment methods. The method was first described in 1832 by American statesman and senator Daniel Webster. In 1842, the method was adopted for proportional allocation of seats in United States congressional apportionment (Act of 25 June 1842, ch 46, 5 Stat. 491). The same method was independently invented in 1910 by the French mathematician André Sainte-Laguë. == Motivation == Proportional electoral systems attempt to distribute seats in proportion to the votes for each political party, i.e. a party with 30% of votes would receive 30% of seats. Exact proportionality is not possible because only whole seats can be distributed. Different apportionment methods, of which the Sainte-Laguë method is one, exist to distribute the seats according to the votes. Different apportionment methods show different levels of proportionality, apportionment paradoxes and political fragmentation. The Sainte-Laguë method minimizes the average seats-to-votes ratio deviation and empirically shows the best proportionality behavior and more equal seats-to-votes ratio for different sized parties among apportionment methods. Among other common methods, the D'Hondt method favours large parties and coalitions over small parties. While favoring large parties reduces political fragmentation, this can be achieved with electoral thresholds as well. The Sainte-Laguë method shows fewer apportionment paradoxes compared to largest remainder methods such as the Hare quota and other highest averages methods such as d'Hondt method. == Description == After all the votes have been tallied, successive quotients are calculated for each party. The formula for the quotient is quotient = V 2 s + 1 {\displaystyle {\text{quotient}}={\frac {V}{2s+1}}} where: V is the total number of votes that party received, and s is the number of seats that have been allocated so far to that party, initially 0 for all parties. Whichever party has the highest quotient gets the next seat allocated, and their quotient is recalculated. The process is repeated until all seats have been allocated. The Webster/Sainte-Laguë method does not ensure that a party receiving more than half the votes will win at least half the seats, which can happen when a party with just over half the vote gets "rounded down" to under half the seats. It also does not ensure that a party with a minority of the vote will not win a majority of the seats, for roughly the same reason. Often there is an electoral threshold; that is, in order to be allocated seats, a minimum percentage of votes must be gained. === Example === In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Since 8 seats are to be allocated, each party's total votes are divided by 1, then by 3, and 5 (and then, if necessary, by 7, 9, 11, 13, and so on by using the formula above) every time the number of votes is the biggest for the current round of calculation. For comparison, the "True proportion" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received. (For example, 100,000/230,000 × 8 = 3.48.) The 8 highest entries (in the current round of calculation) are marked by asterisk: from 100,000 down to 16,000; for each, the corresponding party gets a seat. The below chart is an easy way to perform the calculation: In comparison, the D'Hondt method would allocate four seats to party A and no seats to party D, reflecting the D'Hondt method's overrepresentation of larger parties. == Properties == When apportioning seats in proportional representation, it is particularly important to avoid bias between large parties and small parties to avoid strategic voting. André Sainte-Laguë showed theoretically that the Sainte-Laguë method shows the lowest average bias in apportionment, confirmed by different theoretical and empirical ways.: Sec.5 The European Parliament (Representation) Act 2003 stipulates each region must be allocated at least 3 seats and that the ratio of electors to seats is as nearly as possible the same for each, the Commission found the Sainte-Laguë method produced the smallest standard deviation when compared to the D'Hondt method and Hare quota. === Proportionality under Sainte-Laguë method === The seats-to-votes ratio a i {\displaystyle a_{i}} for a political party i {\displaystyle i} is the ratio between the fraction of seats s i {\displaystyle s_{i}} and the fraction of votes v i {\displaystyle v_{i}} for that party: a i = s i v i {\displaystyle a_{i}={\frac {s_{i}}{v_{i}}}} The Sainte-Laguë method approximates proportionality by optimizing the seats-to-votes ratio among all parties i {\displaystyle i} with the least squares approach. First, the difference between the seats-to-votes ratio for a party and the ideal seats-to-votes ratio is calculated and squared to obtain the error for the party i {\displaystyle i} . To achieve equal representation of each voter, the ideal ratio of seats share to votes share is 1 {\displaystyle 1} . e r r o r i = ( a i − a i d e a l ) 2 = ( s i v i − 1 ) 2 {\displaystyle error_{i}=(a_{i}-a_{ideal})^{2}=\left({\frac {s_{i}}{v_{i}}}-1\right)^{2}} Second, the error for each party is weighted according to the vote share of each party to represent each voter equally. In the last step, the errors for each party are summed up. This error is identical to the Sainte-Laguë Index. e r r o r = ∑ i v i ∗ e r r o r i = ∑ i v i ∗ ( s i v i − 1 ) 2 {\displaystyle error=\sum _{i}v_{i}*error_{i}=\sum _{i}{v_{i}*\left({\frac {s_{i}}{v_{i}}}-1\right)^{2}}} It was shown that this error is minimized by the Sainte-Laguë method. == Modified Sainte-Laguë method == To reduce political fragmentation, some countries, e.g. Nepal, Norway and Sweden, change the quotient formula for parties with no seats (s = 0). These countries changed the quotient from V to V/1.4, though from the general 2018 elections onwards, Sweden has been using V/1.2. That is, the modified method changes the sequence of divisors used in this method from (1, 3, 5, 7, ...) to (1.4, 3, 5, 7, ...). This makes it more difficult for parties to earn only one seat, compared to the unmodified Sainte-Laguë's method. With the modified method, such small parties do not get any seats; these seats are instead given to a larger party. Norway further amends this system by utilizing a two-tier proportionality. The number of members to be returned from each of Norway's 19 constituencies (former counties) depends on the population and area of the county: each inhabitant counts one point, while each km2 counts 1.8 points. Furthermore, one seat from each constituency is allocated according to the national distribution of votes. == History == Webster proposed the method in the United States Congress in 1832 for proportional allocation of seats in United States congressional apportionment. In 1842 the method was adopted (Act of June 25, 1842, ch 46, 5 Stat. 491). It was then replaced by Hamilton method and in 1911 the Webster method was reintroduced. Webster and Sainte-Laguë methods should be treated as two methods with the same result, because the Webster method is used for allocating seats based on states' population, and the Sainte-Laguë based on parties' votes. Webster invented his method for legislative apportionment (allocating legislative seats to regions based on their share of the population) rather than elections (allocating legislative seats to parties based on their share of the votes) but this makes no difference to the calculations in the method. Webster's method is defined in terms of a quota as in the largest remainder method; in this method, the quota is called a "divisor". For a given value of the divisor, the population count for each region is divided by this divisor and then rounded to give the number of legislators to allocate to that region. In order to make the total number of legislators come out equal to the target number, the divisor is adjusted to make the sum of allocated seats after being rounded give the required total. One way to determine the correct value of the divisor would be to start with a very large divisor, so that no seats are allocated after rounding. Then the divisor may be successively decreased until one seat, two seats, three seats and finally the total number of seats are allocated. The number of allocated seats for a given region increases from s to s + 1 exactly when the divisor equals the population of the region divided by s + 1/2, so at each step the next region to get a seat will be the one with the largest value of this quotient. That means that this successive adjustment method for implementing Webster's method allocates seats in the same order to the same regions as the Sainte-Laguë method would allocate them. In 1980 the German physicist Hans Schepers, at the time Head of the Data Processing Group of the German Bundestag, suggested that the distribution of seats according to d'Hondt be modified to avoid putting smaller parties at a disadvantage. German media started using the term Schepers Method and later German literature usually calls it Sainte-Laguë/Schepers. == Threshold for seats == An election threshold can be set to reduce political fragmentation, and any list party which does not receive at least a specified percentage of list votes will not be allocated any seats, even if it received enough votes to have otherwise receive a seat. Examples of countries using the Sainte-Laguë method with a threshold are Germany and New Zealand (5%), although the threshold does not apply if a party wins at least one electorate seat in New Zealand or three electorate seats in Germany. Sweden uses a modified Sainte-Laguë method with a 4% threshold, and a 12% threshold in individual constituencies (i.e. a political party can gain representation with a minuscule representation on the national stage, if its vote share in at least one constituency exceeded 12%). Norway has a threshold of 4% to qualify for leveling seats that are allocated according to the national distribution of votes. This means that even though a party is below the threshold of 4% nationally, they can still get seats from constituencies in which they are particularly popular. == Usage by country == The Webster/Sainte-Laguë method is currently used in Bosnia and Herzegovina, Ecuador, Indonesia, Iraq, Kosovo, Latvia, Nepal, New Zealand, Norway and Sweden. In Germany it is used on the federal level for the Bundestag, and on the state level for the legislatures of Baden-Württemberg, Bavaria, Bremen, Hamburg, North Rhine-Westphalia, Rhineland-Palatinate, Saxony and Schleswig-Holstein. To correct for the deficiency where a party can win a majority of votes but not a majority of seats, in federal elections the law provides such a party will receive extra seats until it has a majority of one. In Denmark it is used for leveling seats in the Folketing, correcting the disproportionality of the D'Hondt method for the other seats. Some cantons in Switzerland use the Sainte-Laguë method for biproportional apportionment between electoral districts and for votes to seats allocation. The Webster/Sainte-Laguë method was used in Bolivia in 1993, in Poland in 2001, and the Palestinian Legislative Council in 2006. The United Kingdom Electoral Commission has used the method from 2003 to 2013 to distribute British seats in the European Parliament to constituent countries of the United Kingdom and the English regions. The method has been proposed by the Green Party in Ireland as a reform for use in Dáil Éireann elections, and by the United Kingdom Conservative–Liberal Democrat coalition government in 2011 as the method for calculating the distribution of seats in elections to the House of Lords, the country's upper house of parliament. == Comparison to other methods == The method belongs to the class of highest-averages methods. It is similar to the Jefferson/D'Hondt method, but uses different divisors. The Jefferson/D'Hondt method favors larger parties while the Webster/Sainte-Laguë method doesn't. The Webster/Sainte-Laguë method is generally seen as more proportional, but risks an outcome where a party with more than half the votes can win fewer than half the seats. When there are two parties, the Webster method is the unique divisor method which is identical to the Hamilton method.: Sub.9.10 == See also == Hagenbach-Bischoff quota == References == == External links == Excel Sainte-Laguë calculator Seats Calculator with the Sainte-Laguë method Java implementation of Webster's method at cut-the-knot Elections New Zealand explanation of Sainte-Laguë Java D'Hondt, Saint-Lague and Hare-Niemeyer calculator
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Wikipedia:André Warusfel#0
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André Warusfel (1 December 1936 – 6 June 2016) was a French mathematician and an alumnus of the École Normale Supérieure. He taught for many years in preparatory classes, mainly in high schools Henri IV and Louis-le-Grand. Inspector General of mathematics from 1994 to 2001, he is Inspector General Emeritus of Mathematics Education. Editor of the Revue de mathématiques spéciales ("Journal of special mathematics") from 1974 until 2007, André Warusfel is also a writer and science journalist specializing in mathematics. As a science journalist, he was, in 1964, one of the reshapers of the journal Atomes ("Atoms"), future journal La Recherche ("Research"). Also passionate about the history of science, he devoted much of his recent research to the mathematical work of René Descartes. He assured a new edition of the text La Géométrie ("Geometry") which was published in 2009 in the third volume of the complete works of Descartes (TEL collection, ed. Gallimard). On the same subject, he defended a thesis at the University of Paris Sorbonne-Paris IV in June 2010, edited by Jean-Luc Marion. He also published a 2009 book on the work of Leonhard Euler, which includes an introductory chapter providing a summary of the evolution of the history of mathematics. He is also the author of several scientific papers on the history and education of mathematics, including a 2005 article on the formation of French mathematicians in the twentieth century, and on the same year on the history of the resolution of algebraic equations. == Works == 1961 : Les Nombres et leurs mystères ("Numbers and their mysteries"), Éditions du Seuil (réédité Collection Points, Le Seuil) 1966 : Dictionnaire raisonné de mathématiques ("Rational dictionary of mathematics"), Éditions du Seuil 1969 : Les mathématiques modernes ("Modern mathematics"), Éditions du Seuil 1971 : Structures algébriques finies ("Finite algebraic structures"), Hachette 1975 : Les Mathématiques ("Mathematics"), collective work in the encyclopedias of modern knowledge 1981 : Réussir Le Rubik's Cube ("Succeed with the Rubik's Cube"), Éditions Denoël, preface by Ernő Rubik 2000 : Les Mathématiques : plaisir et nécessité ("Mathematics: pleasure and necessity"), Éditions Vuibert, with Albert Ducrocq 2009 : Euler : les mathématiques et la vie ("Euler: mathematics and life"), Éditions Vuibert == References ==
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Wikipedia:Andy Liu#0
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Andy Lau Tak-wah (Chinese: 劉德華; Jyutping: Lau4 Dak1 Waa4; born Lau Fook-wing; 27 September 1961), is a Hong Kong actor, singer-songwriter and film producer. He was named the "Fourth Tiger" among the Five Tiger Generals of TVB in the 1980s as well as one of the Four Heavenly Kings in the 1990s. Lau won the Hong Kong Film Award for Best Actor three times, the Golden Horse Award for Best Leading Actor twice, and was entered into the Guinness World Records for the "Most Awards Won by a Cantopop Male Artist" in 2000, with a total of 444 music awards by 2006. In 2018, Lau became a member of the Academy of Motion Picture Arts and Sciences. In 2024, Lau was elected vice chairman of the 11th China Film Association. Over a career of four decades, Lau has been one of the most commercially and critically successful artists in the Chinese world. == Early life == Lau was born Lau Fook-wing in Tai Po, British Hong Kong to fireman Lau Lai (1934–2023). He is the fourth of six siblings and has three elder sisters, one younger sister, and a younger brother named Lau Tak-sing. Although his family was wealthy due to his grandfather being a landowner with farmland and villages, his father moved them to the slums of Diamond Hill when he was six years old so he could receive a bilingual education to improve his English. The area was full of wooden houses, which unfortunately burnt down when he was eleven. During his childhood, Lau had to fetch water for his family up to eight times a day as their house lacked plumbing. He graduated from a Band One secondary school, Ho Lap College in San Po Kong, Kowloon. He also practices Chinese calligraphy and hair styling. He was raised in a nominally Buddhist household and is a follower of the Lingyan Mountain Temple in Taiwan. == Career == === Acting === In 1980, Lau enrolled in TVB's actor training program and graduated the next year, signing a contract with TVB. He was propelled to fame by the TVB series The Emissary (1982). His popularity soared the next year with his role as Yang Guo in the TVB wuxia series The Return of the Condor Heroes; at the end of the year, Lau was featured in the TVB Anniversary Gala Show, alongside Tony Leung, Michael Miu, Felix Wong, and Kent Tong. Since then they were known as the "Five Tiger Generals of TVB". Meanwhile, Lau also started his film career. He made a guest appearance in one of Susanna Kwan's music videos in 1981 and caught the eye of the manager Teddy Robin, who gave Lau his first minor role in the film, Once Upon a Rainbow. Lau then landed a role in Ann Hui's 1982 film, Boat People. In 1983, he had his first leading role in the Shaw Brothers-produced action film, On the Wrong Track. TVB insisted on a binding five-year exclusive contract, which Lau declined to sign, leading to his blacklisting by the network. In the late '80s, Lau departed from TVB and shifted his focus towards films. He established himself for his performances in films such as The Truth (1988), Wong Kar-wai's As Tears Go By (1988) and Benny Chan's film A Moment of Romance (1990). His first major acting prize came with A Fighter's Blues, which was his first Golden Bauhinia Award for Best Actor. He won the Hong Kong Film Award for Best Actor award that year for Running Out of Time. In 2004, he won the Golden Horse Award for Best Leading Actor for his performance in Infernal Affairs III. Since the early 21st century, Lau has started working with filmmakers from China and beyond, notably in Zhang Yimou's House of Flying Daggers (2004) and Feng Xiaogang's A World Without Thieves (2004). In 2006 he starred in the pan-Asian blockbuster A Battle of Wits (2006), followed by a series of big-budget historical films such as The Warlords (2007), Three Kingdoms: Resurrection of the Dragon (2008), and Tsui Hark's Detective Dee and the Mystery of the Phantom Flame (2010). In 2005, Lau received the "No.1 Box office Actor 1985–2005" award of Hong Kong, yielding a box office total of HKD 1,733,275,816 for shooting 108 films in the past 20 years. The aforementioned figure is as compared to the first runner-up Stephen Chow's (HKD 1,317,452,311) and second runner-up Jackie Chan's (HKD 894,090,962). For his contributions, a wax figure of Lau was unveiled on 1 June 2005 at the Madame Tussauds Hong Kong. In 2007, Lau also received the "Nielsen Box Office Star of Asia" award by the Nielsen Company (ACNielsen). April 2017, he starred in the Hong Kong action film Shock Wave, which earned him another Best Actor Award at the 37th Hong Kong Film Awards in 2018. In February 2021, Lau reunited with Tony Leung since the Infernal Affairs series in the action film The Goldfinger. === Film production === In 1991, Lau set up his own film production company, Teamwork Motion Pictures, which in 2002 was renamed to Focus Group Holdings Limited. He was awarded the "Asian Filmmaker of the Year" in the Pusan International Film Festival in 2006. The films Lau has produced include Made in Hong Kong, A Simple Life, A Fighter's Blues, Crazy Stone, Firestorm and Shock Wave. === Music === Lau released his first album Just Know I Only Love You (1985) under the record label Capital Artists to minimal commercial success. However, he achieved mainstream success in 1990 with the release of the album Would It Be Possible which won Lau his first RTHK Top 10 Gold Songs Award. The following year, he released the single "The Days We Spent Together" which topped Hong Kong's music charts and was an international hit across Asia. The song was lauded by Time Out which described its popularity as 'practically a national anthem' and 'one of the most notable hits' in Lau's career. His subsequent albums brought him further recognition spawning hit singles such as "Ice Rain" (1993), "Forget Love Potion" (1994), and "Stupid Fellow" (1998). His popularity as a music artist was such that Lau was dubbed as one of the Cantopop Four Heavenly Kings along with Jacky Cheung, Aaron Kwok, and Leon Lai. His album Love Notes Written in Bone Upon My Heart (1997) is certified 2× Platinum in Taiwan and is one of the best-selling albums with 640,305 copies sold. His other albums Because of Love (1996) and Love is Mysterious (1997) also reached 2× Platinum status there. At the Jade Solid Gold Top 10 Awards, he won the "Most Popular Hong Kong Male Artist" award 7 times and the "Asia Pacific Most Popular Hong Kong Male Artist" award 15 times. By April 2000, he had already won an unprecedented total of 292 awards. That same year, he entered the Guinness World Records for "Most Awards Won by a Cantopop Male Artist" and again in 2021 for "Most Douyin Followers Gained in 24 hours" and "Fastest Time to Reach Ten Million Followers on Douyin". At the 2008 Summer Olympics, Lau sang "Please Stay, Guests From Afar" alongside Jackie Chan and Emil Chau during its closing ceremony. In addition, Lau, who has been supporting disabled athletes in Hong Kong for more than a decade, was appointed as the Goodwill ambassador for the 2008 Summer Paralympics. He led other performers in singing and performing the song "Everyone is No.1" at the Beijing National Stadium before the 2008 Paralympics opening ceremony began. He also sang the theme song "Flying with the Dream" with Han Hong during the Paralympics opening ceremony on 6 September 2008. In 2022, Lau set records when an online concert he held via Douyin attracted more than 350 million viewers. In addition to singing in Cantonese and Mandarin, Lau has also sung in other languages such as English, Japanese and Taiwanese Hokkien. He has held concerts in Asia, North America, Europe, and Oceania, and continues to tour with an upcoming Mainland China leg set for Summer 2024. === Books === Lau has written two books, This Is How I Grew Up (我是這樣長大的) (1995), an autobiography, and My 30 Work Days (我的30個工作天) (2012), a collection of his 30 personal diaries written while working on the 2011 film, A Simple Life. === Art exhibition === In 2023, Lau opened his debut art show titled the 1/X Andy Lau X Art Exhibition, which ran on 25 August at the Freespace venue located in the West Kowloon Cultural District. The exhibit includes a sculpture which Lau designed, a projection of images from his films and concerts, paintings made by him and his daughter, and works where he collaborated with other artists, such as collaborating with Hong Kong artists Sticky Line on a statue of his character from Running on Karma, collaborating with Beijing artist Xu Zhuoer in glass covered film props from A Moment of Romance, and a collaboration with ink painter where Lau showcases his calligraphy. == Philanthropy == In 1994, Lau established the Andy Lau Charity Foundation which helps people in need and promotes a wide range of youth education services. In 1999, he received the Ten Outstanding Young Persons of the World award, being the third person from Hong Kong at that time to receive this distinguished honour. In 2008, Lau took a main role in putting together the Artistes 512 Fund Raising Campaign for relief toward the victims of the 2008 Sichuan earthquake. == Personal life == Lau had two public relationships. In the fall of 1983, while filming Shanghai 13 in Taiwan, Lau was introduced to actress Yu Ke-Hsin. The two began a relationship that lasted for three years. Following the example of Jackie Chan and Joan Lin, they signed a symbolic “marriage certificate” that held no legal validity in Taiwan. Their relationship ended when Carol Chu appeared, and eight years after their breakup, Lau started dating Chu. In 2005, Yu published a memoir in which she detailed her romance with Lau. She revealed that they had agreed to meet again ten years after their breakup and Lau honored the pact by visiting her home in Los Angeles, ringing the doorbell, and claiming that media reports about his relationship with Chu were untrue. This led to a brief rekindling of their relationship. Yu's mother later alleged that all 5,000 copies of the memoir sold in Hong Kong were purchased in bulk to prevent them from reaching store shelves. The books were subsequently returned in full, causing a financial loss of HKD 500,000. In 2008, Lau secretly married Carol Chu in Las Vegas and acknowledged his marriage the following year, ending decades of speculation over their relationship. Both Lau and Chu are vegetarians and Buddhists. On 9 May 2012, Chu gave birth to their daughter Hanna. In January 2017, Lau sustained a serious pelvic injury after being thrown off and stomped on by a horse during a commercial shoot in Thailand. He made a full recovery by the end of the year. == Awards and nominations == == Honors == Lau was noted for his highly positive energy, his hard work and active involvement in charity works throughout his 30 years in showbiz and honoured as a "Justice of Peace" by the Hong Kong SAR government in 2008. In May 2010, he received the "World Outstanding Chinese" award and an "honorary doctorate" from the University of New Brunswick, Canada. On 14 December 2017, Lau was awarded a Doctor of Letters degree from the Hong Kong Shue Yan University, with the citation highlighting his popularity among locals which stated: "His low-key, modest, friendly and approachable personality has endeared him to millions of fans and ordinary folks alike, who also consider him to be a 'heartthrob' and the 'unofficial Chief Executive of Hong Kong'". In 2018, asteroid 55381 Lautakwah, discovered by Bill Yeung at the Desert Eagle Observatory in 2001, was named for Lau. The asteroid measures approximately 8.5 kilometers (5.3 miles) in diameter and is located in the outermost region of the asteroid belt, just inside the Hecuba gap. The official naming citation was published by the Minor Planet Center on 11 July 2018. In 2023, Lau was presented with a Special Tribute award at 2023 Toronto International Film Festival. == Discography == == Filmography == == Concert tours == Andy Lau First Tour (1991) Love's Space Tour (1992) Satchi Tour (1993) True Forever Tour (1995) Reverse the Earth Tour (1996) Love You For Ten Thousand Years Tour (1999) Andy Lau 2000 Tour (2000) Summer Fiesta Tour (2001) Proud of You Tour (2002) Vision Tour (2004–2005) Wonderful World Tour (2007–2009) Unforgettable Tour (2010–2011) Always World Tour (2013) My Love World Tour (2018–2020) Today... is the Day Tour (2024–2025) == See also == Andy Lau filmography == Awards and achievements == == References == == External links == Official website Andy Lau at the Hong Kong Movie Database Andy Lau at IMDb
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Wikipedia:Angela Mihai#0
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Loredana Angela Mihai is an applied mathematician and numerical analyst. Originally from Romania, she is Professor of Applied Mathematics at Cardiff University, and Director of Research and Innovation for the Cardiff University School of Mathematics. She specialises in mathematical modeling of the mechanical properties of soft materials, such as biological tissue. == Education and career == Mihai is from Romania, and graduated with a B.Sc. in Mathematics from the University of Bucharest in 1992. She then worked both as a teaching assistant at the University of Bucharest Faculty of Mathematics and a teacher at a local high school until 1999. In 2000 she completed Part III of the Mathematical Tripos at Cambridge University. Mihai earned her DPhil in numerical analysis at Durham University in 2005. Her dissertation, A class of alternate strip-based domain decomposition methods for elliptic partial differential equations, was supervised by Alan W. Craig. After postdoctoral research at the University of Strathclyde, University of Cambridge, and University of Oxford, she joined the Cardiff University academic staff as a lecturer in 2011. == Professional service == In 2023, Mihai was elected vice-president of the United Kingdom and Republic of Ireland Section of the Society for Industrial and Applied Mathematics. == References == == External links == Angela Mihai publications indexed by Google Scholar
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Wikipedia:Angela Slavova#0
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Angela Slavova is a Bulgarian applied mathematician. She heads the Department of Mathematical Physics in the Institute of Mathematics of the Bulgarian Academy of Sciences, and is former chair of the Bulgarian section of the Society for Industrial and Applied Mathematics. == Education == Slavova graduated from the University of Ruse with M.S. in computer engineering in 1986. From 1992 to 1993, she was a Fulbright scholar at the Florida Institute of Technology, but returned to Bulgaria to obtain her Ph.D. in mathematics in 1995 from her alma mater and in 2005 became Doctor of Science at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. == Career == From 2004 to 2011, Slavova was a head of the Department of Mathematical Physics of the Institute of Mathematics at the Bulgarian Academy of Sciences. Since 2007, she has been a full professor at the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences. Since 2011, she is a head of the Department of Differential Equations and Mathematical Physics at the Institute of Mathematics, Bulgarian Academy of Sciences. She chaired the Bulgarian section Society for Industrial and Applied Mathematics for 2013–2014. == Books == Cellular neural networks: dynamics and modelling (Kluwer, 2003}. Nonlinear waves: An introduction (World Scientific, 2011) Nonlinear waves: A geometrical approach (World Scientific, 2019). == References ==
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Wikipedia:Angelika Steger#0
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Angelika Steger (born 1962) is a mathematician and computer scientist whose research interests include graph theory, randomized algorithms, and approximation algorithms. She is a professor at ETH Zurich. == Education and career == After earlier studies at the University of Freiburg and Heidelberg University, Steger earned a master's degree from Stony Brook University in 1985. She completed a doctorate from the University of Bonn in 1990, under the supervision of Hans Jürgen Prömel, with a dissertation on random combinatorial structures, and earned her habilitation from Bonn in 1994. After a visiting position at the University of Kiel, she became a professor at the University of Duisburg in 1995, moved to the Technical University of Munich in 1996, and moved again to ETH Zurich in 2003. == Books == Steger is the author of a German-language textbook on combinatorics: Steger, Angelika (2007). Diskrete Strukturen Bd. 1. Kombinatorik, Graphentheorie, Algebra / Angelika Steger (in German). Berlin. ISBN 978-3-540-46660-4. OCLC 196449143.{{cite book}}: CS1 maint: location missing publisher (link) and a monograph on the Steiner tree problem: Prömel, Hans Jürgen; Steger, Angelika (2002). The Steiner Tree Problem : a Tour through Graphs, Algorithms, and Complexity. Wiesbaden: Vieweg+Teubner Verlag. ISBN 978-3-322-80291-0. OCLC 851804416. == Recognition == Steger was elected to the Academy of Sciences Leopoldina in 2007. She was an invited speaker at the International Congress of Mathematicians in 2014. == References == == External links == Angelika Steger publications indexed by Google Scholar
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Wikipedia:Angiolo Maria Colomboni#0
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Angiolo Maria Colomboni (1608–1672) was an Italian monk, mathematician, and draughtsman, drawing mainly detailed flowers and birds. He was born in Gubbio in 1608, and joined the monastic order of Olivetans. He applied himself to mathematics. In 1669, while in Bologna, he printed a mathematical text titled Practica Gnomonica. His drawings of flowers and birds, have been compared to those of Giovanni da Udine. In Bologna, he achieved the title of abbot, but returned to Gubbio to indulge his studies. == Sources == Boni, Filippo de' (1852). Biografia degli artisti ovvero dizionario della vita e delle opere dei pittori, degli scultori, degli intagliatori, dei tipografi e dei musici di ogni nazione che fiorirono da'tempi più remoti sino á nostri giorni. Seconda Edizione.. Venice; Googlebooks: Presso Andrea Santini e Figlio. p. 238.
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Wikipedia:Angkana Rüland#0
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Angkana Rüland (born 1987) is a German applied mathematician, a professor in mathematics and holder of a Hausdorff Chair in mathematics at the Hausdorff Center for Mathematics of the University of Bonn. Her research has included work on the mathematical modeling of shape-memory alloys and on the inverse problems arising in animal echolocation. == Education and career == Rüland was born in 1987 in Chiang Mai, but is a German citizen. She grew up in Bonn and was a mathematics student at the University of Bonn. She completed her doctorate in 2014 with the dissertation On Some Rigidity Properties in PDEs supervised by Herbert Koch. After postdoctoral research at the University of Oxford, working there with John M. Ball, she became a researcher at the Max Planck Institute for Mathematics in the Sciences in 2017. She took a professorship at Heidelberg University in 2020 before returning to the University of Bonn in 2023. == Recognition == Rüland is one of the recipients of the 2024 New Horizons in Mathematics Prize, associated with the Breakthrough Prize in Mathematics, "for contributions to applied analysis, in particular the analysis of microstructure in solid-solid phase transitions and the theory of inverse problems". == References == == External links == Research Group PDE and Inverse Problems at the University of Bonn Angkana Rüland publications indexed by Google Scholar
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Wikipedia:Angle of parallelism#0
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In hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism. Given a point not on a line, drop a perpendicular to the line from the point. Let a be the length of this perpendicular segment, and Π ( a ) {\displaystyle \Pi (a)} be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, lim a → 0 Π ( a ) = 1 2 π and lim a → ∞ Π ( a ) = 0. {\displaystyle \lim _{a\to 0}\Pi (a)={\tfrac {1}{2}}\pi \quad {\text{ and }}\quad \lim _{a\to \infty }\Pi (a)=0.} There are five equivalent expressions that relate Π ( a ) {\displaystyle \Pi (a)} and a: sin Π ( a ) = sech a = 1 cosh a = 2 e a + e − a , {\displaystyle \sin \Pi (a)=\operatorname {sech} a={\frac {1}{\cosh a}}={\frac {2}{e^{a}+e^{-a}}}\ ,} cos Π ( a ) = tanh a = e a − e − a e a + e − a , {\displaystyle \cos \Pi (a)=\tanh a={\frac {e^{a}-e^{-a}}{e^{a}+e^{-a}}}\ ,} tan Π ( a ) = csch a = 1 sinh a = 2 e a − e − a , {\displaystyle \tan \Pi (a)=\operatorname {csch} a={\frac {1}{\sinh a}}={\frac {2}{e^{a}-e^{-a}}}\ ,} tan ( 1 2 Π ( a ) ) = e − a , {\displaystyle \tan \left({\tfrac {1}{2}}\Pi (a)\right)=e^{-a},} Π ( a ) = 1 2 π − gd ( a ) , {\displaystyle \Pi (a)={\tfrac {1}{2}}\pi -\operatorname {gd} (a),} where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function. == Construction == János Bolyai discovered a construction which gives the asymptotic parallel s to a line r passing through a point A not on r. Drop a perpendicular from A onto B on r. Choose any point C on r different from B. Erect a perpendicular t to r at C. Drop a perpendicular from A onto D on t. Then length DA is longer than CB, but shorter than CA. Draw a circle around C with radius equal to DA. It will intersect the segment AB at a point E. Then the angle BEC is independent of the length BC, depending only on AB; it is the angle of parallelism. Construct s through A at angle BEC from AB. sin B E C = sinh B C sinh C E = sinh B C sinh D A = sinh B C sin A C D sinh C A = sinh B C cos A C B sinh C A = sinh B C tanh C A tanh C B sinh C A = cosh B C cosh C A = cosh B C cosh C B cosh A B = 1 cosh A B . {\displaystyle \sin BEC={\frac {\sinh {BC}}{\sinh {CE}}}={\frac {\sinh {BC}}{\sinh {DA}}}={\frac {\sinh {BC}}{\sin {ACD}\sinh {CA}}}={\frac {\sinh {BC}}{\cos {ACB}\sinh {CA}}}={\frac {\sinh {BC}\tanh {CA}}{\tanh {CB}\sinh {CA}}}={\frac {\cosh {BC}}{\cosh {CA}}}={\frac {\cosh {BC}}{\cosh {CB}\cosh {AB}}}={\frac {1}{\cosh {AB}}}\,.} See Trigonometry of right triangles for the formulas used here. == History == The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nikolai Lobachevsky. This publication became widely known in English after the Texas professor G. B. Halsted produced a translation in 1891. (Geometrical Researches on the Theory of Parallels) The following passages define this pivotal concept in hyperbolic geometry: The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism) which we will here designate by Π(p) for AD = p.: 13 == Demonstration == In the Poincaré half-plane model of the hyperbolic plane (see Hyperbolic motions), one can establish the relation of Φ to a with Euclidean geometry. Let Q be the semicircle with diameter on the x-axis that passes through the points (1,0) and (0,y), where y > 1. Since Q is tangent to the unit semicircle centered at the origin, the two semicircles represent parallel hyperbolic lines. The y-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle Φ with Q. The angle at the center of Q subtended by the radius to (0, y) is also Φ because the two angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle Q has its center at (x, 0), x < 0, so its radius is 1 − x. Thus, the radius squared of Q is x 2 + y 2 = ( 1 − x ) 2 , {\displaystyle x^{2}+y^{2}=(1-x)^{2},} hence x = 1 2 ( 1 − y 2 ) . {\displaystyle x={\tfrac {1}{2}}(1-y^{2}).} The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, y) : y > 0 } with logarithmic measure. Let the hyperbolic distance from (0, y) to (0, 1) be a, so: log y − log 1 = a, so y = ea where e is the base of the natural logarithm. Then the relation between Φ and a can be deduced from the triangle {(x, 0), (0, 0), (0, y)}, for example: tan ϕ = y − x = 2 y y 2 − 1 = 2 e a e 2 a − 1 = 1 sinh a . {\displaystyle \tan \phi ={\frac {y}{-x}}={\frac {2y}{y^{2}-1}}={\frac {2e^{a}}{e^{2a}-1}}={\frac {1}{\sinh a}}.} == References == Marvin J. Greenberg (1974) Euclidean and Non-Euclidean Geometries, pp. 211–3, W.H. Freeman & Company. Robin Hartshorne (1997) Companion to Euclid pp. 319, 325, American Mathematical Society, ISBN 0821807978. Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press, Oxford (See pages 113 to 118). Béla Kerékjártó (1966) Les Fondements de la Géométry, Tome Deux, §97.6 Angle de parallélisme de la géométry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest.
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Wikipedia:Angles between flats#0
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The concept of angles between lines (in the plane or in space), between two planes (dihedral angle) or between a line and a plane can be generalized to arbitrary dimensions. This generalization was first discussed by Camille Jordan. For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant. These angles are called canonical or principal. The concept of angles can be generalized to pairs of flats in a finite-dimensional inner product space over the complex numbers. == Jordan's definition == Let F {\displaystyle F} and G {\displaystyle G} be flats of dimensions k {\displaystyle k} and l {\displaystyle l} in the n {\displaystyle n} -dimensional Euclidean space E n {\displaystyle E^{n}} . By definition, a translation of F {\displaystyle F} or G {\displaystyle G} does not alter their mutual angles. If F {\displaystyle F} and G {\displaystyle G} do not intersect, they will do so upon any translation of G {\displaystyle G} which maps some point in G {\displaystyle G} to some point in F {\displaystyle F} . It can therefore be assumed without loss of generality that F {\displaystyle F} and G {\displaystyle G} intersect. Jordan shows that Cartesian coordinates x 1 , … , x ρ , {\displaystyle x_{1},\dots ,x_{\rho },} y 1 , … , y σ , {\displaystyle y_{1},\dots ,y_{\sigma },} z 1 , … , z τ , {\displaystyle z_{1},\dots ,z_{\tau },} u 1 , … , u υ , {\displaystyle u_{1},\dots ,u_{\upsilon },} v 1 , … , v α , {\displaystyle v_{1},\dots ,v_{\alpha },} w 1 , … , w α {\displaystyle w_{1},\dots ,w_{\alpha }} in E n {\displaystyle E^{n}} can then be defined such that F {\displaystyle F} and G {\displaystyle G} are described, respectively, by the sets of equations x 1 = 0 , … , x ρ = 0 , {\displaystyle x_{1}=0,\dots ,x_{\rho }=0,} u 1 = 0 , … , u υ = 0 , {\displaystyle u_{1}=0,\dots ,u_{\upsilon }=0,} v 1 = 0 , … , v α = 0 {\displaystyle v_{1}=0,\dots ,v_{\alpha }=0} and x 1 = 0 , … , x ρ = 0 , {\displaystyle x_{1}=0,\dots ,x_{\rho }=0,} z 1 = 0 , … , z τ = 0 , {\displaystyle z_{1}=0,\dots ,z_{\tau }=0,} v 1 cos θ 1 + w 1 sin θ 1 = 0 , … , v α cos θ α + w α sin θ α = 0 {\displaystyle v_{1}\cos \theta _{1}+w_{1}\sin \theta _{1}=0,\dots ,v_{\alpha }\cos \theta _{\alpha }+w_{\alpha }\sin \theta _{\alpha }=0} with 0 < θ i < π / 2 , i = 1 , … , α {\displaystyle 0<\theta _{i}<\pi /2,i=1,\dots ,\alpha } . Jordan calls these coordinates canonical. By definition, the angles θ i {\displaystyle \theta _{i}} are the angles between F {\displaystyle F} and G {\displaystyle G} . The non-negative integers ρ , σ , τ , υ , α {\displaystyle \rho ,\sigma ,\tau ,\upsilon ,\alpha } are constrained by ρ + σ + τ + υ + 2 α = n , {\displaystyle \rho +\sigma +\tau +\upsilon +2\alpha =n,} σ + τ + α = k , {\displaystyle \sigma +\tau +\alpha =k,} σ + υ + α = ℓ . {\displaystyle \sigma +\upsilon +\alpha =\ell .} For these equations to determine the five non-negative integers completely, besides the dimensions n , k {\displaystyle n,k} and ℓ {\displaystyle \ell } and the number α {\displaystyle \alpha } of angles θ i {\displaystyle \theta _{i}} , the non-negative integer σ {\displaystyle \sigma } must be given. This is the number of coordinates y i {\displaystyle y_{i}} , whose corresponding axes are those lying entirely within both F {\displaystyle F} and G {\displaystyle G} . The integer σ {\displaystyle \sigma } is thus the dimension of F ∩ G {\displaystyle F\cap G} . The set of angles θ i {\displaystyle \theta _{i}} may be supplemented with σ {\displaystyle \sigma } angles 0 {\displaystyle 0} to indicate that F ∩ G {\displaystyle F\cap G} has that dimension. Jordan's proof applies essentially unaltered when E n {\displaystyle E^{n}} is replaced with the n {\displaystyle n} -dimensional inner product space C n {\displaystyle \mathbb {C} ^{n}} over the complex numbers. (For angles between subspaces, the generalization to C n {\displaystyle \mathbb {C} ^{n}} is discussed by Galántai and Hegedũs in terms of the below variational characterization.) == Angles between subspaces == Now let F {\displaystyle F} and G {\displaystyle G} be subspaces of the n {\displaystyle n} -dimensional inner product space over the real or complex numbers. Geometrically, F {\displaystyle F} and G {\displaystyle G} are flats, so Jordan's definition of mutual angles applies. When for any canonical coordinate ξ {\displaystyle \xi } the symbol ξ ^ {\displaystyle {\hat {\xi }}} denotes the unit vector of the ξ {\displaystyle \xi } axis, the vectors y ^ 1 , … , y ^ σ , {\displaystyle {\hat {y}}_{1},\dots ,{\hat {y}}_{\sigma },} w ^ 1 , … , w ^ α , {\displaystyle {\hat {w}}_{1},\dots ,{\hat {w}}_{\alpha },} z ^ 1 , … , z ^ τ {\displaystyle {\hat {z}}_{1},\dots ,{\hat {z}}_{\tau }} form an orthonormal basis for F {\displaystyle F} and the vectors y ^ 1 , … , y ^ σ , {\displaystyle {\hat {y}}_{1},\dots ,{\hat {y}}_{\sigma },} w ^ 1 ′ , … , w ^ α ′ , {\displaystyle {\hat {w}}'_{1},\dots ,{\hat {w}}'_{\alpha },} u ^ 1 , … , u ^ υ {\displaystyle {\hat {u}}_{1},\dots ,{\hat {u}}_{\upsilon }} form an orthonormal basis for G {\displaystyle G} , where w ^ i ′ = w ^ i cos θ i + v ^ i sin θ i , i = 1 , … , α . {\displaystyle {\hat {w}}'_{i}={\hat {w}}_{i}\cos \theta _{i}+{\hat {v}}_{i}\sin \theta _{i},\quad i=1,\dots ,\alpha .} Being related to canonical coordinates, these basic vectors may be called canonical. When a i , i = 1 , … , k {\displaystyle a_{i},i=1,\dots ,k} denote the canonical basic vectors for F {\displaystyle F} and b i , i = 1 , … , l {\displaystyle b_{i},i=1,\dots ,l} the canonical basic vectors for G {\displaystyle G} then the inner product ⟨ a i , b j ⟩ {\displaystyle \langle a_{i},b_{j}\rangle } vanishes for any pair of i {\displaystyle i} and j {\displaystyle j} except the following ones. ⟨ y ^ i , y ^ i ⟩ = 1 , i = 1 , … , σ , ⟨ w ^ i , w ^ i ′ ⟩ = cos θ i , i = 1 , … , α . {\displaystyle {\begin{aligned}&\langle {\hat {y}}_{i},{\hat {y}}_{i}\rangle =1,&&i=1,\dots ,\sigma ,\\&\langle {\hat {w}}_{i},{\hat {w}}'_{i}\rangle =\cos \theta _{i},&&i=1,\dots ,\alpha .\end{aligned}}} With the above ordering of the basic vectors, the matrix of the inner products ⟨ a i , b j ⟩ {\displaystyle \langle a_{i},b_{j}\rangle } is thus diagonal. In other words, if ( a i ′ , i = 1 , … , k ) {\displaystyle (a'_{i},i=1,\dots ,k)} and ( b i ′ , i = 1 , … , ℓ ) {\displaystyle (b'_{i},i=1,\dots ,\ell )} are arbitrary orthonormal bases in F {\displaystyle F} and G {\displaystyle G} then the real, orthogonal or unitary transformations from the basis ( a i ′ ) {\displaystyle (a'_{i})} to the basis ( a i ) {\displaystyle (a_{i})} and from the basis ( b i ′ ) {\displaystyle (b'_{i})} to the basis ( b i ) {\displaystyle (b_{i})} realize a singular value decomposition of the matrix of inner products ⟨ a i ′ , b j ′ ⟩ {\displaystyle \langle a'_{i},b'_{j}\rangle } . The diagonal matrix elements ⟨ a i , b i ⟩ {\displaystyle \langle a_{i},b_{i}\rangle } are the singular values of the latter matrix. By the uniqueness of the singular value decomposition, the vectors y ^ i {\displaystyle {\hat {y}}_{i}} are then unique up to a real, orthogonal or unitary transformation among them, and the vectors w ^ i {\displaystyle {\hat {w}}_{i}} and w ^ i ′ {\displaystyle {\hat {w}}'_{i}} (and hence v ^ i {\displaystyle {\hat {v}}_{i}} ) are unique up to equal real, orthogonal or unitary transformations applied simultaneously to the sets of the vectors w ^ i {\displaystyle {\hat {w}}_{i}} associated with a common value of θ i {\displaystyle \theta _{i}} and to the corresponding sets of vectors w ^ i ′ {\displaystyle {\hat {w}}'_{i}} (and hence to the corresponding sets of v ^ i {\displaystyle {\hat {v}}_{i}} ). A singular value 1 {\displaystyle 1} can be interpreted as cos 0 {\displaystyle \cos \,0} corresponding to the angles 0 {\displaystyle 0} introduced above and associated with F ∩ G {\displaystyle F\cap G} and a singular value 0 {\displaystyle 0} can be interpreted as cos π / 2 {\displaystyle \cos \pi /2} corresponding to right angles between the orthogonal spaces F ∩ G ⊥ {\displaystyle F\cap G^{\bot }} and F ⊥ ∩ G {\displaystyle F^{\bot }\cap G} , where superscript ⊥ {\displaystyle \bot } denotes the orthogonal complement. == Variational characterization == The variational characterization of singular values and vectors implies as a special case a variational characterization of the angles between subspaces and their associated canonical vectors. This characterization includes the angles 0 {\displaystyle 0} and π / 2 {\displaystyle \pi /2} introduced above and orders the angles by increasing value. It can be given the form of the below alternative definition. In this context, it is customary to talk of principal angles and vectors. === Definition === Let V {\displaystyle V} be an inner product space. Given two subspaces U , W {\displaystyle {\mathcal {U}},{\mathcal {W}}} with dim ( U ) = k ≤ dim ( W ) := ℓ {\displaystyle \dim({\mathcal {U}})=k\leq \dim({\mathcal {W}}):=\ell } , there exists then a sequence of k {\displaystyle k} angles 0 ≤ θ 1 ≤ θ 2 ≤ ⋯ ≤ θ k ≤ π / 2 {\displaystyle 0\leq \theta _{1}\leq \theta _{2}\leq \cdots \leq \theta _{k}\leq \pi /2} called the principal angles, the first one defined as θ 1 := min { arccos ( | ⟨ u , w ⟩ | ‖ u ‖ ‖ w ‖ ) | u ∈ U , w ∈ W } = ∠ ( u 1 , w 1 ) , {\displaystyle \theta _{1}:=\min \left\{\arccos \left(\left.{\frac {|\langle u,w\rangle |}{\|u\|\|w\|}}\right)\,\right|\,u\in {\mathcal {U}},w\in {\mathcal {W}}\right\}=\angle (u_{1},w_{1}),} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} the induced norm. The vectors u 1 {\displaystyle u_{1}} and w 1 {\displaystyle w_{1}} are the corresponding principal vectors. The other principal angles and vectors are then defined recursively via θ i := min { arccos ( | ⟨ u , w ⟩ | ‖ u ‖ ‖ w ‖ ) | u ∈ U , w ∈ W , u ⊥ u j , w ⊥ w j ∀ j ∈ { 1 , … , i − 1 } } . {\displaystyle \theta _{i}:=\min \left\{\left.\arccos \left({\frac {|\langle u,w\rangle |}{\|u\|\|w\|}}\right)\,\right|\,u\in {\mathcal {U}},~w\in {\mathcal {W}},~u\perp u_{j},~w\perp w_{j}\quad \forall j\in \{1,\ldots ,i-1\}\right\}.} This means that the principal angles ( θ 1 , … , θ k ) {\displaystyle (\theta _{1},\ldots ,\theta _{k})} form a set of minimized angles between the two subspaces, and the principal vectors in each subspace are orthogonal to each other. === Examples === ==== Geometric example ==== Geometrically, subspaces are flats (points, lines, planes etc.) that include the origin, thus any two subspaces intersect at least in the origin. Two two-dimensional subspaces U {\displaystyle {\mathcal {U}}} and W {\displaystyle {\mathcal {W}}} generate a set of two angles. In a three-dimensional Euclidean space, the subspaces U {\displaystyle {\mathcal {U}}} and W {\displaystyle {\mathcal {W}}} are either identical, or their intersection forms a line. In the former case, both θ 1 = θ 2 = 0 {\displaystyle \theta _{1}=\theta _{2}=0} . In the latter case, only θ 1 = 0 {\displaystyle \theta _{1}=0} , where vectors u 1 {\displaystyle u_{1}} and w 1 {\displaystyle w_{1}} are on the line of the intersection U ∩ W {\displaystyle {\mathcal {U}}\cap {\mathcal {W}}} and have the same direction. The angle θ 2 > 0 {\displaystyle \theta _{2}>0} will be the angle between the subspaces U {\displaystyle {\mathcal {U}}} and W {\displaystyle {\mathcal {W}}} in the orthogonal complement to U ∩ W {\displaystyle {\mathcal {U}}\cap {\mathcal {W}}} . Imagining the angle between two planes in 3D, one intuitively thinks of the largest angle, θ 2 > 0 {\displaystyle \theta _{2}>0} . ==== Algebraic example ==== In 4-dimensional real coordinate space R4, let the two-dimensional subspace U {\displaystyle {\mathcal {U}}} be spanned by u 1 = ( 1 , 0 , 0 , 0 ) {\displaystyle u_{1}=(1,0,0,0)} and u 2 = ( 0 , 1 , 0 , 0 ) {\displaystyle u_{2}=(0,1,0,0)} , and let the two-dimensional subspace W {\displaystyle {\mathcal {W}}} be spanned by w 1 = ( 1 , 0 , 0 , a ) / 1 + a 2 {\displaystyle w_{1}=(1,0,0,a)/{\sqrt {1+a^{2}}}} and w 2 = ( 0 , 1 , b , 0 ) / 1 + b 2 {\displaystyle w_{2}=(0,1,b,0)/{\sqrt {1+b^{2}}}} with some real a {\displaystyle a} and b {\displaystyle b} such that | a | < | b | {\displaystyle |a|<|b|} . Then u 1 {\displaystyle u_{1}} and w 1 {\displaystyle w_{1}} are, in fact, the pair of principal vectors corresponding to the angle θ 1 {\displaystyle \theta _{1}} with cos ( θ 1 ) = 1 / 1 + a 2 {\displaystyle \cos(\theta _{1})=1/{\sqrt {1+a^{2}}}} , and u 2 {\displaystyle u_{2}} and w 2 {\displaystyle w_{2}} are the principal vectors corresponding to the angle θ 2 {\displaystyle \theta _{2}} with cos ( θ 2 ) = 1 / 1 + b 2 . {\displaystyle \cos(\theta _{2})=1/{\sqrt {1+b^{2}}}.} To construct a pair of subspaces with any given set of k {\displaystyle k} angles θ 1 , … , θ k {\displaystyle \theta _{1},\ldots ,\theta _{k}} in a 2 k {\displaystyle 2k} (or larger) dimensional Euclidean space, take a subspace U {\displaystyle {\mathcal {U}}} with an orthonormal basis ( e 1 , … , e k ) {\displaystyle (e_{1},\ldots ,e_{k})} and complete it to an orthonormal basis ( e 1 , … , e n ) {\displaystyle (e_{1},\ldots ,e_{n})} of the Euclidean space, where n ≥ 2 k {\displaystyle n\geq 2k} . Then, an orthonormal basis of the other subspace W {\displaystyle {\mathcal {W}}} is, e.g., ( cos ( θ 1 ) e 1 + sin ( θ 1 ) e k + 1 , … , cos ( θ k ) e k + sin ( θ k ) e 2 k ) . {\displaystyle (\cos(\theta _{1})e_{1}+\sin(\theta _{1})e_{k+1},\ldots ,\cos(\theta _{k})e_{k}+\sin(\theta _{k})e_{2k}).} == Basic properties == If the largest angle is zero, one subspace is a subset of the other. If the largest angle is π / 2 {\displaystyle \pi /2} , there is at least one vector in one subspace perpendicular to the other subspace. If the smallest angle is zero, the subspaces intersect at least in a line. If the smallest angle is π / 2 {\displaystyle \pi /2} , the subspaces are orthogonal. The number of angles equal to zero is the dimension of the space where the two subspaces intersect. == Advanced properties == Non-trivial (different from 0 {\displaystyle 0} and π / 2 {\displaystyle \pi /2} ) angles between two subspaces are the same as the non-trivial angles between their orthogonal complements. Non-trivial angles between the subspaces U {\displaystyle {\mathcal {U}}} and W {\displaystyle {\mathcal {W}}} and the corresponding non-trivial angles between the subspaces U {\displaystyle {\mathcal {U}}} and W ⊥ {\displaystyle {\mathcal {W}}^{\perp }} sum up to π / 2 {\displaystyle \pi /2} . The angles between subspaces satisfy the triangle inequality in terms of majorization and thus can be used to define a distance on the set of all subspaces turning the set into a metric space. The sine of the angles between subspaces satisfy the triangle inequality in terms of majorization and thus can be used to define a distance on the set of all subspaces turning the set into a metric space. For example, the sine of the largest angle is known as a gap between subspaces. == Extensions == The notion of the angles and some of the variational properties can be naturally extended to arbitrary inner products and subspaces with infinite dimensions. == Computation == Historically, the principal angles and vectors first appear in the context of canonical correlation and were originally computed using SVD of corresponding covariance matrices. However, as first noticed in, the canonical correlation is related to the cosine of the principal angles, which is ill-conditioned for small angles, leading to very inaccurate computation of highly correlated principal vectors in finite precision computer arithmetic. The sine-based algorithm fixes this issue, but creates a new problem of very inaccurate computation of highly uncorrelated principal vectors, since the sine function is ill-conditioned for angles close to π/2. To produce accurate principal vectors in computer arithmetic for the full range of the principal angles, the combined technique first compute all principal angles and vectors using the classical cosine-based approach, and then recomputes the principal angles smaller than π/4 and the corresponding principal vectors using the sine-based approach. The combined technique is implemented in open-source libraries Octave and SciPy and contributed and to MATLAB. == See also == Singular value decomposition Canonical correlation == References ==
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Wikipedia:Angus MacFarlane-Grieve#0
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Alexander Angus MacFarlane-Grieve, (11 May 1891 – 2 August 1970) was a British academic, mathematician, rower, and decorated British Army officer. He served with the Highland Light Infantry during World War I. He was Master of University College, Durham from 1939 to 1954, and additionally Master of Hatfield College, Durham from 1940 to 1949. == Early life == MacFarlane-Grieve was born on 11 May 1891, and was baptised at St Mary Abbots, an Anglican church in Kensington, London. He was educated at The Perse School, a private school in Cambridge, England. He went on to study mathematics at University College, Durham, and graduated with a Bachelor of Arts (BA) degree in 1913. He rowed for both his college (University College Boat Club) and for the university (Durham University Boat Club). He was President of the DUBC during his final year, from 1912 to 1913. == Military service == On 15 August 1914, having been a member of the Officer Training Corps while at university, MacFarlane-Grieve was commissioned into the 4th Battalion, The Highland Light Infantry, British Army, as a second lieutenant (on probation). In December 1914, his commission and rank were confirmed. In July 1915, he attended the Scottish Command School of Signalling at Peebles, Tweeddale, Scotland. He then became one of three officers commanding the 4th Battalion's signal section. By June 1917, he had been promoted to captain. By the end of the war, he held the acting rank of lieutenant colonel. On 4 August 1923, he was transferred to the Regular Army Reserve of Officers (thereby ending his army career) and was promoted to major with seniority from 4 May 1922. While an academic at Durham, he was an officer of the Durham University contingent of the Officer Training Corps. == Academic career == In 1923, having left the army, MacFarlane-Grieve returned to Durham University to become a lecturer in military subjects. Between 1923 and 1939, he was also Bursar of University College, Durham. In 1939, at the age of 47, he was appointed Master of University College. He was the first head of the college not to be not in Holy Orders: i.e. he was the first layman. With World War II causing a shrinkage in student numbers, he was additionally appointed acting Master of Hatfield College, Durham in 1940. He stood down from that acting appointment in 1949 and was succeeded by Eric Birley. From 1948 to 1953, he was Sub-Warden of the Durham Colleges and therefore the deputy of the Vice-Chancellor and Warden with specific responsibility for the Colleges of Durham University. In 1953, MacFarlane-Grieve inherited the family estate at Edenhall, Scottish Borders, and retired early from Durham. == Honours == In May 1917, MacFarlane-Grieve was awarded the Silver Medal of Military Valor by the King of Italy "for distinguished services rendered during the course of the campaign". In June 1917, he was awarded the Military Cross (MC) "for distinguished service in the field". In February 1932, he was awarded the Efficiency Decoration (TD) for long service. == References == == External links == Portraits of Angus Alexander Macfarlane-Grieve (1891-1970), Lieutenant-Colonel at the National Portrait Gallery, London
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Wikipedia:Anita Hansbo#0
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Anita Hansbo (born 1960) is a Swedish mathematician and academic administrator, the former rector or president of Jönköping University. == Education and early career == Hansbo earned her Ph.D. in 2000 from the University of Gothenburg. Her dissertation, Some Results Related to Smoothing in Discetized Linear Parabolic Equations, was supervised by Vidar Thomée. Before coming to Jönköping University, Hansbo taught at the University of Gothenburg, Chalmers University of Technology, Karlstad University, and University West. She came to University West in the 1990s, and became deputy vice chancellor there in 2004. == Jönköping == She moved to Jönköping University in 2007 as University Director, became acting rector in 2009, and in 2010 was installed as the rector of the university. As head of the university, she presided over its official name change from Högskolan i Jönköping to its English translation, Jönköping University. However, she was criticized for centralizing the university's power structure. In 2016 she went on leave from the university for chronic fatigue, and announced that she would retire as rector, effective June 2018, keeping her position as a mathematics professor. == References == == External links == Anita Hansbo publications indexed by Google Scholar
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Wikipedia:Anja Schlömerkemper#0
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Anja Schlömerkemper (born 1973) is a German mathematician whose research applies mathematical analysis and the calculus of variations to the solutions of partial differential equations modeling problems in materials science. She is Chair of Mathematics in the Natural Sciences at the University of Würzburg, the university's vice president for equal opportunities, career planning and sustainability, and the president of the International Society for the Interaction of Mechanics and Mathematics. == Education and career == Schlömerkemper earned a diploma in physics (the German equivalent of a master's degree) in 1998 at the University of Göttingen. Continuing her studies at the University of Leipzig, she completed a doctorate in mathematics and computer science (Dr. rer. nat.) in 2002. Her doctoral dissertation, Magnetic Forces in Discrete and Continuous Systems, was jointly supervised by Stefan Müller and Andrea Braides. She became a postdoctoral researcher, working with John M. Ball at the Mathematical Institute, University of Oxford, with Alexander Mielke at the Institute for Analysis, Dynamics and Modelling of the University of Stuttgart, and with Stefan Müller at the Max Planck Institute for Mathematics in the Sciences in Leipzig. From 2009 to 2011 she held a temporary professorship at the University of Erlangen-Nuremberg and then a research position at the University of Bonn before taking the Chair of Mathematics in the Natural Sciences in the Institute for Mathematics at the University of Würzburg in 2011. She was named as vice president for equal opportunities, career planning and sustainability at the University of Würzburg in 2021. She also became president of the International Society for the Interaction of Mechanics and Mathematics in 2021. == References == == External links == Mathematics in the Sciences at the University of Würzburg Anja Schlömerkemper publications indexed by Google Scholar
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Wikipedia:Anna Barbara Reinhart#0
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Anna Barbara Reinhart (12 July 1730 – 5 January 1796), was a Swiss mathematician. She was considered an internationally respected mathematician of her era. == Biography == Anna Barbara Reinhart was the third child and first daughter of Councilor Salomon Reinhart (1693 - 1761) and Anna Steiner. Her childhood was overshadowed by an accident when she fell off her horse at a wedding party, which caused her to be confined her to her bed for significant periods of time. Her physician, Dr. Johann Heinrich Hegner, however, noticed her aptitude for mathematics and began to teach her. Henceforth, she studied mathematics using the books of Leonhard Euler, Gabriel Cramer, Pieter van Musschenbroek and Jérôme Lalande. Reinhart corresponded with several mathematicians of the period, such as Christoph Jezler, and also received them as guests. She was active as a teacher of mathematics and was the instructor of Ulrich Hegner and Heinrich Bosshard von Rümikon among others. It is said that she edited the works of several of her contemporaries and wrote a manuscript commenting on the Philosophiae Naturalis Principia Mathematica by Isaac Newton, however her comments were lost after her death, but her letters to Christoph Jezler were preserved. Several contemporaries commended Reinhart in their work, such as Daniel Bernoulli who praised her for expanding and improving the pursuit curve as discussed by Pierre Louis Maupertuis. Reinhart died in 1796 at the age of 66 from gout and the consequences of her childhood accident, from which never fully recovered. == Legacy == In 2003, a street was named after her in her hometown Winterthur in Zurich, Switzerland. == References ==
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Wikipedia:Anna Cartan#0
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Anna Cartan (15 May 1878 – 1923) was a French mathematician, teacher and textbook author who was a student of Marie Curie and Jules Tannery. == Early years == Cartan was the youngest child born to Anne Florentine Cottaz (1841–1927) and Joseph Antoine Cartan (1837–1917), who was the village blacksmith. Anna had an elder sister Jeanne-Marie (1867–1931) who became a dressmaker, a brother Léon (1872–1956) who became a blacksmith working in his father's smithy, and a middle brother Élie Cartan (1869–1951) who became an acclaimed mathematician and sire of a family of mathematicians, notably his first son, Henri Cartan, who later became influential in the field. Anna's brother, Élie Cartan, later recalled that the family was very poor and his childhood had passed under "blows of the anvil, which started every morning from dawn," and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning wheel." == Mathematics career == As the family was poor, it would have been very unusual for any of the children to attend college, but in 1901, partly under Élie's influence, Anna Cartan entered École normale supérieure de jeunes filles in Sèvres near Paris, France. Anna chose to pursue mathematics to become a secondary school teacher. Among the courses she attended were those led by Marie Curie (who taught physics there from 1900 to 1906) and Jules Tannery. One of her friends was the scientist and women's rights activist Eugénie (Feytis) Cotton who would become director of the school in 1936. Another friend was Marthe Baillaud, the niece of Jules Tannery. Anna completed her studies in mathematics in 1904 and taught the subject at the high school in Poitiers, France from 1904 to 1906, and then she taught in Dijon from 1906 to 1908. In 1908, Cartan received a one-year scholarship established by a French philanthropist Albert Kahn to travel around the world to allow her to "enrich her skills and future education by direct knowledge of the world." Among the places she visited were the United States (New York, St. Louis, Chicago, Boston and Niagara Falls), Quebec, Mexico and Cuba. After the tour, she returned to Dijon and taught there until 1916. Then she taught at the Jules Ferry high school in Paris and, until 1920, at the Sèvres application school, annexed to the normal school. In 1912 and 1913, Cartan wrote two books on arithmetic and geometry for girls, and later she co-authored with her brother Élie two more textbooks for both boys and girls. Cartan died of cancer in 1923. == Works == Arithmetic and geometry, first year, secondary education for young girls, by Anna Cartan, 1912 and 1921. Arithmetic, secondary education for young girls, second year, by Anna Cartan, 1913 and 1918. Arithmetic, secondary education, boys and girls, classes of 4th and 3rd, by Anna Cartan and Élie Cartan, 1928 and 1931. Arithmetic, secondary education, boys and girls, grades 6 and 5, by Anna Cartan and Élie Cartan, 1926. == References ==
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Wikipedia:Anna Fino#0
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Anna Maria Fino is an Italian mathematician specializing in differential geometry, complex geometry, and Lie groups. She is a professor of mathematics in the Giuseppe Peano Department of Mathematics at the University of Turin, and founding editor-in-chief of the journal Complex Manifolds. == Education and career == Fino earned a laurea in mathematics in 1992 from the University of Turin. She completed her Ph.D. in 1997 through the Genoa-Turin University Consortium, with a dissertation Geometria e topologia degli spazi omogenei [Geometry and topology of homogeneous spaces] supervised by Simon Salamon. She remained as a researcher at the University of Turin until 2005, when she became an associate professor. She earned a habilitation in 2013 and was promoted to full professor in 2015. She has been editor-in-chief of Complex Manifolds since 2014 when it first began publication, as part of De Gruyter's "Emerging Science Journals" line of open-access journals. == References == == External links == Home page Anna Fino publications indexed by Google Scholar
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Wikipedia:Anna Lawniczak#0
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Anna T. Lawniczak (born 1953) is an applied mathematician known for her work on complex systems including lattice gas automata, a type of cellular automaton used to model fluid dynamics. Educated in Poland and the US, she has worked in the US and Canada, where she is a professor at the University of Guelph. She is the former president of the Canadian Applied and Industrial Mathematics Society. == Education and career == After earning a master's degree in engineering (summa cum laude) from the Wrocław University of Science and Technology in Poland, Lawniczak went to Southern Illinois University in the US for doctoral study in mathematics. She completed her Ph.D. in 1981, supervised by Philip J. Feinsilver. Before taking her current position at the University of Guelph in 1989, Lawniczak was a professor at Louisiana State University in the US, and the University of Toronto in Canada. She was president of the Canadian Applied and Industrial Mathematics Society / Société Canadienne de Mathématiques Appliquées et Industrielles (CAIMS/SCMAI) from 1997 to 2001. As president she guided a 1998 transition that included a new constitution, formal incorporation, a new annual conference, and a change from its former name, the Canadian Applied Mathematics Society / Société Canadienne de Mathématiques Appliquées. == Recognition == The Canadian Applied and Industrial Mathematics Society gave Lawniczak their Arthur Beaumont Distinguished Service Award in 2003. In the same year, the Fields Institute listed her as a Fellow in recognition of her "outstanding contributions to the Fields Institute and its activities". The Engineering Institute of Canada named her as an EIC Fellow in 2018, after a nomination from IEEE Canada, naming her as "an international authority in the discrete modeling & simulation methods like Individually Based Simulation Models, Agent Based Simulations, Cellular Automata and Lattice Gas Cellular Automata, a field of which she is one of the co-developers". == References == == External links == Home page Anna Lawniczak publications indexed by Google Scholar
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Wikipedia:Anna Maria Bigatti#0
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Anna Maria Bigatti is an Italian mathematician specializing in computational methods for commutative algebra. She is a ricercatore in the department of mathematics at the University of Genoa. She is one of the developers of CoCoA, a computer algebra system, and of its core library CoCoALib. == Education and career == Bigatti earned a laurea in mathematics in 1989 from the University of Genoa, and completed her Ph.D. in 1995 at the University of Turin. Her dissertation, Aspetti Combinatorici e Computazionali dell’Algebra Commutativa, was supervised by Lorenzo Robbiano. After postdoctoral study with Robbiano in Genoa, she took her present position as ricercatore in 1997. == Books == Bigatti is an author or co-author of three Italian textbooks, Elementi di matematica - Esercizi con soluzioni per scienze e farmacia (with Grazia Tamone, 2013), Matematica di base (with Lorenzo Robbiano, 2014), and Matematica di base - Esercizi svolti, testi d'esame, richiami di teoria (with Grazia Tamone, 2016). She is also a co-editor of several books in mathematical research including Monomial ideals, computations and applications (Lecture Notes in Mathematics, Springer, 2013) and Computations and Combinatorics in Commutative Algebra (Lecture Notes in Mathematics, Springer, 2017). == References == == External links == Home page Anna Maria Bigatti publications indexed by Google Scholar
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Wikipedia:Anna Mazzucato#0
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Anna Laura Mazzucato is a professor of mathematics, distinguished senior scholar, and associate head of the mathematics department at Pennsylvania State University. Her mathematical research involves functional analysis, function spaces, partial differential equations, and their applications in fluid mechanics and elasticity. == Education and career == Mazzucato earned a master's degree in physics in 1994 from the University of Milan, with a thesis on topological quantum field theory under the supervision of Paolo Cotta-Ramusino. However, during her studies she decided that she preferred the mathematics that she was studying to the physics, and took the advice of Cotta-Ramusino to switch to mathematics for her doctoral studies. She went to the University of North Carolina at Chapel Hill for doctoral study, initially planning to work in quantum cohomology, but switched to functional analysis with Michael E. Taylor as her doctoral advisor. Her dissertation was Analysis of the Navier-Stokes and Other Nonlinear Evolution Equations with Initial Data in Besov-Type Spaces; it studied the Navier–Stokes equations and other nonlinear partial differential equations. After postdoctoral research at the Mathematical Sciences Research Institute (supported by a Liftoff Fellowship from the Clay Mathematics Institute) and the Institute for Mathematics and its Applications, and a term as Gibbs Instructor at Yale University, she became an assistant professor at Pennsylvania State University in 2003. She was promoted to full professor there in 2013. == Recognition == Mazzucato was the winner of the Ruth I. Michler Memorial Prize of the Association for Women in Mathematics for 2011–2012, which she used to fund a research visit to Cornell University. At Cornell, she gave the Michler Lecture on "The Analysis of Incompressible Fluids at High Reynolds Numbers". She was named a SIAM Fellow in the 2021 class of fellows, "for discerning analysis of fundamental problems in partial differential equations and mathematical fluid mechanics including boundary layers, transport, and mixing". == References ==
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Wikipedia:Anna Mullikin#0
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Anna Margaret Mullikin (1893–1975) was an American mathematician who was one of the early investigators of point set theory. She was one of the few women to earn a PhD in math before World War II. == Biography == Anna Margaret Mullikin was born in Baltimore, Maryland, March 7, 1893, as the youngest of four children of Sophia Ridgely (Battee) and William Lawrence Mullikin. Anna received her BA from Goucher College, in Towson, Maryland, in 1915. After she graduated Mullikin taught in private schools including Science Hill School in Shelbyville, Kentucky, (1915–1917) and at Mary Baldwin Seminary in Staunton, Virginia, 1917–1918. Mullikin went on to attend University of Pennsylvania in Philadelphia for doctoral work. She was Robert Lee Moore's third student, graduating with her PhD in 1922 with a dissertation titled Certain Theorems Relating to Plane Connected Point Sets. Her dissertation was published that year in Transactions of the American Mathematical Society and subsequently became the catalyst for significant advances in the field of topology. Her main result was initially referred to as "Miss Mullikin's Theorem" but by 1928 it was called the "Janiszewski-Mullikin theorem." During 1921–1922, while studying for her doctorate, she taught at Oak Lane Country Day School, which served preschool and elementary-aged children in Pennsylvania. As a PhD, she spent her career as a secondary school mathematics teacher, beginning at William Penn High School in Philadelphia for one year (1922–1923) before moving to Germantown High School (Philadelphia). There she became a mentor to Mary-Elizabeth Hamstrom, who would go on to become a student of Moore and professional mathematician herself. At Germantown High School she was named department head of the mathematics in 1952, and she remained there until her retirement in 1959. Anna Mullikin died August 24, 1975, in Philadelphia at 82 and was interred in Mt. Olivet Cemetery in Baltimore, Maryland. == References == == External links == Anna Mullikin at the Mathematics Genealogy Project MacTutor page Green, Judy; LaDuke, Jeanne (2008). Pioneering Women in American Mathematics — The Pre-1940 PhD's. History of Mathematics. Vol. 34 (1st ed.). American Mathematical Society, The London Mathematical Society. ISBN 978-0-8218-4376-5. Biography on p. 448-450 of the Supplementary Material at AMS
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Wikipedia:Anna Panorska#0
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Anna Katarzyna Panorska is a Polish mathematician and statistician who works as a professor in the department of mathematics and statistics at the University of Nevada, Reno. == Research == Panorska's research interests include studying extreme events in the stochastic processes used to model weather, water, and biology. She has also studied the effects of weather conditions on baseball performance, concluding that temperature has a larger effect than wind and humidity. == Education and career == Panorska studied mathematics at the University of Warsaw, completing a degree in 1986. After earning a master's degree in statistics at the University of Texas at El Paso in 1988, she returned to mathematics for her doctoral studies, completing a Ph.D. at the University of California, Santa Barbara in 1992. Her dissertation, Generalized Convolutions, was supervised by Svetlozar Rachev. She became an assistant professor of mathematics at the University of Tennessee at Chattanooga in 1992, but left academia in 1997 to work as a biostatistician for BlueCross BlueShield of Tennessee. After visiting the University of California, Santa Barbara in 1999–2000, she took a research faculty position in 2000 at the Desert Research Institute, associated with the University of Nevada, Reno. In 2002 she became a regular faculty member in mathematics and statistics at the university, and in 2011 she was promoted to full professor. == References == == External links == Home page
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Wikipedia:Anna Romanowska#0
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Anna B. Romanowska is a Polish mathematician specializing in abstract algebra. She is professor emeritus of algebra and combinatorics at the Warsaw University of Technology, and was the first convenor of European Women in Mathematics. == Education and career == Romanowska earned her Ph.D. in 1973 at the Warsaw University of Technology. Her dissertation, Toward an Algebraic Study of the Tone System, was supervised by Tadeusz Traczyk. She became the first convenor of European Women in Mathematics, for 1993–1994. == Books == Romanowska is the coauthor of three books on abstract algebra with Jonathan D. H. Smith: Modal theory: an algebraic approach to order, geometry, and convexity (Heldermann, 1985) Post-modern algebra (Wiley, 1999) Modes (World Scientific, 2002) == References == == External links == Anna Romanowska publications indexed by Google Scholar
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Wikipedia:Anna Sfard#0
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Anna B. Sfard (Hebrew: אנה ספרד) is a retired Israeli psychologist of mathematics education, focusing on the roles of communication and reification in mathematical reasoning. She is a professor emerita of Mathematics Learning Sciences at the University of Haifa. == Education and career == Sfard is the daughter of sociologist and philosopher Zygmunt Bauman. She began studying physics at the University of Warsaw in Poland in 1967. However, before completing her studies there, she moved to Israel with her father during the 1968 Polish political crisis, and entered the Hebrew University of Jerusalem. She earned a bachelor's degree in mathematics and physics there in 1972, a master's degree in mathematics in 1977, and a Ph.D. in 1989. Her doctoral dissertation, Teaching Theory of Algorithms in High-School, was jointly supervised by Menachem Magidor and Michael Maschler. After postdoctoral research in the UK, US, and Canada, she became an assistant professor of mathematics education at the University of Haifa in 1995. She was promoted to full professor there in 2001. She has also been Lappan-Phillips-Fitzgerald Professor of Mathematics Education at Michigan State University from 2003 to 2007, and Chair of Mathematics Education at the University of London from 2007 to 2009. == Selected publications == Sfard is the author of the book Thinking as communicating: Human development, development of discourses, and mathematizing (Cambridge University Press, 2008). She is also the author of two translations and the editor of several edited volumes in mathematics education. Her research articles include: Sfard, Anna (February 1991), "On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin", Educational Studies in Mathematics, 22 (1): 1–36, doi:10.1007/bf00302715 Sfard, Anna; Linchevski, Liora (March 1994), "The gains and the pitfalls of reification – the case of algebra", Educational Studies in Mathematics, 26 (2–3): 191–228, doi:10.1007/bf01273663 Sfard, Anna (March 1998), "On two metaphors for learning and the dangers of choosing just one", Educational Researcher, 27 (2): 4–13, doi:10.3102/0013189x027002004 Sfard, Anna (2001), "There is more to discourse than meets the ears: looking at thinking as communicating to learn more about mathematical learning", Educational Studies in Mathematics, 46 (1–3): 13–57, doi:10.1023/a:1014097416157 Sfard, Anna; Kieran, Carolyn (February 2001), "Cognition as communication: rethinking learning-by-talking through multi-faceted analysis of students' mathematical interactions", Mind, Culture, and Activity, 8 (1): 42–76, doi:10.1207/s15327884mca0801_04 Sfard, Anna; Prusak, Anna (May 2005), "Telling identities: in search of an analytic tool for investigating learning as a culturally shaped activity", Educational Researcher, 34 (4): 14–22, doi:10.3102/0013189x034004014 Sfard, Anna (October 2007), "When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint", Journal of the Learning Sciences, 16 (4): 565–613, doi:10.1080/10508400701525253 == Recognition == Sfard was the recipient of the 2007 Hans Freudenthal Award of the International Commission on Mathematical Instruction, "in recognition of her highly significant and scientifically deep accomplishments within a consistent, long-term research programme focused on objectification and discourse in mathematics education". She was elected as a Fellow of the American Educational Research Association in 2015. She was elected as an international associate of the National Academy of Education in 2016, and as an international honorary member of the American Academy of Arts and Sciences in 2023. She was an invited speaker at the 2022 (virtual) International Congress of Mathematicians. == References ==
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Wikipedia:Anna Sierpińska#0
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Anna Sierpińska (1947 – October 19, 2023) was a Polish-Canadian scholar of mathematics education, known for her investigations of understanding and epistemology in mathematics education. She was a professor emerita of mathematics and statistics at Concordia University. == Education and career == Sierpińska was born in Wrocław in 1947, and lived internationally with her family as a child. She earned a master's degree in 1970 from the University of Warsaw, specializing in commutative algebra. She completed her Ph.D. in mathematics education in 1984 at the Higher School of Pedagogy, Cracow. She was editor-in-chief of Educational Studies in Mathematics from 2001 to 2005. == Recognition == In 2006, Luleå University of Technology in Sweden gave Sierpińska an honorary doctorate. == Selected publications == === Monograph === Sierpinska, Anna (1994), Understanding in Mathematics, The Falmer Press, doi:10.4324/9780203454183, ISBN 978-0-203-45418-3 === Edited volumes === Sierpinska, Anna; Kilpatrick, Jeremy, eds. (1998), Mathematics Education as a Research Domain: A Search for Identity, New ICMI Study Series, vol. 4, Springer Netherlands, doi:10.1007/978-94-011-5470-3, ISBN 978-94-011-5470-3 Steinbring, Heinz; Bussi, Maria G. Bartolini; Sierpinska, Anna, eds. (1998), Language and Communication in the Mathematics Classroom, National Council of Teachers of Mathematics === Articles === Sierpińska, Anna (November 1987), "Humanities students and epistemological obstacles related to limits", Educational Studies in Mathematics, 18 (4): 371–397, doi:10.1007/bf00240986, S2CID 144880659 Sierpinska, Anna (November 1990), "Some remarks on understanding in mathematics", For the Learning of Mathematics, 10 (3): 24–36, 41, JSTOR 40247990 Sierpinska, Anna; Lerman, Stephen (1997), "Epistemologies of mathematics and of mathematics education", in Bishop, Alan J.; Clements, Ken; Keitel, Christine; Kilpatrick, Jeremy; Laborde, Colette (eds.), International Handbook of Mathematics Education, Springer Netherlands, pp. 827–876, doi:10.1007/978-94-009-1465-0_23 Sierpinska, Anna (2000), "On some aspects of students' thinking in linear algebra", in Dorier, Jean-Luc (ed.), On the Teaching of Linear Algebra, Mathematics Education Library, vol. 23, Kluwer Academic Publishers, pp. 209–246, doi:10.1007/0-306-47224-4_8, ISBN 978-0-306-47224-4 Dorier, Jean-Luc; Sierpinska, Anna, "Research into the teaching and learning of linear algebra", in Holton, Derek; Artigue, Michèle; Kirchgräber, Urs; Hillel, Joel; Niss, Mogens; Schoenfeld, Alan (eds.), The Teaching and Learning of Mathematics at University Level, New ICMI Study Series, vol. 7, Kluwer Academic Publishers, pp. 255–273, doi:10.1007/0-306-47231-7_24 == References == == External links == Home page
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Wikipedia:Anna Skripka#0
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Anna Skripka is a Ukrainian-American mathematician whose research topics include noncommutative analysis and probability. She is a professor at the University of New Mexico. == Education and career == Skripka did her undergraduate studies at the V. N. Karazin Kharkiv National University in Ukraine. She completed her Ph.D. at the University of Missouri. After working as a visiting assistant professor at Texas A&M University and as an assistant professor at the University of Central Florida, she joined the University of New Mexico Department of Mathematics and Statistics in 2012, where she is currently a full professor. == Recognition == Skripka is the 2019 winner of the Ruth I. Michler Memorial Prize of the Association for Women in Mathematics. == References == == External links == Home page Anna Skripka publications indexed by Google Scholar
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Wikipedia:Annales Fennici Mathematici#0
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Annales Fennici Mathematici (formerly Annales Academiæ Scientiarum Fennicæ Mathematica and Annales Academiæ Scientiarum Fennicæ) is a peer-reviewed scientific journal published by the Finnish Academy of Science and Letters since 1941. Its founder and editor until 1974 was Pekka Myrberg. It is currently edited by Olli Martio. It publishes research papers in all domains of mathematics, with particular emphasis on analysis. The journal acquired its current name in 2021. == Abstracting and indexing == The journal is indexed and abstracted in the following bibliographic databases: == References == == External links == Official website
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Wikipedia:Anne Bourlioux#0
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Anne Bourlioux is a Canadian mathematician whose research involves the numerical simulation of turbulent combustion. She is a winner of the Richard C. DiPrima Prize, and a professor of mathematics and statistics at the Université de Montréal. She is also a former rugby player for the Berkeley All Blues, and a Canadian national champion and world record holder in indoor rowing. == Education == Bourlioux earned her Ph.D. in 1991 at Princeton University. Her dissertation, Numerical Studies of Unstable Detonations, was supervised by Andrew Majda. She was a Miller Research Fellow at the University of California, Berkeley from 1991 to 1993. == Academic recognition == Bourlioux won the Richard C. DiPrima Prize in 1992. She was a keynote speaker at the 2006 Spring Technical Meeting of the Combustion Institute/Canadian Section, speaking on multiscale modeling of turbulent combustion. == Selected publications == Bourlioux, Anne; Majda, Andrew J. (1992-09-01). "Theoretical and numerical structure for unstable two-dimensional detonations". Combustion and Flame. 90 (3): 211–229. doi:10.1016/0010-2180(92)90084-3. ISSN 0010-2180. Bourlioux, Anne; Majda, Andrew J.; Roytburd, Victor (1991-04-01). "Theoretical and Numerical Structure for Unstable One-Dimensional Detonations". SIAM Journal on Applied Mathematics. 51 (2): 303–343. doi:10.1137/0151016. ISSN 0036-1399. Bourlioux, Anne; Layton, Anita T.; Minion, Michael L. (2003-08-10). "High-order multi-implicit spectral deferred correction methods for problems of reactive flow". Journal of Computational Physics. 189 (2): 651–675. doi:10.1016/S0021-9991(03)00251-1. ISSN 0021-9991. == References ==
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Wikipedia:Anne Broadbent#0
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Anne Lise Broadbent is a mathematician at the University of Ottawa who won the 2016 Aisenstadt Prize for her research in quantum computing, quantum cryptography, and quantum information. As of July 2024, she holds the Tier 1 Canada Research Chair in Quantum Communications and Cryptography. == Early life and education == Broadbent specialised in music at De La Salle High School in Ottawa, graduating in 1997. Her interest in science led her to major in mathematics for her undergraduate degree. Broadbent was a student of Alain Tapp and Gilles Brassard at the Université de Montréal, where she completed her master's in 2004 in the topic of Quantum pseudo-telepathy games, and her Ph.D. in 2008 with a dissertation on Quantum nonlocality, cryptography and complexity. == Career == After postdoctoral studies at the Institute for Quantum Computing at the University of Waterloo, she moved to Ottawa in 2014. She is a Full Professor at the Department of Mathematics and Statistics of the University of Ottawa. From 2014 to 2024, she held the University of Ottawa University Research Chair in Quantum Information Processing. == Awards == Broadbent is the winner of the 2010 John Charles Polanyi Prize in Physics of the Council of Ontario Universities. She was awarded the Aisenstadt Prize by International Scientific Advisory Committee of the Centre de Recherches Mathématiques in 2016 for her leadership and work in quantum information and cryptography. == References == == External links == Anne Broadbent publications indexed by Google Scholar
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Wikipedia:Anne Lemaître#0
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Anne Lemaître (born 1957) is a retired Belgian applied mathematician, formerly of the Université de Namur. She is an expert in orbital mechanics and orbital resonance, and their effects in the Solar System on bodies including asteroids, Mercury, and space debris. == Education and career == Lemaître completed her Ph.D. in 1984 at the Université de Namur. Her dissertation concerned Kirkwood gaps, dips in asteroid density caused by orbital resonances with Jupiter; it was supervised by Jacques Henrard. She is a professor emerita in the mathematics department of the Université de Namur. == Recognition == Minor planet 7330 Annelemaître is named in her honor, "for her pioneering analytic studies of the dynamics of minor planets in mean-motion resonances". == Selected publications == Henrard, J.; Lemaître, A. (1983), "A second fundamental model for resonance", Celestial Mechanics, 30 (2): 197–218, doi:10.1007/BF01234306, MR 0711658 Lemaître, A. (1984), "High-order resonances in the restricted three-body problem", Celestial Mechanics, 32 (2): 109–126, doi:10.1007/BF01231119, MR 0740277 Lemaitre, Anne; D'Hoedt, Sandrine; Rambaux, Nicolas (2006), "The 3:2 spin-orbit resonant motion of Mercury", Celestial Mechanics & Dynamical Astronomy, 95 (1–4): 213–224, doi:10.1007/s10569-006-9032-y, MR 2268199 Lemaître, A.; Delsate, N.; Valk, S. (2009), "A web of secondary resonances for large A/m geostationary debris", Celestial Mechanics & Dynamical Astronomy, 104 (4): 383–402, doi:10.1007/s10569-009-9217-2, MR 2524815 == References ==
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Wikipedia:Anne Sjerp Troelstra#0
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Anne Sjerp Troelstra (10 August 1939 – 7 March 2019) was a professor of pure mathematics and foundations of mathematics at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam. He was a constructivist logician, who was influential in the development of intuitionistic logic With Georg Kreisel, he was a developer of the theory of choice sequences. He wrote one of the first texts on linear logic, and, with Helmut Schwichtenberg, he co-wrote an important book on proof theory. He became a member of the Royal Netherlands Academy of Arts and Sciences in 1976. Troelstra died on 7 March 2019. After his retirement in 2000, Troelstra began a prolific career as the author of books on natural history travel, including the Bibliography of Natural History Travel Narratives, published with Brill in 2017. There are many others in Dutch, including Tijgers op de Ararat. Natuurhistorische reisverhalen 1700-1950 (Tigers on the Ararat. Natural History Travel Narratives 1700-1950), Van Spitsbergen naar Suriname (From Spitsbergen to Surinam). == Notes == == External links == Homepage of A. S. Troelstra : Dead Link - Archived : Homepage of A. S. Troelstra : Retrieved on 27 June 2018 Anne Sjerp Troelstra at the Mathematics Genealogy Project
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Wikipedia:Anne Taormina#0
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Anne Taormina is a Belgian and British mathematical physicist whose research topics include string theory, conformal field theory, and Mathieu moonshine. Beyond mathematical physics, she has also studied the icosahedral symmetry of virus capsids. She was Professor of Theoretical Particle Physics in the Department of Mathematical Sciences at Durham University and served as Head of Department there from January 2014 til December 2018. On 1st September 2024, she moved to King's College London as Head of Department of Mathematics. == Early life and education == Taormina is originally from Mons in Belgium; her parents were schoolteachers, and she has two sisters, one who became a physician and another who became a translator. She earned a license in mathematical sciences in 1980 from the University of Mons, and completed her doctorate in theoretical particle physics at the same university in 1984 under the supervision of Jean Nuyts. == Career == After short-term research positions with the Belgian National Fund for Scientific Research, the laboratory for theoretical physics at the École normale supérieure (Paris) (supported by the French National Centre for Scientific Research), CERN in Geneva, and the University of Chicago, she came to Durham in 1991 as a Science and Engineering Research Council Advanced Fellow. She remained at Durham as a temporary lecturer for 1996–1997, and as a Leverhulme Postdoc from 1997 to 2000, until becoming a lecturer in 2000. She was promoted to reader in 2004 and professor in 2006. Taormina headed the Durham Department of Mathematical Sciences for five years, from 2014 to 2018, and is a member of the council of the London Mathematical Society. == Personal life == Taormina is married to British physicist Nigel Glover, also a professor at Durham. == References == == External links == Home page Anne Taormina publications indexed by Google Scholar
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Wikipedia:Anne Watson (mathematics educator)#0
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Anne Watson is a British mathematics educator. She is a professor emeritus in the department of education at the University of Oxford, where she was a fellow of Linacre College, Oxford. She is a Fellow of the International Society for Design and Development in Education and of the Institute for Mathematics and its Applications. Watson was a comprehensive school teacher before becoming an academic. She has been a Quaker "off and on since the early 1980s" when she was a teacher, and belongs to the Steering Group of the Quaker Values in Education Group of the Society of Friends. Watson has expressed opposition to plans to disallow calculators on the National Curriculum assessment, and to grade the assessment by assigning partial credit to wrong answers using traditional calculation techniques but not for wrong answers using other methods, arguing that this emphasis on rote learning "works against the flexible number sense that we would want all children to develop". == Books == Watson is the co-author of: Inclusive Mathematics 11–18 (with M. Ollerton, Continuum, 2001) Mathematics as a Constructive Activity: Learners Generating Examples (with J. Mason, Erlbaum, 2005) Key Ideas in Teaching Mathematics: Research-based guidance for ages 9–19 (with K. Jones and D. Pratt, Oxford University Press, 2013) With D. Rowe she is the editor of Experience and Faith in Education: essays on Quaker perspectives (Trentham Press, to appear). == References ==
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Wikipedia:Anne van den Nouweland#0
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Anne van den Nouweland is a Dutch-American game theorist specializing in cooperative game theory, the game-based formation of complex networks, and their application in the design of communication networks. She works as a professor of economics at the University of Oregon. == Education and career == Van den Nouweland studied mathematics as an undergraduate at Nijmegen University in the Netherlands, graduating in 1984, and earned a master's degree there in 1989. Her master's thesis research applied intuitionism to the understanding of the Riemann–Stieltjes integral, supervised by Arnoud van Rooij and Wim Veldman. After two more years as a teaching assistant in the mathematics department at Nijmegen, she moved to the econometrics department at Tilburg University, also in the Netherlands, completing her Ph.D. there in 1993. Her doctoral dissertation, Games and Graphs in Economic Situations, was promoted by Stef Tijs. After completing her doctorate, she stayed on at Tilburg as an assistant professor and member of the CentER for Economic Research. She moved to the University of Oregon in 1996, was tenured there as an associate professor in 2001, and was promoted to full professor in 2007. == Book == Van den Nouweland is the coauthor of Social and Economic Networks in Cooperative Game Theory (with Marco Slikker, Kluwer Academic Publishers, 2001). == References == == External links == Home page Anne van den Nouweland publications indexed by Google Scholar
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Wikipedia:Anne-Laure Dalibard#0
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Anne-Laure Dalibard is a French mathematician working on asymptotic behavior of fluid equations occurring in oceanographic models. She works as a staff scientist at the Jacques-Louis Lions Laboratory, a joint research unit between Sorbonne University and the French National Centre for Scientific Research (UMR 7598.) == Education and career == Dalibard earned her PhD from Sorbonne University, working on the homogenization of scalar conservation laws and transport equations under the supervision of Fields medal laureate Pierre-Louis Lions. == Awards and honours == 2023 Mathematics medal of the French Academy of Sciences 2020 Société Mathématique de France Maurice Audin Prize 2018 CNRS bronze medal 2015 European Research Council starting grant 2010 College de France Peccot Prize == External links == Anne-Laure Dalibard – Sorbonne University Anne-Laure Dalibard publications indexed by Google Scholar == References ==
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Wikipedia:Annibale Giordano#0
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Annibale Giuseppe Nicolò Giordano (Ottaviano - San Giuseppe, 20 November 1769 – Troyes, 13 March 1835) was an Italian-French mathematician and revolutionary. == Life == Annibale Giordano was born 20 September 1769 in Ottaviano - San Giuseppe Vesuviano, to an educated middle-class family. His father Michele was a doctor who served both the king Ferdinand I of the Two Sicilies, and the Medici princes of Ottaviano. As a teenager, Annibale Giordano attended the school of Nicolò Fergola, a brilliant mathematician from Naples. In 1789, the year of the French Revolution, he was appointed professor at the Nunziatella Military School, thus becoming a colleague of the chemist Carlo Lauberg, a freemason. In 1790, Giordano and Lauberg established an Accademia di chimica e matematica in Naples, which became a club for Neapolitan progressives and Freemasons; among the members were Mario Pagano, Emanuele De Deo|it, Francesco Lomonaco, Vincenzo De Filippis and Luigi de' Medici di Ottajano, then regent of the Gran Corte della Vicaria court. In 1792 Giordano and Lauberg wrote the Principi analitici delle Matematiche, in which they theorized the political commitment of mathematicians; this essay was Annibale Giordano's last scientific work. In December 1792, Giordano was one of the scholars who met the French admiral Latouche-Tréville; starting from those meetings, a conspiracy began, sketched in the birth in August 1793 of the Società Patriottica Napoletana, a Jacobin association, but structured on the model of Masonic lodges, with a hierarchy such that some secrets were known only by high-ranking members. In February 1794, the Società Patriottica Napoletana split into two clubs. The ROMO (an acronym for Repubblica o Morte, i.e. "Republic or Death" was more radical and led by Andrea Vitaliani, among whose members were also Emanuele De Deo, Vincenzo Galiani and Vincenzo Vitaliani). The LOMO (acronym for "Libertà o Morte", i.e. "Freedom or Death"), was more moderate and willing to accept a constitutional monarchy, and was led by Rocco Lentini, and joined by Annibale Giordano). On 21 March 1794, authorities discovered the organization through a report by a certain Donato Froncillo; in the subsequent trial, some adherents of the ROMO (De Deo, Galiani and Vincenzo Vitaliani) were sentenced to death and executed, while Giordano was sentenced to twenty years and transferred to the Forte spagnolo prison. Many sources state that Annibale Giordano told investigators the secrets of the Società Patriottica Napoletana and that he gave the names of over 250 members, including Luigi de' Medici, who was incarcerated. Back in Naples together with the general Championnet on 5 December 1798, a few days after being released from L'Aquila, Annibale Giordano actively joined the short-lived Neapolitan Republic of 1799 as a member of the military committee and then head of the Navy's accounting service. When the Republic fell (in June 1799), he was again imprisoned by the Bourbon king in Castel Nuovo together with eighteen other revolutionaries including Mario Pagano, Domenico Cirillo and Giuseppe Leonardo Albanese. On 27 January 1800, he was sentenced to death by the junta; but the sentence was commuted to captivity on Favignana island; in July 1801, he left the island together with other political prisoners thanks to the Treaty of Lunéville. The non-execution was explained by many as a reward for Giordano's denunciation; others state that it was due to intercession by his father or Fergola at the Bourbon court. Giordano fled to France where he worked as cadastral surveyor in the French department of Aube; in 1824, he became a naturalized French citizen and changed his surname to Jourdan. == Mathematical advancements == In 1786, Giordano already presented to the Royal Academy of Sciences of Naples a memoir entitled Continuazione del medesimo argomento, which opened the doors of the Academy to him. Shortly thereafter, in 1788, he became famous for solving the following problem: "Given a circle and n points of its plane, inscribe in this circle a polygon whose sides, possibly prolonged, pass, according to a certain order, through the given points"; this problem was a generalization of the "problem of Pappus", which had been already solved for the case of n=3 aligned points, and the "problem of Castillon", solved by the latter in 1776, proposed to him by Cramer, for n=3 points but still arranged in the plane. Carnot thought that "Ottajano", the birthplace of Giordano, was a noble predicate rather than a town, and he called the young mathematician "Ottajano" in his publications; after this, he began to be referred to as "Ottajano" in subsequent scientific publications. == Works == Annibale Giordano; Carlo Lauberg (1792). Principj analitici delle matematiche. Vol. 2. Naples: Gennaro Giaccio. == References == == Bibliography == Fonseca, Giuseppe (2000). "GIORDANO, Annibale Giuseppe Nicolò". Dizionario Biografico degli Italiani, Volume 55: Ginammi–Giovanni da Crema (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6. == External links == "Annibale Giordano - Soluzione del problema di Pappo / Castillon". Archived from the original on 19 December 2009. Retrieved 12 December 2009.
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Wikipedia:Annie Cuyt#0
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Annie A. M. Cuyt (born 1956) is a Belgian computational mathematician known for her work on continued fractions, numerical analysis, Padé approximants, and related topics. She is a professor at the University of Antwerp, and a member of the Royal Flemish Academy of Belgium for Science and the Arts. == Education and career == Cuyt was born on 27 May 1956 in Elizabethstad (now Lubumbashi), in the Belgian Congo. She earned her Ph.D. at the University of Antwerp in 1982. Her dissertation, Padé approximants for operators: theory and applications, was promoted by Luc Wuytack. She was a postdoctoral researcher with support from the Alexander von Humboldt Foundation, and completed a habilitation in 1986. She is a professor in the Department of Mathematics and Computer Science at the University of Antwerp, where she leads the computational mathematics group. == Books == Cuyt is the author or coauthor of: Padé Approximants for Operators: Theory and Applications (Lecture Notes in Mathematics 1065, Springer, 1984) Nonlinear Methods in Numerical Analysis (with Luc Wuytack, North-Holland Mathematics Studies 136, North-Holland, 1987) Handbook of Continued Fractions for Special Functions (with Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones, Springer, 2008) == Recognition == Cuyt was elected to the Royal Flemish Academy of Belgium for Science and the Arts in 2013. The 4th Dolomites Workshop on Constructive Approximation and Applications, in 2016, and a special issue of the Dolomites Research Notes on Approximation, published in 2017, were dedicated to Cuyt in honor of her 60th birthday. == References == == External links == Annie Cuyt publications indexed by Google Scholar
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Wikipedia:Annie MacKinnon#0
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Annie Louise MacKinnon Fitch (June 1, 1868 – September 12, 1940) was a Canadian-born American mathematician who worked with Felix Klein and became a professor of mathematics at Wells College. She was the third woman to earn a mathematics doctorate at an American university. == Early life and education == Annie Louise MacKinnon was born June 1, 1868, in Woodstock, Ontario; her parents were also both originally from Ontario. She moved with her family as an infant to Concordia, Kansas, where her father worked as a realtor and hardware salesman. After graduating from Concordia High School, she became a student at the University of Kansas, which had been coeducational since it was founded in 1866. She graduated in 1889, and remained at the University of Kansas for graduate study in mathematics, becoming the third mathematics graduate student at the university and the first woman. She earned a master's degree in 1891, remaining one more year at the university to work there with Henry Byron Newson. In 1892, MacKinnon transferred to Cornell University. She finished her doctorate there in 1894, supported as an Erastus Brooks fellow. Her dissertation, Concomitant Binary Forms in Terms of the Roots, was supervised by James Edward Oliver, and also thanked James McMahon as a faculty mentor. This made her the third woman to earn a mathematics doctorate at an American university, following Winifred Edgerton Merrill in 1886 at Columbia University in 1886 and Ida Martha Metcalf at Cornell in 1893. From 1894 to 1896 she continued to study mathematics at the University of Göttingen, working there with Felix Klein, supported in the first year by the Association of Collegiate Alumnae European Fellowship and in the second year by the Women's Education Association of Boston European Fellowship. == Career and later life == MacKinnon taught high school mathematics in Lawrence, Kansas from 1890 to 1892. After her return from Europe in 1896, she became professor of mathematics at Wells College, a women's college in Aurora, New York; she was the only mathematician on the faculty. She also served as registrar for the college for 1900–1901. In 1901, MacKinnon married Edward Fitch, an American classics scholar who had been at Göttingen at roughly the same time as MacKinnon, and later taught at Hamilton College. After marrying, she gave up her mathematical career. She died on September 12, 1940, in Clinton, New York. A scholarship in mathematics at Hamilton College was established in her name by her husband. == References ==
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Wikipedia:Annihilating polynomial#0
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A polynomial P is annihilating or called an annihilating polynomial in linear algebra and operator theory if the polynomial considered as a function of the linear operator or a matrix A evaluates to zero, i.e., is such that P(A) = 0. Note that all characteristic polynomials and minimal polynomials of A are annihilating polynomials. In fact, every annihilating polynomial is the multiple of the minimal polynomial of an operator A. == See also == Cayley–Hamilton theorem Minimal polynomial (linear algebra) == References ==
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Wikipedia:Anouar Benmalek#0
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Anouar Benmalek (born January 16, 1956) is an Algerian novelist, journalist, mathematician and poet. After the 1988 riots in Algeria in protest of government policies, he became one of the founders of the Algerian Committee Against Torture. His novel Lovers of Algeria was awarded the Prix Ragid. The novel, The Child of an Ancient People, won the Prix RFO du livre. Benmalek's work has been described as "elegiac, multilayered meditation on Algeria's violent history." He has been compared to Camus and Faulkner. He was born in Casablanca to an Algerian father and a Moroccan mother. == Works == Cortèges d'impatiences, poetry, Éd. Naaman, 1984, Québec La Barbarie, essay, Éd. Enal, 1986, Algiers Rakesh, Vishnou et les autres nouvelles, Éd. Enal, 1985, Algiers Ludmila, novel, Éd. Enal, 1986, Algiers Les amants désunis, novel, Éd. Calmann-Lévy, 1998, Paris; Éd. Livre de Poche, 2000; Prix Mimouni 1999 (translated into 10 languages, sélections Fémina et Médicis). L'enfant du peuple ancien, novel, Jean-Jacques Pauvert, August 2000, Paris; Ed. Livre de Poche, 2002; Prix des auditeurs de la RTBF (Radio Télévision Belge) 2001, Prix RFO du livre 2001, Prix BeurFM-Méditerranée 2001, Prix Millepages 2000 (sélection Fémina, sélection rentrée littéraire 2000 Libraires et lecteurs de la Fnac, sélection du journal Le Soir de Bruxelles, sélection France Télévision, sélection Côté Femmes… translated into 8 languages) L'amour Loup, novel, Éd. Pauvert, February 2002, Éd. Livre de Poche, 2004, Paris Chroniques de l'Algérie amère, Éd. Pauvert, January 2003, Paris Ce jour viendra, novel, Éd Pauvert, September 2003 Ma planète me monte à la tête, poetry, Fayard, January 2005 L'année de la putain, shioet stories, Fayard, 2006 Ô Maria, novel, Fayard, 2006 Vivre pour écrire, interviews, Éd. Sedia, February 2007 Le Rapt, novel, Fayard, 2009 (translated into Italian, Il rapimento, Atmosphere libri, 2014) Tu ne mourras plus demain, narration, Fayard, 2011 Fils du Sheol, novel, Calmann-Levy, 2015 === Collective works === Une journée d'été, Éd. Librio, 2000 Étrange mon étranger, Seloncourt, 2001 Ma langue est mon territoire, Éd. Eden, 2001 Nouvelles d'aujourd'hui, Éd. Écoute, Spotlight Verlag, 2001 Contre offensive, Éd. Pauvert, 2002 Lettres de ruptures, Éd. Pocket, 2002 Des nouvelles d'Algérie, Éd. Métailié, 2005 Le Tour du Mont en 80 pages, Les Lettres européennes, 2005 Nouvelles d'Algérie, Éd. Magellan, 2009 Les Enfants de la balle, Éd. Lattès, 2010 Algérie 50, Éd. Magellan, 2012 == References ==
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Wikipedia:Ant on a rubber rope#0
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The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary, but a typical form is as follows: At first consideration it seems that the ant will never reach the end of the rope, but whatever the length of the rope and the speeds, provided that the length and speeds remain steady, the ant will always be able to reach the end given sufficient time — in the form stated above, it would take 8.9×1043421 years. There are two key principles: first, since the rubber rope is stretching both in front of and behind the ant, the proportion of the rope the ant has already walked is conserved, and, second, the proportional speed of the ant is inversely proportional to the length of the rubber rope, so the distance the ant can travel is unbounded like the harmonic series. == A formal statement of the problem == For sake of analysis, the following is a formalized version of the puzzle. The statement of the puzzle from the introduction corresponds to when c {\displaystyle c} is 1 km, v {\displaystyle v} is 1 km/s, and α {\displaystyle \alpha } is 1 cm/s. == Solutions to the problem == === A discrete mathematics solution === Although solving the problem appears to require analytical techniques, it can actually be answered by a combinatorial argument by considering a variation in which the rope stretches suddenly and instantaneously each second rather than stretching continuously. Indeed, the problem is sometimes stated in these terms, and the following argument is a generalisation of one set out by Martin Gardner, originally in Scientific American and later reprinted. Consider a variation in which the rope stretches suddenly and instantaneously before each second, so that the target-point moves from x = c {\displaystyle x=c} to x = c + v {\displaystyle x=c+v} at time t = 0 {\displaystyle t=0} , and from x = c + v {\displaystyle x=c+v} to x = c + 2 v {\displaystyle x=c+2v} at time t = 1 {\displaystyle t=1} , etc. Many versions of the problem have the rope stretch at the end of each second, but by having the rope stretch before each second we have disadvantaged the ant in its goal, so we can be sure that if the ant can reach the target point in this variation then it certainly can in the original problem, or indeed in variants where the rope stretches at the end of each second. Let θ ( t ) {\displaystyle \theta (t)} be the proportion of the distance from the starting-point to the target point which the ant has covered at time t. So θ ( 0 ) = 0 {\displaystyle \theta (0)=0} . In the first second the ant travels distance α {\displaystyle \alpha } , which is α c + v {\displaystyle {\frac {\alpha }{c+v}}} of the distance from the starting-point to the target-point (which is c + v {\displaystyle c+v} throughout the first second). When the rope stretches suddenly and instantaneously, θ ( t ) {\displaystyle \theta (t)} remains unchanged, because the ant moves along with the rubber where it is at that moment. So θ ( 1 ) = α c + v {\displaystyle \theta (1)={\frac {\alpha }{c+v}}} . In the next second the ant travels distance α {\displaystyle \alpha } again, which is α c + 2 v {\displaystyle {\frac {\alpha }{c+2v}}} of the distance from the starting-point to the target-point (which is c + 2 v {\displaystyle c+2v} throughout that second). So θ ( 2 ) = α c + v + α c + 2 v {\displaystyle \theta (2)={\frac {\alpha }{c+v}}+{\frac {\alpha }{c+2v}}} . Similarly, for any n ∈ N {\displaystyle n\in \mathbb {N} } , θ ( n ) = α c + v + α c + 2 v + ⋯ + α c + n v {\displaystyle \theta (n)={\frac {\alpha }{c+v}}+{\frac {\alpha }{c+2v}}+\cdots +{\frac {\alpha }{c+nv}}} . Notice that for any k ∈ N {\displaystyle k\in \mathbb {N} } , α c + k v ⩾ α k c + k v = ( α c + v ) ( 1 k ) {\displaystyle {\frac {\alpha }{c+kv}}\geqslant {\frac {\alpha }{kc+kv}}=\left({\frac {\alpha }{c+v}}\right)\left({\frac {1}{k}}\right)} , so we can write θ ( n ) ⩾ ( α c + v ) ( 1 + 1 2 + ⋯ + 1 n ) {\displaystyle \theta (n)\geqslant \left({\frac {\alpha }{c+v}}\right)\left(1+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)} . The term ( 1 + 1 2 + ⋯ + 1 n ) {\displaystyle \left(1+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)} is a partial harmonic series, which diverges, so we can find N ∈ N {\displaystyle N\in \mathbb {N} } such that 1 + 1 2 + ⋯ + 1 N ⩾ c + v α {\displaystyle 1+{\frac {1}{2}}+\cdots +{\frac {1}{N}}\geqslant {\frac {c+v}{\alpha }}} , which means that θ ( N ) ⩾ 1 {\displaystyle \theta (N)\geqslant 1} . Therefore, given sufficient time, the ant will complete the journey to the target point. This solution could be used to obtain an upper bound for the time required, but does not give an exact answer for the time it will take. === An analytical solution === A key observation is that the speed of the ant at a given time t > 0 {\displaystyle t>0} is its speed relative to the rope, i.e. α {\displaystyle \alpha } , plus the speed of the rope at the point where the ant is. The target-point moves with speed v {\displaystyle v} , so at time t {\displaystyle t} it is at x = c + v t {\displaystyle x=c+vt} . Other points along the rope move with proportional speed, so at time t {\displaystyle t} the point on the rope at x = X {\displaystyle x=X} is moving with speed v X c + v t {\displaystyle {\frac {vX}{c+vt}}} . So if we write the position of the ant at time t {\displaystyle t} as y ( t ) {\displaystyle y(t)} , and the speed of the ant at time t {\displaystyle t} as y ′ ( t ) {\displaystyle y'(t)} , we can write: y ′ ( t ) = α + v y ( t ) c + v t {\displaystyle y'(t)=\alpha +{\frac {v\,y(t)}{c+vt}}} This is a first order linear differential equation, and it can be solved with standard methods. However, to do so requires some moderately advanced calculus. A much simpler approach considers the ant's position as a proportion of the distance from the starting-point to the target-point. Consider coordinates ψ {\displaystyle \psi } measured along the rope with the starting-point at ψ = 0 {\displaystyle \psi =0} and the target-point at ψ = 1 {\displaystyle \psi =1} . In these coordinates, all points on the rope remain at a fixed position (in terms of ψ {\displaystyle \psi } ) as the rope stretches. At time t ⩾ 0 {\displaystyle t\geqslant 0} , a point at x = X {\displaystyle x=X} is at ψ = X c + v t {\displaystyle \psi ={\frac {X}{c+vt}}} , and a speed of α {\displaystyle \alpha } relative to the rope in terms of x {\displaystyle x} , is equivalent to a speed α c + v t {\displaystyle {\frac {\alpha }{c+vt}}} in terms of ψ {\displaystyle \psi } . So if we write the position of the ant in terms of ψ {\displaystyle \psi } at time t {\displaystyle t} as ϕ ( t ) {\displaystyle \phi (t)} , and the speed of the ant in terms of ψ {\displaystyle \psi } at time t {\displaystyle t} as ϕ ′ ( t ) {\displaystyle \phi '(t)} , we can write: ϕ ′ ( t ) = α c + v t {\displaystyle \phi '(t)={\frac {\alpha }{c+vt}}} ∴ ϕ ( t ) = ∫ α c + v t d t = α v ln ( c + v t ) + κ {\displaystyle \therefore \phi (t)=\int {{\frac {\alpha }{c+vt}}\,dt}={\frac {\alpha }{v}}\ln(c+vt)+\kappa } where κ {\displaystyle \kappa } is a constant of integration. Now, ϕ ( 0 ) = 0 {\displaystyle \phi (0)=0} which gives κ = − α v ln c {\displaystyle \kappa =-{\frac {\alpha }{v}}\ln {c}} , so ϕ ( t ) = α v ln ( c + v t c ) {\displaystyle \phi (t)={\frac {\alpha }{v}}\ln {\left({\frac {c+vt}{c}}\right)}} . If the ant reaches the target-point (which is at ψ = 1 {\displaystyle \psi =1} ) at time t = T {\displaystyle t=T} , we must have ϕ ( T ) = 1 {\displaystyle \phi (T)=1} which gives us: α v ln ( c + v T c ) = 1 {\displaystyle {\frac {\alpha }{v}}\ln {\left({\frac {c+vT}{c}}\right)}=1} ∴ T = c v ( e v / α − 1 ) {\displaystyle \therefore T={\frac {c}{v}}\left(e^{v/\alpha }-1\right)} and hence for the length of the rubber band when the ant catches the target point (i.e. the distance travelled by the ant): y ( T ) = c + v T = c × e v / α . {\displaystyle y\left(T\right)=c+vT=c\times e^{v/\alpha }.} (For the simple case of v = 0, we can consider the limit lim v → 0 T ( v ) {\displaystyle \lim _{v\rightarrow 0}T(v)} and obtain the simple solution T = c α {\displaystyle T={\tfrac {c}{\alpha }}} .) As this gives a finite value T {\displaystyle T} for all finite c {\displaystyle c} , v {\displaystyle v} , α {\displaystyle \alpha } ( v > 0 {\displaystyle v>0} , α > 0 {\displaystyle \alpha >0} ), this means that, given sufficient time, the ant will complete the journey to the target-point. This formula can be used to find out how much time is required. For the problem as originally stated, c = 1 k m {\displaystyle c=1\,\mathrm {km} } , v = 1 k m / s {\displaystyle v=1\,\mathrm {km} /\mathrm {s} } and α = 1 c m / s {\displaystyle \alpha =1\,\mathrm {cm} /\mathrm {s} } , which gives T = ( e 100 000 − 1 ) s ≈ 2.8 × 10 43 429 s {\displaystyle T=(e^{100\,000}-1)\,\mathrm {s} \,\!\approx 2.8\times 10^{43\,429}\,\mathrm {s} } . This is a vast timespan, even compared to the estimated age of the universe, which is only about 4×1017 s. Furthermore, the length of the rope after such a time is similarly huge, 2.8×1043429 km, so it is only in a mathematical sense that the ant can ever reach the end of this particular rope. == Intuition == Consider the situation from the introduction, which is a rope 1 km long being stretched 1 km/s, along which an ant walking is with a relative speed of 1 cm/s. At any moment, we can imagine putting down two marks on the rope: one at the ant's current position, and another 1 mm closer to the target point. If the ant were to stop for a moment, from its point of view the first mark is stationary, and the second mark is moving away at a constant speed of 1 mm/s or less (depending on the starting time). It is clear that the ant will be able to reach this second mark — for a simple over-estimation of the time it takes, imagine that we "turn off" the force that the rope applies to the ant at the exact moment it reaches the first mark (leaving the ant to continue onward at constant velocity). With respect to the reference frame for the first mark at this moment, the ant is moving at 1 cm/s and the second mark is initially 1 mm away and moving at 1 mm/s, and the ant would still reach the mark in 1/9 s. What we need to do is think about the ant's position as a fraction of the length of the rope. The above reasoning shows that this fraction is always increasing, but this is not yet enough. (The ant might asymptotically approach some fraction of the rope and never come close to reaching the target point.) What the reasoning also shows is that every 1/9 s, the fraction of the rope that the ant walks along is (at least as large as a number that is) inversely proportional to the current time, since the target point is moving proportionally to time, and the fraction of the rope this 1 mm interval corresponds to is inversely proportional to that. Quantities that grow at a rate that is inversely proportional to time exhibit logarithmic growth, which grows without bound, however slowly that might be. This means the ant will, eventually, reach the target point. If the speed at which the rope stretches increases through time, then the ant might not reach the target. For example, imagine one end of the rope is attached to a weight that is in free fall in a uniform gravitational field, with the rope applying no force to the weight (in other words, the target point's position is given by a function of the form x = c + 1 2 a t 2 {\displaystyle x=c+{\tfrac {1}{2}}at^{2}} ). If c {\displaystyle c} is 1 m, a {\displaystyle a} is 9.81 m/s2, and the ant moves at 1 cm/s, then the ant won't cover even 0.71% of the length of the rope, despite the fact the ant is always making forward progress. However, if the ant moves at a speed greater than 1.41 m/s it will reach the end of the rope in finite time. Further there are scenarios where the speed of the ant is decreasing exponentially while the length of the rope is increasing exponentially and the ant will also reach the end of the rope in finite time. == See also == Achilles and the tortoise Goodstein's theorem == References ==
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Wikipedia:Anthony Charles Croft#0
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Professor Anthony Charles Croft (Tony), CMath and FIMA, is a British mathematician known for his work in mathematics education including university-wide mathematics and statistics support, engineering mathematics, mathematical education of specialist mathematicians and also non-specialist users of mathematics. He is currently Emeritus Professor of Mathematics Education at Loughborough University. == Early life and education == Croft attended St Thomas Aquinas Grammar School in Leeds. He went on to gain a BSc (First Class Honours) and an M.Phil in Mathematics from the University of Leeds and then a PhD in Numerical Analysis from Keele University. == Career == Croft’s career in mathematics education began by teaching mathematics at Peel Sixth form College in Bury before he moved to Crewe & Alsager College in Cheshire (now part of Manchester Metropolitan University) where he taught mathematics to students studying for Bachelor of Education and Postgraduate Certificate in Education qualifications. In 1988 he took up a lecturing post at Leicester Polytechnic, later De Montfort University. It was in Leicester that he began authoring with his longtime collaborator Robert Davison. In 1996 a further move took him to Loughborough University, initially to establish a support centre to offer supplementary mathematics help to its many thousands of engineering students. In 2002 he was appointed founding director of the Mathematics Education Department at Loughborough University, which today has evolved into one of the UK's leading centres for research into and practice of the learning and teaching of mathematics and statistics. From 2005-2010 with Professor Duncan Lawson from Coventry University he established the sigma Centre for Excellence in University-wide Mathematics and Statistics Support. which was awarded the 2011 Times Higher Education Award for Outstanding Support for Students. Croft and Lawson, in collaboration with Professor Michael Grove from the University of Birmingham have highlighted the need for, and contributed extensively to the academic literature on, the practice of additional mathematics support provision in higher education. This work includes several surveys on the extent of provision and a literature review. Croft, with Lawson and Grove, were lead members of the team that established mathcentre a virtual maths support centre offering resources and advice to students in higher education. To-date hundreds of thousands of students around the world have found this an invaluable resource. Croft is a well-established academic author particularly in the fields of Engineering Mathematics and Foundation Mathematics. The fifth edition of his book Mathematics for Engineers was published in 2019 by Pearson Education with an e-book 6th edition due in 2025. He was awarded a UK National Teaching Fellowship in 2008 in recognition of his pioneering work in mathematics support for students transitioning from school to university. In 2016 he was awarded the Institute of Mathematics and its Applications' Gold Medal jointly with Lawson for "an outstanding contribution to the improvement of the teaching of mathematics" by establishing sigma. Since 2024 he has served on the Council of the Institute of Mathematics and its Applications. == References ==
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Wikipedia:Anthony James Merrill Spencer#0
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Anthony James Merrill Spencer (23 August 1929 — 26 January 2008) FRS was an applied mathematician whose main field of research was in understanding and predicting the mechanical behaviour of advanced materials. == Awards and honours == Spencer was elected a Fellow of the Royal Society (FRS) in 1987. == References ==
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Wikipedia:Antiautomorphism#0
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In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an invertible antihomomorphism, i.e. an antiisomorphism, from a set to itself. From bijectivity it follows that antiautomorphisms have inverses, and that the inverse of an antiautomorphism is also an antiautomorphism. == Definition == Informally, an antihomomorphism is a map that switches the order of multiplication. Formally, an antihomomorphism between structures X {\displaystyle X} and Y {\displaystyle Y} is a homomorphism ϕ : X → Y op {\displaystyle \phi \colon X\to Y^{\text{op}}} , where Y op {\displaystyle Y^{\text{op}}} equals Y {\displaystyle Y} as a set, but has its multiplication reversed to that defined on Y {\displaystyle Y} . Denoting the (generally non-commutative) multiplication on Y {\displaystyle Y} by ⋅ {\displaystyle \cdot } , the multiplication on Y op {\displaystyle Y^{\text{op}}} , denoted by ∗ {\displaystyle *} , is defined by x ∗ y := y ⋅ x {\displaystyle x*y:=y\cdot x} . The object Y op {\displaystyle Y^{\text{op}}} is called the opposite object to Y {\displaystyle Y} (respectively, opposite group, opposite algebra, opposite category etc.). This definition is equivalent to that of a homomorphism ϕ : X op → Y {\displaystyle \phi \colon X^{\text{op}}\to Y} (reversing the operation before or after applying the map is equivalent). Formally, sending X {\displaystyle X} to X op {\displaystyle X^{\text{op}}} and acting as the identity on maps is a functor (indeed, an involution). == Examples == In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism, φ(xy) = φ(y)φ(x) for all x, y in X. The map that sends x to x−1 is an example of a group antiautomorphism. Another important example is the transpose operation in linear algebra, which takes row vectors to column vectors. Any vector-matrix equation may be transposed to an equivalent equation where the order of the factors is reversed. With matrices, an example of an antiautomorphism is given by the transpose map. Since inversion and transposing both give antiautomorphisms, their composition is an automorphism. This involution is often called the contragredient map, and it provides an example of an outer automorphism of the general linear group GL(n, F), where F is a field, except when |F| = 2 and n = 1 or 2, or |F| = 3 and n = 1 (i.e., for the groups GL(1, 2), GL(2, 2), and GL(1, 3)). In ring theory, an antihomomorphism is a map between two rings that preserves addition, but reverses the order of multiplication. So φ : X → Y is a ring antihomomorphism if and only if: φ(1) = 1 φ(x + y) = φ(x) + φ(y) φ(xy) = φ(y)φ(x) for all x, y in X. For algebras over a field K, φ must be a K-linear map of the underlying vector space. If the underlying field has an involution, one can instead ask φ to be conjugate-linear, as in conjugate transpose, below. === Involutions === It is frequently the case that antiautomorphisms are involutions, i.e. the square of the antiautomorphism is the identity map; these are also called involutive antiautomorphisms. For example, in any group the map that sends x to its inverse x−1 is an involutive antiautomorphism. A ring with an involutive antiautomorphism is called a *-ring, and these form an important class of examples. == Properties == If the source X or the target Y is commutative, then an antihomomorphism is the same thing as a homomorphism. The composition of two antihomomorphisms is always a homomorphism, since reversing the order twice preserves order. The composition of an antihomomorphism with a homomorphism gives another antihomomorphism. == See also == Semigroup with involution == References ==
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Wikipedia:Antikythera mechanism#0
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The Antikythera mechanism ( AN-tik-ih-THEER-ə, US also AN-ty-kih-) is an Ancient Greek hand-powered orrery (model of the Solar System). It is the oldest known example of an analogue computer. It could be used to predict astronomical positions and eclipses decades in advance. It could also be used to track the four-year cycle of athletic games similar to an Olympiad, the cycle of the ancient Olympic Games. The artefact was among wreckage retrieved from a shipwreck off the coast of the Greek island Antikythera in 1901. In 1902, it was identified by archaeologist Spyridon Stais as containing a gear. The device, housed in the remains of a wooden-framed case of (uncertain) overall size 34 cm × 18 cm × 9 cm (13.4 in × 7.1 in × 3.5 in), was found as one lump, later separated into three main fragments which are now divided into 82 separate fragments after conservation efforts. Four of these fragments contain gears, while inscriptions are found on many others. The largest gear is about 13 cm (5 in) in diameter and originally had 223 teeth. All these fragments of the mechanism are kept at the National Archaeological Museum, Athens, along with reconstructions and replicas, to demonstrate how it may have looked and worked. In 2005, a team from Cardiff University led by Mike Edmunds used computer X-ray tomography and high resolution scanning to image inside fragments of the crust-encased mechanism and read the faintest inscriptions that once covered the outer casing. These scans suggest that the mechanism had 37 meshing bronze gears enabling it to follow the movements of the Moon and the Sun through the zodiac, to predict eclipses and to model the irregular orbit of the Moon, where the Moon's velocity is higher in its perigee than in its apogee. This motion was studied in the 2nd century BC by astronomer Hipparchus of Rhodes, and he may have been consulted in the machine's construction. There is speculation that a portion of the mechanism is missing and it calculated the positions of the five classical planets. The inscriptions were further deciphered in 2016, revealing numbers connected with the synodic cycles of Venus and Saturn. The instrument is believed to have been designed and constructed by Hellenistic scientists and been variously dated to about 87 BC, between 150 and 100 BC, or 205 BC. It must have been constructed before the shipwreck, which has been dated by multiple lines of evidence to approximately 70–60 BC. In 2022, researchers proposed its initial calibration date, not construction date, could have been 23 December 178 BC. Other experts propose 204 BC as a more likely calibration date. Machines with similar complexity did not appear again until the 14th century in western Europe. == History == === Discovery === Captain Dimitrios Kontos (Δημήτριος Κοντός) and a crew of sponge divers from Symi island discovered the Antikythera wreck in early 1900, and recovered artefacts during the first expedition with the Hellenic Royal Navy, in 1900–01. This wreck of a Roman cargo ship was found at a depth of 45 metres (148 ft) off Point Glyphadia on the Greek island of Antikythera. The team retrieved numerous large objects, including bronze and marble statues, pottery, unique glassware, jewellery, coins, and the mechanism. The mechanism was retrieved from the wreckage in 1901, probably July. It is unknown how the mechanism came to be on the cargo ship. All of the items retrieved from the wreckage were transferred to the National Museum of Archaeology in Athens for storage and analysis. The mechanism appeared to be a lump of corroded bronze and wood. The bronze had turned into atacamite which cracked and shrank when it was brought up from the shipwreck, changing the dimensions of the pieces. It went unnoticed for two years, while museum staff worked on piecing together more obvious treasures, such as the statues. Upon removal from seawater, the mechanism was not treated, resulting in deformational changes. On 17 May 1902, archaeologist Valerios Stais found one of the pieces of rock had a gear wheel embedded in it. He initially believed that it was an astronomical clock, but most scholars considered the device to be prochronistic, too complex to have been constructed during the same period as the other pieces that had been discovered. The German philologist Albert Rehm became interested in the device and was the first to propose that it was an astronomical calculator. Investigations into the object lapsed until British science historian and Yale University professor Derek J. de Solla Price became interested in 1951. In 1971, Price and Greek nuclear physicist Charalampos Karakalos made X-ray and gamma-ray images of the 82 fragments. Price published a paper on their findings in 1974. Two other searches for items at the Antikythera wreck site in 2012 and 2015 yielded art objects and a second ship which may, or may not, be connected with the treasure ship on which the mechanism was found. Also found was a bronze disc, embellished with the image of a bull. The disc has four "ears" which have holes in them, and it was thought it may have been part of the Antikythera mechanism, as a "cog wheel". There appears to be little evidence that it was part of the mechanism; it is more likely the disc was a bronze decoration on a piece of furniture. === Origin === The Antikythera mechanism is generally referred to as the first known analogue computer. The quality and complexity of the mechanism's manufacture suggests it must have had undiscovered predecessors during the Hellenistic period. Its construction relied on theories of astronomy and mathematics developed by Greek astronomers during the second century BC, and it is estimated to have been built in the late second century BC or the early first century BC. In 2008, research by the Antikythera Mechanism Research Project suggested the concept for the mechanism may have originated in the colonies of Corinth, since they identified the calendar on the Metonic Spiral as coming from Corinth, or one of its colonies in northwest Greece or Sicily. Syracuse was a colony of Corinth and the home of Archimedes, and the Antikythera Mechanism Research Project argued in 2008 that it might imply a connection with the school of Archimedes. It was demonstrated in 2017 that the calendar on the Metonic Spiral is of the Corinthian type, but cannot be that of Syracuse. Another theory suggests that coins found by Jacques Cousteau at the wreck site in the 1970s date to the time of the device's construction, and posits that its origin may have been from the ancient Greek city of Pergamon, home of the Library of Pergamum. With its many scrolls of art and science, it was second in importance only to the Library of Alexandria during the Hellenistic period. The ship carrying the device contained vases in the Rhodian style, leading to a hypothesis that it was constructed at an academy founded by Stoic philosopher Posidonius on that Greek island. Rhodes was a busy trading port and centre of astronomy and mechanical engineering, home to astronomer Hipparchus, who was active from about 140–120 BC. The mechanism uses Hipparchus' theory for the motion of the Moon, which suggests he may have designed or at least worked on it. It has been argued the astronomical events on the Parapegma of the mechanism work best for latitudes in the range of 33.3–37.0 degrees north; the island of Rhodes is located between the latitudes of 35.85 and 36.50 degrees north. In 2014, a study argued for a new dating of approximately 200 BC, based on identifying the start-up date on the Saros Dial, as the astronomical lunar month that began shortly after the new moon of 28 April 205 BC. According to this theory the Babylonian arithmetic style of prediction fits much better with the device's predictive models than the traditional Greek trigonometric style. A study by Iversen in 2017 reasons that the prototype for the device was from Rhodes, but that this particular model was modified for a client from Epirus in northwestern Greece; Iversen argues it was probably constructed no earlier than a generation before the shipwreck, a date supported by Jones in 2017. Further dives were undertaken in 2014 and 2015, in the hope of discovering more of the mechanism. A five-year programme of investigations began in 2014 and ended in October 2019, with a new five-year session starting in May 2020. In 2022, researchers proposed the mechanism's initial calibration date, not construction date, could have been 23 December 178 BC. Other experts propose 204 BC as a more likely calibration date. Machines with similar complexity did not appear again until the fourteenth century, with early examples being astronomical clocks of Richard of Wallingford and Giovanni de' Dondi. == Design == The original mechanism apparently came out of the Mediterranean as a single encrusted piece. Soon afterwards it fractured into three major pieces. Other small pieces have broken off in the interim from cleaning and handling, and others were found on the sea floor by the Cousteau expedition. Other fragments may still be in storage, undiscovered since their initial recovery; Fragment F was discovered in that way in 2005. Of the 82 known fragments, seven are mechanically significant and contain the majority of the mechanism and inscriptions. Another 16 smaller parts contain fractional and incomplete inscriptions. Many of the smaller fragments that have been found contain nothing of apparent value, but a few have inscriptions on them. Fragment 19 contains significant back door inscriptions including one reading "... 76 years ..." which refers to the Callippic cycle. Other inscriptions seem to describe the function of the back dials. In addition to this important minor fragment, 15 further minor fragments have remnants of inscriptions on them.: 7 == Mechanics == Information on the specific data obtained from the fragments is detailed in the supplement to the 2006 Nature article from Freeth et al. === Operation === On the front face of the mechanism, there is a fixed ring dial representing the ecliptic, the twelve zodiacal signs marked off with equal 30-degree sectors. This matched with the Babylonian custom of assigning one twelfth of the ecliptic to each zodiac sign equally, even though the constellation boundaries were variable. Outside that dial is another ring which is rotatable, marked off with the months and days of the Sothic Egyptian calendar, twelve months of 30 days plus five intercalary days. The months are marked with the Egyptian names for the months transcribed into the Greek alphabet. The first task is to rotate the Egyptian calendar ring to match the current zodiac points. The Egyptian calendar ignored leap days, so it advanced through a full zodiac sign in about 120 years. The mechanism was operated by turning a small hand crank (now lost) which was linked via a crown gear to the largest gear, the four-spoked gear visible on the front of fragment A, gear b1. This moved the date pointer on the front dial, which would be set to the correct Egyptian calendar day. The year is not selectable, so it is necessary to know the year currently set, or by looking up the cycles indicated by the various calendar cycle indicators on the back in the Babylonian ephemeris tables for the day of the year currently set, since most of the calendar cycles are not synchronous with the year. The crank moves the date pointer about 78 days per full rotation, so hitting a particular day on the dial would be easily possible if the mechanism were in good working condition. The action of turning the hand crank would also cause all interlocked gears within the mechanism to rotate, resulting in the simultaneous calculation of the position of the Sun and Moon, the moon phase, eclipse, and calendar cycles, and perhaps the locations of planets. The operator also had to be aware of the position of the spiral dial pointers on the two large dials on the back. The pointer had a "follower" that tracked the spiral incisions in the metal as the dials incorporated four and five full rotations of the pointers. When a pointer reached the terminal month location at either end of the spiral, the pointer's follower had to be manually moved to the other end of the spiral before proceeding further.: 10 === Faces === ==== Front face ==== The front dial has two concentric circular scales. The inner scale marks the Greek signs of the zodiac, with division in degrees. The outer scale, which is a movable ring that sits flush with the surface and runs in a channel, is marked off with what appear to be days and has a series of corresponding holes beneath the ring in the channel. Since the discovery of the mechanism more than a century ago, this outer ring has been presumed to represent a 365-day Egyptian solar calendar, but research (Budiselic, et al., 2020) challenged this presumption and provided direct statistical evidence there are 354 intervals, suggesting a lunar calendar. Since this initial discovery, two research teams, using different methods, independently calculated the interval count. Woan and Bayley calculate 354–355 intervals using two different methods, confirming with higher accuracy the Budiselic et al. findings and noting that "365 holes is not plausible". Malin and Dickens' best estimate is 352.3±1.5 and concluded that the number of holes (N) "has to be integral and the SE (standard error) of 1.5 indicates that there is less than a 5% probability that N is not one of the six values in the range 350 to 355. The chances of N being as high as 365 are less than 1 in 10,000. While other contenders cannot be ruled out, of the two values that have been proposed for N on astronomical grounds, that of Budiselic et al. (354) is by far the more likely." If one supports the 365 day presumption, it is recognized the mechanism predates the Julian calendar reform, but the Sothic and Callippic cycles had already pointed to a 365+1/4 day solar year, as seen in Ptolemy III's attempted calendar reform of 238 BC. The dials are not believed to reflect his proposed leap day (Epag. 6), but the outer calendar dial may be moved against the inner dial to compensate for the effect of the extra quarter-day in the solar year by turning the scale backward one day every four years. If one is in favour of the 354 day evidence, the most likely interpretation is that the ring is a manifestation of a 354-day lunar calendar. Given the era of the mechanism's presumed construction and the presence of Egyptian month names, it is possibly the first example of the Egyptian civil-based lunar calendar proposed by Richard Anthony Parker in 1950. The lunar calendar's purpose was to serve as a day-to-day indicator of successive lunations, and would also have assisted with the interpretation of the lunar phase pointer, and the Metonic and Saros dials. Undiscovered gearing, synchronous with the rest of the Metonic gearing of the mechanism, is implied to drive a pointer around this scale. Movement and registration of the ring relative to the underlying holes served to facilitate both a 1-in-76-year Callippic cycle correction, as well as convenient lunisolar intercalation. The dial also marks the position of the Sun on the ecliptic, corresponding to the current date in the year. The orbits of the Moon and the five planets known to the Greeks are close enough to the ecliptic to make it a convenient reference for defining their positions as well. The following three Egyptian months are inscribed in Greek letters on the surviving pieces of the outer ring: ΠΑΧΩΝ (Pachon) ΠΑΥΝΙ (Payni) ΕΠΙΦΙ (Epiphi) The other months have been reconstructed; some reconstructions of the mechanism omit the five days of the Egyptian intercalary month. The Zodiac dial contains Greek inscriptions of the members of the zodiac, which is believed to be adapted to the tropical month version rather than the sidereal:: 8 ΚΡΙΟΣ (Krios [Ram], Aries) ΤΑΥΡΟΣ (Tauros [Bull], Taurus) ΔΙΔΥΜΟΙ (Didymoi [Twins], Gemini) ΚΑΡΚΙΝΟΣ (Karkinos [Crab], Cancer) ΛΕΩΝ (Leon [Lion], Leo) ΠΑΡΘΕΝΟΣ (Parthenos [Maiden], Virgo) ΧΗΛΑΙ (Chelai [Scorpio's Claw or Zygos], Libra) ΣΚΟΡΠΙΟΣ (Skorpios [Scorpion], Scorpio) ΤΟΞΟΤΗΣ (Toxotes [Archer], Sagittarius) ΑΙΓΟΚΕΡΩΣ (Aigokeros [Goat-horned], Capricorn) ΥΔΡΟΧΟΟΣ (Hydrokhoos [Water carrier], Aquarius) ΙΧΘΥΕΣ (Ichthyes [Fish], Pisces) Also on the zodiac dial are single characters at specific points (see reconstruction at ref). They are keyed to a parapegma, a precursor of the modern day almanac inscribed on the front face above and beneath the dials. They mark the locations of longitudes on the ecliptic for specific stars. The parapegma above the dials reads (square brackets indicate inferred text): The parapegma beneath the dials reads: At least two pointers indicated positions of bodies upon the ecliptic. A lunar pointer indicated the position of the Moon, and a mean Sun pointer was shown, perhaps doubling as the current date pointer. The Moon position was not a simple mean Moon indicator which would indicate movement uniformly around a circular orbit; rather, it approximated the acceleration and deceleration of the Moon's elliptical orbit, through the earliest extant use of epicyclic gearing. It also tracked the precession of the Moon's elliptical orbit around the ecliptic in an 8.88 year cycle. The mean Sun position is, by definition, the current date. It is speculated that since significant effort was taken to ensure the position of the Moon was correct,: 20, 24 there was likely to have also been a "true sun" pointer in addition to the mean Sun pointer, to track the elliptical anomaly of the Sun (the orbit of Earth around the Sun), but there is no evidence of it among the fragments found. Similarly, neither is there the evidence of planetary orbit pointers for the five planets known to the Greeks among the fragments. But see Proposed gear schemes below. Mechanical engineer Michael Wright demonstrated there was a mechanism to supply the lunar phase in addition to the position. The indicator was a small ball embedded in the lunar pointer, half-white and half-black, which rotated to show the phase (new, first quarter, half, third quarter, full, and back). The data to support this function is available given the Sun and Moon positions as angular rotations; essentially, it is the angle between the two, translated into the rotation of the ball. It requires a differential gear, a gearing arrangement that sums or differences two angular inputs. ==== Rear face ==== In 2008, scientists reported new findings in Nature showing the mechanism not only tracked the Metonic calendar and predicted solar eclipses, but also calculated the timing of panhellenic athletic games, such as the ancient Olympic Games. Inscriptions on the instrument closely match the names of the months that are used on calendars from Epirus in northwestern Greece and with the island of Corfu, which in antiquity was known as Corcyra. On the back of the mechanism, there are five dials: the two large displays, the Metonic and the Saros, and three smaller indicators, the so-called Olympiad Dial, which has been renamed the Games dial as it did not track Olympiad years (the four-year cycle it tracks most closely is the Halieiad), the Callippic, and the exeligmos.: 11 The Metonic dial is the main upper dial on the rear of the mechanism. The Metonic cycle, defined in several physical units, is 235 synodic months, which is very close (to within less than 13 one-millionths) to 19 tropical years. It is therefore a convenient interval over which to convert between lunar and solar calendars. The Metonic dial covers 235 months in five rotations of the dial, following a spiral track with a follower on the pointer that keeps track of the layer of the spiral. The pointer points to the synodic month, counted from new moon to new moon, and the cell contains the Corinthian month names. ΦΟΙΝΙΚΑΙΟΣ (Phoinikaios) ΚΡΑΝΕΙΟΣ (Kraneios) ΛΑΝΟΤΡΟΠΙΟΣ (Lanotropios) ΜΑΧΑΝΕΥΣ (Machaneus, "mechanic", referring to Zeus the inventor) ΔΩΔΕΚΑΤΕΥΣ (Dodekateus) ΕΥΚΛΕΙΟΣ (Eukleios) ΑΡΤΕΜΙΣΙΟΣ (Artemisios) ΨΥΔΡΕΥΣ (Psydreus) ΓΑΜΕΙΛΙΟΣ (Gameilios) ΑΓΡΙΑΝΙΟΣ (Agrianios) ΠΑΝΑΜΟΣ (Panamos) ΑΠΕΛΛΑΙΟΣ (Apellaios) Thus, setting the correct solar time (in days) on the front panel indicates the current lunar month on the back panel, with resolution to within a week or so. Based on the fact that the calendar month names are consistent with all the evidence of the Epirote calendar and that the Games dial mentions the very minor Naa games of Dodona (in Epirus), it has been argued that the calendar on the mechanism is likely to be the Epirote calendar, and that this calendar was probably adopted from a Corinthian colony in Epirus, possibly Ambracia. It has been argued that the first month of the calendar, Phoinikaios, was ideally the month in which the autumn equinox fell, and that the start-up date of the calendar began shortly after the astronomical new moon of 23 August 205 BC. The Games dial is the right secondary upper dial; it is the only pointer on the instrument that travels in an anticlockwise direction as time advances. The dial is divided into four sectors, each of which is inscribed with a year indicator and the name of two Panhellenic Games: the "crown" games of Isthmia, Olympia, Nemea, and Pythia; and two lesser games: Naa (held at Dodona) and the Halieia of Rhodes. The inscriptions on each one of the four divisions are: The Saros dial is the main lower spiral dial on the rear of the mechanism.: 4–5, 10 The Saros cycle is 18 years and 11+1⁄3 days long (6585.333... days), which is very close to 223 synodic months (6585.3211 days). It is defined as the cycle of repetition of the positions required to cause solar and lunar eclipses, and therefore, it could be used to predict them—not only the month, but the day and time of day. The cycle is approximately 8 hours longer than an integer number of days. Translated into global spin, that means an eclipse occurs not only eight hours later, but one-third of a rotation farther to the west. Glyphs in 51 of the 223 synodic month cells of the dial specify the occurrence of 38 lunar and 27 solar eclipses. Some of the abbreviations in the glyphs read: Σ = ΣΕΛΗΝΗ ("Selene", Moon) Η = ΗΛΙΟΣ ("Helios", Sun) H\M = ΗΜΕΡΑΣ ("Hemeras", of the day) ω\ρ = ωρα ("hora", hour) N\Y = ΝΥΚΤΟΣ ("Nuktos", of the night) The glyphs show whether the designated eclipse is solar or lunar, and give the day of the month and hour. Solar eclipses may not be visible at any given point, and lunar eclipses are visible only if the Moon is above the horizon at the appointed hour.: 6 In addition, the inner lines at the cardinal points of the Saros dial indicate the start of a new full moon cycle. Based on the distribution of the times of the eclipses, it has been argued the start-up date of the Saros dial was shortly after the astronomical new moon of 28 April 205 BC. The Exeligmos dial is the secondary lower dial on the rear of the mechanism. The exeligmos cycle is a 54-year triple Saros cycle that is 19,756 days long. Since the length of the Saros cycle is to a third of a day (namely, 6,585 days plus 8 hours), a full exeligmos cycle returns the counting to an integral number of days, as reflected in the inscriptions. The labels on its three divisions are:: 10 Blank or o ? (representing the number zero, assumed, not yet observed) H (number 8) means add 8 hours to the time mentioned in the display Iϛ (number 16) means add 16 hours to the time mentioned in the display Thus the dial pointer indicates how many hours must be added to the glyph times of the Saros dial in order to calculate the exact eclipse times. === Doors === The mechanism has a wooden casing with a front and a back door, both containing inscriptions. The back door appears to be the 'instruction manual'. On one of its fragments is written "76 years, 19 years" representing the Callippic and Metonic cycles. Also written is "223" for the Saros cycle. On another one of its fragments, it is written "on the spiral subdivisions 235" referring to the Metonic dial. === Gearing === The mechanism is remarkable for the level of miniaturisation and the complexity of its parts, which is comparable to that of 14th-century astronomical clocks. It has at least 30 gears, although mechanism expert Michael Wright has suggested the Greeks of this period were capable of implementing a system with many more gears. There is debate as to whether the mechanism had indicators for all five of the planets known to the ancient Greeks. No gearing for such a planetary display survives and all gears are accounted for—with the exception of one 63-toothed gear (r1) otherwise unaccounted for in fragment D. Fragment D is a small quasi-circular constriction that, according to Xenophon Moussas, has a gear inside a somewhat larger hollow gear. The inner gear moves inside the outer gear reproducing an epicyclical motion that, with a pointer, gives the position of planet Jupiter. The inner gear is numbered 45, "ME" in Greek, and the same number is written on two surfaces of this small cylindrical box. The purpose of the front face was to position astronomical bodies with respect to the celestial sphere along the ecliptic, in reference to the observer's position on the Earth. That is irrelevant to the question of whether that position was computed using a heliocentric or geocentric view of the Solar System; either computational method should, and does, result in the same position (ignoring ellipticity), within the error factors of the mechanism. The epicyclic Solar System of Ptolemy (c. 100 AD–c. 170 AD)—hundreds of years after the apparent construction date of the mechanism—carried forward with more epicycles, and was more accurate predicting the positions of planets than the view of Copernicus (1473–1543), until Kepler (1571–1630) introduced the possibility that orbits are ellipses. Evans et al. suggest that to display the mean positions of the five classical planets would require only 17 further gears that could be positioned in front of the large driving gear and indicated using individual circular dials on the face. Freeth and Jones modelled and published details of a version using gear trains mechanically similar to the lunar anomaly system, allowing for indication of the positions of the planets, as well as synthesis of the Sun anomaly. Their system, they claim, is more authentic than Wright's model, as it uses the known skills of the Greeks and does not add excessive complexity or internal stresses to the machine. The gear teeth were in the form of equilateral triangles with an average circular pitch of 1.6 mm, an average wheel thickness of 1.4 mm and an average air gap between gears of 1.2 mm. The teeth were probably created from a blank bronze round using hand tools; this is evident because not all of them are even. Due to advances in imaging and X-ray technology, it is now possible to know the precise number of teeth and size of the gears within the located fragments. Thus the basic operation of the device is no longer a mystery and has been replicated accurately. The major unknown remains the question of the presence and nature of any planet indicators.: 8 A table of the gears, their teeth, and the expected and computed rotations of important gears follows. The gear functions come from Freeth et al. (2008) and for the lower half of the table from Freeth et al. (2012). The computed values start with 1 year per revolution for the b1 gear, and the remainder are computed directly from gear teeth ratios. The gears marked with an asterisk (*) are missing, or have predecessors missing, from the known mechanism; these gears have been calculated with reasonable gear teeth counts. (Lengths in days are calculated assuming the year to be 365.2425 days.) Table notes: There are several gear ratios for each planet that result in close matches to the correct values for synodic periods of the planets and the Sun. Those chosen above seem accurate, with reasonable tooth counts, but the specific gears actually used are unknown. ==== Known gear scheme ==== It is very probable there were planetary dials, as the complicated motions and periodicities of all planets are mentioned in the manual of the mechanism. The exact position and mechanisms for the gears of the planets is unknown. There is no coaxial system except for the Moon. Fragment D that is an epicycloidal system, is considered as a planetary gear for Jupiter (Moussas, 2011, 2012, 2014) or a gear for the motion of the Sun (University of Thessaloniki group). The Sun gear is operated from the hand-operated crank (connected to gear a1, driving the large four-spoked mean Sun gear, b1) and in turn drives the rest of the gear sets. The Sun gear is b1/b2 and b2 has 64 teeth. It directly drives the date/mean sun pointer (there may have been a second, "true sun" pointer that displayed the Sun's elliptical anomaly; it is discussed below in the Freeth reconstruction). In this discussion, reference is to modelled rotational period of various pointers and indicators; they all assume the input rotation of the b1 gear of 360 degrees, corresponding with one tropical year, and are computed solely on the basis of the gear ratios of the gears named. The Moon train starts with gear b1 and proceeds through c1, c2, d1, d2, e2, e5, k1, k2, e6, e1, and b3 to the Moon pointer on the front face. The gears k1 and k2 form an epicyclic gear system; they are an identical pair of gears that do not mesh, but rather, they operate face-to-face, with a short pin on k1 inserted into a slot in k2. The two gears have different centres of rotation, so the pin must move back and forth in the slot. That increases and decreases the radius at which k2 is driven, also necessarily varying its angular velocity (presuming the velocity of k1 is even) faster in some parts of the rotation than others. Over an entire revolution the average velocities are the same, but the fast-slow variation models the effects of the elliptical orbit of the Moon, in consequence of Kepler's second and third laws. The modelled rotational period of the Moon pointer (averaged over a year) is 27.321 days, compared to the modern length of a lunar sidereal month of 27.321661 days. The pin/slot driving of the k1/k2 gears varies the displacement over a year's time, and the mounting of those two gears on the e3 gear supplies a precessional advancement to the ellipticity modelling with a period of 8.8826 years, compared with the current value of precession period of the moon of 8.85 years. The system also models the phases of the Moon. The Moon pointer holds a shaft along its length, on which is mounted a small gear named r, which meshes to the Sun pointer at B0 (the connection between B0 and the rest of B is not visible in the original mechanism, so whether b0 is the current date/mean Sun pointer or a hypothetical true Sun pointer is unknown). The gear rides around the dial with the Moon, but is also geared to the Sun—the effect is to perform a differential gear operation, so the gear turns at the synodic month period, measuring in effect, the angle of the difference between the Sun and Moon pointers. The gear drives a small ball that appears through an opening in the Moon pointer's face, painted longitudinally half white and half black, displaying the phases pictorially. It turns with a modelled rotational period of 29.53 days; the modern value for the synodic month is 29.530589 days. The Metonic train is driven by the drive train b1, b2, l1, l2, m1, m2, and n1, which is connected to the pointer. The modelled rotational period of the pointer is the length of the 6939.5 days (over the whole five-rotation spiral), while the modern value for the Metonic cycle is 6939.69 days. The Olympiad train is driven by b1, b2, l1, l2, m1, m2, n1, n2, and o1, which mounts the pointer. It has a computed modelled rotational period of exactly four years, as expected. It is the only pointer on the mechanism that rotates anticlockwise; all of the others rotate clockwise. The Callippic train is driven by b1, b2, l1, l2, m1, m2, n1, n3, p1, p2, and q1, which mounts the pointer. It has a computed modelled rotational period of 27758 days, while the modern value is 27758.8 days. The Saros train is driven by b1, b2, l1, l2, m1, m3, e3, e4, f1, f2, and g1, which mounts the pointer. The modelled rotational period of the Saros pointer is 1646.3 days (in four rotations along the spiral pointer track); the modern value is 1646.33 days. The Exeligmos train is driven by b1, b2, l1, l2, m1, m3, e3, e4, f1, f2, g1, g2, h1, h2, and i1, which mounts the pointer. The modelled rotational period of the exeligmos pointer is 19,756 days; the modern value is 19755.96 days. It appears gears m3, n1-3, p1-2, and q1 did not survive in the wreckage. The functions of the pointers were deduced from the remains of the dials on the back face, and reasonable, appropriate gearage to fulfill the functions was proposed and is generally accepted. == Reconstruction efforts == === Proposed gear schemes === Because of the large space between the mean Sun gear and the front of the case and the size of and mechanical features on the mean Sun gear, it is very likely that the mechanism contained further gearing that either has been lost in or subsequent to the shipwreck, or was removed before being loaded onto the ship. This lack of evidence and nature of the front part of the mechanism has led to attempts to emulate what the Ancient Greeks would have done and because of the lack of evidence, many solutions have been put forward over the years. But as progress has been made on analyzing the internal structures and deciphering the inscriptions, earlier models have been ruled out and better models developed. Derek J. de Solla Price built a simple model in the 1970s. In 2002 Michael Wright designed and built the first workable model with the known mechanism and his emulation of a potential planetarium system. He suggested that along with the lunar anomaly, adjustments would have been made for the deeper, more basic solar anomaly (known as the "first anomaly"). He included pointers for this "true sun", Mercury, Venus, Mars, Jupiter, and Saturn, in addition to the known "mean sun" (current time) and lunar pointers. Evans, Carman, and Thorndike published a solution in 2010 with significant differences from Wright's. Their proposal centred on what they observed as irregular spacing of the inscriptions on the front dial face, which to them seemed to indicate an off-centre sun indicator arrangement; this would simplify the mechanism by removing the need to simulate the solar anomaly. They suggested that rather than accurate planetary indication (rendered impossible by the offset inscriptions) there would be simple dials for each individual planet, showing information such as key events in the cycle of planet, initial and final appearances in the night sky, and apparent direction changes. This system would lead to a much simplified gear system, with much reduced forces and complexity, as compared to Wright's model. Their proposal used simple meshed gear trains and accounted for the previously unexplained 63 toothed gear in fragment D. They proposed two face plate layouts, one with evenly spaced dials, and another with a gap in the top of the face, to account for criticism that they did not use the apparent fixtures on the b1 gear. They proposed that rather than bearings and pillars for gears and axles, they simply held weather and seasonal icons to be displayed through a window. In a paper published in 2012, Carman, Thorndike, and Evans also proposed a system of epicyclic gearing with pin and slot followers. Freeth and Jones published a proposal in 2012. They proposed a compact and feasible solution to the question of planetary indication. They also propose indicating the solar anomaly (that is, the sun's apparent position in the zodiac dial) on a separate pointer from the date pointer, which indicates the mean position of the Sun, as well as the date on the month dial. If the two dials are synchronised correctly, their front panel display is essentially the same as Wright's. Unlike Wright's model however, this model has not been built physically, and is only a 3-D computer model. The system to synthesise the solar anomaly is very similar to that used in Wright's proposal: three gears, one fixed in the centre of the b1 gear and attached to the Sun spindle, the second fixed on one of the spokes (in their proposal the one on the bottom left) acting as an idle gear, and the final positioned next to that one; the final gear is fitted with an offset pin and, over said pin, an arm with a slot that in turn, is attached to the sun spindle, inducing anomaly as the mean Sun wheel turns. The inferior planet mechanism includes the Sun (treated as a planet in this context), Mercury, and Venus. For each of the three systems, there is an epicyclic gear whose axis is mounted on b1, thus the basic frequency is the Earth year (as it is, in truth, for epicyclic motion in the Sun and all the planets—excepting only the Moon). Each meshes with a gear grounded to the mechanism frame. Each has a pin mounted, potentially on an extension of one side of the gear that enlarges the gear, but doesn't interfere with the teeth; in some cases, the needed distance between the gear's centre and the pin is farther than the radius of the gear itself. A bar with a slot along its length extends from the pin toward the appropriate coaxial tube, at whose other end is the object pointer, out in front of the front dials. The bars could have been full gears, although there is no need for the waste of metal, since the only working part is the slot. Also, using the bars avoids interference between the three mechanisms, each of which are set on one of the four spokes of b1. Thus there is one new grounded gear (one was identified in the wreckage, and the second is shared by two of the planets), one gear used to reverse the direction of the sun anomaly, three epicyclic gears and three bars/coaxial tubes/pointers, which would qualify as another gear each: five gears and three slotted bars in all. The superior planet systems—Mars, Jupiter, and Saturn—all follow the same general principle of the lunar anomaly mechanism. Similar to the inferior systems, each has a gear whose centre pivot is on an extension of b1, and which meshes with a grounded gear. It presents a pin and a centre pivot for the epicyclic gear which has a slot for the pin, and which meshes with a gear fixed to a coaxial tube and thence to the pointer. Each of the three mechanisms can fit within a quadrant of the b1 extension, and they are thus all on a single plane parallel with the front dial plate. Each one uses a ground gear, a driving gear, a driven gear, and a gear/coaxial tube/pointer, thus, twelve gears additional in all. In total, there are eight coaxial spindles of various nested sizes to transfer the rotations in the mechanism to the eight pointers. So in all, there are 30 original gears, seven gears added to complete calendar functionality, 17 gears and three slotted bars to support the six new pointers, for a grand total of 54 gears, three bars, and eight pointers in Freeth and Jones' design. On the visual representation Freeth provides, the pointers on the front zodiac dial have small, round identifying stones. He refers to a quote from an ancient papyrus: ...a voice comes to you speaking. Let the stars be set upon the board in accordance with [their] nature except for the Sun and Moon. And let the Sun be golden, the Moon silver, Kronos [Saturn] of obsidian, Ares [Mars] of reddish onyx, Aphrodite [Venus] lapis lazuli veined with gold, Hermes [Mercury] turquoise; let Zeus [Jupiter] be of (whitish?) stone, crystalline (?)... However, more recent discoveries and research have shown that the above models are not correct. In 2016, the numbers 462 and 442 were found in computed tomography scans of the inscriptions dealing with Venus and Saturn, respectively. These relate to the synodic cycles of these planets, and indicated that the mechanism was more accurate than previously thought. In 2018, based on the CT scans, the Antikythera Mechanism Research Project proposed changes in gearing and produced mechanical parts based on this. In March 2021, the Antikythera Research Team at University College London, led by Freeth, published a new proposed reconstruction of the entire Antikythera Mechanism. They were able to find gears that could be shared among the gear-trains for the different planets, by using rational approximations for the synodic cycles which have small prime factors, with the factors 7 and 17 being used for more than one planet. They conclude that none of the previous models "are at all compatible with all the currently known data", but their model is compatible with it. Freeth has directed a video explaining the discovery of the synodic cycle periods and the conclusions about how the mechanism worked. In 2025 researchers using a computer simulation of the mechanism that took into account the exact shape of the teeth in its gears, concluded that it would have jammed and stopped working after 4 months. === Accuracy === Investigations by Freeth and Jones reveal their simulated mechanism is inaccurate. The Mars pointer is up to 38° wrong in some instances (these inaccuracies occur at the nodal points of Mars' retrograde motion, and the error recedes at other locations in the orbit). This is not due to inaccuracies in gearing ratios in the mechanism, but inadequacies in the Greek theory of planetary movements. The accuracy could not have been improved until c. 160 AD when Ptolemy published his Almagest (particularly by adding the concept of the equant to his theory), then much later by the introduction of Kepler's laws of planetary motion in 1609 and 1619. In short, the Antikythera Mechanism was a machine designed to predict celestial phenomena according to the sophisticated astronomical theories current in its day, the sole witness to a lost history of brilliant engineering, a conception of pure genius, one of the great wonders of the ancient world—but it didn't really work very well! In addition to theoretical accuracy, there is the issue of mechanical accuracy. Freeth and Jones note that the inevitable "looseness" in the mechanism due to the hand-built gears, with their triangular teeth and the frictions between gears, and in bearing surfaces, probably would have swamped the finer solar and lunar correction mechanisms built into it: Though the engineering was remarkable for its era, recent research indicates that its design conception exceeded the engineering precision of its manufacture by a wide margin—with considerable cumulative inaccuracies in the gear trains, which would have cancelled out many of the subtle anomalies built into its design. While the device may have struggled with inaccuracies, due to the triangular teeth being hand-made, the calculations used and technology implemented to create the elliptical paths of the planets and retrograde motion of the Moon and Mars, by using a clockwork-type gear train with the addition of a pin-and-slot epicyclic mechanism, predated that of the first known clocks found in antiquity in medieval Europe, by more than 1000 years. Archimedes' development of the approximate value of pi and his theory of centres of gravity, along with the steps he made towards developing the calculus, suggest the Greeks had enough mathematical knowledge beyond that of Babylonian algebra, to model the elliptical nature of planetary motion. Of special delight to physicists, the Moon mechanism uses a special train of bronze gears, two of them linked with a slightly offset axis, to indicate the position and phase of the moon. As is known today from Kepler's laws of planetary motion, the moon travels at different speeds as it orbits the Earth, and this speed differential is modelled by the Antikythera Mechanism, even though the Ancient Greeks were not aware of the actual elliptical shape of the orbit. == Similar devices in ancient literature == The level of refinement of the mechanism indicates that the device was not unique, and possibly required expertise built over several generations. However, such artefacts were commonly melted down for the value of the bronze and rarely survive to the present day. === Roman world === Cicero's De re publica (54-51 BC), a first century BC philosophical dialogue, mentions two machines that some modern authors consider as some kind of planetarium or orrery, predicting the movements of the Sun, the Moon, and the five planets known at that time. They were both built by Archimedes and brought to Rome by the Roman general Marcus Claudius Marcellus after the death of Archimedes at the siege of Syracuse in 212 BC. Marcellus had great respect for Archimedes and one of these machines was the only item he kept from the siege (the second was placed in the Temple of Virtue). The device was kept as a family heirloom, and Cicero has Philus (one of the participants in a conversation that Cicero imagined had taken place in a villa belonging to Scipio Aemilianus in the year 129 BC) saying that Gaius Sulpicius Gallus (consul with Marcellus's nephew in 166 BC, and credited by Pliny the Elder as the first Roman to have written a book explaining solar and lunar eclipses) gave both a "learned explanation" and a working demonstration of the device. I had often heard this celestial globe or sphere mentioned on account of the great fame of Archimedes. Its appearance, however, did not seem to me particularly striking. There is another, more elegant in form, and more generally known, moulded by the same Archimedes, and deposited by the same Marcellus, in the Temple of Virtue at Rome. But as soon as Gallus had begun to explain, by his sublime science, the composition of this machine, I felt that the Sicilian geometrician must have possessed a genius superior to any thing we usually conceive to belong to our nature. Gallus assured us, that the solid and compact globe, was a very ancient invention, and that the first model of it had been presented by Thales of Miletus. That afterwards Eudoxus of Cnidus, a disciple of Plato, had traced on its surface the stars that appear in the sky, and that many years subsequent, borrowing from Eudoxus this beautiful design and representation, Aratus had illustrated them in his verses, not by any science of astronomy, but the ornament of poetic description. He added, that the figure of the sphere, which displayed the motions of the Sun and Moon, and the five planets, or wandering stars, could not be represented by the primitive solid globe. And that in this, the invention of Archimedes was admirable, because he had calculated how a single revolution should maintain unequal and diversified progressions in dissimilar motions. When Gallus moved this globe, it showed the relationship of the Moon with the Sun, and there were exactly the same number of turns on the bronze device as the number of days in the real globe of the sky. Thus it showed the same eclipse of the Sun as in the globe [of the sky], as well as showing the Moon entering the area of the Earth's shadow when the Sun is in line ... [missing text] [i.e. It showed both solar and lunar eclipses.] Pappus of Alexandria (290 – c. 350 AD) stated that Archimedes had written a now lost manuscript on the construction of these devices titled On Sphere-Making. The surviving texts from ancient times describe many of his creations, some even containing simple drawings. One such device is his odometer, the exact model later used by the Romans to place their mile markers (described by Vitruvius, Heron of Alexandria and in the time of Emperor Commodus). The drawings in the text appeared functional, but attempts to build them as pictured had failed. When the gears pictured, which had square teeth, were replaced with gears of the type in the Antikythera mechanism, which were angled, the device was perfectly functional. If Cicero's account is correct, then this technology existed as early as the third century BC. Archimedes' device is also mentioned by later Roman era writers such as Lactantius (Divinarum Institutionum Libri VII), Claudian (In sphaeram Archimedes), and Proclus (Commentary on the first book of Euclid's Elements of Geometry) in the fourth and fifth centuries. Cicero also said that another such device was built "recently" by his friend Posidonius, "... each one of the revolutions of which brings about the same movement in the Sun and Moon and five wandering stars [planets] as is brought about each day and night in the heavens ..." It is unlikely that any one of these machines was the Antikythera mechanism found in the shipwreck since both the devices fabricated by Archimedes and mentioned by Cicero were located in Rome at least 30 years later than the estimated date of the shipwreck, and the third device was almost certainly in the hands of Posidonius by that date. The scientists who have reconstructed the Antikythera mechanism also agree that it was too sophisticated to have been a unique device. === Eastern Mediterranean and others === This evidence that the Antikythera mechanism was not unique adds support to the idea that there was an ancient Greek tradition of complex mechanical technology that was later, at least in part, transmitted to the Byzantine and Islamic worlds, where mechanical devices which were complex, albeit simpler than the Antikythera mechanism, were built during the Middle Ages. Fragments of a geared calendar attached to a sundial, from the fifth or sixth century Byzantine Empire, have been found; the calendar may have been used to assist in telling time. In the Islamic world, Banū Mūsā's Kitab al-Hiyal, or Book of Ingenious Devices, was commissioned by the Caliph of Baghdad in the early 9th century AD. This text described over a hundred mechanical devices, some of which may date back to ancient Greek texts preserved in monasteries. A geared calendar similar to the Byzantine device was described by the scientist al-Biruni around 1000, and a surviving 13th-century astrolabe also contains a similar clockwork device. It is possible that this medieval technology may have been transmitted to Europe and contributed to the development of mechanical clocks there. In the 11th century, Chinese polymath Su Song constructed a mechanical clock tower that told (among other measurements) the position of some stars and planets, which were shown on a mechanically rotated armillary sphere. == Popular culture and museum replicas == Several exhibitions have been staged worldwide, leading to the main "Antikythera shipwreck" exhibition at the National Archaeological Museum in Athens. As of 2012, the Antikythera mechanism was displayed as part of a temporary exhibition about the Antikythera shipwreck, accompanied by reconstructions made by Ioannis Theofanidis, Derek de Solla Price, Michael Wright, the Thessaloniki University and Dionysios Kriaris. Other reconstructions are on display at the American Computer Museum in Bozeman, Montana, at the Children's Museum of Manhattan in New York, at Astronomisch-Physikalisches Kabinett in Kassel, Germany, at the Archimedes Museum in Olympia, Greece, at the Kotsanas Museum of Ancient Greek Technology in Athens and at the Musée des Arts et Métiers in Paris. The National Geographic documentary series Naked Science dedicated an episode to the Antikythera Mechanism entitled "Star Clock BC" that aired on 20 January 2011. A documentary, The World's First Computer, was produced in 2012 by the Antikythera mechanism researcher and film-maker Tony Freeth. In 2012, BBC Four aired The Two-Thousand-Year-Old Computer; it was also aired on 3 April 2013 in the United States on NOVA, the PBS science series, under the name Ancient Computer. It documents the discovery and 2005 investigation of the mechanism by the Antikythera Mechanism Research Project. A functioning Lego reconstruction of the Antikythera mechanism was built in 2010 by hobbyist Andy Carol, and featured in a short film produced by Small Mammal in 2011. On 17 May 2017, Google marked the 115th anniversary of the discovery with a Google Doodle. The YouTube channel Clickspring documents the creation of an Antikythera mechanism replica using the tools, techniques of machining and metallurgy, and materials that would have been available in ancient Greece, along with investigations into the possible technologies of the era. The film Indiana Jones and the Dial of Destiny (2023) features a plot around a fictionalized version of the mechanism (also referred to as Archimedes' Dial, the titular Dial of Destiny). In the film, the device was built by Archimedes as a temporal mapping system, and sought by a former Nazi scientist as a way to detect time portals in order to travel back in time and help Germany win World War II. A major plot point revolves around the fact that the device did not take Continental drift into account as the theory was unknown in Archimedes' time. On 8 February 2024, a 10X scale replica of the mechanism was built, installed, and inaugurated at the University of Sonora in Hermosillo, Sonora, Mexico. With the name of Monumental Antikythera Mechanism for Hermosillo (MAMH), Dr. Alfonso performed the inauguration. Also attending were Durazo Montaño, Governor of Sonora and Dr. Maria Rita Plancarte Martinez, Chancellor of the Universidad de Sonora, the Ambassador of Greece, Nikolaos Koutrokois, and a delegation from the Embassy. == See also == Ancient technology Archimedes Palimpsest Astrarium Automaton Baghdad Battery Ctesibius Reverse engineering Out-of-place artifact == References == == Further reading == == External links == Bragg, Melvyn (November 2024). "The Antikythera mechanism". BBC, In Our Time programme. New Antikythera mechanism analysis challenges century-old assumption - Arstechnica - Jennifer Ouellette - 7/10/2024 Weibel, Thomas. "The Antikythera Mechanism". Animated model of the Antikythera mechanism in virtual reality. Asimakopoulos, Fivos. "3D model simulation". Manos Roumeliotis's Simulation and Animation of the Antikythera Mechanism page. The Antikythera Mechanism Research Project. The Antikythera Mechanism Research Project. "Videos". YouTube. Retrieved 24 July 2017. "The Antikythera Mechanism Exhibitions". National Hellenic Research Foundation. Archived from the original on 23 April 2012. YAAS – A 3D interactive virtual reality simulator in VRML Wright, M.; Vicentini, M. (25 August 2009). "Virtual Reconstruction of the Antikythera Mechanism". Heritage Key. Archived from the original on 7 November 2021 – via YouTube. Metapage with links December 2021. at antikythera.org Bronze replica 3D engineering manufacturing drawings and operating manual
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Wikipedia:Antilimit#0
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In mathematics, the antilimit is the equivalent of a limit for a divergent series. The concept not necessarily unique or well-defined, but the general idea is to find a formula for a series and then evaluate it outside its radius of convergence. == Common divergent series == == See also == Abel summation Cesàro summation Lindelöf summation Euler summation Borel summation Mittag-Leffler summation Lambert summation Euler–Boole summation and Van Wijngaarden transformation can also be used on divergent series == References == Shanks, Daniel (1949). "An Analogy Between Transients and Mathematical Sequences and Some Nonlinear Sequence-to-Sequence Transforms Suggested by It. Part 1" (PDF). Naval Ordnance Lab White Oak Md. Sidi, Avram (February 2010). Practical Extrapolation Methods. Cambridge University Press. p. 542. doi:10.1017/CBO9780511546815. ISBN 9780511546815.
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Wikipedia:Antilinear map#0
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In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces is said to be antilinear or conjugate-linear if f ( x + y ) = f ( x ) + f ( y ) (additivity) f ( s x ) = s ¯ f ( x ) (conjugate homogeneity) {\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) }}\\f(sx)&={\overline {s}}f(x)&&\qquad {\text{ (conjugate homogeneity) }}\\\end{alignedat}}} hold for all vectors x , y ∈ V {\displaystyle x,y\in V} and every complex number s , {\displaystyle s,} where s ¯ {\displaystyle {\overline {s}}} denotes the complex conjugate of s . {\displaystyle s.} Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity. Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces. == Definitions and characterizations == A function is called antilinear or conjugate linear if it is additive and conjugate homogeneous. An antilinear functional on a vector space V {\displaystyle V} is a scalar-valued antilinear map. A function f {\displaystyle f} is called additive if f ( x + y ) = f ( x ) + f ( y ) for all vectors x , y {\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all vectors }}x,y} while it is called conjugate homogeneous if f ( a x ) = a ¯ f ( x ) for all vectors x and all scalars a . {\displaystyle f(ax)={\overline {a}}f(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.} In contrast, a linear map is a function that is additive and homogeneous, where f {\displaystyle f} is called homogeneous if f ( a x ) = a f ( x ) for all vectors x and all scalars a . {\displaystyle f(ax)=af(x)\quad {\text{ for all vectors }}x{\text{ and all scalars }}a.} An antilinear map f : V → W {\displaystyle f:V\to W} may be equivalently described in terms of the linear map f ¯ : V → W ¯ {\displaystyle {\overline {f}}:V\to {\overline {W}}} from V {\displaystyle V} to the complex conjugate vector space W ¯ . {\displaystyle {\overline {W}}.} === Examples === ==== Anti-linear dual map ==== Given a complex vector space V {\displaystyle V} of rank 1, we can construct an anti-linear dual map which is an anti-linear map l : V → C {\displaystyle l:V\to \mathbb {C} } sending an element x 1 + i y 1 {\displaystyle x_{1}+iy_{1}} for x 1 , y 1 ∈ R {\displaystyle x_{1},y_{1}\in \mathbb {R} } to x 1 + i y 1 ↦ a 1 x 1 − i b 1 y 1 {\displaystyle x_{1}+iy_{1}\mapsto a_{1}x_{1}-ib_{1}y_{1}} for some fixed real numbers a 1 , b 1 . {\displaystyle a_{1},b_{1}.} We can extend this to any finite dimensional complex vector space, where if we write out the standard basis e 1 , … , e n {\displaystyle e_{1},\ldots ,e_{n}} and each standard basis element as e k = x k + i y k {\displaystyle e_{k}=x_{k}+iy_{k}} then an anti-linear complex map to C {\displaystyle \mathbb {C} } will be of the form ∑ k x k + i y k ↦ ∑ k a k x k − i b k y k {\displaystyle \sum _{k}x_{k}+iy_{k}\mapsto \sum _{k}a_{k}x_{k}-ib_{k}y_{k}} for a k , b k ∈ R . {\displaystyle a_{k},b_{k}\in \mathbb {R} .} ==== Isomorphism of anti-linear dual with real dual ==== The anti-linear dualpg 36 of a complex vector space V {\displaystyle V} Hom C ¯ ( V , C ) {\displaystyle \operatorname {Hom} _{\overline {\mathbb {C} }}(V,\mathbb {C} )} is a special example because it is isomorphic to the real dual of the underlying real vector space of V , {\displaystyle V,} Hom R ( V , R ) . {\displaystyle {\text{Hom}}_{\mathbb {R} }(V,\mathbb {R} ).} This is given by the map sending an anti-linear map ℓ : V → C {\displaystyle \ell :V\to \mathbb {C} } to Im ( ℓ ) : V → R {\displaystyle \operatorname {Im} (\ell ):V\to \mathbb {R} } In the other direction, there is the inverse map sending a real dual vector λ : V → R {\displaystyle \lambda :V\to \mathbb {R} } to ℓ ( v ) = − λ ( i v ) + i λ ( v ) {\displaystyle \ell (v)=-\lambda (iv)+i\lambda (v)} giving the desired map. == Properties == The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps. == Anti-dual space == The vector space of all antilinear forms on a vector space X {\displaystyle X} is called the algebraic anti-dual space of X . {\displaystyle X.} If X {\displaystyle X} is a topological vector space, then the vector space of all continuous antilinear functionals on X , {\displaystyle X,} denoted by X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} is called the continuous anti-dual space or simply the anti-dual space of X {\displaystyle X} if no confusion can arise. When H {\displaystyle H} is a normed space then the canonical norm on the (continuous) anti-dual space X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} denoted by ‖ f ‖ X ¯ ′ , {\textstyle \|f\|_{{\overline {X}}^{\prime }},} is defined by using this same equation: ‖ f ‖ X ¯ ′ := sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | for every f ∈ X ¯ ′ . {\displaystyle \|f\|_{{\overline {X}}^{\prime }}~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every }}f\in {\overline {X}}^{\prime }.} This formula is identical to the formula for the dual norm on the continuous dual space X ′ {\displaystyle X^{\prime }} of X , {\displaystyle X,} which is defined by ‖ f ‖ X ′ := sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | for every f ∈ X ′ . {\displaystyle \|f\|_{X^{\prime }}~:=~\sup _{\|x\|\leq 1,x\in X}|f(x)|\quad {\text{ for every }}f\in X^{\prime }.} Canonical isometry between the dual and anti-dual The complex conjugate f ¯ {\displaystyle {\overline {f}}} of a functional f {\displaystyle f} is defined by sending x ∈ domain f {\displaystyle x\in \operatorname {domain} f} to f ( x ) ¯ . {\textstyle {\overline {f(x)}}.} It satisfies ‖ f ‖ X ′ = ‖ f ¯ ‖ X ¯ ′ and ‖ g ¯ ‖ X ′ = ‖ g ‖ X ¯ ′ {\displaystyle \|f\|_{X^{\prime }}~=~\left\|{\overline {f}}\right\|_{{\overline {X}}^{\prime }}\quad {\text{ and }}\quad \left\|{\overline {g}}\right\|_{X^{\prime }}~=~\|g\|_{{\overline {X}}^{\prime }}} for every f ∈ X ′ {\displaystyle f\in X^{\prime }} and every g ∈ X ¯ ′ . {\textstyle g\in {\overline {X}}^{\prime }.} This says exactly that the canonical antilinear bijection defined by Cong : X ′ → X ¯ ′ where Cong ( f ) := f ¯ {\displaystyle \operatorname {Cong} ~:~X^{\prime }\to {\overline {X}}^{\prime }\quad {\text{ where }}\quad \operatorname {Cong} (f):={\overline {f}}} as well as its inverse Cong − 1 : X ¯ ′ → X ′ {\displaystyle \operatorname {Cong} ^{-1}~:~{\overline {X}}^{\prime }\to X^{\prime }} are antilinear isometries and consequently also homeomorphisms. If F = R {\displaystyle \mathbb {F} =\mathbb {R} } then X ′ = X ¯ ′ {\displaystyle X^{\prime }={\overline {X}}^{\prime }} and this canonical map Cong : X ′ → X ¯ ′ {\displaystyle \operatorname {Cong} :X^{\prime }\to {\overline {X}}^{\prime }} reduces down to the identity map. Inner product spaces If X {\displaystyle X} is an inner product space then both the canonical norm on X ′ {\displaystyle X^{\prime }} and on X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} satisfies the parallelogram law, which means that the polarization identity can be used to define a canonical inner product on X ′ {\displaystyle X^{\prime }} and also on X ¯ ′ , {\displaystyle {\overline {X}}^{\prime },} which this article will denote by the notations ⟨ f , g ⟩ X ′ := ⟨ g ∣ f ⟩ X ′ and ⟨ f , g ⟩ X ¯ ′ := ⟨ g ∣ f ⟩ X ¯ ′ {\displaystyle \langle f,g\rangle _{X^{\prime }}:=\langle g\mid f\rangle _{X^{\prime }}\quad {\text{ and }}\quad \langle f,g\rangle _{{\overline {X}}^{\prime }}:=\langle g\mid f\rangle _{{\overline {X}}^{\prime }}} where this inner product makes X ′ {\displaystyle X^{\prime }} and X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} into Hilbert spaces. The inner products ⟨ f , g ⟩ X ′ {\textstyle \langle f,g\rangle _{X^{\prime }}} and ⟨ f , g ⟩ X ¯ ′ {\textstyle \langle f,g\rangle _{{\overline {X}}^{\prime }}} are antilinear in their second arguments. Moreover, the canonical norm induced by this inner product (that is, the norm defined by f ↦ ⟨ f , f ⟩ X ′ {\textstyle f\mapsto {\sqrt {\left\langle f,f\right\rangle _{X^{\prime }}}}} ) is consistent with the dual norm (that is, as defined above by the supremum over the unit ball); explicitly, this means that the following holds for every f ∈ X ′ : {\displaystyle f\in X^{\prime }:} sup ‖ x ‖ ≤ 1 , x ∈ X | f ( x ) | = ‖ f ‖ X ′ = ⟨ f , f ⟩ X ′ = ⟨ f ∣ f ⟩ X ′ . {\displaystyle \sup _{\|x\|\leq 1,x\in X}|f(x)|=\|f\|_{X^{\prime }}~=~{\sqrt {\langle f,f\rangle _{X^{\prime }}}}~=~{\sqrt {\langle f\mid f\rangle _{X^{\prime }}}}.} If X {\displaystyle X} is an inner product space then the inner products on the dual space X ′ {\displaystyle X^{\prime }} and the anti-dual space X ¯ ′ , {\textstyle {\overline {X}}^{\prime },} denoted respectively by ⟨ ⋅ , ⋅ ⟩ X ′ {\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{X^{\prime }}} and ⟨ ⋅ , ⋅ ⟩ X ¯ ′ , {\textstyle \langle \,\cdot \,,\,\cdot \,\rangle _{{\overline {X}}^{\prime }},} are related by ⟨ f ¯ | g ¯ ⟩ X ¯ ′ = ⟨ f | g ⟩ X ′ ¯ = ⟨ g | f ⟩ X ′ for all f , g ∈ X ′ {\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{{\overline {X}}^{\prime }}={\overline {\langle \,f\,|\,g\,\rangle _{X^{\prime }}}}=\langle \,g\,|\,f\,\rangle _{X^{\prime }}\qquad {\text{ for all }}f,g\in X^{\prime }} and ⟨ f ¯ | g ¯ ⟩ X ′ = ⟨ f | g ⟩ X ¯ ′ ¯ = ⟨ g | f ⟩ X ¯ ′ for all f , g ∈ X ¯ ′ . {\displaystyle \langle \,{\overline {f}}\,|\,{\overline {g}}\,\rangle _{X^{\prime }}={\overline {\langle \,f\,|\,g\,\rangle _{{\overline {X}}^{\prime }}}}=\langle \,g\,|\,f\,\rangle _{{\overline {X}}^{\prime }}\qquad {\text{ for all }}f,g\in {\overline {X}}^{\prime }.} == See also == Cauchy's functional equation – Functional equation Complex conjugate – Fundamental operation on complex numbers Complex conjugate vector space – Mathematics conceptPages displaying short descriptions of redirect targets Fundamental theorem of Hilbert spaces Inner product space – Vector space with generalized dot product Linear map – Mathematical function, in linear algebra Matrix consimilarity Riesz representation theorem – Theorem about the dual of a Hilbert space Sesquilinear form – Generalization of a bilinear form Time reversal – Time reversal symmetry in physics == Citations == == References == Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3). Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinear maps are discussed in section 4.6). Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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Wikipedia:Antiunitary operator#0
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In mathematics, an antiunitary transformation is a bijective antilinear map U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}\,} between two complex Hilbert spaces such that ⟨ U x , U y ⟩ = ⟨ x , y ⟩ ¯ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}} for all x {\displaystyle x} and y {\displaystyle y} in H 1 {\displaystyle H_{1}} , where the horizontal bar represents the complex conjugate. If additionally one has H 1 = H 2 {\displaystyle H_{1}=H_{2}} then U {\displaystyle U} is called an antiunitary operator. Antiunitary operators are important in quantum mechanics because they are used to represent certain symmetries, such as time reversal. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem. == Invariance transformations == In quantum mechanics, the invariance transformations of complex Hilbert space H {\displaystyle H} leave the absolute value of scalar product invariant: | ⟨ T x , T y ⟩ | = | ⟨ x , y ⟩ | {\displaystyle |\langle Tx,Ty\rangle |=|\langle x,y\rangle |} for all x {\displaystyle x} and y {\displaystyle y} in H {\displaystyle H} . Due to Wigner's theorem these transformations can either be unitary or antiunitary. === Geometric Interpretation === Congruences of the plane form two distinct classes. The first conserves the orientation and is generated by translations and rotations. The second does not conserve the orientation and is obtained from the first class by applying a reflection. On the complex plane these two classes correspond (up to translation) to unitaries and antiunitaries, respectively. == Properties == ⟨ U x , U y ⟩ = ⟨ x , y ⟩ ¯ = ⟨ y , x ⟩ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle } holds for all elements x , y {\displaystyle x,y} of the Hilbert space and an antiunitary U {\displaystyle U} . When U {\displaystyle U} is antiunitary then U 2 {\displaystyle U^{2}} is unitary. This follows from ⟨ U 2 x , U 2 y ⟩ = ⟨ U x , U y ⟩ ¯ = ⟨ x , y ⟩ . {\displaystyle \left\langle U^{2}x,U^{2}y\right\rangle ={\overline {\langle Ux,Uy\rangle }}=\langle x,y\rangle .} For unitary operator V {\displaystyle V} the operator V K {\displaystyle VK} , where K {\displaystyle K} is complex conjugation (with respect to some orthogonal basis), is antiunitary. The reverse is also true, for antiunitary U {\displaystyle U} the operator U K {\displaystyle UK} is unitary. For antiunitary U {\displaystyle U} the definition of the adjoint operator U ∗ {\displaystyle U^{*}} is changed to compensate the complex conjugation, becoming ⟨ U x , y ⟩ = ⟨ x , U ∗ y ⟩ ¯ . {\displaystyle \langle Ux,y\rangle ={\overline {\left\langle x,U^{*}y\right\rangle }}.} The adjoint of an antiunitary U {\displaystyle U} is also antiunitary and U U ∗ = U ∗ U = 1. {\displaystyle UU^{*}=U^{*}U=1.} (This is not to be confused with the definition of unitary operators, as the antiunitary operator U {\displaystyle U} is not complex linear.) == Examples == The complex conjugation operator K , {\displaystyle K,} K z = z ¯ , {\displaystyle Kz={\overline {z}},} is an antiunitary operator on the complex plane. The operator U = i σ y K = ( 0 1 − 1 0 ) K , {\displaystyle U=i\sigma _{y}K={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}K,} where σ y {\displaystyle \sigma _{y}} is the second Pauli matrix and K {\displaystyle K} is the complex conjugation operator, is antiunitary. It satisfies U 2 = − 1 {\displaystyle U^{2}=-1} . == Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitaries == An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries W θ {\displaystyle W_{\theta }} , 0 ≤ θ ≤ π {\displaystyle 0\leq \theta \leq \pi } . The operator W 0 : C → C {\displaystyle W_{0}:\mathbb {C} \to \mathbb {C} } is just simple complex conjugation on C {\displaystyle \mathbb {C} } W 0 ( z ) = z ¯ {\displaystyle W_{0}(z)={\overline {z}}} For 0 < θ ≤ π {\displaystyle 0<\theta \leq \pi } , the operator W θ {\displaystyle W_{\theta }} acts on two-dimensional complex Hilbert space. It is defined by W θ ( ( z 1 , z 2 ) ) = ( e i 2 θ z 2 ¯ , e − i 2 θ z 1 ¯ ) . {\displaystyle W_{\theta }\left(\left(z_{1},z_{2}\right)\right)=\left(e^{{\frac {i}{2}}\theta }{\overline {z_{2}}},\;e^{-{\frac {i}{2}}\theta }{\overline {z_{1}}}\right).} Note that for 0 < θ ≤ π {\displaystyle 0<\theta \leq \pi } W θ ( W θ ( ( z 1 , z 2 ) ) ) = ( e i θ z 1 , e − i θ z 2 ) , {\displaystyle W_{\theta }\left(W_{\theta }\left(\left(z_{1},z_{2}\right)\right)\right)=\left(e^{i\theta }z_{1},e^{-i\theta }z_{2}\right),} so such W θ {\displaystyle W_{\theta }} may not be further decomposed into W 0 {\displaystyle W_{0}} 's, which square to the identity map. Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces. == References == Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412 Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp.414–416 == See also == Unitary operator Wigner's Theorem Particle physics and representation theory
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Wikipedia:Antivector#0
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An antivector is an element of grade n − 1 in an n-dimensional exterior algebra. An antivector is always a blade, and it gets its name from the fact that its components each involve a combination of all except one basis vector, thus being the opposite of a vector whose components each involve exactly one basis vector. Like a vector, an antivector has n components in n-dimensional space, and this sometimes leads to an inadequate distinction being made between the two types of entities. However, antivectors transform differently with a change of basis than vectors do, which shows that they are different kinds of quantities. In physics, the names pseudovector and axial vector are used to describe vectors that transform in the same way that an antivector transforms. These typically arise as the result of cross products between two vectors. == See also == Exterior algebra Geometric algebra == References ==
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Wikipedia:Antoine André Louis Reynaud#0
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Antoine André Louis Reynaud, 12 September 1771 – 24 February 1844, was a French mathematician. He was a Knight of the Legion of Honour and examiner at the École Polytechnique. == Works == Trigonométrie rectiligne et sphérique. Paris: Courcier. 1818. == See also == Euclidean algorithm == External links == O'Connor, John J.; Robertson, Edmund F., "Antoine-André-Louis Reynaud", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Antoine Meyer#0
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Antoine Meyer, also known as Antun or Tun Meyer (1801–1857) was a Luxembourg-born mathematician and poet who later adopted Belgian nationality. Sometimes referred to as the father of Luxembourgish literature, he is remembered for publishing the very first book in Luxembourgish, a collection of six poems entitled "E' Schrek ob de' lezeburger Parnassus" (A Step up the Luxembourg Parnassus). == Early life == Born on 31 May 1801 in Luxembourg City, Meyer was the son of Hubert Meyer, a shoemaker, and his wife Elisabeth Kirschenbilder who lived in the centre of the old town close to the Place d'Armes. After completing his secondary school education with flying colours at the local Athénée, he studied mathematics at Liège (1817–1823) where he was forced to give private lessons to his fellow students and help out in the library in order to pay for his studies. After receiving his doctorate, he spent an additional year in Paris studying at the Collège de France and at the Sorbonne where he came into contact with Europe's leading mathematicians. == Career == Professionally, Meyer was a brilliant mathematics teacher. In 1826, he taught at the Collège royal at Echternach in Luxembourg before moving to Breda in the Netherlands in 1828 where he worked at the newly opened Royal Military Academy. However, when two years later the Belgians rose up against the Dutch, he had to leave the country. After enormous difficulties, he finally succeeded in finding a job in Belgium at a school in Louvain. He spent a short period at the Institut Gaggia (1834) in Brussels but was then offered a post at the military school before going on to the Université libre de Bruxelles in 1838. In 1849, he became a professor of higher mathematics at the Université de Liège until his death in 1857. For historical reasons, after the separation of Belgian Luxembourg, Meyer acquired Belgian nationality in 1842. The strong recognition he received for his mathematical publications and achievements testify to his full adoption by the Belgians. However, his poems show his lifelong attachment to Luxembourg. == Poetry == Antoine Meyer's publication of "E' Schrek ob de' Lezeburger Parnassus" in 1829 was not received with very much enthusiasm. There were however one or two strong supporters. Félix Thyes, who coincidentally was the first Luxembourger to publish a book in French, commented: "It is Monsieur Antoine Meyer, a mathematics professor at the University of Liège, who has the honour of being the first to rescue this tongue from the indifference and scorn in which it is immersed, creating, as it were, a new literature. The good Luxembourgers were amazed, one morning, when they heard that the learned mathematician had just published a small volume of poems in their own language." The book contains six poems, a love poem: "Uen d'Christine" (To Christine), a meditation on the romantic subject of night: "D'Nuecht" (The Night), a kind of real life painting: "Een Abléck an engem Wiertshaus zu Lëtzebuerg" (A Moment in a Luxembourg Inn), and three fables: "D'porzelains an d'ierde Schierbel" (The Shard of Porcelain and the Earthen Pot), "D'Spéngel an d'Nol" (The Pin and the Needle) and "D'Flou an de Pierdskrécher" (The Fly and the Horse Trough). In regard to the fables, while Aesop and La Fontaine built their stories around animals, Meyer personified inanimate objects. For example, in "D'Spéngel an d'Nol", the well-to-do Miss Needle tries but fails to override the Pin, reflecting the failure of the French aristocracy to prevent the French Revolution. The mathematician went on to write three more poetic works as well as a booklet on spelling rules for Luxembourgish. A number of other poems by Antoine Meyer were published in the press. In his own words, Meyer's objective was to show that "the Luxembourg dialect is not as rough, poor, unregulated, stiff and barbarous as many born Luxembourgers would like to maintain". Félix Thyes commented: "We see in Mr Meyer, that noble pride of the plebeian, that instinct for liberty and often that burning concern for the poorer classes which can be found among all true poets of our times." It is now recognized that he succeeded in initiating the transformation of Luxembourgish into an acceptable literary language. Antoine Meyer died in Liège on 29 April 1857. == Works == 1829: "E' Schrek ob de' lezeburger Parnassus". Collection of poems published in Luxembourg (Lezeburg) 1832: "Jong vum Schrek op de Lezeburger Parnassus". Collection of poems published in Louvain 1845: "Luxemburgische Gedichte und Fabeln". Together with works by Heinrich Gloden. Published in Brussels 1853: "Oilzegt-Kläng". Collection of poems published in Liège (Lüttich) 1854: "Règelbüchelchen vum lezeburger Orthoegraf". Rules of Luxembourg Spelling. Published in Liège == References ==
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Wikipedia:Antoine Song#0
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Antoine Song (born 18 July 1992 in Paris) is a French mathematician whose research concerns differential geometry. In 2018, he proved Yau's conjecture. He was a Clay Research Fellow (2019–2024). He obtained his Ph.D. from Princeton University in 2019 under the supervision of Fernando Codá Marques. He is an assistant professor of mathematics at Caltech. He is a Sloan Fellow. In 2023, together with Conghan Dong, he proved a conjecture from 2001 by G. Huisken and T. Ilmanen on the mathematics of general relativity, about the curvature in spaces with very little mass. He delivered the 2021–2022 Peccot Lectures (in 2022, due to the coronavirus pandemic). == Existence of minimal surfaces == It is known that any closed surface possesses infinitely many closed geodesics. The first problem in the minimal submanifolds section of Yau's list asks whether any closed three-manifold has infinitely many closed smooth immersed minimal surfaces. At the time it was known from Almgren–Pitts min-max theory the existence of at least one minimal surface. Kei Irie, Fernando Codá Marques, and André Neves solved this problem in the generic case and later Antoine Song proved it in full generality. == Selected publications == "Existence of infinitely many minimal hypersurfaces in closed manifolds" (2018), Annals of Mathematics Joint with Marques and Neves: "Equidistribution of minimal hypersurfaces for generic metrics" (2019), Inventiones mathematicae Joint with Conghan Dong: "Stability of Euclidean 3-space for the positive mass theorem" (2023), Inventiones mathematicae == References == == External link == The Geometry of Minimal Surfaces: An Interview with Antoine Song (October 13, 2023)
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Wikipedia:Antoinette Tordesillas#0
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Antoinette A. Tordesillas is an Australian mathematician. She is a professor of applied mathematics at the University of Melbourne, Australia who has helped build a foundational understanding of the dynamics of granular materials. She received the J H Michell Medal in 2000 and her major contributions include research predicting the response of extraterrestrial soil to attempts to build, mine, or drill and a model that can identify the location and time of future landslides or earthquakes by analyzing slope stability changes. == Education and career == Tordesillas attended the University of Adelaide where she majored in applied mathematics and physical and inorganic chemistry and earned a B.S. in 1986. Her honours thesis in applied mathematics involved the development of a model of the hot-dip galvanising process for creating sheet metal. She earned her Ph.D. in solid mechanics in 1992 from the University of Wollongong with a dissertation involving the contact mechanics of roller coating supervised by James Murray Hill. After temporary positions at the University of Colorado at Boulder and Kansas State University, she joined the University of Melbourne department of mathematics and statistics in 1996. She took on a joint position in geomechanics there in 2013 and was promoted to full professor in 2016. At the University of Melbourne, Tordesillas teaches as the senior mathematics and statistics lecturer, heads the Micromechanics of Granular Media Group, and conducts research across the fields of mathematics, engineering, physics, and geophysics. This research has involved international collaborations, multidisciplinary teams, and large-scale projects funded by various international agencies including NASA, the Hong Kong Research Council, the US National Science Foundation, and the US Department of Defense. Much of her research centers around understanding the dynamics of granular materials and applications of this pursuit including predicting seismic activity and preparing for future space travel. == Research == === Understanding martian and lunar soil === Tordesillas, in collaboration with NASA, led a team studying the soil of Mars and the moon with the aim of understanding how their surfaces would respond to attempts to build, mine, or drill. NASA approached Tordesillas at the recommendation of the US Army, who named her as the authority to consult about sand. To tackle the project, Tordesillas and her team at the University of Melbourne used data about extraterrestrial soil and photos collected by orbiters and rovers in conjunction with a study of granule dynamics. This approach involved testing simulated space soil, computer modeling the effects of added pressure, and considering simpler models of highly idealized particles adjusted for differences in the strength of gravity. Tordesillas contemplated highly idealized particles that were round and uniform while questioning how the unique shapes observed on Mars and the moon formed. She recognized that many different local conditions would need to be understood prior to any future landings. She also noted the potential application of this research to combat problems arising from the unpredictable nature of stored granular materials.These materials include the products of important Australian export industries such as wheat, iron ore, and coal. === Predicting seismic activity === Tordesillas received a $750,000 grant from the Australian research fund and the US Army Research Office to develop a model that generates a high-resolution picture of individual granules and allows the visualization of the shear-band microstructures that precede disruptions in granular materials. This represents an important step towards predicting earthquakes and mitigating soil erosion from heavy off-road military vehicles. Tordesillas and her team developed a software tool using applied mathematics and big data analytics to predict the time and location of landslides up to two weeks in advance. This model analyzes large quantities of data to identify sites of future failures, or sites of seismic activity. It flags locations where patterns of motion become ordered because, preceding seismic activity, particles begin moving in similar ways approaching the site of what will become a failure. It uses the expanding capabilities of computer programming and memory to decode big data and convert it into a network conducive to the recognition of hidden patterns. The early detection of subtle changes is key to predicting failures and allows existing data to inform risk assessment and management. This effort culminated after five years of work when Tordesillas and Robin Batterham developed, tested, and patented the Spatiotemporal Slope Stability Analytics for Failure Estimate (SSAFE) model. This model analyzes slope stability data over time to predict the time and location of future failures, combining remote seismic data with the physics of granular failure. The model can be used to predict failures in mines where precise measurements concerning the movement along rock faces. It can also monitor rural areas where satellites collect radar data every few days or weeks, but its ultimate goal is improving early warning systems and mitigating the dangers of landslides in the context of climate change. === Comminution and the removal of liquid from a material === On July 13, 2012, Tordesillas filed a patent application with Peter Joseph Scales, Anthony Dirk Stickland, Robin John Battheram, and the University of Melbourne for the comminution and/or removal of liquid from a material. The World Intellectual Property Organization published that this patent covers a material processing method developed by the four scientists in which a material is fed between oppositely moving surfaces with the result of shearing the material parallel to its direction of flow between the two surfaces. == Awards == Tordesillas was awarded the J H Michell Medal in 2000 by the Australian and New Zealand Industrial and Applied Mathematics Society. This is an annual award for an outstanding new researcher in applied mathematics. == Publications == According to the University of Melbourne website, as of April 2023 Tordesillas has published 167 scholarly articles with publication dates ranging from 1989 to 2023. == References == == External links == Antoinette Tordesillas, candidate biography from election for EMI Board of Governors, ASCE, 2016
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Wikipedia:Anton Davidoglu#0
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Anton Davidoglu (June 30, 1876 – May 27, 1958) was a Romanian mathematician who specialized in differential equations. He was born in 1876 in Bârlad, Vaslui County, the son of Profira Moțoc and Doctor Cleante Davidoglu. His older brother was General Cleante Davidoglu. He studied under Jacques Hadamard at the École Normale Supérieure in Paris, defending his Ph.D. dissertation in 1900. His thesis — the first mathematical investigation of deformable solids — applied Émile Picard's method of successive approximations to the study of fourth order differential equations that model traverse vibrations of non-homogeneous elastic bars. After returning to Romania, Davidoglu became a professor at the University of Bucharest. In 1913, he was founding rector of the Academy of High Commercial and Industrial Studies in Bucharest. He also continued to teach at the University of Bucharest, until his retirement in 1941. Davidoglu was a founding member of the Romanian Academy of Sciences, and was featured on a 1976 Romanian postage stamp. He died in 1958 in Bucharest. == Publications == Davidoglu, Anton (1900). "Sur l'équation des vibrations transversales des verges élastiques" (PDF). Annales Scientifiques de l'École Normale Supérieure. 3rd ser. 17. Paris: 359–444. doi:10.24033/asens.484. JFM 31.0769.01. Davidoglu, Anton (1900). "Sur une application de la méthode des approximations successives". Comptes rendus de l'Académie des Sciences. 130: 1241–1243. JFM 31.0353.02. Davidoglu, Anton (1905). "Étude de l'équation différentielle d 2 [ θ ( x ) d 2 y d x 2 ] d x 2 = k φ ( x ) y {\displaystyle {\frac {d^{2}\left[\theta (x){\frac {d^{2}y}{dx^{2}}}\right]}{dx^{2}}}=k\varphi (x)y} " (PDF). Annales Scientifiques de l'École Normale Supérieure. 3rd ser. 22. Paris: 539–565. doi:10.24033/asens.560. JFM 36.0399.02. Davidoglu, Anton (1936). "Sur une équation des mouvements turbulents". C. R. Acad. Sci. Roumanie (in French). 1: 3–7. JFM 62.1272.02. == References ==
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Wikipedia:Anton Kotzig#0
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Anton Kotzig (22 October 1919 – 20 April 1991) was a Slovak–Canadian mathematician, expert in statistics, combinatorics and graph theory. A number of his mathematical contributions are named after him. These include the Ringel–Kotzig conjecture on graceful labeling of trees (with Gerhard Ringel); Kotzig's conjecture on regularly path connected graphs; Kotzig's theorem on the degrees of vertices in convex polyhedra; as well as the Kotzig transformation. == Biography == Kotzig was born in Kočovce, a village in Western Slovakia. He studied at the secondary grammar school in Nové Mesto nad Váhom, and began his undergraduate studies at the Charles University in Prague. After the closure of Czech universities in 1939, he moved to Bratislava where in 1943, he earned a doctoral degree (RNDr.) in Mathematical Statistics from the Comenius University. He remained in Bratislava working at the Central Bureau of Social Insurance for Slovakia as head of the Department of Mathematical Statistics. Later, he published a book on economy planning. From 1951 to 1959, he lectured at Vysoká škola Ekonomická (today University of Economics in Bratislava), where he served as rector from 1952 to 1958. Thus he spent 20 years in close contact with applications of mathematics. In 1959, he left the University of Economics to become head of the newly-created Mathematical Institute of the Slovak Academy of Sciences, where he remained until 1964. From 1965 to 1969, he was head of the department of the Applied Mathematics on Faculty of the Natural Sciences of Comenius University, where he was also dean for one year. He also earned a habilitation degree (DrSc.) from the Charles University in 1961 for a thesis on Graph Theory (relation and regular relation of finite graphs). Kotzig established the now well-known Slovak School of Graph Theory. One of his first students was Juraj Bosák who was awarded the Czechoslovak State Prize in 1969. In 1969, Kotzig moved to Canada and spent a year at the University of Calgary. He became a researcher at the Centre de recherches mathematiques (CRM) and the University of Montreal in 1970, where he remained until his death. Because of the political situation, he could not travel back to Czechoslovakia, and remained in his adopted country without his books and notes. Although he was separated from his Slovak students, he continued doing mathematics. He died on April 20, 1991 in Montreal, leaving his wife Edita and son Ľuboš. == Contributions == By 1969, the list of his publications already included over 60 articles and 4 books. Many of his results have become classical, including results about graph relations, 1-factors and cubic graphs. As they were published only in Slovak, many of them remained unknown and some of the results were independently rediscovered much later by other mathematicians. In Canada, he wrote more than 75 additional articles. His publications cover a wide range of topics in graph theory and combinatorics: convex polyhedra, quasigroups, special decompositions into Hamiltonian paths, Latin squares, decompositions of complete graphs, perfect systems of difference sets, additive sequences of permutations, tournaments and combinatorial games theory. One of his results, known as Kotzig's Theorem, is the statement that every polyhedral graph has an edge whose two endpoints have total degree at most 13. An extreme case is the triakis icosahedron, where no edge has smaller total degree. Kotzig published the result in Slovakia in 1955, and it was named and popularized in the West by Branko Grünbaum in the mid-1970s. Kotzig published many open problems. One of them is the Ringel–Kotzig conjecture, stating that all trees have a graceful labeling. In 1963, Gerhard Ringel proposed that the complete graph K 2 n + 1 {\displaystyle K_{2n+1}} could be decomposed into isomorphic copies of any given n {\displaystyle n} -vertex tree, and in 1966, Alexander Rosa credited Kotzig with the suggestion that a stronger decomposition always existed, equivalent to the existence of a graceful labeling. The question remains unsolved. == Recognition == In honor of Kotzig's 60th birthday, Alexander Rosa, Gert Sabidussi and Jean Turgeon edited a festschrift, Theory and Practice of Combinatorics: A collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday (Annals of Discrete Mathematics 12, North-Holland, 1982), with contributions from experts from around the world. In 1999, a commemorative plaque was erected on his birth house in Kočovce on the 80th anniversary of his birth. == See also == Kotzig transformations Ringel-Kotzig conjecture == References == == External links == Anton Kotzig, 1919–1991, Mathematica Slovaca 42:3 (1992) 381–383. Prof. Anton Kotzig's biography (in Slovak).
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Wikipedia:Anton Sushkevich#0
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Anton Kazimirovich Sushkevich (Антон Казимирович Сушкевич) (23 January 1889, Borisoglebsk, Russia — 30 August 1961, Kharkiv, Ukraine) was a Russian mathematician and textbook author who expanded group theory to include semigroups and other magmas. Sushkevich attended secondary school in Voronezh and studied in Berlin from 1906 to 1911. There he attended lectures of F. G. Frobenius, Issai Schur, and Hermann Schwarz. Sushkevich studied piano with L. V. Rostropovich, father of Mstislav Rostropovich. In 1906 he was a cello student at Stern Conservatory (now part of Berlin University of the Arts). In 1911 he moved to Saint Petersburg, graduating from the Imperial University in 1913. Moving to Kharkiv, Suskevich taught in secondary education while he pursued a graduate degree at Kharkov State University. His dissertation was The theory of operations as the general theory of groups. Obtaining the degree, he became an assistant professor at the university in 1918, and adjunct professor in 1920. Voronezh State University employed Sushkevich in 1921 as professor of mathematics. He published the first edition of his Higher Algebra (1923). He published a generalization of Cayley's theorem for certain finite semigroups in 1926. The next year he was in Moscow for the Russian Mathematical Congress, and the following year in Bologna for the International Congress of Mathematicians. In Kharkiv, the Ukrainian Scientific Research Institute of Mathematics and Mechanics was established in 1929 with Sushkevich as a member. With a rising interest in abstract algebra, he wrote a second book on algebra: Foundations of Higher Algebra which was published both in Russian and Ukrainian. In 1933 he directed the Algebra & Number Theory section of Kharkov State University's department of mathematics. At that time Stalin caused a famine in Ukraine, the Holodomor, killing millions especially in rural areas. Suskevich survived to edit new editions of his textbook that included "new algebra": fields, integral domains, rings, ideals, and quaternions. His original work, The Theory of Generalized Groups (1937) opened up the area of semigroups. According to biographer Hollings, "He sought to describe his semigroups of interest in terms of certain of their subgroups: from Sushkevich's point of view, groups were objects of known structure.": 46 == Selected works == 1928: "Über die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit", Mathematische Annalen, 99 (1): 30–50, doi:10.1007/BF01459084, hdl:10338.dmlcz/100078, ISSN 0025-5831, MR 1512437. 1929: "On a generalization of the associative law", Transactions of the American Mathematical Society 31(1):204–14 doi:10.1090/S0002-9947-1929-1501476-0 MR1501476 1951: "Materials for the History of Algebra in Russia in the 19th and beginning of the 20th centuries", MR00051749 1954: Theory of Numbers, second edition 1956 MR0091286 == References == Christopher Hollings (2009) "The early development of the algebraic theory of semigroups", Archive for History of Exact Sciences 63(5): 497–536, especially 511–513. O'Connor, John J.; Robertson, Edmund F., "Anton Sushkevich", MacTutor History of Mathematics Archive, University of St Andrews Christopher Hollings Summary of "Finite groups without unique invertibiility" (PDF) via WebCite
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Wikipedia:Antonella Cupillari#0
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Antonella Cupillari (born 1955) is an Italian-American mathematician interested in the history of mathematics and mathematics education. She is an associate professor of mathematics at Penn State Erie, The Behrend College. == Education and career == Cupillari earned a laurea at the University of L'Aquila in 1978, and completed her Ph.D. at the University at Albany, SUNY in 1984. Her dissertation, A Small Boundary for H ∞ {\displaystyle H^{\infty }} on a Strictly Pseudoconvex Domain, concerned functional analysis, and was supervised by R. Michael (Rolf) Range; she also published it in the Proceedings of the American Mathematical Society. Cupillari joined the faculty at Penn State Erie in 1984 and was promoted to associate professor in 1992. == Books == Cupillari is the author of books on mathematics and the history of mathematics including: The Nuts and Bolts of Proofs (Wadsworth, 1989; 2nd & 3rd eds., Harcourt/Academic Press, 2000 & 2005; 4th ed., Academic Press, 2011) Intermediate Algebra in Action (PWS Publishing, 1995) A Biography of Maria Gaetana Agnesi, an Eighteenth-Century Woman Mathematician: With Translations of Some of Her Work from Italian into English (Edwin Mellen Press, 2007) == Recognition == Cupillari was the 2008 winner of the Award for Distinguished College or University Teaching of Mathematics of the Allegheny Mountain Section of the Mathematical Association of America. == References == == External links == Home page
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Wikipedia:Antonella Zanna#0
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Antonella Zanna Munthe-Kaas is an Italian applied mathematician and numerical analyst whose research includes work on numerical integration of differential equations and applications to medical imaging. She is a professor and head of the mathematics department at the University of Bergen in Norway. == Education == Zanna was born in Molfetta, in southern Italy, and earned a degree in mathematics from the University of Bari. She completed her PhD in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 1998. Her dissertation, Numerical Solution of Isospectral Flows, was supervised by Arieh Iserles. == Recognition == Zanna won the Second Prize in the Leslie Fox Prize for Numerical Analysis in 1997. She is a member of the Norwegian Academy of Technological Sciences. == Personal life == Zanna married to Norwegian mathematician Hans Munthe-Kaas in 1997; they have two children and a dog. == References == == Further reading == "Vi er visst helt ute å kjøre: Klokken er tolv. Søstrene Paola og Antonella fra Molfetta helt sør i Italia, der hælen på støvelen begynner, er ikke helt oppdatert på norske stengetider", Bergens Tidende, 24 December 2016 (newspaper story about late Christmas shopping with Zanna and her sister) == External links == Antonella Zanna publications indexed by Google Scholar
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Wikipedia:Antoni Zygmund#0
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Antoni Zygmund Polish pronunciation: [anˈtɔɲi ˈzɘgmunt] (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. == Biography == Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. From 1930 until 1952 he was a member of the Warsaw Scientific Society (TNW), from 1946 of the Polish Academy of Learning (PAU), from 1959 of the Polish Academy of Sciences (PAN), and from 1961 of the National Academy of Sciences in the United States. In 1986 he received the National Medal of Science. In 1935 Zygmund published in Polish the original edition of what has become, in its English translation, the two-volume Trigonometric Series. It was described by Robert A. Fefferman as "one of the most influential books in the history of mathematical analysis" and "an extraordinarily comprehensive and masterful presentation of a ... vast field". Jean-Pierre Kahane called the book "The Bible" of a harmonic analyst. The theory of trigonometric series had remained the largest component of Zygmund's mathematical investigations. His work has had a pervasive influence in many fields of mathematics, mostly in mathematical analysis, and particularly in harmonic analysis. Among the most significant were the results he obtained with Calderón on singular integral operators. George G. Lorentz called it Zygmund's crowning achievement, one that "stands somewhat apart from the rest of Zygmund's work". Zygmund's students included Alberto Calderón, Paul Cohen, Nathan Fine, Józef Marcinkiewicz, Victor L. Shapiro, Guido Weiss, Elias M. Stein and Mischa Cotlar. He died in Chicago. == Family == Antoni Zygmund, who had three sisters, married Irena Parnowska, a mathematician, in 1925. Upon his death he was survived by four grandsons. == Mathematical objects named after Zygmund == Calderón–Zygmund lemma Marcinkiewicz–Zygmund inequality Paley–Zygmund inequality Calderón–Zygmund kernel == Books == Trigonometric Series (Cambridge University Press 1959, 2002) Intégrales singulières (Springer-Verlag, 1971) Trigonometric Interpolation (University of Chicago, 1950) Measure and Integral: An Introduction to Real Analysis, With Richard L. Wheeden (Marcel Dekker, 1977) Analytic Functions, with Stanislaw Saks (Elsevier Science Ltd, 1971) == See also == Calderón–Zygmund lemma Zygmunt Janiszewski Marcinkiewicz–Zygmund inequality Paley–Zygmund inequality List of Poles Centipede mathematics == References == == Further reading == Kazimierz Kuratowski, A Half Century of Polish Mathematics: Remembrances and Reflections, Oxford, Pergamon Press, 1980, ISBN 0-08-023046-6. Gray, Jeremy (1970–1980). "Zygmund, Antoni". Dictionary of Scientific Biography. Vol. 25. New York: Charles Scribner's Sons. pp. 414–416. ISBN 978-0-684-10114-9. == External links == Antoni Zygmund at the Mathematics Genealogy Project Mount Holyoke biography O'Connor, John J.; Robertson, Edmund F., "Antoni Zygmund", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Antonia J. Jones#0
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Antonia Jane Jones (1943 – 2010) was a British mathematician and computer scientist. Her research considered number theory and computer science. == Early life and education == Jones was born in 1943 in Queen Charlotte's and Chelsea Hospital. She was the first member of her family to attend university. Jones contracted polio as a child and lost both of her legs at the age of ten. Jones attended the University of Reading, where she studied mathematics and physics and graduated both with first class honours. She was a doctoral student in number theory at the University of Cambridge, where she completed her PhD in 1969. Jones joined the University of Nottingham after earning her doctorate, before joining Imperial College London as a Senior Lecturer. She spent a year at the Institute for Advanced Study, after which she joined the faculty at the University of Colorado Boulder. == Research and career == Jones returned to the United Kingdom in the 1970s, where she became a lecturer at Royal Holloway, University of London. Her interest switched from mathematics to computing and she started to explore acoustic pattern recognition. Whilst Jones struggled with the early computers, when technology became more accessible for people with physical disabilities she launched her own firm creating random access video controllers. In 1983 Jones joined Brunel University London at a lecturer in Information Technology. Jones later served as Professor of Evolutionary and Neural Computing at Cardiff University. She exposed various security loopholes in banking infrastructure, including identifying significant potential fraud at HSBC. Alongside her scientific research, Jones was involved with science communication and public engagement. She served as an electronic data consultant on the 1986 film Rocinante. She contributed to the 1998 British Science Association Festival of Science. In 2007, Jones retired from Cardiff University. == Selected publications == Kevin E Ashelford; Nadia A Chuzhanova; John C Fry; Antonia J Jones; Andrew J Weightman (1 September 2006). "New screening software shows that most recent large 16S rRNA gene clone libraries contain chimeras". Applied and Environmental Microbiology. 72 (9): 5734–5741. doi:10.1128/AEM.00556-06. ISSN 0099-2240. PMC 1563593. PMID 16957188. Wikidata Q33256604. Kevin E Ashelford; Nadia A Chuzhanova; John C Fry; Antonia J Jones; Andrew J Weightman (1 December 2005). "At least 1 in 20 16S rRNA sequence records currently held in public repositories is estimated to contain substantial anomalies". Applied and Environmental Microbiology. 71 (12): 7724–7736. doi:10.1128/AEM.71.12.7724-7736.2005. ISSN 0099-2240. PMC 1317345. PMID 16332745. Wikidata Q33228706. Stefánsson, Adoalbjörn; Končar, N.; Jones, Antonia J. (1 September 1997). "A note on the Gamma test". Neural Computing & Applications. 5 (3): 131–133. doi:10.1007/BF01413858. ISSN 1433-3058. S2CID 11348191. Jones, A. J. (Antonia Jane), 1943- (2000). Game theory : mathematical models of conflict. Chichester [England]: Horwood Pub. ISBN 1-898563-14-4. OCLC 44654208.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link) == Personal life == Antonia Jones spent many years with her partner Barbara Quinn at their shared farmhouse in the Brecon Beacons. Upon her retirement in 2007, Jones moved to St. Augustine, Florida. Jones died on 23 December 2010. She is survived by Quinn and her sister Jenny Carrl. == References ==
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Wikipedia:Antonio Auffinger#0
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Antonio Auffinger is an American Brazilian mathematician. He works in the area of probability theory and mathematical physics. == Education and career == Auffinger completed his doctorate at the Courant Institute in 2011; his dissertation was supervised by Gerard Ben Arous and was awarded the Francisco Aranda-Ordaz Prize by the Bernoulli Society for Mathematical Statistics and Probability. He was Leonard Eugene Dickson instructor at the University of Chicago before moving in 2014 to Northwestern University, where he is a professor of mathematics. == Book == He co-authored a book on first passage percolation published by the American Mathematical Society. == Recognition == Auffinger won a National Science Foundation CAREER award in 2016. In 2017, he was awarded a gold medal prize by the International Consortium of Chinese Mathematicians for his proof of the uniqueness of the Parisi measure in spin glasses. He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to probability theory and mathematical physics and, in particular, to the study of spin glasses and percolation theory". == References ==
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Wikipedia:Antonio Cagnoli#0
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Antonio Cagnoli (29 September 1743, in Zakynthos – 6 August 1816, in Verona) was an Italian astronomer, mathematician and diplomat in the service of the Republic of Venice. His father Ottavio was chancellor to the Venetian governor of the Ionian Islands. == External links == "OsservatorioMonteBaldo". astrofiliveronesi.it. Retrieved 2015-10-06.
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Wikipedia:Antoon Kolen#0
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Anthonius Wilhelmus Johannes (Antoon) Kolen (22 May 1953 – 3 October 2004) was a Dutch mathematician and Professor at the Maastricht University, in the Department of Quantitative Economics. He is known for his work on dynamic programming, such as interval scheduling and mathematical optimization. == Biography == Born in Tilburg, Kolen obtained his engineering degree from the Eindhoven University of Technology in 1978. In 1982, he obtained his PhD at the Centrum Wiskunde & Informatica, University of Amsterdam under Gijsbert de Leve and Jan Karel Lenstra with the thesis, entitled "Location Problems on Trees and in the Rectilinear Plane." After his graduation, Kolen started his academic career at the Econometric Institute of Erasmus University Rotterdam. Late 1980s he moved to the Maastricht University, where he was appointed Professor at the Department of Quantitative Economics and head of its operations research group. His PhD students at the Erasmus University Rotterdam were Leo Kroon (graduated in 1990), Albert Wagelmans (1990), C. Stan van Hoesel (1991), Wim Pijls (1991), Peter Verbeek (1991), and A. Woerlee (1991); W. Hennen at the Wageningen University and Research Centre (graduated in 1995), and at the Maastricht University Alwin Oerlemans (graduated in 1992), Ron van der Wal (1995), Maarten Oosten (1996), Jons van de Klundert (1996), Robert van de Leensel (1999), Arie Koster (1999), and Alexander Grigoriev (2003). == Selected publications == Antoon Kolen. Location Problems on Trees and in the Rectilinear Plane. PhD thesis, Universiteit van Amsterdam, 1982. Kolen, Antoon WJ, and Arie Tamir. Covering problems. Econometric Institute, 1984. Articles, a selection: Brouwer, Andries E., and Antoon WJ Kolen. "A super-balanced hypergraph has a nest point." Stichting Mathematisch Centrum. Zuivere Wiskunde ZW 146/80 (1980): 1-7. Hoffman, Alan J., A. W. J. Kolen, and Michel Sakarovitch. "Totally-balanced and greedy matrices." SIAM Journal on Algebraic and Discrete Methods 6.4 (1985): 721-730. Kolen, Antoon WJ, A. H. G. Rinnooy Kan, and H. W. J. M. Trienekens. "Vehicle routing with time windows." Operations Research 35.2 (1987): 266-273. Wagelmans, Albert, Stan Van Hoesel, and Antoon Kolen. "Economic lot sizing: an O (n log n) algorithm that runs in linear time in the Wagner-Whitin case." Operations Research 40.1-Supplement - 1 (1992): pp. 145–156. Koster, Arie MCA, Stan P.M. Van Hoesel, and Antoon WJ Kolen. "The partial constraint satisfaction problem: Facets and lifting theorems." Operations research letters 23.3 (1998): 89-97. Kolen, Antoon. "A genetic algorithm for the partial binary constraint satisfaction problem: an application to a frequency assignment problem." Statistica Neerlandica 61.1 (2007): 4-15. == References == == External links == Antoon Kolen in Statistica Neerlandica, 2007
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Wikipedia:Antônio Carbonari Netto#0
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Antônio Carbonari Netto is a Brazilian educator, mathematician, and businessman. He is founder, CEO and one of the main shareholders of Anhanguera Educacional, a holding of 18 institutions of higher education in the state of São Paulo, one of the largest private universities in the country. In addition to his position and CEO of Anhangüera Educacional, he is dean of the Centro Universitário Anhangüera of Leme (Anhangüera University Center), and director general of the following colleges: Faculdades de Valinhos, Faculdade Comunitária de Campinas, Faculdade Politécnica de Jundiaí and Faculdade Politécnica de Matão. Carbonari Netto is a prominent academic and business leader in the private education world in Brazil. He has served in the executive board and councils of several important associations, such as the Associação Brasileira de Mantenedoras do Ensino Superior (Brazilian Association of Higher Education Organizations), Sindicato dos Estabelecimentos do Ensino Superior de São Paulo (Syndicate of the Higher Education Institutions of São Paulo), and others. == References == == External links == Anhangüera Educacional S.A. IFC to finance post-secondary education to lower-income students in Brazil
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Wikipedia:Apala Majumdar#0
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Apala Majumdar is a British applied mathematician specialising in the mathematics of liquid crystals. She is a professor of Applied Mathematics at the University of Strathclyde. == Education and career == Majumdar did her undergraduate studies at the University of Bristol. As a graduate student at Bristol, she also worked with Hewlett Packard Laboratories. She was awarded a PhD in applied mathematics at the University of Bristol in 2006; her dissertation, Liquid crystals and tangent unit-vector fields in polyhedral geometries, was jointly supervised by Jonathan Robbins and Maxim Zyskin. After working as a Royal Commission of the Exhibition of 1851 Research Fellow at the University of Oxford, she moved to the University of Bath in 2012, having been awarded a 5-year EPSRC Career Acceleration Fellowship in 2011. At Bath she became a Reader and the Director of the Centre for Nonlinear Mechanics (2018-2019). In 2019 she was appointed as a professor of Applied Mathematics at the University of Strathclyde. == Recognition == The British Liquid Crystal Society gave Majumdar their Young Scientist Award in 2012. The London Mathematical Society gave her their Anne Bennett Prize in 2015. In 2019 she was the winner of the academic category of the FDM Everywoman in Technology Awards. In 2024, she was elected as a fellow of the Royal Society of Edinburgh. == References ==
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Wikipedia:Apollonian circles#0
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In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned ancient Greek geometer. == Definition == The Apollonian circles are defined in two different ways by a line segment denoted CD. Each circle in the first family (the blue circles in the figure) is associated with a positive real number r, and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r, { X | d ( X , C ) d ( X , D ) = r } . {\displaystyle \left\{X\ {\Biggl |}\ {\frac {d(X,C)}{d(X,D)}}=r\right\}.} For values of r close to zero, the corresponding circle is close to C, while for values of r close to ∞, the corresponding circle is close to D; for the intermediate value r = 1, the circle degenerates to a line, the perpendicular bisector of CD. The equation defining these circles as a locus can be generalized to define the Fermat–Apollonius circles of larger sets of weighted points. Each circle in the second family (the red circles in the figure) is associated with an angle θ, and is defined as the locus of points X such that the inscribed angle ∠CXD equals θ, { X | C X ^ D = θ } . {\displaystyle \left\{X\ {\Bigl |}\ C{\hat {X}}D=\theta \right\}.} Scanning θ from 0 to π generates the set of all circles passing through the two points C and D. The two points where all the red circles cross are the limiting points of pairs of circles in the blue family. == Bipolar coordinates == A given blue circle and a given red circle intersect in two points. In order to obtain bipolar coordinates, a method is required to specify which point is the right one. An isoptic arc is the locus of points X that sees points C, D under a given oriented angle of vectors i.e. isopt ( θ ) = { X | ∡ ( X C → , X D → ) = θ + 2 k π } . {\displaystyle \operatorname {isopt} (\theta )=\left\{X\ {\Biggl |}\ \measuredangle {\biggl (}{\overrightarrow {XC}},{\overrightarrow {XD}}{\biggr )}=\theta +2k\pi \right\}.} Such an arc is contained into a red circle and is bounded by points C, D. The remaining part of the corresponding red circle is isopt(θ + π). When we really want the whole red circle, a description using oriented angles of straight lines has to be used: full red circle = { X | ∡ ( X C → , X D → ) = θ + k π } {\displaystyle {\text{full red circle}}=\left\{X\ {\Biggl |}\ \measuredangle {\biggl (}{\overrightarrow {XC}},{\overrightarrow {XD}}{\biggr )}=\theta +k\pi \right\}} == Pencils of circles == Both of the families of Apollonian circles are pencils of circles. Each is determined by any two of its members, called generators of the pencil. Specifically, one is an elliptic pencil (red family of circles in the figure) that is defined by two generators that pass through each other in exactly two points (C, D). The other is a hyperbolic pencil (blue family of circles in the figure) that is defined by two generators that do not intersect each other at any point. === Radical axis and central line === Any two of these circles within a pencil have the same radical axis, and all circles in the pencil have collinear centers. Any three or more circles from the same family are called coaxial circles or coaxal circles. The elliptic pencil of circles passing through the two points C, D (the set of red circles, in the figure) has the line CD as its radical axis. The centers of the circles in this pencil lie on the perpendicular bisector of CD. The hyperbolic pencil defined by points C, D (the blue circles) has its radical axis on the perpendicular bisector of line CD, and all its circle centers on line CD. == Inversive geometry, orthogonal intersection, and coordinate systems == Circle inversion transforms the plane in a way that maps circles into circles, and pencils of circles into pencils of circles. The type of the pencil is preserved: the inversion of an elliptic pencil is another elliptic pencil, the inversion of a hyperbolic pencil is another hyperbolic pencil, and the inversion of a parabolic pencil is another parabolic pencil. It is relatively easy to show using inversion that, in the Apollonian circles, every blue circle intersects every red circle orthogonally, i.e., at a right angle. Inversion of the blue Apollonian circles with respect to a circle centered on point C results in a pencil of concentric circles centered at the image of point D. The same inversion transforms the red circles into a set of straight lines that all contain the image of D. Thus, this inversion transforms the bipolar coordinate system defined by the Apollonian circles into a polar coordinate system. Obviously, the transformed pencils meet at right angles. Since inversion is a conformal transformation, it preserves the angles between the curves it transforms, so the original Apollonian circles also meet at right angles. Alternatively, the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point X on the radical axis of a pencil P the lengths of the tangents from X to each circle in P are all equal. It follows from this that the circle centered at X with length equal to these tangents crosses all circles of P perpendicularly. The same construction can be applied for each X on the radical axis of P, forming another pencil of circles perpendicular to P. More generally, for every pencil of circles there exists a unique pencil consisting of the circles that are perpendicular to the first pencil. If one pencil is elliptic, its perpendicular pencil is hyperbolic, and vice versa; in this case the two pencils form a set of Apollonian circles. The pencil of circles perpendicular to a parabolic pencil is also parabolic; it consists of the circles that have the same common tangent point but with a perpendicular tangent line at that point. == Physics == Apollonian trajectories have been shown to be followed in their motion by vortex cores or other defined pseudospin states in some physical systems involving interferential or coupled fields, such photonic or coupled polariton waves. The trajectories arise from the Rabi rotation of the Bloch sphere and its stereographic projection on the real space where the observation is made. == See also == Apollonius quadrilateral == Notes == == References == Akopyan, A. V.; Zaslavsky, A. A. (2007), Geometry of Conics, Mathematical World, vol. 26, American Mathematical Society, pp. 57–62, ISBN 978-0-8218-4323-9. Schwerdtfeger, Hans (1962), Geometry of Complex Numbers, University of Toronto Press. Dover reprint, 1979, ISBN 0-486-63830-8. == Further reading == Pfeifer, Richard E.; Van Hook, Cathleen (1993), "Circles, Vectors, and Linear Algebra", Mathematics Magazine, 66 (2): 75–86, doi:10.2307/2691113, JSTOR 2691113. Samuel, Pierre (1988), Projective Geometry, Springer, pp. 40–43. Ogilvy, C. Stanley (1969), Excursions in Geometry, Oxford University Press, esp. Ch. 2 "Harmonic division and Apollonian circles", pp. 13–23. Dover reprint, 1990, ISBN 0-486-26530-7. == External links == Weisstein, Eric W., "Coaxal Circles", MathWorld David B. Surowski: Advanced High-School Mathematics. p. 31
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Wikipedia:Apollonian gasket#0
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In mathematics, an Apollonian gasket, Apollonian net, or Apollonian circle packing is a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. It is named after Greek mathematician Apollonius of Perga. == Construction == The construction of the Apollonian gasket starts with three circles C 1 {\displaystyle C_{1}} , C 2 {\displaystyle C_{2}} , and C 3 {\displaystyle C_{3}} (black in the figure), that are each tangent to the other two, but that do not have a single point of triple tangency. These circles may be of different sizes to each other, and it is allowed for two to be inside the third, or for all three to be outside each other. As Apollonius discovered, there exist two more circles C 4 {\displaystyle C_{4}} and C 5 {\displaystyle C_{5}} (red) that are tangent to all three of the original circles – these are called Apollonian circles. These five circles are separated from each other by six curved triangular regions, each bounded by the arcs from three pairwise-tangent circles. The construction continues by adding six more circles, one in each of these six curved triangles, tangent to its three sides. These in turn create 18 more curved triangles, and the construction continues by again filling these with tangent circles, ad infinitum. Continued stage by stage in this way, the construction adds 2 ⋅ 3 n {\displaystyle 2\cdot 3^{n}} new circles at stage n {\displaystyle n} , giving a total of 3 n + 1 + 2 {\displaystyle 3^{n+1}+2} circles after n {\displaystyle n} stages. In the limit, this set of circles is an Apollonian gasket. In it, each pair of tangent circles has an infinite Pappus chain of circles tangent to both circles in the pair. The size of each new circle is determined by Descartes' theorem, which states that, for any four mutually tangent circles, the radii r i {\displaystyle r_{i}} of the circles obeys the equation ( 1 r 1 + 1 r 2 + 1 r 3 + 1 r 4 ) 2 = 2 ( 1 r 1 2 + 1 r 2 2 + 1 r 3 2 + 1 r 4 2 ) . {\displaystyle \left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}+{\frac {1}{r_{3}}}+{\frac {1}{r_{4}}}\right)^{2}=2\left({\frac {1}{r_{1}^{2}}}+{\frac {1}{r_{2}^{2}}}+{\frac {1}{r_{3}^{2}}}+{\frac {1}{r_{4}^{2}}}\right).} This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three. One or two of the initial circles of this construction, or the circles resulting from this construction, can degenerate to a straight line, which can be thought of as a circle with infinite radius. When there are two lines, they must be parallel, and are considered to be tangent at a point at infinity. When the gasket includes two lines on the x {\displaystyle x} -axis and one unit above it, and a circle of unit diameter tangent to both lines centered on the y {\displaystyle y} -axis, then the circles that are tangent to the x {\displaystyle x} -axis are the Ford circles, important in number theory. The Apollonian gasket has a Hausdorff dimension of about 1.3056867, which has been extended to at least 128 decimal places. Because it has a well-defined fractional dimension, even though it is not precisely self-similar, it can be thought of as a fractal. == Symmetries == The Möbius transformations of the plane preserve the shapes and tangencies of circles, and therefore preserve the structure of an Apollonian gasket. Any two triples of mutually tangent circles in an Apollonian gasket may be mapped into each other by a Möbius transformation, and any two Apollonian gaskets may be mapped into each other by a Möbius transformation. In particular, for any two tangent circles in any Apollonian gasket, an inversion in a circle centered at the point of tangency (a special case of a Möbius transformation) will transform these two circles into two parallel lines, and transform the rest of the gasket into the special form of a gasket between two parallel lines. Compositions of these inversions can be used to transform any two points of tangency into each other. Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent. In a sense, there is therefore only one Apollonian gasket, up to (hyperbolic) isometry. The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group. For Euclidean symmetry transformations rather than Möbius transformations, in general, the Apollonian gasket will inherit the symmetries of its generating set of three circles. However, some triples of circles can generate Apollonian gaskets with higher symmetry than the initial triple; this happens when the same gasket has a different and more-symmetric set of generating circles. Particularly symmetric cases include the Apollonian gasket between two parallel lines (with infinite dihedral symmetry), the Apollonian gasket generated by three congruent circles in an equilateral triangle (with the symmetry of the triangle), and the Apollonian gasket generated by two circles of radius 1 surrounded by a circle of radius 2 (with two lines of reflective symmetry). == Integral Apollonian circle packings == If any four mutually tangent circles in an Apollonian gasket all have integer curvature (the inverse of their radius) then all circles in the gasket will have integer curvature. Since the equation relating curvatures in an Apollonian gasket, integral or not, is a 2 + b 2 + c 2 + d 2 = 2 a b + 2 a c + 2 a d + 2 b c + 2 b d + 2 c d , {\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=2ab+2ac+2ad+2bc+2bd+2cd,\,} it follows that one may move from one quadruple of curvatures to another by Vieta jumping, just as when finding a new Markov number. The first few of these integral Apollonian gaskets are listed in the following table. The table lists the curvatures of the largest circles in the gasket. Only the first three curvatures (of the five displayed in the table) are needed to completely describe each gasket – all other curvatures can be derived from these three. === Enumerating integral Apollonian circle packings === The curvatures ( a , b , c , d ) {\displaystyle (a,b,c,d)} are a root quadruple (the smallest in some integral circle packing) if a < 0 ≤ b ≤ c ≤ d {\displaystyle a<0\leq b\leq c\leq d} . They are primitive when gcd ( a , b , c , d ) = 1 {\displaystyle \gcd(a,b,c,d)=1} . Defining a new set of variables ( x , d 1 , d 2 , m ) {\displaystyle (x,d_{1},d_{2},m)} by the matrix equation [ a b c d ] = [ 1 0 0 0 − 1 1 0 0 − 1 0 1 0 − 1 1 1 − 2 ] [ x d 1 d 2 m ] {\displaystyle {\begin{bmatrix}a\\b\\c\\d\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\-1&1&0&0\\-1&0&1&0\\-1&1&1&-2\end{bmatrix}}{\begin{bmatrix}x\\d_{1}\\d_{2}\\m\end{bmatrix}}} gives a system where ( a , b , c , d ) {\displaystyle (a,b,c,d)} satisfies the Descartes equation precisely when x 2 + m 2 = d 1 d 2 {\displaystyle x^{2}+m^{2}=d_{1}d_{2}} . Furthermore, ( a , b , c , d ) {\displaystyle (a,b,c,d)} is primitive precisely when gcd ( x , d 1 , d 2 ) = 1 {\displaystyle \gcd(x,d_{1},d_{2})=1} , and ( a , b , c , d ) {\displaystyle (a,b,c,d)} is a root quadruple precisely when x < 0 ≤ 2 m ≤ d 1 ≤ d 2 {\displaystyle x<0\leq 2m\leq d_{1}\leq d_{2}} . This relationship can be used to find all the primitive root quadruples with a given negative bend x {\displaystyle x} . It follows from 2 m ≤ d 1 {\displaystyle 2m\leq d_{1}} and 2 m ≤ d 2 {\displaystyle 2m\leq d_{2}} that 4 m 2 ≤ d 1 d 2 {\displaystyle 4m^{2}\leq d_{1}d_{2}} , and hence that 3 m 2 ≤ d 1 d 2 − m 2 = x 2 {\displaystyle 3m^{2}\leq d_{1}d_{2}-m^{2}=x^{2}} . Therefore, any root quadruple will satisfy 0 ≤ m ≤ | x | / 3 {\displaystyle 0\leq m\leq |x|/{\sqrt {3}}} . By iterating over all the possible values of m {\displaystyle m} , d 1 {\displaystyle d_{1}} , and d 2 {\displaystyle d_{2}} one can find all the primitive root quadruples. The following Python code demonstrates this algorithm, producing the primitive root quadruples listed above. ==== The Local-Global Conjecture ==== The curvatures appearing in a primitive integral Apollonian circle packing must belong to a set of six or eight possible residues classes modulo 24, and theoretical results and numerical evidence supported that any sufficiently large integer from these residue classes would also be present as a curvature within the packing. This conjecture, known as the local-global conjecture, was proved to be false in 2023. === Symmetry of integral Apollonian circle packings === There are multiple types of dihedral symmetry that can occur with a gasket depending on the curvature of the circles. ==== No symmetry ==== If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group C1; the gasket described by curvatures (−10, 18, 23, 27) is an example. ==== D1 symmetry ==== Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have D1 symmetry, which corresponds to a reflection along a diameter of the bounding circle, with no rotational symmetry. ==== D2 symmetry ==== If two different curvatures are repeated within the first five, the gasket will have D2 symmetry; such a symmetry consists of two reflections (perpendicular to each other) along diameters of the bounding circle, with a two-fold rotational symmetry of 180°. The gasket described by curvatures (−1, 2, 2, 3) is the only Apollonian gasket (up to a scaling factor) to possess D2 symmetry. ==== D3 symmetry ==== There are no integer gaskets with D3 symmetry. If the three circles with smallest positive curvature have the same curvature, the gasket will have D3 symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2√3 − 3. As this ratio is not rational, no integral Apollonian circle packings possess this D3 symmetry, although many packings come close. ==== Almost-D3 symmetry ==== The figure at left is an integral Apollonian gasket that appears to have D3 symmetry. The same figure is displayed at right, with labels indicating the curvatures of the interior circles, illustrating that the gasket actually possesses only the D1 symmetry common to many other integral Apollonian gaskets. The following table lists more of these almost-D3 integral Apollonian gaskets. The sequence has some interesting properties, and the table lists a factorization of the curvatures, along with the multiplier needed to go from the previous set to the current one. The absolute values of the curvatures of the "a" disks obey the recurrence relation a(n) = 4a(n − 1) − a(n − 2) (sequence A001353 in the OEIS), from which it follows that the multiplier converges to √3 + 2 ≈ 3.732050807. === Sequential curvatures === For any integer n > 0, there exists an Apollonian gasket defined by the following curvatures: (−n, n + 1, n(n + 1), n(n + 1) + 1). For example, the gaskets defined by (−2, 3, 6, 7), (−3, 4, 12, 13), (−8, 9, 72, 73), and (−9, 10, 90, 91) all follow this pattern. Because every interior circle that is defined by n + 1 can become the bounding circle (defined by −n) in another gasket, these gaskets can be nested. This is demonstrated in the figure at right, which contains these sequential gaskets with n running from 2 through 20. == History == Although the Apollonian gasket is named for Apollonius of Perga -- because of its construction's dependence on the solution to the problem of Apollonius -- the earliest description of the gasket is from 1706 by Leibniz in a letter to Des Bosses. The first modern definition of the Apollonian gasket is given by Kasner and Supnick. == See also == Apollonian network, a graph derived from finite subsets of the Apollonian gasket Apollonian sphere packing, a three-dimensional generalization of the Apollonian gasket Sierpiński triangle, a self-similar fractal with a similar combinatorial structure == Notes == == References == Benoit B. Mandelbrot: The Fractal Geometry of Nature, W H Freeman, 1982, ISBN 0-7167-1186-9 Bourgain, Jean; Kontorovich, Alex (2014). "On the Local-Global Conjecture for integral Apollonian gaskets". Inventiones Mathematicae (196): 589–650. arXiv:1205.4416. Paul D. Bourke: "An Introduction to the Apollony Fractal". Computers and Graphics, Vol 30, Issue 1, January 2006, pages 134–136. Fuchs, Elena; Sanden, Katherine (2011-11-28). "Some Experiments with Integral Apollonian Circle Packings". Experimental Mathematics. 20 (4): 380–399. arXiv:1001.1406. doi:10.1080/10586458.2011.565255. ISSN 1058-6458. Graham, Ronald L.; Lagarias, Jeffrey C.; Mallows, Colin L.; Wilks, Alan R.; Yan, Catherine H. (2003). "Apollonian Circle Packings: Number Theory". J. Number Theory. 100 (1): 1–45. Summer Haag; Clyde Kertzer; James Rickards; Katherine E. Stange (2024). "The Local-Global Conjecture for Apollonian circle packings is false". arXiv:2307.02749. Kasner, Edward; Supnick, Fred (1943). "The Apollonian packing of circles". Proceedings of the National Academy of Sciences. 29: 378–384. A.A. Kirillov: A Tale of Two Fractals, Birkhauser, 2013. Kontorovich, Alex (2013). "From Apollonius to Zaremba: Local-global phenomena in thin orbits". Bull. Amer. Math. Soc. 50: 187–228. arXiv:1208.5460. Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks: Beyond the Descartes Circle Theorem, The American Mathematical Monthly, Vol. 109, No. 4 (Apr., 2002), pp. 338–361, (arXiv:math.MG/0101066 v1 9 Jan 2001) David Mumford, Caroline Series, David Wright: Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002, ISBN 0-521-35253-3 Soddy, Frederick (1937), "The bowl of integers and the hexlet", Nature, 139 (3506), London: 77–79, doi:10.1038/139077a0. == External links == Weisstein, Eric W. "Apollonian Gasket". MathWorld. Alexander Bogomolny, Apollonian Gasket, cut-the-knot An interactive Apollonian gasket running on pure HTML5 at the Wayback Machine (archived 2011-05-02) A Matlab script to plot 2D Apollonian gasket with n identical circles Archived 2008-10-07 at the Wayback Machine using circle inversion Online experiments with JSXGraph Apollonian Gasket by Michael Screiber, The Wolfram Demonstrations Project. Interactive Apollonian Gasket Demonstration of an Apollonian gasket running on Java Dana Mackenzie. Computing Science: A Tisket, a Tasket, an Apollonian Gasket. American Scientist, January/February 2010. "Sand drawing the world's largest single artwork", The Telegraph, 16 Dec 2009. Newspaper story about an artwork in the form of a partial Apollonian gasket, with an outer circumference of nine miles. Dynamic apollonian gaskets, Tartapelago by Giorgio Pietrocola, 2014. (in Italian)
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Wikipedia:Apollonian sphere packing#0
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In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume. For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 63.5%. == Classification and terminology == A lattice arrangement (commonly called a regular arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only n vectors to be uniquely defined (in n-dimensional Euclidean space). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as irregular) can still be periodic, but also aperiodic (properly speaking non-periodic) or random. Because of their high degree of symmetry, lattice packings are easier to classify than non-lattice ones. Periodic lattices always have well-defined densities. == Regular packing == === Dense packing === In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centers of the balls in A and one half lies above the hollows of A which were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer which were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures. Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or face-centred cubic, "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP"), where the layers are alternated in the ABAB... sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere touches 12 neighboring spheres, and the average density is π 3 2 ≈ 0.74048. {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}\approx 0.74048.} In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture. Carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014, Hales announced the completion of a formal proof using automated proof checking, removing any doubt. === Other common lattice packings === Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of π 6 ≈ 0.5236 {\displaystyle {\frac {\pi }{6}}\approx 0.5236} , the hexagonal lattice with a density of π 3 3 ≈ 0.6046 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}\approx 0.6046} and the tetrahedral lattice with a density of π 3 16 ≈ 0.3401 {\displaystyle {\frac {\pi {\sqrt {3}}}{16}}\approx 0.3401} . === Jammed packings with a low density === Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or jammed. The strictly jammed (mechanically stable even as a finite system) regular sphere packing with the lowest known density is a diluted ("tunneled") fcc crystal with a density of only π√2/9 ≈ 0.49365. The loosest known regular jammed packing has a density of approximately 0.0555. == Irregular packing == If we attempt to build a densely packed collection of spheres, we will be tempted to always place the next sphere in a hollow between three packed spheres. If five spheres are assembled in this way, they will be consistent with one of the regularly packed arrangements described above. However, the sixth sphere placed in this way will render the structure inconsistent with any regular arrangement. This results in the possibility of a random close packing of spheres which is stable against compression. Vibration of a random loose packing can result in the arrangement of spherical particles into regular packings, a process known as granular crystallisation. Such processes depend on the geometry of the container holding the spherical grains. When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%. Recent research predicts analytically that it cannot exceed a density limit of 63.4% This situation is unlike the case of one or two dimensions, where compressing a collection of 1-dimensional or 2-dimensional spheres (that is, line segments or circles) will yield a regular packing. == Hypersphere packing == The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane. In one dimension it is packing line segments into a linear universe. In dimensions higher than three, the densest lattice packings of hyperspheres are known for 8 and 24 dimensions. Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing. In 2016, Maryna Viazovska announced a proof that the E8 lattice provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the Leech lattice is optimal in 24 dimensions. This result built on and improved previous methods which showed that these two lattices are very close to optimal. The new proofs involve using the Laplace transform of a carefully chosen modular function to construct a radially symmetric function f such that f and its Fourier transform f̂ both equal 1 at the origin, and both vanish at all other points of the optimal lattice, with f negative outside the central sphere of the packing and f̂ positive. Then, the Poisson summation formula for f is used to compare the density of the optimal lattice with that of any other packing. Before the proof had been formally refereed and published, mathematician Peter Sarnak called the proof "stunningly simple" and wrote that "You just start reading the paper and you know this is correct." Another line of research in high dimensions is trying to find asymptotic bounds for the density of the densest packings. It is known that for large n, the densest lattice in dimension n has density θ ( n ) {\displaystyle \theta (n)} between cn ⋅ 2−n (for some constant c) and 2−(0.599+o(1))n. Conjectural bounds lie in between. In a 2023 preprint, Marcelo Campos, Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe announced an improvement to the lower bound of the maximal density to θ ( n ) ≥ ( 1 − o ( 1 ) ) n ln n 2 n + 1 {\displaystyle \theta (n)\geq (1-o(1)){\frac {n\ln n}{2^{n+1}}}} , among their techniques they make use of the Rödl nibble. == Unequal sphere packing == Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing. When many sizes of spheres (or a distribution) are available, the problem quickly becomes intractable, but some studies of binary hard spheres (two sizes) are available. When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space. Even if the large spheres are not in a close-packed arrangement, it is always possible to insert some smaller spheres of up to 0.29099 of the radius of the larger sphere. When the smaller sphere has a radius greater than 0.41421 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the close-packed structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786. Upper bounds for the density that can be obtained in such binary packings have also been obtained. In many chemical situations such as ionic crystals, the stoichiometry is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the Coulomb energy of interacting charges leads to a diversity of optimal packing arrangements. The upper bound for the density of a strictly jammed sphere packing with any set of radii is 1 – an example of such a packing of spheres is the Apollonian sphere packing. The lower bound for such a sphere packing is 0 – an example is the Dionysian sphere packing. == Hyperbolic space == Although the concept of circles and spheres can be extended to hyperbolic space, finding the densest packing becomes much more difficult. In a hyperbolic space there is no limit to the number of spheres that can surround another sphere (for example, Ford circles can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an infinite number of other circles). The concept of average density also becomes much more difficult to define accurately. The densest packings in any hyperbolic space are almost always irregular. Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic n-space where n ≥ 2. In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the horosphere packing of the order-6 tetrahedral honeycomb with Schläfli symbol {3,3,6}. In addition to this configuration at least three other horosphere packings are known to exist in hyperbolic 3-space that realize the density upper bound. == Touching pairs, triplets, and quadruples == The contact graph of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3-cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in n dimensions that the cardinality of the set of n-simplices in the contact graph gives the number of touching (n + 1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples were proved by Karoly Bezdek and Samuel Reid at the University of Calgary. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the "sticky-sphere problem". The maximum is known for n ≤ 11, and only conjectural values are known for larger n. == Other spaces == Sphere packing on the corners of a hypercube (with Hamming balls, spheres defined by Hamming distance) corresponds to designing error-correcting codes: if the spheres have radius t, then their centers are codewords of a (2t + 1)-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error-correcting codes. For example, the binary Golay code is closely related to the 24-dimensional Leech lattice. For further details on these connections, see the book Sphere Packings, Lattices and Groups by Conway and Sloane. == See also == == References == == Bibliography == Aste, T.; Weaire, D. (2000). The Pursuit of Perfect Packing. London: Institute of Physics Publishing. ISBN 0-7503-0648-3. Conway, J. H.; Sloane, N. J. H. (1998). Sphere Packings, Lattices and Groups (3rd ed.). Springer. ISBN 0-387-98585-9. Sloane, N. J. A. (1984). "The Packing of Spheres". Scientific American. 250: 116–125. Bibcode:1984SciAm.250e.116G. doi:10.1038/scientificamerican0584-116. == External links == Dana Mackenzie (May 2002) "A fine mess" (New Scientist) A non-technical overview of packing in hyperbolic space. Weisstein, Eric W. "Circle Packing". MathWorld. "Kugelpackungen (Sphere Packing)" (T. E. Dorozinski) "3D Sphere Packing Applet" Archived 26 April 2009 at the Wayback Machine Sphere Packing java applet "Densest Packing of spheres into a sphere" java applet "Database of sphere packings" (Erik Agrell)
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Wikipedia:Apollonius quadrilateral#0
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In geometry, an Apollonius quadrilateral is a quadrilateral A B C D {\displaystyle ABCD} such that the two products of opposite side lengths are equal. That is, A B ¯ ⋅ C D ¯ = A D ¯ ⋅ B C ¯ . {\displaystyle {\overline {AB}}\cdot {\overline {CD}}={\overline {AD}}\cdot {\overline {BC}}.} An equivalent way of stating this definition is that the cross ratio of the four points is ± 1 {\displaystyle \pm 1} . It is allowed for the quadrilateral sides to cross. The Apollonius quadrilaterals are important in inversive geometry, because the property of being an Apollonius quadrilateral is preserved by Möbius transformations, and every continuous transformation of the plane that preserves all Apollonius quadrilaterals must be a Möbius transformation. Every kite is an Apollonius quadrilateral. A special case of the Apollonius quadrilaterals are the harmonic quadrilaterals; these are cyclic Apollonius quadrilaterals, inscribed in a given circle. They may be constructed by choosing two opposite vertices A {\displaystyle A} and C {\displaystyle C} arbitrarily on the circle, letting E {\displaystyle E} be any point exterior to the circle on line A C {\displaystyle AC} , and setting B {\displaystyle B} and D {\displaystyle D} to be the two points where the circle is touched by the tangent lines to circles through E {\displaystyle E} . Then A B C D {\displaystyle ABCD} is an Apollonius quadrilateral. If A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are fixed, then the locus of points D {\displaystyle D} that form an Apollonius quadrilateral A B C D {\displaystyle ABCD} is the set of points where the ratio of distances to A {\displaystyle A} and C {\displaystyle C} , A D ¯ / C D ¯ {\displaystyle {\overline {AD}}/{\overline {CD}}} , is the fixed ratio A B ¯ / B C ¯ {\displaystyle {\overline {AB}}/{\overline {BC}}} ; this is just a rewritten form of the defining equation for an Apollonius quadrilateral. As Apollonius of Perga proved, the set of points D {\displaystyle D} having a fixed ratio of distances to two given points A {\displaystyle A} and C {\displaystyle C} , and therefore the locus of points that form an Apollonius quadrilateral, is a circle in a family of circles called the Apollonian circles. Because B {\displaystyle B} defines the same ratio of distances, it lies on the same circle. In the case where the fixed ratio is one, the circle degenerates to a line, the perpendicular bisector of A C {\displaystyle AC} , and the resulting quadrilateral is a kite. == See also == Tangential quadrilateral, where sums rather than products of opposite sides are equal == References ==
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Wikipedia:Apotome (mathematics)#0
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In the historical study of mathematics, an apotome is a line segment formed from a longer line segment by breaking it into two parts, one of which is commensurable only in power to the whole; the other part is the apotome. In this definition, two line segments are said to be "commensurable only in power" when the ratio of their lengths is an irrational number but the ratio of their squared lengths is rational. Translated into modern algebraic language, an apotome can be interpreted as a quadratic irrational number formed by subtracting one square root of a rational number from another. This concept of the apotome appears in Euclid's Elements beginning in book X, where Euclid defines two special kinds of apotomes. In an apotome of the first kind, the whole is rational, while in an apotome of the second kind, the part subtracted from it is rational; both kinds of apotomes also satisfy an additional condition. Euclid Proposition XIII.6 states that, if a rational line segment is split into two pieces in the golden ratio, then both pieces may be represented as apotomes. == References ==
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Wikipedia:Applications of sensitivity analysis in epidemiology#0
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Sensitivity analysis studies the relation between the uncertainty in a model-based the inference and the uncertainties in the model assumptions. Sensitivity analysis can play an important role in epidemiology, for example in assessing the influence of the unmeasured confounding on the causal conclusions of a study. It is also important in all mathematical modelling studies of epidemics. Sensitivity analysis can be used in epidemiology, for example in assessing the influence of the unmeasured confounding on the causal conclusions of a study. The use of sensitivity analysis in mathematical modelling of infectious disease is suggested in on the Coronavirus disease 2019 outbreak. Given the significant uncertainty at play, the use of sensitivity analysis to apportion the output uncertainty into input parameters is crucial in the context of Decision-making. Examples of applications of sensitivity analysis to modelling of COVID-19 are and. in particular, the time of intervention time in containing the pandemic spread is identified as a key parameter. == References ==
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Wikipedia:Applications of sensitivity analysis to model calibration#0
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Sensitivity analysis has important applications in model calibration. One application of sensitivity analysis addresses the question of "What's important to model or system development?" One can seek to identify important connections between observations, model inputs, and predictions or forecasts. That is, one can seek to understand what observations (measurements of dependent variables) are most and least important to model inputs (parameters representing system characteristics or excitation), what model inputs are most and least important to predictions or forecasts, and what observations are most and least important to the predictions and forecasts. Often the results are surprising, lead to finding problems in the data or model development, and fixing the problems. This leads to better models. In biomedical engineering, sensitivity analysis can be used to determine system dynamics in ODE-based kinetic models. Parameters corresponding to stages of differentiation can be varied to determine which parameter is most influential on cell fate. Therefore, the most limiting step can be identified and the cell state for most advantageous scale-up and expansion can be determined. Additionally, complex networks in systems biology can be better understood through fitting mass-action kinetic models. Sensitivity analysis on rate coefficients can then be conducted to determine optimal therapeutic targets within the system of interest. == References ==
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Wikipedia:Applied general equilibrium#0
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In mathematical economics, applied general equilibrium (AGE) models were pioneered by Herbert Scarf at Yale University in 1967, in two papers, and a follow-up book with Terje Hansen in 1973, with the aim of empirically estimating the Arrow–Debreu model of general equilibrium theory with empirical data, to provide "“a general method for the explicit numerical solution of the neoclassical model” (Scarf with Hansen 1973: 1) Scarf's method iterated a sequence of simplicial subdivisions which would generate a decreasing sequence of simplices around any solution of the general equilibrium problem. With sufficiently many steps, the sequence would produce a price vector that clears the market. Brouwer's Fixed Point theorem states that a continuous mapping of a simplex into itself has at least one fixed point. This paper describes a numerical algorithm for approximating, in a sense to be explained below, a fixed point of such a mapping (Scarf 1967a: 1326). Scarf never built an AGE model, but hinted that “these novel numerical techniques might be useful in assessing consequences for the economy of a change in the economic environment” (Kehoe et al. 2005, citing Scarf 1967b). His students elaborated the Scarf algorithm into a tool box, where the price vector could be solved for any changes in policies (or exogenous shocks), giving the equilibrium ‘adjustments’ needed for the prices. This method was first used by Shoven and Whalley (1972 and 1973), and then was developed through the 1970s by Scarf’s students and others. Most contemporary applied general equilibrium models are numerical analogs of traditional two-sector general equilibrium models popularized by James Meade, Harry Johnson, Arnold Harberger, and others in the 1950s and 1960s. Earlier analytic work with these models has examined the distortionary effects of taxes, tariffs, and other policies, along with functional incidence questions. More recent applied models, including those discussed here, provide numerical estimates of efficiency and distributional effects within the same framework. Scarf's fixed-point method was a break-through in the mathematics of computation generally, and specifically in optimization and computational economics. Later researchers continued to develop iterative methods for computing fixed-points, both for topological models like Scarf's and for models described by functions with continuous second derivatives or convexity or both. Of course, "global Newton methods" for essentially convex and smooth functions and path-following methods for diffeomorphisms converged faster than did robust algorithms for continuous functions, when the smooth methods are applicable. == AGE and CGE models == AGE models, being based on Arrow–Debreu general equilibrium theory, work in a different manner than CGE models. The model first establishes the existence of equilibrium through the standard Arrow–Debreu exposition, then inputs data into all the various sectors, and then applies Scarf’s algorithm (Scarf 1967a, 1967b and Scarf with Hansen 1973) to solve for a price vector that would clear all markets. This algorithm would narrow down the possible relative prices through a simplex method, which kept reducing the size of the ‘net’ within which possible solutions were found. AGE modelers then consciously choose a cutoff, and set an approximate solution as the net never closed on a unique point through the iteration process. CGE models are based on macro balancing equations, and use an equal number of equations (based on the standard macro balancing equations) and unknowns solvable as simultaneous equations, where exogenous variables are changed outside the model, to give the endogenous results. == References == === Bibliography === Cardenete, M. Alejandro, Guerra, Ana-Isabel and Sancho, Ferran (2012). Applied General Equilibrium: An Introduction. Springer. Scarf, H.E., 1967a, “The approximation of Fixed Points of a continuous mapping”, SIAM Journal on Applied Mathematics 15: 1328–43 Scarf, H.E., 1967b, “On the computation of equilibrium prices” in Fellner, W.J. (ed.), Ten Economic Studies in the tradition of Irving Fischer, New York, NY: Wiley Scarf, H.E. with Hansen, T, 1973, The Computation of Economic Equilibria, Cowles Foundation for Research in economics at Yale University, Monograph No. 24, New Haven, CT and London, UK: Yale University Press Kehoe, T.J., Srinivasan, T.N. and Whalley, J., 2005, Frontiers in Applied General Equilibrium Modeling, In honour of Herbert Scarf, Cambridge, UK: Cambridge University Press Shoven, J. B. and Whalley, J., 1972, "A General Equilibrium Calculation of the Effects of Differential Taxation of Income from Capital in the U.S.", Journal of Public Economics 1 (3–4), November, pp. 281–321 Shoven, J.B. and Whalley, J., 1973, “General Equilibrium with Taxes: A Computational Procedure and an Existence Proof”, The Review of Economic Studies 40 (4), October, pp. 475–89 Velupillai, K.V., 2006, “Algorithmic foundations of computable general equilibrium theory”, Applied Mathematics and Computation 179, pp. 360–69
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Wikipedia:Approximately continuous function#0
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In mathematics, particularly in mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the ordinary limit with an approximate limit. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory. == Definition == Let E ⊆ R n {\displaystyle E\subseteq \mathbb {R} ^{n}} be a Lebesgue measurable set, f : E → R k {\displaystyle f\colon E\to \mathbb {R} ^{k}} be a measurable function, and x 0 ∈ E {\displaystyle x_{0}\in E} be a point where the Lebesgue density of E {\displaystyle E} is 1. The function f {\displaystyle f} is said to be approximately continuous at x 0 {\displaystyle x_{0}} if and only if the approximate limit of f {\displaystyle f} at x 0 {\displaystyle x_{0}} exists and equals f ( x 0 ) {\displaystyle f(x_{0})} . == Properties == A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain. The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem provides a remarkable characterization: Stepanov-Denjoy theorem: A function is measurable if and only if it is approximately continuous almost everywhere. Approximately continuous functions are intimately connected to Lebesgue points. For a function f ∈ L 1 ( E ) {\displaystyle f\in L^{1}(E)} , a point x 0 {\displaystyle x_{0}} is a Lebesgue point if it is a point of Lebesgue density 1 for E {\displaystyle E} and satisfies lim r ↓ 0 1 λ ( B r ( x 0 ) ) ∫ E ∩ B r ( x 0 ) | f ( x ) − f ( x 0 ) | d x = 0 {\displaystyle \lim _{r\downarrow 0}{\frac {1}{\lambda (B_{r}(x_{0}))}}\int _{E\cap B_{r}(x_{0})}|f(x)-f(x_{0})|\,dx=0} where λ {\displaystyle \lambda } denotes the Lebesgue measure and B r ( x 0 ) {\displaystyle B_{r}(x_{0})} represents the ball of radius r {\displaystyle r} centered at x 0 {\displaystyle x_{0}} . Every Lebesgue point of a function is necessarily a point of approximate continuity. The converse relationship holds under additional constraints: when f {\displaystyle f} is essentially bounded, its points of approximate continuity coincide with its Lebesgue points. == See also == Approximate limit Density point Density topology (which serves to describe approximately continuous functions in a different way, as continuous functions for a different topology) Lebesgue point Lusin's theorem Measurable function == References ==
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Wikipedia:Approximation property (ring theory)#0
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In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A. The notion of the approximation property is due to Michael Artin. == See also == Artin approximation theorem Popescu's theorem == Notes == == References == Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal. 104: 85–115. doi:10.1017/S0027763000022698. Rotthaus, Christel (1987). "On the approximation property of excellent rings". Inventiones Mathematicae. 88: 39–63. Bibcode:1987InMat..88...39R. doi:10.1007/BF01405090. Artin, M (1969). "Algebraic approximation of structures over complete local rings". Publications Mathématiques de l'IHÉS. 36: 23–58. doi:10.1007/BF02684596. ISSN 0073-8301. Artin, M (1968). "On the solutions of analytic equations". Inventiones Mathematicae. 5 (4): 277–291. Bibcode:1968InMat...5..277A. doi:10.1007/BF01389777. ISSN 0020-9910.
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Wikipedia:Aram Arutyunov#0
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Aram Arutyunov (Russian: Ара́м Влади́мирович Арутю́нов) (born 1956) is a Russian mathematician, Professor, Dr.Sc., a professor at the Faculty of Computer Science at the Moscow State University and the Peoples' Friendship University of Russia. He defended the thesis «Perturbation of optimal control problems and necessary conditions for the extremum of the first and second order» for the degree of Doctor of Physical and Mathematical Sciences (1988). and was awarded the title of Professor (1991). He has authored seven books and 318 scientific articles. == References == == Literature == Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. 2010. pp. 296–297. ISBN 978-5-211-05838-5 – via Author-compiler Evgeny Grigoriev. == External links == MSU CMC(in Russian) Scientific works of Aram Arutyunov Scientific works of Aram Arutyunov(in English)
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Wikipedia:Arason invariant#0
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In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7). The Rost invariant is a generalization of the Arason invariant to other algebraic groups. == Definition == Suppose that W(k) is the Witt ring of quadratic forms over a field k and I is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from I3 to the Galois cohomology group H3(k,Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –a, –b, ab, -c, ac, bc, -abc (the 3-fold Pfister form«a,b,c») it is given by the cup product of the classes of a, b, c in H1(k,Z/2Z) = k*/k*2. The Arason invariant vanishes on I4, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from I3/I4 to H3(k,Z/2Z). == References == Arason, Jón Kr. (1975), "Cohomologische Invarianten quadratischer Formen", J. Algebra (in German), 36 (3): 448–491, doi:10.1016/0021-8693(75)90145-3, ISSN 0021-8693, MR 0389761, Zbl 0314.12104 Esnault, Hélène; Kahn, Bruno; Levine, Marc; Viehweg, Eckart (1998), "The Arason invariant and mod 2 algebraic cycles", J. Amer. Math. Soc., 11 (1): 73–118, doi:10.1090/S0894-0347-98-00248-3, ISSN 0894-0347, MR 1460391, Zbl 1025.11009 Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5, MR 1999383, Zbl 1159.12311 Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, p. 436, ISBN 0-8218-0904-0, Zbl 0955.16001
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Wikipedia:Arbelos#0
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In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the baseline) that contains their diameters. The earliest known reference to this figure is in Archimedes's Book of Lemmas, where some of its mathematical properties are stated as Propositions 4 through 8. The word arbelos is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. == Properties == Two of the semicircles are necessarily concave, with arbitrary diameters a and b; the third semicircle is convex, with diameter a+b. Let the diameters of the smaller semicircles be BA and AC; then the diameter of the larger semircle is BC. === Area === Let H be the intersection of the larger semicircle with the line perpendicular to BC at A. Then the area of the arbelos is equal to the area of a circle with diameter HA. Proof: For the proof, reflect the arbelos over the line through the points B and C, and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles (with diameters BA, AC) are subtracted from the area of the large circle (with diameter BC). Since the area of a circle is proportional to the square of the diameter (Euclid's Elements, Book XII, Proposition 2; we do not need to know that the constant of proportionality is π/4), the problem reduces to showing that 2 | A H | 2 = | B C | 2 − | A C | 2 − | B A | 2 {\displaystyle 2|AH|^{2}=|BC|^{2}-|AC|^{2}-|BA|^{2}} . The length |BC| equals the sum of the lengths |BA| and |AC|, so this equation simplifies algebraically to the statement that | A H | 2 = | B A | | A C | {\displaystyle |AH|^{2}=|BA||AC|} . Thus the claim is that the length of the segment AH is the geometric mean of the lengths of the segments BA and AC. Now (see Figure) the triangle BHC, being inscribed in the semicircle, has a right angle at the point H (Euclid, Book III, Proposition 31), and consequently |HA| is indeed a "mean proportional" between |BA| and |AC| (Euclid, Book VI, Proposition 8, Porism). This proof approximates the ancient Greek argument; Harold P. Boas cites a paper of Roger B. Nelsen who implemented the idea as the following proof without words. === Rectangle === Let D and E be the points where the segments BH and CH intersect the semicircles AB and AC, respectively. The quadrilateral ADHE is actually a rectangle. Proof: ∠BDA, ∠BHC, and ∠AEC are right angles because they are inscribed in semicircles (by Thales's theorem). The quadrilateral ADHE therefore has three right angles, so it is a rectangle. Q.E.D. === Tangents === The line DE is tangent to semicircle BA at D and semicircle AC at E. Proof: Since ADHE is a rectangle, the diagonals AH and DE have equal length and bisect each other at their intersection O. Therefore, | O D | = | O A | = | O E | {\displaystyle |OD|=|OA|=|OE|} . Also, since OA is perpendicular to the diameters BA and AC, OA is tangent to both semicircles at the point A. Finally, because the two tangents to a circle from any given exterior point have equal length, it follows that the other tangents from O to semicircles BA and AC are OD and OE respectively. === Archimedes' circles === The altitude AH divides the arbelos into two regions, each bounded by a semicircle, a straight line segment, and an arc of the outer semicircle. The circles inscribed in each of these regions, known as the Archimedes' circles of the arbelos, have the same size. == Variations and generalisations == The parbelos is a figure similar to the arbelos, that uses parabola segments instead of half circles. A generalisation comprising both arbelos and parbelos is the f-belos, which uses a certain type of similar differentiable functions. In the Poincaré half-plane model of the hyperbolic plane, an arbelos models an ideal triangle. == Etymology == The name arbelos comes from Greek ἡ ἄρβηλος he árbēlos or ἄρβυλος árbylos, meaning "shoemaker's knife", a knife used by cobblers from antiquity to the current day, whose blade is said to resemble the geometric figure. == See also == Archimedes' quadruplets Bankoff circle Schoch circles Schoch line Woo circles Pappus chain Salinon == References == == Bibliography == Johnson, R. A. (1960). Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Mifflin ed.). New York: Dover Publications. pp. 116–117. ISBN 978-0-486-46237-0. {{cite book}}: ISBN / Date incompatibility (help) Ogilvy, C. S. (1990). Excursions in Geometry. Dover. pp. 51–54. ISBN 0-486-26530-7. Sondow, J. (2013). "The parbelos, a parabolic analog of the arbelos". Amer. Math. Monthly. 120 (10): 929–935. arXiv:1210.2279. doi:10.4169/amer.math.monthly.120.10.929. S2CID 33402874. American Mathematical Monthly, 120 (2013), 929–935. Wells, D. (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 5–6. ISBN 0-14-011813-6. == External links == Media related to Arbelos at Wikimedia Commons The dictionary definition of arbelos at Wiktionary
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Wikipedia:Archibald James Macintyre#0
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Prof Archibald James Macintyre HFRSE (3 July 1908 – 4 August 1967) was a British-born mathematician. == Life == He was born in Sheffield on 3 July 1908, the second child of William Ewart Archibald Macintyre (b.1878) previously of Long Eaton, and his wife, Mary Beatrice Askew. His father was a schoolmaster in Sheffield and his mother was a former teacher. Archibald was educated at the Central Secondary School in Sheffield (previously known as the High Storrs Grammar School). He left school in 1926 and won a place at Magdalene College, Cambridge studying a Mathematics Tripos under Arthur Stanley Ramsey. Fellow students included Donald Coxeter, Raymond Paley and Harold Davenport. He graduated BA as a Wrangler in 1929 then began research under Dr Edward Collingwood. In 1930 he became an assistant lecturer in both applied maths and theoretical physics at Cambridge University. He received his doctorate (PhD) in 1933. In 1936 he accepted a post of lecturer at Aberdeen University. Here he stayed for many years, rising to senior lecturer. In 1947 he was elected an Honorary Fellow of the Royal Society of Edinburgh. His proposers were E. M. Wright, Ivor Etherington, Edward Thomas Copson, Edmund Taylor Whittaker and James Cossar. In 1958 he moved to the University of Cincinnati in the United States, as a visiting professor of mathematics. He was recruited primarily as a reaction to Sputnik. America wanted to increase its role in the sciences and math. His wife stayed in Aberdeen, Scotland where she continued to teach mathematics at King's College. A year later he accepted a permanent position at the University of Cincinnati and sent for his wife who was also given a teaching position as a lecturer in mathematics. They formed a highly unusual husband-wife team. He died in Cincinnati on 4 August 1967, eight years after his wife died of breast cancer. == Family == In 1940, he married Sheila Scott, a noted mathematician in her own right. They had three children: Alister William (February 8, 1944 – May 17, 2017), Douglas who died at age two in 1948, and Susan Elizabeth who currently teaches mathematics for Walnut Hills High School in Cincinnati. == References ==
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Wikipedia:Archibald Read Richardson#0
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Archibald Read Richardson FRS (21 August 1881 – 4 November 1954) was a British mathematician known for his work in algebra. == Career == Richardson collaborated with Dudley E. Littlewood on invariants and group representation theory. They introduced the immanant of a matrix, studied Schur functions and developed the Littlewood–Richardson rule for their multiplication. == Awards and honours == Richardson was elected a Fellow of the Royal Society on 21 March 1946. == See also == Quasideterminant == References ==
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Wikipedia:Archimedean circle#0
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In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and r denotes the radius of any of the inner half circles, then the radius ρ of such an Archimedean circle is given by ρ = 1 2 r ( 1 − r ) , {\displaystyle \rho ={\frac {1}{2}}r\left(1-r\right),} There are over fifty different known ways to construct Archimedean circles. == Origin == An Archimedean circle was first constructed by Archimedes in his Book of Lemmas. In his book, he constructed what is now known as Archimedes' twin circles. === Radius === If a {\displaystyle a} and b {\displaystyle b} are the radii of the small semicircles of the arbelos, the radius of an Archimedean circle is equal to R = a b a + b {\displaystyle R={\frac {ab}{a+b}}} This radius is thus 1 R = 1 a + 1 b {\displaystyle {\frac {1}{R}}={\frac {1}{a}}+{\frac {1}{b}}} . The Archimedean circle with center C {\displaystyle C} (as in the figure at right) is tangent to the tangents from the centers of the small semicircles to the other small semicircle. == Other Archimedean circles finders == === Leon Bankoff === Leon Bankoff constructed other Archimedean circles called Bankoff's triplet circle and Bankoff's quadruplet circle. === Thomas Schoch === In 1978 Thomas Schoch found a dozen more Archimedean circles (the Schoch circles) that have been published in 1998. He also constructed what is known as the Schoch line. === Peter Y. Woo === Peter Y. Woo considered the Schoch line, and with it, he was able to create a family of infinitely many Archimedean circles known as the Woo circles. === Frank Power === In the summer of 1998, Frank Power introduced four more Archimedes circles known as Archimedes' quadruplets. === Archimedean circles in Wasan geometry (Japanese geometry) === In 1831, Nagata (永田岩三郎遵道) proposed a sangaku problem involving two Archimedean circles, which are denoted by W6 and W7 in [3]. In 1853, Ootoba (大鳥羽源吉守敬) proposed a sangaku problem involving an Archimedean circle. == References ==
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Wikipedia:Archimedean spiral#0
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The Archimedean spiral (also known as Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. The term Archimedean spiral is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to Archimedes' spiral (the specific arithmetic spiral of Archimedes). It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation r = b ⋅ θ {\displaystyle r=b\cdot \theta } with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses. Archimedes described such a spiral in his book On Spirals. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon. == Derivation of general equation of spiral == A physical approach is used below to understand the notion of Archimedean spirals. Suppose a point object moves in the Cartesian system with a constant velocity v directed parallel to the x-axis, with respect to the xy-plane. Let at time t = 0, the object was at an arbitrary point (c, 0, 0). If the xy plane rotates with a constant angular velocity ω about the z-axis, then the velocity of the point with respect to z-axis may be written as: | v 0 | = v 2 + ω 2 ( v t + c ) 2 v x = v cos ω t − ω ( v t + c ) sin ω t v y = v sin ω t + ω ( v t + c ) cos ω t {\displaystyle {\begin{aligned}|v_{0}|&={\sqrt {v^{2}+\omega ^{2}(vt+c)^{2}}}\\v_{x}&=v\cos \omega t-\omega (vt+c)\sin \omega t\\v_{y}&=v\sin \omega t+\omega (vt+c)\cos \omega t\end{aligned}}} As shown in the figure alongside, we have vt + c representing the modulus of the position vector of the particle at any time t, with vx and vy as the velocity components along the x and y axes, respectively. ∫ v x d t = x ∫ v y d t = y {\displaystyle {\begin{aligned}\int v_{x}\,dt&=x\\\int v_{y}\,dt&=y\end{aligned}}} The above equations can be integrated by applying integration by parts, leading to the following parametric equations: x = ( v t + c ) cos ω t y = ( v t + c ) sin ω t {\displaystyle {\begin{aligned}x&=(vt+c)\cos \omega t\\y&=(vt+c)\sin \omega t\end{aligned}}} Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation x 2 + y 2 = v ω ⋅ arctan y x + c {\displaystyle {\sqrt {x^{2}+y^{2}}}={\frac {v}{\omega }}\cdot \arctan {\frac {y}{x}}+c} (using the fact that ωt = θ and θ = arctan y/x) or tan ( ( x 2 + y 2 − c ) ⋅ ω v ) = y x {\displaystyle \tan \left(\left({\sqrt {x^{2}+y^{2}}}-c\right)\cdot {\frac {\omega }{v}}\right)={\frac {y}{x}}} Its polar form is r = v ω ⋅ θ + c . {\displaystyle r={\frac {v}{\omega }}\cdot \theta +c.} == Arc length and curvature == Given the parametrization in cartesian coordinates f : θ ↦ ( r cos θ , r sin θ ) = ( b θ cos θ , b θ sin θ ) {\displaystyle f\colon \theta \mapsto (r\,\cos \theta ,r\,\sin \theta )=(b\,\theta \,\cos \theta ,b\,\theta \,\sin \theta )} the arc length from θ1 to θ2 is b 2 [ θ 1 + θ 2 + ln ( θ + 1 + θ 2 ) ] θ 1 θ 2 {\displaystyle {\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\ln \left(\theta +{\sqrt {1+\theta ^{2}}}\right)\right]_{\theta _{1}}^{\theta _{2}}} or, equivalently: b 2 [ θ 1 + θ 2 + arsinh θ ] θ 1 θ 2 . {\displaystyle {\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\operatorname {arsinh} \theta \right]_{\theta _{1}}^{\theta _{2}}.} The total length from θ1 = 0 to θ2 = θ is therefore b 2 [ θ 1 + θ 2 + ln ( θ + 1 + θ 2 ) ] . {\displaystyle {\frac {b}{2}}\left[\theta \,{\sqrt {1+\theta ^{2}}}+\ln \left(\theta +{\sqrt {1+\theta ^{2}}}\right)\right].} The curvature is given by κ = θ 2 + 2 b ( θ 2 + 1 ) 3 2 {\displaystyle \kappa ={\frac {\theta ^{2}+2}{b\left(\theta ^{2}+1\right)^{\frac {3}{2}}}}} == Characteristics == The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmetic spiral". In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. The Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axis will yield the other arm. For large θ a point moves with well-approximated uniform acceleration along the Archimedean spiral while the spiral corresponds to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity (see contribution from Mikhail Gaichenkov). As the Archimedean spiral grows, its evolute asymptotically approaches a circle with radius |v|/ω. == General Archimedean spiral == Sometimes the term Archimedean spiral is used for the more general group of spirals r = a + b ⋅ θ 1 c . {\displaystyle r=a+b\cdot \theta ^{\frac {1}{c}}.} The normal Archimedean spiral occurs when c = 1. Other spirals falling into this group include the hyperbolic spiral (c = −1), Fermat's spiral (c = 2), and the lituus (c = −2). == Applications == One method of squaring the circle, due to Archimedes, makes use of an Archimedean spiral. Archimedes also showed how the spiral can be used to trisect an angle. Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs. The Archimedean spiral has a variety of real-world applications. Scroll compressors, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, involutes of a circle of the same size that almost resemble Archimedean spirals, or hybrid curves. Archimedean spirals can be found in spiral antenna, which can be operated over a wide range of frequencies. The coils of watch balance springs and the grooves of very early gramophone records form Archimedean spirals, making the grooves evenly spaced (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record). Asking for a patient to draw an Archimedean spiral is a way of quantifying human tremor; this information helps in diagnosing neurological diseases. Archimedean spirals are also used in digital light processing (DLP) projection systems to minimize the "rainbow effect", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly. Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter. They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder. Many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine's wheel) are Archimedean. For instance, the star LL Pegasi shows an approximate Archimedean spiral in the dust clouds surrounding it, thought to be ejected matter from the star that has been shepherded into a spiral by another companion star as part of a double star system. == Construction methods == The Archimedean Spiral cannot be constructed precisely by traditional compass and straightedge methods, since the arithmetic spiral requires the radius of the curve to be incremented constantly as the angle at the origin is incremented. But an arithmetic spiral can be constructed approximately, to varying degrees of precision, by various manual drawing methods. One such method uses compass and straightedge; another method uses a modified string compass. The common traditional construction uses compass and straightedge to approximate the arithmetic spiral. First, a large circle is constructed and its circumference is subdivided by 12 diameters into 12 arcs (of 30 degrees each; see regular dodecagon). Next, the radius of this circle is itself subdivided into 12 unit segments (radial units), and a series of concentric circles is constructed, each with radius incremented by one radial unit. Starting with the horizontal diameter and the innermost concentric circle, the point is marked where its radius intersects its circumference; one then moves to the next concentric circle and to the next diameter (moving up to construct a counterclockwise spiral, or down for clockwise) to mark the next point. After all points have been marked, successive points are connected by a line approximating the arithmetic spiral (or by a smooth curve of some sort; see French Curve). Depending on the desired degree of precision, this method can be improved by increasing the size of the large outer circle, making more subdivisions of both its circumference and radius, increasing the number of concentric circles (see Polygonal Spiral). Approximating the Archimedean Spiral by this method is of course reminiscent of Archimedes’ famous method of approximating π by doubling the sides of successive polygons (see Polygon approximation of π). Compass and straightedge construction of the Spiral of Theodorus is another simple method to approximate the Archimedean Spiral. A mechanical method for constructing the arithmetic spiral uses a modified string compass, where the string wraps and winds (or unwraps/unwinds) about a fixed central pin (that does not pivot), thereby incrementing (or decrementing) the length of the radius (string) as the angle changes (the string winds around the fixed pin which does not pivot). Such a method is a simple way to create an arithmetic spiral, arising naturally from use of a string compass with winding pin (not the loose pivot of a common string compass). The string compass drawing tool has various modifications and designs, and this construction method is reminiscent of string-based methods for creating ellipses (with two fixed pins). Yet another mechanical method is a variant of the previous string compass method, providing greater precision and more flexibility. Instead of the central pin and string of the string compass, this device uses a non-rotating shaft (column) with helical threads (screw; see Archimedes’ screw) to which are attached two slotted arms: one horizontal arm is affixed to (travels up) the screw threads of the vertical shaft at one end, and holds a drawing tool at the other end; another sloped arm is affixed at one end to the top of the screw shaft, and is joined by a pin loosely fitted in its slot to the slot of the horizontal arm. The two arms rotate together and work in consort to produce the arithmetic spiral: as the horizontal arm gradually climbs the screw, that arm’s slotted attachment to the sloped arm gradually shortens the drawing radius. The angle of the sloped arm remains constant throughout (traces a cone), and setting a different angle varies the pitch of the spiral. This device provides a high degree of precision, depending on the precision with which the device is machined (machining a precise helical screw thread is a related challenge). And of course the use of a screw shaft in this mechanism is reminiscent of Archimedes’ screw. == See also == == References == == External links == Jonathan Matt making the Archimedean spiral interesting - Video : The surprising beauty of Mathematics - TedX Talks, Green Farms Weisstein, Eric W. "Archimedes' Spiral". MathWorld. archimedean spiral at PlanetMath. Page with Java application to interactively explore the Archimedean spiral and its related curves Online exploration using JSXGraph (JavaScript) Archimedean spiral at "mathcurve"
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Wikipedia:Archimedes' quadruplets#0
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In geometry, Archimedes' quadruplets are four congruent circles associated with an arbelos. Introduced by Frank Power in the summer of 1998, each have the same area as Archimedes' twin circles, making them Archimedean circles. == Construction == An arbelos is formed from three collinear points A, B, and C, by the three semicircles with diameters AB, AC, and BC. Let the two smaller circles have radii r1 and r2, from which it follows that the larger semicircle has radius r = r1+r2. Let the points D and E be the center and midpoint, respectively, of the semicircle with the radius r1. Let H be the midpoint of line AC. Then two of the four quadruplet circles are tangent to line HE at the point E, and are also tangent to the outer semicircle. The other two quadruplet circles are formed in a symmetric way from the semicircle with radius r2. == Proof of congruency == According to Proposition 5 of Archimedes' Book of Lemmas, the common radius of Archimedes' twin circles is: r 1 ⋅ r 2 r . {\displaystyle {\frac {r_{1}\cdot r_{2}}{r}}.} By the Pythagorean theorem: ( H E ) 2 = ( r 1 ) 2 + ( r 2 ) 2 . {\displaystyle \left(HE\right)^{2}=\left(r_{1}\right)^{2}+\left(r_{2}\right)^{2}.} Then, create two circles with centers Ji perpendicular to HE, tangent to the large semicircle at point Li, tangent to point E, and with equal radii x. Using the Pythagorean theorem: ( H J i ) 2 = ( H E ) 2 + x 2 = ( r 1 ) 2 + ( r 2 ) 2 + x 2 {\displaystyle \left(HJ_{i}\right)^{2}=\left(HE\right)^{2}+x^{2}=\left(r_{1}\right)^{2}+\left(r_{2}\right)^{2}+x^{2}} Also: H J i = H L i − x = r − x = r 1 + r 2 − x {\displaystyle HJ_{i}=HL_{i}-x=r-x=r_{1}+r_{2}-x~} Combining these gives: ( r 1 ) 2 + ( r 2 ) 2 + x 2 = ( r 1 + r 2 − x ) 2 {\displaystyle \left(r_{1}\right)^{2}+\left(r_{2}\right)^{2}+x^{2}=\left(r_{1}+r_{2}-x\right)^{2}} Expanding, collecting to one side, and factoring: 2 r 1 r 2 − 2 x ( r 1 + r 2 ) = 0 {\displaystyle 2r_{1}r_{2}-2x\left(r_{1}+r_{2}\right)=0} Solving for x: x = r 1 ⋅ r 2 r 1 + r 2 = r 1 ⋅ r 2 r {\displaystyle x={\frac {r_{1}\cdot r_{2}}{r_{1}+r_{2}}}={\frac {r_{1}\cdot r_{2}}{r}}} Proving that each of the Archimedes' quadruplets' areas is equal to each of Archimedes' twin circles' areas. == References == == More readings == Arbelos: Book of Lemmas, Pappus Chain, Archimedean Circle, Archimedes' Quadruplets, Archimedes' Twin Circles, Bankoff Circle, S. ISBN 1156885493
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Wikipedia:Archives of American Mathematics#0
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The Archives of American Mathematics, located at the University of Texas at Austin, aims to collect, preserve, and provide access to the papers principally of American mathematicians and the records of American mathematical organizations. == History == The Archives began in 1975 at the University of Texas at Austin with the preservation of the papers of Texas mathematicians R.L. Moore and H.S. Vandiver. In 1978, the Mathematical Association of America established the university as the official repository for its archival records and the name "Archives of American Mathematics" was adopted to encompass all of the mathematical archival collections at the university. Originally a part of the Harry Ransom Center, in 1984, the Archives was added to the special collections of the Briscoe Center for American History at the University of Texas at Austin. == Collections == The AAM includes approximately 120 collections. === Notable examples === Mathematical Association of America Records. Thomas F. Banchoff Papers document a career of teaching, writing, and making mathematical films. Marion Walter Photograph Collection includes photographs of A.A. Albert, H.S.M. Coxeter, Paul Erdős, Fritz John, D.H. Lehmer, Alexander Ostrowski, George Polya, Mina Rees, and Olga Taussky-Todd. School Mathematics Study Group Records document the history of the "New Math" movement of the 1960s, and includes the files of the director, Edward G. Begle. Dorothy L. Bernstein Papers reflect both her professional and personal life. Paul R. Halmos Photograph Collection consists of 14,000 photographs Halmos and others took from the 1930s to 2006. Ivor Grattan-Guinness Papers reflect the career of a mathematics historian. Paul Erdős and Carl Pomerance Correspondence Collection consists of 435 letters between Erdős and Pomerance. === Other holdings === == Related collections elsewhere == Significant archives of American mathematicians and their organizations are held by other repositories. The following are examples which include a few Canadian collections with substantial United States connections. For the complete holdings, the catalogs of the individual repositories would need to be consulted. In addition, the archives of academic institutions will typically include administrative records of mathematics departments and clubs as well as the papers of faculty. John Hay Library, Brown University -- American Mathematical Society Records (1888- ); Raymond Clare Archibald (1875-1957); James Glaisher (1848-1928); R. G. D. Richardson (1878-1949); Marshall Harvey Stone (1903-1989); James Joseph Sylvester (1814-1897) also at St. John's College (Cambridge). American Philosophical Society -- Robert Patterson (1743-1824); David Rittenhouse (1732-1796); Robert Adrain (1775-1843); Samuel Stanley Wilks (1906-1964). Amherst College—Ebenezer Strong Snell (1801-1876). Boston Public Library -- Nathaniel Bowditch (1773-1838); Nicholas Pike (1743-1819). Bridgewater State University—L.S. Dederich (1883-1972). College of Charleston Library—Lewis Reeves Gibbes (1810-1894), also Library of Congress; Columbia University -- Arthur Korn (1870-1945); Cassius Jackson Keyser (1862-1947); Christine Franklin (1847-1930) and Fabian Franklin (1853-1939); F. A. P. Barnard (1809-1889); Harold Hotelling (1895-); Henry Seely White (1861-1943); John Howard Van Amringe (1835-1915); Thomas Scott Fiske (1865-1944); David Eugene Smith (1860-1944). Dartmouth College -- George Robert Stibitz (1904-1995). Duke University—Edward Henry Courtenay (1803-1853). Hampshire College—Herman Goldstine (1913-2004). Harvard University -- Benjamin Peirce (1809-1880); Charles Sanders Peirce (1839-1914); Damodar Dharmanand Kosambi (1907–1966); Isaac Greenwood (1702-1745); John Farrar (1779-1853); John Winthrop (1714–1779); Maxime Bôcher (1867-1918); Richard Von Mises (1883-1953); Thomas Hill (1818-1891); George David Birkhoff (1884-1944). Iowa State University -- American Statistical Association (1839-); Herbert Solomon (1919-2004); Edward J. Wegman (1943-); Eugene Lukacs (1906-1987); Ingram Olkin (1924-). Hope College—Albert Eugene Lampen (1887-1963); Jay Erenst Folkert (1916-). Library of Congress -- Andrew Ellicott (1754-1820); George F. Becker (1847-1919); Oswald Veblen (1880-1960); Lewis Reeves Gibbes (1810-1894); John von Neumann (1903-1957), also College of Charleston; Gloria Ford Gilmer (1928-2021). Maryland Historical Society -- John Henry Alexander (1812-1867). McMaster University (Canada) -- Bertrand Russell (1872-1970). Massachusetts Institute of Technology -- John Daniel Runkle (1822-1902); Norbert Wiener (1894-1964). National Research Council (Canada) -- Julius Plücker (1801-1868). New Jersey Historical Society -- Francis Robbins Upton (1852-1921). New York Public Library -- Ferdinand Rudolph Hassler (1770-1843). New York University -- Richard Courant (1888-1972). Northwestern University -- Ernst D. Hellinger (1883-1950); Helen M. Clark (1908-1974); Lois W. Griffiths (1899-1981); Thomas Franklin Holgate (1859-1945). Northwestern State University -- Guy Waldo Dunnington (1906-1974) Ohio History Connection -- Jared Mansfield (1759-1830), also U.S. Military Academy. Princeton University -- Alfred James Lotka (1880-1949); Walter Minto (1753-1796); Eugene Paul Wigner (1902-1995); Henry Dallas Thompson (1864-1927); Kurt Gödel (1906-1978); Nicola Fergola (1757-1824); Sylvester Robins (files: 1880–1899). Rice University—Fred Terry Rogers (1914-1956); Salomon Bochner (1899-1982). Rockefeller University -- Mark Kac (1914-1984). Rutgers University—Edward Albert Bowser (1837-1910); George Washington Coakley (1814-1893). Smith College, Sophia Smith Collection -- Dorothy Maud Wrinch (1895-1976). Stanford University -- Georg Pólya (1887-1985). University of Chicago -- A. Adrian Albert (1905-1972 ); Saunders Mac Lane (1909-2005); E. H. Moore (1862-1932); Alfred L. Putnam (1916-2004); Nicolas Rashevsky (1899-1972); Ernest Julius Wilczynski (1876-1932). University of Illinois at Urbana-Champaign -- Arnold Emch (1871-1959); Arthur Byron Coble (1878-1966); George Abram Miller (1863-1951); George William Meyers (1864-1931); Leonard L. Steimley (1890-1975); Olive C. Hazlett (1890-1974); Robert Daniel Carmichael (1879-1967). University of Michigan—Wooster Woodruff Beman (1850-1922); Louis Allen Hopkins (1881-). State Historical Society of Missouri—Joseph Ficklin (1833-1887). University of North Carolina; U. of Texas -- Charles Scott Venable (1827-1900). University of Oklahoma Library -- Nathan Altshiller Court (1881-1968). University of Toronto -- Kenneth O. May (1915-1977). University of Virginia -- G. T. Whyburn (1904-1969) University of Washington Libraries -- Carl B. Allendoerfer (1911-1974). University of Wisconsin -- Albert C. Schaeffer (files: 1954–1956); Bronson Barlow (Mathematics of Design) (b. 1924); Charles S. Slichter (files: 1891–1941); Cyrus C. MacDuffee (1895-1961); Edward Burr Van Vleck (1863-1943); Ernest B. Skinner (files: 1892–1935); Isaac Schoenberg (files: 1930–1980); Ivan Sokolnikoff (1901-); J. Barkley Rosser (1907-1989); John D. Mayor (?); Mark H. Ingraham (1896-1982); Military Training Programs, WW II (1943-1945); U.S. Naval Research Office (1951-1955); Rudolph E. Langer (1894-1968); Stephen Kleene (1909-1994); Warren Weaver (1894-1978). Virginia Military Institute -- Claudius Crozet (1789-1864). Virginia Polytechnic Institute And State University -- Irving John Good (1916-2009); John Edward Williams (1867-1943). Wake Forest University—John Wesley Sawyer (1916-); Roland L. Gay (1905-1979) Western Reserve Historical Society (Cleveland) -- Elisha Scott Loomis (1852-1940). Yale University Library -- Abraham Robinson (1918-1974); Elias Loomis (1811-1889); Josiah Willard Gibbs (1839-1903). Yeshiva University -- Jekuthiel Ginsburg (1889-1957) == References == == External links == Archives of American Mathematics
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Wikipedia:Arend Heyting#0
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Arend Heyting (Dutch: [ˈaːrənt ˈɦɛitɪŋ]; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. == Biography == Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic. Heyting gave the first formal development of intuitionistic logic in order to codify Brouwer's way of doing mathematics. The inclusion of Brouwer's name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, as Brouwer was opposed in principle to the formalisation of certain intuitionistic principles (and went as far as calling Heyting's work a "sterile exercise"). In 1942 he became a member of the Royal Netherlands Academy of Arts and Sciences. Heyting was born in Amsterdam, Netherlands, and died in Lugano, Switzerland. == Selected publications == Heyting, Arend (1930). "Die formalen Regeln der intuitionistischen Logik". Sitzungsberichte der preußischen Akademie der Wissenschaften, phys.-math. Klasse (in German): 42–56, 57–71, 158–169. OCLC 601568391. (abridged reprint in Berka, Karel; Kreiser, Lothar, eds. (1986). Logik-Texte. De Gruyter. pp. 188–192. doi:10.1515/9783112645826.) — (1934). Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie (in German). Berlin: Springer. — (1941). "Untersuchungen der intuitionistischen Algebra". Verh. Nederl. Akad. Wetensch. Afd. Natuurk. Sect. 1 (in German). 18 (2): 36. — (1956). Intuitionism. An Introduction. Amsterdam: North-Holland Publishing Co. — (1959). "Axioms for intuitionistic plane affine geometry. The axiomatic method. With special reference to geometry and physics". In Henkin, L.; Suppes, P.; Tarski, A. (eds.). Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957–Jan. 4, 1958. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co. pp. 160–173. — (1962). "After thirty years". In Nagel, E. (ed.). Logic, Methodology and Philosophy of Science (Proc. 1960 Internat. Congr.). Stanford, Calif.: Stanford Univ. Press. pp. 194–197. — (1963). Axiomatic projective geometry. Bibliotheca Mathematica. Vol. V. New York; Groningen; Amsterdam: Interscience Publishers John Wiley & Sons, Inc.; P. Noordhoff N.V.; North-Holland Publishing Co. — (1966). Intuitionism: An Introduction (Second revised ed.). Amsterdam: North-Holland Publishing Co. — (1973). "Address to Professor A. Robinson. At the occasion of the Brouwer memorial lecture given by Prof. A. Robinson on the 26th April 1973". Nieuw Arch. Wisk. (3). 21: 134–137. — (1974). Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie (in German) (Reprint ed.). Berlin–New York: Springer-Verlag. — (1980). Axiomatic projective geometry. Bibliotheca Mathematica. Vol. V (Second ed.). Groningen; Amsterdam–New York: Wolters-Noordhoff Scientific Publications, Ltd.; North-Holland Publishing Co. == Notes == == References == Church, Alonzo (1935). "Review: Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie by A. Heyting". Bull. Amer. Math. Soc. (in German). 41: 476–477. doi:10.1090/S0002-9904-1935-06126-9. Digitaal Wetenschapshistorisch Centrum. "Arend Heyting (1898 – 1980)". Royal Netherlands Academy of Arts and Sciences. Retrieved 28 July 2015. Van Stigt, Walter P. (1990). Brouwer's Intuitionism. Amsterdam: North Holland. == External links == O'Connor, John J.; Robertson, Edmund F., "Arend Heyting", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Aretha Teckentrup#0
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Aretha Leonore Teckentrup is a UK-based mathematician, known for her research on uncertainty quantification and numerical analysis. Her work focuses on multilevel Monte Carlo methods for the numerical solution of partial differential equations, Gaussian processes and Bayesian inference. She is a reader in the mathematics of data science at the University of Edinburgh. == Education and career == Teckentrup was a student in mathematics at the University of Bath beginning in 2005. She earned a master's degree there in 2009, and completed her PhD in 2013. Her dissertation, Multilevel Monte Carlo methods and Uncertainty Quantification, was supervised by Robert Scheichl. After postdoctoral research with Max Gunzburger at Florida State University from 2013 to 2014 and with Andrew M. Stuart at the University of Warwick from 2014 to 2016, she became a lecturer at the University of Edinburgh in 2016. == Recognition == Teckentrup was a second-place winner of the Leslie Fox Prize for Numerical Analysis in 2017. In 2018 she became the inaugural winner of the SIAG/Uncertainty Quantification Early Career Prize of the Society for Industrial and Applied Mathematics Activity Group on Uncertainty Quantification. She was one of the 2021 winners of the Whitehead Prize of the London Mathematical Society. == References == == External links == Home page Aretha Teckentrup publications indexed by Google Scholar
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Wikipedia:Ari Laptev#0
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Ari Laptev (born August 10, 1950) is a mathematician working on the spectral theory of partial differential equations. His PhD was obtained in 1978 at Leningrad State University under the supervision of Michael Solomyak. He is currently Professor at both the KTH in Stockholm and Imperial College London. From 2001 to 2003 Laptev served as the President of the Swedish Mathematical Society. In the years 2007–2010 he served as the President of the European Mathematical Society. In April 2007 he was awarded the Royal Society Wolfson Research Merit Award. He was from 2011 to 2018 the Director of Institut Mittag-Leffler. He is Editor-in-Chief of Acta Mathematica, Editor-in-Chief of Arkiv för Matematik, Deputy Chief Editor of the "Journal of Spectral Theory", Editor of the "Bulletin of Mathematical Sciences", Editor of "Problems in Mathematical Analysis", and Editor of "Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika". == References == == Other sources == http://www.ma.ic.ac.uk/~alaptev/CV/Laptev_CV_Imperial.pdf
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Wikipedia:Arie Bialostocki#0
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Arie Bialostocki (Hebrew: אריה ביאלוסטוקי) is an Israeli American mathematician with expertise and contributions in discrete mathematics and finite groups. == Education and career == Arie received his BSc, MSc, and PhD (1984) degrees from Tel-Aviv University in Israel. His dissertation was done under the supervision of Marcel Herzog. After a year of postdoc at University of Calgary, Canada, he took a faculty position at the University of Idaho, became a professor in 1992, and continued to work there until he retired at the end of 2011. At Idaho, Arie maintained correspondence and collaborations with researchers from around the world who would share similar interests in mathematics. His Erdős number is 1. He has supervised seven PhD students and numerous undergraduate students who enjoyed his colorful anecdotes and advice. He organized the Research Experience for Undergraduates (REU) program at the University of Idaho from 1999 to 2003 attracting many promising undergraduates who themselves have gone on to their outstanding research careers. == Mathematics research == Arie has published more than 50 publications. Some of Bialostocki's contributions include: Bialostocki redefined a B {\displaystyle B} -injector in a finite group G to be any maximal nilpotent subgroup B {\displaystyle B} of G {\displaystyle G} satisfying d 2 ( B ) = d 2 ( G ) {\displaystyle d_{2}(B)=d_{2}(G)} , where d 2 ( X ) {\displaystyle d_{2}(X)} is the largest cardinality of a subgroup of G {\displaystyle G} which is nilpotent of class at most 2 {\displaystyle 2} . Using his definition, it was proved by several authors that in many non-solvable groups the nilpotent injectors form a unique conjugacy class. Bialostocki contributed to the generalization of the Erdős-Ginzburg-Ziv theorem (also known as the EGZ theorem). He conjectured: if A = ( a 1 , a 2 , … , a n ) {\displaystyle A=(a_{1},a_{2},\ldots ,a_{n})} is a sequence of elements of Z m {\displaystyle {\mathbb {Z} }_{m}} , then A {\displaystyle A} contains at least ( ⌊ n / 2 ⌋ m ) + ( ⌈ n / 2 ⌉ m ) {\displaystyle {\lfloor {n/2}\rfloor \choose {m}}+{\lceil {n/2}\rceil \choose {m}}} zero sums of length m {\displaystyle m} . The EGZ theorem is a special case where n = 2 m − 1 {\displaystyle n=2m-1} . The conjecture was partially confirmed by Kisin, Füredi and Kleitman, and Grynkiewicz. Bialostocki introduced the EGZ polynomials and contributed to generalize the EGZ theorem for higher degree polynomials. The EGZ theorem is associated with the first degree elementary polynomial. Bialostocki and Dierker introduced the relationship of EGZ theorem to Ramsey Theory on graphs. Bialostocki, Erdős, and Lefmann introduced the relationship of EGZ theorem to Ramsey Theory on the positive integers. In Jakobs and Jungnickel's book "Einführung in die Kombinatorik", Bialostocki and Dierker are attributed for introducing Zero-sum Ramsey theory. In Landman and Robertson's book "Ramsey Theory on the Integers", the number b ( m , k ; r ) {\displaystyle b(m,k;r)} is defined in honor of Bialostocki's contributions to the Zero-sum Ramsey theory. Bialostocki, Dierker, and Voxman suggested a conjecture offering a modular strengthening of the Erdős–Szekeres theorem proving that the number of points in the interior of the polygon is divisible by k {\displaystyle k} , provided that total number of points n ⩾ k + 2 {\displaystyle n\geqslant k+2} . Károlyi, Pach and Tóth made further progress toward the proof of the conjecture. In Recreational Mathematics, Arie's paper on application of elementary group theory to Peg Solitaire is a suggested reading in Joseph Gallian's book on Abstract Algebra. == References ==
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Wikipedia:Arieh Iserles#0
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Arieh Iserles (born 2 September 1947) is a computational mathematician, currently Professor of the Numerical Analysis of Differential Equations at the University of Cambridge and a member of the Department of Applied Mathematics and Theoretical Physics. He studied at the Hebrew University of Jerusalem and Ben-Gurion University of the Negev and wrote his PhD dissertation on numerical methods for stiff ordinary differential equations. His research comprises many themes in computational and applied mathematics: ordinary and partial differential equations, approximation theory, geometric numerical integration, orthogonal polynomials, functional equations, computational dynamics and the computation of highly oscillatory phenomena. He has written a textbook, A First Course in the Numerical Analysis of Differential Equations (Cambridge University Press, 2nd ed. 2009). Iserles is the managing editor of Acta Numerica, editor-in-chief of IMA Journal of Numerical Analysis and an editor of several other mathematical journals. From 1997 to 2000 he was the chair of the Society for the Foundations of Computational Mathematics. From 2010 to 2015 he was a director of the Cambridge Centre for Analysis (CCA), an EPSRC-funded Centre for Doctoral Training in mathematical analysis. In 1999, he was awarded the Onsager Medal by the Norwegian University of Science and Technology. In 2012 he received the David Crighton medal, presented by the Institute of Mathematics and its Applications and London Mathematical Society "for services to mathematics and the mathematics community" and in 2014 he was awarded by the Society for Industrial and Applied Mathematics the SIAM Prize for Distinguished Service to the Profession. In 2012, Iserles was an invited speaker at the 6th European Congress of Mathematics in Kraków. == References == == External links == Professor Iserles' website
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Wikipedia:Arild Stubhaug#0
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Arild Stubhaug (born 25 May 1948) is a Norwegian biographer and poet. He has won several literary awards for his biographies of Norwegian mathematicians. == Early life and education == Stubhaug was born in Naustdal on 25 May 1948, a son educator Lidvald Stubhaug and nurse Borghild Daltveit. He received a cand.mag degree from the University of Bergen with the subjects mathematics, literature and religion. == Literary career == Stubhaug made his literary debut in 1970 with the poetry collection Utkantane. Further poetry collections are Du ber vatn i hendene from 1973, Eld i sol from 1988, and Lemmata from 2008. He has written biographies of the mathematicians Sophus Lie, Niels Henrik Abel and Gösta Mittag-Leffler, Jacob Aall, Conrad Nicolai Schwach, and Stein Rokkan. He received the Brage Prize in 1996 for the biography Et foranskutt lyn. Niels Henrik Abel og hans tid, translated into English under the title Niels Henrik Abel and his Times: Called Too Soon by Flames Afar, He followed up with the 250-year history of the Royal Norwegian Society of Sciences and Letters (2010) and biographies of Jacob Aall (2014) and the social scientist Stein Rokkan (2019). Stubhaug is married to Kari Bøge. == Awards == Brage Prize, 1996 Norsk språkpris, 2001 Norwegian Academy Prize in memory of Thorleif Dahl, 2008 Doblougprisen 2010 == References ==
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Wikipedia:Arithmetic function#0
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In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. == Multiplicative and additive functions == An arithmetic function a is completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n; completely multiplicative if a(1) = 1 and a(mn) = a(m)a(n) for all natural numbers m and n; Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them. Then an arithmetic function a is additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n; multiplicative if a(1) = 1 and a(mn) = a(m)a(n) for all coprime natural numbers m and n. == Notation == In this article, ∑ p f ( p ) {\textstyle \sum _{p}f(p)} and ∏ p f ( p ) {\textstyle \prod _{p}f(p)} mean that the sum or product is over all prime numbers: ∑ p f ( p ) = f ( 2 ) + f ( 3 ) + f ( 5 ) + ⋯ {\displaystyle \sum _{p}f(p)=f(2)+f(3)+f(5)+\cdots } and ∏ p f ( p ) = f ( 2 ) f ( 3 ) f ( 5 ) ⋯ . {\displaystyle \prod _{p}f(p)=f(2)f(3)f(5)\cdots .} Similarly, ∑ p k f ( p k ) {\textstyle \sum _{p^{k}}f(p^{k})} and ∏ p k f ( p k ) {\textstyle \prod _{p^{k}}f(p^{k})} mean that the sum or product is over all prime powers with strictly positive exponent (so k = 0 is not included): ∑ p k f ( p k ) = ∑ p ∑ k > 0 f ( p k ) = f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 5 ) + f ( 7 ) + f ( 8 ) + f ( 9 ) + ⋯ . {\displaystyle \sum _{p^{k}}f(p^{k})=\sum _{p}\sum _{k>0}f(p^{k})=f(2)+f(3)+f(4)+f(5)+f(7)+f(8)+f(9)+\cdots .} The notations ∑ d ∣ n f ( d ) {\textstyle \sum _{d\mid n}f(d)} and ∏ d ∣ n f ( d ) {\textstyle \prod _{d\mid n}f(d)} mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, then ∏ d ∣ 12 f ( d ) = f ( 1 ) f ( 2 ) f ( 3 ) f ( 4 ) f ( 6 ) f ( 12 ) . {\displaystyle \prod _{d\mid 12}f(d)=f(1)f(2)f(3)f(4)f(6)f(12).} The notations can be combined: ∑ p ∣ n f ( p ) {\textstyle \sum _{p\mid n}f(p)} and ∏ p ∣ n f ( p ) {\textstyle \prod _{p\mid n}f(p)} mean that the sum or product is over all prime divisors of n. For example, if n = 18, then ∑ p ∣ 18 f ( p ) = f ( 2 ) + f ( 3 ) , {\displaystyle \sum _{p\mid 18}f(p)=f(2)+f(3),} and similarly ∑ p k ∣ n f ( p k ) {\textstyle \sum _{p^{k}\mid n}f(p^{k})} and ∏ p k ∣ n f ( p k ) {\textstyle \prod _{p^{k}\mid n}f(p^{k})} mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then ∏ p k ∣ 24 f ( p k ) = f ( 2 ) f ( 3 ) f ( 4 ) f ( 8 ) . {\displaystyle \prod _{p^{k}\mid 24}f(p^{k})=f(2)f(3)f(4)f(8).} == Ω(n), ω(n), νp(n) – prime power decomposition == The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: n = p 1 a 1 ⋯ p k a k {\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}} where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.) It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then νp(n) = ai, otherwise it is zero. Then n = ∏ p p ν p ( n ) . {\displaystyle n=\prod _{p}p^{\nu _{p}(n)}.} In terms of the above the prime omega functions ω and Ω are defined by To avoid repetition, formulas for the functions listed in this article are, whenever possible, given in terms of n and the corresponding pi, ai, ω, and Ω. == Multiplicative functions == === σk(n), τ(n), d(n) – divisor sums === σk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number. σ1(n), the sum of the (positive) divisors of n, is usually denoted by σ(n). Since a positive number to the zero power is one, σ0(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or τ(n) (for the German Teiler = divisors). σ k ( n ) = ∏ i = 1 ω ( n ) p i ( a i + 1 ) k − 1 p i k − 1 = ∏ i = 1 ω ( n ) ( 1 + p i k + p i 2 k + ⋯ + p i a i k ) . {\displaystyle \sigma _{k}(n)=\prod _{i=1}^{\omega (n)}{\frac {p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}}=\prod _{i=1}^{\omega (n)}\left(1+p_{i}^{k}+p_{i}^{2k}+\cdots +p_{i}^{a_{i}k}\right).} Setting k = 0 in the second product gives τ ( n ) = d ( n ) = ( 1 + a 1 ) ( 1 + a 2 ) ⋯ ( 1 + a ω ( n ) ) . {\displaystyle \tau (n)=d(n)=(1+a_{1})(1+a_{2})\cdots (1+a_{\omega (n)}).} === φ(n) – Euler totient function === φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n. φ ( n ) = n ∏ p ∣ n ( 1 − 1 p ) = n ( p 1 − 1 p 1 ) ( p 2 − 1 p 2 ) ⋯ ( p ω ( n ) − 1 p ω ( n ) ) . {\displaystyle \varphi (n)=n\prod _{p\mid n}\left(1-{\frac {1}{p}}\right)=n\left({\frac {p_{1}-1}{p_{1}}}\right)\left({\frac {p_{2}-1}{p_{2}}}\right)\cdots \left({\frac {p_{\omega (n)}-1}{p_{\omega (n)}}}\right).} === Jk(n) – Jordan totient function === Jk(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. It is a generalization of Euler's totient, φ(n) = J1(n). J k ( n ) = n k ∏ p ∣ n ( 1 − 1 p k ) = n k ( p 1 k − 1 p 1 k ) ( p 2 k − 1 p 2 k ) ⋯ ( p ω ( n ) k − 1 p ω ( n ) k ) . {\displaystyle J_{k}(n)=n^{k}\prod _{p\mid n}\left(1-{\frac {1}{p^{k}}}\right)=n^{k}\left({\frac {p_{1}^{k}-1}{p_{1}^{k}}}\right)\left({\frac {p_{2}^{k}-1}{p_{2}^{k}}}\right)\cdots \left({\frac {p_{\omega (n)}^{k}-1}{p_{\omega (n)}^{k}}}\right).} === μ(n) – Möbius function === μ(n), the Möbius function, is important because of the Möbius inversion formula. See § Dirichlet convolution, below. μ ( n ) = { ( − 1 ) ω ( n ) = ( − 1 ) Ω ( n ) if ω ( n ) = Ω ( n ) 0 if ω ( n ) ≠ Ω ( n ) . {\displaystyle \mu (n)={\begin{cases}(-1)^{\omega (n)}=(-1)^{\Omega (n)}&{\text{if }}\;\omega (n)=\Omega (n)\\0&{\text{if }}\;\omega (n)\neq \Omega (n).\end{cases}}} This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.) === τ(n) – Ramanujan tau function === τ(n), the Ramanujan tau function, is defined by its generating function identity: ∑ n ≥ 1 τ ( n ) q n = q ∏ n ≥ 1 ( 1 − q n ) 24 . {\displaystyle \sum _{n\geq 1}\tau (n)q^{n}=q\prod _{n\geq 1}(1-q^{n})^{24}.} Although it is hard to say exactly what "arithmetical property of n" it "expresses", (τ(n) is (2π)−12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function) it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σk(n) and rk(n) functions (because these are also coefficients in the expansion of modular forms). === cq(n) – Ramanujan's sum === cq(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity: c q ( n ) = ∑ gcd ( a , q ) = 1 1 ≤ a ≤ q e 2 π i a q n . {\displaystyle c_{q}(n)=\sum _{\stackrel {1\leq a\leq q}{\gcd(a,q)=1}}e^{2\pi i{\tfrac {a}{q}}n}.} Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixed value of n it is multiplicative in q: If q and r are coprime, then c q ( n ) c r ( n ) = c q r ( n ) . {\displaystyle c_{q}(n)c_{r}(n)=c_{qr}(n).} === ψ(n) – Dedekind psi function === The Dedekind psi function, used in the theory of modular functions, is defined by the formula ψ ( n ) = n ∏ p | n ( 1 + 1 p ) . {\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right).} == Completely multiplicative functions == === λ(n) – Liouville function === λ(n), the Liouville function, is defined by λ ( n ) = ( − 1 ) Ω ( n ) . {\displaystyle \lambda (n)=(-1)^{\Omega (n)}.} === χ(n) – characters === All Dirichlet characters χ(n) are completely multiplicative. Two characters have special notations: The principal character (mod n) is denoted by χ0(a) (or χ1(a)). It is defined as χ 0 ( a ) = { 1 if gcd ( a , n ) = 1 , 0 if gcd ( a , n ) ≠ 1. {\displaystyle \chi _{0}(a)={\begin{cases}1&{\text{if }}\gcd(a,n)=1,\\0&{\text{if }}\gcd(a,n)\neq 1.\end{cases}}} The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n): ( a n ) = ( a p 1 ) a 1 ( a p 2 ) a 2 ⋯ ( a p ω ( n ) ) a ω ( n ) . {\displaystyle \left({\frac {a}{n}}\right)=\left({\frac {a}{p_{1}}}\right)^{a_{1}}\left({\frac {a}{p_{2}}}\right)^{a_{2}}\cdots \left({\frac {a}{p_{\omega (n)}}}\right)^{a_{\omega (n)}}.} In this formula ( a p ) {\displaystyle ({\tfrac {a}{p}})} is the Legendre symbol, defined for all integers a and all odd primes p by ( a p ) = { 0 if a ≡ 0 ( mod p ) , + 1 if a ≢ 0 ( mod p ) and for some integer x , a ≡ x 2 ( mod p ) − 1 if there is no such x . {\displaystyle \left({\frac {a}{p}}\right)={\begin{cases}\;\;\,0&{\text{if }}a\equiv 0{\pmod {p}},\\+1&{\text{if }}a\not \equiv 0{\pmod {p}}{\text{ and for some integer }}x,\;a\equiv x^{2}{\pmod {p}}\\-1&{\text{if there is no such }}x.\end{cases}}} Following the normal convention for the empty product, ( a 1 ) = 1. {\displaystyle \left({\frac {a}{1}}\right)=1.} == Additive functions == === ω(n) – distinct prime divisors === ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function). == Completely additive functions == === Ω(n) – prime divisors === Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function). === νp(n) – p-adic valuation of an integer n === For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive. === Logarithmic derivative === ld ( n ) = D ( n ) n = ∑ p prime p ∣ n v p ( n ) p {\displaystyle \operatorname {ld} (n)={\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}} , where D ( n ) {\displaystyle D(n)} is the arithmetic derivative. == Neither multiplicative nor additive == === π(x), Π(x), ϑ(x), ψ(x) – prime-counting functions === These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive. π(x), the prime-counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers. π ( x ) = ∑ p ≤ x 1 {\displaystyle \pi (x)=\sum _{p\leq x}1} A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/k on integers which are the kth power of some prime number, and the value 0 on other integers. Π ( x ) = ∑ p k ≤ x 1 k . {\displaystyle \Pi (x)=\sum _{p^{k}\leq x}{\frac {1}{k}}.} ϑ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x. ϑ ( x ) = ∑ p ≤ x log p , {\displaystyle \vartheta (x)=\sum _{p\leq x}\log p,} ψ ( x ) = ∑ p k ≤ x log p . {\displaystyle \psi (x)=\sum _{p^{k}\leq x}\log p.} The second Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below. === Λ(n) – von Mangoldt function === Λ(n), the von Mangoldt function, is 0 unless the argument n is a prime power pk, in which case it is the natural logarithm of the prime p: Λ ( n ) = { log p if n = 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , … = p k is a prime power 0 if n = 1 , 6 , 10 , 12 , 14 , 15 , 18 , 20 , 21 , … is not a prime power . {\displaystyle \Lambda (n)={\begin{cases}\log p&{\text{if }}n=2,3,4,5,7,8,9,11,13,16,\ldots =p^{k}{\text{ is a prime power}}\\0&{\text{if }}n=1,6,10,12,14,15,18,20,21,\dots \;\;\;\;{\text{ is not a prime power}}.\end{cases}}} === p(n) – partition function === p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different: p ( n ) = | { ( a 1 , a 2 , … a k ) : 0 < a 1 ≤ a 2 ≤ ⋯ ≤ a k ∧ n = a 1 + a 2 + ⋯ + a k } | . {\displaystyle p(n)=\left|\left\{(a_{1},a_{2},\dots a_{k}):0<a_{1}\leq a_{2}\leq \cdots \leq a_{k}\;\land \;n=a_{1}+a_{2}+\cdots +a_{k}\right\}\right|.} === λ(n) – Carmichael function === λ(n), the Carmichael function, is the smallest positive number such that a λ ( n ) ≡ 1 ( mod n ) {\displaystyle a^{\lambda (n)}\equiv 1{\pmod {n}}} for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n. For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n: λ ( n ) = { ϕ ( n ) if n = 2 , 3 , 4 , 5 , 7 , 9 , 11 , 13 , 17 , 19 , 23 , 25 , 27 , … 1 2 ϕ ( n ) if n = 8 , 16 , 32 , 64 , … {\displaystyle \lambda (n)={\begin{cases}\;\;\phi (n)&{\text{if }}n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots \\{\tfrac {1}{2}}\phi (n)&{\text{if }}n=8,16,32,64,\dots \end{cases}}} and for general n it is the least common multiple of λ of each of the prime power factors of n: λ ( p 1 a 1 p 2 a 2 … p ω ( n ) a ω ( n ) ) = lcm [ λ ( p 1 a 1 ) , λ ( p 2 a 2 ) , … , λ ( p ω ( n ) a ω ( n ) ) ] . {\displaystyle \lambda (p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega (n)}^{a_{\omega (n)}})=\operatorname {lcm} [\lambda (p_{1}^{a_{1}}),\;\lambda (p_{2}^{a_{2}}),\dots ,\lambda (p_{\omega (n)}^{a_{\omega (n)}})].} === h(n) – class number === h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples. === rk(n) – sum of k squares === rk(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different. r k ( n ) = | { ( a 1 , a 2 , … , a k ) : n = a 1 2 + a 2 2 + ⋯ + a k 2 } | {\displaystyle r_{k}(n)=\left|\left\{(a_{1},a_{2},\dots ,a_{k}):n=a_{1}^{2}+a_{2}^{2}+\cdots +a_{k}^{2}\right\}\right|} === D(n) – Arithmetic derivative === Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that D ( n ) = 1 {\displaystyle D(n)=1} if n prime, and D ( m n ) = m D ( n ) + D ( m ) n {\displaystyle D(mn)=mD(n)+D(m)n} (the product rule) == Summation functions == Given an arithmetic function a(n), its summation function A(x) is defined by A ( x ) := ∑ n ≤ x a ( n ) . {\displaystyle A(x):=\sum _{n\leq x}a(n).} A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0. Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right: A 0 ( m ) := 1 2 ( ∑ n < m a ( n ) + ∑ n ≤ m a ( n ) ) = A ( m ) − 1 2 a ( m ) . {\displaystyle A_{0}(m):={\frac {1}{2}}\left(\sum _{n<m}a(n)+\sum _{n\leq m}a(n)\right)=A(m)-{\frac {1}{2}}a(m).} Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x. A classical example of this phenomenon is given by the divisor summatory function, the summation function of d(n), the number of divisors of n: lim inf n → ∞ d ( n ) = 2 {\displaystyle \liminf _{n\to \infty }d(n)=2} lim sup n → ∞ log d ( n ) log log n log n = log 2 {\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)\log \log n}{\log n}}=\log 2} lim n → ∞ d ( 1 ) + d ( 2 ) + ⋯ + d ( n ) log ( 1 ) + log ( 2 ) + ⋯ + log ( n ) = 1. {\displaystyle \lim _{n\to \infty }{\frac {d(1)+d(2)+\cdots +d(n)}{\log(1)+\log(2)+\cdots +\log(n)}}=1.} An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that g is an average order of f if ∑ n ≤ x f ( n ) ∼ ∑ n ≤ x g ( n ) {\displaystyle \sum _{n\leq x}f(n)\sim \sum _{n\leq x}g(n)} as x tends to infinity. The example above shows that d(n) has the average order log(n). == Dirichlet convolution == Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges): F a ( s ) := ∑ n = 1 ∞ a ( n ) n s . {\displaystyle F_{a}(s):=\sum _{n=1}^{\infty }{\frac {a(n)}{n^{s}}}.} Fa(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is ζ(s) the Riemann zeta function. The generating function of the Möbius function is the inverse of the zeta function: ζ ( s ) ∑ n = 1 ∞ μ ( n ) n s = 1 , ℜ s > 1. {\displaystyle \zeta (s)\,\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}=1,\;\;\Re s>1.} Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The product Fa(s)Fb(s) can be computed as follows: F a ( s ) F b ( s ) = ( ∑ m = 1 ∞ a ( m ) m s ) ( ∑ n = 1 ∞ b ( n ) n s ) . {\displaystyle F_{a}(s)F_{b}(s)=\left(\sum _{m=1}^{\infty }{\frac {a(m)}{m^{s}}}\right)\left(\sum _{n=1}^{\infty }{\frac {b(n)}{n^{s}}}\right).} It is a straightforward exercise to show that if c(n) is defined by c ( n ) := ∑ i j = n a ( i ) b ( j ) = ∑ i ∣ n a ( i ) b ( n i ) , {\displaystyle c(n):=\sum _{ij=n}a(i)b(j)=\sum _{i\mid n}a(i)b\left({\frac {n}{i}}\right),} then F c ( s ) = F a ( s ) F b ( s ) . {\displaystyle F_{c}(s)=F_{a}(s)F_{b}(s).} This function c is called the Dirichlet convolution of a and b, and is denoted by a ∗ b {\displaystyle a*b} . A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function: g ( n ) = ∑ d ∣ n f ( d ) . {\displaystyle g(n)=\sum _{d\mid n}f(d).} Multiplying by the inverse of the zeta function gives the Möbius inversion formula: f ( n ) = ∑ d ∣ n μ ( n d ) g ( d ) . {\displaystyle f(n)=\sum _{d\mid n}\mu \left({\frac {n}{d}}\right)g(d).} If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. == Relations among the functions == There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions. Here are a few examples: === Dirichlet convolutions === ∑ δ ∣ n μ ( δ ) = ∑ δ ∣ n λ ( n δ ) | μ ( δ ) | = { 1 if n = 1 0 if n ≠ 1 {\displaystyle \sum _{\delta \mid n}\mu (\delta )=\sum _{\delta \mid n}\lambda \left({\frac {n}{\delta }}\right)|\mu (\delta )|={\begin{cases}1&{\text{if }}n=1\\0&{\text{if }}n\neq 1\end{cases}}} where λ is the Liouville function. ∑ δ ∣ n φ ( δ ) = n . {\displaystyle \sum _{\delta \mid n}\varphi (\delta )=n.} φ ( n ) = ∑ δ ∣ n μ ( n δ ) δ = n ∑ δ ∣ n μ ( δ ) δ . {\displaystyle \varphi (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta =n\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta }}.} Möbius inversion ∑ d ∣ n J k ( d ) = n k . {\displaystyle \sum _{d\mid n}J_{k}(d)=n^{k}.} J k ( n ) = ∑ δ ∣ n μ ( n δ ) δ k = n k ∑ δ ∣ n μ ( δ ) δ k . {\displaystyle J_{k}(n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\delta ^{k}=n^{k}\sum _{\delta \mid n}{\frac {\mu (\delta )}{\delta ^{k}}}.} Möbius inversion ∑ δ ∣ n δ s J r ( δ ) J s ( n δ ) = J r + s ( n ) {\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)} ∑ δ ∣ n φ ( δ ) d ( n δ ) = σ ( n ) . {\displaystyle \sum _{\delta \mid n}\varphi (\delta )d\left({\frac {n}{\delta }}\right)=\sigma (n).} ∑ δ ∣ n | μ ( δ ) | = 2 ω ( n ) . {\displaystyle \sum _{\delta \mid n}|\mu (\delta )|=2^{\omega (n)}.} | μ ( n ) | = ∑ δ ∣ n μ ( n δ ) 2 ω ( δ ) . {\displaystyle |\mu (n)|=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)2^{\omega (\delta )}.} Möbius inversion ∑ δ ∣ n 2 ω ( δ ) = d ( n 2 ) . {\displaystyle \sum _{\delta \mid n}2^{\omega (\delta )}=d(n^{2}).} 2 ω ( n ) = ∑ δ ∣ n μ ( n δ ) d ( δ 2 ) . {\displaystyle 2^{\omega (n)}=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d(\delta ^{2}).} Möbius inversion ∑ δ ∣ n d ( δ 2 ) = d 2 ( n ) . {\displaystyle \sum _{\delta \mid n}d(\delta ^{2})=d^{2}(n).} d ( n 2 ) = ∑ δ ∣ n μ ( n δ ) d 2 ( δ ) . {\displaystyle d(n^{2})=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)d^{2}(\delta ).} Möbius inversion ∑ δ ∣ n d ( n δ ) 2 ω ( δ ) = d 2 ( n ) . {\displaystyle \sum _{\delta \mid n}d\left({\frac {n}{\delta }}\right)2^{\omega (\delta )}=d^{2}(n).} ∑ δ ∣ n λ ( δ ) = { 1 if n is a square 0 if n is not square. {\displaystyle \sum _{\delta \mid n}\lambda (\delta )={\begin{cases}&1{\text{ if }}n{\text{ is a square }}\\&0{\text{ if }}n{\text{ is not square.}}\end{cases}}} where λ is the Liouville function. ∑ δ ∣ n Λ ( δ ) = log n . {\displaystyle \sum _{\delta \mid n}\Lambda (\delta )=\log n.} Λ ( n ) = ∑ δ ∣ n μ ( n δ ) log ( δ ) . {\displaystyle \Lambda (n)=\sum _{\delta \mid n}\mu \left({\frac {n}{\delta }}\right)\log(\delta ).} Möbius inversion === Sums of squares === For all k ≥ 4 , r k ( n ) > 0. {\displaystyle k\geq 4,\;\;\;r_{k}(n)>0.} (Lagrange's four-square theorem). r 2 ( n ) = 4 ∑ d ∣ n ( − 4 d ) , {\displaystyle r_{2}(n)=4\sum _{d\mid n}\left({\frac {-4}{d}}\right),} where the Kronecker symbol has the values ( − 4 n ) = { + 1 if n ≡ 1 ( mod 4 ) − 1 if n ≡ 3 ( mod 4 ) 0 if n is even . {\displaystyle \left({\frac {-4}{n}}\right)={\begin{cases}+1&{\text{if }}n\equiv 1{\pmod {4}}\\-1&{\text{if }}n\equiv 3{\pmod {4}}\\\;\;\;0&{\text{if }}n{\text{ is even}}.\\\end{cases}}} There is a formula for r3 in the section on class numbers below. r 4 ( n ) = 8 ∑ 4 ∤ d d ∣ n d = 8 ( 2 + ( − 1 ) n ) ∑ 2 ∤ d d ∣ n d = { 8 σ ( n ) if n is odd 24 σ ( n 2 ν ) if n is even , {\displaystyle r_{4}(n)=8\sum _{\stackrel {d\mid n}{4\,\nmid \,d}}d=8(2+(-1)^{n})\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d={\begin{cases}8\sigma (n)&{\text{if }}n{\text{ is odd }}\\24\sigma \left({\frac {n}{2^{\nu }}}\right)&{\text{if }}n{\text{ is even }}\end{cases}},} where ν = ν2(n). r 6 ( n ) = 16 ∑ d ∣ n χ ( n d ) d 2 − 4 ∑ d ∣ n χ ( d ) d 2 , {\displaystyle r_{6}(n)=16\sum _{d\mid n}\chi \left({\frac {n}{d}}\right)d^{2}-4\sum _{d\mid n}\chi (d)d^{2},} where χ ( n ) = ( − 4 n ) . {\displaystyle \chi (n)=\left({\frac {-4}{n}}\right).} Define the function σk*(n) as σ k ∗ ( n ) = ( − 1 ) n ∑ d ∣ n ( − 1 ) d d k = { ∑ d ∣ n d k = σ k ( n ) if n is odd ∑ 2 ∣ d d ∣ n d k − ∑ 2 ∤ d d ∣ n d k if n is even . {\displaystyle \sigma _{k}^{*}(n)=(-1)^{n}\sum _{d\mid n}(-1)^{d}d^{k}={\begin{cases}\sum _{d\mid n}d^{k}=\sigma _{k}(n)&{\text{if }}n{\text{ is odd }}\\\sum _{\stackrel {d\mid n}{2\,\mid \,d}}d^{k}-\sum _{\stackrel {d\mid n}{2\,\nmid \,d}}d^{k}&{\text{if }}n{\text{ is even}}.\end{cases}}} That is, if n is odd, σk*(n) is the sum of the kth powers of the divisors of n, that is, σk(n), and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n. r 8 ( n ) = 16 σ 3 ∗ ( n ) . {\displaystyle r_{8}(n)=16\sigma _{3}^{*}(n).} Adopt the convention that Ramanujan's τ(x) = 0 if x is not an integer. r 24 ( n ) = 16 691 σ 11 ∗ ( n ) + 128 691 { ( − 1 ) n − 1 259 τ ( n ) − 512 τ ( n 2 ) } {\displaystyle r_{24}(n)={\frac {16}{691}}\sigma _{11}^{*}(n)+{\frac {128}{691}}\left\{(-1)^{n-1}259\tau (n)-512\tau \left({\frac {n}{2}}\right)\right\}} === Divisor sum convolutions === Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series: ( ∑ n = 0 ∞ a n x n ) ( ∑ n = 0 ∞ b n x n ) = ∑ i = 0 ∞ ∑ j = 0 ∞ a i b j x i + j = ∑ n = 0 ∞ ( ∑ i = 0 n a i b n − i ) x n = ∑ n = 0 ∞ c n x n . {\displaystyle \left(\sum _{n=0}^{\infty }a_{n}x^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}x^{n}\right)=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}x^{i+j}=\sum _{n=0}^{\infty }\left(\sum _{i=0}^{n}a_{i}b_{n-i}\right)x^{n}=\sum _{n=0}^{\infty }c_{n}x^{n}.} The sequence c n = ∑ i = 0 n a i b n − i {\displaystyle c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}} is called the convolution or the Cauchy product of the sequences an and bn. These formulas may be proved analytically (see Eisenstein series) or by elementary methods. σ 3 ( n ) = 1 5 { 6 n σ 1 ( n ) − σ 1 ( n ) + 12 ∑ 0 < k < n σ 1 ( k ) σ 1 ( n − k ) } . {\displaystyle \sigma _{3}(n)={\frac {1}{5}}\left\{6n\sigma _{1}(n)-\sigma _{1}(n)+12\sum _{0<k<n}\sigma _{1}(k)\sigma _{1}(n-k)\right\}.} σ 5 ( n ) = 1 21 { 10 ( 3 n − 1 ) σ 3 ( n ) + σ 1 ( n ) + 240 ∑ 0 < k < n σ 1 ( k ) σ 3 ( n − k ) } . {\displaystyle \sigma _{5}(n)={\frac {1}{21}}\left\{10(3n-1)\sigma _{3}(n)+\sigma _{1}(n)+240\sum _{0<k<n}\sigma _{1}(k)\sigma _{3}(n-k)\right\}.} σ 7 ( n ) = 1 20 { 21 ( 2 n − 1 ) σ 5 ( n ) − σ 1 ( n ) + 504 ∑ 0 < k < n σ 1 ( k ) σ 5 ( n − k ) } = σ 3 ( n ) + 120 ∑ 0 < k < n σ 3 ( k ) σ 3 ( n − k ) . {\displaystyle {\begin{aligned}\sigma _{7}(n)&={\frac {1}{20}}\left\{21(2n-1)\sigma _{5}(n)-\sigma _{1}(n)+504\sum _{0<k<n}\sigma _{1}(k)\sigma _{5}(n-k)\right\}\\&=\sigma _{3}(n)+120\sum _{0<k<n}\sigma _{3}(k)\sigma _{3}(n-k).\end{aligned}}} σ 9 ( n ) = 1 11 { 10 ( 3 n − 2 ) σ 7 ( n ) + σ 1 ( n ) + 480 ∑ 0 < k < n σ 1 ( k ) σ 7 ( n − k ) } = 1 11 { 21 σ 5 ( n ) − 10 σ 3 ( n ) + 5040 ∑ 0 < k < n σ 3 ( k ) σ 5 ( n − k ) } . {\displaystyle {\begin{aligned}\sigma _{9}(n)&={\frac {1}{11}}\left\{10(3n-2)\sigma _{7}(n)+\sigma _{1}(n)+480\sum _{0<k<n}\sigma _{1}(k)\sigma _{7}(n-k)\right\}\\&={\frac {1}{11}}\left\{21\sigma _{5}(n)-10\sigma _{3}(n)+5040\sum _{0<k<n}\sigma _{3}(k)\sigma _{5}(n-k)\right\}.\end{aligned}}} τ ( n ) = 65 756 σ 11 ( n ) + 691 756 σ 5 ( n ) − 691 3 ∑ 0 < k < n σ 5 ( k ) σ 5 ( n − k ) , {\displaystyle \tau (n)={\frac {65}{756}}\sigma _{11}(n)+{\frac {691}{756}}\sigma _{5}(n)-{\frac {691}{3}}\sum _{0<k<n}\sigma _{5}(k)\sigma _{5}(n-k),} where τ(n) is Ramanujan's function. Since σk(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences for the functions. See Ramanujan tau function for some examples. Extend the domain of the partition function by setting p(0) = 1. p ( n ) = 1 n ∑ 1 ≤ k ≤ n σ ( k ) p ( n − k ) . {\displaystyle p(n)={\frac {1}{n}}\sum _{1\leq k\leq n}\sigma (k)p(n-k).} This recurrence can be used to compute p(n). === Class number related === Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol. An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4 ≡ 2 or 3 (mod 4). Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol: ( a 2 ) = { 0 if a is even ( − 1 ) a 2 − 1 8 if a is odd. {\displaystyle \left({\frac {a}{2}}\right)={\begin{cases}\;\;\,0&{\text{ if }}a{\text{ is even}}\\(-1)^{\frac {a^{2}-1}{8}}&{\text{ if }}a{\text{ is odd. }}\end{cases}}} Then if D < −4 is a fundamental discriminant h ( D ) = 1 D ∑ r = 1 | D | r ( D r ) = 1 2 − ( D 2 ) ∑ r = 1 | D | / 2 ( D r ) . {\displaystyle {\begin{aligned}h(D)&={\frac {1}{D}}\sum _{r=1}^{|D|}r\left({\frac {D}{r}}\right)\\&={\frac {1}{2-\left({\tfrac {D}{2}}\right)}}\sum _{r=1}^{|D|/2}\left({\frac {D}{r}}\right).\end{aligned}}} There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then r 3 ( | D | ) = 12 ( 1 − ( D 2 ) ) h ( D ) . {\displaystyle r_{3}(|D|)=12\left(1-\left({\frac {D}{2}}\right)\right)h(D).} === Prime-count related === Let H n = 1 + 1 2 + 1 3 + ⋯ + 1 n {\displaystyle H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}} be the nth harmonic number. Then σ ( n ) ≤ H n + e H n log H n {\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\log H_{n}} is true for every natural number n if and only if the Riemann hypothesis is true. The Riemann hypothesis is also equivalent to the statement that, for all n > 5040, σ ( n ) < e γ n log log n {\displaystyle \sigma (n)<e^{\gamma }n\log \log n} (where γ is the Euler–Mascheroni constant). This is Robin's theorem. ∑ p ν p ( n ) = Ω ( n ) . {\displaystyle \sum _{p}\nu _{p}(n)=\Omega (n).} ψ ( x ) = ∑ n ≤ x Λ ( n ) . {\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n).} Π ( x ) = ∑ n ≤ x Λ ( n ) log n . {\displaystyle \Pi (x)=\sum _{n\leq x}{\frac {\Lambda (n)}{\log n}}.} e θ ( x ) = ∏ p ≤ x p . {\displaystyle e^{\theta (x)}=\prod _{p\leq x}p.} e ψ ( x ) = lcm [ 1 , 2 , … , ⌊ x ⌋ ] . {\displaystyle e^{\psi (x)}=\operatorname {lcm} [1,2,\dots ,\lfloor x\rfloor ].} === Menon's identity === In 1965 P Kesava Menon proved ∑ gcd ( k , n ) = 1 1 ≤ k ≤ n gcd ( k − 1 , n ) = φ ( n ) d ( n ) . {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\gcd(k-1,n)=\varphi (n)d(n).} This has been generalized by a number of mathematicians. For example, B. Sury ∑ gcd ( k 1 , n ) = 1 1 ≤ k 1 , k 2 , … , k s ≤ n gcd ( k 1 − 1 , k 2 , … , k s , n ) = φ ( n ) σ s − 1 ( n ) . {\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},n)=1}}\gcd(k_{1}-1,k_{2},\dots ,k_{s},n)=\varphi (n)\sigma _{s-1}(n).} N. Rao ∑ gcd ( k 1 , k 2 , … , k s , n ) = 1 1 ≤ k 1 , k 2 , … , k s ≤ n gcd ( k 1 − a 1 , k 2 − a 2 , … , k s − a s , n ) s = J s ( n ) d ( n ) , {\displaystyle \sum _{\stackrel {1\leq k_{1},k_{2},\dots ,k_{s}\leq n}{\gcd(k_{1},k_{2},\dots ,k_{s},n)=1}}\gcd(k_{1}-a_{1},k_{2}-a_{2},\dots ,k_{s}-a_{s},n)^{s}=J_{s}(n)d(n),} where a1, a2, ..., as are integers, gcd(a1, a2, ..., as, n) = 1. László Fejes Tóth ∑ gcd ( k , m ) = 1 1 ≤ k ≤ m gcd ( k 2 − 1 , m 1 ) gcd ( k 2 − 1 , m 2 ) = φ ( n ) ∑ d 2 ∣ m 2 d 1 ∣ m 1 φ ( gcd ( d 1 , d 2 ) ) 2 ω ( lcm ( d 1 , d 2 ) ) , {\displaystyle \sum _{\stackrel {1\leq k\leq m}{\gcd(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},} where m1 and m2 are odd, m = lcm(m1, m2). In fact, if f is any arithmetical function ∑ gcd ( k , n ) = 1 1 ≤ k ≤ n f ( gcd ( k − 1 , n ) ) = φ ( n ) ∑ d ∣ n ( μ ∗ f ) ( d ) φ ( d ) , {\displaystyle \sum _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}f(\gcd(k-1,n))=\varphi (n)\sum _{d\mid n}{\frac {(\mu *f)(d)}{\varphi (d)}},} where ∗ {\displaystyle *} stands for Dirichlet convolution. === Miscellaneous === Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of quadratic reciprocity: ( m n ) ( n m ) = ( − 1 ) ( m − 1 ) ( n − 1 ) / 4 . {\displaystyle \left({\frac {m}{n}}\right)\left({\frac {n}{m}}\right)=(-1)^{(m-1)(n-1)/4}.} Let D(n) be the arithmetic derivative. Then the logarithmic derivative D ( n ) n = ∑ p prime p ∣ n v p ( n ) p . {\displaystyle {\frac {D(n)}{n}}=\sum _{\stackrel {p\mid n}{p{\text{ prime}}}}{\frac {v_{p}(n)}{p}}.} See Arithmetic derivative for details. Let λ(n) be Liouville's function. Then | λ ( n ) | μ ( n ) = λ ( n ) | μ ( n ) | = μ ( n ) , {\displaystyle |\lambda (n)|\mu (n)=\lambda (n)|\mu (n)|=\mu (n),} and λ ( n ) μ ( n ) = | μ ( n ) | = μ 2 ( n ) . {\displaystyle \lambda (n)\mu (n)=|\mu (n)|=\mu ^{2}(n).} Let λ(n) be Carmichael's function. Then λ ( n ) ∣ ϕ ( n ) . {\displaystyle \lambda (n)\mid \phi (n).} Further, λ ( n ) = ϕ ( n ) if and only if n = { 1 , 2 , 4 ; 3 , 5 , 7 , 9 , 11 , … (that is, p k , where p is an odd prime) ; 6 , 10 , 14 , 18 , … (that is, 2 p k , where p is an odd prime) . {\displaystyle \lambda (n)=\phi (n){\text{ if and only if }}n={\begin{cases}1,2,4;\\3,5,7,9,11,\ldots {\text{ (that is, }}p^{k}{\text{, where }}p{\text{ is an odd prime)}};\\6,10,14,18,\ldots {\text{ (that is, }}2p^{k}{\text{, where }}p{\text{ is an odd prime)}}.\end{cases}}} See Multiplicative group of integers modulo n and Primitive root modulo n. 2 ω ( n ) ≤ d ( n ) ≤ 2 Ω ( n ) . {\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}.} 6 π 2 < ϕ ( n ) σ ( n ) n 2 < 1. {\displaystyle {\frac {6}{\pi ^{2}}}<{\frac {\phi (n)\sigma (n)}{n^{2}}}<1.} c q ( n ) = μ ( q gcd ( q , n ) ) ϕ ( q gcd ( q , n ) ) ϕ ( q ) = ∑ δ ∣ gcd ( q , n ) μ ( q δ ) δ . {\displaystyle {\begin{aligned}c_{q}(n)&={\frac {\mu \left({\frac {q}{\gcd(q,n)}}\right)}{\phi \left({\frac {q}{\gcd(q,n)}}\right)}}\phi (q)\\&=\sum _{\delta \mid \gcd(q,n)}\mu \left({\frac {q}{\delta }}\right)\delta .\end{aligned}}} Note that ϕ ( q ) = ∑ δ ∣ q μ ( q δ ) δ . {\displaystyle \phi (q)=\sum _{\delta \mid q}\mu \left({\frac {q}{\delta }}\right)\delta .} c q ( 1 ) = μ ( q ) . {\displaystyle c_{q}(1)=\mu (q).} c q ( q ) = ϕ ( q ) . {\displaystyle c_{q}(q)=\phi (q).} ∑ δ ∣ n d 3 ( δ ) = ( ∑ δ ∣ n d ( δ ) ) 2 . {\displaystyle \sum _{\delta \mid n}d^{3}(\delta )=\left(\sum _{\delta \mid n}d(\delta )\right)^{2}.} Compare this with 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... + n)2 d ( u v ) = ∑ δ ∣ gcd ( u , v ) μ ( δ ) d ( u δ ) d ( v δ ) . {\displaystyle d(uv)=\sum _{\delta \mid \gcd(u,v)}\mu (\delta )d\left({\frac {u}{\delta }}\right)d\left({\frac {v}{\delta }}\right).} σ k ( u ) σ k ( v ) = ∑ δ ∣ gcd ( u , v ) δ k σ k ( u v δ 2 ) . {\displaystyle \sigma _{k}(u)\sigma _{k}(v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{k}\sigma _{k}\left({\frac {uv}{\delta ^{2}}}\right).} τ ( u ) τ ( v ) = ∑ δ ∣ gcd ( u , v ) δ 11 τ ( u v δ 2 ) , {\displaystyle \tau (u)\tau (v)=\sum _{\delta \mid \gcd(u,v)}\delta ^{11}\tau \left({\frac {uv}{\delta ^{2}}}\right),} where τ(n) is Ramanujan's function. == First 100 values of some arithmetic functions == == Notes == == References == Tom M. Apostol (1976), Introduction to Analytic Number Theory, Springer Undergraduate Texts in Mathematics, ISBN 0-387-90163-9 Apostol, Tom M. (1989), Modular Functions and Dirichlet Series in Number Theory (2nd Edition), New York: Springer, ISBN 0-387-97127-0 Bateman, Paul T.; Diamond, Harold G. (2004), Analytic number theory, an introduction, World Scientific, ISBN 978-981-238-938-1 Cohen, Henri (1993), A Course in Computational Algebraic Number Theory, Berlin: Springer, ISBN 3-540-55640-0 Edwards, Harold (1977). Fermat's Last Theorem. New York: Springer. ISBN 0-387-90230-9. Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and work, Providence RI: AMS / Chelsea, hdl:10115/1436, ISBN 978-0-8218-2023-0 Hardy, G. H.; Wright, E. M. (1979) [1938]. An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN 0-19-853171-0. MR 0568909. Zbl 0423.10001. Jameson, G. J. O. (2003), The Prime Number Theorem, Cambridge University Press, ISBN 0-521-89110-8 Koblitz, Neal (1984), Introduction to Elliptic Curves and Modular Forms, New York: Springer, ISBN 0-387-97966-2 Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea William J. LeVeque (1996), Fundamentals of Number Theory, Courier Dover Publications, ISBN 0-486-68906-9 Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950 Elliott Mendelson (1987), Introduction to Mathematical Logic, CRC Press, ISBN 0-412-80830-7 Nagell, Trygve (1964), Introduction to number theory (2nd Edition), Chelsea, ISBN 978-0-8218-2833-5 {{citation}}: ISBN / Date incompatibility (help) Niven, Ivan M.; Zuckerman, Herbert S. (1972), An introduction to the theory of numbers (3rd Edition), John Wiley & Sons, ISBN 0-471-64154-5 Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766 Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2076-6 Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, vol. 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002 == Further reading == Schwarz, Wolfgang; Spilker, Jürgen (1994), Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties, London Mathematical Society Lecture Note Series, vol. 184, Cambridge University Press, ISBN 0-521-42725-8, Zbl 0807.11001 == External links == "Arithmetic function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Matthew Holden, Michael Orrison, Michael Varble Yet another Generalization of Euler's Totient Function Huard, Ou, Spearman, and Williams. Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions Archived 2021-01-16 at the Wayback Machine László Tóth, Menon's Identity and arithmetical sums representing functions of several variables
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Wikipedia:Arithmetica#0
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Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the implementation of binary arithmetic on computers. Some arithmetic systems operate on mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic. Arithmetic operations form the basis of many branches of mathematics, such as algebra, calculus, and statistics. They play a similar role in the sciences, like physics and economics. Arithmetic is present in many aspects of daily life, for example, to calculate change while shopping or to manage personal finances. It is one of the earliest forms of mathematics education that students encounter. Its cognitive and conceptual foundations are studied by psychology and philosophy. The practice of arithmetic is at least thousands and possibly tens of thousands of years old. Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs. The ancient Indians developed the concept of zero and the decimal system, which Arab mathematicians further refined and spread to the Western world during the medieval period. The first mechanical calculators were invented in the 17th century. The 18th and 19th centuries saw the development of modern number theory and the formulation of axiomatic foundations of arithmetic. In the 20th century, the emergence of electronic calculators and computers revolutionized the accuracy and speed with which arithmetic calculations could be performed. == Definition, etymology, and related fields == Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and logarithm. The term arithmetic has its root in the Latin term arithmetica which derives from the Ancient Greek words ἀριθμός (arithmos), meaning 'number', and ἀριθμητική τέχνη (arithmetike tekhne), meaning 'the art of counting'. There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers. However, the more common view is to include operations on integers, rational numbers, real numbers, and sometimes also complex numbers in its scope. Some definitions restrict arithmetic to the field of numerical calculations. When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations. Arithmetic is closely related to number theory and some authors use the terms as synonyms. However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality. Traditionally, it is known as higher arithmetic. == Numbers == Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different numeral systems to represent them. === Kinds === The main kinds of numbers employed in arithmetic are natural numbers, whole numbers, integers, rational numbers, and real numbers. The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as { 1 , 2 , 3 , 4 , . . . } {\displaystyle \{1,2,3,4,...\}} . The symbol of the natural numbers is N {\displaystyle \mathbb {N} } . The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as { 0 , 1 , 2 , 3 , 4 , . . . } {\displaystyle \{0,1,2,3,4,...\}} and have the symbol N 0 {\displaystyle \mathbb {N} _{0}} . Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers. The set of integers encompasses both positive and negative whole numbers. It has the symbol Z {\displaystyle \mathbb {Z} } and can be expressed as { . . . , − 2 , − 1 , 0 , 1 , 2 , . . . } {\displaystyle \{...,-2,-1,0,1,2,...\}} . Based on how natural and whole numbers are used, they can be distinguished into cardinal and ordinal numbers. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?". A number is rational if it can be represented as the ratio of two integers. For instance, the rational number 1 2 {\displaystyle {\tfrac {1}{2}}} is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are 3 4 {\displaystyle {\tfrac {3}{4}}} and 281 3 {\displaystyle {\tfrac {281}{3}}} . The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is Q {\displaystyle \mathbb {Q} } . Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to 3 10 {\displaystyle {\tfrac {3}{10}}} , and 25.12 is equal to 2512 100 {\displaystyle {\tfrac {2512}{100}}} . Every rational number corresponds to a finite or a repeating decimal. Irrational numbers are numbers that cannot be expressed through the ratio of two integers. They are often required to describe geometric magnitudes. For example, if a right triangle has legs of the length 1 then the length of its hypotenuse is given by the irrational number 2 {\displaystyle {\sqrt {2}}} . π is another irrational number and describes the ratio of a circle's circumference to its diameter. The decimal representation of an irrational number is infinite without repeating decimals. The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is R {\displaystyle \mathbb {R} } . Even wider classes of numbers include complex numbers and quaternions. === Numeral systems === A numeral is a symbol to represent a number and numeral systems are representational frameworks. They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number. Numeral systems are either positional or non-positional. All early numeral systems were non-positional. For non-positional numeral systems, the value of a digit does not depend on its position in the numeral. The simplest non-positional system is the unary numeral system. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers. Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks. Egyptian hieroglyphics had a more complex non-positional numeral system. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000. A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the Hindu–Arabic numeral system, the radix is 10. This means that the first digit is multiplied by 10 0 {\displaystyle 10^{0}} , the next digit is multiplied by 10 1 {\displaystyle 10^{1}} , and so on. For example, the decimal numeral 532 stands for 5 ⋅ 10 2 + 3 ⋅ 10 1 + 2 ⋅ 10 0 {\displaystyle 5\cdot 10^{2}+3\cdot 10^{1}+2\cdot 10^{0}} . Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits. Another positional numeral system used extensively in computer arithmetic is the binary system, which has a radix of 2. This means that the first digit is multiplied by 2 0 {\displaystyle 2^{0}} , the next digit by 2 1 {\displaystyle 2^{1}} , and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for 1 ⋅ 2 3 + 1 ⋅ 2 2 + 0 ⋅ 2 1 + 1 ⋅ 2 0 {\displaystyle 1\cdot 2^{3}+1\cdot 2^{2}+0\cdot 2^{1}+1\cdot 2^{0}} . In computing, each digit in the binary notation corresponds to one bit. The earliest positional system was developed by ancient Babylonians and had a radix of 60. == Operations == Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output. The most important operations in arithmetic are addition, subtraction, multiplication, and division. Further operations include exponentiation, extraction of roots, and logarithm. If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations. Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0. There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in 13 + 4 − 4 = 13 {\displaystyle 13+4-4=13} . Defined more formally, the operation " ⋆ {\displaystyle \star } " is an inverse of the operation " ∘ {\displaystyle \circ } " if it fulfills the following condition: t ⋆ s = r {\displaystyle t\star s=r} if and only if r ∘ s = t {\displaystyle r\circ s=t} . Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, 7 + 9 {\displaystyle 7+9} is the same as 9 + 7 {\displaystyle 9+7} . Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since ( 5 × 4 ) × 2 {\displaystyle (5\times 4)\times 2} is the same as 5 × ( 4 × 2 ) {\displaystyle 5\times (4\times 2)} . === Addition and subtraction === Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is + {\displaystyle +} . Examples are 2 + 2 = 4 {\displaystyle 2+2=4} and 6.3 + 1.26 = 7.56 {\displaystyle 6.3+1.26=7.56} . The term summation is used if several additions are performed in a row. Counting is a type of repeated addition in which the number 1 is continuously added. Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is − {\displaystyle -} . Examples are 14 − 8 = 6 {\displaystyle 14-8=6} and 45 − 1.7 = 43.3 {\displaystyle 45-1.7=43.3} . Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance 14 − 8 = 14 + ( − 8 ) {\displaystyle 14-8=14+(-8)} . This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations. The additive identity element is 0 and the additive inverse of a number is the negative of that number. For instance, 13 + 0 = 13 {\displaystyle 13+0=13} and 13 + ( − 13 ) = 0 {\displaystyle 13+(-13)=0} . Addition is both commutative and associative. === Multiplication and division === Multiplication is an arithmetic operation in which two numbers, called the multiplier and the multiplicand, are combined into a single number called the product. The symbols of multiplication are × {\displaystyle \times } , ⋅ {\displaystyle \cdot } , and *. Examples are 2 × 3 = 6 {\displaystyle 2\times 3=6} and 0.3 ⋅ 5 = 1.5 {\displaystyle 0.3\cdot 5=1.5} . If the multiplicand is a natural number then multiplication is the same as repeated addition, as in 2 × 3 = 2 + 2 + 2 {\displaystyle 2\times 3=2+2+2} . Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the quotient. The symbols of division are ÷ {\displaystyle \div } and / {\displaystyle /} . Examples are 48 ÷ 8 = 6 {\displaystyle 48\div 8=6} and 29.4 / 1.4 = 21 {\displaystyle 29.4/1.4=21} . Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its reciprocal. The reciprocal of a number is 1 divided by that number. For instance, 48 ÷ 8 = 48 × 1 8 {\displaystyle 48\div 8=48\times {\tfrac {1}{8}}} . The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, 13 × 1 = 13 {\displaystyle 13\times 1=13} and 13 × 1 13 = 1 {\displaystyle 13\times {\tfrac {1}{13}}=1} . Multiplication is both commutative and associative. === Exponentiation and logarithm === Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are 2 4 = 16 {\displaystyle 2^{4}=16} and 3 {\displaystyle 3} ^ 3 = 27 {\displaystyle 3=27} . If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in 2 4 = 2 × 2 × 2 × 2 {\displaystyle 2^{4}=2\times 2\times 2\times 2} . Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of 1 2 {\displaystyle {\tfrac {1}{2}}} and the cube root of a number is the same as raising the number to the power of 1 3 {\displaystyle {\tfrac {1}{3}}} . Examples are 4 = 4 1 2 = 2 {\displaystyle {\sqrt {4}}=4^{\frac {1}{2}}=2} and 27 3 = 27 1 3 = 3 {\displaystyle {\sqrt[{3}]{27}}=27^{\frac {1}{3}}=3} . Logarithm is the inverse of exponentiation. The logarithm of a number x {\displaystyle x} to the base b {\displaystyle b} is the exponent to which b {\displaystyle b} must be raised to produce x {\displaystyle x} . For instance, since 1000 = 10 3 {\displaystyle 1000=10^{3}} , the logarithm base 10 of 1000 is 3. The logarithm of x {\displaystyle x} to base b {\displaystyle b} is denoted as log b ( x ) {\displaystyle \log _{b}(x)} , or without parentheses, log b x {\displaystyle \log _{b}x} , or even without the explicit base, log x {\displaystyle \log x} , when the base can be understood from context. So, the previous example can be written log 10 1000 = 3 {\displaystyle \log _{10}1000=3} . Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in 14 1 = 14 {\displaystyle 14^{1}=14} . However, exponentiation does not have a general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative. == Types == Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers. === Integer arithmetic === Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers. Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an addition table or a multiplication table. Other common methods are verbal counting and finger-counting. For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method addition with carries, the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added. Other methods used for integer additions are the number line method, the partial sum method, and the compensation method. A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative. A basic technique of integer multiplication employs repeated addition. For example, the product of 3 × 4 {\displaystyle 3\times 4} can be calculated as 3 + 3 + 3 + 3 {\displaystyle 3+3+3+3} . A common technique for multiplication with larger numbers is called long multiplication. This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with the rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers. Other techniques used for multiplication are the grid method and the lattice method. Computer science is interested in multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm, the Schönhage–Strassen algorithm, and the Toom–Cook algorithm. A common technique used for division is called long division. Other methods include short division and chunking. Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. One way to ensure that the result is an integer is to round the result to a whole number. However, this method leads to inaccuracies as the original value is altered. Another method is to perform the division only partially and retain the remainder. For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions. A simple method to calculate exponentiation is by repeated multiplication. For instance, the exponentiation of 3 4 {\displaystyle 3^{4}} can be calculated as 3 × 3 × 3 × 3 {\displaystyle 3\times 3\times 3\times 3} . A more efficient technique used for large exponents is exponentiation by squaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation 3 65 {\displaystyle 3^{65}} can be written as ( ( ( ( ( 3 2 ) 2 ) 2 ) 2 ) 2 ) 2 × 3 {\displaystyle (((((3^{2})^{2})^{2})^{2})^{2})^{2}\times 3} . By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than the 64 operations required for regular repeated multiplication. Methods to calculate logarithms include the Taylor series and continued fractions. Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer. ==== Number theory ==== Number theory studies the structure and properties of integers as well as the relations and laws between them. Some of the main branches of modern number theory include elementary number theory, analytic number theory, algebraic number theory, and geometric number theory. Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include divisibility, factorization, and primality. Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like how prime numbers are distributed and the claim that every even number is a sum of two prime numbers. Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of fields and rings, as in algebraic number fields like the ring of integers. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane. Further branches of number theory are probabilistic number theory, which employs methods from probability theory, combinatorial number theory, which relies on the field of combinatorics, computational number theory, which approaches number-theoretic problems with computational methods, and applied number theory, which examines the application of number theory to fields like physics, biology, and cryptography. Influential theorems in number theory include the fundamental theorem of arithmetic, Euclid's theorem, and Fermat's Last Theorem. According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the number 18 is not a prime number and can be represented as 2 × 3 × 3 {\displaystyle 2\times 3\times 3} , all of which are prime numbers. The number 19, by contrast, is a prime number that has no other prime factorization. Euclid's theorem states that there are infinitely many prime numbers. Fermat's Last Theorem is the statement that no positive integer values exist for a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} that solve the equation a n + b n = c n {\displaystyle a^{n}+b^{n}=c^{n}} if n {\displaystyle n} is greater than 2 {\displaystyle 2} . === Rational number arithmetic === Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have the same denominator then they can be added by adding their numerators and keeping the common denominator. For example, 2 7 + 3 7 = 5 7 {\displaystyle {\tfrac {2}{7}}+{\tfrac {3}{7}}={\tfrac {5}{7}}} . A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find a common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, 1 3 + 1 2 = 1 ⋅ 2 3 ⋅ 2 + 1 ⋅ 3 2 ⋅ 3 = 2 6 + 3 6 = 5 6 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{2}}={\tfrac {1\cdot 2}{3\cdot 2}}+{\tfrac {1\cdot 3}{2\cdot 3}}={\tfrac {2}{6}}+{\tfrac {3}{6}}={\tfrac {5}{6}}} . Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in 2 3 ⋅ 2 5 = 2 ⋅ 2 3 ⋅ 5 = 4 15 {\displaystyle {\tfrac {2}{3}}\cdot {\tfrac {2}{5}}={\tfrac {2\cdot 2}{3\cdot 5}}={\tfrac {4}{15}}} . Dividing one rational number by another can be achieved by multiplying the first number with the reciprocal of the second number. This means that the numerator and the denominator of the second number change position. For example, 3 5 : 2 7 = 3 5 ⋅ 7 2 = 21 10 {\displaystyle {\tfrac {3}{5}}:{\tfrac {2}{7}}={\tfrac {3}{5}}\cdot {\tfrac {7}{2}}={\tfrac {21}{10}}} . Unlike integer arithmetic, rational number arithmetic is closed under division as long as the divisor is not 0. Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm. One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the nth root of the result based on the denominator of the exponent. For example, 5 2 3 = 5 2 3 {\displaystyle 5^{\frac {2}{3}}={\sqrt[{3}]{5^{2}}}} . The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ Newton's method, which uses a series of steps to gradually refine an initial guess until it reaches the desired level of accuracy. The Taylor series or the continued fraction method can be utilized to calculate logarithms. The decimal fraction notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers 1 10 {\displaystyle {\tfrac {1}{10}}} , 371 100 {\displaystyle {\tfrac {371}{100}}} , and 44 10000 {\displaystyle {\tfrac {44}{10000}}} are written as 0.1, 3.71, and 0.0044 in the decimal fraction notation. Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions. Not all rational numbers have a finite representation in the decimal notation. For example, the rational number 1 3 {\displaystyle {\tfrac {1}{3}}} corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of repeating decimal is 0.3. Every repeating decimal expresses a rational number. === Real number arithmetic === Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and π. Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1. Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like 2 + π {\displaystyle {\sqrt {2}}+\pi } or e ⋅ 3 {\displaystyle e\cdot {\sqrt {3}}} . In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding. For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number π has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number π is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to π than 3.141. These methods allow computers to efficiently perform approximate calculations on real numbers. === Approximations and errors === In science and engineering, numbers represent estimates of physical quantities derived from measurement or modeling. Unlike mathematically exact numbers such as π or 2 {\displaystyle {\sqrt {2}}} , scientifically relevant numerical data are inherently inexact, involving some measurement uncertainty. One basic way to express the degree of certainty about each number's value and avoid false precision is to round each measurement to a certain number of digits, called significant digits, which are implied to be accurate. For example, a person's height measured with a tape measure might only be precisely known to the nearest centimeter, so should be presented as 1.62 meters rather than 1.6217 meters. If converted to imperial units, this quantity should be rounded to 64 inches or 63.8 inches rather than 63.7795 inches, to clearly convey the precision of the measurement. When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly considered to be non-significant. For example, the numbers 0.056 and 1200 each have only 2 significant digits, but the number 40.00 has 4 significant digits. Representing uncertainty using only significant digits is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach. In the example, the person's height might be represented as 1.62 ± 0.005 meters or 63.8 ± 0.2 inches. In performing calculations with uncertain quantities, the uncertainty should be propagated to calculated quantities. When adding or subtracting two or more quantities, add the absolute uncertainties of each summand together to obtain the absolute uncertainty of the sum. When multiplying or dividing two or more quantities, add the relative uncertainties of each factor together to obtain the relative uncertainty of the product. When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to the leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors. (See Significant figures § Arithmetic.) More sophisticated methods of dealing with uncertain values include interval arithmetic and affine arithmetic. Interval arithmetic describes operations on intervals. Intervals can be used to represent a range of values if one does not know the precise magnitude, for example, because of measurement errors. Interval arithmetic includes operations like addition and multiplication on intervals, as in [ 1 , 2 ] + [ 3 , 4 ] = [ 4 , 6 ] {\displaystyle [1,2]+[3,4]=[4,6]} and [ 1 , 2 ] × [ 3 , 4 ] = [ 3 , 8 ] {\displaystyle [1,2]\times [3,4]=[3,8]} . It is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude. The precision of numerical quantities can be expressed uniformly using normalized scientific notation, which is also convenient for concisely representing numbers which are much larger or smaller than 1. Using scientific notation, a number is decomposed into the product of a number between 1 and 10, called the significand, and 10 raised to some integer power, called the exponent. The significand consists of the significant digits of the number, and is written as a leading digit 1–9 followed by a decimal point and a sequence of digits 0–9. For example, the normalized scientific notation of the number 8276000 is 8.276 × 10 6 {\displaystyle 8.276\times 10^{6}} with significand 8.276 and exponent 6, and the normalized scientific notation of the number 0.00735 is 7.35 × 10 − 3 {\displaystyle 7.35\times 10^{-3}} with significand 7.35 and exponent −3. Unlike ordinary decimal notation, where trailing zeros of large numbers are implicitly considered to be non-significant, in scientific notation every digit in the significand is considered significant, and adding trailing zeros indicates higher precision. For example, while the number 1200 implicitly has only 2 significant digits, the number 1.20 × 10 3 {\displaystyle 1.20\times 10^{3}} explicitly has 3. A common method employed by computers to approximate real number arithmetic is called floating-point arithmetic. It represents real numbers similar to the scientific notation through three numbers: a significand, a base, and an exponent. The precision of the significand is limited by the number of bits allocated to represent it. If an arithmetic operation results in a number that requires more bits than are available, the computer rounds the result to the closest representable number. This leads to rounding errors. A consequence of this behavior is that certain laws of arithmetic are violated by floating-point arithmetic. For example, floating-point addition is not associative since the rounding errors introduced can depend on the order of the additions. This means that the result of ( a + b ) + c {\displaystyle (a+b)+c} is sometimes different from the result of a + ( b + c ) {\displaystyle a+(b+c)} . The most common technical standard used for floating-point arithmetic is called IEEE 754. Among other things, it determines how numbers are represented, how arithmetic operations and rounding are performed, and how errors and exceptions are handled. In cases where computation speed is not a limiting factor, it is possible to use arbitrary-precision arithmetic, for which the precision of calculations is only restricted by the computer's memory. === Tool use === Forms of arithmetic can also be distinguished by the tools employed to perform calculations and include many approaches besides the regular use of pen and paper. Mental arithmetic relies exclusively on the mind without external tools. Instead, it utilizes visualization, memorization, and certain calculation techniques to solve arithmetic problems. One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating 85 − 47 {\displaystyle 85-47} , one calculates 85 − 50 {\displaystyle 85-50} which is easier because it uses a round number. In the next step, one adds 3 {\displaystyle 3} to the result to compensate for the earlier adjustment. Mental arithmetic is often taught in primary education to train the numerical abilities of the students. The human body can also be employed as an arithmetic tool. The use of hands in finger counting is often introduced to young children to teach them numbers and simple calculations. In its most basic form, the number of extended fingers corresponds to the represented quantity and arithmetic operations like addition and subtraction are performed by extending or retracting fingers. This system is limited to small numbers compared to more advanced systems which employ different approaches to represent larger quantities. The human voice is used as an arithmetic aid in verbal counting. Tally marks are a simple system based on external tools other than the body. This system relies on mark making, such as strokes drawn on a surface or notches carved into a wooden stick, to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read. The abacus is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several beads. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result. Related aids include counting boards, which use tokens whose value depends on the area on the board in which they are placed, and counting rods, which are arranged in horizontal and vertical patterns to represent different numbers. Sectors and slide rules are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations. Printed tables were particularly relevant as an aid to look up the results of operations like logarithm and trigonometric functions. Mechanical calculators automate manual calculation processes. They present the user with some form of input device to enter numbers by turning dials or pressing keys. They include an internal mechanism usually consisting of gears, levers, and wheels to perform calculations and display the results. For electronic calculators and computers, this procedure is further refined by replacing the mechanical components with electronic circuits like microprocessors that combine and transform electric signals to perform calculations. === Others === There are many other types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division. Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals. Vector arithmetic and matrix arithmetic describe arithmetic operations on vectors and matrices, like vector addition and matrix multiplication. Arithmetic systems can be classified based on the numeral system they rely on. For instance, decimal arithmetic describes arithmetic operations in the decimal system. Other examples are binary arithmetic, octal arithmetic, and hexadecimal arithmetic. Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min. Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like 1 + 1 = 1 {\displaystyle 1+1=1} and 2 + 2 = 5 {\displaystyle 2+2=5} . They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation 1 + 1 = 1 {\displaystyle 1+1=1} can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one. == Axiomatic foundations == Axiomatic foundations of arithmetic try to provide a small set of laws, called axioms, from which all fundamental properties of and operations on numbers can be derived. They constitute logically consistent and systematic frameworks that can be used to formulate mathematical proofs in a rigorous manner. Two well-known approaches are the Dedekind–Peano axioms and set-theoretic constructions. The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by Richard Dedekind and later refined by Giuseppe Peano. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and successor. The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts. 0 is a natural number. For every natural number, there is a successor, which is also a natural number. The successors of two different natural numbers are never identical. 0 is not the successor of a natural number. If a set contains 0 and every successor then it contains every natural number. Numbers greater than 0 are expressed by repeated application of the successor function s {\displaystyle s} . For example, 1 {\displaystyle 1} is s ( 0 ) {\displaystyle s(0)} and 3 {\displaystyle 3} is s ( s ( s ( 0 ) ) ) {\displaystyle s(s(s(0)))} . Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add 2 {\displaystyle 2} to any number is the same as applying the successor function two times to this number. Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set ∅ {\displaystyle \varnothing } . Each subsequent number can be defined as the union of the previous number with the set containing the previous number. For example, 1 = 0 ∪ { 0 } = { 0 } {\displaystyle 1=0\cup \{0\}=\{0\}} , 2 = 1 ∪ { 1 } = { 0 , 1 } {\displaystyle 2=1\cup \{1\}=\{0,1\}} , and 3 = 2 ∪ { 2 } = { 0 , 1 , 2 } {\displaystyle 3=2\cup \{2\}=\{0,1,2\}} . Integers can be defined as ordered pairs of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number −9. Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number 3 7 {\displaystyle {\tfrac {3}{7}}} . One way to construct the real numbers relies on the concept of Dedekind cuts. According to this approach, each real number is represented by a partition of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest. Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result. == History == The earliest forms of arithmetic are sometimes traced back to counting and tally marks used to keep track of quantities. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed. However, a basic sense of numbers may predate these findings and might even have existed before the development of language. It was not until the emergence of ancient civilizations that a more complex and structured approach to arithmetic began to evolve, starting around 3000 BCE. This became necessary because of the increased need to keep track of stored items, manage land ownership, and arrange exchanges. All the major ancient civilizations developed non-positional numeral systems to facilitate the representation of numbers. They also had symbols for operations like addition and subtraction and were aware of fractions. Examples are Egyptian hieroglyphics as well as the numeral systems invented in Sumeria, China, and India. The first positional numeral system was developed by the Babylonians starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient. Abacuses have been utilized as hand-operated calculating tools since ancient times as efficient means for performing complex calculations. Early civilizations primarily used numbers for concrete practical purposes, like commercial activities and tax records, but lacked an abstract concept of number itself. This changed with the ancient Greek mathematicians, who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems. Another novel feature was their use of proofs to establish mathematical truths and validate theories. A further contribution was their distinction of various classes of numbers, such as even numbers, odd numbers, and prime numbers. This included the discovery that numbers for certain geometrical lengths are irrational and therefore cannot be expressed as a fraction. The works of Thales of Miletus and Pythagoras in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics. Diophantus was an influential figure in Greek arithmetic in the 3rd century BCE because of his numerous contributions to number theory and his exploration of the application of arithmetic operations to algebraic equations. The ancient Indians were the first to develop the concept of zero as a number to be used in calculations. The exact rules of its operation were written down by Brahmagupta in around 628 CE. The concept of zero or none existed long before, but it was not considered an object of arithmetic operations. Brahmagupta further provided a detailed discussion of calculations with negative numbers and their application to problems like credit and debt. The concept of negative numbers itself is significantly older and was first explored in Chinese mathematics in the first millennium BCE. Indian mathematicians also developed the positional decimal system used today, in particular the concept of a zero digit instead of empty or missing positions. For example, a detailed treatment of its operations was provided by Aryabhata around the turn of the 6th century CE. The Indian decimal system was further refined and expanded to non-integers during the Islamic Golden Age by Middle Eastern mathematicians such as Al-Khwarizmi. His work was influential in introducing the decimal numeral system to the Western world, which at that time relied on the Roman numeral system. There, it was popularized by mathematicians like Leonardo Fibonacci, who lived in the 12th and 13th centuries and also developed the Fibonacci sequence. During the Middle Ages and Renaissance, many popular textbooks were published to cover the practical calculations for commerce. The use of abacuses also became widespread in this period. In the 16th century, the mathematician Gerolamo Cardano conceived the concept of complex numbers as a way to solve cubic equations. The first mechanical calculators were developed in the 17th century and greatly facilitated complex mathematical calculations, such as Blaise Pascal's calculator and Gottfried Wilhelm Leibniz's stepped reckoner. The 17th century also saw the discovery of the logarithm by John Napier. In the 18th and 19th centuries, mathematicians such as Leonhard Euler and Carl Friedrich Gauss laid the foundations of modern number theory. Another development in this period concerned work on the formalization and foundations of arithmetic, such as Georg Cantor's set theory and the Dedekind–Peano axioms used as an axiomatization of natural-number arithmetic. Computers and electronic calculators were first developed in the 20th century. Their widespread use revolutionized both the accuracy and speed with which even complex arithmetic computations can be calculated. == In various fields == === Education === Arithmetic education forms part of primary education. It is one of the first forms of mathematics education that children encounter. Elementary arithmetic aims to give students a basic sense of numbers and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division. It is usually introduced in relation to concrete scenarios, like counting beads, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in early arithmetic education are number lines, addition and multiplication tables, counting blocks, and abacuses. Later stages focus on a more abstract understanding and introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers. They further cover more advanced numerical operations, like exponentiation, extraction of roots, and logarithm. They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of algorithms and calculators to solve complex arithmetic problems. === Psychology === The psychology of arithmetic is interested in how humans and animals learn about numbers, represent them, and use them for calculations. It examines how mathematical problems are understood and solved and how arithmetic abilities are related to perception, memory, judgment, and decision making. For example, it investigates how collections of concrete items are first encountered in perception and subsequently associated with numbers. A further field of inquiry concerns the relation between numerical calculations and the use of language to form representations. Psychology also explores the biological origin of arithmetic as an inborn ability. This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation. One of the concepts studied by psychology is numeracy, which is the capability to comprehend numerical concepts, apply them to concrete situations, and reason with them. It includes a fundamental number sense as well as being able to estimate and compare quantities. It further encompasses the abilities to symbolically represent numbers in numbering systems, interpret numerical data, and evaluate arithmetic calculations. Numeracy is a key skill in many academic fields. A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, by misunderstanding mortgage plans and insurance policies. === Philosophy === The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and ontological status of numbers, the relation of arithmetic to language and logic, and how it is possible to acquire arithmetic knowledge. According to Platonism, numbers have mind-independent existence: they exist as abstract objects outside spacetime and without causal powers. This view is rejected by intuitionists, who claim that mathematical objects are mental constructions. Further theories are logicism, which holds that mathematical truths are reducible to logical truths, and formalism, which states that mathematical principles are rules of how symbols are manipulated without claiming that they correspond to entities outside the rule-governed activity. The traditionally dominant view in the epistemology of arithmetic is that arithmetic truths are knowable a priori. This means that they can be known by thinking alone without the need to rely on sensory experience. Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of rational intuition through which mathematical truths can be apprehended. A more recent alternative view was suggested by naturalist philosophers like Willard Van Orman Quine, who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences. === Others === Arithmetic is relevant to many fields. In daily life, it is required to calculate change when shopping, manage personal finances, and adjust a cooking recipe for a different number of servings. Businesses use arithmetic to calculate profits and losses and analyze market trends. In the field of engineering, it is used to measure quantities, calculate loads and forces, and design structures. Cryptography relies on arithmetic operations to protect sensitive information by encrypting data and messages. Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations. Algebra relies on arithmetic principles to solve equations using variables. These principles also play a key role in calculus in its attempt to determine rates of change and areas under curves. Geometry uses arithmetic operations to measure the properties of shapes while statistics utilizes them to analyze numerical data. Due to the relevance of arithmetic operations throughout mathematics, the influence of arithmetic extends to most sciences such as physics, computer science, and economics. These operations are used in calculations, problem-solving, data analysis, and algorithms, making them integral to scientific research, technological development, and economic modeling. == See also == Algorism Expression (mathematics) Finite field arithmetic Outline of arithmetic Plant arithmetic == References == === Notes === === Citations === === Sources === == External links ==
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Wikipedia:Arithmetization of analysis#0
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The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century which aimed to abolish all geometric intuition from the proofs in analysis. For the followers of this program, the fundamental concepts of calculus should also not make references to the ideas of motion and velocity. This ideal was pursued by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, among others, who considered that Isaac Newton's calculus lacked rigor. == History == Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work. == Research program == The highlights of this research program are: the various (but equivalent) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field; the epsilon-delta definition of limit; and the naïve set-theoretic definition of function. == Legacy == An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus. The arithmetization of analysis had several important consequences: the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems continued unabated, as documented by P. Ehrlich; the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in the way mathematics is taught today; it made possible the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert; it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis. == Quotation == "God created the natural numbers, all else is the work of man." — Kronecker == References == Torina Dechaune Lewis (2006) The Arithmetization of Analysis: From Eudoxus to Dedekind, Southern University. Carl B. Boyer, Uta C. Merzbach (2011) A History of Mathematics John Wiley & Sons. Arithmetization of analysis at Encyclopedia of Mathematics. James Pierpont (1899) "On the arithmetization of mathematics", Bull. Amer. Math. Soc. 5(8): 394–406.
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Wikipedia:Arjan van der Schaft#0
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Abraham Jan (Arjan) van der Schaft (born 1955) is emeritus professor of systems and control theory at the Bernoulli Institute of Mathematics Computer Science and Artificial Intelligence, University of Groningen. He is notable for his contributions to network modelling and control of complex physical systems, specifically in the areas of Port-Hamiltonian systems, passivity-based control, nonlinear H∞ control, hybrid systems, and port-thermodynamical systems. He is a Fellow of the IEEE. == Career == Arjan van der Schaft received the undergraduate and Ph.D. degrees in Mathematics from the University of Groningen, Netherlands, in 1979 and 1983, respectively. In 1982 he joined the Department of Applied Mathematics, University of Twente, Enschede, where he was appointed as a full professor in Mathematical Systems and Control Theory in 2000. In September 2005 he returned to Groningen as a full professor in Mathematics. As of June 2021, he became emeritus professor at the same university. In 2006 he was an invited speaker at the International Congress of Mathematicians in Madrid. Van der Schaft has served as Associate Editor for Systems & Control Letters, Journal of Nonlinear Science, SIAM Journal on Control and Optimization, and the IEEE Transactions on Automatic Control. Currently he is Associate Editor for the Journal of Geometric Mechanics, and Editor-at-Large for the European Journal of Control. == Books == System Theoretic Descriptions of Physical Systems (1984) Variational and Hamiltonian Control Systems (1987, with P.E. Crouch) Nonlinear Dynamical Control Systems (1990, with H. Nijmeijer) L2-Gain and Passivity Techniques in Nonlinear Control (2000) An Introduction to Hybrid Dynamical Systems (2000, with J.M. Schumacher) Modeling and Control of Complex Physical Systems: the Port-Hamiltonian Approach (Geoplex Consortium, 2009) == Honors and awards == Fellow of the Institute of Electrical and Electronics Engineers since 2002. The paper "A.J. van der Schaft, L2-gain analysis of nonlinear systems and nonlinear state feedback H∞ control, TAC. AC-37, pp. 770–784, 1992" was the Dutch research paper in international scientific journals within the Technical Sciences that obtained the largest number of citations during the evaluation period 1994–1998. Invited semi-plenary speaker at the International Congress of Mathematicians, Madrid, 22–30 August 2006: From networks models to geometry: a new view on Hamiltonian systems. SICE Takeda Best Paper Prize 2008 for "An approximation method for the stabilizing solution of the Hamilton-Jacobi equation for integrable systems", a Hamiltonian perturbation approach', Transactions of the Society of Instrument and Control Engineers (SICE), 43, pp. 572–580, 2007. == References ==
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Wikipedia:Arjen Lenstra#0
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Arjen Klaas Lenstra (born 2 March 1956, in Groningen) is a Dutch mathematician, cryptographer and computational number theorist. He is a professor emeritus from the École Polytechnique Fédérale de Lausanne (EPFL) where he headed of the Laboratory for Cryptologic Algorithms. == Career == He studied mathematics at the University of Amsterdam. He is a former professor at the EPFL (Lausanne), in the Laboratory for Cryptologic Algorithms, and previously worked for Citibank and Bell Labs. == Research == Lenstra is active in cryptography and computational number theory, especially in areas such as integer factorization. With Mark Manasse, he was the first to seek volunteers over the internet for a large scale volunteer computing project. Such projects became more common after the Factorization of RSA-129 which was a high publicity distributed factoring success led by Lenstra along with Derek Atkins, Michael Graff and Paul Leyland. He was also a leader in the successful factorizations of several other RSA numbers. Lenstra was also involved in the development of the number field sieve. With coauthors, he showed the great potential of the algorithm early on by using it to factor the ninth Fermat number, which was far out of reach by other factoring algorithms of the time. He has since been involved with several other number field sieve factorizations including the current record, RSA-768. Lenstra's most widely cited scientific result is the first polynomial time algorithm to factor polynomials with rational coefficients in the seminal paper that introduced the LLL lattice reduction algorithm with Hendrik Willem Lenstra and László Lovász. Lenstra is also co-inventor of the XTR cryptosystem. On 1 March 2005, Arjen Lenstra, Xiaoyun Wang, and Benne de Weger of Eindhoven University of Technology demonstrated construction of two X.509 certificates with different public keys and the same MD5 hash, a demonstrably practical hash collision. The construction included private keys for both public keys. == Distinctions == Lenstra is the recipient of the RSA Award for Excellence in Mathematics 2008 Award. == Private life == Lenstra's brother and co-author Hendrik Lenstra is a professor in mathematics at Leiden University and his brother Jan Karel Lenstra is a former director of Centrum Wiskunde & Informatica (CWI). == See also == L-notation General number field sieve Schnorr–Seysen–Lenstra algorithm == References == == External links == Web page on Arjen Lenstra at EPFL
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Wikipedia:Arkadi Nemirovski#0
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Arkadi Nemirovski (born March 14, 1947) is a professor at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology. He has been a leader in continuous optimization and is best known for his work on the ellipsoid method, modern interior-point methods and robust optimization. == Biography == Nemirovski earned a Ph.D. in Mathematics in 1974 from Moscow State University and a Doctor of Sciences in Mathematics degree in 1990 from the Institute of Cybernetics of the Ukrainian Academy of Sciences in Kiev. He has won three prestigious prizes: the Fulkerson Prize, the George B. Dantzig Prize, and the John von Neumann Theory Prize. He was elected a member of the U.S. National Academy of Engineering (NAE) in 2017 "for the development of efficient algorithms for large-scale convex optimization problems", and the U.S National Academy of Sciences (NAS) in 2020. In 2023, Nemirovski and Yurii Nesterov were jointly awarded the 2023 WLA Prize in Computer Science or Mathematics "for their seminal work in convex optimization theory, including the theory of self-concordant functions and interior-point methods, a complexity theory of optimization, accelerated gradient methods, and methodological advances in robust optimization." == Academic work == Nemirovski first proposed mirror descent along with David Yudin in 1983. His work with Yurii Nesterov in their 1994 book is the first to point out that the interior point method can solve convex optimization problems, and the first to make a systematic study of semidefinite programming (SDP). Also in this book, they introduced the self-concordant functions which are useful in the analysis of Newton's method. == Books == co-authored with Yurii Nesterov: Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics. 1994. ISBN 978-0898715156. co-authored with Aharon Ben-Tal: Lectures on Modern Convex Optimization. Society for Industrial and Applied Mathematics. 2001. ISBN 978-0-89871-491-3. co-authored with A. Ben-Tal and L. El Ghaoui: Robust Optimization. Princeton University Press. 2009. ISBN 978-0-691-14368-2. == References == == External links == Arkadi Nemirovski, Ph.D. – ISyE Archived 2015-03-03 at the Wayback Machine Arkadi Nemirovski's website Archived 2022-12-09 at the Wayback Machine Arkadi Nemirovski – Technion https://web.archive.org/web/20160513155431/https://www.informs.org/Recognize-Excellence/INFORMS-Prizes-Awards/John-von-Neumann-Theory-Prize
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Wikipedia:Arkady Onishchik#0
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Arkady L'vovich Onishchik (Russian: Арка́дий Льво́вич Они́щик, born 14 November 1933 in Moscow; died 12 February 2019) was a prominent Soviet and Russian mathematician, who worked on Lie groups and their geometrical applications. Onishchik was a student of Eugene Dynkin, under whose guidance he got his PhD at Moscow State University in 1960. In 1962 Onishchik received the Prize of the Moscow Mathematical Society for young mathematicians. In 1970 he got Habilitation (Russian title of Doctor of Sciences). Since 1975 Onishchik was a professor of Yaroslavl State University. Onishchik introduced new homotopy invariants of homogeneous spaces and classified factorizations of connected simple compact Lie groups into the product of two connected Lie subgroups. In complex analysis, the Matsushima–Onishchik theorem describes homogeneous spaces of complex reductive groups that are Stein manifolds. In addition to Lie groups and algebras, Onishchik also worked on nonabelian cohomology and on supermanifolds. == Books == with E. B. Vinberg: Lie groups and algebraic groups, Springer-Verlag 1990. Topology of transitive transformations groups. Barth, Leipzig 1994. with E. B. Vinberg: (Eds.): Lie groups and Lie algebras. 3 volumes. Encyclopaedia of Mathematical Sciences, Springer-Verlag, of which by him: with E. B. Vinberg: Foundations of Lie theory, in Vol. 1, 1997. with V. V. Gorbatsevich: Lie transformation theory, in Vol. 1. with V. V. Gorbatsevich and E. B. Vinberg: Structure of Lie groups and Lie algebras, in Vol. 3, 1994. Lectures on real semisimple Lie algebras and their representations. European Mathematical Society, Zürich 2004. with Rolf Sulanke: Projective and Cayley-Klein Geometries. Springer-Verlag, 2006. == References == 1. D. N. Akhiezer, È. B. Vinberg, V. V. Gorbatsevich, V. G. Durnev, R. Sulanke, L, S. Kazarin, D. A. Leites, V. V. Serganova, V. M. Tikhomirov. Arkadiĭ Lʹvovich Onishchik (on the occasion of his seventieth birthday). (Russian) Uspekhi Mat. Nauk 58 (2003), no. 6(354), 193–200 (DOI: https://doi.org/10.4213/rm695). English translation in Russian Math. Surveys 58 (2003), no. 6, 1245–1253 (DOI: http://dx.doi.org/10.1070/RM2003v058n06ABEH000695). 2. D. N. Akhiezer, È. B. Vinberg, V. V. Gorbatsevich, L. S. Kazarin, D. A. Leites, A. M. Lukatskii, A. N. Shchetinin. Arkady L'vovich Onishchik (obituary). (Russian) Uspekhi Mat. Nauk 75 (2020), no. 4(454), 195–206 (DOI: https://doi.org/10.4213/rm9941). English translation in Russian Math. Surveys 75 (2020), no. 4, 765–777 (DOI: http://dx.doi.org/10.1070/RM9941). == External links == Dynkin Collection Mathnet.ru Autor's profile Arkady L’vovich Onishchik at zbMATH
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Wikipedia:Arnaldo Garcia#0
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Arnaldo Leite Pinto Garcia (born 1950) is a Brazilian mathematician working on algebraic geometry and coding theory. He is a titular researcher at the IMPA. Garcia is a titular member of the Brazilian Academy of Sciences and has received Brazil's National Order of Scientific Merit. He obtained his Ph.D. at the IMPA in 1980 under the guidance of Karl-Otto Stöhr. == Selected writings == A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound On the asymptotic behaviour of some towers of function fields over finite fields On subfields of the Hermitian function field On maximal curves == References ==
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Wikipedia:Arne Sletsjøe#0
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Arne Bernhard Sletsjøe (sometimes shown as Arne Slettsjø, born 8 April 1960) is a Norwegian mathematician and retired canoe sprinter who competed internationally in the mid to late 1980s. He won two medals in the K-4 10000 m event at the ICF Canoe Sprint World Championships with a gold in 1987 and a silver in 1983. Sletsjøe also competed in two Summer Olympics in the K-4 1000 m event. At the 1984 Summer Olympics in Los Angeles, he was eliminated in the semifinals. Four years later in Seoul, Sletsjøe and his teammates made the semifinals, but did not finish. Sletsjøe later served as president of the Norwegian Canoe/ Kayak Federation. He finished his secondary education at Oslo Cathedral School in 1977, graduated from the University of Oslo with the cand.real. degree in 1983 and took the doctorate in 1989. His academic advisor was Arnfinn Laudal. Sletsjøe still works at the university as an associate professor, having started out as a research fellow. He is a son of violist Arne Sletsjøe. He is married to Ingeborg Rasmussen, has two children and resides at Jar. == References == == External links == Arne Sletsjøe at Olympedia Arne Sletsjøe at Olympics.com
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Wikipedia:Arnfinn Laudal#0
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Olav Arnfinn Laudal (born 19 June 1936) is a Norwegian mathematician. == Early life and education == O.A. Laudal was born in Kirkenes as the son of teachers Trygve Laudal (1896–1964) and Agnes Mønnesland (1898–1982). He finished his secondary education in 1954 in Mandal, and enrolled in the University of Oslo in the same year. He studied at École Normale Supérieure from 1957, but in 1958 he was back in Oslo and took the cand.real. degree. == Mathematical career == Laudal was a research fellow at Columbia University and Institut Henri Poincaré between 1959 and 1962. He was appointed as lecturer at the University of Oslo in 1962, was promoted to docent in 1964 and was a professor from 1985 to 2003. His most notable book is 1979's Formal Moduli of Algebraic Structures. He was among the founders of the Abel Prize, and has been involved in the International Centre for Theoretical Physics. == Organizational memberships == Laudal is a member of the Norwegian Academy of Science and Letters. He has been a deputy member of Bærum municipal council for the Socialist Left Party. == References ==
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